B. Consider the same set-up as in exercise 27.1, but now suppose that each individual assumes the government will balance its budget and therefore anticipates the impact his giving has on the tax rate t when the subsidy s is greater than zero.

a. The problem is again symmetric in the sense that all individuals are the same, so in equilibrium, all individuals will end up giving the same amount to the public good. Suppose all 1N 2 12 individuals other than n give z when the subsidy is s. Express the budget-balancing tax rate as a function of s assuming person n gives zn while everyone else gives z.

b. Individual n knows that his after-tax income will be 11 2 t2I while his cost of giving zn is 11 2 s2zn

. Using your answer from (a), express individual n’s private good consumption as a function of s and zn

(given everyone else gives z).

c. Set up the utility maximization problem for individual n to determine his best response giving function (given that everyone else gives z). Then solve for zn as a function of z and s. (The problem is easiest to solve if it is set up as an unconstrained optimization problem with only z1 as the choice variable, and with utility expressed as the log of the Cobb–Douglas functional form.)

d. Use the fact that zn has to be equal to z in equilibrium to solve for the equilibrium individual contribution z eq as a function of s. 1You should be able to simplify the denominator of your expression to 11 1 a1N 2 12 11 2 s2 2.2

e. If everyone gave an equal share of the efficient level of the public good funding, how much would each person contribute? Use this to derive the optimal level of s. Does it depend on N?

f. True or False: When individuals take into account the tax implications of governmentsubsidized private giving, the optimal subsidy rate is the same regardless of N and equal to what it is when N gets large for the case when people do not consider the impact of subsidized giving on tax rates (as explored in exercise 27.1).