Marco Del Negro, Teoria y Politica Monetaria, Fall 1999

Problem set 2
Solutions

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:

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

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.

a) Let us substitute the constraint in the objective function:

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

and notice that the term s_t appears in the sum above only in the two terms:
$\beta^t u(y_t-q_t (s_t-s_{t-1}),(1-\delta)s_{t-1}) + \\ \beta^{t+1} u(y_{t+1}-q_{t+1}(s_{t+1}-s_{t}),(1-\delta)s_t)$

Taking derivatives, the first order condition is:

$$\beta^t q_t u_c(c_t,(1-\delta)s_{t-1})=\beta^{t+1} [(1-\delta)u_s(c_{t+1},(1-\delta)s_t)+q_{t+1} u_c(c_{t+1},(1-\delta)s_t)]$$

where $u_c\equiv\frac{du}{dc}$, and $u_s \equiv \frac{du}{ds}$
which can be rewritten as:

$$q_t u_c(c_t,(1-\delta)s_{t-1}) =\beta [(1-\delta)u_s(c_{t+1},(1-\delta)s_t)+q_{t+1} u_c(c_{t+1},(1-\delta)s_t)]$$

The marginal cost (benefit) from decreasing (increasing) durable goods holdings today is

$$q_t u_c(c_t,(1-\delta)s_{t-1})$$

where q_t is the decrease (increase) in consumption, and $u_c(c_t,(1-\delta)s_{t-1})$ is the decrease (increase) in today's utilty. The marginal benefit (cost) from increasing (decreasing) durable goods holdings tomorrow comes from the additonal resources that can be spent from consumption,

$$\beta q_{t+1} u_c(c_{t+1},(1-\delta)s_t)$$

plus the service in terms of utility that comes from holding the durable good:

$$\beta (1-\delta)u_s(c_{t+1},(1-\delta)s_t)$$

Following what we did in class, we can write the FOC for periods t+1,..,t+T, substitute again and again for q_t+1 u_c(c_t+1,s_t),...
and obtain the following expression:

$$q_t= (1-\delta) \sum_{i=1}^T \beta^i \frac{u_s(c_{t+i},(1-\de... ...{t+T},(1-\delta)s_{t+T-1})}{u_c(c_t,(1-\delta)s_{t-1})} q_{t+T}$$

Assume that $lim_{T \rightarrow \infty}\beta^T \frac{u_c(c_{t+T},(1-\delta)s_{t+T-1})}{u_c(c_t,(1-\delta)s_{t-1})} q_{t+T}=0$, and we obtain:

$$q_t=(1-\delta)\sum_{i=1}^{\infty} \beta^i \frac{u_s(c_{t+i},(1-\delta)s_{t+i-1})} {u_c(c_t,(1-\delta)s_{t-1})}$$

b) in equilibrium it must be that in each period c_t=y_t and s_t=1

the pricing formula then becomes:

$$q_t=(1-\delta)\sum_{i=1}^{\infty} \frac{u_s(y_{t+i},(1-\delta))}{u_c(y_t,(1-\delta))}$$