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

Problem set 1; Solutions

1) the budget constraint:

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

can be interpreted as follows. On the right hand side the term y_t represent the income that the agent receives, and the term q_t s_t-1 represents the wealth of the agent. The two together represent the total amount of resources available to the agent at the beginning of period t, which can be spent in consumption c_t, or re-invested in assets q_t s_t.
2) The first order condition:

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

can be interpreted as saying that at the optimum the marginal cost q_t u_c(c_t,s_t-1) from increasing the amount of saving in durable goods by an amount $\Delta s$ ( $q_t\Delta s$ is the amount of consumption the agent gives up, and $q_t u_c(c_t,s_{t-1})\Delta s$ is the cost in terms of utility) has to be equal to the marginal benefit $\beta [q_{t+1}u_c(c_{t+1},s_t)+u_s(c_{t+1},s_t)]$, where ( $q_{t+1}\Delta s$ is the increase in consumption in the next period, and $\beta q_{t+1}u_c(c_{t+1},s_t) \Delta s$ is the increase in utility from consumption, while $u_s(c_{t+1},s_t)) \Delta s$ is the increase in utility from the services provided by the durable good.
3) In this problem set you were asked to study the intertemporal asset pricing problem when households are subject to non-negativity constraints on their holdings of assets (that is, when households are not allowed to increase their consumption by taking a `short' position on the stock market, i.e., by `borrowing' on the stock market).
The problem of the household is the following:
$max_{\{c_t\}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u(c_t)\\ \mbox{ subject to } q_t s_t+c_t \leq (q_t+d_t)s_{t-1} \mbox{, } s_t\geq0 \mbox{, given} s_0=1$
where the constraints hold for each t=0,1,..
As in the model studied in class, the intertemporal budget constraint at the optimum will hold with = sign, so we can substitute the constraint in the objective function:
$max_{{s_t}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u((q_t+d_t)s_{t-1}-q_t s_t)\\ \mbox{ subject to } s_t\geq0 \mbox{, given} s_0=1$
This is a constrained optimization problem. There are two ways to go about it (both are considered good). The first one, more formal, is to use Lagrange multipliers. Notice that there is one non-negativity constraint $s_t \geq 0$ for each period t, so that the Lagrangian will be:
$max_{{s_t}_{t=0}^\infty} \sum_{t=0}^\infty [\beta^t u((q_t+d_t)s_{t-1}-q_t s_t) +\lambda_t s_t]$
where $\lambda_t$ is the Lagrange multiplier associated with the constraint $s_t \geq 0$
Let us focus on the first order condition (FOC) with respect to s_t. Notice that the term s_t appears in the sum above only in the terms:
$\beta^t u((q_t+d_t)s_{t-1}-q_t s_t) + \beta^{t+1} u((q_{t+1}+d_{t+1})s_t-q_{t+1} s_{t+1}) +\lambda_t s_t$
so the first order condition is:
$\beta^t u'(c_t)q_t=\beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})+\lambda_t$
Since we know that $\lambda_t \geq 0$, then at the optimum it must be that:
$\beta^t u'(c_t)q_t \geq \beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})$
The second way to proceed, less formal but perhaps more intuitive, is the following. Let us first solve the problem where the constraint does not bind, that is, assuming the problem is as in figure A. Then we know that the first order condition is the same one we studied in class:

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

Second, let us solve the problem where the constraint is binding, that is, as in figure B, where the maximum is reached in the `forbidden' area s_t<0: the agent would like to hold a negative amount of stock, but is not allowed to. The best thing to do, in that case, is to hold as less stock as possible, that is, s_t=0. If the objective function is strictly concave and differentiable, the then at the constrained solution s_t=0 the objective function is downward sloping, as shown in figure B. Then it must be that the derivative of the objective function at the constrained optimum is negative:

$$\beta^t u'(c_t)q_t -\beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})<0 \nonumber$$

or

$$\beta^t u'(c_t)q_t >\beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})$$ (2)

From 1 and 3 we obtain that at the optimum it must be:

$$\beta^t u'(c_t)q_t \geq \beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})$$ (3)

In class equation 1 was interpreted as saying that at the optimum the marginal cost beta^t u'(c_t)q_t from increasing the amount of saving by an amount $\Delta s$ has to be equal to the marginal benefit $\beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})$. This is the same as saying that at the optimum the marginal benefit from decreasing the amount of saving (that is, dis-saving) by an amount $\Delta s$, beta^t u'(c_t)q_t, has to be equal to the marginal cost $\beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})$. This interpretation of the first order condition is valid also in the problem considered here when the borrowing constraint is not binding (that is, when equation 4 holds with the = sign, figure A). When the borrowing constraint is binding the agent would like to dis-save more if she only could, but she cannot. At the constrained solution s_t=0 the household's `desire' to dis-save is still not satisfied: the marginal benefit from dis-saving by an amount $\Delta s$, beta^t u'(c_t)q_t is greater than the marginal cost $\beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})$ (figure B). This is the interpretation of equation 4 when it holds with the < sign.
4 can be rewritten as: $u'(c_t)q_t \geq \beta u'(c_{t+1}) (q_{t+1}+d_{t+1})$
writing the FOC for periods t+1,.., substituting, and imposing the transversality condition (that is, following the same steps discussed in class) we obtain the following expression (assuming u'(c_t)>0):
$q_t \geq \sum_{i=1}^{\infty} \beta^i \frac{u'(c_{t+i})}{u'(c_t)} d_{t+i}$
Does the presence of the borrowing constraint affect the equilibrium? From class we know that at the equilibrium price implied by the model without borrowing constraints, namely:
$q_t=\sum_{i=1}^{\infty} \beta^i \frac{u'(d_{t+i})}{u'(d_t)} d_{t+i}$
the demand for stock is equal to 1 (this is indeed the definition of equilibrium: prices are such that demand equals supply, which is 1). This is to say that when $q_t=\sum_{i=1}^{\infty} \beta^i \frac{u'(d_{t+i})}{u'(d_t)} d_{t+i}$ the agent will never want to hold a negative quantity of s_t: therefore we always are in the case of figure A, where the constraint is not binding. We can conclude that:
$q_t=\sum_{i=1}^{\infty} \beta^i \frac{u'(d_{t+i})}{u'(d_t)} d_{t+i}$
is still an equilibrium of the model, even when there are borrowing constraints.
One may wonder why, when d_t is particularly low with respect to d_t+1, the agent is not trying to smooth consumption by borrowing and going short in s_t. The answer is that the equilibrium price is such that she would not want to do so: the equilibrium return on the asset
$\frac{u'(d_t)}{\beta u'(d_{t+1})}=\frac{(q_{t+1}+d_{t+1})}{q_t}$
is so high that she is happy holding the existing supply of stocks, and she would not want to borrow.