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Competitive Equilibrium and Planner's Problem (50 pts) Consider a simple one-period, closed economy model where the representative consumer has a
Competitive Equilibrium and Planner’s Problem (50 pts)
Consider a simple one-period, closed economy model where the representative consumer has a utility function u (C, l) where C denotes consumption while l is leisure. Assume that the representative consumer hash available hours to divide between work and leisure. The representative firm has technology given byY = zFK, Nd. There is a government that sets its expenditure level at value G.
(a.) (5 pts) Considering the case of u (C, l) = C1/2l1/2 =√Cl, h = 20 and taking as given the wage rate w,the firm’s profit, π and the tax, T. Solve the consumer’s problem, specifically, find out the FOC whichspecifies the relationship between C and l (given of w, π and T).
(b.) (5 pts) Considering the case of Y = zK1/3Nd2/3, z = 1, K = 1000 and taking as given the wage ratew, solve the firm’s optimal choice of Nd, as a function of w.
(c.) (10 pts) Assume that G = 0, so that the market clearing condition in good market isY = Ccombine with your answers in Consumer’s and Firm’s problem above, as well as the fact that Nd =Ns = h − l and T = G, what is the equilibrium labor supply (or demand)?
(d.) (10 pts) Given your answer in last section, what the equilibrium consumption, C, the equilibrium wagerate, w and equilibrium firm’s problem, π?
(e.) (10 pts) Consider instead, the social planner problem, which is
(f.) (10 pts) Does your solution of C and l under the social planner’s problem similar to your solution of Cand l under competitive equilibrium (subquestions 1-4)?
