Teoria y Politica Monetaria, Spring 2000; Instructor: Marco Del Negro

Problem set 2

1) Consider the 'price of durable goods' model studied in class, and assume that the household benefits only from a fraction $(1-\delta)$ of the durable goods she owns. Specifically, the household's problem is:

$$max_{{c_t}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u(c_t,(1-\delta)s_{t-1})$$

subject to the budget constraint (which holds in each period t):

$$q_t s_t+c_t \leq y_t + q_t s_{t-1}$$

and the initial condition s_-1. (Remember that s_t-1 and s_t represent the amount of durable goods held by the household at the end of periods t and t+1 respectively, q_t is the price of the durable goods in period t, y_t and c_t are endowment and consumption at time t respectively. All variables are expressed in units of the consumption good) The supply of the durable goods is 1 in each period.
a) Find the first order condition of the agent with respect to s_t and interpret it from an economic standpoint.
b) Find the equilibrium values for c_t, s_t, and q_t.