# Explain the economic rationale for the difference between the central planner’s solution and the decentralized, voluntary solution.

Political Economy – W4370

Fall 2022

Problem Set 7

Question 1.

This question asks whether voting can play a role in solving free-riding problems in public good provision, and if so why.

Imagine a small economy composed of three individuals, 1, 2, and 3. Each of them has utility function:

Ui = ln(ci) + i

2 ln(G)

where i ∈ {1, 2, 3}, ci is the consumption of individual i, and G is a public good.

All individuals have the same income y that can be immediately converted either in the consumption good or in the public good (or equivalently, all prices are equal to 1).

Suppose first that the public good is voluntarily provided. Call gi ≥ 0

individual i’s provision of the public good. Hence: G = g1 + g2 + g3.

(2 points). What is i’s budget constraint?

(10 points) What are g1, g2, and g3? You are asked for the precise numerical values. (Hint: After solving the problem, verify whether your solution is feasible for all individuals—neither ci nor gi can be negative. If your preliminary solution gives you a negative value, set it to 0.

(5 points) What is G? What are c1, c2, and c3?

(10 points) How do these numbers compare to the numbers chosen by a benevolent central planner who chooses G and c1, c2, c3 so as to maximize the sum of utilities? In particular: what is the level of G chosen by the central planner? And c1, c2, c3? (Remember that the central planner can allocate the total resources of the economy as he sees fit).

(5 points) Does the central planner’s solution you found satisfy Samuel- son’s condition? verify.

(3 points) Explain the economic rationale for the difference between the central planner’s solution and the decentralized, voluntary solution.

The benevolent, all-knowing central planner does not exist. What exists instead is a political system. In an effort to improve public good provision, the three individuals in this society decide to introduce a proportional income tax and devote all tax revenues to the public good. Thus ci = (1 − t)y and G = ty1 + ty2 + ty3 = t(3y). There is no other contribution to the public good.

Each individual proposes his ideal tax rate t, and the decision is then made by majority voting: if one of the three tax rates is preferred by a majority to both of the others, it is implemented.

(5 points) Please write each individual’s utility as function of the tax rate

(Hint: Using the budget constraint, what is i’s consumption of the private good ci now?)

(5 points). What is the value of t preferred by each individual i? Call that ti for individual i.

(5 points) Suppose the only three alternatives are the three values of t preferred by the three individuals.Fill the following table with their preference rankings, as I did below for individual 1.

1 2 3

t1

t2

t3

(3 points) Does this problem satisfy the conditions of the median voter theorem? Why?

(3 points) What value of t is the Condorcet winner? What is G then?

What are c1, c2, and c3?

(4 points) How do these numbers compare to the numbers you find when the public good is voluntarily provided? And to the central planner’s solution? Do you expect these results to be general? Why?

Question 2. (80 points).

It is often argued that one advantage of deferring decisions to committees is that committees are less prone to ”capture”, the undue influence of lobbyists.

The logic is simply that in a committee several individuals will need to be influenced, and that will require more time and money than influencing a single individual. But is this logic correct? (This question is based on ”Bribing Voters” by Ernesto Dal Bo’, American Journal of Political Science, 2007).

Consider the following scenario. A lobbyist wants to obtain the approval of a proposal X. The lobbyist is not budget constrained and can afford a large expense to ensure approval, but would prefer to do so at as little cost as possible.

In all that follows, think of this as a one-shot game with no future repetitions and no reputation effects. However, the lobbyist always respects his promises.

(5 points). Imagine first that approval of the policy depends on a single policy-maker. The policy-maker is opposed to the proposal and derives a disutility that, expressed in dollar terms, equals θ, if the proposal is approved.

θ is publicly known. On the other hand, the policy-maker values the transfer received from the lobbyist. If we call it b, the policy-maker’s utility is b − θ if the proposal is approved, and 0 otherwise. What is the minimum cost to the lobbyist of ensuring approval?

Suppose now that the proposal is decided by a committee of 5 voters. The decision is taken by simple majority, and all voters will vote simultaneously.

All 5 voters oppose the proposal and each of them derives disutility θ from its passing (again this fact and the value of θ are publicly known). All 5 voters would value positively transfers received from the lobbyist, and their utility if the proposal passes or does not pass is identical to the utility of the single policy-maker described above.

As in real-life legislatures, all votes are observable–i.e. the lobbyist will be able to observe not only the final number of votes on each side but also who cast which vote. He offers each of the 5 voters the following contract: ”I will pay you θ + ε if you vote in favor of the proposal and your vote is pivotal” where ε is a small positive amount.

When deciding how to vote, a voter does not know whether or not he is pivotal. He needs then to consider the different possible scenarios.

(i) (10 points). Identify in which scenarios the voter is indifferent between voting Yes or No, and in which scenarios the voter has a strict preference over voting Yes or No, given the contract offered by the lobbyist.

(ii) (10 points) For the rest of this question, you can assume that when indifferent the voter votes as if he were pivotal. How will the voter vote then?

(iii) (5 points). Given your answer to (ii) above, how many votes will then be cast in favor of the proposal, and how many against?

(iv) (10 points). How much does the lobbyist need to pay? How does this amount compare to the cost of lobbying a single policy-maker?

(10 points). After having received the lobbyist’s offer but before voting, the committee members can meet and discuss how to vote. Would communication alone allow them to coordinate on a different outcome?

(10 points). Suppose now that the committee members’ preferences are not publicly known. What is publicly known however is that the maximal disutility any committee member can suffer from the proposal is some known value θ. Can the lobbyist offer a new contract that will allow him to obtain a favorable vote now? How much will it cost him?

Afraid of the possibility of capture, the committee decides to raise the threshold for passing the proposal to 4 positive votes. Let’s go back for simplicity to the scenario of point 3–a commonly known disutility θ that each voter would suffer from the proposal passing.

(i) (10 points). Will this change the outcome?

(ii) (10 points). Would requiring unanimity change the outcome?

Explain the economic rationale for the difference between the central planner’s solution and the decentralized, voluntary solution.