Marco Del Negro, Teoria y Politica Monetaria, Spring 2000

Problem set 5

Consider an economy where the government budget constraint is:

$$M^s_{t+1}+B^s_{t+1}=M^s_t+B^s_t(1+R_t)+p_tg_t-p_t\tau_t$$

as in the previous problem set. The household solves the problem:

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

subject to

$$M_{t+1}+B_{t+1}+p_tc_t=M_t+B_t(1+R_t)+p_ty_t-p_t\tau_t$$

and $B_t\geq -B$, where B is some positive number, and where

$$y^*_t=y_t-\frac{\tau^2_t}{y_t}.$$

Assume that the utility function is of the form:

$$u(c_t,\frac{M_t}{p_t})=c_t+\log(\frac{M_t}{p_t}),$$

that $\beta=\frac12$, and that:

$$y_t=1 \mbox{, }g_t=0 \mbox{, }B^s_t=0 \mbox{, }M^s_t=M^s_0 (1+\mu)^t \mbox{, all }t.$$

(Note that the equilibrium is the same one found in the previous problem set.)
a) Find the equilibrium real money balances as a function of $\mu$. Show that the equilibrium real money balances are a decreasing function of $\mu$. Find equilibrium consumption.
b) Find the optimal inflation, that is, the level of $\mu$ that maximizes welfare. (Hint: substitute equilibrium real money balances and consumption into the objective function of the agent and...).
c) Show that Friedman's rule does not hold in this economy. Discuss why this the case.