UNIVERSITY OF MARYLAND Department of Economics Economics 754 Topics in Political Economy Fall 2005 Allan Drazen. Exercise Set I

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1 UNIVERSITY OF MARYLAND Department of Economics Economics 754 Topics in Political Economy Fall 005 Allan Drazen Exercise Set I The first four exercises are review of what we did in class on 8/31. The next few (taken from Persson and Tabellini give further practice. Questions 8 and following are related to the lecture of 9/7. You should do problem 8 before the lecture. 1. Show that if each individual has single-peaked preferences as in the Guns versus Butter example in PEM p. 75-6, the budget allocation for guns preferred by the median voter is a Condorcet winner (that is, it cannot be defeated by any other allocation in pairwise voting).. Show that if preferences of voters satisfy the single crossing property as discussed in class, the policy preferred by the voter with the median preference is a Condorcet winner. 3. Consider the Meltzer-Richard model of labor taxation. An individual s utility is u (c.z), where c is consumption and z is leisure, where z =1 l, where l is labor supply. Individual s differ in their endowment ξ, where labor income is y = ξl. Consumption is c =(1 τ) lξ + v, wherev is a lump-sum transfer, identical across all individuals. In equilibrium v = τ y, where y is average labor income over the population. An individual of type ξ chooses l as a function of τ and v, givenhisξ. Call the solution l = L (τ; ξ). (This problem is discussed and solved in section 8. of PEM.) The indirect utility function of an individual of type ξ maythenbewritten V (τ; ξ) u ((1 τ) ξl(τ; ξ)+τy, 1 L (τ; ξ)) The indirect utility function V (τ; ξ) is not well-behaved even when the indifference curves generated by by u (c.z) are strictly concave. Show, however, that the preferences represented by u (c.z) satisfy the single crossing property as discussed in class. 4. Show the sense in which the preferences represented in Table 3.1 on p. 7 of PEM are single-peaked. Why then does no Condorcet winner exist? 5. Consider three voters indexed by i {1,, 3}, each characterized by an intrinsic parameter α i, where α 1 <α <α 3. Agent i derives a utility W (q j ; α i )overpolicyq j. Three possible policies q j {q 1,q,q 3 } can be implemented. A policy is selected by simple majority rule. a. The preferences of agent i {1,, 3} are such that W (q 1 ; α 1 ) >W(q 3 ; α 1 ) >W(q ; α 1 ) W (q ; α ) >W(q 1 ; α ) >W(q 1 ; α ) W (q 3 ; α 3 ) >W(q ; α 3 ) >W(q 1 ; α 3 ) Prove that no Condorcet winner exists under majority rule. Discuss. b. Suppose that agents have the same preferences as in (a) but agent 1 is the agenda setter. He selects two rounds in which all agents vote sincerely. What is the optimal agenda from the perspective of agent 1

2 1? Suppose now that agent 1 sets the agenda and agents and 3 vote sincerely. Can agent 3 improve his welfare by voting strategically? Discuss. c. Suppose that the agents have the following preferences: W (q 1 ; α 1 ) >W(q ; α 1 ) >W(q 3 ; α 1 ) W (q ; α ) >W(q 1 ; α ) >W(q 3 ; α ) W (q 3 ; α 3 ) >W(q ; α 3 ) >W(q 1 ; α 3 ) with q 1 <q <q 3. Is there a Condorcet winner? Explain. d. Suppose that the preferences of agent are such that: W (q ; α ) >W(q 1 ; α ) >W(q 3 ; α ) with q 1 <q <q 3. Construct the preferences (ordering) of agents 1 and 3 so that they verify the single-crossing property. Then show that the median voter is a Condorcet winner. 6. Consider a society inhabited by a continuum of citizens and normalize the size of the population to 1. Suppose that the preferences of agent i over a publicly provided good y and a privately provided good c i are: ω i = c i + α i V (y), where V (.) is a concave, well-behaved function and α i is an intrinsic parameter of agent i distributed according to F ( ) with mean α. Assume, in addition, that all individuals have initial resources in private good e i =1for all i. Suppose also that one unit of private good is required to produce one unit of public good. Last, suppose that to finance the production of the public good, the government raises a tax q on each individual so that agent i 0 s budget constraint is c i 1 q. a. What is the (utiliarian) social optimum in this economy? b. Compute each individual s policy preferences. What is the preferred policy q(α i ) of agent i? c. Under majority rule, what is the selected policy? Compare this to the social optimum. When does the social optimum coincide with the equilibrium policy? d. Suppose now that each agent s preferences are given by: ω i = c i +(α i â) V (y), where ˆα is given value of α i. Again, compute the social optimum as well as the policy preferences of individuals. Do we reach the same conclusions as in part c? 7. Consider an economy in which the preferences of individual i are quasi-linear, namely: ω i = c i + V (x i ), where c i represents the individual s consumption and x i his leisure. Moreover, V ( ) is increasing and concave in x i. The private budget constraint of each agent is given by: (1 + qc)c i (1 q L ) i + f, where q L istheincometaxrate,q c the consumption tax rate, and f a fixed subsidy from the government. The real wage of each agent is exogenous and normalized to unity. Furthermore, each agent has a

3 private productivity parameter α i, so that agents have different amounts of effective time available. More precisely, they face the following time constraint: 1+α i x i + i. Assume that α i is drawn from a distribution with mean α and median α m. a. Compute each individual s optimal labor supply. What effects does an increase in q L (respectively, q c ) have on the individual labor supply? Discuss the result. b. Write the government budget constraint and derive the level of the subsidy as a function of q = (q L,q c ). Compute the policy preferences W (q; α i ) of individual i. c. Does a Condorcet winner exist in that case? If yes, who is the Condorcet winner? d. Compute the utiliarian welfare and determine the socially optimal policy. What is the winning policy q when α i = α for all i? What happens if agents are heterogenous? 8. A pair of platforms (τ A,τ B ) is an equilibrium if: (i) τ A = argmaxp(τ A,τ B )χ; and (ii) τ B = arg max(1 p(τ A,τ B ))χ, wherep(τ A,τ B ) is the probability of A winning the election and χ is the value of holding office. For the case of non-probabilistic voting in a multidimensional issue space, prove the following proposition: A pair of platforms (τ A,τ B ) is an equilibrium if and only if, for each candidate J, τ J is a Condorcet winner. 9. Assume a one-dimensional policy space of τ (0, 1), wherethepayoff to voter i from policy τ J is u i τ J, eτ i = τ J eτ i and the payoff to the candidates is u P ³τ ³ J, eτ P = χ τ J eτ P for P = L, R. eτ L =1 eτ R = t, wheret< 1. Candidates believe that the ideal point of the median voter µ is uniformly distributed on 1 ε, 1 + ε,whereε 0, 1 t. If the candidates choose platforms τ L and τ R, the indifferent voter is one whose most preferred policy is at the midpoint τ mid τ L,τ R = τ L +τ R, so that L wins if µ<τ mid τ L,τ R. Therefore, assuming that τ L +τ R 1 ε, 1 + ε, the probability that candidate L wins is π τ L,τ R ³ = Pr µ<τ mid ( ) = τ mid ( ) 1 ε 1 + ε 1 ε = 1 + τ L + τ R 1 4ε A pair of ideologies τ L,τ R is an equilibrium if τ L = argmaxπ τ L,τ R h χ τ L t i 1 π τ L,τ R h τ R t i and τ R = argmax π τ L,τ R τ L (1 t ) + 1 π τ L,τ R χ τ R (1 t ) 3

4 a. Show that if χ<ε (1 t), the equilibrium is symmetric with τ L =1 τ R = χ + 1 t (1 ε) ε +1 t b. Show that the degree to which candidates converge is positively related to the value of holding office χ and negatively related to the degree of uncertainty in the location of the median voter, represented by ε. 10. Consider the economy described problem 6. Agent i 0 s preferences over a publicly provided good y and a privately provided good c i is expressed by: ω i = c i + α i V (y), where V ( ) is a concave, well-behaved function and α i is the intrinsic parameter of agent i that is drawn from distribution F ( ) with mean α. Again, all individuals have initial resources only in the private good e i =1for all i, and one unit of private good is required to produce one unit of public good. To finance the public-good production, the government raises a tax q on each individual so that agent i 0 s budget constraint is c i 1 q. a. Derive the policy preferences of each agent W (q; α i ) as well as the social optimum in this economy. Suppose that two politicians P = A, B select platforms Q A and q B. Assume that each maximizes the expected value of some exogenous rent R. Call π P the vote share for politician P ; then P 0 s probability of winning the election is p P =Pr(π P 1 ) and his expected utility is then p P R. First, the two candidates announce their platforms simultaneously and noncooperatively. Then elections are held. Last, the elected politician implements his announced policy. b. Assume that α i = α. Determine the candidates probability of winning. What are the announced platforms and which one is implemented? Discuss. c. Determine each candidate s probability of winning when agents are heterogeneous. What are the selected platforms in that case? Which one is implemented? d. What are the model s economic predictions? Discuss. 11. Consider the same model as in problem 10, but assume that three factors affect voter i 0 s voting strategy: (1) the economic policy implemented, q, () his individual ideological bias σ i towardh candidate i B, and (3) the popularity δ of politician B. We assume that σ i is uniformly distributed on 1 φ, 1 φ. Moreover, δ is the same for all voters and is drawn from the uniform distribution on: 1 ψ, 1. ψ The distributions are common knowledge, but only agent i observes his own parameter σ i. Then, i 0 s preferences over the policy implemented by A are summarized by W (q A ; α i ), whereas the preferences over the policy implemented by politician B take the final form: W (q B,α i )+σ i + δ. The timing is as follows: First, each voter observes σ i, and politicians simultaneously and noncooperatively announce platforms q A and q B. Second, δ is realized. Third, elections take place, and last, the announced policy is implemented. 4

5 a. Give an interpretation of σ i. Characterize the agent who is indifferent between voting for politician A and voting for politician B for given policies Q A and q B. Suppose that α i = α. Deduce candidate A 0 s vote share as well as his probability of winning. b. Which platforms do the politicians select? Which one is implemented? Discuss. c. Suppose that agents are heterogeneous. What does this imply for the equilibrium? d. Discuss your results and compare them with the results obtained in problem Assume that the indirect utility function of tax policy is described by: ω i = (τ τ i ). There are two political parties, one with the preferred tax rate zero, the other with preferred tax rate one. The parties can commit to a party platform that will be implemented should the party win the election. The political parties are uncertain about the most preferred tax rate τ M of the median voter and assign a uniform probability distribution between ( 1 α) and ( 1 + α) to τ m. The parameter α lies in the interval (0, 1). The parties are exclusively policy motivated. Let τ 0 and τ 1 be the policies proposed by parties 0 and 1, respectively. a. Show that the parties will never choose their bliss points and will never converge completely. b. Solve for the equilibrium policy, given that it is symmetric, that is, τ 0 =1 τ 1. Discuss how the equilibrium policies depend on the level of uncertainty, as described by α. c. Show that the equilibrium must be of the form τ 0 =1 τ Consider the following economy. Agent i 0 s preferences over a publicly provided good g and a privately provided good c i is expressed by: u i = c i + α i v(g), where v ( ) is a concave, well-behaved function and α i is the parameter that characterizes agent i s type that is drawn from distribution F ( ) with mean ᾱ. All individuals have initial resources of 1, where the technology is that one unit of private good is required to produce one unit of public good. To finance the public-good production, the government levies a tax τ on each individual so that agent i s budget constraint is c i 1 τ. a. Derive the utility function over policy for each agent w(τ; α i ) as well as the social optimum in this economy. Suppose that two politicians P = A, B select platforms τ A and τ B. Assume that each maximizes the expected value of some exogenous rent χ. Call N P thevoteshareforpoliticianp ; then P 0 s probability of winning the election is ρ P = Pr(N P 1 ) and his expected utility is then ρ P χ. First, the two candidates announce their platforms simultaneously. Then elections are held. Last, the elected politician implements his announced policy. b. Suppose firstthatallagentshaveα i =ᾱ. Determine the candidates probability of winning. What are the announced platforms and which one is implemented? Discuss. c. Determine each candidate s probability of winning when agents are heterogeneous in α. Whatare the selected platforms in that case? Which one is implemented? Now suppose that in addition to τ and α i, a voter s decision also h depends i on his individual ideological bias η i toward candidate B, whereη i is uniformly distributed on φ 1, φ 1. The distribution is common knowledge, but only agent i observes his own parameter η i. That is, i s preferences over the policy 5

6 implemented by A are summarized by w(τ A ; α i ), whereas the preferences over the policy implemented by politician B take the final form: w(τ B,α i )+η i The timing is as follows: First, each voter observes η i, and politicians simultaneously announce platforms τ A and τ B. Then, elections take place, and last, the announced policy is implemented. d. Give an interpretation of η i. Characterize the agent who is indifferent between voting for politician A and voting for politician B for given policies τ A and τ B. Suppose that α i =ᾱ. Deduce candidate A 0 s vote share as well as his probability of winning. e. Which platforms do the politicians select? Which one is implemented? Discuss. f. Suppose that agents are heterogeneous. What does this imply for the equilibrium? g. Discuss your results and compare them with the results obtained in parts a b and c. 14. Consider the incumbent discipline model considered in class, with two types of incumbents, ω 1 and ω,whereω >ω 1. Politician type is unobserved, where the prior on probability of the incumbent being type 1 is π>0. Higher types like more effort and get more utility from holding office for any given effort a. A politician not in office gets a per-period payoff of 0. Suppose also that there are just two outcome levels y t {y L,y H } where y L <y H.Assumecontinuous effort choice and let the probability that y t = y H be F (a) where F 0(a) > 0 and F 00 (a) < 0. Assume incumbents effort is such that a t = γ t and a 1 t = γ 1 t,wherethatγ t >γ 1 t,sothatf γ t >F γ 1 t. The voter s strategy can be summarized is a probability of voting for incumbent when y t = y L,y H namely (σ(y L ),σ(y H )), eachbetween0 and 1. a. Using Bayes Rule, characterize voters beliefs about the probability that ω = ω conditional on observing y t {y L,y H }. b. Consider a cut-off rule (σ(y L )=0,σ(y H )=1). Justify the form of this rule. c. Characterize an incumbent s optimal behavior in a two period example, given the voter s optimal rule (σ(y L )=0,σ(y H )=1). d. Given the politician s optimal behavior in response to (σ(y L )=0,σ(y H )=1), discuss the voter s incentive to deviate from this behavior. 6