Bernanke -- Problems -- Chapter 7 -- Price Level and Inflation

### Bernanke -- Problems -- Chapter 7 -- The Price Level and Inflation

1. Computing the CPI

 Good Year 1 Year 2 --- Quantity Price Quantity Price Pizzas 20 \$ 10 30 \$ 11 Rent 1 \$ 600 2 \$ 640 Car 1 \$ 100 4 \$ 120 Phone 1 \$ 50 0.5 \$ 40
It is easy to compute the NOMINAL spending in each year: multiply prices and quantities and add them up.

YEAR 1:

• Expenditure on pizza = \$ 200
• Expenditure on rent = \$ 600
• Expenditure on car = \$ 100
• Expenditure on phone = \$ 50

• Total NOMINAL expenditure = \$ 950

YEAR 2:

• Expenditure on pizza = \$ 330
• Expenditure on rent = \$ 1280
• Expenditure on car = \$ 480
• Expenditure on phone = \$ 20

• Total NOMINAL expenditure = \$ 2110

To compute a CPI, we must first choose a base year. Let's assume Year 1 is the base year.

1. Find the price of the consumption basket in the base year.

2. Next, multiply the prices and BASE YEAR quantities in the next year and add them up:
 Good Price Quantity Expenditure Pizza \$ 11 20 \$220 Rent \$ 640 1 \$640 Car \$ 120 1 \$120 Phone \$ 40 1 \$40 SUM \$1020

The CPI for any year is given by the formula:

Price of BASE YEAR consumption basket in any given year
--------------------------------------------------
Price of BASE YEAR consumption basket in the BASE year

Applying this formula to the second year, we get

CPI = 1020 / \$ 950 = 1.0737

Note that this CPI would be reported by the Bureau of Economic Analysis as 107.37 , since it is conventional to multiply the ratio of the baskets by 100. You may think of the 107.37 number as saying that the consumption basket in the second year costs 107 percent of the price of the basket in the base year.

The inflation rate is the percent change in the CPI . Here, it would be 7.37 percent .

2. Finding the inflation rate.

In order to find the inflation rate, we repeatedly apply the formula for percentage change to the inflation rate:

NEW value of CPI - OLD value of CPI
------------------------------------------------------------------- times 100
OLD value of CPI

Note that we cannot calculate the first value, since we don't have an old value.
I corrected the text value for 1999 (the number was revised since the text appeared), and added the later data. The results for the period since 1990 are:
 Year CPI Inflation rate 1990 130.7 --- 1991 136.2 4.21 1992 140.3 3.01 1993 144.5 2.99 1994 148.2 2.56 1995 152.4 2.83 1996 156.9 2.95 1997 160.5 2.29 1998 163.0 1.96 1999 166.2 2.21 2000 172.1 3.36 2001 177.6 2.85

3. Finding the real income of a family of four

In order to meaningfully compare incomes (or anything else) over time, we must convert nominal incomes to real incomes . The appropriate formula is:

REAL INCOME = NOMINAL INCOME divided by the CONSUMER PRICE INDEX

Applying the formula to the data given, we find

 YEAR Nominal income CPI REAL INCOME 1980 24, 332 82.4 29,529 1985 32, 777 107.6 30,462 1990 41, 451 130.7 31,715 1997 53, 350 160.5 33,240 2000 59, 346 172.2 34,503

If the Boskin Committee was correct in claiming that the CPI overstated the true rise in the cost of living, the "real income" measured in cost of living terms would be greater than reported above.

Note that I have added a value for the year 2000; this is for "all married couples", rather than a "family of four" (data for which is no longer reported), so it is not strictly comparable.

4. Real and nominal wages of college graduates

The question requires a bit of thought: we are given the 8 percent decline in the REAL wage from 1990 to 1997, and are given the NOMINAL wage of \$ 13.65.

Using the data from problem 2, we find that the CPI in 1990 was 130.7 and in 1997 was 160.5.
To solve the problem

1. Find the real wage in 1997 or

Nominal wage divided by CPI = 13.65 / 1.605 = 8.5047

2. Find the real wage in 1990 :
My notation is RW90 = real wage in 1990 (not known) and RW97 = real wage in 1997 = 8.5047 (just found)

Use the percent change formula in the form (note again in computations percent changes must be in decimal form):

RW97 - RW90
------------------------ = - .08
RW90

8.5047 - RW90
------------------------ = - .08
RW90

(8.5047 / RW90) - 1.00 = - .08

8.5047 / RW90 = .92

RW90 = 8.5047 divided by .92 = 9.2442

3. Find the nominal wage in 1990. Since real wage = nominal wage DIVIDED by CPI, the

Nominal wage = real wage TIMES CPI

Here, 9.2442 times 1.307 = 12.08

Note that although the nominal wage rose over the period from 12.08 to 13.65 or 13 percent,
the real wage fell by 8 percent.

5. Indexing tax brackets to prevent "Inflation creep"

In order to keep the real tax brackets the same, we will have to adjust the borders of the nominal dividing lines.
We will have to translate the given nominal values to real values by dividing by 1.75 since the CPI for 2000 was 175;
then, we will have to translate the real values we find to new nominal values by multiplying by 1.85 (the CPI for 2001 is 185).
The results are shown in the following table:

 Nominal Real New Nominal 20,000 11,429 21,143 30,000 17,143 31,714 50,000 28,571 52,857 80,000 45,714 84,571

6. The "cost of eating" index and substitution bias

The first step in the problem is to compute the cost of the specified basket in the two years:

• In the base year, 30 chickens at \$ 3.00 + 10 hams at \$ 6.00 + 10 steaks at \$ 8.00 yield a basket value of:
30 (3.00) + 10 (6.00) + 10 (8.00) = 90.00 + 60.00 + 80.00 = \$ 230.00

• In the next year, 30 chickens at \$ 5.00 + 10 hams at \$ 7.00 + 10 steaks at \$ 8.00 yield a basket value of:
30 (5.00) + 10 (7.00) + 10 (8.00) = 150.00 + 70.00 + 80.00 = \$ 300.00

The CPI can now be calculated as the ratio of the cost of the market basket now to the cost of the market basket in the base year:

300.00 / 230.00 = 1.3043

(note that this would be reported as 130.43)

The inflation rate would be reported as 30.43 percent.

However, consumers will note that the relative price of chicken has gone up (compared to ham)

Old relative price of chicken to ham: \$ 3 / \$ 6 = 0.5000

New relative price of chicken to ham: \$ 5 / \$ 7 = 0.7143

They will substitute away from chicken and toward ham . Given the conditions described in the problem, they will find themselves buying no chicken at all. Since they are indifferent between two chickens (which now cost \$10) and one ham (which now costs \$ 7), they could buy 25 hams and 10 steaks and be just as well off as before .

The "cost of living," here meaning the cost of maintaining the same standard of living , will have risen from the initial \$230 to \$7 (25 hams) + \$8 (10 steaks) = \$175 + \$80 = \$255.

If we use this new cost of living to create a "cost of eating" index, we would report an index of:

Price of basket giving same utility
--------------------------------------------------
Price of consumption basket in the BASE year

Cost of living index = \$255 / \$230 = 1.1087

This means that the utility-based calculation sees a rise in the cost of living of only 10.87 percent ;
the CPI overstates the rise in the cost of living by almost 20 percentage points .

7. Nominal and real gas prices

The calculation is straightforward: divide the nominal gas price by the CPI to et the real price of gas.

 Year Gas price CPI Real price 1978 0.663 0.652 1.017 1979 0.901 0.726 1.241 1980 1.269 0.824 1.540 1981 1.391 0.909 1.530 1982 1.309 0.965 1.357 1983 1.277 0.996 1.282 1984 1.229 1.039 1.183 1985 1.241 1.076 1.153 1986 0.955 1.136 1.136

8. Woodrow's hardware, revisited

In the text example, Woodrow withdrew \$25,000 at the beginning of the week, so that he would have his required \$5000 for the last day of the week. With a 10 percent average interest rate, and an average balance of \$15,000 on any working day, the foregone interest means that Woodrow is paying an opportunity cost of ten percent of \$ 15,000, or \$1500 per year, to maintain his cash in the cash register. [If he did not keep it on hand, he could himself take the opportunity to lend out \$15,000 at 10 percent].

In the problem as revised, we have three sets of changes

1. The rate of interest falls from 10 percent to 5 percent.
The cost of keeping money on hand is now only 5 percent of \$15,000 per year, or \$750 dollars.
If Woodrow made daily trips to the bank for 200 extra days (note that he would go every week -- 50 times a year -- in any case) at a cost of \$ 4 per trip, the effort to save money would cost him \$ 800 -- and would not save him the full \$750, since he would still have an average daily balance of \$5000, and hence would still be paying \$250 in interest. There will be no shoe-leather costs , since there are no extra trips to the bank.

2. If we combine with the fall in interest rate a fall in the cost of a trip to the bank (perhaps gas prices have fallen as well) to only \$2 a trip, the trips would cost Woodrow \$ 500.
The cost of the trips would be less than the \$500 savings (\$750 in foregone interest from keeping \$15,000 on hand minus \$250 from keeping \$5,000 on hand), so he will make the trips and incur the shoe-leather costs.
The moral of the story here is that if all prices change by the same percentage, real behavior does not change or put another way that only relative prices matter

3. The rate of inflation is 10 percent and a trip to the bank costs \$4 and Woodrow needs \$10,000 per day.
Note that he would have to begin the week by withdrawing \$50,000, so that he would have \$40,000 on Tuesday, \$30,000 on Wednesday, \$20,000 on Thursday and \$10,000 on Friday

The cost of keeping the average of \$30,000 on hand is \$3000 a year.
The cost of keeping the daily requirement of \$10,000 on hand is \$1000.
Hence trips to the bank would result in net savings of \$2000, and at \$4 each they cost \$800 a year for 200 trips.
He would continue the trips and incur \$800 in shoe-leather costs

9. Albert's Assets

Albert places \$1000 on deposit for three years at 6 percent interest, leaving the sum plus accrued interest on deposit for the entire time.
The sums of money he has in the bank, their real value (money divided by CPI) and the
real interest rate , calculated as the percentage change in real value
are given by the following table:
 Date On deposit CPI Real value Real interest in percent Jan 2000 \$1000.00 1.00 \$ 1000.00 --- Jan 2001 \$1060.00 1.05 \$ 1009.52 0.952 Jan 2002 \$1123.60 1.10 \$ 1021.45 1.1817 Jan 2003 \$1191.02 1.18 \$ 1009.33 -1.1865

Note that the results are not quite the same as they would be if we had used the formula on page 172,

Real interest rate = nominal interest rate MINUS the inflation rate

Since the rate of interest is always 6 percent and the inflation rate would be calculated on the basis of the CPI given as 5 percent in 2000, 4.76 percent in 2001 and 7.27 percent in 2002, we would have the following results:

 Nominal interest Inflation rate Real interest in percent 6 5.00 1.00 6 4.76 1.24 6 7.27 -1.277

The values are close for Albert to use in getting an idea of what is happening to his thousand dollars.
Bill Gates might prefer the first set of calculations if he has a billion dollars on deposit.

10. Frank and Sarah and Irving
Irving Fisher is the real hero of the Frank and Sarah problem.
His treatment of the relation between nominal and real interest rates and inflation suggests that if Frank wants a two percent real interest rate , the nominal rate that he must charge is the rate of inflation plus two percent . Since in this problem everyone expects a ten percent rate of inflation, the nominal interest rate which would guarantee a two percent real return is twelve percent .

In part (b), with the rate of inflation uncertain, an indexing scheme such that the interest payment was 2 percent more than the rate of inflation could be used to guarantee a two percent real return .

11. A multi-good CPI
The inflation rate calculated through the CPI is a weighted average of the inflation rates of all individual goods, with the weights being given by the expenditure shares in the base year (which are given by the table).

Since the inflation rate is zero for all goods except food, housing and medical services, the overall inflation rate will be:

.178 (10) + .428 (5) + .057 (10)

1.78 + 2.14 + .57 = 4.49

If the CPI in the base year is 100.00 (as it always is), the CPI in the current year will be 104.49.