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

Δ???? =

1

1 − ????(1 − ????)

Δ????

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)

2

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

reserves?

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.

Δ????

????

− 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.

3

relationship between the inflation rate and real economic activity). Obviously, it works the

other direction, as well. So the sacrifice ratio is

???????????????????????????????????? ???????????????????? =

Δ????

????

Δ????

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

NAIRU.

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%

4

• 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

by

????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

holds:

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

5

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

move on).

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.







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