CLASS 5: INFLATION AND THE GROWTH RATE OF MONEY SUPPLY
we will study a model in which money supply grows at a constant
rate
we will see that different values of money growth imply different values of:
inflation
the nominal interest rate
real money holdings
welfare of households
the bottom line of this class will be trying to understand
the effect of different monetary policy regimes on inflation, interest rates, real money
holdings, and the welfare of individuals
for the time being, the rate of growth in money supply is supposed to be
constant at a given level forever
or
introduce the government budget constraint
why? if the Central Bank is printing money, then it is also collecting seignorage
what is seignorage? the amount of real resources obtained by the Central Bank by
printing money:
seignorage needs to be spent somehow. we will assume that it is given back to the
household through lump sum transfers Tt
notice that if money creation is negative (the Central Bank is buying money back)
lump sum transfers become lump sum taxes
the government budget constraint therefore is:
Mt+1s-Mts=Tt
the problem faced by the household is the same as before:
subject to
and subject to an initial given quantity of money, M0
except that we now have lump sum transfers in the intertemporal budget constraint:
since this constraint will hold with the = sign at the optimum,
we can substitute it into the objective function, which becomes:
notice that the presence of transfers does not affect at all the first order condition
with respect to Mt+1, which is:
Using the same arguments of the previous class (including transversality) we obtain
the condition:
Equilibrium
in equilibrium it must be that ct=yt, all t
assume that the endowment is constant: yt=y. Therefore consumption is constant.
as discussed in the introduction, the money supply is growing at a constant
rate, which we denote as
(which can possibly be negative):
in equilibrium, money demand equals money supply. it follows that
substituting for ct and Mt in the pricing formula, we obtain:
let us use the following trick: let us call
and
substitute in the above formula, obtaining:
or, simplifying:
notice that the pricing formula is again stationary, but this time
as a function of
and not of p:
depends on
in the same way that
depends on
consequently, a constant price level,
,
is a solution
what is the implication? If
is constant over time in
equilibrium, since by definition
,
it must be that
prices also grow at the constant rate ,
that is:
in this model inflation is equal to the growth rate in money supply
to find the equilibrium, we still have to determine what the value of
is
Notice that when the price level grows at the constant rate ,
real money balances
are constant at a level
with constant prices and constant real money balances the pricing formula
becomes:
or, rearranging terms:
Is there an equilibrium level of real money balances m* such that
notice that if we find an equilibrium level of real money balances
then we have also found the the value of ,
which would be:
the argument of why an equilibrium value of real money balances exists and is unique
is the same as in the previous class
also, under the same set of conditions, we have that the equilibrium real money balances
increase with an increase in income
How does the equilibrium real money demand change with changes in
the growth rate of money supply ?
this can be inferred from the study of:
or:
if
increases, the right hand term decreases. In order to restore
equilibrium, m* has to change. How?
If real money and consumption are complement, and if the marginal utility from
real money balances is decreasing in m, a decrease in m* decreases the left hand side
uc(y,m*), and increases the term
um(y,m*), thereby restoring equilibrium
so, if
increases, m* has to decrease to restore equilibrium
we found that an increase in the rate growth of money supply implies an
equal increase in inflation (growth rate of the price level) and a decrease in real money
holdings (Bresciani-Turroni effect)
Table:
Bresciani-Turroni effect
Country
Period
Inflation
real money as % of
initial value
Greece
1944
85,000,000 %
0.7 %
Hungary
1946
42,000,000,000,000 %
0.25 %
Bolivia
1985
100% monthly
25% (1980)
Notice that low real money demand means tat people do not have enough currency
to carry over transactions (that is, they have a lot of pieces of paper, but the
real value of those pieces of paper is very low). Is printing pieces of
paper at a faster rate a solution? the answer we get from our model is no, because
real money balances would go down even further.
The Optimal Quantity of Money
what is the optimal rate at which the central bank should be printing money?
notice that from our model we obtain that the utility of the household in
equilibrium is:
therefore agents utility is larger in equilibrium the larger the amount of
real money balances they hold
we saw that real money balances are a negative function of .
How can we make real money balances as high as possible? by choosing
as low as possible
what is the lowest possible level of
compatible with the existence
of an equilibrium?
From the equilibrium condition, which can be rewritten as
, given that both uc and um are non negative, we obtain that:
in particular, if when
the demand for money can be satiated,
that is,
(remember that equilibrium real money balances are
a function of )
is such that
,
then we can have an equilibrium
with
:
this is the Friedman rule
from the government budget constraint
Mt+1s-Mts=Tt
one can see that the Friedman rule (or any negative growth of money) means that
the Central bank should be taxing the household in order to reduce existing money supply.
Why is this optimal? Because equilibrium consumption is not affected anyway
(remember
,
and in equilibrium ct=yt)
and real money balances are higher