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| 22.6.5. Party Competition and the Downsian Policy Convergence Theorem.The focus so far has been on voting between two alternative policies or on open agenda vot- ing, which can be viewed as an extreme form of “direct democracy”. The MVT becomes potentially more relevant and more powerful when applied in the context of indirect democ- racy, that is, when combined with a simple model of party competition. We now give a brief |
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direct democracy |
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| 22.6.6. Beyond Single-Peaked Preferences. Single-peaked preferences played a very important role in the results of Theorem 22.2 by ensuring the existence of a Condorcet winner.However, single peakedness is a very strong assumption and does not have a natural analog in situations in which voting is over more than one policy choice. When there are multiple policy choices (or even voting over “functions” such as nonlinear taxation), much more structure needs to be imposed over voting procedures and agenda setting to determine equilibrium policies. Those issues are beyond the scope of our treatment here. Nevertheless, it is possible to relax the assumption of single-peaked preferences, and also introduce a set of preferences that are “close” to single-peaked in multidimensional spaces. The latter task would take us |
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functions” such as nonlinear taxation), much more structureneeds to be imposed over voting procedures and agenda setting to determine equilibriumpolicies. Those issues are beyond the scope of our treatment here. Nevertheless, it is possibleto relax the assumption of single-peaked preferences, and also introduce a set of preferencesthat are “close |
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| 4 But there is nothing in the analysis that implies that these functions have to be differentiable; in fact, equilibria with non-differentiable functions are easy to construct. Nevertheless, it is generally thought that equilibria with non-differentiable functions are more “fragile” and thus less relevant. See the discussion in Grossman and Helpman (1994) and Bernheim and Whinston (1986) |
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fragile |
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1986 |
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| 22.7. Distributional Conflict and Economic Growth: Heterogeneity and the Median VoterLet us now return to the model of Section 22.2 with linear preferences, but relax the assumption that political power is in the hands of an elite. Instead, we will now introduce heterogeneity among the agents and then apply the tools from the previous section, in partic- ular, the Median Voter Theorem, Theorems 22.2 and 22.5, to analyze the political economy of this model. Recall that these theorems show that if there is a one-dimensional policy choice and individuals have single-peaked preferences (or preferences over the menu of policies that satisfy the single-crossing property), then the political equilibrium will coincide with the most preferred policy of the median voter.To focus on the main issues in the simplest possible way, I will modify the environment from Section 22.2 slightly. First, there are no longer any elites. Instead, economic decisions will be made by democratic voting among all the agents. Second, to abstract from political conflict between entrepreneurs and workers, I will also assume that there are no workers (recall Exercise 22.3 for why having only entrepreneurs simplifies the analysis; see Exercise 22.31 for an economy where individuals differ both in terms of their productivity and occupation) |
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Distributional Conflict and Economic Growth: Heterogeneity and the Median Voter |
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| (4) In light of your answers to 2 and 3 above, explain why the political equilibrium might involve the use of inefficient fiscal instruments, even when more efficient alternatives exist.Exercise 22.16. * Prove Proposition 22.18.Exercise 22.17. * Prove Proposition 22.19.Exercise 22.18. Consider an environment with concave preferences as in Section 22.5. As- sume that there is full depreciation (i.e., δ = 1), citizens are yeoman-producers only using their own labor and have access to a production technology for producing the unique final good given by Y i (t) = AK i (t) α , where K i (t) is the capital holding of producer i. Both citizens and elites have logarithmic preferences. Characterize the MPE in this environment.[Hint: conjecture a policy rule that depends only on the current (average) net output, so that the tax rate for next period is τ (t + 1) = τ ¡Y N (t) ¢, where Y N (t) = (1 − τ (t)) AK (t) α , where K (t) is the common capital stock of all producers].Exercise 22.19. * Prove that if individual preferences are reflexive, complete and transitive, then they can be represented by a real-valued utility function.Exercise 22.20. * |
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| (4) Now consider a society consisting of three individuals, with preferences given by:1 a  b  c 2 c  a  b 3 b  c  aConsider a series of pairwise votes between the alternatives. Show that when agents vote sincerely, the resulting social ordering will be “intransitive”. Relate this to the Theorem 22.1 |
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| (3) In this model with two taxes, now suppose that agents first vote over the capital income tax, and then taking the capital income tax as given, they vote on the labor income tax. Does a voting equilibrium exist? Explain. If an equilibrium exists, how does the equilibrium tax rate change when k increases? How does it change when λ increases?Exercise 22.32. Derive expression (22.51).Exercise 22.33. (1) Show that V ˜ i¡ τ 0 | p t+1 ¢defined in (22.54) is not necessarily quasi- concave.(2) Show that V ˜ i¡ τ 0 | p t+1 ¢satisfies the single-crossing property in Definition 22.3.Exercise 22.34. Consider an economy consisting of three groups, a fraction θ p poor agents each with income y p , a fraction θ m middle-class agents with income y r > y m , and the remain- ing fraction θ r = 1 − θ p − θ m rich agents with income y r > y p . Suppose that both θ p and θ r are less than 1/2, so that the individual with the median income (the “median voter”) is a middle-class individual |
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| 22.6.7.1. Probabilistic Voting and Swing Voters. Let the society consist of G distinct groups of voters, with all voters within a group having the same economic characteristics and preferences. As in the Downsian model, there is electoral competition between two parties, A and B, and let π g P be the fraction of voters in group g voting for party P where P = A, B, and let λ g be the share of voters in group g and naturally P Gg=1 λ g = 1. Then the expected |
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| (22.41) U ˜ i g (p, P ) = U g (p) + ˜ σ g i (P)when party P comes to power, where p is the vector of economic policies chosen by the party in power. We assume that p ∈ P ⊂ R K , where K is an integer, possibly greater than 1. Thus p ≡ ¡p 1 , ..., p K ¢is a potentially multi-dimensional vector of policies. In addition, U g (p) is the indirect utility of agents in group g as before (previously denoted by U (p; α i ) for individual i) and captures their economic interests. In addition, the term σ ˜ g i (P ) captures the non-policy related benefits that the individual will receive if party P comes to power. The most obvious source of these preferences would be ideological. So this model allows individuals within the same economic group to have different ideological preferences.Let us normalize σ ˜ g i (A) = 0, so that |
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| (22.42) U ˜ i g (p, A) = U g (p), and U ˜ i g (p, B) = U g (p) + ˜ σ g i In that case, the voting behavior of individual i can be represented as (22.43) v g i (p A , p B ) =⎧ ⎨⎩1 if U g (p A ) − U g (p B ) > σ ˜ g i12 if U g (p A ) − U g (p B ) = ˜ σ g i 0 if U g (p A ) − U g (p B ) < σ ˜ g i,where v g i (p A , p B ) denotes the probability that the individual will vote for party A, p A is the platform of party A and p B is the platform of party B, and as above, we have assumed that if an individual is indifferent between the two parties (inclusive of the ideological benefits), he randomizes his vote.Let us now assume that the distribution of non-policy related benefits ˜ σ g i for individual i in group g is given by a smooth cumulative distribution function H g defined over ( −∞ , + ∞ ), with the associated probability density function h g . The draws of σ ˜ g i across individuals are independent. Consequently, the vote share of party A among members of group g isπ g A = H g (U g (p A ) − U g (p B )) |
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| 22.6.7.2. Lobbying. Consider next a very different model of policy determination, a lob- bying model. In a lobbying model, different groups make campaign contributions or pay money to politicians in order to induce them to adopt a policy that they prefer. With lob- bying, political power comes not only from voting, but also from a variety of other sources, including whether various groups are organized, how much resources they have available, and their marginal willingness to pay for changes in different policies. Nevertheless, the most important result for us will be that even with lobbying, equilibrium policies will look like the solution to a weighted utilitarian social welfare maximization problem |
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| (3) There are no profitable deviations for any lobby, g = 1, 2, .., G 0 , i.e., (22.48)p ∗ ∈ arg maxp⎛⎝λ g (U g (p) − γ ˆ g (p)) +G 0Xg 0 =1λ g 0 ˆ γ g 0 (p) + a X G g 0 =1λ g 0 U g 0 (p)⎞⎠ for all g = 1, 2, .., G 0 . (4) There exists a policy p g for every lobby g = 1, 2, .., G 0 such thatp g ∈ arg maxp⎛⎝G 0Xg 0 =1λ g 0 γ ˆ g 0 (p) + a X G g 0 =1λ g 0 U g 0 (p)⎞⎠ |
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| (1) there is voting over a linear tax rate on output τ (t + 1) ∈ [0, 1] that will apply to all entrepreneurs in the next period (at t + 1). We assume that the voting is between two parties with policy commitment, so that Theorems 22.2 and 22.5 (and Theorems 22.4 and 22.6) apply |
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| (2) the proceeds of the taxation from time t+ 1 are redistributed as a lump-sum transfer to all agents, denoted by T (t + 1) ≥ 0.We will focus on the Markov Perfect Political Economy Equilibrium of this game.The important assumption here is that at each stage voting is over the tax rate that will apply in the next period only (with the lump-sum transfer determined from the budget constraint). Moreover, given the linear preferences, each individual takes future taxes as given (independent of current tax decisions and the current capital stock) and only cares about the current tax rate when making its current decisions. Thus, despite the fact that the economy involves an infinite sequence of taxes, the MVT can be applied to the tax decision at each date, provided the other conditions of the theorem are satisfied. We next show that this is the case |
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| (22.54) V ˜ i ¡τ 0 | p t+1 ¢= − A i ˆ k ¡ τ 0 ¢+ β h¡1 − τ 0 ¢ A i f ³ˆ k ¡ τ 0 ¢´+ τ 0 Af ¯ ³ k ˆ ¡τ 0 ¢´+ ˜ V i ¡ p t+2 ¢i, where τ 0 denotes the tax rate announced for date t + 1 and I have used the notation V ˜ i to distinguish this value function defined over the current tax rate from the value function V idefined in (22.50). In addition, V ˜ i ¡ p t+1 ¢is defined as the continuation value from the end of date t + 1 onwords and I have substituted for the transfer T (t + 1) from (22.53).We can obtain the most preferred tax rate for entrepreneur i, from the expression for V ˜ i¡ τ 0 | p t+1 ¢. However, it can be verified easily that V ˜ i¡ τ 0 | p t+1 ¢is not necessarily quasi- concave in τ 0 , thus preferences are not single peaked (see Exercise 22.33). However, we have:Proposition 22.22. Preferences given by V ˜ i ¡τ 0 | p t+1 ¢in (22.54) over the policy menu τ 0 ∈ [0, 1] satisfy the single crossing property in Definition 22.3.Proof. See Exercise 22.33. ¤In view of Proposition 22.22, we can apply Theorems 22.5 and 22.6, and conclude that at each date, the tax rate most preferred by the entrepreneur with the median productivity will be implemented. Let this median productivity be denoted by A m . From (22.54), this most preferred tax rate satisfies the following first-order condition |
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| (22.55) ¡ A ¯ − A m ¢ f ³k ˆ ¡ τ 0 ¢´+ τ 0 A ¯³ f 0 ³k ˆ (τ 0 ) ´´ 2(1 − τ 0 ) f 00 ³ˆ k (τ 0 ) ´ ≤ 0 and τ 0 ≥ 0,with complementary slackness. In writing this expression, we have made use of condition (22.52) to simplify the expression and also to express the derivative of k ˆ 0 (τ 0 ), asˆ k 0 ¡ τ 0 ¢=f 0 ³ k ˆ (τ 0 ) ´ (1 − τ 0 ) f 00 ³k ˆ (τ 0 ) ´ .This derivative is strictly negative since f 00 < 0. Therefore, as in Section 22.2, higher taxes lead to lower capital-labor ratios and lower output (higher distortions). The emphasis on complementary slackness in (22.55) is important here, since the most preferred tax rate of the median voter (entrepreneur) may not satisfy the first-order condition as equality, instead corresponding to a corner solution of τ 0 = 0. The next proposition shows that this is in fact relevant for a range of distributions of productivity among the entrepreneurs |
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| (22.59) A (t) = A f b ≡ β 1/(φ − 1)(1 − α) and the first-best levels of the capital-labor ratio and output arek f b ≡ β φ/(φ − 1)(1 − α) and Y f b ≡ 1α β (φα+1 − α)/(φ − 1)(1 − α) .Let us next focus on the Markov Perfect Equilibrium (MPE) of this game. As usual, a MPE is defined as a set of strategies at each date t, such that these strategies only depend on the current (payoff-relevant) state of the economy, A (t), and on prior actions within the same date according to the timing of events above. Thus, a MPE can be represented by³τ (A (t)) , [k i (A (t))] i ∈ [0,1] , G (A (t)) ´, where, by definition of a MPE, the key actions, which consist of the tax rate on output, τ , the capital-labor ratio decision of each entrepreneur [k i ] i ∈ [0,1] , and the government expenditure on public good, G, are conditioned on the current payoff-relevant state variable, A (t). Clearly, since each yeoman-entrepreneur employs only himself, the capital-labor ratio, k i , and the total capital stock, K i , of each entrepreneur are identical.It is clear that in any MPE, the unique equilibrium tax rate for the political elite will be(22.60) τ (t) = ¯ τ for all t,since investment decisions are already sunk at the time the elite set the taxes.Next, the capital-labor ratio of entrepreneurs is again given by (22.18), and thus can be written as |
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| (22.62) T (A (t)) = (β (1 − ¯ τ )) α/(1 − α) ¯ τ A (t)α .Finally, the elite will choose public investment, G (t) to maximize his consumption. To characterize this, let us write the discounted net present value of the elite as(22.63) V e (A (t)) = maxA(t+1)ẵT (A (t)) − 1 − ααφ A (t + 1) φ + βV e (A (t + 1))ắ ,which simply follows from writing the discounted payoff of the elite recursively, after substi- tuting for their consumption, C E (t), as equal to taxes given by (22.62) minus their spending on public goods from equation (22.58).Since, for φ > 1, the instantaneous payoff of the elite is bounded, continuously differ- entiable and concave in A, so Theorems 6.3, 6.4 and 6.6 in Chapter 6 imply that the value function V e ( ã ) is concave and continuously differentiable. Hence, the first-order condition of the ruler in choosing A (t + 1) can be written as:(22.64) 1 − αα A (t + 1) φ − 1 = β (V e ) 0 (A (t + 1)) ,where (V e ) 0 denotes the derivative of the value function of the elite. This equation links the marginal cost of greater investment in public goods to the greater value that will follow from this. To make further progress, I use the standard envelope condition, which is obtained by differentiating (22.63) with respect to A (t) |
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| (22.65) (V e ) 0 (A (t + 1)) = T 0 (A (t)) = (β (1 − ¯ τ)) α/(1 − α) τ ¯α .The value of greater public goods for the elite is the additional tax revenue that this will generate, which is given by the expression in (22.65).Combining these conditions, we obtain the unique Markov Perfect Equilibrium choice of the elite as |
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| (22.66) A (t + 1) = A [¯ τ ] ≡ ³β 1/(1 − α) (1 − α) − 1 (1 − τ ¯ ) α/(1 − α) ¯ τ ´ φ 1− 1,which also defines A [¯ τ ] as an expression that will be useful below. Substituting (22.66) into (22.63) yields a simple form of the elite’s value function |
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