Formal Theory for Comparative Politics

Transcription

1 Formal Theory for Comparative Politics Avidit Acharya January 2013 I taught this material in the first half of a class called Formal Modeling in Comparative Politics to second year graduate students in the political science and economics departments at the University of Rochester. The second half of the class was mainly student presentations. This material was taught to University of Rochester graduate students in political science and economics in the spring of Assistant Professor of Political Science and Economics at the University of Rochester. 1

2 Contents 1 Political Economy Redistributive Politics (Meltzer & Richard 1981) Non-distortionary Taxation Targetted Transfers (Lindbeck & Weibull 1987) Corruption and Party Bias (Acharya, Roemer & Somanathan 2012) Progressive Income Taxation (Roemer 1999) Political Economy with Strategic Voting (Acharya 2012) Comparative Politics and Public Finance (Persson, Roland & Tabellini 2000) 19 2 Dynamic Methods, etc Infinite Horizon Dynamic Programming Dynamic Game Theory Weak and Strong States (Acemoglu 2005) Political Transitions (Acemoglu & Robinson 2001) Global Games (Carlsson & van Damme 1993) The Global Game of Revolution (Acharya 2009) Persistent Effects of Colonization in Africa (Nunn 2007) Behavioral Political Economy Ideology (Benabou 2008) Social Identity Equilibrium (Shayo 2009)

3 1 Political Economy I assume that you already know about single-peaked preferences, the single-crossing property, sincere vs. strategic voting, and Downsian parties. You should also be somewhat familiar with the basics of the probabilistic voting model. Finally, you should be familiar with the difficulties in generating existence of Nash equilibrium in pure strategies when parties are Downsian and the policy-space is multidimensional. If you do not know about these things, please review your notes from previous classes and find out about them quickly! 1.1 Redistributive Politics (Meltzer & Richard 1981) There is a unit mass continuum of individuals. Each agent has a taste -parameter θ. The distribution of the taste parameter is given by the function F, which has a continuous density f on its support [θ, θ + ]. The distribution is skewed so that the median taste parameter θ m is strictly less than the mean θ, which is quite large. Specifically, we assume (i) 0 < θ < θ + < 2θ and (ii) θ m < θ (A1) What does the taste parameter capture? Assume that all individuals can produce y(e) = e units of output by exerting e 0 units of effort, but they each dislike putting in effort. In particular, an individual with taste parameter θ has preferences over consumption and effort given by u(c, e θ) = c e2 (1) 2θ where c is his consumption and e 0 is his effort. Thus, θ is a parameter that reflects an agent s propensity to work: If θ is small, then the individual has a large distaste for work, but if it is large then he doesn t. Because θ > 0 all agents have some degree of distaste for work. Now, suppose that once the individuals make their effort decisions, the government taxes their output at the rate τ and redistributes all of its revenue as (undirected) lumpsum transfers. If all individuals of type θ exert e(τ, θ) units of effort when the tax rate is τ, then the government collects T = θ + θ τy(e(τ, θ))f(θ)dθ τy (2) units of revenue. Here, Y is the total output produced. Since there is a unit mass of agents, this is also the amount that each agent receives in return from the government. Therefore, an individual of type θ who exerts e units of effort gets to consume c = (1 τ)y(e) + T = (1 τ)e + τy (3) 3

4 units of output. Note that because each agent is infinitesimal, he is negligible: the effort decision e has no effect on T (or Y ). How is the tax rate determined? We assume that prior to the agents exerting their effort, there is a majoritarian election in which two Downsian political parties, A and B, announce (and commit to) a tax rate τ [0, 1]. Thus, these parties compete over the issue of fiscal policy. All voters vote sincerely for the party whose policy will give them a higher utility. Voters who are indifferent toss a coin. If the election is tied, then each party comes to power with probability 1/2. The party that comes to power implements its announced policy. So, the timing of events is as follows. 1. Parties announce τ A and τ B. 2. Voting takes place, and the winner is determined. 3. All agents exert effort and produce output. 4. The party in power implements the policy that it announced, so tax and redistribution take place. 5. Consumption takes place. A voter of type θ has preferences over tax and effort given by (1 τ)e + τy e2 2θ (4) This expression is strictly concave in effort, so we can find the optimal effort by solving the first order condition. The solution is e (τ, θ) = (1 τ)θ (5) Here we have used the fact that the agent s effort decision does not affect total output Y. We can see that the agent s optimal effort is increasing in his taste parameter θ and decreasing in τ. The reason it is decreasing in τ is because taxes are distortionary: when agents get to keep less of their output and their effort does not affect how much they get from the government, they work less. We can now use the optimal effort decisions to compute total output θ + Y (τ) = y(e (τ, θ))f(θ)dθ = (1 τ)θ (6) θ This reflects the distortionary effect of taxation on the aggregate economy. As the tax rate τ increases, total output decreases because agents exert less effort. 4

5 Figure 1: τ (θ) Now let us substitute back the optimal effort e (τ, θ) and total output Y (τ) into the agent s preferences in (4). This gives us the the agent s preference over the tax rate τ: v(τ θ) = θ 2 (1 τ)2 + τ(1 τ)θ (7) By assumption A1(i), the above expression is strictly concave in τ for all agent types θ. This means that we can find type θ s most preferred tax rate τ (θ) by solving the first order condition to the maximization of (7) subject to τ [0, 1]. The solution is τ (θ) = { θ θ 2θ θ if θ < θ 0 if θ θ (8) One can see that by assumption A1(i), the optimal feasible tax rate satisfies τ (θ) < 1 for all θ [θ, θ + ] and is positive only if θ < θ. By plotting τ (θ) one can see that the optimal tax rate is decreasing in θ. (See Figure 1.) Since voters preferences are single peaked in τ for all θ, you should immediately know that both parties will announce the most preferred tax rate of a voter of type θ m. Thus, Downsian convergence takes place and the implemented tax rate is always τ (θ m ) = θ θm > 0. (9) 2θ θm The ratio of incomes of any two types θ and θ is simply y(e (τ, θ))/y(e (τ, θ )) = θ/θ, which is invariant to the tax rate. Therefore, fiscal policy does not change the relative positions of any two individuals. Then, observe that the equilibrium policy τ (θ m ) is decreasing with the ratio of median to mean income θ m /θ. Since this ratio is taken to be an inverse 5

6 measure of inequality, the model argues that inequality and redistribution covary, though the relationship is not causal. The equilibrium tax rate τ (θ m ) differs from the efficient tax rate τ fb = 0 (10) that maximizes total output Y (τ). Thus, we have seen the first of many models that argue that economic distortions have political causes. Here, the political cause is the class-conflict between those who dislike to work (low θ agents) versus those who have a greater proclivity to work (high θ agents). Exercise. Consider the following modification to the model above. Suppose that now tax revenue can be used for two things: to provide untargeted lump-sum transfers (as above) and to finance a public good g that results in an additional payoff of H(g) = 1 α (g)α for every agent. Thus, agent θ s preferences are now given by v(c, e, g θ) = u(c, e θ) + H(g) (11) where u(c, e θ) is the function in (1). Parties compete over the linear tax rate τ and the fraction z of total tax revenue T that is used to create the public good. Assume that g units of the public good can be created using exactly g units of the tax revenue. Compute the optimal z as a function of T for an agent of type θ, and show that it is independent of θ. Then show that there exists a Nash equilibrium to the policy announcement game in which both Downsian parties propose the same tax rate as in equation (9). 1.2 Non-distortionary Taxation We could have, instead, started with an even simpler model in which taxes are nondistortionary. Suppose that a unit continuum of agents are indexed by their income y which is distributed by a function G with continuous density g on the support [y, y + ] R +. The mean income y is strictly greater than the median y m. Again, we have a government that taxes at a linear rate τ and redistributes all proceeds as untargeted lump-sum transfers. Note that tax revenues are T = y + y τyg(y)dy = τy (12) Thus an agent with income y has preferences over τ given by (1 τ)y+τy. These preferences are single peaked in τ for every y. So, again, Downsian convergence holds and the optimal tax rate of the median income earner is implemented. Because y m < y this optimal tax rate is τ (y m ) = 1. 6

7 The assumption that median income is lower than the mean income is one that is empirically true for every country in the world. That we don t observe linear tax rates of 100% may be because of the distortionary effects of taxation that Meltzer and Richard (1981) formalized using a model similar to that of the previous section. But many empirical studies have estimated that the distortion must be enormous to explain why the tax rate is as low as it is in many countries where the gap between mean and median income is very large. (Think of Latin America.) A large literature in political economy is concerned with explaining why more redistribution does not take place. Indeed, some scholars consider it to be the most important question in political economy. 1.3 Targetted Transfers (Lindbeck & Weibull 1987) The Meltzer-Richard model assumes that transfers are lump-sum and untargeted. When transfers are targeted across multiple, say three, groups then the policy space becomes multidimensional. Here we outline a popular way to deal with the problem of existence of pure strategy Nash equilibrium when the policy space is multidimensional: probabilistic voting. We also assume that the two parties are maximizing vote share, which simplifies the analysis but is not crucial to probabilistic voting models. Assume that there is a continuum of unit mass citizens partitioned into three groups j = 1, 2, 3, with population shares λ j > 0. Two political parties must decide how to target a resource of 1 unit across these groups. Every individual within a group receives the same amount as every other individual in the group. Thus a policy is triple x = (x 1, x 2, x 3 ) with x j representing the amount that a member of group j receives. The parties propose (and commit to) feasible policies x A and x B. A feasible policy x is a nonnegative vector that satisfies the budget constraint λ j x j = 1 Members of group j have the following preferences over policy j u j (x j ) = (x j ) 1 α (BC) where α (0, 1) is a parameter that reflects diminishing marginal utility. In addition a voter from group j receives a preference shock ε in favor of party B. The shock is drawn uniformly and independently across voters from the interval [ 1 2φ j, 1 2φ j ]. We will assume that these intervals are quite large, in particular φ j < 1 2 j 7

8 So, a voter from group j votes for party A if and (essentially) only if (x A j ) 1 α > (x B j ) 1 α + ε or, ε < (x A j ) 1 α (x B j ) 1 α j (x A j, x B j ) Note that j (x A j, xb j ) is strictly concave in xa j and strictly convex in x B j. Also, for all feasible distributions x A j, xb j, the difference j(x A j, xb j ) lies in the interval [ 1 ] 1 2φ j, 2φ j since j (x A j, xb j ) [ 1, 1] [ 1 ] 1 2φ j, 2φ j. So, the vote share of party A from group j is φ j j (x A j, x B j ) This is because there is a continuum of voters in each group and voters are identical within groups, so the probability that an individual voter votes for party A is also the vote share of party A from that group. The total vote share for party A is then V (x A, x B ) = j λ j φ j j (x A j, x B j ) which is strictly concave in each x A j since j (x A j, xb j ) is strictly concave in xa j. The vote share of party B is 1 V (x A, x B ), which is strictly concave in each x B j. Therefore, the best response of party A to a policy x B of party B can then be found by solving the first order conditions of the maximization of V (x A, x B ) with respect to x A, and subject to the budget constraint (BC). These conditions are simply λ j φ j (1 α)(x A j ) α = µλ j, j = 1, 2, 3 where µ is the Lagrange multiplier associated with the budget constraint (BC). These conditions imply that the ratios of distributions satisfy x A i x A j = ( φi φ j ) 1/α, i, j = 1, 2, 3 So, along with the budget constraint, we have three equations in three unknowns. There are two ratio equations, above, since the others are redundant. (For example, the ratios x A 1 /xa 2 and xa 2 /xa 3 already define the ratio xa 1 /xa 3, and the reciprocal ratios.) And there is the budget constraint (BC). Solving these three equations we find the unique best response for party A x A j = (φ j) 1/α i λ, j = 1, 2, 3 i(φ i ) 1/α which is independent of party B s policy. By a symmetric argument, party B s best response is the same quantity. These mutual best responses give us the unique Nash equilibrium of the electoral competition game between parties. 8

9 A few things to note about the equilibrium. Groups that are more responsive to policy receive more from both parties. Group j is more responsive to policy when φ j is higher, in which case the preference shocks for that groups members are small and voting behavior is mostly determined by the relative comparison of policy. Exercise. The model above assumes that each voter in group j has a bias toward party B if his realized ε is larger than 0, and a bias toward party A if the realized ε is smaller than 0. But since there is a continuum of voters in each group, the average bias across voters in any given group j is 0. This is because we assumed the distribution of ε to be uniform on an interval [ 1 ] 1 2φ j, 2φ j that is centered at 0. Suppose instead that for a voter in group j, the bias ε is drawn uniformly from an interval [ b j 1 2φ j, b j + 1 ] 2φ j, b j 0. Now, on average, members of group j have a bias b j toward party B. Allow b j to be positive or negative, so that they may in fact have a positive average bias towards party A (when b j < 0). Show that there is still a unique (interior) Nash equilibrium in pure strategies when the magnitudes of the average biases b j are not too large. Show that in fact the equilibrium is the same for all small enough values of the biases b j, including the benchmark case b j = 0 for all j. Conclude that there is policy convergence even when there is aggregate party bias. 1.4 Corruption and Party Bias (Acharya, Roemer & Somanathan 2012) Now we provide a probabilistic voting model with policy divergence. To do this, we assume that there is party bias as in the exercise above, and that the parties are maximizing something different from their vote shares. Consider a polity consisting of a set of citizens who belong to two different ethnic groups, called 1 and 2. Fraction λ of the population is of group 1 while the remaining 1 λ is of group 2. There are two political parties, labeled A and B. Party A is identified by the voters with group 1 and party B with group 2. If party i = A, B wins a fraction V i of the total vote share, then it controls exactly that fraction of the government s budget. It can do two things with the budget: spend it on the populace, and take rents for itself. The distribution to the citizens, however, cannot be targeted. This means that every citizen (regardless of ethnic group) must receive the same amount of the good. Thus a party s policy is uni-dimensional. It is represented by a number x i, which is is the fraction of money under party i s control that distributes to the populace, and 1 x i is the fraction that it steals. The fact that parties are associated, by the voters, with ethnic groups means that 9

10 (on average) voters of each ethnic group have biases in favor of their associated party. Specifically, the payoff u i j ( ) to a voter of ethnic group j from voting for party i is u A 1 (x A ) = x A + b 1 + ε 1 u A 2 (x A ) = x A u B 1 (x B ) = x B u B 2 (x B ) = x B + b 2 + ε 2 where b 1, b 2 > 0 are the average biases towards the favored parties in groups 1 and 2, and ε 1, ε 2 are the realizations of random variables independently drawn across voters. We suppose that each ε j is uniform over an interval [ (1 d), (1 d) ], 0 < d < 1 and we will denote this uniform c.d.f. by F. Facing a policy pair (x A, x B ), a citizen of ethnic group 1 votes for party A if and (essentially) only if u A 1 (x A ) > u B 1 (x B ) ε 1 < x A x B + b 1 1 (x A, x B ) and a citizen of ethnic group 2 votes for party A if and (essentially) only if u A 2 (x A ) > u B 2 (x B ) ε 2 < x A x B b 2 2 (x A, x B ) Assuming that 1 (x A, x B ) and 2 (x A, x B ) both lie in the interval [ (1 d), (1 d) ], the vote share of party A (and the fraction of government budget that it controls) is V A (x A, x B ) = λf( 1 (x A, x B )) + (1 λ)f( 2 (x A, x B )) = ( λ 1 (x A, x B ) + (1 λ) 2 (x A, x B ) ) 2(1 d) while the vote share of party B is V B (x A, x B ) = 1 V A (x A, x B ). Now, instead of assuming that the parties wish to maximize their vote shares, we assume that they are venal, and wish only to maximize the rents that they take. Then the the payoff function for party i is Π i (x A, x B ) = (1 x i ) V i (x A, x B ). Notice that party i s payoff is strictly concave in its own policy x i. So, we can compute the best response of party i to policy x i of the other party i by taking the first order condition for the maximization of Π i (x A, x B ). For the moment, we ignore the budget constraint that x i must be a number between 0 and 1. The first order conditions for the two parties are V i (x A, x B ) + (1 x i ) V i (x A, x B ) x i = 0, i = A, B These are two equations in two unknowns. If the solution to these equations are numbers x A, x B [0, 1], and the values of 1 (x A, x B ) and 2 (x A, x B ) both lie in the interior of 10

11 [ (1 d), (1 d) ] at the solution, then we have found a local Nash equilibrium (LNE). 1 The solutions are x A = d λb 1 (1 λ)b 2 3 x B = d + λb 1 (1 λ)b 2. 3 If b 1 and b 2 are small, then both of these numbers lie between 0 and 1 and 1 (x A, x B ) and 2 (x A, x B ) will lie inside [ (1 d), (1 d) ]. So, we have an LNE. Notice that the equilibrium value of x A is decreasing in b 1 and increasing in b 2 while x B is increasing in b 1 and decreasing in b 2. As the ethnic party bias grows for the members of group j = 1, 2, the corruption level of their party increases while the corruption level of the other party decreases. Exercise. The model above assumes that the parties cannot target their distribution to the two groups as in the model of the previous section. Assume now that the parties can target their distribution. In particular, party i can give x i j to each member of group j, keeping 1 λx i 1 (1 λ)xi 2 for itself. Find conditions under which a local Nash equilibrium exists with venal parties, and compute the equilibrium. Which features of the equilibrium are noteworthy? 1.5 Progressive Income Taxation (Roemer 1999) There is a continuum of voters of unit mass. Income is distributed among voters according to a continuous (atomless) probability distribution F on the support [0, 1]. A tax policy is a triple of real numbers (a, b, c). If this policy is implemented, then a voter with income w receives after-tax income aw 2 + bw + c. Taxes are purely redistributive, so the budget balance condition is (aw 2 + bw + c)df (w) = wdf (w) µ. This implies that c = aµ 2 bµ + µ where µ 2 w 2 df (w). So the budget balance condition implies that a tax policy can be denoted simply (a, b) where c is determined residually. The after-tax income (and utility) of a voter with income w at tax policy (a, b) is u(a, b, w) = a(w 2 µ 2 ) + b(w µ) + µ 1 I could in fact make the equilibrium global by assuming that the payoff were instead φu i j ( ), where φ > 0 is a small number in comparison to d. 11

12 Figure 2: The triangle OUV Thus, a voter with income w µ is indifferent between two policies (a, b) and (a, b ) iff aφ(w) + b = a φ(w) + b, where φ(w) = w2 µ 2 w µ So, the indifference curves of a voter with income w µ are straight lines in (a, b)-space of slope φ(w). The indifference curves of a voter with income µ are vertical straight lines. We further restrict a tax-policy (a, b) to satisfy the following assumptions that are maintained throughout. (1.) a(w 2 µ 2 ) + b(w µ) + µ 0 for all w (2.) 2aw + b 0 for all w The first assumption says that no voter can be left with a negative income. The second condition says that after-tax income must be a nondecreasing function of income. This is an incentive compatibility constraint that says that no voter wants to burn money. Exercise. First, show that the conditions (1.) and (2.) above imply that the set of admissible tax policies (a, b) coincides with the triangle OU V in Figure 2. Second, define a tax policy to be (weakly) progressive if after-tax income as a percent of income is a (weakly) decreasing function of income. A (weakly) regressive policy is one in which after-tax income as a percent of income is a (weakly) increasing function of income. A proportional (or linear) tax is one in which after-tax income as a percent of income is constant in income. Show that a tax policy (a, b) is weakly progressive iff it lies in the triangle OUT, weakly regressive iff it lies in the triangle OV T, and proportional iff it lies on the segment OT. Note that the 12

13 Figure 3: The function φ(w) policy O is full redistribution to the mean and T is laissez faire. The tax rate is linearly and decreasing as we move from O to T. Last, prove the following properties of the function φ: (i) φ(0) = µ 2 /µ, and φ is strictly increasing asymptotically to + on interval [0, µ). (ii) φ is strictly increasing, beginning at the value, in the interval (µ, 1], and cuts the income axis at the value w = µ 2. (iii) 2 > φ(1) > φ(0). Conclude that φ is correctly depicted in Figure 3. Now examine Figures 2 and Figure 3. Figure 3 defines the critical income levels w and w. It is straightforward to show that O is the ideal policy of all working class types w < w, V is the ideal policy all wealthy types w > w, and U is the ideal policy of all middle class types in between, w < w < w. To see this, note that Figure 3 implies that the indifference curves of types w < w are negatively-sloped straight lines in (a, b)-space that are less steep than OU and utility is increasing in the south-west direction. So, the best that these types can do is obtain the policy O. As income w increases the indifference curves rotate clockwise. They continue to rotate, becoming positively sloped at middle levels of income, but by the time w = 1 they are again negatively sloped albeit flatter than 13

14 Figure 4: The triangle OUV with policies L and R the segment OU since φ(1) < 2 (which is a fact depicted in Figure 3 and proven by you in the Exercise above). We suppose that there are two parties, called Left and Right, that propose policies t L = (a, b) and t R = (a, b ). Let Ω(t L, t R ) be the set of voters that strictly prefer policy t L to t R, and thus vote for the Left party. We assume that voters who are indifferent mix in an arbitrary way, but as long as Left and Right propose different policies, this set of voters is always measure zero so their behavior will not matter in determining vote shares. The vote share of Left is thus π(t L, t R ) = F (Ω(t L, t R )) and the vote share of Right is 1 π(t L, t R ). Now, we assume that some members of each party are policy-motivated. Specifically, some members of Left have the interest of a voter of income w L < w in mind, whereas some members of Right have the interest of a voter of income w R > w in mind. We call such factions the Militants. The militants of party j want to propose a policy t j that maximizes u(t j, w j ). We also assume that some members of the parties are purely office-motivated in that they care only about maximizing vote-share. We call these fractions the Opportunists. The policy pair (t L, t R ) is a Roemer-Nash equilibrium (RNE) if it is admissible (i.e. each policy lies in the triangle OUV ) and (L) there is no admissible policy t L for Left such that π(t L, t R) π(t L, t R ) and u(t L, w L) u(t L, w L ) with at least one inequality strict; and (R) there is no admissible policy t R for Right such that 1 π(t L, t R ) 1 π(t L, t R ) and u(t R, w R) u(t R, w R ) with at least one inequality strict. 14

15 These state that neither Left nor Right can find deviations that make their Militants better off without making their Opportunists worse off, or make their Opportunists better off without making their Militants worse off. We now have the following result. Theorem 1: There exists an RNE. In particular, there is one such that Left and Right propose distinct strictly progressive policies. Proof: Pick a point L on the interior of segment OU and a point R on the interior of segment UV such that the slope of LR is nearly 2, in particular pick L close to O. Look at Figure 4 for a depiction. Suppose Left proposes the policy L and Right proposes the policy R. Note that the set of types that vote for Left is precisely [0, w + ε) for some small ε. Next, since the income type w L is smaller than w, its indifference curve through the point L is a straight line that is flatter in slope than OU. Moreover, utility is increasing in the south-west direction. Therefore, the set of policies that make income type w L weakly better off (than under policy L) are the policies that lie in the small shaded triangle LOQ. But a deviation to any other policy in this triangle would strictly decrease Left s vote share since it would make the line LR less steep, reducing the set of types that vote for Left from [0, w + ε) to [0, w) with w < w + ε. (Note that the triangle LOQ can really be made as small as one wants by choosing L close enough to O.) Therefore, the equilibrium condition (L) is satisfied. In fact, the only deviations that would increase Left s vote share without reducing w L s utility, or increase w L s utility without reducing Left s vote share are in the shaded cone below the point L in Figure 4; but these are not admissible deviations. A similar argument establishes that the analogously profitable deviations for Right are in the shaded cone above point R; but again these are not admissible. Hence the equilibrium condition (R) is also satisfied, and (L, R) is and RNE. In addition, L and R are strictly progressive policies and L R. Under some slightly stronger assumptions, and a weak refinement of RNE called strong RNE, one can show that Left and Right propose weakly progressive policies in all strong RNE. But the key point to highlight is that by modeling the fact of intra-party competition between Opportunist factions that care about winning and Militant factions that care about protecting the base, one can generate existence of equilibrium in multidimensional political competition while also enhancing a model s realism. 1.6 Political Economy with Strategic Voting (Acharya 2012) There are 2n + 1 voters. With probability λ (0, 1/2) a voter is a high income earner, and with probability 1 λ he is a low income earner. There are two periods and two policies. In the first period, a status quo policy is in effect. Then, an election is held to decide whether 15

16 to continue with the status quo or to switch to a more redistributive policy. The policy that wins the election is implemented in the second period. Under the more redistributive policy, each high income voter receives a payoff y h while each low income voter receives a payoff y l. Under the status quo policy, each high income voter has payoff y h, while each low income voter has probability δ of receiving an opportunity to become a high income voter. If a low income voter receives the opportunity to climb the economic ladder, and he is talented, then he receives a payoff y h. If he is not talented, or if he does not receive the opportunity, then his payoff is y l. Each low income voter has prior probability p (1/2, 1) of being talented. If a voter becomes a high income earner in the first period, he remains a high income earner in the second period. Voters who receive an opportunity to become high income earners in the first period and are unsuccessful learn that they are untalented, and would also be unsuccessful in the following period even if they got the opportunity again. Voters who do not receive the opportunity in the first period do not learn whether or not they are talented. Voters do not directly observe δ, nor does any voter observe the consequences of the first period policy for any other voter. Instead, voters believe that δ is a random variable with continuous density f. Assume that there is a number ν > 0 for which f(δ) > ν δ [0, 1] (A2) If the status quo policy is re-elected, then the value of δ in the second period is the same as in the first. Let δ denote the expected value of δ according to f. Conditional on δ, the expected payoff from the status quo policy for a low income voter is given by y(δ) = (1 δp)y l + δpy h (13) Therefore, the unconditional expected payoff to re-electing the status quo policy for a low income voter who did not receive the economic opportunity is y( δ). His expected payoff to electing the redistributive policy is simply y l. We will assume that (i) y l < y l y h < y h and (ii) y( δ) < y l < y(1) (A3) A strategy for a voter is the probability with with he votes to re-elect the status quo policy. The equilibrium concept is symmetric Bayes Nash equilibrium in weakly un-dominated strategies. All voters are rational, so each voter casts his ballot as if he believes that his vote will decide the election, i.e. he conditions his vote on the event that his vote is pivotal. Call the set of voters that started off as high income earners H, the set of voters who got the opportunity and climbed the socio-economic ladder (because they were talented) L +, the set of voters who got the opportunity but failed to climb the socio-economic ladder 16

17 (because they were not talented) L and the set of voters who did not get the opportunity L 0. The following observation is straightforward. Lemma 1: In every equilibrium of the game, voters in H and L + vote to re-elect the status quo while voters in L vote to switch to the more redistributive alternative. Proof. Voters in H and L + have a weakly dominant strategy to vote for the status quo, while voters in L have a weakly dominant strategy to vote for the more redistributive alternative. Since the equilibrium concept rules out equilibria in weakly dominated strategies, the result follows. Let x denote the probability with which an L 0 voter votes to re-elect the status quo. Lemma 1 implies that equilibria can be identified by entirely by x. Given any symmetric strategy x [0, 1], the unconditional probability of casting a vote for the status quo policy (when H, L + and L voters play their symmetric equilibrium strategy) is π(δ, x) = λ + (1 λ) ( δp + (1 δ)x ) The probability that an L 0 voter is pivotal is ( ) 2n φ(δ x, n) = (π(δ, x)) n (1 π(δ, x)) n n The distribution of the state variable δ given that an L 0 voter is pivotal (viewed as a function of x and n) is f piv φ(δ x, n)(1 δ)f(δ) (δ x, n) = 1 0 φ(ω x, n)(1 ω)f(ω)dω The expectation of δ for an L 0 voter, conditional on pivotal, is δ piv (x, n) = Now, the next result is also straightforward. 1 0 δf piv (δ x, n)dδ Lemma 2: There exists a cutoff δ (0, 1) such that the symmetric strategy x = 0 is part of an equilibrium iff δ piv (0, n) δ, the symmetric strategy x = 1 is part of an equilibrium iff δ piv (1, n) δ, and the symmetric strategy x (0, 1) is part of an equilibrium iff δ piv (x, n) = δ. Proof. Follows immediately from (A2), (13), (A3)(ii) and Lemma 1. Now note that for fixed values of x p, the function π(δ, x) is a strictly monotone in δ on the interval [0, 1]. Therefore, there is a unique number δ (x) that minimizes π(δ, x) 1/2. Now, the following mathematical result will be extremely useful. 17

18 Theorem 2: If x p then for all ε > 0 there is N such that n N implies δ piv (x, n) δ (x) ε If x = p then δ piv (x, n) is a constant number between 0 and 1, for all n. Proof. The proof of the x = p part is straightforward. For the first part, define the function h : [0, 1] R by h( π) = π(1 π) Since π(δ, x) is continuous in its arguments, the composite function h(π(, )) : [0, 1] 2 R is continuous. Moreover, for all x p, the function h(π(, x)) : [0, 1] R is single peaked and maximized by the value of δ [0, 1] that minimizes π(δ, x) 1/2. Thus δ (x) maximizes h(π(, x)). Fix x p and define ε (x) = {δ [0, 1] : δ δ (x) ε} to be an ε-neighborhood of δ (x). Fix any ε. If ε is small enough, then ε (x) [0, 1]. In this case, there exists a small number η ε (0, 1) such that Furthermore, define the set h(π(δ (x), x)) η ε > sup h(π(δ, x)) δ / ε(x) Ω ε (x) = {ω ε (x) : h(π(δ (x), x)) h(π(ω, x)) η ε /2 Note that the set Ω ε (x) must contain a small interval I [0, 1] that in turn contains the number δ (x). Let µ be the length of this interval. We will prove that there is a number N large enough that for all n N we have f piv (δ x, n) > 1 ε. δ ε(x) To see this, note that if ε (x) = [0, 1] then δ ε(x) f piv (δ x, n) = 1, so the above inequality trivially holds. If ε (x) [0, 1] then f piv δ / (δ x, n)dδ = ɛ(x) φ(δ x, n)(1 δ)f(δ)dδ δ / 1 ɛ(x) φ(δ x, n)(1 δ)f(δ)dδ δ / ɛ(x) 0 φ(ω x, n)(1 ω)f(ω)dω ω Ω ɛ(x) φ(ω x, n)(1 ω)f(ω)dω sup δ / ɛ(x) φ(δ x, n) δ / ɛ(x) (1 δ)f(δ)dδ ( ) supδ / ɛ(x) φ(δ x, n) 1 inf ω Ωε(x) φ(ω x, n) ω Ω ε(x) (1 ω)f(ω)dω inf ω Ωε(x) φ(ω x, n) ν ω Ω ε(x) (1 ω)dω ( h(π(δ ) n ( ) (x), x)) η ɛ 1 1 n ηɛ 1 h(π(δ (x), x)) η ɛ /2 1 η ɛ /2 1 2 µ2 ν µ2 ν.

19 So we can choose N large enough such that n N implies that the integral on the left above is smaller than ε. Finally, to prove the theorem, write δ piv (x, n) = δf piv (δ x, n)dδ + δ / ɛ(x) δ ɛ(x) δf piv (δ x, n)dδ. and observe that for n N, this quantity is bounded above by ɛ + (δ (x) + ɛ) = δ (x) + 2ɛ and bounded below by (δ (x) ɛ)(1 ɛ) > δ (x) 2ɛ. Thus, the conditional expectation δ piv ( x, n) is within 2ɛ of δ (x). But ε was arbitrary. The following is an immediate corollary of Theorem 2. Proposition 1: There exists N such that for all n N, it is (part of) an equilibrium for all L 0 voters to vote for the status quo policy with probability 1. Proof. Suppose x = 1. Then δ (1) = 1. So Theorem 2 implies that for n large enough δ piv (x, n) is close to 1. So by Lemma 2, x = 1 is part of an equilibrium. Exercise. Assume that n is large, and characterize all of the equilibria of the model. Bonus Exercise. (Only to be done by those who took Sourav s class last semester and know what information aggregation means.) Which of the equilibria aggregate information and which do not? 1.7 Comparative Politics and Public Finance (Persson, Roland & Tabellini 2000) The exposition follows that of Persson & Tabellini, Ch. 10, with some minor changes in notation. There are three voters j = 1, 2, 3 residing in different districts. Each voter j is represented by a legislator l = 1, 2, 3. Voter j has preferences given by w j = y τ + f j + H(g) where y is his income, τ is the amount he pays in taxes, f j is the pork he receives and H(g) is the value of a public good g, one unit of which can be produced by one unit of tax revenue. H is a strictly concave function that you can assume to be H(g) = z 1 α (g)1 α 19

20 Figure 5: The Legislative Game Suppose that each legislator l captures rents r l from the budget. Then, the budget constraint can be written 3τ = g + f j + r l (BC) j l ) A policy vector is denoted q = (τ, g, (f j ) j, (r l ) l, where all components are nonnegative. ) The status quo (or disagreement ) policy vector is the nasty one q 0 = ( r, 0, (0) j, ( r) l, where r > 0. Legislators have preferences represented by v l = r l + p l R where p l is the probability with which they are re-elected and R represents exogenously set future rents from re-election. We assume that voters can pre-commit to threshold re-election rules given by p l = { 1 if w j (q) w j, j = l 0 otherwise In other words voter j asks his representative to deliver him at least utility w j. If he gets this amount, then he re-elects the legislator; otherwise, he kicks him out of office. The timing of the game is given by Figure 5, and summarized as follows. Nature selects a legislator a to be the agenda setter. Voters pre-commit to the thresholds w j. The agenda setter proposes a policy q. The entire legislature votes; if a majority vote for q then q passes, otherwise q 0 is implemented. The election is held. We will make the following three assumptions 1. R r, so the future has greater value than that of sabotaging policy today 2. R + r (z) 1/α > 0, which will imply that pork transfer to the agenda setters district are positive in equilibrium 20

21 3. 3y R r > 0, so that there is lots of money in the economy 4. 3y > (3z) 1/α, so the socially optimal level of the public good is affordable We first characterize the socially optimum level of public goods from the perspective of the voters. To do this, define the social welfare function (over policies) to be the sum of only the voters utility W (q) = j w j = 3(y τ) + j f j + 3H(g) The social planner maximizes this function. Proposition 2: At the solution to the maximization of the planner s objective function W (q) subject to the budget constraint (BC), we have g sp = (3z) 1/α and r sp l = 0 for all l. Taxes τ must be weakly greater than g sp, but otherwise taxes and transfers, f j, j = 1, 2, 3, are indeterminate. Proof. Substituting the budget constraint (BC) into W (q), we get W (q) = 3y g + l r l + 3H(g) Clearly r sp l = 0 for all l. To find the optimal g, we solve the first order condition 3H (g) = 3z(g) α = 1 which gives us g sp = (3z) 1/α. This is called the Samuelson level of the public good, and by Assumption 4 above, it is affordable with a tax rate τ sp = g sp /3. Note that taxes may be higher so long as they are given back to the districts in the form of transfers. Consequently, the level of taxes and transfers is indeterminate. We can now compare the equilibrium level of the public good with the planner s optimal level. An equilibrium of the game is a subgame perfect equilibrium in which legislators use weakly un-dominated voting strategies. Then, the equilibrium level of the public good is characterized in the following proposition. Proposition 3: On every equilibrium path we have the following g = (z) 1/α τ = y r a = 3y R r r l = 0, l a f a = R + r g f j = 0, j a w a = H(g ) + f a w j = H(g ), j a 21

22 Proof sketch. I will sketch the argument that these are sequentially optimal on the path of play, and you will (in the Bonus Exercise below) close the argument by proving that there are equilibria that support this path of play (and indeed all of them support this path of play). We work backwards. Note that because all legislators use weakly un-dominated voting strategies, in every equilibrium all legislators vote to pass any proposal q that gives them a payoff least r whenever it is the case that if the proposal ( fails, then they ) will not be re-elected. So, suppose the agenda setter proposes q = τ, g, (f j ) j, (rl ) l when voters use the thresholds w j. Then all of the legislators must vote to pass the proposal. If the proposal does not pass, then q 0 is implemented and voters in all districts receive utility H(0) = 0 < w j, all j. Moreover, both legislators l a receives a payoff r instead of the payoff he would receive from q which is rl + R = R > r by Assumption 1. The legislator a receives a payoff r instead of the payoff ra + R > r by Assumptions 1 and 3, which states that r a > 0. So, all vote to pass the proposal q. Now, we show that it is optimal for the agenda setter a to propose q when the voters use the thresholds w j. Suppose a proposes a policy that gives voter j = a a payoff lower than w a. Then he forgoes re-election, receiving a payoff at most 3y r. This is because his best deviation from q is the policy τ = 1, g = f j = 0, all j and r a = 3y r, r b = r, r c = 0, where b is either of the other two legislators and c is the third. But by proposing q he gets exactly 3y R r + R = 3y r, so he is indifferent. On the other hand, it cannot be optimal for him to give voter j = a a payoff higher than w a, since this eats into his own resources (and we will show below that g and f a have been chosen by voter j = a optimally). Finally, note that it is optimal for w b = w c = H(g ). If either b or c ask for more, then q will pass anyway. Likewise, it does not make sense to ask for less. Finally, voter j = a sets w j according to the following problem max w a = y τ + f a + H(g) q subject to (BC) and r a 3y R r In other words, he maximizes his payoff subject to the budget constraint and the incentive constraint that legislator a wants to propose a policy that gets him re-elected. The solutions are given in the statement of the proposition. Pork transfers to district a are positive, and the equilibrium level of the public good is affordable by Assumptions 2 and 4 respectively. By Assumption 3, legislator a captures positive rents. We immediately have the following corollary. 22

23 Figure 6: The Congressional Committees Game Corollary 1: The equilibrium level of the public good is lower than the Samuelson level (social planner s optimum): g < g sp Bonus Exercise. (This exercise is optional.) Complete the proof of Proposition 3, i.e. show that all equilibria of the game are path-equivalent. Exercise. (Congressional Committees) The congressional policy game is like the game above, except that there are two different agenda setters aτ and ag, the finance committee and the expenditure committee, that are selected by nature at the start of the game. Voters set their utility cutoffs wj as before. aτ proposes the tax rate τ and an up or down vote is held. If the proposal fails then an exogenous τ0 (0, r ), r < y. Then ag proposes g, (f j )j and (rl )l subject to the budget constraint (BC) where τ here is the tax proposal that passes or τ0 if the proposal fails to pass. Then again a vote takes place on ag s proposal. If the proposal fails then g = 0, f j = τ rl 0, all j, and rl = r, all l, is implemented. Finally, elections are held. (See Figure 6.) Maintain Assumptions 1-4 and characterize the equilibria of the Congressional Committees game, showing in particular that the equilibrium level of the public good is unique and under-provided, as before. Specifically, g = (z)1/α Exercise. (Parliamentary Regimes) Now consider a parliamentary model depicted in Figure 7. Nature selects the expenditure and finance ministers, ag and aτ. Voters set their The finance minister, aτ, proposes τ. The expenditure reservation utilities w j as before. j minister ag proposes g, (f )j, (rl )l subject to (BC) given the proposed tax rate τ. Either 23

24 Figure 7: The Parliamentary Game member of government a g or a τ can veto the aggregate proposal q = (τ, g, (f j ) j, (r l ) l ). If neither does, then the proposal passes, and elections are held. If one of them vetoes, government collapses and a legislator is selected randomly (i.e. each with probability 1/3) to form a caretaker government. Voters ( then set new reservation ) utilities w j. The caretaker makes an entire budget proposal q = τ, g, (f j ) j, (r l ) l which passes or fails by simple majority. If it fails then the following default policy is implemented: τ 0 = r g 0 = 0 r l = r, l f j = 0, j Maintain Assumptions 1-4 and show that there is an equilibrium in which the level of the public good supplied is larger than the level supplied in the congressional committees regime, but lower than the Samuelson level, the sum total of equilibrium rents in the parliamentary regime is larger than the sum total of equilibrium rents in the congressional committees regime, and taxes are higher under the parliamentary regime than under the congressional committees regime. 24

25 2 Dynamic Methods, etc. I assume that you have some familiarity with this material. If you do not, please consult a good textbook, such as Mailath and Samuelson s (2006) Repeated Games and Reputations and Daron Acemoglu s (2008) Modern Economic Growth. 2.1 Infinite Horizon Dynamic Programming We consider an infinite horizon stationary discounted dynamic programming problem as follows. Here, I simply state the main results of dynamic programming, leaving it to you to discover the proofs if necessary. Suppose time is discrete and indexed by t = 0, 1, 2...,. Let δ [0, 1) be a discount factor. Let X R m be the set of control variables. Let G : X 2 X be a correspondence from X to itself and u : X X R an instantaneous reward or payoff function. The canonical stationary dynamic program is max U({x t } t=0) δ t u(x t, x t+1 ) {x t} t=0 t=0 subject to x t+1 G(x t ) for all t 0 and x 0 = x X A solution to this problem may or may not exist. If it exists, then label it {x t } t=0 and let U be the value of U attained at this solution, i.e. U = U({x t } t=0 ). For any t 0, it will be useful to define the set of feasible plans starting with an initial value x t X as Φ(x t ) = {{x s } s=t : x s+1 G(x s ), s = t, t + 1,..., } Now define a completely new function V : X R such that x X V (x) = max {u(x, y) + δv (y)} (B) y G(x) Again, such a function may or may not exist. The equality in (B) is called a Bellman equation and the function V is called a value function. Consider the following assumption. Assumption A: X is compact; G is continuous, nonempty-, convex-, and compactvalued, and it is monotone in the sense that x x implies G(x) G(x ); and u is continuously differentiable on int X X, strictly concave, and strictly increasing in its first m arguments; and, for all x 0 X and plans {x t } t=0 Φ(x 0), the following limit exists and is finite: lim n n δ t u(x t, x t+1 ) t=0 25 (P)

26 Then we immediately have the following result. Theorem A: Suppose Assumption A holds. Then: (i) A plan {x t } t=0 solving (P) exists for all starting values x 0 can be expressed as a recursion = x X, and this plan x t+1 = π(x t ) where π : X X is a continuous function (called a policy function ). (ii) There exists a unique, continuous, strictly concave and bounded function V that satisfies the Bellman equation (B) and is strictly increasing in all of its arguments. In particular, {x t } t=0 is a solution to (P) if and only if V (x t ) = u(x t, x t+1) + δv (x t+1) And, moreover, let π be a policy function as in (i) above. If x int X and π(x) int G(x) then V is differentiable at x with gradient given by V (x) = x u(x, π(x)) ( x is the gradient operator applied to x, i.e. the first m arguments of u.) The theorem implies that under Assumption A, the maximization problem in (B) is strictly concave and the maximand is differentiable. So for any interior solution y G(x), the first order conditions are necessary and sufficient for an optimum. Optimal solutions are characterized by the Euler equation: y u(x, y ) + δ V (y ) = 0 (Euler) where y is the optimal value and y is the gradient operator applied to y, i.e. the last m arguments of u. The Euler equation is simply a first order condition. We could use it to solve for the optimal policy y if we knew the functional form of V ; but the theorem does not tell us that. Fortunately, we can differentiate the Bellman equation (B) with respect to x to obtain an Envelope condition for dynamic programming. (This is essentially the same as the standard Envelope condition that you know from optimization theory.) The optimal value y in (B) will depend on x; in particular, the theorem implies y = π(x). Then, apply the implicit function theorem to (B) to get V (x) = x u(x, π(x)) + [ y u(x, π(x)) + δ V (π(x))] π(x) 26

27 But by the Euler equation above, the term in square brackets is 0, so we are left with V (x) = x u(x, π(x)) (Envelope) which is the Envelope condition. This condition must hold for all x int X, in particular it must hold at π(x). So, we have V (π(x)) = x u(π(x), π(π(x))) We can combine this with the Euler equation to get y u(x, π(x)) + δ x u(π(x), π(π(x))) = 0 This condition characterizes the optimal policy function π in terms of the one period reward function u. This condition is, however, only necessary. For sufficiency, we also need what is called a transversality condition but we will not discuss that here. 2.2 Dynamic Game Theory Time is discrete and indexed by t = 0, 1, 2,...,. There are I players. In each period, each player i takes an action from the set A i (s) A i R n. The action set is written to depend on the state s S R m. The action taken by i in period t is written a i t. The action profile is a t = (a 1 t,..., a I t ). As usual, a i t denotes the action profile of all players other than i. Each player i has an instantaneous utility function u i (s, a) over states s and action profiles a. Each player s objective at time t is to maximize the discounted payoff Ut i (s t ) = E t δ j u i (s t+j, a t+j ) j=0 (Obj) where s t+j is the state at time t + j and a t+j is the action profile at that time, and E t is the expectations operator conditional on information available at time t. Since we only work with models where all information is public, we do not index the expectation by i. We take this expectation because we allow the state to evolve stochastically. δ (0, 1) is the discount factor. The state evolves according to a Markov process in which q(s t+1 s t, a t ) denotes the probability (or density) that the state is s t+1 given that the state was s t today and players chose actions a t. If players use pure strategies, then a history at time t is simply h t = (s 0, a 0, s 1, a 1,..., s t 1, a t 1 ) with the convention h 0 =. The set of all possible histories at time t is denoted H t, and H = t Ht. Each history corresponds to a particular subgame. A (pure) strategy for player 27