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

Midterm Exam, Solutions

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?
Real money balances enter the indirect utility function in the Sidrauski-Brock model because, for a given amount of total consumption c, having real money balances gives the agent a chance to purchase cash goods. Without real money balances, the agent cannot consume any cash goods, and has to spend all c on credit goods. If the government, all of a sudden, prevents by law the use of credit cards, a number of goods that were before considered `credit goods' become `cash goods'. People would demand more real money balances, as they need them to purchase goods (formally, u_m would rise, shifting down the f(m) curve discussed in class). For given money supply, the price level would fall.
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 ?
Friedman's rule states that monetary authorities should lower the rate of money creation, and therefore the rate of inflation, until the nominal interest rate equals zero. Formally, $\mu=-(1-\beta)$. By doing so, the monetary authorities would eliminate the opportunity cost of holding money. Hence individuals will hold as much real money balances as they would like. This in turn eases their transactions and therefore increases their utility. A rate of money growth below Friedman's rule implies a negative nominal interest rate. But the nominal interest rate cannot be negative, as it is not an equilibrium. The equilibrium condition states that:

Ru_c=u_m

if R is negative the right hand side is negative (u_c is always non-negative), but the left hand side cannot be negative, since u_m is always non-negative. Money is ``too'' attractive an asset when nominal interest rates are negative for an equilibrium to exist.
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.
Here you have to solve for the intratemporal problem, which is:

$$max_{\{c_{1},c_{2}\}} \delta \ln(c_{1})+c_{2}$$

subject to the constraints:

$$\begin{array}{l} c_{1}\leq m\\ c_{1}+c_{2} \leq c \end{array}$$

where $m\equiv \frac{M}{p}$. Just as in class, you can tackle this problem by forgetting about the `cash-in-advance' constraint, and solving:

$$max_{\{c_{1},c_{2}\}} \ln(c_{1})+c_{2}$$

$$c_{1}+c_{2} \leq c$$

or

$$max_{\{c_{1}\}} \ln(c_{1})+c-c_{1}$$

The FOC is:

$$-1+\delta\frac1{c_1}=0$$

and the solution is:

$$c_1=\delta$$

$$c_2=c-\delta$$

Of course, this is `the' solution only if it satisfies the `cash-in-advance' constraint. Otherwise, if $\delta>m$, the solution is:

c₁=m

c₂=c-m

(note that the FOC for $c_2=m<\delta$ is positive! The agent would like to spend more on c₁ if she could). So the indirect utility function is:

$$u(c,m)=\{ \begin{array}{c} \delta \ln{m}+c-m \mbox{ if }\del... ...ta \ln(\delta)+c-\delta \mbox{ if } \delta \leq m \end{array}$$

and the intertemporal problem is:

$$max_{\{c_t,M_t\}_{t=0}^{\infty}} \{ \begin{array}{c} \sum_{t... ...t-\delta) \mbox{ if } \delta \leq \frac{M_t}{p_t} \end{array}$$

s.t.

$$M_{t+1} \leq M_t- p_t c_{t} +p_t y_{t}+T_t.$$

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 $\delta$.
a) Dividing the budget constraint of the government by the price level one obtains, after imposing the condition B^s_t=0:

$$\tau_t=g_t-\frac{M^s_{t+1}-M^s_t}{p_t}$$

Since money growth is constant and equal to $\mu$, real seignorage is $\mu m^*(\mu)$ and equilibrium taxes are:

$$\tau_t=g-\mu m^*(\mu).$$

b) Using the individual and the government's budget constraint one obtains equilibrium consumption:

c_t=y-g.

The equilibrium condition for real money balances is:

$$\frac{1+\mu-\beta}{\beta}u_c=u_m$$

which becomes:

$$\frac{1+\mu-\beta}{\beta}=\delta-m$$

or

$$m=\delta-\frac{1-\beta}{\beta}-\frac1{\beta}\mu$$

which shows that equilibrium real money balances are a decreasing function of $\mu$.
c) Recognizing that equilibrium inflation is $\mu$, that is:

$$1+\mu=\ln(p_{t+1})-\ln(p_t)$$

the above formula becomes:

$$m=\delta-\frac{1-\beta}{\beta}-\frac1{\beta}(\ln(p_{t+1})-\ln(p_t)-1)$$

or

$$m=\delta+1-\frac1{\beta}(\ln(p_{t+1})-\ln(p_t))$$

so

$$\alpha_0=\delta+1$$

$$\alpha_1=-\frac1{\beta}$$