The Economic Journal

Ticchi,, Davide;Vindigni,, Andrea

doi: 10.1111/j.1468-0297.2009.02309.xpmid: N/A

Abstract We present a theory of the choice of alternative democratic constitutions, a majoritarian or a consensual one, in an unequal society. We show that a consensual system turns out to be preferred by society when ex ante income inequality is relatively low, while a majoritarian system is chosen when income inequality is relatively high. We also find that consensual democracies should be expected to be ruled more often by centre‐left coalitions while the right should have an advantage in majoritarian constitutions. The implications for the relationship between inequality and redistribution are discussed. Historical evidence and a cross‐sectional analysis support our results. In recent years, there has been an increasing interest in the political economics literature for the role of key constitutional norms in shaping fiscal policy outcomes. Important theoretical contributions include, among others, Myerson (1993), Persson et al. (1997, 2000), Persson and Tabellini (2000), Austen‐Smith (2000), Lizzeri and Persico (2001) and Milesi‐Ferretti et al. (2002), and generally predict that proportional electoral systems and parliamentary regimes should be associated with more provision of public goods and universalistic welfare programmes, as well as with larger size governments. These results have been confirmed by some recent empirical works. Persson and Tabellini (2003, 2004) present cross‐country evidence suggesting that a switch from proportional to majoritarian elections reduces total government spending by almost 5% of GDP and welfare spending by 2–3% of GDP, and obtain similar results for a switch from a parliamentary to a presidential regime. Milesi‐Ferretti et al. (2002), Lijphart (1999) and other works provide empirical evidence going in the same direction. These contributions are based on the premise of taking political institutions as given. But if different constitutional provisions lead to different fiscal policies and, therefore, generate different benefits for the various groups in the society, we should expect individuals to have different preferences over constitutions and take this into account at the time of the constitutional choice. In this article, we start from the consideration that the various groups in the society may have conflicting interests over constitutions and provide an economic theory of the choice of a democratic constitution on the base of the (pre‐tax) distribution of income within a society.1 In other words, we recognise that constitutions are endogenous and their choice affected by economic fundamentals. Whereas constitutional models differ along several dimensions, in our analysis we focus on the electoral system as this is generally considered one of the most important constitutional provisions, at least from a fiscal policy outcome perspective. As pointed out by Lijphart (1999), in parliamentary regimes the electoral system leads to two distinctive types of democratic models. One is the majoritarian model characterised by a majoritarian (plurality rule or first‐past‐the‐post) electoral system that generally leads to the formation of a two‐party system and to the concentration of power in the hands of the prime minister. The other is the consensual (or proportional) model which is characterised by a proportional electoral rule leading to a multi‐party system and, therefore, to coalition governments.2 While our theoretical model represents precisely these two types of parliamentary regimes, at some point we argue that our model of majoritarian democracy may also well describe presidential regimes where the president has relatively large legislative powers. Our main result is that a majoritarian constitution is more likely to be chosen when the degree of income inequality is relatively high, while consensual democracy is more likely to arise in relatively homogeneous societies. We present a simple public finance model where fiscal policy is about the provision of some public goods, financed with proportional taxation of income. The public goods considered are local, or group‐specific, in the sense that each of them is desired by one and only one of the three social groups (or ‘classes’) which compose the society and that are identified by their level of pre‐tax income: the poor, the middle class and the rich. We go on by characterising the political equilibrium of the model in a majoritarian and in a consensual democracy respectively. A key assumption that we maintain in each constitutional environment is that politicians are citizen‐candidates who have a direct interest in the policy implemented and cannot credibly commit to implementing any policy different from their preferred one. We assume that fiscal policy in a majoritarian democracy is decided by a ‘leader’ elected directly by the people through a majority voting process. We demonstrate that in equilibrium the winner is always a rich citizen‐candidate as the rich enjoy a natural advantage over the other two classes, arising from the interaction of their relative fiscal conservatism and the majoritarian electoral law, and conclude that the structure of majoritarian democracy biases policy outcomes in favour of this group. In a consensual democracy fiscal policy is decided by a coalition government formed as the outcome of some legislative bargaining process among the members of a parliament elected with a proportional electoral law. We show that the government coalition depends on the distribution of income. According to our model, in a consensual democracy a middle class and rich (middle class and poor), or centre‐right (centre‐left), government coalition is more likely to be formed when the distribution of income is more (less) polarised. We also show that taxation and the size of government in a consensual democracy under a centre‐left coalition are higher than under a centre‐right one, and that they are generally higher in a consensual than in a majoritarian democracy. Finally, we evaluate the welfare implications of the two types of political institutions from the point of view of the different groups of citizens and let individuals vote in an ‘original position’ in the absence of any veil of ignorance on which constitution to adopt. We find that a society with high income inequality prefers a majoritarian constitution while consensual democracy is preferred when inequality is lower. The intuition behind this result is simple. The rich prefer the majoritarian system because they always get to power. The middle class prefer the consensual model as it is always part of the government coalition. Hence, the poor are the swing voters. When inequality is low, they end up being part of the government in consensual democracy and will therefore prefer this political system. When inequality is high, the poor are not part of the government in the consensual system and then prefer the majoritarian one. They do not get their specific public good in any case but the rich under majoritarian democracy tax them less than the middle class and rich coalition. The main prediction of our article, that more unequal societies are expected to choose a majoritarian constitution while more equal countries opt for a consensual one, is tested with a cross‐sectional analysis in a sample of 57 democracies and in a sub‐sample of 31 parliamentary democracies. To avoid problems of endogeneity, we use the income inequality at the time of (or before) the adoption of the constitution and find that, as predicted by our theory, the degree of inequality is a highly significant determinant of constitutional choice. The model has also other important implications concerning the politico‐economic outcomes in representative democracies. In accordance with the existing literature, we obtain that taxation and the size of government in consensual democracies are higher than in majoritarian ones. However, the mechanism generating this result in our model is new and due to a selection bias in the composition of the government coalition. Consensual democracies should be expected to be ruled relatively often by centre‐left coalitions, more willing to tax and redistribute income, while the more fiscally conservative right should have an advantage in majoritarian countries. We also emphasise that our constitution selection theory may shed some light on the relationship between inequality and redistribution, suggesting that the former not only affects fiscal policy for a given constitution, but it also influences the choice of the constitution itself. As a result, the relationship between income inequality and redistribution may well be absent or negative as suggested by the empirical evidence (Perotti, 1996) and contrary to the results obtained by models based on the median voter.3 As explained above, our theory also provides a clear prediction about the preferences of the social groups for the type of constitution. We present some historical evidence that is in accordance with our findings. In particular, we present evidence that the constitution of the US has been drafted to reflect essentially the interests of the economic elite of the time, and something similar happened in other majoritarian democracies such as the UK and Chile. We also discuss the adoption of consensual constitutions by several continental European countries and argue that, in accordance with our theory, in most cases this choice was made by centre‐left forces in a period of low and/or declining income inequality. Although this article contains various interesting results, its main contribution to the literature is to provide a theory of constitutional choice, highlighting the endogeneity of political institutions and the role played by economic fundamentals in this choice. Such fundamentals, in our case income inequality, are themselves affected by the constitutional rules through their effect on fiscal policy outcomes. By clarifying the endogeneity and the factors behind the choice of a constitution, our theory may offer guidelines for future empirical research to the literature on the economic consequences of constitutions. Our article is also related to the recent literature about the choice of voting rules (Aghion and Bolton, 2003; Barbera and Jackson, 2004; Messner and Polborn, 2004) and other political institutions (Acemoglu and Robinson, 2000; Aghion et al, 2004). The article is organised as follows. Section 1 describes the basic economic setting and the public finance problem we focus on. Sections 2 and 3 present the political equilibrium of the model in majoritarian and consensual democracy respectively. Section 4 characterises the properties of the political equilibrium within and across constitutions. Section 5 deals with the key issue of the endogeneity of the constitution and its relation with the distribution of income. In Section 6 we discuss the intermediate results of our theory and present evidence supporting them. Section 7 shows the results of our cross‐sectional analysis on the relationship between income inequality and constitutional choice. Section 8 concludes. 1. A Simple Model of Public Finance: Basic Setup We consider a simple model of ‘local’ (that is, group specific) public goods provision based on Persson and Tabellini (2000, ch. 7). A society is made up by N > 1 groups of individuals. For convenience, we focus on the case where N = 3. Group j ∈ ℑ ≡ {p, b, r} has size (measure) mj and each individual of that group has an exogenous pre‐tax income equal to yj. Total population is made by a continuum of unitary measure ∑j ∈ ℑmj = 1 and with no loss of generality we assume that max{mp,mr } < mb < 1/2 and that yp < yb < yr. This means that group b is the largest one and has an intermediate level of income, so that it is natural to identify it with the ‘middle class’. Group p and group r correspond to the ‘poor’ and to the ‘rich’ people. The absolute majority of votes is reached by the combination of any pair of groups. Notice also that the above assumptions are sufficient to ensure that the voter with median income (i.e. the median voter if preferences are single‐crossing in income) belongs to group b. Finally, we assume that : the voter with median income is poorer then the (virtual) mean voter, which means that the distribution of income is skewed to the left consistently with the empirical evidence. We assume that the utility function of each member of group j has the following quasi‐linear form (1) where cj denotes the consumption of a private good and gj the level of the type j public good provided. H(·) is a smooth, increasing and concave function that satisfies the Inada conditions.4 We also assume that H(0) = 0. The Inada conditions guarantee that at the optimum each group will always strictly prefer to have some taxation and some provision of its desired public good to the alternative of no taxation and no public good. All the above properties are satisfied by the constant elasticity functional form H(gj) = A(gj)α, where A is a constant and α ∈ (0,1). At some point we will use such preference specification to obtain some analytical and numerical results. Each group is perfectly homogeneous. Heterogeneity is only between groups and is related to the differences in the pre‐tax income level and to group‐specific preferences on the public good to be provided. The specification of preferences in (1) implies that each group values one particular public good only and there are as many kinds of public goods as the groups of people. The local public goods can be interpreted as publicly provided private goods, like education, health and housing, on which different income groups have different preferences.5 However, the important implication of our specification of individual preferences is that redistribution can be targeted toward specific social groups.6 Income is taxed at a proportional rate τ ∈ [0,1] that will be determined later as a part of the political equilibrium of the model. Therefore, the budget constraint of the agents of group j is simply cj = (1 − τ)yj. We also assume that the government can finance public expenditures only out of the revenues generated by income taxation. In equilibrium, gj is positive only when group j is part of the government. If we incorporate this result in the public sector budget constraint, the latter can be rewritten as , where Ω ≡ {j ∈ ℑ : j is part of the government}. As will become clear later, the quasi‐linearity of preferences and the assumption of no tax discrimination simplify the analysis. Our results generalise as long as higher income levels translate into a preference for lower tax rates and groups in power cannot raise tax rates on other groups only. Our results would no longer go through if the group in power could impose a tax rate on other groups that is independent of the one that applies to itself (in this case the group in power could fully expropriate any other groups). In the next two Sections, we derive and characterise the political equilibrium of our model, namely the tax rate τ, the overall level of public expenditure and its composition G ≡ (gp, gb, gr), in the case of both a majoritarian constitution and a consensual one. Since the constitution is at this stage still taken as given, these equilibria can be considered as partial political equilibria. Then, we characterise the general political equilibrium where the constitution will be itself endogenous and chosen by the society. We assume that voting is sincere in any constitutional environment and model the political process going on within a majoritarian or a consensual democracy drawing on the citizen‐candidate apparatus of Osborne and Slivinsky (1996) and Besley and Coate (1997).7 We adopt a model of endogenous political candidacy since we want to emphasise the link existing between individual preferences (of citizens as well as of politicians) and individual income. Moreover, a key advantage of this model is to allow for the existence of an equilibrium even when individual preferences fail to be single‐peaked.8 2. Majoritarian Democracy We assume that in a majoritarian democracy fiscal policy is decided by a ‘leader’ elected directly by the people through a majority voting process among the menu of citizen‐candidates participating to the election. With this assumption, we mean to capture the winner‐takes‐all nature of political competition going on within a majoritarian democracy. Thus, the model represents both a parliamentary democracy where the legislature is elected with a majoritarian electoral rule (e.g. the UK), a presidential regime with a legislative assembly elected with a majoritarian electoral law (e.g. the US) and a presidential regime where the president plays a very important role in the legislative process (which is typically the case in Latin America).9 The menu of candidates is endogenous and one individual runs for office if and only if, in equilibrium, the net gain of doing so (the difference between the utility he gets if does‐does not run) exceeds the exogenous cost of running. The winner of the election is the candidate gaining the plurality of votes and he alone decides on fiscal policy. To characterise the political equilibrium under a majoritarian constitution, it is useful to start from the benchmark case of the unconstrained preferred policy of each social group. Then, suppose that a member of group j (which one is irrelevant given the assumption of perfect within group homogeneity) could act as a dictator and implement his preferred policy (‘dictatorial policy’). It is clear that he would not spend anything in any public good other than his preferred one, so that gi = 0, i ≠ j and . Hence, he would maximise the following utility . The (unique) optimal dictatorial tax rate of group j that solves this problem is10 (2) It is straightforward to verify that ∂τj/∂yj < 0. The richer a group j member is (for a given mean level of income), the higher is the marginal cost of public good provision he faces and the lower is his demand for his preferred public good. Hence, the dictatorial tax rates for the three groups can be ordered as: τr < τb < τp. No commitment technology is assumed to be available and, therefore, candidates cannot announce credibly before the election to pursue, if elected, any policy different from their preferred one. Let k denote some private benefit of being in office, which is either a psychological benefit or a non‐taxed monetary income, and ɛ be the cost of running. Both are exogenous and equal for everybody with k  ɛ. Now, we can state the main result of this Section. Proposition 1. The model has a unique political equilibrium with the following features. Only rich citizen‐candidates run for office, only the public good preferred by the rich is provided and the dictatorial tax rate of the rich τr is implemented. Proof. See Appendix. There are four elements of the model that are important for the results of this Proposition. First, no one group has the majority of the votes alone. Second, the utility function is chosen in such a way that the rich, as dictator, is the group that prefers the lowest taxes. Third, the winner‐takes‐all nature of the electoral process: in two‐candidate contests between the rich and another group, the rich always win since they prefer less taxation.11 Fourth, if a group expects to lose an election, no candidate is forthcoming. 3. Consensual Democracy In a consensual democracy voters do not elect a leader directly but rather elect their representatives to the parliament. We assume the existence of a parliament composed by a continuum of measure ρ ∈ (0,1) of members which are elected with a pure proportional electoral rule in a single nation‐wide electoral district. The government is formed as the outcome of a process of legislative bargaining among the representatives of the different groups and it expresses a certain parliamentary majority.12 We also assume that the plurality of parliamentary votes is sufficient to form a government. The policy formation process corresponds to the following three‐stages game: the entry of candidates stage; the voting stage; the legislative bargaining stage. Assuming that there are three groups in the parliament and that no group has the absolute majority of parliamentary members (which will be the case in equilibrium), events take place at the legislative bargaining stage according to the following protocol. Round 1 of the bargaining game: the head of the representatives (appointed at random) of the group having the relative majority of seats in the parliament is called to make a policy proposal to the head of the representatives of another group of his choice. Given that ‘buying’ votes is costly, only two groups coalitions will be observed and a version of Riker’s minimum size coalition principle will apply. If the proposal is accepted, the government coalition is formed and the agreed policy is implemented. Round 2 of the bargaining game: if the proposal is not approved, a second agenda setter is appointed randomly by nature between the representatives of the two groups of which no member was agenda setter at round 1. More precisely, a member of either of these groups is appointed as agenda setter at round 2 with probability equal to the share of the parliamentary seats of his group, relative to the total number of seats of the two groups. Then, the second agenda setter has the opportunity to form a government and formulates a coalition proposal to another group of his choice. If no proposal is approved at round 2, the game ends and the status quo policy is implemented. We assume that the status quo policy corresponds to no taxation and no public goods provision. Notice that our modelling of the policy‐making process in a consensual democracy is innovative in at least two dimensions. First, we study a legislative bargaining process between citizen‐candidates representatives. Second, we analyse how the distribution of income shapes fiscal policy outcomes through the non‐standard channel of the bargaining power of the different classes, which is endogenous and turns out to depend on the income distribution itself. 3.1. Entry of Candidates, Voting and Bargaining The equilibrium of the policy formation game must be sequentially rational, which means that the Nash equilibrium at each stage of the game must rationally anticipate its subsequent equilibrium path. The assumption of sincere voting and the citizen‐candidate structure imply that each individual will simply vote for a candidate from his own social group. This fact and the proportionality of the electoral law imply that group j elects a total of ρ mj representatives.13 That is, the parliament is a mirror‐image of society in the sense that the distribution of seats across the three groups exactly reflects the distribution of the population across these groups. The agenda setter at round 1 is a representative of the middle class, which (being the largest class) has the largest number of seats in the parliament. Moreover, if the middle class fails to form a government, the second agenda setter is appointed randomly by nature and chosen between the representatives of the poor and the rich. By assumption, the probability that a poor (rich) representative will be the agenda setter at the second round (conditional of the game reaching it) is equal to the share of the seats of the poor (rich) of the combined number of seats of the poor and of the rich. Hence, is the probability that a poor is appointed as agenda setter at round 2. It is clear that φ can be interpreted as an index of the bargaining power of the poor: the higher is the number of the poor mp, the higher is φ, the higher is the probability that the poor are the agenda setter at the second round, the higher is their expected utility at that stage of the game and, therefore, the higher will be the public good that the middle class agenda setter (at round 1) provides them for any given level of taxation so to accept her government coalition proposal. Moreover, as we will show later, φ is also a measure of income inequality: other things equal, a higher φ corresponds to a more unequal income distribution.14 The legislative bargaining game has a unique (subgame‐perfect) Nash equilibrium. The first agenda setter (from the middle class) formulates a coalition formation proposal based on a fiscal policy programme to another group only, given that no more than that is needed to reach a parliamentary majority. The coalition formation offer leaves the group receiving it indifferent between accepting and rejecting it and the offer is accepted. Therefore, the question we need to answer is: which group (among the poor and the rich) is the cheapest to buy? To answer this question, we first solve our bargaining game by backward induction starting from the second round. We denote the group of the agenda setter with h, the other group part of the government with l and the stage of the game with s. Therefore, τs,h,l is the tax rate proposed to group l by the agenda setter h at round s of the game. The correspondent level of public good received by the group i will be . Similarly, the level of utility of the group i is . 3.1.1. Round 2 of the bargaining game Lemma 1. At round 2 of the bargaining game, the poor are always part of the government coalition; the middle class is so only if the agenda setter is a poor and the rich only if the agenda setter is a rich. Proof. At round 2, the outside option of each group is its status quo utility, i.e. its gross income. Since the agenda setter optimises giving to the coalition partner what is strictly necessary to induce it to accept the policy proposed, the policy menu offered from the agenda setter h to group l satisfies the condition . Consider the schedule defined implicitly by this equation. Holding constant τ2,h,l, this schedule is such that . This means that the richer a group is, the more it has to be compensated in terms of public good provision for any level of taxation. Thus, if the rich representative is appointed agenda setter at the second round, he will always prefer the poor to the middle class as coalition partner. Alternatively, if the second round agenda setter is poor, the middle class will be cheaper to buy than the rich. The legislative bargaining protocol implies that the middle class is never the agenda setter at round 2. Therefore, if at round 2 the agenda setter is rich, then by Lemma 1 the poor will be the coalition partner. The substitution of the equilibrium government budget constraint () in the utility function of the rich implies that their maximisation problem is (3) subject to the participation constrain of the poor . If at round 2 the agenda setter is poor, he makes a coalition with the middle class and the maximisation problem is derived in a similar way. 3.1.2. Round 1 of the bargaining game While the poor are always part of the government coalition if the game reaches round 2 (an off‐equilibrium event), this does not need be the case at round 1. At this stage of the game, the middle class agenda setter will form the government coalition with the group that allows her to reach the highest level of utility from the implemented policy. This policy will be such to leave the group receiving the offer just indifferent between accepting it and going to the second round. As we will see, the expected utility of each group at round 2 depends positively on its probability of being agenda setter at that stage. Hence, the higher is the probability φ of the poor of being agenda setter at round 2, the higher is their expected utility at this stage of the game, the more costly it is for the middle class to buy their vote at round 1 and, therefore, the less likely that they are part of the government coalition. In what follows, we establish a global result which identifies the winning coalition in terms of a critical value of φ. To proceed in this direction, we first define the maximisation problems of the middle class under the two possible coalitions. If the government coalition is made up by the middle class and the rich, the participation constraint of the rich at round 1 is (4) The left hand side of (4) represents the utility of the rich if the middle class’s policy proposal at round 1 is implemented, while the right hand side is their expected utility conditional on the game reaching round 2. Substituting the equilibrium government budget constraint () in the utility function of the middle class, their maximisation problem becomes (5) subject to the participation constraint of the rich (4). This constraint may not be always binding however. This is the case when the dictatorial policy of the middle class gives to the rich a higher utility than their expected utility at round 2. In this situation the consensual democracy equilibrium is equivalent to the dictatorship of the middle class which obtains the maximum level of utility by implementing her unconstrained preferred policy.15 When the participation constraint of the rich is binding, the first order condition relative to the maximisation problem (5) is (6) and this equation allows us to get τ1,b,r and if combined with (4). Then, is obtained from the government budget constraint. If the coalition government is made up by the middle class and the poor, the participation constraint of the poor at round 1 is (7) The left hand side of (7) is the utility of the poor at round 1 if the middle class’s policy proposal is implemented, while the right hand side corresponds to their expected utility if the game reaches the second round. This participation constraint is always binding (if yp > 0), and therefore hold with the equality sign, because the expected utility of the poor at round 2 is at least equal to their level of income (i.e. what they get if the status quo policy is implemented) given that they are always part of the government coalition at this stage. The substitution of the equilibrium government budget constraint () in the utility function of the middle class implies that their maximisation problem can be written as (8) subject to the participation constraint of the poor (7). The first order condition of this problem reads (9) From (9) and (7) we obtain τ1,b,p and , while is derived from the government budget constraint. The next Proposition characterises the outcome of the coalition formation process at the first round of the legislative bargaining game. Proposition 2. There exists a threshold value of φ, φ* ∈  (0,1) such that (i) if φ < φ*, the government coalition is made by the middle class and by the poor; (ii) if φ > φ*, the government coalition is made by the middle class and by the rich. Proof. See Appendix. The intuition for this result is straightforward. When φ is relatively low (φ < φ*), the probability that the poor are the agenda setter at round 2 is also low and so is their expected utility at the second round. This means that their vote is relatively cheap to buy at round 1. On the other hand, when φ is relatively small, 1 − φ is relatively high, and so is the probability of the rich being the agenda setter at round 2, which in turn implies that their expected utility at the second round is high and their vote is costly to buy at round 1. Therefore, there exists a level of φ sufficiently small that the middle class prefers to make a government coalition with the poor because their vote is cheaper to buy (than that of the rich). Clearly, the opposite is true when φ is relatively high (φ > φ*). At the threshold φ*, the middle class is just indifferent between forming a coalition with the rich or with the poor.16 Two issues are worth mentioning. The first is that the sincere voting assumption is important for the result in Proposition 2, as otherwise the poor could vote for the rich when they anticipate that their representatives will not be part of the government. The second is that our result is instead independent on the status quo policy assumed. It is clear that changing the status quo may imply a variation in the government coalitions observed at round 2 with the poor not being always the cheapest to buy. However, what is key for our results is that the expected utility of poor and rich at round 2 is increasing in their probability of being the agenda setter at that stage of the game. And this is always the case because the utility of any group (poor or rich) when it is the agenda setter is higher than when it is not (independently that it belongs or not to the ruling coalition). The status quo may affect the absolute level of the expected utility of each class at round 2 for a given level of φ and, therefore, the level of the threshold φ*. However, the middle class will prefer the poor to the rich when φ is lower than this threshold and vice versa. 4. The Size of Government Across Constitutions and Coalitions While the tax rate chosen under a majoritarian constitution is only a function of the income of the rich (relative to the average one), the tax rate in a consensual democracy under the two possible coalitions is a function of the income distribution, i.e. of both the incomes of the classes () and the value of φ. A comparison of these tax rates is not straightforward due to the strong non‐linearity present in the first order conditions defining them. However, by making some assumptions on the levels of income of the classes and the utility function of the individuals we can state the results presented and discussed below. First, we assume a power function specification for the utility derived from the public good: H(gj) = A(gj)α, with α ∈ (0,1) and A > 0. Result 1. If yp =  0 and the income of the rich is sufficiently high relative to the average income, then the taxes set in equilibrium can be ordered as follows: τr < τ 1 ,b,r < τ 1 ,b,p.17 Proof. See Appendix. Without the above assumptions it is not possible to derive further analytical results and, therefore, we have run several numerical simulations. The simulations show that the two assumptions on the income of the poor and the rich, which guarantee that Result 1 holds and can be proved analytically, are not necessary. Nevertheless, these assumptions provide an insight into the characteristics of the income distribution that lead to that result. In particular, the result that τr < τ1,b,r < τ1,b,p for all φ is easy to obtain when the income of the poor yp and the income of the rich yr are respectively low and high with respect to the average income or, in other words, when there is enough dispersion in the income of the three classes. From the numerical analysis we have obtained two interesting results. First, a level of yr sufficiently high relative to is enough to obtain the taxation ranking of Result 1 (τr < τ1,b,r < τ1,b,p) even when yp and yb are both very close to . Second, with an extremely equal income distribution, taxation in majoritarian democracy is always higher than taxation in consensual democracy (regardless of the ruling coalition). These results can be illustrated using two parameterisations for the income of the three classes.18 The first is: yp = 0.9, yb = 0.95, , yr=1.1. In the second one we just change the income of the rich and use yr = 4. The first parameterisation corresponds to a very equal society given that the income of the poor is only 10% lower than the average income while the income of the rich is only 10% higher than the mean.19Figure 2 shows the tax rates obtained with these two parameterisations. In this case the tax rate set in consensual democracy is lower than the tax rate set in majoritarian democracy. The increase in the income of the rich from 1.1 to 4 leads to a reduction in the tax rate set in majoritarian democracy higher than the reduction in the tax rate of the middle class and rich coalition up to the point that the ranking of tax rates of Result 1 generally holds.20 The explanation for this result is the following. Fig. 2. Open in new tabDownload slide Taxation with yp =  0.9, yb =  0.95, ,yr =  1.1
Note: The Schedules with (new ) and φ** refer to yr = 4. Fig. 2. Open in new tabDownload slide Taxation with yp =  0.9, yb =  0.95, ,yr =  1.1
Note: The Schedules with (new ) and φ** refer to yr = 4. The fiscal policy of a single group government should involve, other things equal (i.e. if all groups have a similar income as it is in the first parameterisation), a higher tax rate and total expenditure than the policy of a two groups government coalition regardless of how the tax revenues are divided among the public goods provided. Indeed, recall that a public good is provided only if the group which likes it is part of the government coalition and the optimal tax rate of a group is such that the marginal cost of taxation equals the marginal benefit from the public good provision. While the marginal cost of taxation is independent of the number of public goods that are financed with the tax revenues (i.e. the number of groups in the government coalition), the marginal benefit from that increase in taxation decreases with the number of public goods among which this increase in taxation is split. This implies that the tax rate should decrease with the number of groups in the government coalition, which in turn implies that, other things equal, the tax rate in majoritarian democracy should be higher than tax rate in consensual democracy. However, we now need to explain why Result 1 and most numerical simulations lead to the opposite result, namely that generally the tax rate in consensual democracy is higher than the tax rate in majoritarian democracy. This result is due to the fact that in majoritarian democracy fiscal policy is decided by the group with the highest level of income (the rich), while in consensual democracy fiscal policy is chosen by a government coalition representing two groups with an average level of income lower than the income of the rich. Similarly, the middle class and poor coalition taxes and spends more than the middle class and rich one exactly because it contains a group (the poor) with lower income. If the incomes of the three classes are sufficiently spread, then the latter effect more than compensates the effect (described above) generated by the number of groups in the government leading to the taxation ranking of Result 1 (τr < τ1,b,r < τ1,b,p). There are also other features of the relationship between the tax rates across constitutions for different income distributions that deserve to be analysed more deeply. Our numerical simulations have shown that taxation in majoritarian democracy is lower than taxation under the middle class and poor coalition even if the dispersion in the income levels of the three classes is very small.21 A slightly higher spread in this distribution is necessary if we want the tax rate of the middle class and rich coalition to be higher than the tax rate in majoritarian democracy for all φ > φ*. As it will be made clear in the next Section, we are interested in the case where τr < τ1,b,r when φ > φ* and the numerical simulations suggest that this result holds if there is a minimum degree of dispersion in the income levels of the three groups. Indeed, using the previous parameterisation (yp = 0.9, yb = 0.95, ) and increasing the income of the rich to 1.6 is enough to have τr < τ1,b,r for all φ > φ* and a further increase in yr implies that τr < τ1,b,r also for most values of φ < φ*.22 The result of the numerical simulation with such parameters is reported in Figure 3, while Figure 4 presents an example with a different parameterisation.23 Fig. 4. Open in new tabDownload slide Taxation with yp =  0.3,yb = 0.8, ,yr =  1.6
Note: The Schedules with (new ) and φ** refer to yr =  2.4. Fig. 4. Open in new tabDownload slide Taxation with yp =  0.3,yb = 0.8, ,yr =  1.6
Note: The Schedules with (new ) and φ** refer to yr =  2.4. Fig. 3. Open in new tabDownload slide Taxation with yp = 0.9,yb = 0.95, , yr = 1.6
Note:The Schedules with (new ) and φ** refer to yr = 2.5. Fig. 3. Open in new tabDownload slide Taxation with yp = 0.9,yb = 0.95, , yr = 1.6
Note:The Schedules with (new ) and φ** refer to yr = 2.5. 5. Income Inequality and Constitutional Choice Having characterised the political equilibrium of the model under the two possible institutional arrangements, we now turn to the question of which of them would be chosen ex ante by society, when the constitution is endogenous. We assume that the process of constitutional choice takes place in an original position where individuals know their class‐status and preferences and before any other politico‐economic interaction. We also assume that the decision is taken by simple majority voting and that the available alternatives are the two constitutional regimes we have considered. Given that there is no veil of ignorance and uncertainty, individuals correctly anticipate what their level of utility would be under the two possible constitutions and vote consequentially. Finally, we assume that there is a minimum degree of dispersion in the income levels of the three groups so that the tax rates ranking of Result 1 holds, or at least that the tax rate set under the middle class and rich coalition is higher than the tax rate under majoritarian democracy when φ > φ*. Proposition 3. If φ < φ*, society chooses consensual democracy while it prefers majoritarian democracy when φ > φ*. Proof. It is clear that for the rich and the middle class the constitutional choice has a trivial, albeit opposite, solution. Since majoritarian democracy expresses the dictatorship of the rich, they will prefer it unconditionally. Similarly, given that the middle class has the relative majority of votes, which allows it to have the first agenda setter in the legislative bargaining game, she will prefer the consensual constitution unconditionally.24 The most interesting decision is the one of the poor, who turn out to be the swing voters. The poor do gain from the higher political inclusion which is typical of consensual democracy, only if they are part of the government coalition as partner of the middle class agenda setter. In this case they are clearly better off than they are in a majoritarian setting.25 However, we know that this need not always be the case, since the ruling coalition does not include them whenever φ ∈ (φ*, 1). In this instance the poor are actually worse‐off in a consensual democracy: they do pay higher taxes (see the discussion in the previous Section) but also get no provision at all of their specific public good. Therefore, if φ ∈ (0, φ*) the majority prefers consensual democracy while majoritarian democracy is chosen when φ ∈ (φ*, 1). We now present two mean preserving spreads of the distribution of income which show that an increase in income inequality makes the adoption of a majoritarian democracy more likely. Mean Preserving Spread 1. We first consider a transformation of the income distribution that affects the size of the three classes. We suppose that mp and mr increase and that mb decreases in such a way that both the size of the population and the average level of income remain constant. Then, society has a smaller middle class, more rich and more poor; that is, it is more unequal. We now show that, whilst the threshold φ* is not affected by this transformation, the value of φ necessarily increases. Therefore, it becomes more likely that φ belongs to the interval (φ*,1) and that a majoritarian democracy is chosen. To this end, let us consider the definition of the average income, , and divide both sides by (1−mb)=(mp+mr) taking into account that φ = mp/(mp + mr). Then, rearranging terms we get (10) A decrease in mb implies a reduction in the right hand side of (10) as . Therefore, φ must increase given that yr > yp. This means that φ can also be interpreted as a measure of income inequality. Again, our model predicts that when income inequality is relatively low (φ < φ*) society prefers a consensual democracy, while it chooses a majoritarian system when income inequality is relatively high (φ > φ*). Mean Preserving Spread 2. Another mean preserving spread we consider is generated by a transformation of both the size of the classes and the income of the rich. In particular, we analyse the effect on constitutional choice of an increase in inequality caused by the increase in the income of the rich, accompanied by an equi‐proportional increase in the number of the rich and of the poor (and a reduction of the middle class) so that both φ and are unaffected.26 Given that φ remains constant, we need to determine how φ* changes to understand which constitution is more likely to be chosen by the society. To this end, we need to find out how the two schedules representing the utility of the middle class under the two possible coalitions vary as yr increases. First, observe that the utility of the middle class in the coalition with the poor does not depend on the income of the rich yr and therefore it is not affected by any variation of it. The utility of the middle class in the government coalition with rich is instead a function of yr. We are not able to show analytically how this schedule changes with the income of the rich but the numerical simulations we discussed in the previous Section point out that it shifts upward (i.e. increases) as yr goes up leading to a reduction in φ*.27 This means that this increase in income inequality leads to a reduction of the range (0, φ*) where consensual democracy is chosen and, therefore, it makes more likely the adoption of a majoritarian constitution. Finally, it is worth emphasising that, although the model is purely static, it contains forces that would make it dynamically stable. If income inequality is relatively high, society should choose a majoritarian constitution. The fiscal policy in this constitutional system should favour the rich and therefore not reduce (or even increase) the initial degree of inequality. Then, society should continue to prefer a majoritarian constitution. Conversely, if income inequality is relatively low, society will prefer a consensual democracy under which fiscal policy generally reflects the preferences of middle class and poor. This should lead to the provision of public goods that reduce or maintain the initial level of inequality and, therefore, a majority for a consensual constitution. 6. Some Evidence on Class Politics and Constitutional Choice The theory presented in this article has many interesting predictions. The main result is that income inequality has an important effect on the choice of the constitution. While a test of this prediction is reported in the next Section, we here discuss other results of our theory that are worth emphasising in relation to the available evidence. A first prediction of our theory is that consensual democracies should have bigger governments than majoritarian ones and a larger part of government expenditure should go to the advantage of a greater number of social groups and, in particular, to lower income individuals (the poor and the middle class). As emphasised in the Introduction, these results are in line with the existing theoretical literature (Persson and Tabellini, 2000; Lizzeri and Persico, 2001) and with the empirical results of Persson and Tabellini (2003, 2004), Milesi‐Ferretti et al. (2002) and Lijphart (1999). The novelty of our theory is in the specific mechanism leading to this result. It is usually argued that public expenditure is higher under proportional electoral systems than under majoritarian ones because the former favours the representation of many groups and the formation of multi‐party coalition governments, which in turn spend more because they need to please broader and more diverse constituencies than single‐party executives. Instead, in our model, the same prediction is due to a selection bias in the composition of the government coalition as consensual democracies should be expected to be ruled relatively more often by centre‐left coalitions, while the fiscally conservative right should have an advantage in majoritarian constitutions. While we do not pursue an empirical analysis on this point here (although interesting, this is not the key finding of our article), it is worth noting that such a result is consistent with the existent empirical evidence about the effects of the government’s ideology on fiscal policy outcomes suggesting that left‐wing executives are willing to tax and spend more than right‐wing ones; see on this Alesina et al. (1997) and the references cited therein, as well as Perotti and Kontopoulos (2002). Our mechanism is also in accordance with the findings of Powell (2002) which, in a sample of 17 advanced democracies for the period 1978–94, show that 58% of the governments in PR systems were leftist, while this fraction was only 37% in majoritarian democracies. A similar result can be obtained using an index of partisanship of the government on the left–right dimension compiled by Cusack (1997). The index ranges from 1 (extreme left) to 4 (extreme right) and it is provided for 16 OECD countries for the periods 1950–9, 1960–9, 1970–9, 1980–91.28 Over the whole period 1950‐91, the average value of the Cusack index is 3.49 in majoritarian democracies (Australia, Canada, UK, US and France) and 2.92 in consensual democracies (Austria, Belgium, Denmark, Finland, Germany, Italy, Norway, the Netherlands, Sweden, Switzerland and Japan) and this difference is statistically significant at 5% level. 29 These results corroborate the prediction of our model according to which consensual democracies should be ruled relatively more often by centre‐left governments whereas majoritarian democracy should advantage conservative parties.30 Our model has the prediction that the poor may vote for the rich in the majoritarian systems. This finding is in accordance with previous results obtained in models of public provision of private goods (Epple and Romano, 1996) where ‘extreme coalitions’ (i.e. coalitions made up by the rich and the poor) emerge in equilibrium and offer a rational explanation for the ‘Reagan Democrat’ phenomenon (Greenberg, 1996), namely for the working‐class voter supporting low tax and socially conservative policies.31 The results of our model can also help us to understand the relationship between the distribution of income and fiscal policy outcomes better. The standard result of the political economy models based on the theory of the median voter (Meltzer and Richard, 1981) is that higher (pre‐tax) inequality should be expected to generate political support for a larger fiscal redistribution of resources. This theory, however, does not seem to be corroborated by the data (Perotti, 1996; Bénabou, 2000) which, if anything, indicate that more equal societies tend to redistribute more rather then less. Our theory suggests that the absence of a negative relationship between inequality and redistribution may be due to the fact that more unequal societies are more likely to choose a majoritarian system, which in turn generates lower redistribution.32 Moreover, income inequality may also have ambiguous effects for a given constitution. In a consensual democracy, an increase in inequality generally leads to an increase in taxation and redistribution for a given government coalition but has the opposite effect if it leads to the formation of a centre‐right majority instead than a centre‐left one. In a majoritarian democracy, more inequality affects taxation and redistribution only if it is associated to a variation in the income of the rich. Another prediction of our model concerns the preferences of the social classes for the constitution. The rich should prefer the majoritarian democracy, the middle class the consensual one, while the attitude of the poor should depend on the level of income inequality. They are supposed to prefer the consensual system for low levels of inequality and vice versa. Identifying the preferences of the various social classes for the type of constitution is not an easy task and it goes beyond the scope of this article to provide a deep analysis on this point. However, we now present some historical examples of constitutional choice that appear to be in accordance with our theory. In particular, we first discuss three cases (US, UK and Chile) of choice of majoritarian constitutions that support the idea that the rich prefer this political system. Then, we analyse the adoption of consensual constitutions in continental European countries in the first half of the twentieth century and argue that this choice was generally influenced by centre‐left forces in a period of low and/or declining income inequality. The US has the oldest written constitution of the world, dating from 1787. It was drafted by a Constitutional Convention of delegates from all States (with the exception of Rhode Island). Apart from some relatively minor changes, it has remained essentially the same up today. An economic analysis of the American constitution has been provided by Beard (1913), which shares with ours the basic premise that key constitutional principles ought to be interpreted as reflecting the interests of particular social groups or classes as opposed to the ‘public good’. In particular, Beard points out that the economic interests of the members of the Convention essentially corresponded only to those of the rich (the commercial and financial elite as well as of the landlords), who were concerned to secure individual property rights and guarantee the best possible institutional framework for private economic activity. Correspondingly, the middle and the lower classes had very little if any voice at all in the Convention, due to both the strong franchise restriction of the time and the ‘class‐consciousness’ of many of the less well‐off among the enfranchised. Beard also demonstrates the extraordinary awareness of the economic elite and of its political and intellectual leadership, about the nature of its interests in the process of constitution making.33 Remarkably, in the tenth number of The Federalist, Madison argues that: ‘The first object of government is the protection of the diversity in the faculties of men, from which the rights of property originate’, and that this requires the creation of an institutional framework to prevent the exploitation of the rich by the lower classes. England has the oldest unwritten constitution of the world. It consists in a collection of different documents including the Magna Carta of 1215, the Bill of Rights of 1689, commonly observed practices and conceptions, as well as some laws. Despite experiencing significant transformations over time, including a gradual extension of franchise since the first decades of the nineteenth century (1832, 1867 and 1884), the constitution always remained majoritarian. The making of the English constitution, like that one of the US, occurred over a period of time during which the political voice of the upper classes was overwhelming and is another example of the preference of the rich for the majoritarian model. The rich have occasionally been able to impose their constitutional preferences even in much more recent times and, as suggested by our theory, they have opted for the majoritarian model. A notable example is the majoritarian constitution of Chile implemented under the leadership of General Pinochet, who arguably represented the political and economic interests of the elites. A politico‐economic analysis of this constitution has been offered by Baldez and Carey (1999), who assert (p. 52) that: ‘The Chilean military regime consciously crafted the 1980 Constitution, the budget process, and the electoral system in order to constrain the policy outcomes generated by elected civilian politicians. To a certain extent, the institutional engineers appear to have succeeded – particularly in designing an electoral system that has disproportionately rewarded the right and a budget process that generally encourages fiscal austerity.’ The model of consensual democracy has been adopted extensively in continental Europe in the first half of the twentieth century. Our theory predicts that this should have been the choice of the centre and leftist parties in the presence of low income inequality. Both conditions are not easy to identify precisely for many countries but, as we now show, the available historical evidence is broadly consistent with this result of our theory. Proportional electoral systems have been first introduced in Nordic countries (Denmark, Norway, Sweden and Finland) and elsewhere in Northern Europe (the Netherlands) between the nineteenth and the early twentieth century. By 1921, all Nordic countries had adopted some form of proportional representation (PR) none of which has been discarded afterwards, even for a short period; see for instance Lakeman and Lambert (1955). Other European countries, such as France, Germany and Italy, adopted PR in 1919, after World War I. This system was abandoned at some point during the interwar period and then reintroduced immediately after World War II.34 The first question to address is whether inequality was low in the above cited countries. Although the available data on income inequality are limited and not very precise, it can be argued this seems to have been the case. Piketty (2007) (see also the references therein) documents the occurrence in France of a sharp fall in inequality associated to a sensible drop in the top percentile income share during the period 1914–45.35 This trend has been driven primarily by the shocks represented by the two World Wars (especially the second one) and by the events of the interwar period (inflation and the Great Depression). As we said, in the immediate aftermath of both wars, France switched to PR. Piketty (2007) makes clear that this reduction in income inequality was not limited to France but common to many European countries, especially those that were hit by the two World Wars.36 This is consistent with countries, such as Italy and Germany, switching to PR after the end of both wars.37 For European countries that chose their constitution at the beginning of the twentieth century, we can obtain some data on income distribution from Morrisson (2000). He suggests that income inequality in Denmark at the time of the adoption of PR (1920) was overall relatively low.38 For Norway, Morrisson (Table 2, p. 224) documents a fall in the Gini coefficient from 0.68 in 1855 to 0.34 in 1920, a low level of inequality consistent with the adoption of PR in 1921. The Netherlands adopted a proportional electoral law as early as 1917. While no precise evidence at the time of the adoption of the constitution is available, income inequality seems to have generally decreased in the Netherlands since the end of the nineteenth century. Table 2 
Income Distribution and Constitution Selection (a) Income distribution in parliamentary democracies . . 1st quartile 21.5≤gini≤26.98 . 2nd quartile 30.06≤gini≤40.58 . 3rd quartile 41.71≤gini≤45.49 . 4th quartile 46.02≤gini≤57.4 . Total .  Majoritarian Ukraine Nepal, Bangladesh, Malaysia India, Bahamas, Barbados Trinidad & T., Japan, Singapore, France, Thailand, Jamaica, Botswana  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia Taiwan, Spain, Italy, Germany, Portugal Greece, Estonia, Fiji, Senegal Turkey  Majoritarian 1 3 3 7 14  Consensual 7 5 4 1 17  Total 8 8 7 8 31 Below the median Above the median Majoritarian 4 10 14 Consensual 12 5 17 Total 16 15 31 (a) Income distribution in parliamentary democracies . . 1st quartile 21.5≤gini≤26.98 . 2nd quartile 30.06≤gini≤40.58 . 3rd quartile 41.71≤gini≤45.49 . 4th quartile 46.02≤gini≤57.4 . Total .  Majoritarian Ukraine Nepal, Bangladesh, Malaysia India, Bahamas, Barbados Trinidad & T., Japan, Singapore, France, Thailand, Jamaica, Botswana  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia Taiwan, Spain, Italy, Germany, Portugal Greece, Estonia, Fiji, Senegal Turkey  Majoritarian 1 3 3 7 14  Consensual 7 5 4 1 17  Total 8 8 7 8 31 Below the median Above the median Majoritarian 4 10 14 Consensual 12 5 17 Total 16 15 31 (b) Income distribution in the whole sample. Classification of majoritarian and consensual based on the electoral system . . 1st quartile 21.5≤gini≤35.45 . 2nd quartile 37.11≤gini≤43 . 3rd quartile 45≤gini≤50.11 . 4th quartile 51≤gini≤68.6 . Total .  Majoritarian Ukraine, Belarus, Nepal, Bangladesh Pakistan, Malaysia, Ghana, India Bahamas, Barbados, Trinidad & T., Philippines Japan, Uganda, Singapore, France, Zambia Thailand, Jamaica, Botswana, Chile, Zimbabwe, Malawi  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia, Taiwan, South Korea, Sri Lanka Spain, Italy, Paraguay, Germany, Portugal, Uruguay, Greece, Argentina, Estonia, Fiji, Senegal Dominican R., Venezuela, El Salvador, Bolivia, Guatemala Turkey, Peru, Brazil, Mexico, Nicaragua, Colombia, Honduras, Ecuador  Majoritarian 4 4 9 6 23  Consensual 10 11 5 8 34  Total 14 15 14 14 57 Below the median Above the median  Majoritarian 8 15 23  Consensual 21 13 34  Total 29 28 57  Majoritarian Ukraine, Belarus, Nepal, South Korea, Sri Lanka, Bangladesh Pakistan, Malaysia, Paraguay, Ghana, Uruguay, Argentina, India Dominican R., Bahamas, Barbados, Trinidad & T., Venezuela, Philippines, Japan, Uganda, Singapore, El Salvador, France, Bolivia, Guatemala, Zambia Thailand, Jamaica, Peru, Brazil, Mexico, Nicaragua, Botswana, Chile, Colombia, Honduras, Zimbabwe, Ecuador, Malawi  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia, Taiwan Spain, Italy, Germany, Portugal, Greece, Estonia, Fiji, Senegal Turkey  Majoritarian 6 7 14 13 40  Consensual 8 8 0 1 17  Total 14 15 14 14 57 Below the median Above the median  Majoritarian 13 27 40  Consensual 16 1 17  Total 29 28 57 (b) Income distribution in the whole sample. Classification of majoritarian and consensual based on the electoral system . . 1st quartile 21.5≤gini≤35.45 . 2nd quartile 37.11≤gini≤43 . 3rd quartile 45≤gini≤50.11 . 4th quartile 51≤gini≤68.6 . Total .  Majoritarian Ukraine, Belarus, Nepal, Bangladesh Pakistan, Malaysia, Ghana, India Bahamas, Barbados, Trinidad & T., Philippines Japan, Uganda, Singapore, France, Zambia Thailand, Jamaica, Botswana, Chile, Zimbabwe, Malawi  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia, Taiwan, South Korea, Sri Lanka Spain, Italy, Paraguay, Germany, Portugal, Uruguay, Greece, Argentina, Estonia, Fiji, Senegal Dominican R., Venezuela, El Salvador, Bolivia, Guatemala Turkey, Peru, Brazil, Mexico, Nicaragua, Colombia, Honduras, Ecuador  Majoritarian 4 4 9 6 23  Consensual 10 11 5 8 34  Total 14 15 14 14 57 Below the median Above the median  Majoritarian 8 15 23  Consensual 21 13 34  Total 29 28 57  Majoritarian Ukraine, Belarus, Nepal, South Korea, Sri Lanka, Bangladesh Pakistan, Malaysia, Paraguay, Ghana, Uruguay, Argentina, India Dominican R., Bahamas, Barbados, Trinidad & T., Venezuela, Philippines, Japan, Uganda, Singapore, El Salvador, France, Bolivia, Guatemala, Zambia Thailand, Jamaica, Peru, Brazil, Mexico, Nicaragua, Botswana, Chile, Colombia, Honduras, Zimbabwe, Ecuador, Malawi  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia, Taiwan Spain, Italy, Germany, Portugal, Greece, Estonia, Fiji, Senegal Turkey  Majoritarian 6 7 14 13 40  Consensual 8 8 0 1 17  Total 14 15 14 14 57 Below the median Above the median  Majoritarian 13 27 40  Consensual 16 1 17  Total 29 28 57 Open in new tab Table 2 
Income Distribution and Constitution Selection (a) Income distribution in parliamentary democracies . . 1st quartile 21.5≤gini≤26.98 . 2nd quartile 30.06≤gini≤40.58 . 3rd quartile 41.71≤gini≤45.49 . 4th quartile 46.02≤gini≤57.4 . Total .  Majoritarian Ukraine Nepal, Bangladesh, Malaysia India, Bahamas, Barbados Trinidad & T., Japan, Singapore, France, Thailand, Jamaica, Botswana  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia Taiwan, Spain, Italy, Germany, Portugal Greece, Estonia, Fiji, Senegal Turkey  Majoritarian 1 3 3 7 14  Consensual 7 5 4 1 17  Total 8 8 7 8 31 Below the median Above the median Majoritarian 4 10 14 Consensual 12 5 17 Total 16 15 31 (a) Income distribution in parliamentary democracies . . 1st quartile 21.5≤gini≤26.98 . 2nd quartile 30.06≤gini≤40.58 . 3rd quartile 41.71≤gini≤45.49 . 4th quartile 46.02≤gini≤57.4 . Total .  Majoritarian Ukraine Nepal, Bangladesh, Malaysia India, Bahamas, Barbados Trinidad & T., Japan, Singapore, France, Thailand, Jamaica, Botswana  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia Taiwan, Spain, Italy, Germany, Portugal Greece, Estonia, Fiji, Senegal Turkey  Majoritarian 1 3 3 7 14  Consensual 7 5 4 1 17  Total 8 8 7 8 31 Below the median Above the median Majoritarian 4 10 14 Consensual 12 5 17 Total 16 15 31 (b) Income distribution in the whole sample. Classification of majoritarian and consensual based on the electoral system . . 1st quartile 21.5≤gini≤35.45 . 2nd quartile 37.11≤gini≤43 . 3rd quartile 45≤gini≤50.11 . 4th quartile 51≤gini≤68.6 . Total .  Majoritarian Ukraine, Belarus, Nepal, Bangladesh Pakistan, Malaysia, Ghana, India Bahamas, Barbados, Trinidad & T., Philippines Japan, Uganda, Singapore, France, Zambia Thailand, Jamaica, Botswana, Chile, Zimbabwe, Malawi  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia, Taiwan, South Korea, Sri Lanka Spain, Italy, Paraguay, Germany, Portugal, Uruguay, Greece, Argentina, Estonia, Fiji, Senegal Dominican R., Venezuela, El Salvador, Bolivia, Guatemala Turkey, Peru, Brazil, Mexico, Nicaragua, Colombia, Honduras, Ecuador  Majoritarian 4 4 9 6 23  Consensual 10 11 5 8 34  Total 14 15 14 14 57 Below the median Above the median  Majoritarian 8 15 23  Consensual 21 13 34  Total 29 28 57  Majoritarian Ukraine, Belarus, Nepal, South Korea, Sri Lanka, Bangladesh Pakistan, Malaysia, Paraguay, Ghana, Uruguay, Argentina, India Dominican R., Bahamas, Barbados, Trinidad & T., Venezuela, Philippines, Japan, Uganda, Singapore, El Salvador, France, Bolivia, Guatemala, Zambia Thailand, Jamaica, Peru, Brazil, Mexico, Nicaragua, Botswana, Chile, Colombia, Honduras, Zimbabwe, Ecuador, Malawi  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia, Taiwan Spain, Italy, Germany, Portugal, Greece, Estonia, Fiji, Senegal Turkey  Majoritarian 6 7 14 13 40  Consensual 8 8 0 1 17  Total 14 15 14 14 57 Below the median Above the median  Majoritarian 13 27 40  Consensual 16 1 17  Total 29 28 57 (b) Income distribution in the whole sample. Classification of majoritarian and consensual based on the electoral system . . 1st quartile 21.5≤gini≤35.45 . 2nd quartile 37.11≤gini≤43 . 3rd quartile 45≤gini≤50.11 . 4th quartile 51≤gini≤68.6 . Total .  Majoritarian Ukraine, Belarus, Nepal, Bangladesh Pakistan, Malaysia, Ghana, India Bahamas, Barbados, Trinidad & T., Philippines Japan, Uganda, Singapore, France, Zambia Thailand, Jamaica, Botswana, Chile, Zimbabwe, Malawi  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia, Taiwan, South Korea, Sri Lanka Spain, Italy, Paraguay, Germany, Portugal, Uruguay, Greece, Argentina, Estonia, Fiji, Senegal Dominican R., Venezuela, El Salvador, Bolivia, Guatemala Turkey, Peru, Brazil, Mexico, Nicaragua, Colombia, Honduras, Ecuador  Majoritarian 4 4 9 6 23  Consensual 10 11 5 8 34  Total 14 15 14 14 57 Below the median Above the median  Majoritarian 8 15 23  Consensual 21 13 34  Total 29 28 57  Majoritarian Ukraine, Belarus, Nepal, South Korea, Sri Lanka, Bangladesh Pakistan, Malaysia, Paraguay, Ghana, Uruguay, Argentina, India Dominican R., Bahamas, Barbados, Trinidad & T., Venezuela, Philippines, Japan, Uganda, Singapore, El Salvador, France, Bolivia, Guatemala, Zambia Thailand, Jamaica, Peru, Brazil, Mexico, Nicaragua, Botswana, Chile, Colombia, Honduras, Zimbabwe, Ecuador, Malawi  Consensual Slovak R., Hungary, Romania, Bulgaria, Czech R., Poland, Latvia, Taiwan Spain, Italy, Germany, Portugal, Greece, Estonia, Fiji, Senegal Turkey  Majoritarian 6 7 14 13 40  Consensual 8 8 0 1 17  Total 14 15 14 14 57 Below the median Above the median  Majoritarian 13 27 40  Consensual 16 1 17  Total 29 28 57 Open in new tab The second question we need to answer is whether the demand of adoption of PR came from centre‐left parties as our theory suggests. A discussion of this issue is presented in Alesina and Glaeser (2004) where they argue that the diffusion of PR across Continental Europe (for the above cited countries) in the twentieth century was related to the political strength enjoyed by the left and by the workers’ movement that succeeded in imposing it on the conservative forces. An exception to this pattern is represented by Sweden where the choice of a proportional electoral system in 1907 was made by the right that feared to lose representation after the extension of franchise. This case is in contradiction with our theory and in accordance with Rokkan (1970) who explains the introduction of PR in Europe as a strategic choice of the traditional nineteenth‐century liberal parties to preserve part of their political power in spite of the gradual franchise extension, and of the consequent increasing importance of new mass political parties, in particular Socialist ones. While the historical example of Sweden, and probably of other countries,39 may not be in accordance with our theory, the broad picture that emerges from the above discussion is that there is some historical evidence supporting our theoretical results. The recent history on constitutional choice also provides some useful information. For example, at the beginning of the 1990s, the Eastern European countries of the former Soviet Block have made a transition to democracy. These countries had an equal income distribution at that time (see Table 2 for details) and have generally chosen a consensual constitution. On the other hand, there are also consensual democracies in Europe that have recently experienced an increase in inequality. For these countries, it is natural to ask whether PR will come in to question or be abandoned. Up until now, there have been very few changes of electoral rules in advanced democracies without any clear pattern and, therefore, to answer this question we will probably need some more time. It is important to keep in mind, however, that an increase in inequality may not be enough to produce a constitutional change, as this requires inequality increasing above a certain threshold. And this may in turn be anticipated by centre‐left governments that have an interest in this not taking place. 7. Empirical Evidence on Inequality and Constitutional Choice In this Section we test the main prediction of our article that more unequal countries are expected to choose a majoritarian democracy while equal societies should prefer a consensual constitution. We test this prediction by analysing the relationship between the type of constitution adopted and the income inequality of the country at the time, or before, the constitution was chosen in order to avoid problems of endogeneity.40 Given that constitutional reforms are rare, a feature potentially representing an equilibrium outcome according to our model, we perform a cross‐sectional analysis using the dataset compiled by Persson and Tabellini (2003, 2004) (PT from now on) to analyse the economic effects of constitutions. As a measure of income inequality, we employ the Gini index and use the dataset compiled by Deininger and Squire (1996). PT report the variables maj and pres that define the electoral system and the form of government for 85 countries that have been selected on the base that they can be classified as free or partly free democracies for the period 1990–8.41 PT classify the electoral rule and the form of government within the countries at the beginning of the 1990s. We have taken this classification and identified the first year when this constitution (represented by maj and pres) was first introduced and summarised it in the variable yearcons. Then, we have selected a Gini coefficient for each country from the Deininger and Squire dataset according to the following rules. We have first considered the high quality data and selected the Gini coefficient correspondent to the year when the constitution was introduced, which is given by the variable yearcons. If there were no data in that year, we have gone backward and selected the first Gini coefficient available. When there were no data before the year of the constitution, we have taken the first Gini coefficient available after that year with the constraint that it was not more than 5 years older.42 Following this procedure, we have obtained a high quality Gini coefficient for 43 countries. When we did not find a high quality Gini coefficient with this procedure, we have relied on the other (low quality) data available in this dataset following the same procedure. When more than one coefficient was available in the same year, we have taken the average. This has allowed us to obtain a Gini coefficient for a total of 57 countries.43 The data from Deininger and Squire specify whether the Gini coefficient is computed using information on income or on expenditure, if the income is gross or net of taxes and if the recipient unit is an individual or a household. Deininger and Squire argue that the most important distinction is between the Gini coefficients that are based on information on income and those based on expenditure. In order to ensure intertemporal and international comparability, they strongly suggest adjusting for differences between income‐based and expenditure‐based coefficients by adding 6.6 points to the latter. We have therefore made this adjustment and denoted by giniycai the Gini coefficient with this correction. An important point that we need to address concerns the classification of the countries into majoritarian and consensual democracies. As explained before, this classification is straightforward for parliamentary systems given that the executive is accountable to the parliament so that the electoral rule, majoritarian versus proportional, is enough to classify the type of democracy. In presidential regimes fiscal policy is the outcome of a bargaining process between the president and the legislative assembly. If the assembly is elected with plurality rule, then it is reasonable to classify these systems as majoritarian democracies given that both subjects that have a role in fiscal policy decisions are elected with a winner‐takes‐all process and are, therefore, expected to have the same fiscally conservative preferences. Instead, when the congress is elected with PR, it is important to understand the power of the president in fiscal policy decisions (see footnote 9). In our sample, the presidential systems with PR are characterised by a relative powerful president and should, therefore, be considered majoritarian democracies. This is confirmed by three considerations. First, PT classify as presidential only those regimes where the confidence of the assembly is not necessary for the executive. Second, the presidential countries with PR in our dataset are classified as ‘direct presidential’ in the Database of Political Institutions.44 Finally, it is worth emphasising that almost all of these countries in our sample are located in Latin America and while the exact distribution of the legislative power varies across countries, according to many scholars of comparative politics the president in Latin American countries typically plays a key role in the legislative process regarding fiscal policy.45 Based on the above considerations, we have structured our empirical analysis as follows. First, we have analysed the relationship between income inequality and electoral system in the sub‐sample of parliamentary democracies. This choice is justified by the fact that our model describes precisely this system and also allows us to consider a more homogenous sample. Second, we have considered the whole sample, classifying the type of democracy on the base of the electoral system only. In this case, the presidential systems with PR enter as consensual democracies. As we will see, this is the worst specification for our theory because the income distribution in these countries is generally very unequal. Finally, we develop the empirical work with the classification that we consider more reasonable and closer to our model, namely by classifying the presidential regimes with PR as majoritarian systems. Table 1 provides descriptive statistics for income inequality in various samples. Among the parliamentary democracies, the countries with a majoritarian electoral rule are more unequal than those with PR. The difference in the average Gini coefficient is almost 10 points and it is statistically significant at the 1% level. The unconditional correlation between the Gini coefficient and a majoritarian electoral system is 0.485. We observe that the average income inequality in presidential regimes is high (the Gini index is about 49) and independent on the electoral formula of the assembly. In our view, this is consistent with the majoritarian characteristics of both systems. When we consider the whole sample and distinguish on the base of the electoral rule, we find that countries with plurality rule have higher income inequality than those with PR (significant at 10% level) but this difference is lower with respect to the sample of parliamentary democracies. The unconditional correlation between the Gini coefficient and the electoral system is positive but not very high (it is 0.174). Finally, the last part of the Table shows the statistics when presidential systems with PR are classified as majoritarian democracies. Majoritarian democracies are much more unequal than consensual ones and the difference in average inequality is about 13 points with a statistically significance at 1% level (the correlation between Gini coefficient and majoritarian democracy is also high, 0.532). Table 2(a) reports the majoritarian and consensual countries in each quartile of the income distribution for the sample of parliamentary democracies. The first quartile contains just one majoritarian country and seven consensual ones, while we observe the opposite pattern in the last quartile of the distribution. The number of majoritarian (consensual) countries is also monotonically increasing (decreasing) in the degree of income inequality. Panel (b) shows the results of the same analysis for the whole sample when we consider the classification based on the electoral system only. As expected from the above discussion, we observe a positive relationship between the degree of income inequality and number of majoritarian democracies, though this is weaker than in the sample of parliamentary democracies. The positive link between inequality and majoritarian electoral system can be appreciated if we consider two groups, countries with inequality below and above the median. The number of majoritarian countries with income inequality higher than the median is the double of those with inequality lower than the median. Finally, Panel (c) provides the results of this analysis when presidential regimes with an assembly elected with PR are classified as majoritarian systems. The positive relationship between income inequality and majoritarian democracy is very strong as, with the exception of one country, there are no consensual democracies with inequality above the median. We can therefore conclude that the unconditional relationship between income inequality and constitution is consistent with our theory. We now present the results of logit regressions when the relationship between inequality and constitution is analysed conditioning for different variables that may potentially affect the choice of the constitution. The most important variable is ethnic fragmentation whose importance for constitutional choice has been emphasised by Lijphart (1999) and Aghion et al. (2004). We always control for it by including the variables ethnic and language from Alesina et al. (2003). This allows us to disentangle different kinds of heterogeneity as the first one is based on a broad measure of ethnicity while the second is strictly based on language. We also control for religious fractionalisation using the variable religion taken from the same authors, but we do not include it in the baseline specification because Alesina et al. (2003) argue that the endogeneity problems for this variable may be more serious.46 We control for country size and level of development by including the log of population in 1960 (lpop_60) and the GDP per capita expressed relative to the US in 1960 (y_60) taken from the Penn World Table 6.1.47 We consider three regional dummies to take into account the role of geography. In particular, we use a dummy variable for continental location in Africa (africa), in eastern and southern Asia (asiae) and southern and central America including the Caribbean (laam). To take into account the possible effects of religious and cultural factors, we control for the percentage of the population in each country professing the Protestant religion in 1980 (prot80), the percentage of the population belonging to the Roman Catholic religion in 1980 (catho80) and, if the majority of population is Confucian‐Buddhist‐Zen (confu). PT argue that ex‐British colonies tend to be parliamentary and majoritarian while all former Spanish–Portuguese colonies are presidential. To take into account these effects, we control for British (col_uk), for Spanish–Portuguese (col_esp) and other (col_oth) colonial origins. All the above variables are taken from the PT dataset. Table 3 presents the relationship between constitution and income inequality in the sample of parliamentary democracies. Column (1) shows the results for our baseline specification where we consider the degree of income inequality and of ethnic fragmentation. Higher inequality is positively correlated with a majoritarian electoral system and this relationship is statistically significant at 1%. The quantitative effect of an increase in income inequality on the probability of adopting a majoritarian system is also very high as the marginal effect at the mean is 0.03.48 In columns (2) and (3) we control for the population and level of income respectively. In column (4), we include dummies for Africa and Asia while the dummy for Latin America is omitted because the four parliamentary democracies in this region are all majoritarian. Column (5) shows the results when we add the religious characteristics of each country that are likely to be correlated with cultural factors. In all specifications, we find that the relationship between inequality and constitution is unchanged with respect to the baseline one.49 In column (6), we control for colonial origins and find that the correlation between income inequality and majoritarian democracy has the expected sign but it is not statistically significant at standard levels. This is due to the inclusion of the dummy variable for former British colonies.50 In this sample, 8 out of 9 ex‐British colonies have a majoritarian electoral rule. It is worth noting that these former British colonies are also very unequal. The mean Gini coefficient for these 9 countries is 46.58 and the most equal one has a coefficient of 38.1. This in turn implies that all former British colonies have a degree of inequality above average (see Table 1). Such collinearity between the Gini coefficient and British colonial origin, combined with measurement errors in inequality, could be at the root of inequality becoming insignificant. In other words, while controlling for colonial origins substantially weakens our results, it is not clear to us whether former colonies have chosen a certain type of democracy to imitate the coloniser’s institutions or because these type of institutions suited well with their economic fundamentals. Another interpretation is that British colonies borrowed not only the majoritarian electoral law but also the pro‐market attitude of England. As the latter might induce high inequality, this may explain the correlation between British colonial origins and high inequality observed in the data.51 We have therefore included in the regression a variable of regulatory quality, taken from Kaufmann et al. (2005), which measures the incidence of market‐unfriendly policies.52 We have used the average value of the first two years available in the dataset, 1996 and 1998. Given that many constitutions have been chosen much before the 1990s, we are here assuming that this variable has not been subject to large time variations. As there could be endogeneity problems with this specification, we have not included it in Table 3. The results (available from the authors) are encouraging. The Gini coefficient is now very similar to all other specifications (0.029) and its p‐value is 5.6%. The dummy variable for former British colonies is still statistically significant while the regulatory quality variable is not. Finally, column (7) shows that the results are robust when we control for religious fractionalisation. Table 3 
Constitution Selection and Income Inequality in Parliamentary Democracies Sample: Parliamentary democracies – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.03 0.031 0.03 0.033 0.032 0.02 0.033 (0.012)*** (0.012)*** (0.012)** (0.016)** (0.016)* (0.015) (0.012)*** ethnic 0.16 0.469 0.152 0.762 0.339 −0.517 −0.129 (0.649) (0.911) (0.637) (0.715) (0.676) (0.818) (0.703) language 0.233 0.031 0.096 −0.758 −0.2 0.561 0.486 (0.504) (0.634) (0.531) (0.759) (0.596) (0.711) (0.506) lpop_60   [0.626]           y_60     [0.555]         regional dummies       [0.215]       prot80, catho80, confu         [0.234]     colonial origins           [0.238]   religion             [0.173] Observations 31 31 31 31 31 31 31 Pseudo R‐squared 0.2 0.21 0.21 0.31 0.29 0.31 0.26 Sample: Parliamentary democracies – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.03 0.031 0.03 0.033 0.032 0.02 0.033 (0.012)*** (0.012)*** (0.012)** (0.016)** (0.016)* (0.015) (0.012)*** ethnic 0.16 0.469 0.152 0.762 0.339 −0.517 −0.129 (0.649) (0.911) (0.637) (0.715) (0.676) (0.818) (0.703) language 0.233 0.031 0.096 −0.758 −0.2 0.561 0.486 (0.504) (0.634) (0.531) (0.759) (0.596) (0.711) (0.506) lpop_60   [0.626]           y_60     [0.555]         regional dummies       [0.215]       prot80, catho80, confu         [0.234]     colonial origins           [0.238]   religion             [0.173] Observations 31 31 31 31 31 31 31 Pseudo R‐squared 0.2 0.21 0.21 0.31 0.29 0.31 0.26 Notes: Robust standard errors in parentheses. * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Logit coefficients are marginal effects on the probability of having a majoritarian system evaluated at the mean. The table includes p‐values for Chi‐square test of joint significance of the control sets. Regional dummies include africa and asiae. Regional dummy laam has not been included because all four parliamentary democracies in Latin America are majoritarian. Colonial origins include col_uk and col_oth but not col_esp as there are no Spanish‐Portuguese colonies in the sample. Open in new tab Table 3 
Constitution Selection and Income Inequality in Parliamentary Democracies Sample: Parliamentary democracies – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.03 0.031 0.03 0.033 0.032 0.02 0.033 (0.012)*** (0.012)*** (0.012)** (0.016)** (0.016)* (0.015) (0.012)*** ethnic 0.16 0.469 0.152 0.762 0.339 −0.517 −0.129 (0.649) (0.911) (0.637) (0.715) (0.676) (0.818) (0.703) language 0.233 0.031 0.096 −0.758 −0.2 0.561 0.486 (0.504) (0.634) (0.531) (0.759) (0.596) (0.711) (0.506) lpop_60   [0.626]           y_60     [0.555]         regional dummies       [0.215]       prot80, catho80, confu         [0.234]     colonial origins           [0.238]   religion             [0.173] Observations 31 31 31 31 31 31 31 Pseudo R‐squared 0.2 0.21 0.21 0.31 0.29 0.31 0.26 Sample: Parliamentary democracies – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.03 0.031 0.03 0.033 0.032 0.02 0.033 (0.012)*** (0.012)*** (0.012)** (0.016)** (0.016)* (0.015) (0.012)*** ethnic 0.16 0.469 0.152 0.762 0.339 −0.517 −0.129 (0.649) (0.911) (0.637) (0.715) (0.676) (0.818) (0.703) language 0.233 0.031 0.096 −0.758 −0.2 0.561 0.486 (0.504) (0.634) (0.531) (0.759) (0.596) (0.711) (0.506) lpop_60   [0.626]           y_60     [0.555]         regional dummies       [0.215]       prot80, catho80, confu         [0.234]     colonial origins           [0.238]   religion             [0.173] Observations 31 31 31 31 31 31 31 Pseudo R‐squared 0.2 0.21 0.21 0.31 0.29 0.31 0.26 Notes: Robust standard errors in parentheses. * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Logit coefficients are marginal effects on the probability of having a majoritarian system evaluated at the mean. The table includes p‐values for Chi‐square test of joint significance of the control sets. Regional dummies include africa and asiae. Regional dummy laam has not been included because all four parliamentary democracies in Latin America are majoritarian. Colonial origins include col_uk and col_oth but not col_esp as there are no Spanish‐Portuguese colonies in the sample. Open in new tab Table 1 
Descriptive Statistics of Income Inequality ( Variable: Giniycai ) . Obs . Mean . Std. Dev. . Min . Max . Sample: PARLIAMENTARY DEMOCRACIES  Majoritarian 14 43.936 8.933 25.71 57.4  Proportional 17 34.179 9.132 21.5 51  Total 31 38.585 10.17 21.5 57.4 Sample: PRESIDENTIAL DEMOCRACIES  Majoritarian 9 48.95 12.723 28.53 68.6  Proportional 17 49.534 9.33 33.64 65.38  Total 26 49.332 10.373 28.53 68.6 Sample: WHOLE – classification: based on the electoral system  Majoritarian 23 45.898 10.596 25.71 68.6  Consensual 34 41.856 11.974 21.5 65.38  Total 57 43.487 11.515 21.5 68.6 Sample: WHOLE – classification: presidential systems with PR enter as majoritarian  Majoritarian 40 47.443 10.118 25.71 68.6  Consensual 17 34.179 9.132 21.5 51  Total 57 43.487 11.515 21.5 68.6 . Obs . Mean . Std. Dev. . Min . Max . Sample: PARLIAMENTARY DEMOCRACIES  Majoritarian 14 43.936 8.933 25.71 57.4  Proportional 17 34.179 9.132 21.5 51  Total 31 38.585 10.17 21.5 57.4 Sample: PRESIDENTIAL DEMOCRACIES  Majoritarian 9 48.95 12.723 28.53 68.6  Proportional 17 49.534 9.33 33.64 65.38  Total 26 49.332 10.373 28.53 68.6 Sample: WHOLE – classification: based on the electoral system  Majoritarian 23 45.898 10.596 25.71 68.6  Consensual 34 41.856 11.974 21.5 65.38  Total 57 43.487 11.515 21.5 68.6 Sample: WHOLE – classification: presidential systems with PR enter as majoritarian  Majoritarian 40 47.443 10.118 25.71 68.6  Consensual 17 34.179 9.132 21.5 51  Total 57 43.487 11.515 21.5 68.6 Open in new tab Table 1 
Descriptive Statistics of Income Inequality ( Variable: Giniycai ) . Obs . Mean . Std. Dev. . Min . Max . Sample: PARLIAMENTARY DEMOCRACIES  Majoritarian 14 43.936 8.933 25.71 57.4  Proportional 17 34.179 9.132 21.5 51  Total 31 38.585 10.17 21.5 57.4 Sample: PRESIDENTIAL DEMOCRACIES  Majoritarian 9 48.95 12.723 28.53 68.6  Proportional 17 49.534 9.33 33.64 65.38  Total 26 49.332 10.373 28.53 68.6 Sample: WHOLE – classification: based on the electoral system  Majoritarian 23 45.898 10.596 25.71 68.6  Consensual 34 41.856 11.974 21.5 65.38  Total 57 43.487 11.515 21.5 68.6 Sample: WHOLE – classification: presidential systems with PR enter as majoritarian  Majoritarian 40 47.443 10.118 25.71 68.6  Consensual 17 34.179 9.132 21.5 51  Total 57 43.487 11.515 21.5 68.6 . Obs . Mean . Std. Dev. . Min . Max . Sample: PARLIAMENTARY DEMOCRACIES  Majoritarian 14 43.936 8.933 25.71 57.4  Proportional 17 34.179 9.132 21.5 51  Total 31 38.585 10.17 21.5 57.4 Sample: PRESIDENTIAL DEMOCRACIES  Majoritarian 9 48.95 12.723 28.53 68.6  Proportional 17 49.534 9.33 33.64 65.38  Total 26 49.332 10.373 28.53 68.6 Sample: WHOLE – classification: based on the electoral system  Majoritarian 23 45.898 10.596 25.71 68.6  Consensual 34 41.856 11.974 21.5 65.38  Total 57 43.487 11.515 21.5 68.6 Sample: WHOLE – classification: presidential systems with PR enter as majoritarian  Majoritarian 40 47.443 10.118 25.71 68.6  Consensual 17 34.179 9.132 21.5 51  Total 57 43.487 11.515 21.5 68.6 Open in new tab Table 4 reports the results for the same analysis using the whole sample with the classification based on the electoral rule. While the unconditional correlation between income inequality and electoral system in this sample is not very high, the correlation becomes strong when we control for ethnic fragmentation. In fact, in the baseline specification reported in column (1) we find that the marginal effect evaluated at the mean of an increase in inequality on the probability of adopting a majoritarian electoral rule is 0.016 and this is statistically significant at 5% level. The linguistic fractionalisation coefficient has also a positive statistically significant effect. The same result is obtained controlling for population and income (columns (2) and (3)). The Gini coefficient has the same magnitude but it is less precisely estimated when we include the three regional dummies. The p‐value of the coefficient is 10.7% however. In columns (5) and (6) we control for religious and for colonial origins respectively. The estimated Gini coefficient is pretty large (about 0.03) and significant at 5% in both specifications. Results are also robust when controlling for religious fractionalisation which is positively correlated with plurality rule (see column (7)). Table 4 
Constitution Selection and Income Inequality in the Whole Sample (Classification Based on the Electoral Rule) Whole sample – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.016 0.016 0.017 0.015 0.032 0.029 0.023 (0.007)** (0.007)** (0.007)** (0.009) (0.014)** (0.013)** (0.009)*** ethnic −0.755 −0.757 −0.729 −0.617 −0.81 −1.089 −0.782 (0.615) (0.615) (0.629) (0.57) (0.511) (0.525)** (0.504) language 1.344 1.344 1.414 0.843 1.081 1.277 1.403 (0.498)*** (0.498)*** (0.499)*** (0.542) (0.402)*** (0.398)*** (0.411)*** lpop_60   [0.980]         y_60     [0.665]         regional dummies       [0.518]       prot80, catho80, confu         [0.027]     colonial origins           [0.004]   religion             [0.012] Observations 56 56 56 56 56 56 56 Pseudo R‐squared 0.21 0.21 0.21 0.24 0.39 0.43 0.34 Whole sample – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.016 0.016 0.017 0.015 0.032 0.029 0.023 (0.007)** (0.007)** (0.007)** (0.009) (0.014)** (0.013)** (0.009)*** ethnic −0.755 −0.757 −0.729 −0.617 −0.81 −1.089 −0.782 (0.615) (0.615) (0.629) (0.57) (0.511) (0.525)** (0.504) language 1.344 1.344 1.414 0.843 1.081 1.277 1.403 (0.498)*** (0.498)*** (0.499)*** (0.542) (0.402)*** (0.398)*** (0.411)*** lpop_60   [0.980]         y_60     [0.665]         regional dummies       [0.518]       prot80, catho80, confu         [0.027]     colonial origins           [0.004]   religion             [0.012] Observations 56 56 56 56 56 56 56 Pseudo R‐squared 0.21 0.21 0.21 0.24 0.39 0.43 0.34 Notes: Robust standard errors in parentheses. * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Logit coefficients are marginal effects on the probability of having a majoritarian system evaluated at the mean. The table includes p‐values for Chi‐square test of joint significance of the control sets. Regional dummies include africa, asiae, laam. Colonial origins include col_uk, col_esp, col_oth. Open in new tab Table 4 
Constitution Selection and Income Inequality in the Whole Sample (Classification Based on the Electoral Rule) Whole sample – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.016 0.016 0.017 0.015 0.032 0.029 0.023 (0.007)** (0.007)** (0.007)** (0.009) (0.014)** (0.013)** (0.009)*** ethnic −0.755 −0.757 −0.729 −0.617 −0.81 −1.089 −0.782 (0.615) (0.615) (0.629) (0.57) (0.511) (0.525)** (0.504) language 1.344 1.344 1.414 0.843 1.081 1.277 1.403 (0.498)*** (0.498)*** (0.499)*** (0.542) (0.402)*** (0.398)*** (0.411)*** lpop_60   [0.980]         y_60     [0.665]         regional dummies       [0.518]       prot80, catho80, confu         [0.027]     colonial origins           [0.004]   religion             [0.012] Observations 56 56 56 56 56 56 56 Pseudo R‐squared 0.21 0.21 0.21 0.24 0.39 0.43 0.34 Whole sample – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.016 0.016 0.017 0.015 0.032 0.029 0.023 (0.007)** (0.007)** (0.007)** (0.009) (0.014)** (0.013)** (0.009)*** ethnic −0.755 −0.757 −0.729 −0.617 −0.81 −1.089 −0.782 (0.615) (0.615) (0.629) (0.57) (0.511) (0.525)** (0.504) language 1.344 1.344 1.414 0.843 1.081 1.277 1.403 (0.498)*** (0.498)*** (0.499)*** (0.542) (0.402)*** (0.398)*** (0.411)*** lpop_60   [0.980]         y_60     [0.665]         regional dummies       [0.518]       prot80, catho80, confu         [0.027]     colonial origins           [0.004]   religion             [0.012] Observations 56 56 56 56 56 56 56 Pseudo R‐squared 0.21 0.21 0.21 0.24 0.39 0.43 0.34 Notes: Robust standard errors in parentheses. * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Logit coefficients are marginal effects on the probability of having a majoritarian system evaluated at the mean. The table includes p‐values for Chi‐square test of joint significance of the control sets. Regional dummies include africa, asiae, laam. Colonial origins include col_uk, col_esp, col_oth. Open in new tab Table 5 shows the results when presidential regimes with PR are classified as majoritarian systems. The relationship between inequality and majoritarian democracy is very strong. The marginal effect on the probability of adopting a majoritarian democracy evaluated at the mean is large, about 0.024, and statistically significant at 1% level in all specifications.53 Table 5 
Constitution Selection and Income Inequality When Presidential Regimes With PR are Classified as Majoritarian Democracies Whole sample – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.024 0.024 0.023 0.025 0.025 0.022 0.025   (0.006)*** (0.006)*** (0.006)*** (0.008)*** (0.007)*** (0.008)*** (0.007)*** ethnic 0.153 0.222 0.125 0.339 0.31 0.06 0.12   (0.283) (0.299) (0.267) (0.329) (0.311) (0.283) (0.296) language 0.162 0.129 0.028 ‐0.071 0.072 0.054 0.137   (0.199) (0.227) (0.222) (0.358) (0.214) (0.229) (0.208) lpop_60   [0.538]         y_60     [0.167]         regional dummies       [0.297]       prot80, catho80, confu         [0.198]     colonial origins           [0.401]   religion             [0.261] Observations 56 56 56 56 56 56 56 Pseudo R‐squared 0.28 0.29 0.30 0.33 0.34 0.32 0.31 Whole sample – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.024 0.024 0.023 0.025 0.025 0.022 0.025   (0.006)*** (0.006)*** (0.006)*** (0.008)*** (0.007)*** (0.008)*** (0.007)*** ethnic 0.153 0.222 0.125 0.339 0.31 0.06 0.12   (0.283) (0.299) (0.267) (0.329) (0.311) (0.283) (0.296) language 0.162 0.129 0.028 ‐0.071 0.072 0.054 0.137   (0.199) (0.227) (0.222) (0.358) (0.214) (0.229) (0.208) lpop_60   [0.538]         y_60     [0.167]         regional dummies       [0.297]       prot80, catho80, confu         [0.198]     colonial origins           [0.401]   religion             [0.261] Observations 56 56 56 56 56 56 56 Pseudo R‐squared 0.28 0.29 0.30 0.33 0.34 0.32 0.31 Notes: Robust standard errors in parentheses. * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Logit coefficients are marginal effects on the probability of having a majoritarian system evaluated at the mean. The Table includes p‐values for Chi‐square test of joint significance of the control sets. Regional dummies include africa and asiae. Regional dummy laam has not been included because all nineteen countries in Latin America enter as majoritarian. Colonial origins include col_uk and col_esp but not col_esp because all fifteen former Spanish‐Portuguese colonies in the sample enter as majoritarian democracies. Open in new tab Table 5 
Constitution Selection and Income Inequality When Presidential Regimes With PR are Classified as Majoritarian Democracies Whole sample – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.024 0.024 0.023 0.025 0.025 0.022 0.025   (0.006)*** (0.006)*** (0.006)*** (0.008)*** (0.007)*** (0.008)*** (0.007)*** ethnic 0.153 0.222 0.125 0.339 0.31 0.06 0.12   (0.283) (0.299) (0.267) (0.329) (0.311) (0.283) (0.296) language 0.162 0.129 0.028 ‐0.071 0.072 0.054 0.137   (0.199) (0.227) (0.222) (0.358) (0.214) (0.229) (0.208) lpop_60   [0.538]         y_60     [0.167]         regional dummies       [0.297]       prot80, catho80, confu         [0.198]     colonial origins           [0.401]   religion             [0.261] Observations 56 56 56 56 56 56 56 Pseudo R‐squared 0.28 0.29 0.30 0.33 0.34 0.32 0.31 Whole sample – Logit estimates . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . Dep. var. maj maj maj maj maj maj maj giniycai 0.024 0.024 0.023 0.025 0.025 0.022 0.025   (0.006)*** (0.006)*** (0.006)*** (0.008)*** (0.007)*** (0.008)*** (0.007)*** ethnic 0.153 0.222 0.125 0.339 0.31 0.06 0.12   (0.283) (0.299) (0.267) (0.329) (0.311) (0.283) (0.296) language 0.162 0.129 0.028 ‐0.071 0.072 0.054 0.137   (0.199) (0.227) (0.222) (0.358) (0.214) (0.229) (0.208) lpop_60   [0.538]         y_60     [0.167]         regional dummies       [0.297]       prot80, catho80, confu         [0.198]     colonial origins           [0.401]   religion             [0.261] Observations 56 56 56 56 56 56 56 Pseudo R‐squared 0.28 0.29 0.30 0.33 0.34 0.32 0.31 Notes: Robust standard errors in parentheses. * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Logit coefficients are marginal effects on the probability of having a majoritarian system evaluated at the mean. The Table includes p‐values for Chi‐square test of joint significance of the control sets. Regional dummies include africa and asiae. Regional dummy laam has not been included because all nineteen countries in Latin America enter as majoritarian. Colonial origins include col_uk and col_esp but not col_esp because all fifteen former Spanish‐Portuguese colonies in the sample enter as majoritarian democracies. Open in new tab Finally, we have performed the following three robustness checks whose results are available from the authors. First, we have estimated the relationship between inequality and constitution also with probit and linear probability models. This yielded similar results. Second, we have used three other Gini coefficients: the unadjusted Gini; the Gini adjusted if the data are based on net income rather than gross (along with the adjustment for income versus expenditure); the Gini coefficient with a further adjustment if the information is based on households instead than individuals. Using these Gini coefficients yields very similar estimates in all samples. Third, we have tried to understand if the quality of the Gini coefficients could play a role in our estimate even though, in principle, the inclusion of lower quality data should work against our hypothesis. To this purpose, we have first restricted the sample to high quality data only and the estimates are similar to those presented above. A second check was to add a dummy variable for the quality (high or low) of the Gini coefficient in each specification. Again, the estimate for income inequality remained basically unchanged while the estimate of the variable for the quality of the Gini coefficient was always statistically insignificant. Therefore, we conclude that there is a strong positive relationship between income inequality and majoritarian democracy as predicted by our model. 8. Conclusions This article shares with other recent contributions in political economics the premise that constitutional principles are of great importance in shaping fiscal policy outcomes in representative democracies. We show that generally consensual democracies have higher taxation and bigger governments than majoritarian ones. But additionally we demonstrate that, once institutions are viewed as endogenous, consensual democracy is more difficult to sustain politically in a more unequal society since greater inequality tends to undermine the stability of the coalition supporting it. The results of our empirical analysis are consistent with the theoretical finding that more unequal societies are expected to adopt a majoritarian constitution, while relatively equal countries are more likely to choose a consensual constitution. The results of our model concerning the effects of inequality on institutional choice, as well as on fiscal policy, appear to be not only significant per se but, for the guidelines they offer for future empirical research on the economic consequences of constitutions. In particular, they provide a firm theoretical background for the intuitive claim that the assignment of constitutions across countries is not random but reflects a number of ‘fundamentals’, among which the extent of income inequality is of special importance. Future constitutional choices and new data on income inequality will provide further information to test our theory. On the one hand, it will be interesting to observe the constitutions chosen by new countries escaping from dictatorship and moving towards democracy. On the other hand, it will also be possible to see whether democratic countries experiencing a sensible increase or reduction in income inequality will question or change their current constitutional framework. Footnotes 1 " The focus on the distribution of income is motivated by the insights provided by the positive political economics theory of taxation and redistribution in democracies (Romer, 1975; Roberts, 1977; Meltzer and Richard, 1981), which stresses the importance of this variable in shaping fiscal policy outcomes when individual preferences are aggregated directly by majority voting. The class conflict perspective over different institutional frameworks and the key role played by income inequality have been first emphasised by Acemoglu and Robinson (2000) in their analysis of democratic transitions. 2 " It is worth emphasising that the features of the majoritarian model, typically represented by the UK, also lead to a dominance of the executive over the legislative power, while in consensual democracies, widely observed in continental Europe, the distribution of power between the executive and the legislative power is more balanced. 3 " Our theory implies that more unequal countries are expected to choose a majoritarian model where redistribution is low, while more unequal societies are expected to adopt consensual constitutions that lead to more redistributive outcomes. 4 " This means that Hg(·) > 0, Hg(·) < 0 and limgj→0Hg(gj)=∞. 5 " See for example Besley and Coate (1991) who show that, allowing for different quality levels of the public goods, a de jure universal provision scheme does not imply that it is de facto universal and explain why some publicly provided private goods like health care may go to the advantage of the poor and not to the rich. Fernandez and Rogerson (1995) also discuss the case of higher education and emphasise how the public provision of it can benefit higher‐income individuals at the expense of the poor. 6 " It can be shown that the main results of the article are robust to a more general specification of individual preferences and of the menu of the fiscal policy instruments available. If, for example, the members of a social group obtain some utility also from the provision of the other public goods, or there is a general public good providing utility to all individuals, the main results of the article would still hold as long as the taste for the group‐specific public good is strong enough; see for further details Ticchi and Vindigni (2003). 7 " Osborne and Slivinsky (1996) assume sincere voting while in Besley and Coate (1997) individuals are strategic. In this sense our model is closer to the first one. 8 " This is a potentially serious problem in our model since the policy space is multidimensional and, thus, voting cycles may occur. The citizen‐candidate model allows us to avoid the problem of non‐existence of an equilibrium. At the same time, we are able to show that the main drawback of it, namely the generic multiplicity of political equilibria, is not an issue in our economy. 9 " Our theory suggests that because in the US both the legislature and the president are elected with the same electoral law, their preferences over fiscal policy should be relatively similar and ‘divided government’ (Alesina and Rosenthal, 1996) should not be of central importance in this respect. Also, in an extension of our model (available from the authors), we show that essentially the same political equilibrium carries over in an institutional environment where, as in Latin America, the president has relatively large legislative powers (including setting the agenda and vetoing bills) and the assembly is elected with proportional representation (PR). We therefore conclude that fiscal policy outcomes in presidential regimes with PR depend primarily on the nature of the electoral law of the president. 10 " It is immediate to verify that the second order condition is satisfied. The Inada conditions imposed on H(·) imply that the tax rate is always strictly positive. 11 " This implies that, off‐equilibrium, it is possible to observe an ‘extreme coalition’ made up by the rich and the poor. As it is clear from the proof of Proposition 1, in case there is a rich and a middle class candidate, the poor prefer to vote for the rich since the latter’s fiscal conservatism is the best alternative they have. 12 " In modelling the legislative bargaining process taking place in a consensual democracy, we partly draw on the agenda‐setting model of Romer and Rosenthal (1979), as well as on the model of legislative decision making in a three‐party proportional representation system offered by Austen‐Smith and Banks (1988). 13 " It is straightforward to verify that the supply of candidates from each group is not lower than ρ mj. Since the parliament is large, the policy outcome does not depend on whether any individual does or does not run for office. Therefore, a citizen‐candidate of group j runs for office if and only if pk  ɛ, where p is the probability of being elected, pinned down at p = ɛ/k ∈ (0,1] by the assumption of free entry of candidates. 14 " Finally notice that the probability φ ∈ (0,1) because mp and mr are always positive. However, in the analysis presented below we also consider the limit cases where φ = 0 and φ = 1 because they allows us to define the tax rates and utilities in the closed interval [0,1]. 15 " From an inspection of (4), it is immediate to verify that this is always the case whenever, as φ approaches one, the middle class dictatorial tax rate τb is lower than τ2,p,b. For example, it is easy to prove analytically that τb < τ2,p,b at φ = 1 if yp = 0. Indeed, under these conditions τ2,p,b is defined by the equation and the dictatorial tax rate of the middle class by , which imply that . From follows that τb < τ2,p,b. Our numerical simulations, that we discuss in the next Section, show that this result also holds for values of φ different from one or when the income of the poor is positive. Clearly, the lower is the income of the poor and the higher is τ2,p,b, which in turn makes more likely that the participation constraint of the rich is not binding. 16 " The result in Proposition 2 can be regarded as an application of the general principle by which, in coalition formation games, it can be advantageous to be in a relatively weak bargaining position as that increases the likelihood of becoming a member of the coalition. This is the opposite of what happens in a Nash bargaining process where a lower bargaining power only reduces the share of the surplus of which one can appropriate. 17 " A graphical representation of the tax rates is provided in Figure 1. (1) [Taxation Across Constitutions and Coalitions when yp =  0 and yr is High Relative to ] 18 " In the numerical simulations we have used the power function specification for H(·) with A = 1 and α = 0.5. 19 " In this case the variation in the size of the classes and, therefore, of φ cannot change the fact that the distribution is very equal even when there are many poor (φ high). 20 " Notice that the tax rate of the middle class and poor coalition is not affected by the income of the rich. 21 " For example, with the above parameterisation (yp = 0.9, yb = 0.95, ) an income level of the rich higher than 1.3 is enough to obtain τr < τ1,b,p for all φ ∈ (0,1). 22 " Moreover, notice that an increase in the income of the rich always leads to a reduction in φ*. 23 " The results presented above are robust to all parameterisations we have used. Additional numerical simulations are available from the authors. It is also worth noting that all simulations confirms that the utility of the middle class in the government coalition with the rich is monotonically increasing in φ, while the utility of the middle class in the coalition with the poor is strictly monotonically decreasing in φ, so that φ* is unique. 24 " It is trivial to deduce that the middle class is always better off in a consensual democracy than in a majoritarian one. Indeed, notice that in consensual democracy the middle class would always have the option of replicating the majoritarian outcome by offering to the rich of forming a coalition and implementing their own preferred policy. 25 " If the poor are part of the government coalition they get a utility at least as high as their income level, while in majoritarian democracy they always get a lower level of utility because of positive taxation and no provision of their public good. 26 " According to our theory, this particular mean preserving spread corresponds to the shocks driving the cases of constitutional change in France, Germany and Italy that we discuss next. 27 " More precisely, for certain parameterisations it is possible to observe that shifts upward for almost all φ except for values close to one where it shifts down. However, this is always irrelevant because it happens for values of φ very far from φ*. 28 " The index is based on the computation of the ‘political centre of gravity’ of a government, defined as the average ideological collocation of the parties of the coalition on the left‐right continuum, weighted by the share of seats of each of them; see Cusack (1997, Table 1 and pp. 381–2) for more details. 29 " The results do not change if we consider the subperiods, where it is also possible to observe that the variability of this index over time is rather small. For consensual democracies the index gets the following average values: 2.95 in 1950–9, 2.91 in 1960–9, 2.76 in 1970–9 and 3.06 in 1980–91. The average values for majoritarian democracies are: 3.72 in 1950–9, 3.46 in 1960‐9, 3.41 in 1970–9 and 3.39 in 1980–91. 30 " Our results also provide a theoretical ground to the finding of Scheve and Stasavage (2007) that, in a sample of 13 countries for the period 1976–2000, government of the right (left) is associated with more (less) income inequality. 31 " We thank an anonymous referee for pointing this to our attention. 32 " Our model of endogenous selection of the constitution may also help explaining why some recent studies, such as Scheve and Stasavage (2007), do not find an effect of PR on the reduction of income inequality. Most equal societies or those that tend more rapidly towards equalisation are likeliest to have adopted and retained PR. 33 " This also confirms that our assumption on the absence of a veil of ignorance in choosing the constitution is appropriate. 34 " In particular, Denmark adopted PR in 1920, Norway in 1921, Sweden in 1907, Finland in 1906, the Netherlands in 1917, France in 1919 and 1946, Germany in 1919 and 1945, Italy in 1919 and 1948. France abandoned PR after the 1924 election with the reversion to the second ballot in single‐member constituencies that was used until the war. The suspension of PR in Italy and Germany followed instead the rise of fascism and nazism respectively. 35 " The reduction of income inequality generated by a reduction in the top percentile income share corresponds to a change in the income distribution equivalent to our mean preserving spread 2 that we have analysed in the previous Section. 36 " For example, the available evidence from Morrison (2000, p. 232) also confirms that income inequality in Germany strongly decreased from 1913 to 1926 and that it was low after World War II. 37 " Austria also adopted PR at the end of World War I. 38 " The data on income inequality in Denmark are based on the maximum equalisation coefficient (MEC), which indicates the share of total income which has to be transferred from the population with income above the average to those below in order to achieve an equal distribution. This index falls sharply from 0.50 to 0.35 between 1870 and 1900, and was 0.36 in 1925. 39 " On the other hand, we admit that there is no unanimous consensus among political scientists about the Alesina and Glaeser’s view. 40 " In fact, our theory predicts that income inequality determines the choice of the constitution, which in turn affects the distribution of income through its effects on fiscal policy outcomes. 41 " In particular, maj is a dummy variable equal to 1 if all the lower house is elected under plurality rule and 0 under a proportional (or mixed) electoral system, while pres is a dummy variable for the form of government, equal to 1 in presidential regimes and 0 in parliamentary democracies; see PT (2003, ch. 4) for details and clarifications. 42 " The choice of taking a Gini coefficient that refers to few years later the constitutional choice is based on the same rationale that has led PT to exclude the reforms that some countries have made during the 1990s in their cross‐sectional analysis for the period 1990–8. They argue that it takes some time before constitutional reforms have an impact on fiscal policy outcomes (see p. 88). However, if this is true (as we believe), then there will also need some time before the constitution has an impact on income inequality given that this effect should mainly work through fiscal policy. 43 " Cyprus and South Africa are not in the dataset although we had a Gini coefficient for these countries. Cyprus has been excluded because we had a Gini coefficient for one year only and two constitutional choices very close to that year. While the more reasonable constitution would in principle favour our hypothesis, including Cyprus with either constitutions leaves our results unchanged. Details can be supplied by the authors in the form of a data appendix. South Africa has been excluded because it is an outlier and an influential observation. A further appendix, available from the authors, also shows that the inclusion of this observation in the sample has a very large impact on the estimates. 44 " In this dataset, compiled by Beck et al. (2000), three elements are considered: (a) veto power: president can veto legislation and the parliament needs a supermajority to override the veto; (b) appoint prime minister: president can appoint and dismiss prime minister and/or other ministers; (c) dissolve parliament: president can dissolve parliament and call for new elections. The system is a ‘direct presidential’ if (a) is true, or if (b) and (c) are true. 45 " For instance, Morgenstern (2002) argues that Latin American legislatures are only ‘reactive’, namely they only have the ability to amend/veto legislative proposals made by the president. Such prerogatives are also limited as presidents facing a hostile legislature typically have many ways to bypass it by using their ‘unilateral powers’, such as various types of decrees and regulatory rule making, as well as their own veto powers on parliamentary deliberations. According to Cox and Morgenstern (2002 p. 461)‘…presidents in Latin America regularly make policy decisions almost unilaterally. Presidents in Bolivia, Brazil, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay and Venezuela have tremendous advantages in structuring the budget process, as the legislatures there are constitutionally restricted from making significant changes.’ As a result, according to the same authors, ‘Latin American legislatures are hindered by a lack of time, resources and experience. This combination of constitutional and organisational limits has converted many Latin American presidents into virtual budget dictators.’ 46 " The variables ethnic, language and religion take values in the range between zero and one that are increasing in the degree of ethnic, linguistic and religious fractionalisation respectively. There are no data for the variable language for El Salvador. 47 " There are a few countries for which the data on the level of income per capita relative to the US and the size of population are not available in 1960. In these cases, we have taken the first year available. This is equivalent to assuming that the ratio between the GDP per capita of these countries and that one of the US has remained unchanged between 1960 and the first year available. As a robustness check, we have also run regressions with the log of the GDP per capita in 1960. For the countries where GDP per capita in 1960 was not available, we have multiplied their per capita income relative to the US in the first year available with US per capita income in 1960. Using the log of this variable instead of the GDP per capita expressed relative to US yields basically the same results. 48 " Recall that the Gini coefficient employed is measured in a scale from 0 to 100, which means that an increase of one point in inequality increases the probability of adopting a majoritarian system by 3%. 49 " The Gini coefficient in column (5) is less precisely estimated but the p‐value is 5.2%. 50 " There are no Spanish–Portuguese colonies in the sample and the inclusion of col_oth alone would not change the results with respect to the baseline specification. When we control only for col_uk, the Gini estimate is 0.02 and the p‐value 12.4%. 51 " We thank an anonymous referee for bringing this to our attention. 52 " In particular, this variable takes into account price controls or inadequate bank supervision, as well as perceptions of the burdens imposed by excessive regulation in areas such as foreign trade and business development. 53 " We have also replicated the same analysis excluding the presidential regimes with PR. The results are very close to those obtained for the sample of parliamentary democracies reported in Table 3. 54 " The participation constraint of the rich is always binding at φ = 0 as their expected utility at round 2 is greater than yr. 55 " Formally, the maximisation problem when φ = 1 is the following: and it is immediate to verify that . 56 " We recall that τ1,b,r = τb when the participation constraint is not binding. 57 " We are using the fact that τ2,r,p = τr, with . 58 " The first component of (23) and (24) represents the net gain (with respect to the status quo) of the rich in utility terms when they are agenda setter at round 2 multiplied by (1 − φ). Clearly, this component goes to zero as φ tends to one. 59 " Using the fact that the dictatorial policy of the rich is implemented at φ = 0 and therefore , it is immediate to verify that the left‐hand side of (24) is equal to zero at φ = 0. Moreover, notice that the left hand side of (24) is more likely to be positive when φ is small. In this case τ1,b,r would be decreasing in φ. When yr is not sufficiently high, the numerical simulations confirm that τ1,b,r is decreasing for values of φ low and then it becomes increasing when φ is big enough. References Acemoglu , D. and Robinson , J. A. ( 2000 ). ‘Why did the West extend the franchise? Democracy, inequality, and growth in historical perspective’ , Quarterly Journal of Economics , vol. 115 ( 4 ), pp. 1167 – 99 . Google Scholar Crossref Search ADS WorldCat Aghion , P. and Bolton , P. ( 2003 ). ‘Incomplete social contracts’ , Journal of the European Economic Association , vol. 1 ( 1 ), pp. 38 – 67 . Google Scholar Crossref Search ADS WorldCat Aghion , P. , Alesina , A. and Trebbi , F. ( 2004 ). ‘Endogenous political institutions’ , Quarterly Journal of Economics , vol. 119 ( 2 ), pp. 565 – 611 . Google Scholar Crossref Search ADS WorldCat Alesina , A ., Devleeschauwer , A., Easterly , W., Kurlat , S. and Wacziarg , R. ( 2003 ). ‘ Fractionalization ’, Journal of Economic Growth , vol. 8 , pp. 155 – 94 . Google Scholar Crossref Search ADS WorldCat Alesina , A . and Glaeser , E. ( 2004 ). Fighting Poverty in the US and Europe: A World of Difference , Oxford: Oxford University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Alesina , A. and Rosenthal , H. ( 1996 ). ‘A theory of divided government’ , Econometrica , vol. 64 ( 6 ), pp. 1311 – 42 . Google Scholar Crossref Search ADS WorldCat Alesina , A. , Roubini N. and Cohen , G. D. ( 1997 ). Political Cycles and the Macroeconomy , Cambridge, MA: The MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Austen‐Smith , D. ( 2000 ). ‘Redistributing income under proportional representation’ , Journal of Political Economy , vol. 108 ( 6 ), pp. 1235 – 69 . Google Scholar Crossref Search ADS WorldCat Austen‐Smith , D. and Banks , J. S. ( 1988 ). ‘Elections, coalitions, and legislative outcomes’ , American Political Science Review , vol. 82 , pp. 405 – 22 . Google Scholar Crossref Search ADS WorldCat Baldez , L. and Carey , J.M. ( 1999 ). ‘Executive agenda control and spending policy: lessons from general Pinochet’s constitution’ , American Political Science Review , vol. 43 , pp. 29 – 55 . Google Scholar Crossref Search ADS WorldCat Barbera , S. and Jackson , M. O. ( 2004 ). ‘Choosing how to choose: self‐stable majority rules and constitutions’ , Quarterly Journal of Economics , vol. 119 ( 3 ), pp. 1011 – 48 . Google Scholar Crossref Search ADS WorldCat Beard , C. A. ( 1913 ). An Economic Interpretation of the Constitution of the United States , New York: The Macmillan Company . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Beck , T. , Clark , G., Groff , A., Keefer , P. and Walsh , P. ( 2000 ). ‘ New tools and new tests in comparative political economy: the database of political institutions ’, World Bank Research Working Paper No. 2283. Bénabou , R. ( 2000 ). ‘Unequal societies: income distribution and the social contract’ , American Economic Review , vol. 90 ( 1 ), pp. 96 – 129 . Google Scholar Crossref Search ADS WorldCat Besley , T. and Coate , S. ( 1991 ). ‘Public provision of private goods and the redistribution of income’ , American Economic Review , vol. 81 ( 4 ), pp. 979 – 84 . OpenURL Placeholder Text WorldCat Besley , T. and Coate , S. ( 1997 ). ‘An economic model of representative democracy’ , Quarterly Journal of Economics , vol. 112 ( 1 ), pp. 85 – 114 . Google Scholar Crossref Search ADS WorldCat Cox , G. W. and Morgenstern , S. ( 2002 ). ‘Epilogue: Latin America’s reactive assemblies and proactive presidents’. in ( S. Morgenstern and B. Nacif, eds.), Legislative Politics in Latin America , pp. 446 – 68 , Cambridge: Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Cusack , T . ( 1997 ). ‘ Partisan politics and public finance: changes in public spending in the industrialized democracies, 1955‐1989 ’, Public Choice , vol. 91 ( 3/4 ), pp. 375 – 95 . Google Scholar Crossref Search ADS WorldCat Deininger , K. and Squire , L. ( 1996 ). ‘A new data set measuring income inequality’ , World Bank Economic Review , vol. 10 , pp. 565 – 91 . Google Scholar Crossref Search ADS WorldCat Epple , D. and Romano , R. E. ( 1996 ). ‘Ends against the middle: determining public service provision when there are private alternatives’ , Journal of Public Economics , vol. 62 ( 3 ), pp. 297 – 325 . Google Scholar Crossref Search ADS WorldCat Fernandez , R. and Rogerson , R. ( 1995 ). ‘On the political economy of education subsidies’ , Review of Economic Studies , vol. 62 ( 2 ), pp. 249 – 62 . Google Scholar Crossref Search ADS WorldCat Greenberg , S. B. ( 1996 ). Middle Class Dreams: Politics and Power of the New American Majority . New Haven: Yale University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Kaufmann , D. , Kraay , A. and Mastruzzi , M. ( 2005 ). ‘ Governance matters IV: governance indicators for 1996‐2004 ’. The World Bank Policy Research Paper No. 3630. Lakeman , E. and Lambert , J. D. ( 1963 ). Voting in Democracies. A Study of Majority and Proportional Systems , London: Faber and Faber . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Lijphart , A. ( 1999 ). Patterns of Democracy , New Haven: Yale University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Lizzeri , A. and Persico , N. ( 2001 ). ‘The provision of public goods under alternative political regimes’ , American Economic Review , vol. 91 ( 1 ), pp. 225 – 39 . Google Scholar Crossref Search ADS WorldCat Meltzer , A. H. and Richard , S. F. ( 1981 ). ‘A rational theory of the size of government’ , Journal of Political Economy , vol. 89 ( 5 ), pp. 914 – 27 . Google Scholar Crossref Search ADS WorldCat Messner , M. and Polborn , M. K. ( 2004 ). ‘Voting on majority rules’ , Review of Economic Studies , vol. 71 ( 1 ), pp. 115 – 32 . Google Scholar Crossref Search ADS WorldCat Milesi‐Ferretti , G. M. , Perotti , R. and Rostagno , M. ( 2002 ). ‘Electoral systems and public spending’ , Quarterly Journal of Economics , vol. 117 ( 2 ), pp. 609 – 57 . Google Scholar Crossref Search ADS WorldCat Morgenstern , S. ( 2002 ). ‘Explaining legislative politics in Latin America’. in ( S. Morgenstern and B. Nacif, eds.), Legislative Politics in Latin America , ch. 14, Cambridge: Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Morrisson , C. ( 2000 ). ‘Historical perspectives on income distribution: the case of Europe’. in ( A. B. Atkinson and F. Bourguignon, eds.), Handbook of Income Distribution , vol.1, pp. 217 – 60 , Amsterdam: North‐Holland . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Myerson , R. B. ( 1993 ). ‘Incentives to cultivate favored minorities under alternative electoral systems’ , American Political Science Review , vol. 87 ( 4 ), pp. 856 – 69 . Google Scholar Crossref Search ADS WorldCat Osborne , M. J. and Slivinsky , A. ( 1996 ). ‘A model of political competition with citizen‐candidates’ , Quarterly Journal of Economics , vol. 111 ( 1 ), pp. 65 – 96 . Google Scholar Crossref Search ADS WorldCat Perotti , R. ( 1996 ). ‘Growth, income distribution, and democracy: what the data say’ , Journal of Economic Growth , vol. 1 ( 2 ), pp. 149 – 87 . Google Scholar Crossref Search ADS WorldCat Perotti , R. and Kontopoulos , Y. ( 2002 ). ‘Fragmented fiscal policy’ , Journal of Public Economics , vol. 86 ( 2 ), pp. 191 – 222 . Google Scholar Crossref Search ADS WorldCat Persson , T. and Tabellini , G. ( 2000 ). Political Economics: Explaining Economic Policy , Cambridge, MA: The MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Persson , T. and Tabellini , G. ( 2003 ). The Economic Effects of Constitutions , Cambridge, MA: The MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Persson , T. and Tabellini , G. ( 2004 ). ‘Constitutional rules and fiscal policy outcomes’ , American Economic Review , vol. 94 ( 1 ), pp. 25 – 45 . Google Scholar Crossref Search ADS WorldCat Persson , T. , Roland , G. and Tabellini , G. ( 1997 ). ‘Separation of powers and political accontability’ , Quarterly Journal of Economics , vol. 112 ( 4 ), pp. 1163 – 202 . Google Scholar Crossref Search ADS WorldCat Persson , T. , Roland , G. and Tabellini , G. ( 2000 ). ‘Comparative politics of public finance’ , Journal of Political Economy , vol. 108 ( 6 ), pp. 1121 – 61 . Google Scholar Crossref Search ADS WorldCat Piketty , T. ( 2007 ). ‘Top incomes over the twentieth century: a summary of main findings’, in ( A.B. Atkinson and T. Piketty, eds.), Top Incomes Over the Twentieth Century , Oxford: Oxford University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Powell , G. B. Jr. ( 2002 ). ‘ PR, the median voter, and economic policy: an exploration ’. mimeo, University of Rochester. Roberts , K. ( 1977 ). ‘Voting over income tax schedules’ , Journal of Public Economics , vol. 8 ( 3 ), pp. 329 – 40 . Google Scholar Crossref Search ADS WorldCat Rokkan , S . ( 1970 ). Citizens, Elections, Parties. Approaches to the Comparative Study of the Process of Development , Oslo: Universitetsforlaget . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Romer , T. ( 1975 ). ‘Individual welfare, majority voting, and the properties of a linear income tax’ , Journal of Public Economics , vol. 4 ( 2 ), pp. 163 – 85 . Google Scholar Crossref Search ADS WorldCat Romer , T. and Rosenthal , H. ( 1979 ). ‘Bureaucrats versus voters: on the political economy of resource allocation by direct democracy’ , Quarterly Journal of Economics , vol. 93 ( 4 ), pp. 563 – 87 . Google Scholar Crossref Search ADS WorldCat Scheve , K. and Stasavage , D. ( 2007 ). ‘ Political institutions, partisanship, and inequality in the long run ’. mimeo, Yale University. Ticchi , D. and Vindigni , A. ( 2003 ). ‘ Endogenous constitutions ’, IIES Seminar Paper No. 726, Stockholm University. Appendix A.1. Proof of Proposition 1 We first prove that only the rich run for office and then that the set of citizen‐candidates running for office is not empty. Step 1. Assume that at least one rich candidate runs for office. Would anyone else run for office? The answer is no. If a middle class agent also runs, he would be defeated by the rich candidate because both the rich and the poor would vote for the rich. The poor find it convenient to vote for the rich candidate because in their eyes he is the less bad of the two of them: he offers to the poor none of their preferred public good but demands lower taxes. Similarly, if a poor agent runs for office against a rich candidate, he would be defeated by the vote of the middle class and the rich. Lastly, notice it cannot be the case that a candidate from each group runs for office. Indeed, the middle class candidate would win the election with certainty in that instance (recall that mb > max{mp,mr} by assumption), and therefore neither a poor nor a rich candidate would run against him. Step 2. We now demonstrate that the set of citizen‐candidates running for office is not empty, i.e. that at least one rich candidate runs for office. Let p denote the probability of victory for a rich citizen‐candidate (in a symmetric equilibrium, this will be the same across identical citizen‐candidates). A rich agent wants to run for office if and only if the expected gain of running exceeds its cost, namely if , where denotes the policy vector implemented if he does not run and the term in the graph parenthesis is the expected utility if he runs, given that only rich candidates do so. To show that the set of citizen‐candidates running for office is not empty, it is sufficient to demonstrate that this condition is satisfied when only one rich person runs for office for all . To see this, observe that in this case p = 1 and the participation constraint of the rich candidate reads . Since the policy vector (τr, Gr) maximises the welfare of the rich, , for all with , and given that k  ɛ, the participation constraint of the rich candidate always holds with strict inequality when p = 1. A.2. Proof of Proposition 2 In this Appendix we show that a value of φ exists, that we call φ*, such that the utility derived by the middle class from the government coalition with the poor is higher than the corresponding utility from the coalition with the rich if φ < φ*, and that the opposite is true whenever φ > φ*. To this end, we first show that is monotonically increasing in φ, while is strictly monotonically decreasing in φ. Then, to prove that there is a single crossing between these two schedules in the range where φ ∈ (0,1), we show that is higher than at φ = 0 and that the opposite holds at φ = 1. The utility of the middle class in the government coalition with the rich is defined by the maximisation problem (5) subject to the participation constraint of the rich (4). Differentiating (5) with respect to φ and applying the envelope theorem we get that (11) Indeed, assuming that (4) is binding and applying the implicit function theorem we get (12) given that the numerator is positive because it is the difference between the utility of the rich when they are agenda setter at round 2 (which is greater than yr) and their utility under the poor and middle class government coalition (which is lower than yr). In this case d is strictly positive. Instead, if the participation constraint of the rich (4) is not binding, then is at its global maximum, and d. Hence, the result in (11) shows that is monotonically increasing in φ. The utility of the middle class in the government coalition with the poor is defined by the maximisation problem (8) subject to the participation constraint of the poor (7). If we differentiate (8) with respect to φ and apply the envelope theorem we obtain that (13) given that from (7) (14) Indeed, the numerator on the right hand side of (14) is negative because it is the difference between yp and the utility of the poor when they are agenda setter at a round 2 (which is greater than yp). The result in (13) means that is strictly monotonically decreasing in φ. Then, it remains to show that and that . We prove the first inequality by showing that the participation constraints of the rich and the poor at φ = 0 imply that, for any given level of tax rate chosen by the middle class, the rich have to be compensated with a greater amount of their preferred public good. Indeed, the participation constraint of the poor at φ = 0 implies that (15) while from the participation constraint of the rich we get54 (16) The expression in the square brackets in the right hand side of (16) is the utility of the rich when they are agenda setter at round 2 and it is clearly greater than yr. Hence, it is of course the case that (17) By combining (15) and (17) and taking into account also the fact that yr > yp, one can easily verify that for any given level of tax rate τ1,b,r = τ1,b,p = τ we have that . This in turn implies that the middle class agenda setter obtains a higher level of utility by making a coalition with the poor instead than with the rich, i.e. . At φ = 1 the utility of the middle class under the government coalition with the poor is equal to yb. This result can be obtained by observing that at φ = 1 the maximisation problem of the middle class is subject to the participation constraint of the poor where they are agenda setter with probability one at round 2 and maximise their utility subject to the constraint of giving to the middle class a level of utility equal to the status quo (which corresponds to their level of income yb).55 From the maximisation problem of the middle class, when the agenda setter forms a government coalition with the rich, it is immediate to verify that is always greater than yb at φ = 1. Indeed, from the participation constraint of the rich at φ = 1, i.e. , we know that the middle class could implement the following policy: 0 < τ1,b,r  τ2,p,b, , . This policy satisfies the participation constraint of the rich and gives to the middle class agenda setter a higher utility than her income level yb. Given that the optimal policy gives to the middle class a higher utility than this policy and, therefore, of yb, it is clear that . A.3. Proof of Result 1 In this Appendix we show that under the assumptions stated in the main text the following inequalities hold for all φ: τr < τ1,b,r < τ1,b,p. To this aim, we first prove that τr < τ1,b,r  τb by showing that τ1,b,r = τr at φ = 0, τ1,b,r = τb at φ = 1 and that τ1,b,r is monotonically increasing in φ. Next, we show that τb < τ1,b,p. At φ = 0 the rich are the agenda setter with probability one at round 2, and they form the government coalition with the poor. If the income of the poor is equal to zero, the fiscal policy implemented at the second round corresponds to the dictatorial policy of the rich as the poor do not need to be compensated with a positive amount of their preferred public good. Thus, to form a government coalition at round 1, the middle class has to propose a policy to the rich such that their level of utility is the same they obtain at round 2. Given that the latter is equal to its global unconstrained maximum, the middle class can only implement the dictatorial policy of the rich, that is: τ1,b,r = τ2,r,p = τr, and . In other words, the policy implemented (and the tax rate chosen) by the government coalition formed by the middle class and the rich at φ = 0 (if yp = 0) is the same as the one in majoritarian democracy. We have previously shown that τ1,b,r = τb for all values of φ such that the participation constraint of the rich is not binding and that this is always the case at φ = 1 if yp = 0. To prove that τ1,b,r is monotonically increasing in φ, we differentiate the first order condition (6) with respect to φ by taking into consideration the fact that from the participation constraint of the rich (4), and .56 After some algebra we obtain (18) which allows us to determine the sign of ∂τ1,b,r/∂φ. Indeed, notice that the term in graph parenthesis that multiplies ∂τ1,b,r/∂φ is always negative and from (12) we know that . Therefore, the sign of the relationship between τ1,b,r and φ depends on the sign of the term in the second graph parenthesis of (18) which multiplies . In particular, ∂τ1,b,r/∂φ  0 if and only if (19) and vice versa. Even though in general it is not possible to give a definite sign to the left hand side of (19), assuming a power function for the utility of the public good it turns out that inequality (19) is satisfied if (20) where the numerator of the left hand side of (20) is the utility that the rich get from their preferred public good provided and the denominator represents the taxes they pay. It is useful to rewrite inequality (20) as (21) By using a power function specification for H(·) and subtracting yr to both sides, we can rewrite the participation constraint of the rich (4) as (22) The substitution of (22) into (21) leads to the following weak inequality (23) Now, if we take into consideration the fact that the dictatorial policy of the rich is implemented at round 2 under the rich and poor coalition (see the discussion above), we are able to rewrite (23) as57 (24) Notice that only the first component of the left hand side of (24) is positive and this term decreases until zero as yr increases.58 Therefore, there exists a yr sufficiently high relatively to such that (24) is always satisfied, which in turn means that ∂τ1,b,r/∂φ  0.59 At this point we know that τ1,b,r is monotonically increasing in φ, that τ1,b,r(0) = τr and that τ1,b,r(1) = τb and this implies that τr < τ1,b,r  τb. We now want to prove that τ1,b,p is always higher than τb and to this aim we show that τ1,b,p = τb at φ = 0 and that τ1,b,p is increasing in φ. The first point is easily shown by observing that the middle class agenda setter in the coalition with the poor can implement her dictatorial policy at φ = 0 if yp = 0 because the participation constraint of the poor (7) is not binding. To prove the second point we differentiate the first order condition (9) with respect to φ by taking into account that from the participation constraint of the poor (7), and . After rearranging terms we get (25) From (14) we know that and, given that the term in the first graph parenthesis is always negative, the sign of ∂τ1,b,p/∂φ depends on the sign of the term in the second graph parenthesis that multiplies . If (26) then ∂τ1,b,p/∂φ > 0, and vice versa. In this case we do not need to use a power function specification for H(·) because using the fact that yp = 0 and for all φ > 0, inequality (26) becomes , which is always satisfied given that Hg(·) < 0. This implies that τ1,b,p is monotonically increasing in φ and its minimum level is equal to τb at φ = 0. Author notes " Part of this article was written while Davide Ticchi was at University Pompeu Fabra whose hospitality is gratefully acknowledged. We thank two anonymous referees, Daron Acemoglu, Alberto Alesina, Roland Bénabou, Walter Garcia‐Fontes, José Garcia Montalvo, Gene Grossman, Arend Lijphart, Massimo Morelli, Elias Papaioannou, Torsten Persson, Gérard Roland, Stefano Sacchi, Gilles Saint‐Paul, Davide Sala, James Snyder, Guido Tabellini, Jonathan Temple, Ernesto Villanueva, Fabrizio Zilibotti and seminar participants at Princeton University, Brown University, MIT, Toulouse University, University of Turin, University of Ancona, IGIER‐Bocconi, Yale University, University of Modena, University of Urbino and The World Bank for useful comments. We are heavily indebted to Antonio Ciccone and Howard Rosenthal for long discussions, encouragements, and many valuable comments and suggestions. The usual disclaimers apply. © The Author(s). Journal compilation © Royal Economic Society 2009

Bardsley,, Nicholas;Mehta,, Judith;Starmer,, Chris;Sugden,, Robert

doi: 10.1111/j.1468-0297.2009.02304.xpmid: N/A

Abstract This article reports experimental tests of two alternative explanations of how players use focal points to select equilibria in one‐shot coordination games. Cognitive hierarchy theory explains coordination as the result of common beliefs about players’ pre‐reflective inclinations towards the relevant strategies; the theory of team reasoning explains it as the result of the players’ using a non‐standard form of reasoning. We report two experiments. One finds strong support for team reasoning; the other supports cognitive hierarchy theory. In the light of additional questionnaire evidence, we conclude that players’ reasoning is sensitive to the decision context. It is well known that the players of one‐shot coordination games are often successful to a degree that classical game theory cannot explain. In these games, particular Nash equilibria seem to constitute ‘focal points’ on which the players’ expectations converge. The existence of focal points was first demonstrated by Schelling (1960); his informal experiments have been replicated under controlled conditions (Mehta et al., 1994). Although the concept of a focal point has been routinely used in game theory for many years, there is still no generally‐accepted explanation of how, in reality, real people manage to reach these equilibria. Two alternative lines of explanation have developed. One approach, first suggested by Lewis (1969), rests on assumptions about ‘primary salience’– that is, players’ psychological propensities to play particular strategies by default, when there are no other reasons for choice. More recently, this approach has been formalised as level‐n theory (Stahl and Wilson, 1995; Bacharach and Stahl, 2000) and, in a simplified form, as cognitive hierarchy theory (Camerer et al., 2004).1 The other approach, arguably implicit in some parts of Schelling’s own analysis, assumes that each player chooses the decision rule which, if used by all players, would be optimal for each of them. This has since been formalised as the theory of team reasoning (Sugden, 1993, 1995; Bacharach, 1999, 2006). In this article, we report two experiments designed to discriminate between these approaches to explaining coordination. We begin by setting out the two approaches and showing that, for certain classes of coordination games, they make different predictions (Section 1). We describe an experimental design which allows two tests of these predictions. The first test uses pure coordination games in which strategies are distinguished by labels, and compares the behaviour of subjects in three treatments: ‘pickers’, who choose between labels without any incentive to choose one rather than another, ‘guessers’, who guess how pickers have behaved, and ‘coordinators’, who try to coordinate with one another. Because the relevant cross‐treatment comparisons can be made for any given set of labels, this test does not depend on prior assumptions about the salience of the labels used. This is a great advantage. There is a widespread perception among economists that when salience is culturally dependent, as it is in the most famous of Schelling’s games, it is resistant to decision‐theoretic analysis. In our cross‐treatment comparisons, the cultural determinants of salience are held constant, permitting direct tests of game‐theoretic hypotheses about play in these games. The second test uses another type of coordination game discussed by Schelling (1960), in which both players’ payoffs are higher in some Nash equilibria than in others (Section 2). We implemented this design in two experiments, which differed only in apparently small details. Surprisingly, the results of one experiment seem to support the theory of team reasoning, while those of the other seem to support cognitive hierarchy theory (Sections 3, 4 and 5). We investigated the reasons for this difference by using questionnaires to elicit perceptions of salience from members of the two subject pools (Section 6). Reviewing our experimental results in the light of the questionnaire responses, we conclude that modes of reasoning similar to those modelled by each of the two theoretical approaches are at work; which of them is used is sensitive to subtle differences in the specifications of coordination tasks. The implication, we suggest, is that one should be pessimistic about finding any simple, unified theory of focal points. We suggest that this conclusion is consistent with Schelling’s own analysis, which emphasises the diversity of the methods by which focal points are found (Section 7). We are conscious that this article does not have a simple story line. It would have been easy for us to provide one, either by reporting each of the experiments separately, or by reporting only one of the two tests that the experiments were designed to conduct; but such a strategy would have given a false representation of what we know to be the case. As we shall explain, neither the differences between the results of the two experiments nor the differences between those of the two tests can be understood as revealing lack of control in the experimental design. We have carried out what we believe to be a well‐controlled investigation of the two leading approaches to explaining a phenomenon that has puzzled game theorists for nearly half a century. Our findings may be disappointing to readers who are looking for a simple and general game‐theoretic explanation of focal points, but – as Schelling warned from the outset – the whole idea of such an explanation may be no more than a mirage. If that is the case, it is important to know. 1. Theory 1.1. The Framework Throughout this article, we are concerned only with one‐shot coordination games.2 Our aim is to understand how human players actually coordinate. For simplicity, we confine our attention to two‐player games. Our definition of a coordination game refers both to its normal form – which is how it is represented in classical game theory – and to the mechanism of ‘labelling’ which allows players to distinguish between strategies. Described by its normal form, a coordination game is a game for players 1 and 2. Player 1 chooses a strategy from the set S1 = {s11, …, s1n} where n ≥ 2; player 2 chooses from S2 = {s21, …, s2n}. Payoffs are defined in terms of a vector of strictly positive utility indices U1,…,Un. If, for some j, the chosen strategies are s1j and s2j (that is, if both players choose strategies with the same index j), then each player receives the payoff Uj; otherwise, each receives zero. The case in which U1 = U2 = ??? = Un is a pure coordination game; otherwise there is a Hi‐Lo game.3 In the normal form of a pure coordination game, the n strategies of each player are completely symmetrical with one another; correspondingly, there are n symmetrical Pareto‐efficient pure‐strategy Nash equilibria (s1j, s2j). In classical game theory, each Nash equilibrium is treated as a candidate ‘solution’; the players’ problem is to ‘select’ one equilibrium from the set of candidates. However, if we consider only the normal form of the game, it is not clear what selection of a pure‐strategy equilibrium can mean. If one equilibrium is to be singled out by the players, it must be distinguished from the others in some way that both of them can recognise; but if the n pure‐strategy equilibria are completely symmetrical with one another, what distinguishes one from another? Indeed, some theorists claim that, in a world of ideal rationality, these equilibria are indistinguishable and, hence, that the only rational solution to a pure coordination game is the mixed strategy equilibrium which assigns a probability of 1/n to each pure strategy (Harsanyi and Selten, 1988). In theoretical analyses of focal points in pure coordination games, the problem of indistinguishability is usually overcome by making explicit assumptions about the labelling of strategies (Mehta et al., 1994; Bacharach and Stahl, 2000; Bacharach, 1993; Casajus, 2001). Following this approach, we make it part of the definition of a coordination game that there is a set L = {l1,…,ln} of distinct labels, common to both players; these may be words, numbers, pictures, rows or columns in a matrix, or anything else that players can recognise. In cases in which labels consist of strings of characters, we denote this by enclosing the relevant strings in the symbols ≪ and ≫; thus the coordination game in which players name ‘heads’ or ‘tails’ can be denoted by L = {l1,l2} with l1 = ≪heads≫ and l2 = ≪tails≫. Each player knows L, and registers her strategy choice by choosing a label from this set. Labels are tied to strategies so that, if player i chooses label lj, she thereby chooses the strategy denoted sij in the normal form. By using the concept of labelling, it is possible to talk meaningfully about equilibrium selection in pure coordination games. But the problem remains of explaining the remarkable success with which human players choose the same labels in these games. Since we are trying to explain this success, it is useful to have an operational measure of it. We adapt a measure proposed by Mehta et al. (1994). Consider a coordination game with the label set L = {l1,…,ln}, and any set of N individuals, each of whom plays that game once with an anonymous co‐player. For each label lj, let mj be the number of individuals who choose it. Then the coordination indexc is given by: (1) This index measures the probability that two distinct individuals, chosen at random without replacement from the set of N individuals, choose the same label. It takes the value 1 if all individuals choose the same label, and 0 if everyone chooses a different label. If labels are chosen at random, the expected value of the index is 1/n. When making comparisons between games with different numbers of labels, it is clarifying to use the normalised coordination index (NCI ) defined by c* = cn. This can be interpreted as the ratio of c, the probability that two randomly‐chosen individuals choose the same label, to 1/n, the corresponding probability if labels are chosen at random. For example, Schelling (1960, pp. 54–8) reports an experiment in which 42 people were asked how they would choose in a pure coordination game with L = {≪heads≫, ≪tails≫}; 36 chose ≪heads≫. This implies c = 0.75, while random picking would imply the expected value c = 0.5. Thus c* = 1.50. We will say that, among a population of players of a pure coordination game, the distribution of label choices is more concentrated (or, equivalently, less dispersed), the higher the value of c*. Clearly, any explanation of why NCIs in pure coordination games are consistently higher than 1 must take some account of the content of the labels: it must show how some labels are more attractive or choiceworthy than others. We now consider two alternative theoretical approaches, each of which allows choices to be influenced by labelling. 1.2. Primary and Secondary Salience in Pure Coordination Games: Cognitive Hierarchy Theory The idea that focal points can be explained in terms of ‘primary’ and ‘secondary’ salience was first proposed by Lewis (1969, pp. 24–36).4 It is now possible to formulate this hypothesis more rigorously in terms of cognitive hierarchy theory. We apply this theoretical approach to the case of a two‐player coordination game. In such a game, each player chooses from the same set of labels L. A player’s behaviour can be represented by a probability distribution over these labels, typically denoted by p = (p1,…,pn). (As we shall not need to refer to specific players, we dispense with the indices ‘1’ and ‘2’ which identify the players.) The theory postulates a hierarchy of cognitive levels 0, 1,… . Each player has a specific cognitive level, representing the degree to which he can reason about other players. Players are uncertain about the cognitive levels of their opponents. For each level k, the relative frequency of level k players in the population of potential players of the game is qk; it is required that q0 > 0. Level 0 reasoners do not use game‐theoretic reasoning, but simply randomise between labels according to some exogenously given probability distribution p0. Each level 1 reasoner believes that his opponent reasons at level 0 and, hence (if that distribution is assumed to be common knowledge – an assumption that we will reconsider later) that the opponent acts according to p0. The level 1 reasoner chooses whichever label l * maximises his expected utility, given this belief. Each level 2 reasoner has the following beliefs about her opponent: With probability q0/(q0 + q1), the opponent reasons at level 0 and hence chooses according to p0. With probability q1/(q0 + q1), the opponent reasons at level 1, and hence chooses l * with probability 1. The level 2 reasoner chooses whichever label l ** maximises her expected utility relative to these beliefs.5 And similarly for the higher cognitive levels. It is a fundamental feature of cognitive hierarchy theory that, at each level, a player believes that his opponent’s level is lower than his own; by means of this assumption, the theory generates determinate solutions rather than equilibrium conditions. In the versions of cognitive hierarchy theory proposed by Stahl and Wilson (1995) and Camerer et al. (2004), p0 is assumed to be a uniform distribution. With this assumption, the theory predicts that all strategies in a pure coordination game are chosen with equal probability. However, a theory of focal points can be generated if p0 is allowed to be non‐uniform and is interpreted as describing the tendency of players to opt for the various labels when responding to them in some non‐rational or non‐strategic way. A very simple (but, we shall argue later, empirically inadequate) theory can be generated by assuming p0 to be common knowledge among players of level 1 or above. Given this assumption, level 0 reasoners choose according to p0 and all higher‐level reasoners choose the label l * with the highest value of (assuming that to be uniquely defined). If, as in Crawford and Iriberri’s (2007) analysis of hide‐and‐seek games, the assumed properties of p0 are justified only by appeals to intuitions about salience, a theory of this kind provides a framework for organising data but has little substantive content: it predicts that players tend to choose ‘salient’ labels but does not explain what ‘salience’ is. One way of going further is to develop a theory of the behaviour of level 0 reasoners. Bacharach and Stahl (2000) propose a theory of this kind, which rests on strong assumptions about the formal structure of labels and about how these are perceived by players. These assumptions allow a game to be re‐described in terms of the ‘options’ that players perceive; level 0 reasoners are assumed to choose each option with equal probability but this can induce non‐uniform probabilities for strategies. For our purposes, however, it is not necessary to make any particular assumptions about p0. The core idea of the cognitive hierarchy approach, that focal points are induced by non‐uniformities in p0, can be tested without making any prior assumptions about what those non‐uniformities might be. To do this, we follow Mehta et al. (1994) in defining p0 empirically, as measuring the actual frequencies with which the different labels are chosen in a picking task – that is, an experimental task in which players are required to ‘just pick’ one label from L in the absence of any strategic or payoff‐related reasons to choose one rather than another. The tendency for a given label to be picked in such a task is its degree of primary salience. This empirical definition of p0 is consistent with the logic of cognitive hierarchy theory. In those versions of the theory in which p0 is uniform, the underlying idea is that level 0 ‘reasoners’ have no perception of reasons for choosing one strategy rather than another: they just pick. Because (by assumption) strategies are perceived as symmetrical with one another, each strategy is picked with equal probability. If the theory is to be generalised to allow level 0 reasoners to take account of labels in pure coordination games, it is natural to assume that such players behave as if they were facing a picking task. It would be possible to stop at this point and specify a cognitive hierarchy model by assuming that p0, defined empirically by behaviour in a picking task, is common knowledge among reasoners of level 1 and above. But, in the context of one‐shot coordination games of the kind studied by Schelling (1960), that assumption seems implausible. For example, consider a coordination game played among university students in 2008, in which L = {≪1950≫, ≪1951≫,…, ≪2000≫}. On the basis of previous research, a theorist of focal points might predict that, in a picking task, most respondents would pick either their own birth years or ≪2000≫ (Mehta et al., 1994). But would a typical respondent know that? And even if she did, would she be able to predict the relative frequencies of the two types of answer? And would she know the distribution of birth years among her co‐players? It seems more realistic to allow for the possibility that different individuals have different beliefs about p0 and hence about which label has the greatest primary salience (that is, which label is chosen with the highest probability at level 0). We can define a probability distribution p1 over labels such that each is the probability that a randomly‐selected player of any of the levels 1, 2,… believes that lj is the label with the greatest primary salience. We will say that each is a measure of the secondary salience of the corresponding label lj. Notice that p1 is not a belief that can be attributed to any player, or to players in general. It is a probability distribution over players’ (possibly different) beliefs about primary salience. What relationship should we expect to find between p1 and p0? Consider any player i of level 1 or above. By imagining herself ‘just picking’, she can simulate the behaviour of a level 0 reasoner. If this simulation of picking is not simply the application of a random device, and if it is governed by the same mental process as governs actual picking, the result of the simulation – the simulated pick of a particular strategy – provides i with some information about the behaviour of level 0 reasoners. If this were the only relevant information available to i, the label that she picked would also be the label that she believed to have the greatest primary salience (i.e. the label that she believed to be modal in p0). If this were true for all players, we would have p1 = p0. But in coordination games of the kind described by Schelling, in which labels have distinct and meaningful descriptions in terms of the players’ own language, culture or experience, it is reasonable to suppose that some players do have additional information. Games in which labels are meaningful in this sense will be called describable. To understand the nature of this information, take the case of the game in which L = {≪1950≫, ≪1951≫, …, ≪2000≫}. Consider a player who happens to be considerably older than most of her fellow‐students. She imagines herself picking and picks ≪1973≫. On reflection, she realises that the special feature of this label is that she was born in 1973. Combining the information that she has picked her birth year with her imperfect background knowledge of the age distribution of university students, she might form the belief that the mode of p0 is the modal birth year of current students, and that this is 1988. If everyone behaves in this way (that is, picking her own birth year but attributing greatest primary salience to the year she believes to be the most common birth year) and if errors are random, p0 will have the same distribution as actual birth years; the distribution of p1 will have the same mode but will be less dispersed. Generalising from this example, whenever players have some understanding of their own propensities to pick some labels rather than others, we should expect that p1 and p0 have the same mode and that p1 is less dispersed. In principle, this line of analysis could be extended indefinitely. The next step would be to ask what players of levels 2 and above believe about p1. Just as in the case of p0, it is implausible to assume that p1 is common knowledge among these higher‐level reasoners. We might define a probability distribution p2 over labels such that each is the probability that a randomly‐selected player of any of the levels 2, 3,… believes that lj is the label with the greatest secondary salience; each measures the third‐order salience of the corresponding label lj. And so on. For typical experimental applications, however, it seems unlikely that third and higher‐order salience differ from secondary salience. Consider again the mature student in the coordination game with L = {≪1950≫, ≪1951≫,…, ≪2000≫}. Her own simulated pick is ≪1973≫; she believes that other subjects pick their birth years and that the modal birth year is 1988; so she predicts (or ‘guesses’) that the modal pick is ≪1988≫. We are now asking what prediction she would make about the modal guess of other subjects. If she attributes to them the same mode of reasoning that she used in her own guessing, she will predict that (except for random error) they all guess ≪1988≫. In principle, it is conceivable that she has background knowledge about the reasoning process by which subjects make such guesses, and that she knows that her own reasoning about this matter is atypical; but such knowledge is far more esoteric than the analogous knowledge about picking. It seems reasonable to assume that subjects do not have such information and, hence, that each subject’s prediction of the modal guess of other subjects is the same as her own guess. This implies p1 = p2. Repeating this argument for successively higher levels of reasoning, we have p1 = p2 = p3 = …. In other words, the behaviour of all players of levels 1, 2, … is predicted by p1. If this result holds, behaviour in a coordination game can be predicted using the information generated by a guessing task with the following structure: individuals are asked to guess which label from L was chosen by an unknown other subject in a picking task, and they are rewarded for guessing correctly. The inclusion of a guessing treatment is one of the key innovations of the experiments reported in this article. For players of level 1 or above, the guessing task can be interpreted as asking for a judgement about which label is primarily salient (that is, modal in p0); thus, we should expect the responses of such players to be predicted by p1. Given the logic of the cognitive hierarchy hypothesis, with its implicit assumption that level 0 reasoners do not consider how their opponents might behave, it is natural to assume that, when guessing, such individuals merely report what they themselves would have chosen in the picking treatment and, hence, that their responses are predicted by p0. But these assumptions imply a distribution of responses to the guessing task that is exactly the same as the distribution predicted for the coordination game. (Consider a randomly‐selected player in the coordination game. With probability 1 –q0 she reasons at level 1 or above and so her behaviour is predicted by p1; with probability q0 she is a level 0 reasoner and so her behaviour is predicted by p0.) Summing up, the preceding analysis has generated two hypotheses about pure coordination (‘PC’) games which can be tested in an experiment with counterbalanced picking, guessing and coordination treatments: Hypothesis PC1: In any pure coordination game, the distribution of responses is at least as concentrated for guessers as it is for pickers. If the game is describable, the distribution is more concentrated for guessers. Hypothesis PC2: In any pure coordination game, the distribution of responses is the same for coordinators as for guessers. These hypotheses decompose Mehta et al.’s (1994) finding that coordinators’ responses are more concentrated than pickers’. Hypothesis PC1 is implied by plausible assumptions about guessers’ information on the factors that influence pickers; it is not specific to cognitive hierarchy theory. In contrast, hypothesis PC2 is a distinctive implication of cognitive hierarchy theory and provides a test of that theory’s explanation of focal points. Notice that these hypotheses do not depend on any prior assumptions about the relative salience of different labels. Thus, tests of these hypotheses can use coordination games in which salience is subjective and culturally dependent. 1.3. Schelling Salience in Pure Coordination Games: the Theory of Team Reasoning We use the term Schelling salience in the same sense as Mehta et al. (1994), who quote Schelling’s explanation of why, in a pure coordination game with the instruction ‘Name a positive number’, the number 1 is the modal choice, even though it is not the most common response when people are asked just to pick a number. Schelling says: ‘If one… asks what number, among all positive numbers, is most clearly unique, or what rule of selection would lead to unambiguous results, one may be struck by the fact that the universe of all positive numbers has a ‘first’ or ‘smallest’ number’ (1960, p. 94, italics in original). The implication is that a ‘rule of selection’ is a criterion that a player can use to choose a label from the relevant set L; in the case of choosing from a set of integers, examples might include ‘Choose the smallest number’, ‘Choose your favourite number’, ‘Choose the number with the largest number of prime factors’ and so on. As these examples suggest, rules differ both in their probability of being recognised and (given that they are recognised and followed by both players) in their probability of leading to coordination. Schelling’s idea seems to be that the players look for a rule which clearly outperforms its rivals on these criteria; such a rule has ‘Schelling salience’. This idea has been developed using the concept of team reasoning by Sugden (1993, 1995) and Michael Bacharach (1999, 2006). An individual i team‐reasons with respect to a group G if she works out which profile of options for members of G would give the best results for G, and then chooses her component of that team‐optimal profile. Roughly, the individual asks ‘What should we do?’ and acts upon the answer in the expectation that other members of the group think and behave analogously. Whether the players of a particular game actually use team reasoning may depend on the nature of the game. Bacharach (2006) proposes that coordination games are particularly likely to prompt team reasoning, because the players’ interests are aligned and there are opportunities for mutual gain. If the profiles of ‘options’ over which players optimise are interpreted as strategy profiles, the team‐reasoning hypothesis has the same implications for behaviour in coordination games as the hypothesis that players use payoff dominance as an equilibrium selection device, as proposed by Harsanyi and Selten (1988).6 One of those implications is that, in pure coordination games, strategies are chosen at random. If the theory of team reasoning is to explain focal points in pure coordination games, an ‘option’ must be interpreted similarly to a ‘rule of selection’ in Schelling’s analysis, with no requirement of a one‐to‐one correspondence between options and strategies. Some theorists have followed the approach that Bacharach and Stahl (2000) use in conjunction with cognitive hierarchy theory – that is, to use assumptions about the formal structure of labels to re‐describe games in terms of the options that the players themselves perceive and then to assume that players optimise over profiles of such options (Bacharach, 1999, 2006; Casajus, 2001; Janssen, 2001).7 Another approach is to assume that players observe independent realisations of some payoff‐irrelevant process which, with non‐uniform probabilities, picks out (or ‘mentions’) labels from the set L; this allows players to use rules of selection such as ‘Choose the most‐frequently mentioned label’ (Sugden, 1995). For our purposes, however, there is no need to presuppose a particular formal model of rule selection because, as we now explain, our design will allow us to test more general implications of the team reasoning approach by comparing responses to guessing and coordination treatments. We start from the observation that primary and secondary salience can themselves be used as rules of selection. ‘Choose a label as if you were just picking’ (or ‘Choose the label with the greatest immediate appeal to you’) seems a credible rule of selection and corresponds with primary salience. ‘Choose the label most likely to be picked by someone who is just picking’ (or ‘Choose the label most likely to have immediate appeal to an average person’), is equally credible and corresponds with secondary salience. For experimental subjects confronting pure coordination games for the first time, the sheer oddness of having to choose from a set of apparently arbitrary labels seems likely to cue thoughts about just picking. Thus, one might expect subjects who are capable of team reasoning to be aware of these two rules. For the reasons explained in Section 1.2, two co‐players will generally have a greater probability of coordinating if they both follow the secondary salience rule than if they both follow the primary salience rule. Hence, if a team‐reasoning player cannot find a rule of selection which gives a higher probability of coordination than the secondary salience rule, she will follow the latter rule. In this case, secondary salience and Schelling salience coincide. Thus, the theory of team reasoning is not disconfirmed if, as predicted by cognitive hierarchy theory, guessing and coordination treatments generate the same distribution of responses. But if the two distributions are different, we can ask whether the differences have the characteristics that would be expected, were the theory of team reasoning correct. Since team reasoners look for a team‐optimal rule of selection, they should reject the secondary salience rule only in favour of rules which give at least as great a probability of coordination. If guessing and coordination treatments generate different distributions of responses, the distribution from the latter treatment should be at least as concentrated as that from the former. Thus, the theory of team reasoning implies the following hypothesis: Hypothesis PC3: In any pure coordination game, if the guessing and coordination treatments generate different distributions of responses, the distribution from the coordination treatment is at least as concentrated as that from the guessing treatment. 1.4. Nondescript Hi‐Lo Games The principles underlying the two rival hypotheses can be tested in another way, by adapting an example discussed by Schelling (1960, pp. 295–6). Schelling considers a Hi‐Lo game with n = 4, U1 = 10, U2 = 10, U3 = 10, and U4 = 9. If we consider only the normal form of the game, there are three completely symmetrical pure‐strategy Nash equilibria and one further such equilibrium, distinguished from the others by giving a lower payoff to both players. Schelling asks us to assume that ‘the strategies occur in a way that makes ordering them intellectually impossible for rational players’. In our framework, in which it is a matter of definition that every strategy has a unique label, the closest approximation to Schelling’s assumption is to make the differences between the labels nondescript– that is, such that, although normal players are aware that the labels are not the same, they do not have any readily available way of describing those differences, even to themselves.8 If all the labels in a coordination game are nondescript, we will say that the game itself is ‘nondescript’. Schelling claims of his game: ‘[I]f no better means of coordination can be discerned, the “solution” may be the strategy pair… with payoffs of 9 apiece’. This conclusion follows from a straightforward extension of the team‐reasoning analysis in Section 1.3. Because the labels are nondescript, there is no obvious rule which unambiguously picks out one of the labels by virtue of its standing out. However, there is an apparently obvious rule which, if followed by both players, would lead them both to choose l4 by virtue of the corresponding payoff . This is the rule ‘Choose the label attached to the payoff that is the odd one out’. The opposite rule, ‘Pick one of the labels attached to the highest payoff’, is sub‐optimal in the team‐reasoning sense (on the assumption that players seek jointly to maximise expected utility). Thus, the hypothesis of team reasoning implies that l4 is chosen. In contrast, consider the implications of cognitive hierarchy theory. The first step in applying this theory is to specify p0, the distribution of the responses of level 0 reasoners. Recall that level 0 reasoners are people who do not engage in any kind of strategic reasoning; they act as if unaware that they are interacting with anyone. One possible assumption is that these individuals are completely unaware of the significance of payoffs and so just pick among labels (as, in our analysis, they do in pure coordination games). If the labels are nondescript, this is equivalent to picking at random. Given that level 0 reasoners can be expected to behave in this way, level 1 reasoners are indifferent between l1, l2 and l3 (each of which they believe will give an expected utility of 10/4) but strictly prefer each of these to l4 (which they believe will give 9/4). If level 1 reasoners randomise between l1, l2 and l3, level 2 reasoners are also indifferent between these three labels and prefer each of them to l4 and so on. The overall implication is that l4 is chosen with probability q0/4, while each other label is chosen with probability q0/4 + (1 − q0)/3. Alternatively, in specifying p0, we might assume that level 0 reasoners take some account of payoffs but in a non‐strategic way. It is natural to assume that, for a player who is not thinking strategically, higher payoffs have a stronger tendency to prompt positive affective responses than lower payoffs do – in the same sense that, in a picking task, ≪Porsche≫ is a more attractive label than ≪Volkswagen≫. If, as we conjecture, primary salience is associated with pre‐reflective attractiveness, level 0 reasoners will choose l4 with probability less than 1/4 and higher‐level reasoners will not choose it at all, with the result that its overall probability of being chosen is less than q0/4.9 This analysis can be extended to the general class of nondescript Hi‐Lo games. Consider any such game. Suppose that there are n1 labels for which the payoff is x1, n2 labels for which the payoff is x2,…, and nm labels for which the payoff is xm, where x1 > x2 > ??? > xm. Then the rule ‘Pick one of the labels associated with a payoff of xk’, if followed by both players, would give each an expected payoff of xk/nk. Each of the nk labels associated with the k that maximises the value of xk/nk is team‐optimal. Team reasoning requires each player to pick from the set of team‐optimal labels. For example, in a game in which there are six labels associated with payoffs 10, 10, 10, 9, 8, 7, the optimal rule is to choose the label with the payoff 9; in a game in which there are five labels and payoffs 10, 10, 10, 10, 1, the optimal rule is to pick from the set of labels with payoff 10 (giving an expected payoff of 2.5). In contrast, cognitive hierarchy theory implies that every player of level 1 or above randomises among the labels associated with the highest payoff, while level 0 players randomise among all labels (possibly giving greater weight to labels associated with higher payoffs). Thus, averaging across players of all levels, the choice probability for each of the labels associated with the highest payoff x1 is at least q0/n + (1 − q0)/n1. The analysis in the preceding paragraph assumes that players of level 1 or above maximise expected utility (and that this is common knowledge) and that utility payoffs are common knowledge. In applying this analysis to games in which payoffs are described in material units such as money, some allowance must be made for players’ attitudes to risk, and for these attitudes not being common knowledge. However, it seems reasonable to assume it to be common knowledge that players’ attitudes to risk are not pathologically distant from risk neutrality. Thus, in the first example discussed in the preceding paragraph, if payoffs are in sterling, it seems uncontroversial to assume that the certainty of £9 is preferred to a 0.33 chance of £10. In the second example, it is probably safe to assume that a 0.25 chance of £10 is preferred to the certainty of £1. Summing up, we have generated two rival hypotheses about behaviour in the coordination treatment of Hi‐Lo (‘HL’) games. (The formulations we use below allow for random error in players’ choices.) Hypothesis HL1 is implied by cognitive hierarchy theory, while HL2 is implied by team reasoning: Hypothesis HL1: In any nondescript Hi‐Lo game, the choice probability for each of the labels associated with the highest payoff is greater than that for every label associated with a lower payoff. Hypothesis HL2: In any nondescript Hi‐Lo game, the choice probability for each team‐optimal label is greater than that for every other label. 2. Experimental Design 2.1. Features Common to Both Experiments We implemented two versions of the same design, conducted in March 2001 using subjects recruited from the general student populations of the University of Amsterdam in the Netherlands (for one experiment) and the University of Nottingham in the UK (for the other).10 In each case, subjects faced a series of tasks. In each task, the subject was presented with a set of objects and was required to choose one. Each object was associated with a specified number of points. There were three treatments. In the picking treatment, the subject was simply asked to choose one object and scored the number of points specified for that object. In the guessing treatment, the subject was paired with a randomly‐selected anonymous partner in the picking treatment and was asked to guess which object her partner had chosen; this pairing was the same for all tasks. If this guess was correct, the guesser scored the number of points associated with the relevant object; otherwise, she scored nothing. In the coordination treatment, the subject was paired with a randomly‐selected anonymous partner facing the same task in the same treatment; again, the pairing was the same for all tasks. If the two partners chose the same object, both scored the number of points associated with it; otherwise, both scored nothing. In all treatments, subjects were unable to communicate with one another. No feedback was given until the end of the experiment, when subjects were paid in proportion to the total number of points scored. In implementing these three treatments, we used tasks of two types. In a text task, all objects carry the same number of points (10 in all such tasks in both experiments) but each has a distinct label in the form of a string of text. For example, one text task in the Amsterdam experiment contains four objects with the labels ≪Jaguar≫, ≪Ford≫, ≪Porsche≫, ≪Ferrari≫. In a number task, the objects may carry different numbers of points but in other respects they are (as far as possible) nondescript. Notice that, when presented in the coordination treatment, text tasks are describable pure coordination games. Number tasks in which all objects carry the same number of points are nondescript pure coordination games. Other number tasks are nondescript Hi‐Lo games.11 Subjects were allocated at random between coordination and picking/guessing sessions. In the coordination sessions, all subjects faced the whole set of text and number tasks in the coordination treatment. In picking/guessing sessions, subjects first faced half of the set of text tasks and half of the set of number tasks in the picking treatment. They then faced the remaining tasks in the guessing treatment. Within each treatment, the order in which tasks were presented to subjects was randomised. The design was counterbalanced so that each task was faced in each of the three treatments, in each case by a different set of subjects. Subjects were allocated to sessions so as to generate approximately equal numbers of responses for the three treatments. Because picking was always done before guessing and because the instructions for the guessing tasks were not given until the picking tasks had been completed, pickers had no reason to think of their responses as having any effect on other subjects. However, we hoped that guessers’ prior experience of picking would help them to understand the picking task that their partners had faced. Because coordinators were not aware of the picking and guessing tasks, reasoning in the coordination treatment could not be cued by ideas suggested by the other two treatments. In the Amsterdam experiment, 164 subjects were randomly allocated to 15 sequential sessions. Three observations were lost through computer crashes, resulting in sample sizes of 53, 52 and 56 subjects for the picking, guessing and coordination treatments respectively. Subjects were paid at a pre‐announced rate of 15 Dutch cents per point ($0.06 at the exchange rate of the time), in addition to a fixed show‐up fee of 5 guilders ($2.11); average earnings were 30.12 guilders ($12.05) per subject. In the Nottingham experiment, 134 subjects took part in three simultaneous sessions, resulting in sample sizes of 45, 45 and 44 for the three treatments. Subjects were told at the start of the experiment that payment would be at a constant rate per point, to be calculated ex post to ensure an average payment of £7 ($10.43) per subject for the experiment as a whole. Although the two experiments shared a common basic design, there were some differences in the presentation of tasks to subjects and different sets of tasks were used. These features of the experiments are described in the following two Sections. 2.2. Presentation of Tasks The Amsterdam experiment was computerised. The objects from which a choice had to be made were presented as discs moving within a rectangular field on the computer screen. Each disc moved in a straight line until it collided with a border or another disc, in which case it rebounded in a randomly perturbed direction. The subject selected a disc by clicking on it with the mouse. In a text task, both the relevant piece of text and the number of points was written on each disc. In a number task, only the number of points was shown. Figures 1 and 2 show examples of the two types of task, as represented in this display. (Lines have been added to indicate movement; these did not appear in the experiment.) Since the pattern of movement of the discs in any given task was the same for both members of any given pair of co‐players, the movement of each disc gave it a distinct label, even in a number task in which two or more discs carried the same number of points. (To avoid confounds, patterns of movement were varied across sessions.) We expected that this kind of labelling would be perceived as nondescript by most subjects, while text differences would be immediately obvious. Fig. 2. Open in new tabDownload slide Display for Amsterdam Number Tasks Fig. 2. Open in new tabDownload slide Display for Amsterdam Number Tasks Fig. 1. Open in new tabDownload slide Display for Amsterdam Text Tasks Fig. 1. Open in new tabDownload slide Display for Amsterdam Text Tasks Instructions were given both orally (to all subjects in the session together, to ensure common knowledge) and on subjects’ computer screens. The relevant instructions in the picking treatment (described to subjects as ‘part 1’ of the experiment) were: In this part of the experiment, your earnings are determined by your decisions alone. There are fourteen tasks in part 1. Each task shows a set of moving objects, with a number on each one. The display in each task can be thought of as a short ‘film’. For each task, you have to click on one object in each film, using your mouse. … For that task, you will earn the number of points shown on the object. The corresponding instructions in the guessing treatment were: Each ‘film’ in part 2 is one that your partner had during part 1, in which he or she just clicked on an object and received the number of points written on it. Again, you have to click on one object for each task, and confirm your decision. This time, though, you have to guess what your partner did during part 1. If you click on the same object as your partner, you will receive the number of points indicated on that object. If not, you will receive nothing for that task. In the coordination treatment, the instructions were: Each task shows a set of moving objects, with a number on each one. The display in each task can be thought of as a short ‘film’. Your partner has the same set of films. For each task, you have to click on one object in each film, using your mouse. … If you click on the same object as your partner, you will both receive the number of points indicated on that object. If not, neither of you will receive anything for that task. In the Nottingham experiment, tasks were presented in booklets. Each task appeared as a row of five objects, and the subject selected one by marking a tick below it. Subjects who had been paired with one another saw the same five objects, but the order in which these were displayed from left to right was randomised across subjects (and subjects were told this). Each object was represented as a box, subdivided into two parts. The lower part stated the number of points associated with the object. In text tasks, the upper part of each box contained a distinct string of text; Figure 3 shows a typical example. In number tasks, the upper part of each box contained a distinct pattern of symbols; Figure 4 shows an example. These five patterns were generated by separate runs of a common computer program which included a random component. Patterns were generated independently for each pair of subjects. Our intention was that these patterns, although clearly constituting distinct labels, would be perceived as nondescript. Fig. 4. Open in new tabDownload slide Display for Nottingham Number Tasks Fig. 4. Open in new tabDownload slide Display for Nottingham Number Tasks Fig. 3. Open in new tabDownload slide Display for Nottingham Text Tasks Fig. 3. Open in new tabDownload slide Display for Nottingham Text Tasks The relevant instructions (given both orally, to all participants together, and in print) were: [Picking treatment] Your objective is the same for each task: to pick one of the boxes. You are required to indicate which box you have chosen by putting a tick just below the box. … For each of the sixteen tasks, you will be awarded the number of points specified in the box you have picked. The total number of points awarded to you for all the tasks determines how much money you win in this part of the experiment. [Guessing treatment] There is an even number of people taking part in this room, and we have randomly divided you into pairs for the duration of this part of the experiment. …. What you see in your second booklet is the same as your partner saw in their first booklet when you were all asked to pick one of the five boxes for each task. So for each task in your second booklet, your partner has already chosen one of the five boxes and scored the corresponding number of points, which they keep regardless of what you do next. Your objective for each task now is: to guess which of the boxes your unknown partner picked. You are required to indicate which box you think this is by putting a tick just below the box. … If you correctly guess which box your partner picked, then you will be awarded the number of points specified in the box. If you fail to guess which box your partner picked, you will not receive any points for that task. The total number of points awarded to you for these sixteen tasks determines how much money you win in this part of the experiment. [Coordination treatment] There is an even number of people taking part in this room, and we have randomly divided you into pairs for the duration of the experiment. ….Your objective is the same for each task: to choose the same box as that of your unknown partner. You are required to indicate which box you have chosen by putting a tick just below the box. … If the pair of you choose the same box, then you as an individual will be awarded the number of points specified in the box. If the pair of you fail to choose the same box, you will not receive any points for that task. The total number of points awarded to you for all the tasks determines how much money you win. 2.3. Text Tasks Each experiment used fourteen text tasks, denoted TA1–TA14 (for Amsterdam) and TN1–TN14 (for Nottingham). These tasks used the following sets of labels (the string symbols ≪ and ≫ are omitted to reduce clutter): Amsterdam text tasks12 TA1: {grijs, indigo, karmozijn, magenta, turkoois} TA2: {Ferrari, Ford, Jaguar, Porsche} TA3: {Berlin, Brussel, Lissabon, Madrid, Mannheim} TA4: {almond, cashew, peanut, walnut} TA5: {diamond, emerald, glass, sapphire} TA6: {chrome, copper, iron, plastic, steel} TA7: {bread, curry, pizza, steak} TA8: {beer, sherry, water, whisky, wine} TA9: {Carlsberg, Corsendonk, Grimbergen, Rochefort, Westmalle}13 TA10: {frog, leopard, panther, tiger} TA11: {aeroplane, bicycle, helicopter, hovercraft} TA12: {chess, football, squash, tennis, volleyball} TA13: {Barbados, Bern, Florida, Honolulu} TA14: {jogging, running, sitting, walking} Nottingham text tasks TN1: {Friday lunchtime, Monday morning, Saturday night, Sunday night, Wednesday evening} TN2: {Earth, Mars, Mercury, Saturn, Venus} TN3: {Ford, Mercedes, Pontiac, Porsche, Volkswagen} TN4: {cheese omelette, ham omelette, mushroom omelette, plain omelette, prawn omelette} TN5: {1, 2, 7, 10, 15} TN6: {deck chair, dining chair, easy chair, rocking chair, stool} TN7: {Colorado, Florida, Louisiana, Nevada, Ontario} TN8: {jogging, sitting, sunbathing, swimming, walking} TN9: {1978, 1979, 1980, 1981, 2000} TN10: {David, John, Michael, Robert, Steven} TN11: {win champagne, win chocolate, win money, win nothing, win trophy} TN12: {blue, green, orange, purple, red} TN13: {apple juice, carrot juice, grapefruit juice, mango juice, pineapple juice} TN14: {Berlin, Calais, Paris, Prague, Rome} In composing these sets of labels, we tried to ensure that Schelling salience and secondary salience would diverge, so as to increase the potential for team reasoning, if operative, to generate differences between the responses of guessers and coordinators. However, we emphasise again that our formal hypothesis tests apply to any set of labels; they are not conditional on any particular characteristics of our tasks. Our aim (which, as will emerge later, we achieved with varying degrees of success) was that in each task, one of the labels, say l1, would be unambiguously picked out by some obvious rule of selection other than primary or secondary salience; we will call this label the intended salient. In the Amsterdam tasks, the relevant rule was always ‘Choose the odd one out’. The Nottingham tasks were composed with the intention that each of them would evoke one or other of the rules that seemed to have been used by subjects in Mehta et al.’s (1994) coordination treatment; in addition to ‘Choose the odd one out’, these included ‘Choose the archetype’ and ‘Choose the status quo’.14 We intended that each of the other labels l2, .., ln should be roughly equal in the kind of immediate appeal which is likely to induce primary salience, while l1 should have either the same or less appeal. For example, consider the set of labels used in TA3: {≪Berlin≫, ≪Brussel≫,≪Lisbon≫, ≪Madrid≫, ≪Mannheim≫}. Here, ≪Mannheim≫ is the odd one out: all the other cities are national capitals. We conjecture that immediate appeal is determined by a person’s affective response to whatever ideas are suggested by the labels. In this case, one might expect each of the four capital cities to evoke ideas of national or cultural significance, or of attractiveness as a tourist destination; for any individual, the relative force of these ideas would depend on matters of taste, culture, nationality and personal association. By comparison, Mannheim is not generally credited with comparable positive qualities. Thus, one might expect p0 to have a dispersed distribution. Since people will find it difficult to judge which label is modal in p0, the distribution of p1 will be dispersed too and so co‐players who follow the rule of secondary salience will be relatively unsuccessful. In particular, because ≪Mannheim≫ is so obviously the odd one out in L, they will be less successful than they would be by following the rule ‘Choose the odd one out’. To allow readers to test their own intuitions, the intended salient for each task is identified only in a footnote.15 If the reader’s intuitions sometimes differ from ours, he or she should remember that the intended salient plays no role in our hypothesis tests. 2.4. Number Tasks The Amsterdam experiment included fourteen number tasks, NA1–NA14. The Nottingham experiment included eighteen such tasks, NN1–NN18. For our purposes, the main characteristic of a number task is the array of points carried by the set of objects from which the subject must choose. The following arrays were used: Amsterdam number tasks Type 1 NA1: (10, 10, 10, 9) NA2: (10, 10, 10, 10, 10, 9) NA3: (10, 10, 10, 9, 8, 7) NA4: (10, 10, 10, 9, 9, 8) NA5: (10, 10, 10, 10, 9, 9) Type 2 NA6: (10, 9) NA7: (10, 10, 10, 9, 9, 9) NA8: (10, 1) NA9: (10, 10, 10, 1) NA10: (10, 10, 10, 10, 10, 1) Type 3 NA11: (10, 10) NA12: (10, 10, 10, 10) NA13: (10, 10, 10, 10, 10) NA14: (10, 10, 10, 10, 10, 10) Nottingham number tasks Type 1 NN1, NN2, NN3, NN4, NN5, NN6: (10, 10, 10, 10, 9) Type 2 NN7, NN8, NN9, NN10, NN11, NN12: (10, 10, 10, 10, 1) Type 3 NN13, NN14, NN15, NN16, NN17, NN18: (10, 10, 10, 10, 10) Coordination‐treatment responses that are consistent with the theory of team reasoning (under credible assumptions about risk attitudes)16 are shown in bold. In all cases, the cognitive hierarchy hypothesis implies that subjects who reason at level 1 or above choose 10‐point options. These tasks are divided into three types. In type 1 tasks, the coordination treatment is a nondescript Hi‐Lo game in which the team reasoning hypothesis implies that players will choose an option which does not carry 10 points. These tasks allow a direct comparison between the two hypotheses. In type 2 tasks, the coordination treatment is a nondescript Hi‐Lo game in which both hypotheses have the same implications for behaviour. These tasks are significant because they subject the team reasoning hypothesis to an additional test. In almost all type 1 tasks, the team‐optimal response is also the option with the lowest number of points, and, in terms of points, is the odd one out. In type 2 tasks, however, choosing options with these characteristics is contrary to the team reasoning hypothesis. In type 3 tasks, all options carry 10 points, and so the coordination treatment is a pure coordination game. These tasks allow us to test the background assumption that the labels associated with the options are nondescript. If that assumption were true, the NCI for the coordination treatment of a type 3 task would equal 1 (plus or minus random noise). In the Nottingham experiment, each of three arrays of points (one for each of the task types) occurs in six different tasks. Recall that each number task involved a set of five randomly‐generated patterns. We used six different pattern‐generating programs; each program was paired with each array of points in a factorial design. This allowed us to investigate whether subjects’ responses were influenced by the kinds of patterns they were shown. In fact, we found no pattern‐specific effects. 3. Results for Text Tasks 3.1. Presentation of Results Table 1 reports the frequency distribution of responses for each of the 28 text tasks and for each of the three treatments. For each distribution, the NCI is shown at the bottom of the relevant column. For each task, Table 1 reports four tests. Table 1 
Responses to Text Tasks . pick . guess . coordinate . Amsterdam tasks TA1  grijs 11 17 35  indigo 12 8 8  karmozijn 9 9 5  magenta 10 11 1  turkoois 11 7 7  NCI 0.935 1.040 2.125  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA2  Ford 10 11 31  Ferrari 13 22 11  Jaguar 20 7 5  Porsche 10 12 9  NCI 1.040 1.124 1.472  significance wrt:   pick ns **   guess *  chisq wrt guess ** TA3  Mannheim 12 9 25  Berlin 3 2 4  Brussels 8 13 9  Lisbon 15 8 7  Madrid 15 20 11  NCI 1.115 1.255 1.355  significance wrt:   pick ns ns   guess ns  chisq wrt guess * TA4  peanut 13 13 12  almond 13 20 18  cashew 20 14 15  walnut 7 5 11  NCI 1.064 1.112 0.984  significance wrt:   pick ns #   guess #  chisq wrt guess ns TA5  glass 11 14 30  diamond 24 28 21  emerald 7 8 3  sapphire 11 2 2  NCI 1.180 1.504 1.684  significance wrt:   pick ns *   guess ns  chisq wrt guess * TA6  plastic 15 16 36  chrome 11 11 7  copper 9 10 2  iron 11 6 6  steel 7 9 5  NCI 0.985 1.020 2.200  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA7  bread 8 6 8  curry 12 9 23  pizza 10 16 17  steak 23 21 8  NCI 1.136 1.148 1.156  significance wrt:   pick ns ns   guess ns  chisq wrt guess ** TA8  water 20 15 38  beer 13 26 11  sherry 4 1 0  whisky 6 6 5  wine 10 4 2  NCI 1.210 1.700 2.495  significance wrt:   pick * **   guess **  chisq wrt guess ** TA9  Carlsberg 25 23 37  Corsendonk 5 3 2  Grimbergen 8 13 2  Rochefort 9 4 11  Westmalle 6 9 4  NCI 1.410 1.420 2.365  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA10  frog 17 17 41  leopard 11 11 5  panther 7 4 5  tiger 18 20 5  NCI 1.060 1.168 2.208  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA11  bicycle 18 18 37  aeroplane 19 18 15  helicopter 6 6 2  hovercraft 10 10 2  NCI 1.116 1.104 2.008  significance wrt:   pick # **   guess **  chisq wrt guess ** TA12  chess 18 15 36  football 11 30 14  squash 5 1 0  tennis 16 6 3  volleyball 3 0 3  NCI 1.235 2.095 2.360  significance wrt:   pick ** **   guess ns  chisq wrt guess ** TA13  Bern 11 12 29  Barbados 13 10 4  Florida 21 17 12  Honolulu 8 13 11  NCI 1.076 0.980 1.384  significance wrt:   pick # *   guess **  chisq wrt guess * TA14  sitting 16 21 39  jogging 6 5 5  running 20 15 10  walking 11 11 2  NCI 1.104 1.148 2.072  significance wrt:   pick ns **   guess **  chisq wrt guess ** Nottingham tasks TN1  Friday lunchtime 13 2 4  Monday morning 6 2 3  Saturday night 17 36 34  Sunday night 4 1 0  Wednesday evening 5 4 3  NCI 1.235 3.220 3.030  significance wrt:   pick ** **   guess #  chisq wrt guess ns TN2  Earth 18 25 33  Mars 5 6 3  Mercury 9 2 1  Saturn 8 4 3  Venus 5 8 4  NCI 1.195 1.770 2.855  significance wrt:   pick * **   guess **  chisq wrt guess ns TN3  Ford 3 4 2  Mercedes 8 11 13  Pontiac 4 1 0  Porsche 21 29 26  Volkswagen 9 0 3  NCI 1.430 2.360 2.150  significance wrt:   pick ** *   guess #  chisq wrt guess ns TN4  plain omelette 5 9 7  cheese omelette 19 19 21  ham omelette 6 11 3  mushroom omelette 8 2 7  prawn omelette 7 4 6  NCI 1.235 1.360 1.425  significance wrt:   pick ns ns   guess ns  chisq wrt guess ns TN5  1 4 3 5  2 5 0 2  7 9 16 6  10 11 7 8  15 16 19 23  NCI 1.145 1.590 1.625  significance wrt:   pick * *   guess ns  chisq wrt guess ns TN6  stool 4 3 5  deck chair 11 8 7  dining chair 4 2 1  easy chair 10 24 15  rocking chair 16 8 16  NCI 1.170 1.695 1.355  significance wrt:   pick * ns   guess #  chisq wrt guess ns TN7  Ontario 9 3 4  Colorado 8 6 5  Florida 18 33 33  Louisiana 4 0 0  Nevada 6 3 2  NCI 1.200 2.775 2.880  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN8  sitting 3 2 2  jogging 11 4 4  sunbathing 13 30 26  swimming 11 5 7  walking 7 4 5  NCI 1.070 2.315 1.920  significance wrt:   pick ** **   guess #  chisq wrt guess ns TN9  2000 13 23 27  1978 3 1 2  1979 3 1 2  1980 11 7 5  1981 15 13 8  NCI 1.230 1.780 2.065  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN10  John 9 10 14  David 11 8 9  Michael 9 13 12  Robert 9 6 5  Steven 7 8 4  NCI 0.930 0.980 1.105  significance wrt:   pick ns ns   guess ns  chisq wrt guess ns TN11  win nothing 4 2 1  win champagne 8 4 1  win chocolate 6 1 0  win money 22 38 41  win trophy 5 0 1  NCI 1.465 3.585 4.335  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN12  red 9 15 8  blue 17 16 23  green 7 3 4  orange 3 1 3  purple 9 10 6  NCI 1.170 1.380 1.610  significance wrt:   pick ns *   guess ns  chisq wrt guess ns TN13  carrot juice 5 6 2  apple juice 10 19 29  grapefruit juice 7 4 3  mango juice 12 7 5  pineapple juice 11 9 5  NCI 0.995 1.260 2.275  significance wrt:   pick ns **   guess **  chisq wrt guess ns TN14  Calais 5 1 4  Berlin 2 3 1  Paris 10 23 27  Prague 8 7 2  Rome 20 11 10  NCI 1.385 1.675 2.130  significance wrt:   pick ns **   guess ns  chisq wrt guess ns . pick . guess . coordinate . Amsterdam tasks TA1  grijs 11 17 35  indigo 12 8 8  karmozijn 9 9 5  magenta 10 11 1  turkoois 11 7 7  NCI 0.935 1.040 2.125  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA2  Ford 10 11 31  Ferrari 13 22 11  Jaguar 20 7 5  Porsche 10 12 9  NCI 1.040 1.124 1.472  significance wrt:   pick ns **   guess *  chisq wrt guess ** TA3  Mannheim 12 9 25  Berlin 3 2 4  Brussels 8 13 9  Lisbon 15 8 7  Madrid 15 20 11  NCI 1.115 1.255 1.355  significance wrt:   pick ns ns   guess ns  chisq wrt guess * TA4  peanut 13 13 12  almond 13 20 18  cashew 20 14 15  walnut 7 5 11  NCI 1.064 1.112 0.984  significance wrt:   pick ns #   guess #  chisq wrt guess ns TA5  glass 11 14 30  diamond 24 28 21  emerald 7 8 3  sapphire 11 2 2  NCI 1.180 1.504 1.684  significance wrt:   pick ns *   guess ns  chisq wrt guess * TA6  plastic 15 16 36  chrome 11 11 7  copper 9 10 2  iron 11 6 6  steel 7 9 5  NCI 0.985 1.020 2.200  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA7  bread 8 6 8  curry 12 9 23  pizza 10 16 17  steak 23 21 8  NCI 1.136 1.148 1.156  significance wrt:   pick ns ns   guess ns  chisq wrt guess ** TA8  water 20 15 38  beer 13 26 11  sherry 4 1 0  whisky 6 6 5  wine 10 4 2  NCI 1.210 1.700 2.495  significance wrt:   pick * **   guess **  chisq wrt guess ** TA9  Carlsberg 25 23 37  Corsendonk 5 3 2  Grimbergen 8 13 2  Rochefort 9 4 11  Westmalle 6 9 4  NCI 1.410 1.420 2.365  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA10  frog 17 17 41  leopard 11 11 5  panther 7 4 5  tiger 18 20 5  NCI 1.060 1.168 2.208  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA11  bicycle 18 18 37  aeroplane 19 18 15  helicopter 6 6 2  hovercraft 10 10 2  NCI 1.116 1.104 2.008  significance wrt:   pick # **   guess **  chisq wrt guess ** TA12  chess 18 15 36  football 11 30 14  squash 5 1 0  tennis 16 6 3  volleyball 3 0 3  NCI 1.235 2.095 2.360  significance wrt:   pick ** **   guess ns  chisq wrt guess ** TA13  Bern 11 12 29  Barbados 13 10 4  Florida 21 17 12  Honolulu 8 13 11  NCI 1.076 0.980 1.384  significance wrt:   pick # *   guess **  chisq wrt guess * TA14  sitting 16 21 39  jogging 6 5 5  running 20 15 10  walking 11 11 2  NCI 1.104 1.148 2.072  significance wrt:   pick ns **   guess **  chisq wrt guess ** Nottingham tasks TN1  Friday lunchtime 13 2 4  Monday morning 6 2 3  Saturday night 17 36 34  Sunday night 4 1 0  Wednesday evening 5 4 3  NCI 1.235 3.220 3.030  significance wrt:   pick ** **   guess #  chisq wrt guess ns TN2  Earth 18 25 33  Mars 5 6 3  Mercury 9 2 1  Saturn 8 4 3  Venus 5 8 4  NCI 1.195 1.770 2.855  significance wrt:   pick * **   guess **  chisq wrt guess ns TN3  Ford 3 4 2  Mercedes 8 11 13  Pontiac 4 1 0  Porsche 21 29 26  Volkswagen 9 0 3  NCI 1.430 2.360 2.150  significance wrt:   pick ** *   guess #  chisq wrt guess ns TN4  plain omelette 5 9 7  cheese omelette 19 19 21  ham omelette 6 11 3  mushroom omelette 8 2 7  prawn omelette 7 4 6  NCI 1.235 1.360 1.425  significance wrt:   pick ns ns   guess ns  chisq wrt guess ns TN5  1 4 3 5  2 5 0 2  7 9 16 6  10 11 7 8  15 16 19 23  NCI 1.145 1.590 1.625  significance wrt:   pick * *   guess ns  chisq wrt guess ns TN6  stool 4 3 5  deck chair 11 8 7  dining chair 4 2 1  easy chair 10 24 15  rocking chair 16 8 16  NCI 1.170 1.695 1.355  significance wrt:   pick * ns   guess #  chisq wrt guess ns TN7  Ontario 9 3 4  Colorado 8 6 5  Florida 18 33 33  Louisiana 4 0 0  Nevada 6 3 2  NCI 1.200 2.775 2.880  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN8  sitting 3 2 2  jogging 11 4 4  sunbathing 13 30 26  swimming 11 5 7  walking 7 4 5  NCI 1.070 2.315 1.920  significance wrt:   pick ** **   guess #  chisq wrt guess ns TN9  2000 13 23 27  1978 3 1 2  1979 3 1 2  1980 11 7 5  1981 15 13 8  NCI 1.230 1.780 2.065  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN10  John 9 10 14  David 11 8 9  Michael 9 13 12  Robert 9 6 5  Steven 7 8 4  NCI 0.930 0.980 1.105  significance wrt:   pick ns ns   guess ns  chisq wrt guess ns TN11  win nothing 4 2 1  win champagne 8 4 1  win chocolate 6 1 0  win money 22 38 41  win trophy 5 0 1  NCI 1.465 3.585 4.335  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN12  red 9 15 8  blue 17 16 23  green 7 3 4  orange 3 1 3  purple 9 10 6  NCI 1.170 1.380 1.610  significance wrt:   pick ns *   guess ns  chisq wrt guess ns TN13  carrot juice 5 6 2  apple juice 10 19 29  grapefruit juice 7 4 3  mango juice 12 7 5  pineapple juice 11 9 5  NCI 0.995 1.260 2.275  significance wrt:   pick ns **   guess **  chisq wrt guess ns TN14  Calais 5 1 4  Berlin 2 3 1  Paris 10 23 27  Prague 8 7 2  Rome 20 11 10  NCI 1.385 1.675 2.130  significance wrt:   pick ns **   guess ns  chisq wrt guess ns Open in new tab Table 1 
Responses to Text Tasks . pick . guess . coordinate . Amsterdam tasks TA1  grijs 11 17 35  indigo 12 8 8  karmozijn 9 9 5  magenta 10 11 1  turkoois 11 7 7  NCI 0.935 1.040 2.125  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA2  Ford 10 11 31  Ferrari 13 22 11  Jaguar 20 7 5  Porsche 10 12 9  NCI 1.040 1.124 1.472  significance wrt:   pick ns **   guess *  chisq wrt guess ** TA3  Mannheim 12 9 25  Berlin 3 2 4  Brussels 8 13 9  Lisbon 15 8 7  Madrid 15 20 11  NCI 1.115 1.255 1.355  significance wrt:   pick ns ns   guess ns  chisq wrt guess * TA4  peanut 13 13 12  almond 13 20 18  cashew 20 14 15  walnut 7 5 11  NCI 1.064 1.112 0.984  significance wrt:   pick ns #   guess #  chisq wrt guess ns TA5  glass 11 14 30  diamond 24 28 21  emerald 7 8 3  sapphire 11 2 2  NCI 1.180 1.504 1.684  significance wrt:   pick ns *   guess ns  chisq wrt guess * TA6  plastic 15 16 36  chrome 11 11 7  copper 9 10 2  iron 11 6 6  steel 7 9 5  NCI 0.985 1.020 2.200  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA7  bread 8 6 8  curry 12 9 23  pizza 10 16 17  steak 23 21 8  NCI 1.136 1.148 1.156  significance wrt:   pick ns ns   guess ns  chisq wrt guess ** TA8  water 20 15 38  beer 13 26 11  sherry 4 1 0  whisky 6 6 5  wine 10 4 2  NCI 1.210 1.700 2.495  significance wrt:   pick * **   guess **  chisq wrt guess ** TA9  Carlsberg 25 23 37  Corsendonk 5 3 2  Grimbergen 8 13 2  Rochefort 9 4 11  Westmalle 6 9 4  NCI 1.410 1.420 2.365  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA10  frog 17 17 41  leopard 11 11 5  panther 7 4 5  tiger 18 20 5  NCI 1.060 1.168 2.208  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA11  bicycle 18 18 37  aeroplane 19 18 15  helicopter 6 6 2  hovercraft 10 10 2  NCI 1.116 1.104 2.008  significance wrt:   pick # **   guess **  chisq wrt guess ** TA12  chess 18 15 36  football 11 30 14  squash 5 1 0  tennis 16 6 3  volleyball 3 0 3  NCI 1.235 2.095 2.360  significance wrt:   pick ** **   guess ns  chisq wrt guess ** TA13  Bern 11 12 29  Barbados 13 10 4  Florida 21 17 12  Honolulu 8 13 11  NCI 1.076 0.980 1.384  significance wrt:   pick # *   guess **  chisq wrt guess * TA14  sitting 16 21 39  jogging 6 5 5  running 20 15 10  walking 11 11 2  NCI 1.104 1.148 2.072  significance wrt:   pick ns **   guess **  chisq wrt guess ** Nottingham tasks TN1  Friday lunchtime 13 2 4  Monday morning 6 2 3  Saturday night 17 36 34  Sunday night 4 1 0  Wednesday evening 5 4 3  NCI 1.235 3.220 3.030  significance wrt:   pick ** **   guess #  chisq wrt guess ns TN2  Earth 18 25 33  Mars 5 6 3  Mercury 9 2 1  Saturn 8 4 3  Venus 5 8 4  NCI 1.195 1.770 2.855  significance wrt:   pick * **   guess **  chisq wrt guess ns TN3  Ford 3 4 2  Mercedes 8 11 13  Pontiac 4 1 0  Porsche 21 29 26  Volkswagen 9 0 3  NCI 1.430 2.360 2.150  significance wrt:   pick ** *   guess #  chisq wrt guess ns TN4  plain omelette 5 9 7  cheese omelette 19 19 21  ham omelette 6 11 3  mushroom omelette 8 2 7  prawn omelette 7 4 6  NCI 1.235 1.360 1.425  significance wrt:   pick ns ns   guess ns  chisq wrt guess ns TN5  1 4 3 5  2 5 0 2  7 9 16 6  10 11 7 8  15 16 19 23  NCI 1.145 1.590 1.625  significance wrt:   pick * *   guess ns  chisq wrt guess ns TN6  stool 4 3 5  deck chair 11 8 7  dining chair 4 2 1  easy chair 10 24 15  rocking chair 16 8 16  NCI 1.170 1.695 1.355  significance wrt:   pick * ns   guess #  chisq wrt guess ns TN7  Ontario 9 3 4  Colorado 8 6 5  Florida 18 33 33  Louisiana 4 0 0  Nevada 6 3 2  NCI 1.200 2.775 2.880  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN8  sitting 3 2 2  jogging 11 4 4  sunbathing 13 30 26  swimming 11 5 7  walking 7 4 5  NCI 1.070 2.315 1.920  significance wrt:   pick ** **   guess #  chisq wrt guess ns TN9  2000 13 23 27  1978 3 1 2  1979 3 1 2  1980 11 7 5  1981 15 13 8  NCI 1.230 1.780 2.065  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN10  John 9 10 14  David 11 8 9  Michael 9 13 12  Robert 9 6 5  Steven 7 8 4  NCI 0.930 0.980 1.105  significance wrt:   pick ns ns   guess ns  chisq wrt guess ns TN11  win nothing 4 2 1  win champagne 8 4 1  win chocolate 6 1 0  win money 22 38 41  win trophy 5 0 1  NCI 1.465 3.585 4.335  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN12  red 9 15 8  blue 17 16 23  green 7 3 4  orange 3 1 3  purple 9 10 6  NCI 1.170 1.380 1.610  significance wrt:   pick ns *   guess ns  chisq wrt guess ns TN13  carrot juice 5 6 2  apple juice 10 19 29  grapefruit juice 7 4 3  mango juice 12 7 5  pineapple juice 11 9 5  NCI 0.995 1.260 2.275  significance wrt:   pick ns **   guess **  chisq wrt guess ns TN14  Calais 5 1 4  Berlin 2 3 1  Paris 10 23 27  Prague 8 7 2  Rome 20 11 10  NCI 1.385 1.675 2.130  significance wrt:   pick ns **   guess ns  chisq wrt guess ns . pick . guess . coordinate . Amsterdam tasks TA1  grijs 11 17 35  indigo 12 8 8  karmozijn 9 9 5  magenta 10 11 1  turkoois 11 7 7  NCI 0.935 1.040 2.125  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA2  Ford 10 11 31  Ferrari 13 22 11  Jaguar 20 7 5  Porsche 10 12 9  NCI 1.040 1.124 1.472  significance wrt:   pick ns **   guess *  chisq wrt guess ** TA3  Mannheim 12 9 25  Berlin 3 2 4  Brussels 8 13 9  Lisbon 15 8 7  Madrid 15 20 11  NCI 1.115 1.255 1.355  significance wrt:   pick ns ns   guess ns  chisq wrt guess * TA4  peanut 13 13 12  almond 13 20 18  cashew 20 14 15  walnut 7 5 11  NCI 1.064 1.112 0.984  significance wrt:   pick ns #   guess #  chisq wrt guess ns TA5  glass 11 14 30  diamond 24 28 21  emerald 7 8 3  sapphire 11 2 2  NCI 1.180 1.504 1.684  significance wrt:   pick ns *   guess ns  chisq wrt guess * TA6  plastic 15 16 36  chrome 11 11 7  copper 9 10 2  iron 11 6 6  steel 7 9 5  NCI 0.985 1.020 2.200  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA7  bread 8 6 8  curry 12 9 23  pizza 10 16 17  steak 23 21 8  NCI 1.136 1.148 1.156  significance wrt:   pick ns ns   guess ns  chisq wrt guess ** TA8  water 20 15 38  beer 13 26 11  sherry 4 1 0  whisky 6 6 5  wine 10 4 2  NCI 1.210 1.700 2.495  significance wrt:   pick * **   guess **  chisq wrt guess ** TA9  Carlsberg 25 23 37  Corsendonk 5 3 2  Grimbergen 8 13 2  Rochefort 9 4 11  Westmalle 6 9 4  NCI 1.410 1.420 2.365  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA10  frog 17 17 41  leopard 11 11 5  panther 7 4 5  tiger 18 20 5  NCI 1.060 1.168 2.208  significance wrt:   pick ns **   guess **  chisq wrt guess ** TA11  bicycle 18 18 37  aeroplane 19 18 15  helicopter 6 6 2  hovercraft 10 10 2  NCI 1.116 1.104 2.008  significance wrt:   pick # **   guess **  chisq wrt guess ** TA12  chess 18 15 36  football 11 30 14  squash 5 1 0  tennis 16 6 3  volleyball 3 0 3  NCI 1.235 2.095 2.360  significance wrt:   pick ** **   guess ns  chisq wrt guess ** TA13  Bern 11 12 29  Barbados 13 10 4  Florida 21 17 12  Honolulu 8 13 11  NCI 1.076 0.980 1.384  significance wrt:   pick # *   guess **  chisq wrt guess * TA14  sitting 16 21 39  jogging 6 5 5  running 20 15 10  walking 11 11 2  NCI 1.104 1.148 2.072  significance wrt:   pick ns **   guess **  chisq wrt guess ** Nottingham tasks TN1  Friday lunchtime 13 2 4  Monday morning 6 2 3  Saturday night 17 36 34  Sunday night 4 1 0  Wednesday evening 5 4 3  NCI 1.235 3.220 3.030  significance wrt:   pick ** **   guess #  chisq wrt guess ns TN2  Earth 18 25 33  Mars 5 6 3  Mercury 9 2 1  Saturn 8 4 3  Venus 5 8 4  NCI 1.195 1.770 2.855  significance wrt:   pick * **   guess **  chisq wrt guess ns TN3  Ford 3 4 2  Mercedes 8 11 13  Pontiac 4 1 0  Porsche 21 29 26  Volkswagen 9 0 3  NCI 1.430 2.360 2.150  significance wrt:   pick ** *   guess #  chisq wrt guess ns TN4  plain omelette 5 9 7  cheese omelette 19 19 21  ham omelette 6 11 3  mushroom omelette 8 2 7  prawn omelette 7 4 6  NCI 1.235 1.360 1.425  significance wrt:   pick ns ns   guess ns  chisq wrt guess ns TN5  1 4 3 5  2 5 0 2  7 9 16 6  10 11 7 8  15 16 19 23  NCI 1.145 1.590 1.625  significance wrt:   pick * *   guess ns  chisq wrt guess ns TN6  stool 4 3 5  deck chair 11 8 7  dining chair 4 2 1  easy chair 10 24 15  rocking chair 16 8 16  NCI 1.170 1.695 1.355  significance wrt:   pick * ns   guess #  chisq wrt guess ns TN7  Ontario 9 3 4  Colorado 8 6 5  Florida 18 33 33  Louisiana 4 0 0  Nevada 6 3 2  NCI 1.200 2.775 2.880  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN8  sitting 3 2 2  jogging 11 4 4  sunbathing 13 30 26  swimming 11 5 7  walking 7 4 5  NCI 1.070 2.315 1.920  significance wrt:   pick ** **   guess #  chisq wrt guess ns TN9  2000 13 23 27  1978 3 1 2  1979 3 1 2  1980 11 7 5  1981 15 13 8  NCI 1.230 1.780 2.065  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN10  John 9 10 14  David 11 8 9  Michael 9 13 12  Robert 9 6 5  Steven 7 8 4  NCI 0.930 0.980 1.105  significance wrt:   pick ns ns   guess ns  chisq wrt guess ns TN11  win nothing 4 2 1  win champagne 8 4 1  win chocolate 6 1 0  win money 22 38 41  win trophy 5 0 1  NCI 1.465 3.585 4.335  significance wrt:   pick ** **   guess ns  chisq wrt guess ns TN12  red 9 15 8  blue 17 16 23  green 7 3 4  orange 3 1 3  purple 9 10 6  NCI 1.170 1.380 1.610  significance wrt:   pick ns *   guess ns  chisq wrt guess ns TN13  carrot juice 5 6 2  apple juice 10 19 29  grapefruit juice 7 4 3  mango juice 12 7 5  pineapple juice 11 9 5  NCI 0.995 1.260 2.275  significance wrt:   pick ns **   guess **  chisq wrt guess ns TN14  Calais 5 1 4  Berlin 2 3 1  Paris 10 23 27  Prague 8 7 2  Rome 20 11 10  NCI 1.385 1.675 2.130  significance wrt:   pick ns **   guess ns  chisq wrt guess ns Open in new tab The first of these tests whether, as predicted by hypothesis PC1, guessers’ responses are more concentrated than those of pickers. We use a bootstrap method (Efron, 1979). We start with the null hypothesis that guessers’ responses are drawn from a distribution with the same relative frequencies as the actual responses of pickers. We obtain critical values of the NCI that would be generated by repeated sampling, with a sample size equal to the actual number of guessers, if the null hypothesis were true. Our estimates of these critical values are constructed from 20,000 simulated samples. For example, consider task TA1. The NCI is 0.935 for pickers and 1.040 for guessers. In 5% of our simulations, the NCI for guessers is greater than 1.130. Since 1.040 < 1.130, we cannot reject the null hypothesis at the 5% level in a one‐tail test. This finding, that the NCI is not significantly greater for guessers than for pickers, is reported by the entry ‘ns’ in the ‘guess’ column against ‘significance wrt pick’. Cases in which the null hypothesis can be rejected at the 5% level or 1% level are reported as * or **; cases in which the observed difference is in the ‘wrong’ direction are recorded as #. Using the same method, we test whether, as in the data reported by Mehta et al. (1994), coordinators’ responses are more concentrated than pickers’. The results of these tests are reported against ‘significance wrt pick’ in the ‘coordinate’ column. And, most importantly, we test whether coordinators’ responses are more concentrated than guessers’; the results of these tests are reported against ‘significance wrt guess’ in the ‘coordinate’ column. Recall that hypothesis PC2, implied by cognitive hierarchy theory, predicts that the two distributions are the same (and hence equally concentrated), while PC3, implied by team reasoning, predicts that if the two distributions are different, coordinators’ responses are at least as concentrated as guessers’. Finally, we report a chi‐squared test of the null hypothesis that coordinators’ and guessers’ responses are drawn from the same population distribution. The result of this test is reported against ‘chisq wrt guess’ in the ‘coordinate’ column; rejection of the null at the 5% or 1% level is denoted by * or **.17 This is a direct test of PC2. For some purposes, it is convenient to work with summary statistics which aggregate across tasks. For each experiment, Table 2 reports five such statistics concerning pickers, guessers and coordinators. (The entries in the other rows and columns will be explained later.) The entry in the (pick, pick) cell is the NCI for pickers (denoted NCIPP), averaged across the relevant experiment.18 The entries in the (guess, guess) and (coordinate, coordinate) cells are the corresponding average NCIs for guessers and coordinators, denoted NCIGG and NCICC. The other entries are averages of cross‐group NCIs, defined as follows. Consider a label set L = {l1,…,ln} and two disjoint groups of individuals, one with N members and one with N ′ members. Each individual chooses one label from L. Each label lj is chosen by mj individuals from the first group and by from the second. The cross‐group coordination index, , measures the probability that an individual drawn at random from one group will choose the same label as an individual drawn at random from the other. Since this index takes the value 1/n when individuals in one or both groups choose at random, we can multiply it by n to arrive at the cross‐group NCI. Cross‐group NCIs provide information about how far the responses of subjects in different treatments are concentrated on the same labels. Table 2 
Normalised Coordination Indices . pick . guess . coordinate . stand out . favourite . Amsterdam text tasks pick 1.116 1.172 1.191 1.164 1.165 guess 1.259 1.267 1.227 1.202 coordinate 1.819 1.521 1.072 stand out 1.334 1.181 favourite 1.397 Nottingham text tasks pick 1.204 1.416 1.470 1.187 1.345 guess 1.982 2.067 1.492 1.756 coordinate 2.197 1.565 1.838 stand out 1.398 1.438 favourite 1.713 Nottingham number tasks stand out 1.274 1.073 favourite 1.095 . pick . guess . coordinate . stand out . favourite . Amsterdam text tasks pick 1.116 1.172 1.191 1.164 1.165 guess 1.259 1.267 1.227 1.202 coordinate 1.819 1.521 1.072 stand out 1.334 1.181 favourite 1.397 Nottingham text tasks pick 1.204 1.416 1.470 1.187 1.345 guess 1.982 2.067 1.492 1.756 coordinate 2.197 1.565 1.838 stand out 1.398 1.438 favourite 1.713 Nottingham number tasks stand out 1.274 1.073 favourite 1.095 Open in new tab Table 2 
Normalised Coordination Indices . pick . guess . coordinate . stand out . favourite . Amsterdam text tasks pick 1.116 1.172 1.191 1.164 1.165 guess 1.259 1.267 1.227 1.202 coordinate 1.819 1.521 1.072 stand out 1.334 1.181 favourite 1.397 Nottingham text tasks pick 1.204 1.416 1.470 1.187 1.345 guess 1.982 2.067 1.492 1.756 coordinate 2.197 1.565 1.838 stand out 1.398 1.438 favourite 1.713 Nottingham number tasks stand out 1.274 1.073 favourite 1.095 . pick . guess . coordinate . stand out . favourite . Amsterdam text tasks pick 1.116 1.172 1.191 1.164 1.165 guess 1.259 1.267 1.227 1.202 coordinate 1.819 1.521 1.072 stand out 1.334 1.181 favourite 1.397 Nottingham text tasks pick 1.204 1.416 1.470 1.187 1.345 guess 1.982 2.067 1.492 1.756 coordinate 2.197 1.565 1.838 stand out 1.398 1.438 favourite 1.713 Nottingham number tasks stand out 1.274 1.073 favourite 1.095 Open in new tab 3.2. Amsterdam Results We begin by looking at the aggregated data, shown in Table 2. Notice that NCIPP =1.116, NCIGG = 1.259 and NCICC = 1.819. That is, relative to the benchmark case of subjects who choose at random, a pair of pickers is only 12% more likely to give matched responses, while the corresponding figures for guessers and coordinators are 26% and 82% respectively. The marked difference between the concentration of pickers’ and coordinators’ responses replicates the main finding of Mehta et al. (1994) that guessers’ responses are more concentrated than pickers’ is consistent with PC1. That coordinators’ responses are much more concentrated than guessers’ is contrary to PC2 (and cognitive hierarchy theory) but consistent with PC3 (and team reasoning). Notice also that NCIPC ≈ NCIPG and NCIGC ≈ NCIGG : guessers are almost as successful as coordinators in matching the responses of both pickers and (other) guessers. The implication is that coordinators’ differential success in matching one another is not the result of their responses being highly concentrated on the modal responses of pickers and/or guessers, which again is suggestive of team reasoning. We now consider the data for individual tasks, shown in Table 1. Guessers’ responses are more concentrated than pickers’ in 12 of the 14 tasks, as predicted by PC1; in two of these tasks, the difference between the two NCIs is statistically significant. The distributions of responses of coordinators and guessers are significantly different from one another, contrary to PC2 and cognitive hierarchy theory, in 13 out of 14 tasks. These 13 cases can be used for tests of PC3. In all 13 cases, consistently with PC3 and team reasoning, NCICC > NCIGG; the difference is statistically significant in nine cases. Modal choices are different in seven cases, which again is suggestive of team reasoning. In twelve of the fourteen tasks, the modal choice of coordinators is the intended salient. (The exceptions are ≪almond≫ in TA4 and ≪curry≫ in TA7.) Taken together, these findings suggest that coordinators are using a mode of reasoning which is different from that used by guessers and which generates responses which are both more concentrated and more skewed in favour of the ‘odd one out’. We interpret all this as strong evidence against cognitive hierarchy theory, and as supportive of the theory of team reasoning. 3.3. Nottingham Results Aggregating across tasks, NCIPP = 1.204, NCIGG = 1.982 and NCICC = 2.197 (see Table 2). Again, Mehta et al.’s (1994) main result is replicated. Guessers’ responses are much more concentrated than pickers’, strongly supporting PC1. In contrast to the Amsterdam results, however, coordinators’ responses are only slightly more concentrated than guessers. In none of the fourteen tasks is there a significant difference between the distributions of responses for coordinators and guessers. These findings are entirely consistent with PC2 and cognitive hierarchy theory. Since there is no evidence of systematic differences between the responses of coordinators and guessers, PC3 does not apply. It seems that the Nottingham coordinators are responding as if they were guessers. 4. Results for Number Tasks 4.1. Presentation of Results Table 3 reports the frequency distribution of responses for each of the type 1 and type 2 number tasks, for each of the three treatments. Responses are classified only by the number of points carried by the options chosen. For example, task NA1 has four options with the array of points (10, 10, 10, 9) but we disaggregate responses into ‘10‐point options’ (denoted ‘10×3’ to signify that there are three options, each carrying 10 points) and ‘9‐point options’ (denoted ‘9×1’). Our designs preclude further disaggregation. As explained in Section 2.2, the procedures by which individual options were labelled (the ‘films’ in the Amsterdam experiment and the ‘patterns’ in the Nottingham experiment) were randomised between sessions (in Amsterdam) or between subject pairs (in Nottingham). Table 3 
Responses to Type 1 and Type 2 Number Tasks . . pick . guess . coordinate . significance . Type 1 tasks: Amsterdam NA1 10 × 3 50 48 8 # 9 × 1 3 4 48 ** NA2 10 × 5 49 46 10 # 9 × 1 4 6 46 ** NA3 10 × 3 51 50 10 # 9 × 1 1 1 43 ** 8 × 1 0 0 1 7 × 1 1 1 2 NA4 10 × 3 52 48 15 # 9 × 2 0 1 0 8 × 1 1 3 41 ** NA5 10 × 4 49 51 21 # 9 × 2 4 1 35 ** Type 1 tasks: Nottingham NN1 10 × 4 42 44 35 # 9 × 1 3 1 9 ns NN2 10 × 4 44 44 33 # 9 × 1 1 1 11 ns NN3 10 × 4 43 44 34 # 9 × 1 2 1 10 ns NN4 10 × 4 43 43 34 # 9 × 1 2 2 10 ns NN5 10 × 4 44 40 33 # 9 × 1 1 5 11 ns NN6 10 × 4 44 42 32 # 9 × 1 1 3 12 ns Type 2 tasks: Amsterdam NA6 10 × 1 52 51 54 ** 9 × 1 1 1 2 NA7 10 × 3 52 49 50 ** 9 × 3 1 3 6 NA8 10 × 1 52 51 54 ** 1 × 1 1 1 2 NA9 10 × 3 49 51 41 # 1 × 1 4 1 15 NA10 10 × 5 47 51 37 # 1 × 1 6 1 19 Type 2 tasks: Nottingham NN7 10 × 4 44 45 41 * 1 × 1 1 0 3 NN8 10 × 4 44 44 40 * 1 × 1 1 1 4 NN9 10 × 4 41 45 42 ** 1 × 1 4 0 2 NN10 10 × 4 44 44 38 # 1 × 1 1 1 6 NN11 10 × 4 45 43 39 # 1 × 1 0 2 5 NN12 10 × 4 45 43 39 # 1 × 1 0 2 5 . . pick . guess . coordinate . significance . Type 1 tasks: Amsterdam NA1 10 × 3 50 48 8 # 9 × 1 3 4 48 ** NA2 10 × 5 49 46 10 # 9 × 1 4 6 46 ** NA3 10 × 3 51 50 10 # 9 × 1 1 1 43 ** 8 × 1 0 0 1 7 × 1 1 1 2 NA4 10 × 3 52 48 15 # 9 × 2 0 1 0 8 × 1 1 3 41 ** NA5 10 × 4 49 51 21 # 9 × 2 4 1 35 ** Type 1 tasks: Nottingham NN1 10 × 4 42 44 35 # 9 × 1 3 1 9 ns NN2 10 × 4 44 44 33 # 9 × 1 1 1 11 ns NN3 10 × 4 43 44 34 # 9 × 1 2 1 10 ns NN4 10 × 4 43 43 34 # 9 × 1 2 2 10 ns NN5 10 × 4 44 40 33 # 9 × 1 1 5 11 ns NN6 10 × 4 44 42 32 # 9 × 1 1 3 12 ns Type 2 tasks: Amsterdam NA6 10 × 1 52 51 54 ** 9 × 1 1 1 2 NA7 10 × 3 52 49 50 ** 9 × 3 1 3 6 NA8 10 × 1 52 51 54 ** 1 × 1 1 1 2 NA9 10 × 3 49 51 41 # 1 × 1 4 1 15 NA10 10 × 5 47 51 37 # 1 × 1 6 1 19 Type 2 tasks: Nottingham NN7 10 × 4 44 45 41 * 1 × 1 1 0 3 NN8 10 × 4 44 44 40 * 1 × 1 1 1 4 NN9 10 × 4 41 45 42 ** 1 × 1 4 0 2 NN10 10 × 4 44 44 38 # 1 × 1 1 1 6 NN11 10 × 4 45 43 39 # 1 × 1 0 2 5 NN12 10 × 4 45 43 39 # 1 × 1 0 2 5 Open in new tab Table 3 
Responses to Type 1 and Type 2 Number Tasks . . pick . guess . coordinate . significance . Type 1 tasks: Amsterdam NA1 10 × 3 50 48 8 # 9 × 1 3 4 48 ** NA2 10 × 5 49 46 10 # 9 × 1 4 6 46 ** NA3 10 × 3 51 50 10 # 9 × 1 1 1 43 ** 8 × 1 0 0 1 7 × 1 1 1 2 NA4 10 × 3 52 48 15 # 9 × 2 0 1 0 8 × 1 1 3 41 ** NA5 10 × 4 49 51 21 # 9 × 2 4 1 35 ** Type 1 tasks: Nottingham NN1 10 × 4 42 44 35 # 9 × 1 3 1 9 ns NN2 10 × 4 44 44 33 # 9 × 1 1 1 11 ns NN3 10 × 4 43 44 34 # 9 × 1 2 1 10 ns NN4 10 × 4 43 43 34 # 9 × 1 2 2 10 ns NN5 10 × 4 44 40 33 # 9 × 1 1 5 11 ns NN6 10 × 4 44 42 32 # 9 × 1 1 3 12 ns Type 2 tasks: Amsterdam NA6 10 × 1 52 51 54 ** 9 × 1 1 1 2 NA7 10 × 3 52 49 50 ** 9 × 3 1 3 6 NA8 10 × 1 52 51 54 ** 1 × 1 1 1 2 NA9 10 × 3 49 51 41 # 1 × 1 4 1 15 NA10 10 × 5 47 51 37 # 1 × 1 6 1 19 Type 2 tasks: Nottingham NN7 10 × 4 44 45 41 * 1 × 1 1 0 3 NN8 10 × 4 44 44 40 * 1 × 1 1 1 4 NN9 10 × 4 41 45 42 ** 1 × 1 4 0 2 NN10 10 × 4 44 44 38 # 1 × 1 1 1 6 NN11 10 × 4 45 43 39 # 1 × 1 0 2 5 NN12 10 × 4 45 43 39 # 1 × 1 0 2 5 . . pick . guess . coordinate . significance . Type 1 tasks: Amsterdam NA1 10 × 3 50 48 8 # 9 × 1 3 4 48 ** NA2 10 × 5 49 46 10 # 9 × 1 4 6 46 ** NA3 10 × 3 51 50 10 # 9 × 1 1 1 43 ** 8 × 1 0 0 1 7 × 1 1 1 2 NA4 10 × 3 52 48 15 # 9 × 2 0 1 0 8 × 1 1 3 41 ** NA5 10 × 4 49 51 21 # 9 × 2 4 1 35 ** Type 1 tasks: Nottingham NN1 10 × 4 42 44 35 # 9 × 1 3 1 9 ns NN2 10 × 4 44 44 33 # 9 × 1 1 1 11 ns NN3 10 × 4 43 44 34 # 9 × 1 2 1 10 ns NN4 10 × 4 43 43 34 # 9 × 1 2 2 10 ns NN5 10 × 4 44 40 33 # 9 × 1 1 5 11 ns NN6 10 × 4 44 42 32 # 9 × 1 1 3 12 ns Type 2 tasks: Amsterdam NA6 10 × 1 52 51 54 ** 9 × 1 1 1 2 NA7 10 × 3 52 49 50 ** 9 × 3 1 3 6 NA8 10 × 1 52 51 54 ** 1 × 1 1 1 2 NA9 10 × 3 49 51 41 # 1 × 1 4 1 15 NA10 10 × 5 47 51 37 # 1 × 1 6 1 19 Type 2 tasks: Nottingham NN7 10 × 4 44 45 41 * 1 × 1 1 0 3 NN8 10 × 4 44 44 40 * 1 × 1 1 1 4 NN9 10 × 4 41 45 42 ** 1 × 1 4 0 2 NN10 10 × 4 44 44 38 # 1 × 1 1 1 6 NN11 10 × 4 45 43 39 # 1 × 1 0 2 5 NN12 10 × 4 45 43 39 # 1 × 1 0 2 5 Open in new tab For each task and each treatment, we report a one‐tail binomial test of the hypothesis that 10‐point options are chosen with greater probability than if choice were random (that is, that this probability is greater than the proportion of options which carry 10 points). For the coordination treatment of type 1 tasks, this is a test of HL1, which is implied by cognitive hierarchy theory. For the coordination treatment of type 2 tasks, it is a test of both HL1 and HL2, and so does not discriminate between the two theories. For the coordination treatment of type 1 tasks, we report a corresponding test of whether team‐optimal options are chosen with greater probability than if choice is random. This is a test of HL2, which is implied by team reasoning. Cases in which the null hypothesis is rejected at the 5% (1%) level are denoted by * (**) in the final column of Table 3. Cases in which the options relevant for the test are chosen with lower frequency than would be implied by random choice are denoted by #. Our tests of team reasoning are premised on the assumption that, in number tasks, subjects perceive labels as nondescript. One check on the validity of this assumption is to look at type 3 tasks (that is, the tasks in which all options carried 10 points) and to measure how far subjects who saw the same labels gave matched responses. For the Amsterdam experiment, in which labels were randomised by session, we can calculate an average within‐session NCI for each treatment. This is a weighted average of NCIs which have been calculated separately for each session. For the Nottingham experiment, in which labels were randomised separately for each pair of subjects, we count the number of cases in which the responses of paired subjects matched one another, and express this as a ratio of the expected number of matches under the assumption of random choice. If (consciously or unconsciously) subjects use labels as a means of matching, these measures will be greater than 1. Table 4 presents these measures, averaged across all type 3 tasks, for the two experiments and the three treatments. It is clear that, for subjects who saw the same labels, matches were more frequent than would have been generated by random choice.19 Surprisingly, in both experiments, pickers were more ‘successful’ in matching on the (intendedly) nondescript labels of the number tasks than on the apparently more distinguishable labels of the text tasks (compare the NCIPP measures in Table 2). However, in contrast to the text tasks, there is little evidence to suggest that the responses of guessers or coordinators are more concentrated than those of pickers. This seems to be a case in which matching, even among coordinators, is attributable to primary salience. In any event, these data suggest that if two partners both use labels as their method of trying to coordinate, their probability of success is only about 30% or 40% greater than if they choose at random. Unless team‐reasoning individuals have highly unrealistic expectations of their ability to discriminate between (what were intended as) nondescript labels, team‐optimal choices in both type 1 and type 2 tasks will be as specified in Section 2.4. Table 4 
Normalised Frequency of Matching on Type 3 Number Tasks . pick . guess . coordinate . Amsterdam 1.242 1.286 1.336 Nottingham 1.364 1.515 1.402 . pick . guess . coordinate . Amsterdam 1.242 1.286 1.336 Nottingham 1.364 1.515 1.402 Open in new tab Table 4 
Normalised Frequency of Matching on Type 3 Number Tasks . pick . guess . coordinate . Amsterdam 1.242 1.286 1.336 Nottingham 1.364 1.515 1.402 . pick . guess . coordinate . Amsterdam 1.242 1.286 1.336 Nottingham 1.364 1.515 1.402 Open in new tab 4.2. Amsterdam Results The results for type 1 tasks, shown in Table 3, are extremely sharp. There are five such tasks. In every case, as one would expect, almost all pickers and guessers chose 10‐point options. HL1 predicts that coordinators will choose 10‐point options more frequently than if choices were made at random. In all cases, the opposite is true, contrary to cognitive hierarchy theory. HL2 identifies some other response as team‐optimal and predicts that coordinators will choose this more frequently than if choices were made at random. In every case, more than 60% of coordinators chose the team‐optimal response, while if choices were random, the expected frequency would be only 0.167 or 0.25. In every case, the null hypothesis of random choice is rejected at the 1% level. These results give very strong support to the team reasoning hypothesis. In the five type 2 tasks, we again find that almost all pickers and guessers chose 10‐point options. For these tasks, HL1 and HL2 make the same prediction, namely that coordinators will choose 10‐point options more frequently than if choices were made at random. In three tasks (NA6, NA7 and NA8), overwhelming majorities of coordinators chose 10‐point options and the null hypothesis is rejected at the 1% level. However, this prediction is not as successful for the other two tasks (NA9 and NA10). In these latter tasks, although 10‐point options were chosen by large majorities of subjects (73% and 66% respectively), the frequency of such choices was less than if subjects had acted at random. Notice that, in each of the five type 2 tasks, there are just two numbers of points, ‘high’ (10) and ‘low’ (9 or 1, depending on the task). In NA6, NA7 and NA8, there are exactly as many high‐point options as there are low‐point ones; thus, apart from any differences in the labels themselves, the only distinguishing feature of the low‐point options is that they carry fewer points. In NA9 and NA10, in contrast, there are more high‐point options than low‐point ones, so a low‐point option can be perceived as an odd one out. We speculate that a minority of subjects favoured the odd one out, even in cases in which this was not team‐optimal. 4.3. Nottingham Results In the Nottingham experiment, as in the Amsterdam one, almost all pickers and guessers chose 10‐point options in both type 1 and type 2 tasks. However, the responses that are relevant for our tests are those for coordinators; and here the Nottingham results are rather different. In all six type 1 tasks, coordinators chose 10‐point options slightly less frequently (and, correspondingly, chose the team‐optimal 9‐point option slightly more frequently) than if choices had been made at random but, in each case, the null hypothesis of random choice cannot be rejected. Summing over all type 1 tasks, the 9‐point option accounted for 23.9% of choices, compared with the random‐choice benchmark of 20%. Formally, neither HL1 nor HL2 is supported by the evidence. However, an examination of behaviour at the level of the individual subject shows that coordinators’ choices were not random. Of the 63 instances of subjects choosing 9‐point options, 46 were attributable to just 8 of the 44 coordinators, while 25 coordinators never chose any 9‐point option. We suggest that the most credible interpretation is that, on type 1 tasks, the majority of Nottingham subjects behaved roughly in accordance with cognitive hierarchy theory, while a minority behaved in similarly rough accordance with the theory of team reasoning. In every type 2 task, the frequency with which coordinators chose 10‐point options was higher than the random‐choice benchmark, as predicted by both HL1 and HL2. The null hypothesis of random choice can be rejected in three cases out of six. An analysis of behaviour at the individual level shows that the instances in which 1‐point options were chosen were not random errors. All 25 of these instances were attributable to just 8 subjects, all of whom also chose at least one 9‐point option in a type 1 task. The implication is that a small minority of subjects may have been attracted to odd‐one‐out options. 5. Taking Stock As an aide‐memoire, the results from the two experiments are summarised in Table 5. The Amsterdam results, for both text and number tasks, support the theory of team reasoning rather than cognitive hierarchy theory. The Nottingham results, for both text and number tasks, seem to point in the opposite direction. Table 5 
Summary of Results . cognitive hierarchy theory predicts . team reasoning theory predicts . Amsterdam result . Nottingham result . tests using pure coordination games: compare NCIGG and NCIPP (test PC1) NCIGG > NCIPP NCIGG > NCIPP NCIGG > NCIPP (difference small) NCIGG > NCIPP (difference large) compare distributions of responses of coordinators and guessers (test PC2) coordinators’ and guessers’ responses have same distribution no prediction distributions different no significant differences between distributions if distributions different, compare NCICC and NCIGG (test PC3) not applicable NCICC > NCIGG NCICC > NCIGG not applicable tests using nondescript Hi‐Lo games: type 1 tasks (high‐payoff and team‐optimal labels are different: tests HL1 and HL2) high‐payoff labels chosen with greater‐than‐random frequency team‐optimal labels chosen with greater‐than‐random frequency team‐optimal labels chosen with greater‐than‐random frequency both types of label chosen with approximately random frequency type 2 tasks (high‐payoff labels are also team‐optimal: tests HL1 and HL2) high‐payoff labels chosen with greater‐than‐random frequency high‐payoff labels chosen with greater‐than‐random frequency high‐payoff labels usually chosen with greater‐than‐random frequency high‐payoff labels chosen with greater‐than‐random frequency . cognitive hierarchy theory predicts . team reasoning theory predicts . Amsterdam result . Nottingham result . tests using pure coordination games: compare NCIGG and NCIPP (test PC1) NCIGG > NCIPP NCIGG > NCIPP NCIGG > NCIPP (difference small) NCIGG > NCIPP (difference large) compare distributions of responses of coordinators and guessers (test PC2) coordinators’ and guessers’ responses have same distribution no prediction distributions different no significant differences between distributions if distributions different, compare NCICC and NCIGG (test PC3) not applicable NCICC > NCIGG NCICC > NCIGG not applicable tests using nondescript Hi‐Lo games: type 1 tasks (high‐payoff and team‐optimal labels are different: tests HL1 and HL2) high‐payoff labels chosen with greater‐than‐random frequency team‐optimal labels chosen with greater‐than‐random frequency team‐optimal labels chosen with greater‐than‐random frequency both types of label chosen with approximately random frequency type 2 tasks (high‐payoff labels are also team‐optimal: tests HL1 and HL2) high‐payoff labels chosen with greater‐than‐random frequency high‐payoff labels chosen with greater‐than‐random frequency high‐payoff labels usually chosen with greater‐than‐random frequency high‐payoff labels chosen with greater‐than‐random frequency Open in new tab Table 5 
Summary of Results . cognitive hierarchy theory predicts . team reasoning theory predicts . Amsterdam result . Nottingham result . tests using pure coordination games: compare NCIGG and NCIPP (test PC1) NCIGG > NCIPP NCIGG > NCIPP NCIGG > NCIPP (difference small) NCIGG > NCIPP (difference large) compare distributions of responses of coordinators and guessers (test PC2) coordinators’ and guessers’ responses have same distribution no prediction distributions different no significant differences between distributions if distributions different, compare NCICC and NCIGG (test PC3) not applicable NCICC > NCIGG NCICC > NCIGG not applicable tests using nondescript Hi‐Lo games: type 1 tasks (high‐payoff and team‐optimal labels are different: tests HL1 and HL2) high‐payoff labels chosen with greater‐than‐random frequency team‐optimal labels chosen with greater‐than‐random frequency team‐optimal labels chosen with greater‐than‐random frequency both types of label chosen with approximately random frequency type 2 tasks (high‐payoff labels are also team‐optimal: tests HL1 and HL2) high‐payoff labels chosen with greater‐than‐random frequency high‐payoff labels chosen with greater‐than‐random frequency high‐payoff labels usually chosen with greater‐than‐random frequency high‐payoff labels chosen with greater‐than‐random frequency . cognitive hierarchy theory predicts . team reasoning theory predicts . Amsterdam result . Nottingham result . tests using pure coordination games: compare NCIGG and NCIPP (test PC1) NCIGG > NCIPP NCIGG > NCIPP NCIGG > NCIPP (difference small) NCIGG > NCIPP (difference large) compare distributions of responses of coordinators and guessers (test PC2) coordinators’ and guessers’ responses have same distribution no prediction distributions different no significant differences between distributions if distributions different, compare NCICC and NCIGG (test PC3) not applicable NCICC > NCIGG NCICC > NCIGG not applicable tests using nondescript Hi‐Lo games: type 1 tasks (high‐payoff and team‐optimal labels are different: tests HL1 and HL2) high‐payoff labels chosen with greater‐than‐random frequency team‐optimal labels chosen with greater‐than‐random frequency team‐optimal labels chosen with greater‐than‐random frequency both types of label chosen with approximately random frequency type 2 tasks (high‐payoff labels are also team‐optimal: tests HL1 and HL2) high‐payoff labels chosen with greater‐than‐random frequency high‐payoff labels chosen with greater‐than‐random frequency high‐payoff labels usually chosen with greater‐than‐random frequency high‐payoff labels chosen with greater‐than‐random frequency Open in new tab Given the similarities of design between the two experiments, the differences between their results are surprising. In this and the following Section, we consider possible explanations for these differences. In thinking about these explanations, it is important to keep in mind that the two experiments support independent tests of well‐defined hypotheses derived from two recognised theories. The validity of those tests is unaffected by the analysis which follows. However, an investigation of the differences between the two sets of results may give some clues for further theoretical work. One possibility is that there was a difference between the two subject pools with respect to their modes of reasoning about coordination and Hi‐Lo games in general. Without completely ruling out this explanation, we judged it unlikely. It seemed implausible to suppose that the modes of reasoning used by Dutch university students are fundamentally different from those used by their English counterparts. Another possibility is that differences between the displays may be having some effect. In the number tasks, the labels used in the Nottingham display (randomly generated patterns) may seem less nondescript than those of the Amsterdam display (positions of moving discs). This may have prompted Nottingham subjects to focus on labels rather than payoffs. But the normalised frequencies of matching in type 3 number tasks suggest that, in fact, the Nottingham labelling was only slightly less nondescript than the Amsterdam labelling (see Section 4.1 and Table 4). A further difference is that the Nottingham display allows subjects to label objects by their positions in the row, using concepts such as ‘first’, ‘middle’ and ‘last’. Because this layout is randomised independently for each co‐player, position cannot be used as a coordinating device; but it might still influence behaviour in the picking treatment. To the extent that picking is influenced by position (which pickers’ co‐players cannot observe), successful guessing is made more difficult, as is coordination if players act according to cognitive hierarchy theory. In fact, pickers’ choices were distributed almost uniformly among the five positions and Nottingham responses were more consistent with cognitive hierarchy theory than were Amsterdam responses. We judged it most likely that the difference in results reflected some difference(s) in the content of the labels used in the two designs. Looking for clues in subjects’ responses to text tasks, we tried to find common features in those labels that were the modal choices of coordinators. For the Amsterdam coordinators, the most obvious common feature seemed to be that of standing out from the other labels by virtue of some significant but not necessarily desirable characteristic. For the Nottingham coordinators, choices seemed to be concentrated on the labels whose content was most liked by, or was the favourite of, most students. But these interpretations were merely conjectures, based on our intuitions about the cultural significance of different labels. As we have said repeatedly, our investigative strategy is to avoid appeals to intuitions about salience and instead to use cross‐treatment comparisons in which cultural variables are held constant. To test our conjectures, we used two further treatments in a questionnaire study, which we now describe. 6. The Questionnaire Study We administered a questionnaire to independent samples of respondents recruited from students at the Universities of Amsterdam and Nottingham. The aim was to investigate respondents’ perceptions of the labels used in the original experiment in relation to the criteria of standing out and favouriteness. The study was carried out in 2002; respondents were rewarded by being entered in a lottery with a cash prize. The questionnaire had two variants, which were administered to different, randomised samples. One variant investigated perceptions of standing out, the other perceptions of favouriteness. Each questionnaire contained a mixture of Amsterdam text tasks, Nottingham text tasks and Nottingham number tasks in randomised order. (Because the questionnaire was a pen‐and‐paper exercise, the computerised displays of the Amsterdam experiment could not be replicated.) For each task, the questionnaire displayed a row of four or five ‘items’; these were the labels used in the relevant task in the original experiments (i.e. the strings of text which appeared on the ‘discs’ or in the upper ‘boxes’ of the text task displays, or the patterns in the upper boxes in the Nottingham number tasks; points were not shown). Respondents were given the following instructions (text which differed between the two variants of the questionnaire is shown in square brackets): In each task, you are shown a row of four or five ‘items’. In each task, you must do one of two things. First, show [which of the items is your favourite for you/which of the items stands out from the others]. You show this by circling one of the items. You must circle exactly one item in each task. Even if you do not feel strongly that any of the items [is your favourite/ stands out], please circle one of them; the second part of the task will allow you to tell us how strong your feelings are. Second, show how strongly you feel that the item you have circled [is your favourite/ stands out from the others]. You show this by marking a point on a scale from 0 (‘not at all’) to 5 (‘very strongly’). By using questionnaires with different sets of tasks, we were able to collect around 95 responses for each of the 28 text tasks in the original experiments and for six representative examples of Nottingham number tasks. For each task, whichever experiment it was taken from, we collected approximately equal numbers of questionnaire responses in Amsterdam and Nottingham. This allowed us to check for subject pool effects. In fact, we found no systematic differences between responses collected in the two locations.20 We therefore combined the two sets of responses. Table 6 reports respondents’ average strengths of feeling about standing‐outness and favouriteness for the Amsterdam text tasks, Nottingham text tasks and Nottingham number tasks. The most striking feature of these data is the lower strength of feeling for number tasks than for text tasks. This is consistent with our intention that the labels for number tasks be nondescript. A further feature is that standing‐outness was more pronounced than favouriteness in the Amsterdam tasks, while the opposite was true of the Nottingham tasks. Hypotheses of no difference between reported strength of feeling between the two sets of text tasks are rejected at the 5% level (in two‐tailed Wilcoxon signed rank tests, p = 0.026 for standing‐outness and p < 0.01 for favouriteness). Table 6 
Strength of Standing‐outness and Favouriteness (on 0‐5 Scale) . standing‐outness . favouriteness . Amsterdam text tasks 3.45 3.39 Nottingham text tasks 3.31 3.54 Nottingham number tasks 2.23 1.92 . standing‐outness . favouriteness . Amsterdam text tasks 3.45 3.39 Nottingham text tasks 3.31 3.54 Nottingham number tasks 2.23 1.92 Open in new tab Table 6 
Strength of Standing‐outness and Favouriteness (on 0‐5 Scale) . standing‐outness . favouriteness . Amsterdam text tasks 3.45 3.39 Nottingham text tasks 3.31 3.54 Nottingham number tasks 2.23 1.92 . standing‐outness . favouriteness . Amsterdam text tasks 3.45 3.39 Nottingham text tasks 3.31 3.54 Nottingham number tasks 2.23 1.92 Open in new tab Using questionnaire responses to questions about standing out (S) and favourites (F) in conjunction with the actual responses of pickers (P), guessers (G) and coordinators (C) in the experiments, we can calculate within‐group NCIs for S‐responses (denoted NCISS) and F‐responses (NCIFF), and a cross‐group NCI for each pair of distinct response groups. These statistics are shown in Table 2. Two features of these data, specific to text tasks, are of particular interest. First, consider NCISS and NCIFF. For the Amsterdam tasks, the two indices have similar values. The value of NCISS for Nottingham tasks (1.398) is similar to the corresponding Amsterdam value but NCIFF (1.713) is much greater. The implication is that two co‐players, trying to coordinate on a Nottingham task, would be much more likely to succeed by using the primary‐salience rule ‘Choose your favourite’ than ‘Choose the label which stands out most for you’. Presumably they would be even more likely to succeed by using secondary salience (‘Choose the object you believe to be most people’s favourite’). It seems that, for the Nottingham tasks, choosing according to secondary salience may also be the best rule in Schelling’s sense: cognitive hierarchy theory and team reasoning make the same predictions about coordination. Now consider the values of NCICS and NCICF. For the Amsterdam tasks, NCICS = 1.521 and NCICF = 1.072. The implication is that, on these tasks, coordinators tended to choose labels that were generally perceived as standing out, rather than ones that were generally perceived as favourites. For the Nottingham tasks, the opposite is true: NCICS = 1.565 and NCICF = 1.838. To test this account of the behaviour of coordinators more rigorously, we estimated (separately for each experiment) an equation in which the dependent variable is the frequency with which each label was chosen in the coordination treatment, and the independent variables are the frequencies with which the same label was named as the stander‐out (‘standout’) and as the favourite (‘favourite’). Since the frequency with which the nth option is chosen is a residual, where n is the number of options in a task, we dropped one observation for each task. Since the dependent variable is a proportion, we use a GLM specification with logistic link function, binomial error structure and robust standard errors (Papke and Wooldridge, 1996). Results are given in Table 7. These regressions confirm that coordinators were attracted to standing‐out labels in the Amsterdam experiment but to favourite labels in the Nottingham experiment. Table 7 
Proportion of Coordinators Choosing an Option Regressed on its Favouriteness and Standing‐out Score . Coefficient (SE) . ∂y/∂x (SE) . Amsterdam experiment standout 0.089** (0.005) 0.014** (0.001) favourite −0.026** (0.003) −0.004** (0.001) constant −2.829** (0.179) LogPseudoL: −15.06; AIC: 0.75; BIC −13.27 Nottingham experiment standout 0.018 (0.017) 0.002 (0.002) favourite 0.057** (0.012) 0.008** (0.002) constant −3.092** (0.200) LogPseudoL: −15.98; AIC: 0.68; BIC: −4.45 . Coefficient (SE) . ∂y/∂x (SE) . Amsterdam experiment standout 0.089** (0.005) 0.014** (0.001) favourite −0.026** (0.003) −0.004** (0.001) constant −2.829** (0.179) LogPseudoL: −15.06; AIC: 0.75; BIC −13.27 Nottingham experiment standout 0.018 (0.017) 0.002 (0.002) favourite 0.057** (0.012) 0.008** (0.002) constant −3.092** (0.200) LogPseudoL: −15.98; AIC: 0.68; BIC: −4.45 Notes: 1. ** denotes significance at the 1% level. 2. ∂y/∂x calculated at mean values of the independent variables. Open in new tab Table 7 
Proportion of Coordinators Choosing an Option Regressed on its Favouriteness and Standing‐out Score . Coefficient (SE) . ∂y/∂x (SE) . Amsterdam experiment standout 0.089** (0.005) 0.014** (0.001) favourite −0.026** (0.003) −0.004** (0.001) constant −2.829** (0.179) LogPseudoL: −15.06; AIC: 0.75; BIC −13.27 Nottingham experiment standout 0.018 (0.017) 0.002 (0.002) favourite 0.057** (0.012) 0.008** (0.002) constant −3.092** (0.200) LogPseudoL: −15.98; AIC: 0.68; BIC: −4.45 . Coefficient (SE) . ∂y/∂x (SE) . Amsterdam experiment standout 0.089** (0.005) 0.014** (0.001) favourite −0.026** (0.003) −0.004** (0.001) constant −2.829** (0.179) LogPseudoL: −15.06; AIC: 0.75; BIC −13.27 Nottingham experiment standout 0.018 (0.017) 0.002 (0.002) favourite 0.057** (0.012) 0.008** (0.002) constant −3.092** (0.200) LogPseudoL: −15.98; AIC: 0.68; BIC: −4.45 Notes: 1. ** denotes significance at the 1% level. 2. ∂y/∂x calculated at mean values of the independent variables. Open in new tab In the light of the questionnaire data, we conjecture that the crucial difference between the two experiments is to be found in the specifications of the labels for the text tasks. In the Amsterdam tasks, the odd‐one‐out labels were perceived as standing out from the others. These labels were used as focal points, contrary to the predictions of cognitive hierarchy theory. The Nottingham tasks did not have such obvious odd‐ones‐out, while being more effective than the Amsterdam ones in priming ideas of relative desirability and favouriteness.21 As a result (and consistent with both theories), favourites were focal points. We conjecture that there was some tendency for the modes of reasoning used in the text tasks to ‘spill over’ to the number tasks. In the Amsterdam experiment, coordinators consistently chose team‐optimal options in number tasks, whether or not those options carried the highest number of points. In the Nottingham experiment, the majority of coordinators responded to number tasks by using the same favourite‐based reasoning as they used in text tasks, with only a small minority choosing according to team optimality. 7. Discussion The main aim of the two experiments is to test cognitive hierarchy theory and the theory of team reasoning as rival explanations of behaviour in pure coordination and Hi‐Lo games. Formally, our conclusion must be that each theory failed at least one test. Cognitive hierarchy theory was disconfirmed in the Amsterdam text and number tasks, while both theories were disconfirmed in the Nottingham number tasks. However, it would be equally true to say that each theory had some success. Whenever one of the theories failed, the disconfirming evidence revealed a regularity in behaviour for which the other theory provided an explanation. Our experiments seem to have identified two modes of reasoning, each of which is sometimes used by players of coordination and Hi‐Lo games. Which of these modes of reasoning is brought into play may be sensitive to the decision context. In thinking about the relationship between the two theories, it is useful to step back and compare their main properties. In the context of coordination and Hi‐Lo games, perhaps the most important feature of cognitive hierarchy theory is the role it gives to players’ pre‐reflective inclinations – the non‐rational choice propensities that generate primary salience, and that are modelled in the behaviour of ‘level 0’ players. The workings of the theory are such that equilibrium selection is strongly influenced by these inclinations. In contrast, the theory of team reasoning gives no role to primary salience. It models the decisions of agents who are fully rational, but in the special sense that they optimise over profiles of strategies assessed from the viewpoint of the players as a collective, rather than over individual strategies assessed from the viewpoints of individual players. Intuitively, it seems that each of these approaches captures a significant aspect of focal points. On the one hand, many apparently obvious focal points seem to be identified by pre‐reflective inclinations. For example, think of the pure coordination game in which the set of labels is {≪heads≫, ≪tails≫}. Why do most players choose ≪heads≫? Even if we assume it to be common cultural knowledge that ≪heads≫ takes priority over ≪tails≫, this does not provide a reason for the players (individually or collectively) to choose ≪heads≫. Ultimately, it seems, an explanation has to appeal to a pre‐reflective association between the idea of priority and the idea of choosing: it is psychologically more natural to choose the more important than to choose the less. On the other hand, there are equally obvious cases in which focal points seem to be identified by optimising over strategy profiles. Consider a pure coordination game in which each player has to point to one of four cubes on a tray; three are red and one is green. What makes the choice of the green cube the focal point? Here, it seems implausible to appeal to a pre‐reflective propensity to pick the odd one out. The most obvious answer is that ‘Choose the green cube’ is a better rule for the two players together than ‘Pick a red cube’. Our experimental strategy was to create games in which these two ways of trying to identify focal points pull in different directions. In designing the text tasks, we tried to ensure that subjects’ pre‐reflective inclinations would attract them towards desirable or favourite labels, while thoughts about the best rule for the two co‐players together would attract them to less desirable odd ones out. In the ‘type 1’ number tasks, the objects with the highest numbers of points are the most immediately desirable, but the best rule for the co‐players together is to choose an object with fewer points. It seems that both of these forces were at work in our experiments, and that their relative strength was sensitive to details of experimental design. We seem to have found a class of coordination problems which, because they prime opposing modes of reasoning, are particularly difficult for people to solve.22 One of the most remarkable features of our experiments is the success with which participants overcame this difficulty. Within each of our experiments, a large majority of subjects used a common mode of reasoning for identifying focal points. That common mode of reasoning was different in the two experiments but, as far we can tell, this difference was not attributable to differences between the subject pools. The implication is that our subjects were able to use subtle features of the experimental environment to solve the problem of coordinating on a common mode of reasoning. This behaviour reveals an ability to solve coordination problems at a conceptual level above that of the theories of cognitive hierarchy and team reasoning that we have been examining. Each of those theories captures certain aspects of focal‐point reasoning but some essential feature of the human ability to solve coordination problems seems to have escaped formalisation. However disheartening this conclusion may be for game theorists, it ought not to be too surprising to readers of Schelling. In Strategy of Conflict, Schelling repeatedly insists on the diversity of the methods by which people find focal points, and rejects any suggestion that these methods can be reduced to a single formal theory. Although there are hints of team reasoning in his idea that players search for the ‘best rule’, he allows this search to range over a much wider domain than that represented in game theory (whether as practised in 1960, or as practised now). His list of methods or rules includes ‘analogy’, ‘precedent’, ‘aesthetic or geometric configuration’, ‘casuistic reasoning’, and ‘whimsy’ (1960, p. 57); he even suggests that some methods use ‘excuses’ and ‘pretences’ in place of reasons and beliefs (p. 298). In the context of a Nash demand game, he says: The basic intellectual premise, or working hypothesis, for rational players in this game seems to be the premise that some rule must be used if success is to exceed coincidence, and that the best rule to be found, whatever its rationalization, is consequently a rational rule. (p. 283, italics added) Schelling is advising us not to expect a unified theoretical rationalisation of focal points.23 Nearly half a century later, we are beginning to understand some of the methods by which focal points are found but so far the search for a unified theory has been unsuccessful. We suspect that Schelling is not surprised by this state of affairs. Footnotes 1 " From now on, we will use the term ‘cognitive hierarchy theory’ to refer to this approach in general, and not merely to the specific model proposed by Camerer et al. 2 " Repeated coordination games provide additional, confounding means of communication, which are not part of our subject matter. This class of games is analysed by Crawford and Haller (1990). 3 " This term is due to Bacharach (2006). Bacharach uses it only for cases in which there is some j such that Uj > Uk for all k ≠ j, but (as we shall show in Section 1.4) his analysis can be extended to the wider class of games encompassed by our definition. 4 " Lewis presented his ideas relatively informally, at a time when some of what are now seen as fundamental principles of game theory had not been developed. Mehta et al. (1994) discuss Lewis’s theory of salience. For a fuller discussion of Lewis’s game theory, see Cubitt and Sugden (2003). 5 " In an alternative formulation, proposed by Crawford and Iriberri (2007), a level 2 player believes that her opponent reasons at level 1, a level 3 player believes that his opponent reasons at level 2 and so on. For the games analysed in this article, the implications of the two versions of the theory are essentially the same. 6 " The two hypotheses have different implications when the profile of strategies that is best for the group is not a Nash equilibrium, as in the Prisoner’s Dilemma. 7 " Using this approach as their representation of focal‐point reasoning, Binmore and Samuelson (2006) develop an evolutionary model in which ‘monitoring’ of labels is costly; selection induces an equilibrium in which the degree of monitoring is less than optimal. 8 " This concept of ‘nondescriptness’ is due to Bacharach and Bernasconi (1997). An alternative implementation of Schelling’s idea, discussed by Crawford and Haller (1990), is to assign labels to the two co‐players by independent random draws from a given distribution. We take the view that, if the Crawford–Haller labelling system is used, Schelling’s game is no longer the 4×4 Hi‐Lo game that the payoff matrix purports to represent. Instead, there is a 4×4 payoff matrix in which one cell has the payoff profile (9, 9) and nine cells have the payoff profile (10/3, 10/3). Since we are investigating classic coordination games, we use nondescript labelling rather the Crawford–Haller method. 9 " A different possibility is suggested by Crawford and Iriberri’s (2007) assumption that level 0 reasoners choose options that, in some intuitive sense, stand out. Conceivably, the fact that l4 is an odd‐one‐out payoff might it more likely to be chosen by level 0 reasoners. In fact, our results do not support this version of cognitive hierarchy theory. (In relation to pure coordination games, the best evidence of behaviour in accordance with cognitive hierarchy theory comes from the Nottingham experiment. In that experiment, in the nondescript Hi‐Lo games most similar to the present example, the low‐payoff label does not act as a focal point: see the results for tasks NN1 to NN6 in Table 3.) 10 " The Amsterdam experiment was carried out by Bardsley, the Nottingham one by Mehta, Starmer and Sugden. In the early stages of the development of the design, all four authors were working together in the UK. The design process bifurcated when Bardsley moved to the Netherlands. As a result, the two experiments have a common basic design, implemented in slightly different ways. 11 " When different objects carry different numbers of points, the picking treatment cannot be interpreted as eliciting p0. If cognitive hierarchy theory holds, reasoners of level 1 and above will not perceive tasks in the picking treatment as ‘just picking’: they will recognise the rationality of choosing an object with the maximum number of points. 12 " Labels were written in English whenever the relevant words would be familiar to Dutch students. Subjects were given a list of translations between English and Dutch for all labels. The relevant Dutch‐to‐English translations are: grijs = grey, karmozijn = crimson, turkoois = turquoise, Lissabon = Lisbon. 13 " Carlsberg is a popular and widely available brand of beer. Corsendonk, Grimbergen, Westmalle and Rochefort are specialist Trappist‐style beers, brewed in Belgium. 14 " In the experiment of Mehta et al., most questions were open‐ended (e.g. ‘Name any car manufacturer’), rather than requiring a closed choice from a finite set of pre‐specified labels. With the benefit of hindsight, we now think that the two types of question may prompt different rules of selection. For example, Ford (the archetypal car manufacturer) was the clear focal point for Mehta et al.’s open‐ended task; but, facing a finite list of car manufacturers, subjects may perceive ‘Choose the most glamorous’ as a more obvious rule. 15 " The intended salients for TA1, …, TA14 were: ≪grijs≫, ≪Ford≫, ≪Mannheim≫, ≪peanut≫, ≪glass≫, ≪plastic≫, ≪bread≫, ≪water≫, ≪Carlsberg≫, ≪frog≫, ≪bicycle≫, ≪chess≫, ≪Bern≫, and ≪sitting≫ (all odd ones out). For TN2, …, TN14 they were: ≪Earth≫ (status quo), ≪Ford≫ (archetype), ≪plain omelette≫ (archetype), ≪1≫ (smallest), ≪stool≫ (odd one out), ≪Ontario≫ (odd one out), ≪sitting≫ (status quo), ≪2000≫ (round number, most talked about), ≪John≫ (archetype), ≪win nothing≫ (odd one out), ≪red≫ (archetype), ≪carrot juice≫ (odd one out), and ≪Calais≫ (odd one out). TN1 did not have an intended salient. The booklets were originally prepared for an experiment carried out in 2000 at the University of East Anglia on a Friday lunchtime, making this the status quo in TN1. In that experiment, subjects in the picking treatment of the number tasks distributed their choices approximately randomly between options, irrespective of the points assigned to them. Since the most credible explanation of this behaviour was that subjects had not understood the role of points in the experiment, we revised the instructions and re‐ran the experiment in Nottingham, using the same booklets. The Nottingham experiment took place on a Wednesday lunchtime. The results of the East Anglia experiment are available from the authors on request. 16 " Our claims about team reasoning require the assumptions (i ) that the safer of two lotteries is preferred when its expected value is greater than that of the riskier lottery and (ii ) that the riskier lottery is preferred when its expected value is at least twice that of the safer one. 17 " If, for a given label, the expected number of choices is less than 5, we combine choices for that label with those for the label with the next smallest expected number of choices, and so on until the expected number of entries in each cell is at least 5. 18 " This ‘average NCI’ is calculated as follows. For each task independently, we calculate the probability that two pickers, drawn at random without replacement, choose the same option. We sum these probabilities across all tasks to arrive at the expected number of ‘same choices’ in the whole experiment, per pair of pickers. We then divide this by the expected number of ‘same choices’ if pickers choose at random. 19 " The method of averaging is the same as that used to generate the summary statistics shown in Table 2. For the Amsterdam experiment, we carried out a bootstrap test, separately for each of the four type 3 tasks and for each treatment, of whether the average within‐session NCI was greater than 1. A statistically significant difference (always at the 1% level) was found for two tasks in the picking treatment, two tasks in the guessing treatment, and three tasks in the coordination treatment. In the Nottingham experiment, the number of pairs of coordinators (22) and the number of matching responses (on average, about 6 per task) is too small for powerful statistical tests at the pair or task level. Aggregating over the 22 pairs and the 6 tasks (i.e. 132 cases), there were 37 matched responses, compared with an expectation of 26.4 from random choice. The null hypothesis that all coordinators chose randomly can be rejected at the 5% level in a binomial test. 20 " Statistically significant differences (at the 5% level, using a chi‐squared test) between responses in Amsterdam and Nottingham were found for only 3 of the 34 standing‐out questions and for only 5 of the 34 questions about favourites. 21 " This effect may have been enhanced by the fact that the Nottingham instructions used the word ‘choose’ to refer to the act of selecting a label, while the Amsterdam instructions used the more neutral expression ‘click on’. Choosing has stronger connotations of liking (and hence of favouriteness) than clicking on. 22 " There is perhaps some parallel here with the results of an experiment reported by Crawford et al. (2008), in which small asymmetries in payoffs between players disrupt focal‐point reasoning. In that experiment, however, there seems to be a motivational conflict between seeking to coordinate with one’s co‐player and seeking advantage (or avoiding disadvantage) relative to her. In our experiments, the conflict is between opposing modes of reasoning about how to coordinate. 23 " This reading of Schelling is defended by Sugden and Zamarrón (2006). References Bacharach , M. ( 1993 ). ‘Variable universe games’ , in ( K. Binmore, A. Kirmanand and P. Tani, eds.), Frontiers of Game Theory , pp. 255 – 76 , Cambridge , MA: MIT Press. OpenURL Placeholder Text WorldCat Bacharach , M. ( 1999 ). ‘Interactive team reasoning: a contribution to the theory of cooperation’ , Research in Economics , vol. 53 ( 2 ), pp. 117 – 47 . Google Scholar Crossref Search ADS WorldCat Bacharach , M. ( 2006 ). Beyond Individual Choice: Teams and Frames in Game Theory , ( N. Gold and R. Sugden, eds), Princeton, NJ: PrincetonUniversity Press . Bacharach , M. and Bernasconi , M. ( 1997 ). ‘The variable frame theory of focal points: an experimental study’ , Games and Economic Behavior , vol. 19 ( 1 ), pp. 1 – 45 . Google Scholar Crossref Search ADS WorldCat Bacharach , M. and Stahl , D.O. ( 2000 ). ‘Variable‐frame level‐n theory’ , Games and Economic Behavior , vol. 33 ( 2 ), pp. 220 – 46 . Google Scholar Crossref Search ADS WorldCat Binmore , K. and Samuelson , L. ( 2006 ). ‘The evolution of focal points’ , Games and Economic Behavior , vol. 55 ( 1 ), pp. 21 – 42 . Google Scholar Crossref Search ADS WorldCat Camerer , C.F. , Ho , T.H. and Chong , J.K. ( 2004 ). ‘A cognitive hierarchy model of games’ , Quarterly Journal of Economics , vol. 119 ( 3 ), pp. 861 – 98 . Google Scholar Crossref Search ADS WorldCat Casajus , A. ( 2001 ). Focal Points in Framed Games: Breaking the Symmetry , Berlin: Springer‐Verlag . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Crawford , V.P. and Haller , H. ( 1990 ). ‘Learning how to cooperate: optimal play in repeated coordination games’ , Econometrica , vol. 58 ( 3 ), pp. 571 – 95 . Google Scholar Crossref Search ADS WorldCat Crawford , V.P. and Iriberri , N. ( 2007 ). ‘Fatal attraction: salience, naïveté, and sophistication in experimental ‘‘hide‐and‐seek’’ games’ , American Economic Review , vol. 97 ( 5 ), pp. 1731 – 50 . Google Scholar Crossref Search ADS WorldCat Crawford , V.P. , Gneezy , U. and Rottenstreich , Y. ( 2008 ). ‘The power of focal points is limited: even minute payoff asymmetry may yield large coordination failures’ , American Economic Review , vol. 98 ( 4 ), pp. 1443 – 58 . Google Scholar Crossref Search ADS WorldCat Cubitt , R.P. and Sugden , R. ( 2003 ). ‘Common knowledge, salience and convention: a reconstruction of David Lewis’s game theory’ , Economics and Philosophy , vol. 19 ( 2 ), pp. 175 – 210 . Google Scholar Crossref Search ADS WorldCat Efron , B. ( 1979 ). ‘Bootstrap methods: another look at the jacknife’ , Annals of Statistics , vol. 7 ( 1 ), pp. 1 – 26 . Google Scholar Crossref Search ADS WorldCat Harsanyi , J.C. and Selten , R. ( 1988 ). A General Theory of Equilibrium Selection in Games , Cambridge, MA: MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Janssen , M.C.W. ( 2001 ). ‘Rationalising focal points’ , Theory and Decision , vol. 50 ( 2 ), pp. 119 – 48 . Google Scholar Crossref Search ADS WorldCat Lewis , D.K. ( 1969 ). Convention: A Philosophical Study , Cambridge, MA: Harvard University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Mehta , J. , Starmer , C. and Sugden , R. ( 1994 ). ‘The nature of salience: an experimental investigation’ , American Economic Review , vol. 84 ( 3 ), pp. 658 – 73 . OpenURL Placeholder Text WorldCat Papke , L.E. and Wooldridge , J.M. ( 1996 ). ‘Econometric methods for fractional response variables with an application to 401(k) plan participation rates’ , Journal of Applied Econometrics , vol. 11 ( 6 ), pp. 619 – 32 . Google Scholar Crossref Search ADS WorldCat Schelling , T.C. ( 1960 ). The Strategy of Conflict , Cambridge, MA: Harvard University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Stahl , D.O. and Wilson , P.W. ( 1995 ). ‘On players’ models of other players’ , Games and Economic Behavior , vol. 10 ( 1 ), pp. 218 – 54 . Google Scholar Crossref Search ADS WorldCat Sugden , R. ( 1993 ). ‘Thinking as a team: towards an explanation of non‐selfish behavior’ , Social Philosophy and Policy , vol. 10 ( 1 ), pp. 69 – 89 . Google Scholar Crossref Search ADS WorldCat Sugden , R. ( 1995 ). ‘A theory of focal points’ . Economic Journal , vol. 105 ( 430 ), pp. 533 – 50 . Google Scholar Crossref Search ADS WorldCat Sugden , R. and Zamarrón , I.E. ( 2006 ). ‘Finding the key: the riddle of focal points’ , Journal of Economic Psychology , vol. 27 ( 5 ), pp. 609 – 21 . Google Scholar Crossref Search ADS WorldCat Author notes " The Amsterdam experiment was conducted while Bardsley was affiliated with CREED, Universiteit van Amsterdam; it was programmed by Jos Theelen and financed by EU TMR project ‘ENDEAR’, FMRX‐CT98‐0238. The Nottingham experiment and Starmer and Sugden’s subsequent work were supported by the Leverhulme Trust; Jacinto Braga, Steve Humphrey and Henrik Orzen helped in running the experiment. We also thank Vincent Crawford, Peter Moffatt, Tassos Magdalinos, David Myatt, David Rojo‐Arjona, Thomas Schelling, Ignacio Zamarrón, two anonymous referees and participants at the 2001 Barcelona meeting of the Economic Science Association for comments. The idea of using a ‘guessing’ treatment was first suggested by a student participant at a seminar at the University of Oxford, shortly after the publication of Mehta et al. (1994); we regret that we do not know her name. © The Author(s). Journal compilation © Royal Economic Society 2009

Harrison, Glenn, W.;Humphrey, Steven, J.;Verschoor,, Arjan

doi: 10.1111/j.1468-0297.2009.02303.xpmid: N/A

Abstract We review experimental evidence collected from risky choice experiments using poor subjects in Ethiopia, India and Uganda. Using these data we estimate that just over 50% of our sample behaves in accordance with expected utility theory and that the rest subjectively weight probability according to prospect theory. Our results show that inferences about risk aversion are robust to whichever model we adopt when we estimate each model separately. However, when we allow both models to explain portions of the data simultaneously, we infer risk aversion for subjects behaving according to expected utility theory and risk‐seeking behaviour for subjects behaving according to prospect theory. How do individuals in developing countries make choices under uncertainty? Economic theory now provides a rich array of theories to explain this type of behaviour. To answer this question we evaluate two competing theories using data collected from artefactual field experiments1 conducted in three developing countries (India, Ethiopia and Uganda). Our data consist of real choices from 531 subjects in very poor locales, along with information on individual demographic characteristics. Our primary objectives are to assess the weight of evidence for the two major received theories of choice under uncertainty, expected utility theory (EUT) and prospect theory (PT), and to assess whether they lead to different inferences about the risk attitudes of our subjects. The importance for development policy of characterising choice behaviour, and hence risk attitudes, is well established. Welfare evaluation of any proposed policy with uncertain outcomes should take into account the aversion that some individuals may have to risk, and the manner in which risk‐coping strategies mitigate exposure to risks (Fafchamps, 2004). Furthermore, it is clear that producers and consumers in developing countries face extraordinarily risky environments in general (Collier and Gunning 1999). The rural poor that make up our subject pool are no different. Fafchamps (2004; p.196) concludes a book‐length treatment as follows: We have learned that risk affects the rural poor in numerous and profound ways. The magnitude and range of shocks that affect rural populations of the Third World is without comparison in developed economies. Perhaps the only way to describe it to people who have never been there is to compare it to a war economy: death strikes at random a large proportion of the population, especially children; the provision of health services is either non‐existent or insufficient; trade with the rest of the world is difficult so that many commodities are rationed or unavailable and local prices are erratic; food is at times very scarce; and steady wage employment is non‐existent so that people must make a living from self‐employment in little jobs. To deal with such a harsh environment, people are equipped with very little in terms of advanced technology and accumulated assets. Financial institutions are either absent or inefficient and expensive, and in many places, inflation is rife so that the cost of hoarding money is high. Thus it is a high priority to obtain accurate characterisations of the risk attitudes of the rural poor. To do so, as we will show, one must also obtain an accurate characterisation of the manner in which choices under uncertainty are made. Our analysis builds on an experimental tradition that uses field experiments focused on behaviour towards risk, started in India by Binswanger (1980, 1981, 1982). Later contributions include experiments in Zimbabwe by Barr and Genicot (2008), in Chile and Peru by Barr and Packard (2003, 2005), in India, Ethiopia and Uganda by Humphrey and Verschoor (2004a,b), and in Timor‐Leste by Botelho et al. (2005). Humphrey and Verschoor (2004a,b) conclude that the behaviour they observed is inconsistent with expected utility maximisation and exhibits subjective probability weighting. They recommend that models of choice under uncertainty in developing countries should replace EUT with a version of PT. This model could then be used in conjunction with the experimental data to evaluate and quantify specific features of behaviour such as attitudes towards risk. This conclusion echoes calls made on the basis of data collected from numerous experiments conducted in the developed world (Camerer, 1998). However, it is founded on a questionable premise: that the research agenda should establish the single best account of behaviour, and that ‘best’ should be defined in terms of the single model that explains the data most accurately. What if some subjects are best characterised by one model of choice under uncertainty and other subjects are best characterised by another model of choice under uncertainty, and the two models imply different risk attitudes? It is clear that such a scenario makes it more difficult to inform policy interventions than when one assumes just one model of choice. To investigate this possibility, and contrary to the approach adopted in conventional studies of risky choice, we take two of the major competing models of risky choice in the literature and allow the data to determine the fraction of behaviour described by each model. Using a ‘finite mixture model’ approach, we estimate the parameters of each competing model and contrast the results with those emerging from conventional estimates that assume either EUT or PT describe behaviour but not both. We conclude that there is, in fact, support for each model in our data, so that there is no single, correct model that explains all of the data. Furthermore, as conjectured above, we show that the inferences about parameters of each model differ when one estimates the flexible specification that allows the data to determine the fraction of the choices explained by each model. In particular, under the flexible specification our data point to a concave (risk averse) utility of income function if you assume EUT but a convex (risk‐seeking) utility of income function if you assume PT. These results are consistent with the views of Humphrey (2000) and Harrison and Rutström (forthcoming), who argue that it may be inappropriate to search for a single model of risky choice because behaviour is sufficiently heterogeneous that it cannot be described by a single theory. Moreover, this is not the sort of heterogeneity that one can assume to be correlated with observable characteristics of the individuals, although the statistical approach we employ does allow for that. We review the design of the experimental tasks in Section 1. Each subject made eight choices between pairs of lotteries with real monetary consequences, with outcomes that were substantial in terms of their income and wealth. Each subject was given these experimental tasks as part of a larger household survey, so we also have a set of characteristics to describe the individual and their household. In Section 2 we specify statistical models for these data which allow choices to be made consistently with EUT or PT. We consider all binary choices jointly in order to characterise the decision processes used by our subjects across a range of tasks. The parameters of each theory are allowed to be linear functions of observed individual characteristics, as well as experimental treatments and locations, so we do not assume that every subject has the same utility function or probability weighting function. In Section 2 we also specify a finite mixture model in which observed choices can may be generated by either EUT decision‐makers or PT decision‐makers. Section 3 presents our results and Section 4 concludes. 1. Experiments The experimental design provided subjects with an array of choices over monetary lotteries, just as one finds in traditional laboratory experiments in developed countries. Table 1 summarises the design and parameters; an Appendix provides details on procedures and the complete design. All values here are in terms of US dollars and are expressed in cents. We discuss the local currency equivalents, and scaling for purchasing power parity, below. Table 1 
Experimental Design Task Type . Task Number . Lottery A . Lottery B . Outcome 1 . Outcome 2 . Outcome 3 . Outcome 1 . Outcome 2 . Outcome 3 . Common Consequence Effect 1 250; 1/4 0; 1/4 100; 1/2 100; 1 2 250; 1/4 0; 3/4 100; 1/2 0; 1/2 3 250; 3/4 0; 1/4 250; 1/2 100; 1/2 Cyclical Choice 4 550; 1/2 0; 1/2 250; 3/4 0; 1/4 5 250; 3/4 0; 1/4 250; 1/2 200; 1/4 0; 1/4 6 550; 1/2 0; 1/2 250; 1/2 200; 1/4 0; 1/4 Preference Reversal 7 500; 1/4 0; 3/4 150; 3/4 0; 1/4 Repeat (one of three possible tasks) 8 250; 3/4 0; 1/4 250; 1/2 100; 1/2 250; 3/4 0; 1/4 250; 1/2 200; 1/4 0; 1/4 500; 1/4 0; 3/4 150; 3/4 0; 1/4 Task Type . Task Number . Lottery A . Lottery B . Outcome 1 . Outcome 2 . Outcome 3 . Outcome 1 . Outcome 2 . Outcome 3 . Common Consequence Effect 1 250; 1/4 0; 1/4 100; 1/2 100; 1 2 250; 1/4 0; 3/4 100; 1/2 0; 1/2 3 250; 3/4 0; 1/4 250; 1/2 100; 1/2 Cyclical Choice 4 550; 1/2 0; 1/2 250; 3/4 0; 1/4 5 250; 3/4 0; 1/4 250; 1/2 200; 1/4 0; 1/4 6 550; 1/2 0; 1/2 250; 1/2 200; 1/4 0; 1/4 Preference Reversal 7 500; 1/4 0; 3/4 150; 3/4 0; 1/4 Repeat (one of three possible tasks) 8 250; 3/4 0; 1/4 250; 1/2 100; 1/2 250; 3/4 0; 1/4 250; 1/2 200; 1/4 0; 1/4 500; 1/4 0; 3/4 150; 3/4 0; 1/4 Open in new tab Table 1 
Experimental Design Task Type . Task Number . Lottery A . Lottery B . Outcome 1 . Outcome 2 . Outcome 3 . Outcome 1 . Outcome 2 . Outcome 3 . Common Consequence Effect 1 250; 1/4 0; 1/4 100; 1/2 100; 1 2 250; 1/4 0; 3/4 100; 1/2 0; 1/2 3 250; 3/4 0; 1/4 250; 1/2 100; 1/2 Cyclical Choice 4 550; 1/2 0; 1/2 250; 3/4 0; 1/4 5 250; 3/4 0; 1/4 250; 1/2 200; 1/4 0; 1/4 6 550; 1/2 0; 1/2 250; 1/2 200; 1/4 0; 1/4 Preference Reversal 7 500; 1/4 0; 3/4 150; 3/4 0; 1/4 Repeat (one of three possible tasks) 8 250; 3/4 0; 1/4 250; 1/2 100; 1/2 250; 3/4 0; 1/4 250; 1/2 200; 1/4 0; 1/4 500; 1/4 0; 3/4 150; 3/4 0; 1/4 Task Type . Task Number . Lottery A . Lottery B . Outcome 1 . Outcome 2 . Outcome 3 . Outcome 1 . Outcome 2 . Outcome 3 . Common Consequence Effect 1 250; 1/4 0; 1/4 100; 1/2 100; 1 2 250; 1/4 0; 3/4 100; 1/2 0; 1/2 3 250; 3/4 0; 1/4 250; 1/2 100; 1/2 Cyclical Choice 4 550; 1/2 0; 1/2 250; 3/4 0; 1/4 5 250; 3/4 0; 1/4 250; 1/2 200; 1/4 0; 1/4 6 550; 1/2 0; 1/2 250; 1/2 200; 1/4 0; 1/4 Preference Reversal 7 500; 1/4 0; 3/4 150; 3/4 0; 1/4 Repeat (one of three possible tasks) 8 250; 3/4 0; 1/4 250; 1/2 100; 1/2 250; 3/4 0; 1/4 250; 1/2 200; 1/4 0; 1/4 500; 1/4 0; 3/4 150; 3/4 0; 1/4 Open in new tab These particular lotteries were chosen with two considerations in mind. First, we wanted to have pairs of lotteries that presented subjects with several of the classical tests of EUT. These are the tests that have been used in developed country lab experiments to evaluate EUT (Camerer, 1995), and we wanted to have comparable tasks in developing countries. These tests are carefully calibrated to provide fertile grounds for violation of EUT. Some might argue that they are ‘booby trap’ or ‘trip‐wire’ tasks, that are so finely calibrated as to make it impossible for EUT to account for behaviour. This is incorrect: if one drew lotteries at random, one would find in many cases that EUT and PT predict exactly the same choice. So all we have done is use our experimental control over the choice of lotteries to focus attention on the domain of (paired) tasks where the data can best discriminate between them. The second consideration underlying these lottery choices is that we were concerned with literacy, since some of our subjects were bound to be illiterate. We also expected, correctly, that they had no experience with experiments and we wanted a design that we could be as confident as possible that they would understand. For this reason we chose ‘salient probabilities’ of 0, 1/4, 1/2, 3/4 and 1. This made it easier to explain the operationalisation of risk to the subject, and thereby enhanced credibility. Each outcome was a ball of a particular colour, so a probability of 1/2 would have been, say, 2 green and 2 red balls in a bag and a probability of 1/4 would have been 1 green and 3 red, and so on. Consider the implementation of task 1 from Table 1 in Uganda as a specific example. Lottery A was represented as a red bag containing four coloured marbles. The experiment organiser (Verschoor) placed one yellow marble into the bag and explained that, should this bag be selected and the yellow marble subsequently drawn, it would be worth the equivalent in Ugandan Shillings (Ush) of USD 2.50. This is shown as Outcome 1. Similarly, two green marbles (each worth the local equivalent of USD 1) and one blue marble (worth nothing) were added to the red bag. Lottery B was represented by a blue bag of four green marbles, each worth the local equivalent of USD 1. Subjects were provided with a corresponding piece of paper, which showed the contents of the red and blue bags, with appropriate values attached to each differently coloured marble. The subject’s task was to indicate which bag of marbles they preferred. The eight tasks were all binary choice lotteries. In each task the subject picked either lottery A or lottery B. At the end of the experiment one of the eight tasks was selected at random for each subject and the lottery chosen in that task was played‐out for real money. This procedure motivates subjects to consider each choice carefully as if it were for real money, rewards them for participation in the experiment and controls for wealth effects. To control for possible order effects, roughly one half of the subjects had the tasks presented in one order and the other half in reverse order. There were 531 subjects in all. In India there were 223 subjects, drawn from two villages (108 in Vepur and 115 in Guddi). In Uganda there were 208 subjects, again drawn from two villages (107 in Sironko and 101 in Bufumbo). In Ethiopia there were 103 subjects. Mosley and Verschoor (2005) provide more details on these regions and descriptive characteristics of the sample. We have a total of 4,248 actual choices, allowing for some missing responses. With minor variations, the task and procedures were identical across each sample. In all cases the lotteries were presented in terms of local currency which approximately matched the values shown for the outcomes in Table 1. Our statistical analysis converts the local currency units into US dollars and cents using purchasing power parity (PPP) conversion rates. Thus the statistical analysis is undertaken using our best estimate of the local purchasing power of each monetary outcome. In 2000 the PPP rates for Ethiopia, India and Uganda were 8.22, 44.94 and 1644.47, in terms of the rate at which the local currency converted to one US dollar (Heston et al. 2002). The PPP exchange rates actually used for our experiments are close to these: 8.75, 50.33 and 1750, respectively, and differ due to differences in exchange rates prevailing at the exact time of each experiment.2 We recognise that there can be significant differences in purchasing power within regions of developing countries, reflecting differences in patterns of consumption and local prices (Deaton, 1997; §5.2). Although definitions of poverty differ, there can nonetheless be no doubt that a large fraction of our subjects were closer to absolute poverty lines than conventionally encountered in experiments of this type. The outcomes in our experiment also represented substantial amounts of money to our subjects. Humphrey and Verschoor (2004b; p.422) note that the payoffs were 250%, 339% and 278% of prevailing daily wages in Uganda, India and Ethiopia, respectively; the experiments lasted roughly 3 hours for each individual. 2. Alternative Theories We assume just two competing theories of choice under uncertainty to explain these data: EUT and PT. There are several major alternative theories and many parametric variants of these theories but we take these two theories to be major competitors in the literature.3 We adopt relatively flexible functional forms to implement each theory. One of the proposed models is a simple EUT specification which assumes a constant relative risk aversion (CRRA) utility function defined over the ‘final monetary prize’ that the subject would receive if the lottery were played out. That is, the argument of the utility function is the prize in the lottery, which is always non‐negative.4 The other model is a popular specification of prospect theory (PT) due to Kahneman and Tversky (1979), in which the utility function is defined over gains and losses separately and a probability weighting function converts the underlying probabilities of the lottery into subjective probabilities. The three critical features of the PT model are (i) that the arguments of the utility function be gains or losses relative to some reference point, taken here to be zero; (ii) that losses loom larger than gains in the utility function; and (iii) that there be a nonlinearity in the transformed probabilities. The first and second points are irrelevant here since all lottery choices were in the gain domain.5 A more complete test of EUT and PT would include loss frames but we did not want to complicate the design of our experiments in the field. Furthermore, many of the early, classic tests of EUT against PT conducted by experimental economists refer solely to the gain domain: for example, see Camerer (1989), Starmer (1992), Harless and Camerer (1994) and Hey and Orme (1994). Thus our differentiation between EUT and PT focuses solely on the properties of the nonlinear probability weighting function, which we discuss more formally below. Since we do not consider losses, EUT is a special case of PT in which there is no probability weighting. Hence we could just estimate a PT model and test the EUT parametric restriction. This is true, and relevant if one wanted to characterise the data in terms of only one model of choice behaviour. However, our interest is precisely in allowing the observed choice data to be generated by two latent models of choice behaviour and where we do not know a priori that observable individual demographic characteristics will allow us to differentiate EUT and PT decision‐makers crisply. 2.1. Expected Utility Specification We assume that utility of income is defined by U(x) = (x1−r)/(1 − r) where x is the lottery prize and r is a parameter to be estimated. With this CRRA specification, r = 0 indicates risk neutrality, r > 0 indicates risk aversion and r < 0 indicates risk loving. Probabilities for each outcome k, p(k), are those that are induced by the experimenter, so expected utility (EU) is simply the probability weighted utility of each outcome in each lottery. Since there were up to 3 implicit outcomes in each lottery i, EUi = ∑k[p(k) × U(k)] for k = 1, 2, 3. A simple stochastic specification is used to specify likelihoods conditional on the model. The EU for each lottery pair is calculated for a candidate estimate of r and the difference ∇EU = EUR − EUL calculated, where EUL is the EU of the left lottery in the display and EUR is the EU of the right lottery. A stochastic choice EUT can then be specified by assuming some cumulative probability distribution function, Ψ(·), such as the logistic.6 Thus the likelihood, conditional on the EUT model being true, depends on the estimates of r given the above specification and the observed choices. The conditional log‐likelihood is where I(·) is the indicator function, yi = 1(0) denotes the choice of the right (left) lottery in task i and X is a vector of individual characteristics. We allow each parameter to be a linear function of the observed individual characteristics of the subject. This is the X vector referred to above. We consider six characteristics. Four are binary variables to identify the order of the task, the country, females, and subjects that reported having some secondary education or more. We also included age in years and the number of people living in the household. The estimates of each parameter in the above likelihood function entails estimation of the coefficients of a linear function of these characteristics. So the estimate of , would actually be where is the estimate of the constant, normalised on India in terms of countries. If we collapse this specification by dropping all individual characteristics and country dummies, we would simply be estimating the constant terms for r. The estimates allow for the possibility of correlation between responses by the same subject, so the standard errors on estimates are corrected for the possibility that the eight responses are clustered for the same subject. The use of clustering to allow for ‘panel effects’ from unobserved individual effects is common in the statistical survey literature.7 Our estimates also allow for the stratification of observations by village (and hence also country). 2.2. Prospect Theory Specification There are two components to the PT specification, the utility function and the probability weighting function. We use the same CRRA functional form as specified for EUT:U(x) = (x1−α)/(1 − α). We do not have any losses in the lotteries considered here, so we drop the part of the utility function in PT that is defined for losses. Our evaluation of EUT and PT is therefore restricted to the gain domain, which is the domain over which most of the initial tests of PT were conducted. There are two variants of PT, depending on the manner in which the probability weighting function is combined with utilities. The original version proposed by Kahneman and Tversky (1979) posits some weighting function which is separable in outcomes, and has been usefully termed Separable Prospect Theory (PT) by Camerer and Ho (1994; p. 185). The alternative version, proposed by Tversky and Kahneman (1992), posits a weighting function defined over cumulative probability distributions. In either case, the weighting function proposed by Tversky and Kahneman (1992) has been widely used. It is assumed to have well‐behaved endpoints such that w(0) = 0 and w(1) = 1 and to imply weights w(p) = p γ/[p γ + (1 − p)γ]1/γ for 0 < p < 1. The normal assumption, backed by a substantial amount of evidence reviewed by Gonzalez and Wu (1999), is that 0 < γ < 1. This gives the weighting function an ‘inverse S‐shape’, characterised by a concave section signifying the overweighting of small probabilities up to a crossover‐point where w(p) = p, beyond which there is then a convex section signifying underweighting. If γ > 1 the function takes the less conventional ‘S‐shape’, with convexity for smaller probabilities and concavity for larger probabilities. Assuming that PT is the true model, prospective utility (PU) is defined in much the same manner as when EUT is assumed to be the true model. The PT utility function is used instead of the EUT utility function and w(p) is used instead of p but the steps are otherwise essentially identical (Harrison and Rutström, 2008; §3.1). The difference in prospective utilities is defined similarly as ∇PU = PUR − PUL. Thus the likelihood, conditional on the PT model being true, depends on the estimates of α and γ given the above specification and the observed choices. The conditional log‐likelihood is The parameters α and γ can again be estimated as linear functions of the vector X. The use of probability weighting introduces another way in which individuals might be averse to risk, quite apart from the implied aversion to risk from having a concave utility function. The idea that one could use non‐linear transformations of the probabilities in a lottery when weighting outcomes, instead of non‐linear transformations of the outcome into utility, was most sharply presented by Yaari (1987). To illustrate the point clearly, he assumed a linear utility function, in effect ruling out any risk aversion or risk seeking from the shape of the utility function per se. Instead, concave (convex) probability weighting functions would imply risk seeking (risk aversion). In general, in PT one can have aversion to risk from either or both of probability weighting and utility curvature. We follow EUT convention and refer to ‘risk aversion’ or ‘risk attitude’ solely as a property of the curvature of the utility function, even when we refer to PT estimates, but this semantic point should be kept in mind and we return to it when it affects the interpretation of results. 2.3. A Mixture Model Specification If we let πEUT denote the probability that the EUT model is correct, and πPT = (1 − πEUT) denote the probability that the PT model is correct, the grand likelihood can be written as the probability weighted average of the conditional likelihoods. Thus the likelihood for the overall model estimated is defined by This log‐likelihood can be maximised to find estimates of the parameters. Just as we allowed the parameters for EUT and PT to be estimated as linear functions of the observables X, we could do so in this case. However, the sample is not sufficiently large to allow robust estimation of the mixture model with the full set of covariates, so we restrict analysis to including the country dummies for πEUT. 3. Results 3.1. Estimates of the EUT Specification Table 2 collates the estimates from our data assuming that EUT is the sole theory explaining behaviour. Panel (a) presents estimates assuming no covariates and panel (b) extends this by including covariates. Figure 1 displays the predicted distribution of risk attitudes, using the estimated model that includes covariates. This distribution reflects the predicted values of the CRRA coefficient r, where the prediction depends on the characteristics of the individual, the location of the experiments, and the order in which the tasks were presented. These results point to moderate risk aversion over these stakes, with virtually no evidence of any risk‐loving behaviour in the sample as a whole. In Panel (a) the coefficient of CRRA is estimated to be 0.536, remarkably close to estimates obtained with comparable experiments and statistical methods in developed countries.8 Fig. 1. Open in new tabDownload slide Risk Attitudes Assuming EUT Fig. 1. Open in new tabDownload slide Risk Attitudes Assuming EUT Table 2 
Maximum Likelihood Estimates of EUT Model of Choices Coefficient . Variable . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) No Covariates r Constant 0.536 0.024 0.000 0.488 0.583 (b) Including Covariates r Constant 0.841 0.091 0.000 0.662 1.021 Ethiopia 0.050 0.074 0.502 −0.095 0.195 Uganda 0.169 0.070 0.015 0.032 0.306 Order of tasks −0.063 0.059 0.283 −0.178 0.052 Age in years −0.006 0.002 0.002 −0.010 −0.002 Female −0.085 0.046 0.068 −0.176 0.006 Some secondary education 0.056 0.059 0.345 −0.060 0.172 Number in household −0.013 0.010 0.178 −0.033 0.006 Coefficient . Variable . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) No Covariates r Constant 0.536 0.024 0.000 0.488 0.583 (b) Including Covariates r Constant 0.841 0.091 0.000 0.662 1.021 Ethiopia 0.050 0.074 0.502 −0.095 0.195 Uganda 0.169 0.070 0.015 0.032 0.306 Order of tasks −0.063 0.059 0.283 −0.178 0.052 Age in years −0.006 0.002 0.002 −0.010 −0.002 Female −0.085 0.046 0.068 −0.176 0.006 Some secondary education 0.056 0.059 0.345 −0.060 0.172 Number in household −0.013 0.010 0.178 −0.033 0.006 Open in new tab Table 2 
Maximum Likelihood Estimates of EUT Model of Choices Coefficient . Variable . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) No Covariates r Constant 0.536 0.024 0.000 0.488 0.583 (b) Including Covariates r Constant 0.841 0.091 0.000 0.662 1.021 Ethiopia 0.050 0.074 0.502 −0.095 0.195 Uganda 0.169 0.070 0.015 0.032 0.306 Order of tasks −0.063 0.059 0.283 −0.178 0.052 Age in years −0.006 0.002 0.002 −0.010 −0.002 Female −0.085 0.046 0.068 −0.176 0.006 Some secondary education 0.056 0.059 0.345 −0.060 0.172 Number in household −0.013 0.010 0.178 −0.033 0.006 Coefficient . Variable . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) No Covariates r Constant 0.536 0.024 0.000 0.488 0.583 (b) Including Covariates r Constant 0.841 0.091 0.000 0.662 1.021 Ethiopia 0.050 0.074 0.502 −0.095 0.195 Uganda 0.169 0.070 0.015 0.032 0.306 Order of tasks −0.063 0.059 0.283 −0.178 0.052 Age in years −0.006 0.002 0.002 −0.010 −0.002 Female −0.085 0.046 0.068 −0.176 0.006 Some secondary education 0.056 0.059 0.345 −0.060 0.172 Number in household −0.013 0.010 0.178 −0.033 0.006 Open in new tab From Panel (b) we observe that, compared to the model with no covariates, estimated risk aversion is slightly higher on average in India (0.841), which is the implicit country captured by the constant term. It is 0.050 higher in Ethiopia but this effect only has a p‐value of 0.502; it is 0.169 higher in Uganda and this effect has a p‐value of 0.015. The order of experimental tasks had a mild effect on elicited risk attitudes. Women appear to be slightly less risk averse than men, although the quantitative effect is small (−0.085) and barely significant (p‐value = 0.068). There is a statistically significant effect from age but it is quantitatively small: every 10 years of age is associated with a decline in risk aversion by 0.06. 3.2. Estimates of the PT Specification Table 3 collates estimates of the data assuming that PT is the sole model explaining the data, and that we use the probability weighting function proposed by Tversky and Kahneman (1992). Again, Panel (a) shows estimates that apply one model to all subjects and Panel (b) includes covariates for each parameter. From Panel (a) we see that the estimates of risk attitudes are considerably lower than under EUT (α = 0.464 < 0.536 = r), although still consistent with risk aversion since α > 0. The explanation, of course, is that probability weighting is allowed and that substitutes for some of the concavity of the utility function when explaining the data. The estimate of γ from Panel (a) of Table 3 is 1.384. This implies an S‐shaped probability weighting function which would entail underweighting of low probabilities and overweighting of large probabilities. This result contrasts starkly with the empirical claims from data collected in experimental laboratories in developed countries (Gonzalez and Wu, 1999). The left panel in Figure 2 illustrates the function implied by this estimate: it clearly has underweighting for probabilities below 0.6 but the extent of overweighting for higher probabilities is not great.9 We return to consider more flexible functions below. Fig. 2. Open in new tabDownload slide Alternative Probability Weighting Functions Fig. 2. Open in new tabDownload slide Alternative Probability Weighting Functions Table 3 
Maximum Likelihood Estimates of PT Model of Choices With Tversky‐Kahneman Probability Weighting Function Coefficient . Variable . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) No Covariates α Constant 0.464 0.036 0.000 0.393 0.535 γ Constant 1.384 0.070 0.000 1.246 1.522 (b) Including Covariates α Constant 0.896 0.128 0.000 0.645 1.147 Ethiopia 0.033 0.126 0.792 −0.215 0.281 Uganda 0.195 0.097 0.045 0.004 0.385 Order of tasks −0.056 0.098 0.566 −0.248 0.136 Age in years −0.007 0.003 0.031 −0.014 −0.001 Female −0.104 0.071 0.144 −0.245 0.036 Some secondary education 0.061 0.081 0.448 −0.097 0.219 Number in household −0.027 0.016 0.094 −0.060 0.005 γ Constant 0.690 0.375 0.066 −0.046 1.427 Ethiopia 0.045 0.396 0.909 −0.734 0.824 Uganda 0.208 0.169 0.219 −0.124 0.540 Order of tasks 0.211 0.165 0.201 −0.113 0.535 Age in years 0.004 0.012 0.742 −0.019 0.027 Female 0.002 0.187 0.993 −0.366 0.370 Some secondary education 0.076 0.159 0.634 −0.236 0.387 Number in household 0.060 0.036 0.092 −0.010 0.131 Coefficient . Variable . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) No Covariates α Constant 0.464 0.036 0.000 0.393 0.535 γ Constant 1.384 0.070 0.000 1.246 1.522 (b) Including Covariates α Constant 0.896 0.128 0.000 0.645 1.147 Ethiopia 0.033 0.126 0.792 −0.215 0.281 Uganda 0.195 0.097 0.045 0.004 0.385 Order of tasks −0.056 0.098 0.566 −0.248 0.136 Age in years −0.007 0.003 0.031 −0.014 −0.001 Female −0.104 0.071 0.144 −0.245 0.036 Some secondary education 0.061 0.081 0.448 −0.097 0.219 Number in household −0.027 0.016 0.094 −0.060 0.005 γ Constant 0.690 0.375 0.066 −0.046 1.427 Ethiopia 0.045 0.396 0.909 −0.734 0.824 Uganda 0.208 0.169 0.219 −0.124 0.540 Order of tasks 0.211 0.165 0.201 −0.113 0.535 Age in years 0.004 0.012 0.742 −0.019 0.027 Female 0.002 0.187 0.993 −0.366 0.370 Some secondary education 0.076 0.159 0.634 −0.236 0.387 Number in household 0.060 0.036 0.092 −0.010 0.131 Open in new tab Table 3 
Maximum Likelihood Estimates of PT Model of Choices With Tversky‐Kahneman Probability Weighting Function Coefficient . Variable . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) No Covariates α Constant 0.464 0.036 0.000 0.393 0.535 γ Constant 1.384 0.070 0.000 1.246 1.522 (b) Including Covariates α Constant 0.896 0.128 0.000 0.645 1.147 Ethiopia 0.033 0.126 0.792 −0.215 0.281 Uganda 0.195 0.097 0.045 0.004 0.385 Order of tasks −0.056 0.098 0.566 −0.248 0.136 Age in years −0.007 0.003 0.031 −0.014 −0.001 Female −0.104 0.071 0.144 −0.245 0.036 Some secondary education 0.061 0.081 0.448 −0.097 0.219 Number in household −0.027 0.016 0.094 −0.060 0.005 γ Constant 0.690 0.375 0.066 −0.046 1.427 Ethiopia 0.045 0.396 0.909 −0.734 0.824 Uganda 0.208 0.169 0.219 −0.124 0.540 Order of tasks 0.211 0.165 0.201 −0.113 0.535 Age in years 0.004 0.012 0.742 −0.019 0.027 Female 0.002 0.187 0.993 −0.366 0.370 Some secondary education 0.076 0.159 0.634 −0.236 0.387 Number in household 0.060 0.036 0.092 −0.010 0.131 Coefficient . Variable . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) No Covariates α Constant 0.464 0.036 0.000 0.393 0.535 γ Constant 1.384 0.070 0.000 1.246 1.522 (b) Including Covariates α Constant 0.896 0.128 0.000 0.645 1.147 Ethiopia 0.033 0.126 0.792 −0.215 0.281 Uganda 0.195 0.097 0.045 0.004 0.385 Order of tasks −0.056 0.098 0.566 −0.248 0.136 Age in years −0.007 0.003 0.031 −0.014 −0.001 Female −0.104 0.071 0.144 −0.245 0.036 Some secondary education 0.061 0.081 0.448 −0.097 0.219 Number in household −0.027 0.016 0.094 −0.060 0.005 γ Constant 0.690 0.375 0.066 −0.046 1.427 Ethiopia 0.045 0.396 0.909 −0.734 0.824 Uganda 0.208 0.169 0.219 −0.124 0.540 Order of tasks 0.211 0.165 0.201 −0.113 0.535 Age in years 0.004 0.012 0.742 −0.019 0.027 Female 0.002 0.187 0.993 −0.366 0.370 Some secondary education 0.076 0.159 0.634 −0.236 0.387 Number in household 0.060 0.036 0.092 −0.010 0.131 Open in new tab Including covariates in the PT specification leads to qualitative conclusions about risk attitudes that are similar to those obtained under EUT. Subjects in Ethiopia are estimated to be slightly more risk averse than those in India (+0.033) but the effect is not statistically significant (p‐value = 0.792). However, those in Uganda are estimated to be much more risk averse on average (+0.195), and the effect is significant (p‐value = 0.045). Women are again slightly less risk averse than men and the effect is barely significant (p‐value = 0.144). The effect of age is roughly the same as for EUT. There does not appear to be major differences in the extent of probability weighting across countries. The size of the household does affect the average extent of probability weighting. There are some limitations of the conventional Tversky and Kahneman (1992) probability weighting function. It does not allow independent specification of location and curvature; it has a fixed point, where p = w(p) at p = 1/e = 0.37 for γ < 1 and at p = 1 − 0.37 = 0.63 for γ > 1; and it is not even increasing in p for small values of γ. Prelec (1998) offers a two‐parameter probability weighting function that exhibits more flexibility than the Tversky and Kahneman (1992) function. The Prelec (1998) function is w(p) =  exp [−η(− ln pφ)], which is defined for 0 < p < 1, η > 0 and 0 < φ < 1. Rieger and Wang (2006; Proposition 2) offer a two‐parameter polynomial of third degree which is defined for 0 ≤ p ≤ 1, unlike the Prelec (1998) function: w(p) = p + [(3 − 3b)/(a2 − a + 1)][p3 − (a + 1)p2 + ap], where 0 < a < 1 and 0 < b < 1. The parameter restrictions on a and b ensure that the function is concave for lower values of p and then convex for larger values of p. Values of b larger than 1 would allow convex and then concave shapes, which we want to allow a priori given the findings of Humphrey and Verschoor (2004a,b).10 Table 4 reports estimates of the PT model assuming these two alternative functional forms; Figures 2 and 3 display the effects on the shape of the probability weighting function and elicited risk attitudes. Both of the alternatives confirm the presence of significant underweighting of probabilities over a wide range of probabilities. In fact, both of the two‐parameter probability weighting functions are weakly well‐behaved with respect to the conventional empirical wisdom that there should be a concave and then convex (‘inverse‐S’) shape. These shapes, in fact, are quite close to the original form sketched by Kahneman and Tversky (1979; p. 283), which exhibited considerable under‐weighting of probabilities for virtually the whole range of p. Fig. 3. Open in new tabDownload slide Alternative Risk Attitudes Assuming Prospect Theory Fig. 3. Open in new tabDownload slide Alternative Risk Attitudes Assuming Prospect Theory Table 4 
Maximum Likelihood Estimates of PT Model of Choices With Different Probability Weighting Function Coefficient . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) Tversky & Kahneman Probability Weighting Function: w(p) = p γ/[p γ + (1 − p) γ]1/γ α 0.464 0.036 0.000 0.393 0.535 γ 1.384 0.070 0.000 1.246 1.522 (b) Prelec Probability Weighting Function: w(p) =  exp [−η(− ln pφ)] α 0.504 0.033 0.000 0.439 0.569 η 1.202 0.053 0.000 1.097 1.307 φ 0.963 0.076 0.000 0.814 1.113 (c) Rieger & Wang Probability Weighting Function: w(p) = p + [(3 − 3b)/(a2 − a + 1)][p3 − (a + 1)p2 + ap] α 0.546 0.025 0.000 0.496 0.596 a 0.000 † † † † b 0.775 0.048 0.000 0.680 0.870 (d) Estimates of Risk Aversion Parameter α Using the Rieger & Wang Probability Weighting Function Constant 0.823 0.102 0.000 0.622 1.024 Ethiopia 0.055 0.081 0.500 −0.104 0.213 Uganda 0.198 0.093 0.034 0.015 0.381 Order of tasks −0.057 0.065 0.382 −0.186 0.071 Age in years −0.006 0.002 0.012 −0.011 −0.001 Female −0.084 0.050 0.098 −0.183 0.015 Some secondary education 0.053 0.062 0.394 −0.069 0.175 Number in household −0.015 0.013 0.265 −0.040 0.011 Coefficient . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) Tversky & Kahneman Probability Weighting Function: w(p) = p γ/[p γ + (1 − p) γ]1/γ α 0.464 0.036 0.000 0.393 0.535 γ 1.384 0.070 0.000 1.246 1.522 (b) Prelec Probability Weighting Function: w(p) =  exp [−η(− ln pφ)] α 0.504 0.033 0.000 0.439 0.569 η 1.202 0.053 0.000 1.097 1.307 φ 0.963 0.076 0.000 0.814 1.113 (c) Rieger & Wang Probability Weighting Function: w(p) = p + [(3 − 3b)/(a2 − a + 1)][p3 − (a + 1)p2 + ap] α 0.546 0.025 0.000 0.496 0.596 a 0.000 † † † † b 0.775 0.048 0.000 0.680 0.870 (d) Estimates of Risk Aversion Parameter α Using the Rieger & Wang Probability Weighting Function Constant 0.823 0.102 0.000 0.622 1.024 Ethiopia 0.055 0.081 0.500 −0.104 0.213 Uganda 0.198 0.093 0.034 0.015 0.381 Order of tasks −0.057 0.065 0.382 −0.186 0.071 Age in years −0.006 0.002 0.012 −0.011 −0.001 Female −0.084 0.050 0.098 −0.183 0.015 Some secondary education 0.053 0.062 0.394 −0.069 0.175 Number in household −0.015 0.013 0.265 −0.040 0.011 †The point estimate for a is 1.60e−28. It is not possible to calculate estimates of the standard error because of the lack of numerical precision at such extreme values. Parameter a is estimated by estimating a non‐linear transform κ ∈ (−∞,+∞), where a = 1/[1 +  exp (κ)]. Then the point estimates and standard errors of a are recovered from the estimates for κ using the ‘delta method’, which requires that derivatives be calculated in the neighbourhood of the point estimate. For certain extreme values of these point estimates, these numerical derivatives become unstable and the estimated standard error unreliable. Open in new tab Table 4 
Maximum Likelihood Estimates of PT Model of Choices With Different Probability Weighting Function Coefficient . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) Tversky & Kahneman Probability Weighting Function: w(p) = p γ/[p γ + (1 − p) γ]1/γ α 0.464 0.036 0.000 0.393 0.535 γ 1.384 0.070 0.000 1.246 1.522 (b) Prelec Probability Weighting Function: w(p) =  exp [−η(− ln pφ)] α 0.504 0.033 0.000 0.439 0.569 η 1.202 0.053 0.000 1.097 1.307 φ 0.963 0.076 0.000 0.814 1.113 (c) Rieger & Wang Probability Weighting Function: w(p) = p + [(3 − 3b)/(a2 − a + 1)][p3 − (a + 1)p2 + ap] α 0.546 0.025 0.000 0.496 0.596 a 0.000 † † † † b 0.775 0.048 0.000 0.680 0.870 (d) Estimates of Risk Aversion Parameter α Using the Rieger & Wang Probability Weighting Function Constant 0.823 0.102 0.000 0.622 1.024 Ethiopia 0.055 0.081 0.500 −0.104 0.213 Uganda 0.198 0.093 0.034 0.015 0.381 Order of tasks −0.057 0.065 0.382 −0.186 0.071 Age in years −0.006 0.002 0.012 −0.011 −0.001 Female −0.084 0.050 0.098 −0.183 0.015 Some secondary education 0.053 0.062 0.394 −0.069 0.175 Number in household −0.015 0.013 0.265 −0.040 0.011 Coefficient . Estimate . Standard Error . p‐value . 95% Confidence Intervals . (a) Tversky & Kahneman Probability Weighting Function: w(p) = p γ/[p γ + (1 − p) γ]1/γ α 0.464 0.036 0.000 0.393 0.535 γ 1.384 0.070 0.000 1.246 1.522 (b) Prelec Probability Weighting Function: w(p) =  exp [−η(− ln pφ)] α 0.504 0.033 0.000 0.439 0.569 η 1.202 0.053 0.000 1.097 1.307 φ 0.963 0.076 0.000 0.814 1.113 (c) Rieger & Wang Probability Weighting Function: w(p) = p + [(3 − 3b)/(a2 − a + 1)][p3 − (a + 1)p2 + ap] α 0.546 0.025 0.000 0.496 0.596 a 0.000 † † † † b 0.775 0.048 0.000 0.680 0.870 (d) Estimates of Risk Aversion Parameter α Using the Rieger & Wang Probability Weighting Function Constant 0.823 0.102 0.000 0.622 1.024 Ethiopia 0.055 0.081 0.500 −0.104 0.213 Uganda 0.198 0.093 0.034 0.015 0.381 Order of tasks −0.057 0.065 0.382 −0.186 0.071 Age in years −0.006 0.002 0.012 −0.011 −0.001 Female −0.084 0.050 0.098 −0.183 0.015 Some secondary education 0.053 0.062 0.394 −0.069 0.175 Number in household −0.015 0.013 0.265 −0.040 0.011 †The point estimate for a is 1.60e−28. It is not possible to calculate estimates of the standard error because of the lack of numerical precision at such extreme values. Parameter a is estimated by estimating a non‐linear transform κ ∈ (−∞,+∞), where a = 1/[1 +  exp (κ)]. Then the point estimates and standard errors of a are recovered from the estimates for κ using the ‘delta method’, which requires that derivatives be calculated in the neighbourhood of the point estimate. For certain extreme values of these point estimates, these numerical derivatives become unstable and the estimated standard error unreliable. Open in new tab We observe from Table 4 and Figure 3 that the implied risk attitudes are mildly sensitive to the use of the two flexible probability weighting functions. The Prelec (1998) function leads to estimates that are mid‐way between those obtained with the Tversky and Kahneman (1992) function and EUT, and the Rieger and Wang (2006) function leads to estimates that are closer to EUT. These differences derive from seemingly small differences in the probability weighting functions shown in Figure 2: the Prelec (1998) function exhibits the greatest under‐weighting for lower probabilities and the Rieger and Wang (2006) function exhibits the greatest under‐weighting for all probabilities. Since it matters for inferences about risk attitudes, how should one select from these two flexible functional forms? The Prelec (1998) function is implied by a series of properties that it is claimed that the function has to satisfy, many of which have been inferred from previous experimental tasks rather than from theoretical considerations. There is nothing wrong with this procedure, apart from the fact that it rests as a logical matter on those prior empirical inferences being valid. The Rieger and Wang (2006) function, on the other hand, is derived as the simplest polynomial to satisfy some theoretically attractive properties, most notably that it be strictly increasing and continuously differentiable on p ∈ [0,1]. Thus it does not depend, for it is a priori validity, on the validity of prior empirical tests, and for our purposes is preferable.11 Figure 4 compares directly the distribution of elicited risk attitudes in our sample under EUT or PT. These distributions are based on predictions from estimated models that include the full set of covariates for each parameter. Thus we use the predictions from the estimates shown in Panel (b) of Table 2 and Panel (d) in Table 4. Although there is a slight increase in estimates of risk aversion under PT, the results are remarkably similar, and the slight non‐linearity of the probability weighting function would not change that conclusion. Fig. 4. Open in new tabDownload slide Risk Attitudes Under EUT or PT Fig. 4. Open in new tabDownload slide Risk Attitudes Under EUT or PT 3.3. Estimates of the Mixture Model Finally, we extend the comparison of the two models to consider the mixture model that allows both to play a role in explaining observed behaviour. We employ the EUT model and the PT model with the Rieger and Wang (2006) probability weighting function. Maximum likelihood estimates are reported in Table 5. Table 5 
Maximum Likelihood Estimates of Mixture Model Coefficient . Estimate . Standard Error . p‐value . 95% Confidence Intervals . α −0.195 0.061 0.001 −0.315 −0.076 a 9.11e−08 † † † † b 4.09e−07 † † † † r 0.796 0.035 0.000 0.727 0.866 πEUT 0.461 0.050 0.000 0.363 0.559 Coefficient . Estimate . Standard Error . p‐value . 95% Confidence Intervals . α −0.195 0.061 0.001 −0.315 −0.076 a 9.11e−08 † † † † b 4.09e−07 † † † † r 0.796 0.035 0.000 0.727 0.866 πEUT 0.461 0.050 0.000 0.363 0.559 † It is not possible to calculate estimates of the standard error of a and b because of the lack of numerical precision at such extreme values. Parameter a is estimated by estimating a non‐linear transform κ ∈ (−∞,+∞), where a = 1/[1 +  exp (κ)]; a similar transform is used for parameter b. Then the point estimates and standard errors of a are recovered from the estimates for κ using the ‘delta method’, which requires that derivatives be calculated in the neighbourhood of the point estimate. For certain extreme values of these point estimates, these numerical derivatives become unstable and the estimated standard error unreliable. Open in new tab Table 5 
Maximum Likelihood Estimates of Mixture Model Coefficient . Estimate . Standard Error . p‐value . 95% Confidence Intervals . α −0.195 0.061 0.001 −0.315 −0.076 a 9.11e−08 † † † † b 4.09e−07 † † † † r 0.796 0.035 0.000 0.727 0.866 πEUT 0.461 0.050 0.000 0.363 0.559 Coefficient . Estimate . Standard Error . p‐value . 95% Confidence Intervals . α −0.195 0.061 0.001 −0.315 −0.076 a 9.11e−08 † † † † b 4.09e−07 † † † † r 0.796 0.035 0.000 0.727 0.866 πEUT 0.461 0.050 0.000 0.363 0.559 † It is not possible to calculate estimates of the standard error of a and b because of the lack of numerical precision at such extreme values. Parameter a is estimated by estimating a non‐linear transform κ ∈ (−∞,+∞), where a = 1/[1 +  exp (κ)]; a similar transform is used for parameter b. Then the point estimates and standard errors of a are recovered from the estimates for κ using the ‘delta method’, which requires that derivatives be calculated in the neighbourhood of the point estimate. For certain extreme values of these point estimates, these numerical derivatives become unstable and the estimated standard error unreliable. Open in new tab The first result is that the estimated probability for the EUT model is 0.461, and that this estimate is significantly different from 0 or 1 (p‐value < 0.001). A test of the null hypothesis that πEUT = 1/2 has a p‐value of 0.44 and the upper and lower bounds of the 95% confidence interval for πEUT are 0.36 and 0.56. Thus we might be inclined to conclude that the weight of the evidence supports PT over EUT by a (quantum) nose, but that would be an invalid inference for reasons explained earlier. Instead, we conclude that the data is consistent with each model playing a roughly equal role as a data generating process. We believe that this finding is of more general significance. For example, Humphrey (2000; p.260) draws the following conclusion from some common consequence tests with subjects from developed countries that resonates well with our findings: The data are not explained by any of the generalised expected utility models which were developed to explain observed violations of expected utility theory in decision‐problems of exactly the type used in this experiment. More worrying, perhaps, is that minor changes in problem representation seemingly impart large changes in choices. It is not surprising, therefore, that any single model is descriptively inadequate. Starmer (1992; p. 829) suggests that individual choice behaviour is ‘more subtle and complex’ than decision theorists have generally conveyed in their models. If so, this may render the induction of theories from sub‐sets of experimental evidence problematic. […] This conclusion depends upon the perceived role of theory. If a single theory should explain as much (as parsimoniously) as possible, the volumes of diverse observed influences on decision‐making behaviour seemingly condemn this task to inevitable failure. If, however, risky choice is recognised as being too complex to be captured by any single theory and that the role of a single theory is to capture a facet of behaviour in a specific context, then it may be necessary to accept that slightly different contexts will invoke additional facets of behaviour and overall explanations of data will require more than one model. Although the behaviour observed in this experiment might, with sufficient ingenuity, be explained by a single model involving a complex probability weighting function, experience suggests that any such function will be limited in its application to other types of decision problem. Our results are also consistent with more elaborate mixture models applied to larger databases of choice under uncertainty by Harrison and Rutström (forthcoming). The second result from Table 5 is that the estimated risk attitudes and shape of the estimated probability weighting function change significantly under PT. In fact, the sample exhibits significant risk‐loving behaviour (a convex utility function) to the extent that it follows the PT data generating process (α = −0.195) but remains risk averse to the extent that it follows the EUT data generating process (r = +0.796). In interpreting this result it is important to emphasise that, as discussed above, this result refers only to the curvature of the utility function. Moreover, since the influence of the convex probability weighting function in Figure 5 will be to partially offset this risk seeking, the net effect on behaviour will be to make it more similar to that of the EUT decision‐makers than the different curvatures of the utility function would imply. Fig. 5. Open in new tabDownload slide Effect of Mixture Model on Probability Weighting Fig. 5. Open in new tabDownload slide Effect of Mixture Model on Probability Weighting This second point can be illustrated by using the parameters of the PT model to infer a risk premium in the traditional sense and, hence, infer a certainty‐equivalent (Levy and Levy, 2002). That certainty‐equivalent can then be compared directly to the standard certainty‐equivalent under EUT. Using the estimates from Panel (a) of Tables 2 and 3, one can show that these certainty‐equivalents only differ by about 6% for most of the lotteries in our experiment. An Appendix shows the results of a simulation of this method using the estimates from Panel (a) and Panel (b) of Tables 2 and 3. If, as these simulation results suggest, the revealed behaviour of EUT decision‐makers is broadly similar to that of PT decision‐makers, then the question arises as to how one might interpret the results of the mixture model analysis in a behavioural sense. We suggest that a natural interpretation is that the mixture model estimates are identifying the different decision‐making processes that are used by different individuals to arrive at these, broadly similar, choices. Although the mixture model applied to our data does not allow us to identify the precise nature of these decision‐making processes, there are several plausible possibilities suggested in the literature. Brandstätter et al. (2002) argue that emotions are important in determining the shape of the probability weighting function. The underweighting of probabilities that we observe, for example, is consistent with pessimism regarding chance events. The mixture model could then be interpreted, for any given and broadly similar behaviour, as sorting out the pessimistic decision‐makers from those who are neither pessimistic nor optimistic. The heterogeneity in PT decision‐makers’ utility functions might then be subsequently interpreted as reflecting ‘residual’ and different influences on risk‐attitude. For example, against the background of a convex probability weighting function, PT decision‐maker A may be influenced by a factor x which is reflected in a convex utility function and still behave in the same manner as PT decision‐maker B who is influenced by a different factor y which is reflected in a concave utility function. The only requirement is that PT decision‐maker A’s convex utility function does not offset the convexity of the probability weighting function to the extent which would causes their behaviour to deviate from PT decision‐maker B ’s. This example also suggests that it is perhaps unsurprising that PT decision‐makers are more heterogeneous in their utility functions than EUT decision‐makers. The only difference between our PT specification and EUT specification is that the former involves an additional process which is captured by the probability weighting function. For given behaviour, this allows heterogeneity in utility for PT decision‐makers which is simply not possible for EUT decision‐makers. Indeed, PT allows for four possible effects on risk attitudes from these two distinct processes: agents can be more or less risk averse because of the shape of their utility function, ceteris paribus their decision weights, and they can be more or less risk averse because of the shape of their probability weighting function, ceteris paribus their utility function.12 Thus we have some basis for believing that the heterogeneity in risk attitudes we see with our PT specification should be a general outcome. The effect of allowing for the mixture model estimates on the probability weighting function are illustrated in Figure 5. The panel on the left shows the estimated function when PT was assumed to be the sole data generating process and ‘had to’ explain 100% of these data.13 The panel on the right of Figure 5 shows the estimated function when PT only has to account for 54% of these data and EUT is allowed to explain the other 46% of these data. It exhibits the same qualitative shape as the function estimated conditional on PT being the sole data generating process but with a marked increase in the underweighting of probabilities. These results also force one to pay attention to the choice of parametric models for utility and probability weighting. The Reiger and Wang (2006) function is actually ‘well‐behaved’ with the parameter values in Table 5: even though it has a proximately flat region for probabilities between 0.2 and 0.4, it is strictly increasing, weakly concave for the lowest probabilities and then sharply convex for most of the probabilities used in the lotteries. We also extended the mixture model to include binary dummies for Ethiopia and Uganda for the πEUT parameter. The results indicate that there is least support for the EUT model on average in India (0.35) than in Ethiopia (0.57) or Uganda (0.51). We can reject the hypothesis that the EUT and PT models have equal explanatory power in India (p‐value = 0.014) but not in Ethiopia (p‐value = 0.38) or Uganda (p‐value = 0.85). However, even in the case of India, the 95% confidence intervals for the support of the EUT model are between 0.23 and 0.47. The changes in results under PT are striking as we move from the original specification to the mixture model. Consider the underweighting of probabilities. Underweighting means that when subjects are told that some outcome has a 50% chance of occurring that they behave as if it has much less chance of occurring. This appears to be true for all of the probabilities in our lotteries, which range from 1/4 to 3/4. One possible explanation for this observation is that our mixture model estimates reflect the pessimism of PT subjects. Our subjects might behave pessimistically because of the general economic conditions prevailing at the time of the experiments. The regions we visited in India and Ethiopia were experiencing droughts. If this served to engender a general pessimism about uncertain events, this might account for our results. There is some support for this interpretation from psychological studies of the same phenomenon. Hertwig et al. (2004) review evidence from a range of psychology experiments, some with real rewards and without deception (Barron and Erev, 2003), that provide striking evidence of underweighting at low probabilities. They argue that this is characteristic of tasks that involve ‘decisions from experience’, where the probabilities in question derive from the subject’s previous experience with comparable events.14 They also argue that evidence of overweighting at low probabilities derives from tasks that involve ‘decision from description’, where the task description itself provides the probabilities. In turn, they argue that the underweighting behaviour might derive from reliance on relatively small samples of information and the overweighting of recently sampled information. Our experiments do not allow us to discriminate between these lines of argument but the significant underweighting we observe does suggest that our subjects viewed the probabilities in the task description in terms of their experience, perhaps from naturally‐occurring risk in their environment. When viewed in the light of the findings of existing studies, providing an interpretation of our evidence of the underweighting of probabilities brings into focus the complex issue of the influence of experience on decision‐making. List (2004) argues that evidence from field experiments shows that decisions taken by experienced decision‐makers can be organised by standard neoclassical economic theory, specifically, Hicksian consumer preferences. On the other hand, he argues that organisation of the data he observes from inexperienced decision‐makers requires a reference‐dependent utility function of the type employed in prospect theory. List (2004) studies decisions over whether to trade mugs for chocolate bars; if this evidence was applied to the risky lottery choices which we study, then it would imply that our prospect theory decision‐makers, for whom we observe the underweighting of probabilities, may simply be less experienced in risky decision‐making than the EU decision‐makers. This implication would be at odds with the previous interpretation suggested by Hertwig et al. (2004) and is not something we can resolve satisfactorily with the present data. It does point to the value of an important extension of our design to build in natural controls for experience with the type of decision being studied, as stressed in the general literature on field experiments (Harrison and List, 2004) as well as specific field experiments examining risk attitudes (Bohm and Lind, 1993; Harrison, List and Towe, 2007). Another possible explanation for underweighting is that it reflects subjects not believing that the random process was actually fair, despite the fact that we used no deception whatsoever, used transparent physical randomisation devices and saw no evidence that our subjects were concerned with being cheated. However, if the subjects believe that the experimenter had a way of making the outcome actually go against them, then one might expect to see behaviour of this kind. Such concerns are always a part of any experiment, of course, and are the reason that many experimental economists use physical randomising devices rather than rely on computers whenever possible. But it is distinctly possible that cultural beliefs about certain physical randomising devices, and experiences with being ‘cheated’ in such interactions, are different in developing countries.15 Now consider the qualitative change in risk attitudes under the PT model when one moves from assuming it to be the only data‐generating process to being one of two possible data‐generating process. One might ask if this result is an artefact of the use of a mixture model. Intuitively, if EUT can explain about 50% of the sample data and, if all of these subjects happen to be risk averse, one might ask whether the mixture model simply assigns the risk‐loving subjects to the PT model since there is no alternative model for them to be assigned to. Thus what appears to be a change in risk attitudes under PT is, according to this view, just due to the risk lovers being ‘residually’ assigned to the PT model. Although difficult to state formally, this is a good question, which goes to the heart of the use of statistical models to simultaneously identify parameters and alternative models. This question is in effect a comment on the potential dangers of assuming that the EUT decision‐maker and the PT decision‐maker are each homogeneous: they can have different risk attitudes in the specification estimated in Table 5, but if you are an EUT decision‐maker you have to have the same risk attitude as every other EUT decision‐maker. A complete response to this question would require that one include individual characteristics of the respondents in the parameters of the mixture model, in order to identify any heterogeneity within the subset of EUT or PT decision‐makers. We do not have sufficient degrees of freedom to do this for the whole model but we make the following two observations. First, we can identify heterogeneity within the group of PT decision‐makers with respect to risk aversion coefficient α. The average value for this coefficient is −0.16, consistent with the estimate from the homogeneous‐PT model in Table 5. But these estimates show a significant variation in the risk attitudes within the subset of PT decision‐makers. Those in India are the most risk‐loving on average (−0.32), with those in Uganda being risk‐neutral on average (0.03) and those in Ethiopia being in‐between and risk‐loving (−0.16).16 The presence of some secondary education is associated with a significantly higher aversion to risk at the margin (+0.14, with a p‐value of 0.045) and there is a dramatic effect of age. Every additional year lowers risk aversion by 0.033 and this a marginal effect that is statistically significant (p‐value = 0.002). The effect of age can be seen in Figure 6, which stratifies the predicted risk aversion coefficient α under PT for each subject. Younger subjects tend to be risk averse under PT and older subjects tend to be risk loving under PT. Whether or not the same effect is observed under EUT, Figure 6 dramatically illustrates that there is considerable variation in risk attitudes within the subset of PT decision‐makers. Fig. 6. Open in new tabDownload slide Effect of Age on Estimated Risk Aversion of PT Decision‐Makers Fig. 6. Open in new tabDownload slide Effect of Age on Estimated Risk Aversion of PT Decision‐Makers By way of contrast, Figure 7 shows a comparable display of the association of age on risk attitudes within the subset of EUT decision‐makers. Although there is a similarly declining marginal effect on risk attitudes (−0.005 per year of age), the effect is not statistically significant (p‐value = 0.56). Thus we see that there is considerable sensitivity of the demographic pattern of risk attitudes to the type of choice theory that best explains behaviour. Thus reliable policy inferences about age and risk attitude should condition on the heterogeneity of the type of decision‐making model being used as well as the observable characteristic age.17 Fig. 7. Open in new tabDownload slide Effect of Age on Estimated Risk Aversion of EUT Decision‐Makers Fig. 7. Open in new tabDownload slide Effect of Age on Estimated Risk Aversion of EUT Decision‐Makers Second, as discussed earlier, the estimates of certainty equivalents for the lotteries used in our experiments show that our subjects evaluated the lotteries broadly similarly in a quantitative sense, irrespective of whether they were assumed to be PT or EUT decision‐makers. The fact that the PT certainty equivalents were all lower, albeit marginally so, than the EUT certainty equivalents implies that the latter group made riskier choices than the former group. This implication follows from the mixture model sorting subjects into two groups which correspond to the two latent processes we assume, and according to the heterogeneity or homogeneity of their utility functions. That is, given the significant underweighting manifest in w(·), the PT utility function necessarily exhibits greater heterogeneity if the theory as a whole is to explain the broadly similar evaluations of lotteries which EUT requires only a single utility function to organise. In this respect the insight of the mixture model approach is that the estimates of the PT risk aversion parameter α are substantially more heterogeneous than the quantitative evaluations of the lotteries might suggest. Of course, there are many extensions of our approach possible before one can draw definitive conclusions. More data always helps but for statistical inferences based on mixture models it is more than normally true since one remains ‘agnostic’ about which data‐generating process dominates. This need for more data would only become more severe if one admitted more than two data‐generating processes.18 In addition, we would want to examine alternatives to EUT that have some theoretically attractive properties in comparison to the separable PT considered here. In this vein, it might be useful to examine some of the popular stochastic error specifications that have been proposed. We would also be particularly interested in extensions to explicitly consider outcomes frames as losses, to assess the effect of loss aversion on choices under uncertainty among the rural poor in developing countries. Finally, we would encourage examination of risk‐taking behaviour in the broader economics context appropriate for the country, village and individual: there is considerable evidence and theory to suggest that ‘background risk’ can influence risk‐taking behaviour over ‘foreground risk’ (Harrison, List and Towe, 2007). Examples of this type of background risk include the effects of weather conditions, violence, corruption, reliability of government infrastructure, the efficacy of risk‐sharing and risk‐pooling arrangements at a social level and even norms for responses to individuals experiencing good luck or bad luck. We would expect there to be unusually wide variations in the extent of background risk and its composition in developing countries. One objective of our foreground risk task being standardised was to provide a controlled baseline against which these influences can be measured in future research. 4. Concluding Remarks Our results show how important it is to be clear about the theoretical and statistical assumptions underlying inferences from observed data. For example, our results point to the dangers of drawing inferences about risk attitudes when one incorrectly assumes that behaviour is generated by only one data‐generating process. When we do that and assume PT or EUT, we infer risk aversion and relatively little heterogeneity of risk attitudes. Moreover, our inferences about the degree of risk aversion do not appear to be affected by which of the two models we adopt. But when we allow some of the data to be explained by EUT and some to be explained by PT, we infer relatively homogeneous risk aversion for the subjects following EUT and considerable heterogeneity of risk attitudes for the subjects following PT. In fact, the heterogeneity of risk attitudes for the subjects following PT spans a significant fraction of risk‐loving, risk‐neutral and risk‐averse individual behaviour. The average subject following PT exhibits risk‐loving behaviour but the key finding here is heterogeneity of risk attitudes. The differences obtained when one allows multiple data‐generating processes is a general point that is true for developed countries as well as developing countries. It may be more significant in developing countries where one might expect more noise in the data due to the relative unfamiliarity of the tasks and the adoption of a wider range of heuristics to make decisions. Substantively, we conclude that there is roughly equal support for the two major models of choice under uncertainty considered here. It is not the case that EUT or PT wins but that the data is consistent with each playing some roughly equal role, as in Harrison and Rutström (forthcoming) for comparable laboratory tasks with university students. Thus, substituting PT for EUT would be tantamount to replacing one ‘half wrong’ assumption with another. This conclusion implies that policies should not be designed under the assumption that one or other theory explains all behaviour. Policy‐makers are therefore faced with the substantial challenge of examining the sensitivity of the predicted impact of policy interventions to the replacement of single key assumptions with a probability distribution over competing assumptions, as suggested by estimates emerging from mixture model analyses based on experimental data. There is a clear a rationale for closer collaboration between experimentalists and policy‐makers in developing countries than has hitherto been the case. Footnotes 1 " Our experiments are ‘artefactual field experiments’ in the terminology of Harrison and List (2004). That is, they involve taking procedures from the laboratory and applying them in the field. The use of field experiments in developing countries has grown dramatically in recent years and is reviewed by Cardenas and Carpenter (2008) and Duflo (2006). 2 " To use the specific example from task 1 from Uganda, the non‐zero prizes were Ush 5,000 and Ush 2,000, which translated into 286 US cents and 114 US cents, respectively. This is as close as we could come to 250 cents and 100 cents with prize values that were rounded in local currency units. The validity of these lotteries as tests of EUT does not depend on the local currency units exactly matching the values shown in Table 1 but rather on the local currency equivalent of 250 cents being the same across tasks for the same subject. 3 " Starmer (2000) provides an excellent review of the major alternatives. He concludes that if EUT is to be replaced as the dominant theory of risky choice in economics, the evidence points to Tversky and Kahneman’s (1992) PT as being the best candidate. Although we reject the notion of completely replacing EUT, as explained later, we accept his view that PT should be viewed as the strongest contender. Since we restrict attention to the gain domain, and typically only have two final outcomes in our lotteries, our implementation of PT is virtually identical to the rank‐dependent utility model of Quiggin (1982). 4 " Some take the view that EUT requires that utility be defined over terminal wealth and not income. This is false, as discussed by Cox and Sadiraj (2006) and Rubinstein (2002). 5 " An extension would be to consider parametric reference points that differ from zero and evaluate the structural model implied. This approach is illustrated by Harrison and Rutström (2008; §3.2) in the context of laboratory experiments framed as gains and losses relative to some exogenous endowment. They demonstrate that assuming that subjects use a positive reference point, rather than the one presented in the frame presented by the experimenter, can significantly affect estimates of core structural parameters, such as the extent of loss aversion. 6 " One could extend this analysis to include a behavioural specification of errors conditional on each theoretical model. There are several alternative specifications: see Harless and Camerer (1994), Hey and Orme (1994) and Loomes and Sugden (1995) for the first wave of empirical studies including some formal stochastic specification in the version of EUT tested. There are several species of ‘errors’ in use, reviewed by Loomes and Sugden (1995). Some place the error at the final choice between one lottery or the other after the subject has decided deterministically which one has the higher expected utility; some place the error earlier, on the comparison of preferences leading to the choice; and some place the error even earlier, on the determination of the expected utility of each lottery. 7 " Clustering commonly arises in national field surveys from the fact that physically proximate households are often sampled to save time and money but it can also arise from more homely sampling procedures. For example, Williams (2000; p. 645) notes that it could arise from dental studies that ‘collect data on each tooth surface for each of several teeth from a set of patients’ or ‘repeated measurements or recurrent events observed on the same person.’ The procedures for allowing for clustering allow heteroscedasticity between and within clusters, as well as autocorrelation within clusters. They are closely related to the ‘generalised estimating equations’ approach to panel estimation in epidemiology (Liang and Zeger, 1986) and generalise the ‘robust standard errors’ approach popular in econometrics (Rogers, 1993). Wooldridge (2003) reviews some issues in the use of clustering for panel effects, in particular noting that significant inferential problems may arise with small numbers of panels. 8 " For example, Holt and Laury (2002) and Harrison, Johnson, McInnes and Rutström (2005) for college students in the US and Harrison, Lau and Rutström (2007) and Harrison, Lau, Rutström and Sullivan (2005) for the adult population in Denmark. 9 " The bottom axis of each panel in Figure 2 shows the probability that was presented to the subject in a task and the vertical axis shows the estimated weighted probability that the subject used. Overweighting means that the subject has a w(p) estimate that is greater than the p it corresponds to; underweighting is the reverse situation in which w(p) < p. 10 " In this case b must not exceed 1 + 1/3[(a2 − a + 1)/(1/2 + |a − 1/2|)] or the function becomes non‐increasing (Marc Oliver Rieger; personal communication). 11 " Underlying this perspective is some agnosticism with respect to assertions that previous tests of EUT provide clear conclusions that are independent of complications. From different perspectives, Harrison (1994), Humphrey (2000) and Harrison and Rutström (forthcoming) illustrate our concerns. 12 " If one also allowed for loss aversion, in a design that allowed for losses, there would be a third process that could additionally allow for two additional effects on risk attitudes. 13 " The panel on the left of Figure 5 is the same function shown on the far right panel of Figure 2. 14 " This experience can also come from simply providing feedback to the subject about the mechanism used to make random draws. We did not provide such feedback in our experiments. 15 " The outcomes in our experiment were substantial and, in terms of PPP, above those typically offered in experiments in developed countries. In experiments similar to ours but conducted in developed countries, large incentives might be considered an attractive design feature: they motivate careful consideration of decisions and may offset some ‘induced’ risk seeking stemming from subjects ‘gambling with the house money’. In our experiments large incentives may have induced the underweighting of probabilities if subjects believed that the opportunity to gamble with such large sums of ‘house money’ was too good to be true. Such belief may have undermined confidence in the authenticity of the random devices employed to resolve risk. This possibility is likely to be mitigated in developed countries where experimental subjects, who are usually undergraduate students, are immersed in an environment where being paid non‐trivial sums to complete simple experimental tasks for research purposes is likely to be viewed less suspiciously. 16 " These estimates are average effects, including variations in age and sex, for example, in each country. 17 " For example, one of the most important risky choices that our subjects make in practice is whether or not to take up the production of lucrative but input‐intensive cash crops, such as tomatoes and cabbages. Agricultural extension workers told us that it was primarily young men who gamble all on such crops, for a season or two, often to finance their eventual migration to a town or city. Our findings are certainly inconsistent with the presumption that this behaviour is due to risk‐loving tendencies among the young and may have broader implications for micro‐finance schemes that consider financing the growing of lucrative, input‐intensive crops and for the targeting of agricultural extension. 18 " Intuitively, the need for data is relatively less severe if the alternative models have sharply different predictions and relatively more severe if the alternative models have similar predictions. We therefore doubt that one can easily discriminate between the competitors to EUT without significantly more controlled data, since they have many ‘family similarities’ (Starmer, 2000). References Barr , A. and Genicot , G. ( 2008 ). ‘Risk sharing, commitment, and information: an experimental analysis’ , Journal of the European Economic Association , vol. 6 ( 6 ), pp. 1151 – 85 . Google Scholar Crossref Search ADS WorldCat Barr , A. and Packard , T. ( 2002 ). ‘Revealed preference and self insurance: can we learn from the self employed in Chile’ , Policy Research Working Paper No. 2754, World Bank, Washington DC. Barr , A. and Packard , T. G. ( 2005 ). ‘Seeking solutions to vulnerability in old age: preferences, constraints, and alternatives for coverage under Peru’s pension system’ , Working Paper No. 2005‐05, Centre for the Study of African Economies, Department of Economics, University of Oxford. Barron , G. and Erev , I. ( 2003 ). ‘Small feedback‐based decisions and their limited correspondence to description‐based decisions’ , Journal of Behavioural Decision Making , vol. 16 , pp. 215 – 33 . Google Scholar Crossref Search ADS WorldCat Binswanger , H. P. ( 1980 ). ‘Attitudes toward risk: experimental measurement in rural India’ , American Journal of Agricultural Economics , vol. 62 (August) pp. 395 – 407 . Google Scholar Crossref Search ADS WorldCat Binswanger , H. P. ( 1981 ). ‘Attitudes toward risk: theoretical implications of an experiment in rural India’ , Economic Journal , vol. 91 (December) pp. 867 – 90 . Google Scholar Crossref Search ADS WorldCat Binswanger , H. P. ( 1982 ). ‘Empirical estimation and use of risk preferences: discussion’ , American Journal of Agricultural Economics , vol. 64 (May) pp. 391 – 3 . Google Scholar Crossref Search ADS WorldCat Bohm , P. and Lind , H. ( 1993 ). ‘Preference reversal, real‐world lotteries, and lottery‐interested subjects’ , Journal of Economic Behavior and Organization , vol. 22 , pp. 327 – 48 . Google Scholar Crossref Search ADS WorldCat Botelho , A. , Harrison , G. W., Pinto , L. M. C., Rutström , E. E. and Veiga , P. ( 2005 ). ‘Discounting in developing countries: a pilot experiment in Timor‐Leste’ , Working Paper No. 05‐31, Department of Economics, College of Business Administration, University of Central Florida. Brandstätter , E. , Kühberger , A. and Schneider , F. ( 2002 ). ‘A cognitive‐emotional account of the shape of the probability weighting function’ , Journal of Behavioral Decision Making , vol. 15 , pp. 19 – 100 . Google Scholar Crossref Search ADS WorldCat Camerer , C. F. ( 1989 ). ‘An experimental test of several generalized utility theories’ , Journal of Risk and Uncertainty , vol. 2 , pp. 61 – 104 . Google Scholar Crossref Search ADS WorldCat Camerer , C. F. ( 1995 ). ‘Individual Decision Making’, in ( J.H. Kagel and A.E. Roth, eds.), The Handbook of Experimental Economics , pp. 587 – 704 , Princeton: Princeton University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Camerer , C. ( 1998 ). ‘Bounded rationality in individual decision making’ , Experimental Economics , vol. 1 , pp. 163 – 83 . Google Scholar Crossref Search ADS WorldCat Camerer , C. and Ho T. ( 1994 ). ‘Violations of the betweenness axiom and nonlinearity in probabilities’ , Journal of Risk and Uncertainty , vol. 8 , pp. 167 – 96 . Google Scholar Crossref Search ADS WorldCat Cardenas , J. C. and Carpenter , J. P. ( 2008 ). ‘Behavioural development economics: lessons from field labs in the developing world’ , Journal of Development Studies , vol. 44 ( 3 ), pp. 311 – 38 . Google Scholar Crossref Search ADS WorldCat Collier , P. and Gunning , J. W. ( 1999 ). ‘Explaining African economic performance’ , Journal of Economic Literature , vol. 37 , pp. 64 – 111 . Google Scholar Crossref Search ADS WorldCat Cox , J. C. and Sadiraj , V. ( 2006 ). ‘Small‐ and large‐stakes risk aversion: implications of concavity calibration for decision theory’ , Games & Economic Behavior , vol. 56 ( 1 ), pp. 45 – 60 . Google Scholar Crossref Search ADS WorldCat Deaton , A. ( 1997 ). The Analysis of Household Surveys: A Microeconometric Approach to Development Policy , Baltimore: Johns Hopkins University Press and the World Bank . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Duflo , E. ( 2006 ). ‘Field experiments in development economics’, in ( R. Blundell, W. Newey, and T. Persson, eds.), Advances in Economics and Econometrics , vol. 2 , pp. 322 – 48 , New York: Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Fafchamps , M. ( 2004 ). Rural Poverty, Risk and Development , Northampton, MA : Edward Elgar. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Gonzalez , R. and Wu , G. ( 1999 ). ‘On the shape of the probability weighting function’ , Cognitive Psychology , vol. 38 , pp. 129 – 66 . Google Scholar Crossref Search ADS PubMed WorldCat Harless , D. W. and Camerer , C. F. ( 1994 ). ‘The predictive utility of generalized expected utility theories’ , Econometrica , vol. 62 , pp. 1251 – 89 . Google Scholar Crossref Search ADS WorldCat Harrison , G. W. ( 1994 ). ‘Expected utility theory and the experimentalists’ , Empirical Economics , vol. 19 ( 2 ), pp. 223 – 53 . Google Scholar Crossref Search ADS WorldCat Harrison , G. W. , Johnson , E., McInnes , M. M. and Rutström , E. E. ( 2005 ). ‘Risk aversion and incentive effects: comment’ , American Economic Review , vol. 95 ( 3 ), pp. 897 – 901 . Google Scholar Crossref Search ADS WorldCat Harrison , G. W. , Lau , M. I. and Rutström , E. E. ( 2007 ). ‘Estimating risk attitudes in Denmark: a field experiment’ , Scandinavian Journal of Economics , vol. 109 ( 2 ), pp. 341 – 68 . Google Scholar Crossref Search ADS WorldCat Harrison , G. W. , Lau , M. I., Rutström , E. E. and Sullivan , M. B. ( 2005 ). ‘Eliciting risk and time preferences using field experiments: some methodological issues’, in (  J. Carpenter, G.W. Harrison and J.A. List, eds.), Field Experiments in Economics, Research in Experimental Economics , vol. 10, pp. 125 – 218 , Greenwich, CT: JAI Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Harrison , G. W. and List , J. A. ( 2004 ). ‘Field experiments’ , Journal of Economic Literature , vol. 42 ( 4 ), pp. 1013 – 59 . Google Scholar Crossref Search ADS WorldCat Harrison , G. W. , List , J. A. and Towe , C. ( 2007 ). ‘Naturally occurring preferences and exogenous laboratory experiments: a case study of risk aversion’ , Econometrica , vol. 75 ( 2 ), pp. 433 – 58 . Google Scholar Crossref Search ADS WorldCat Harrison , G. W. and Rutström , E. E. ( 2008 ). ‘Risk aversion in the laboratory’, in ( J. Cox and G.W. Harrison, eds.), Risk Aversion in Experiments, Research in Experimental Economics , vol. 12, pp. 41 – 196 , Bingley: Emerald . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Harrison , G. W. and Rutström , E. E. (forthcoming). ‘Representative agents in lottery choice experiments: one wedding and a decent funeral’ , Experimental Economics . OpenURL Placeholder Text WorldCat Hertwig , R. , Barron , G., Weber , E. U. and Erev , I. ( 2004 ). ‘Decisions from experience and the effect of rare events in risky choice’ , Psychological Science , vol. 15 ( 8 ), pp. 534 – 9 . Google Scholar Crossref Search ADS PubMed WorldCat Heston , A. , Summers , R. and Aten , B. ( 2002 ). Penn World Table Version 6.1 , Center for International Comparisons at the University of Pennsylvania (CICUP) , data accessed at http://pwt.econ.upenn.edu/php_site/pwt61_form.php. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Hey , J. D. and Orme , C. ( 1994 ). ‘Investigating generalizations of expected utility theory using experimental data’ , Econometrica , vol. 62 ( 6 ), pp. 1291 – 326 . Google Scholar Crossref Search ADS WorldCat Holt , C. A. and Laury , S. K. ( 2002 ). ‘Risk aversion and incentive effects’ , American Economic Review , vol. 92 ( 5 ), pp. 1644 – 55 . Google Scholar Crossref Search ADS WorldCat Humphrey , S. J. ( 2000 ). ‘The common consequence effect: testing a unified explanation of recent mixed evidence’ , Journal of Economic Behaviour & Organization , vol. 41 , pp. 239 – 62 . Google Scholar Crossref Search ADS WorldCat Humphrey , S. J. and Verschoor , A. ( 2004a ). ‘Decision‐making under risk among small farmers in East Uganda’ , Journal of African Economies , vol. 13 ( 1 ), pp. 44 – 101 . Google Scholar Crossref Search ADS WorldCat Humphrey , S. J. and Verschoor , A. ( 2004b ). ‘The probability weighting function: experimental evidence from Uganda, India and Ethiopia’ , Economics Letters , vol. 84 , pp. 419 – 25 . Google Scholar Crossref Search ADS WorldCat Kahneman , D. and Tversky , A. ( 1979 ). ‘Prospect theory: an analysis of decision under risk’ , Econometrica , vol. 47 , pp. 263 – 91 . Google Scholar Crossref Search ADS WorldCat Levy , H. and Levy , M. ( 2002 ). ‘Arrow‐Pratt risk aversion, risk premium and decision weights’ , Journal of Risk & Uncertainty , vol. 25 ( 3 ), pp. 265 – 90 . Google Scholar Crossref Search ADS WorldCat Liang , K‐Y. and Zeger , S.L. ( 1986 ). ‘Longitudinal data analysis using generalized linear models’ , Biometrika , vol. 73 , pp. 13 – 22 . Google Scholar Crossref Search ADS WorldCat List , J. A. ( 2004 ). ‘Neoclassical theory versus prospect theory: evidence from the marketplace’ , Econometrica , vol. 72 ( 2 ), pp. 615 – 25 . Google Scholar Crossref Search ADS WorldCat Loomes , G. and Sugden , R. ( 1995 ). ‘Incorporating a stochastic element into decision theories’ , European Economic Review , vol. 39 , pp. 641 – 8 . Google Scholar Crossref Search ADS WorldCat Mosley , P. and Verschoor , A. ( 2005 ). ‘Risk attitudes and the ‘‘vicious circle of poverty’’ ’ , European Journal of Development Research , vol. 17 ( 1 ), pp. 59 – 88 . Google Scholar Crossref Search ADS WorldCat Prelec , D. ( 1998 ). ‘The probability weighting function’ , Econometrica , vol. 6 , pp. 497 – 527 . Google Scholar Crossref Search ADS WorldCat Quiggin , J. ( 1982 ). ‘A theory of anticipated utility’ , Journal of Economic Behaviour & Organization , vol. 3 ( 4 ), pp. 323 – 43 . Google Scholar Crossref Search ADS WorldCat Rieger , M. O. and Wang , M. ( 2006 ). ‘Cumulative prospect theory and the St. Petersburg paradox’ , Economic Theory , vol. 28 , pp. 665 – 79 . Google Scholar Crossref Search ADS WorldCat Rogers , W. H. ( 1993 ). ‘Regression standard errors in clustered samples’ , Stata Technical Bulletin , vol. 13 , pp. 19 – 23 . OpenURL Placeholder Text WorldCat Rubinstein , A. ( 2002 ). ‘Comments on the risk and time preferences in economics’, mimeo , Department of Economics, Princeton University . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Starmer , C. ( 1992 ). ‘Testing new theories of choice under uncertainty using the common consequence effect’ , Review of Economic Studies , vol. 59 , pp. 813 – 30 . Google Scholar Crossref Search ADS WorldCat Starmer , C. ( 2000 ). ‘Developments in non‐expected utility theory developments in non‐expected utility theory: the hunt for a descriptive theory of choice under risk’ , Journal of Economic Literature , vol. 38 (June) pp. 332 – 82 . Google Scholar Crossref Search ADS WorldCat Tversky , A. and Kahneman , D. ( 1992 ). ‘Advances in prospect theory: cumulative representation of uncertainty’ , Journal of Risk and Uncertainty , vol. 5 , pp. 297 – 323 . Google Scholar Crossref Search ADS WorldCat Williams , R. L. ( 2000 ). ‘A note on robust variance estimation for cluster‐correlated data’ , Biometrics , vol. 56 (June) pp. 645 – 6 . Google Scholar Crossref Search ADS PubMed WorldCat Wooldridge , J. ( 2003 ). ‘Cluster‐sample methods in applied econometrics’ , American Economic Review (Papers & Proceedings) , vol. 93 (May), pp. 133 – 8 . Google Scholar Crossref Search ADS WorldCat Yaari , M. E. ( 1987 ). ‘The dual theory of choice under risk’ , Econometrica , vol. 55 ( 1 ), pp. 95 – 115 . Google Scholar Crossref Search ADS WorldCat Appendix Appendix A: Measures of Risk Premia Levy and Levy (2002) show that Pratt’s measure of the risk premium is given by the ρ which solves Σ(pi)u(xi) = u[Σw(pi)xi − ρ]. We can calculate this measure for the lotteries used in our experiment by implementing this formula with the CRRA utility function and the Tversky and Kahneman (1992) probability weighting function we assume. In this case the estimates of α and γ from Table 3 are used (columns A and B). We do the same on the assumption of EUT but in this case impose the restriction γ = 1 (column B) and use the estimate of r (column A) from Table 2. We can then add these calculated risk premia to the expected value of each lottery to provide a certainty equivalent value for that lottery, under the respective assumptions of EUT and PT (column D). Expressing the difference between these two certainty equivalents (column E) as a proportion of the EUT certainty equivalent provides a measure of the evaluation of each lottery by our PT decision makers in relation to our EUT decision makers (column F). We restrict this exercise to two outcome lotteries and report the results in the table below. Pratt’s Measure of Risk Premium Lottery (Table 1) . A . B . C . D . E . F . α or r . γ . ρ . pixi+ρ . (pixi+ρ)EUT−(pixi+ρ)PT . . 2A 0.464 1.384 33.29 95.79 0.536 1 49.91 112.41 16.62 14.8% 3A 0.464 1.384 38.23 225.73 0.536 1 53.29 240.79 15.07 6.3% 4A 0.464 1.384 123.70 398.70 0.536 1 151.53 426.53 27.82 6.5% 7A 0.464 1.384 66.57 191.57 0.536 1 99.79 224.78 33.21 14.8% 2B 0.464 1.384 22.53 72.54 0.536 1 27.55 77.55 5.01 6.5% 3B 0.464 1.384 7.83 182.83 0.536 1 9.13 184.13 1.3 0.7% 7B 0.464 1.384 22.95 135.45 0.536 1 31.84 144.34 8.89 6.2% Lottery (Table 1) . A . B . C . D . E . F . α or r . γ . ρ . pixi+ρ . (pixi+ρ)EUT−(pixi+ρ)PT . . 2A 0.464 1.384 33.29 95.79 0.536 1 49.91 112.41 16.62 14.8% 3A 0.464 1.384 38.23 225.73 0.536 1 53.29 240.79 15.07 6.3% 4A 0.464 1.384 123.70 398.70 0.536 1 151.53 426.53 27.82 6.5% 7A 0.464 1.384 66.57 191.57 0.536 1 99.79 224.78 33.21 14.8% 2B 0.464 1.384 22.53 72.54 0.536 1 27.55 77.55 5.01 6.5% 3B 0.464 1.384 7.83 182.83 0.536 1 9.13 184.13 1.3 0.7% 7B 0.464 1.384 22.95 135.45 0.536 1 31.84 144.34 8.89 6.2% Open in new tab Pratt’s Measure of Risk Premium Lottery (Table 1) . A . B . C . D . E . F . α or r . γ . ρ . pixi+ρ . (pixi+ρ)EUT−(pixi+ρ)PT . . 2A 0.464 1.384 33.29 95.79 0.536 1 49.91 112.41 16.62 14.8% 3A 0.464 1.384 38.23 225.73 0.536 1 53.29 240.79 15.07 6.3% 4A 0.464 1.384 123.70 398.70 0.536 1 151.53 426.53 27.82 6.5% 7A 0.464 1.384 66.57 191.57 0.536 1 99.79 224.78 33.21 14.8% 2B 0.464 1.384 22.53 72.54 0.536 1 27.55 77.55 5.01 6.5% 3B 0.464 1.384 7.83 182.83 0.536 1 9.13 184.13 1.3 0.7% 7B 0.464 1.384 22.95 135.45 0.536 1 31.84 144.34 8.89 6.2% Lottery (Table 1) . A . B . C . D . E . F . α or r . γ . ρ . pixi+ρ . (pixi+ρ)EUT−(pixi+ρ)PT . . 2A 0.464 1.384 33.29 95.79 0.536 1 49.91 112.41 16.62 14.8% 3A 0.464 1.384 38.23 225.73 0.536 1 53.29 240.79 15.07 6.3% 4A 0.464 1.384 123.70 398.70 0.536 1 151.53 426.53 27.82 6.5% 7A 0.464 1.384 66.57 191.57 0.536 1 99.79 224.78 33.21 14.8% 2B 0.464 1.384 22.53 72.54 0.536 1 27.55 77.55 5.01 6.5% 3B 0.464 1.384 7.83 182.83 0.536 1 9.13 184.13 1.3 0.7% 7B 0.464 1.384 22.95 135.45 0.536 1 31.84 144.34 8.89 6.2% Open in new tab Author notes " We are grateful to the UK Department for International Development (award no. R7617) and to the US National Science Foundation (NSF/HSD 0527675 and NSF/SES 0616746) for financial support. Supporting data and statistical code are stored in the ExLab Digital Library at http://exlab.bus.ucf.edu. We are grateful to Robin Cubitt, Oliver Morrissey and two referees for comments. © The Author(s). Journal compilation © Royal Economic Society 2009

Garcia‐Milà,, Teresa;Marcet,, Albert;Ventura,, Eva

doi: 10.1111/j.1468-0297.2009.02333.xpmid: N/A

Abstract We evaluate the effect on welfare of shifting the burden of capital income taxes to labour taxes in a dynamic equilibrium model with heterogeneous agents and constant tax rates. We calibrate and simulate the economy; we find that lowering capital taxes has two effects: it increases efficiency in terms of aggregate production and it redistributes wealth in favour of those agents with a low wage/wealth ratio. When the parameters of the model are calibrated to match the distribution of income in terms of the wage/wealth ratio, the redistributive effect dominates, and agents with a high wage/wealth ratio would experience a large loss in utility if capital income taxes were eliminated. A large part of the literature on dynamic taxation in equilibrium models with rational expectations has reached the conclusion that capital taxes should be abolished or, at the very least, severely reduced. Chamley (1986) showed that in a dynamic equilibrium model with proportional taxes, full commitment and time‐varying taxes, it was optimal to suppress capital taxes in the long run. This reduction in capital taxes would promote aggregate investment, increase production and consumption in the long run. This result has been shown to be robust to many extensions.1 In particular, it is robust to the introduction of heterogeneity: even if agents are heterogeneous optimal policies drive capital taxes to zero in the long run.2 In this way the study of capital taxation in dynamic rational expectations models has provided rigorous ground for an old idea in economics: a decrease in capital taxes would increase the size of the pie and, perhaps, make everybody better off. The reduction of capital taxes, is not just a purely academic issue, it has been at the forefront of policy discussions. Some countries have recently reduced capital gains taxes or corporate taxes. To mention a few, Spain, France, Sweden and the US. The economic success of Ireland is often linked to lower capital taxes. Most empirical measures of capital taxes show that these were extremely high but that they have been going down in the last two decades. Carey and Tchilinguirian (2000), with estimates for the OECD countries for the period 1980–97, conclude that there has been a shift in the relative tax burden from capital to labour, with an average annual decrease of −0.2% in capital taxes, and an increase of 0.3% on labour taxes. For the US these rates are −0.5% and 0.2% respectively. Chamley’s result is only about long‐run tax rates: it is well known that optimal capital taxes are not zero in the transition to the steady state. As shown in Jones et al. (1993) the transition of optimal taxes shows very large oscillations through time. Optimal taxes can take extreme values in different periods, and the exact shape of the transition is highly dependent on the exact model at hand, making it difficult to implement the correct Ramsey tax policy in the real world. Therefore it is of interest to study the effect of implementing policies with simpler dynamics, in particular, policies with constant tax rates. Inspired by the long‐run results of Chamley, one could consider the effect of abolishing capital taxes and to set labour taxes to a new constant level, high enough to keep the same level of government spending. Lucas (1990) performed exactly this experiment in a neoclassical dynamic model of capital accumulation and he found that abolishing capital taxes and shifting the burden of tax revenue to labour taxes was welfare improving. Cooley and Hansen (1992) confirmed these results even when considering inflation tax and consumption taxes. Lucas (1990), and Cooley and Hansen (1992) used a model with homogeneous agents. Therefore, they could not address issues of equity and redistribution that immediately come to mind when discussing capital vs. labour taxes. The object of the current article is to study the effects of abolishing capital taxes in a model with heterogeneous agents. In this way we can address both issues of efficiency and equity. We keep the model as close as possible to that of Chamley. Therefore we consider a model of capital accumulation, infinitely‐lived agents, flexible prices, proportional capital and labour taxes, complete markets and competitive equilibrium. We rule out redistributive lump sum taxes, as these would render the redistributive issue irrelevant and such taxes are impossible to implement in the real world. We also consider agents that can both save and work, as in the data the vast majority of agents (excluding retired) do so. We calibrate our model to observed heterogeneity of agents in a relevant way for the exercise at hand. We find the usual result that a reduction in capital taxes enhances economic activity: wages, aggregate investment, aggregate consumption and aggregate output all increase by a significant amount. Nevertheless, abolishing capital taxes also changes the distribution of wealth since it increases the disposable income of capital‐rich agents in a major way; the redistributive effect is so important that the utility of agents with a high wage/wealth ratio decreases dramatically; only consumers with a low wage/wealth ratio are better off. The effects on individual welfare are very large: the lowest quintile of the population would suffer a loss of between 20% and 60% (depending on the calibration). Furthermore, depending on the calibration, either 40% or 60% of the population would lose from the reform. Some papers have shown how it may be difficult to implement Ramsey policies due to time inconsistency. For example, Klein et al. (2008) show how a time consistent policy under balanced budget would involve capital taxes that are quite high in the long run. One possible conclusion from these observations is that issues such as lowering capital taxes should be written in the constitution. Our results say the new constitution would have to be written very carefully in order to be approved, since it would have to implement the actual transition of optimal taxes under full commitment. The median voter is likely to disagree with a change in the constitution stating that capital taxes are immediately suppressed, and a significant part of the population would very strongly disagree. Since we are extrapolating the behaviour of the economy into an area where no observations are available, the answer we find is highly dependent on both theoretical and empirical elements introduced in the analysis. In the article we provide a careful discussion of how to capture the features of the joint distribution of labour and capital income across agents that are relevant for the exercise at hand. Also, we discuss carefully the effects of different assumptions on the elasticity of labour. In the empirical literature on inequality it is standard to focus on either the distribution of wealth or the distribution of income. We argue that the joint distribution of wealth and labour income across the population is what matters and, in particular, that the relevant dimension of this distribution for us is the dispersion of the wage/wealth ratio across agents.3 Our approach is to match the observed distribution of the wage/wealth ratio. Another key aspect in the calibration is the parameter values and functional forms that relate to the elasticity of labour, since these will influence the efficiency cost of the higher labour taxes that are needed to compensate for the lost capital tax revenue. We argue that the standard neoclassical model does not allow us to match both the variability of hours worked across time and across agents. Since we are particularly concerned about agents’ heterogeneity we choose a highly inelastic labour supply to roughly match the cross‐section observations. This is a revised version of our working paper that was first circulated in 1995.4 Other papers have analysed related issues since our first working paper came out. Correia (1999) shows the source of redistributive effects analytically in a model with aggregation, Domeij and Heathcote (2004) use a model with incomplete markets and focus on the effects of idiosyncratic uncertainty.5Flodén (2009) studies a model where the transition of capital taxes is optimal from the point of view of one of the agents in the model. Maliar and Maliar (2001) derive aggregation results, calibrate the model with 8 heterogeneous groups of agents and compare the results with those of our 1995 working paper. Conesa et al. (2009) find that in a model with overlapping generations and idiosyncratic uninsurable risk it is often optimal ex ante to have high capital taxes. A summary of these papers is that our main finding is very robust: suppressing capital taxes has large redistributive effects that would strongly decrease the welfare of large parts of the population under many extensions of our model. Our article still is the closest one to Chamley’s framework, so it shows the effect of heterogeneity in isolation and in a simple model. Some available work has used models where aggregation obtains. For example, Correia (1999), Domeij and Heathcote (2004) and Flodén (2009) use Greenwood Herkowitz and Huffman (GHH) preferences. In those cases one can solve first for the aggregate solution and then disaggregate the results. But in a model with growth these preferences imply zero hours worked in steady state and, therefore, in the status quo economy. GHH also presents some problems in matching volatility of hours. Therefore we use a model without aggregation and solve explicitly for the disaggregated choices of each type of agent. However, given our approach this only increases the computation costs mildly relative to a homogeneous agent model. Since under GHH preferences one has to resort to numerical solutions for the aggregate variables anyway, the increase in computational cost from having a model without aggregation is quite minor. Along the way, we re‐examine the result of Chari et al. (1994) that suppressing capital taxes would be undesirable in a model with a representative agent and high risk aversion. We find that if the calibration maintains a roughly plausible capital output ratio, suppressing capital taxes is beneficial for a representative agent even with high risk aversion. However, in a model with heterogeneous agents and high risk aversion, the redistributive effects of suppressing capital taxes are even higher. The layout of the article is as follows. The model is presented in Section 1. Section 2 discusses issues pertaining to parameter calibration using data from the US economy. Section 3 presents the results derived from the simulations. Section 4 performs sensitivity analysis. The conclusion ends the main article. Appendix A discusses the details of the calibration using PSID data set and Appendix B discusses the numerical algorithm in detail. 1. The Model In this Section we describe a simple neoclassical growth model with heterogeneous agents, endogenous production, labour choice, exogenous deterministic growth6 and government spending. Government can only use distortionary capital and labour taxes. Agents differ both in terms of their human, and non‐human wealth. 1.1. Consumer, Firm, and Government Behaviour Assume that n infinitely‐lived consumer types indexed by j = 1,2,…,n derive utility from consumption and leisure, and they are endowed with one unit of time every period. The number of each type of agents is normalised to 1/n. They receive income from working and from renting their capital. All agents can work and accumulate (or divest) capital. Agents are heterogeneous in the productivity of their endowment of labour hours and initial capital stock. Income from labour and capital are taxed at constant rates τl and τk. Consumers of type j solve the following maximisation problem: (1) where are the choice variables of the consumer. We assume separability in time and in the consumption–leisure decision. Here, cj,t, kj,t, lj,t denote consumption, capital stock and hours worked of agent j at time t; wt denotes the wage for efficiency units of work, rt capital rental, these prices normalised in terms of the consumption good of the period. The wage obtained per hour worked in period t by agent j is φjμtwt, reflecting the fact that this agent produces φj efficiency units per hour worked and that labour productivity grows exogenously at the rate μ.7 Since we concentrate our study on issues of distribution, our agents only differ in their initial wealth kj,−1 and their efficiency of labour φj > 0; these are normalised so that . Parameters δ,d are in the interval [0,1], they stand for the discount factor of future utility and the depreciation rate of capital. Notice that only the capital income net of depreciation allowances is taxable. Functions u and v are differentiable and satisfy the appropriate Inada conditions to insure interior solutions; u(·) and v(·, μ) are strictly concave; u(·) and v(l, ·) are strictly increasing and v(·, μ) is strictly decreasing. Individual capital holdings could be negative if the agent holds some debt.8 There is one representative firm that maximises period‐by‐period profits; it manages a production technology, rents capital at a price rt and hires efficiency units of labour at a wage wt to solve (2) where yt represents output, kt−1 the demand of capital, and et the demand for efficiency units of labour. F is the production function gross of depreciation, strictly concave and homogeneous of degree one. Since total supply of efficiency units of labour is all variables grow at a rate μ in the steady state except labour, which is constant in steady state. Normalising the group size to 1/n together with guarantees that by setting φi = φj and ki,−1 = kj,−1 for all i,j = 1,2,…,n we are back to the homogeneous agent model in Lucas (1990). We now discuss the constraints of government fiscal policy. Government spending is exogenous and grows at the same rate as output, so the sequence of government consumption is given by gt ≡ μtg for a given constant g.9 Tax revenues accrue from constant capital and labour tax rates τk, τl. Government can save or dissave by borrowing or lending at equilibrium interest rates. As is well known this is equivalent with assuming that the government has (possibly negative) capital stock holdings . This amounts to the following budget constraint at period‐t (3) Initial government savings are given. 1.2. Equilibrium We assume competitive equilibrium. As usual, an equilibrium is a sequence for prices and allocations and a government policy (g, τk, τl), such that when consumers maximise utility and firms maximise profits taking prices and government policy as given, they choose equilibrium allocations that clear all markets and the budget constraint of the government is satisfied. The equations determining equilibrium are as follows. Market clearing in capital, labour and consumption good are given, for all t, by (4) (5) (6) For interior solutions, the first order conditions for capital and labour in the consumer’s problem are (7) (8) for all t and j. Here, v ′ ≡ ∂v/∂l. These are familiar conditions, setting the intertemporal marginal rate of substitution of consumption (between leisure and consumption) equal to the price of capital (labour) net of taxes. As usual, equilibrium factor prices equal marginal product to set rt = Fk(kt−1, et) and wt = Fe(kt−1, et). It is easy to see that these equilibrium conditions can be summarised in the following way. Equation (7) implies that for some constants λj (9) For constant relative risk aversion (CRRA) utility of consumption this is the familiar condition that under complete markets and common discount factors the share of consumption is constant through time. Substituting (7) and (8), and substituting for individual savings in the consumer budget constraint we obtain the present value formulation of the consumers’ budget constraints (10) The budget constraint of the government is guaranteed by Walras’ law and, therefore, can be ignored. It is easy to see that given a policy (g, τk, τl) necessary and sufficient conditions for to be an equilibrium sequence are10 for all t = 0,1,… the following equations hold: (6), (7) for j = n, (8) for j = n, and (9) for some λ1,…,λn−1. (10) for j = 1,…,n This reduces the number of variables and equations that need to be found to compute an equilibrium, since (7) and (8) for j = 2,…,n, period‐t budget constraints (1) and (3) can be ignored. Notice that the way we formulate the problem involves finding the individual variables directly, we do not use any aggregation result, as there is no aggregation result that holds for this model. An algorithm is described in detail in Appendix B which in this model implies negligible increase in computational costs due to heterogeneity. 2. Calibration, Stylised Facts, Analytic Results and an Algorithm For our calibration we assume the following functional form of the utility function: (11) for γc,γl < 0 and B > 0, and we assume that hours worked satisfy 0 ≤ lj,t ≤ 1. Notice that, since we choose γc = −1 in the benchmark calibration the term μ disappears from the utility function in that case. As usual we use a Cobb‐Douglas production function F(kt−1,et) = μαA. The effects of a tax reform are highly dependent on parameter values. Therefore, we need to use parameter values that can arguably represent the behaviour of actual economies in the dimensions that are relevant for our exercise. We now describe the criteria that guided our choice of parameter values in the benchmark economy. 2.1. Preference, Technology and Policy Parameters To insure comparability with the rest of the literature and to match various empirical regularities that are successfully explained by neoclassical growth models many parameters are chosen in a standard way. The values we use are summarised in Table 1. Table 1 
Benchmark Calibration. Technology, Utility and Policy Parameters α 0.36 τl 0.23 δ 0.99 τk 0.57 d 0.02 μ 1.004   γc −1.0 kg,−1 −2.0 γl −10.0 A 1 α 0.36 τl 0.23 δ 0.99 τk 0.57 d 0.02 μ 1.004   γc −1.0 kg,−1 −2.0 γl −10.0 A 1 Open in new tab Table 1 
Benchmark Calibration. Technology, Utility and Policy Parameters α 0.36 τl 0.23 δ 0.99 τk 0.57 d 0.02 μ 1.004   γc −1.0 kg,−1 −2.0 γl −10.0 A 1 α 0.36 τl 0.23 δ 0.99 τk 0.57 d 0.02 μ 1.004   γc −1.0 kg,−1 −2.0 γl −10.0 A 1 Open in new tab We choose log utility, γc = −1. This represents a low level of risk aversion but it is the value most commonly found in studies of fiscal policy. In this case we see from (9) that λj gives exactly the consumption ratio relative to agent n: (12) As usual, B is chosen so that the representative agent works 1/3 of his time endowment in the steady state corresponding to the status quo. Also, α is chosen to match the labour share of income. Depreciation rate, discount rate of utility, parameter A and growth rate are set to the usual values for quarterly data. As for policy parameters (τl, τk, g), tax rates are chosen to match measured average effective marginal tax rates. There is a long literature on this measurement. Papers vary in the method employed to measure these taxes, in the sample used, in the introduction of depreciation allowances and growth. We use McGrattan et al. (1997) estimates of τk = 0.57 and τl = 0.23 for the period 1947–87, who follow the procedure of Joines (1981). These values are not too different from the ones estimated for the US in Carey and Tchilinguirian (2000), who, updating the Mendoza et al. (1994) methodology, obtain estimates of around 0.5 for capital tax and 0.22 for labour tax for the period 1980‐97.11 We discuss in detail the sensitivity of our results to the value of τk. Government spending g is selected to balance the government budget constraint in status quo steady state.12 Initial aggregate capital is set at the steady state of the status quo policy.13 Initial government debt is set to . Since output is close to 1 and the model is calibrated to quarters this amounts to choosing a yearly debt/output ratio of about fifty per cent in the status quo. 2.2. Heterogeneity Parameters The parameters that determine agents’ heterogeneity, namely the productivity of labour φj and initial levels of wealth kj.−1, are key to the outcome of the policy reform under study. Therefore it is important to calibrate these parameters so as to capture appropriately the actual joint distribution of wage and wealth across agents. We focus on those aspects of this distribution that are key for the policy outcome. We argue that the relevant dimension to be matched is the distribution of wage/wealth ratios across agents. Two agents with the same wage/wealth ratio are likely to be affected in the same way by a tax reform, even if one of them has a much higher total income than the other. The following concrete example demonstrates this point. Consider the case where the wage/wealth ratio is constant across all agents: (13) That is, an agent who is twice as productive is also twice as wealthy. Also, for simplicity, consider μ = 1 and . It can be easily checked that for any set of tax rates equilibrium allocations in this example satisfy In other words, all agents work the same but an agent twice as productive (and, under (13), twice as wealthy) consumes and saves twice as much each period. It is clear that, in this case, the ratio λj is equal to φj/φn, therefore this ratio is independent of tax rates. It follows that any gain or loss from alternative tax policies affects equally the profile of consumption and leisure of all agents. If agent i consumes twice that of agent j before the reform, agent i will continue to consume twice that of agent j after the reform. If (13) was a good approximation to the actual distribution of wealth and wages all agents would experience a similar gain from the tax reform we consider. In this case introducing heterogeneity in the model provides no new insights. On the other hand, if we find a lot of dispersion of wage/wealth ratios in actual data some agents may gain and others may lose from suppressing capital taxes. Therefore, we should examine if (13) is a good approximation to the empirical distribution of income. For this purpose we examine the joint distribution of wealth and wages in actual data. Figure 1 plots wages against wealth for different households computed from the PSID.14 Each dot represents the wage and wealth of a family in the sample. If (13) was a good approximation to actual data most dots would be located near a straight line going through the origin (a ‘ray’). It is obvious, however, that the actual distribution is not grouped along one ray. The dispersion of wage/wealth ratios is very high and therefore abolishing capital taxes may affect different agents differently. Fig. 1. Open in new tabDownload slide Sample Wages and Wealth Fig. 1. Open in new tabDownload slide Sample Wages and Wealth The issue is, then, how to introduce the relevant aspects of the distribution of wages and wealth in the model in a parsimonious way. Agents located either in the upper left corner or in the lower right corner of this figure are both ‘rich’ but those agents in the upper left corner are likely to lose from the abolition of capital taxes because most of their income comes from labour, which will be taxed more heavily after the reform. Agents with a similar wage/wealth ratio either all gain or all lose, regardless of their total wealth. To give some names to the situation: it is not that important for us to distinguish between a very highly qualified and a low qualified worker if their levels of wealth are both low. These workers might have a very different level of income but both have a high wage/wealth ratio. It is important, however, to distinguish between a very highly qualified worker and a large landowner who has zero labour income: they both have a high total income but they have very different wage/wealth ratios. In most studies of the wealth distribution the usual criterion is to classify agents according to their total income or total wealth, so that the large landowner and the highly qualified worker would be lumped together incorrectly, because the first is likely to gain from the reform we consider while the latter is likely to lose. In order to capture the observed distribution of wealth/wage ratios we rank all households by their wage/wealth ratio and find the quintiles of this distribution. Each type in the model will represent one of the quintiles. Graphically, the split in quintiles would be represented by four rays in Figure 1 such that each of the five areas separated by the rays contains 20% of households. The more traditional criterion of classifying families by total income would correspond instead to splitting the sample with four negatively sloped lines, each line representing a given level of total income. The other traditional criterion of classifying by total wealth would correspond to splitting the sample using four vertical lines. Another complication stems from the fact that our measures are affected by a pure life cycle effect, something that our model does not take into account. For example, older people are usually wealthier than younger people and they are likely to be retired, which corresponds to φ = 0 in our model. Almost all of them would belong to group 1, thus confusing the life‐cycle effect with the wealth effect. We try to remove this effect from our measures by splitting the sample into six age groups and dividing each age group into five quintiles according to their wage/wealth ratio. The wage of type 1 agents is then calculated with a weighted average of the observed wages of households in the lowest wage/wealth ratio across age groups; the weights given to each age group correspond to percentages of US population as reported by the Census.15,16 To summarise, in the benchmark case heterogeneity parameters φj, kj,−1 are obtained by matching each type of agents in the model to the average of each quintile of the distribution of wage/wealth ratios, eliminating the life‐cycle effects. In the Section on robustness exercises we also calculate the heterogeneity parameters splitting the sample with a pure wealth criterion (i.e., splitting the sample by means of vertical lines). The statistics obtained from these two possible criteria are reported in Table 2. Table 2 
Means and Ratios by Quintiles, PSID Sample Wage/Wealth partition . Type . Means by type . Ratios of type i over type 5 . Hours . Wage . Income . Hours . Wage . Income . 1 2,708.03 7.89 58,611.94 1.315 1.048 3.241 2 2,837.86 11.11 50,397.86 1.378 1.475 2.787 3 2,468.28 9.72 37,822.32 1.199 1.291 2.092 4 2,333.49 9.4 31,790.4 1.133 1.248 1.758 5 2,059.41 7.53 18,083.11 – – – Wealth partition 1 3,031.43 15.04 84,644.67 1.597 2.549 5.708 2 2,858.14 10.31 45,058.34 1.505 1.747 3.039 3 2,520.16 7.99 31,277.28 1.327 1.354 2.109 4 2,098.94 6.48 21,047.11 1.106 1.098 1.419 5 1,898.61 5.9 14,828.54 – – – Wage/Wealth partition . Type . Means by type . Ratios of type i over type 5 . Hours . Wage . Income . Hours . Wage . Income . 1 2,708.03 7.89 58,611.94 1.315 1.048 3.241 2 2,837.86 11.11 50,397.86 1.378 1.475 2.787 3 2,468.28 9.72 37,822.32 1.199 1.291 2.092 4 2,333.49 9.4 31,790.4 1.133 1.248 1.758 5 2,059.41 7.53 18,083.11 – – – Wealth partition 1 3,031.43 15.04 84,644.67 1.597 2.549 5.708 2 2,858.14 10.31 45,058.34 1.505 1.747 3.039 3 2,520.16 7.99 31,277.28 1.327 1.354 2.109 4 2,098.94 6.48 21,047.11 1.106 1.098 1.419 5 1,898.61 5.9 14,828.54 – – – Type 1 corresponds to households with a lower wage/wealth ratio or a higher wealth Open in new tab Table 2 
Means and Ratios by Quintiles, PSID Sample Wage/Wealth partition . Type . Means by type . Ratios of type i over type 5 . Hours . Wage . Income . Hours . Wage . Income . 1 2,708.03 7.89 58,611.94 1.315 1.048 3.241 2 2,837.86 11.11 50,397.86 1.378 1.475 2.787 3 2,468.28 9.72 37,822.32 1.199 1.291 2.092 4 2,333.49 9.4 31,790.4 1.133 1.248 1.758 5 2,059.41 7.53 18,083.11 – – – Wealth partition 1 3,031.43 15.04 84,644.67 1.597 2.549 5.708 2 2,858.14 10.31 45,058.34 1.505 1.747 3.039 3 2,520.16 7.99 31,277.28 1.327 1.354 2.109 4 2,098.94 6.48 21,047.11 1.106 1.098 1.419 5 1,898.61 5.9 14,828.54 – – – Wage/Wealth partition . Type . Means by type . Ratios of type i over type 5 . Hours . Wage . Income . Hours . Wage . Income . 1 2,708.03 7.89 58,611.94 1.315 1.048 3.241 2 2,837.86 11.11 50,397.86 1.378 1.475 2.787 3 2,468.28 9.72 37,822.32 1.199 1.291 2.092 4 2,333.49 9.4 31,790.4 1.133 1.248 1.758 5 2,059.41 7.53 18,083.11 – – – Wealth partition 1 3,031.43 15.04 84,644.67 1.597 2.549 5.708 2 2,858.14 10.31 45,058.34 1.505 1.747 3.039 3 2,520.16 7.99 31,277.28 1.327 1.354 2.109 4 2,098.94 6.48 21,047.11 1.106 1.098 1.419 5 1,898.61 5.9 14,828.54 – – – Type 1 corresponds to households with a lower wage/wealth ratio or a higher wealth Open in new tab Calibrating the initial wealth of agents in the model with the initial wealth of the quintiles in the data seems problematic, because different assets in the data yield different returns and agents with large wealth are often able to access higher returns. Instead we calibrate λ to the ratio of consumption that can be sustained by total labour and capital income of each agent given the actual assets and the actual returns of these assets for the agents in the sample. For a detailed description on how we compute total capital income see Appendix A. The ratios are reported in Table 2. From these consumption ratios we find the initial wealth of each group in the model consistent with steady state and the calibrated consumption ratios in the status quo tax rates. The heterogeneity parameters found in this way and used in the model are reported in Table 3.17 Table 3 
Heterogeneity Parameters (Benchmark Economy) Wage/Wealth Partition . Wealth Partition .   φ1/φ5 1.05 φ1/φ5 2.55     φ2/φ5 1.48 φ2/φ5 1.75   φ3/φ5 1.29 φ3/φ5 1.35   φ4/φ5 1.25 φ4/φ5 1.10   k1,−1/k−1 5.54 k1,−1/k−1 10.39   k2,−1/k−1 1.76 k2,−1/k−1 0.87   k3,−1/k−1 0.35 k3,−1/k−1 −0.85   k4,−1/k−1 −0.63 k4,−1/k−1 −2.76 Wage/Wealth Partition . Wealth Partition .   φ1/φ5 1.05 φ1/φ5 2.55     φ2/φ5 1.48 φ2/φ5 1.75   φ3/φ5 1.29 φ3/φ5 1.35   φ4/φ5 1.25 φ4/φ5 1.10   k1,−1/k−1 5.54 k1,−1/k−1 10.39   k2,−1/k−1 1.76 k2,−1/k−1 0.87   k3,−1/k−1 0.35 k3,−1/k−1 −0.85   k4,−1/k−1 −0.63 k4,−1/k−1 −2.76 Open in new tab Table 3 
Heterogeneity Parameters (Benchmark Economy) Wage/Wealth Partition . Wealth Partition .   φ1/φ5 1.05 φ1/φ5 2.55     φ2/φ5 1.48 φ2/φ5 1.75   φ3/φ5 1.29 φ3/φ5 1.35   φ4/φ5 1.25 φ4/φ5 1.10   k1,−1/k−1 5.54 k1,−1/k−1 10.39   k2,−1/k−1 1.76 k2,−1/k−1 0.87   k3,−1/k−1 0.35 k3,−1/k−1 −0.85   k4,−1/k−1 −0.63 k4,−1/k−1 −2.76 Wage/Wealth Partition . Wealth Partition .   φ1/φ5 1.05 φ1/φ5 2.55     φ2/φ5 1.48 φ2/φ5 1.75   φ3/φ5 1.29 φ3/φ5 1.35   φ4/φ5 1.25 φ4/φ5 1.10   k1,−1/k−1 5.54 k1,−1/k−1 10.39   k2,−1/k−1 1.76 k2,−1/k−1 0.87   k3,−1/k−1 0.35 k3,−1/k−1 −0.85   k4,−1/k−1 −0.63 k4,−1/k−1 −2.76 Open in new tab 2.3. Elasticity of Labour The choice of γl is quite important since it governs the elasticity of labour and it will be crucial in determining hours worked after the reform and the impact on welfare of the higher labour taxes. Ideally we would use a parameter value that matched some basic facts concerning the variability of hours worked. Let us point to two basic well‐known facts: (a) Across time variability of aggregate hours worked is higher than variability of aggregate consumption. (b) Across individuals variability of hours worked is lower than variability of consumption. These observations have been documented by many authors. Fact (a) has been emphasised by a number of papers, for example Hansen (1985) and Rogerson (1986). Fact (b) has been documented in several contributions and it is confirmed within our calibration of heterogeneity reported in Table 2: the fourth column indicates that agents with the highest number of hours worked (type j = 2) work 40% more than type j = 5 but they consume almost three times as much. Similar conclusions are derived from the wealth partition. Fact (a) has to do with the reaction of hours worked to a temporal shock to aggregate wealth, while fact (b) has to do with the elasticity of hours worked to changes in wealth and wage. The policy experiment that we are considering will cause both a change over time of aggregate hours worked and a redistribution of wealth so that, ideally, we would like to use a model and parameter values that agree with both facts mentioned. Unfortunately, this cannot be done within the standard neoclassical dynamic model. To see this, we first argue that low values of |γl| help to explain fact (a), but they are incompatible with fact (b). Consider the model with linear utility of leisure, so γl = 0, and assume that agents only differ in their initial wealth, so that φi = φ for all i. Hansen (1985) and Rogerson (1986) showed that fact (a) above can be explained under these assumptions. But (8) implies that in this case Therefore, linear utility of leisure contradicts fact (b) above, because consumption is constant across agents of different wealth. Conversely, we can see that high values of |γl| fail to explain fact (a) but they are compatible with fact (b). It is easy to see that in a stochastic model for our choice of B, for all j and t. This is because for high |γl| agents are so averse to changes in hours worked that they are likely to choose low volatility of hours across time and they will choose to adapt to fluctuations in income by higher volatility of consumption. Therefore, high values of |γl| are likely to generate nearly constant hours worked across time in a model with aggregate uncertainty. Hence high |γl| matches fact (b) but it spoils fact (a) in a stochastic model. For our purposes, it seems particularly important to capture fact (b) and to have a model where hours worked do not react very strongly to changes in policy. For this reason, we choose γl = −10 in the benchmark case which implies a very low wage elasticity of labour. This calibration is incompatible with fact (a).18 As with many other parameters, we will check robustness of our main results to this choice. 2.4. Numerical Issues Since before the reform the economy is at the steady state it is trivial to find the equilibrium g. After the reform, there will be a transition period as allocations converge to the new steady state in deviations from trend. The difficulty is, therefore, finding the transition along with the labour tax rate and the ratios λ that will balance the budget constraints after the reform. Since analytic solutions under the benchmark parameters are not known we resort to numerical simulation. Details on the algorithm and on the model in deviations from trend are given in Appendix B. Since there is no aggregation in the model, we need to solve for the aggregate variables jointly with the individual variables. Therefore, aggregate variables are solved jointly with the ratios λ. In Appendix B we show that adding the ratios λ to the list of variables to be computed implies a small additional computational cost relative to a model with aggregation. 3. Results We first show that in a homogeneous agent version of our model suppressing capital taxes causes a small improvement in welfare. This confirms the results of Lucas (1990) and Cooley and Hansen (1992) in our slightly different model and calibration. Furthermore, relative to the literature we find these gains are more robust: we find that, contrary to past results, there is an improvement in welfare even for very high values of risk aversion −γc. We then go on to show the results for the heterogeneous agent case. 3.1. Homogeneous Agent 3.1.1. Replicating homogeneous agent results We use the benchmark parameters of Table 1. Steady state values are shown in Table 4. The first column shows values for the status quo, while the second column displays the values after the reform. As expected the level of capital, labour productivity and even the wage net of taxes are higher in the long run if the reform takes place. The labour tax has to increase from 23% to 37% in order to finance the capital tax cut. Table 4 
Steady State, Homogeneous Agent, Before and After Reform Variable . Status Quo . Zero capital Tax . τk 0.57 0 k 6.72 13.21 invest 0.16 0.32 GNP 0.98 1.25 l 0.333 0.331 c 0.57 0.68 w 1.89 2.41 r 0.05 0.03 τl 0.23 0.37 w(1 − τl) 1.46 1.52 g 0.25 0.25 πH 5.90% Variable . Status Quo . Zero capital Tax . τk 0.57 0 k 6.72 13.21 invest 0.16 0.32 GNP 0.98 1.25 l 0.333 0.331 c 0.57 0.68 w 1.89 2.41 r 0.05 0.03 τl 0.23 0.37 w(1 − τl) 1.46 1.52 g 0.25 0.25 πH 5.90% Open in new tab Table 4 
Steady State, Homogeneous Agent, Before and After Reform Variable . Status Quo . Zero capital Tax . τk 0.57 0 k 6.72 13.21 invest 0.16 0.32 GNP 0.98 1.25 l 0.333 0.331 c 0.57 0.68 w 1.89 2.41 r 0.05 0.03 τl 0.23 0.37 w(1 − τl) 1.46 1.52 g 0.25 0.25 πH 5.90% Variable . Status Quo . Zero capital Tax . τk 0.57 0 k 6.72 13.21 invest 0.16 0.32 GNP 0.98 1.25 l 0.333 0.331 c 0.57 0.68 w 1.89 2.41 r 0.05 0.03 τl 0.23 0.37 w(1 − τl) 1.46 1.52 g 0.25 0.25 πH 5.90% Open in new tab Higher output in the long run does not necessarily imply that suppressing capital taxes should lead to higher welfare. Consumption and leisure are lower immediately after the reform (to allow for higher investment and the accumulation of capital), which is a cost of the reform that is ignored in steady state calculations. Therefore the transition has to be analysed explicitly. The welfare benefits of changing the tax system are evaluated, separately for each agent. We use the standard measure given by the permanent increase in consumption that would leave each individual indifferent between the status quo and the reform, keeping leisure unchanged. More precisely, letting and be the equilibrium allocations before and after the reform, the welfare gain for agent j is given by πj that satisfies The last line of Table 4 shows that we find a welfare gain for the homogeneous agent of πH = 5.9%. This gain is similar to the one reported in previous papers, slightly larger due to the higher capital taxes in our benchmark parameterisation. 3.1.2. Emphasising the efficiency gains of suppressing capital taxes It has been pointed out that the benefits of suppressing capital taxes in a homogeneous agent model may disappear if the curvature of the utility function with respect to consumption is sufficiently high. To the extent that we are not sure about the true curvature, this brings a word of caution to the efficiency benefits of actually suppressing capital taxes. We re‐examine this result and we find that, under homogeneous agents, if the capital/output ratio is kept constant, there is a gain from suppressing capital taxes even for high risk aversion. This reinforces the view that suppressing capital taxes is a good policy from the point of view of aggregate efficiency and it will be important for the robustness exercises that we perform in Section 4. The reason that higher curvature in the utility function may limit the benefits of suppressing capital taxes is the following. Increasing −γc has two effects: first, it causes labour to be more elastic, increasing the costs of a higher labour tax after the reform; second, the initial drop in consumption caused by the cut in capital taxes is more costly if u has more curvature. Indeed, Chari et al. (1994) show that if relative risk aversion is γc = −8 suppressing capital taxes would cause a loss in utility in a homogeneous agent case. We find a similar result in Table 5: even though γc = −8 still shows a small gain in utility due to our slightly different model and calibration, a utility loss is experienced from suppressing capital taxes when γc = −11. Table 5 
Utility Gain from Suppressing Capital Taxes, Homogeneous Agent, Varyingγc −γc . kstst . g . τl . πH (%) . 0.5 7.77 0.26 0.35 6.34 1 6.72 0.25 0.37 5.90 3 4.17 0.21 0.45 4.25 5 2.88 0.17 0.51 2.97 8 1.87 0.12 0.61 1.39 11 1.33 0.08 0.70 −0.17 −γc . kstst . g . τl . πH (%) . 0.5 7.77 0.26 0.35 6.34 1 6.72 0.25 0.37 5.90 3 4.17 0.21 0.45 4.25 5 2.88 0.17 0.51 2.97 8 1.87 0.12 0.61 1.39 11 1.33 0.08 0.70 −0.17 The first column refers to the parameter varied. Columns 2–5 indicate how the calibration and results change for the homogeneous agent case. τl is the labour tax rate after suppressing capital taxes in this case, while πH measures the welfare gain when agents are homogeneous. Open in new tab Table 5 
Utility Gain from Suppressing Capital Taxes, Homogeneous Agent, Varyingγc −γc . kstst . g . τl . πH (%) . 0.5 7.77 0.26 0.35 6.34 1 6.72 0.25 0.37 5.90 3 4.17 0.21 0.45 4.25 5 2.88 0.17 0.51 2.97 8 1.87 0.12 0.61 1.39 11 1.33 0.08 0.70 −0.17 −γc . kstst . g . τl . πH (%) . 0.5 7.77 0.26 0.35 6.34 1 6.72 0.25 0.37 5.90 3 4.17 0.21 0.45 4.25 5 2.88 0.17 0.51 2.97 8 1.87 0.12 0.61 1.39 11 1.33 0.08 0.70 −0.17 The first column refers to the parameter varied. Columns 2–5 indicate how the calibration and results change for the homogeneous agent case. τl is the labour tax rate after suppressing capital taxes in this case, while πH measures the welfare gain when agents are homogeneous. Open in new tab But increasing −γc and leaving all other parameters constant has some undesirable effects for the calibration of the economy. In the model in deviations from trend the effective discount factor becomes (see Appendix B). Therefore the effective discount factor is lower as −γc increases and the capital output ratio goes down if all remaining parameters are left unchanged. Table 5 shows that the steady state capital for γc = −11 is about one fifth of the capital for log utility. This means that for γc = −11 labour at the status quo is much less productive than in the log utility case and it explains why the labour tax rate needs to be raised much more (to 70% instead of 37%) in order to compensate for suppressing capital taxes when γc = −11. Therefore changing −γc relative to the benchmark case not only influences the elasticity of labour and the utility cost of the transition but it also increases the size of the distortion that labour has to suffer if capital taxes disappear. In order to analyse the effects of increasing risk aversion in isolation we prefer to increase risk aversion without modifying the capital output ratio. For this purpose we change the scaling constant A in the production function to keep the same capital output ratio for different γc. The results are shown in Table 6. We now find that the gains from suppressing capital taxes are indeed lower for high risk aversion but the homogeneous consumer never loses utility from suppressing capital taxes, even for very high risk aversion. Table 6 
Utility Gain from Suppressing Capital Taxes, Homogeneous Agent, Varyingγc, KeepingK/LConstant −γc . kstst . g/y . τl . πH (%) . 0.5 6.72 0.25 0.35 6.31 1 6.72 0.25 0.37 5.90 3 6.72 0.27 0.44 4.52 4 6.72 0.27 0.46 4.05 5 6.72 0.28 0.47 3.69 8 6.72 0.28 0.50 3.06 11 6.72 0.29 0.52 2.71 14 6.72 0.29 0.53 2.20 18 6.72 0.30 0.54 0 22 6.72 0.30 0.55 0 −γc . kstst . g/y . τl . πH (%) . 0.5 6.72 0.25 0.35 6.31 1 6.72 0.25 0.37 5.90 3 6.72 0.27 0.44 4.52 4 6.72 0.27 0.46 4.05 5 6.72 0.28 0.47 3.69 8 6.72 0.28 0.50 3.06 11 6.72 0.29 0.52 2.71 14 6.72 0.29 0.53 2.20 18 6.72 0.30 0.54 0 22 6.72 0.30 0.55 0 Notes: See Table 5 Open in new tab Table 6 
Utility Gain from Suppressing Capital Taxes, Homogeneous Agent, Varyingγc, KeepingK/LConstant −γc . kstst . g/y . τl . πH (%) . 0.5 6.72 0.25 0.35 6.31 1 6.72 0.25 0.37 5.90 3 6.72 0.27 0.44 4.52 4 6.72 0.27 0.46 4.05 5 6.72 0.28 0.47 3.69 8 6.72 0.28 0.50 3.06 11 6.72 0.29 0.52 2.71 14 6.72 0.29 0.53 2.20 18 6.72 0.30 0.54 0 22 6.72 0.30 0.55 0 −γc . kstst . g/y . τl . πH (%) . 0.5 6.72 0.25 0.35 6.31 1 6.72 0.25 0.37 5.90 3 6.72 0.27 0.44 4.52 4 6.72 0.27 0.46 4.05 5 6.72 0.28 0.47 3.69 8 6.72 0.28 0.50 3.06 11 6.72 0.29 0.52 2.71 14 6.72 0.29 0.53 2.20 18 6.72 0.30 0.54 0 22 6.72 0.30 0.55 0 Notes: See Table 5 Open in new tab In summary, the example discussed by Chari et al. certainly serves their purpose, namely, to show how ignoring the transition for optimal capital and labour taxes can result in an even lower utility than at the status quo. But suppressing capital taxes is always beneficial in terms of aggregate efficiency if the calibration is adjusted appropriately. 3.2. Heterogeneous Agents The main goal of this article is to study the welfare effects of eliminating capital taxes when agents are heterogeneous. Since this is a model where there is no aggregation it is not obvious that suppressing capital taxes will lead to higher aggregate output as it did in the model with homogeneous agents. However, probably because of the presence of complete markets, aggregate variables in the heterogeneous agent case behave in a similar way as in the homogeneous agent model of the previous subsection. Therefore, output, investment, capital, gross wages and wages net of taxes increase in steady state under heterogeneity. This can be seen in Figure 2, representing the evolution of some variables after the reform. Capital nearly doubles and it is halfway through the new steady state in about 30 quarters. Investment is much higher than in the status quo, as it is even higher in the first few periods than in the new steady state after the reform. Wages increase by about 25%. As expected consumption is very low in the initial periods. Hours worked are higher at the beginning of the transition, showing that the effect of the reform is to induce a higher labour supply. The last two graphs show how consumption and hours worked are very different for agents 1 and 5. Fig. 2. Open in new tabDownload slide Sample Paths for (a) Capital and Investment; (b) Consumption, Wages and Hours Worked Fig. 2. Open in new tabDownload slide Sample Paths for (a) Capital and Investment; (b) Consumption, Wages and Hours Worked But under heterogeneous agents abolishing capital taxes also has a redistributive effect. Lower capital taxes mean that a larger part of the tax bill in present discounted terms is paid by agents with a high wage/wealth ratio. This may offset the gains from the higher aggregate efficiency for these agents. Since we labelled j = 5 the agent with the highest wage/wealth ratio, a reduction in capital taxes is likely to lower the relative consumption of agent j = 5. Therefore, according to (12), suppressing capital taxes is likely to increase the ratios λj = cj,t/c5,t for j = 1,…,4. Table 7 shows the effects of this redistribution of wealth by reporting equilibrium ratios of consumption and labour for different capital taxes, with labour taxes adjusted to maintain the same level of government spending in all cases. The first row corresponds to the status quo capital tax, so it simply describes the equilibrium consumption ratios λj = cj,t/cn,t and labour ratios before the reform. As expected λj is lower for higher j, as we consider agents with a higher wage/wealth ratio. As in the data the cross‐sectional variation of hours worked is much smaller than the cross‐section variation of consumption, justifying our choice of a large −γl to match fact (b) in subsection 3.3.19 Table 7 
Consumption and Labour Ratios New τk . Wage/Wealth Partition . . . . . . . . . 0.57 3.23 2.77 2.10 1.77 0.95 0.94 0.92 0.89 0.456 3.57 3.00 2.21 1.82 0.95 0.93 0.92 0.88 0.342 3.85 3.11 2.31 1.88 0.95 0.93 0.91 0.88 0.228 4.34 3.43 2.47 2.00 0.94 0.93 0.91 0.87 0.114 4.76 3.67 2.62 2.10 0.94 0.92 0.90 0.86 0 5.56 4.11 2.94 2.28 0.94 0.92 0.90 0.85 New τk . Wage/Wealth Partition . . . . . . . . . 0.57 3.23 2.77 2.10 1.77 0.95 0.94 0.92 0.89 0.456 3.57 3.00 2.21 1.82 0.95 0.93 0.92 0.88 0.342 3.85 3.11 2.31 1.88 0.95 0.93 0.91 0.88 0.228 4.34 3.43 2.47 2.00 0.94 0.93 0.91 0.87 0.114 4.76 3.67 2.62 2.10 0.94 0.92 0.90 0.86 0 5.56 4.11 2.94 2.28 0.94 0.92 0.90 0.85 Open in new tab Table 7 
Consumption and Labour Ratios New τk . Wage/Wealth Partition . . . . . . . . . 0.57 3.23 2.77 2.10 1.77 0.95 0.94 0.92 0.89 0.456 3.57 3.00 2.21 1.82 0.95 0.93 0.92 0.88 0.342 3.85 3.11 2.31 1.88 0.95 0.93 0.91 0.88 0.228 4.34 3.43 2.47 2.00 0.94 0.93 0.91 0.87 0.114 4.76 3.67 2.62 2.10 0.94 0.92 0.90 0.86 0 5.56 4.11 2.94 2.28 0.94 0.92 0.90 0.85 New τk . Wage/Wealth Partition . . . . . . . . . 0.57 3.23 2.77 2.10 1.77 0.95 0.94 0.92 0.89 0.456 3.57 3.00 2.21 1.82 0.95 0.93 0.92 0.88 0.342 3.85 3.11 2.31 1.88 0.95 0.93 0.91 0.88 0.228 4.34 3.43 2.47 2.00 0.94 0.93 0.91 0.87 0.114 4.76 3.67 2.62 2.10 0.94 0.92 0.90 0.86 0 5.56 4.11 2.94 2.28 0.94 0.92 0.90 0.85 Open in new tab The last row of Table 7 corresponding to τk = 0 shows the effects of suppressing capital taxes. We see that all groups j = 1,…,4 will consume more and work less, relative to agent 5, after the reform. Furthermore, the one who benefits the most is agent j = 1 with the lowest wage/wealth ratio: while his consumption ratio increases by 70% (it goes from 3.23 before the reform to 5.56) the consumption ratio of the agent in the middle quintile, j = 3, only increases by about 40% (from 2.1 before the reform to 2.94). It is clear, therefore, that lowering capital taxes has a redistributive effect and it lowers the relative consumption of agents with a high wage/wealth ratio such as agents j = 5. This shows that the reform redistributes wealth in favour of the agents with a low wage/wealth ratio. The middle rows of Table 7 report the effect of four less radical reforms, each reform consisting of cutting the capital tax rate by an additional 20%. We see the effect is monotone: all λs increase as capital taxes decrease. These rows will serve to understand the next Table. It is clear from Table 7 that lowering capital taxes increases inequality. But since there is a gain in aggregate efficiency, as shown in Figure 2, it could happen that less wealthy agents experience a net gain from suppressing capital taxes. To resolve this issue we consider the change in welfare for each agent of suppressing capital taxes. Table 8 shows the gains in utility from each of the possible reforms considered in the previous Table. If capital taxes were completely suppressed (last row) 40% of the population would be worse off. Perhaps more importantly, agents of type 5 would experience a very large loss in welfare of 32%. Agents of type 1, on the other hand, benefit greatly from the reform. Table 8 
Welfare Gains in Benchmark Case New τk . π1 (%) . π2 (%) . π3 (%) . π4 (%) . π5 (%) . 0.456 6.67 3.56 2.22 0.70 −4.05 0.342 12.38 5.88 3.08 −0.07 −9.90 0.228 17.52 7.44 3.08 −1.79 −16.86 0.114 22.33 8.50 2.54 −4.13 −24.51 0 26.98 9.26 1.62 −6.89 −32.60 New τk . π1 (%) . π2 (%) . π3 (%) . π4 (%) . π5 (%) . 0.456 6.67 3.56 2.22 0.70 −4.05 0.342 12.38 5.88 3.08 −0.07 −9.90 0.228 17.52 7.44 3.08 −1.79 −16.86 0.114 22.33 8.50 2.54 −4.13 −24.51 0 26.98 9.26 1.62 −6.89 −32.60 Open in new tab Table 8 
Welfare Gains in Benchmark Case New τk . π1 (%) . π2 (%) . π3 (%) . π4 (%) . π5 (%) . 0.456 6.67 3.56 2.22 0.70 −4.05 0.342 12.38 5.88 3.08 −0.07 −9.90 0.228 17.52 7.44 3.08 −1.79 −16.86 0.114 22.33 8.50 2.54 −4.13 −24.51 0 26.98 9.26 1.62 −6.89 −32.60 New τk . π1 (%) . π2 (%) . π3 (%) . π4 (%) . π5 (%) . 0.456 6.67 3.56 2.22 0.70 −4.05 0.342 12.38 5.88 3.08 −0.07 −9.90 0.228 17.52 7.44 3.08 −1.79 −16.86 0.114 22.33 8.50 2.54 −4.13 −24.51 0 26.98 9.26 1.62 −6.89 −32.60 Open in new tab We can see that even with a small reduction in capital taxes (first row of Table 8) group j = 5 with the highest wage/wealth ratio would lose welfare, although the rest of the population would benefit. These welfare comparisons confirm that eliminating capital income taxation at the expense of labour income taxation is not Pareto improving. If capital taxes were suppressed, the distributional issues dominate the gain in aggregate efficiency in the sense that they are not Pareto improving and a large part of the population may experience a loss in utility. The loss in welfare for these agents is very high, specially if compared with those reported on the aggregate effects of changes in fiscal or monetary policy using dynamic equilibrium models. We will see in Section 4 that these features are very robust to changes in parameter values. In Table 8 the median voter (agent j = 3) does gain from any permanent reduction in capital taxes but this hardly suggests that suppressing capital taxes at the expense of labour taxes is likely to occur in a modern democracy. First of all because given the very large loss in utility experienced by a large part of the population the reform we consider would be difficult to implement. In modern democracies it is not only the median voter’s opinion that matters, as it is difficult to implement a reform in practice if it hurts a sufficiently large part of the population significantly. Second, in the robustness experiments of Section 4 we will find that for slightly different parameter values the median voter often loses from suppressing capital taxes. Therefore it is not clear ex ante that even the median voter will favour such a reform. 4. Sensitivity Analysis Table 9 shows the welfare gains of all agents from suppressing capital taxes when several parameters of the benchmark case are changed one at a time. In all cases we adjust B so that the hours worked are one third of total time endowment. The column labelled kstst refers to the capital steady state before the reform. The next column shows government spending over output before the reform. Column τl contains the labour tax that would operate after the reform. Table 9 
Sensitivity Analysis: Effects of Parameter Variations on Calibration and Welfare Gains of Fully Suppressing Capital Taxes . kstst . g/y . τl . πH (%) . π1 (%) . π2 (%) . π3 (%) . π4 (%) . π5 (%) . −γc Risk aversion 0.5 6.72 0.25 0.35 6.31 20.27 7.94 3.01 −1.92 −18.56 1 6.72 0.25 0.37 5.90 26.98 9.26 1.62 −6.89 −32.60 3 6.72 0.27 0.44 4.52 51.09 17.19 −2.53 −19.18 −60.48 4 6.72 0.27 0.46 4.05 73.28 22.12 −3.77 −22.88 −66.64 −γl Labour disutility 15 6.72 0.25 0.37 6.05 26.12 9.03 1.74 −6.36 −30.98 10 6.72 0.25 0.37 5.90 26.98 9.26 1.62 −6.89 −32.60 1 6.72 0.25 0.38 4.32 57.07 9.72 −6.82 −23.15 −61.58 τk Status quo capital taxation 40 9.09 0.23 0.33 1.74 12.58 3.41 −0.55 −4.99 −18.74 30 10.31 0.21 0.30 0.74 7.78 1.81 −0.78 −3.68 −12.75 20 11.41 0.20 0.27 0.24 4.39 0.86 −0.67 −2.39 −7.79 Wealth partition 6.72 0.25 0.37 5.9 36.91 5.48 −8.08 −37.67 −49.38 . kstst . g/y . τl . πH (%) . π1 (%) . π2 (%) . π3 (%) . π4 (%) . π5 (%) . −γc Risk aversion 0.5 6.72 0.25 0.35 6.31 20.27 7.94 3.01 −1.92 −18.56 1 6.72 0.25 0.37 5.90 26.98 9.26 1.62 −6.89 −32.60 3 6.72 0.27 0.44 4.52 51.09 17.19 −2.53 −19.18 −60.48 4 6.72 0.27 0.46 4.05 73.28 22.12 −3.77 −22.88 −66.64 −γl Labour disutility 15 6.72 0.25 0.37 6.05 26.12 9.03 1.74 −6.36 −30.98 10 6.72 0.25 0.37 5.90 26.98 9.26 1.62 −6.89 −32.60 1 6.72 0.25 0.38 4.32 57.07 9.72 −6.82 −23.15 −61.58 τk Status quo capital taxation 40 9.09 0.23 0.33 1.74 12.58 3.41 −0.55 −4.99 −18.74 30 10.31 0.21 0.30 0.74 7.78 1.81 −0.78 −3.68 −12.75 20 11.41 0.20 0.27 0.24 4.39 0.86 −0.67 −2.39 −7.79 Wealth partition 6.72 0.25 0.37 5.9 36.91 5.48 −8.08 −37.67 −49.38 Notes: See Table 5. Open in new tab Table 9 
Sensitivity Analysis: Effects of Parameter Variations on Calibration and Welfare Gains of Fully Suppressing Capital Taxes . kstst . g/y . τl . πH (%) . π1 (%) . π2 (%) . π3 (%) . π4 (%) . π5 (%) . −γc Risk aversion 0.5 6.72 0.25 0.35 6.31 20.27 7.94 3.01 −1.92 −18.56 1 6.72 0.25 0.37 5.90 26.98 9.26 1.62 −6.89 −32.60 3 6.72 0.27 0.44 4.52 51.09 17.19 −2.53 −19.18 −60.48 4 6.72 0.27 0.46 4.05 73.28 22.12 −3.77 −22.88 −66.64 −γl Labour disutility 15 6.72 0.25 0.37 6.05 26.12 9.03 1.74 −6.36 −30.98 10 6.72 0.25 0.37 5.90 26.98 9.26 1.62 −6.89 −32.60 1 6.72 0.25 0.38 4.32 57.07 9.72 −6.82 −23.15 −61.58 τk Status quo capital taxation 40 9.09 0.23 0.33 1.74 12.58 3.41 −0.55 −4.99 −18.74 30 10.31 0.21 0.30 0.74 7.78 1.81 −0.78 −3.68 −12.75 20 11.41 0.20 0.27 0.24 4.39 0.86 −0.67 −2.39 −7.79 Wealth partition 6.72 0.25 0.37 5.9 36.91 5.48 −8.08 −37.67 −49.38 . kstst . g/y . τl . πH (%) . π1 (%) . π2 (%) . π3 (%) . π4 (%) . π5 (%) . −γc Risk aversion 0.5 6.72 0.25 0.35 6.31 20.27 7.94 3.01 −1.92 −18.56 1 6.72 0.25 0.37 5.90 26.98 9.26 1.62 −6.89 −32.60 3 6.72 0.27 0.44 4.52 51.09 17.19 −2.53 −19.18 −60.48 4 6.72 0.27 0.46 4.05 73.28 22.12 −3.77 −22.88 −66.64 −γl Labour disutility 15 6.72 0.25 0.37 6.05 26.12 9.03 1.74 −6.36 −30.98 10 6.72 0.25 0.37 5.90 26.98 9.26 1.62 −6.89 −32.60 1 6.72 0.25 0.38 4.32 57.07 9.72 −6.82 −23.15 −61.58 τk Status quo capital taxation 40 9.09 0.23 0.33 1.74 12.58 3.41 −0.55 −4.99 −18.74 30 10.31 0.21 0.30 0.74 7.78 1.81 −0.78 −3.68 −12.75 20 11.41 0.20 0.27 0.24 4.39 0.86 −0.67 −2.39 −7.79 Wealth partition 6.72 0.25 0.37 5.9 36.91 5.48 −8.08 −37.67 −49.38 Notes: See Table 5. Open in new tab Column πH indicates welfare improvement in the representative agent version of the model. This can be thought of as a rough measure of the aggregate efficiency gain of suppressing capital taxes for each set of parameters. Columns πj for j = 1, …, 5 show the utility gains of each agent. We first consider changes in relative risk aversion −γc. Robustness in this dimension is relevant because relative risk aversion is often thought to be larger than one, with values between 2 and 4 much more widely accepted in the literature. For each γc we adjust the constant A in the production function so as to keep the capital stock constant for the reasons explained in Section 3.1.2.20 Recall that the row for γc = −1 corresponds to the benchmark case. We find that the pattern of gains and losses across agents is similar to the one of the benchmark case but the size of welfare gains or losses is exaggerated by increasing risk aversion.21 Gains of agents j = 1, 2, and losses of agents j = 4, 5, are much larger as −γc increases. Now agent 5 loses 60% of his utility for γc = −3. In addition we find that the median voter j = 3 experiences a mild utility loss for reasonable values of relative risk aversion such as 3 or 4. We conclude that for more reasonable values of risk aversion the redistributive effects of suppressing capital taxes are much larger than for log utility and that the median voter will be against the reform for likely values of risk aversion.22 It is intuitive that higher risk aversion should increase the inequality effects of suppressing capital taxes. First of all there is the standard effect of making the initial drop in consumption more costly, which means that the efficiency gain is even lower and there is less welfare to gain from suppressing capital taxes. But it is also well known that the wage elasticity of labour is higher for higher risk aversion. This means that for higher −γc labour goes down more steeply for a given increase in labour taxes and in order to meet the budget constraint the government needs a larger labour tax hike after the reform. As can be seen from Table 9, for a risk aversion of 1 we have τl = 0.37 after the reform but for risk aversion of 4 we have τl = 0.46. Agents with high wage/wealth ratio have to pay more taxes when risk aversion is higher and they lose relatively more. Also, since labour is more elastic for high risk aversion, the increase in labour taxes is more distortionary and more costly in terms of welfare for the reasons usually considered in public finance taxation. We also consider robustness to the value of γl. As we explained in Section 3 the choice for the benchmark case is questionable because it would fail to account for the variability of hours worked across time in a stochastic version of the model; furthermore, it implies a wage elasticity of about 0.1 which is lower than usually estimated for the aggregate economy. We see from Table 9 that lower values of −γl (and, therefore, closer to those used in the RBC literature) only exaggerate the inequality generated by suppressing capital taxes. As is well known, lower −γl implies higher wage elasticity of labour and the same discussion as in the previous paragraph justifies the results. Again, for a sufficiently high elasticity the median voter j = 3 would now be against the reform. There is much disagreement about the relevant level of average marginal capital tax rates (see discussion in footnote 11 for references), so we also study the sensitivity to the tax levels in the status quo. The third panel of Table 9 considers different values for the capital tax before the reform. A lower value for τk in status quo causes the redistributive effect to be smaller: agents with high (low) wage/wealth ratio lose (gain) less for lower status quo capital taxes. But it is also true that the aggregate gain represented by πH is smaller if initially the capital tax was not very high. These results are intuitive: if the capital tax is low to begin with the redistributive effect is lower but there is less to be gained from the reform at an aggregate level. The median voter, again, would be marginally against the reform. Crucial to our results were the heterogeneity parameters determining φ and initial wealth of each type of agent. These we calibrated by splitting our sample according to quintiles of the wage/wealth ratio and by removing effects from life cycle. Since this is a relatively non‐standard criterion to measure inequality it is worthwhile to explore the effects of the reform using the more traditional criterion of wealth inequality and without adjusting for life cycle. We use the data in the second panel of Table 2 and report the results for this calibration in the fourth panel of Table 9. Again, the large changes in utility are reinforced and the median voter would be against the reform. It is clear that the results are very robust. If anything, the benchmark calibration understates the redistributive effects of suppressing capital taxes. 5. Conclusion The Chamley (1986) and Judd (1985, 1987) results say that in a model with heterogeneous agents and distortionary taxes all Pareto optimal allocations have the property that capital taxes disappear in the long run, even if the planner cares mostly about workers. One may wonder if these long‐run results could be implemented immediately and if suppressing capital taxes could benefit all agents. We explore whether this is the case in a model with heterogeneous agents. Our model is as close as possible to that of Chamley (1986) and Lucas (1990) so as to explore in isolation the effects of heterogeneity. We find that if capital taxes were suppressed and the lost revenue was compensated by higher labour taxes the welfare of at least 20% of the population would go down dramatically. For all the experiments we have performed 40% of the population would be worse off. This happens despite the fact that there is always an aggregate efficiency gain from suppressing capital taxes. This result is robust to different parameter values and to the criterion for splitting the sample. For some parameter values, including reasonable values of relative risk aversion, agents in the lowest quintile of the population lose 60% of their utility. The effect of suppressing capital taxes on the median voter (our type 3 agent) is always quite small. In fact, whether the median voter would gain or lose from the tax reform depends very much on the parameter values chosen for the model. We find that for reasonable levels of risk aversion the median voter would lose from the reform but for log utility it would gain. Therefore, from the vantage point of traditional political economy the model does not give strong predictions about whether such tax reform would be approved in a once‐and‐for‐all referendum. In any case, the loss in welfare for the lowest quintile is so large that it is not surprising that such a reform has not even been considered in actual policy discussions. We find that there is an aggregate efficiency gain even with very high risk aversions but that, in this case, the redistributive effect is even larger. In this sense, for the issue of capital taxation, the problem of distribution of wealth is several orders of magnitude more important than other traditional topics of macroeconomics. We think that research on distributive and efficiency issues in dynamic equilibrium models is, therefore, a very promising avenue for research. Capital taxes in the real world are indeed very high, it is probably the case that if capital taxes are lowered this may result in a widespread gain in efficiency. But transferring the burden to labour taxes is unlikely to be implemented in democratic societies, where large minorities have a strong influence in blocking reforms. Dynamic fiscal policy analysis with equilibrium models should help to find ways that capital taxes can be lowered, thereby achieving higher aggregate efficiency and, at the same time, insuring that most of the population can benefit from such a reform. In addressing the calibration of the model we argue that the relevant dimension is not the distribution of total wealth but the wage/wealth ratio across agents. Therefore the heterogeneity parameters in our model attempt to reproduce the features of the distribution of wage/wealth ratios. Our intention is to examine the effect of heterogeneity in isolation, therefore we stay as close as possible to the model of Chamley throughout the article. Along the way we find a number of empirical issues that this model does not address and that should be resolved in order to examine the effects of reforms in factor taxation. For example, we point out that the standard neoclassical model cannot simultaneously match the observed volatility of hours worked and consumption across time and the variation of these variables across agents. Several modifications of the model may help in resolving this puzzle such as introducing time non‐separability in leisure, endogenous human capital accumulation, the introduction of both an intensive and extensive margin in a model with uninsurable risk. These are left for future research. Other issues in the calibration of heterogeneity demand a more careful analysis. We treated all families in the same way but the propensity to consume and work of a family with two children is not the same as that of a single person. A better modelling of families of different types would be crucial. Finally, the model has a difficult time explaining total wealth held by all agents and total capital income, due to the fact that all assets in our model yield very similar returns. This indicates that there is enormous scope for future research in studying tradeoff between efficiency and equity when considering changes in the tax code with equilibrium models and heterogeneous agents. Footnotes 1 " For a review of the extensions see the relevant chapters of Ljunqvist and Sargent (2004) and Chari and Kehoe (1999). 2 " The result is obtained in Chamley (1986) and Judd (1985, 1987). A proof for the model considered in the current article where no lump sum transfers are available is found in Atkeson et al. (1999). 3 " Few papers have stressed the importance of the joint distribution of wealth and wage earnings. Krusell and Ríos‐Rull (1999) note how results in a model of political economy are sensitive to whether consumers are ranked according to wealth distribution or to earnings distribution. Domeij and Heathcote (2004) also discuss the correlation of earnings and wealth across agents in the data. 4 " Some differences with that version are that we have now five agents instead of two, we now only consider a deterministic model, there are many more robustness checks and we have added the analysis for the high risk aversion case. 5 " The ‘no‐earnings‐risk’ economy of Domeij and Heathcote (2004) amounts to redoing our exercise for GHH preferences and without growth. 6 " Introducing growth explicitly is important in order to quantify the effect of depreciation allowances. This is because in the stationary version of the model total investment is no longer equal to gross investment, therefore the size of the tax base is not the same as if the analysis was based on the stationary version of the model. This is made explicit in Appendix B where we show the equations for the model in deviations from trend. 7 " Introducing the trend of labour productivity (μt ) in the utility function is a standard way to insure a non‐degenerate solution for hours worked in the long run in the presence of growth. This formulation has been controversial. Some economists have argued that this is artificial, while others have argued that it is consistent with assuming that higher human capital yields higher utility from leisure. This controversy is not relevant for our benchmark calibration with log utility of consumption, where the term μt drops out. We only need the term μt in the utility function for the high risk aversion cases considered in the robustness exercises in Section 4. 8 " As usual, some additional lower bound on (possibly negative) capital holding has to be introduced in order to rule out Ponzi schemes. The same will be true for the budget constraint of the government. 9 " Since we maintain g constant across policy experiments, the equilibrium computed and the welfare gains discussed in Sections 3 and 4 are consistent with a model where government spending enters the utility function or the production function. To keep notation simple, we write the article as if government spending has no productive use. 10 " For details see Appendix 5 of the 1995 working paper version. That paper presents the case with uncertainty which encompasses the certainty case. 11 " The rate of τk = 0.57 is not as high as it may appear, since it is applied to income after depreciation allowances and since this is the sum of all taxes on capital income paid by consumers and firms. In any case, there is considerable disagreement on the relevant level of labour and income taxes, specially on the level of the capital tax. Feldstein et al. (1983) obtain estimates of τk that range between 0.55 and 0.85 for the period 1953–79. Cooley and Hansen use a lower tax rate, setting τk = 0.5 (this number is based on Joines (1981) with the data ending in 1979) and they do not subtract growth from the depreciation allowances; Chari et al. use τk = 0.27; Lucas (1990) considers capital and labour taxes of 0.4; Greenwood et al. (1995) set τk = 0.70. 12 " Since we are interested in the effects of substituting capital taxes by labour taxes, and in keeping with the practice in Lucas (1990) and Cooley and Hansen (1992), we will only consider government spending that is financed from these two taxes. Therefore, total government spending in our model will be lower than the one actually observed. 13 " Table 4 shows the values of capital and output. The capital/output ratio in status quo is about seven, lower than the values of ten or twelve that are often used for a quarterly model. This lower capital/output ratio is due to the large capital taxes combined with the standard A = 1. Changing A so as to match the capital/output ratio does not change the results significantly. 14 " The details on how this Figure has been constructed are in Appendix A. 15 " The six age groups are as follows: less than 25 years old (14.40% of the US population), from 25 to 34 (23.32%), from 35 to 44 (20.30%), 45 to 54 (13.62%), 55 to 64 (11.43%) and older than 64 (16.89%). 16 " Conesa et al., (2009) explore the effect of capital taxes in an overlapping generations model. Therefore, they are better able to match income through the life cycle. 17 " As can be seen from Table 3 the consumption ratios that we find can only be sustained if wealth of some of the agents is higher than total capital. This happens because, in the real world, assets such as land play a very important role in the portfolios of individuals, while land is not present in our model. An alternative approach would be to introduce land that delivers returns and services of consumption. 18 " We check that this is the case in a model with heterogeneous agents, taxes and aggregate uncertainty in the 1995 working paper version. 19 " Notice, however, that the level of hours worked across agents does not reproduce the data: in the model hours increase with j but they decrease with j in the data. Ideally one would study the effect of suppressing capital taxes with a model that matches this basic observation but this would mean going away from the standard neoclassical model so we leave this exercise for future research. The differences of hours worked across agents, in any case, are not large so one would not expect large changes in the results on the gains from suppressing capital taxes. 20 " The following caveat is in order. While it is clear that for log utility the wage/wealth ratio is the relevant criterion for splitting the sample, with higher risk aversions this is not strictly speaking correct, since consumption may less than double when wage and wealth double. Nevertheless we maintain the calibration of heterogeneity parameters based on wage/wealth ratios. This is for three reasons: (i) comparability, (ii) simplicity, (iii) because this is probably a reasonable approximation to the actual equivalent agents. Probably, capturing the relevant joint distribution exactly with high risk aversion requires a more elaborate criterion than the one used in the rest of the article. 21 " Only the results up to γc = −4 are reported because the algorithm failed to converge for higher levels of risk aversion. We do not know if this is a failure of the algorithm or, more likely, this happens because there is no equilibrium with zero capital taxes, that is, there is no way to collect enough from only labour taxes in order to maintain g. 22 " For the case considered in Chari et al. (1994) where A is constant for all levels of relative risk aversion we obtain even larger welfare losses for low wealth agents. For example, for γc = −3. we find π1 = 64.65%,π2 = 22.12%,π3 = −3.15%,π4 = −22.88%,π5 = −68.49%. 23 " All rates of return or price series were extracted from CITIBANK. 24 " As the difference between real estate value and principal mortgage remaining. References Atkeson , A. , Chari , V.V. and Kehoe , P.J. ( 1999 ). ‘Taxing capital income: a bad idea’ , Federal Reserve Bank of Minneapolis Quarterly Review , vol. 23 ( 3 ), pp. 3 – 17 . OpenURL Placeholder Text WorldCat Carey , D. , and Tchilinguirian , H. ( 2000 ). ‘ Average effective tax rates on capital, labour and consumption ’, OECD Economics Department Working Paper No. 258. Chamley , C. ( 1986 ). ‘Optimal taxation of capital income in general equilibrium with infinite lives’ , Econometrica , vol. 54 ( 3 ), pp. 607 – 22 . Google Scholar Crossref Search ADS WorldCat Chari , V.V. , Christiano , L.J. and Kehoe , P.J. ( 1994 ). ‘Fiscal policy in a business cycle model’ , Journal of Political Economy , vol. 102 , pp. 617 – 52 . Google Scholar Crossref Search ADS WorldCat Chari , V.V. and Kehoe P.J. ( 1999 ). ‘Optimal fiscal and monetary policy’, in ( J. Taylor and M. Woodford, eds.), Handbook of Macroeconomics , pp. 1671 – 745 , Amsterdam: North Holland . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Conesa J.C. , Kitao , S. and Krueger , D. ( 2009 ). ‘Taxing capital? Not a bad idea after all!’ , American Economic Review , vol. 99 ( 1 ), pp. 25 – 48 . Google Scholar Crossref Search ADS WorldCat Cooley , T.F. and Hansen , G.D. ( 1992 ). ‘Tax distortions in a neoclassical monetary economy’ , Journal of Economic Theory , vol. 58 ( 2 ), pp. 290 – 316 . Google Scholar Crossref Search ADS WorldCat Correia , I.H. ( 1999 ) ‘On the efficiency and equity trade‐off’ , Journal of Monetary Economics , vol. 44 ( 3 ), pp. 581 – 603 . Google Scholar Crossref Search ADS WorldCat Domeij , D. and Heathcote , J. ( 2004 ). ‘On the distributional effects of reducing capital taxes’ , International Economic Review , vol. 45 ( 2 ), pp. 523 – 54 . Google Scholar Crossref Search ADS WorldCat Feldstein , M.S. , Dicks‐Mireaux , L. and Poterba , J. ( 1983 ). ‘The effective tax rate and the pretax rate of return’ , Journal of Public Economics , vol. 21 , pp. 129 – 58 . Google Scholar Crossref Search ADS WorldCat Flodén , M. ( 2009 ). ‘Why are capital income taxes so high?’ , Macroeconomic Dynamics , vol. 13 ( 3 ), pp. 279 – 304 . Google Scholar Crossref Search ADS WorldCat Garcia‐Milà , T. , Marcet , A. and Ventura , E. ( 1995 ). ‘ Supply side interventions and redistribution ’, Working Paper Universitat Pompeu Fabra, No. 115. Greenwood , J. , Rogerson , R. and Wright , R. ( 1995 ). ‘Household production in real business cycle theory’, in ( T.F. Cooley, ed.), Frontiers of Business Cycle Research , pp. 157 – 74 , Princeton: Princeton University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Hansen , G.D. ( 1985 ). ‘Indivisible labour and the business cycle’ , Journal of Monetary Economics , vol. 16 ( 3 ), pp. 309 – 28 . Google Scholar Crossref Search ADS WorldCat Joines , D.H ( 1981 ). ‘Estimates of effective marginal tax rates on factor incomes’ , Journal of Business , vol. 54 ( 2 ), pp. 191 – 226 . Google Scholar Crossref Search ADS WorldCat Jones , L.E. , Manuelli , R.E. and Rossi , P.E. ( 1993 ). ‘Optimal taxation in models of endogenous growth’ , Journal of Political Economy , vol. 101 ( 3 ), pp. 485 – 517 . Google Scholar Crossref Search ADS WorldCat Judd , K.L. ( 1985 ). ‘Redistributive taxation in a simple perfect foresight model’ , Journal of Public Economics , vol. 28 ( 1 ), pp. 59 – 83 . Google Scholar Crossref Search ADS WorldCat Judd , K.L. ( 1987 ). ‘The welfare cost of factor taxation in a perfect foresight model’ , Journal of Political Economy , vol. 95 , pp. 675 – 709 . Google Scholar Crossref Search ADS WorldCat Klein P , Krusell , P. and Ríos‐Rull , V. ( 2008 ). ‘Time consistent fiscal policy’ , Review of Economic Studies , vol. 75 ( 3 ), pp. 789 – 808 . Google Scholar Crossref Search ADS WorldCat Krusell , P. and Ríos‐Rull , V. ( 1999 ). ‘On the size of U.S. government: political economy in the neoclassical growth model’ , American Economic Review , vol. 89 ( 5 ), pp. 1156 – 81 . Google Scholar Crossref Search ADS WorldCat Ljungqvist , L. and Sargent , T.J. ( 2004 ). Recursive Macroeconomic Theory , 2nd edition, Cambridge MA: MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Lucas , R.E. Jr. ( 1990 ). ‘ Supply‐side economics: an analytical review ’, Oxford Economic Papers , vol. 42 ( 2 ), pp. 293 – 316 . Google Scholar Crossref Search ADS WorldCat Maliar , L. and Maliar , S. ( 2001 ). ‘Heterogeneity in capital and skills in a neoclassical stochastic growth model’ , Journal of Economic Dynamics and Control , vol. 25 ( 9 ), pp. 1367 – 97 . Google Scholar Crossref Search ADS WorldCat McGrattan , E. , Rogerson , R. and Wright , R. ( 1997 ). ‘An equilibrium model of the business cycle with household production and fiscal policy’ , International Economic Review , vol. 38 ( 2 ), pp. 267 – 90 . Google Scholar Crossref Search ADS WorldCat Mendoza , E. , Razin , A. and Tesar , L. ( 1994 ). ‘Effective tax rates in macroeconomics. Cross country estimates of tax rates on factor incomes and consumption’ , Journal of Monetary Economics , vol. 34 ( 3 ), pp. 297 – 323 . Google Scholar Crossref Search ADS WorldCat Rogerson , R. ( 1986 ). ‘Indivisible labour, lotteries and equilibrium’ , Journal of Monetary Economics , vol. 21 ( 1 ), pp. 3 – 16 . Google Scholar Crossref Search ADS WorldCat Rosenthal , S.S. ( 1988 ). ‘A residence time model of housing markets’ , Journal of Public Economics , vol. 36 , pp. 87 – 109 . Google Scholar Crossref Search ADS WorldCat Appendices Appendix A Calibration of Heterogeneity Parameters We have used the Panel Study of Income Dynamics (PSID) to obtain several distributive measures involved in the calibration of the model. This is a well‐known data set that collects information on families and their offspring. We select families that were interviewed and that kept the same head from 1984 to 1989. Agents in the model are interpreted as households in the data, not the different individuals that compose each household. The variables we want to calibrate are the efficiency parameters φj, and the value of the initial capital stocks kj,−1 for each family. For this purpose we look at wages and assets. The PSID provides measures for average hourly wages, labour income, and several categories of non‐human wealth and asset income. These are reported in Figure 1. From these measures we obtain five quintiles in the distribution of φj/kj,−1 ratios. For the actual calibration we need to estimate the relative permanent consumption of different types of agent. For this purpose we compare the total labour and capital income of different groups and identify the ratio of income to the ratio of consumption. PSID provides data on labour income. To measure capital income of each family we use the reported measures of asset returns whenever these are available, averaging asset income or rates of return over the last five years of the sample period. Otherwise we multiply each asset’s value by average long‐run net rate of return as reported in several studies. In what follows we specify how we find the return of each particular component of non‐human wealth. 1 Types of assets for which the PSID reports actual asset returns. Net value of Business or Farms, market and gardening activities, or rooming and boarding activities. Cash assets (savings and checking accounts, CDs, IRAs etc.) and dividends. 2 Types of assets for which we impute an asset return.
Here we multiply the current value of the asset held by an average (over five years) real rate of return. The following is a list of these assets and the return series we use. Net value of Bonds, Insurance Policies and Collectible Goods: Moody’s average corporate bond yield23. Stocks, Mutual Funds: S&Ps common stock price index. (Dividends are reported as asset income in the category of ‘cash assets’.) Total real estate:24 we use the value calculated in Rosenthal (1988, p 95). Rents perceived by the families are already embedded in that rate of return, therefore we do not use the rents reported in the PSID, as to avoid double counting. Pensions and Annuities: we use the US Government Security Yield, 10 years or more, Treasury compiled. Other Debts: we use the secondary market yields on FHA mortgages since this is composed, mostly, of second mortgages. We deflate these nominal returns or rates by the wholesale consumer price index. The PSID also reports the net value of cars, mobile homes etc. We do not impute any rent for this category. Appendix B Numerical Algorithm We describe in detail here how we solve for the equilibrium quantities after the reform that suppresses capital taxes. At the end of Section 1 we show the equations that characterise the sequence . To allow for a numerical solution we need to convert the model in deviations from trend, in this way a steady state can exist and we can find transitions to this steady state. Let deviations from trend be given by and so on. Standard algebra shows that these satisfy (14) (15) (16) for and . Notice that does not subtitute the original depreciation rate d everywhere. In particular, in the FOC with respect to capital, we have d/μ instead. The present value budget constraints can be rewritten in terms of deviations from trend as (17) Finally, for the welfare calculations we use the equality The numerical problem can be further simplified by noting that, for candidate values λ1,…,λn−1 we can use (9) to substitute out consumption and labour in (16) for agents j = 1,…,n − 1 in terms of and the λs. Therefore the numerical problem at hand reduces to the following: given τk and g, find three sequences , plus n constants (λ1,…,λn−1,τl) such that (14), (15), (16) hold for all t and (17) hold for all j. We convert this into a finite problem by fixing large T and computing a sequence that satisfies: (a) (14), (16), (15) for t = 0,…,T − 1 (b) (17) for j = 1,…,n (c) Variables dated t > T − 1 are set at steady state. Notice that (a) provides 3T equations and (b) provides n additional equations. We have 3T unknowns in plus n unknowns in (τl,λ1,…,λn−1). This gives 3T + n unknowns and the same number of equations. We know this system of equations cannot be solved exactly, for cannot be at steady state unless the initial capital is at steady state but the system can be solved approximately by various numerical solution methods for solving non‐linear systems of equations. As T → ∞ we can potentially obtain an arbitrarily accurate approximation. We use T = 200 and check with 250 for robustness. From the graphs in Figure 2 we see that this allows the solution to reach steady state. Notice that conditional on the model being at steady state after T periods infinite discounted sums involved in the calculations can be computed exactly. It should be clear, therefore, that we do not use any aggregation result: aggregate capital and consumption are determined jointly with the λ’s. Notice that adding heterogeneity means having to solve for 3T + n variables instead of 3T + 1 in the homogeneous agent case. Therefore, despite the lack of aggregation, the increase in the computational cost from adding heterogeneity is negligible. Author notes " We thank the editor, A. Scott, and two anonymous referees. N. Guner, V. Ríos‐Rull, J. Geweke, E. McGrattan, A. Braun, E. Prescott, L. Christiano, O. Attanasio, M. Reiter, N. Roubini, D. Epple and C. Sims provided helpful suggestions. Katharina Greulich has provided invaluable research assistance. Part of Marcet’s work was done while he was a visitor at the Federal Reserve Bank of Minneapolis. Support from the Ministry of Science and Technology of Spain grants SEJ2007‐64340, SEJ2006‐13537, Instituto de Estudios Fiscales, CREI and Barcelona GSE Research Network and of the Government of Catalonia. All errors are our own. © The Author(s). Journal compilation © Royal Economic Society 2009

Nakov,, Anton;Pescatori,, Andrea

doi: 10.1111/j.1468-0297.2009.02302.xpmid: N/A

Abstract We assess the extent to which the greater US macroeconomic stability since the mid‐1980s can be accounted for by changes in oil shocks and the oil elasticity of gross output. We estimate a DSGE model and perform counterfactual simulations. We nest two popular explanations for the Great Moderation: smaller (non‐oil\link real shocks and better monetary policy. We find that oil played an important role in the stabilisation. Around half of the reduced volatility of inflation is explained by better monetary policy alone, and 57% of the reduced volatility of GDP growth is attributed to smaller TFP shocks. Oil related effects explain around a third. For more than a decade since Hamilton’s (1983) seminal article the relevance of oil as a source of macroeconomic fluctuations was viewed as conventional wisdom. Yet Hooker (1999) pointed to a break in the oil price–GDP relationship and Hooker (2002) found a parallel break in the oil price–inflation relationship, both around 1981.1 This break date roughly coincides with (but precedes) the beginning of a period of remarkable macroeconomic stability, dubbed by some economists as the ‘Great Moderation’, and reflected in a sharp decline in the volatility (and sometimes the persistence) of key macroeconomic variables in a number of industrialised economies, including the US (see Table 1 and Figure 1).2 Fig. 1. Open in new tabDownload slide US Volatility Moderation and the Price of Oil Fig. 1. Open in new tabDownload slide US Volatility Moderation and the Price of Oil Table 1 
US Volatility Reduction Since 1984 . Standard deviation (×100) . Volatility reduction % . . 1970:I–1983:IV . 1984:I–2007:IV . Inflation 0.57 0.25 57 GDP growth 1.20 0.52 57 Interest rate 0.88 0.57 35 Real oil price 19.0 13.0 31 . Standard deviation (×100) . Volatility reduction % . . 1970:I–1983:IV . 1984:I–2007:IV . Inflation 0.57 0.25 57 GDP growth 1.20 0.52 57 Interest rate 0.88 0.57 35 Real oil price 19.0 13.0 31 Open in new tab Table 1 
US Volatility Reduction Since 1984 . Standard deviation (×100) . Volatility reduction % . . 1970:I–1983:IV . 1984:I–2007:IV . Inflation 0.57 0.25 57 GDP growth 1.20 0.52 57 Interest rate 0.88 0.57 35 Real oil price 19.0 13.0 31 . Standard deviation (×100) . Volatility reduction % . . 1970:I–1983:IV . 1984:I–2007:IV . Inflation 0.57 0.25 57 GDP growth 1.20 0.52 57 Interest rate 0.88 0.57 35 Real oil price 19.0 13.0 31 Open in new tab Since oil shocks3 are likely to affect many oil‐importing countries in a similar way, a reduction in oil sector volatility or a dampening of the transmission of that volatility to the rest of the world economy is a natural candidate (perhaps working alongside other factors) for explaining the rise of macroeconomic stability in the advanced world. One possibility is that major oil shocks have become less frequent in the period after 1984; another is that diversification towards less oil‐intensive sectors and increased energy efficiency may have diminished the importance of oil shocks, by reducing the ‘share of oil in GDP’.4 We assess the extent to which the macroeconomic moderation in the US can be explained by changes in oil shocks and in the oil elasticity of output, by simulating the model of Nakov and Pescatori (2007) estimated via Bayesian techniques for the periods pre and post 1984. In doing so, we nest two popular explanations for the Great Moderation: ‘good luck’ in the form of a shift in the distribution of TFP and other (non‐oil) real shocks, as claimed for example by Ahmed et al. (2004) and Stock and Watson (2002); and an improvement in the conduct of monetary policy, as argued by Clarida et al. (2000) and Boivin and Giannoni (2006). We do not control for other possible explanations, such as better inventory management (McConnell and Perez‐Quiros, 2000) or financial innovation (Dynan et al., 2005). We find that oil played an important role in the stabilisation, especially of inflation. In particular, the diminished reliance on oil can explain around a third of the reduced volatility of inflation and 18% of the lower volatility of GDP growth. In turn, smaller oil shocks alone can explain around 17% of the lower inflation volatility and 11% of the reduced volatility of GDP growth. This notwithstanding, around half of the reduced volatility of inflation is explained by better monetary policy alone, while 57% of the reduced volatility of GDP growth is attributed to smaller TFP shocks. Related to this, we find evidence that, due to the smaller oil elasticity of output, the inflation–output gap tradeoff has become more benign after 1984, making it easier for the central bank to stabilise both variables. More generally, oil sector productivity and capacity shocks have become less important for US macroeconomic fluctuations relative to US‐originating shocks to TFP, preferences and monetary policy. The rest of the article is organised as follows. The next Section puts our work in the context of the related literature; Section 2 presents the stylised volatility facts; Section 3 sketches a log‐linearised version of the oil pricing model of Nakov and Pescatori (2007) and illustrates how different factors could lead to reduced volatility; Section 4 covers the data and estimation methodology; Section 5 describes our priors and the estimation results; Section 6 contains counterfactual analyses decomposing the volatility moderation into contributions by each factor and discusses the implied changes in the Phillips curve; Section 7 relates our results to those of the literature and the last Section concludes. 1. Related Literature Our article is related to several distinct lines of research. One is the empirical literature on the link between oil and the macroeconomy starting with Hamilton (1983), who argued that most US recessions were (Granger) caused by increases in the price of crude oil. Bernanke et al. (1997) challenged this claim, documenting that essentially all US recessions in the postwar period were preceded by both oil price increases and a tightening of monetary policy. Using a modified VAR methodology they argued that the systematic monetary policy response to inflation (presumably caused by the oil price increases) accounted for the bulk of the depressing effects of oil price shocks on the real economy. What is more, Barsky and Kilian (2001) and Kilian (2008) argued that even the major oil price increases in the 1970s were not an essential part of the mechanism that generated stagflation and that the latter is attributable instead to monetary factors. Unlike these studies, our analysis is based on a structural model featuring optimal oil price setting, estimated with Bayesian methods. This allows us to disentangle the contribution of policy from the effects of oil shocks and the oil elasticity of production. Another strand of research deals with theoretical models of the link between oil and the macroeconomy. Some of the more recent contributions include Kim and Loungani (1992), Rotemberg and Woodford (1996), Finn (1995, 2000), Leduc and Sill (2004) and Carlstrom and Fuerst (2005). While these studies differ in the way oil is employed in the economy (as a consumption good, as a standard productive input or as a factor linked to capital utilisation) and hence in the implications of oil shocks, they all make the assumption that either the oil price or oil supply is exogenous and, hence, unrelated to any economic fundamentals. This is both unappealing from a theoretical point of view as argued by Kilian (forthcoming) and inconsistent with the evidence presented in Kilian (forthcoming), Mabro (1998) and Hamilton (1983).5 Moreover, with an exogenous (or, for that matter, a perfectly competitive) oil sector and without any real rigidities (e.g. real wage rigidities as in Blanchard and Gali, (2007), there is no meaningful trade‐off between inflation and output gap stabilisation, implying that full price stability is optimal even in the face of oil‐sector shocks. The fact that inflation in the 1970s was highly volatile suggests that either policy was very far from optimal or that, indeed, there was an important policy trade‐off. Different from the existing contributions, in our model both the oil price and oil supply are endogenous general equilibrium variables, responding to any of the exogenous shocks. The model features a dominant oil exporter (OPEC) that charges an optimally varying oil price markup, which enters the Phillips curve as a ‘cost‐push’ term and induces a trade‐off between the output gap and inflation (Nakov and Pescatori, 2007). The shocks in our model include structural disturbances to productivity of the oil‐importing region, technology in the oil sector and the capacity of the competitive fringe of (non‐OPEC) oil producers. Finally, our article is related to the literature on the Great Moderation, starting with Kim and Nelson (1999) and McConnell and Perez‐Quiros (2000). With some simplification, most of the explanations for the increased stability can be classified into three broad categories: (a) ‘good practices’, that is, changes in private sector behaviour unrelated to stabilisation policy, for instance improved inventory management (McConnell and Perez‐Quiros, 2000) or financial innovation (Dynan et al., 2005); (b) ‘good policy’, notably better monetary policy as argued by Clarida et al. (2000), Boivin and Giannoni (2006), and Gali and Gambetti (2009); and (c) ‘good luck’, meaning a favourable shift in the distribution of real shocks, as in Ahmed et al. (2004), Stock and Watson (2002) and Justiniano and Primiceri (2006). Explanations of ‘good luck’ in particular often give smaller oil shocks as an example (Summers, 2005).6 Our framework allows us to separate oil from non‐oil factors, while nesting the ‘better policy’ and ‘smaller non‐oil shocks’ explanations. In this respect, our work is most closely related to Leduc and Sill (2007) who assess the role played by monetary policy relative to TFP and oil shocks in the Great Moderation. The main advantage of our approach lies in modelling the oil sector from optimising first principles rather than assuming an exogenous process for oil supply. Another difference is that we estimate most of the model’s parameters separately for each sample with Bayesian techniques which allows us to fit better the volatility reduction facts compared to Leduc and Sill who calibrate their model. In addition, compared to their paper, we put a special focus on the role played by the reduced oil elasticity of production and not only on oil shocks. 2. Volatility Reduction Facts Table 1 shows the standard deviations of three quarterly US macro series: GDP growth, deflator inflation and the federal funds rate, for two subsamples, pre and post 1984. ‘The Great Moderation’ refers to the pronounced decline in the volatility of these (and other) macro variables in the post‐1984 sample. In particular, the volatility of GDP growth and inflation declined by about 57% each, and of the nominal interest rate by around 35%. For comparison, the last row of the Table shows the standard deviation of the quarterly percentage change in the real price of oil. While the reduction in its volatility by 31% is somewhat less pronounced than that of GDP growth or inflation, the difference in volatility between the two samples is statistically very significant (at the 1% level using three standard tests for equality of the variance). Clearly, the volatility reduction facts reported in Table 1 are not insensitive to the choice of break year. Different studies have estimated different break dates for the different variables but usually they lie in the range from 1982 to 1986. Redoing the calculations with 1982:I as the break date, we obtain volatility reductions of 51%, 48%, 27% and 37%, respectively. And doing the same with 1986:I, we obtained 60%, 56%, 38% and 24%. While the differences are non‐trivial, by and large all three sample splits tell the same story. The aim of this article is to evaluate the contribution of oil sector volatility and its propagation empirically, and compare it with alternative explanations for the volatility reduction (better monetary policy and non‐oil related ‘good luck’). While the Great Moderation is sometimes associated also with a reduction in the persistence of macro variables (Canova et al., 2007), we will not attempt to replicate this phenomenon or attribute it to the various factors. 3. The Log‐Linearised Model We base our empirical analysis on the model of Nakov and Pescatori (2007), outlined for convenience in the Appendix. In this model the oil industry is represented by a dominant producer (OPEC) and a fringe of competitive oil suppliers (non‐OPEC), who are small individually but collectively can restrain the market power of the cartel. Our choice of modelling of the oil market in this way is motivated by the simple observation that OPEC today produces about the same amount of oil that it produced back in 1973 (slightly over 30 million barrels per day), even though it sits on the largest and lowest‐cost known oil fields on the planet. At the same time, since the 1970s, the higher‐cost and less oil‐rich non‐OPEC countries have almost doubled their output (see Figure 2), as can be expected from competitive suppliers facing secular growth in demand. While throughout the years non‐OPEC output was growing (except between 1988 and 1992), on a number of peace‐time occasions, OPEC’s output actually declined. As Adelman (2002) aptly puts it, ‘for lower‐cost output to fall or stagnate, while higher‐cost output rises, is like water flowing uphill. Some special explanation is needed’. Unlike any previous general equilibrium model that we are aware of, our setup is able to generate such a negative (conditional) correlation between OPEC and non‐OPEC supply, as the profit‐maximising reaction of OPEC to a sudden increase in non‐OPEC productive capacity. Fig. 2. Open in new tabDownload slide OPEC and non‐OPEC Supply(thousands of barrels per day)and OPEC Market Share Source. EIA (2008) Fig. 2. Open in new tabDownload slide OPEC and non‐OPEC Supply(thousands of barrels per day)and OPEC Market Share Source. EIA (2008) Our view of the oil market is consistent with the empirical evidence in Griffin (1985), Jones (1990) and Dahl and Yucel (1991) who find that OPEC’s behaviour is closer to that of a cartel than a confederation of competitive suppliers. At the same time we acknowledge that there are alternative views of the oil market, such as those held by Kilian (forthcoming) or Almoguera and Herrera (2007), who are more sceptical of OPEC’s role as a cartel. In this Section we sketch a compact representation of the more important equations of our model, expressed in terms of log‐deviations from the efficient equilibrium. In order to treat the household sector equally with the other four types of agents (final goods firms, monetary authority, OPEC and non‐OPEC producers), we include a shock to the time discount factor as an additional source of aggregate fluctuation. 3.1. Dynamic IS Curve Log‐linearising the consumer’s Euler equation, replacing consumption with final goods value added (that is, GDP), and casting the resulting expression in deviation from the efficient allocation, we obtain (1) where is the (log) distance between actual value added and its efficient level (we refer to it as the ‘output gap’ for simplicity). The IS curve thus relates the current output gap positively to its expected future level, and negatively to the distance between the ex ante real interest rate and the efficient real interest rate , defined as the expected growth rate of efficient GDP. In equilibrium, the latter is given by the expression (2) which depends negatively on shocks to TFP, , and productivity in the oil sector, , and positively on the shock to the discount factor , where so is the oil elasticity of gross output. The shocks and are assumed to follow independent stationary AR(1) processes (3) where and ρb are persistence parameters, and and are iid innovations to US total factor productivity, oil sector production technology and the time discount factor, all of them mean zero and with standard deviations σa, σz and σb, respectively. As a robustness check, we will also estimate our model with a hybrid backward‐ and forward‐looking IS curve of the form (4) The latter equation is obtained by assuming that households form habits in consumption, where h ∈ [0,1] is an ‘external habit’ parameter. When h = 0, the above equation reduces to the more standard forward‐looking IS curve (1). 3.2. Phillips Curve Aggregating the optimal staggered price‐setting decision of final goods firms, we obtain the following first‐order approximation to the dynamics of inflation around the deterministic steady‐state with zero inflation (5) where πt denotes inflation, the output gap, is the optimal oil price markup (determined below), β is the mean time discount factor; and parameter λ is related to the structural parameters of the underlying model as follows (6) where ψ is the inverse of the Frisch labour supply elasticity, μ is the average markup in the final goods sector, 1 − θ is the frequency of price adjustment and sl is the labour elasticity of gross output. Notice that the oil price markup enters the Phillips curve like a ‘cost‐push’ term. Namely, a rise in the oil price markup leads to a rise in inflation and/or a fall of the output gap, implying a trade‐off between the two policy objectives. This is in contrast with the case of perfect competition in the oil sector (or exogenous oil price), in which oil price shifts are necessarily associated with an opposite movement in the efficient level of output and imply no tension between inflation and output gap stabilisation; for more details we refer the reader to Nakov and Pescatori (2007). Iterating the Phillips curve forward, we obtain the expression (7) which shows that inflation is a weighted average of current and expected future output gaps and oil price markups. 3.3. Monetary Policy The central bank follows a Taylor‐type rule of the form (8) where πt is inflation, is the output gap, is a zero mean iid monetary policy shock, and φi, φ and φy are policy reaction coefficients. Note that we allow for monetary policy to react to the output gap besides inflation, which in our model is an appropriate objective for a central bank concerned with the welfare of the representative household. 3.4. Oil Sector In Nakov and Pescatori (2007) we model OPEC as a dominant supplier of oil which seeks to maximise the welfare of its owners, internalising the effect of its pricing decision on global output and oil demand. Operating alongside a competitive fringe of price‐taking oil suppliers, the dominant oil exporter sells its output to an oil importing country (the US), which uses it to produce final goods. A first‐order approximation of the optimal oil price setting rule of the dominant oil supplier takes the form (9) where is a vector collecting all exogenous shocks and γ is a row vector of non‐linear functions of the structural parameters of the model. Notice that while the behaviour of households and firms of the oil importer is fully forward‐looking in the model, the optimal commitment solution of OPEC’s problem is history‐dependent. In particular, it is a function of past value added, , and nominal interest rate, , both of which are state variables; in addition, it depends on past promises about future oil supply, captured by the vector of Lagrange multipliers. Competitive fringe producers seek to maximise profits while taking the oil price as given. In Nakov and Pescatori (2007) we show that, in equilibrium, competitive fringe output is an increasing function of the oil price , oil‐sector technology , and the shock to fringe capacity (10) The total capacity of competitive fringe producers is allowed to vary according to a stationary AR(1) process with persistence ρ, (11) where is an iid innovation with mean zero and standard deviation σ. We make this allowance to capture the fact that some oil fields of the fringe are used up, while new ones are discovered and so the total amount of oil recoverable by the competitive fringe is not constant over time. In Nakov and Pescatori (2007) we evaluate the effects of a transitory change in the availability of oil outside OPEC’s control on the equilibrium oil price and macroeconomic aggregates. As we show there, it is the only shock in the model which induces a negative correlation between the supply of OPEC and the output of the competitive fringe, a feature of the data which is prominent in the 1980s and early 1990s (see Figure 2). 3.5. What Factors Could Lead to Reduced Volatility? We illustrate how different factors may contribute to the volatility moderation of different variables based on the above model. Perhaps the simplest explanation could be that the distribution of real disturbances hitting the economy has changed so that real shocks have become smaller on average. Notice that smaller real shocks would reduce the volatility of , while smaller oil sector shocks in particular are likely to diminish the variance of the oil price markup, . Since these are the two main driving variables in our model, for any given monetary policy and oil elasticity of production, the volatility of output, inflation and the interest rate would be reduced. An alternative (or complementary) explanation has to do with better monetary policy. This includes smaller monetary surprises ( shocks), as well as a more stabilising policy rule. Smaller monetary shocks reduce the volatility of the interest rate, which is transmitted through the IS and Phillips curves to actual output and inflation. At the same time stronger systematic reaction of the policy instrument to inflation and output deviations from target result in better stabilisation of these variables over the cycle.7 Finally, part of the moderation may be due to the fact that oil – perhaps once an important source of volatility – now accounts for a smaller fraction of output compared to the past. The latter can be due to increased energy efficiency and diversification away from oil‐intensive sectors. The oil elasticity of production affects the volatility of as well as the coefficient on the cost‐push term in the Phillips curve. Other things equal, a smaller oil elasticity of production is likely to reduce the volatility of output and the pass‐through from the oil price to inflation. To see how the oil elasticity of production affects the inflation–output gap tradeoff, notice that a policy of strict price stability (πt = 0) implies (12) while a policy aimed at strict output gap stability implies, (13) In both cases the extent to which stabilising one variable induces inefficient fluctuations in the other is a function of the oil elasticity of production. Finally, the oil elasticity of production affects the elasticity of demand for OPEC’s oil and thus the volatility of the oil price markup itself. 4. Data and Methodology We assess the extent to which the macroeconomic moderation in the US can be explained by changes in oil shocks and the oil elasticity of production, by simulating counterfactually the model of Nakov and Pescatori (2007) estimated via Bayesian techniques for the periods pre and post 1984. Our estimation methodology is similar to Rabanal and Rubio‐Ramírez (2005), Gali and Rabanal (2004), An and Schorfheide (2007) and Smets and Wouters (2007). The observable variables (whose volatility change we want to explain) are US GDP growth, inflation, the nominal interest rate and the percentage change of the real price of oil. Quarterly data on real GDP, the GDP deflator, the Federal Funds rate and the West Texas Intermediate oil price from 1970:I to 2007:IV are taken from FRED II (2007).8 GDP growth and inflation are computed as quarterly percentage changes of real GDP and the GDP deflator;9 the nominal interest rate is converted to quarterly frequency to render it consistent with the model; the oil price is deflated by the GDP deflator and cast in quarterly percentage changes. The resulting series are demeaned by their sub‐sample means prior to estimation. Since our model is meant to describe the behaviour of OPEC we start our sample in 1970 which is when the cartel started gaining more power. Note, however, that the market power of OPEC is endogenous in the model, that is, OPEC’s market share and price markup fluctuate in response to fundamental shocks. The model is therefore particularly well suited to account for shifts between periods with more competitive and periods with more monopolistic oil markets. The sample is split in 1984:I. This corresponds to the estimated break in US output volatility by McConnell and Perez‐Quiros (2000), Cecchetti et al. (2006) and others. A break in inflation volatility was found around that date as well (Kahn et al., 2002); a break in the oil–GDP link (Hooker, 1999) and the oil–inflation relationship (Hooker, 2002) was identified around 1981; and a break in the conduct of monetary policy around 1979–82 (Clarida et al. 2000). We fix several parameters of the model based on historical averages over the full sample (as in the case of the time discount factor), or on values which are standard in the literature (as with the elasticity of substitution among final goods). These calibrated parameter values are given in Table 2. Table 2 
Calibrated Parameters Parameter . . . Based on . Quarterly discount factor β 0.9926 Aver. annual real rate 3% Elast. of subst. among varieties ε 7.66 Aver. markup 15% Mean of non‐OPEC capacity 4.93e−3 Aver. OPEC share 40% Production function parameters (for gross output) Capital elasticity of output sk 1/3 Aver. cost share of capital Oil elasticity of output, 1970‐1983 so 0.0472 Aver. cost share of oil Oil elasticity of output, 1984‐2007 so 0.0264 Aver. cost share of oil Parameter . . . Based on . Quarterly discount factor β 0.9926 Aver. annual real rate 3% Elast. of subst. among varieties ε 7.66 Aver. markup 15% Mean of non‐OPEC capacity 4.93e−3 Aver. OPEC share 40% Production function parameters (for gross output) Capital elasticity of output sk 1/3 Aver. cost share of capital Oil elasticity of output, 1970‐1983 so 0.0472 Aver. cost share of oil Oil elasticity of output, 1984‐2007 so 0.0264 Aver. cost share of oil Open in new tab Table 2 
Calibrated Parameters Parameter . . . Based on . Quarterly discount factor β 0.9926 Aver. annual real rate 3% Elast. of subst. among varieties ε 7.66 Aver. markup 15% Mean of non‐OPEC capacity 4.93e−3 Aver. OPEC share 40% Production function parameters (for gross output) Capital elasticity of output sk 1/3 Aver. cost share of capital Oil elasticity of output, 1970‐1983 so 0.0472 Aver. cost share of oil Oil elasticity of output, 1984‐2007 so 0.0264 Aver. cost share of oil Parameter . . . Based on . Quarterly discount factor β 0.9926 Aver. annual real rate 3% Elast. of subst. among varieties ε 7.66 Aver. markup 15% Mean of non‐OPEC capacity 4.93e−3 Aver. OPEC share 40% Production function parameters (for gross output) Capital elasticity of output sk 1/3 Aver. cost share of capital Oil elasticity of output, 1970‐1983 so 0.0472 Aver. cost share of oil Oil elasticity of output, 1984‐2007 so 0.0264 Aver. cost share of oil Open in new tab 4.1. Calibration of the Oil Elasticity of Gross Output One of the parameters which we calibrate is the oil elasticity of gross output, that is, the exponent of oil input in a Cobb‐Douglas production function for final goods (denoted by so in (28) in the Appendix). In our model, this elasticity need not be equal (or even proportional) to the cost share of oil in production. Cobb‐Douglas technology together with cost minimisation by monopolistic firms imply that the constant oil elasticity of production times the current marginal cost of firms must equal the cost share of oil in gross output, (14) where PotOt is the nominal value of oil inputs, PtQt is nominal gross output, and mct is the time‐varying marginal cost of final goods firms.10 Note that the formula is consistent with a constant oil elasticity of production so, as long as movements in marginal cost shadow the fluctuations in the oil cost share observed in the data. Since this relationship holds in every period, it holds also on average, that is (15) where average marginal cost equals the inverse of the steady‐state markup, mc ≡ E(mct) = μ−1 = (ε − 1)/ε and ε is the constant elasticity of substitution among product varieties (which we fix at 7.66 in both samples). Note also that nominal gross output in our model is equal to value added plus the dollar value of oil inputs: PQ = PY + PoO. Thus, given data on nominal GDP (PY), the oil price (Po) and the quantity of oil used in the US (O),11 we can infer the constant oil elasticity of production in each sub‐sample based on the sub‐sample average of the cost share of oil in nominal gross output, (16) where t runs from 1970 to 1983 in the first sample and from 1984 to 2007 in the second. In this way we obtain an elasticity of 0.0472 in the first period and 0.0264 in the second, which we fix prior to estimation. Our calibrated elasticity in the second sample is close to the value of 0.02 obtained by Rotemberg and Woodford (1996) using a related but somewhat different approach. These authors add up the average nominal value added in oil extraction and the average value of petroleum imports as a share of GDP, obtaining a share of 0.034.12 They then round up this number to 0.04 to account for other energy inputs that might be close substitutes to oil. Assuming that materials account for 50% of total cost, they infer an oil elasticity of gross output of 0.02. Evidence of a declining cost share of oil in GDP and in consumption is found in a number of studies (Blanchard and Galí, 2007; Edelstein and Kilian, 2007a,b). In particular, Edelstein and Kilian (2007b) show the evolution of the energy share in GDP, which declined from around 5% in 1981, to 1% in 1998, before rising to 3.3% in 2005. These movements in the energy cost share are not inconsistent with a constant oil elasticity of production over the sub‐samples, as shown by (14) and (15) above. While the cost share in our model, as in the data, fluctuates over time (being affected among other things by the price of oil), it does so around different sub‐sample means. Under our assumptions, the latter is enough to infer the oil elasticity of production, which is the structural parameter of interest. 4.2. Estimation Procedure The above procedure leaves us with fourteen parameters to estimate: the frequency of price adjustment (θ), the Frisch labour supply elasticity (ψ), the parameters of the monetary policy rule (φi, φ, φy), the shocks’ autoregressive parameters (ρa, ρb, ρz, ρ) and standard deviations of the innovations (σa, σb, σz, σ, σr). In the case with a hybrid IS curve, there is an additional parameter h, measuring the degree of external habit formation by households. We approximate our model to first‐order and solve it with a standard method for linear rational expectations models (Sims, 2002; Klein, 2000). Given the state‐space representation, we use the Kalman filter to evaluate the likelihood of the four observable variables. From Bayes’ rule the posterior density function is proportional to the product of the likelihood and the prior density of the parameters. We use a random walk Metropolis‐Hastings algorithm to obtain 1,000,000 draws from the posterior distribution. We follow Benati’s (2008) approach to obtain a scale for the jumping distribution which yields an acceptance rate of around 0.23. The posterior distributions are obtained by discarding the first two‐thirds of the draws and then keeping one draw for every 100 of the remaining draws to break the serial correlation. Once we obtain the estimates for each sample period, we perform counterfactual simulations isolating the effect of a change in a single factor (e.g. the oil elasticity of production) on the volatility moderation. 5. Priors and Estimation Results 5.1. Choice of Priors The first four columns of Table 3 shows the assumed prior densities for the parameters whose posterior distributions we want to characterise. We use the same prior densities for each parameter in both samples, except for the parameter on inflation in the monetary policy rule. For this parameter we assume a normal (1.5, 0.5) distribution in the second sample but a gamma prior with mean 1.1 and a standard deviation of 0.5 in the first sample. Following Lubik and Schorfheide (2004) and Justiniano and Primiceri (2007), this assigns roughly equal probability on the inflation coefficient being either less or greater than one, while restricting it to be positive.13 Table 3 
Prior and Posterior Distributions Parameter . Prior distribution . Posterior distribution . Density and domain . Mean . Std . Mean . Std . Mode . (a) 1970–83 θ Beta [0,1) 0.60 0.10 0.614 0.068 0.622 ψ Gamma 1.00 0.25 0.961 0.224 0.897 φi Normal 0.60 0.10 0.542 0.079 0.537 φ Gamma 1.10 0.50 2.438 0.359 2.224 φy Normal 0.50 0.125 0.545 0.108 0.531 ρa Beta [0,1) 0.90 0.05 0.958 0.017 0.969 ρb Beta [0,1) 0.90 0.05 0.890 0.035 0.896 ρz Beta [0,1) 0.90 0.05 0.917 0.032 0.927 ρ Beta [0,1) 0.90 0.05 0.926 0.031 0.937 100σa Inv. Gamma 0.70 ∞ 1.359 0.127 1.331 100σb Inv. Gamma 0.70 ∞ 2.762 0.599 2.207 100σz Inv. Gamma 10.0 ∞ 21.27 2.436 21.33 100σ Inv. Gamma 10.0 ∞ 34.95 6.763 30.73 100σr Inv. Gamma 0.10 ∞ 0.530 0.068 0.494 (b) 1984–2007 θ Beta [0,1) 0.60 0.10 0.473 0.063 0.477 ψ Gamma 1.00 0.25 1.070 0.238 1.009 φi Normal 0.60 0.10 0.676 0.059 0.691 φ Normal 1.50 0.50 3.191 0.295 3.100 φy Normal 0.50 0.125 0.535 0.098 0.539 ρa Beta [0,1) 0.90 0.05 0.978 0.010 0.983 ρb Beta [0,1) 0.90 0.05 0.951 0.015 0.951 ρz Beta [0,1) 0.90 0.05 0.881 0.033 0.882 ρ Beta [0,1) 0.90 0.05 0.954 0.018 0.960 100σa Inv. Gamma 0.70 ∞ 0.630 0.045 0.620 100σb Inv. Gamma 0.70 ∞ 2.133 0.516 1.862 100σz Inv. Gamma 10.0 ∞ 14.92 1.596 15.18 100σ Inv. Gamma 10.0 ∞ 25.43 4.721 23.40 100σr Inv. Gamma 0.10 ∞ 0.231 0.034 0.212 Parameter . Prior distribution . Posterior distribution . Density and domain . Mean . Std . Mean . Std . Mode . (a) 1970–83 θ Beta [0,1) 0.60 0.10 0.614 0.068 0.622 ψ Gamma 1.00 0.25 0.961 0.224 0.897 φi Normal 0.60 0.10 0.542 0.079 0.537 φ Gamma 1.10 0.50 2.438 0.359 2.224 φy Normal 0.50 0.125 0.545 0.108 0.531 ρa Beta [0,1) 0.90 0.05 0.958 0.017 0.969 ρb Beta [0,1) 0.90 0.05 0.890 0.035 0.896 ρz Beta [0,1) 0.90 0.05 0.917 0.032 0.927 ρ Beta [0,1) 0.90 0.05 0.926 0.031 0.937 100σa Inv. Gamma 0.70 ∞ 1.359 0.127 1.331 100σb Inv. Gamma 0.70 ∞ 2.762 0.599 2.207 100σz Inv. Gamma 10.0 ∞ 21.27 2.436 21.33 100σ Inv. Gamma 10.0 ∞ 34.95 6.763 30.73 100σr Inv. Gamma 0.10 ∞ 0.530 0.068 0.494 (b) 1984–2007 θ Beta [0,1) 0.60 0.10 0.473 0.063 0.477 ψ Gamma 1.00 0.25 1.070 0.238 1.009 φi Normal 0.60 0.10 0.676 0.059 0.691 φ Normal 1.50 0.50 3.191 0.295 3.100 φy Normal 0.50 0.125 0.535 0.098 0.539 ρa Beta [0,1) 0.90 0.05 0.978 0.010 0.983 ρb Beta [0,1) 0.90 0.05 0.951 0.015 0.951 ρz Beta [0,1) 0.90 0.05 0.881 0.033 0.882 ρ Beta [0,1) 0.90 0.05 0.954 0.018 0.960 100σa Inv. Gamma 0.70 ∞ 0.630 0.045 0.620 100σb Inv. Gamma 0.70 ∞ 2.133 0.516 1.862 100σz Inv. Gamma 10.0 ∞ 14.92 1.596 15.18 100σ Inv. Gamma 10.0 ∞ 25.43 4.721 23.40 100σr Inv. Gamma 0.10 ∞ 0.231 0.034 0.212 Open in new tab Table 3 
Prior and Posterior Distributions Parameter . Prior distribution . Posterior distribution . Density and domain . Mean . Std . Mean . Std . Mode . (a) 1970–83 θ Beta [0,1) 0.60 0.10 0.614 0.068 0.622 ψ Gamma 1.00 0.25 0.961 0.224 0.897 φi Normal 0.60 0.10 0.542 0.079 0.537 φ Gamma 1.10 0.50 2.438 0.359 2.224 φy Normal 0.50 0.125 0.545 0.108 0.531 ρa Beta [0,1) 0.90 0.05 0.958 0.017 0.969 ρb Beta [0,1) 0.90 0.05 0.890 0.035 0.896 ρz Beta [0,1) 0.90 0.05 0.917 0.032 0.927 ρ Beta [0,1) 0.90 0.05 0.926 0.031 0.937 100σa Inv. Gamma 0.70 ∞ 1.359 0.127 1.331 100σb Inv. Gamma 0.70 ∞ 2.762 0.599 2.207 100σz Inv. Gamma 10.0 ∞ 21.27 2.436 21.33 100σ Inv. Gamma 10.0 ∞ 34.95 6.763 30.73 100σr Inv. Gamma 0.10 ∞ 0.530 0.068 0.494 (b) 1984–2007 θ Beta [0,1) 0.60 0.10 0.473 0.063 0.477 ψ Gamma 1.00 0.25 1.070 0.238 1.009 φi Normal 0.60 0.10 0.676 0.059 0.691 φ Normal 1.50 0.50 3.191 0.295 3.100 φy Normal 0.50 0.125 0.535 0.098 0.539 ρa Beta [0,1) 0.90 0.05 0.978 0.010 0.983 ρb Beta [0,1) 0.90 0.05 0.951 0.015 0.951 ρz Beta [0,1) 0.90 0.05 0.881 0.033 0.882 ρ Beta [0,1) 0.90 0.05 0.954 0.018 0.960 100σa Inv. Gamma 0.70 ∞ 0.630 0.045 0.620 100σb Inv. Gamma 0.70 ∞ 2.133 0.516 1.862 100σz Inv. Gamma 10.0 ∞ 14.92 1.596 15.18 100σ Inv. Gamma 10.0 ∞ 25.43 4.721 23.40 100σr Inv. Gamma 0.10 ∞ 0.231 0.034 0.212 Parameter . Prior distribution . Posterior distribution . Density and domain . Mean . Std . Mean . Std . Mode . (a) 1970–83 θ Beta [0,1) 0.60 0.10 0.614 0.068 0.622 ψ Gamma 1.00 0.25 0.961 0.224 0.897 φi Normal 0.60 0.10 0.542 0.079 0.537 φ Gamma 1.10 0.50 2.438 0.359 2.224 φy Normal 0.50 0.125 0.545 0.108 0.531 ρa Beta [0,1) 0.90 0.05 0.958 0.017 0.969 ρb Beta [0,1) 0.90 0.05 0.890 0.035 0.896 ρz Beta [0,1) 0.90 0.05 0.917 0.032 0.927 ρ Beta [0,1) 0.90 0.05 0.926 0.031 0.937 100σa Inv. Gamma 0.70 ∞ 1.359 0.127 1.331 100σb Inv. Gamma 0.70 ∞ 2.762 0.599 2.207 100σz Inv. Gamma 10.0 ∞ 21.27 2.436 21.33 100σ Inv. Gamma 10.0 ∞ 34.95 6.763 30.73 100σr Inv. Gamma 0.10 ∞ 0.530 0.068 0.494 (b) 1984–2007 θ Beta [0,1) 0.60 0.10 0.473 0.063 0.477 ψ Gamma 1.00 0.25 1.070 0.238 1.009 φi Normal 0.60 0.10 0.676 0.059 0.691 φ Normal 1.50 0.50 3.191 0.295 3.100 φy Normal 0.50 0.125 0.535 0.098 0.539 ρa Beta [0,1) 0.90 0.05 0.978 0.010 0.983 ρb Beta [0,1) 0.90 0.05 0.951 0.015 0.951 ρz Beta [0,1) 0.90 0.05 0.881 0.033 0.882 ρ Beta [0,1) 0.90 0.05 0.954 0.018 0.960 100σa Inv. Gamma 0.70 ∞ 0.630 0.045 0.620 100σb Inv. Gamma 0.70 ∞ 2.133 0.516 1.862 100σz Inv. Gamma 10.0 ∞ 14.92 1.596 15.18 100σ Inv. Gamma 10.0 ∞ 25.43 4.721 23.40 100σr Inv. Gamma 0.10 ∞ 0.231 0.034 0.212 Open in new tab We should stress that the conditions for local determinacy of equilibria in our model are not the standard ones. In particular, φ > 1 is not a necessary condition for local uniqueness, and indeed there is a large region of determinacy for values of φ below 1. The reason is that, different from the standard three equation New Keynesian framework, in our model the Phillips curve includes an additional term – the oil price markup – which responds (optimally) to other endogenous variables, and in particular to past output gaps. This explains why we can solve and estimate our model for values of φ below 1. For the other parameters of the monetary policy rule we use normal prior densities in both samples. For the price adjustment probability we assume a beta prior with mean 0.6 and standard deviation of 0.1.14 For the inverse Frisch labour supply elasticity we assume a gamma prior with mean 1 and standard deviation of 0.25.15 The autocorrelation coefficients of the shocks are assumed to be distributed beta with mean 0.9 and standard deviation of 0.05. And for the standard deviation of the innovations we assume an inverted gamma distribution (which ensures non‐negativity) fixing the means around the calibrated values in Nakov and Pescatori (2007). 5.2. Estimation Results Comparing the two sets of estimated posterior modes in Table 3 we notice several important parameter shifts. First, the mode of the inflation coefficient of the monetary policy rule is larger in the second sample, implying that monetary policy was reacting more strongly to inflation compared to the first period. At the same time, the estimated standard deviation of the interest rate innovation in the pre‐1984 sample is more than double that in the post‐1984 sample, suggesting that policy was more erratic in the first period. Secondly, the mode of the Calvo (1983) parameter governing the frequency of price adjustment is smaller in the post‐1984 period, suggesting that prices have become more flexible. At first sight this may seem counterintuitive given that inflation was higher in the first period, which – other things equal – should call for more frequent price changes. However, the real cost of price adjustment itself may well have decreased in the second sample, owing to improvements in the technology of calculating and posting new prices, making prices more flexible. An alternative explanation is that the fall of the Calvo parameter is a way for the model to capture the reduced inflation persistence in the second period, given that the model lacks price indexation.16 Third, there is evidence of changes in the volatility (and persistence) of real shocks. In particular, the volatility of the US technology innovation was cut by half in the post‐1984 period, while preference shocks became more persistent. Finally, oil sector shocks became smaller in the latter period. Table 4 shows that the estimated model does quite a good job at matching the second moments and the post‐1984 volatility reduction of the variables of interest. To be more precise, although it slightly overestimates the volatility of GDP growth and underestimates the volatility of the oil price in both periods, the model matches the post‐1984 reduction in the volatility of both variables quite well. And while the volatility moderation of the nominal interest rate is somewhat overestimated, the volatility of inflation in both periods (and hence its reduction) is matched pretty well. Table 4 
Second Moments of Model and Data . 1970–83 . 1984–2007 . Volat. reduction . . Data . Model . Data . Model . Data . Model . Inflation 0.57 0.61 0.25 0.25 57% 58% GDP growth 1.20 1.64 0.52 0.72 57% 56% Interest rate 0.88 0.89 0.57 0.43 35% 51% Real oil price 19.0 16.6 13.0 12.0 31% 28% . 1970–83 . 1984–2007 . Volat. reduction . . Data . Model . Data . Model . Data . Model . Inflation 0.57 0.61 0.25 0.25 57% 58% GDP growth 1.20 1.64 0.52 0.72 57% 56% Interest rate 0.88 0.89 0.57 0.43 35% 51% Real oil price 19.0 16.6 13.0 12.0 31% 28% Open in new tab Table 4 
Second Moments of Model and Data . 1970–83 . 1984–2007 . Volat. reduction . . Data . Model . Data . Model . Data . Model . Inflation 0.57 0.61 0.25 0.25 57% 58% GDP growth 1.20 1.64 0.52 0.72 57% 56% Interest rate 0.88 0.89 0.57 0.43 35% 51% Real oil price 19.0 16.6 13.0 12.0 31% 28% . 1970–83 . 1984–2007 . Volat. reduction . . Data . Model . Data . Model . Data . Model . Inflation 0.57 0.61 0.25 0.25 57% 58% GDP growth 1.20 1.64 0.52 0.72 57% 56% Interest rate 0.88 0.89 0.57 0.43 35% 51% Real oil price 19.0 16.6 13.0 12.0 31% 28% Open in new tab 6. Macroeconomic Implications 6.1. What Factors Explain the Great Moderation? In this Section we attribute the volatility reduction implied by the model (the last column of Table 4) to counterfactual changes in each factor in isolation, including: the oil elasticity of production; oil shocks; the monetary policy rule; monetary policy shocks; total factor productivity shocks; and other factors (including shifts in the frequency of price adjustment, and in the Frisch labour supply elasticity, as well as a residual due to the interaction of all factors). Table 5 reports the percentage contribution to the total volatility reduction achieved by a change in a single factor keeping the rest of the parameters at their pre‐1984 values.17 For instance, had the oil elasticity of production in the period 1970–83 been at its post‐1984 value (that is, 0.0264 instead of 0.0472), inflation would have been 18% less volatile, while GDP growth would have been 10% less volatile. Expressed as a percentage of the predicted volatility reduction for these variables (the last column of Table 4), the diminished reliance on oil alone can explain around 32% of the reduced volatility of inflation and 18% of the smaller volatility of GDP growth. By the same token, the diminished incidence of major oil shocks is responsible for 17% of the inflation volatility moderation and 11% of the reduced volatility of GDP growth. This suggests that oil‐related factors have played an important role in the stabilisation, especially of inflation. Table 5 
Contributions to the Reduced Volatility (%) . Oil . Monet. policy . TFP shock . Other factors . share . shocks . rule . shock . . Inflation 32 17 40 11 2 −2 GDP growth 18 11 0 4 57 10 Interest rate 12 3 37 4 8 36 Real oil price −3 101 0 0 0 2 . Oil . Monet. policy . TFP shock . Other factors . share . shocks . rule . shock . . Inflation 32 17 40 11 2 −2 GDP growth 18 11 0 4 57 10 Interest rate 12 3 37 4 8 36 Real oil price −3 101 0 0 0 2 Open in new tab Table 5 
Contributions to the Reduced Volatility (%) . Oil . Monet. policy . TFP shock . Other factors . share . shocks . rule . shock . . Inflation 32 17 40 11 2 −2 GDP growth 18 11 0 4 57 10 Interest rate 12 3 37 4 8 36 Real oil price −3 101 0 0 0 2 . Oil . Monet. policy . TFP shock . Other factors . share . shocks . rule . shock . . Inflation 32 17 40 11 2 −2 GDP growth 18 11 0 4 57 10 Interest rate 12 3 37 4 8 36 Real oil price −3 101 0 0 0 2 Open in new tab Nevertheless, we find that better monetary policy played the biggest role in reducing the volatility of inflation. In particular, the more aggressive policy reaction to inflation after 1984 accounts for 40% of its volatility decrease. In addition, smaller monetary shocks have contributed another 11% to reducing the volatility of inflation. However, we find that monetary policy played only a minor role in the reduced volatility of GDP growth (4%). Our calculations lead us to conclude that the main factor for the reduced volatility of GDP growth (contributing around 57%) has been a favourable shift in the distribution of productivity shocks in the US. By contrast, we find only a trivial role of productivity shocks in reducing the volatility of inflation (by 2%). The bottom line of this analysis is that the smaller ‘oil share’ and oil shocks have played an important role in the reduced volatility, especially of inflation, even if the other two factors – better monetary policy and smaller TFP shocks – have played the dominant role in the stabilisation of inflation and GDP growth respectively. 6.2. Changes in the Phillips Curve Hooker (2002) finds evidence of a break in standard (backward‐looking) core US inflation Phillips curves regressions, with oil price changes making a substantial contribution to core inflation before 1981 but little or no pass‐through since that time. Similarly, estimating the standard New Keynesian model via maximum likelihood, Ireland (2004) finds that ‘cost push’ shocks have become smaller since the 1980s. Our findings are in broad agreement with these claims (see Table 6). Indeed, they point to the decrease in the oil elasticity of production as a likely cause for the improvement in the Phillips curve tradeoff as inflation and the output gap have become more aligned with each other and less sensitive to oil price fluctuations. In particular, the last column of Table 6 shows that conditional on a 44% reduction of the oil elasticity of production from 4.7% to 2.6% (and keeping all other factors unchanged), the volatility of the output gap is reduced by around 40% and the coefficient of ‘pass‐through’ from the cost‐push term to inflation declines by around 45%. Thus, the decrease in the oil elasticity of production alone explains a 18% decline in the volatility of inflation (around a third of the total reduction). Table 6 
Changes in the Phillips Curve . . 1970–83 . 1984–2007 . Counter factual so . Elast. of oil in production so 0.047 0.026 0.026 Common slope coefficient λ 0.668 1.748 0.664 Oil markup pass‐through soλ 0.032 0.046 0.018 Oil markup volatility std 0.238 0.175 0.233 Oil markup persistence 0.934 0.939 0.933 Output gap coefficient (1 − so)λ 0.637 1.702 0.647 Output gap volatility std 0.012 0.005 0.007 Output gap persistence 0.886 0.913 0.779 . . 1970–83 . 1984–2007 . Counter factual so . Elast. of oil in production so 0.047 0.026 0.026 Common slope coefficient λ 0.668 1.748 0.664 Oil markup pass‐through soλ 0.032 0.046 0.018 Oil markup volatility std 0.238 0.175 0.233 Oil markup persistence 0.934 0.939 0.933 Output gap coefficient (1 − so)λ 0.637 1.702 0.647 Output gap volatility std 0.012 0.005 0.007 Output gap persistence 0.886 0.913 0.779 Open in new tab Table 6 
Changes in the Phillips Curve . . 1970–83 . 1984–2007 . Counter factual so . Elast. of oil in production so 0.047 0.026 0.026 Common slope coefficient λ 0.668 1.748 0.664 Oil markup pass‐through soλ 0.032 0.046 0.018 Oil markup volatility std 0.238 0.175 0.233 Oil markup persistence 0.934 0.939 0.933 Output gap coefficient (1 − so)λ 0.637 1.702 0.647 Output gap volatility std 0.012 0.005 0.007 Output gap persistence 0.886 0.913 0.779 . . 1970–83 . 1984–2007 . Counter factual so . Elast. of oil in production so 0.047 0.026 0.026 Common slope coefficient λ 0.668 1.748 0.664 Oil markup pass‐through soλ 0.032 0.046 0.018 Oil markup volatility std 0.238 0.175 0.233 Oil markup persistence 0.934 0.939 0.933 Output gap coefficient (1 − so)λ 0.637 1.702 0.647 Output gap volatility std 0.012 0.005 0.007 Output gap persistence 0.886 0.913 0.779 Open in new tab In addition, thanks mostly to smaller oil shocks, the volatility of the oil price markup itself has decreased by around 27% in the period after 1984. This, together with a stronger reaction of monetary policy to inflation since the mid‐1980s, has made it possible for monetary policy to stabilise better both the output gap and inflation. 6.3. Changes in the Relative Importance of Shocks Table 7 shows the asymptotic variance decomposition of the four variables of interest in the first and the second sample.18 Table 7 
Variance Decomposition, 1970–83, 1984–2007 . US shocks . Oil shocks . . Real . Nom. . . . . . . . . . (a) Inflation 2.74 33.3 15.7 12.2 36.0 GDP growth 69.1 1.10 4.99 22.7 2.13 Interest rate 8.70 69.3 4.44 0.27 17.3 Real oil price 0.05 0.00 0.04 87.1 12.8 (b) Inflation 0.80 37.4 41.5 1.18 19.0 GDP growth 77.8 0.42 2.13 17.5 2.14 Interest rate 3.30 87.4 1.10 1.08 7.11 Real oil price 0.02 0.00 0.01 87.7 12.3 . US shocks . Oil shocks . . Real . Nom. . . . . . . . . . (a) Inflation 2.74 33.3 15.7 12.2 36.0 GDP growth 69.1 1.10 4.99 22.7 2.13 Interest rate 8.70 69.3 4.44 0.27 17.3 Real oil price 0.05 0.00 0.04 87.1 12.8 (b) Inflation 0.80 37.4 41.5 1.18 19.0 GDP growth 77.8 0.42 2.13 17.5 2.14 Interest rate 3.30 87.4 1.10 1.08 7.11 Real oil price 0.02 0.00 0.01 87.7 12.3 Open in new tab Table 7 
Variance Decomposition, 1970–83, 1984–2007 . US shocks . Oil shocks . . Real . Nom. . . . . . . . . . (a) Inflation 2.74 33.3 15.7 12.2 36.0 GDP growth 69.1 1.10 4.99 22.7 2.13 Interest rate 8.70 69.3 4.44 0.27 17.3 Real oil price 0.05 0.00 0.04 87.1 12.8 (b) Inflation 0.80 37.4 41.5 1.18 19.0 GDP growth 77.8 0.42 2.13 17.5 2.14 Interest rate 3.30 87.4 1.10 1.08 7.11 Real oil price 0.02 0.00 0.01 87.7 12.3 . US shocks . Oil shocks . . Real . Nom. . . . . . . . . . (a) Inflation 2.74 33.3 15.7 12.2 36.0 GDP growth 69.1 1.10 4.99 22.7 2.13 Interest rate 8.70 69.3 4.44 0.27 17.3 Real oil price 0.05 0.00 0.04 87.1 12.8 (b) Inflation 0.80 37.4 41.5 1.18 19.0 GDP growth 77.8 0.42 2.13 17.5 2.14 Interest rate 3.30 87.4 1.10 1.08 7.11 Real oil price 0.02 0.00 0.01 87.7 12.3 Open in new tab Notably, the last two columns of both parts of the table reveal that the contribution of oil sector shocks to US inflation and GDP growth variability was stronger in the first sample and weaker in the second. In particular, oil productivity and fringe capacity shocks ( and ) together contributed to as much as 48% of inflation volatility and 25% of growth volatility in the period 1970–83. By contrast, in the period 1984–2007 oil shocks contributed less: around 20% of inflation volatility and 20% of growth volatility. Interestingly, the shock to oil sector productivity () turns out to be more important for the volatility of GDP growth (and the oil price), while the fringe capacity shock () is more relevant for the volatility of inflation (and the nominal interest rate). Turning to US‐originating disturbances, the shock to TFP () which accounts for the bulk of GDP growth volatility before 1984 has become even more important for GDP growth after that year (but has decreased its impact on inflation and the interest rate). The preference shock () was important for inflation and the nominal interest rate before 1984 and has become even more relevant for both variables since then; and the interest rate shock () has increased its relative importance for inflation (but has become less relevant for GDP growth and the interest rate). Finally, Figures 3 and 4 shows the imputed structural innovations. The shocks are signed so that a positive value is associated with an increase in the oil price. Interestingly, Figure 4 suggests that the recent persistent climb of the oil price (starting after the Asian crisis in 1997 and interrupted temporarily around the recession of 2001), reflects to a greater extent fringe capacity shocks (that is, a strengthening of OPEC due to reduced availability of oil outside its control) rather than changes in the marginal cost of oil production (which for much of the past 10 years seems to have decreased rather than increased). Fig. 4. Open in new tabDownload slide Imputed Structural Innovations, 1984–2007 Fig. 4. Open in new tabDownload slide Imputed Structural Innovations, 1984–2007 Fig. 3. Open in new tabDownload slide Imputed Structural Innovations, 1970–83 Fig. 3. Open in new tabDownload slide Imputed Structural Innovations, 1970–83 6.4. Robustness of the Results with a Hybrid IS Curve Some of the empirical DSGE literature has found an important role for habit formation in matching the strong autocorrelation of the output gap, mitigating the need for persistent structural shocks (Galí and Rabanal, 2004; Smets and Wouters, 2007). In this Section we test the sensitivity of our results to allowing individual household utility to depend on its consumption relative to an external habit, proportional to average consumption in the previous period. Formally, utility of consumption at time t is a function u(Ct − Ht), where Ht = hCt−1 and h is an external habit parameter. This assumption results in a log‐linearised IS curve of the form (17) with weight h/(1 + h) on the lagged output gap. Notice that with h = 0 the above equation reduces to the more standard forward‐looking IS curve (1). We assume a beta prior for h with mean 0.5 and standard deviation 0.2. Our posterior mode estimate is 0.21 for the first sample and 0.26 for the second. Thus, while we find a role for external habit formation in consumption, it is somewhat less important in our model than in the model of Galí and Rabanal (2004), who obtain a posterior mean for h of 0.4 assuming a uniform prior between 0 and 1. Our main conclusions are robust to assuming a hybrid IS curve with the estimated degree of habit persistence. Namely, we find that the smaller oil elasticity of production was responsible for 31%, and smaller oil shocks for 11%, of the reduced volatility of inflation (compared to 32% and 17% without habit formation). Likewise, we find that the smaller oil elasticity of production contributed 16%, and smaller oil shocks 10%, to reducing the volatility of output growth (compared to 18% and 11% before). As before, smaller TFP shocks are responsible for the bulk of the reduction in GDP growth volatility (55%). And better monetary policy explains around half of the reduced volatility of inflation. Overall, we conclude that our main findings are not affected by considering a hybrid IS curve instead of the more traditional purely forward‐looking IS equation. 7. Comparison of the Results with the Literature Compared to Clarida et al. (2000), our analysis ascribes to monetary policy only a minor role in the stabilisation of GDP growth (but an important role in stabilising inflation). This could be for several reasons. One is the proximity of our model to the RBC paradigm: apart from nominal price rigidities (with a Calvo parameter estimated around 0.6 in the first period and 0.5 in the second period) and imperfect competition in oil, our model features no other imperfections or real rigidities (e.g. as in Blanchard and Gali, 2007) that would enhance the importance of the interest rate channel. Second, we assume that the central bank reacts to the output gap (and not to output growth), which in our model is a relevant target variable for a central bank concerned with the welfare of the representative household. Given this rule, however, better monetary policy does not necessarily imply smaller output volatility, especially if real disturbances imply large fluctuations in the efficient level of output. Third, the estimated reaction to the output gap is quite similar across the two samples (it is the reaction to inflation which increases substantially in the second period), so even if the fluctuations of efficient output were not large, the post‐1984 rule may not have stabilised output much better than the pre‐1984 one. While it may not be very appealing intellectually to attribute the stabilisation of growth to unexplained changes in productivity, a similar conclusion has been reached in a number of other studies, e.g., by Stock and Watson (2002), Justiniano and Primiceri (2006) and Leduc and Sill (2007). Based on a calibrated model with exogenous oil supply, Leduc and Sill (2007) in particular conclude that improved monetary policy can account for 45% of the decline in inflation volatility but only 5% to 10% of the reduction in output volatility, the bulk of which can be explained by smaller TFP shocks. These findings are similar to ours. However, our results are distinct when it comes to ascribing the volatility moderation to oil‐related factors. While we find that smaller oil shocks have contributed to 17% of the diminished inflation volatility and 11% of the reduced GDP growth volatility, Leduc and Sill claim that oil shocks became larger after 1984 and hence pushed in the direction of raising overall volatility. This discrepancy is due to the different way in which Leduc and Sill identify oil shocks by treating oil supply as constant except for four episodes of military conflict, with larger average production drops after 1984. This is in contrast to our modelling of the oil sector from first principles, and identifying oil shocks as structural disturbances to oil productivity or fringe capacity. In addition, we find that the reduced oil elasticity of production can explain about 32% of the reduced volatility of inflation and 18% of the decrease in volatility of GDP growth, a question which is not addressed by Leduc and Sill. Blanchard and Gali (2007) introduce real wage rigidities to generate an inflation–output gap trade‐off. They demonstrate how a reduction in the oil share in consumption and production shifts inward the policy frontier and goes some way towards explaining the observed reduction in inflation and output volatility. Our model in comparison generates a policy tradeoff by assuming imperfect competition in the oil market while ignoring real wage rigidities. We also attempt to quantify more precisely the contribution of each factor by estimating the model with Bayesian techniques and performing counterfactual simulations. Canova (2007) investigates the causes of the Great Moderation in the US by estimating the benchmark small scale New Keynesian model with Bayesian techniques over rolling samples. He finds that even though changes in the parameters of the private sector are largest, they cannot account by themselves for the full decline in volatility of output and inflation, while changes in the parameters of the policy rule and the covariance of the shocks can. Our findings are similar to Canova in that the bulk of the reduced volatility of GDP growth is attributed to smaller real shocks, while half of the inflation volatility moderation is due to better monetary policy. Yet we find that as much as a third of the inflation volatility moderation and 18% of the volatility reduction of output growth is attributable to the smaller oil elasticity of production, which is not directly measurable in the model estimated by Canova. Gali and Gambetti (2009) look for the sources of the Great Moderation using a VAR with time‐varying coefficients and stochastic volatility. They show that a significant fraction of the observed changes in co‐movements and impulse‐responses can be accounted for by a stronger reaction of monetary policy to inflation and an apparent end of short‐run increasing returns to labour. Herrera and Pesavento (2009) estimate a VAR in the spirit of Bernanke et al. (2004) identifying oil shocks through exclusion restrictions. They find that better monetary policy was responsible for around half of the reduced volatility of inflation and a quarter of the lower volatility of GDP growth, while oil shocks played only a minor role. On the other hand, using a VAR with time‐varying coefficients identified through sign restrictions, Canova and Gambetti (forthcoming) find no evidence that there was an increase in the response of the interest rate to inflation and overall conclude that monetary policy was marginally responsible for the Great Moderation. Indeed, recent work by Benati and Surico (forthcoming) casts doubt on the ability of VARs to distinguish between the ‘good policy’ and ‘good luck’ explanations for the Great Moderation. Finally, there are a number of alternative (or complementary) explanations for the reduced macroeconomic volatility. Kilian (forthcoming) and Kilian and Park (2007), for example, argue that the reduced impact of oil price shocks on the economy in recent years reflects mostly changes in the composition of oil price shocks. Herrera and Pesavento (2009) point to changes in inventory behaviour as an explanation for the reduced output volatility. Edelstein and Kilian (2007a,b) and Kilian (2008) identify declines in the employment and output share of the US automobile industry since the 1980s as a key factor. This latter explanation in particular is not inconsistent with the notion of a declining oil elasticity of gross output discussed in the present article. Compared with the above studies, our analysis based on a structural model assigns an important role to monetary policy, especially in reducing the volatility of inflation. At the same time we find a significant contribution of the reduced dependence on oil and of smaller structural disturbances in the oil sector to the stabilisation, especially of inflation. 8. Conclusions We assess the extent to which the increased macroeconomic stability in the US after 1984 can be accounted for by changes in oil shocks and the oil elasticity of production by taking the model of Nakov and Pescatori (2007) to the data with Bayesian techniques and performing counterfactual simulations. In doing so we nest two popular explanations for the Great Moderation, namely smaller non‐oil shocks, and better monetary policy. Our estimates indicate that oil played an important role in the volatility reduction, especially of inflation. In particular, we find that the diminished reliance on oil can explain around a third of the reduced volatility of inflation and 18% of the lower volatility of GDP growth. In turn, oil sector shocks alone can explain around 17% of the lower inflation volatility, and 11% of the reduced volatility of GDP growth. At the same time, around half of the reduced volatility of inflation is explained by better monetary policy alone, while 57% of the reduced volatility of GDP growth is attributable to smaller TFP shocks. Footnotes 1 " Specifically, Hooker (1999) found that two widely used transformations of the oil price do not Granger cause output in the post‐1980 period, while Hooker (2002) identified a structural break in core US inflation Phillips curves such that oil prices contributed substantially to core inflation before 1981 but since that time the pass‐through has been negligible. 2 " The ‘Great Moderation’ was noticed by Kim and Nelson (1999) and McConnell and Perez‐Quiros (2000) and its beginning is usually dated around 1984. Cecchetti et al. (2006) find evidence of volatility moderation in 16 out of 25 industrialised countries and Stock and Watson (2002) report similar evidence for 6 of the G‐7 countries; on the other hand, see Canova et al. (2007) for evidence that the Great Moderation has been more of an Anglo‐Saxon phenomenon. 3 " In our model ‘oil shocks’ refer to: (1) structural disturbances to productivity of the oil sector, and (2) disturbances to the capacity of non‐OPEC suppliers. We do not call them ‘oil price’, ‘oil supply’, or ‘oil demand’ shocks because in the model these are endogenous variables, responding simultaneously to any shock. See Section 3 for details. 4 " The structural parameter of interest is the oil elasticity of gross output, that is, the exponent of oil input in a Cobb‐Douglas production function for gross output (denoted by so in (28) in the Appendix). Under standard assumptions about firms’ objectives and technology, this parameter is related to the cost share of oil in GDP (see Section 5.1 for details). 5 " When testing the null hypothesis that the oil price is not Granger‐caused collectively by US output, unemployment, inflation, wages, money and import prices, Hamilton (1983) obtained a rejection at the 6% significance level. 6 " Not all studies fit the above classification. For example, Canova et al. (2007) claim that it is impossible to account for both the Great Inflation of the 1970s and the strong output growth in the 1990s with a single explanation. Using a different approach, Canova (2007) finds that the fall in variances of output and inflation had different causes, and that the quest for a single explanation is likely to be misplaced. See Section 7 for more on this. 7 " Strictly speaking, stronger reaction to the output gap would result in better alignment of output with its efficient level, which need not imply smaller volatility of the growth rate of output, especially if real shocks are large. 8 " The original series names are GDPC96, GDPDEF, FEDFUNDS and OILPRICE. 9 " Our model makes no difference between GDP deflator and CPI inflation. 10 " A similar formula obtains with a more general CES production function. 11 " Annual data on US petroleum consumption (in barrels per day) from 1960 to 2007 are available from the Energy Information Administration, US Dept. of Energy, http://www.eia.doe.gov/aer/txt/ptb1110.html. 12 " It is not clear what historical period Rotemberg and Woodford (1996) have used for the calculation of these average shares. 13 " The estimation results turn out to be almost identical if instead we assume the same normal prior density for the coefficient on inflation in both samples. 14 " This is consistent with the 13.9% median monthly frequency of regular price changes that Klenow and Kryvtsov (2008) find in US micro data. The mean frequency of regular price changes found by these authors is 29.9% per month. 15 " We base our estimation on the full model in which the Frisch labour supply elasticity enters in several equations independently from the Calvo parameter. Hence, we are able to identify these two parameters separately, unlike in the three equation New Keynesian model. 16 " We are grateful to a referee for making this point. 17 " We do not model transition dynamics here; Canova and Gambetti (forthcoming) propose an alternative method of performing counterfactual simulations based on re‐estimating all the model’s parameters conditional on the chosen counterfactual value for any given parameter. Essentially this amounts to treating all model parameters as reduced‐form rather than deep behavioural parameters independent of the experiment. While this can be a useful alternative methodology, we stick to the more standard approach of Stock and Watson (2002) treating our parameters as behavioural. 18 " This is obtained by solving the equation Σy = AΣyA′ + BΣuB′ in Σy, the unconditional variance of y, where yt is the solution to the linear rational expectations model of the form yt = Ayt−1 + But. It is thus the decomposition of the unconditional variance of endogenous variables, given that shocks occur in every period from today to infinity. References Adelman , M. ( 2002 ). ‘World oil production & prices, 1947‐2000’ , The Quarterly Review of Economics and Finance , vol. 42 ( 2 ), pp. 169 – 91 . Google Scholar Crossref Search ADS WorldCat Ahmed , S. , Levin , A. and Wilson , B. ( 2004 ). ‘Recent US macroeconomic stability: good policies, good practices, or good luck?’ , Review of Economics and Statistics , vol. 86 ( 3 ), pp. 824 – 32 . Google Scholar Crossref Search ADS WorldCat Almoguera , P. and Herrera , A.M. ( 2007 ). ‘Testing for the cartel in OPEC: noncooperative collusion or just noncooperative?’ , mimeo, Wayne State University . An , S. and Schorfheide , F. ( 2007 ). ‘Bayesian analysis of DSGE models’ , Econometric Reviews , vol. 26 ( 2‐4 ), pp. 113 – 72 . Google Scholar Crossref Search ADS WorldCat Barsky , R. B. and Kilian , L. ( 2001 ). ‘ Do we really know that oil caused the great stagflation? A monetary alternative ’, NBER Working Papers No. 8389 (July). Benati , L. ( 2008 ). ‘Investigating inflation persistence across monetary regimes’ , Quarterly Journal of Economics , vol. 123 ( 3 ), pp. 1005 – 60 . Google Scholar Crossref Search ADS WorldCat Benati , L. and Surico , P. (forthcoming). ‘ VAR analysis and the Great Moderation ’, American Economic Review . OpenURL Placeholder Text WorldCat Bernanke , B. S. , Gertler , M. and Watson , M. ( 1997 ). ‘Systematic monetary policy and the effects of oil price shocks’ , Brookings Papers on Economic Activity , 1997 ( 1 ), pp. 91 – 157 . Google Scholar Crossref Search ADS WorldCat Bernanke , B. S. , Gertler , M. and Watson , M. ( 2004 ). ‘Oil shocks and aggregate macroeconomic behavior: the role of monetary policy: a reply’ , Journal of Money, Credit and Banking , vol. 36 ( 2 ), pp. 287 – 91 . Google Scholar Crossref Search ADS WorldCat Blanchard , O. and Galí , J. ( 2007 ). ‘ The macroeconomic effects of oil price shocks: why are the 2000s so different from the 1970s? ’, NBER Working Paper No. 13368. Boivin , J. and Giannoni , M. ( 2006 ). ‘Has monetary policy become more effective?’ , Review of Economics and Statistics , vol. 88 ( 3 ), pp. 445 – 62 . Google Scholar Crossref Search ADS WorldCat Calvo , G. A. ( 1983 ). ‘Staggered prices in a utility‐maximizing framework’ , Journal of Monetary Economics , vol. 12 ( 3 ), pp. 383 – 98 . Google Scholar Crossref Search ADS WorldCat Canova , F. ( 2007 ). ‘ Sources of structural changes in the US economy ’, mimeo, Universitat Pompeu Fabra. Canova , F. and Gambetti , L. (forthcoming). ‘ Structural changes in the US economy: is there a role for monetary policy? ’, Journal of Economic Dynamics and Control . OpenURL Placeholder Text WorldCat Canova , F. , Gambetti , L. and Pappa , E. ( 2007 ). ‘The structural dynamics of output growth and inflation: some international evidence’ , Economic Journal , vol. 117 , pp. 167 – 91 . Google Scholar Crossref Search ADS WorldCat Carlstrom , C. T. and Fuerst , T. S. ( 2005 ). ‘ Oil prices, monetary policy, and counterfactual experiments ’, Federal Reserve Bank of Cleveland, Working Paper No. 0510. Cecchetti , S. , Flores‐Lagunes , A. and Krause , S. ( 2006 ). ‘ Assessing the sources of changes in the volatility of real growth ’, NBER Working Paper No. 11946. Clarida , R. , Gali , J. and Gertler , M. ( 2000 ). ‘Monetary policy rules and macroeconomic stability: evidence and some theory’ , Quarterly Journal of Economics , vol. 115 ( 1 ), pp. 147 – 80 . Google Scholar Crossref Search ADS WorldCat Dahl , C. and Yücel , M. ( 1991 ). ‘Testing alternative hypotheses of oil producer behavior’ , The Energy Journal , vol. 12 ( 4 ), pp. 117 – 38 . Google Scholar Crossref Search ADS WorldCat Dynan , K. , Elmendorf , D. and Sichel , D. ( 2005 ). ‘ Can financial innovation help to explain the reduced volatility of economic activity? ’ FEDs Working Paper No. 2005‐54. Edelstein , P. and Kilian , L. ( 2007a ). ‘ The response of business fixed investment to changes in energy prices: a test of some hypotheses about the transmission of energy price shocks ’, CEPR Discussion Paper No. 6507. Edelstein , P. and Kilian , L. ( 2007b ). ‘ Retail energy prices and consumer expenditures ’, CEPR Discussion Paper No. 6255. EIA ( 2008 ). ‘ Monthly energy review ’, Report, Energy Information Administration, US Department of Energy. Finn , M. ( 1995 ). ‘Variance properties of Solow’s productivity residual and their cyclical implications’ , Journal of Economic Dynamics and Control , vol. 19 ( 5 ), pp. 1249 – 81 . Google Scholar Crossref Search ADS WorldCat Finn , M. ( 2000 ). ‘Perfect competition and the effects of energy price increases on economic activity’ , Journal of Money, Credit and Banking , vol. 32 ( 3 ), pp. 400 – 16 . Google Scholar Crossref Search ADS WorldCat FRED II ( 2007 ). ‘ Federal reserve economic data ’, Database, Federal Reserve Bank of St. Louis. Galí , J. and Gambetti , L. ( 2009 ). ‘ On the sources of the great moderation ’, American Economic Journal: Macroeconomics , vol. 1 ( 1 ), pp. 26 – 57 . Google Scholar Crossref Search ADS WorldCat Galí , J. and Rabanal , P. ( 2004 ). ‘ Technology shocks and aggregate fluctuations: how well does the RBC model fit postwar US data? ’, NBER Macroeconomics Annual , vol. 19 , pp. 225 – 88 . Google Scholar Crossref Search ADS WorldCat Griffin , J. M. ( 1985 ). ‘OPEC behavior: a test of alternative hypotheses’ , American Economic Review , vol. 75 ( 5 ), pp. 954 – 63 . OpenURL Placeholder Text WorldCat Hamilton , J. D. ( 1983 ). ‘Oil and the macroeconomy since world war II’ , Journal of Political Economy , vol. 91 ( 2 ), pp. 228 – 48 . Google Scholar Crossref Search ADS WorldCat Herrera , A. M. and Pesavento , E. ( 2009 ). ‘ Oil price shocks, systematic monetary policy and the great moderation ’, Macroeconomic Dynamics , vol. 13 , pp. 107 – 37 . Google Scholar Crossref Search ADS WorldCat Hooker , M. A. ( 2002 ). ‘Are oil shocks inflationary? Asymmetric and nonlinear specifications versus changes in regime’ , Journal of Money, Credit and Banking , vol. 34 ( 2 ), pp. 540 – 61 . Google Scholar Crossref Search ADS WorldCat Hooker , M. A. and Board of Governors of the Federal Reserve System ( 1999 ). Oil and the Macroeconomy Revisited , Washington DC: Divisions of Research & Statistics and Monetary Affairs, Federal Reserve Board . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Ireland , P. ( 2004 ). ‘Technology shocks in the New Keynesian model’ , Review of Economics and Statistics , vol. 86 ( 4 ), pp. 923 – 36 . Google Scholar Crossref Search ADS WorldCat Jones , C. ( 1990 ). ‘OPEC behavior under falling prices: implications for cartel stability’ , The Energy Journal , vol. 11 ( 3 ), pp. 117 – 29 . Google Scholar Crossref Search ADS WorldCat Justiniano , A. and Primiceri , G. ( 2006 ). ‘ The time varying volatility of macroeconomic fluctuations ’, NBER Working Paper No. 9127. Kahn , J. , McConnell , M. and Perez‐Quiros , G. ( 2002 ). ‘On the causes of the increased stability of the US economy’ , Federal Reserve Bank of New York Economic Policy Review , vol. 8 ( 1 ), pp. 183 – 202 . OpenURL Placeholder Text WorldCat Kilian , L. ( 2008 ). ‘ Exogenous oil supply shocks: how big are they and how much do they matter for the US economy? ’, Review of Economics and Statistics , vol. 90 ( 2 ), pp. 216 – 40 . Google Scholar Crossref Search ADS WorldCat Kilian , L. (forthcoming). ‘ Not all oil price shocks are alike: disentangling demand and supply shocks in the crude oil market ’, American Economic Review . OpenURL Placeholder Text WorldCat Kilian , L. and Park , C. ( 2007 ). ‘ The impact of oil price shocks on the U.S. stock market ’, CEPR Discussion Paper No. 6166. Kim , C. and Nelson , C. ( 1999 ). ‘Has the US economy become more stable? A Bayesian approach based on a Markov‐switching model of the business cycle’ , Review of Economics and Statistics vol. 81 ( 4 ), pp. 608 – 16 . Google Scholar Crossref Search ADS WorldCat Kim , I. M. and Loungani , P. ( 1992 ). ‘The role of energy in real business cycle models’ , Journal of Monetary Economics , vol. 29 ( 2 ), pp. 173 – 89 . Google Scholar Crossref Search ADS WorldCat Klein , P. ( 2000 ). ‘Using the generalized schur form to solve a multivariate linear rational expectations model’ , Journal of Economic Dynamics and Control , vol. 24 ( 10 ), pp. 1405 – 23 . Google Scholar Crossref Search ADS WorldCat Klenow , P. J. and Kryvtsov , O. ( 2008 ). ‘State‐dependent or time‐dependent pricing: does it matter for recent U.S. inflation?’ . Quarterly Journal of Economics , vol. 123 ( 3 ), pp. 863 – 904 . Google Scholar Crossref Search ADS WorldCat Leduc , S. and Sill , K. ( 2004 ). ‘A quantitative analysis of oil‐price shocks, systematic monetary policy, and economic downturns’ , Journal of Monetary Economics , vol. 51 ( 4 ), pp. 781 – 808 . Google Scholar Crossref Search ADS WorldCat Leduc , S. and Sill , K. ( 2007 ). Monetary Policy, Oil Shocks, and TFP: Accounting for the Decline in US Volatility , Washington DC: Board of Governors of the Federal Reserve System . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Lubik , T. and Schorfheide , F. ( 2004 ). ‘Testing for indeterminacy: an application to US monetary policy’ , American Economic Review , vol. 94 ( 1 ), pp. 190 – 217 . Google Scholar Crossref Search ADS WorldCat Mabro , R. ( 1998 ). ‘OPEC behaviour 1960‐1998: a review of the literature’ , Journal of Energy Literature , vol. 4 ( 1 ), pp. 3 – 27 . OpenURL Placeholder Text WorldCat McConnell , M. and Perez‐Quiros , G. ( 2000 ). ‘Output fluctuations in the United States: what has changed since the early 1980’s?’ , American Economic Review , vol. 90 ( 5 ), pp. 1464 – 76 . Google Scholar Crossref Search ADS WorldCat Nakov , A. and Pescatori , A. ( 2007 ). ‘ Inflation‐output gap tradeoff with a dominant oil supplier ’, Banco de España, Working Paper No. 0723. Rabanal , P. and Rubio‐Ramírez , J. ( 2005 ). ‘Comparing New Keynesian models of the business cycle: a Bayesian approach’ , Journal of Monetary Economics , vol. 52 ( 6 ), pp. 1151 – 66 . Google Scholar Crossref Search ADS WorldCat Rotemberg , J. J. and Woodford , M. ( 1996 ). ‘Imperfect competition and the effects of energy price increases on economic activity’ , Journal of Money, Credit and Banking , vol. 28 ( 4 ), pp. 550 – 77 . Google Scholar Crossref Search ADS WorldCat Sims , C. ( 2002 ). ‘Solving linear rational expectations models’ , Computational Economics , vol. 20 ( 1 ), pp. 1 – 20 . Google Scholar Crossref Search ADS WorldCat Smets , F. and Wouters , R. ( 2007 ). ‘Shocks and frictions in us business cycles: a Bayesian DSGE approach’ , American Economic Review , vol. 97 ( 3 ), pp. 586 – 606 . Google Scholar Crossref Search ADS WorldCat Stock , J. and Watson , M. ( 2002 ). ‘ Has the business cycle changed and why?’ , NBER Working Paper No. 9127. Summers , P. M. ( 2005 ). ‘ What caused the great moderation? Some cross‐country evidence ’, Economic Review (Q III) , pp. 5 – 32 . OpenURL Placeholder Text WorldCat Appendix A.1.Model Equations The dominant oil exporter seeks to maximise the present discounted utility of the household‐owner (18) subject to the constraints imposed by the optimal behaviour of the competitive fringe (19) of households and final goods firms in the oil importing country (20) (21) (22) (23) (24) (25) (26) (27) (28) the rule followed by the monetary authority (29) and the global resource constraint (30) We assume that OPEC can commit to the optimal rule that brings about the equilibrium which maximises (18). Furthermore, we restrict our attention to Markovian stochastic processes for all exogenous variables and to optimal decision rules which are time‐invariant functions of the state of the economy. A.2. First‐order Conditions (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) The set of equations (19) to (42), together with the laws of motion of the exogenous states (3) and (11), constitute a full description of the model. Author notes " We are grateful for comments to Jordi Galí, James Hamilton, Fabio Canova, Pau Rabanal, Gabriel Perez‐Quiros, Wouter den Haan, Morten Ravn, Bruce Preston, Thijs van Rens and two anonymous referees. We also thank seminar and conference participants at the New York Fed, Bank of England, RES, Warwick, ES, Pittsburgh, SNDE, San Francisco, CEF, Paris and EEA, Milan. © The Author(s). Journal compilation © Royal Economic Society 2009

Biderman,, Ciro;, De Mello, João M P;Schneider,, Alexandre

doi: 10.1111/j.1468-0297.2009.02299.xpmid: N/A

Abstract We use a difference‐in‐differences design to estimate the causal impact of the adoption of dry laws in the São Paulo Metropolitan Area (SPMA) on violent behaviour. Dry laws cause a 10% reduction in homicides. Similar impacts were found on battery and deaths by car accidents. The empirical literature shows that alcohol consumption causes all sorts of social maladies. In this article, we study the impact of social consumption of alcohol on murder, the utmost form of violence. Specifically, we estimate the causal effect on homicide of restricting the recreational consumption of alcohol, which is mandatory night closing hours for bars and restaurants (dry laws, hereafter). We evaluate the impact of dry laws on homicides by taking advantage of a unique empirical opportunity. Between March 2001 and August 2004, 16 out of 39 municipalities in the São Paulo Metropolitan Area (SPMA, hereafter) adopted dry laws. We estimate the reduced form effect of dry laws and find that they cause a 10% drop in homicides. Similar impacts are found on battery and deaths by car accident. Our article relates to several pieces of literature. First, and rather generally, our results pertain to the literature on alcohol consumption and violence. Experimental studies in psychology suggest that alcohol suppresses inhibition, impairs judgment and induces aggressive behaviour (McClelland et al., 1972). However, the literature with non‐experimental data has had difficulty documenting a convincing link. Omission of common determinants such as child abuse and mental problems is one issue; see Currie and Terkin (2006) on child abuse and alcohol consumption. Non‐random selection plagues studies that use arrest or victim data because sober offenders or victims are less likely to get caught or be victimised (Martin, 2001). Overall, the epidemiological literature has not settled the issue of causality (Lipsey et al., 1997). In this context of weak documentation of the causal effects of alcohol consumption, our work relates to a few recent articles that employ sharper identification strategies. Arguably, the most convincing work is Carpenter and Dobkin (2008). They exploit the exogenous variation provided by the 21‐year‐old legal drinking age in the US to show that alcohol consumption causes car accident deaths and youth suicide. The cost of their high internal validity is losing some external validity: the result concerns only youth drinking. In addition, they do not look at violent crime. Somewhat different from our results, Carpenter (2007) finds that youth drinking increases property crime but has no impact on violent crime. The contrast between results in Carpenter (2007) and ours may be due to the fact that the SPMA dry laws only restrict the recreational consumption of alcohol. As expected, such restrictions caused a reduction in bar consumption only partially substituted by consumption at home. At bars, mental impairment and reduction of inhibition combine with altercations that sometimes grow into fights. Settling scores when intoxicated is perhaps the perfect recipe for disaster. Additionally, there is less reason to believe that the impact of social consumption of alcohol on property crime is stronger than alcohol consumption in general. Previous empirical evidence on the link from social consumption to violence is unconvincing. Stockwell et al. (1993), in a survey of Western Australian adults, found that bars were the preferred venue of alcohol consumption prior to committing violent crimes. But bars could be the preferred venue in general. Roncek and Maier (1991) and Scribner et al. (1995) find similar results in other empirical settings; see Martin (2001) for a survey. In contrast, Gorman et al. (1998), using data on New Jersey cities, cannot link bar density to crime after controlling for demographics. These articles employ only cross‐section variation and thus cannot convincingly control for common determinants of bar presence and violence. Directly related to our article is Duailibi et al. (2007), which uses only time‐series variation from Diadema, one of the 16 adopting cities in our sample. Their results are in line with ours but they cannot infer causality because of the lack of cross‐section variation in adoption. With a difference‐in‐differences design, we have a sharper identification strategy. Even if a causal link from alcohol (not necessarily consumed socially) to violence was well established, policy implications are equivocal. The economics of crime literature paints an ambiguous picture of outright prohibition and taxation. On the one hand, Miron and Zwiebel (1991, 1995), for instance, argue that prohibition does not reduce alcohol consumption. Miron (1998) also argues that price‐oriented interventions (e.g., taxation) are equally ineffective because the price‐elasticity of the demand for alcohol is (presumably) quite inelastic. Perhaps reflecting the relative inefficacy of taxation, Markowitz (2005) finds puzzling results using victimisation data: higher beer taxes reduce assaults but have no impact on rape, a set of results hard to rationalise. However, the literature is not consensual as to the low price elasticity of alcohol demand. Although the Grossman et al. (1993) survey shows evidence that the long‐run alcohol demand is somewhat elastic, they concede that the demand for alcohol is ‘fairly inelastic in the short run’ (pp. 220), possibly because alcohol is an addictive good (Becker and Murphy, 1988). In another survey, Chaloupka et al. (2002) argue that most of the literature confirms the price responsiveness of alcohol consumption. They do recognise, however, that the literature covered in their survey does not control for endogeneity, making it difficult to infer ‘cause‐and‐effect relationships from the study findings’ (pp. 23). Indeed, Chaloupka et al. (2002) quote a study from Dee (1999) showing that when state fixed effects are added to the model, ‘beer excise tax no longer had a significant effect on consumption’. In addition to ineffectiveness, making alcohol illegal altogether has perverse effects. One is violence induced by the impossibility of settling contracts through the formal judicial system (Miron and Zweibel, (1991, 1995). Another is a substitution effect: illegality levels alcohol with illicit psychotropic drugs and reduces the relative price of moving to ‘stronger’ drugs (Thornton, 1998). Colin et al. (2005) use county‐level variation in alcohol consumption prohibition in Texas to show that access to alcohol reduces crime associated with illicit drugs. Nevertheless, the consequences of this ‘substitution effect’ for policy are unclear: should we facilitate the access to alcohol in order to fight drug use? In light of the literature, targeted sales restrictions are interesting from a policy perspective. Because dry laws are less radical than prohibition, they are less likely to trigger substitution effects and contract‐enforcement crime. Because they are focused at circumstances in which the effects of alcohol are magnified by social interaction, dry laws are relatively economical from a welfare perspective. Figure 1 summarises the story of the article. Not surprisingly, adopting cities were more violent than non‐adopting before adoption but homicides were dropping at about the same rate before adoption. Around the year 2002, when most cities adopted the dry law (see Table 1), homicides started to drop much faster in adopting cities. While in 2001 homicides in adopting were 15% higher than in non‐adopting cities, rates were the same in 2004. Fig. 1. Open in new tabDownload slide Evolution of Homocide Rates (Adopting and non‐adopting Cities over 1991–2004) Source. Secretaria de Segurança do Estado de São Paulo and Municipal Laws. 
Total number of homicide over the year at the city level was aggregated to the group level, adopting and non‐adopting cities. Fig. 1. Open in new tabDownload slide Evolution of Homocide Rates (Adopting and non‐adopting Cities over 1991–2004) Source. Secretaria de Segurança do Estado de São Paulo and Municipal Laws. 
Total number of homicide over the year at the city level was aggregated to the group level, adopting and non‐adopting cities. Table 1 
Month of Dry Law Adoption City . Date of Adoption . Closing Hours . Population in year 2004 . Barueri Mar‐01 11pm–6am all week 250,385 Jandira Aug‐01 11pm–6am all week 105,024 Itapevi Jan‐02 11pm–6am all week 193,475 Diadema Mar‐02 11pm–6am all week 389,354 Juquitiba May‐02 11pm–6am weekdays, 2am–6am Fridays, Saturdays, Sundays and Holidays 28,353 São Lourenço da Serra Jun‐02 11pm–6am all week 14,915 Suzano Jun‐02 11pm–5am all week 267,769 Itapecerica July 02 11pm–6am all week 149,977 Mauá July 02 11pm–6am all week 396,717 Ferraz de Vasconcelos Sep‐02 11pm–6am all week 167,583 Embu Dec‐02 11pm–5am all week 239,144 Osasco Dec‐02 0am‐5am all week 684,079 Embu – Guaçu Apr‐03 11pm–6am weekdays, 1am–6am Fridays, Saturdays, 0am‐6am Sundays and Holidays 60,696 Vargem Grande Paulista Dec‐03 11pm–5am all week 40,083 São Caetano July 04 11pm–6am weekdays, 0am–6am Fridays, Saturdays, Sundays and Holidays 142,692 Poá Aug‐04 11pm–4am all week 104,328 City . Date of Adoption . Closing Hours . Population in year 2004 . Barueri Mar‐01 11pm–6am all week 250,385 Jandira Aug‐01 11pm–6am all week 105,024 Itapevi Jan‐02 11pm–6am all week 193,475 Diadema Mar‐02 11pm–6am all week 389,354 Juquitiba May‐02 11pm–6am weekdays, 2am–6am Fridays, Saturdays, Sundays and Holidays 28,353 São Lourenço da Serra Jun‐02 11pm–6am all week 14,915 Suzano Jun‐02 11pm–5am all week 267,769 Itapecerica July 02 11pm–6am all week 149,977 Mauá July 02 11pm–6am all week 396,717 Ferraz de Vasconcelos Sep‐02 11pm–6am all week 167,583 Embu Dec‐02 11pm–5am all week 239,144 Osasco Dec‐02 0am‐5am all week 684,079 Embu – Guaçu Apr‐03 11pm–6am weekdays, 1am–6am Fridays, Saturdays, 0am‐6am Sundays and Holidays 60,696 Vargem Grande Paulista Dec‐03 11pm–5am all week 40,083 São Caetano July 04 11pm–6am weekdays, 0am–6am Fridays, Saturdays, Sundays and Holidays 142,692 Poá Aug‐04 11pm–4am all week 104,328 Sources. Municipal Laws and IBGE. Open in new tab Table 1 
Month of Dry Law Adoption City . Date of Adoption . Closing Hours . Population in year 2004 . Barueri Mar‐01 11pm–6am all week 250,385 Jandira Aug‐01 11pm–6am all week 105,024 Itapevi Jan‐02 11pm–6am all week 193,475 Diadema Mar‐02 11pm–6am all week 389,354 Juquitiba May‐02 11pm–6am weekdays, 2am–6am Fridays, Saturdays, Sundays and Holidays 28,353 São Lourenço da Serra Jun‐02 11pm–6am all week 14,915 Suzano Jun‐02 11pm–5am all week 267,769 Itapecerica July 02 11pm–6am all week 149,977 Mauá July 02 11pm–6am all week 396,717 Ferraz de Vasconcelos Sep‐02 11pm–6am all week 167,583 Embu Dec‐02 11pm–5am all week 239,144 Osasco Dec‐02 0am‐5am all week 684,079 Embu – Guaçu Apr‐03 11pm–6am weekdays, 1am–6am Fridays, Saturdays, 0am‐6am Sundays and Holidays 60,696 Vargem Grande Paulista Dec‐03 11pm–5am all week 40,083 São Caetano July 04 11pm–6am weekdays, 0am–6am Fridays, Saturdays, Sundays and Holidays 142,692 Poá Aug‐04 11pm–4am all week 104,328 City . Date of Adoption . Closing Hours . Population in year 2004 . Barueri Mar‐01 11pm–6am all week 250,385 Jandira Aug‐01 11pm–6am all week 105,024 Itapevi Jan‐02 11pm–6am all week 193,475 Diadema Mar‐02 11pm–6am all week 389,354 Juquitiba May‐02 11pm–6am weekdays, 2am–6am Fridays, Saturdays, Sundays and Holidays 28,353 São Lourenço da Serra Jun‐02 11pm–6am all week 14,915 Suzano Jun‐02 11pm–5am all week 267,769 Itapecerica July 02 11pm–6am all week 149,977 Mauá July 02 11pm–6am all week 396,717 Ferraz de Vasconcelos Sep‐02 11pm–6am all week 167,583 Embu Dec‐02 11pm–5am all week 239,144 Osasco Dec‐02 0am‐5am all week 684,079 Embu – Guaçu Apr‐03 11pm–6am weekdays, 1am–6am Fridays, Saturdays, 0am‐6am Sundays and Holidays 60,696 Vargem Grande Paulista Dec‐03 11pm–5am all week 40,083 São Caetano July 04 11pm–6am weekdays, 0am–6am Fridays, Saturdays, Sundays and Holidays 142,692 Poá Aug‐04 11pm–4am all week 104,328 Sources. Municipal Laws and IBGE. Open in new tab Is Figure 1 indisputable evidence that dry laws caused a reduction on homicides? The answer is no because adoption is a choice of cities. Adopting cities may have implemented other crime‐fighting policies, which is all more likely because adoption occurred in violent cities. We control for a long list of ‘other suspects’, but it is always possible that the dry law is confounded with other unobserved policies. Furthermore, adopting and non‐adopting cities may differ in time‐varying dimensions. For example, homicides could be following different secular trends prior to adoption, although Figure 1 suggests otherwise. Finally, mean reversion could produce the results mechanically. The article is organised as follows. Information on data sources is in Section 1. Section 2 describes the empirical setting and narrates the chronology of the events. Section 3 contains an extensive description of the empirical strategy designed to address the difficulties raised by the non‐random adoption of dry laws. Results are presented in Section 4, which also contains an extensive robustness analysis, as well as validation and falsification tests. Section 5 concludes. 1. Data Data come from several sources. Crime and enforcement data are from the Secretaria Estadual de Segurança Pública de São Paulo (Secretaria hereafter), the state‐level enforcement authority. Crime data are at monthly frequency. Homicides and vehicle robbery data run from April 1999 to December 2004. Other crime categories are available from January 2001. Police, incarceration and arms apprehension data are only available with annual frequency and starting in 2001. Deaths by car accidents are from DATASUS, a hospital database from the Ministry of Health. Also from Secretaria, we have report‐level data from INFOCRIM, a compustat crime‐tracking system. INFOCRIM started in 1999 in the São Paulo City. Implementation in other cities in the SPMA was gradual, as precincts were slowly incorporated in the system. Cities enter the sample as INFOCRIM was implemented at its precincts but not all precincts within a city enter at the same time. Thus levels are not comparable over time. Still, with INFOCRIM we can compute the distribution of crime during the day, which is useful for corroboration purposes. Although crime data usually suffer from under‐reporting, our two main dependent variables – homicide and vehicle robbery – are well measured. Under‐reporting is negligible for homicides because an investigation is mandatory as long as a body is produced.1 Vehicle robbery is well measured for three reasons: avoiding receiving traffic tickets; avoiding having one’s name involved in criminal activities related to the subsequent use of a stolen car; and for insurance purposes. A few remarks on under‐reporting are needed because we use other crime categories such as battery as corroborative evidence. Most crime statistics suffer from serious under‐reporting in Brazil, stemming from historical lack of confidence in authorities. Under‐reporting per se does not invalidate the use of other categories, but extra caution must be exercised because reporting improved over the sample period. Institutional innovations in the state‐level bureaucracy reduced the costs of reporting. Among them are: (i) the creation of Poupa‐Tempo, whose claque is ‘time‐saver’, which are offices where all bureaucratic errands, including reporting crimes, may be done; (ii) Delegacia Eletrônica (electronic police station) for on‐line reporting; and (iii) Delegacias da Mulher, police stations specialised in domestic violence. Recorded crime rates hint that reporting improved over time. Figure 2 shows three categories: homicides, vehicle theft/robbery and common theft/robbery (all except vehicle). In 1999 vehicle and common theft/robbery rates were similar, an evidence of under‐reporting. In the US, recorded common theft/robbery is three times higher than vehicle theft/robbery (Mueller, 2006). Overtime, homicides and vehicle theft/robbery follow a similar pattern of reduction, reflecting the general drop in crime in the SPMA (see Section 2). In contrast, common theft/robbery increased during the period, which is hard to rationalise except for improvements in reporting. Fig. 2. Open in new tabDownload slide Evolution of Crime by Categories Source. Secretaria Estadual de Segurança do Estado de São Paulo. Common theft/robbery includes all categories except vehicle. Both theft/robbery categories are plot on the right axis. Homicides are plotted on the left axis. Fig. 2. Open in new tabDownload slide Evolution of Crime by Categories Source. Secretaria Estadual de Segurança do Estado de São Paulo. Common theft/robbery includes all categories except vehicle. Both theft/robbery categories are plot on the right axis. Homicides are plotted on the left axis. An additional problem is that reporting did not improve simultaneously across cities. Poupa‐Tempo started in São Paulo City. Delegacia Eletrônica was available across the state, but internet penetration varied wildly both across cities and over time. For all these reasons, under‐reported categories are used only as additional evidence and with caution. Demographic data are from Instituto Brasileiro de Geografia e Estatística (IBGE), the Brazilian Bureau of Statistics. We have annual city‐level income per capita, population and male population in ages 15 to 30 years, which are interpolated to obtain monthly frequencies. From Fundação SEADE, a state government think‐tank, comes information on municipal‐level policies such as the date of establishment of a municipal police force (if any), its size, spending on education and welfare, and the creation date of a municipal secretary of justice (if any). Information on the dry laws comes from the text of the law, which we collected on‐line or requested from the city council by telephone. Alcohol consumption data are from Pesquisa de Orçamento Familiar (POF, hereafter), a household income and consumption survey conducted by IBGE. POF was conducted twice: 1995/6 and 2002/3. Thirteen municipalities enacted the law between March 2001 and July 2003, and 89% of the adopting cities’ population was in cities that adopted before January 2003 (see Table 1). Thus, most dry laws were effective when interviews were conducted. POF has consumption by type of outlet, i.e., bars and restaurants versus supermarkets and grocery stores, allowing us to measure not only the impact of the dry laws on bar consumption but also substitution effects from bar to supermarket purchases. One caveat is that the public file does not identify the municipality where the household is located, only whether the household is located at the São Paulo City or at any other municipality in the SPMA. Still, we can compare a group of cities that contains adopting cities to a group without adopting cities. 2. The Empirical Setting and the Chronology of Events With roughly 19 million inhabitants in 2005, the SPMA is the largest contiguous urban area in South America and the third largest worldwide. Politically, it is defined as an administrative region in the state of São Paulo. It is composed of 39 independent municipalities, each with its own mayor and city council. City sizes vary widely, from Santa Isabel with a population of 11,000 to São Paulo City with its 11 million inhabitants in 2005. Despite a recent reduction in crime, the SPMA is a violent place. In our 69‐month sample, more than 45,000 people were murdered, which gives a monthly rate of 3.65 homicides per 100,000 inhabitants. For comparison, in New York City at its 1990 peak the rate was 3.56. Figure 1 shows homicides increasing steadily through the 1990s and reaching a peak in 1999. Since then they fell sharply, a reversion comparable to that of New York in the 1990s. Several factors contributed to this reversion. For example, De Mello and Schneider (2007) show the role demography: the proportion of youngsters rose in the 1990s and fell in the 2000s. In reaction to the sharp increase in crime during the 1990s policy interventions took place at every level of government. The most famous are: (i) the Lei do Desarmamento (LD) (December 2003), a strict federal legislation on firearms’ possession; and (ii) INFOCRIM, a compustat‐like system that improved police intelligence at the state level. It is likely that both contributed to the decline in homicide depicted in Figures 1 and 2; see Marinho de Sousa et al. (2007) on the impact of the LD. For our purposes, however, the relevant fact is that these policy interventions cannot be confounded with the dry laws because they were either too broad (LD) or too restricted (INFOCRIM). Municipalities have jurisdiction over the regulation of local commerce. This allowed Barueri to pass in March 2001 legislation imposing mandatory closing hours for bars and restaurants, from 11pm to 6am all week long. The law allowed for exceptions under certain circumstances. In Barueri, less than 60 bars and restaurants out of roughly 4,000 were exempt.2 Several cities followed suit and, as of December 2004, 16 out of 39 cities in the SPMA had adopted similar legislation. Table 1 has the adoption dates, the closing hours and the population in 2004 for all adopting cities. Figure 3 depicts the geographical distribution of adoption. Laws varied somewhat in strictness, with a few adopting cities having laxer rules during weekends. Still, 71.68% of the population in adopting cities were in municipalities where the curfew at 11pm was in place all week; only Osasco has a midnight curfew during weekdays. Adopting cities’ population was 3.2 million in 2004, representing 17% of the SPMA (40% excluding São Paulo City). Prior to dry laws, no restrictions in opening hours were in place. Bars typically worked on ‘last client served’ basis and opened between 6am and 7am. Fig. 3. Open in new tabDownload slide Geographical Distribution of Adoption Fig. 3. Open in new tabDownload slide Geographical Distribution of Adoption Anecdotal evidence suggests that the laws worked. One newspaper story is illustrative. The owner of a bar in Diadema, a particularly violent adopting city, reports that ‘…before [adoption] it was a little messy here. The law is good because it avoids fights.’3 In weak institutional settings such as Brazil, it is not obvious that dry laws were actually enforced, i.e., whether bar consumption of alcohol dropped following of adoption. For example, Romano et al. (2007) find that despite the minimum drinking (18 years old) adolescents find it easy to purchase alcohol. Anecdotal evidence again suggests that the laws were effective. In the same newspaper story, the husband of the bar owner reports that ‘…sales have fallen after the law was passed’. We confirm this anecdotal evidence using household consumption data. We measure the impact of dry law on the consumption of two alcoholic beverages: beer and cachaça, which represent roughly 82% of total alcohol consumption in value (figures are from POF).4 The model is: (1) where SPMAi is 1 if household i lived in the SPMA excluding the city of São Paulo, 0 otherwise. 2003t is 1 if the interview was done in the 2002–3 POF, 0 otherwise. Controls include gender, age, years of schooling and household income of the respondent. Alcoholit is the total household consumption in bars or in grocery stores, in reais (R$). We use sampling weights to make observations representative of the population. Estimated standard errors should be viewed with caution because we only have three cross‐section units and thus we lack degrees of freedom to estimate the standard errors properly (Donald and Lang, 2007). With this caveat in mind, we proceed to interpret results in Table 2. Table 2 
The Mechanism and Substitution Effects Dependent variable: total monthly consumption of alcohol by type (in R$) . . All sample . Only 15–30 year‐old males . . Beer . Cachaça . Beer . Cachaça . . (1) . (2) . (3) . (4) . (5) . (6) . . In bars . In stores . In bars . In stores . In bars . In bars . SPMA × 2003 −28.554 11.572 −2.176 0.238 −66.210 −2.324 (6.382)*** (4.601)*** (1.073)** (0.367) (41.675) (1.370)* No. of Observations 5,294 5,294 4,810 4,810 721 638 Dependent variable: total monthly consumption of alcohol by type (in R$) . . All sample . Only 15–30 year‐old males . . Beer . Cachaça . Beer . Cachaça . . (1) . (2) . (3) . (4) . (5) . (6) . . In bars . In stores . In bars . In stores . In bars . In bars . SPMA × 2003 −28.554 11.572 −2.176 0.238 −66.210 −2.324 (6.382)*** (4.601)*** (1.073)** (0.367) (41.675) (1.370)* No. of Observations 5,294 5,294 4,810 4,810 721 638 Source. Pesquisa de Orçamento Familiar (POF). Robust standard errors in parentheses. Omitted regressors are: dummy for SPMA excluding São Paulo City, dummy for 2003, age in years, log of income, years of schooling and dummy for gender. *** = significant at the 1% level. ** = significant at the 5%. * = significant at the 10%. Open in new tab Table 2 
The Mechanism and Substitution Effects Dependent variable: total monthly consumption of alcohol by type (in R$) . . All sample . Only 15–30 year‐old males . . Beer . Cachaça . Beer . Cachaça . . (1) . (2) . (3) . (4) . (5) . (6) . . In bars . In stores . In bars . In stores . In bars . In bars . SPMA × 2003 −28.554 11.572 −2.176 0.238 −66.210 −2.324 (6.382)*** (4.601)*** (1.073)** (0.367) (41.675) (1.370)* No. of Observations 5,294 5,294 4,810 4,810 721 638 Dependent variable: total monthly consumption of alcohol by type (in R$) . . All sample . Only 15–30 year‐old males . . Beer . Cachaça . Beer . Cachaça . . (1) . (2) . (3) . (4) . (5) . (6) . . In bars . In stores . In bars . In stores . In bars . In bars . SPMA × 2003 −28.554 11.572 −2.176 0.238 −66.210 −2.324 (6.382)*** (4.601)*** (1.073)** (0.367) (41.675) (1.370)* No. of Observations 5,294 5,294 4,810 4,810 721 638 Source. Pesquisa de Orçamento Familiar (POF). Robust standard errors in parentheses. Omitted regressors are: dummy for SPMA excluding São Paulo City, dummy for 2003, age in years, log of income, years of schooling and dummy for gender. *** = significant at the 1% level. ** = significant at the 5%. * = significant at the 10%. Open in new tab Columns (1) and (3) show the impact of dry law adoption on alcohol consumption. Monthly household consumption of beer drops by R$28, which represents 70% of the average bar consumption. For cachaça, the drop is R$2.2, which represents 58% of the mean household bar consumption. Since male youngsters are the main perpetrators of homicides, we restrict the sample to households headed by males age 15–30. Results are similar (columns (5) and (6)). In columns (2) and (4) we measure possible substitution effects. For beer there is a substitution effect smaller than the direct effect: consumption in stores increases by R$11. For cachaça, no substitution effect arises, confirming the perception that cachaça is a bar drink. In summary, household data show that dry laws reduced bar consumption, with a small substitution for grocery purchases in the case of beer. 3. The Empirical Strategy Our identification strategy is based on six pillars. First, with a difference‐in‐differences strategy we control for all time‐invariant heterogeneity across cities, a necessary condition for causal inference. Several common determinants of crime and alcohol (ab)use – such as child abuse, poverty and psychological disturbances – are not observable and remain fairly constant over short periods of time. Second, the staggered nature of adoption provides additional identifying variation. Different adoption periods allow us to compare early adopting cities with late adopting cities, mitigating the problems posed by endogenous adoption. Third, dry laws should have different impacts on different types of crimes. Thus, other crime categories provide the basis for validation and falsification tests. Fourth, if dry laws have an impact on homicides, then the distribution of homicides during the day must have changed in response to the restriction in bar opening hours. Fifth, we evaluate the empirical determinants of the adoption of dry laws and show that adoption of dry laws is not explained by the adoption of other observable municipal and state level policies. Finally, we conduct an extensive sensitivity analysis to probe the robustness of our results. 3.1. Summary Statistics: Adopting and Non‐adopting Cities Summary statistics on adopting and non‐adopting cities are in Table 3. Observations are weighted by city population. Non‐adopting cities resemble adopting in demographics, a desirable feature of a control group. They have similar percentages of male population between 15 and 30 years old, income per capita and school attainment measured both by the number of years of schooling and by the high‐school drop‐out rate. Non‐adopting cities seem larger than adopting ones but the difference is due to São Paulo City, which represents roughly 60% of the population of the SPMA. Excluding São Paulo, average population is similar across groups. Table 3 
Summary Statistics, Adopting and Non‐adopting Cities . Adopting (16 cities) . non‐Adopting (23 cities) . non‐Adopting excl. São Paulo . . 6‐month period pre‐adoption . 6‐month period post‐adoption . 6‐month period pre‐adoption . 6‐month period post‐adoption . 6‐month period pre‐adoption . 6‐month period post‐adoption . Monthly Crime Rate per 100,000 inhabitants Homicide 4.83 2.24 4.29 2.40 3.89 2.23 (3.00) (1.11) (0.94) (0.56) (1.66) (0.97) Vehicle Robbery 31.78 18.85 44.51 30.80 42.00 25.43 (16.96) (12.11) (17.13) (12.95) (31.43) (22.81) Battery 26.82 26.69 24.07 30.19 28.43 32.29 (7.03) (11.14) (6.40) (7.22) (10.42) (12.61) Deaths by Car Accident 0.72 0.48 0.56 0.60 0.42 0.41 (0.79) (0.52) (0.33) (0.35) (0.58) (0.59) Cargo Robbery 1.00 1.40 1.37 2.49 1.38 1.61 (0.94) (1.32) (0.74) (0.98) (1.31) (1.31) Bank Robbery 0.01 0.05 0.07 0.14 0.03 0.05 (0.04) (0.23) (0.15) (0.12) (0.26) (0.16) Demographics Population (in thousands) 176 201 639 683 199 227 (156) (167) (208) (216) (260) (292) %Male Population, age 15–30 14.63 14.15 13.99 13.14 14.35 14.05 (0.67) (0.92) (0.41) (0.72) (0.62) (0.76) Educational Attainment (in year 2000) High‐school drop‐out rate (in %) 11.01 10.08 9.89 (2.87) (1.23) (2.23) Average number of years of schooling (age 15–64) 7.19 8.10 7.47 (0.75) (0.60) (0.77) Income in 2004 reais Income per capita 10,045 13,165 10,233 13,023 8,811 11,484 (6,425) (6,990) (2,242) (9,317) (3,778) (5,523) . Adopting (16 cities) . non‐Adopting (23 cities) . non‐Adopting excl. São Paulo . . 6‐month period pre‐adoption . 6‐month period post‐adoption . 6‐month period pre‐adoption . 6‐month period post‐adoption . 6‐month period pre‐adoption . 6‐month period post‐adoption . Monthly Crime Rate per 100,000 inhabitants Homicide 4.83 2.24 4.29 2.40 3.89 2.23 (3.00) (1.11) (0.94) (0.56) (1.66) (0.97) Vehicle Robbery 31.78 18.85 44.51 30.80 42.00 25.43 (16.96) (12.11) (17.13) (12.95) (31.43) (22.81) Battery 26.82 26.69 24.07 30.19 28.43 32.29 (7.03) (11.14) (6.40) (7.22) (10.42) (12.61) Deaths by Car Accident 0.72 0.48 0.56 0.60 0.42 0.41 (0.79) (0.52) (0.33) (0.35) (0.58) (0.59) Cargo Robbery 1.00 1.40 1.37 2.49 1.38 1.61 (0.94) (1.32) (0.74) (0.98) (1.31) (1.31) Bank Robbery 0.01 0.05 0.07 0.14 0.03 0.05 (0.04) (0.23) (0.15) (0.12) (0.26) (0.16) Demographics Population (in thousands) 176 201 639 683 199 227 (156) (167) (208) (216) (260) (292) %Male Population, age 15–30 14.63 14.15 13.99 13.14 14.35 14.05 (0.67) (0.92) (0.41) (0.72) (0.62) (0.76) Educational Attainment (in year 2000) High‐school drop‐out rate (in %) 11.01 10.08 9.89 (2.87) (1.23) (2.23) Average number of years of schooling (age 15–64) 7.19 8.10 7.47 (0.75) (0.60) (0.77) Income in 2004 reais Income per capita 10,045 13,165 10,233 13,023 8,811 11,484 (6,425) (6,990) (2,242) (9,317) (3,778) (5,523) Source. Secretaria de Segurança do Estado de São Paulo, Fundação SEADE and Municipal Laws. Except for population, all means are computed using population as a weight. Standard deviations in parentheses. Pre‐adoption period is July 1999/December 1999; post‐adoption period is July 2004/December 2004. The observation from Poá in July 2004 was excluded from the post‐adoption in adopting cities. Open in new tab Table 3 
Summary Statistics, Adopting and Non‐adopting Cities . Adopting (16 cities) . non‐Adopting (23 cities) . non‐Adopting excl. São Paulo . . 6‐month period pre‐adoption . 6‐month period post‐adoption . 6‐month period pre‐adoption . 6‐month period post‐adoption . 6‐month period pre‐adoption . 6‐month period post‐adoption . Monthly Crime Rate per 100,000 inhabitants Homicide 4.83 2.24 4.29 2.40 3.89 2.23 (3.00) (1.11) (0.94) (0.56) (1.66) (0.97) Vehicle Robbery 31.78 18.85 44.51 30.80 42.00 25.43 (16.96) (12.11) (17.13) (12.95) (31.43) (22.81) Battery 26.82 26.69 24.07 30.19 28.43 32.29 (7.03) (11.14) (6.40) (7.22) (10.42) (12.61) Deaths by Car Accident 0.72 0.48 0.56 0.60 0.42 0.41 (0.79) (0.52) (0.33) (0.35) (0.58) (0.59) Cargo Robbery 1.00 1.40 1.37 2.49 1.38 1.61 (0.94) (1.32) (0.74) (0.98) (1.31) (1.31) Bank Robbery 0.01 0.05 0.07 0.14 0.03 0.05 (0.04) (0.23) (0.15) (0.12) (0.26) (0.16) Demographics Population (in thousands) 176 201 639 683 199 227 (156) (167) (208) (216) (260) (292) %Male Population, age 15–30 14.63 14.15 13.99 13.14 14.35 14.05 (0.67) (0.92) (0.41) (0.72) (0.62) (0.76) Educational Attainment (in year 2000) High‐school drop‐out rate (in %) 11.01 10.08 9.89 (2.87) (1.23) (2.23) Average number of years of schooling (age 15–64) 7.19 8.10 7.47 (0.75) (0.60) (0.77) Income in 2004 reais Income per capita 10,045 13,165 10,233 13,023 8,811 11,484 (6,425) (6,990) (2,242) (9,317) (3,778) (5,523) . Adopting (16 cities) . non‐Adopting (23 cities) . non‐Adopting excl. São Paulo . . 6‐month period pre‐adoption . 6‐month period post‐adoption . 6‐month period pre‐adoption . 6‐month period post‐adoption . 6‐month period pre‐adoption . 6‐month period post‐adoption . Monthly Crime Rate per 100,000 inhabitants Homicide 4.83 2.24 4.29 2.40 3.89 2.23 (3.00) (1.11) (0.94) (0.56) (1.66) (0.97) Vehicle Robbery 31.78 18.85 44.51 30.80 42.00 25.43 (16.96) (12.11) (17.13) (12.95) (31.43) (22.81) Battery 26.82 26.69 24.07 30.19 28.43 32.29 (7.03) (11.14) (6.40) (7.22) (10.42) (12.61) Deaths by Car Accident 0.72 0.48 0.56 0.60 0.42 0.41 (0.79) (0.52) (0.33) (0.35) (0.58) (0.59) Cargo Robbery 1.00 1.40 1.37 2.49 1.38 1.61 (0.94) (1.32) (0.74) (0.98) (1.31) (1.31) Bank Robbery 0.01 0.05 0.07 0.14 0.03 0.05 (0.04) (0.23) (0.15) (0.12) (0.26) (0.16) Demographics Population (in thousands) 176 201 639 683 199 227 (156) (167) (208) (216) (260) (292) %Male Population, age 15–30 14.63 14.15 13.99 13.14 14.35 14.05 (0.67) (0.92) (0.41) (0.72) (0.62) (0.76) Educational Attainment (in year 2000) High‐school drop‐out rate (in %) 11.01 10.08 9.89 (2.87) (1.23) (2.23) Average number of years of schooling (age 15–64) 7.19 8.10 7.47 (0.75) (0.60) (0.77) Income in 2004 reais Income per capita 10,045 13,165 10,233 13,023 8,811 11,484 (6,425) (6,990) (2,242) (9,317) (3,778) (5,523) Source. Secretaria de Segurança do Estado de São Paulo, Fundação SEADE and Municipal Laws. Except for population, all means are computed using population as a weight. Standard deviations in parentheses. Pre‐adoption period is July 1999/December 1999; post‐adoption period is July 2004/December 2004. The observation from Poá in July 2004 was excluded from the post‐adoption in adopting cities. Open in new tab Average characteristics may disguise time‐series heterogeneity. For a clean, seasonality‐free pre‐ and post‐treatment periods comparison, we use the six‐month periods July 1999/December 1999 and July‐2004/December 2004 for homicides, vehicle robbery, deaths by car accidents and the demographics.5 For the other crime categories we compare six‐month periods July 2001/December 2001 and July 2004/December 2004, and drop Barueri and Jandira, who adopted in 2001. Start with the demographics. Nominal per capita income rose by 31% and 27% in adopting and non‐adopting cities, respectively. The proportion of population in the crime‐prime age (male in the 15–30 age bracket) dropped by the same magnitude in both groups. Population growth is also similar. Excluding São Paulo City from the non‐adopting group does not change any conclusion. Homicides evolved differently in the adopting and non‐adopting cities. In the post‐adoption period, the average six‐month rate was 2.24 in adopting cities. This is 54% lower than the 4.83 rate in July 1999 to December 1999. In non‐adopting cities the reduction was less pronounced: 44%. Reported battery rates increased in the SPMA area as a whole. In adopting cities, however, they fell slightly, suggesting that dry laws also had an impact on assault. Finally, while deaths by car accident dropped markedly in adopting cities, they stayed flat in non‐adopting ones. Results are not sensitive to the presence of the São Paulo City in the non‐adopting group. In line with Figure 1, pre‐ and post‐treatment average comparison suggest that dry laws reduced the violent crime and deaths by car accident. In contrast, no marked pre‐post difference arises for the bank, cargo and vehicle robbery. We argue below that one should not expect these categories to be affected by the dry law. In fact we will use them as falsification tests. Before proceeding to confirm the suggestion of the difference in means, we do an in depth investigation of the determinants of the decision to adopt the dry law. 3.2. Investigating the Decision to Adopt the Law Endogenous adoption of dry laws poses two threats to causal inference. First, if adoption occurred in reaction to surges in homicides, then it is likely that other unobserved policies were adopted concurrently. Second, if observed policies explain dry law adoption, then it is likely that all policies – observed and unobserved – were adopted in bundle. We estimate a duration model for the probability of transiting from non‐adoption to adoption and evaluate the empirical relevance of the two threats (Jenkins, 1995). The following factors are included in the duration analysis: • Municipal and state‐level policy variables. Policies are divided into two sets:  (a) municipal enforcement policies, such as the presence of a municipal secretary of justice, of a municipal police force, their adoption time if they were established during the sample period, and the size (in personnel) of the municipal police force and policy choices that are arguably related to crime prevention, such as the municipal expenditures on welfare (social assistance), education and cultural activities; (b) state‐level enforcement variables (at the city level): number of police officers per capita, arrests per capita and firearms apprehended per capita. By constitutional mandate, enforcement is mostly done at the state‐level in Brazil. • Recent dynamics of homicide. This allows us to test the hypothesis that dry law adoption was related to recent shocks to homicides. We also include the average homicides in 2000 as a baseline measure of homicides to evaluate if overall violence affects the decision to adopt. • Demographic controls. Income, population and male population between 15 and 30 are included because they may affect homicides and the decision to adopt dry laws (a younger constituency may oppose the adoption). In some specifications a polynomial of time is included to account for time varying hazard rates. Adoption occurs over time and homicides are declining in the sample period. • Number of adopting neighbours. Figure 3 shows that adoption is clustered geographically, suggesting that emulation or fear of spillover effects may be important drivers of adoption. Table 4 has the results. The first column has the results of a stripped‐down model. Neither the dynamics of homicide nor competing municipal or state‐level policies are included.6 In line with descriptive statistics, demographics are unrelated to the adoption of dry laws. Time explains adoption, but only weakly (the p‐value on log(time) is 22.2%). Base line homicides in 2000 increase the hazard rate of adoption, i.e., more violent cities were more prone to adopt earlier. Finally, the number of adopting neighbours explains adoption. Taken together, these variables explain less than 9% of variation in the timing of adoption. In column (2) we include the municipal and state‐level policies, the competing explanations. Only the size of the state police force has an impact on adoption. However, it has the wrong sign: an increase in the number of state police officers in the city retards adoption. In column (3) the lags of homicide are included. They are neither individually nor jointly significant. Relative to column (2), the dynamics of homicides explain only one additional percentage point of the variation in adoption. Thus, dry law adoption did not occur as a reaction to a recent increase in homicides. In column (4) we exclude all policy variables. They explain no more than 6% of the variation in adoption above and beyond the variables included in column (3). Lastly, the model in column (5) excludes the base line homicides. The dynamics of homicide are still unrelated to adoption. Table 4 
Log Normal Duration Regression of Adoption of Dry Law . (1) . (2) . (3) . (4) . (5) . . Marginal Effects . Dynamics of Homicides Homicidest − 1 0.035 0.045 0.086 (0.061) (0.083) (0.079) Homicidest − 2 −0.083 −0.090 −0.067 (0.079) (0.103) (0.099) Homicidest − 3 −0.040 −0.036 −0.005 (0.069) (0.092) (0.088) Homicidest − 4 0.002 0.008 0.051 (0.063) (0.084) (0.080) Competing Municipal Policies Dummy for the Presence of a Municipal Police Force 0.517 0.512 0.829 (0.401) (0.384) (0.488)* Dummy for the Presence of a Municipal Secretary of Justice −0.107 −0.068 −0.205 (0.325) (0.322) (0.480) Log(Size of Police Force per capita) 0.041 0.031 0.062 (0.075) (0.069) (0.091) Log(Education Spending per capita) −0.215 −0.200 −0.283 (0.303) (0.283) (0.036) Log(Welfare Spending per capita) 0.342 0.335 0.411 (0.002) (0.221) (0.259)* Competing State Policies Log(Prison per capita) 0.284 0.286 −0.003 (0.403) (0.376) (0.488) Log(Number of Policemen per capita) −0.448 −0.440 −0.515 (0.206)** (0.199)*** (0.235)** Log(Guns Aprehended per capita) −0.147 −0.109 0.128 (0.337) (0.316) (0.462) Demographic controls Log(City Level GDP per capita) 0.510 0.324 0.257 0.478 0.162 (0.346) (0.336) (0.320) (0.339) (0.401) Log(Population) 1.772 2.612 2.124 1.311 0.330 (3.964) (3.196) (2.997) (3.954) (3.779) Log(Male Population, 15 and 30 years) −2.145 −2.972 −2.416 −1.629 −0.545 (4.040) (3.266) (3.068) (4.041) (3.843) Time Trends Time −15.958 −10.265 −8.832 −15.271 −14.959 (12.808) (9.933) (9.300) (12.589) (0.118) (Time)2 0.027 0.017 0.015 0.026 0.025 (0.022) (0.017) (0.016) (0.022) (0.021) Log(Time) 1185.235 766.387 661.159 1134.459 1110.041 (919.776) (714.433) (669.410) (904.265) (852.040) Number of AdoptingNeighbours 0.196 0.173 0.161 0.184 0.128 (0.117)* (0.126) (0.971)* (0.113)* (0.150) Time Invariant Controls Base Line Homicides 0.389 0.300 0.311 0.406 (0.140)*** (0.124)*** (0.135)*** (0.158)*** Pseudo‐R2 0.088 0.148 0.159 0.095 0.112 . (1) . (2) . (3) . (4) . (5) . . Marginal Effects . Dynamics of Homicides Homicidest − 1 0.035 0.045 0.086 (0.061) (0.083) (0.079) Homicidest − 2 −0.083 −0.090 −0.067 (0.079) (0.103) (0.099) Homicidest − 3 −0.040 −0.036 −0.005 (0.069) (0.092) (0.088) Homicidest − 4 0.002 0.008 0.051 (0.063) (0.084) (0.080) Competing Municipal Policies Dummy for the Presence of a Municipal Police Force 0.517 0.512 0.829 (0.401) (0.384) (0.488)* Dummy for the Presence of a Municipal Secretary of Justice −0.107 −0.068 −0.205 (0.325) (0.322) (0.480) Log(Size of Police Force per capita) 0.041 0.031 0.062 (0.075) (0.069) (0.091) Log(Education Spending per capita) −0.215 −0.200 −0.283 (0.303) (0.283) (0.036) Log(Welfare Spending per capita) 0.342 0.335 0.411 (0.002) (0.221) (0.259)* Competing State Policies Log(Prison per capita) 0.284 0.286 −0.003 (0.403) (0.376) (0.488) Log(Number of Policemen per capita) −0.448 −0.440 −0.515 (0.206)** (0.199)*** (0.235)** Log(Guns Aprehended per capita) −0.147 −0.109 0.128 (0.337) (0.316) (0.462) Demographic controls Log(City Level GDP per capita) 0.510 0.324 0.257 0.478 0.162 (0.346) (0.336) (0.320) (0.339) (0.401) Log(Population) 1.772 2.612 2.124 1.311 0.330 (3.964) (3.196) (2.997) (3.954) (3.779) Log(Male Population, 15 and 30 years) −2.145 −2.972 −2.416 −1.629 −0.545 (4.040) (3.266) (3.068) (4.041) (3.843) Time Trends Time −15.958 −10.265 −8.832 −15.271 −14.959 (12.808) (9.933) (9.300) (12.589) (0.118) (Time)2 0.027 0.017 0.015 0.026 0.025 (0.022) (0.017) (0.016) (0.022) (0.021) Log(Time) 1185.235 766.387 661.159 1134.459 1110.041 (919.776) (714.433) (669.410) (904.265) (852.040) Number of AdoptingNeighbours 0.196 0.173 0.161 0.184 0.128 (0.117)* (0.126) (0.971)* (0.113)* (0.150) Time Invariant Controls Base Line Homicides 0.389 0.300 0.311 0.406 (0.140)*** (0.124)*** (0.135)*** (0.158)*** Pseudo‐R2 0.088 0.148 0.159 0.095 0.112 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws. Sample period is January 2001 to December 2004; all five specifications have 1,469 observations. Standard errors in parentheses. ***significant at the 1% level, **significant at the 5% level, *significant at the 10% level. All variables divided by 100. Open in new tab Table 4 
Log Normal Duration Regression of Adoption of Dry Law . (1) . (2) . (3) . (4) . (5) . . Marginal Effects . Dynamics of Homicides Homicidest − 1 0.035 0.045 0.086 (0.061) (0.083) (0.079) Homicidest − 2 −0.083 −0.090 −0.067 (0.079) (0.103) (0.099) Homicidest − 3 −0.040 −0.036 −0.005 (0.069) (0.092) (0.088) Homicidest − 4 0.002 0.008 0.051 (0.063) (0.084) (0.080) Competing Municipal Policies Dummy for the Presence of a Municipal Police Force 0.517 0.512 0.829 (0.401) (0.384) (0.488)* Dummy for the Presence of a Municipal Secretary of Justice −0.107 −0.068 −0.205 (0.325) (0.322) (0.480) Log(Size of Police Force per capita) 0.041 0.031 0.062 (0.075) (0.069) (0.091) Log(Education Spending per capita) −0.215 −0.200 −0.283 (0.303) (0.283) (0.036) Log(Welfare Spending per capita) 0.342 0.335 0.411 (0.002) (0.221) (0.259)* Competing State Policies Log(Prison per capita) 0.284 0.286 −0.003 (0.403) (0.376) (0.488) Log(Number of Policemen per capita) −0.448 −0.440 −0.515 (0.206)** (0.199)*** (0.235)** Log(Guns Aprehended per capita) −0.147 −0.109 0.128 (0.337) (0.316) (0.462) Demographic controls Log(City Level GDP per capita) 0.510 0.324 0.257 0.478 0.162 (0.346) (0.336) (0.320) (0.339) (0.401) Log(Population) 1.772 2.612 2.124 1.311 0.330 (3.964) (3.196) (2.997) (3.954) (3.779) Log(Male Population, 15 and 30 years) −2.145 −2.972 −2.416 −1.629 −0.545 (4.040) (3.266) (3.068) (4.041) (3.843) Time Trends Time −15.958 −10.265 −8.832 −15.271 −14.959 (12.808) (9.933) (9.300) (12.589) (0.118) (Time)2 0.027 0.017 0.015 0.026 0.025 (0.022) (0.017) (0.016) (0.022) (0.021) Log(Time) 1185.235 766.387 661.159 1134.459 1110.041 (919.776) (714.433) (669.410) (904.265) (852.040) Number of AdoptingNeighbours 0.196 0.173 0.161 0.184 0.128 (0.117)* (0.126) (0.971)* (0.113)* (0.150) Time Invariant Controls Base Line Homicides 0.389 0.300 0.311 0.406 (0.140)*** (0.124)*** (0.135)*** (0.158)*** Pseudo‐R2 0.088 0.148 0.159 0.095 0.112 . (1) . (2) . (3) . (4) . (5) . . Marginal Effects . Dynamics of Homicides Homicidest − 1 0.035 0.045 0.086 (0.061) (0.083) (0.079) Homicidest − 2 −0.083 −0.090 −0.067 (0.079) (0.103) (0.099) Homicidest − 3 −0.040 −0.036 −0.005 (0.069) (0.092) (0.088) Homicidest − 4 0.002 0.008 0.051 (0.063) (0.084) (0.080) Competing Municipal Policies Dummy for the Presence of a Municipal Police Force 0.517 0.512 0.829 (0.401) (0.384) (0.488)* Dummy for the Presence of a Municipal Secretary of Justice −0.107 −0.068 −0.205 (0.325) (0.322) (0.480) Log(Size of Police Force per capita) 0.041 0.031 0.062 (0.075) (0.069) (0.091) Log(Education Spending per capita) −0.215 −0.200 −0.283 (0.303) (0.283) (0.036) Log(Welfare Spending per capita) 0.342 0.335 0.411 (0.002) (0.221) (0.259)* Competing State Policies Log(Prison per capita) 0.284 0.286 −0.003 (0.403) (0.376) (0.488) Log(Number of Policemen per capita) −0.448 −0.440 −0.515 (0.206)** (0.199)*** (0.235)** Log(Guns Aprehended per capita) −0.147 −0.109 0.128 (0.337) (0.316) (0.462) Demographic controls Log(City Level GDP per capita) 0.510 0.324 0.257 0.478 0.162 (0.346) (0.336) (0.320) (0.339) (0.401) Log(Population) 1.772 2.612 2.124 1.311 0.330 (3.964) (3.196) (2.997) (3.954) (3.779) Log(Male Population, 15 and 30 years) −2.145 −2.972 −2.416 −1.629 −0.545 (4.040) (3.266) (3.068) (4.041) (3.843) Time Trends Time −15.958 −10.265 −8.832 −15.271 −14.959 (12.808) (9.933) (9.300) (12.589) (0.118) (Time)2 0.027 0.017 0.015 0.026 0.025 (0.022) (0.017) (0.016) (0.022) (0.021) Log(Time) 1185.235 766.387 661.159 1134.459 1110.041 (919.776) (714.433) (669.410) (904.265) (852.040) Number of AdoptingNeighbours 0.196 0.173 0.161 0.184 0.128 (0.117)* (0.126) (0.971)* (0.113)* (0.150) Time Invariant Controls Base Line Homicides 0.389 0.300 0.311 0.406 (0.140)*** (0.124)*** (0.135)*** (0.158)*** Pseudo‐R2 0.088 0.148 0.159 0.095 0.112 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws. Sample period is January 2001 to December 2004; all five specifications have 1,469 observations. Standard errors in parentheses. ***significant at the 1% level, **significant at the 5% level, *significant at the 10% level. All variables divided by 100. Open in new tab We interpret these results as follows. Violent cities adopted dry laws as a measure to fight crime and neighbours followed suit, perhaps because of anecdotal evidence that dry laws worked or for fear of spillovers. Thus, the two threats posed by endogenous adoption are not relevant empirically. 3.3. The Empirical Model We estimate several versions of the following model: (2) where i is a city in the SPMA, and t is a month. AdoptLawit is a dummy variable that assumes the value 1 if the dry law was in place in city i at period t, and 0 otherwise. Hence, for non‐adopting cities, it assumes only the value 0. We test whether the parameter β1 is negative, i.e., whether dry laws reduced homicides. Montht is a full set of period dummies. Their inclusion is important because homicides were falling in the SPMA as a whole. If period specific effects are not accounted for, AdoptLawit will capture aggregate shocks because it assumes more values of 1 at the end of the sample period. Cityi is a full set of city dummies to control for city fixed‐effects. Although model (2) discards all pure cross‐sectional and time‐series variation, objections to causal interpretation still arise. First, the procedure does not account for all time‐varying heterogeneity, which is true in any policy evaluation but poses a more serious threat when policy adoption is a choice. Controlsit are the most direct way to account for time‐varying heterogeneity. They include income, population and the percentage of population between 15 and 30 years, a problematic age bracket. These demographic variables affect homicide and are observed at the annual frequency. Figure 1 suggests that results are not driven by different secular trends in homicides. Nevertheless, we play it safe and we implement two procedures to account for this possibility. In most specifications Controlsit includes several lags of the homicide as explanatory variables. We have no specific theoretical reason to believe that past homicides cause present homicides, after time and city dummies are included. However, a rich dynamic model serves the dual purpose of controlling for different secular trends and proxying for possible unobserved policy reactions. Alternatively, we estimate a ‘city‐specific trends’ model in which each city has its own linear trend θit. Finally, Controlsit also includes a long list of policies that may compete with dry laws. They are the same in the duration model: (i) municipal spending in education and welfare, the presence of a municipal secretary of justice, the presence of a municipal police force and its size (if any); (ii) state‐level enforcement variables, which are the size of the police force in the city, the number of arrests and the number of guns apprehended. The state‐level enforcement variables are particularly important because the state is the main law enforcer by constitutional mandate and the empirical literature has established the link from enforcement to crime (Marvell and Moody, 1996; Corman and Mocan, 2000; Di Tella and Schargrodsky, 2004; Levitt, 2002). We weight observations by population, which serves two purposes. First, it emulates a regression at the individual level, i.e., weighting observations provides estimates closer to a random sample in the SPMA. Second, homicides are not a common occurrence and observations from small cities are much noisier than those from larger cities (the variance of ɛit decreases with population). Thus variation from smaller cities should be discounted. In order to avoid giving more weight to observations in the later part of the sample, the weight is the city population in 2000. Finally, observations are clustered at the city level. Thus, all estimated standard errors are robust to within city correlation, an important feature in light of results in Bertrand et al. (2004). 4. Results 4.1. Main Estimates Table 5 shows estimates of several versions of model (2). For conciseness, only is reported. All models include a full set of city and period dummies. Start in panel (a). Column (1) shows the estimates of a stripped‐down model, with no controls besides period and city dummies. The estimated coefficient on the variable AdoptLaw ( is −0.616, and it is reasonably well estimated (p‐value = 5.73%). Considering the homicide rate in adopting cities in the period July 1999 to December 1999 (4.83 in Table 3), means a 13% drop in homicides per 100,000 inhabitants, a significant reduction. In terms of lives, had the law been adopted in the city of São Paulo (10 million inhabitants), 740 lives would have been saved annually (0.616 × 100 × 12). Table 5 
Main Estimates Dependent Variable: Homicides per 100,000 inhabitants . . Full Sample . January 01 to December 04 . . (a) adopting and non‐adopting cities . . (1) . (2) . (3) . (4) . AdoptLaw −0.616 −0.490 −0.605 −0.613 (0.342)* (0.210)** (0.252)** (0.245)** Covariates?† No Yes Yes Yes 4 Lags of Homicide? No Yes Yes Yes Enforcement Variables?‡ No No No Yes no of Observations 2,535 2,535 1,872 1,872 (b) Only adopting cities AdoptLaw −0.877 −0.668 −0.649 −0.654 (0.309)*** (0.291)** (0.362)* (0.381)* Covariates?† No Yes Yes Yes 4 Lags of Homicide? No Yes Yes Yes Enforcement Variables?‡ No No No Yes no of Observations 1,040 1,040 768 768 Dependent Variable: Homicides per 100,000 inhabitants . . Full Sample . January 01 to December 04 . . (a) adopting and non‐adopting cities . . (1) . (2) . (3) . (4) . AdoptLaw −0.616 −0.490 −0.605 −0.613 (0.342)* (0.210)** (0.252)** (0.245)** Covariates?† No Yes Yes Yes 4 Lags of Homicide? No Yes Yes Yes Enforcement Variables?‡ No No No Yes no of Observations 2,535 2,535 1,872 1,872 (b) Only adopting cities AdoptLaw −0.877 −0.668 −0.649 −0.654 (0.309)*** (0.291)** (0.362)* (0.381)* Covariates?† No Yes Yes Yes 4 Lags of Homicide? No Yes Yes Yes Enforcement Variables?‡ No No No Yes no of Observations 1,040 1,040 768 768 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws. *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. In all specifications, observations are weighted according to population. Standard errors in parentheses are clustered at the city level. Period of Analysis is May 1999 to December 2004, unless otherwise noted. All specifications contain a full set of period (month) and city dummies. † Covariates include: logs of population, of income per capita, of the number of 15–30 year‐old males, the number of neighbouring cities that adopted the law, a dummy for the presence of a municipal secretary of justice, a dummy for the presence of a municipal police force and log of its size, the log of the municipal per capita spending on education, and the log of the municipal per capita spending on welfare programmes. ‡ Yearly data on the number of guns apprehended per capita, the number of prisons per capita and the number of police officers per capita. Open in new tab Table 5 
Main Estimates Dependent Variable: Homicides per 100,000 inhabitants . . Full Sample . January 01 to December 04 . . (a) adopting and non‐adopting cities . . (1) . (2) . (3) . (4) . AdoptLaw −0.616 −0.490 −0.605 −0.613 (0.342)* (0.210)** (0.252)** (0.245)** Covariates?† No Yes Yes Yes 4 Lags of Homicide? No Yes Yes Yes Enforcement Variables?‡ No No No Yes no of Observations 2,535 2,535 1,872 1,872 (b) Only adopting cities AdoptLaw −0.877 −0.668 −0.649 −0.654 (0.309)*** (0.291)** (0.362)* (0.381)* Covariates?† No Yes Yes Yes 4 Lags of Homicide? No Yes Yes Yes Enforcement Variables?‡ No No No Yes no of Observations 1,040 1,040 768 768 Dependent Variable: Homicides per 100,000 inhabitants . . Full Sample . January 01 to December 04 . . (a) adopting and non‐adopting cities . . (1) . (2) . (3) . (4) . AdoptLaw −0.616 −0.490 −0.605 −0.613 (0.342)* (0.210)** (0.252)** (0.245)** Covariates?† No Yes Yes Yes 4 Lags of Homicide? No Yes Yes Yes Enforcement Variables?‡ No No No Yes no of Observations 2,535 2,535 1,872 1,872 (b) Only adopting cities AdoptLaw −0.877 −0.668 −0.649 −0.654 (0.309)*** (0.291)** (0.362)* (0.381)* Covariates?† No Yes Yes Yes 4 Lags of Homicide? No Yes Yes Yes Enforcement Variables?‡ No No No Yes no of Observations 1,040 1,040 768 768 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws. *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. In all specifications, observations are weighted according to population. Standard errors in parentheses are clustered at the city level. Period of Analysis is May 1999 to December 2004, unless otherwise noted. All specifications contain a full set of period (month) and city dummies. † Covariates include: logs of population, of income per capita, of the number of 15–30 year‐old males, the number of neighbouring cities that adopted the law, a dummy for the presence of a municipal secretary of justice, a dummy for the presence of a municipal police force and log of its size, the log of the municipal per capita spending on education, and the log of the municipal per capita spending on welfare programmes. ‡ Yearly data on the number of guns apprehended per capita, the number of prisons per capita and the number of police officers per capita. Open in new tab Results in column (2) show that the estimated impact of dry law adoption is robust to the inclusion of controls. Although the estimated coefficient is a little smaller in magnitude (−0.490), it is still quite significant practically, and more precisely estimated (p – value = 2.6%). In column (3) we restrict the sample to January 2001 to December 2004, the period for which we have data on the enforcement variables. Results are stronger than in column (2). In column (4) the enforcement variables are included. Results are, if anything, slightly stronger. Since including enforcement variables restrict the sample but does not change results significantly, our benchmark estimate is −0.490 (column (4)), the point estimate from the most complete model whose sample is full (May 1999 to December 2004). In panel (b) we restrict the sample to adopting cities. Since adoption did not occur simultaneously across cities, we may use the staggered nature of adoption as the source of identifying variation. The control group is now adopting cities before adoption. Restricting the attention to adopting cities involves a variance‐bias trade‐off. On the one hand, excluding non‐adopting cities discards relevant variation and increases variance. On the other hand, restricting the sample to adopters reduces potential bias for two reasons. First, late adopters have a very high ‘propensity’ to adopt, given that they eventually adopted. Thus, concentrating on them helps to ‘homogenise’ the control and treatment groups. Second, it reduces the risk of capturing potential unobserved policies. It may be that late adopters adopted unobserved policies later and the effects would still be confounded. However, the ‘unobserved policies bias’ story now needs a very fine tuning of timing to work. Incidentally, when attention is restricted to adopting cities, São Paulo City is excluded. This is important for robustness purposes because observations are weighted by population and 60% of the population of the SPMA live in the São Paulo City. Within a column, estimates should be compared across panels. Comparing the stripped‐down models results are, if anything, stronger (−0.877 versus −0.616 in column (1)). In terms of the benchmark model results are again stronger (−0.668 versus −0.490). In column (3) we include the state‐level enforcement variables. Results are again unchanged. Table 6 has a long list of robustness checks. Column (1) has the benchmark estimate for comparison (Table 5, Panel (a), column (2)). In column (2) we estimate the model by OLS without weights to check whether the weighting procedure is driving results. The point estimate is similar but the estimated standard errors are larger under OLS, confirming the efficiency of the weighting scheme. Table 6 
Robustness Checks Dependent Variable: . Homicides per 100,000 inhabitants . Log of Homicides . WLS . OLS . Arellano‐Bond . WLS . WLS . WLS . (1) . (2) . (3) . (4) . (5) . (6) . AdoptLaw −0.490 −0.406 −0.536 −0.583 −0.433 −0.152  (0.210)**  (0.245)* (0.206)***  (0.291)**  (0.244)*  (0.059)*** 4 Lags of homicide? Yes Yes Yes No Yes Yes City‐specific Trends?§ No No No Yes Yes No no of Observations 2,535 2,535 2,496 2,535 2,535 1,573 Dependent Variable: . Homicides per 100,000 inhabitants . Log of Homicides . WLS . OLS . Arellano‐Bond . WLS . WLS . WLS . (1) . (2) . (3) . (4) . (5) . (6) . AdoptLaw −0.490 −0.406 −0.536 −0.583 −0.433 −0.152  (0.210)**  (0.245)* (0.206)***  (0.291)**  (0.244)*  (0.059)*** 4 Lags of homicide? Yes Yes Yes No Yes Yes City‐specific Trends?§ No No No Yes Yes No no of Observations 2,535 2,535 2,496 2,535 2,535 1,573 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws. *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. WLS = Observations weighted by population as in Table 5. OLS = Observations un‐weighted. Standard Errors in parentheses are clustered at the city level Period of Analysis is May 1999 to December 2004. Arellano‐Bond GMM procedure, four lags included (p), dependent variable and regressors are first‐differences, one‐stage standard deviations, Ti − p − 2 lags of the dependent variable used as instruments. No weights included. All specifications include the set of covariates as in Table 5. All specifications contain a full set of period (month) and city dummies. § : One linear trend (θit) for each city i in the sample (city dummies interacted with time). Open in new tab Table 6 
Robustness Checks Dependent Variable: . Homicides per 100,000 inhabitants . Log of Homicides . WLS . OLS . Arellano‐Bond . WLS . WLS . WLS . (1) . (2) . (3) . (4) . (5) . (6) . AdoptLaw −0.490 −0.406 −0.536 −0.583 −0.433 −0.152  (0.210)**  (0.245)* (0.206)***  (0.291)**  (0.244)*  (0.059)*** 4 Lags of homicide? Yes Yes Yes No Yes Yes City‐specific Trends?§ No No No Yes Yes No no of Observations 2,535 2,535 2,496 2,535 2,535 1,573 Dependent Variable: . Homicides per 100,000 inhabitants . Log of Homicides . WLS . OLS . Arellano‐Bond . WLS . WLS . WLS . (1) . (2) . (3) . (4) . (5) . (6) . AdoptLaw −0.490 −0.406 −0.536 −0.583 −0.433 −0.152  (0.210)**  (0.245)* (0.206)***  (0.291)**  (0.244)*  (0.059)*** 4 Lags of homicide? Yes Yes Yes No Yes Yes City‐specific Trends?§ No No No Yes Yes No no of Observations 2,535 2,535 2,496 2,535 2,535 1,573 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws. *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. WLS = Observations weighted by population as in Table 5. OLS = Observations un‐weighted. Standard Errors in parentheses are clustered at the city level Period of Analysis is May 1999 to December 2004. Arellano‐Bond GMM procedure, four lags included (p), dependent variable and regressors are first‐differences, one‐stage standard deviations, Ti − p − 2 lags of the dependent variable used as instruments. No weights included. All specifications include the set of covariates as in Table 5. All specifications contain a full set of period (month) and city dummies. § : One linear trend (θit) for each city i in the sample (city dummies interacted with time). Open in new tab Column (3) deals with the econometric challenges posed by including the lags of the dependent variable as regressors. The fixed‐effect transformation does not work if N is large and T small, unless the error term is strictly exogenous, which rules out unobserved serial correlation. Since in our case N is small and T is large, OLS has small bias but Monte Carlo experiments suggest that both large N and very large T are necessary. Despite complications in identifying models with fixed‐effects and lagged dependent variables, we implement a GMM procedure that instruments for the lags of homicide with further lags of homicide (Arellano and Bond, 1991).7 Results are stronger than the benchmark. Thus, any bias caused by inclusion of lags of the dependent variable is towards zero, if anything. Adopting cities were more violent than in non‐adopting cities around the period of adoption. Thus mean reversion may be driving results. In columns (4) and (5) we allow each city to have its own linear trend θit. Results are again similar, both with and without dynamics. Finally, results are similar when the model is estimated in logs (column (6)): dry laws cause a 15% reduction in homicides. Figure 4 presents the coefficients of a different specification. Treatment is coded as a set of dummies for the number of months to the introduction of the law. A total of 36 dummy coefficients are estimated, 18 for the months before and 18 for after the law. The sample is restricted to 18 months before and after adoption. Two patterns arise. Before adoption, the estimated dummies are all zero, except for the 12th month before adoption, a positive outlier. At the month of adoption, we estimate a big negative coefficient on the dummy. For subsequent months, estimated dummy coefficients fluctuate around −1, in line with the hypothesis that dry law had a causal impact on homicide. Fig. 4. Open in new tabDownload slide Impact of Dry Laws: Dummies for Months To and From Adoption Source. Secretaria de Segurança do Estado de São Paulo, Fundação SEADE and Municipal Laws. Homicides are regressed on covariates (listed in Table 5), four lags of homicides, city‐specific trends and a treatment variable. Treatment is coded as a set of 37 dummies for 18 months before the law, the month of adoption and 18  months subsequent to the adoption of the law. The figure shows the dummy coefficient estimates. Only 18 months before and after adoption included in sample for this regression. Only adopting cities included in this regression. Fig. 4. Open in new tabDownload slide Impact of Dry Laws: Dummies for Months To and From Adoption Source. Secretaria de Segurança do Estado de São Paulo, Fundação SEADE and Municipal Laws. Homicides are regressed on covariates (listed in Table 5), four lags of homicides, city‐specific trends and a treatment variable. Treatment is coded as a set of 37 dummies for 18 months before the law, the month of adoption and 18  months subsequent to the adoption of the law. The figure shows the dummy coefficient estimates. Only 18 months before and after adoption included in sample for this regression. Only adopting cities included in this regression. 4.2. Distribution of Crime over the Day Report‐level data from INFOCRIM provide additional evidence that dry laws worked. Cities enter the sample as INFOCRIM was implemented at its precincts but not all precincts within a city enter at the same time. Thus levels are not comparable over time. For this reason, we use report‐level data to compare the distribution of crime throughout the day in adopting and non‐adopting cities before and after adoption. We have INFOCRIM data for 10 adopting cities (Barueri, Diadema, Embu, Embu‐Guaçu, Ferraz de Vasconcelos, Itapecerica, Jandira, Mauá, Osasco e Suzano). São Paulo City is the control group. The estimation strategy is as follows. An observation is a homicide (indexed by j). Let i be a city, and d be a day. The dependent variable is multinomial: ALjid is 1 if city i had a dry law in effect in day d. We run a multinomial logit regression of Hjid on ALjid with baseline category being 3. We then compute the predicted probabilities for AL = 1 and 0. Typically, curfews are from 11:00pm to 6:00am. Thus, we expect that the proportion of homicides committed in the late night‐early morning period to fall following adoption. We also expect the proportion of homicides in the evening (7pm to 22:59pm) to increase because these are now the busiest bar hours. Results are in Table 7. Panel (a) shows that the presence of the dry law reduces by 7.5% the probability that the homicide was committed between 11pm and 6:59am. This impact is significant at the 10% level. The proportion of homicides in the evening increases (5.2%) but the impact is not precisely estimated. In panel (b) São Paulo City is excluded. It is not surprising that the number of observations is dramatically reduced. Nonetheless, results are stronger, if anything. Now both expected effects arise: the share of late night‐early morning homicides drops and evening share increases. Table 7 
Impact of Dry Law Adoption on the Distribution of Crime over the Day Dependent Variable: Hour of the Day = 0, 1, 2 and 3. Baseline category = 3 . . Multinomial Logit Effect on the Predicted Probabilities . (a) São Paulo included 0 (between 11:00pm and 6:59am) −0.075 (0.045)* 1 (between 7:00am and12:59pm) 0.016 (0.036) 2 (between 13:00pm and 6:59pm) 0.007 (0.039) 3 (between 7:00pm and 10:59pm) 0.052 (0.048) Observations 23,885 (b) São Paulo excluded 0 (between 11:00pm and 6:59am) −0.118 (0.073)* 1 (between 7:00am and12:59pm) −0.043 (0.066) 2 (between 13:00pm and 6:59pm) 0.019 (0.067) 3 (between 7:00pm and 10:59pm) 0.142 (0.761)* Observations 145 Dependent Variable: Hour of the Day = 0, 1, 2 and 3. Baseline category = 3 . . Multinomial Logit Effect on the Predicted Probabilities . (a) São Paulo included 0 (between 11:00pm and 6:59am) −0.075 (0.045)* 1 (between 7:00am and12:59pm) 0.016 (0.036) 2 (between 13:00pm and 6:59pm) 0.007 (0.039) 3 (between 7:00pm and 10:59pm) 0.052 (0.048) Observations 23,885 (b) São Paulo excluded 0 (between 11:00pm and 6:59am) −0.118 (0.073)* 1 (between 7:00am and12:59pm) −0.043 (0.066) 2 (between 13:00pm and 6:59pm) 0.019 (0.067) 3 (between 7:00pm and 10:59pm) 0.142 (0.761)* Observations 145 Source. INFOCRIM and Municipal Laws. Coefficients represent the difference in predicted probabilities with and without the presence of the dry law that a homicide occurred in a given hour of the day. *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. Standard errors in parentheses. An observation is a homicide. Sample is composed of observations from Barueri, Diadema, Embu, Embu‐Guaçu, Ferraz de Vasconcelos, Itapecerica, Jandira, Mauá, Osasco e Suzano and the city of São Paulo. AL = AdoptLaw. Baseline category is H = 3 (hours between 7pm and 10:59pm) Open in new tab Table 7 
Impact of Dry Law Adoption on the Distribution of Crime over the Day Dependent Variable: Hour of the Day = 0, 1, 2 and 3. Baseline category = 3 . . Multinomial Logit Effect on the Predicted Probabilities . (a) São Paulo included 0 (between 11:00pm and 6:59am) −0.075 (0.045)* 1 (between 7:00am and12:59pm) 0.016 (0.036) 2 (between 13:00pm and 6:59pm) 0.007 (0.039) 3 (between 7:00pm and 10:59pm) 0.052 (0.048) Observations 23,885 (b) São Paulo excluded 0 (between 11:00pm and 6:59am) −0.118 (0.073)* 1 (between 7:00am and12:59pm) −0.043 (0.066) 2 (between 13:00pm and 6:59pm) 0.019 (0.067) 3 (between 7:00pm and 10:59pm) 0.142 (0.761)* Observations 145 Dependent Variable: Hour of the Day = 0, 1, 2 and 3. Baseline category = 3 . . Multinomial Logit Effect on the Predicted Probabilities . (a) São Paulo included 0 (between 11:00pm and 6:59am) −0.075 (0.045)* 1 (between 7:00am and12:59pm) 0.016 (0.036) 2 (between 13:00pm and 6:59pm) 0.007 (0.039) 3 (between 7:00pm and 10:59pm) 0.052 (0.048) Observations 23,885 (b) São Paulo excluded 0 (between 11:00pm and 6:59am) −0.118 (0.073)* 1 (between 7:00am and12:59pm) −0.043 (0.066) 2 (between 13:00pm and 6:59pm) 0.019 (0.067) 3 (between 7:00pm and 10:59pm) 0.142 (0.761)* Observations 145 Source. INFOCRIM and Municipal Laws. Coefficients represent the difference in predicted probabilities with and without the presence of the dry law that a homicide occurred in a given hour of the day. *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. Standard errors in parentheses. An observation is a homicide. Sample is composed of observations from Barueri, Diadema, Embu, Embu‐Guaçu, Ferraz de Vasconcelos, Itapecerica, Jandira, Mauá, Osasco e Suzano and the city of São Paulo. AL = AdoptLaw. Baseline category is H = 3 (hours between 7pm and 10:59pm) Open in new tab 4.3. Spillover Effects Adoption in a city may shift bar drinking to its non‐adopting neighbours. Thus, the control group could be affected by the treatment, introducing additional challenges for causal inference. Table 8 shows several specifications that measure the spillover effect and assess its consequences. Columns (1) to (3) present direct evidence on spillovers. The sample is restricted to non‐adopting cities and adopting cities before adoption, the ‘control group’. The main variable of interest is the intensity of neighbour adoption, which is measured as: Table 8 
Spillover Effects Dependent Variable: Homicides per 100,000 inhabitants . . non‐adopting and adopting before . Whole Sample . Population >100,000 . Population >200,000 . Largest Areas . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . AdoptLaw −0.735 −0.573 −0.812 −0.432 (0.258)*** (0.231)** (0.298)** (0.189)** Interaction 0.238 (0.204) Number of Adopting Neighbours −0.028 −0.028 (0.022) (0.048) % Adopting Neighbours 0.004 (0.003) % Adopting Neighbouring Population 0.001 (0.002) no of Observations 1,495 1,495 1,495 2,535 1,008 528 1,536 Dependent Variable: Homicides per 100,000 inhabitants . . non‐adopting and adopting before . Whole Sample . Population >100,000 . Population >200,000 . Largest Areas . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . AdoptLaw −0.735 −0.573 −0.812 −0.432 (0.258)*** (0.231)** (0.298)** (0.189)** Interaction 0.238 (0.204) Number of Adopting Neighbours −0.028 −0.028 (0.022) (0.048) % Adopting Neighbours 0.004 (0.003) % Adopting Neighbouring Population 0.001 (0.002) no of Observations 1,495 1,495 1,495 2,535 1,008 528 1,536 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws. *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. Weighted Least Squares procedure with population as weights. In columns (1) through (3) the period of Analysis is May 1999 to December 2004. In columns (4) to (7) they it is May 1999 to December 2004 Observations are clustered at the city level. City and period (month) dummies, four lags of homicides and covariates as defined in Table 5 included in all specifications. Open in new tab Table 8 
Spillover Effects Dependent Variable: Homicides per 100,000 inhabitants . . non‐adopting and adopting before . Whole Sample . Population >100,000 . Population >200,000 . Largest Areas . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . AdoptLaw −0.735 −0.573 −0.812 −0.432 (0.258)*** (0.231)** (0.298)** (0.189)** Interaction 0.238 (0.204) Number of Adopting Neighbours −0.028 −0.028 (0.022) (0.048) % Adopting Neighbours 0.004 (0.003) % Adopting Neighbouring Population 0.001 (0.002) no of Observations 1,495 1,495 1,495 2,535 1,008 528 1,536 Dependent Variable: Homicides per 100,000 inhabitants . . non‐adopting and adopting before . Whole Sample . Population >100,000 . Population >200,000 . Largest Areas . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . AdoptLaw −0.735 −0.573 −0.812 −0.432 (0.258)*** (0.231)** (0.298)** (0.189)** Interaction 0.238 (0.204) Number of Adopting Neighbours −0.028 −0.028 (0.022) (0.048) % Adopting Neighbours 0.004 (0.003) % Adopting Neighbouring Population 0.001 (0.002) no of Observations 1,495 1,495 1,495 2,535 1,008 528 1,536 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws. *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. Weighted Least Squares procedure with population as weights. In columns (1) through (3) the period of Analysis is May 1999 to December 2004. In columns (4) to (7) they it is May 1999 to December 2004 Observations are clustered at the city level. City and period (month) dummies, four lags of homicides and covariates as defined in Table 5 included in all specifications. Open in new tab (i) number of adopting neighbours, (ii) % of adopting neighbours and (iii) % of adopting neighbour population. In all three cases, spillover effects are small and statistically insignificant. In column (4) the sample is full again. We interact the number of adopting neighbours with the presence of the dry law in the city. If spillovers are relevant, then whether a dry law neighbour comes across the boundary to drink will depend on whether the receiving city adopted the dry law. We expect the own law effect to be negative, the neighbours’ law positive (since it captures spillovers from neighbours if one does not have a law) and the interaction negative (undoing the positive neighbour effect). Only the own effect has the expected negative sign. The coefficient on the interaction is positive but insignificant. Moreover, the number of adopting neighbours seems to reduce homicides, although the coefficient is small in magnitude and statistically insignificant. Again, results suggest that spillover effects are not relevant. Despite their absence, we assess whether results are affected by spillovers. In columns (5) to (7) the sample is restricted to large cities, where it is more costly for drinkers go to bars in non‐adopting neighbouring cities. In columns (5) and (6) the criteria for staying in the sample is population. Results are, if anything, stronger. However, physical size may be a better measure of the cost of moving around. In column (7) the estimated coefficient is slightly a smaller (−0.432) but still statistically and practically significant. In summary, spillovers do not affect our estimates. 4.4. Validation Tests Arguably, dry laws should have an impact on other outcome variables. As a validation exercise we measure the impact of dry law adoption on battery and deaths by car accidents. 4.4.1. Impact of dry laws on battery The newspaper story suggests that dry laws reduced fights. Thus, we expect them to reduce violent crimes other than murder. We test this conjecture by estimating the impact of dry laws on battery.8Table 9 presents some of the models in Table 5. Columns (1) through (3) show that dry laws reduced battery, regardless of the inclusion of controls. Consider the full model estimate −2.175 in column (3). The coefficient means an 8% reduction in batteries due to adoption (see Table 3), which resembles the impact on homicides. Results are robust to including state‐level enforcement variables and to using only the staggered nature of adoption (columns (4) and (5), respectively). Table 9 
Battery Dependent Variable: Battery per 100,000 inhabitants . . All Sample . All Sample . All Sample . All Sample . Only Adopters . . (1) . (2) . (3) . (4) . (5) . AdoptLaw −4.419 −3.601 −2.175 −2.301 −2.159 (2.292)* (1.359)*** (0.664)*** (0.642)*** (0.708)*** 4 Lags of Battery? No No Yes Yes Yes Covariates? No Yes Yes Yes Yes Enforcement Variables? No No No Yes Yes no of Observations 1,716 1,716 1,716 704 704 Dependent Variable: Battery per 100,000 inhabitants . . All Sample . All Sample . All Sample . All Sample . Only Adopters . . (1) . (2) . (3) . (4) . (5) . AdoptLaw −4.419 −3.601 −2.175 −2.301 −2.159 (2.292)* (1.359)*** (0.664)*** (0.642)*** (0.708)*** 4 Lags of Battery? No No Yes Yes Yes Covariates? No Yes Yes Yes Yes Enforcement Variables? No No No Yes Yes no of Observations 1,716 1,716 1,716 704 704 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. Weighted Least Squares with population as weights. Observations are clustered at the city level. Covariates are as defined in table 5. All specifications include city and period dummies. Sample is May‐2001/December–2004 unless otherwise noted. Open in new tab Table 9 
Battery Dependent Variable: Battery per 100,000 inhabitants . . All Sample . All Sample . All Sample . All Sample . Only Adopters . . (1) . (2) . (3) . (4) . (5) . AdoptLaw −4.419 −3.601 −2.175 −2.301 −2.159 (2.292)* (1.359)*** (0.664)*** (0.642)*** (0.708)*** 4 Lags of Battery? No No Yes Yes Yes Covariates? No Yes Yes Yes Yes Enforcement Variables? No No No Yes Yes no of Observations 1,716 1,716 1,716 704 704 Dependent Variable: Battery per 100,000 inhabitants . . All Sample . All Sample . All Sample . All Sample . Only Adopters . . (1) . (2) . (3) . (4) . (5) . AdoptLaw −4.419 −3.601 −2.175 −2.301 −2.159 (2.292)* (1.359)*** (0.664)*** (0.642)*** (0.708)*** 4 Lags of Battery? No No Yes Yes Yes Covariates? No Yes Yes Yes Yes Enforcement Variables? No No No Yes Yes no of Observations 1,716 1,716 1,716 704 704 Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. Weighted Least Squares with population as weights. Observations are clustered at the city level. Covariates are as defined in table 5. All specifications include city and period dummies. Sample is May‐2001/December–2004 unless otherwise noted. Open in new tab 4.4.2. Impact of dry laws on deaths by car accident Table 10 shows results for deaths by car accidents. The estimated coefficient in column (1) (−0.055) represents a 7% reduction in car accident deaths, an impact comparable to the one on homicides. However, the effect is not precisely estimated, which is not surprising for several reasons. Table 10 
Deaths in Car Accidents Dependent Variable: Deaths by Car Accidents per 100,000 inhabitants . . Whole Sample . Only largest Adopters and all non‐adopters . Only largest adopters and non‐adopters . Only largest adopters and all non‐adopters . Only largest adopters and non‐adopters . . (1) . (2) . (3) . (4) . (5) . AdoptLaw −0.055 −0.116 −0.108 −0.119 −0.110 (0.048) (0.053)** (0.071) (0.082) (0.086) 4 Lags of Deaths by Car Accident? Yes Yes Yes Yes Yes Covariates? Yes Yes Yes Yes Yes Enforcement Variables? No No No Yes Yes no of Observations 2,535 2,080 1,430 1,536† 1,056† Dependent Variable: Deaths by Car Accidents per 100,000 inhabitants . . Whole Sample . Only largest Adopters and all non‐adopters . Only largest adopters and non‐adopters . Only largest adopters and all non‐adopters . Only largest adopters and non‐adopters . . (1) . (2) . (3) . (4) . (5) . AdoptLaw −0.055 −0.116 −0.108 −0.119 −0.110 (0.048) (0.053)** (0.071) (0.082) (0.086) 4 Lags of Deaths by Car Accident? Yes Yes Yes Yes Yes Covariates? Yes Yes Yes Yes Yes Enforcement Variables? No No No Yes Yes no of Observations 2,535 2,080 1,430 1,536† 1,056† Source. DATASUS, Fundação SEADE and Municipal Laws. All specifications include a full set of city and period dummies. Sample period runs from January 1999 to December 2004. † Sample runs from January 2001 to December 2004 *** = significant at the 1% level, ** = significant 5%, * = significant 10%. Weighted Least Squares with population as weights. Observations are clustered at the city level. Covariates are as defined in Table 5. Open in new tab Table 10 
Deaths in Car Accidents Dependent Variable: Deaths by Car Accidents per 100,000 inhabitants . . Whole Sample . Only largest Adopters and all non‐adopters . Only largest adopters and non‐adopters . Only largest adopters and all non‐adopters . Only largest adopters and non‐adopters . . (1) . (2) . (3) . (4) . (5) . AdoptLaw −0.055 −0.116 −0.108 −0.119 −0.110 (0.048) (0.053)** (0.071) (0.082) (0.086) 4 Lags of Deaths by Car Accident? Yes Yes Yes Yes Yes Covariates? Yes Yes Yes Yes Yes Enforcement Variables? No No No Yes Yes no of Observations 2,535 2,080 1,430 1,536† 1,056† Dependent Variable: Deaths by Car Accidents per 100,000 inhabitants . . Whole Sample . Only largest Adopters and all non‐adopters . Only largest adopters and non‐adopters . Only largest adopters and all non‐adopters . Only largest adopters and non‐adopters . . (1) . (2) . (3) . (4) . (5) . AdoptLaw −0.055 −0.116 −0.108 −0.119 −0.110 (0.048) (0.053)** (0.071) (0.082) (0.086) 4 Lags of Deaths by Car Accident? Yes Yes Yes Yes Yes Covariates? Yes Yes Yes Yes Yes Enforcement Variables? No No No Yes Yes no of Observations 2,535 2,080 1,430 1,536† 1,056† Source. DATASUS, Fundação SEADE and Municipal Laws. All specifications include a full set of city and period dummies. Sample period runs from January 1999 to December 2004. † Sample runs from January 2001 to December 2004 *** = significant at the 1% level, ** = significant 5%, * = significant 10%. Weighted Least Squares with population as weights. Observations are clustered at the city level. Covariates are as defined in Table 5. Open in new tab Bar drinking relates to traffic fatalities more tenuously than it relates to homicides. Most bars are in the periphery, whose dwellers are poor and use the public transportation system. Thus, for the majority of bar drinkers car accidents are irrelevant simply because they do not own a car. The geography of the relationship between bar drinking and deaths by car accident is also unfavourable. It is unclear whether an accident will happen at the city where the bar is located or somewhere else. The odds that the homicide will be committed nearby are higher because committing homicides do not imply driving. Hospital data is also problematic. The victim may end up in hospital in a city other than where the bar is located or the accident took place. Finally, if a victim is declared dead at the scene, she goes directly to the morgue and does not show up in the hospital data.9 To mitigate the fact that accidents may happen outside the adopting cities limits, we discard the half of adopting cities that are smallest in terms of area. Results are now stronger and precisely estimated. In column (3), we also discard the same group of non‐adopting cities. Results are similar but precision is lost due to the small number of observations. Including state‐level enforcement variables does not change any conclusion. 4.5. Falsification Tests Some crimes should not be affected by the dry law. If they are, we would suspect that the estimated impact of the dry law is spurious and may be attributed to other unobserved policies. Thus they serve as falsification tests. We use three categories: vehicle, bank and cargo robbery. 4.5.1. Impact of dry laws on vehicle robbery Vehicle robbery is our preferred falsification category for several reasons. First, it does not suffer from under‐reporting. Accuracy, however, does not imply that it is a good falsification category. If it was an impulsive crime it would be affected by dry laws. It is hard to argue that the dampening inhibition effect of alcohol does not induce all sorts of bad behaviours. Differently from homicides, however, alcohol consumed socially should not have a pronouncedly larger impact on vehicle robbery. It is well known (but hard to quantify) that in the SPMA vehicle robbery is a professional crime, driven by the secondary market for parts and, to a less extent, by smuggling to neighbouring states and countries, which is hardly an impulsive type of crime. The same argument applies for vehicle theft but robbery is a better falsification category because, by definition, it involves an imminent threat to life, normally with the presence of weapon. Thus, the victim must be present and the crime occurs mainly during hours when people are circulating in the streets. Panel (a) of Figure 5 shows that only 20% of robberies occur during the hours in which the dry laws are ‘binding’. Most vehicle robberies occur in the evening rush hour when dry laws are not binding. In contrast, panel (b) shows that 36% vehicle theft occur during the dry law hours (11pm–6am), which is also the mode of the distribution. This is unsurprising because theft does not require threat, and the typical target is a vehicle parked in a dark empty street, i.e., late night and early morning, when dry laws are binding. Fig. 5. Open in new tabDownload slide Distribution of Vehicle Theft and Bank/Cargo/Vehicle Robbery Over the Day Source. Secretaria de Segurança Pública do Estado de São Paulo, INFOCRIM. Sample is composed of all homicides committed in the SPMA and recorded by INFOCRIM between 1999 and 2003. Fig. 5. Open in new tabDownload slide Distribution of Vehicle Theft and Bank/Cargo/Vehicle Robbery Over the Day Source. Secretaria de Segurança Pública do Estado de São Paulo, INFOCRIM. Sample is composed of all homicides committed in the SPMA and recorded by INFOCRIM between 1999 and 2003. Panel (a) of Table 11 shows some of the models we estimated for homicides. In column (1) we report the stripped down model. The impact of dry law is negative but insignificant statistically and practically (compare the point estimate with the means in Table 3). In column (2) we add covariates and state‐level enforcement variables. The impact is now positive but again insignificant statistically and practically. Column (3) adds the lags of homicide and again we find no impact on vehicle robbery. Column (4) has the un‐weighted OLS estimate, with similar results. Table 11 
Falsification Tests . WLS . WLS . WLS . OLS . . (1) . (2) . (3) . (4) . (a) Dependent Variable: Vehicle Robbery per 100,000 inhabitants AdoptLaw −0.260 1.896 0.735 0.125 (1.781) (1.774) (0.753) (0.854) 4 Lags of Vehicle Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes (b) Dependent Variable: Bank Robbery per 100,000 inhabitants (1) (2) (3) (4) AdoptLaw −0.008 0.015 0.010 0.043 (0.018) (0.029) (0.025) (0.043) 4 Lags of Bank Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes (c) Dependent Variable: Cargo Robbery per 100,000 inhabitants AdoptLaw −0.205 0.046 0.035 0.137 (0.243) (0.166) (0.114) (0.135) 4 Lags of Cargo Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes . WLS . WLS . WLS . OLS . . (1) . (2) . (3) . (4) . (a) Dependent Variable: Vehicle Robbery per 100,000 inhabitants AdoptLaw −0.260 1.896 0.735 0.125 (1.781) (1.774) (0.753) (0.854) 4 Lags of Vehicle Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes (b) Dependent Variable: Bank Robbery per 100,000 inhabitants (1) (2) (3) (4) AdoptLaw −0.008 0.015 0.010 0.043 (0.018) (0.029) (0.025) (0.043) 4 Lags of Bank Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes (c) Dependent Variable: Cargo Robbery per 100,000 inhabitants AdoptLaw −0.205 0.046 0.035 0.137 (0.243) (0.166) (0.114) (0.135) 4 Lags of Cargo Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. Observations are clustered at the city level. Covariates as defined in Table 5. For all specifications the number of observations is 1,716 in all specifications Open in new tab Table 11 
Falsification Tests . WLS . WLS . WLS . OLS . . (1) . (2) . (3) . (4) . (a) Dependent Variable: Vehicle Robbery per 100,000 inhabitants AdoptLaw −0.260 1.896 0.735 0.125 (1.781) (1.774) (0.753) (0.854) 4 Lags of Vehicle Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes (b) Dependent Variable: Bank Robbery per 100,000 inhabitants (1) (2) (3) (4) AdoptLaw −0.008 0.015 0.010 0.043 (0.018) (0.029) (0.025) (0.043) 4 Lags of Bank Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes (c) Dependent Variable: Cargo Robbery per 100,000 inhabitants AdoptLaw −0.205 0.046 0.035 0.137 (0.243) (0.166) (0.114) (0.135) 4 Lags of Cargo Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes . WLS . WLS . WLS . OLS . . (1) . (2) . (3) . (4) . (a) Dependent Variable: Vehicle Robbery per 100,000 inhabitants AdoptLaw −0.260 1.896 0.735 0.125 (1.781) (1.774) (0.753) (0.854) 4 Lags of Vehicle Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes (b) Dependent Variable: Bank Robbery per 100,000 inhabitants (1) (2) (3) (4) AdoptLaw −0.008 0.015 0.010 0.043 (0.018) (0.029) (0.025) (0.043) 4 Lags of Bank Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes (c) Dependent Variable: Cargo Robbery per 100,000 inhabitants AdoptLaw −0.205 0.046 0.035 0.137 (0.243) (0.166) (0.114) (0.135) 4 Lags of Cargo Robbery? No No Yes Yes Covariates? No Yes Yes Yes Enforcement Variables? No Yes Yes Yes Source. Secretaria Estadual de Segurança Pública de São Paulo, Fundação SEADE and Municipal Laws *** = significant at the 1% level, ** = significant at the 5%, * = significant at the 10%. Observations are clustered at the city level. Covariates as defined in Table 5. For all specifications the number of observations is 1,716 in all specifications Open in new tab 4.5.2. Impact of dry laws on bank and cargo robbery Besides vehicle robberies, we have monthly data from January 2001 onwards on bank and cargo robbery, two good categories for falsification tests. Bank and cargo robbery should not be affected by dry laws because both are professional crimes. Bank robberies are complex ventures, which involve planning. Cargo robbers need a network of contacts to dispose of the merchandise in the market. Both bank and cargo robberies tend to be well measured because of insurance reasons. Finally, both categories occur mainly during the daytime. Panel (c) of Figure 4 shows that 92% of bank robberies occur between 7am and 10pm, and 82% between 7am and 6pm. This is expected because by definition robberies must involve threat and, thus, should almost always happen during bank opening hours. Cargo robberies have a similar distribution during the day (Figure 4, panel (d)). Relative to vehicle robbery, bank and (to lesser extent) cargo robbery have the disadvantage of being less frequent, which reduces the power of the test. Panels (b) and (c) in Table 11 have the estimates. Start in panel (b). The impact of dry laws on bank robberies is never different from zero statistically and the estimated coefficient is erratic, with oscillating sign. Bank robberies are very infrequent and the failure to estimate the impact of dry laws on bank robbery may be due to the low power of the test. Panel (c) has the estimated impact of dry laws on cargo robbery, which are more frequent than bank robbery. Again, we never reject the null hypothesis that the impact of dry laws is zero. The estimated coefficient in column (1), −0.205, is large when compared to the mean of cargo robbery in adopting cities before adoption (1.00 see Table 3) but it is not statistically significant. Furthermore, the estimates are not robust to the inclusion of controls: in all other three columns the impact of cargo robbery is insignificant in practice as well as statistically. 5. Conclusion At our benchmark estimate, dry laws cause monthly homicide rates per 100,000 inhabitants to fall by almost 0.5, which means a 10% reduction. To the best of our knowledge, this is the first estimate of the impact of alcohol restrictions on bars and restaurants on violent crime accounting for endogeneity and that cannot be confounded with other policies or secular trends. Restricting opening hours has the advantage of being easily enforceable. Consider the enforcement of the minimum drinking age: it is much harder to monitor whether a bar sells alcohol to minors then verifying whether it is opened at certain hours. Our results provide a guarded support for policies that restrain the recreational consumption of alcohol. We use the word ‘guarded’ because in different institutional settings results may not arise. Furthermore, our results are silent with respect to the welfare cost of dry laws. Finally, we have no data to assess potentially perverse effects of the law. In the UK, for example, police report data suggest an increase in violent behaviour right after 11pm, as pubs were closing (Finey, 2004). A full cost–benefit analysis should be conducted in order to assert confidently that opening hour restrictions are worth implementing as a public policy. Extrapolation to general alcohol consumption is not warranted. In fact, our results are not in contradiction with previous results in the economics of crime literature. Prohibition and taxation fail because they do not reduce consumption and may shift consumption to heavier ‘psychotropic’ substances. Restricting recreational consumption is less radical and more targeted than prohibition. The purpose is not to prevent people from drinking, but to make it difficult to do so in particularly dangerous settings. Footnotes 1 " Homicides are attributed to a city if the crime was committed in that city (or if the dead body was found within the city limits and the investigation cannot determine where the crime was committed). Some ‘miscoding’ happens because the dead body could be moved. Except for very elaborate stories, this only introduces noise in the homicide data. Incidentally, reporting is mandatory in the case of deaths by car accident. 2 " Conditions for exemption included not being located near schools, being outside ‘crime zones’ or zones without nuisance complaints. The presence of acoustic isolation and of private security in front of bar was also a necessary condition. See http://www.propagandasembebida.org.br, in Portuguese. 3 " This story is at Globo Online, the electronic version of O Globo, the second largest circulation newspaper in Brazil. In Portuguese at http://g1.globo.com/Noticias/SaoPaulo/0,,AA1359613‐5605,00.html. The Economist, 20/10/2005, reporting a story on Diadema, lists dry laws as an important factor contributing for the decline in murder rates starting in 2001. In an interview to O Globo, Barueri’s Municipal Secretary of Communication claims that homicides ‘fell up to 70%’ after the city implemented the dry law. 4 " Cachaça is the national liquor, distilled from fermented sugar cane. Its alcohol strength ranges from 38%/Vol. to 48%/Vol. 5 " We drop the observation from Poá in July 2004 when computing the post‐adoption means for adopting cities because Poá adopted in August 2004. 6 " The sample is restricted to the period January 2001 to December 2004 to include the state‐level enforcement variables. Although we lose observations between January 1999 and December 2000, no adoption occurred during this period. Thus, for the duration model it does not make much difference if we include 1999 and 2000 since adoption occurred in this period. 7 " A wide range of possible specifications for the Arellano‐Bond estimator is available. For conciseness and because this is only one of the many robustness checks, we do not dwell into the several implications of different estimation methods. We implement the standard version on the STATA package. All variables are first‐differenced, the one‐step estimator for the standard deviation is used and Ti − p − 2 lags are used as instruments for the p included lagged dependent variable. Only one slight modification: four lags (the p) of the dependent variable are included (instead of two). 8 " Battery is actual physical violence. Assault is defined as the threat of violence. The Brazilian Penal Code does not have the assault category, only Lesão Corporal Dolosa (‘Bodily Injury with Intent’), which in Common Law is battery. 9 " Adams and Cotti (2008) show that smoking restrictions in the US caused an increase in deaths by car accidents because people drove longer distances to go to bars in counties without smoking restrictions. The same could apply here, although this effect is second order because most bar drinkers do not drive in the SPMA. References Adams , S. and Cotti , C. ( 2008 ). ‘Drunk driving after the passage of smoking bans in bars’ , Journal of Public Economics , vol. 92 , pp. 1288 – 305 . Google Scholar Crossref Search ADS WorldCat Arellano , M. and Bond , S. ( 1991 ). ‘Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations’ , Review of Economic Studies , vol. 58 , pp. 277 – 197 . Google Scholar Crossref Search ADS WorldCat Becker , G. and Murphy , K. ( 1988 ). ‘A theory of rational addiction’ , Journal of Political Economy , vol. 96 , pp. 675 – 700 . Google Scholar Crossref Search ADS WorldCat Bertrand , M. , Duflo , E. and Mullainathan , S. ( 2004 ). ‘How much should we trust difference‐in‐differences estimates?’ , Quarterly Journal of Economics , vol. 119 , pp. 249 – 75 . Google Scholar Crossref Search ADS WorldCat Carpenter , C. ( 2007 ). ‘Heavy alcohol use and crime: evidence from underage drunk‐driving laws’ , Journal of Law and Economics , vol. 50 , pp. 539 – 58 . Google Scholar Crossref Search ADS WorldCat Carpenter , C. and Dobkin , C. ( 2009 ). ‘The effect of alcohol consumption on mortality: regression discontinuity evidence from the minimum drinking age’ , American Economic Journal of Applied Economics , vol. 1 ( 10 ). pp. 164 – 82 . Google Scholar Crossref Search ADS WorldCat Chaloupka , F. , Grossman , M. and Saffer , H. ( 2002 ) ‘The effects of price on alcohol consumption and alcohol‐related problems’ , Alcohol Research and Health , vol. 26 , pp. 22 – 34 . Google Scholar PubMed OpenURL Placeholder Text WorldCat Colin , M. , Dickert‐Colin , S. and Pepper , J. ( 2005 ). ‘The effect of alcohol prohibition on illicit drug related crimes: an unintended consequence of regulation’ , Journal of Law and Economics , vol. 48 , pp. 215 – 34 . Google Scholar Crossref Search ADS WorldCat Cook , P. and Moore , M. ( 2002 ). ‘The economics of alcohol abuse and alcohol‐control policies’ , Health Affairs , vol. 21 , pp. 120 – 33 . Google Scholar Crossref Search ADS PubMed WorldCat Corman , H. and Mocan , N. ( 2000 ). ‘A time‐series analysis of crime, deterrence and drug abuse in New York City’ , American Economic Review , vol. 90 , pp. 584 – 604 . Google Scholar Crossref Search ADS WorldCat Currie , J. and Terkin , E. ( 2006 ). ‘ Does child abuse cause crime? ’, NBER Working Paper No. 12171. De Mello , J. and Schneider , A. ( 2007 ). ‘ Age structure explaining a large shift in homicides: the case of the state of São Paulo’ , PUC‐RIO: Texto para Discussão No. 549 . Dee , T.S. ( 1999 ). ‘State alcohol policies, teen drinking and traffic accidents’ , Journal of Public Economics , vol. 72 , pp. 289 – 315 . Google Scholar Crossref Search ADS WorldCat Di Tella , R. and Schardrosky , E. ( 2004 ). ‘Do police reduce crime? Estimates using the allocation of police forces after a terrorist attack’ , American Economic Review , vol. 94 , pp. 115 – 33 . Google Scholar Crossref Search ADS WorldCat Donald , S. and Lang , K. ( 2007 ). ‘Inference with difference‐in‐differences and other panel data’ , Review of Economics and Statistics , vol. 89 , pp. 221 – 33 . Google Scholar Crossref Search ADS WorldCat Duailibi , S. , Ponicki , W., Grube , J., Pinsky , I., Laranjeira , R. and Raw , M. ( 2007 ). ‘The effect of opening hours on alcohol related violence’ , American Journal of Public Health , vol. 97 , pp. 2276 – 80 . Google Scholar Crossref Search ADS PubMed WorldCat Finey , A ( 2004 ). ‘Violence in the night‐time economy: key findings from the research’, Findings 214 , Research Development and Statistics Division , London: Home Office. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Gorman D. , Speer , P. Labouvie , E. and Subaiya , A. ( 1998 ). ‘Risk of assaultive violence and alcohol availability in New Jersey’ , American Journal of Public Health , vol. 88 ( 1 ), pp. 97 – 100 . Google Scholar Crossref Search ADS PubMed WorldCat Grossman , M. , Sindelar , J., Mullahy , J. and Anderson , R. ( 1993 ) ‘Alcohol and cigarette taxes’ , Journal of Economic Perspectives , vol. 7 , pp. 211 – 22 . Google Scholar Crossref Search ADS WorldCat Jenkins , S. ( 1995 ). ‘Easy estimation methods for discrete‐time duration models’ , Oxford Bulletin of Economics and Statistics , vol. 57 , pp. 129 – 38 . Google Scholar Crossref Search ADS WorldCat Levitt , S. ( 2002 ). ‘Using electoral cycles in police hiring to estimate the effects of police on crime: reply’ , American Economic Review , vol. 92 , pp. 1244 – 50 . Google Scholar Crossref Search ADS WorldCat Lipsey , M. , Wilson , D. and Cohen , M. ( 1997 ). ‘Is there a causal relationship between alcohol use and violence? A synthesis of the evidence’, in ( M. Galanter, ed.), Recent Developments in Alcoholism , pp. 245 – 82 , New York: Plenum Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Marinho de Sousa , M. , Macinko , J., Aencar , A. Malta , D. and Morais Neto , O. ( 2007 ). ‘Reductions in firearm‐related mortality and hospitalization in Brazil after gun control’ , Health Affairs , vol. 26 , pp. 575 – 84 . Google Scholar Crossref Search ADS PubMed WorldCat Markowitz , S. ( 2005 ). ‘Alcohol, drugs and violent crime’ , International Review of Law and Economics , vol. 25 , pp. 20 – 44 . Google Scholar Crossref Search ADS WorldCat Martin , S. ( 2001 ). ‘The links between alcohol, crime and the criminal justice system: explanations, evidence and interventions’ , American Journal of Addiction , vol. 10 , pp. 136 – 58 . Google Scholar Crossref Search ADS WorldCat Marvell , T. and Moody C. ( 1996 ). ‘Police levels, crime rates and specification problems’ , Criminology , vol. 34 , pp. 609 – 46 . Google Scholar Crossref Search ADS WorldCat McClelland , D. , Davis , W. Kalin , R. and Wanner , E. ( 1972 ). The Drinking Man: Alcohol and Human Motivation . New York: The Free Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Miron , J. ( 1998 ). ‘An economic analysis of alcohol prohibition’ , Journal of Drug Issues , vol. 28 , pp. 741 – 50 . Google Scholar Crossref Search ADS WorldCat Miron , J. and Zwiebel , J. ( 1991 ). ‘Alcohol consumption during prohibition’ , American Economic Review (Articles and Proceedings), vol. 81 , pp. 741 – 62 . OpenURL Placeholder Text WorldCat Miron , J. and Zwiebel , J. ( 1995 ). ‘The economic case against drug prohibition’ , Journal of Economic Perspectives , vol. 9 , pp. 175 – 92 . Google Scholar Crossref Search ADS WorldCat Mueller , R.S. III ( 2006 ) Preliminary Annual Crime Report , Washington DC: Federal Bureau of Investigation, United States Department of Justice . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Romano , M. , Dualibi , S., Pinsky , I. and Laranejra , R. ( 2007 ). ‘Alcohol purchase survey by adolescents in two cities of State of São Paulo, Southeastern Brazil’ , Revista de Saúde Pública , vol. 41 , pp. 495 – 501 . Google Scholar Crossref Search ADS PubMed WorldCat Roncek D. and Maier , R. ( 1991 ). ‘Bars, blocks, and crimes revisited: linking the theory of routing activities to the empiricism of ‘‘hot spots’’’ , Criminology , vol. 29 , pp. 725 – 54 . Google Scholar Crossref Search ADS WorldCat Scribner , R. , MacKinnon , D. and Dwyer , J. ( 1995 ). ‘The risk of assaultive violence and alcohol availability in Los Angeles County’ , American Journal of Public Health , vol. 85 , pp. 335 – 40 . Google Scholar Crossref Search ADS PubMed WorldCat Stockwell , T. , Lang , E. and Rydon , P. ( 1993 ). ‘High risk drinking settings: the association of serving and promotional practices with harmful drinking’ , Addiction , vol. 88 , pp. 1519 – 26 . Google Scholar Crossref Search ADS PubMed WorldCat Thornton M. ( 1998 ). ‘The potency of illegal drugs’ , Journal of Drug Issues , vol. 28 , pp. 725 – 40 . Google Scholar Crossref Search ADS WorldCat Author notes " Lilia Konishe, Edson Macedo, Mariano Lima, Euripedes Oliveira and Michel Azulai provided excellent research assistance. We thank Tulio Kahn from the Secretaria de Segurança de São Paulo for sharing the data. We also thank Paulo Arvate, Paulina Achurra, Claudio Ferraz and seminar participants at PUC‐Rio, IPEA‐RJ, EPGE‐FGV and at the 11th Annual Meeting of LACEA for comments. Finally, the article was much improved by the invaluable suggestions of three anonymous referees and the editor Jörn‐Steffen Pischke. The usual disclaimer applies. Biderman acknowledges funding generously provided by FAPESP grant 2004/03327‐1. Schneider stresses that opinions expressed here are solely his and not the official position of the Municipality of São Paulo. © The Author(s). Journal compilation © Royal Economic Society 2009

Bloemen, Hans, G.

doi: 10.1111/j.1468-0297.2009.02292.xpmid: N/A

Abstract I present a structural empirical model of collective household labour supply that includes the non‐participation decision. I specify a simultaneous model for hours, participation and wages of husband and wife. I discuss the problems of identification and statistical coherency that arise in the application of the collective household labour supply model. The model includes random effects and it is estimated using a panel data set of Dutch couples. The estimates allow me to check the underlying regularity conditions on individual preferences and to obtain estimates of the sharing rule that governs the division of household income between husband and wife. The literature on collective models of household labour supply is growing. Traditionally, the unitary model has been used to explain the labour supply behaviour of households. The unitary model is the direct application of the neoclassical labour supply model for individual labour supply behaviour to the household as a decision unit. This approach has been criticised for several reasons. One of the fundaments of micro‐economic theory is that the individual is the decision making unit. The unitary model assumes that a utility function exists for the household as a whole and leaves the underlying preferences of the household members unspecified. An implication of the unitary model is income pooling, which means that the explanation of working hours of household members is based on the pooled household income, while the source of the income components does not matter. This implication places the income pooling restrictions on the labour supply functions of individual household members, which are often rejected in empirical work. Furthermore, the unitary model implies that the Slutsky conditions hold. Empirical studies often reject the Slutsky conditions for households with more than one member. These considerations have led to the development of models of household decision making that allow for individual preferences and bargaining between household members. McElroy and Horney (1981) introduced Nash bargaining in a household decision model. Apps and Rees (1988) do not impose a specific bargaining rule but assume efficiency. The collective model of household labour supply is explicitly based on individual preferences of the household members. Chiappori (1988; 1992) shows that under the assumption of Pareto efficiency of the household decision process and the assumption that individual preferences are of the egoistic or caring type it is possible to derive testable implications for the labour supply behaviour of individual household members. The household decision process can be represented by a two‐stage budgeting process:1 in the first stage the non‐labour income of household members is divided between the members and in the second stage, conditional on this division, individuals make their labour supply decision. Chiappori (1988) shows that under the assumptions mentioned it is possible to recover the sharing rule of non‐labour income between household members non‐parametrically, up to an additive constant. Fortin and Lacroix (1997) empirically analyse the collective household labour supply model. For a Canadian cross‐section dataset of two‐earner households, they estimate a collective household labour supply model with a flexible functional form. They carry out a complete set of tests of restrictions implied by the collective model and by the unitary model. However, as opposed to what is usual in the labour supply literature since the work by Heckman (1979), the specification of Fortin and Lacroix (1997) does not incorporate non‐participation and corner solutions. The reason for this is that the modelling of the participation decision in the collective model is much more complicated than in the unitary model. Since the wage rate enters the labour supply function both directly, as in the unitary model and, indirectly, through the sharing rule, the derivation of a unique reservation wage is non‐trivial and requires the placement of additional restrictions on the preferences of household members. However, for empirical work on collective household labour supply the inclusion of non‐participants is very important since restricting the sample to couples with working spouses may lead to selectivity bias in the estimates of the labour supply functions and the sharing rule. Recently, the inclusion of non‐participants in the collective labour supply model has been dealt with theoretically, both by Donni (2003) and by Blundell et al. (2007). Donni (2003) leaves the hours choice of husband and wife unrestricted: both household members can choose not to work or to work any positive amount of working hours. Blundell et al. (2007) assume that men can only choose to work 40 hours a week or choose not to work at all. Donni (2003) shows that the identification result by Chiappori (1988) for the sharing rule can be extended to the case of a non‐participating household member. In this study I expand on the empirical literature of collective household labour supply. I specify a model that allows for non‐participation. My model includes information on working hours of both husband and wife to estimate the labour supply functions for both spouses. A serious complication that arises in modelling both the hours and the participation decision is the problem of statistical coherence. I derive parameter restrictions that are sufficient for the coherence of my model. The model that I specify contains equations for the working hours of husband and wife and for the wages of both spouses. In the econometric specification I allow for the endogeneity of wage rates and for correlation between working hours of husband and wife due to unobserved heterogeneity. In the estimation, I use simulation methods to integrate over unobserved wage rates: I estimate the parameters of the hours and participation equations and the wage distribution simultaneously by simulated maximum likelihood (SML). Estimation of the model gives estimates of the sharing rule and of the labour supply functions of husband and wife. Since the marital status may influence the relative bargaining position of husband and wife, I follow an estimation strategy that allows the parameters of the sharing rule to be fully flexible by marital status. Using a minimum distance estimation criterion, I test for equality of the labour supply and sharing rule parameters by marital status. I use data from the Dutch Socio Economic Panel for the years 1990–2001. This is a panel survey among Dutch households. I select a panel data set of Dutch couples who are living without children. The data contain detailed information about the labour market status, working hours, wage rates and various background characteristics of both husband and wife. I exploit the panel feature of the data by allowing for random effects in the labour supply equations and the wage equations. Section 1 describes the collective model of labour supply which provides the necessary background for my model. Section 2 specifies the empirical model. Here I address problems of identification and statistical coherence. Section 3 contains a description of the available data. Section 4 presents the results of estimation. Section 5 concludes. 1. The Collective Model: Theoretical Framework This Section describes the collective model of labour supply. I formulate the model and discuss the identification results of the sharing rule, as derived by Chiappori (1988; 1992), and the extended results that allow for corner solutions by Blundell et al. (2007) and Donni (2003). In what follows I consider the labour supply decision of a household, consisting of husband and wife. Working hours, consumption and the wage rate of the husband and wife are denoted by (hm, Cm, wm) and (hf, Cf, wf), respectively, while y denotes the household’s non‐labour income. The identification results of the sharing rule require household preferences to be of the egoistic (or caring) type. Moreover, the sharing rule representation of the collective labour supply problem requires the absence of public goods within the household.2 In this Section I assume preferences are egoistic.3 The utility level of household member i, working hi hours and consuming Ci, is denoted by ui(1 − hi,Ci). I assume that the utility function satisfies the usual regularity conditions. The collective approach assumes that households only take Pareto efficient decisions. Formally, for any combination of wages and non‐labour income (wm, wf, y) a utility level exists such that (hm, Cm, hf, Cf) is the solution to the following decision problem:4 (1) The utility level can be interpreted as the outcome of a bargaining process between husband and wife, and the wage rates of husband and wife may influence the bargaining power of the household members. The bargaining process itself is left unspecified. An alternative representation of (1) follows from applying the second fundamental theorem of welfare economics. A pair of labour supply functions () is collectively rational (i.e. satisfies (1)) if a combination () and a pair of functions (ρm,ρf) exists such that () is a solution to (2) Thus, the collective rational household labour supply decision can be represented by a two‐stage budgeting problem. In the first stage the share ρi,i = m, f, of non‐labour income y of each household member is determined. The share depends on the bargaining power of the household members, which in turn depends on the wages of the members and the non‐labour income, and I denote ρm = ρ(wm, wf, y) and ρf = y − ρ(wm, wf, y).5 Note that ρ is not restricted to be positive so transfers from one household member to the other are possible. The decision problem (2) is the basis for deriving the restrictions that the collective approach imposes on the labour supply function. Let H i(wi, ρi) denote the ‘structural’ labour supply function conditional on ρi that follows from solving (2), and let denote the ‘reduced form’ labour supply function. Then it follows that (3) Thus, from (3) it follows that the wage rate of the other household member enters the labour supply function through the sharing rule. The relationship in (3) is the basis for the identification result of Chiappori (1988; 1992) which reads that, given information on working hours, wages and non‐labour income of both household members, in the absence of corner solutions, the sharing rule ρ(wm, wf, y) can be recovered up to an additive constant. This means that it is possible to recover marginal effects of the respective wage rates of husband and wife and of the household’s non‐labour income on the sharing rule.6 The sharing rule can be recovered by taking first and second order partial differentials of (3) which constitutes a system of differential equations from which the sharing rule can be solved. The approach described so far does not account for the presence of corner solutions. In empirical models of collective labour supply corner solutions are usually ignored by restricting the sample to working couples. In contrast, in empirical models of individual labour supply the option of choosing zero working hours usually is accounted for to prevent selectivity bias. In models of individual labour supply, the reservation wage follows naturally from the model as the wage rate at which optimal working hours are exactly equal to zero, or, equivalently, as the wage rate that is equal to the marginal rate of substitution between consumption and labour supply, evaluated at zero working hours. In collective models of labour supply the reservation wage does not follow naturally from the model as the wage of the male affects male labour supply both directly, in the same way as in individual models of labour supply and, indirectly, through the sharing rule and, moreover, it affects female labour supply by the sharing rule. Both Donni (2003) and Blundell et al. (2007) address this problem in a different way. Donni (2003) takes as point of departure the model outlined above. This assumes that working hours of husband and wife both vary continuously. Blundell et al. (2007) assume that the hours of the wife vary continuously, whereas the weekly working hours of the husband are restricted to either zero or forty. Since my empirical model is based on the framework by Donni (2003), I will give an outline of the result by Donni (2003). First Donni (2003) defines the function ωi(wm, wf, y) by the marginal rate of substitution between leisure and consumption of household member i: (4) In (4) the subscripts h and c indicate partial derivatives with respect to labour supply and consumption, respectively. Now the reservation wage rate for household member i is defined as the wage wi for which wi = ωi(wm, wf, y). An additional assumption needs to be made to ensure uniqueness of the reservation wage rate. To this purpose Donni (2003) formulates Assumption R1. For any and preferences and the sharing rule are such that (5) This condition implies that the system of equations (4) is a contraction mapping with respect to the variables wm and wf. Intuitively, (5) requires that the impact of wm and wf on the shares ρm and ρf should not be ‘too large’. Condition (5) ensures uniqueness of the reservation wage in two senses. First, a unique pair of reservation wages and exists such that at wage rates wi with both household members prefer to work. Second, for household member i there exists a reservation wage γi(wj, y), given the wage wj of the other household member, such that household member i prefers to work if wi > γi(wj, y). Both Donni (2003) and Blundell et al. (2007) provide a proof of the identification of the sharing rule (up to an additive constant) by extending the identification result by Chiappori (1988). The original identification result for the case in which both husband and wife participate is based on the labour supply equations in (3). Taking first and second order derivatives of (3) results in a system of partial differential equations, from which the sharing rule can be recovered up to an additive constant. Suppose that household member j works at wage wj while household member i does not. In this situation, there is only one partial differential equation, namely the equation for the participating spouse j to identify the sharing rule from. Define the participation frontier of household member i as for i,j = m, f, i ≠ j. Donni (2003) shows that the sharing rule can be identified from this differential equation if wi approaches the participation frontier γi(wj, y). 2. Specification 2.1. Preferences, Labour Supply and the Sharing Rule For the empirical implementation of the model, I specify the following labour supply function: (6) The labour supply function (6) is flexible enough to allow for backward bending as it is quadratic in the wage rate.7 Hausman and Ruud (1984) showed that the underlying utility function of the labour supply function (6) is (7) with (8) for household member j, j = m, f. The parameters of (7), (8) and (6) are related by . Note that βj < 0 and γj > 0 imply that and . Now I assume that the sharing rule ρ(wm, wf, y) has the following form:8 (9) The sharing rule is specified as a quadratic function of the wage rates of both spouses and of the household’s non‐labour income. In empirical applications of the collective model, often a measure for the relative wage rates of the spouses is included to represent their relative bargaining power. Therefore the sharing rule includes the husband’s wage rate expressed as a fraction of the sum of the wage rates of both spouses.9 Marital status may influence the relative bargaining position of husband and wife. In the empirical analysis, the parameters of the sharing rule (9) are allowed to vary with marital status b, and I may write κj = κj(b), j = 1,…,7. Hence, the relative bargaining position of partners within married and unmarried couples can be different. In Section 2 I discuss the estimation strategy to allow for variation in the sharing rule by marital status. For the moment it is sufficient to realise that if the sharing rule parameters vary with martial status, the parameters of the resulting labour supply function vary with marital status as well. In the remainder of the Section I suppress the dependence of the parameters on marital status in my notation. In my analysis I treat marital status as an explanatory variable. Becker (1973) and Manser and Brown (1980) explicitly model marriage, or couple formation, as a choice variable that is closely related to issues as time allocation and specialisation in home production by household members.10 Inserting the sharing rule (9) in the ‘structural’ labour supply functions (6) (with ym = ρ and yf = y − ρ), results in the ‘reduced form’ labour supply functions: (10) The parameters of the sharing rule kl, l = 2,…,7 can be identified from the parameters of the labour supply functions (10): (11) The parameters of the structural labour supply function (6) can be expressed in terms of the parameters in (10) as (12) A requirement for the identification of the parameters of the sharing rule is that Δ ≠ 0. From the restrictions above it follows that if Δ = 0 then and the sharing rule would not even enter the labour supply equations. Note that the ‘reduced form’ labour supply functions (10) may be consistent with more general specifications of preferences. However, the possibility of recovering the sharing rule parameters (11) separately from the preference parameters (12) stems from the assumption of egoistic preferences and from the assumption that the wage rate fraction enters the labour supply equations by the sharing rule. Leisure is a normal good for both husband and wife if . A necessary condition for this is that the parameters a4 and b4 in the ‘reduced form’ labour supply functions (10) have the opposite sign, as can be seen from (12). From (12) it can be seen that the conditions are not trivial and therefore constitute a test of the theory. Note that it is potentially important to allow for backward bending of the ‘structural’ labour supply functions. For instance, an increase in the husband’s wage rate has a direct effect on his supply of labour, by the coefficients of the labour supply function (6) and an indirect effect, that runs through the sharing rule (9). Not allowing for backward bending will bias the estimates of the effect that runs through the sharing rule. For future reference I introduce the vector notation φm=(a1, a2, a3, a4, a5, a6, a7)′, φf  = (b1, b2, b3, b4, b5, b6, b7)′ and . The relations (11) between the labour supply parameters in (10) and the parameters of the sharing rule (9) hold for the case in which both husband and wife are working and can choose any number of working hours. The analysis by Donni (2003) shows how the labour supply model can be extended to the case in which the husband works and the wife does not. Donni (2003) proposes a switching regression model for labour supply of the husband. If the wife does not work, male labour supply becomes: (13) The underlying reason for this ‘switch’ is a difference in the sharing rule due to the different female labour market state. If the wife works, her wage income actually affects the total household income and consequently influences the sharing rule. Apart from that, the wife’s wage rate affects the sharing rule due to her bargaining power. If the wife does not work, only the effect of the wife’s (potential) wage rate on the bargaining power remains. As shown by Donni (2003) the parameters φm and Φm of male labour supply in case of a working and a non‐working wife must satisfy restrictions for the labour supply function to be continuous along the participation frontier of the wife. This continuity holds for the following relation: (14) In (14) sm is a parameter. Restriction (14) implies that if the wife is on the participation frontier (which means that female labour supply is equal to zero) the two regressions coincide. Blundell et al. (2007) give a more fundamental explanation of this continuity restriction postulated by Donni (2003). They introduce the ‘double indifference’ assumption. This assumption states that if the wife is indifferent between working or not working the husband should be indifferent as well (or, in above terms, there should be no discrete ‘jump’ in the labour supply function of the husband). Blundell et al. (2007) show that if there were a discrete ‘jump’ in the husband’s preferences, there would be scope for a Pareto improving reallocation of the household’s resources. This contradicts the assumption of Pareto efficiency. Similarly, I denote the sharing rule in case of a non‐working wife by (15) Letting k′W represent the vector notation of the right‐hand side of (9). Continuity on the participation frontier imposes the following restriction (16) In (16) rm is a parameter. The following relation between the parameters sm and rm from (14) and (16) is implied by the structure of the collective setting: (17) In a similar way, I may introduce a switching regression for the wife if the husband does not work. Let the wife’s labour supply function in this case be presented by (18) and let the sharing rule be (19) The equivalents of (14) and (16) become (20) and (21) and the collective framework implies the following restriction between the parameters rf and sf : (22) 2.2. Heterogeneity and Coherence 2.2.1. Introducing heterogeneity So far I have abstained from introducing observed and unobserved heterogeneity. The identification results by Donni (2003) are derived in the absence of heterogeneity. The theoretical framework by Blundell et al. (2007) does not incorporate heterogeneity but, in their empirical model, they include additive observed and unobserved heterogeneity in the labour supply functions and the sharing rule. The additive form of the heterogeneity ensures that the derived results for the identification of the sharing rule remain valid, as the additive constant of the sharing rule is not identified. It can easily be checked that a labour supply function that is additive in observed and unobserved heterogeneity is obtained if the preference parameter ϑj in (8) is specified as a linear combination of observed and unobserved heterogeneity. Alternatively, additive heterogeneity may come from additive heterogeneity in the sharing rule. Therefore, it is not possible to identify whether the additive heterogeneity in the labour supply function comes from heterogeneity in preferences for leisure or from heterogeneity in the sharing rule. The available data form a panel. With subscript i and t I denote household i and year t, respectively. Since the panel is unbalanced, the number of time periods varies with i and I denote Ti. The number of households is indicated by N. Let zitm and zitf be vectors of observed characteristics affecting male and female preferences, respectively. Let νitj, j = m, f denote unobserved heterogeneity, varying across households and years, and θij, j = m, f denote unobserved time invariant heterogeneity for household member j. I extend (10) by writing (23) for i = 1,…, N, t = 1,…,Ti. The analysis so far assumed the observability of wages, even if someone is not working. For the empirical analysis, I specify the following wage equation for i =1, …, N, t  = 1,…,Ti: (24) with xitj and uitj observed and unobserved characteristics, and ωij an unobserved random effect. Combining (23) with (14), (20) and (24) and denoting ditj = 1 if household member j participates, ditj = 0 otherwise, I may summarise the conditions for participation, conditional on the wages, for households i = 1, …,  N and time periods t = 1,…,Ti as11 (25) The equations for male and female working hours become: (26) 2.2.2. Coherence of the model In the estimation of the model a complication arises that so far has not been addressed in any discussion of the collective model. The system (25) is not necessarily coherent: without any further restrictions, for given values of (Wit, zitm, zitf, θim, θif, νitm, νitf) the model in (25) may generate multiple outcomes for the participation of husband and wife in a household. For instance, the conditions for (ditm = 0, ditf = 1) and for (ditm =1, ditf = 0) in (25) may not be exclusive. This arises because in the collective model with non‐participation, the labour market participation status of the other partner enters the equation for labour market participation. This problem has not been addressed by Donni (2003) or Blundell et al. (2007).12 A statistical consequence of incoherence is that the probabilities of the four combinations of male and female participation may add up to an amount larger than 1. Estimation of a model that is incoherent in parts of the parameter space can lead to inconsistent estimates. In general, imposing coherence in a model like this is either quite complicated – see, for instance, Kooreman (1994)– or greatly reduces the generality of the model; see Heckman (1978). For the system (25), however, it can be shown that, apart from the trivial case smsf = 0, coherence holds for more combinations of sm and sf. In the Appendix I show that the model is coherent for all combinations of sm and sf for which |smsf| < 1. 2.2.3. Estimation by (simulated) maximum likelihood Now the complete model is defined by (24)–(26). I estimate the model parameters by maximum likelihood. For the time varying error terms (νitm, νitf, uitm, uitf)′ I assume that they are independently and identically distributed across households and across time. Moreover, I assume them to be uncorrelated with the random effects (θim, θif, ωim, ωif)′ and uncorrelated with the observed characteristics. The random effects are independently distributed across households and uncorrelated with the observed characteristics. Throughout I assume that the errors of the wage equations and the labour supply equations follow a joint normal distribution: (27) Note that I allow for non‐zero correlation between the errors of the labour supply functions (parameter σmf). Thus I allow for common, household specific, unobserved characteristics in, say, preferences. Furthermore, I allow for correlation between the male (female) labour supply equation and the male (female) wage rates to correct for possible selectivity bias for the fact that wages are only observed for participants. Note that I restricted the correlation between the errors of the husband’s (wife’s) labour supply equation and the wife’s (husband’s) wage equation to zero.13 Finally, I allow for non‐zero correlation between the errors of the wages of husband and wife. This may capture common unobserved components in the wages of both partners, like for instance the effect of region, or a matching effect. I assume that the random effects are normally distributed: (28) The construction of the likelihood function consists of the following steps. From (27), (28) and (26) the likelihood contributions for working hours, conditional on the wages witm and witf and random effects (θim, θif, ωim, ωif)′ can be constructed. Next, the density function of wages, conditional on random effects, can be formed. To complete the likelihood contribution I integrate over random effects. For observations without observed values for the wages, which includes the non‐participating men and women, I integrate over wages to complete the likelihood contribution. In the application I employ the smooth simulated maximum likelihood method (Börsch‐Supan and Hajivassiliou, 1993) with 60 replications to integrate over wages and random effects. The coherence constraint |smsf| < 1 will be imposed in the estimation of the model. The model formulated in (26) and (25) may be estimated with information on both participation and working hours of husband and wife. In this respect, the model expands on the work by Blundell et al. (2007) and Donni (2003): both approaches suggest modelling collective labour supply by a switching regression model, using information on both working hours and participation of one partner and only on participation of the other partner. Thus, variation in working hours by the other partner is ignored. 2.2.4. Sharing rule parameters that vary with marital status: estimation strategy In Section 2.1 I discussed the idea that the parameters of the sharing rule (9) may differ by marital status, and I may write κj = κj(b), j = 1,…,7, distinguishing married couples and unmarried couples. If preferences do not vary with marital status,14 the parameters do not depend on marital status. The parameters of the reduced form labour supply function, (10), will depend on marital status. Now that I have introduced heterogeneity, I can say more about the implementation of an estimation procedure that allows the parameters to vary with marital status. The additive heterogeneity can be attributed to both heterogeneity in preferences and heterogeneity in the level of the sharing rule κ1: it cannot be identified, because the intercept of the sharing rule is not identified separately from preferences. However, if the parameters of the sharing rule are allowed to depend on marital status, in general the parameters of the wage rates and non‐labour income, φj, j = m, f, vary not only with marital status but also with the parameters that affect heterogeneity. Not allowing for this flexibility may lead to spurious effects in the estimation of the parameters φj, j = m, f and hence inconsistent estimates of the structural parameters and κl(b), l = 2,…,7. To allow for this flexibility, I have adopted the following approach. I stratify the sample and estimate the model separately for married and unmarried couples. Next, I use a minimum distance estimator to estimate the parameters of the structural labour supply function and the parameters of the sharing rule. Also, I can use the minimum distance criterion to test the hypothesis that the structural labour supply parameters do not depend on marital status. Moreover, with the same parameter estimates of the full system (25), I can obtain different minimum distance estimates for the parameters κl, l = 2,…,7: estimates that do and estimates that do not vary with marital status. Thus, I can test the hypothesis of constant sharing rule parameters. 3. The Data I use data from the Socio‐Economic Panel (SEP). The SEP is a household survey collected by Statistics Netherlands. I use data for the years 1990 to 2001. In this period, households were interviewed on a yearly basis, every May. An obvious advantage of this household survey is that not only is the head of the household interviewed but detailed information is obtained about the partner as well. I made several selections to arrive at the dataset that I use in my analysis. For each year, I selected couples living together (either married or unmarried) without children, with the male in the age range of 22 to 60 and the female no older than 60. I excluded households in which either the husband or the wife was self‐employed. Furthermore, I require the availability of information on the labour market state of both household members, the non‐labour income and information on the level of schooling and the sector of education. I use information on hourly wage rates and working hours of both partners but I also use information on individuals with missing working hours and wage rates.15 The pooled dataset contains 8,049 observations (‘couple‐years’). Table 1 contains descriptive statistics for the data. It shows that 84.4% of the male respondents are employed as are 70.3% of their female partners. In interpreting these numbers please recall that couples without children were selected. Therefore, the percentages of males and females working are relatively high in this sample. 64.3% of the households are dual earner couples. In 20.2% of the households the husband is the sole breadwinner. For 9.5% of the households none of the members is working, whereas in 6.0% of the households only the wife works. Table 1 
Descriptive Statistics Variable . Pooled data . Married couples . Unmarried couples . 8,049 Observations . 5,558 Observations . 2,491 Observations . Husband . Wife . Husband . Wife . Husband . Wife . Employment status  Employed 84.4% 70.3% 80.4% 62.0% 93.3% 88.7%  Not Employed 15.6% 29.7% 19.6% 38.0% 6.7% 11.3% Education level  Primary 7.3% 11.2% 9.2% 14.3% 3.0% 4.3%  Lower vocational 16.0% 23.4% 16.0% 26.7% 16.2% 16.0%  Intermediate 49.3% 42.5% 49.9% 40.7% 48.1% 46.6%  Higher Vocational 20.0% 18.2% 19.8% 15.1% 20.3% 25.3%  University degree 7.0% 4.4% 4.8% 3.1% 11.9% 7.4% Education sector  Technical 34.4% 5.3% 34.1% 4.5% 35.1% 7.1%  Economic/administrative 25.9% 24.5% 24.8% 23.8% 28.5% 26.1%  General (not specialised) 18.1% 30.2% 19.0% 32.7% 16.1% 24.7%  Services 21.5% 40.0% 22.1% 39.0% 20.2% 42.1% Weekly working hours  No. of observations 6,618 5,408 4,350 3,300 2,268 2,108  Mean 39.4 30.9 39.3 28.9 39.6 34.2  (Standard deviation) (7.9) (10.8) (8.0) (11.3) (7.8) (9.2) Hourly wage rates  No. of observations 6,258 4,997 4,113 3,009 2,145 1,988  Mean (Guilders) 21.8 18.2 23.0 18.7 19.1 17.4  (Standard deviation) (7.7) (6.4) (8.0) (6.7) (6.5) (6.0) Age  Mean 40.8 38.7 45.1 43.1 31.2 28.8  (Standard deviation) (12.4) (12.5) (11.6) (11.7) (7.7) (7.9) Household level variables Non‐labour income  Household level, weekly  Mean (guilders) 37.7 36.8 39.8  Standard deviation (94.8) (93.2) (98.2) Employment status  Both partners working 64.3% 55.4% 84.0%  Husband working, wife not 20.2% 25.1% 9.3%  Wife working, husband not 6.0% 6.6% 4.7%  Both not working 9.5% 12.9% 2.0% Marital status  Married 69.1% 100% 0% Variable . Pooled data . Married couples . Unmarried couples . 8,049 Observations . 5,558 Observations . 2,491 Observations . Husband . Wife . Husband . Wife . Husband . Wife . Employment status  Employed 84.4% 70.3% 80.4% 62.0% 93.3% 88.7%  Not Employed 15.6% 29.7% 19.6% 38.0% 6.7% 11.3% Education level  Primary 7.3% 11.2% 9.2% 14.3% 3.0% 4.3%  Lower vocational 16.0% 23.4% 16.0% 26.7% 16.2% 16.0%  Intermediate 49.3% 42.5% 49.9% 40.7% 48.1% 46.6%  Higher Vocational 20.0% 18.2% 19.8% 15.1% 20.3% 25.3%  University degree 7.0% 4.4% 4.8% 3.1% 11.9% 7.4% Education sector  Technical 34.4% 5.3% 34.1% 4.5% 35.1% 7.1%  Economic/administrative 25.9% 24.5% 24.8% 23.8% 28.5% 26.1%  General (not specialised) 18.1% 30.2% 19.0% 32.7% 16.1% 24.7%  Services 21.5% 40.0% 22.1% 39.0% 20.2% 42.1% Weekly working hours  No. of observations 6,618 5,408 4,350 3,300 2,268 2,108  Mean 39.4 30.9 39.3 28.9 39.6 34.2  (Standard deviation) (7.9) (10.8) (8.0) (11.3) (7.8) (9.2) Hourly wage rates  No. of observations 6,258 4,997 4,113 3,009 2,145 1,988  Mean (Guilders) 21.8 18.2 23.0 18.7 19.1 17.4  (Standard deviation) (7.7) (6.4) (8.0) (6.7) (6.5) (6.0) Age  Mean 40.8 38.7 45.1 43.1 31.2 28.8  (Standard deviation) (12.4) (12.5) (11.6) (11.7) (7.7) (7.9) Household level variables Non‐labour income  Household level, weekly  Mean (guilders) 37.7 36.8 39.8  Standard deviation (94.8) (93.2) (98.2) Employment status  Both partners working 64.3% 55.4% 84.0%  Husband working, wife not 20.2% 25.1% 9.3%  Wife working, husband not 6.0% 6.6% 4.7%  Both not working 9.5% 12.9% 2.0% Marital status  Married 69.1% 100% 0% Open in new tab Table 1 
Descriptive Statistics Variable . Pooled data . Married couples . Unmarried couples . 8,049 Observations . 5,558 Observations . 2,491 Observations . Husband . Wife . Husband . Wife . Husband . Wife . Employment status  Employed 84.4% 70.3% 80.4% 62.0% 93.3% 88.7%  Not Employed 15.6% 29.7% 19.6% 38.0% 6.7% 11.3% Education level  Primary 7.3% 11.2% 9.2% 14.3% 3.0% 4.3%  Lower vocational 16.0% 23.4% 16.0% 26.7% 16.2% 16.0%  Intermediate 49.3% 42.5% 49.9% 40.7% 48.1% 46.6%  Higher Vocational 20.0% 18.2% 19.8% 15.1% 20.3% 25.3%  University degree 7.0% 4.4% 4.8% 3.1% 11.9% 7.4% Education sector  Technical 34.4% 5.3% 34.1% 4.5% 35.1% 7.1%  Economic/administrative 25.9% 24.5% 24.8% 23.8% 28.5% 26.1%  General (not specialised) 18.1% 30.2% 19.0% 32.7% 16.1% 24.7%  Services 21.5% 40.0% 22.1% 39.0% 20.2% 42.1% Weekly working hours  No. of observations 6,618 5,408 4,350 3,300 2,268 2,108  Mean 39.4 30.9 39.3 28.9 39.6 34.2  (Standard deviation) (7.9) (10.8) (8.0) (11.3) (7.8) (9.2) Hourly wage rates  No. of observations 6,258 4,997 4,113 3,009 2,145 1,988  Mean (Guilders) 21.8 18.2 23.0 18.7 19.1 17.4  (Standard deviation) (7.7) (6.4) (8.0) (6.7) (6.5) (6.0) Age  Mean 40.8 38.7 45.1 43.1 31.2 28.8  (Standard deviation) (12.4) (12.5) (11.6) (11.7) (7.7) (7.9) Household level variables Non‐labour income  Household level, weekly  Mean (guilders) 37.7 36.8 39.8  Standard deviation (94.8) (93.2) (98.2) Employment status  Both partners working 64.3% 55.4% 84.0%  Husband working, wife not 20.2% 25.1% 9.3%  Wife working, husband not 6.0% 6.6% 4.7%  Both not working 9.5% 12.9% 2.0% Marital status  Married 69.1% 100% 0% Variable . Pooled data . Married couples . Unmarried couples . 8,049 Observations . 5,558 Observations . 2,491 Observations . Husband . Wife . Husband . Wife . Husband . Wife . Employment status  Employed 84.4% 70.3% 80.4% 62.0% 93.3% 88.7%  Not Employed 15.6% 29.7% 19.6% 38.0% 6.7% 11.3% Education level  Primary 7.3% 11.2% 9.2% 14.3% 3.0% 4.3%  Lower vocational 16.0% 23.4% 16.0% 26.7% 16.2% 16.0%  Intermediate 49.3% 42.5% 49.9% 40.7% 48.1% 46.6%  Higher Vocational 20.0% 18.2% 19.8% 15.1% 20.3% 25.3%  University degree 7.0% 4.4% 4.8% 3.1% 11.9% 7.4% Education sector  Technical 34.4% 5.3% 34.1% 4.5% 35.1% 7.1%  Economic/administrative 25.9% 24.5% 24.8% 23.8% 28.5% 26.1%  General (not specialised) 18.1% 30.2% 19.0% 32.7% 16.1% 24.7%  Services 21.5% 40.0% 22.1% 39.0% 20.2% 42.1% Weekly working hours  No. of observations 6,618 5,408 4,350 3,300 2,268 2,108  Mean 39.4 30.9 39.3 28.9 39.6 34.2  (Standard deviation) (7.9) (10.8) (8.0) (11.3) (7.8) (9.2) Hourly wage rates  No. of observations 6,258 4,997 4,113 3,009 2,145 1,988  Mean (Guilders) 21.8 18.2 23.0 18.7 19.1 17.4  (Standard deviation) (7.7) (6.4) (8.0) (6.7) (6.5) (6.0) Age  Mean 40.8 38.7 45.1 43.1 31.2 28.8  (Standard deviation) (12.4) (12.5) (11.6) (11.7) (7.7) (7.9) Household level variables Non‐labour income  Household level, weekly  Mean (guilders) 37.7 36.8 39.8  Standard deviation (94.8) (93.2) (98.2) Employment status  Both partners working 64.3% 55.4% 84.0%  Husband working, wife not 20.2% 25.1% 9.3%  Wife working, husband not 6.0% 6.6% 4.7%  Both not working 9.5% 12.9% 2.0% Marital status  Married 69.1% 100% 0% Open in new tab The males in the sample are on average more highly educated than the females. I also have information about educational criteria and there are some typical differences between males and females. There are few women with a technical type of education whereas the majority of the men followed a technical education. Most women are educated for the service sector. There are also more women without specialisation in education. The mean age for males is about 2 years higher than for females, which is quite common for married couples. The mean weekly number of working hours is 39 for males and 31 for females. The males have an hourly wage rate that on average is almost 4 guilders higher than the wage rate of females. The non‐labour income includes interest income, income out of real estate, rent subsidy, income out of life insurance (‘lijfrente’), gifts by family, income out of profits and scholarships. In the survey it is measured on a yearly basis and in Table 1 it is converted to guilders per week. The average is about 34 guilders a week, and there is quite some variation in it, with some households reporting much higher amounts, and some households reporting not to have received any non‐labour income. A comparison of the sub sample of the married and the unmarried shows that the unmarried are on average younger, there are more dual earner couples among the unmarried, education levels are higher among the unmarried and education differences between men and women are on average smaller for unmarried couples. The pooled data contain observations on 5,558 married couples and 2,491 unmarried couples. The sub sample of married couples contains 1,514 different households and the sub sample of unmarried couples consists of 875 different households. Households appear at least once and at most 12 times in the sample. About 70% of the households appear more than once. For various reasons only a limited number of the households appear in all of the 12 years. These reasons include attrition (households either leave the panel or do not satisfy my selection criteria anymore), item non‐response (relevant information is missing), wave non‐response, or households may be new entrants in the panel. Throughout I maintain the assumption that non‐response is independent of hours, wage rates and employment status (as it is defined in this analysis). Note that attrition of households may also occur if a couple splits up: a household is considered the same through different waves if both of the partners remain the same two persons. 4. Results I have estimated the model defined in (24)–(28) by the method of simulated maximum likelihood (SML). I use 60 replications to simulate wages (for individuals without an observed wage) and the random effects. To allow the underlying parameters of the sharing rule to differ by marital status, I estimate the model separately for unmarried and married couples. Note that additive heterogeneity in the (reduced form) labour supply function can come both from (observed and unobserved) heterogeneity in preferences and from additive heterogeneity in the sharing rule. Therefore, unless one is willing to make arbitrary exclusion restrictions about the source of heterogeneity, a sharing rule that differs by marital status requires that all parameters of the labour supply functions are different for the married and the unmarried.16 I use a minimum distance estimator to impose and test equality of the parameters of the structural labour supply function (6) by marital status. I also use a minimum distance estimator to impose and test equality of the parameters of the sharing rule (9) by marital status. 4.1. Parameter Estimates of the ‘Reduced Form’ Labour Supply Function Table 2 presents the estimates of the parameters of the labour supply function of husband and wife by marital status. These are the estimates of the parameters aj and bj, j = 1,…,7 in (10).17 The parameters of the remaining coefficients are presented in the Appendix. Table A1 shows the coefficients of the observed characteristics included in the labour supply function, Table A2 shows the coefficients of the wage equations, and Table A3 presents the coefficients of the covariance matrices. I use the information on education sectors as an exclusion restriction in the wage equations. Thus, I assume that the individual preferences for leisure do not vary by sector. However, wages are likely to differ by sector. The service sector is the reference category. In the wage equation, the highest level of education is the reference level. In the labour supply functions I have merged the two higher education levels to one category, since the highest education group is small. I have also included time dummies in the wage equations. Table 2 
Simulated Maximum Likelihood Estimates: Labour Supply Equations Variable . Unmarried . Married . Men . Women . Men . Women . Intercept −11.829 31.319** 16.468** 30.374** (9.562) (11.121) (8.271) (6.411) Wage husband 0.291 0.342 0.192 0.192 (0.394) (0.543) (0.274) (0.241) Wage wife 1.278** 0.102 0.422 0.900** (0.457) (0.543) (0.305) (0.270) wm/(wm + wf) 50.619** −38.265** 8.425 −39.761** (17.408) (22.468) (14.168) (11.400) Wage husband squared/10 −0.120** −0.016 −0.085** 0.004 (0.043) (0.061) (0.027) (0.024) Wage wife squared/10 −0.132** −0.064 −0.054 −0.128** (0.063) (0.067) (0.033) (0.034) Non‐labour income/1000 −9.728** −7.157** −4.709** 5.902** (2.140) (2.304) (2.165) (1.407) sj(j = m,f) 118.095** −0.005 40.950** −0.024 (42.895) (0.069) (3.556) (0.075) Variable . Unmarried . Married . Men . Women . Men . Women . Intercept −11.829 31.319** 16.468** 30.374** (9.562) (11.121) (8.271) (6.411) Wage husband 0.291 0.342 0.192 0.192 (0.394) (0.543) (0.274) (0.241) Wage wife 1.278** 0.102 0.422 0.900** (0.457) (0.543) (0.305) (0.270) wm/(wm + wf) 50.619** −38.265** 8.425 −39.761** (17.408) (22.468) (14.168) (11.400) Wage husband squared/10 −0.120** −0.016 −0.085** 0.004 (0.043) (0.061) (0.027) (0.024) Wage wife squared/10 −0.132** −0.064 −0.054 −0.128** (0.063) (0.067) (0.033) (0.034) Non‐labour income/1000 −9.728** −7.157** −4.709** 5.902** (2.140) (2.304) (2.165) (1.407) sj(j = m,f) 118.095** −0.005 40.950** −0.024 (42.895) (0.069) (3.556) (0.075) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table 2 
Simulated Maximum Likelihood Estimates: Labour Supply Equations Variable . Unmarried . Married . Men . Women . Men . Women . Intercept −11.829 31.319** 16.468** 30.374** (9.562) (11.121) (8.271) (6.411) Wage husband 0.291 0.342 0.192 0.192 (0.394) (0.543) (0.274) (0.241) Wage wife 1.278** 0.102 0.422 0.900** (0.457) (0.543) (0.305) (0.270) wm/(wm + wf) 50.619** −38.265** 8.425 −39.761** (17.408) (22.468) (14.168) (11.400) Wage husband squared/10 −0.120** −0.016 −0.085** 0.004 (0.043) (0.061) (0.027) (0.024) Wage wife squared/10 −0.132** −0.064 −0.054 −0.128** (0.063) (0.067) (0.033) (0.034) Non‐labour income/1000 −9.728** −7.157** −4.709** 5.902** (2.140) (2.304) (2.165) (1.407) sj(j = m,f) 118.095** −0.005 40.950** −0.024 (42.895) (0.069) (3.556) (0.075) Variable . Unmarried . Married . Men . Women . Men . Women . Intercept −11.829 31.319** 16.468** 30.374** (9.562) (11.121) (8.271) (6.411) Wage husband 0.291 0.342 0.192 0.192 (0.394) (0.543) (0.274) (0.241) Wage wife 1.278** 0.102 0.422 0.900** (0.457) (0.543) (0.305) (0.270) wm/(wm + wf) 50.619** −38.265** 8.425 −39.761** (17.408) (22.468) (14.168) (11.400) Wage husband squared/10 −0.120** −0.016 −0.085** 0.004 (0.043) (0.061) (0.027) (0.024) Wage wife squared/10 −0.132** −0.064 −0.054 −0.128** (0.063) (0.067) (0.033) (0.034) Non‐labour income/1000 −9.728** −7.157** −4.709** 5.902** (2.140) (2.304) (2.165) (1.407) sj(j = m,f) 118.095** −0.005 40.950** −0.024 (42.895) (0.069) (3.556) (0.075) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab To evaluate the full effect of a change in the husband’s wage rate on his supply of labour I computed the marginal effect a2 + a4wf/(wm + wf)2 + 2a5wm for all couples with observed wage rates in the sample (separately for the married and the unmarried). For unmarried men, this effect is negative for 7.4% of the cases, while for married men, the percentage with a negative wage effect is 67.7%. The latter indicates a backward bending labour supply curve for married men.18 The marginal effect of the wife’s wage rate on the husband’s labour supply is determined by a3 − a4wm/(wm + wf)2 + 2a6wf. For the married, this is positive for 98% of the couples, suggesting that married men tend to increase working hours upon a wage increase of the wife. For the unmarried, the percentage with a positive effect of the wife’s wage rate on the husband’s labour supply is 79%. Non‐labour income has a significant negative effect on the husband’s supply of labour, both for married and unmarried men. The marginal effect of a change in the wife’s wage rate on her labour supply is b3 − b4wm/(wm + wf)2 + 2b5wf. Both for married and unmarried women, this is positive at mean wage rates but for high wage rates the negative backward bending effect dominates. Counting frequencies, the backward bending effect dominates for 2.2% of the married women and for 0.8% of the unmarried. Both for married and unmarried women I find that the marginal effect of the husband’s wage rate on their labour supply is negative for about 98% of the wage rates in the sample. For married women non‐labour income has a negative effect on the wife’s labour supply for unmarried but a positive effect for the married. The estimates of a4 and b4, the coefficients of the wage rate fraction, have the opposite sign, as required by the underlying collective model. The parameter estimates of sm and sf, that determine the husband’s (wife’s) labour supply if the wife (husband) does not work, are positive and negative, respectively. In the estimated values, the coherence constraint |smsf | < 1 is satisfied. The parameter sm is significant and, moreover, its impact is much larger than sf, which is not estimated significantly. I comment on these differences when I discuss the results of the sharing rule.19 4.2. Estimates of Structural Labour Supply and the Sharing Rule Using the estimates of the parameters of the reduced form labour supply equations from Table 2, the parameters of the individual labour supply functions (6) and the sharing rule (9) can be computed using the expressions in (11) and (12). Thus, the total effects of the wage rates and non‐labour income on labour supply can be decomposed into effects arising from individual preferences and into the effects arising from the sharing rule.20 This decomposition can be done separately for the married and unmarried. First I assume that the parameters of the wage rates and non‐labour income of the structural labour supply equations, (6), are the same for the married and the unmarried21 but the sharing rule parameters differ by marital status. I impose this restriction and obtain parameter estimates using a minimum distance estimator. Next, I use a second minimum distance estimator to obtain estimates under the additional assumption that the parameters k2 to k7 of the sharing rule are the same for the married and the unmarried. Imposing restrictions may lead to more precise estimates but only make sense if the restrictions imposed are valid. The value of the objective function of the minimum distance estimator follows the chi‐squared distribution and can be used to test the restrictions. Tables 3 and 4 show the parameters estimates of the individual labour supply functions (6) and the estimates of the sharing rule (9), respectively. They contain both the estimates that do not impose any restrictions between the married and the unmarried, and the estimates that impose restrictions on the preferences parameters and the sharing rule. The chi‐squared statistic, obtained in the minimum distance estimation procedure is 56 if only the parameters of the wage rates and non‐labour income in the labour supply equations are restricted to be equal for the married and the unmarried (6 restrictions). It takes the value 96 if the parameters of the sharing rule are restricted to be equal as well (14 restrictions in total). In both cases, the restrictions are rejected.22 Therefore, I prefer the estimates obtained by separate estimation for the married and unmarried. A comparison with the restricted estimates may reveal the possibly distorting influence of the imposition of restrictions between the married and unmarried on key outcomes. Table 4 
Parameters of the Sharing Rule by Marital Status (see Equation 9) Variable . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Unmarried . Married . Unmarried . Married . All . Men Wage husband, (k2) 24 12 76* 26 36 (32) (34) (41) (18) (24) Wage wife, (k3) −66** −122 −160** −86 −130** (29) (171) (54) (31) (47) wm/(wm + wf), (k4) −2,637** −2,436 −5,578** −3,308** −5,088** (1,096) (5,617) (2,124) (1,249) (1,984) Sqr. wage husband, (k5) −0.11 0.022 −0.56 −0.10 −0.087 (0.4) (0.14) (0.42) (0.17) (0.22) Sqr. wage wife, (k6) 0.69** 1.6** 1.5** 0.91** 1.2** (0.38) (0.2) (0.6) (0.39) (0.5) Non‐labour income, (k7) 0.51** 1.4* 0.51** 1.4** 1.3** (0.17) (0.8) (0.20) (0.15) (0.18) rm −6,152 −11,841 −19,278** −6,138** −5,735** (5,507) (15,558) (7,460) (1,368) (1,259) rf −0.34 −1.5 −0.16 −1.8 −0.18 (4.7) (5.7) (4.7) (5.1) (5.3) Variable . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Unmarried . Married . Unmarried . Married . All . Men Wage husband, (k2) 24 12 76* 26 36 (32) (34) (41) (18) (24) Wage wife, (k3) −66** −122 −160** −86 −130** (29) (171) (54) (31) (47) wm/(wm + wf), (k4) −2,637** −2,436 −5,578** −3,308** −5,088** (1,096) (5,617) (2,124) (1,249) (1,984) Sqr. wage husband, (k5) −0.11 0.022 −0.56 −0.10 −0.087 (0.4) (0.14) (0.42) (0.17) (0.22) Sqr. wage wife, (k6) 0.69** 1.6** 1.5** 0.91** 1.2** (0.38) (0.2) (0.6) (0.39) (0.5) Non‐labour income, (k7) 0.51** 1.4* 0.51** 1.4** 1.3** (0.17) (0.8) (0.20) (0.15) (0.18) rm −6,152 −11,841 −19,278** −6,138** −5,735** (5,507) (15,558) (7,460) (1,368) (1,259) rf −0.34 −1.5 −0.16 −1.8 −0.18 (4.7) (5.7) (4.7) (5.1) (5.3) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table 4 
Parameters of the Sharing Rule by Marital Status (see Equation 9) Variable . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Unmarried . Married . Unmarried . Married . All . Men Wage husband, (k2) 24 12 76* 26 36 (32) (34) (41) (18) (24) Wage wife, (k3) −66** −122 −160** −86 −130** (29) (171) (54) (31) (47) wm/(wm + wf), (k4) −2,637** −2,436 −5,578** −3,308** −5,088** (1,096) (5,617) (2,124) (1,249) (1,984) Sqr. wage husband, (k5) −0.11 0.022 −0.56 −0.10 −0.087 (0.4) (0.14) (0.42) (0.17) (0.22) Sqr. wage wife, (k6) 0.69** 1.6** 1.5** 0.91** 1.2** (0.38) (0.2) (0.6) (0.39) (0.5) Non‐labour income, (k7) 0.51** 1.4* 0.51** 1.4** 1.3** (0.17) (0.8) (0.20) (0.15) (0.18) rm −6,152 −11,841 −19,278** −6,138** −5,735** (5,507) (15,558) (7,460) (1,368) (1,259) rf −0.34 −1.5 −0.16 −1.8 −0.18 (4.7) (5.7) (4.7) (5.1) (5.3) Variable . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Unmarried . Married . Unmarried . Married . All . Men Wage husband, (k2) 24 12 76* 26 36 (32) (34) (41) (18) (24) Wage wife, (k3) −66** −122 −160** −86 −130** (29) (171) (54) (31) (47) wm/(wm + wf), (k4) −2,637** −2,436 −5,578** −3,308** −5,088** (1,096) (5,617) (2,124) (1,249) (1,984) Sqr. wage husband, (k5) −0.11 0.022 −0.56 −0.10 −0.087 (0.4) (0.14) (0.42) (0.17) (0.22) Sqr. wage wife, (k6) 0.69** 1.6** 1.5** 0.91** 1.2** (0.38) (0.2) (0.6) (0.39) (0.5) Non‐labour income, (k7) 0.51** 1.4* 0.51** 1.4** 1.3** (0.17) (0.8) (0.20) (0.15) (0.18) rm −6,152 −11,841 −19,278** −6,138** −5,735** (5,507) (15,558) (7,460) (1,368) (1,259) rf −0.34 −1.5 −0.16 −1.8 −0.18 (4.7) (5.7) (4.7) (5.1) (5.3) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table 3 
Parameters Structural Labour Supply by Marital Status Variable . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Unmarried . Married . All . All . Men Non‐lab. inc() −0.019** −0.0035 −0.0066** −0.0072** (0.0056) (0.0028) (0.0014) (0.0015) (Own) Wage () 0.74 0.23 0.43** 0.26* (0.45) (0.20) (0.11) (0.15) (Own) wage squared () −0.014** −0.0084** −0.010** −0.0080** (0.068) (0.0026) (0.0015) (0.0020) Women Non‐lab. inc() −0.015** −0.016 −0.015** −0.0094** (0.0046) (0.035) (0.0041) (0.0035) (Own) Wage () 1.1 2.9 1.9** 1.9** (1.3) (2.3) (0.21) (0.14) (Own) wage squared () −0.016 −0.038 −0.024** −0.024** (0.23) (0.14) (0.0034) (0.0021) Variable . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Unmarried . Married . All . All . Men Non‐lab. inc() −0.019** −0.0035 −0.0066** −0.0072** (0.0056) (0.0028) (0.0014) (0.0015) (Own) Wage () 0.74 0.23 0.43** 0.26* (0.45) (0.20) (0.11) (0.15) (Own) wage squared () −0.014** −0.0084** −0.010** −0.0080** (0.068) (0.0026) (0.0015) (0.0020) Women Non‐lab. inc() −0.015** −0.016 −0.015** −0.0094** (0.0046) (0.035) (0.0041) (0.0035) (Own) Wage () 1.1 2.9 1.9** 1.9** (1.3) (2.3) (0.21) (0.14) (Own) wage squared () −0.016 −0.038 −0.024** −0.024** (0.23) (0.14) (0.0034) (0.0021) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table 3 
Parameters Structural Labour Supply by Marital Status Variable . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Unmarried . Married . All . All . Men Non‐lab. inc() −0.019** −0.0035 −0.0066** −0.0072** (0.0056) (0.0028) (0.0014) (0.0015) (Own) Wage () 0.74 0.23 0.43** 0.26* (0.45) (0.20) (0.11) (0.15) (Own) wage squared () −0.014** −0.0084** −0.010** −0.0080** (0.068) (0.0026) (0.0015) (0.0020) Women Non‐lab. inc() −0.015** −0.016 −0.015** −0.0094** (0.0046) (0.035) (0.0041) (0.0035) (Own) Wage () 1.1 2.9 1.9** 1.9** (1.3) (2.3) (0.21) (0.14) (Own) wage squared () −0.016 −0.038 −0.024** −0.024** (0.23) (0.14) (0.0034) (0.0021) Variable . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Unmarried . Married . All . All . Men Non‐lab. inc() −0.019** −0.0035 −0.0066** −0.0072** (0.0056) (0.0028) (0.0014) (0.0015) (Own) Wage () 0.74 0.23 0.43** 0.26* (0.45) (0.20) (0.11) (0.15) (Own) wage squared () −0.014** −0.0084** −0.010** −0.0080** (0.068) (0.0026) (0.0015) (0.0020) Women Non‐lab. inc() −0.015** −0.016 −0.015** −0.0094** (0.0046) (0.035) (0.0041) (0.0035) (Own) Wage () 1.1 2.9 1.9** 1.9** (1.3) (2.3) (0.21) (0.14) (Own) wage squared () −0.016 −0.038 −0.024** −0.024** (0.23) (0.14) (0.0034) (0.0021) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab The first two columns of Table 3 contain the parameters of the individual labour supply functions (6) for the unmarried and married.23 Not all the parameters are estimated precisely. However, regularity conditions are satisfied. Since the coefficient of non‐labour income is negative (both for married and unmarried men and women) the estimates imply that leisure is a normal good. For unmarried men, the backward bending labour supply effect becomes dominant from wage rates with values from around the sample mean on, whereas for married men the backward bending effect is also dominant for wage rates below the sample mean. For women, the backward bending effect is much weaker and there is a positive effect of wage rates for almost all women in the sample. Table 4 contains the estimates of the sharing rule (9). The estimates in the first two columns (for the married and the unmarried) have been directly computed from the reduced form parameters in Table 2. Thus, no restrictions are imposed between any of the parameters of the married and the unmarried. The third and fourth columns are obtained by the first minimum distance estimator, that restricts the parameters of the wage rates and non‐labour income in the structural labour supply function to be equal for the married and the unmarried. However, the parameters of the sharing rule differ by marital status. The final column shows the results obtained with the second minimum distance estimator. Both the parameters of the structural labour supply function and the parameters of the sharing rule are restricted to be equal for the married and the unmarried. In general, coefficients have the same sign for the different methods of estimation, but the quantitative impact differs. For the unmarried, the coefficient of non‐labour income is around 0.5, indicating that unmarried couples seem to share their non‐labour income fifty‐fifty. For the married, I find a coefficient slightly larger than, but not always significantly different from, 1. This suggests that in married couples a substantial part of the non‐labour income is assigned to the husband.24 This difference between the married and the unmarried is not revealed if the coefficients of the sharing rule are restricted to be equal: the final column of Table 4 shows a coefficient of non‐labour income that is not significantly different from one. The marginal effect of a change in wm on the sharing rule is k2 + k4wf/(wm + wf)2 + 2k5 wm. I computed these marginal effects for all couples with observed wage rates in the sample. I counted the percentage of households for which the effect is positive. Depending on which parameter estimates and subsample from Table 4 are considered, the percentage with a negative impact ranges from 96 to 98. Thus, the wife’s share increases upon an increase in the husband’s wage rate. The consequence of the increase in the wife’s share is that her supply of labour will diminish since leisure is a normal good to her. The marginal effect of the wife’s wage rate on the sharing rule is k3 − k4wm/(wm + wf)2 + 2k6wf. This is negative for the majority of the observations. Depending on which parameter estimates and subsample from Table 4 are considered, the percentage with a negative impact ranges from 79 to 98. Thus, an increase in the wife’s wage rate apparently leads to a decrease in the husband’s share for most of the cases. The negative impact of the wife’s wage rate on the share of the husband is easily interpretable in terms of an increase in the wife’s bargaining power as a result of the increase in her wage rate. The husband’s resources decrease and he increases his supply of labour to compensate for that. The negative impact of the husband’s wage rate on his share is beneficial to the wife. Here a household income effect dominates, allowing the wife to consume more leisure. The total effect of the increase of the husband’s wage rate on his welfare level is ambiguous. The decrease in his share decreases his leisure and therefore decreases his utility level. On the other hand, the higher wage rate has, at given working hours, a positive impact on his private consumption, which increases his utility level. A possibly backward bending labour supply curve may increase his leisure, with a positive consequence for his utility level. To evaluate the eventual impact on utility, I have computed the marginal effects of the wage rates and the non‐labour income on the indirect utility function, substituting the appropriate share for non‐labour income.25 I computed the marginal effects for all the observations with observed wage rates and for 60 replications of the random effects.26Table 5 reports the sample percentage of men and women experiencing an increase in utility due to a change in either of the wage rates or the non‐labour income. For almost all men in the sample, the utility level increases if the men’s wage rate goes up, irrespective of the estimates looked at. Also for almost all women the utility level increases as a result of an increase in the husband’s wage rate, which reflects the fact that the wife’s share increases for almost all women. Table 5 
Sample Percentages of Individuals with an Increase in Utility . Estimation method . . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Increase in . Percentage men with increase in utility Unmarried Married Unmarried Married Unmarried Married Wage rate husband 99.4 100 100 100 99.9 100 Wage rate wife 20.9 2.4 5.4 15.1 15.1 11.3 Non‐labour income 100 100 100 100 100 100 Increase in . Percentage women with increase in utility Unmarried Married Unmarried Married Unmarried Married Wage rate husband 97.8 97.5 95.9 97.1 97.7 96.2 Wage rate wife 99.9 100 99.3 100 100 100 Non‐labour income 100 0 100 0 0 0 . Estimation method . . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Increase in . Percentage men with increase in utility Unmarried Married Unmarried Married Unmarried Married Wage rate husband 99.4 100 100 100 99.9 100 Wage rate wife 20.9 2.4 5.4 15.1 15.1 11.3 Non‐labour income 100 100 100 100 100 100 Increase in . Percentage women with increase in utility Unmarried Married Unmarried Married Unmarried Married Wage rate husband 97.8 97.5 95.9 97.1 97.7 96.2 Wage rate wife 99.9 100 99.3 100 100 100 Non‐labour income 100 0 100 0 0 0 Sample Percentages of Individuals with an Increase in Utility in Response to an Increase in Wage Rates and Non‐labour Income. Computations for couples with observed wages. Open in new tab Table 5 
Sample Percentages of Individuals with an Increase in Utility . Estimation method . . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Increase in . Percentage men with increase in utility Unmarried Married Unmarried Married Unmarried Married Wage rate husband 99.4 100 100 100 99.9 100 Wage rate wife 20.9 2.4 5.4 15.1 15.1 11.3 Non‐labour income 100 100 100 100 100 100 Increase in . Percentage women with increase in utility Unmarried Married Unmarried Married Unmarried Married Wage rate husband 97.8 97.5 95.9 97.1 97.7 96.2 Wage rate wife 99.9 100 99.3 100 100 100 Non‐labour income 100 0 100 0 0 0 . Estimation method . . Directly computed from estimates in Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Increase in . Percentage men with increase in utility Unmarried Married Unmarried Married Unmarried Married Wage rate husband 99.4 100 100 100 99.9 100 Wage rate wife 20.9 2.4 5.4 15.1 15.1 11.3 Non‐labour income 100 100 100 100 100 100 Increase in . Percentage women with increase in utility Unmarried Married Unmarried Married Unmarried Married Wage rate husband 97.8 97.5 95.9 97.1 97.7 96.2 Wage rate wife 99.9 100 99.3 100 100 100 Non‐labour income 100 0 100 0 0 0 Sample Percentages of Individuals with an Increase in Utility in Response to an Increase in Wage Rates and Non‐labour Income. Computations for couples with observed wages. Open in new tab An increase in the wife’s wage rate leads to an increase in the husband’s utility level for only a small fraction of the sample. This suggests that the wife’s bargaining power increases as her wage rate increases. The utility level increases for almost every woman in the sample. The largest difference between the unmarried and the married is shown by an increase in the household’s non‐labour income. Both for the unmarried and the married, the husband’s share is increased as a result of an increase in the non‐labour income. Since leisure is a normal good, his supply of labour decreases, which has a positive direct effect on his utility. His total earnings will decrease due to the decrease in working hours. The total effect is an increase in utility for all unmarried and married men. For unmarried women, the share is also increased due to an increase in the household’s non‐labour income. I also find an increase in the utility of all unmarried women. For married women, the share decreases. It should be noted, however, that since the coefficient of the non‐labour income is not always significantly different from one, the women’s share may not be affected significantly. The utility of all married women decreases. This suggests that unmarried women have a stronger bargaining position than married women. Assigning the same sharing rule parameters to married and unmarried women leads to the wrong conclusion that the utility level of unmarried women decreases as well. From estimates in Table 4 I also compute the sharing rule for the case in which the husband (wife) works and the wife (husband) not (see (16) and (21)). I have done this explicitly for the unrestricted estimates. The results are in Table 6. Notably the sharing rule for a working husband and a non‐working wife is different from the rule in Table 4, which is due to the large value of the estimate of rm in (16). The estimate of rf in (21) is much smaller and is not significantly different from zero. Reservation wages of non‐working males may be much lower and show less variation than reservation wages of non‐working females. This may explain why the sharing rule for working males with non‐working wives differs much more from the sharing rule in Table 4. I have determined that for almost all wage rates in the sample range an increase in the wage rate of the husband now leads to an increase in the ‘share’ of the husband. This may reflect a higher bargaining power of the husband if the wife is not working. Moreover, a non‐working woman cannot reduce working hours if the husband transfers his wage increase to his wife. The latter may be a motivation for a husband with a working wife to increase her share upon an increase in her wage rate. The share of the husband decreases if the wage rate of the non‐working wife increases, both for married and unmarried couples. Thus, the wage rate of a non‐working wife may still function as a threat point. However, the relatively large standards error indicate that it is hard to obtain accurate estimates of the sharing rule for households with a non‐working wife. This may be related to the small fraction of these households in the sample. Table 6 
The Sharing Rule Outside the Participation Frontier Variable . Husband works Wife not Equation (16) . Wife works Husband not Equation (21) . Unmarried . Married . Unmarried . Married . Wage husband −2,079 −2,267 23** 12 (4,992) (5,398) (7) (80) Wage wife −695 −10,777 −67** −123 (2,936) (12,175) (3) (172) wm/(wm + wf) 232,772 468,391 2,654** −2,449 (337,906) (728,362) (1,100) (5,638) Sqr. wage husband 10 −4 −0.11 0.034 (44) (25) (24) (5) Sqr. wage wife 40 153 0.69* 1.6 (33) (182) (0.38) (2.0) Non‐labour income 45 −69 0.51** 1.4* (41) (91) (0.17) (0.8) Variable . Husband works Wife not Equation (16) . Wife works Husband not Equation (21) . Unmarried . Married . Unmarried . Married . Wage husband −2,079 −2,267 23** 12 (4,992) (5,398) (7) (80) Wage wife −695 −10,777 −67** −123 (2,936) (12,175) (3) (172) wm/(wm + wf) 232,772 468,391 2,654** −2,449 (337,906) (728,362) (1,100) (5,638) Sqr. wage husband 10 −4 −0.11 0.034 (44) (25) (24) (5) Sqr. wage wife 40 153 0.69* 1.6 (33) (182) (0.38) (2.0) Non‐labour income 45 −69 0.51** 1.4* (41) (91) (0.17) (0.8) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Estimates computed from Table 2. Open in new tab Table 6 
The Sharing Rule Outside the Participation Frontier Variable . Husband works Wife not Equation (16) . Wife works Husband not Equation (21) . Unmarried . Married . Unmarried . Married . Wage husband −2,079 −2,267 23** 12 (4,992) (5,398) (7) (80) Wage wife −695 −10,777 −67** −123 (2,936) (12,175) (3) (172) wm/(wm + wf) 232,772 468,391 2,654** −2,449 (337,906) (728,362) (1,100) (5,638) Sqr. wage husband 10 −4 −0.11 0.034 (44) (25) (24) (5) Sqr. wage wife 40 153 0.69* 1.6 (33) (182) (0.38) (2.0) Non‐labour income 45 −69 0.51** 1.4* (41) (91) (0.17) (0.8) Variable . Husband works Wife not Equation (16) . Wife works Husband not Equation (21) . Unmarried . Married . Unmarried . Married . Wage husband −2,079 −2,267 23** 12 (4,992) (5,398) (7) (80) Wage wife −695 −10,777 −67** −123 (2,936) (12,175) (3) (172) wm/(wm + wf) 232,772 468,391 2,654** −2,449 (337,906) (728,362) (1,100) (5,638) Sqr. wage husband 10 −4 −0.11 0.034 (44) (25) (24) (5) Sqr. wage wife 40 153 0.69* 1.6 (33) (182) (0.38) (2.0) Non‐labour income 45 −69 0.51** 1.4* (41) (91) (0.17) (0.8) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Estimates computed from Table 2. Open in new tab 4.3. Elasticities The qualitative implications of the estimation results obtained by different methods do not deviate much from each other, except for the specification with a common sharing rule for the married and the unmarried. However, the restrictions imposed in the minimum distance estimation have been rejected. Computation of the elasticities for the various estimates may reveal the quantitative implications of imposing these restrictions. Tables 7 and 8 report elasticities of labour supply with respect to the wage rates of both spouses and non‐labour income. The elasticities are computed separately for the married and the unmarried, and evaluated in the sample means of the corresponding subsamples. Table 7 shows total effects of preferences and the sharing rule. The elasticities for the married and the unmarried in the first two columns are directly based on the reduced form estimates in Table 2. To compute the elasticities of labour supply for the minimum distance estimates, I have inserted the sharing rule estimates from Table 4 in the structural labour supply function, based on the corresponding estimates in Table 3. Table 8 
Elasticities of the Structural Labour Supply Function . Estimation method . . Reduced form estimates from Table 2 . Minimum distance, same preferences different sharing rule . Minimum distance, same preferences same sharing rule . Men: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.098 −0.085* 0.013 −0.026 −0.021 −0.058*  husband (0.099) (0.050) (0.026) (0.027) (0.035) (0.035) Non‐labour −0.019** −0.0031 −0.0065** −0.0059** −0.0071** −0.0064**  income (0.006) (0.0025) (0.0014) (0.0012) (0.0014) (0.0013) Women: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.25 0.95 0.56** 0.67** 0.54** 0.65**  wife (0.73) (0.87) (0.02) (0.06) (0.004) (0.05) Non‐labour −0.017** −0.020 −0.017** −0.018** −0.011** −0.011**  income (0.005) (0.043) (0.005) (0.003) (0.004) (0.004) . Estimation method . . Reduced form estimates from Table 2 . Minimum distance, same preferences different sharing rule . Minimum distance, same preferences same sharing rule . Men: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.098 −0.085* 0.013 −0.026 −0.021 −0.058*  husband (0.099) (0.050) (0.026) (0.027) (0.035) (0.035) Non‐labour −0.019** −0.0031 −0.0065** −0.0059** −0.0071** −0.0064**  income (0.006) (0.0025) (0.0014) (0.0012) (0.0014) (0.0013) Women: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.25 0.95 0.56** 0.67** 0.54** 0.65**  wife (0.73) (0.87) (0.02) (0.06) (0.004) (0.05) Non‐labour −0.017** −0.020 −0.017** −0.018** −0.011** −0.011**  income (0.005) (0.043) (0.005) (0.003) (0.004) (0.004) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table 8 
Elasticities of the Structural Labour Supply Function . Estimation method . . Reduced form estimates from Table 2 . Minimum distance, same preferences different sharing rule . Minimum distance, same preferences same sharing rule . Men: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.098 −0.085* 0.013 −0.026 −0.021 −0.058*  husband (0.099) (0.050) (0.026) (0.027) (0.035) (0.035) Non‐labour −0.019** −0.0031 −0.0065** −0.0059** −0.0071** −0.0064**  income (0.006) (0.0025) (0.0014) (0.0012) (0.0014) (0.0013) Women: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.25 0.95 0.56** 0.67** 0.54** 0.65**  wife (0.73) (0.87) (0.02) (0.06) (0.004) (0.05) Non‐labour −0.017** −0.020 −0.017** −0.018** −0.011** −0.011**  income (0.005) (0.043) (0.005) (0.003) (0.004) (0.004) . Estimation method . . Reduced form estimates from Table 2 . Minimum distance, same preferences different sharing rule . Minimum distance, same preferences same sharing rule . Men: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.098 −0.085* 0.013 −0.026 −0.021 −0.058*  husband (0.099) (0.050) (0.026) (0.027) (0.035) (0.035) Non‐labour −0.019** −0.0031 −0.0065** −0.0059** −0.0071** −0.0064**  income (0.006) (0.0025) (0.0014) (0.0012) (0.0014) (0.0013) Women: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.25 0.95 0.56** 0.67** 0.54** 0.65**  wife (0.73) (0.87) (0.02) (0.06) (0.004) (0.05) Non‐labour −0.017** −0.020 −0.017** −0.018** −0.011** −0.011**  income (0.005) (0.043) (0.005) (0.003) (0.004) (0.004) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table 7 
Elasticities: Total Effects of Wage Rates and Non‐labour Income on Labour Supply . Estimation method . . Reduced form estimates from Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Men: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.24** −0.057* 0.071 0.028 0.092 0.036  husband (0.04) (0.032) (0.093) (0.058) (0.086) (0.090) Wage rate 0.038 0.052** 0.078** 0.023 0.040 0.052  wife (0.030) (0.021) (0.023) (0.062) (0.067) (0.078) Non‐labour −0.0096** −0.0042** −0.0033** −0.0082** −0.0092** −0.0083**  income (0.0021) (0.0019) (0.0017) (0.0016) (0.0014) (0.0013) Women: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate −0.12** −0.18** −0.15** −0.16** −0.17** −0.17**  husband (0.04) (0.03) (0.04) (0.03) (0.03) (0.03) Wage rate 0.22** 0.61** 0.36 0.60** 0.48** 0.55**  wife (0.07) (0.04) (0.31) (0.19) (0.14) (0.16) Non‐labour −0.0081** 0.0071** −0.0080** 0.0071** 0.0033 0.0035  income (0.0023) (0.0013) (0.0044) (0.0032) (0.0026) (0.0023) . Estimation method . . Reduced form estimates from Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Men: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.24** −0.057* 0.071 0.028 0.092 0.036  husband (0.04) (0.032) (0.093) (0.058) (0.086) (0.090) Wage rate 0.038 0.052** 0.078** 0.023 0.040 0.052  wife (0.030) (0.021) (0.023) (0.062) (0.067) (0.078) Non‐labour −0.0096** −0.0042** −0.0033** −0.0082** −0.0092** −0.0083**  income (0.0021) (0.0019) (0.0017) (0.0016) (0.0014) (0.0013) Women: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate −0.12** −0.18** −0.15** −0.16** −0.17** −0.17**  husband (0.04) (0.03) (0.04) (0.03) (0.03) (0.03) Wage rate 0.22** 0.61** 0.36 0.60** 0.48** 0.55**  wife (0.07) (0.04) (0.31) (0.19) (0.14) (0.16) Non‐labour −0.0081** 0.0071** −0.0080** 0.0071** 0.0033 0.0035  income (0.0023) (0.0013) (0.0044) (0.0032) (0.0026) (0.0023) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table 7 
Elasticities: Total Effects of Wage Rates and Non‐labour Income on Labour Supply . Estimation method . . Reduced form estimates from Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Men: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.24** −0.057* 0.071 0.028 0.092 0.036  husband (0.04) (0.032) (0.093) (0.058) (0.086) (0.090) Wage rate 0.038 0.052** 0.078** 0.023 0.040 0.052  wife (0.030) (0.021) (0.023) (0.062) (0.067) (0.078) Non‐labour −0.0096** −0.0042** −0.0033** −0.0082** −0.0092** −0.0083**  income (0.0021) (0.0019) (0.0017) (0.0016) (0.0014) (0.0013) Women: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate −0.12** −0.18** −0.15** −0.16** −0.17** −0.17**  husband (0.04) (0.03) (0.04) (0.03) (0.03) (0.03) Wage rate 0.22** 0.61** 0.36 0.60** 0.48** 0.55**  wife (0.07) (0.04) (0.31) (0.19) (0.14) (0.16) Non‐labour −0.0081** 0.0071** −0.0080** 0.0071** 0.0033 0.0035  income (0.0023) (0.0013) (0.0044) (0.0032) (0.0026) (0.0023) . Estimation method . . Reduced form estimates from Table 2 . Minimum distance, same labour supply, different sharing rule . Minimum distance, same labour supply, same sharing rule . Men: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate 0.24** −0.057* 0.071 0.028 0.092 0.036  husband (0.04) (0.032) (0.093) (0.058) (0.086) (0.090) Wage rate 0.038 0.052** 0.078** 0.023 0.040 0.052  wife (0.030) (0.021) (0.023) (0.062) (0.067) (0.078) Non‐labour −0.0096** −0.0042** −0.0033** −0.0082** −0.0092** −0.0083**  income (0.0021) (0.0019) (0.0017) (0.0016) (0.0014) (0.0013) Women: elasticities of labour supply Elasticity w.r.t. Unmarried Married Unmarried Married Unmarried Married Wage rate −0.12** −0.18** −0.15** −0.16** −0.17** −0.17**  husband (0.04) (0.03) (0.04) (0.03) (0.03) (0.03) Wage rate 0.22** 0.61** 0.36 0.60** 0.48** 0.55**  wife (0.07) (0.04) (0.31) (0.19) (0.14) (0.16) Non‐labour −0.0081** 0.0071** −0.0080** 0.0071** 0.0033 0.0035  income (0.0023) (0.0013) (0.0044) (0.0032) (0.0026) (0.0023) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab None of the estimation methods shows much sensitivity of male labour supply to either the husband’s wage rate or the wife’s wage rate. The unrestricted estimates show that married men are less sensitive with respect to non‐labour income than unmarried men. I do not find this when restrictions are imposed. For married women, their wage elasticities, both with respect to their own wage rate and their husbands’, are quite stable across estimation methods. The wage elasticity of the unmarried women’s own wage rate is affected most by imposing restrictions by marital status. The largest deviation due to imposing restrictions is found for the elasticities of female labour supply with respect to non‐labour income. Overall, the labour supply of the wife is more sensitive with respect to changes in the wage rates than the labour supply of the husband. The order of magnitude of the wage elasticities for the women is comparable to the values usually found in the empirical literature.27 Women are more responsive to changes in the partner’s wage rate than men. Table 8 shows the elasticities of the structural labour supply functions. These are conditional elasticities, since they are conditional on the ‘share’. As the ‘share’ itself is not observed and identified only up to an additive constant, I used the household’s non‐labour income as a base value to compute the elasticity of labour supply with respect to the ‘share’. The elasticities show the sensitivity of labour supply that can be assigned to preferences. The difference in the elasticities between Tables 7 and 8 is the effect that is due to the sharing rule. The estimates of the structural labour supply function obtained with the minimum distance estimator are the same for the unmarried and the married. The difference in the elasticities comes from differences in sample means for the married and the unmarried. The elasticities of labour supply of men with respect to their wage rate again reveal the insensitivity of male labour supply with respect to the wage rate. For female labour supply, the wage elasticities are not precisely estimated if no restrictions are imposed. The elasticities of female labour supply with respect to non‐labour income does not differ much across estimation methods. 5. Conclusions I have specified an empirical model of collective household labour supply that allows for non‐participation of husband and wife. In my model I use information on both the participation and the working hours of husband and wife. To make this possible, particular attention needs to be paid to the coherence of the model. For my specification I derive a restriction on the parameter space such that coherence of the model is satisfied. As a basis for my model I specify the labour supply function, based on individual preferences, as a function that is quadratic in wage rates and linear in non‐labour income. The specification is flexible with in wage rates and allows for a backward bending labour supply function. I estimate the model with data from the Dutch Socio Economic Panel (SEP) over the period 1990–2002. I allow the parameters of the sharing rule to be different for married and unmarried couples. With a minimum distance estimator I impose and test restrictions between the parameters of the married and the unmarried. The estimates show that labour supply is backward bending for married men and to a lesser extent also for unmarried men. For women, I do not find a backward bending labour supply curve. The estimates satisfy some basic regularity conditions: leisure is a normal good for men and women, irrespective of their marital status, and a cross equation condition on the parameters that is implied by the collective model is satisfied. The share of the husband decreases if the wage rate of the husband increases. This allows the wife to consume more leisure. For married men, the backward bending effect on labour supply is stronger, so the husband of a married couple may increase his leisure as well. For unmarried men, the labour supply is more likely to increase. I have shown that the utility level of both husband and wife increases as a result of the increase in the husband’s wage rate. Apparently the increase in the husband’s wage rate causes an increase in total household income and makes everyone in the household better off. The share of the husband (wife) decreases (increases) if the wage rate of the wife increases. The husband will supply more labour. For women, the net result is also an increase in the supply of labour. The utility of the husband decreases and the utility of the wife increases due to an increase in the wife’s wage rate. This is consistent with the interpretation that the wife’s threat point increases due to an increase in her wage rate, and an improvement of her bargaining position. These qualitative effects of changes in wage rates on the sharing rule are the same for the married and the unmarried. The effect of non‐labour income differs by marital status. An increase in non‐labour income is split between husband and wife in unmarried couples, whereas it is assigned to the husband in married couples. This suggests that unmarried women have a better bargaining position than married women. The results show that wrong conclusions are drawn for unmarried women if the sharing rule parameters are restricted to be the same for married and unmarried women. These restrictions are also formally rejected by the minimum distance criterion. I have restricted the parameters of the wage rates and non‐labour income in the structural labour supply function to be equal for the married and the unmarried. This leads to more precise estimates of the parameters. But these restrictions are formally rejected as well. I can derive the parameters of the sharing rule of households with one non‐employed member. The sharing rule of a working wife and a non‐working husband is not significantly different from the sharing rule of a dual earner couple. The reason for this may be that reservation wages of men tend to be very low and show little variation. The sharing rule for a working husband with a non‐working wife differs considerably from the sharing rule for working partners. An increase in the husband’s wage rate now leads to an increase in the husband’s ‘share’. This suggests an increase in bargaining power of the husband if the wife is not working. An increase in the wife’s wage rate still leads to a decrease in the husband’s share. Computation of elasticities shows that male labour supply is less sensitive to changes in wage rates than female labour supply. Imposing restrictions between the parameters of the married and the unmarried may lead to underestimation of the wage elasticity of unmarried men. The wage elasticities of married women are quite stable, irrespective of whether or not restrictions between the married and the unmarried are imposed. Some comments are in order. Although the reduced form labour supply functions can be consistent with more general preferences, the decomposition into the structural labour supply equations and the sharing rule is based on the assumptions of egoistic preferences and the absence of public goods. Here I followed the theoretical results by Chiappori (1988) and Donni (2003). But egoistic preferences rule out that leisure time of spouses can be complements, or that non‐market time of spouses can be substitutes. The model does not incorporate fixed costs of work, or the influence of demand side conditions on employment. The latter implies that no distinction is made between non‐participation and involuntary unemployment. Incorporating fixed costs will make it harder to develop a statistical model that is continuous around the participation frontier. In estimating the sharing rule, I use non‐labour income at the household level. Thus, I have not used information on ownership of underlying income sources to identify the shares assigned to each spouse. Models of labour supply and marriage (or matching) suggest the existence of public or shared goods. Becker (1973) argues that economies to scale in household production may lead to specialisation in time allocation between spouses. My empirical analysis allows for different outcomes by marital status but I do not model couple formation itself.28 These considerations suggest that there is room for extensions of the basic model in several directions. In empirical work, identification of alternative models is a key issue, which sometimes requires the imposition of alternative assumptions or restrictions, but, in the better case, may be achieved by collecting the appropriate information in household surveys. Footnotes 1 " An underlying assumption for this two‐stage procedure is that consumption of household members can be represented by a Hicks aggregate commodity and that there is no public good within the household. Children are often mentioned as an example of a public good that affects the division of time between household members. As a consequence, in empirical studies of collective household labour supply the sample is either restricted to couples without children (Blundell et al, 2007), or to couples without young children (Fortin and Lacroix, 1997). 2 " In reality, there are many potential public goods in a household. But within the literature on collective labour supply models, children are considered as collective goods that have a potentially large influence on the division of time between household members. For this reason the analysis is usually restricted to childless couples; see e.g. Fortin and Lacroix (1997). Recently, Chiappori et al. (2005) extended the collective model to allow for public goods. However, an empirical application of this approach to children in the household requires the availability of information on household expenditures on children. Another problem is that children are related to domestic production. In empirical labour supply models non‐work time is usually interpreted as leisure time, ignoring domestic production. In a survey article Apps (2003) discusses the problems that arise in empirical labour supply models due to neglecting the domestic production and alternative time uses. 3 " The assumption of egoistic, or caring, preferences is used by Chiappori (1988; 1992) to identify preference parameters from sharing rule parameters. The assumption of egoistic preferences places restrictions on the behaviour of households. For instance, it excludes the possibility of complementarities in leisure between spouses, or substitution in non‐market time of spouses. 4 " Various alternative representations of the decision problem are possible. For example, the objective function can be written as a weighted average of the utility functions of husband and wife, with weights that depend on (wm,wf,y). This can directly be seen by applying Kuhn‐Tucker to (1). See Vermeulen (2002) for an overview. 5 " This formulation relates the shares of spouses to aggregate household non‐labour income. It excludes the possibility that shares of household non‐labour income are assigned to either spouse on basis of ownership. 6 " This possibility for recovering the marginal effects depends on the assumption of egoistic, or caring, preferences. As discussed before, this assumption imposes restrictions on the preferences of spouses. 7 " A similar specification is used in Bloemen and Kapteyn (2008) in an analysis of female labour supply and by Kapteyn et al. (1990) in the context of a household labour supply model. The latter study shows evidence of a backward bending labour supply function for men. 8 " Note that before I have specified the share of the husband as ρm = ρ(wm, wf, y) and the share of the wife as ρf = y − ρ(wm, wf, y). Thus, additivity of the husband’s and wife’s share to the household’s non‐labour income y is imposed. 9 " I have experimented with a sharing rule with the higher order term wm/(wm + wf)y but this additional term turned out not to be significant. Browning et al. (1994) first recognised the importance of including some measure of the relative earning capacity of husband and wife. They discuss the identification of the sharing rule and stress that identification may depend on the non‐linear shape the measure. I have done a sensitivity analysis with an alternative measure of relative wage rates of the spouses, the logarithm of wage rate ration of husband and wife. Both qualitative and quantitative outcomes were hardly affected by this choice. 10 " Manser and Brown (1980) state that the utility that each spouse within a couple derives from his or her consumption and leisure, should be at least as large as the utility he or she can obtain as a single. Becker (1973) argued that the gains from marriage come from the production of home produced goods with a production technology that exhibits increasing returns to time inputs, such that it can be beneficial to have specialisation by spouses in different market or non‐market activities. The analysis of Manser and Brown (1980) also shows that it is empirically hard to test the model implications as wage rates of individuals both affect the utility of being single and influence the bargaining position of spouses within a couple. In my analysis, I only include marital status as an explanatory variable and thus I ignore the issue of formation of marriages. Keeley (1979) uses a search framework to describe the process of marriage formation. From the analysis by Manser and Brown (1980), it follows that marital status potentially may enter the analysis by two routes: it may influence the bargaining position of spouses within households and there may be heterogeneity in preferences that define the single individual threat point. Both may determine the marital status of a couple. 11 " The equations (25) adjust the specifications (14) and (20) include additive observed and unobserved heterogeneity. 12 " In Blundell et al. (2007) the problem is implicitly circumvented as their model describes working hours and participation for the wife, but only participation of the husband. Therefore the labour market state of the wife does not enter the participation equation of the husband. 13 " Note that this restriction is not necessary for identification but I did not find any a priori reason for allowing this correlation coefficient to vary. Also note that this restriction does not imply that, say, the distribution of νm, conditional on (νf,um,uf) does not depend on uf, since I allow for correlation between νm and νf and for correlation between νf and uf. 14 " The discussion in Section 2.1 showed that preferences may depend on marital status as the individual utility level is a determinant of the (net) gains of marriage. Thus, married individuals may have selected themselves into this state because of their preferences. 15 " In the estimation, likelihood contributions will be adjusted accordingly. 16 " This way, I aim to avoid spurious effects: if only the parameters of the wage rates and non‐labour income would differ by marital status, these parameters may just react to neglected flexibility in heterogeneity. 17 " Note that due to normalisations, Table 2 presents the coefficients 1000a7,1000b7,10a5,10a6,10b5, and 10b6. 18 " With different data from a different time period, Kapteyn et al. (1990) also find evidence of a backward bending labour supply curve for men, whereas backward bending is absent for women. 19 " For reasons of comparison, I also obtained estimates of the reduced form parameters ignoring non‐participation: I re‐estimated the model using dual earner couples only. The key results are in Appendix Table A4, which is the equivalent of Table 2. I find different coefficient estimates for the parameters of the wage rates and non‐labour income. For instance, the coefficients of non‐labour income in Table A4 have the same sign as in Table 2 but the estimates seem to be biased towards zero. 20 " Recall that this decomposition is based on the assumption of egoistic preferences, while I have assumed that the wage fraction enters the sharing rule but not the structural labour supply function. 21 " To be more precise, I assume that the parameters do not differ by marital status. I do not impose any restrictions on the intercept and the parameters of observed and unobserved heterogeneity. 22 " In Section 2.1 I argued that there may be two reasons why outcomes may differ by marital status. Marital status can influence the bargaining power within the household but the decision to marry is also influenced by the utility obtained while being single. The rejection of the restrictions shows that not only the sharing rule parameters differ by marital status, but also the preference parameters that enter the individual labour supply functions. 23 " Squared wages and non‐labour income are, unlike in Table 2, not normalised in Tables 4 and 5. 24 " Note that this result is similar to an earlier version of the article, where married and unmarried were pooled in the estimation, and the coefficient of non‐labour income was about 1 as well. The present coefficients reveal heterogeneity by marital status. 25 " The indirect utility function belonging to the specification (7) reads . The parameters can be computed from the estimates, as indicated below (8). Additive heterogeneity enters by ϑj. 26 " Thereby assuming that the random effects are part of preferences and/or the sharing rule. 27 " For instance, Bloemen and Kapteyn (2008) estimate labour supply elasticities for married women with a Dutch data set, using a wide variety of estimation methods. 28 " See e.g. Manser and Brown (1980). 29 " For identification I need sm ≠ 1/sf, but that is a different story from coherence. References Apps , P. ( 2003 ). ‘ Gender, time use and models of the household ’, IZA Discussion Paper No. 796. Apps , P. and Rees , R. ( 1988 ). ‘ Taxation and the household ’, Journal of Public Economics , vol. 35 ( 3 ) (April), pp. 355 – 69 . Google Scholar Crossref Search ADS WorldCat Becker , G. ( 1973 ). ‘ A theory of marriage: part I ’, Journal of Political Economy , vol. 81 ( 4 ) (July–August), pp. 813 – 46 . Google Scholar Crossref Search ADS WorldCat Bloemen , H.G. and Kapteyn , A. ( 2008 ). ‘ The estimation of utility consistent labor supply models by means of simulated scores ’, Journal of Applied Econometrics , vol. 23 ( 5 ) (June), pp. 395 – 422 . Google Scholar Crossref Search ADS WorldCat Blundell , R. , Chiappori , P.A., Magnac , T. and Meghir , C. ( 2007 ). ‘ Collective labor supply: heterogeneity and nonparticipation ’, Review of Economic Studies , vol. 74 ( 2 ) (April), pp. 417 – 45 . Google Scholar Crossref Search ADS WorldCat Börsch‐Supan , A. and Hajivassiliou , V. ( 1993 ). ‘ Smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variable models ’, Journal of Econometrics , vol. 58 ( 3 ) (August), pp. 347 – 68 . Google Scholar Crossref Search ADS WorldCat Browning , M. , Bourguignon , F., Chiappori , P.A. and Lechene , V. ( 1994 ). ‘ Incomes and outcomes: a structural model of intrahousehold allocation ’, Journal of Political Economy , vol. 102 ( 6 ) (December), pp. 1067 – 96 . Google Scholar Crossref Search ADS WorldCat Chiappori , P.A. ( 1988 ). ‘ Rational household labor supply ’, Econometrica , vol. 56 ( 1 ) (January), pp. 63 – 89 . Google Scholar Crossref Search ADS WorldCat Chiappori , P.A. ( 1992 ). ‘ Collective labor supply and welfare ’, Journal of Political Economy , vol. 100 ( 3 ) (June), pp. 437 – 67 . Google Scholar Crossref Search ADS WorldCat Chiappori , P.A. , Blundell , R. and Meghir , C. ( 2005 ). ‘ Collective labor supply with children ’, Journal of Political Economy , vol. 113 ( 6 ) (December), pp. 1277 – 306 . Google Scholar Crossref Search ADS WorldCat Donni , O. ( 2003 ). ‘ Collective household labor supply: nonparticipation and income taxation ’, Journal of Public Economics , vol. 87 ( 5‐6 ) (May), pp. 1179 – 98 . Google Scholar Crossref Search ADS WorldCat Fortin , B. and Lacroix , G. ( 1997 ). ‘ A test of the unitary and collective models of household labor supply ’, Economic Journal , vol. 107 ( 443 ) (July), pp. 933 – 55 . Google Scholar Crossref Search ADS WorldCat Hausman , J. and Ruud , P. ( 1984 ). ‘ Family labor supply with taxes ’, American Economic Review , vol. 74 ( 2 ) (May), pp. 242 – 8 . OpenURL Placeholder Text WorldCat Heckman , J.J. ( 1978 ). ‘ Dummy endogenous variables in a simultaneous equation system ’, Econometrica , vol. 46 ( 4 ) (July), pp. 931 – 60 . Google Scholar Crossref Search ADS WorldCat Heckman , J.J. ( 1979 ). ‘ Sample selection bias as a specification error ’, Econometrica , vol. 47 ( 1 ) (January), pp. 153 – 61 . Google Scholar Crossref Search ADS WorldCat Kapteyn , A. , Kooreman , P. and Van Soest , A. ( 1990 ). ‘ Quantity rationing and concavity in a flexible household labor supply model ’, Review of Economics and Statistics , vol. 72 ( 1 ) (February), pp. 55 – 62 . Google Scholar Crossref Search ADS WorldCat Keeley , M. ( 1979 ). ‘ An analysis of the age pattern of first marriage ’, International Economic Review , vol. 20 ( 2 ) (June), pp. 527 – 44 . Google Scholar Crossref Search ADS WorldCat Kooreman , P. ( 1994 ). ‘ Estimation of econometric models of some discrete games ’, Journal of Applied Econometrics , vol. 9 ( 3 ) (July‐September), pp. 255 – 68 . Google Scholar Crossref Search ADS WorldCat Manser , M. and Brown , M. ( 1980 ). ‘ Marriage and household decision‐making: a bargaining analysis ’, International Economic Review , vol. 21 ( 1 ) (October), pp. 31 – 44 . Google Scholar Crossref Search ADS WorldCat McElroy , M.B. and Horney , M.J ( 1981 ). ‘ Nash‐bargained household decisions: towards a generalization of the theory of demand ’, International Economic Review , vol. 22 ( 2 ) (June), pp. 333 – 49 . Google Scholar Crossref Search ADS WorldCat Vermeulen , F. ( 2002 ). ‘ Collective household models: principles and main results ’, Journal of Economic Surveys , vol. 16 ( 4 ) (September), pp. 533 – 64 . Google Scholar Crossref Search ADS WorldCat Appendix Appendix 1. On the Coherence of the Model In this Appendix I show how I may derive restrictions on the parameters sm and sf in order to get coherence. To simplify notation, I denote the right hand side of (23) by for the purpose of this exercise: (29) Thus, I may rewrite model (25) as (30) First, let me explain what coherence implies in the context of this model. Consider the case in which both spouses choose to work: dm = 1, df = 1. The model in (30) generates this outcome if both and . The model would be incoherent if the model can generate a second outcome as well, for instance, dm = 0, df = 0 by and . Now, concentrate on this possible source of coherence, since the other outcomes, dm = 1, df = 0 and dm = 0, df = 1 cannot be generated by the model simultaneously with dm = 1, df = 1. Consider when the events (31) and (32) may occur simultaneously. Note that (31) and (32) cannot occur together if either sm ≥ 0 or sf ≥ 0.29 The two events could occur, though, if both sm < 0 and sf < 0. Combining (31) and (32) yields two conditions: (33) and (34) Combining (33) and (34) yields sm < 1/sf. To prevent this from happening I need the reverse to hold, so sm > 1/sf. So if sm < 0 and sf < 0, a condition to ensure that the model does not generate the two outcomes dm = 1, df = 1 and dm = 0, df = 0 at the same time is smsf < 1. Next, note that the model outcomes dm = 1, df = 0 and dm = 0, df = 1 are not exclusive a priori either. The model generates these two outcomes, respectively if (35) and (36) Again, note that (35) and (36) cannot occur simultaneoulsy if either sm > 0 or sf > 0. If sm < 0 and sf < 0, combining (35) and (36) again yields the conditions (33) and (34). So again, I have smsf < 1 if sm < 0 and sf < 0 to impose coherence. Finally, reconsider (32). I may rewrite these conditions as (37) with and . First, assume that sm > 0 and sf > 0. Then (37) implies (38) This case does not imply incoherency, but I do see that there is a identification problem since parameters will lead to the same probability as (sm,sf). Imposing smsf < 1 for all positive values of sm and sf clearly excludes . Next, assume that sm < 0 and sf > 0. Then (37) implies (39) Thus, the outcome generated by the model is not unique. I have a combined coherence‐identification problem here, which can easily be excluded by assuming |smsf| < 1. The same holds for the case sm > 0, sf < 0. The final case sm < 0, sf < 0 has already been covered above. Summarising, I have the following result: a sufficient condition for coherence of the model is |smsf| < 1. Note that it is not necessary to restrict either sm or sf to zero, as in the model by Heckman (1978) for limited dependent variables with endogenous right hand side variables, which in my notation reads (40) Using similar arguments as above, it is readily established that (40) requires smsf = 0 for coherence, as was shown by Heckman (1978). Table A1 
Simulated Maximum Likelihood Estimates, Labour Supply Equations Parameters of Observed Characteristics Variable . Unmarried . Married . Men . Women . Men . Women . log(age husb/17) 42.811** 4.621 59.777** 35.703** (8.051) (7.585) (9.736) (9.188) log(age husb/17) squared −30.302** −6.173 −49.004** −17.329** (5.549) (5.279) (5.381) (5.144) log(age wife/17) −27.154** 61.545** −13.186 −3.339 (6.603) (7.165) (8.447) (7.911) log(age wife/17) squared 14.507** −49.510** 22.376** −21.150** (4.916) (4.691) (4.945) (4.706) Educ level husb. 1 −1.971 −4.055** −8.634** −3.209** (2.203) (1.443) (0.900) (0.893) Educ level husb. 2 1.757* −0.909 −4.005** −1.164 (1.012) (1.010) (0.765) (0.718) Educ level husb. 3 0.381 0.118 −3.176** −0.706 (0.774) (0.750) (0.561) (0.573) Educ level wife 1 3.551** −8.120** −0.555 −5.549** (1.135) (1.503) (0.853) (0.857) Educ level wife 2 1.324 −2.044* 1.579** −2.907** (1.116) (1.147) (0.622) (0.721) Educ level wife 3 0.632 −2.808** 1.595** −0.137 (0.611) (0.792) (0.539) (0.618) Variable . Unmarried . Married . Men . Women . Men . Women . log(age husb/17) 42.811** 4.621 59.777** 35.703** (8.051) (7.585) (9.736) (9.188) log(age husb/17) squared −30.302** −6.173 −49.004** −17.329** (5.549) (5.279) (5.381) (5.144) log(age wife/17) −27.154** 61.545** −13.186 −3.339 (6.603) (7.165) (8.447) (7.911) log(age wife/17) squared 14.507** −49.510** 22.376** −21.150** (4.916) (4.691) (4.945) (4.706) Educ level husb. 1 −1.971 −4.055** −8.634** −3.209** (2.203) (1.443) (0.900) (0.893) Educ level husb. 2 1.757* −0.909 −4.005** −1.164 (1.012) (1.010) (0.765) (0.718) Educ level husb. 3 0.381 0.118 −3.176** −0.706 (0.774) (0.750) (0.561) (0.573) Educ level wife 1 3.551** −8.120** −0.555 −5.549** (1.135) (1.503) (0.853) (0.857) Educ level wife 2 1.324 −2.044* 1.579** −2.907** (1.116) (1.147) (0.622) (0.721) Educ level wife 3 0.632 −2.808** 1.595** −0.137 (0.611) (0.792) (0.539) (0.618) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Reference category of education: levels 4 and 5 (highest). Open in new tab Table A1 
Simulated Maximum Likelihood Estimates, Labour Supply Equations Parameters of Observed Characteristics Variable . Unmarried . Married . Men . Women . Men . Women . log(age husb/17) 42.811** 4.621 59.777** 35.703** (8.051) (7.585) (9.736) (9.188) log(age husb/17) squared −30.302** −6.173 −49.004** −17.329** (5.549) (5.279) (5.381) (5.144) log(age wife/17) −27.154** 61.545** −13.186 −3.339 (6.603) (7.165) (8.447) (7.911) log(age wife/17) squared 14.507** −49.510** 22.376** −21.150** (4.916) (4.691) (4.945) (4.706) Educ level husb. 1 −1.971 −4.055** −8.634** −3.209** (2.203) (1.443) (0.900) (0.893) Educ level husb. 2 1.757* −0.909 −4.005** −1.164 (1.012) (1.010) (0.765) (0.718) Educ level husb. 3 0.381 0.118 −3.176** −0.706 (0.774) (0.750) (0.561) (0.573) Educ level wife 1 3.551** −8.120** −0.555 −5.549** (1.135) (1.503) (0.853) (0.857) Educ level wife 2 1.324 −2.044* 1.579** −2.907** (1.116) (1.147) (0.622) (0.721) Educ level wife 3 0.632 −2.808** 1.595** −0.137 (0.611) (0.792) (0.539) (0.618) Variable . Unmarried . Married . Men . Women . Men . Women . log(age husb/17) 42.811** 4.621 59.777** 35.703** (8.051) (7.585) (9.736) (9.188) log(age husb/17) squared −30.302** −6.173 −49.004** −17.329** (5.549) (5.279) (5.381) (5.144) log(age wife/17) −27.154** 61.545** −13.186 −3.339 (6.603) (7.165) (8.447) (7.911) log(age wife/17) squared 14.507** −49.510** 22.376** −21.150** (4.916) (4.691) (4.945) (4.706) Educ level husb. 1 −1.971 −4.055** −8.634** −3.209** (2.203) (1.443) (0.900) (0.893) Educ level husb. 2 1.757* −0.909 −4.005** −1.164 (1.012) (1.010) (0.765) (0.718) Educ level husb. 3 0.381 0.118 −3.176** −0.706 (0.774) (0.750) (0.561) (0.573) Educ level wife 1 3.551** −8.120** −0.555 −5.549** (1.135) (1.503) (0.853) (0.857) Educ level wife 2 1.324 −2.044* 1.579** −2.907** (1.116) (1.147) (0.622) (0.721) Educ level wife 3 0.632 −2.808** 1.595** −0.137 (0.611) (0.792) (0.539) (0.618) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Reference category of education: levels 4 and 5 (highest). Open in new tab Table A2 
Simulated Maximum Likelihood Estimates: The Wage Equations Variable . Unmarried . Married . Men . Women . Men . Women . intercept 2.591** 2.323** 3.052** 2.591** (0.059) (0.054) (0.053) (0.049) log(age/17) 0.675** 1.368** −0.119 0.724** (0.157) (0.136) (0.129) (0.121) log(age/17) squared 0.053 −0.697** 0.419** −0.364** (0.114) (0.113) (0.077) (0.075) Education level 1 −0.225** −0.301** −0.343** −0.358** (0.059) (0.055) (0.025) (0.031) Education level 2 −0.303** −0.244** −0.360** −0.338** (0.034) (0.035) (0.021) (0.027) Education level 3 −0.218** −0.146** −0.323** −0.272** (0.029) (0.030) (0.018) (0.027) Education level 4 −0.092** −0.038 −0.227** −0.121** (0.028) (0.031) (0.019) (0.026) Technical 0.042* 0.062** 0.022* −0.055* (0.023) (0.026) (0.012) (0.033) Econ./adm. 0.040* 0.060** 0.039** 0.042** (0.022) (0.020) (0.013) (0.012) General −0.010 0.064** 0.041** 0.073** (0.022) (0.021) (0.016) (0.013) 1992 0.046 0.014 0.017 0.051* (0.035) (0.031) (0.025) (0.031) 1993 0.058* 0.082** 0.052** 0.074** (0.030) (0.032) (0.024) (0.026) 1994 0.049* 0.027 0.019 0.088** (0.029) (0.028) (0.022) (0.023) 1995 0.073** 0.074** 0.066** 0.097** (0.036) (0.030) (0.022) (0.025) 1996 0.083** 0.115** 0.070** 0.122** (0.027) (0.030) (0.020) (0.020) 1997 0.096** 0.123** 0.087** 0.152** (0.024) (0.028) (0.022) (0.025) 1998 0.134** 0.176** 0.116** 0.193** (0.036) (0.039) (0.019) (0.024) 1999 0.138** 0.167** 0.105** 0.246** (0.033) (0.033) (0.022) (0.023) 2000 0.183** 0.201** 0.146** 0.251** (0.029) (0.027) (0.024) (0.028) 2001 0.248** 0.332** 0.221** 0.364** (0.030) (0.032) (0.023) (0.024) Variable . Unmarried . Married . Men . Women . Men . Women . intercept 2.591** 2.323** 3.052** 2.591** (0.059) (0.054) (0.053) (0.049) log(age/17) 0.675** 1.368** −0.119 0.724** (0.157) (0.136) (0.129) (0.121) log(age/17) squared 0.053 −0.697** 0.419** −0.364** (0.114) (0.113) (0.077) (0.075) Education level 1 −0.225** −0.301** −0.343** −0.358** (0.059) (0.055) (0.025) (0.031) Education level 2 −0.303** −0.244** −0.360** −0.338** (0.034) (0.035) (0.021) (0.027) Education level 3 −0.218** −0.146** −0.323** −0.272** (0.029) (0.030) (0.018) (0.027) Education level 4 −0.092** −0.038 −0.227** −0.121** (0.028) (0.031) (0.019) (0.026) Technical 0.042* 0.062** 0.022* −0.055* (0.023) (0.026) (0.012) (0.033) Econ./adm. 0.040* 0.060** 0.039** 0.042** (0.022) (0.020) (0.013) (0.012) General −0.010 0.064** 0.041** 0.073** (0.022) (0.021) (0.016) (0.013) 1992 0.046 0.014 0.017 0.051* (0.035) (0.031) (0.025) (0.031) 1993 0.058* 0.082** 0.052** 0.074** (0.030) (0.032) (0.024) (0.026) 1994 0.049* 0.027 0.019 0.088** (0.029) (0.028) (0.022) (0.023) 1995 0.073** 0.074** 0.066** 0.097** (0.036) (0.030) (0.022) (0.025) 1996 0.083** 0.115** 0.070** 0.122** (0.027) (0.030) (0.020) (0.020) 1997 0.096** 0.123** 0.087** 0.152** (0.024) (0.028) (0.022) (0.025) 1998 0.134** 0.176** 0.116** 0.193** (0.036) (0.039) (0.019) (0.024) 1999 0.138** 0.167** 0.105** 0.246** (0.033) (0.033) (0.022) (0.023) 2000 0.183** 0.201** 0.146** 0.251** (0.029) (0.027) (0.024) (0.028) 2001 0.248** 0.332** 0.221** 0.364** (0.030) (0.032) (0.023) (0.024) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Reference level of education: level 5 (highest); sector: services; year: 2002. Open in new tab Table A2 
Simulated Maximum Likelihood Estimates: The Wage Equations Variable . Unmarried . Married . Men . Women . Men . Women . intercept 2.591** 2.323** 3.052** 2.591** (0.059) (0.054) (0.053) (0.049) log(age/17) 0.675** 1.368** −0.119 0.724** (0.157) (0.136) (0.129) (0.121) log(age/17) squared 0.053 −0.697** 0.419** −0.364** (0.114) (0.113) (0.077) (0.075) Education level 1 −0.225** −0.301** −0.343** −0.358** (0.059) (0.055) (0.025) (0.031) Education level 2 −0.303** −0.244** −0.360** −0.338** (0.034) (0.035) (0.021) (0.027) Education level 3 −0.218** −0.146** −0.323** −0.272** (0.029) (0.030) (0.018) (0.027) Education level 4 −0.092** −0.038 −0.227** −0.121** (0.028) (0.031) (0.019) (0.026) Technical 0.042* 0.062** 0.022* −0.055* (0.023) (0.026) (0.012) (0.033) Econ./adm. 0.040* 0.060** 0.039** 0.042** (0.022) (0.020) (0.013) (0.012) General −0.010 0.064** 0.041** 0.073** (0.022) (0.021) (0.016) (0.013) 1992 0.046 0.014 0.017 0.051* (0.035) (0.031) (0.025) (0.031) 1993 0.058* 0.082** 0.052** 0.074** (0.030) (0.032) (0.024) (0.026) 1994 0.049* 0.027 0.019 0.088** (0.029) (0.028) (0.022) (0.023) 1995 0.073** 0.074** 0.066** 0.097** (0.036) (0.030) (0.022) (0.025) 1996 0.083** 0.115** 0.070** 0.122** (0.027) (0.030) (0.020) (0.020) 1997 0.096** 0.123** 0.087** 0.152** (0.024) (0.028) (0.022) (0.025) 1998 0.134** 0.176** 0.116** 0.193** (0.036) (0.039) (0.019) (0.024) 1999 0.138** 0.167** 0.105** 0.246** (0.033) (0.033) (0.022) (0.023) 2000 0.183** 0.201** 0.146** 0.251** (0.029) (0.027) (0.024) (0.028) 2001 0.248** 0.332** 0.221** 0.364** (0.030) (0.032) (0.023) (0.024) Variable . Unmarried . Married . Men . Women . Men . Women . intercept 2.591** 2.323** 3.052** 2.591** (0.059) (0.054) (0.053) (0.049) log(age/17) 0.675** 1.368** −0.119 0.724** (0.157) (0.136) (0.129) (0.121) log(age/17) squared 0.053 −0.697** 0.419** −0.364** (0.114) (0.113) (0.077) (0.075) Education level 1 −0.225** −0.301** −0.343** −0.358** (0.059) (0.055) (0.025) (0.031) Education level 2 −0.303** −0.244** −0.360** −0.338** (0.034) (0.035) (0.021) (0.027) Education level 3 −0.218** −0.146** −0.323** −0.272** (0.029) (0.030) (0.018) (0.027) Education level 4 −0.092** −0.038 −0.227** −0.121** (0.028) (0.031) (0.019) (0.026) Technical 0.042* 0.062** 0.022* −0.055* (0.023) (0.026) (0.012) (0.033) Econ./adm. 0.040* 0.060** 0.039** 0.042** (0.022) (0.020) (0.013) (0.012) General −0.010 0.064** 0.041** 0.073** (0.022) (0.021) (0.016) (0.013) 1992 0.046 0.014 0.017 0.051* (0.035) (0.031) (0.025) (0.031) 1993 0.058* 0.082** 0.052** 0.074** (0.030) (0.032) (0.024) (0.026) 1994 0.049* 0.027 0.019 0.088** (0.029) (0.028) (0.022) (0.023) 1995 0.073** 0.074** 0.066** 0.097** (0.036) (0.030) (0.022) (0.025) 1996 0.083** 0.115** 0.070** 0.122** (0.027) (0.030) (0.020) (0.020) 1997 0.096** 0.123** 0.087** 0.152** (0.024) (0.028) (0.022) (0.025) 1998 0.134** 0.176** 0.116** 0.193** (0.036) (0.039) (0.019) (0.024) 1999 0.138** 0.167** 0.105** 0.246** (0.033) (0.033) (0.022) (0.023) 2000 0.183** 0.201** 0.146** 0.251** (0.029) (0.027) (0.024) (0.028) 2001 0.248** 0.332** 0.221** 0.364** (0.030) (0.032) (0.023) (0.024) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Reference level of education: level 5 (highest); sector: services; year: 2002. Open in new tab Table A3 
Simulated Maximum Likelihood Estimates of the Covariance Matrices Parameter . Unmarried . Married . σm 11.132** 10.715** (0.230) (0.139) σf 11.880** 12.164** (0.270) (0.149) σmf −0.243** −0.435** (0.024) (0.015) τm 0.244** 0.214** (0.003) (0.002) τf 0.247** 0.234** (0.004) (0.002) ρ 0.162** 0.127** (0.024) (0.018) ρmu −0.498** −0.312** (0.030) (0.026) ρfu −0.423** −0.389** (0.042) (0.020) σθ,m 8.761** 16.357** (0.275) (0.240) ρθ,mf 0.233** −0.393** (0.052) (0.029) σθ,f 8.205** 10.147** (0.371) (0.229) σω,m 0.191** 0.220** (0.008) (0.005) ρω,mf −0.177** 0.299** (0.076) (0.028) σω,f 0.134** 0.234** (0.010) (0.007) ρθω,m −0.282** −0.506** (0.041) (0.026) ρθω,fm −0.307** −0.660** (0.088) (0.022) ρθω,mf 0.064 0.037 (0.048) (0.031) ρθω,f −0.004 0.322** (0.091) (0.031) Parameter . Unmarried . Married . σm 11.132** 10.715** (0.230) (0.139) σf 11.880** 12.164** (0.270) (0.149) σmf −0.243** −0.435** (0.024) (0.015) τm 0.244** 0.214** (0.003) (0.002) τf 0.247** 0.234** (0.004) (0.002) ρ 0.162** 0.127** (0.024) (0.018) ρmu −0.498** −0.312** (0.030) (0.026) ρfu −0.423** −0.389** (0.042) (0.020) σθ,m 8.761** 16.357** (0.275) (0.240) ρθ,mf 0.233** −0.393** (0.052) (0.029) σθ,f 8.205** 10.147** (0.371) (0.229) σω,m 0.191** 0.220** (0.008) (0.005) ρω,mf −0.177** 0.299** (0.076) (0.028) σω,f 0.134** 0.234** (0.010) (0.007) ρθω,m −0.282** −0.506** (0.041) (0.026) ρθω,fm −0.307** −0.660** (0.088) (0.022) ρθω,mf 0.064 0.037 (0.048) (0.031) ρθω,f −0.004 0.322** (0.091) (0.031) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table A3 
Simulated Maximum Likelihood Estimates of the Covariance Matrices Parameter . Unmarried . Married . σm 11.132** 10.715** (0.230) (0.139) σf 11.880** 12.164** (0.270) (0.149) σmf −0.243** −0.435** (0.024) (0.015) τm 0.244** 0.214** (0.003) (0.002) τf 0.247** 0.234** (0.004) (0.002) ρ 0.162** 0.127** (0.024) (0.018) ρmu −0.498** −0.312** (0.030) (0.026) ρfu −0.423** −0.389** (0.042) (0.020) σθ,m 8.761** 16.357** (0.275) (0.240) ρθ,mf 0.233** −0.393** (0.052) (0.029) σθ,f 8.205** 10.147** (0.371) (0.229) σω,m 0.191** 0.220** (0.008) (0.005) ρω,mf −0.177** 0.299** (0.076) (0.028) σω,f 0.134** 0.234** (0.010) (0.007) ρθω,m −0.282** −0.506** (0.041) (0.026) ρθω,fm −0.307** −0.660** (0.088) (0.022) ρθω,mf 0.064 0.037 (0.048) (0.031) ρθω,f −0.004 0.322** (0.091) (0.031) Parameter . Unmarried . Married . σm 11.132** 10.715** (0.230) (0.139) σf 11.880** 12.164** (0.270) (0.149) σmf −0.243** −0.435** (0.024) (0.015) τm 0.244** 0.214** (0.003) (0.002) τf 0.247** 0.234** (0.004) (0.002) ρ 0.162** 0.127** (0.024) (0.018) ρmu −0.498** −0.312** (0.030) (0.026) ρfu −0.423** −0.389** (0.042) (0.020) σθ,m 8.761** 16.357** (0.275) (0.240) ρθ,mf 0.233** −0.393** (0.052) (0.029) σθ,f 8.205** 10.147** (0.371) (0.229) σω,m 0.191** 0.220** (0.008) (0.005) ρω,mf −0.177** 0.299** (0.076) (0.028) σω,f 0.134** 0.234** (0.010) (0.007) ρθω,m −0.282** −0.506** (0.041) (0.026) ρθω,fm −0.307** −0.660** (0.088) (0.022) ρθω,mf 0.064 0.037 (0.048) (0.031) ρθω,f −0.004 0.322** (0.091) (0.031) ** (*): significant at 5% (10%) level. Standard errors in parentheses. Open in new tab Table A4 
Simulated Maximum Likelihood Estimates Labour Supply Equations: Dual Earner Couples Only Variable . Unmarried . Married . Men . Women . Men . Women . Intercept −9.399 47.082** 17.018** 75.932** (7.152) (5.742) (5.487) (5.233) Wage husband −0.606** 0.391 1.343** 1.635** (0.285) (0.354) (0.201) (0.213) Wage wife 1.048** −0.704** 0.113 −1.039** (0.320) (0.303) (0.216) (0.210) wm/(wm + wf) 70.498** −33.411** −6.037 −89.776** (12.424) (12.136) (9.121) (9.542) Wage husband squared/10 0.021 −0.043 −0.119** −0.147** (0.038) (0.048) (0.022) (0.023) Wage wife squared/10 −0.088** 0.029 −0.018 0.086** (0.044) (0.036) (0.027) (0.022) Non‐labour income/1000 −2.093 −2.947 −1.352 0.630 (1.533) (1.863) (1.719) (1.694) Variable . Unmarried . Married . Men . Women . Men . Women . Intercept −9.399 47.082** 17.018** 75.932** (7.152) (5.742) (5.487) (5.233) Wage husband −0.606** 0.391 1.343** 1.635** (0.285) (0.354) (0.201) (0.213) Wage wife 1.048** −0.704** 0.113 −1.039** (0.320) (0.303) (0.216) (0.210) wm/(wm + wf) 70.498** −33.411** −6.037 −89.776** (12.424) (12.136) (9.121) (9.542) Wage husband squared/10 0.021 −0.043 −0.119** −0.147** (0.038) (0.048) (0.022) (0.023) Wage wife squared/10 −0.088** 0.029 −0.018 0.086** (0.044) (0.036) (0.027) (0.022) Non‐labour income/1000 −2.093 −2.947 −1.352 0.630 (1.533) (1.863) (1.719) (1.694) ** (*): significant at 5% (10%) level. Standard errors in parentheses. The table only displays the coefficients of wages rates and non‐labour income. Open in new tab Table A4 
Simulated Maximum Likelihood Estimates Labour Supply Equations: Dual Earner Couples Only Variable . Unmarried . Married . Men . Women . Men . Women . Intercept −9.399 47.082** 17.018** 75.932** (7.152) (5.742) (5.487) (5.233) Wage husband −0.606** 0.391 1.343** 1.635** (0.285) (0.354) (0.201) (0.213) Wage wife 1.048** −0.704** 0.113 −1.039** (0.320) (0.303) (0.216) (0.210) wm/(wm + wf) 70.498** −33.411** −6.037 −89.776** (12.424) (12.136) (9.121) (9.542) Wage husband squared/10 0.021 −0.043 −0.119** −0.147** (0.038) (0.048) (0.022) (0.023) Wage wife squared/10 −0.088** 0.029 −0.018 0.086** (0.044) (0.036) (0.027) (0.022) Non‐labour income/1000 −2.093 −2.947 −1.352 0.630 (1.533) (1.863) (1.719) (1.694) Variable . Unmarried . Married . Men . Women . Men . Women . Intercept −9.399 47.082** 17.018** 75.932** (7.152) (5.742) (5.487) (5.233) Wage husband −0.606** 0.391 1.343** 1.635** (0.285) (0.354) (0.201) (0.213) Wage wife 1.048** −0.704** 0.113 −1.039** (0.320) (0.303) (0.216) (0.210) wm/(wm + wf) 70.498** −33.411** −6.037 −89.776** (12.424) (12.136) (9.121) (9.542) Wage husband squared/10 0.021 −0.043 −0.119** −0.147** (0.038) (0.048) (0.022) (0.023) Wage wife squared/10 −0.088** 0.029 −0.018 0.086** (0.044) (0.036) (0.027) (0.022) Non‐labour income/1000 −2.093 −2.947 −1.352 0.630 (1.533) (1.863) (1.719) (1.694) ** (*): significant at 5% (10%) level. Standard errors in parentheses. The table only displays the coefficients of wages rates and non‐labour income. Open in new tab Author notes " I am greatly indebted to Statistics Netherlands for providing the data. I thank five anonymous referees, Martin Browning and participants of the CEMFI Madrid Micro Econometrics Conference 2003 for their comments. All remaining errors are mine. © The Author(s). Journal compilation © Royal Economic Society 2009

Alós‐Ferrer,, Carlos;Kirchsteiger,, Georg;Walzl,, Markus

doi: 10.1111/j.1468-0297.2009.02297.xpmid: N/A

Abstract We study competition among market designers who create new trading platforms, when boundedly rational traders learn to select among them. We ask whether ‘Walrasian’ platforms, leading to market‐clearing trading outcomes, will dominate the market in the long run. If several market designers compete, we find that traders learn to select non‐market clearing platforms with prices systematically above the market‐clearing level, provided at least one such platform is introduced by a market designer. This in turn leads market designers to introduce non‐market clearing platforms. Hence platform competition induces non‐competitive market outcomes. Markets are not only characterised by demand and supply but also by the rules that govern the trading process. The ‘institutional’ framework determines the set of market participants, their available options and the matching and information structure of the market. In reality we observe a huge variety of different market frameworks, even for trading the very same good. Real estate, for example, is traded at auctions as well as by personal bargaining. There is also a large amount of evidence that these characteristics are crucial for the resulting trading outcome and for the realised prices. Since the impact of the trading rules on market outcomes is difficult to investigate with time‐series of real‐life market data, for a more detailed discussion see Friedman (1993), empirical evidence mainly relies on laboratory experiments, for an overview of the evidence see, e.g., Plott (1982) and Holt (1995); in the context of financial markets see also Friedman (1993). While double auctions typically tend to generate market clearing prices and quantities, posted‐offer markets establish prices that tend to be above the market‐clearing level, whereas the prices on posted‐bid markets seem to be below the Walrasian level (Plott and Smith, 1978). As a consequence, some gains of trade are not realised on these trading platforms and inefficiencies occur due to the design of the trading platform. In a similar way, Dutch or first‐price auctions are notorious for inducing overbidding and creating inefficient allocations compared to second‐price formats (Kagel, 1995). In a field study, Roth and Ockenfels (2002) show that fixed ending‐rules (‘hard‐close’) in online auctions lead to late bidding (‘sniping’); see also Ockenfels and Roth (2006). A laboratory experiment by Ariely et al. (2005) confirmed this finding, and also showed that fixed ending‐rules lead to lower revenues for the seller (and less efficient allocations) than automatic extensions of the auction (‘soft ending’). All these studies suggest that socially desirable features of market outcomes such as unbiased (market‐clearing) prices and efficient allocations are rather sensitive to details of the respective market institution. Moreover, there seems to be a trade‐off between efficiency and a price bias in favour of one of the market sides. A remarkable example for the coexistence of a variety of trading institutions is provided by Business to Business (B2B) trading platforms, for a comprehensive analysis see, e.g., Lucking‐Reiley and Spulber (2001). Recent decades have seen a proliferation of B2B platforms, and despite the burst of the internet bubble there were more than 1000 B2B marketplaces active in Europe in 2003 (European Commission, 2003). While most of the public and scientific attention is devoted to e‐marketplaces targeting consumers (like e‐bay or Yahoo), about 95% of the e‐commerce is actually B2B (UNCTAD, 2002). In 2004 B2B had an estimated volume of $1 trillion (The Economist, 2004). In contrast with Business to Consumers or Consumer to Consumer platforms, large quantities of relatively standardised products are traded at B2B exchanges. Agents seem to act either as buyers or as sellers on these platforms but not as both (European Commission, 2003). B2B e‐commerce is organised in three different ways. The predominant modus in the early days of e‐commerce were platforms opened by buyers or sellers (or respective umbrella organisations). An example is MetalSite, a platform organised by steel producers that suspended operations in 2001. Currently, B2B e‐commerce is typically organised either as e‐procurement1 (where sellers use standardised software and exchange opportunities offered by platforms such as Ariba or CommerceOne to design and allocate procurement contracts) or via institutions operated by a third party (this holds, e.g., for CheMatch – a trade platform for chemical products – or for a large part of the product portfolio offered at EnronOnline – a multi‐commodity exchange run by Enron until 2002). Of all firms active on B2B platforms, about one third operates on such platforms run by third parties (European Commission, 2003). Both e‐procurement software and market designs of third parties show a variety of institutional arrangements. EnronOnline, for instance, was organised as a posted offer market while competing platforms such as AltraEnergy (or on the chemical sector CheMatch) are exchange platforms that work like double auctions. The software solutions offered by Ariba and CommerceOne include various institutional arrangements such as Dutch auctions or proxy‐bidding (with hard and soft ending rules).2 Platform designs seem to exist with different propensities to generate market clearing outcomes in B2B e‐commerce(see the experimental literature cited above). Given the variety of different market institutions, and the variety of their efficiency properties, one wonders which type(s) of trading institutions will be observed in the long run. In particular, our article investigates whether institutions promoting efficient, market clearing outcomes will dominate less efficient trading platforms in the long run. To answer this question, we are led to investigate the evolution of market institutions. It is useful to distinguish between two aspects of this evolution, namely the selection between existing institutions by the traders and the emergence of new institutions. New market institutions can either be introduced on purpose by a market designer, or be the (unintended) by‐product of the actions of the traders. In what follows we focus on market platforms introduced on purpose.3 If a trading platform is introduced by a market designer who demands user fees, the design of a new platform by the designer and the selection among existing ones by the traders are closely interlinked. The market designer will try to introduce a new platform with characteristics that attract many traders. This attractiveness in turn determines the long‐run survival of the platform. In this article we analyse this interplay between the creation of new and the selection among existing trading platforms and we investigate the characteristics of the resulting platforms with respect to their ability to achieve market clearing outcomes. Trading platforms are created by profit‐maximising, risk‐neutral market designers. The designers compete with each other through platform designs. Each designer chooses a trading fee that he demands from the traders for the use of his platform. To capture the trade‐off between efficiency and a price bias for one market side that has been observed in laboratory studies (see above), we allow designers to choose platform designs with systematic price biases, above or below the market clearing price. Hence, through the trading fee, each designer decides upon his share of the surplus created through trade at his platform. But he can also favour one type of trader with the introduction of a price bias. Any bias reduces the surplus generated at the platform (and thereby ceteris paribus the revenue for the designer) but may also make it more attractive for the favoured type of trader, which in turn may enhance the platform’s survival probability. To analyse this trade‐off, we model competition between two market designers and compare the results of this setting with the benchmark case of a monopolistic market designer. After the platforms have been designed, traders decide which platform they want to be active on (for the monopolistic case, there is of course no real choice – traders just trade on the only existing platform). The role of a trader (buyer or seller) is exogenously given. Sellers are assumed to be firms with a constant returns to scale production technology.4 Buyers are characterised by their demand functions, and might be either consumers or other firms. For given platform characteristics, the selection by traders gives rise to a coordination game. If each trader opts for a particular platform no trader has an incentive to deviate from this platform – independently of the design alternatives offered by the competing platform. If traders were fully rational, we would have established a standard two stage game (Stage 1: Market Design; Stage 2: Traders’ platform choice) with (network) externalities in the Stage 2‐subgame. Such a game typically exhibits a multiplicity of (subgame perfect) equilibria. As in the battle of sexes, coordination of all traders on each platform is clearly an equilibrium of the 2nd stage, next to a mixed strategy equilibrium where traders are indifferent between platforms. To select among these equilibria, we drop the assumption of fully rational traders and instead assume that traders are boundedly rational but may learn to coordinate on a particular platform. Following the game‐theoretic learning literature (Young, 1993; Kandori et al., 1993; Ellison, 2000), we use a Markovian model to analyse the platform choice of the traders. We assume that the traders’ behaviour depends on the market outcomes generated by the different platforms and thereby on the characteristics of all feasible platforms. We are interested in the long‐term properties of this learning process, i.e. in its (limit) invariant distribution. This invariant distribution in turn determines the payoffs of the market designers. Hence, we establish a link between designer revenues and the characteristics of all feasible platforms. For the case of competing platforms we find that, in the long run, traders will always coordinate on a platform with prices above the market clearing level, provided that such a platform has been introduced by at least one designer. This forces designers to introduce platforms that are not market clearing but that have a price bias in favour of the sellers. On the other hand we find that a monopolistic designer will always introduce a market‐clearing platform. Therefore competition at the designers’ level turns out to be detrimental for a competitive outcome at the traders’ level. We regard this result as paradoxical. The present article is related to three strands of the literature. First, since we investigate the role of trading platforms with exogenously given buyers and sellers, our article is to some extent related to the two‐sided markets literature; see Rochet and Tirole (2006) for an overview. This literature is based on the assumption of network externalities. It analyses the impact of these externalities and of platform competition on the structure of the fees demanded by the market designers (Armstrong, 2006; Belleflamme and Toulemonde, 2004; Caillaud and Julien, 2003; Rochet and Tirole, 2003). In contrast, we want to investigate whether traders learn to coordinate on market‐clearing trading platforms, if such platforms are feasible. Therefore we explicitly model the learning behaviour of the traders, whereas the two‐sided market literature assumes rational traders. Further, we ask whether platform competition induces market designers to establish platforms with characteristics that achieve market‐clearing outcomes. Consequently, we abstract from any network externalities that are not internalised by the price at which trade takes place. In our model trading fees demanded by the market designers are neutral insofar as the market outcome is only influenced by the total fee imposed on both market sides but not on the distribution of the fees on the two market sides.5 Second, our article is also related to the literature on competition between exogenously given trading institutions. Ellison and Fudenberg (2003) and Ellison et al. (2004) analyse the circumstances under which different market institutions can coexist in equilibrium. Due to their different research questions these papers do not allow for institutions with systematic price biases. Kugler et al. (2006) investigate the case of centralised versus decentralised trading institutions. All of these papers rely on the assumption of rational traders and do not allow for learning. In terms of traders’ behaviour, the learning model of Gerber and Bettzüge (2007) is relatively close to our article. But since they focus on the possibility of multiplicity of active trading platforms, they consider neither non‐market‐clearing platforms nor market designers. The paper most closely related to the one at hand is that of Alós‐Ferrer and Kirchsteiger (2008), which also analyses the learning behaviour of traders who face the choice between different, not necessarily market‐clearing platforms. That paper, however, deals only with the selection among different, exogenously given institutions and does not consider competition between market designers. In our model, rational market designers are confronted with boundedly rational, learning traders.6 Hence, our article belongs to a small but growing literature that we call ‘asymmetric rationality’, where fully rational firms or otherwise sophisticated agents are confronted with a population of boundedly rational ones. The basic motivation is that consumers and small traders do not have the resources to obtain all the relevant information and fully optimise their behaviour, often relying on behavioural rules of thumb instead. However, large firms, market designers, etc. can be taken as comparatively sophisticated. Schlag (2004), Gabaix and Laibson (2006), Hopkins (2007) and Spiegler (2006) apply this approach to the analysis of industries facing boundedly rational consumers. See Ellison (2006) for an overview of this literature. The article is organised as follows. Section 1 presents the basic model. Section 2 discusses the traders’ platform choice of the traders. Section 3 analyses the design of the platform. Section 4 concludes. All proofs are in Appendix A. In Appendix B we analyse the robustness of our results with respect to boundedly rational designers and with respect to decreasing returns to scale in production. 1. The Model We study the trade of a homogenous good at different market platforms, which are set up by profit‐maximising market designers. For simplicity, we restrict our attention to two competing market designers (referred to as competitive market design). As a benchmark, we also analyse the case where only one market designer can set up a trading platform (referred to as monopolistic market design). In this Section, we introduce the trading rules that are at the market designer’s disposal and analyse trade and profits for a given choice of trading rules by the designers and a given platform choice by buyers and sellers. 1.1. Market Platforms’ Design Before trade takes place, market designers decide upon the set of trading rules under which their respective platforms operate, and the trading fees they demand from the traders. We do not aim at a complete description of the different sets of rules the designers can introduce. Rather, we characterise them by their ability to establish market clearing. Market designers may choose to design platforms such that market clearing is guaranteed, or they may pick platforms where the price is systematically biased above or below the market clearing price. Denote by the market‐clearing price if at least one seller and at least one buyer choose this platform and by βi > 0 the bias of platform i = 1,2. The actual price at which trade takes place at platform i is then given by . If the actual price is not market clearing (i.e. βi ≠ 1), the quantity traded is determined by the short market side, and traders on the long market side are rationed. Sellers are rationed equally if βi > 1. We do not specify any rationing rule for the buyers. The common set of feasible biases is assumed to be a finite, regular grid B = {βmin,βmin +δ, …,1, …βmax −δ, βmax}, where 0 < βmin < 1 < βmax and δ is the step of the grid. To understand why institutions with different price biases are feasible for the designers, recall the experimental and empirical results mentioned in the introduction. In our framework posted offer markets or first price auctions are characterised by β > 1, posted bid markets or proxy‐auctions with ‘hard‐close’ by a β < 1, while double auctions can be represented by β = 1. We refer to the platform with β = 1 as the market‐clearing platform and we assume that such a platform is always feasible. |B| denotes the number of feasible biases. After the platforms are set up, traders will use their observations and experience to learn eventually which platform to use. Formally, we analyse a learning process with an infinite number of trading rounds. The designers’ long‐run payoffs are the expected per round charges. Furthermore, we assume that the charges of designer i are a fixed share of the revenue generated by trade on i’s platform.7 Denote by fi the trading fee demanded by designer i, and by ERi the expected per round revenue generated on platform i. Then market designer i’s profits are given by πD,i = fiERi. The set of feasible fees is the same for both designers. For simplicity we assume that it is given by a finite, regular grid F = {fmin,fmin+γ,…fmax −γ,fmax}, where 0 < fmin < fmax < 1.8|F | denotes the number of feasible fees. The trading fee can be imposed on the sellers’ side, on the buyers’ side, or divided between both sides. However, the market‐clearing price, the realised price at which trade is conducted and the traded quantities depend only on the total fee and not on the distribution of the fee over the two market sides. Buyers at platform i pay pi for each unit, market designers receive fi pi and sellers ultimately receive (1 − fi)pi. Hence, we do not need to specify on which market side the fee is imposed. The characteristics of a platform i are denoted by si = (βi,fi), and the set of feasible characteristics by S = B × F. 1.2. Traders The good is supplied by a finite set M of at least two profit‐maximising firms (sellers) that use the same constant returns to scale technology with marginal costs of c > 0.9 When deciding upon his supply, a seller takes into account the trading fee of the platform at which he operates. Hence, sellers supply a strictly positive but finite quantity if and only if the price net of trading fee is equal to c. As shown later, the assumption of a constant returns to scale technology allows us to derive results for a very general class of learning models. That is, by focusing on this case, we obtain results that are robust to the details of the learning process. In Appendix B.2. we illustrate that for strictly decreasing returns to scale the results depend on the details of the learning model. In particular, the results of the constant returns to scale case can be replicated also for strictly decreasing returns to scale, but not for the whole class of learning models we analyse here. The good is demanded by a finite set N of buyers with |N | > 1. Each buyer n ∈ N is endowed with a demand function dn(p) which might be different for different buyers. All the demand functions are assumed to be strictly decreasing in p. Furthermore, 0 < dn(p) < ∞ for all p, n. To avoid discontinuities in the designers’ profit functions we also assume that  limp→∞ pdn(p) = 0 for all n ∈ N.10 We call a platform active if both sellers and buyers are present and positive quantities are traded, and inactive if not. The presence of both types of traders does not ensure that the platform is active. To see this, note that due to the assumption of a constant returns to scale technology the market‐clearing price of a platform i where both sellers and buyers are present is given by . The realised price at which trade is conducted on platform i is then (1) If βi < 1, the net price received by the sellers is below the marginal costs. Hence, supply is zero, and platform i is inactive despite both types of traders being present on platform i. Denote by Ni the set of buyers who choose platform i, and by Mi the set of sellers who choose platform i. Platform i is active if and only if |Ni| > 0, |Mi| > 0, and βi ≥ 1. Let (2) denote the total demand at platform i. The quantities traded by a buyer n ∈ Ni, qn,i(Ni,Mi,si), and by a seller m ∈ Mi, qm,i(Ni, Mi, si), are given by (3) (4) In the single‐designer case, traders cannot choose between different platforms but have to use platform i. Hence, Ni = N, Mi = M and the market outcome is only determined by the platform characteristics si. If there is competition between market designers, trade can take place at different platforms and the outcome depends also on the way traders learn which platform to use. This learning process is driven by the market outcomes of both platforms (see above) and by the individual evaluations of these outcomes. For the latter part note that if buyers trade strictly positive amounts, they are strictly better off than without trade. Hence, inactive platforms are worse for buyers than active ones. Furthermore, whenever a buyer trades a strictly positive quantity, he is not rationed at all. It is thus natural to assume that buyers’ evaluation of active platforms is monotonically decreasing in the price. Therefore, buyers’ evaluation of platform i could be represented e.g. by11 (5) If both platforms i and j are active (i.e. positive amounts are traded), (6) This implies in particular that if βi = βj = 1 and fi < fj, then πn,i(si) > πn,j(sj). The sellers’ evaluations of the platforms are determined by the respective profits. An inactive platform gives of course zero profits. Furthermore, whenever βi > 1, sellers trading on platform i are on the long market side and equally rationed. Hence, the sellers’ evaluation of platform i is given by (7) Note that for βi > 1 the sellers’ profits are strictly positive provided that platform i is active. On the other hand, for βj = 1, πm,j(Nj, Mj, sj) = 0 irrespective of whether platform j is active or not. So as long as there is an active non‐market clearing platform, its outcome is always strictly better for the sellers than the outcome of a market clearing platform. That is, for all fi, fj, (8) 2. The Traders’ Platform Choice In our model, market designers first choose their platforms’ characteristics and then buyers and sellers decide which platform to join. If there is only one market designer, traders’ choices are trivial – they simply opt for the existing platform. With more than one market designer, traders have to choose between the two platforms. For any given si, sj, the choice of platform constitutes a coordination game. If all traders choose platform i, no trader has an incentive to deviate to the other platform j. Furthermore, if βi and βj are strictly larger than 1, full coordination on any platform is even a strict Nash equilibrium. Hence, nothing guarantees coordination on any particular platform and therefore traders have to learn which platform to use. In this Section, we introduce the learning process and analyse long‐run trading patterns and platform revenues for a given configuration of designs. 2.1. The Learning Process We consider a social learning process defined by (i) the information available to each trader, (ii) the way traders revise their platform choices whenever they have the opportunity, (iii) the opportunities to revise a platform choice, and (iv) the way traders make mistakes when choosing a platform. The information available to a trader is not only his own experience (as it would be in a reinforcement learning model, for instance). Rather, each trader observes the prices and the quantities of both platforms (including the observation of the inactiveness of a platform). We also assume that an individual trader does not have enough information on other traders or is not able to perform all the necessary computations in order to predict the future behaviour of the other traders. Hence, individual traders cannot accurately predict the future outcomes of the platforms. Furthermore, they also lack the capability necessary to always compute an exact (but myopic) best reply to the current choices of all other traders. What can a trader do in such a situation? From his individual, myopic standpoint, if he considers himself to be small relative to market size, the best thing he can do is to evaluate the outcomes of both platforms and switch to the other platform if he perceives the other platform’s outcome as better. A trader can perceive this behaviour as approximately rational, since when he switches, the implied changes in prices and traded quantities will most of the time be small and hence this behaviour is close to the best reply. Of course, this could also be interpreted as imitation of successful traders of the own market type. We want to stress, though, that the behaviour described does not require the observation of any evaluation conducted by other traders but merely the observation of prices and traded quantities in both platforms. With this learning rule the switching decision of each trader depends on the trading outcomes of both platforms in the last trading round. These trading outcomes depend on the distribution of the traders over both platforms. Hence, the distribution of traders over platforms depends on the last period’s distribution of traders over platforms. A state ω specifies which trading platform is chosen by each buyer and each seller. The state space is given by Ω = {1,2}|N | × {1,2}|M |, and trader k’s platform choice in state ω ∈ Ω is denoted by ω(k) ∈ {1,2}. The following notation will prove convenient: (9) (10) i.e. Ni(ω) ⊆ N is the set of buyers who are on platform i in state ω and Mi(ω) ⊆ M the set of sellers who are on platform i in state ω. By definition, all those traders who are not on platform i have to be on the other platform j. The state of the process at time t = 0, 1, 2,… is given by ω(t) ∈ Ω. That is, ω(t)(k) ∈ {1,2} denotes the platform chosen by trader k at time t. 2.1.1. Unperturbed learning process We first concentrate on the unperturbed learning process, where traders switch platforms only because of learning but not because of experimentation (experimentation is introduced in the next subsection). If an agent is able to revise his choice for a given period t + 1, he takes the new market outcomes of both platforms in period t and evaluates them. As explained above, we postulate the following learning rule: Assumption A. A trader, who gets the opportunity to revise, observes the outcomes of both platforms in the last period. He chooses the platform whose outcome he evaluates as best. In case of indifference, he stays with his old platform. Whenever trader k receives a revision opportunity at period t, he will choose the platform with the period t − 1 outcome that he evaluates highest. If, by chance, the outcomes of both platforms are equally evaluated, the trader sticks to his former platform choice. For instance, in the case in which one platform is inactive and the other is active but yields zero profits for the sellers, sellers do not switch. This assumption could be justified by small but positive switching costs.12 But when are agents allowed to revise their choices? It is common in learning models to introduce some inertia explicitly allowing for the possibility that not all agents are able to revise strategies simultaneously (or, for instance, accounting for idiosyncratic switching costs). Different specifications of how revision opportunities arrive give rise to different dynamics and often affect the results. Rather than adopting a specific formulation, here we follow Alós‐Ferrer and Kirchsteiger (2008) and postulate a general class of dynamics encompassing the standard examples (and many others).13 This general dynamics is defined by the following assumptions. Assumption B1. For every agent k and state ω there is strictly positive probability that agent k is the only trader of his own market side who is able to revise his platform choice. Notice that B1 implies that every agent has a strictly positive probability of being able to revise at any given state. It also allows the revision probability to depend on the state ω and on the identity of the trader, k. Since we have two clearly differentiated populations, we introduce a weak form of independence between the revision opportunities in those populations (it can actually be considered as an anonymity requirement). Assumption B2. For every agent k and state ω, if k is the only one of his own market side who is allowed to revise, either no trader of the other market side is able to revise his platform choice, or there is a strictly positive probability for each trader of the other market side to be allowed to revise. This Assumption explicitly excludes non‐anonymous situations where, say, whenever seller number 17 gets the opportunity to revise, buyers 3 and 6 also get the opportunity to revise. Assumptions B1 and B2 are rather general. They are fulfilled by the standard models considered in the literature of learning in games. In these models, revision opportunities are either modelled through independent probabilities (a case we call independent inertia; see e.g. Samuelson (1994); Kandori and Rob (1995)) or assumed to arrive in an asynchronous way (a case we term asynchronous learning; see Blume (1995), Binmore and Samuelson (1997) and Benaïm and Weibull (2003)).14 That is, our formulation covers the following standard examples. Independent Inertia. For each agent k and each state ω there is an exogenous, equal, independent and strictly positive probability ρ < 1 that agent k does not get a revision opportunity. Asynchronous Learning. Each period, only one agent (i.e. either a buyer or a seller) is (randomly) selected and allowed to revise his strategy. Asynchronous Learning within Types. In our case, it is natural to conceive of a dynamic where, in every period, only one buyer and one seller are selected (randomly and independently) and given the opportunity to revise. Obviously, B1 and B2 are fulfilled by all these types of learning. The specification above allows for more general learning processes than those described by independent inertia or asynchronous learning. Since the revision probability is allowed to be a function of the state ω, it might depend, for example, on the difference between the evaluation of the outcomes of both platforms (so that unsatisfied traders are more likely to revise), or on idiosyncratic characteristics of the currently chosen platform. Assumptions A, B1, and B2 define a stationary Markov chain on the (finite) state space Ω. Given two states ω,ω′ ∈ Ω, denote by P 0(ω,ω′) the probability of transition from ω to ω′ in one period for the unperturbed learning process. The transition matrix is given by P 0 = [P 0(ω,ω′)]ω,ω′∈ Ω. An absorbing set of the unperturbed dynamics is a minimal subset of states which, once entered, is never abandoned. An absorbing state is an element which forms a singleton absorbing set, i.e. P 0(ω,ω) = 1.15 As a first step in the analysis of long‐run trading patterns, we determine the absorbing sets of the unperturbed learning dynamics. Depending on the design of the two platforms, there exist multiple such absorbing sets. The reason is that no trader ever switches to a platform which does not have an agent of each market side and/or has a bias below 1 and is therefore inactive. Moreover, indifferent traders do not switch. In particular, sellers never switch to a market‐clearing platform as it does not offer a positive profit for them. These considerations lead to the following results. Lemma 1. Assume A, B1, and B2. Let i ≠ j, and . All absorbing sets of the unperturbed dynamics are singletons. Depending on platforms’ characteristics, the absorbing states are as follows. (a) Ifβi > 1 andβj > 1, the monomorphic statessuch thatand every state inΩ0 = {ω | πm,i = πm,j, πn,i = πn,j}.16 (b) Ifβi > 1 andβj = 1, the monomorphic state, the cross‐statewith, , and every state in (which includesand). (c) Ifβi = βj = 1, the elements inand, plus, if and only ifp(si) = p(sj), every state with two active platforms. (d) Ifβi > 1 andβj < 1, the monomorphic state, and all states in which platformiis inactive ( forβj < 1, platformjis always inactive). (e) Ifβi = 1 andβj < 1, the elements of and all states in which platformiis inactive. ( f ) Ifβi < 1 andβj < 1, all statesω ∈ Ω. 2.1.2. Perturbed learning process In order to select among the multiple absorbing states, we now turn to the analysis of the stability properties of the platforms with respect to experimentation. The dynamics are enriched with a perturbation in the form of experiments (or mistakes) in the following way. With an independent, small probability ɛ > 0, each agent, in each round, might experiment (or make a mistake or ‘mutate’), and simply pick a platform at random,17 independently of other considerations. The dynamics with experimentation is called perturbed learning process. Its transition matrix is denoted by P ɛ. Since experiments make transitions between any two states possible, the perturbed process has a single absorbing set formed by the whole state space (i.e. the process is irreducible) and there is a unique probability distribution over states μ ∈ Δ(Ω) which, if taken as initial condition, would be reproduced in probabilistic terms after updating (more precisely, μP ɛ = μ). This μ is called the invariant distribution of P ɛ. For the perturbed dynamics P ɛ the limit invariant distributionμ*= limɛ→0 μ exists and is an invariant distribution of the unperturbed process P 0 (see e.g. Kandori et al. 1993; Young 1993; Ellison 2000). It singles out a stable prediction of the unperturbed dynamics, in the sense that, for any ɛ > 0 small enough, the play approximates that described by μ* in the long run. Thereby μ*(ω) is the probability that (for small ɛ) the process will be in state ω in the long‐run. The states in the support of μ*, i.e. {ω ∈ Ω | μ*(ω) > 0} are called stochastically stable states or long‐run equilibria. The set of stochastically stable states is the union of some absorbing sets of the original, unperturbed chain (ɛ = 0). We call a platform active in the long‐run if there is a positive probability for trade at this platform in the long‐run, i.e., if there is a stochastically stable state with platform i being active.18 Theorem 2. Assume A, B1, and B2. (a) Supposeβi < 1. Then, platformiis not active in the long run. (b) Supposeβi = βj = 1. Then, platformsiandjare active in the long run. (c) Supposeβi > 1 andβj ≤ 1. Then, platformiis active andjis inactive in the long run. The intuition for this theorem is straightforward. Since there is no trade on a platform with βi < 1, it will never be active. Furthermore, on a market clearing platform the sellers’ profits are always zero. Hence, sellers do not care at which platform they are if they have to choose between two market clearing platforms. Consequently, both platforms are active in the long run if both are market clearing. Finally, a seller is never worse off at platform i with βi > 1 than at platform j with βj ≤ 1, even if there are no buyers at i. A buyer, on the other hand, is worse off at j than at i when he finds no seller at j. So sellers have an unambiguous tendency to learn to use i, whereas buyers do not always have a tendency towards j. As a consequence, all traders will coordinate on the non‐market clearing platform i in the long run. On the level of the platform design, it is thus easy to compete with a market clearing platform by introducing a platform design with a positive price bias. 2.2. The Long‐run Trading Patterns We now proceed to analyse long‐run trading patterns (i.e. the stochastic stability of platforms) for a given design configuration si = (βi, fi) and sj = (βj, fj). As a benchmark, we start with the case of identical platform design. To analyse platforms with identical characteristics (si = sj), we observe that, for every state ω ∈ Ω we can uniquely define a so‐called mirror state by changing the platform affiliation of all traders, that is, is the only state such that and . Then, Lemma 3. Suppose si = sj. Then, the distribution of traders over the platforms is symmetric in the long run, i.e., . Theorem 2 already identifies the set of long‐run active platforms (i.e. stochastically stable states with active platforms) whenever at least one platform i has a price bias βi ≤ 1. Hence we are left with design configurations si and sj where both price biases favour sellers (i.e. βi,βj > 1). There, Lemma 1(a) implies that full coordination on each platform and states with indifference of both buyers and sellers are the only candidates for stochastically stable states. To pin down stochastic stability, it proves useful to distinguish the two platforms with respect to their prices. Lemma 4. Suppose βi,βj > 1 and pi = βic/(1 − fi) < βjc/(1 − fj) = pj. Then, (a) only monomorphic states can be stochastically stable. (b) is stochastically stable. According to Lemma 4, the platform with trade at a lower price is always stochastically stable as it is preferred by buyers as long as it is active. The only other candidate for stochastic stability is coordination on the high price institution. While all our previous results did not depend on the modelling details such as (i) absolute population size of buyers and sellers, (ii) the relative size of these populations, (iii) the heterogeneity of buyers, (iv) the price elasticity of demand, (v) the grid size δ, and (vi) details of the learning process (e.g. adjustment speed, asymmetries between buyers and sellers), these details do matter now as the following results illustrate. Lemma 5. Suppose βi,βj > 1, pi = βic/(1−fi) < βjc/(1 − fj) = pj, so that is stochastically stable. (a) In a dynamics with independent inertia,is also stochastically stable if and only if there is at least one buyersuch that (11) (b) In a dynamics with asynchronous learning,is also stochastically stable if and only if there is at least one buyersuch that (12) The condition in part (a) is violated whenever buyers are identical, | N | ≥ | M|, and the price elasticity of demand is sufficiently high. It can be satisfied for βj > βi whenever buyers are sufficiently heterogeneous (i.e. such that ), or buyers are identical and |M| >> |N|, or d(p) is sufficiently inelastic. The stochastic stability of is harder to establish if the dynamics is slow as e.g. under asynchronous learning. The condition in part (b) is violated whenever buyers are identical and the price elasticity of demand is sufficiently high (in contrast to the case of independent inertia, this holds independently of the sizes of populations |M| and |N|). The condition can be fulfilled for βj > βi if buyers are sufficiently heterogeneous or buyers are identical and demand is sufficiently inelastic. Remark 1. The proofs of the previous lemmata (see Appendix A) rely on transition paths involving at most two simultaneous mutations. Thus the speed of convergence is relatively high. Since the number of required mutations does not increase with population size, our dynamics escapes the well‐known critique that for large populations the long run may actually be ‘too long’ to be relevant (Kandori et al. 1993; Ellison 1993). We find this point important, because real‐world designers are not infinitely long‐lived, and some market institutions (e.g. online platforms) appear to go out of business relatively quickly if lacking customers. Since the speed of convergence of the trader‐learning process is quick, we think that our results remain relevant. 2.3. Platform Revenues and Designers’ Profits Till now we have analysed the learning dynamics of the traders and the resulting long run pattern of trades. Next we turn to the revenues generated by the platforms, which in turn determine the profits of the market designers. When analysing the market designers’ choice of the characteristics of the trading platforms we will assume that platform designers are long‐lived, patient, and (relatively) rational agents when compared with individual buyers or sellers. Hence, the designers consider a platform profitable if it is active in the long run and they ignore revenues made during the adjustment process to the limit invariant distribution.19 Given the platform characteristics s = (si,sj), the long‐run expected revenues per round ERi(s) depend on the limit invariant distribution. The profits of designer i are given by πD,i(s) = fiERi(s) implying that πD,i(s) ≥ 0 for all s. Consider first a platform i with βi < 1. Lemma 6. Suppose βi < 1. Then πD,i((βi,fi),sj) = 0 for all feasible fi,sj. Unsurprisingly, a platform with βi < 1 does not generate any profit for the designer as it is always inactive. Hence, we are left with platform configurations (si,sj) where both platforms have a price bias weakly larger than one. In this case expected revenues at platform i depend not only on the design of platform i but also on the design of the other platform as the following results indicate. Lemma 7. Consider a platform configuration s = (si,sj) with si = (βi,fi), sj=(βj,fj) and prices pi = βi[c/(1−fi)], pj = βj[c/(1−fj)]. (a) Ifsi = sjandβi,βj ≥ 1, then,fork = 1,2. (b) Ifβi = βj = 1 andfi < fj, thenfkpkDN(pk) > πD,k(si,sj) > 0 fork = 1,2. (c) Ifβi > 1 andβj ≤ 1, thenπD,i(si,sj) = fipiDN(pi) andπD,j(sj,si) = 0. The first part of this Lemma shows that for identical platforms the designers’ long‐run profits are identical and strictly positive. This follows from the symmetry of the limit invariant distribution (Lemma 3). With two market clearing institutions, none of the platforms can reap all long‐term revenues even if the fees differ. Finally and most importantly, when a non‐market clearing and a market‐clearing platform compete, the designer of the former makes strictly positive profits, whereas the profits of a designer of a market‐clearing institution are zero, because all traders coordinate on the non‐market clearing platform in the long run (see Theorem 2(c)). 3. The Platform Design We now compare the design choices by a monopolistic designer and by two competing designers. 3.1. Monopolistic Market Design As a benchmark, we briefly consider the case where only one platform is available, with characteristics s = (β, f ). In this case traders have no choice but to use this platform and designer’s profits are given by: (13) It follows that a monopolistic designer introduces a market‐clearing platform – as long as the grid of feasible fees is fine enough. Proposition 8. Suppose there is only one platform available. Then, the designer chooses a market‐clearing platform, i.e. β* = 1, if γ is sufficiently small. The intuitive reason for this result is as follows. Suppose revenues pDN(p) are maximised at price p *. Note that this price can be attained with different (β,f) combinations and that p * = βc/(1 − f) is increasing both in β and f. Since the monopolist designer’s profits are fpDN(p), he will try to reach p* with that (β,f) combination that has the highest fee and hence the lowest possible β ≥ 1. 3.2. Competitive Market Design In order to reflect that platform designers are ‘more rational’ than individual buyers and sellers, we simply consider them rational players in the normal‐form game defined by their payoff functions.20 That is, both designers choose their platforms simultaneously and payoffs are given as in Section 2.3. The sets of pure strategies of designer i and j are given by Si = Sj = B × F. We also allow designers to use mixed strategies, i.e. choose a probability distribution over S rather than picking up a particular characteristic for sure. Denote by σi the (mixed) strategy of designer i. The expected payoff of i is (14) Since the sets of pure strategies are finite, a Nash equilibrium of the designers’ game always exists (possibly in mixed strategies). To characterise these equilibria, we need the following Lemma. Lemma 9. Let be a Nash equilibrium (possibly in mixed strategies). Then, for any pure strategy si = (βi, fi) of player i such that , it holds that βi ≥ 1. Lemma 9 is an immediate consequence of the fact that there is no trade at a platform with β < 1. Therefore, only platforms weakly biased in favour of the sellers will be chosen in equilibrium. We now show that, actually, in any equilibrium, both designers will introduce platforms that lead to prices strictly above the market clearing level–platforms that lead to market clearing prices will not be designed in equilibrium. This result holds as long as the grid of feasible biases is fine enough. Theorem 10. Let be a Nash equilibrium (possibly in mixed strategies). For any pure strategy si = (βi,fi) of player i such that , it holds that βi > 1 if δ is sufficiently small. We have thus established the paradoxical result that competition among platform designers will induce them to select biased platforms which implement non‐competitive market outcomes. As we have seen, competition between a market clearing and a non‐market clearing platform leads to full coordination of the traders on the latter platform (see Theorem 2(c)). Hence, designers do not introduce a market clearing platform when facing platform competition. In general, nothing more can be said about the specific characteristics of the Nash equilibria. A brief examination of Lemma 5 should convince the reader that a full characterisation of the Nash equilibria will depend on the exact shape of the limit invariant distribution, and not only on its support. This distribution in turn depends on the details of the dynamics, e.g. whether learning opportunities arise simultaneously among traders or asynchronously. In contrast, the last theorem holds for any specification of the learning dynamics satisfying Assumptions B1 and B2. Still, one might suspect that competition leads to platforms close to the market‐clearing one, i.e. to platforms with βi = 1 + δ. If this hypothesis would be correct, the chosen platforms would nearly resemble market‐clearing ones as long as the grid of feasible biases is fine enough. The next Proposition, however, shows that this hypothesis is in general false. For simplicity, consider identical buyers with a demand function d(p) and denote the price elasticity of demand by ɛp(p) = −pd′(p)/d(p). Proposition 11. Assume independent inertia, identical buyers and |M | = |N |. If δ and γ are sufficiently small and ɛp is not much larger than one, then there exists no Nash equilibrium (neither in pure nor in mixed strategies) where both designers introduce only platforms with βi = βj = 1 + δ. Beyond the features highlighted in Theorem 10, equilibrium designs are rather sensitive to details of the economy and the learning process. It cannot be excluded that in equilibrium designers choose ‘near market‐clearing’ platform characteristics for some specifications of the learning dynamics and/or of the demand functions. But in general the equilibrium choices are not ‘near market‐clearing’ platforms. 4. Discussion We have shown that if several trading platforms are available, traders will learn to coordinate on a platform with prices systematically above the market‐clearing level, if such a platform is feasible. This forces competing market designers to create such non‐market‐clearing platforms. On the other hand a monopolistic market designer will always introduce a market‐clearing platform in order to maximise his profits. Hence, we derive the paradoxical result that platform competition induces non‐competitive market outcomes. This result could also explain why so many B2B platforms exhibit institutional designs that are notorious for biased (non‐market‐clearing) prices (e.g. posted offer markets, proxy auctions with ‘hard‐close’ or Dutch auctions). The results of our article depend of course on our key assumptions. First, we have focused on boundedly rational traders who choose platforms myopically. For fully rational designers and traders, our set‐up would correspond to a two stage game where, in stage 1, designers (simultaneously) choose platform designs, and in stage 2, traders coordinate on platforms. The second stage thereby resembles a coordination problem or a game with network externalities. Naturally, this structure induces a multiplicity of equilibria – in particular, there is a subgame perfect equilibrium where all traders coordinate on a market‐clearing platform (with monopolistic trading fees). Our analysis of the coordination problem in stage 2 using a learning dynamics can be interpreted as an equilibrium selection device, which rejects the above‐described equilibrium and selects only configurations with non‐market‐clearing platform designs. Second, we assumed asymmetric rationality in the sense that designers are more sophisticated than traders. Furthermore, by focusing on long‐run profits we have implicitly assumed that it is much harder for designers to change the properties of their platforms than for traders to switch trading platforms, or that platform providers are much more patient than traders. Cases such as the downfall of Enron Online or the bankruptcy of CommerceOne illustrate, however, that this assumption may not be fully justified. Sometimes platform providers indeed suffer rather quickly from a lack of traders and are removed from the market at short notice. But our modelling framework can cope with cases where platform designers and traders revise their decisions with equal speed. In Appendix B we investigate the case of boundedly rational designers who have to learn how to design a platform through a regular (trial‐and‐error) design revision process (i.e., designers are as myopic or impatient as traders). Our main results carry over to such a setting. Hence, while asymmetric rationality is a crucial ingredient of our model, it is not the driving force behind the emergence of non‐market clearing institutions. Furthermore, our results also hold when the traders’ learning process shows a relatively high speed of convergence. In real life this would imply that a market designer can enter and remain in the market with a superior design, because traders can coordinate on the respective platform at short notice. In this sense our model allows for the ‘free entry’ of superior platform designs and still non‐market clearing institutions emerge in equilibrium. Third, we have assumed sellers to be producers endowed with a technology with constant returns to scale. Although this is a focal, economically meaningful case, it clearly simplifies the analysis and allows for a clear‐cut derivation of the results. Under production technologies exhibiting decreasing returns to scale, the results are less strong and a characterisation of the limit invariant distribution requires both a further specification of the learning behaviour of the traders and a further specification of demand and supply. In Appendix B we provide an extended example with decreasing returns to scale where our main result still holds. It shows, however, that the optimality of a price bias is no longer independent of details like learning velocities. Nonetheless, this clearly illustrates that the scope of the paradox identified here goes beyond the constant returns to scale case. These robustness checks show that neither the assumption of constant returns to scale nor that of rational designers drive our results. Rather, it is indeed platform competition that leads to the emergence of non‐market‐clearing trading platforms. Footnotes 1 " For a recent discussion of the adoption of e‐procurement in B2B and an overview of market designs see Davila et al. (2003). 2 " Interestingly, CommerceOne applied for Chapter 11 bankruptcy in 2004 and was bought out later. In general, entry and exit are still a frequent phenomenon in the market for B2B platforms, suggesting that this industry is still at a relatively early stage of development. However, business analysts identify a development from the creation of new platforms (the common business model in the 1990s) to buy‐outs, which indicates some degree of maturity (Keys, 2002). 3 " For an analysis of markets as a by‐product of traders’ actions, see Kirchsteiger et al. (2005). 4 " In Appendix B we investigate the robustness of our results with respect to decreasing returns to scale. 5 " Rochet and Tirole (2006) define two‐sided markets by the non‐neutrality of the fees. In their terminology we model a one‐sided market. 6 " In Appendix B we show the robustness of our results with respect to learning designers. 7 " Our results would not change if we assume quantity‐dependent charges instead of revenue‐dependent charges. 8 " The assumption that the fees are strictly positive can be justified by (unmodelled) setup costs for the market designers. 9 " The assumption of identical sellers might seem restrictive at the first sight. Within our framework, firms without access to the lowest cost technology would face zero market demand. Hence, our assumptions only rule out the case where exactly one firm has access to the lowest‐cost technology. 10 " Our results do not depend on the assumption that the value of demand goes to zero when the price approaches infinity. However, the presentation is simplified by this assumption. 11 " We do not use this particular payoff function. If demand is derived from utility maximisation, though, the realised (indirect) utility must be a strictly monotone transformation thereof. 12 " See Oechssler (1997) for a discussion. In Alós‐Ferrer et al. (2006), we investigate a different tie‐breaking rule. Indifferent traders randomise their platform choice, i.e., every platform is chosen with a strictly positive but not necessarily identical probability. All our results (in particular Theorem 10) are robust towards such a modification of the tie‐breaking rule. 13 " See Alós‐Ferrer (2003) for a discussion. Learning processes fulfilling B1 correspond to ‘regular’ learning processes in Alós‐Ferrer and Netzer (2007). 14 " The reason we explicitly choose Assumptions B1, B2 is that, in the literature of learning in games, many models are not robust to minute changes in the dynamic assumptions. We want to make explicit that our model is not so sensitive to the details of the dynamics. 15 " Our analysis of Markov chains as defined by the learning dynamics uses the methods and concepts introduced in Kandori et al. (1993) and Young (1993). Detailed overviews can be found e.g. in Ellison (2000), Fudenberg and Levine (1998) or Samuelson (1997). 16 " Ω0 is non‐empty as it always contains cross‐states (see (b)). 17 " We mean that an institution is picked up according to a pre‐specified probability distribution having full support. The exact distribution does not affect the results, as long as it has full support and does not depend on ɛ. 18 " In the following, whenever we say absorbing sets or states, we refer to the unperturbed dynamics. Since the perturbed dynamics is irreducible, no confusion should arise. 19 " Otherwise, market designers’ payoffs would depend on the initial distribution of the traders over the platforms. In the absence of a plausible theory on the initial distribution, the results would be arbitrary. Further, as pointed out in Remark 1, convergence to full coordination is fast and hence the assumption is, to some extent, justified. 20 " As shown in Appendix B.1, our main results do not change if we consider boundedly rational market designers who learn the same way as traders do. 21 " The grid can be assumed to be ex ante fine enough by a uniform continuity argument. 22 " For simplicity, we assume that designers randomise over S with full support. 23 " This specification also allows for different learning speeds for traders and designers, allowing for example, for the likely situation where buyers and sellers revise with a larger probability than designers. 24 " We assume for simplicity from now on that δ is such that 21/3 ∈ B. 25 " A proof of these claims and more detailed exposition of the material discussed in Appendix B is available on request. References Alós‐Ferrer , C. ( 2003 ). ‘Finite population dynamics and mixed equilibria’ , International Game Theory Review , vol. 5 ( 3 ), pp. 263 – 90 . Google Scholar Crossref Search ADS WorldCat Alós‐Ferrer , C. and Kirchsteiger , G. ( 2008 ). ‘ Learning and market clearing ’, mimeo, available at http://www.ecares.org/kirchsteiger.html. Alós‐Ferrer , C. , Kirchsteiger , G. and Walzl , M. ( 2006 ). ‘ On the evolution of market institutions: the platform design paradox ’, CEPR Discussion Paper No. 5538. Alós‐Ferrer , C. and Netzer , N. ( 2007 ). ‘ The logit‐response dynamics ’, TWI Research Paper Series No 28. Ariely , D. , Ockenfels , A. and Roth , A. ( 2005 ). ‘An experimental analysis of ending rules in internet auctions’ , RAND Journal of Economics , vol. 36 , pp. 790 – 809 . OpenURL Placeholder Text WorldCat Armstrong , M. ( 2006 ). ‘Competition in two‐sided markets’ , Rand Journal of Economics , vol. 37 , pp. 668 – 91 Google Scholar Crossref Search ADS WorldCat Belleflamme , P. and Toulemonde , E. ( 2004 ). ‘ Emergence and entry of B2B marketplaces ’, CORE Discussion Paper No. 78. Benaïm . M. Weibull , J. ( 2003 ). ‘Deterministic approximation of stochastic evolution in games’ , Econometrica , vol. 71 , pp. 878 – 903 . Google Scholar Crossref Search ADS WorldCat Binmore , K. and Samuelson , L. ( 1997 ). ‘Muddling through: noisy equilibrium selection’ , Journal of Economic Theory , vol. 74 , pp. 235 – 65 . Google Scholar Crossref Search ADS WorldCat Blume , L. ( 1995 ). ‘The statistical mechanics of best‐response strategy revision’ , Games and Economic Behavior , vol. 11 , pp. 111 – 45 . Google Scholar Crossref Search ADS WorldCat Caillaud , B. and Julien , B. ( 2003 ). ‘Chicken and egg: competition among intermediation service providers’ , RAND Journal of Economics , vol. 34 , pp. 521 – 52 . Google Scholar Crossref Search ADS WorldCat Davila , A. , Gupta , M. and Palmer , R. ( 2003).Moving procurement systems to the internet: the adoption and use of e–procurement technology models’ , European Management Journal , vol. 21 ( 1 ), pp. 11 – 23 . Google Scholar Crossref Search ADS WorldCat The Economist ( 2004 ). ‘E‐commerce takes off’ , May 13th. OpenURL Placeholder Text WorldCat Ellison , G. ( 1993 ). ‘Learning, local interaction, and coordination’ . Econometrica , vol. 61 , pp. 1047 – 71 . Google Scholar Crossref Search ADS WorldCat Ellison , G. ( 2000 ). ‘Basins of attraction, long‐run stability, and the speed of step‐by‐step evolution’ , Review of Economic Studies , vol. 67 , pp. 17 – 45 . Google Scholar Crossref Search ADS WorldCat Ellison , G. ( 2006 ). ‘Bounded rationality in industrial organization’, in ( R. Blundell, W.K. Newy and T. Persson, eds.), Advances in Economics and Econometrics: Theory and Applications , pp. 142 – 80 , Ninth World Congress, Cambridge: Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Ellison , G. and Fudenberg , D. ( 2003 ). ‘Knife‐edge or plateau: when do markets models tip?’ , Quarterly Journal of Economics , vol. 118 , pp. 1249 – 78 . Google Scholar Crossref Search ADS WorldCat Ellison , G. , Fudenberg , D. and Möbius , M. ( 2004 ). ‘Competing auctions’ , Journal of the European Economic Association , vol. 2 , pp. 30 – 66 . Google Scholar Crossref Search ADS WorldCat European Commission, Enterprise Directorate General ( 2003 ). ‘ Report of the Expert Group on B2B Internet Trading Platforms ’, final report Freidlin , M.I. and Wentzell , A.D. ( 1984 ). Random Perturbations of Dynamical Systems , New York: Springer Verlag . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Friedman , D. ( 1993 ). ‘How trading institutions affect financial market performance: some laboratory evidence’ , Economic Inquiry , vol. 31 , pp. 410 – 35 . Google Scholar Crossref Search ADS WorldCat Fudenberg , D. and Levine , D. ( 1998 ). The Theory of Learning in Games . Cambridge, MA: MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Gabaix , X. and Laibson , D. ( 2006 ). ‘Shrouded attributes, consumer myopia, and information suppression in competitive markets’ , Quarterly Journal of Economics , vol. 121 ( 2 ), pp. 505 – 40 . Google Scholar Crossref Search ADS WorldCat Gerber , A. and Bettzüge , M.O. ( 2007 ). ‘Evolutionary choice of markets’ , Economic Theory , vol. 30 , pp. 453 – 72 . Google Scholar Crossref Search ADS WorldCat Holt , C. (1995). ‘Industrial organization’, in ( J. Kagel and A. Roth, eds.), Handbook of Experimental Economics , pp. 349 – 443 , Princeton: Princeton University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Hopkins , E. ( 2007 ). ‘Adaptive learning models of consumer behavior’ , Journal of Economic Behavior and Organization , vol. 64 , pp. 348 – 68 . Google Scholar Crossref Search ADS WorldCat Kagel , J. ( 1995 ). ‘Auctions: a survey of experimental research’, in ( J. Kagel and A. Roth, eds.), Handbook of Experimental Economics , pp. 501 – 86 , Princeton, NJ: Princeton University Press, Princeton. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Kandori , M. , Mailath , G. and Rob , R. ( 1993 ). ‘Learning, mutation, and long‐run equilibria in games’ , Econometrica , vol. 61 , pp. 29 – 56 . Google Scholar Crossref Search ADS WorldCat Kandori , M. and Rob , R. ( 1995 ). ‘Evolution of equilibria in the long run: a general theory and applications’ , Journal of Economic Theory , vol. 65 , pp. 383 – 414 . Google Scholar Crossref Search ADS WorldCat Keys , S. ( 2002 ). ‘ Online trading platforms: to build or to buy? ’, mimeo, available where?. Kirchsteiger , G. , Niederle , M. and Potters , J. ( 2005 ). ‘Endogenizing market institutions: an experimental approach’ , European Economic Review , vol. 49 ( 7 ), pp. 1827 – 53 . Google Scholar Crossref Search ADS WorldCat Kugler , T. , Neeman , Z. and Vulkan , N. ( 2006 ). ‘Markets versus negotiations: an experimental investigation’ , Games and Economic Behavior , vol. 56 , pp. 121 – 34 . Google Scholar Crossref Search ADS WorldCat Lucking‐Reiley , D. and Spulber , D. ( 2001 ). ‘Business to business electronic commerce’ , Journal of Economic Perspectives , vol. 15 , pp. 55 – 68 . Google Scholar Crossref Search ADS WorldCat Ockenfels , A. and Roth , A. ( 2006 ). ‘Late and multiple bidding in second‐price internet auctions: theory and evidence concerning different rules for ending an auction’ , Games and Economic Behavior , vol. 55 , pp. 297 – 320 . Google Scholar Crossref Search ADS WorldCat Oechssler , J. ( 1997 ). ‘An evolutionary interpretation of mixed‐strategy equilibria’ , Games and Economic Behavior , vol. 21 , pp. 203 – 37 . Google Scholar Crossref Search ADS WorldCat Plott , C. ( 1982 ). ‘Industrial organization: theory and experimental economics’ , Journal of Economic Literature , vol. 20 , pp. 1485 – 587 . OpenURL Placeholder Text WorldCat Plott , C. and Smith , V. ( 1978 ). ‘An experimental examination of two exchange institutions’ , Review of Economic Studies , vol. 45 , pp. 133 – 53 . Google Scholar Crossref Search ADS WorldCat Rochet , J. and Tirole , J. ( 2003 ). ‘Platform competition in two‐sided markets’ , Journal of the European Economic Association , vol. 1 , pp. 990 – 1029 . Google Scholar Crossref Search ADS WorldCat Rochet , J. and Tirole , J. ( 2006 ). ‘Two‐sided markets: an overview’ , RAND Journal of Economics , vol. 35 ( 3 ), pp. 645 – 67 . Google Scholar Crossref Search ADS WorldCat Roth , A. and Ockenfels , A. ( 2002 ). ‘Last‐minute bidding and the rules for ending second‐price auctions: evidence from eBay and Amazon on the internet’ , American Economic Review , vol. 92 ( 4 ), pp. 1093 – 103 . Google Scholar Crossref Search ADS WorldCat Samuelson , L. ( 1994 ). ‘Stochastic stability in games with alternative best replies’ , Journal of Economic Theory , vol. 64 , pp. 35 – 65 . Google Scholar Crossref Search ADS WorldCat Samuelson , L. ( 1997 ). Evolutionary Games and Equilibrium Selection , Cambridge, MA: MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Schlag , K . ( 2004 ). ‘ Competing for boundedly rational consumers ’, mimeo, Universitat Pompeu Fabra. Spiegler , R. ( 2006 ). ‘Competition over agents with boundedly rational expectations’ , Theoretical Economics , vol. 1 , pp. 207 – 31 . OpenURL Placeholder Text WorldCat UNCTAD ( 2002 ). E-Commerce and Development Report 2002  , Geneva: United Nations. Young , P. ( 1993 ). ‘The evolution of conventions’ , Econometrica , vol. 61 , pp. 57 – 84 . Google Scholar Crossref Search ADS WorldCat Appendices Appendix A. Proofs The proofs rely on the well‐known characterisation of the invariant distribution of a Markow Chain developed by Freidlin and Wentzell (1984) and its implications for stochastic stability as discussed by Kandori et al. (1993) and Young (1993), which we briefly review here. Lemma 3.1 from Freidlin and Wentzell (1984) states the following. Fix an ω ∈ Ω. An ω−tree T is a tree in Ω with root ω, i.e. a graph on Ω such that for every state ω′ ≠ ω there exists a unique directed path from ω′ to ω. Let be the set of all ω−trees and define (i.e. q is the product of all transition probabilities on a given ω−tree summed over all ω−trees). Then, the invariant distribution for fixed ɛ is given by μ(ω) = q/∑ω∈Ωq. Relying on this result, Kandori et al. (1993) or Young (1993) observe that in the limit as ɛ→0, μ*(ω) is determined by those ω−trees that imply the smallest possible number of mutations necessary to form a tree in Ω. Given two states ω and ω′, let c(ω,ω′) (the transition cost from ω to ω′) denote the minimal number of experiments necessary for a transition (or link) from ω to ω′ along a positive probability path starting in ω and leading to ω′. The cost of an ω−tree is the sum of all costs along links in it. Let γ(ω) (the stochastic potential of ω) be the minimal cost of an ω−tree. A state ω is stochastically stable if and only if its stochastic potential is minimal, i.e. γ(ω) ≤ γ(ω′) for all ω′ ∈ Ω. Let A be the set of absorbing sets. If X ∈ A, all states in X have the same stochastic potential, denoted γ(X ). Proof of Lemma 1 (a) Let β1 > 1 and β2 > 1. Monomorphic states are absorbing because at the corresponding platform both buyers and sellers make strictly positive profits and the other platform is inactive. Thus traders stay at the active one. Moreover, elements of Ω0 are absorbing because traders do not switch if the respective profits are identical on both platforms. It is now enough to show that there exists a positive probability path from any other to a monomorphic state or a state in Ω0. Consider . At least one platform has to be active. If only platform i is active, the monomorphic state is reached with positive probability because buyers and sellers receive positive profits at platform i and zero profits at platform j. Hence, suppose that both platforms are active. Suppose that p(si) ≠ p(sj) and (without loss of generality) that p(si) < p(sj). Then, buyers strictly prefer platform i to platform j. By B1 and B2, there is positive probability that all buyers at j receive revision opportunity in successive periods and only sellers at platform j receive revision opportunity. Then, buyers will switch away from j and no new seller will switch to j. Thus, either the monomorphic state is eventually reached, or j becomes inactive Now suppose that p(si) = p(sj). Buyers are indifferent and will never switch. As ω ∉ Ω0, sellers prefer one platform and there is a positive probability path to a state with an inactive platform or a state in Ω0. (b) Let βj = 1 and βi > 1. Monomorphic states are absorbing, because at the corresponding platform both buyers and sellers make weakly positive profits and the other platform is inactive. Thus traders do not switch. Cross states are absorbing because traders do not switch if profits are identical on both platforms. Finally, sellers receive zero profits at platform j. Hence, they are indifferent between an inactive platform i and platform j, so they never switch. Buyers strictly prefer an active platform j to an inactive platform due to strictly positive profits. Therefore, all states in are absorbing. It remains to show that there exists a positive probability path from any to a monomorphic state or a state in . In such a state ω, at least one platform has to be active. If only platform i is active, buyers and sellers strictly prefer i to j and is reached with positive probability. If only platform j is active, buyers strictly prefer j to i while sellers do not switch at all. Hence, a state in is reached with positive probability. If both platforms are active, sellers strictly prefer i and, by B1 and B2, there is positive probability that all sellers at j but only buyers at i receive revision opportunities in successive periods. Hence, sellers will switch away from j and no new buyer will switch to j. Thus, either the monomorphic state is eventually reached, or j becomes inactive. (c) Let βi = βj = 1. Then, sellers never switch. Suppose first that p(si) ≠ p(sj). Buyers strictly prefer an active to an inactive platform and are indifferent between two inactive platforms. Hence, states in and are absorbing. It is enough to show that there exists a positive probability path from any state outside to a state inside this joint set. Consider . In ω, at least one platform has to be active. If only one is active, buyers strictly prefer this platform to the other. Hence, a state in or (depending on which was the active platform) is reached with positive probability. If both platforms are active, suppose without loss of generality that p(si) < p(sj). Then, buyers strictly prefer platform i and there is a positive probability path to an element in . Consider now the case p(si) = p(sj). States in and are absorbing, because buyers strictly prefer an active to an inactive platform. Also, every state with two active platforms is absorbing, because buyers are indifferent between active platforms. Consider a state ω with exactly one active platform (i, say) which is not in . Then, buyers strictly prefer i to j and there is a positive probability path to a state in . (d) Let βi > 1 and βj < 1. Then, platform j is always inactive. If platform i is active, buyers and sellers strictly prefer platform i to platform j. Hence, is absorbing. If platform i is inactive, buyers and sellers do not switch at all. Hence, every state with an inactive platform i is absorbing. To complete the proof consider a non‐monomorphic state ω with an active platform i. Then, buyers and seller strictly prefer platform i to platform j and there is a positive probability path to . (e) Let βi = 1 and βj < 1. Then, sellers never switch and platform j is always inactive. If platform i is active, buyers strictly prefer i to j. Hence, every state in is absorbing. If platform i is inactive, buyers and sellers do not switch at all. Hence, every state with an inactive platform i is absorbing. Consider a state with an active platform i. Then, buyers strictly prefer platform i to platform j and there is a positive probability path to a state in . (f) If βi, βj < 1, neither buyers nor sellers switch, hence every state is absorbing. Proof of Theorem 2 (a) For βi < 1, no trade occurs at platform i independent of the number of buyers or sellers at platform i. (b) Let X ∈ A, then γ(X) ≥ |A| − 1 as at least one mistake is needed for a transition between any two absorbing sets. A transition between any two absorbing states ω, ω′ with ‖Mi(ω)| − |Mi(ω′)‖ + ‖Ni(ω)| − |Ni(ω′)‖ = 1 is possible with one mistake. Also a transition from a cross‐state to is possible with one mistake: Consider and a buyer who (by mistake) switches to platform j. As platform i is inactive and buyers receive strictly positive profits at platform j, by Assumptions B1 and B2 there is a positive probability path of the unperturbed dynamics to . Hence γ(X) = |A| − 1 for all X ∈ A and every state in A is stochastically stable. (c) Let βj = 1. The absorbing states are and the states in . A transition from to an element in needs at least two mistakes because buyers and sellers receive strictly positive profits at i. Hence, γ(ω) > |A| − 1 for . A transition between any two states with ‖Mj(ω)| − |Mj(ω′)‖ = 1 is possible with one mistake. A transition from the cross‐state to is possible with one mistake: Consider and a buyer who (by mistake) switches to platform i. As platform j is inactive and buyers receive strictly positive profits at platform i, by Assumptions B1 and B2 there is a positive probability path of the unperturbed dynamics to . In the same way, one can also construct a positive probability path (with one mistake by a seller) from to . Hence, and is the only stochastically stable state. Let βj < 1. Then, A consists of and the states without active platforms. A transition from to a state with two inactive platforms needs at least two mistakes because buyers and sellers receive strictly positive profits at i. Hence, γ(ω) > |A| − 1 for . A transition between any two states with ‖Mi(ω)| − |Mi(ω′)‖+‖Ni(ω)| − |Ni(ω′)‖ = 1 is possible with one mistake. A transition from the cross‐state to is possible with one mistake as well, exactly as in the case βj = 1. Hence, and is the only stochastically stable state. Proof of Lemma 3 Follows directly from ∀ω, ω′ ∈ Ω which holds for si < sj due to platform symmetry (recall Assumption A). Proof of Lemma 4 If pi < pj, buyers strictly prefer platform i to platform j whenever it is active. Lemma 1(a) implies that the states , , and form the only absorbing sets. A single experiment suffices for a transition from a cross state to a monomorphic state: consider without loss of generality and suppose a buyer switches (by mistake). Then, platform j is active and platform i is not. Buyers and sellers strictly prefer platform j and there is a positive probability path to . In contrast, at least two experiments are necessary for a positive probability path from a monomorphic state to another monomorphic state or a cross‐state: consider without loss of generality . As long as only one trader switches (by mistake), platform j remains inactive and platform i is strictly preferred by buyers and sellers. As a consequence, , , , and . There is a positive probability path with two experiments from to : Consider and suppose that a buyer and a seller switch (by mistake) to platform i. Then, platform i is active and buyers strictly prefer platform i to platform j. By Assumptions B1 and B2, there is a positive probability that only buyers and sellers at j receive revision opportunity. But then, is reached with positive probability. Hence, and which shows Part (a). Part (b) follows from . Proof of Lemma 5 By the proof of Lemma 4, if βi, βj > 1 and pi < pj. By Lemma 1(a), only the monomorphic states can be stochastically stable. Hence, a necessary and sufficient condition for the stochastic stability of is . Since pi < pj, buyers never switch to platform j as long as i is active. Hence, has to be reached through switching of all sellers to platform j and a subsequent switch of all buyers to the only remaining active platform. In case (a), under independent inertia there is positive probability that all sellers at platform j simultaneously receive the opportunity to revise. If one seller and buyer are already present at platform j, sellers will switch to j if Hence, this condition is sufficient for the stochastic stability of . To see that it is also necessary suppose that it is violated. Then no seller will switch to j after one seller and any buyer induced trade on this platform. As a consequence, more than 2 experiments are needed to reach . Consider part (b) (asynchronous learning). Suppose one seller and buyer switch to platform j by mutation. By B1 and B2, with positive probability in the subsequent rounds only sellers and buyers at platform i receive the opportunity to revise. If it follows that for all Mj with 1 ≤ |Mj| ≤ |M| − 1. Hence, sellers prefer platform j whenever it is active and there are at least one and less than |M| sellers already there. Thus, there is a positive probability path with just two mutations from to where first all sellers move to platform j and subsequently all buyers switch to j as it is the only active platform. Hence, the condition displayed in the Lemma is sufficient for the stochastic stability of . To see that it is also necessary, suppose it is not fulfilled. Then a seller at platform i prefers to stay there if all other sellers are at platform j together with any buyer . Under asynchronous learning this implies that at least a third mutation is needed to reach , which implies that this state cannot be stochastically stable by Theorem 2(a). Proof of Lemma 6 Follows from the fact that trade is never possible on platforms with βi < 1 (see Theorem 2(a)). Proof of Lemma 7 (a) follows from Lemma 3 as the price and the traded quantity in ω at platform i are identical to those in at platform j. (b) follows from Theorem 2(b). (c) follows from Theorem 2(c). Proof of Proposition 8 For β < 1 the profits for a monopolistic designer are zero, whereas for β ≥ 1 and for 0 < f < 1 the profits are strictly positive. Hence, β* ≥ 1. Now assume for a moment that β and f are continuous variables with f ∈ [0,1] and β ∈ [1,∞). Denote p = βc/(1−f) and recall that limp→∞dn(p)p = 0 for all n ∈ N. Hence, it must hold that 0 < f * < 1. Differentiating the designer’s profits yields (for β ≥ 1) where Let the optimal price be p* = β*c/(1−f  *). Since 0 < f  * < 1, the first order conditions for the designer’s optimum imply that thus . This implies that hence the designer’s profits are maximised at the corner solution β* = 1. In our model β and f are not continuous variables. However, if the grid of feasible fees is fine enough, the optimal fee approximates the one of the continuous case, and hence the optimal β is 1 also in the discontinuous case. Hence we conclude that a monopolistic market designer would introduce a market clearing platform. Proof of Lemma 9 Assume to the contrary that there exists a pure strategy with and . By Lemma 6 this pure strategy gives designer i a profit of zero against all strategies of j. Hence, and, since is an equilibrium strategy, . Suppose that, in equilibrium, j chooses only platforms with βj < 1. That is, βj < 1 for all sj = (βj,fj) ∈ S with . If designer i chooses with certainty a platform with , Lemma 7(c) implies that . Since , this contradicts that is a Nash equilibrium. Thus, there must exist an with such that . Then, if designer i deviates to the pure strategy , . Since by Lemma 7(a) E, we conclude that , again contradicting that is a Nash equilibrium. Proof of Theorem 10 By the previous Lemma, only platforms with β ≥ 1 will be designed in equilibrium. Assume by contradiction that there exist some pure strategies si = (βi, fi) with and βi = 1. Denote a strategy of this type by and let . Denote the carrier or support of by . Let πD,i(si, σj) denote the expected payoff if designer i chooses si for sure and j chooses the probability distribution σj. Suppose that, for all we actually had that βj > 1. This implies by Lemma 7(c) that . Take any . By Lemma 7(a), and recalling that πD,i(si , sj) ≥ 0 for all si,sj, we obtain that Hence, player i would have an incentive to deviate from , a contradiction. We conclude that there exists some with βj = 1. Let . Notice that, since , we have by Lemma 7(c) that for all with βj > 1. Then, by Lemma 7(a, b), However, for any with and , due to Lemma 7(c) (the inequality follows from the fact that for all sj). This latter expression is continuous in . Thus, for approaching one from above, . Hence, if the grid is fine enough21 player i has an incentive to deviate from to an institution with but close enough to 1, a contradiction. Proof of Proposition 11 Lemma 5 (a) implies that if βic/(1−fi) < βjc/(1−fj), is stochastically stable iff (A.1) Part (b) follows from (a). To see (a), assume to the contrary that β = 1 + δ for all platform characteristics in the support of and . Denote by and the highest fee of a platform in the support of and , respectively. Without loss of generality assume that . We can distinguish three cases: (i) : Condition (A.1) shows that full coordination on platform is not stochastically stable vis‐à‐vis any platform characteristics in the support of . Hence, this strategy earns designer j zero profits and, since it is assumed to be in the support of j’s equilibrium strategy, j’s equilibrium profits would be zero. But j could always guarantee himself a strictly positive profit by playing the same (possibly mixed) strategy as i. Hence, case (i) is inconsistent with Nash equilibrium. (ii) . Condition (A.1) shows that full coordination on platform is not stochastically stable vis‐à‐vis any platform characteristics in the support of but platform . Furthermore, Lemma 3 implies that if is chosen by i and is chosen by j. Therefore But choosing the alternative platform design with , and implies that Hence, again by (A.1) if is chosen by i and is chosen by j, which yields If the grid of F is fine enough, i.e. if γ is small enough, this implies , a contradiction with Nash equilibrium. (iii ) – both designers choose for sure. Then Condition (A.1) guarantees the existence of a and a such that platform j is stochastically stable vis‐à‐vis if the grid F is sufficiently fine, i.e. if γ is sufficiently small. Furthermore, if ɛp is not much larger than 1, designer j’s profits from full coordination on his platform with design , is strictly larger than the respective profit from choosing , It remains to show that no decrease in overcompensates this effect. To see this suppose that in such a way that (feasible if δ is sufficiently small). Then sellers prefer platform j with characteristics whenever it is active, while buyers prefer platform i with characteristics . With independent inertia and |M| = |N|, this establishes symmetry of the transition matrix P and hence . Therefore, choosing in this case and does not reduce while it strictly increases revenue in . Hence, with and can not be a best response to . Appendix B. Robustness of the Results We have derived our results under two crucial assumptions. First, designers are assumed to be rational while traders are not (asymmetric rationality). Even though this assumption seems to be justified in a wide range of applications, one might be interested in the robustness or our results with respect to the (bounded) rationality of designers. We discuss the case of learning designers in Section B.1. Second, we assumed that sellers have a constant‐returns‐to‐scale technology. In Section B.2. we analyse an example with decreasing returns that illustrates the robustness of our findings. B.1. Boundedly Rational Designers To account for learning designers, we have to extend the state space by the feasible design configurations, and we have to redefine the (unperturbed learning process). The state space is given by Ω = {1,2}n × {1,2}m × S2. A state ω ∈ Ω denotes the location of buyers and sellers and the design of both platforms. Traders learn according to Assumption A. The learning process of designers is defined as follows. Assumption C. A designer who gets the opportunity to revise, observes the revenues and designs of platforms in the last period. If revenues differ, he chooses the design which led to a higher revenue (imitation). If both platforms generate zero profits, he randomises with positive probability in the next round over all possible design alternatives (innovation).22 In case of identical positive profits at both platforms designers stick to their former choice (inertia). We further assume B1 and B2 (on the enlarged state space and for three instead of two different types of players).23 The perturbations are defined as in Section 2.1. The perturbed process is again irreducible. We now prove the counterpart of Theorem 2 in the modified learning model. Proposition 12. Assume A, B1, B2, and C. Then, βi > 1 for i = 1,2 in every stochastically stable state. Proof. Throughout this proof, we adopt the convention that for a given platform i the other platform is denoted by −i. First, observe that the monomorphic state with βi ≥ 1 (and β−i = βi) is an absorbing state of the unperturbed process. Second, each cross‐state with designers randomising over all designs forms a (non‐singleton) absorbing set. A platform i with a positive number of buyers and sellers cannot have a price bias βi < 1 in any absorbing set. To see this, consider a state ω with Ni(ω) ≠ φ and Mi(ω) ≠ φ and βi < 1. Case 1: Suppose β−i ≥ 1 and platform −i is active. Then, there is a positive probability that i imitates −i (inducing βi ≥ 1) without any migration of buyers and sellers. Both platforms yield positive profits and have the same design. For this case Assumption C implies that designs can only change if a cross state is reached. But a cross state (with randomising designers) forms an absorbing set. Hence, the unperturbed process never reaches a state with βi < 1 and a positive number of buyers and sellers at i again. Case 2: Suppose β−i ≥ 1 and platform −i is inactive or β−i < 1. Then, both platforms generate profits of zero and designers randomise. With positive probability, platform i will have βi > 1 and generate positive profits for designers, buyers and sellers while β−i < 1. Then, with βi > 1 is reached with positive probability. Moreover, a state where both platforms are active and one has bias βi > 1 and the other bias β−i = 1 cannot be part of an absorbing set. To see this, observe that from such a state there is always a positive‐probability path to because sellers strictly prefer platform i and there is a positive probability that they are the only ones with an opportunity to revise for sufficiently many periods. Hence, absorbing sets are of three kinds. First, states where all active platforms have price biases β > 1. Denote the set of such absorbing sets by A0 and the set of all other absorbing sets by A1. Second, states where either two active platforms have price bias β = 1, or the only active platform has price bias β = 1. Last, cross states where no platform has a positive number of buyers and sellers. To prove the Proposition, we compare the stochastic potential of absorbing sets in A0 and A1. First, observe that it takes at least two mistakes to leave an absorbing set in A0. If only one platform i is active, the corresponding absorbing state will be . Because both types of traders receive positive profits, mistakes by both types of traders are needed to induce another active platform. It takes also more than one mistake to reach a cross state. If both platforms are active, designers generate positive profits and designs will only change to β ≤ 1 if a cross state is reached (inertia and imitation do not lead to platforms with β ≤ 1; and designers innovate only in cross‐states). However, from a state with two active platforms, it takes more than one mistake by traders to reach a cross‐state. If there is only one active platform i, with bias βi = 1, then platform −i also has β−i = 1 due to imitation. At i all buyers and at least one seller are assembled. The different absorbing states differ only with respect to the number of sellers and can be connected to a (restricted) tree with one mistake per absorbing set (see the proof of Theorem 2). This tree can be connected to the cross states with one mistake. If two active platforms have price bias β = 1, prices have to be the same at both platforms and sellers do not make any profits on both platforms. The respective absorbing sets can be connected with one mistake (i.e., a switching seller) per absorbing set and the resulting tree can again be connected to the cross states with one mistake. Hence, there is a tree with root in some cross state connecting all absorbing sets in A1 with one mistake per absorbing set. But a cross state can be left with one mistake towards a monomorphic state in A0 (due to innovation, there is a positive probability that there is a platform i with βi > 1 and, with one mistake, this platform becomes active and is strictly preferred by all traders). Denote by the state which minimises the number of mistakes needed to form a tree restricted to A0. For a tree in Ω, the minimal cost of a ‐tree is where is the minimal number of mistakes needed to connect all states in A0 to the respective ‐subtree. Now consider a state . To construct a minimal‐cost ‐tree, we take the ‐tree, delete the outgoing link of (which has cost one), and add a (least‐resistance) link from to a state in A1 (of cost larger than one). Hence, . In every stochastically stable state traders are therefore located at a platform i with βi > 1. We conclude that boundedly rational platform designers exhibit qualitatively the same behaviour as rational ones: Platform competition forces them to introduce non‐market clearing platforms only. B.2. Decreasing Returns to Scale Consider the following example. Two identical sellers produce with costs given by . For given prices (pi) and fees (fi) at a platform i, their profit is and maximisation leads to the supply function s(pi) = (1−fi)pi. Two identical buyers, each with income of one unit, consume q units of the commodity traded at the platforms and x units of a second commodity which price is normalised to 1. The buyers’ utility is given by and utility maximisation for a given price pi at the respective platform yields the buyer’s demand function . Equating demand and supply gives the market clearing price at platform i in state ω (with ri(ω) = |Ni(ω)|/|Mi(ω)|). Traders’ and designers’ profits depend on state and design and are calculated the same way as before. For our purposes it suffices to note that sellers are not rationed whenever βi ≤ 1 and their corresponding profit is increasing in βi and decreasing in fi. Analogously, sellers are rationed for βi > 1 and profits amount to which is also monotonically decreasing in fi but reaches a (global) maximum at βi = 21/3.24 For expositional ease we further specify the learning model and assume independent inertia within types. That is, in every round any seller m ∈ M is allowed to revise his location decision with probability ρS ∈ ]0,1[ while every buyer is allowed to revise with probability ρB ∈ ]0,1[. Full coordination on any platform is an absorbing state (both types of traders get strictly positive profits on any active platform). Analogously to Lemma 1, it is easy to see that the monomorphic states and the states in Ω0 = {ω | πm,i = πm,j,πm,i = πm,j} form the only absorbing sets. Moreover, only the monomorphic states can be stochastically stable. The designers’ profits, though, depend not only on the support of the limit invariant distribution μ* but also on its weights for the different (monomorphic) states. Hence, analysing this setting requires a direct application of Lemma 3.1 in Freidlin and Wentzell (1984). To economise on notation, identify a state with a pair (k, l ) where k is the number of selelrs and l the number of buyers present at platform i. Let and . It is easy to see that a minimal‐cost tree in is as follows. State (1,1) is connected to , with transition probability . States (0,0), (0,1), and (1,0) are connected to (1,1) with transition probabilities P((0,0),(1,1)) = ɛ2, P((0,1),(1,1)) = ɛ(1 − ρB)2, and P((1,0),(1,1)) = ɛ(1 − ρS)2. All other states are directly connected to , with transition probabilities , , , and . Hence, the product of transition probabilities in this tree is . The probabilities for can be derived by a permutation of indices B and S, thus the corresponding product is . In the limit, is determined by the quotient of these two products, which simplifies to . This leads to the following useful result (as sellers and buyers are identical profits only depend on the number of sellers and buyers at a platform). Lemma 13. Supposeπm,i(1,1,si) > πm,j(1,1,sj). Then for everyκ > 0there is a such that for all . Proof. It is easy to see that is the only stochastically stable state if and only if πm,i(1,1,si) > πm,j(1,1,sj) and πn,i(1,1,si) > πn,j(1,1,sj) (as it then needs more than 2 mistakes to get from to ). If πn,i(1,1,si) = πn,j(1,1,sj), and . If πn,i(1,1,si) < πn,j(1,1,sj), and . Hence, in both cases approaches zero if ρS→ 1 and . Proposition 14. There exists a such that, for all , the platform profile with and is a strict Nash equilibrium. Proof. As sellers’ profits decrease in fi and reach their global maximum at β = 21/3 it is clear that for any . Suppose both and some institution are available, and let ω be the state where all traders are at si. By Lemma 13, for any κ > 0 there exists such that μ* (ω) < κ for all . Hence, for κ small enough (and since there are finitely many strategies), we obtain for all ; thus is a strict Nash equilibrium. Intuitively, if sellers learn much faster then buyers, only the platform that offers higher revenues to sellers will survive with a positive probability if both platforms are active. This induces the following strict Nash‐Equilibrium. It can be shown that, for ρS large enough, βi ≤ 1 is not chosen by any designer in any pure strategy equilibrium. Moreover, in any mixed strategy equilibrium there is a least one designer i where implies that βi > 1.25 Note also that the condition on ρS is a sufficient but not a necessary one. For a smaller ρS we cannot characterise the limit invariant distribution and hence do not know the equilibrium behaviour of the designers. The result might hold even for , depending on the details of the demand and supply conditions. These results illustrate that the ‘Platform Design Paradox’, i.e. the fact that competition between market designers might lead to the design of non‐market clearing institutions, also appears in the case of decreasing returns to scale. Author notes " We are grateful to Dan Friedman, Fei Shi, seminar participants in Berlin, UL Brussels, University College London, Louvain‐la‐Neuve and Maastricht, and an anonymous referee for helpful comments. The first author acknowledges funding by the Austrian Science Fund (FWF) under Project P18141‐G09, the second author by the Banque Nationale de Belgique and the third author by the Dutch Science Foundation (NWO). © The Author(s). Journal compilation © Royal Economic Society 2009

Basu, Arnab, K.;Chau, Nancy, H.;Kanbur,, Ravi

doi: 10.1111/j.1468-0297.2009.02298.xpmid: N/A

Abstract In many countries, non‐compliance with minimum wage legislation is widespread and authorities may be seen as having turned a blind eye to legislation they have themselves passed. We show that turning a blind eye can indeed be an equilibrium phenomenon with ex post credibility, in a model of minimum wage policy with imperfect competition, imperfect enforcement and imperfect commitment. Since credible enforcement requires costly ex post transfer of income from employers to workers, a government concerned only with efficiency but not with distribution is shown, paradoxically, to be unable to credibly elicit efficiency improvements via a minimum wage reform. The comparative statics of a minimum wage have inspired a vast empirical literature and vigorous policy debates. The tradeoffs associated with a minimum wage hike are typically articulated in efficiency and equity terms, depending in particular on the competitiveness of the labour market.1 Arguments based on a standard competitive model of the labour market imply sharp efficiency and equity tradeoffs, as employment is predicted to fall with a well‐enforced and binding minimum wage.2 In contrast, if the relevant frame is of the monopsonistic variety, predicted employment response runs in the opposite direction, so long as the minimum wage is not too high (Stigler, 1946). Consequently, both efficiency and equity improvements may be brought about at once provided the minimum wage is ‘skillfully‐set’3 and perfectly enforced. A standing assumption in both these archetypal settings is perfect enforcement, and by implication, full compliance with the minimum wage. This assumption is at odds with a growing body of empirical evidence however, which finds non‐compliance with minimum wage legislation to be widespread. Specifically, the pioneering work of Ashenfelter and Smith (1979) found compliance rate in the US in 1973 to be around 65%.4 Non‐compliance is also found to prevail in an accumulating list of developing countries, including, for example, Brazil (Lemos, 2004, 2006), Costa Rica (Gindling and Terrell, 1995), Honduras (Gindling and Terrell, 2006), Indonesia (Harrison and Scorse, 2004), Mexico (Bell, 1997), Peru (Baanante, 2005), Trinidad and Tobago (Strobl and Walsh, 2001), and a selection of Latin American countries (Maloney and Nunez, 2004).5 Evidently, not only is it the case that compliant and non‐compliant employers co‐exist, there are also broad ranges of non‐compliance, which come typically in the form of a spike at the official minimum, alongside a dispersion of subminimum wages in covered sectors.6 This emerging evidence underscores that the legislated wage floor and the intensity of enforcement are two indispensable arms of a minimum wage policy. Meanwhile, the same evidence also raises two issues that have so far evaded rigorous scrutiny. First, can a simple deviation from perfect to imperfect enforcement alone be sufficient to overturn the predicted impacts of a minimum wage hike when non‐compliance is now a genuine possibility? Equally important, and backtracking one step, what are some of the reasons behind the pairing of lax enforcement but a high minimum wage, enough to provoke non‐compliance to begin with? More formally, the first question deals with the comparative static properties of a minimum wage hike at a given but less than perfect level of enforcement. The second deals instead with the issue of endogenous enforcement and questions the underlying determinants of enforcement imperfection. Answers to these questions contribute to the minimum wage policy debate in a number of ways. First, we study the full set of equilibrium labour market implications of an imperfectly enforced minimum wage hike in an imperfectly competitive labour market,7 when labour supply is taken to respond endogenously to the extent of enforcement imperfection, given any minimum wage.8 Whether employment response to a rise in minimum wage should be expected to be (i) positive within the standard range, consistent with the familiar monopsonistic frame with perfect enforcement, or (ii) negative, thus consistent with a competitive labour market response also with perfect enforcement, or (iii) muted because precisely of enforcement imperfection, are clearly key but nevertheless open questions.9 Second, while there has been extensive discussion on the efficiency and equity tradeoffs associated with a perfectly enforced minimum wage (Freeman, 1996; Fields and Kanbur, 2007), a symmetric treatment of the tradeoffs associated with, and thus some of the underlying determinants of the enforcement of such a wage has not received equal attention. This is despite the fact that enforcement has been noted to differ widely between developed and developing countries (Neumark and Wascher, 2007). Within countries, enforcement is also known to differ across employers in different geographical locations (Ashenfelter and Smith, 1979) and in different industries (Weil, 2004). Further, the familiar distinction between covered and uncovered sectors (Maloney and Nunez, 2004; Gindling and Terrell, 2006) may also effectively be seen as a legislated distinction between sectors where there may be some enforcement, and others with no enforcement by design. As a third contribution to the minimum wage debate, the combination of imperfect and endogenous enforcement can open up new ways to understand how the labour market responds, or fails to respond to minimum wage legislation, depending ultimately on whether the minimum wage is expected to be enforced ex post. The potential insights that this combination can yield was first pointed out by Ashenfelter and Smith (1979) in the context of the minimum wage provisions of the Fair Labour Standards Act in the US. Somewhat unexpectedly, the study finds that compliance rates were higher in the southern states of the US where wages were typically lower, while higher compliance rates also prevailed among employers of female workers compared to males. This counterintuitive finding can indeed be understood, as Ashenfelter and Smith (1979) reasoned, by recognising that government compliance efforts were either concentrated in handling reports of actual violations, or were otherwise devoted to the inspection of sectors where the potential for violation was the greatest. The result was a skewed enforcement resource allocation, with added weight put towards sectors where violations are in fact prevalent. With rational expectations, this anticipated bias should, in turn, be expected to influence equilibrium compliance and employment responses to a minimum wage.10 Based on these observations, in this article we develop an incentive compatible equilibrium model of a minimum wage policy, incorporating imperfect competition and imperfect enforcement of the minimum wage. We find this setting to yield results that are consistent with the stylised facts already noted. For the same minimum wage policy, there can be co‐existence of compliant and non‐compliant employers; a clustering/spike of employer types that uniformly comply; and a dispersion of firm‐specific equilibrium subminimum wages.11 With respect to our first question at the outset, having to do with how employment response to a minimum wage may change with an imperfect but fixed, as opposed to a perfect level of enforcement, we find that the possibilities run the gamut from no change at all, to a class of cases where there is a sharp reversal in sign from positive to negative, and then further to cases where there is a muted response. We show that each of these cases can prevail within well‐defined ranges of minimum wages and enforcement intensities. In addition, these minimum wage thresholds and enforcement intensities are themselves specific to the characteristics of the labour market in question, including demand and supply side parameters. We then turn to our next question and consider the decision of a planner who is at liberty to choose a minimum wage and a level of enforcement12 and who harbours a variable degree of concern for efficiency versus distribution. The concern for efficiency addresses underemployment in the face of an imperfectly competitive labour market, while the concern for distribution addresses earning shortfalls relative to the minimum wage on the part of workers attached to non‐compliant employers, along with those who are unemployed. In view of the documentation of ex post complaints‐driven enforcement resource allocation as in Ashenfelter and Smith (1979), we contrast the case of commitment, where both the minimum wage and enforcement levels are fixed ex ante, with the case of discretion, where the choice of enforcement intensities is determined ex post, depending in particular on whether there is in fact non‐compliance with the minimum wage. Interestingly, given our formulation of a social welfare function combining both efficiency and distributional concerns, we find that non‐compliance can be a rational expectation equilibrium with ex post discretion but not with commitment. Commitment rules out non‐compliance in our setting because if non‐compliance is expected for any chosen level of enforcement, the minimum wage can always be adjusted downwards to alleviate the scale of any earnings shortfall. Meanwhile, full compliance cannot be a rational expectation equilibrium with discretion, because the ex post optimal level of costly enforcement will certainly be nil if there is literally nothing to enforce, which is of course the case when there is in fact full compliance. Finally, we show that the endogenous level of enforcement subject to ex post credibility exhibits a number of interesting characteristics. First, credible enforcement is indeed need‐based, in the sense that all else equal, enforcement will be higher when the incidence and severity of non‐compliance is likely the greatest, due, for example, to low labour productivity. Ex post enforcement also rises with the minimum wage, provided that the planner espouses a sufficiently high degree of concern for distribution. Put simply, a government more concerned about the earnings shortfall relative to the minimum wage will have a higher ex post incentive to enforce. Finally, since ex post enforcement of a minimum wage is but a costly income transfer from a non‐compliant employer to workers, a planner who cares only about efficiency, and who attaches no intrinsic value to the earning shortfalls of workers relative to the minimum wage, will be rendered least capable of enforcing a minimum wage in a rational expectation equilibrium. In this setting where a higher minimum wage can be used to raise employment and improve efficiency, we end with an intriging result: a planner who cares only about efficiency cannot credibly elicit efficiency‐improving minimum wage reforms. 1. The Model Consider an employer who draws labour input from a population of heterogeneous workers and who possesses control over wages and employment within this population. The associated revenue is R(ℓ) = (a − bℓ/2)ℓ, where ℓ denotes the number of workers employed, and a > 0, and b ≥ 0 are technological parameters respectively capturing labour productivity, and diminishing marginal product. The implied inverse labour demand schedule is therefore of the form Workers differ according to a mobility cost of employment t ∈ [0,T ],13 and are distributed uniformly along the [0,T] interval. The utility of a worker with mobility cost t and employed at wage w is u(t,w) = w − t. The reservation utility of every worker is given by . Labour supply facing the employer, at given wage offer w, is accordingly made up of the sum of the individual labour supplies from workers with mobility cost not high enough to deter them from employment , or . This implies an inverse labour supply schedule of the familiar form: Two benchmarks can now be singled out, respectively the competitive outcome associated with a wage‐taking employer and the monopsonistic outcome as in Stigler (1946).14 So long as the reservation utility is not too high, , the competitive outcome is given by {w*,ℓ*}: (1) where the marginal revenue product of labour coincides with the prevailing wage wd(ℓ*) = w* = ws(ℓ*). Thus, there is less than full employment () if and only if mobility costs are large enough: . Henceforth, we focus on labour markets in which this mobility cost driven lack of full employment is a genuine concern, and assume that . Now let W(ℓ) ≡ ws(ℓ)ℓ denote total labour cost, and be the associated marginal labour cost. The monopsonistic labour market outcome {wdo,wso,ℓo} is (2) with strict inequalities whenever τ > 0. Thus, a strictly positive mobility cost and asymmetric bargaining power favouring the employer jointly implies that equilibrium marginal revenue product (wdo) strictly exceeds the corresponding equilibrium wage (wso), as the employer takes advantage of per worker wage savings that come only with lower employment.15 The result is a lower level of employment compared to the competitive benchmark ℓo < ℓ* and unemployed workers constitute a select group with some of the highest mobility costs. 2. Minimum Wage with Imperfect Enforcement A minimum wage policy is the combination of a minimum wage and an enforcement intensity based on a likelihood λ of inspection and discovery. The timing of the policy is: The government announces and commits to and λ.16 Employment and wage decisions are made . Employer inspections are carried out with likelihood λ. If the employer chooses not to comply, a penalty equaling the shortfall , to be transferred to the worker, follows if inspection occurs. Otherwise, both the worker and the employer will be unaffected; Workers strictly prefer receiving the minimum wage directly from a compliant employer as wage payment up front, as opposed to receiving any shortfall as settlement ex post. The transaction cost expended in the process of wage settlement upon inspection is given by a fraction 1 − σ ∈ (0,1] of the settlement, , forgone.17 For employers, imperfect enforcement implies three classes of options: over‐compliance, exact compliance and non‐compliance. Imperfect enforcement also implies that the income of workers attached to a non‐complying employer now depends explicitly on enforcement and discovery. We consider each of these in turn. 2.1. Imperfect Enforcement and Labour Supply Let be the maximal labour supply corresponding to the minimum wage . Given λ, the expected utility of a worker facing a subminimum wage offer w is where and w are respectively labour income with and without inspection, while λ σ denotes the transaction cost adjusted intensity of enforcement. The adjusted weight λ σ is strictly less than λ itself whenever σ < 1 and depends jointly on the enforcement intensity and the cost wage settlement σ. Comparing and the reservation utility , the corresponding enforcement adjusted labour supply and inverse labour supply ( and ) schedules are: (3) (4) for a non‐compliant employer with for (3) and thus for (4). In contrast, a compliant employer and his hired workers are unaffected by inspection. A worker’s expected utility is thus w − t as before. Meanwhile, labour supply reduces to ℓs(w), and to for a compliant employer. From (3), enforcement adjusted labour supply exceeds the unregulated benchmark at constant contracted wage w, whenever there is a positive likelihood of income gains subsequent to employer inspection λ σ > 0. Also from (3), either an increase in , or an increase in λ further increases labour supply at given subminimum wage w. Thus, imperfect enforcement gives rise to shifts in the labour supply in response to changes in the minimum wage policy whenever there is non‐compliance, to be accounted for in the employer’s decision problem below. 2.2. Imperfect Enforcement and Expected Labour Cost The expected profit of the employer is: (5) where the last expression reflects the per worker wage cost conditional on inspection. For a non‐compliant employer, . Using (4), expected profit in (5) simplifies to (6) where ψ adjusts the weight given to the minimum wage as part of the expected labour cost per worker by accounting for (4). Further, ψ is less than λ itself, lies between zero and unity, and is monotonically increasing in λ: In contrast, for an employer that over or exactly complies, expected profit is simply (7) Taken together, expected labour cost and the corresponding expected marginal labour cost where the derivative exists, , are given by (8) (9) Figure 1 illustrates. As shown, is increasing and piecewise continuous in ℓ. Furthermore, for a non‐complying employer with , is a weighted average. As λ tends to 1 (and accordingly ψ to 1), is perfectly elastic at for , as in the perfect enforcement setting in Stigler (1946). In contrast, in the complete absence of enforcement so that ψ = λ = 0, the marginal labour cost schedule is independent of the minimum wage and coincides instead with the unregulated marginal labour cost. Finally, expected marginal labour cost is truncated exactly at whenever λ ∈ (0,1]. Beyond , and thus for an employer that overcomplies, expected marginal hiring cost coincides with the no‐intervention benchmark. Fig. 1. Open in new tabDownload slide Expected Marginal Labour Cost with Imperfect Enforcement Fig. 1. Open in new tabDownload slide Expected Marginal Labour Cost with Imperfect Enforcement One of the key insights of Stigler (1946) is that a perfectly enforced, binding minimum wage in the appropriate range can encourage hiring by lowering the marginal labour cost of hiring. Equation (9) echoes and extends this insight to cases with imperfect enforcement. Consider therefore a binding minimum wage , henceforth taken to mean . Evaluating expected marginal labour cost at the no‐intervention employment level, ℓo, (10) Stricter enforcement via an increase in λ, and thus ψ, lowers expected marginal cost at ℓo if and only if . Put differently, (10) shows that even in an environment of imperfect enforcement, in which the government turns a blind eye to the possibility of non‐compliance with regular frequency (1 − λ), raising enforcement continues to lower the expected marginal cost of hiring relative to the no intervention baseline in the range . Outside the range, however, with greater than the marginal revenue product of labour at is now strictly increasing in λ. These observations suggest that the comparative statics of a minimum wage policy will likely depend crucially on the size of the minimum wage relative to the thresholds wso and wdo. We turn to these next. 2.3. Labour Market Equilibrium and Minimum Wage Thresholds A labour market equilibrium consistent with expected profit maximisation, expected utility decision making and a binding but imperfectly enforced minimum wage can be shown to exhibit a number of possible configurations, separated by three distinct minimum wage thresholds. Two of which have already been singled out: wso and wdo. A third threshold is endogenous and depends on enforcement. divides labour market equilibria into those that are to the right, left, or exactly at the point where the expected marginal labour cost schedule truncates:18 11 Thus at given enforcement, expected marginal labour cost exceeds marginal value product, evaluated at the maximal labour supply available at the minimum wage, as soon as the minimum wage exceeds . As such, puts an upper bound on the range of minimum wages, beyond which equilibrium employment will be scaled back to a level less than the available labour supply at the minimum wage. Using (1), (2) and (11), it is of interest to note that lies between the unregulated wage wso and the competitive wage w*,19 or whenever λ < 1. These are illustrated in Figure 2 for λ ∈ (0,1). The corresponding labour market equilibria can be divided into: Over‐compliance: If , the minimum wage is non‐binding, and there is equilibrium over‐compliance. Thus, and . Consequently, employment, expected utility, and the equilibrium wage are all independent of small changes in the minimum wage policy In Figure 3a, inverse demand schedule a1 − bℓ is consistent with this regime. Fig. 3. Open in new tabDownload slide (a) Compliance (b) Non‐compliance Fig. 3. Open in new tabDownload slide (a) Compliance (b) Non‐compliance Fig. 2. Open in new tabDownload slide Minimum Wage Thresholds Fig. 2. Open in new tabDownload slide Minimum Wage Thresholds Exact Compliance: Raise the minimum wage until . There is thus a binding minimum wage but the marginal revenue product of labour is greater than the expected marginal labour cost evaluated at , since is less than the threshold . Labour market equilibrium is accordingly supply‐constrained (a2 − bℓ in Figure 3a). Equilibrium employment and wages are determined based purely on supply side considerations, with: (12) since from (5). The elasticities of equilibrium employment, expected utility, and equilibrium wage with respect to a rise in the minimum wage are all positive. In addition, since such an employer is already in strict compliance with the minimum wage legislation, a further increase in the intensity of enforcement has no further impact on equilibrium hiring or wage. Non‐compliance: As noted, for a minimum wage , labour market equilibrium is demand constrained (a3 − bℓ and a4 − bℓ in Figure 3b).20 Equilibrium employment is determined by the intersection of the marginal revenue product and the expected marginal labour cost: , while . As long as there is positive employment, it follows from (2) and (9) that (13) (14) Thus, employment and expected utility exceed (are less than) their no‐intervention benchmarks whenever is less than (greater than) wdo. Also, from (4), (11) and (13), (15) if and only if and λ > 0. Thus, equilibrium non‐compliance, , is synonymous with a binding demand constraint. From (15), equilibrium subminimum wage depends on employer productivity and labour supply conditions, since In particular, a more productive employer (a), a smaller population of workers to draw from and a high mobility cost (T) are associated with a higher subminimum wage. Finally, since equilibrium employment is demand constrained in this range, (13)–(15) show that a further rise in the minimum wage decreases equilibrium employment, workers’ expected utility and equilibrium wage at constant enforcement intensity. The preceding discussion suggests three sets of issues of particular empirical relevance, examined in greater detail below. 2.3.1. Imperfect enforcement and comparative statics Figure 4 summarises and compares the relationship between equilibrium employment and the minimum wage for the case of perfect enforcement, λ = 1, and imperfect enforcement λ′ < 1, based on (12)–(15). As shown, the predicted comparative statics responses of a minimum wage are highly sensitive to imperfect enforcement and accommodate cases ranging from Fig. 4. Open in new tabDownload slide Employment and Minimum Wage With Imperfect Enforcement
A: No Change; B: Sign Reversal; C: Muted Response Fig. 4. Open in new tabDownload slide Employment and Minimum Wage With Imperfect Enforcement
A: No Change; B: Sign Reversal; C: Muted Response (i) no change, (ii) a sign reversal and (iii) a muted response, depending systematically on the size of the minimum wage relative to the minimum wage thresholds already discussed. For minimum wages less than the endogenously determined , there is either over or strict compliance despite imperfect enforcement. Equilibrium employment is accordingly independent of the intensity of enforcement, as in the perfect enforcement case. Next, for minimum wages in the range , there is now non‐compliance. Contrary to the case of perfect enforcement, a further rise in the minimum wage now decreases, rather than increases employment, even though the minimum wage is strictly less than the competitive baseline w*. Finally, for minimum wages greater than the competitive wage w*, employment continues to fall with respect to a rise in , albeit at a muted rate because of imperfect enforcement. The same Figure also shows that at a given minimum wage, the relationship between enforcement and equilibrium employment is nuanced. In particular, for a binding minimum wage in the range [wso,wdo], an increase in λ (from λ′ to 1) leaves employment unchanged for a firm in strict compliance, raise employment for any other (newly compliant or non‐compliant) firms. In Figure 3b, inverse demand a3 − bℓ is consistent with this regime. The intuition follows from (10), where the expected marginal labour cost at ℓo is shown to decrease with enforcement intensity in this range. Outside this range, with higher than wdo, stricter enforcement can only decrease hiring by a non‐compliant employer (a4 − bℓ in Figure 3b). 2.3.2. Compliant clusters and wage dispersion It is worth reiterating that each of the minimum wage thresholds is endogenous, with and wdo all positively associated with the productivity of labour a, the reservation utility , and supply side parameters, . Thus, the minimum wage thresholds can be re‐expressed to give the combinations of labour demand and supply conditions consistent with over‐compliance, strict compliance, and non‐compliance, given the same minimum wage policy. Figure 5 illustrates in space. Area A characterises the cluster of employers types and labour supply conditions consistent with exact compliance.21 Any employer in this area responds uniformly to the same minimum wage policy by paying exactly the minimum wage. As labour productivity falls, or when the pool of available workers () increases, for example, area B applies. Area B characterises the case of non‐compliance with positive employment response to stricter enforcement.22 Finally, area C corresponds to the case of non‐compliance with negative employment response to stricter enforcement.23 Both areas B and C admit a continuous range of subminimum wages depending systematically on combinations of a and τ via (15). Fig. 5. Open in new tabDownload slide Labour Demand and Supply Determinants and Equilibrium Configurations
A: Strict Compliance; B: Effective Enforcement; C: Ineffective Enforcement Fig. 5. Open in new tabDownload slide Labour Demand and Supply Determinants and Equilibrium Configurations
A: Strict Compliance; B: Effective Enforcement; C: Ineffective Enforcement These observations are consistent with a number of well‐known empirical findings already noted in the Introduction. The clustering of compliant employers as in Area A is consistent with the oft noted spike at the minimum wage along the wage distribution. Workers earning the minimum wage co‐exist with other subminimum wage earners along a dispersed subminimum wage distribution, consistent with Areas B and C. In addition, an increase in the minimum wage can now be seen to give rise to two distinct sets of effects on wages. A pure wage effect works through (12) and (15), indicating respectively a positive wage impact on those who exactly comply and a negative impact on those who do not. But the same rise in the minimum wage also embodies a composition effect, which now accommodates an endogenous switch from compliance to non‐compliance. In Figure 5, such an increase in the minimum wage moves areas A, B and C upwards. The combined wage and composition effect of a minimum wage hike is thus ambiguous in general. Not surprisingly, then, co‐movements of the legislated minimum and subminimum wage have also been observed in the empirical literature but these have so far come up with mixed findings on the direction of observed co‐movements (Card and Krueger, 1995; Lemos, 2004; Baanante, 2005; Strobl and Walsh, 2001). 2.3.3. Endogenous enforcement and turning a blind eye Implicit in our findings so far is that imperfect enforcement need not be associated with non‐compliance. Indeed, the threshold in (11) gives the largest minimum wage that can be applied without triggering non‐compliance, for any λ ∈ [0,1]. Furthermore, can be fine‐tuned by adjusting λ. Routine differentiation with respect to λ gives an intuitive answer: the threshold can be raised by increasing λ. As λ tends to 1, approaches w*, coinciding with the perfect enforcement case. But as λ approaches zero, now tends to wso implying in contrast that no employer will comply when a strictly binding minimum wage is not enforced. Because of the monotonicity of in λ, (11) can be used to retrieve the minimum enforcement intensity, , required to elicit compliance for any given minimum wage : (16) Naturally, this is the dual to (11), which seeks the maximum minimum wage that can be imposed without eliciting non‐compliance, given λ.24 Interestingly, the minimal level of enforcement also responds to the underlying productivity and supply conditions of the labour market. Using (11), is strictly decreasing respectively in a and in . In other words, the level of enforcement required to elicit compliance relates systematically to whether non‐compliance should be expected to begin with. Of course, also rises with a higher minimum wage . As such, (16) provides one possible endogenous link between the two components of a minimum wage policy, applicable whenever a minimum wage policy combines a legislated wage floor and the minimal enforcement required to elicit the market payment of this wage, . All these prompt two important questions: what is the nature of the comparative statics of a minimum wage with endogenous enforcement, represented by the pair (), when it is common knowledge that the government is committed to choosing enforcement systematically based on ? Meanwhile, under what conditions will such a minimum wage policy be consistent with social welfare maximisation? The first question points to an endogeneity problem, when enforcement is now dependent on the size of the minimum wage itself. To appreciate the potential scale of this problem, note from (12) that for reduces to , and . As such, the intricacies of the comparative statics of a minimum wage policy with imperfect (but fixed) enforcement are effectively cancelled out with this endogenous enforcement scheme. The predicted comparative statics of a minimum wage hike thus coincides with that of a world with perfect enforcement, even when imperfect enforcement is clearly still in play and is strictly less than unity. At the heart of the second question is the issue: why do governments turn a blind eye to the minimum wage law that they have themselves passed? A minimum wage policy like accordingly sets out a baseline, and turning a blind eye simply means a level of enforcement that is less than , given . In what follows, we examine the nature of the comparative statics of a minimum wage policy with an endogenous and social‐welfare‐maximising level of enforcement. 3. Minimum Wage and the Credibility of Enforcement Starting with the announcement of a minimum wage , let λ be the expected intensity of enforcement, held by both employers and workers when employment contracts are made, as we have done up to now. Let p, in contrast, denote the actual intensity of enforcement carried out post contract negotiations. Thus, p also determines the fraction of contracted workers earning less than the minimum wage, who ultimately receive wage settlement net of transaction costs, . Consider therefore a social welfare function made up of three components, taking as given the expectation λ: (i) the sum of the profit of the employer, the income net of mobility cost for all workers along the interval, and the reservation income equivalent of the unemployed,25 (ii) a strictly increasing and strictly convex cost of employer inspection C(p)ℓm, where c ≡ Cp(0) ≥ 0, denotes the marginal cost of raising enforcement evaluated at p = 0,26 and (iii) a loss function which captures the government’s distributional concern D(·), with respect to employer wage decisions that deviate from the legislated minimum.27 The first two components of the government’s objective function capture efficiency concerns in the standard way. The third distributional concern component is captured by a loss function. With non‐compliance, the loss function is With full (exact or over) compliance, the loss function is just Within the distributional realm, D(·) gives the number of workers receiving less than the minimum wage target, weighted by the corresponding proportional income shortfall. D(·) may also be interpreted as analogous to one of the Foster et al. (1984) measures of poverty, where the minimum wage serves as the government’s definition of who is poor. The parameter γ > 0 measures the government’s concern for distribution relative to efficiency overall and represents the marginal social welfare cost of a small change in the poverty measure D(·). The social welfare function is thus: (17) In what follows, we focus on minimum wages in the range and do so for two reasons. First, and as has been discussed, is required for the minimum wage to bind. Meanwhile, always reduces employment relative to the unregulated benchmark and is thus inferior to no regulation at all both on efficiency grounds, and on equity grounds based on Ω(·). Second, from (13)–(15), stricter enforcement of a minimum wage greater than wdo cannot improve social welfare since it raises enforcement costs but serves only to decrease employment, expected utility, and subminimum wages even further. As such, it will be not at all surprising that a government turns a blind eye to minimum wages greater than wdo. 3.1. Commitment In this regime, the government simultaneously commits to a minimum wage and a corresponding level of enforcement to maximise the social welfare function . Let be such a minimum wage policy with ex ante commitment. Rational expectation in the case of commitment implies p = λc. We can thus denote social welfare as . We now show that social‐welfare‐maximising minimum wage policy with commitment exhibits two important characteristics: it must be the case that (i) there is full compliance: , and (ii) the cost minimising level of enforcement is undertaken, as in (16). To see this, suppose in contrast that maximises the social welfare function but ), and and are the corresponding equilibrium employment and market wage. There are two possibilities: (i) the hypothetical λ′ is strictly greater than and (ii) λ′ is strictly less than . Suppose first that . It follows by definition of that there is full compliance, with employment and wage respectively at: In other words, both employment and market wage are at their full compliance levels consistent with the minimum wage , but the cost of enforcement C(λ′) is strictly higher than . Thus, . Intuitively, therefore, there is no reason to raise enforcement beyond the minimal level required to guarantee full compliance. Suppose instead that and enforcement is insufficient to guarantee full compliance. Equilibrium employment and the (subminimum) market wage are then: which follows since and stricter enforcement increases employment from (13). Also from (13), a reduction in the minimum wage from to can thus raise employment and ensure full compliance at the same enforcement cost. It follows therefore that and is dominated by . The intuition here is that if enforcement is too low to guarantee compliance, the minimum wage should be adjusted downwards since doing so can in fact raise employment. Thus, using (11) and , we have Proposition 1. With ex ante commitment, if a minimum wage policy maximises the social welfare function : There is equilibrium compliance achieved at the lowest possible cost, with , and . Given is independent of γ. Given is higher in a labour market with lower productivity a, lower mobility cost T, or a larger available work force . is strictly increasing in . Equilibrium employment and expected workers’utility are both strictly increasing in . We have thus set out a benchmark and confirmed that there is little justification for governments to turn a blind eye to a minimum wage legislation that they themselves have passed in this setting, so long as they can commit ex ante and carry through ex post both parameters of the minimum wage law . As shown, since endogenous enforcement is indeed given by the minimum enforcement required to guarantee compliance, , demand conditions such as labour productivity and labour supply parameters such as T and govern the size of but not the degree of government distributional concerns γ. Finally, the last item of Proposition 1 highlights the comparative statics implications of the endogenous enforcement scheme . Importantly, it suggests that the comparative statics of a minimum wage with a non‐compliant employer are out of equilibrium phenomena in this case with commitment and occur only when starting with a minimum wage and/or an intensity of enforcement that are not optimally set. With Proposition 1, the problem of the government can be simplified as essentially the choice of an optimal level of minimum wage, for full compliance implies that . The maximisation problem of the government can therefore be rewritten via a change of variable (replacing by by and λ by ). We assume henceforth that the enforcement cost is sufficiently convex, so that the revised objective function is strictly concave in . The first order condition for an interior optimum requires: (18) where a subscript denotes partial derivative and ηℓ denotes the elasticity of equilibrium employment with respect to the minimum wage . The expression measures the efficiency and distributional gains from raising employment, to be balanced against the marginal cost of enforcement. Evaluating the first order condition above at , social welfare maximisation implies a strictly binding minimum wage if (19) This follows since the degree of monopsonistic labour market distortion is given by , while from (11) and (16), and also since . Thus, Proposition 2. With commitment, a binding minimum wage and the associated endogenous enforcement improve social welfare beyond the no‐intervention benchmark if labour productivity a and the degree of distributional concern γ are both sufficiently high relative to the cost of enforcement c so that inequality (19) holds. Put differently, a minimum wage legislation may no longer be welfare maximising in sectors where labour productivity is sufficiently low, since the costs required to enforce such a minimum wage outweigh benefits. Thus, even within the same country and hence arguably the same government objective function, the co‐existence of covered and uncovered sectors (Proposition 2), and full compliance in covered sectors (Proposition 1) are consistent with social welfare maximisation with ex ante commitment. 3.2. Credible Enforcement We turn now to the case where ex ante commitment is not feasible.28 For any minimum wage announcement , enforcement credibility requires that p is determined ex post taken as given expectation λ, and hence :29 (20) A rational expectation equilibrium level of enforcement is thus given by the fixed point (21) Turning a blind eye in a rational expectation equilibrium requires We proceed first by taking as given the announcement of the minimum wage and an expectation λ ∈ [0,1). The ex post welfare implications of enforcement can be determined via It follows immediately that Proposition 3. for any and λ ∈ [0,1) if (22) Thus, there will be no enforcement of the minimum wage ex post, or , if any one of the following holds: (i) the cost of enforcement is too high, (ii) the government harbours only efficiency concerns (γ = 0); (iii) the labour market exhibits full compliance to begin with, or , since there is literally nothing to enforce given full compliance; and (iv) a transaction cost 1 − σ that is sufficiently large, even when γ > 0 and non‐compliance is known to exist, . The latter applies since the transfer of wage settlement is costly whenever σ < 1. Thus, as much as there may be a desire to enforce the minimum wage on equity grounds, enforcement alone may be made ineffectual if the resulting income gain for workers, net of transaction costs, is too low. This is reflected in the difference . Suppose instead that (22) is not satisfied. The ex post optimal and thus credible level of enforcement implicitly solves , or (23) Three issues of particular interest here are the uniqueness, slope and existence of the rational expectation equilibrium enforcement for . To this end, we continue to assume that the cost of enforcement C(Λ) is sufficiently convex, while γ is large enough relative to the transaction cost of wage settlement , such that the left‐hand side of (23) is strictly decreasing in Λ and increasing in . Intuitively, these require that (i) the marginal welfare gains from raising enforcement ex post in a rational expectation equilibrium is diminishing in λD and (ii) the transaction cost of wage settlement is never high enough to offset the distributional gains from enforcing the minimum wage. It follows then by standard arguments that , if it exists, is uniquely determined and strictly increasing in . The credibility constraint (23) additionally implies that the range of credible minimum wage is bounded from below, since ex post incentives to enforce depend critically on the severity of violations, as measured by . This new lower bound can be obtained by identifying the minimum wage consistent with λD = 0 in (23), (24) so that there is no enforcement and thus λD = 0 exactly solves (23). It follows from (24) that as long as enforcement cost c is strictly positive, the new lower bound is strictly greater than the monopsonistic wage wso. Also from (24), this new lower bound is decreasing in the degree of distributional concern γ. The range of feasible minimum wages that satisfies the credibility criterion but still capable of improving employment outcomes relative to no‐intervention is thus from (13). As a sufficient condition for existence, therefore, we assume henceforth that γ is large enough, so that and the range is accordingly non‐empty. With these observations in mind, (23) gives rise to a new set of comparative static responses to a minimum wage hike, in which it is expected that enforcement is too low to guarantee full compliance but the expected degree of imperfect enforcement systematically changes with the minimum wage itself:30 Proposition 4. A rational expectation equilibrium level of enforcement that solves (23) has the following characteristics: There is equilibrium non‐compliance: , and . Given rises with γ. Given is higher in a labour market with lower employer productivity a, lower mobility cost T , or a larger available work force . If γ is sufficiently large, is strictly increasing in , while equilibrium employment and expected workers’utility are likewise strictly increasing in . Thus, turning a blind eye survives the credibility criterion laid out in (23), whereas full compliance does not.31 Interestingly, a government that is only concerned with efficiency (γ = 0) and, hence, , is guaranteed the least efficient (monopsonistic) labour market outcome ℓo, since the credible level of enforcement rises with γ and is equal to zero for γ = 0. This is indeed striking – governments espousing only efficiency concerns cannot credibly implement efficiency‐improving minimum wage reforms. In addition, the ex post credible enforcement intensity is once again need‐based, with higher in labour markets where labour productivity is relatively low and where market wages are low driven by an excess supply of labour and a low mobility cost. The last item illustrates the role of enforcement endogeneity on equilibrium employment via two sets of forces. From (13), an exogenous increase in the minimum wage negatively impacts employment whenever there is non‐compliance. However, from (23), a re‐definition of who is non‐compliant relative to the new minimum wage increases the severity of the violation at constant λ. This raises the ex post incentive to enforce, as , and runs contrary to the first round employment impact of a minimum wage increase since and enforcement raises employment in this range from (13). As shown, the net outcome on employment will depend on the strength of the distributional concern parameter γ. Finally, turning to the question of whether welfare maximisation involves a binding minimum wage policy even with non‐compliance, note that the government’s problem now involves maximising by choice of an appropriate minimum wage , subject to . The first order condition requires that (25) where ℓD denotes employment and denotes the elasticity of ℓD with respect to the minimum wage. From Proposition 4, εℓ > 0 provided that γ is sufficiently large. Meanwhile, denotes the elasticity of the proportional income shortfall with respect to the minimum wage. It can be readily confirmed from (23) that εw is also positive. The choice of a welfare‐maximising minimum wage thus involves equating the anticipated marginal benefits, [wd(ℓD) − ws(ℓD) + γ]εℓ, with marginal costs. The latter is now made up of three parts. The first term is analogous to the commitment case and expresses the marginal cost required to credibly enforce a rise in the minimum wage. The second term denotes the ex post gains from enforcing the minimum wage, as has already been seen in (23). The final part expresses the efficiency and distributional losses associated with a rise in an imperfectly enforced minimum wage. Specifically, γ(1 − λDσ)εℓ and γ(1 − λDσ)εw respectively show the distributional losses associated with an increase in the number of subminimum wage earners and an increase in the proportional income shortfall as the minimum wage increases. Finally, the expression λD(1 − σ)εw represents efficiency losses via the transaction costs incurred by workers in the wage settlement process. Making use of (23) and evaluating (25) at the lower bound in (24), or , a social‐welfare‐maximising minimum wage policy involves if (26) As before, since monopsonistic labour market distortion is strictly increasing in labour productivity, a, we have thus: Proposition 5. With ex post discretion, a minimum wage and the associated ex post credible level of enforcement improves social welfare beyond the no‐intervention benchmark if labour productivity a and distributional concern γ are both large enough so that (26) is satisfied. Thus, within the same country and given the same government objective function the co‐existence of (relatively productive) covered and (relatively less productive) uncovered sectors (Proposition 5) along with less than full compliance in covered sectors (Propositions 3, 4), are consistent with social welfare maximisation with ex post discretion.32 4. Conclusion There is now extensive evidence particularly from developing country labour markets that non‐compliance with minimum wage legislations in covered sectors is pervasive. The stylised facts reviewed in this article include: (i) co‐existence of compliant and non‐compliant employers, (ii) a spike at the minimum wage, (iii) a dispersion of subminimum wages and (iv) co‐movements of minimum and subminimum wages. We have argued that all of these stylised facts can be consistent with a setting in which there is imperfect competition, imperfect enforcement, and imperfect commitment. But beyond this, our comparative static analysis further underscores additional insights that may be gained by taking seriously the issue of imperfect enforcement of a minimum wage. Indeed, a simple deviation from perfect to imperfect enforcement is shown to be sufficient for standard comparative static predictions to be overturned, with equilibrium employment now predicted to respond negatively to a minimum wage hike in an imperfectly competitive labour market, for cases when the standard Stigler model would yield a positive response to the same minimum wage hike when enforcement is perfect. The key message that can be drawn is that observed empirical relationship between minimum wage and employment can no longer serve as the litmus test of the competitiveness of a labour market when enforcement is imperfect. We have also reviewed evidence that enforcement varies systematically across countries and across geographic regions, or industries within countries. This suggests a need for a theory of the endogeneous determination of enforcement. Consistent with the pioneering study of Ashenfelter and Smith (1979), the endogenous enforcement we derive does indeed exhibit the characteristic that enforcement of any given minimum wage is higher in labour markets where non‐compliance is likely to be prevalent, as would be the case where productivity is low or where the pool of available labour force, given labour demand, is large enough. The endogenous level of enforcement derived here is further shown to vary with the minimum wage itself. For a government that cares sufficiently about distribution, a rise in the minimum wage can signal a rise in enforcement. Indeed, equilibrium employment and expected workers’ welfare can now rise with a minimum wage hike but for reasons that have to do with endogenous enforcement rather than with imperfect labour market competition per se. Following on from the contribution to positive analysis, the article also analyses the behaviour of governments with and without the ability to commit ex ante to both the wage and enforcement dimensions of a minimum wage policy. Since ex post enforcement of a minimum wage is but a costly transfer of income from employers to workers, a government’s concern for distribution is shown to interact in interesting ways with the problem of credible commitment on enforcement intensity. Simply put, a government that cares more about distribution will care more about violations of the minimum wage and can therefore signal a commitment to enforce. By the same token, a government that does not care at all about distribution cannot improve efficiency. The basic setup that we work with should be seen as a beginning, with a number of useful extensions that await further exploration. These can be framed under four main categories. An obvious line of research is to consider the issue of enforcement and credibility with alternative specifications of distributional concerns. This can be accomplished in two ways. First, rather than a concern for those whose income fall below a specified threshold (in our case, the minimum wage), an alternative specification can also incorporate measures of income inequality, including possibly income disparity between employers and workers. Second, additional insights may also be obtained by introducing variable poverty aversion parameters associated with the proportional income shortfall (Foster et al. 1994). Regardless of how the distributional concern component of government objective is alternatively measured, however, our main conclusion that credible enforcement cannot come with a complete disregard for the intrinsic value of income transfer from employers to subminimum wage earners, will remain intact. A second promising research agenda involves a re‐examination of the different components of a minimum wage policy package, including particularly the role of fines and penalty in addition to the compensation of any income shortfall. These penalties may be officially sanctioned by legislation (Lott and Roberts, 1995), or more indirectly through the loss of goodwill upon discovery (Harrison and Scorse, 2004), for example. Of particular interest is whether a minimum wage policy package can be further fine‐tuned to encourage compliance through the threat of fines and how the comparative statics of a minimum wage will depend on the credibility of such a threat. In a variety of situations, the minimum wage can arguably be thought of as an exogenously imposed labour standard and the government is left with only the choice of an appropriate enforcement strategy. Such an exogenous minimum wage may correspond to an exogenous poverty line, a minimum labour standard imposed extra‐nationally as a condition for exports, or a result of a political process, separate from the decision to allocate public funds to enforce the minimum wage, for example (Sobel, 1999). The issue of enforcement and credibility takes on a different type of significance here, particularly when the government may be held accountable not just to its own efficiency and equity concerns but also to the very set of forces that introduced the minimum wage as a labour standard to begin with. Finally, the comparison between commitment and discretion highlighted here can potentially shed new light on a range of policies other than the minimum wage, where enforcement and compliance are central. Some examples include tax evasion and enforcement, as well as financial bailouts and ex post enforcement of bailout conditions.33 In Basu et al. (2009), the distinction between commitment and discretion is further shown to be instrumental in understanding the labour market consequences of employment guarantee programmes, where the degree of distributional concern of the government is shown to directly impact the perceived credibility of the employment guarantee. Footnotes 1 " See for example Freeman (1996), Card and Krueger (1995), Bhaskar et al. (2002), Fields and Kanbur (2007), Neumark and Wascher (2007). 2 " Bhaskar et al. (2002) examine the usefulness and empirical relevance of the competitive, monopsonistic, oligopsonistic and monopolistically competitive frames. In the context of search theory of unemployment (Fershtman and Fishman, 1994; Mortensen and Pissarrides, 1994), a minimum wage has also been shown to be efficiency enhancing as equilibrium employment and equilibrium job retention rates rise with the legislated minimum wage. 3 " As Stigler (1946) notes, such an optimal minimum wage is endogenously determined, and should vary with occupation, among firms, and through time. As such, ‘a national minimum wage, infrequently changed, is wholly unsuited to these diversities of conditions’ (p. 361). 4 " There have also been exceptions. For example, Machin et al. (2003) observed little non‐compliance with the national minimum wage of 1999 in the residential care homes industry in the UK. 5 " Also see Saget (2001) for a survey of evidence in other developing countries. 6 " Kernel density plots and/or wage histograms of dispersed subminimum wage distributions and associated spikes at or about a binding minimum wage have been shown for many countries. See, for a few examples, Bell (1997) for Colombia, Maloney and Nunez (2004) for eight Latin American countries, Cardoso and Portugal (2002) for Portugal, Gindling and Terrell (2006) for Honduras, Strobl and Walsh (2001) for Trinidad and Tobago, Lemos (2004) for both the formal and informal sectors in Brazil. 7 " A theoretical literature modifies the effects of a minimum wage under different specifications of the enforcement and penalty regime based on a competitive labour market (Ashenfelter and Smith, 1979; Chang and Ehrlich, 1985; Grenier, 1982; Yaniv, 2001; Squire and Suthiwart‐Narueput, 1997; Harrison and Leamer, 1997). This literature takes a single market determined subminimum wage as given, and does not address the issue of subminimum wage dispersion. Relatedly, Eckstein et al. (2006) examine a search model with exogenous exemptions to a minimum wage, but full compliance in the covered sector. 8 " Like our setting, Yaniv (1988) is an exception in the literature of minimum wage compliance in that it too departs from the competitive labour market assumption and examines the compliance behaviour of a monopsonistic firm with an imperfectly enforced minimum wage. But unlike our setting, labour supply is taken as exogenously given and, as such, it does not accommodate rational and endogenous labour supply response to any change in the likelihood of discovery or the possibility of subsequent wage settlement – the determinants of which are the central themes of this article. With a focus on compliance behaviour, Yaniv (1988) does not undertake a full‐fledged comparison of the difference in employment comparative statics response that imperfect enforcement makes given any minimum wage, or an examination of the possible sources of enforcement endogeneity. 9 " See Neumark and Wascher (2007) for a survey of the recent empirical literature of minimum wage and employment in developed and developing countries. 10 " Harrison and Scorse (2004) examines empirically the issue of endogenous minimum wage compliance in Indonesia. Taking the average of the observed dispersion of subminimum wages as a proxy for the market determined subminimum wage in the competitive frame, it is shown that foreign ownership and the corresponding emphasis on enforcement are associated with a firm‐level employment increases subsequent to a minimum wage. 11 " With perfect competition in the standard sense, namely that of costless job search and wage‐taking employers, there is by definition a market determined subminimum wage that employers then take as given, (Ashenfelter and Smith, 1979; Chang and Ehrlich, 1985; Grenier, 1982; Yaniv, 2001; Squire and Suthiwart‐Narueput, 1997; Harrison and Leamer, 1997). 12 " As we show later, if the minimum wage is an exogenously imposed standard rather than optimally chosen, then it may come as no surprise at all that insufficient enforcement will be given to uphold the standard, and non‐compliance naturally follows. 13 " t should be interpreted as any employment deterring transaction costs or barriers that drive a wedge between the supply price and a worker’ reservation utility. These include: geographical distance; lack of information; the cost of correcting skill mismatch; or other worker‐specific disutility of employment depending on the conditions of work. 14 " We note that the functional forms assumed here allow us to derive closed‐form solutions, and are otherwise not necessary for our comparative statics findings. In particular, a revenue function R(ℓ) satisfying diminishing marginal product and any non‐degenerate distribution on [0,T] with positive density can alternatively be used without changing the qualitative findings. 15 " With zero mobility cost, the monopsonistic and competitive labour market equilibria coincide, in which the single employer faces a take‐it‐or‐leave‐it offer of from every worker. 16 " The question of whether such a commitment is credible is the subject of Section 4. 17 " As another useful interpretation, σ can also parameterise the effectiveness of the judicial system and government bureaucracy. See Flanagan (1989) for an empirical analysis of the role of substantial time lags, among other things, on employers’ decision to comply with the US National Labor Relations Act. 18 " Since the is monotonically decreasing in is well defined and unique by standard arguments. In particular, with a binding minimum wage, if and only if . Rearranging terms and using yields as shown. 19 " To see this, note that from (2) and (11) . Meanwhile by inspection of (11). 20 " It can also be checked using (5) that whenever , non‐compliance strictly dominates compliance with employment rationing at , where . To see this, expected profit maximisation implies whenever from (5). 21 " From (11), strict compliance requires, 22 " This requires that 23 " This final class requires , since a minimum wage greater than induces zero employment upon rearranging (13). 24 " For , full compliance will always require perfect enforcement. 25 " This first component of the government welfare function is given by: where the expression denotes the transaction cost of enforcing the minimum wage policy when non‐compliance is in place. 26 " We assume that the government finance enforcement activities through lump sum taxation. Any additional costs of the use of public funds incurred in the process are subsumed in the enforcement cost function C(p). 27 " We suppress the arguments of equilibrium employment and wage and whenever there is no risk of confusion. 28 " The approach to time inconsistency problems adopted here has been applied in a variety of policy settings (Kydland and Prescott, 1977; Chau, 2001). Our contribution here lies in pointing out the importance of credibility in the minimum wage setting, particularly when perfect enforcement cannot be costlessly guaranteed. 29 " The second order condition of the maximisation problem in (20) is always satisfied. 30 " Since the left‐hand side of (23) is strictly decreasing in Λ with C(Λ) sufficiently convex, the Proposition follows by noting that the left hand side of (23) is increasing in γ and decreasing in a and by (15). The last part of the Proposition follows from differentiating (13) using the implicit relationship defined in (23). 31 " Of course, another possibility that may give rise to equilibrium non‐compliance may simply be because of a lack of full information concerning labour market conditions, including demand and supply side parameters. Proposition 4 shows that even in the absence of such information asymmetries, a case for imperfect enforcement induced equilibrium non‐compliance can still be made. 32 " In Basu et al. (2009), we point out that an employment guarantee scheme which directly generates employment through public funds can be an alternative but viable policy option for imperfectly competitive labour markets with or without the ability to commit, precisely in labour markets where labour productivity is sufficiently low. 33 " Anderlini et al. (2007) examine another intriguing distinction between ex ante and ex post decision making of contract enforcing courts. In a setup that, unlike our setting, features asymmetric information between contracting partieis, it is shown that a court can improve the welfare of contracting parties by voiding, ex post, voluntary contracts that these same parties would like the court to enforce ex ante. In our context, the issue of minimum wage enforcement in the presence of information asymmetry (between workers and employers, or between employers and the government) is yet another promising avenue for future research. References Anderlini , L. , Felli , L. and Postlewaite , A. ( 2007 ). ‘Courts of law and unforeseen contingencies’ , Journal of Law, Economics and Organization , vol. 23 ( 3 ), pp. 662 – 84 . Google Scholar Crossref Search ADS WorldCat Ashenfelter , O. and Smith , R. S. ( 1979 ). ‘Compliance with the minimum wage law’ , Journal of Political Economy , vol. 87 ( 2 ), pp. 333 – 50 . Google Scholar Crossref Search ADS WorldCat Baanante , M. J. ( 2005 ). ‘Minimum wage effects under endogenous compliance. evidence from Peru’ , available at http://www.grade.org.pe/download/docs/Minimum%20wages%20in%20Peru%20M.%20Jaramillo.pdf. Basu , A. K. , Chau , N. H. and Kanbur , R. ( 2009 ). ‘A theory of employment guarantee schemes: contestability, credibility and distributional concerns’ , Journal of Public Economics , vol. 93 ( 3–4 ), pp. 482 – 97 . Google Scholar Crossref Search ADS WorldCat Bell , L. ( 1997 ). ‘The impact of minimum wages in Mexico and Columbia’ , Journal of Labor Economics , vol. 15 ( 3 ), pp. S102 – 35 . Google Scholar Crossref Search ADS WorldCat Bhaskar , V. , Manning , A. and To , T. ( 2002 ). ‘Oligopsony and monopolistic competition in labor markets’ , Journal of Economic Perspectives , vol. 16 ( 2 ), pp. 155 – 74 . Google Scholar Crossref Search ADS WorldCat Card , D. and Krueger , A. B. ( 1995 ). Myth and Measurement: The New Economics of the Minimum Wage , Princeton: Princeton University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Cardoso , A. R. and Portugal , P. ( 2002 ). ‘Disentangling the minimum wage puzzle: an analysis of worker accessions and separations’ , IZA Discussion Paper No. 544. Chang , Y‐M. and Ehrlich , I. ( 1985 ). ‘On the economics of compliance with the minimum wage law’ , Journal of Political Economy , vol. 93 ( 1 ), pp. 84 – 91 Google Scholar Crossref Search ADS WorldCat Chau , N. H. ( 2001 ). ‘Strategic amnesty and credible immigration reform’ , Journal of Labor Economics , vol. 19 , pp. 604 – 34 . Google Scholar Crossref Search ADS WorldCat Eckstein , Z. , Ge , S. and Petrongolo , B. ( 2006 ). ‘Job and wage mobility in a search model with non‐compliance (exemptions) with the minimum wage’ . IZA Discussion Paper No. 2076. Fershtman , C. and Fishman , A. ( 1994 ). ‘The perverse effects of wage and price controls in search markets’ , European Economic Review , vol. 38 , pp. 1099 – 112 . Google Scholar Crossref Search ADS WorldCat Fields , G. and Kanbur , R. ( 2007 ). ‘Minimum wage and poverty’ , Journal of Economic Inequality , vol. 5 ( 2 ), pp. 135 – 47 . Google Scholar Crossref Search ADS WorldCat Flanagan , R. J. ( 1989 ). ‘Compliance and enforcement decisions under the National Labor Relations Act’ , Journal of Labor Economics , vol. 7 ( 3 ), pp. 257 – 80 . Google Scholar Crossref Search ADS WorldCat Foster , J. , Greer , J. and Thorbecke , E. ( 1994 ). ‘A class of decomposable poverty measures’ , Econometrica , vol. 52 ( 3 ), pp. 761 – 66 . Google Scholar Crossref Search ADS WorldCat Freeman , R. ( 1996 ). ‘The minimum wage as a redistributive tool’ , Economic Journal , vol. 106 , pp. 639 – 49 . Google Scholar Crossref Search ADS WorldCat Gindling , T. and Terrell , K. ( 1995 ). ‘The nature of minimum wages and their effectiveness as a wage floor in Costa Rica, 1976‐1991’ , World Development , vol. 23 , 1439 – 58 . Google Scholar Crossref Search ADS WorldCat Gindling , T. and Terrell , K. ( 2006 ). ‘Minimum wages, globalization and poverty in Honduras’ , IZA Discussion Paper No. 2497. Grenier , G. ( 1982 ). ‘On compliance with the minimum wage law’ , Journal of Political Economy , vol. 90 ( 1 ), pp. 184 – 87 . Google Scholar Crossref Search ADS WorldCat Harrison , A. and Leamer , E. ( 1997 ). ‘Labor markets in developing countries: an agenda for research’ , Journal of Labor Economics , vol. 15 ( S3 ), pp. S1 – 9 . Google Scholar Crossref Search ADS WorldCat Harrison , A. and Scorse , J. ( 2004 ). ‘The impact of globalization on compliance with labor standards: a plant‐ level study’ , in ( S. Collins and D. Rodrik eds.) Brookings Trade Forum 2003 , pp. 83 – 94 , Washington DC: Brookings Institution Press . OpenURL Placeholder Text WorldCat Kydland , F. and Prescott , E. ( 1977 ). ‘Rules rather than discretion: the inconsistency of optimal plans’ , Journal of Political Economy , vol. 85 , pp. 473 – 90 . Google Scholar Crossref Search ADS WorldCat Lemos , S. ( 2004 ). ‘The effects of the minimum wage in the formal and informal sectors in Brazil’ , IZA Discussion Paper No. 1089. Lemos , S. ( 2006 ). ‘Minimum wage effects in a developing country’ , mimeo, University of Leicester . Lott , J. R. and Roberts , R. D. ( 1995 ). ‘The expected penalty for committing a crime: an analysis of minimum wage violations’ , Journal of Human Resources , vol. 30 ( 2 ), pp. 397 – 408 . Google Scholar Crossref Search ADS WorldCat Machin , S. , Manning , A. and Rahman , L. ( 2003 ). ‘Where the minimum wage bites hard: introduction of minimum wages to a low wage sector’ , Journal of the European Economic Association , vol. 1 ( 1 ), pp. 154 – 80 . Google Scholar Crossref Search ADS WorldCat Maloney , W. and Nunez , J. ( 2004 ). ‘Measuring the impact of minimum wages: evidence from Latin America’ , in ( J. Heckman and C. Pagès eds) Law and Employment. Lessons from Latin America and the Carribean . pp. 109 – 30 , Cambridge, MA: NBER . OpenURL Placeholder Text WorldCat Mortensen , D. and Pissarides , C. ( 1994 ). ‘Job creation and job destruction in the theory of unemployment’ , Review of Economic Studies , vol. 61 ( 3 ), pp. 397 – 415 . Google Scholar Crossref Search ADS WorldCat Neumark , D. and Wascher , W. ( 2007 ). ‘Minimum wages and employment’ , IZA Discussion Paper No. 2570. Saget , C. ( 2001 ). ‘Is the minimum wage an effective tool to promote decent work and reduce poverty? The experience of selected developing countries’, Employment Paper No. 2001/13 , Geneva: Employment Strategy Department, International Labour Office . Sobel , R. S. ( 1999 ). ‘Theory and evidence on the political economy of the minimum wage’ , Journal of Political Economy , vol. 107 ( 4 ), pp. 761 – 85 . Google Scholar Crossref Search ADS WorldCat Squire , L. and Suthiwart‐Narueput , S. ( 1997 ). ‘The impact of labor market regulations’ , World Bank Economic Review , vol. 11 ( 1 ), pp. 119 – 43 . Google Scholar Crossref Search ADS WorldCat Stigler , G. J. ( 1946 ). ‘The economics of minimum wage legislation’ , American Economic Review , vol. 36 ( 3 ), pp. 358 – 65 . OpenURL Placeholder Text WorldCat Strobl , E. and Walsh , F. ( 2001 ). ‘Minimum wage and compliance: the case of Trinidad and Tobago’ , Economic Development and Cultural Change , vol. 51 ( 2 ), pp 427 – 50 . Google Scholar Crossref Search ADS WorldCat Weil , D. ( 2004 ). ‘Compliance with the minimum wage: can government make a difference’ , mimeo, Boston University School of Management . Yaniv , G. ( 1988 ). ‘Enforcement and monopsonistic compliance with the minimum wage law’ , Southern Economic Journal , vol. 55 ( 2 ), pp. 505 – 9 . Google Scholar Crossref Search ADS WorldCat Yaniv , G. ( 2001 ). ‘Minimum wage noncompliance and the employment decision’ , Journal of Labor Economics vol. 19 ( 3 ), pp. 596 – 603 Google Scholar Crossref Search ADS WorldCat Author notes " We thank Gary Fields, Mick Keen, Russ Krelove, Jonathan Thomas, seminar participants at Edinburgh, the IMF, and Institute for the Study of Labor (IZA), members of the Cornell‐SEWA‐WIEGO Exposure and Dialogue Group, an Editor of this Journal and two anonymous referees for helpful comments and stimulating discussions. The usual disclaimer applies. © The Author(s). Journal compilation © Royal Economic Society 2009

Simonsohn,, Uri

doi: 10.1111/j.1468-0297.2009.02296.xpmid: N/A

Abstract Does current utility bias predictions of future utility for high stakes decisions? Here I provide field evidence consistent with such Projection Bias in one of life’s most thought‐about decisions: college enrolment. After arguing and documenting with survey evidence that cloudiness increases the appeal of academic activities, I analyse the enrolment decisions of 1,284 prospective students who visited a university known for its academic strengths and recreational weaknesses. Consistent with the notion that current weather conditions influence decisions about future academic activities, I find that an increase in cloudcover of one standard deviation on the day of the visit is associated with an increase in the probability of enrolment of 9 percentage points. When making decisions about future consumption, people must make predictions about the utility they will derive from it. While economic theory typically assumes away any difficulty in making such predictions, abundant empirical work has shown that predicting future utility is actually quite difficult; for a review see Loewenstein and Schkade (1999). A particularly robust finding in this literature is that people tend to bias their estimates of future utility towards their current utility, a phenomenon labelled Projection Bias by Loewenstein et al. (2003). While it is sensible to base predictions about the future on the present, Projection Bias is a bias because predictions of future utility are systematically off in the direction of current utility, i.e. they are predictably wrong. In general, Projection Bias will lead to systematic errors when decisions are made in the presence of factors that influence current but not future utility. Prior research, for example, has shown that grocery shoppers buy more items if shopping while hungry (Gilbert et al., 2002), that current arousal influences predictions about future sexual behaviour (Ariely and Loewenstein, 2006) and that catalogue orders for winter clothing are more likely to be returned if ordered on colder days (Conlin et al., 2007). As with all deviations from rationality, an important question regarding biases in the prediction of future utility is whether they play a role in high stakes decisions – when people are highly motivated to ‘getting it right’– or whether they are only present in hypothetical and low‐stake decisions. This article provides evidence consistent with Projection Bias playing a role in one of life’s most thought‐about decisions: which college to enrol in. Although college enrolment decisions are made during a long period of time and hence identifying the current utility which could potentially contaminate predictions about future utility is difficult, one aspect of the college decision process lends itself particularly well to studying projection bias: college visits. Many college applicants visit the schools they are applying to prior to making enrolment decisions, providing a specific instance in which a particular experienced utility (that enjoyed during the visit) may influence predicted future utility (that to be enjoyed from attending the visited school over several years). Importantly, if visitors fall prey to Projection Bias, then transient factors that influence the utility that would be experienced if the prospective student belonged to the visited university on the day of the visit will influence the predicted utility of belonging to such institution in the future. For example, prospective students visiting a school well known for its party life would have a more positive assessment of the utility associated with attending that school if they visited it, say, on a Friday after having worked hard during the preceding week. Because partying that day would be particularly appealing, the projected utility of being able to do so often during the next four years would probably be exaggerated. For a school whose forte was academic, on the other hand, transient factors that made engaging in academic activities more appealing on the day of the visit would increase the predicted utility of engaging in those activities in the future and hence possibly increase enrolment rates. One factor likely to influence the appeal of engaging in academic activities is the weather. Intuition and survey evidence presented in Section 2 suggest that cloudy weather, probably due to the sadder mood it induces and the reduced opportunity cost of outdoor activities it creates, is more inviting to academic activities than sunny weather is. Visitors of an academically demanding institution, then, may be more prone to enrolling in such an institution after visiting it on a cloudier day, since academic work would have seemed more inviting during their visit and hence been projected as more inviting into the future as well.1 In this article I test this prediction, analysing enrolment decisions of 1,284 prospective college students who visited a university well known for its extremely challenging academic environment.2 Consistent with the logic put forward above, I find that cloudiness during visits has a statistically and practically significant impact on enrolment rates: an increase in cloudiness of one standard deviation on the day of a visit is associated with an increase in the enrolment probability of around 9 percentage points. Adding controls for average weather conditions for the calendar date of the visit and month dummies leaves the results unchanged, ruling out the possibility that this pattern arises as a result of a time‐of‐year confound. Employing the admission rather than the enrolment decision as the dependent variable, there is no impact of cloudcover, suggesting the pattern is not due to self‐selection into interviews as a function of weather on the day of the visit. There is an important difference between the main finding of this article and existing studies documenting Projection Bias. Considering that visitors are probably not deciding whether to enrol in the visited school during their visit, cloudiness must be influencing college decisions through memory. Notably, visitors are influenced by an incidental factor which is no longer present at the time they are making a decision. Rather than projecting current utility, people appear to be projecting their remembered utility. Such a process is consistent with the findings from the seminal paper by Dutton and Aron (1974), where men who met a woman in a situation of high arousal (after having just crossed a suspension bridge) were more likely to call her, at a future time, than those who met her in a situation of normal arousal (at least 10 minutes after having crossed that same bridge). In relation to the classical example of shopping on an empty stomach, these results are equivalent to demonstrating that foods tasted for the first time on an empty stomach are remembered as more enjoyable and might hence be disproportionately likely to be purchased again. The remainder of the article is organised as follows: Section 1 presents survey evidence consistent with cloudy weather increasing the appeal of academic activities, Section 2 reports the results from the visits and enrolment data and Section 3 concludes. 1. Cloudiness and Academics Academic related activities and goods are likely to be more appealing under cloudier weather for at least two reasons. First, cloudiness induces sad moods (Cunningham, 1979; Hirshleifer and Shumway, 2003; Rind, 1996), making mellow activities like reading or studying more appealing. Secondly, sunny weather increases the appeal of outdoor activities like practising sports or hiking, increasing the opportunity cost of engaging in academic activities. To assess the validity of the hypothesised link between current cloudiness and the appeal of academic activities empirically I included two questions in separate surveys carried out at an Ivy League University. The first directly asked people to reveal whether they find studying more appealing on cloudy or sunny days and the second asked a different set of respondents to indicate on which of two days, a cloudy or a sunny one, they would prefer to complete a 4‐hour long school assignment. In particular, the question inserted into one survey was (N = 37): Recent studies show that some people find it more appealing to study and do homework on cloudy days while others find it more appealing on sunny days. In your personal experience, when do you find school work more appealing (or less aversive)? Figure 1 shows the distribution of answers to this question (on a 1–7 scale, where 1 is definitely more appealing on sunny days and 7 is definitely more appealing on sunny days). The results are consistent with cloudiness increasing the marginal utility of studying; the average answer was M =5.13, significantly greater than the neutral answer of four (t(36) = 4.13, p = 0.0002). Similarly, the majority of respondents, 78%, chose a number greater than four compared to just 14% choosing a number smaller than four. Fig. 1. Open in new tabDownload slide Distribution of Answers from Student Survey (N = 37) to Question ‘Do you find it more appealing to study on cloudy or sunny days’? Fig. 1. Open in new tabDownload slide Distribution of Answers from Student Survey (N = 37) to Question ‘Do you find it more appealing to study on cloudy or sunny days’? The question inserted into the other survey (N = 137) was: Suppose you have a project due for class which would take 4 hours of work (including reading from books, writing quick summaries and searching for information on the internet), and you can do it either tomorrow or the day after tomorrow. In making the decision, consider that [tomorrow/the day after tomorrow] is forecast to be a dark cloudy day, and [the day after tomorrow/tomorrow] a bright and sunny one. On which of the two days you think you would prefer to do the work? (Subjects responded using a 1–10 scale where 1 is Definitely Tomorrow and 10 is Definitely Day After Tomorrow.) Half the subjects were asked to imagine ‘tomorrow’ would be sunny and ‘the day after’ cloudy, and half the subjects were asked to imagine the opposite. The distribution of answers is reported in Figure 2. Fig. 2. Open in new tabDownload slide Distribution of Answers from Student Survey (N = 137) Asking Students to Decide Whether to Conduct 4 Hours of School‐work‘tomorrow’or‘the day after tomorrow’, Randomly Varying Which of the Two Days Would be Sunny or Cloudy Fig. 2. Open in new tabDownload slide Distribution of Answers from Student Survey (N = 137) Asking Students to Decide Whether to Conduct 4 Hours of School‐work‘tomorrow’or‘the day after tomorrow’, Randomly Varying Which of the Two Days Would be Sunny or Cloudy Respondents expressed a clear preference for completing the work ‘tomorrow’ when ‘tomorrow’ was going to be a cloudy day (M = 2.9) but the reversed preference was obtained when ‘tomorrow’ was going to be a sunny day (M = 6.4), t(151) = 9.08, p < 0.0001.3 Similarly, 33% of subjects in the ‘cloudy tomorrow’ condition chose ‘1‐Definitely tomorrow’ compared to just 9% in the ‘sunny tomorrow’ condition, a statistically significant difference χ2 = 13.1, p = 0.0003. The survey evidence, therefore, strongly suggests that current cloudiness increases the (relative) marginal utility of studying. Additional evidence of a complementarity between cloudiness and taste for academic ‘goods’ comes from a recent paper where I analyse admission decisions made by university admission reviewers, finding that reviewers accept academically stronger candidates when reviewing applications on cloudier rather than sunnier days (Simonsohn, 2007). 2. College Visits Data 2.1. Visits Data The college visits data was provided by the admissions office of a private university in the northeastern US, which is, as mentioned in the Introduction, well known for its academic strengths and recreational weaknesses. The dataset consists of the university’s records of 1,284 interviews with undergraduate applicants. Interviews are conducted by an admission specialist from the university and are designed primarily to help students learn more about the school they are applying to. They are voluntary and are not part of the admission process per se. Students typically sign up for interviews ahead of time, combining them with a campus visit. The dataset includes information on the date of the campus visit, whether the applicant was admitted to the university, and (conditional on being accepted) whether s/he chose to enrol or not. In total the dataset contains information on 1,284 visitors, 562 of which (44%) were admitted, 259 of which (46%) enrolled. Not surprisingly, given the self‐selection involved in deciding to visit campus, both of these rates are higher than those for the full pool of applicants. 2.2. Weather Data Weather data on temperature, wind speed, precipitation and cloudcover were obtained from the National Oceanic and Atmospheric Administration’s (NOAA) website for the academic year for which the visits data are available and for the preceding 5 years. The historical weather data were used to construct average weather conditions for each calendar date of the year, providing useful time‐of‐year controls. Although all weather variables just listed are utilised in the analyses that follow, the variable of primary interest is cloudcover and hence it is useful to provide some additional information on it. Cloudcover is measured, in the weather station of the city of interest, on a discrete 0‐clear skies to 10‐complete overcast scale.4Figure 3 shows a histogram with the relative frequencies of the 11 possible values, with visits (rather than calendar dates) as the unit of observation. It shows ample variation within the sample (M = 6.77, SD = 2.79). Fig. 3. Open in new tabDownload slide Distribution of Cloudcover Across Days Applicants in Sample Visited Campus (Each visit (rather than calendar date) is an Observation) Fig. 3. Open in new tabDownload slide Distribution of Cloudcover Across Days Applicants in Sample Visited Campus (Each visit (rather than calendar date) is an Observation) An interesting feature of cloudcover is that it experiences significant short‐term fluctuations. Table 1 shows correlations of cloudcover on day t with cloudcover on days t − 1, t − 2 and t − 3. Column (1) shows raw correlations and Column (2) partials out month fixed effects (i.e. it reports the correlations for residuals from regressions with month dummies as the only predictors of cloudcover). The Table shows that cloudcover varies sufficiently during short periods of time that the correlation between cloudcover today and cloudcover in 3 days is not statistically different from 0. When month fixed effects are partialled out, the correlation between cloudcover in t and in t − 2 is not statistically significant (r = 0.02). This already suggests that any association between cloudcover and enrolment is unlikely to be caused by a time‐of‐year confound. Table 1 
Correlations of Cloudover (0–10 scale) Across Proximate Days . Raw correlations with cloudcover on day t . Correlations with cloudcover on day t net of month fixed effects . Cloudcover on day t 1 1 Cloudcover on day t − 1 0.41*** 0.36*** Cloudcover on day t − 2 0.10** 0.02 Cloudcover on day t − 3 0.05 −0.04 . Raw correlations with cloudcover on day t . Correlations with cloudcover on day t net of month fixed effects . Cloudcover on day t 1 1 Cloudcover on day t − 1 0.41*** 0.36*** Cloudcover on day t − 2 0.10** 0.02 Cloudcover on day t − 3 0.05 −0.04 Table reports correlations in cloudcover across proximate days (cloudcover is measured in a discrete 0–10 scale by the local weather station). Column 1 reports raw correlations while column 2 partialing out monthly fixed effects. *,**,*** indicates significance at the 10%, 5% and 1% level respectively Open in new tab Table 1 
Correlations of Cloudover (0–10 scale) Across Proximate Days . Raw correlations with cloudcover on day t . Correlations with cloudcover on day t net of month fixed effects . Cloudcover on day t 1 1 Cloudcover on day t − 1 0.41*** 0.36*** Cloudcover on day t − 2 0.10** 0.02 Cloudcover on day t − 3 0.05 −0.04 . Raw correlations with cloudcover on day t . Correlations with cloudcover on day t net of month fixed effects . Cloudcover on day t 1 1 Cloudcover on day t − 1 0.41*** 0.36*** Cloudcover on day t − 2 0.10** 0.02 Cloudcover on day t − 3 0.05 −0.04 Table reports correlations in cloudcover across proximate days (cloudcover is measured in a discrete 0–10 scale by the local weather station). Column 1 reports raw correlations while column 2 partialing out monthly fixed effects. *,**,*** indicates significance at the 10%, 5% and 1% level respectively Open in new tab 2.3. Regression Analyses To assess the impact of cloudcover during a visit on subsequent enrolment decisions, a linear probability model was estimated with visitors as the unit of observation, enrolment (1‐yes, 0‐no) as the dependent variable and all of the weather variables experienced during the visit as independent variables (neither qualitatively nor in significance do the results change if a logistic regression is estimated instead). Since counterfactual enrolment decisions by visitors who were not admitted are not observed, the sample is restricted to the 562 visitors who were admitted. The results are presented on Table 2. Column 1 presents the baseline specification where only the key independent variable, cloudcover, is included in the regression. The point estimate is positive and significant at the 5% level. It indicates that in response to a one point change in cloudcover experienced during the college visit, the probability of enrolment, conditioning on acceptance, changes by around 1.8 percentage points. Table 2 
Impact of Cloudcover on Enrolment and Admission (OLS) Dependent variable (1‐yes, 0‐no) . (1) (2) (3) (4) (5) Enrolled? Enrolled? Enrolled? Enrolled? Admitted? Baseline Adds other weather variables Adds Average weather conditions Predicts with weather from two days prior to visit Same as (3) but with admission decision as dependent variable Intercept 0.342*** 0.180 −0.013 0.407*** 0.538** (0.055) (0.164) (0.353) (0.137) (0.210) Cloud Cover on day of visit (0‐clear skies to 10‐overcast) 0.018** 0.027** 0.032** – 0.004 (0.008) (0.011) (0.012) – (0.008) Cloud Cover two days prior to visit – – – 0.001 – – – – (0.000) – Maximum Temperature (max) – 0.004 0.003 0.000 0.000 – (0.004) (0.004) (0.004) (0.003) Minimum Temperature (min) – −0.002 −0.005 0.001 −0.002 – (0.004) (0.005) (0.004) (0.003) Wind Speed (miles per hour) – −0.004 −0.005 0.002 −0.003 – (0.003) (0.004) (0.004) (0.002) Rain precipitation (in inches) – −0.056 −0.024 −0.076 0.026 – (0.091) (0.119) (0.144) (0.078) Snow precipitation (in inches) – 0.008 0.009 0.002 0.007 – (0.008) (0.009) (0.008) (0.006) Average weather conditions for calendar date (DF = 6) No No Yes No Yes Month dummies No No Yes No Yes Number of Observations 562 562 562 562 1,284 R2 0.0096 0.0146 0.0573 0.0018 0.0279 Dependent variable (1‐yes, 0‐no) . (1) (2) (3) (4) (5) Enrolled? Enrolled? Enrolled? Enrolled? Admitted? Baseline Adds other weather variables Adds Average weather conditions Predicts with weather from two days prior to visit Same as (3) but with admission decision as dependent variable Intercept 0.342*** 0.180 −0.013 0.407*** 0.538** (0.055) (0.164) (0.353) (0.137) (0.210) Cloud Cover on day of visit (0‐clear skies to 10‐overcast) 0.018** 0.027** 0.032** – 0.004 (0.008) (0.011) (0.012) – (0.008) Cloud Cover two days prior to visit – – – 0.001 – – – – (0.000) – Maximum Temperature (max) – 0.004 0.003 0.000 0.000 – (0.004) (0.004) (0.004) (0.003) Minimum Temperature (min) – −0.002 −0.005 0.001 −0.002 – (0.004) (0.005) (0.004) (0.003) Wind Speed (miles per hour) – −0.004 −0.005 0.002 −0.003 – (0.003) (0.004) (0.004) (0.002) Rain precipitation (in inches) – −0.056 −0.024 −0.076 0.026 – (0.091) (0.119) (0.144) (0.078) Snow precipitation (in inches) – 0.008 0.009 0.002 0.007 – (0.008) (0.009) (0.008) (0.006) Average weather conditions for calendar date (DF = 6) No No Yes No Yes Month dummies No No Yes No Yes Number of Observations 562 562 562 562 1,284 R2 0.0096 0.0146 0.0573 0.0018 0.0279 Notes. Table reports point estimates in linear probability models with college visitors as the unit of observation. In columns 1–4 the dependent variable equals 1 if the visitor enrolled and 0 if she did not. In column 5 the dependent variable equals 1 if the visitor was admitted and 0 otherwise. Standard errors reported below parameter estimates. Average weather conditions correspond to averages for each of the weather variables presented in the Table over the 5 preceding years. *,**,*** Indicates significance at the 10%, 5% and 1% level respectively. Open in new tab Table 2 
Impact of Cloudcover on Enrolment and Admission (OLS) Dependent variable (1‐yes, 0‐no) . (1) (2) (3) (4) (5) Enrolled? Enrolled? Enrolled? Enrolled? Admitted? Baseline Adds other weather variables Adds Average weather conditions Predicts with weather from two days prior to visit Same as (3) but with admission decision as dependent variable Intercept 0.342*** 0.180 −0.013 0.407*** 0.538** (0.055) (0.164) (0.353) (0.137) (0.210) Cloud Cover on day of visit (0‐clear skies to 10‐overcast) 0.018** 0.027** 0.032** – 0.004 (0.008) (0.011) (0.012) – (0.008) Cloud Cover two days prior to visit – – – 0.001 – – – – (0.000) – Maximum Temperature (max) – 0.004 0.003 0.000 0.000 – (0.004) (0.004) (0.004) (0.003) Minimum Temperature (min) – −0.002 −0.005 0.001 −0.002 – (0.004) (0.005) (0.004) (0.003) Wind Speed (miles per hour) – −0.004 −0.005 0.002 −0.003 – (0.003) (0.004) (0.004) (0.002) Rain precipitation (in inches) – −0.056 −0.024 −0.076 0.026 – (0.091) (0.119) (0.144) (0.078) Snow precipitation (in inches) – 0.008 0.009 0.002 0.007 – (0.008) (0.009) (0.008) (0.006) Average weather conditions for calendar date (DF = 6) No No Yes No Yes Month dummies No No Yes No Yes Number of Observations 562 562 562 562 1,284 R2 0.0096 0.0146 0.0573 0.0018 0.0279 Dependent variable (1‐yes, 0‐no) . (1) (2) (3) (4) (5) Enrolled? Enrolled? Enrolled? Enrolled? Admitted? Baseline Adds other weather variables Adds Average weather conditions Predicts with weather from two days prior to visit Same as (3) but with admission decision as dependent variable Intercept 0.342*** 0.180 −0.013 0.407*** 0.538** (0.055) (0.164) (0.353) (0.137) (0.210) Cloud Cover on day of visit (0‐clear skies to 10‐overcast) 0.018** 0.027** 0.032** – 0.004 (0.008) (0.011) (0.012) – (0.008) Cloud Cover two days prior to visit – – – 0.001 – – – – (0.000) – Maximum Temperature (max) – 0.004 0.003 0.000 0.000 – (0.004) (0.004) (0.004) (0.003) Minimum Temperature (min) – −0.002 −0.005 0.001 −0.002 – (0.004) (0.005) (0.004) (0.003) Wind Speed (miles per hour) – −0.004 −0.005 0.002 −0.003 – (0.003) (0.004) (0.004) (0.002) Rain precipitation (in inches) – −0.056 −0.024 −0.076 0.026 – (0.091) (0.119) (0.144) (0.078) Snow precipitation (in inches) – 0.008 0.009 0.002 0.007 – (0.008) (0.009) (0.008) (0.006) Average weather conditions for calendar date (DF = 6) No No Yes No Yes Month dummies No No Yes No Yes Number of Observations 562 562 562 562 1,284 R2 0.0096 0.0146 0.0573 0.0018 0.0279 Notes. Table reports point estimates in linear probability models with college visitors as the unit of observation. In columns 1–4 the dependent variable equals 1 if the visitor enrolled and 0 if she did not. In column 5 the dependent variable equals 1 if the visitor was admitted and 0 otherwise. Standard errors reported below parameter estimates. Average weather conditions correspond to averages for each of the weather variables presented in the Table over the 5 preceding years. *,**,*** Indicates significance at the 10%, 5% and 1% level respectively. Open in new tab Column 2 adds other weather variables for the day of the visit. None of them is statistically significant, and a joint test of all non‐cloudcover variables equalling zero is not rejected (p = 0.83). Controlling for other weather variables the point estimate for cloudcover increases in both size and significance. 2.4. Time‐of‐year as a Possible Confound Heterogeneity in ex ante likelihood of enrolling across visitors coming at different times of the year is the most plausible confound for the documented relationship between cloudcover and enrolment rates. The specifications reported in columns 3 and 4 of Table 2 attempt to address this time‐of‐year concern. In particular, Column 3 adds as controls average weather conditions for the calendar date of the visit from the five previous years. For example, when attempting to predict the enrolment decision of a visitor on October 15th of a given year, the regression controls not only for all other weather variables on that day but also for the average of all these variables, including cloudcover, on October 15th of the preceding 5 years. Column 3 also adds month dummies to further take into account any systematic time‐of‐year variation in enrolment probability. Contrary to what would be expected in the presence of a time‐of‐year confound, the point estimate for cloudcover in column (3) increases slightly in both size and significance with respect to column (2). The point estimate from column (3) indicates than a change in cloudcover of one standard deviation is associated with a change in probability of enrolling, conditional on being accepted, of roughly 9 percentage points. Column (4) takes an alternative approach for ruling out the time‐of‐year confound. It regresses enrolment decisions not on cloudcover on the day of the visit but on cloudcover two days prior to it. Any reasonable time‐of‐year confound story would predict a very similar effect for columns (4) and (3). The point estimate for cloudcover in column (4), however, is very small and not statistically significant, further suggesting it is actual cloudcover on the day of the visit rather than time‐of‐year proxied by such variable that is leading to the significant relationship that is documented. 2.5. Self‐selection into the Sample as Possible Confound A spurious relationship between cloudcover and enrolment could also be the result of the following selection bias: students’ex ante likelihood of enrolling in the visited school affects how they respond to weather conditions on the day of their interview and, in particular, students with lower inclination to enrol are less likely to show up for their interview on a bad‐weather day. If this type of self‐selection occurred, days of bad weather would over‐represent enthusiastic students (because the dataset includes only students who show up for interviews) generating a spurious relationship between cloudcover and enrolment. There are several reasons to doubt that such a process could be behind the results. First of all, the process itself seems implausible: although one may imagine that in the presence of extreme weather conditions (e.g. snowstorms or high winds) students may refrain from attending a pre‐scheduled interview, it is unlikely that students would cancel appointments because the sky is ‘too grey’. If weather correlates with enrolment rates because unenthusiastic students cancel appointments on bad weather days, temperature, wind and precipitation would be expected to be the strongest predictors of behaviour, not cloudcover. To test this alternative explanation empirically, it would be desirable to obtain data on students who scheduled interviews and do not show up for them but such data are regrettably unavailable. Through personal conversations with the admissions office, however, I was assured that interviews are very rarely cancelled. There are other predictions arising from the self‐selection story that can be tested with the available data. One of them relies on the admission decision. If students who choose to attend an interview conditional on it being a cloudy day differ from those that do so on sunny days, cloudcover might predict whether a student is admitted to the university. To test this possibility column 5 in Table 2 reports the results from a linear probability model where admission (1‐yes, 0‐otherwise) is the dependent variable and weather conditions on the day of the visit were the explanatory ones. Neither cloudcover nor any other weather variable on the day of the visit correlate with the chances of being admitted (all p‐values greater than 0.4). Furthermore, the joint test of all coefficients being zero, including cloudcover, cannot be rejected (p = 0.54). Perhaps most tellingly, if cloudiness affects students’ decisions to show up to their interview, total number of interviews per day and cloudcover should be correlated, yet they are not (r = 0.04, p = 0.58). 2.6. Correlation with Cloudcover at Competing Universities The last concern I address is the plausibility and possible consequences of cloudcover being correlated across different schools visited by the same student. The fact that cloudcover within the same city has a correlation of 0 between two non‐consecutive days (see Table 1) already suggests that a significant correlation in cloudcover across different schools is unlikely, as students would almost certainly visit different schools on different days. Once one considers that visitors differ in the set of schools they visit, in the order in which they visit them and in how much time they place between visits, it becomes virtually impossible that there might be a significant correlation between the cloudcover visitors to the school providing the data experienced and what they experienced in the other schools they visited (recall that the regressions partial out seasonal variation in cloudcover). However implausible, it is worth considering the possible consequences of cloudcover being correlated across different schools. To this end consider a stylised linear decision model for student i deciding between schools j and k, where the probability of enrolling in school j, Ei,j, is a function of cloudcover experienced while visiting j and k, Ci,j and Ci,k respectively, as in: If Ci,k is not included in the regression, then no longer equals b, but rather . The sign of the bias in the estimation of b depends on the sign of d and of Cov(Ci,j, Ci,k). The most plausible sign for d is the opposite of b’s; if higher cloudcover in j increases enrolment into j, then it reduces enrolment into k. Since should be negative. In terms of the correlation in cloudcover between cities; if contrary to the evidence from Table 1 Cov(Ci,j, Ci,k) ≠ 0, its most plausible sign is positive; schools visited by a given student over a short period of time will tend to be geographically close and hence affected by similar, rather than dissimilar, weather conditions over short periods of time. If as is argued here d < 0 and Cov(Ci,j, Ci,k) > 0, the net effect of omitting Ci,k from the regression would be to bias towards 0. In sum, concerns about possible omitted variable bias caused by unobserved cloudcover conditions in other schools seems unwarranted: cloudcover across school visits is almost certainly uncorrelated and, if it was correlated, by far the most likely consequence would be attenuation of the estimated impact of cloudcover on enrolment decisions. 3. Conclusions Economic models assume away any difficulty in predicting future utility for making intertemporal decisions. Abundant empirical work, however, has shown that predicting future utility is actually quite difficult and that, in particular, people tend to exaggerate the degree to which their future utility will resemble current utility, a phenomenon referred to as Projection Bias. In this article I assess whether such prediction errors are detectable in one of life most thought‐about decisions, college choice. I find that prospective students visiting the campus of a very competitive university showed a greater tendency to enrol in such university the cloudier the weather was during their visit. This is consistent with the proposition that because cloudiness makes belonging to an academically challenging institution more appealing (less aversive?) today, it biases upwards the estimated future utility of attending that university. Unlike previous studies of Projection Bias which focus on the impact of current utility on predicted utility, here it is most likely the impact of remembered utility that drives the effect. The fact a decision as important as which college to enrol in can be influenced by such trivial and transparently irrelevant transient factor as cloudcover on a single day, suggests that projection bias is likely to be a rather general phenomenon, probably playing a role in an important share of intertemporal decisions. Economists interested in predicting future consumption, or in inferring consumers’ future preferences based upon consumers’ current decisions involving future consumption, should take into account factors that influence current utility, particularly if such factors are likely to change between the moment when a decision is made and when its consequences will be experienced. Footnotes 1 " Note that in order for cloudiness to increase enrolment rates it is not necessary for the visited school to be the academically strongest option in the applicant’s choice set, as would be the case if assessing the impact of cloudcover experienced during the day in which applicants make final enrollment decisions. Because here I study the impact of cloudcover during a school visit, a much weaker condition is required, simply that the visited school’s academic attributes be more favorable than its non‐academic ones, such that greater cloudiness during the visit increases the appeal of the university’s forte. 2 " Although the identity of the school that facilitated the enrolment data cannot be disclosed, it is informative that a recent college guide describes it with ‘sleep, friends, work, choose two’, and that online reviews by its alumni include in its pros: ‘strong education’, ‘great professional concentration’ and ‘terrific academics’ and its cons ‘socialising [is] difficult’ and ‘get[s] boring’). 3 " The difference is also significant at the 0.0001 level in a non‐parametric sign test. 4 " Other weather stations, and this weather station in previous years, measure cloudcover as the percentage of all daylight minutes in the day in which there was sunshine. References Ariely , D. and Loewenstein , G. ( 2006 ). ‘The heat of the moment: the effect of sexual arousal on sexual decision making’ , Journal of Behavioral Decision Making , vol. 19 , pp. 87 – 98 . Google Scholar Crossref Search ADS WorldCat Conlin , M. , O’Donoghue , T. and Vogelsang , J. ( 2007 ). ‘Projection bias in catalog orders’ , American Economic Review , vol. 97 ( 4 ), pp. 1 – 33 . Google Scholar Crossref Search ADS WorldCat Cunningham , M.R. ( 1979 ). ‘Weather, mood, and helping behavior: quasi experiments with the sunshine samaritan’ . Journal of Personality & Social Psychology , vol. 37 ( 11 ), pp. 1947 – 56 . Google Scholar Crossref Search ADS WorldCat Dutton , D.G. and Aron , A.P. ( 1974 ). ‘Some evidence for heightened sexual attraction under conditions of high anxiety’ , Journal of Personality and Social Psychology , vol. 30 ( 4 ), pp. 510 – 7 . Google Scholar Crossref Search ADS PubMed WorldCat Gilbert , D.T. , Gill , M.J. and Wilson , T.D. ( 2002 ). ‘The future is now: temporal correction in affective forecasting’ , Organizational Behavior and Human Decision Processes , vol. 88 ( 1 ), pp. 430 – 44 . Google Scholar Crossref Search ADS WorldCat Hirshleifer , D. and Shumway , T. ( 2003 ). ‘Good day sunshine: stock returns and the weather’ , Journal of Finance , vol. 58 ( 3 ), pp. 1009 – 32 . Google Scholar Crossref Search ADS WorldCat Loewenstein , G. , O’Donoghue , T. and Rabin , M. ( 2003 ). ‘Projection bias in predicting future utility’ , Quarterly Journal of Economics , vol. 118 ( 4 ), pp. 1209 ‐ 48 . Google Scholar Crossref Search ADS WorldCat Loewenstein , G. and Schkade , D. ( 1999 ) ‘Wouldn’t it be nice? Predicting future feelings’, in ( Kahneman D., Diener E. and Schwartz N. eds.), Understanding Well Being: The Foundations of Hedonic Psychology , pp. 85 – 105 , New York: Russell Sage Foundation . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Rind , B. ( 1996 ). ‘Effect of beliefs about weather conditions on tipping’ , Journal of Applied Social Psychology , vol. 26 ( 2 ), pp. 137 – 47 . Google Scholar Crossref Search ADS WorldCat Simonsohn , U. ( 2007 ). ‘Clouds make nerds look good’ , Journal of Behavioral Decision Making , vol. 20 , pp. 143 – 52 . Google Scholar Crossref Search ADS WorldCat © The Author(s). Journal compilation © Royal Economic Society 2009