Macroeconomic Theory (AS.440.602) Fall 2022 Problem Set 5
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Macroeconomic Theory (AS.440.602)
Fall 2022
Problem Set 5
Due: Thursday, November 3, 2022
1) The representative agent lives for two periods (1 and 2) and receives exogenous incomes of 
and
, respectively. The second-period income equals
with probability q and
with probability (1-q). In the class, we analyzed the cases with 
and
. You will now need to analyze the case with 
.
The agent is allowed to save or borrow at the real interest rate
, 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. Solve the optimization problem of the agent using the lifetime budget constraint. In particular, show the Euler equation.
c. Assume that 
. Will the actual saving of the agent in period 1 be more than, less than or equal to her saving under certainty equivalence. Show your work for 
You do not need to use any specific functional form.
2) The representative agent lives for infinite periods (0, 1, 2, …) and receives exogenous incomes of
, respectively. As assumed in class, the agent faces uncertainty about the future stream of her incomes. The lifetime present discounted value of utility is given by:
with
being the discount factor. The agent is allowed to save or borrow at the real interest rate
, 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 corresponding Euler equation.
c. Using your answer for part b, show the Euler equation when 
and 
d. Assume now
. Show the Euler equation when 
e. Compare your results from part d with your results from part c.
f. Use the quadratic utility function of part d and assume
Will the agent be engaged in consumption smoothing? Explain and show your work.
2023-12-19