1. (36 points) A government wants to raise revenue to fund a public good. It can do so by taxing widgets, which are produced in a perfectly competitive market with no pre-existing distortions. The widget supply and demand curves are PS(Q) = Q1 PDQ) = 4-Q The total benefit from spending R in tax revenue is B(R) = 2 x R. For this problem, assume that the public good can only be funded through a per-unit commodity tax on widgets. You may also assume there are no income effects. 1 (a) (12 points) Find the equilibrium quantity, revenue, and deadweight loss as a function of the tax rate T. Assume the government's objective is to maximize the total benefit 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 first-order condition with respect to the tax rate. (c) (6 points) Solve for the optimal tax rate T*. Compare the total benefit B(R(T*)) to the dead- weight loss DWL(T*) and revenue R(T*) 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 T, QM(T). It may be helpful to first derive the monopolist's marginal revenue curve, MR(Q). (e) (6 points) Solve for the optimal tax rate T under monopoly. Compare the total benefit B(RM(M)) to the revenue RM(TM) and deadweight loss due to the tax. (f) (3 points) Comment on any similarities and differences between your answers to (d) and (f)