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CLASS 4: THE VALUE OF MONEY: THE SIDRAUSKI-BROCK MODEL
- we will study the determinants of the price level in an economy
where the money supply is constant
- we will also point out at analogies, but also at crucial differences,
between money and durable goods
The Sidrauski-Brock Model
we will start exactly where we left last time; that is, the
problem faced by the household is as follows:
subject to
and subject to an initial given quantity of money, M0
notice that the intertemporal budget constraint can be rewritten as:
since this constraint will hold with the = sign at the optimum,
we can substitute it into the objective function, which becomes:
since the term Mt+1 appears only in the following two terms of the
infinite sum:
the FOC with respect to Mt+1 is:
As in the previous classes, we use the FOC's for
Mt+2,Mt+3,.. to
substitute repeatedly for
and obtain:
if
,
and if the transversality condition
holds true (we will discuss it later),
we obtain a formula for (the inverse of) the price level:
- the similarities between this formula and the one for
the price of durable goods is striking: both the value
of durable goods and the value of money (the inverse of the price
level) is a present discounted value of all the future services
that they respectively provide.
- however, there are also notable differences between the durable goods
and money: while the service provided by a durable good does not depend
on their price (the service provided by a refrigerator does not
depend on its price), the service provided by money balances depends
crucially on the future price levels (with a given amount of nominal money
balances you can buy more consumption goods if prices are low than if
prices are high)
- moreover, one can see that in a finite horizon economy money is
worthless, while durable goods are not.
Let us suppose that the economy ends at time T. Then at time T money is
worthless. But that will imply that at time T-1 nobody will accept money
as a mean of payments. Because of this, money is worthless at time T-1
as well. So nobody will accept it at time T-2, and so on... the conclusion
is that money is worthless in all periods (why is this not the case for
a durable good?)
Equilibrium
In equilibrium it must be that ct=yt, all t
For the time being let us assume that the Central Bank keeps the money
supply constant at the level Ms in all periods. since in equilibrium
money demand must equal money supply: Mt=Ms, all t.
Substituting for ct and Mt in the pricing formula, we obtain:
Unlike in the previous models, we are not home yet! the formula
does not deliver a solution for the price level, because the price in
period t depends on the prices in all future periods.
In order to obtain a formula for the price level we have to make the
further assumption that output is constant in all periods:
yt=y all t.
under this assumption, the pricing formula becomes:
Notice that the pricing formula is stationary, in the sense
that pt depends on
pt+1,pt+2,.. in the same way that
pt+1 depends on
pt+2,pt+3,..
consequently, a constant price level, pt=p, is a solution (it may not be the only solution.
However, we will focus on stationary equilibria).
Notice also that when the price level is constant, real money balances
are constant as well at a level
.
How is the equilibrium price level determined?
with constant prices and constant real money balances the pricing formula
becomes:
rearranging terms this formula becomes:
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 equilibrium price level, given that:
To answer this question let us draw (figure 1) the function:
If the function crosses the horizontal axis at least once, then there
is an equilibrium level for real money balances. if it crosses the
axis only once, then the equilibrium is unique.
- when real money balances tend to zero, the marginal utility from
holding money is likely to be higher (possibly
)
than the
marginal utility from income, therefore:
- conversely, when real money balances are infinity, the
marginal utility from holding money is likely to be lower
(possibly 0, remember the ``cash in advance'' model) than the
marginal utility from income, therefore:
- if the function f(.) is continuous, it must be that it crosses
the x-axis at least once
- furthermore, if the function f(.) is always increasing, that is,
if:
then it crosses the axis only once, and the equilibrium is unique
(one can show that this condition has to do with convexity of the
utility function)
- How does the equilibrium real money demand change with changes in
income y (figure 2)?
if real money and consumption are complement, then an increase in y
increases the marginal utility of money (increase um(y,m), for
given m). on the other hand, if the marginal utility from consumption
is decreasing, an increase in income will decrease the marginal utility
of consumption (decrease in uc(y,m)). the overall effect is that the
function f(.) shift down: the equilibrium real money balances
increase with an increase in income
- finally, notice that the equilibrium real money balance do not
depend on the level of the money supply. so doubling money supply in
this model (assuming that it remains constant afterwards) only implies
that the price level double, since
What have we got?
- the Quantity Theory of Money
- when money supply doubles, the price level doubles
- when income increases real money balances increase as well.
For given money supply, this means that prices decline
- perhaps more importantly, this model will be the workhorse to study
more interesting economies that the one we analyzed in this class
(constant money supply, constant price level): this model provides us
with the tools to understand inflations, currency crises, and other
interesting phenomena
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Marco Del Negro
2000-01-24