CPI and Inflation Worked Example: Basket, Real Wages, CPI Bias

To calculate the consumer price index (CPI), price a fixed market basket of goods in the current year, divide by the cost of the same basket in the base year, and multiply by 100. To calculate the inflation rate, take the percentage change in the CPI between two years: (new CPI minus old CPI) / old CPI x 100. The base year always has a CPI of exactly 100, so a CPI of 115 means the basket costs 15 percent more than it did in the base year.

This guide works both formulas through a full three-year example, shows how to use the CPI to turn nominal values into real ones, and explains why the CPI tends to overstate inflation. It is written for AP Macro Topic 2.4, price indices and inflation.

What the CPI measures

A price index tracks the cost of a set of goods over time relative to a reference period. The CPI does this for the things households buy.

• The market basket is a fixed list of goods and services, in fixed quantities, meant to represent what a typical urban consumer buys: food, housing, transportation, medical care, clothing, entertainment and so on.
• The base year is the reference period. The basket's cost in that year is set equal to 100.
• Because the quantities in the basket stay fixed, any change in the basket's cost comes from prices alone.

In the United States, the Bureau of Labor Statistics publishes the CPI every month. Its official CPI uses 1982 to 1984 as the reference period equal to 100, which is why published CPI values are well above 100 today.

The two formulas

• CPI = (cost of the basket in the current year / cost of the basket in the base year) x 100
• Inflation rate = ((CPI this year - CPI last year) / CPI last year) x 100

The first formula turns dollar costs into an index. The second turns two index values into a percentage change. The CPI calculator and the inflation rate calculator run each one.

Worked example: a three-year basket

Here is a hypothetical basket for a student. Year 1 is the base year.

Step 1: price the basket each year. Multiply each quantity by that year's price and add.

• Year 1: (10 x 10.00) + (5 x 12.00) + (20 x 2.00) = 100 + 60 + 40 = $200
• Year 2: (10 x 11.00) + (5 x 12.00) + (20 x 2.50) = 110 + 60 + 50 = $220
• Year 3: (10 x 11.50) + (5 x 13.00) + (20 x 2.55) = 115 + 65 + 51 = $231

Step 2: convert each cost to a CPI. Divide by the base-year cost of $200 and multiply by 100.

• Year 1: 200 / 200 x 100 = 100
• Year 2: 220 / 200 x 100 = 110
• Year 3: 231 / 200 x 100 = 115.5

Step 3: compute the inflation rate between years.

• Year 1 to Year 2: (110 - 100) / 100 x 100 = 10 percent
• Year 2 to Year 3: (115.5 - 110) / 110 x 100 = 5.5 / 110 x 100 = 5 percent

Notice the trap in Step 3. The CPI rose by 5.5 index points from Year 2 to Year 3, but inflation was 5 percent, not 5.5 percent, because you divide by last year's CPI (110), not by 100. Index points and percentages only match when the starting value is exactly 100.

You can check the whole example a second way. The basket cost rose from $220 to $231, and 11 / 220 = 0.05, which is the same 5 percent. The CPI is just the basket cost rescaled, so percentage changes in the CPI equal percentage changes in the basket's cost.

Using the CPI to convert nominal to real

A nominal value is measured in the dollars of its own year. A real value is adjusted for the price level so you can compare purchasing power across years. The conversion into base-year dollars is:

• Real value = (nominal value / CPI) x 100

Continue the example. Suppose the student earned a nominal wage of $15.00 an hour in Year 1 and $17.00 an hour in Year 3.

1. Year 1 real wage = 15.00 / 100 x 100 = $15.00 in base-year dollars.

2. Year 3 real wage = 17.00 / 115.5 x 100 = about $14.72 in base-year dollars.

The paycheck rose by $2.00, a nominal raise of about 13.3 percent (2 / 15). But prices rose 15.5 percent over the same two years, so the student can buy slightly less than before. The real wage fell even though the nominal wage went up. The real wage calculator does this conversion, and the cost-of-living adjustment calculator shows the raise needed to keep purchasing power unchanged: 15.00 x 115.5 / 100 = $17.33.

The same idea applies to GDP and any other dollar figure (real GDP usually divides by the GDP deflator instead of the CPI). Interest rates work differently: the real interest rate equals the nominal interest rate minus the inflation rate. The real vs nominal guide covers each case.

Why the CPI can overstate inflation

Because the CPI prices a fixed basket, it tends to report a slightly higher inflation rate than the true rise in the cost of living. The AP course expects you to name the reasons.

Substitution bias. When the price of one good rises faster than others, consumers buy less of it and more of cheaper substitutes. A fixed basket keeps buying the original quantity of the expensive good, so it overstates how much more people actually need to spend. In the example above, gas rose 25 percent in Year 2; a real student would likely drive a little less, but the basket still buys 20 gallons.

Quality changes. When a product gets better at the same price, consumers are getting more for their money, which is effectively a price cut. If the index records only the sticker price, it misses that improvement. If a better product costs more, part of the higher price pays for higher quality rather than inflation, and the index can count all of it as a price increase.

New goods. New products, and the price drops that often follow their launch, enter the basket only after a delay. Consumers gain from new choices before the index reflects them.

Statistical agencies adjust for some of these effects, for example by updating the basket's weights, but a fixed-basket index still leans toward overstating inflation. The GDP deflator vs CPI guide shows the substitution effect in numbers, because the deflator's changing basket captures substitution that the CPI misses.

Inflation, deflation, and what causes them

A positive inflation rate means the price level is rising. A negative rate means it is falling, which is deflation (see deflation explained). A falling but still positive rate, such as 10 percent then 5 percent in the example, is disinflation: prices are still rising, just more slowly. Economists also track core inflation, which leaves out food and energy prices because they swing so much from month to month.

The CPI tells you how much prices rose. It does not tell you why. For the causes, demand-pull and cost-push, read what causes inflation.

How it shows up on the AP exam

CPI and inflation calculations sit in AP Macro Topic 2.4 (the price indices and inflation topic page has a summary). Questions on it tend to ask you to:

• Calculate a CPI from a basket with prices in two or more years.
• Calculate the inflation rate between two CPI values.
• Convert a nominal wage, price, or income into real terms and say whether purchasing power rose or fell.
• Explain why the CPI may overstate the rise in the cost of living.
• Identify the base year from a table (it is the year with an index of 100).

Common mistakes:

• Dividing by the wrong year. The CPI divides by the base-year cost. The inflation rate divides by the earlier year's CPI.
• Reading index points as percentages. A move from 110 to 115.5 is 5 percent, not 5.5 percent.
• Changing the quantities. The CPI basket uses the same quantities every year. Only prices change.
• Treating a lower inflation rate as falling prices. Inflation dropping from 10 to 5 percent means prices still rose. For the difference between the index and the rate, see the CPI vs inflation rate comparison.
• Deflating backwards. Divide the nominal value by the CPI and multiply by 100. Multiplying by the CPI inflates instead.

Summary

Price the same basket each year, divide by the base-year cost, and multiply by 100 to get the CPI. Take the percentage change between two CPI values to get the inflation rate, always dividing by the earlier year. Divide any nominal value by the CPI and multiply by 100 to express it in base-year dollars. And remember the fixed basket is the source of the CPI's upward bias, through substitution, quality changes, and new goods.