Consider an economy with a representative consumer, a representative firm and a government. The consumer owns K unit of capital that she can rent to the firms at a price r, and can work for up to h hours at an hourly rate of w. She only gets utility from consumption and does not care about how much she works. Her utility is given by U (C) = ln c. Her capital rent is taxed by the government, so that her budget constraint is:
C = wNs + (1 − τ ) rKs + π
where Ns and Ks are the supply of labor and capital by the consumer. She faces following extra two constraints:
N s ≤ h Ks ≤ K
since she cannot work more than h hours, or rent more than the K units of capital she owns. The firm operates a CRS technology that uses labor (Nd) and capital (Kd) to produce. So total output is given by:
Y = zF(Kd , Nd) = z (Kd)^α (Nd)^1−α
The firm seeks to maximize profits choosing capital and labor demand. The government taxes capital rent at a rate τ that it chooses so as to finance an exogenously given level of spending G.1−α
(a) Define an equilibrium for this economy.
(b) Find the labor and capital supply. Draw them in two diagrams.
(c) Pose the firm’s problem and get the firm’s FOC. What are firm’s profits in equilibrium?
(d) Define what it means for an equilibrium of THIS economy to be Pareto efficient.
(e) Pose the planner’s problem for this economy. Interpret the planner’s constraints.
(f) Is the equilibrium Pareto efficient? Show why or why not using the planner’s problem.