TEORIA Y POLITICA MONETARIA, SPRING 2000; INSTRUCTOR: MARCO DEL NEGRO

Midterm Exam

Rules:

• you are not allowed to consult any notes or books
• you are not allowed to talk to each other
• any infraction of these rules will negatively affect your grade in the exam by at least one point
• the exam will last 1 hour and 20 minutes
• please write your name in every sheet you use
• please hand in only the final version of your answers
• you may use a hand calculator (although there is no need for it)
• you are allowed to answer in Spanish, if you prefer
• the number in square brackets ([.]) are the points assigned to each question, as well as the estimated amount of time needed to complete the answer (they sum up to 80)

1) [15] Explain why, in general, real money balances enter the indirect utility function in the Sidrauski-Brock model. Assume that the government, all of a sudden, prevents by law the use of credit cards. Other things being equal (money supply and output) what happens to real money balances, and the price level?
2) [15] Express Friedman's rule (the optimal rate of money growth) and explain it. What would be wrong with a rate of money growth below Friedman's rule ?
3) [25] The household's problem is:

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

subject to the constraints:

$$\begin{array}{l} p_t c_{1t}\leq M_t\\ y_{1t}+y_{2t} \leq y... ...t- p_t c_{1t}) +p_t y_{1t}+p_t (y_{2t}-c_{2t})+T_t \end{array}$$

where M_t+1 is the amount of money held by the households at the end of period t, c_1t and c_2t represent the consumption of cash and credit goods respectively in period t, p_t is the price of both goods in terms of money, y_t represents the endowment of the household at the beginning of period t, which can be used to produce both cash goods (y_1t) and credit goods (y_2t), and T_t represents nominal transfers from the government. Solve the intratemporal problem and find the indirect utility function. Write the objective function and the constraints for the intertemporal problem.
4) [35] Consider an economy where the government budget constraint is:

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

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. Assume that the utility function is of the form:

$$u(c_t,\frac{M_t}{p_t})= c_t+\delta\frac{M_t}{p_t}-\frac12(\frac{M_t}{p_t})^2,$$

where $\delta$ and $\gamma$ are positive numbers, and that:

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

a) Find equilibrium taxes ($\tau_t$). Discuss whether Friedman's rule is optimal in this economy.
b) Find equilibrium consumption. Find the equilibrium real money balances as a function of $\mu$, $\beta$, and $\delta$. Show that the equilibrium real money balances are a decreasing function of $\mu$.
c) The following is a standard specification of the money demand equation:

$$\frac{M_t}{p_t}=\alpha_0+\alpha_1(\ln(p_{t+1})-\ln(p_t)).$$

Show that you can derive this money demand equation from the model you just solved, and find the values of $\alpha_0$ and $\alpha_1$ in terms of the parameters $\beta$ and $\delta$.