1. (22 points) Consider a two-period economy where the representative firm uses a production technology given by Yi = ziF (Ki, Ni) for i = 1, 2, where Ki is the capital input, that depreciates at the rate 6, and Ni is the labor input used by the firm in period i to produce Yi.
Suppose that in period i, for i = 1, 2, the government levies a proportional tax, " i,
where " i 2 (0, 1) on firms revenue Yi. So the taxes paid by the representative firm are " iYi. The tax revenues are rebated lump-sum to households. Let TRi denote the lump sum transfer that the representative consumer receives.
The representative firm owns capital, pays a wage rate w per unit of labor hired and can borrow or lend at the market real interest rate r. It decides on how much labor to hire in each period and how much capital to accumulate for the following period in order to maximize the present discounted value of after-tax profits:
(a) 4 points) Write the equations describing the profit (pre-tax and after-tax) of the representative firm in both periods.
(b) (4 points) Set-up the representative firm’s problem, and derive the corresponding optimality conditions.
(c) (2 points) Assume that the government balances its budget in every period. Write the government’s budget constraints:
Consider now that in this economy the representative consumer maximizes
U = u(C1)+ #u(C2)
where Ct is per capita consumption. 0 < # < 1. The representative consumer receives disposable income in each period: (w1N1 + N1 + TR1) in period 1 and (w2N2 + N2 + TR2) in period 2, supplies N1=N2=1 units of labor in each period and can borrow or lend at the market interest rate, r. TRi for i=1, 2 denote the lump sum transfer that the representative consumer receives in each period as described previously. Assume the consumer allocates resources across both periods consumption ac- cording to the optimality condition uC (C1)=(1 + r) #uC (C2) (d) (4 points) Write the system of equations that defines the competitive equilib- rium for this economy. List all the variables that you are solving for and make sure you have enough equations to solve for all the variables Assume that the production function takes the form, F (K, N )=K!N 1!!, while 6=1 and the momentary utility takes the following functional form: u(C)=log C. (e) (6 points) Solve for the competitive equilibrium level of capital accumulation, K2. (f) (6 points)How does capital accumulation respond to an increase in the tax rates, " t for t = 1, 2? How does consumption respond in each period? Explain intuitively
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