In exercise 27.1 we extended our analysis of subsidized voluntary giving from 2 to N people. In the process, we simply assumed the government would set t to cover its costs — and that individuals would take t as given when they make their decision on how much to give. We now explore how the strategic setting changes when individuals predict how their giving will translate into taxes.
A: Consider again the case where N identical people enjoy the public good.
(a) First, suppose N = 2 and suppose the government subsidizes private giving at a rate of s. If individual n gives yn to the public good, what fraction of the resulting tax to cover the subsidy on his giving will he have to pay?
(b) Compare the case where the individual does not take the tax effect of his giving into account to the case where he does. What would you expect to happen to n’s best response function for giving to the public good in the former case relative to the latter case? In which case would you expect the equilibrium response to a subsidy s to be greater?
(d) Given your answer to (c) (and given that the optimal subsidy level when N = 2 in exercise 27.1 was 0.5), what do you think s would have to be to achieve the efficient level of the public good now that individuals think about balanced-budget tax consequences?
(e) Next suppose N is very large. Explain why it is now a good approximation to assume that individual n takes t as given when he chooses his contribution level to the public good (as he did in exercise 27.1).
(f) True or False: The efficient level of the subsidy is the same when N = 2 as when N is very large if individuals take into account the tax implication of increasing their giving to the subsidized public good.
(g) Finally, suppose we start with N = 2 and raise N. What happen to the degree to which n’s giving decisions impact n’s tax obligations as N increases? What happens to the size of the free rider problem as N increases? In what sense do these introduce offsetting forces as we think about the equilibrium level of private contributions?
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 (N −1) 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 (1− t) I while his cost of giving zn is (1−s) zn. 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 zeq as a function of s. (You should be able to simplify the denominator of your expression to (1+α (N −1) (1−s).)
(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 government subsidized 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).