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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,
and
.
If
then
we have that
pT<pT-1.
3) a) The first order conditions with respect to money and bonds are the same ones discussed in class, namely:
![\begin{displaymath}\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}})]
\end{displaymath}](img4.gif) |
(1) |
and
![\begin{displaymath}\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}})
\end{displaymath}](img5.gif) |
(2) |
The first order condition with respect to st+1 is:
![\begin{displaymath}\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}})
\end{displaymath}](img6.gif) |
(3) |
b) Dividing
by
one obtains:
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 st=1, one obtains:
As in the model with bonds only, the first order conditions for money are unchanged. This implies that the inflation
rate is
,
the nominal interest rates is
,
and real money balances are determined from
the expression:
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,
.
With this definition equation
becomes:
which is the same equation studied in class 1, and whose solution is:
Substituting the equilibrium values for c and m we obtain:
4) a) The intratemporal problem was solved in class, and we know that the indirect utility function is of the form:
so that the intertemporal problem of the agent is:
subject to:
b) The first order condition for money balances are as usual:
![\begin{displaymath}\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}})]
\end{displaymath}](img4.gif) |
(4) |
and the equilibrium consumption is:
Let us argue that when Friedman's rule is implemented equilibrium real money balances are:
Notice that if real money balances are constant, then prices must grow at the same rate of money supply,
.
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 ct=y. The first order condition
with respect to money becomes:
or (using
)
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
the right hand side is zero as well.
In conclusion, the first order conditions are satisfied and an equilibrium does exist with
.
Note that in fact an equilibrium exists for any value of real money balances
:
households can hold real money balances above satiation - it certainly does not harm them, and there is no cost in
doing that.
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Marco Del Negro
2000-02-26