Bernanke -- Problems -- Chapter 7 -- Price Level and Inflation
|Pizzas||20||$ 10||30||$ 11|
|Rent||1||$ 600||2||$ 640|
|Car||1||$ 100||4||$ 120|
|Phone||1||$ 50||0.5||$ 40|
To compute a CPI, we must first choose a base year. Let's assume Year 1 is the base year.
The CPI for any year is given by the formula:
Applying this formula to the second year, we get
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 .
In order to find the inflation rate, we repeatedly apply the formula for percentage change to the inflation rate:
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:
In order to meaningfully compare incomes (or anything else) over time, we must convert nominal incomes to real incomes . The appropriate formula is:
Applying the formula to the data given, we find
|YEAR|| Nominal |
|CPI|| REAL |
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.
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
Nominal wage divided by CPI = 13.65 / 1.605 = 8.5047
Use the percent change formula in the form (note again in computations percent changes must be in decimal form):
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.
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 |
The first step in the problem is to compute the cost of the specified basket in the two years:
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)
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:
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 .
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|
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
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 |
|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,
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 |
| Inflation |
| Real interest |
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.
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 .
Since the inflation rate is zero for all goods except food, housing and medical services, the overall inflation rate will be:
If the CPI in the base year is 100.00 (as it always is), the CPI in the current year will be 104.49.