Political Economy Problem Set2

Political Economy

Due: 5 pm, Wednesday, October 30, 2013

Submission: Upload to the course website.

1.

Austen-Smith and Banks 4.1. (10 points)

a) Prove: If is odd then plurality rule is transitive or all [Hint: use Exercise 1.1 and Theorem 4.1]

{Exercise 1.1 a) Prove that binary relation is transitive on if and only if the following three conditions hold for all:

1.

2.

3. and

b) Prove that if is complete (but not necessarily transitive), then (1) and (2) imply (3).}

{Theorem 4.1: Let be a voting rule; then for all, is quasi-transitive}

b) Prove: if is single-peaked with respect to and then

2. Austen-Smith and Banks 5.5. (10 points)

Suppose is a strong simple rule and. Prove that implies whenever is a convex set in.

3. Austen-Smith and Banks 5.6. (10 points)

Suppose each has Euclidean preferences with ideal point. Provide examples to show that there is no equivalence relation between any pair from where:

A) is a multidimensional median iff,

B) is a partial median iff is a multidimensional median for some set of orthogonal basis vectors;

C) is a total median iff , .

4. Consider a policy space where there are three agents, whose preferences are as in Example 5.1 in Austen-Smith and Banks. Agent 1’s most favored point is (0, 0), Agent 2’s most favored point is (0, 1), and Agent 3’s most favored point is (1, 0). The status quo is given by. Show that, if all three agents vote sincerely (with indifference to be decided at your discretion), Agent 3, by only making policy proposals that are Pareto optimal, can set a finite agenda so that for any, the resulting policy is within of his ideal point. (15 points)

{Example 5.1: Let; and assume is Euclidean; that is, can be represented by the utility functions, where is‘s most preferred alternative. Set and: see Figure 5.1. Then for any, there exists such that. }

5. A three person legislature has to choose both the level and distribution of a budget over a two-dimensional issue space, the set of non-negative vectors in. (It’s not possible to budget less than $0 for an issue). Assume each legislator i = 1, 2, 3 has Euclidean policy preferences over X with ideal point. The horizontal axis measures dollars allocated to issue 1 and the vertical axis measures dollars allocated to issue 2.

Consider two procedures for reaching a decision. In the budget process, the legislature first determines the total budget and then, having fixed, chooses the distribution of expenditures between the two issues, such that. In the appropriations process, the legislature first determines the level of expenditure on issue 1 and then, having fixed, chooses the level of expenditure on issue 2. The total budget is then defined by summing the two allocation decisions. All decision are made via majority rule with an open rule.

Legislators only care about the final policy outcome and vote strategically, in the following sense. In the budget process, when voting on a legislator anticipates the division of the budget between and that will be chosen in the second vote. Similarly, in the appropriations process, a legislator voting on anticipates the level of expenditure on that will be chosen once is fixed.

a) Illustrate examples of distributions of ideal points that result in. (7 points)

b) Does it matter that the legislators vote strategically rather than sincerely (e.g. in the appropriations process, we could say that a legislator votes sincerely if she assumes that her most preferred will be chosen and takes this as given when voting on )? Why or why not? (8 points).

6. We consider a model of a politician P and an agency A. The politician chooses a policy which results in an outcome. There exists a bijective map which maps the policy choice to the outcome. However, the map is completely unknown to the politician, except for the status quo policy 0 which maps to the status quo outcome 0; this is better than choosing the outcome randomly for the politician. There also exists an agency who knows the map completely. Both the politician and the agency are endowed with a smooth strictly concave utility function over the outcome, and these utility functions are common knowledge.

a) Assume that the agency can choose a policy for which they can reveal to the politician. What will be the equilibrium outcome? (8 points)

b) Now assume that the agency can choose a policy and tell the politician what outcome it maps to, but the politician has no way to ensure the agency is telling the truth. What is the equilibrium outcome now? (7 points)

7. We consider a model where there are two public goods, and h. There exists a set of voters where where is odd. Each agent is endowed with wealth and has a utility function of the following form

where is a head tax levied on each agent to fund the public good. Assume that are smooth, strictly increasing and strictly concave functions. A policy is an ordered pair. The policy space is .

a) Show that agents’ preferences satisfy an order restriction. (8 points)

b) Characterize the core outcome under majority rule. (5 points)

c) Now consider a game where two agencies, and h, simultaneously make budget proposals for the associated public goods, and the combined budget proposal is then put before the voters. Agency proposes a tax. Both agencies wish to maximize their individual budget. Let the status quo allocation be (0, 0). Voters then vote on the combined proposal versus the status quo. You may assume that voters always vote truthfully, or ‘as-if-pivotal’; that is, if they have a strict preference between the proposal and the status quo they will vote for the one they prefer. Characterize the set of Nash equilibria. (7 points)

d) Assume the agencies can collude, with the goal of maximizing the total budget. Can they obtain a higher total budget than the maximum budget in the set of Nash equilibria? (5 points)