Macroeconomics III: Problem Set 3 - Consumer Behavior with Durable Goods and Labor Supply, Assignments of Local Government Studies

Download Macroeconomics III: Problem Set 3 - Consumer Behavior with Durable Goods and Labor Supply and more Assignments Local Government Studies in PDF only on Docsity! Core Macroeconomics III Professor Nicolas Magud Problem Set 3 Spring 2006 Due Date: 04/27/05 Problem 1: The Life-Cycle Hypothesis and Durable Goods So far in this term we have looked at models in which consumption goods yield utility only in the period in which they are bought. This assumption may be appropriate for nondurable goods and services, but it is clearly not appropriate for consumer durable goods such as autos and houses, since a durable good purchased in period t will typically yield flow utility at time t and for several periods thereafter. Assume that a consumer’s utility depends on the flow of services from a stock of durable goods. The consumer problem is max ∞∑ t=0 βtu(kt) (1) s.t. kt = ct + (1− δ)kt−1 (2) at+1 = (1 + r)(at + yt − ct) (3) a0; k0 (4) aT+1 = 0(NPG) (5) where kt is the consumer’s stock of durable goods, ct is purchases of durables, yt is labor income, at is financial wealth, and δ ∈ (0, 1) is the depreciation rate of durables. Assume that yt is random as of t− 1, that the riskless interest rate r is constant, and that there is no risky asset. We can restate the consumer’s problem in dynamic programming form, as follows (note that kt is not a state variable at t, but kt−1 is): Vt (at, kt−1) = max ct {u(kt) + βEt [Vt+1(at+1, kt)]} (6) s.t. (2) and (3). a) What additional assumptions would you have to make in order for the value function Vt(at, kt−1) to be time -invariant? b) Use the Bellman Equation (6) to establish the following Euler Equation for the stock of durable goods: 1 u′(kt) = β(1 + r)Etu′(kt+1) (7) Hints: the first order condition for c will involve u′(k) and the expected derivatives of V with respect to both a and k. Evaluate the derivatives of V using the envelope theorem. Show that Vk = (1 − δ)Va, and use this to eliminate the term involving dV/dk from the first order condition for c. Then show that Va = (1 + r) u′(k) r+δ . c) Now assume that u(k) is quadratic and that β(1 + r) = 1. Show that the stock of durable follows a random walk: kt+1 = kt + ²t+1 (8) where Et²t+1 = 0. Show that durables purchases obey ct+1 = ct + ²t+1 − (1− δ)²t (9) where ²’s in (8) and (9) are identical. d) Makiw (1982, on your reading list) finds that aggregate durables purchases (ct) in the post war US follow a random walk; given ct, variable such as ct−1, (ct − ct−1), kt, and (kt − kt−1) do not help predict ct+1. Is this result consistent with (8) and (9)? (Note: for consumer durables, a reasonable value for δ is 0.3). Problem 2: Consumption with Variable Labor Supply Consider the problem of a household that has to choose both consumption and labor supply in a stochastic dynamic environment. The household problem is V0 = max ct,lt E0 T∑ t=0 βtu(ct, lt) (10) s.t. at+1 = (1 + r)(at + wtlt + It − ct) (11) and a0 is given, and aT+1 = 0, where lt is the household’s hours worked at t, wt is the wage rate, It is transfer income, and within-period preferences are given by u(ct, lt) = log(ct)− B γ lγt (12) where B > 0 (so the household gets disutility from working) and γ > 1. The household regards l as a choice variable, while w and I are seen as exogenous. Assume that the household knows wt and It at time t, but not at time t− 1 (in other words, w and I are stochastic). For convenience, note that there is no risky assets and that the interest rate is constant; you may go further to assume that β(1 + r) = 1. a) Warm up question: show that γ > 1 insures that the marginal disutility of work is increasing in hours worked. Does this assumption makes sense? 2