Problem
Pareto Optimal Allocation and Competitive Equilibrium with Proportional Income Tax . Consider an economy with the representative consumer, representative firm, and government. For simplicity, suppose that G = 0. The consumer's preferences over the consumption good and leisure are given by the utility function U(C, l) = 2 ln C + ln l. Let h = 18. The firm's production function is Y = zK1/2N1/2 , where z = 2 and K = 1.
(a) Write down the social planner's maximization problem and the social planner's optimality condition. (Recall: MRSl,C = ?U/?l ?U/?C and MPN = ?zF(K, N)/?N.)
(b) Derive the Pareto optimal allocation for leisure, labor, output, and consumption.
(c) Suppose the government imposes a proportional income tax t = .5 on the representative consumer, and distributes the revenue back to them as a lump-sum transfer (subsidy). That is, the consumer's budget constraint is c = w(1 - t)(h - l) + p + S, where S is the subsidy. Compute the amount of labor in the competitive equilibrium, and show how it compares to the one you found in part (b).
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.