(a) (12 points) Find the equilibrium quantity, revenue, and deadweight loss as a function of the tax rate Ir. Assume the government's objective is to maximize the total benet from funding the public good, minus the loss in product market surplus due to the commodity taxes. (b) (3 points) Write down the government's objective function and take the rst-order condition with respect to the tax rate. (c) (6 points) Solve for the optimal tax rate 7*. Compare the total benet B (R('r*]] to the dead- weight loss DWL(-r*) and revenue ROW\") under the optimal widget tax. Now suppose that widgets are produced by a monopolist. (d) (6 points) Solve for the equilibrium quantity under monopoly as a function of the tax rate 'r, QM(T). It may be helpful to rst derive the monopolist's marginal revenue curve, MR(Q). (e) (6 points) Solve for the optimal tax rate 71"} under monopoly. Compare the total benefit B(RM(7;J)] to the revenue RM (734) and deadweight loss due to the tan; (1') (3 points) Comment on any similarities and differences between your answers to (d) and (f). 2. (32 points) There are two agents in the economy, agent 1 and agent 2. Both agents can earn a wage of $24 per hour. However, they have different preference-3 over consumption and hours worked: 1 1 1 m _ _ 7 : _ i 2 u1(c,h) 2h1(c) + 2h1(T h) m(c,h) 242 c h Note that we have defmed the agents' utility functions in terms of hours worked rather than leisure. Both agents can work a maximum of T : 24 hours per day. There is a government which would like to maximize the sum of the two agents' utilities: SU : \"1(65 h) + \"2(Cs h) (a) {8 points) Show that if each agent works the privately optimal amount and consumes her own income, they will choose to work the same number of hours. (b) (8 points) Compare the two agents' marginal utilities of consumption and marginal disutilities of work in the privately optimal allocation. Would the government want to make any transfers between the two agents or adjust their hours of work in the rst-best allocation? Justify your answer. You do not need to explicitly solve for the rstabest. For the rest of this problem, assume that agent 2's utility function is instead given by u2(c,h) : 5:3 c2>