Econ 6022 Macroeconomic Analysis Problem Set 2 Fall 2022
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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?
2022-09-26