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

Problem set 1

1) Let us assume that s_t-1 and s_t represent the amount of stocks held by the household at the end of periods t and t+1 respectively; d_t and q_t are the amount of dividends paid by one unit of stocks in period t and the price of the stock in period t respectively, and c_t is consumption at time t. All variables are expressed in units of the consumption good. Interpret the intertemporal budget constraint by explaining the meaning of each of the terms that appear in it, and why the left hand side has to be less or equal than the right hand side:

$$q_t s_t+c_t\leq (q_t+d_t)s_{t-1}$$

2) The expression:

$$q_tu'(c_t)=\beta(q_{t+1}+d_{t+1})u'(c_{t+1})$$

represents the first order condition of the asset pricing model seen in class (all variables are defined as in question (1), and u'(c_t) represents the marginal utility of consumption at time t). Interpret such condition from an economic point of view.
3) In the infinite horizon asset pricing model discussed in class (Class 1) assume that households are not allowed to hold negative amounts of stocks. In other words, the problem of the household becomes:

$$max_{\{c_t\}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u(c_t)$$

given s₀=1, subject to the budget constraint (which holds in each period t):

$$q_t s_t+c_t\leq (q_t+d_t)s_{t-1}$$

and subject to the extra constraint $s_t \geq 0$, all t a) Write the first order condition (FOC) with respect to s_t and discuss its meaning from the economic point of view b) Show that the pricing formula for the agent is:

$$q_t \geq \sum_{i=1}^{\infty} \beta^i \frac{u'(c_{t+i})}{u'(c_t)} d_{t+i}$$

and discuss the meaning of the above formula. c) argue that the equilibrium of this model and of the model studied in class is the same. (remember that in equilibrium it must be that in each period c_t=d_t and that s_t=1)