Teoria y Politica Monetaria
Instructor: Marco Del Negro

Solutions; Problem set 6

1) From the cash-in-advance model the amount of cash goods the household can consume in each period cannot exceed the amount of real money balances held at the beginning of the period. Unless cash and credit goods are prefect substitutes in consumption, real money balances will enter the household's indirect utility function, as they make it possible to relax the cash-in-advance constraint. if cash and credit goods are perfect substitutes in consumption, the household will not care about the composition of total consumption, and the cash-in-advance constraint will have no impact on individual's welfare. Of course, once the cash-in-advance constraint is no longer binding, real money balances will no longer enter the utility function. Also, it is real and not nominal real money balances that enter the indirect utility function because the household cares about the ``real" amount of cash goods consumed, which depends on the real amount of money balances, as it is apparent from the cash-in-advance constraint. 2) When a stabilization is announced and is credible, the demand for real money balances increases: future inflation is expected to be low, and therefore the future services from holding real money balances are perceived to be high. The increase in real money balances means that money demand increases: if it increases more than money supply, then prices may decline.
Formally, $p_{T-1}=\frac{M^s_{T-1}}{m(\mu)}$ and $p_{T}=\frac{M^s_{T}}{m(0)}$. If $\frac{m(0)}{m(\mu)}>\frac{M^s_{T}}{M^s_{T-1}}$ then we have that p_T<p_T-1.
3) a) The first order conditions with respect to money and bonds are the same ones discussed in class, namely:

$$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{1}{p_{t+1... ...c{M_{t+1}}{p_{t+1}})+ u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})]$$ (1)

and

$$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{(1+R_{t+1})}{p_{t+1}}u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$ (2)

The first order condition with respect to s_t+1 is:

$$\frac{q_t}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{(q_{t+1}+d p_{t+1})}{p_{t+1}}u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$ (3)

b) Dividing by one obtains:

$$\frac{(q_{t+1}+d p_{t+1})}{q_t}=1+R_{t+1}$$

that is, the return on stocks and bonds should be equal. This is no surprise: since we are in a world without uncertainty, bonds and stocks are the same asset.
c) From the budget constraints of households and government, and from the condition s_t=1, one obtains:

$$c_t=d \mbox{ all }t$$

As in the model with bonds only, the first order conditions for money are unchanged. This implies that the inflation rate is $\mu$, the nominal interest rates is $R=\frac{1+\mu-\beta}{\beta}$, and real money balances are determined from the expression:

$$(1+\mu-\beta)u_c(y,m^*)=\beta u_m(y,m^*)$$

From b) the return on stock is equal to the return on bonds, which is an increasing function of inflation.
d) Call q^* the value of stock in real terms, that is, $q^*_t=\frac{q_t}{p_t}$. With this definition equation becomes:

$$q^*_t u_c(c_t,m_t)= \beta (q^*_{t+1}+d )u_c(c_{t+1},m_{t+1})$$

which is the same equation studied in class 1, and whose solution is:

$$q^*_t=\sum_{i=1}^{\infty} \beta^i \frac{u'(c_{t+i},m_{t+i})}{u'(c_t,m_t)} d$$

Substituting the equilibrium values for c and m we obtain:

$$q^*_t=\sum_{i=1}^{\infty} \beta^i \frac{u'(d,m^*)}{u'(d,m^*)} d=\frac{\beta}{1-\beta}d$$

4) a) The intratemporal problem was solved in class, and we know that the indirect utility function is of the form:
$u(c_t,m_t)=\{ \begin{array}{c} \ln{m_t}+\delta \ln{c_t-m_t} \mbox{ if }\frac{1... ...\delta}{1+\delta}_t} \mbox{ if } \frac{1}{1+\delta}c_t \leq m_t \end{array}$
so that the intertemporal problem of the agent is:

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

subject to:

$$M_{t+1} \leq M_t- p_t c_t +p_t y_t+T_t$$

b) The first order condition for money balances are as usual:

$$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{1}{p_{t+1... ...c{M_{t+1}}{p_{t+1}})+ u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})]$$ (4)

and the equilibrium consumption is:

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

Let us argue that when Friedman's rule is implemented equilibrium real money balances are:

$$m^*_t=\frac{1}{1+\delta}y \mbox{ all }t$$

Notice that if real money balances are constant, then prices must grow at the same rate of money supply, $\mu$. To see whether this is really an equilibrium, we have to check whether it satisfies all first order conditions and the budget constraint. The budget constraint is clearly satisfied for c_t=y. The first order condition with respect to money becomes:

$$\frac{1}{p_t} u_c(y,m^*)= \beta \frac{1}{p_{t+1}}[u_m(y,m^*)+u_c(y,m^*)]$$

or (using $\frac{p_{t+1}}{p_t}=1+\mu$)

$$\frac{1+\mu-\beta}{\beta} u_c(y,m^*)= u_m(y,m^*)$$

The left hand side of the above expression is equal to zero when Friedman's rule is implemented. From the indirect utility function one can see that for $m^*_t=\frac{1}{1+\delta}y$ the right hand side is zero as well. In conclusion, the first order conditions are satisfied and an equilibrium does exist with $m^*_t=\frac{1}{1+\delta}y$. Note that in fact an equilibrium exists for any value of real money balances $m^*_t \geq \frac{1}{1+\delta}y$: households can hold real money balances above satiation - it certainly does not harm them, and there is no cost in doing that.