Teoria y Politica Monetaria, Instructor: Marco Del Negro

Problem Set 10, Mock Final Exam

1) Assume that all of a sudden a number of goods which could be previously purchased only which cash now can be purchased with credit cards. Other things being equal (money supply and output) what happens to real money balances and to the price level?
2) If taxes are distortive, following the Friedman rule may not be optimal. Explain. Can a negative rate of inflation still be optimal? Explain.
3) Introduce real denominated bonds (UDIs) in the Sidrauski-Brock model studied in class. Specifically, assume that the objective function of the representative agent is:
$max_{\{M_{t+1},c_t\}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t u(c_t,\frac{M_t}{p_t})$
and that the budget constraint of the household is:
$M_{t+1} +B_{t+1}+p_t c_{t}+p_t b_{t+1}\leq M_t+B_t(1+R_t)+T_t+p_t y+p_t (1+r_t) b_t$
where b_t is the amount of UDIs held by the household at the beginning of period t, r_t is the return to holding UDIs (defined in real terms), in period t. All other variables are defined as usual (notice that output in real terms is constant). The budget constraint of the government is:
M_t+1^s+B_t+1^s+p_tb_t+1^s=M_t^s+B_t^s(1+R_t)+T_t+p_t (1+r_t) b_t^s
a) find the first order conditions with respect to M_t+1, B_t+1, and b_t+1.
b) using the first order conditions show that the Fisher Equation, $R=r+\pi$, has to hold in this model (remember that $\log(1+x) \simeq x$ for x close to 0)
c) find equilibrium consumption, real money balances, inflation, and nominal interest rate, assuming that money supply grows at the constant rate $\mu$. (Also, assume that government's debt, both in terms of nominal and real bonds, is zero: b_t^s=B_t^s=0) d) Show that the return on UDIs does not depend on inflation.
4) Let us assume that the aggregate supply equation is:

$$y=\bar{y}+a(\pi-\pi^e)+e$$

where e is an aggregate supply shock, and let us postulate the following link between inflation and the growth rate of money supply:

$$\pi=\mu.$$

The objective function of the Central Bank is:

$$min_\mu \frac12\lambda(y-\bar{y}-k)^2+\frac12 \pi^2$$

subject to the two equations above, where k is a positive number (that is, the Central Bank would like to stabilize output around the level $\bar{y}+k$, and also dislikes inflation).
The timing of events is the same one described in class: 1) the private sector forms inflationary expectations 2) the supply shock e is realized 3) the Central Bank sets $\mu$ (can respond to e).
a) Find the equilibrium inflation under discretion (call it $\pi^d$)
b) Find the equilibrium inflation under commitment (call it $\pi^c$)