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The first problem addresses the difference between the simple fiscal multiplier and what we have termed the "total multiplier.
The first problem addresses the difference between the simple fiscal multiplier and what we
have termed the "total multiplier." This is the multiplier using the model where taxes are
modeled explicitly as a tax on income, or GDP. So tax revenue collected by the government is tY,
where t is the tax rate. The consumption function is given by
???? = ???? + ???? 1 − ???? ????
If Δ???? is the change in government spending, then the simple fiscal multiplier is given by
1 − ????(1 − ????)
but, of course, this multiplier effect can be mitigated by three factors that we discussed in class.
1. Suppose that economists observe that an increase in government spending of $10 billion
raises real GDP by $20 billion. Assume that the economy-wide tax rate on income is t = 25%.
A. If these economists ignore the possibility of crowding out or other factors that mitigate the
multiplier effect, what would they estimate the marginal propensity to consume (MPC) to be?
B. Now suppose that the economists realize that they have neglected the influence of mitigating
effects like crowding out and/or Ricardian Equivalence Would their new estimate of the
MPC be larger or smaller than their initial estimate?
C. Briefly explain why the multiplier effects of an increase in government spending (or a tax
cut) are greater the higher is the marginal propensity to consume, and the lower is the
income tax rate.
Here is some practice using the money multiplier as it would occur under two separate scenarios.
2. The following values describe the initial scenario:
M1 = 3,500
Cu = 2,000 (currency in circulation)
rr = 5% (reserve requirement)
ER = 0 (banks are holding no excess reserves)
A. Calculate the M1 money multiplier.
B. When the Fed engages in an open market sale of Treasury securities in the amount of $500,
what will be the new level of M1 once this feeds through the banking system?
C. Now assume that all the same values hold, but banks are holding $2,000 in excess reserves
above and beyond those they are required to hold.1 What would the size of the open market
purchase required to get the same impact on M1 as in part B, when banks held no excess
D. Comment on the difference in results. When would we expect each of these scenarios to
prevail. And generally speaking, what is necessary in order for the money multiplier to be
less than one?
This problem addresses the money multiplier and the effects of monetary policy on the economy.
To answer the problem fully to the end, two fairly simple macroeconomic relations are
necessary in addition to the Phillips Curve. Both of these constitute "shortcuts" to evaluate the
ultimate effects of changes in policy on the real economy (real GDP and the unemployment rate,
for our purposes). Then, we will specify a simple equation for the Phillips Curve that will allow
you to quantify the relationship between the inflation rate and the unemployment rate.
Okun's Law gives the relationship between the growth rate of real GDP and the unemployment
rate. Common estimates of Okun's Law gives the following relationship, assuming that the
growth rate of real GDP is 3% per year (that's the "3" on the left side). Given the recent
slowdown in the economy, more recent estimates may turn out lower, and we'll use a lower
figure in the exercise below.
− 3 = −2Δ????
Okun's Law in this form says that the unemployment rate increases or decreases depending on
whether real GDP growth is less than or more than 3%. It also tells us by how much
unemployment increases or decreases. So, for example, if real GDP growth is 4%, then the
unemployment rate is predicted to fall by 0.5 percentage point, since 4 - 3 = 1 = -2(-.5).
Next, the sacrifice ratio gives the percentage loss in real GDP for each percentage point
reduction in the rate of inflation (note that this is similar to the Phillips Curve, as it gives a
1 Some of those excess reserves will be the result of previous reserve injections (open market purchases), by the Fed.
Some will have been accumulated do to this particular injection, as well. The Fed will predict the effect of it injections
on holdings of excess reserves, so the new multiplier includes this prediction. But it is really the excess reserve ratio
that matters for the multiplier, not just the level of reserves.
relationship between the inflation rate and real economic activity). Obviously, it works the
other direction, as well. So the sacrifice ratio is
???????????????????????????????????? ???????????????????? =
where the numerator is the growth rate of real GDP and the denominator is the change in the
inflation rate rate (which is already measured as a proportion or percentage).
Finally, the Phillips Curve is given by ???? = ????! − ???? ???? − ????! , where ????! is the expected rate of
inflation and ????! is the natural rate of. Now, this reflects the modern theory of the Phillips Curve
(the Expectations Augmented Phillips Curve), which says that firms make their decisions on how
much to produce and therefore how many workers to hire based on the inflation rate expected
to prevail in the future (the expected inflation rate, ????!). If aggregate demand is higher than
expected, then actual inflation will be higher than originally expected, and the unemployment
rate will have a tendency to move below the NAIRU, un (the economy could be in danger of
overheating). And if the aggregate demand turns out lower than expected, inflation will be
below what was expected and the unemployment rate will be above the NAIRU.
Rewriting the Phillips Curve as
???? − ????! = −???? ???? − ????!
makes clear negative relationship between actual inflation and the unemployment rate, which
is the basic "stylized fact" that underlies the theory of the Phillips Curve. Note that when
???? = ????!, it must be true that ???? = ????!. Essentially, this implies that the Phillips Curve is flat when
unemployment is above the NAIRU, and then it becomes vertical once unemployment falls to the
3. You are an analyst at a financial institution. Assume that the economy is growing at the long
run average growth rate of potential GDP of 2%, but Fed officials believe that there are
signals that AD is about to "take off" and the economy is in danger of overheating. All of the
variables are the same as in the second scenario of the previous problem. The current M1
money supply is $3,500. But now, the Fed is about to engage in an open market sale of
securities in the amount of $200 to decrease the money supply. Your job as an analyst is to
discern the change in the M1 money supply, along with its effect on "the" nominal interest
rate, inflation, the unemployment rate, and real GDP. To make things closer to reality,
assume that the figures are in billions. and The initial level of real GDP is $19,000.
• Initial M1 = 3,500. Fed undertakes an open market sale of $200
• Assume that the current nominal interest rate is i = 3.0%
• The expected rate of inflation is ????! = 1.5%. Let ????! be the ex ante real interest rate.
• Initial real GDP = Y = 19,000
• Investment, I, is 20% of GDP.
• Long-run growth rate of potential GDP = 2%
• Initial unemployment rate = u = 5.0% = un (since the economy is growing at a rate equal to
the growth rate of potential GDP, unemployment must be at the NAUIRU).
A. When the Fed engages in an open market sale of Treasury securities in the amount of $200,
what will be the new value of M1?
B. Assume that money demand function (liquidity preference, or the demand for M1) is given
????1 = 5,000 − .002???? + .75????
where i is the nominal interest rate in percentage terms (i.e., i = 2 for i = 2.0%), and Y is real
GDP, changes in which will shift the money demand curve, as we discussed in class. In more
comprehensive analysis, such as that done by investment banks, you may see a more
complex specification for money demand that includes other financial variables. I just want
you to be aware that although we will just rely on the relationship between money demand
on nominal interest rates, you could see more complex and realistic specifications in things
you read or rely upon in the future.
Since we assume that the money supply is set by the Fed, we have the following equilibrium
condition (supply and demand curves intersect at equilibrium):
????1! = ????1! = ????1
So we can drop the superscripts and now write the relationship as
????1 = 5,000 − 200???? + .75????
Now, to further simplify the problem and avoid issues of simultaneity, we need to assume
that Y is not going to change in response to a change in the money supply. That could be
realistic if we are only interested in the very short run impact, because the effects of
monetary policy may only "kick in" wit a lag. Or, it could be that we are in a world where
monetary policy is not particularly effective. But beyond that, it is purely a simplifying
assumption. But by assuming that Y stays fixed, we then know the following relationship
Δ????1 = −200Δ????
So now, your simple task is to calculate the change in the nominal interest rate associated
with the change in the M1. This is obviously a simple exercise mathematically, so the lesson
here is that you will need to rely upon some estimate of the effect of monetary policy on
interest rates in order to evaluate the overall policy effects.
C. One of the monetary transmission mechanisms that we discuss in class is the interest rate
effect, which depends on the liquidity effect of monetary policy. The interest rate effect states
that real investment expenditure is negatively related to the real interest rate (it could be
real consumption expenditure, as well, but we will assume that only investment is influenced
by changes in the real interest rate). Now, in reality, it is long-term interest rates that will
matter for investment, and monetary policy as it is normally conducted is intended to
influence short term rates. But using the theories of the term structure of interest rates that
you learn in other classes, you can deduce that normally, when short-term rates rise, so will
long-term rates. Although there are exceptions to this (see Greenspan's "bond market
conundrum" circa 2005), historically, it has been the norm. So long as that is true, assuming
a single interest rate is at least somewhat justified.
To make the model as realistic as possible with our relatively simple confines, I will just walk
you through the next step, which is unlike anything we did in class (but we need this step to
In our simple world, the only source of growth of real GDP would be growth in investment
expenditure, so we could specify an investment function that relates the real interest rate to
the growth rate of investment expenditure. To fully specify a real investment function, we
would need to include everything that influences firm's investment decisions. For example,
corporate profits depend on expectations of future profitability. But we are really only
interested in the interest rate effect here, so I will omit a fuller specification of the growth of
real investment expenditure and skip straight to the component of that specification that
matters to us, the real interest rate:
Δ????! = −2Δ????!
where Δ????! is measured in percentage terms. This is extremely simple, and just says that the
change in growth rate of investment will be equal to twice the decline in the real interest
rate. And since this is the only effect on aggregate demand and real GDP (i.e., there are no
subsequent spending multiplier effects, it is a simple exercise to determine the final impact
on real GDP by relying on the fact that investment is 20% of GDP, so the growth rate of GDP
1/5 the growth rate of investment expenditure.
Assume that there is no change in expected inflation (or, more realistically, that expectations
are adaptive, and it will take time for inflationary expectation to adjust). What is the change
in the real interest rate, the growth rate of investment, and therefore the growth rate of GDP?
D. Using Okun's Law, what will be the effect on the unemployment rate from the Fed's policy?
What will be the new unemployment rate?
E. Using historical data on the Phillips Curve, your economists estimate ???? = 0.75. Using the
Phillips Curve, calculate the inflation rate before and after the Fed's policy, assuming no
change in expected inflation.
F. Finally, calculate the sacrifice ratio implied by the values that you have calculated.