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Macroeconomic Theory (AS.440.602)

Fall 2022

Problem Set 6

Due: Thursday, November 17, 2022

1) The representative agent lives for two periods (1 and 2). Each period, the agent consumes (ct) and supplies labor (lt). The real wage per unit of labor is wt and the agent does not pay taxes. The lifetime present discounted value of utility is given by:

u(c1,l1) +  βu(c2,l2)

with β(< 1) being the discount factor.

The agent is allowed to save or borrow at the real interest rate r, but she cannot die with debt or wealth. Assume also that the initial wealth is zero.

a. Find the lifetime budget constraint of the agent.

b. Solve the optimization problem of the agent using the period by period budget constraints. In particular, show the Euler equation.

c. Assume that  and . Show the Euler equation in consumption.

d. Find the labor supply condition for each period.

e. Show the ratios and  Will consumption be smoothed? Will labor supply be smoothed? Discuss!

f. Find the elasticity of labor supply with respect to the real wage in period 1.

2) The representative agent lives for infinite periods (0,1,2,…). Each period, the agent consumes (ct) and supplies labor (lt). The pre-tax real wage per unit of labor is wt and the agent pays a tax rate of τt. The lifetime present discounted value of utility is given by:

with β(< 1) being the discount factor. The agent is allowed to save or borrow at the real interest rate rt, but she cannot die with debt or wealth. Assume also that the initial wealth is zero.

a. Solve the optimization problem of the agent and show the Euler equation.

b. Assume that . Show the Euler equations in consumption and labor supply between time t and time t+1.

c. Show the labor supply condition for time t. How does the tax rate affect labor supply?

3) The representative agent lives for infinite periods (0,1,2,…). Each period, the agent consumes (ct) and supplies labor (lt). The real wage per unit of labor is wt and the agent does not pay taxes rate. The lifetime present discounted value of utility is given by:

with β(< 1) being the discount factor.

The agent is allowed to save or borrow at the real interest rate rt, but she cannot die with debt or wealth. Assume also that the initial wealth is zero.

a. Solve the optimization problem of the agent using the period by period budget constraints. In particular, show the Euler equation.

b. Assume that . Show the labor supply condition.

c. Show the labor supply condition when σ = 1 and φ = 1.

d. Show the labor supply condition when σ = 0.

e. Assume now the following GHH utility function: . Find the labor supply condition.

f. Compare your answers to part d and part e.

g. 

4) Assume the following: u(ct) = ln(ct), β(1+r) = 1 and no uncertainty.

Write the full problem of the consumer with the borrowing constraint, derive the Euler equation and then express the Lagrange multiplier on the borrowing constraint (μt) as a function of consumption levels.