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Macroeconomic Analysis

Econ 6022

Problem Set 2

Fall 2022

1    Marginal Propensity to Consume

In Lecture 2, we saw an example where the household only lives for two periods.  Now suppose that the

N

household lives for N periods.  Therefore, the life time utility function is U =  对 βt−1u(ct ).  The income

t=1

1. Write down the objective function of the household.

2. Write down the inter-temporal budget constraint.

3. Write down the Euler equation between Period t and t\ , where t\  t. (Hint: You may find the Lagrange method useful in this case.)

4. For simplicity, we assume that the discount factor β = 1 and interest rate i = 0. Solve for the optimal consumption plan for the N periods, ct , where t = 1, 2, 3, ··· N .

5. What’s the marginal propensity to consume in period t = 1 ? (You can explain in words.)

6. What’s the marginal propensity to consume in period t = 2 ? (You can explain in words.)

2    Precautionary Savings

In a two period model, suppose the agent’s lifetime utility function is U(c1 ,c2 ) = u(c1 ) + βu(c2 ), where u(·) is a concave function. The market interest rate is constant, r . The agent’s income are y1  and y2  in Period 1 and 2, respectively. The initial wealth endowment is w0 .

1. Derive the Euler equation in this case.

2. Further assume that (r + 1) · β = 1 and r = 0, solve for the optimal consumption (c1(∗) ,c2(∗)) in Period 1 and 2.

3. Further assume that income in Period 2 is a random variable, y˜2 , which takes two values, yh  and yl , with equal probability, i.e., E[y˜2] =   = y2 .  What is the agent’s optimal consumption in Period 1, if the utility function takes the quadratic form, i.e.  u(c) = η · c − c2 ?  Is there any precautionary saving? Why or why not?

4.  (Optional) Now assume there is uncertainty in agent’s income in period 2. The agent’s income in Period

2 can be y2(h)  and y2(l)  with equal probability.  Moreover, y2(h)  = y1 + σ > 0 and y2(l)  = y − σ < 0 and then  = y2  = y1 . Specifically, we assume that σ = ^ y1 . Compare the consumption with the previous problem.

5.  (Optional) If the utility function is log utility ln(c), what is the optimal consumption in Period 1?