Teoria y Politica Monetaria, Instructor: Marco Del Negro.

Problem set 8, Solutions

a) The FOCs with respect to M_t+1, A_t+1, and B_t+1 are respectively:

$$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{1}{p_{t+... ...frac{M_{t+1}}{p_{t+1}})+ u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})]$$

$$\frac{e_t}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{e_{t+1}}{p_{t+1}}(1+r^w)u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$

$$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{1}{p_{t+1}}(1+R_{t+1})u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$

b) Dividing the second FOC by the third one one obtains precisely:

$$(1+R_{t+1})=(1+r^w)\frac{e_{t+1}}{e_{t}}$$

or

$$(1+R_t)=(1+r^w_t)\frac{e_t}{e_{t-1}}$$

c) Substituting the budget constraint of the government into the budget constraint of the household we obtain equilibrium consumption:

$$c_t=y+\frac1p^w (A^p+A^g) r^w.$$

Dividing the second FOC by the first one and rearranging (as done in class) one obtains:

$$R= \frac{u_m(c_t,\frac{M_t}{p_t})}{u_c(c_t,\frac{M_t}{p_t})}$$

Since we know that from the money demand equation (obtained as usual from the FOC with respect to money holdings)

$$u_c(c,m)-\frac{\beta}{1+\mu-\beta}u_m(c,m)=0$$

then

$$R=\frac{1+\mu-\beta}{\beta}$$

As usual, the price level is given by:

$$p_t=\frac{M^s_t}{m^*(\mu)}$$

From the Law of One Price we have:

$$p_t=e_t p^w \mbox{, all } t$$

which implies that the level of the exchange rate is:

$$e_t=\frac{M^s_t}{p^w m^*(\mu)}$$

and the equilibrium rate of depreciation is $\mu$.