Marco Del Negro, Teoria y Politica Monetaria, Spring 2000

Problem Set 3

In the cash-in-advance model studied in class assume that people are able to invest not only in nominal money balances, but also in bonds. That is, the household's problem is now:

$$max_{{c_{1t},c_{2t}}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t v(c_{1t},c_{2t})$$

subject to:

$$\begin{eqnarray*}& p_t c_{1t}\leq M_t \\ & y_{1t}+y_{2t} \leq y_t\\ & M_{t+1... ...- p_t c_{1t}) +p_t y_{1t}+p_t (y_{2t}-c_{2t})+(1+R_t)+B_{t}+T_t \end{eqnarray*}$$

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 budget constraint of the government is:

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

Show that the reduced form of this model is the same as in the Sidrauski-Brock model studied in class 7, which is:

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

subject to

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