Marco Del Negro, Teoria y Politica Monetaria, Spring 2000

Problem set 3
Solutions

The household's problem is:
$max_{{c_{1t},c_{2t}}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t v(c_{1t},c_{2t})$
subject to:
$p_t c_{1t}\leq M_t\\ y_{1t}+y_{2t} \leq y_t\\ M_{t+1} +B_{t+1}\leq (M_t- p_t c_{1t}) +p_t y_{1t}+p_t (y_{2t}-c_{2t})+(1+R_t)+B_{t}+T_t$
where B_t^s is the supply of nominal (one period) bonds, R_t is the it nominal interest rate, and T_t are transfers from the government.
The question gives the budget constraint of the government as well:

M_t+1^s+B_t+1^s=M_t^s+B_t^s(1+R_t)+T_t

.
Note that the savings of the household depend on total consumption only, and not on the division between cash and credit goods. Therefore, if we call
c_t=c_1t+c_2t
we can rewrite the intertemporal budget constraint as:

$$M_{t+1} +B_{t+1}+p_t c_{t}\leq M_t+B_t(1+R_t)+p_t y_{t}+T_t$$

As we did in class, we can divide the household's problem in two steps, an intratemporal problem and an intertemporal problem:
Step 1 (intratemporal): Given the initial quantity of money M_t and the price level p_t, and for a given choice of c_t, the household decides how to split consumption between cash and credit goods. The problem is then:
$max_{c_{1t},c_{2t}} v(c_{1t},c_{2t})\\ \mbox{subject to } c_{1t}\leq \frac{M_t}{p_t}, c_{1t}+c_{2t} \leq c_t$
The solution to this problem is a function of the two variables determining the constraints, namely c_t and $\frac{M_t}{p_t}$. we can then define an indirect utility function,
$u(c_t,\frac{M_t}{p_t})$
which is defined as:
$u(c_t,\frac{M_t}{p_t}) = max_{c_{1t},c_{2t}} v(c_{1t},c_{2t})\\ \mbox{subject to } c_{1t}\leq \frac{M_t}{p_t}, c_{1t}+c_{2t} \leq c_t$ Notice that the introduction of bonds does not change the intratemporal problem. Once c_t is taken as given, the portfolio choice between bonds and money is irrelevant for the intratemporal problem. Of course, the portfolio choice becomes relevant for the intertemporal problem, which becomes:

$$max_{\{M_{t+1},c_t\}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t u(c_t,\frac{M_t}{p_t})$$

subject to

$$M_{t+1} +B_{t+1}+p_t c_{t}\leq M_t+B_t(1+R_t)+p_t y_{t}+T_t$$

In order to solve for the equilibrium we now need the government budget constraint. But this is another problem.