We discussed in the text the basic externality problem that we face when we rely on private giving to public projects. In this exercise, we consider how this changes as the number of people involved increases.

A: Suppose that there are N individuals who consume a public good.

(a) Begin with the best response function in panel (a) of Graph 27.3 — i.e. the best response of one person’s giving to another person’s giving when N = 2. Draw the 45 degree line into your graph of this best response function.

(b) Now suppose that all N individuals are the same — just as we assumed the 2 individuals in Graph 27.3 are the same. Given the symmetry of the problem (in terms of everyone being identical), how must the contributions of each person relate to one another in equilibrium?

(c) In your graph, replace y2 — the giving by person 2, with y — and let y be the giving that each person other than person 1 undertakes (assuming they all give the same amount). As N increases, what happens to the best response function for person 1? Explain, and relate your answer to the free rider problem.

(d) Given your answers to (b) and (c), what happens to person 1’s equilibrium contribution as N increases?

(e) When N = 2, how much of the overall benefit from his contribution is individual 1 taking into account as he determines his level of giving? How does this change when N increases to 3 and 4? How does it change as N gets very large?

(f) What does your answer imply for the level of subsidy s that is necessary to get people to contribute to the efficient level of the public good as N increases? (Define s as the level of subsidy that will cause a $1 contribution to the public good to cost the individual only $(1−s).)

(g) Explain how, as N becomes large, the optimal subsidy policy becomes pretty much equivalent to the government simply providing the public good.

B: In Section 27B.2.2, we considered how two individuals respond to having the government subsidize their voluntary giving to the production of a public good. Suppose again that individuals have preferences that are captured by the utility function u(x, y) = x^αy^(1−α) where x is dollars worth of private consumption and y is dollars spent on the public good. All individuals have income I, and the public good is financed by private contributions denoted z_n for individual n. The government subsidizes private contributions at a rate of s ≤ 1 and finances this with a tax t on income.

(a) Suppose there are N individuals. What is the efficient level of public good funding?

(b) Since individuals are identical, the Nash equilibrium response to any policy (t, s) will be symmetric— i.e. all individuals end up giving the same in equilibrium. Suppose all individuals other than n give z. Derive the best response function z_n (t, s,z) for individual n. (As in the text, this is most easily done by defining n’s optimization as an unconstrained optimization problem with only z_n as the choice variable and the Cobb-Douglas utility function written in log form.)

(c) Use your answer to (b) to derive the equilibrium level of individual private giving z^eq (t, s). How does it vary with N?

(d)What is the equilibrium quantity of the public good for policy (t, s)?

(e) For the policy (t, s) to result in the optimal level of public good funding, what has to be the relationship between t and s if the government is to cover the cost of the subsidy with the tax revenues it raises?

(f) Substitute your expression for t from (e) into your answer to (d). Then determine what level of s is necessary in order for private giving to result in the efficient level of output you determined in (a).

(g) Derive the optimal policy (t∗, s∗) that results in efficient levels of public good provision through voluntary giving. What is the optimal policy when N = 2? (Your answer should be equal to what we calculated for the 2-person case in Section 27B.2.2.) What if N = 3 and N = 4?

(h) Can you explain s^∗ when N is 2, 3, and 4 in terms of how the externality changes as N increases? Does s^∗ for N = 1make intuitive sense?

(i) What does this optimal policy converge to as N gets large? Interpret what this means.