Chapter 6 Homework From That Equation We can Some Algebra See

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ADDITIONAL ISSUES FOR CLASSROOM DISCUSSION

1. Students often find it difficult to work quickly through the logic of the

following question: Would you rather be a borrower or lender if inflation

turned out to be higher than expected? Rather than focusing on the

2. The discussion about inflation and the tax system is crucial to our

discussion of monetary policy in Chapter 17. It may be helpful to go

through a number of examples of how the numbers work, so they can

see that the tax on nominal income translates into a large tax rate on

real income, with the real tax rate rising with inflation.

3. You can get an interesting discussion going regarding the best way to

avoid the distortion from the interaction of inflation and the tax system.

Students are intrigued about the di’erent possibilities, including

improving the tax system to make it neutral with respect to inflation. But

if you talk about the complications, especially when it comes to figuring

out the portion of capital gains that is attributable to inflation, they will

see why it has not yet been done. The idea that, if inflation were zero,

the distortion would go away is important to our discussion of monetary

policy in Chapter 17.

SOLUTION TO TEXTBOOK NUMERICAL

EXERCISES AND ANALYTICAL PROBLEMS

Numerical Exercises

Chapter 6: Real Interest Rates 53

12. a. In expansions,

b. In recessions,

c. In expansions,

13. Use the equations

Chapter 6: Real Interest Rates 54

Note that faster economic growth raises all three interest rates.

Chapter 6: Real Interest Rates 55

e. The nominal interest rate increases when the growth rate of the

economy rises, the tax rate increases, and the inflation rate increases.

Chapter 6: Real Interest Rates 56

c. 2013: Hans pays taxes on his nominal interest income of C2,000.

Since his tax rate is 25 percent, he must pay taxes of

Analytical Problems

16. Both the homeowner and the bank share the risk of changes in the

price level in nominal terms, but they both reduce their risk in real terms.

Thus, in real terms, they both gain. The contract eliminates the risk from

inflation by indexing to the inflation rate. Such a mortgage would be

beneficial to both borrowers and lenders.

17. Since both the supply and demand for bonds depend only on the

after-tax expected real interest rate, which is independent of the

Chapter 6: Real Interest Rates 57

expected inflation rate, higher expected inflation will not a’ect supply or

demand. For the after-tax expected real interest rate to remain constant,

and since ra = [(1− t ) × i ] − πe , the higher expected inflation rate must

lead to an increase in the nominal interest rate. From that equation, we

18. When inflation is unexpectedly high, banks’ profits decline. The rise in

inflation will lead nominal interest rates to rise, so the bank will have to

increase the interest rates on its deposit accounts. But, the bank is

locked in to making many long-term loans at interest rates that cannot

rise (although it will raise the interest rate it charges on new loans). As a

result, the banks’ profits are squeezed by having to pay out more

interest to depositors but not getting more interest on its loans. As profits

fall, the banks’ stock prices fall.

19. You might make a loan even if there were a negative after-tax

expected real interest rate because alternatives might be worse. For

example, if you expected to lose 1 percent in real after-tax terms from

such a loan, that deal might be better than holding cash if the inflation

rate were high or investing in other assets whose value was declining.

Historically, there have been times when many investment returns were

negative at the same time, such as the 1970s.

ADDITIONAL TEACHING NOTES

The Concept of the Real Yield Curve

Because of changes in expected inflation over time, the yield curve drawn

with real interest rates may be more useful in forecasting the economy than

the yield curve drawn with nominal interest rates. Because business firms

base their decisions on expected real interest rates, not nominal interest

Chapter 6: Real Interest Rates 58

Discussion of Alternative Tax System Based on Real Income Rather

than Nominal Income

Recall the example in which Tom invested $1,000 at 6 percent interest. He

had a 30 percent tax rate and the inflation rate was 3 percent. Under the

modified tax system that is based on real income, the steps to calculate

Tom’s taxes are:

1. Nominal interest income = i × P

Chapter 6: Real Interest Rates 59

Suppose P is the principal amount invested, i is the nominal interest

rate, t is the (real) tax rate, and πe is the expected inflation rate.

Nominal interest income equals the interest rate times the principal

value:

Nominal interest income = i × P.

Chapter 6: Real Interest Rates 60

The asterisks (*) in equations (14) and (15) mean that these are the

real interest rates under an alternative tax system that taxes only real

income, not nominal income. In both equations (14) and (15) you can

see that there is now no interaction between inflation and taxes; the

tax rate applies to the real interest rate in both cases. Note the

di’erence between these equations and equations (12) and (13),

where the tax rate was applied to the nominal interest rate, then

expected inflation was subtracted.

If we repeat the exercise from Table 6.1 with this new tax scheme, we

get the results shown in Table 6.3.

Table 6.3: How the After-Tax Real Interest Rate Varies with the

Expected inflation Rate,

When the Real Interest Rate Is Constant and Taxes Are Based on

Real Income

Real interest

rate,

Expected inflation

rate, πe

Nominal interest rate,

After-tax real interest

rate, r*a

3.0 0.0 3.0 2.1

3.0 3.0 6.0 2.1

3.0 6.0 9.0 2.1

3.0 9.0 12.0 2.1

…

.Note: All interest rates and inflation rates are in percent. The table

takes as given the data in the first two columns on the real interest

rate (r) and expected inflation rate (πe), then generates values for the

nominal interest rate (i) using equation (2) [i = r + πe], and the

after-tax real interest rate (r*) when taxes are based on real interest

Chapter 6: Real Interest Rates 61

would change with the expected inflation rate for a fixed after-tax real

interest rate. Equation (14) is:

r*a = (1− t ) × (1 − πe)

Divide both sides of the equation by 1 − t to get: ..

Using equation (16) to derive a comparable table to Table 6.2 under

the new tax scheme, we find the results shown in Table 6.4.

Table 6.4: How the Real Interest Rate Varies with the Expected

inflation Rate,

When the After-Tax Real Interest Rate Is Constant and Taxes Are

Based on Real Income

After-tax real

interest rate,

r*a

Expected

in2ation rate, πe

Nominal

interest rate, i

Real interest

rate, r

2.0 0.0 3.0 3.0

2.0 3.0 6.0 3.0

2.0 6.0 9.0 3.0

2.0 9.0 12.0 3.0

Note: All interest rates and inflation rates are in percent. The table

takes as given the data in the first two columns on the after-tax

expected real interest rate (r*a) and the expected inflation rate (πe).

Values for the nominal interest rate (i) are generated using equation

(16)