Teoria y Politica Monetaria, Spring 2000
Instructor: Marco Del Negro

Problem set 6 - Mock 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) [10] Explain why, in general, real money balances enter the indirect utility function in the Sidrauski-Brock model. Explain why `real' and not `nominal' money balances enter the indirect utility function. Explain also why, if cash and credit goods were perfect substitutes in consumption, real money balances would not enter the indirect utility function in the Sidrauski-Brock model.
2) [10] When hyperinflations end due to a successful stabilization program (change in regime) it may happen that prices decrease slightly after the program is announced in spite of the fact that the monetary base is constant or even increasing. Explain why this may happen first intuitively, and then using the formula $p_t=\frac{M^s_t}{m_t}$ explained in class (where m_t is the demand for real money balances)
3) [30] Introduce the stock market in the Sidrauski-Brock model studied in class. Specifically, assume that the preferences of individuals are:

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

and that the budget constraint of the household is:

$$M_{t+1} +B_{t+1}+p_t c_{t}+q_t s_{t+1}\leq M_t+B_t(1+R_t)+T_t+ ( q_t+p_t d) s_t$$

where s_t is the amount of stocks held by the household at the beginning of period t, d is the dividend in real terms, which is constant over time, and q_t is the price of the stock in period t. All other variables are defined as usual. The budget constraint of the government is:

M_t+1^s+B_t+1^s=M_t^s+B_t^s(1+R_t)+T_t

a) find the first order conditions with respect to M_t+1, B_t+1, and s_t+1.
b) using the first order conditions show that the return on stocks, which is defined as $\frac{(q_{t+1}+d p_{t+1})}{q_t}$ has to be equal to the return on bonds, 1+R_t
c) find equilibrium consumption, real money balances, inflation, and nominal interest rate, assuming that the supply of stocks is constant over time and equal to 1, and that money supply grows at the constant rate $\mu$. Show that the return on stocks in equilibrium is an increasing function of inflation.
d) find the price of stocks q_t in equilibrium (which will be a function of equilibrium real money balances)
4) [30] The household's problem is:

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

subject to the constraints:

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

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.
The budget constraint of the government is:

M_t+1 -M_t=T_t

Assume that the growth rate of money supply is constant:

$$M_t^s=M_0^s(1+\mu)^t\mbox{ all }t$$

and the endowment is constant as well:

$$y_t=y \mbox{ all }t$$

a)[10] Solve the intratemporal problem and find the indirect utility function.
b)[20] Show that there is an equilibrium if the government implements Friedman's rule, that is, $\mu=-(1-\beta)$.