## Index Numbers

The numerical results of a calculation based on a basket of goods can get a little messy. The simplified example in this table has only three goods and the prices are in even dollars, not numbers like 79 cents or $124.99. If the list of products was much longer, and more realistic prices were used, the total quantity spent over a year might be some messy-looking number like $17,147.51 or $27,654.92.

To simplify the task of interpreting the price levels for more realistic and complex baskets of goods, the price level in each period is typically reported as an **index number**, rather than as the dollar amount for buying the basket of goods. Price indices are created to calculate an overall average change in relative prices over time. To convert the money spent on the basket to an index number, economists arbitrarily choose one year to be the **base year**, or starting point from which we measure changes in prices. The base year, by definition, has an index number equal to 100. This sounds complicated, but it is really a simple math trick. In the example above, say that time period 3 is chosen as the base year. Since the total amount of spending in that year is $107, we divide that amount by itself ($107) and multiply by 100. Mathematically, that is equivalent to dividing $107 by 100, or $1.07. Doing either will give us an index in the base year of 100. Again, this is because the index number in the base year *always* has to have a value of 100. Then, to figure out the values of the index number for the other years, we divide the dollar amounts for the other years by 1.07 as well. Note also that the dollar signs cancel out so that index numbers have no units.

Calculations for the other values of the index number, based on the example presented in this table are shown in this table. Because the index numbers are calculated so that they are in exactly the same proportion as the total dollar cost of purchasing the basket of goods, the inflation rate can be calculated based on the index numbers, using the percentage change formula. So, the inflation rate from period 1 to period 2 would be

\(\begin{array}{rclll}\cfrac{(\text{99.5 – 93.4})}{\text{93.4}}& \text{=}& \text{0.065}& \text{=}& \text{6.5%}\end{array}\)

This is the same answer that was derived when measuring inflation based on the dollar cost of the basket of goods for the same time period.

**Calculating Index Numbers When Period 3 is the Base Year**

Total Spending | Index Number | Inflation Rate Since Previous Period | |
---|---|---|---|

Period 1 | $100 | \(\begin{array}{rcl}\cfrac{\text{100}}{\text{1.07}}& \text{=}& \text{93.4}\end{array}\) | |

Period 2 | $106.50 | \(\begin{array}{rcl}\cfrac{\text{106.50}}{\text{1.07}}& \text{=}& \text{99.5}\end{array}\) | \(\begin{array}{rclll}\cfrac{(\text{99.5 – 93.4})}{\text{93.4}}& \text{=}& \text{0.065}& \text{=}& \text{6.5%}\end{array}\) |

Period 3 | $107 | \(\begin{array}{rcl}\cfrac{\text{107}}{\text{1.07}}& \text{=}& \text{100.0}\end{array}\) | \(\begin{array}{rclll}\cfrac{\text{100 – 99.5}}{\text{99.5}}& \text{=}& \text{0.005}& \text{=}& \text{0.5%}\end{array}\) |

Period 4 | $117.50 | \(\begin{array}{rcl}\cfrac{\text{117.50}}{\text{1.07}}& \text{=}& \text{109.8}\end{array}\) | \(\begin{array}{rclll}\cfrac{\text{109.8 – 100}}{\text{100}}& \text{=}& \text{0.098}& \text{=}& \text{9.8%}\end{array}\) |

If the inflation rate is the same whether it is based on dollar values or index numbers, then why bother with the index numbers? The advantage is that indexing allows easier eyeballing of the inflation numbers. If you glance at two index numbers like 107 and 110, you know automatically that the rate of inflation between the two years is about, but not quite exactly equal to, 3%. By contrast, imagine that the price levels were expressed in absolute dollars of a large basket of goods, so that when you looked at the data, the numbers were $19,493.62 and $20,009.32. Most people find it difficult to eyeball those kinds of numbers and say that it is a change of about 3%. However, the two numbers expressed in absolute dollars are exactly in the same proportion of 107 to 110 as the previous example. If you’re wondering why simple subtraction of the index numbers wouldn’t work, read the following Clear It Up feature.

### Why do you not just subtract index numbers?

A word of warning: When a price index moves from, say, 107 to 110, the rate of inflation is not *exactly* 3%. Remember, the inflation rate is not derived by subtracting the index numbers, but rather through the percentage-change calculation. The precise inflation rate as the price index moves from 107 to 110 is calculated as (110 – 107) / 107 = 0.028 = 2.8%. When the base year is fairly close to 100, a quick subtraction is not a terrible shortcut to calculating the inflation rate—but when precision matters down to tenths of a percent, subtracting will not give the right answer.

Two final points about index numbers are worth remembering. First, index numbers have no dollar signs or other units attached to them. Although index numbers can be used to calculate a percentage inflation rate, the index numbers themselves do not have percentage signs. Index numbers just mirror the proportions found in other data. They transform the other data so that the data are easier to work with.

Second, the choice of a base year for the index number—that is, the year that is automatically set equal to 100—is arbitrary. It is chosen as a starting point from which changes in prices are tracked. In the official inflation statistics, it is common to use one base year for a few years, and then to update it, so that the base year of 100 is relatively close to the present. But any base year that is chosen for the index numbers will result in exactly the same inflation rate. To see this in the previous example, imagine that period 1, when total spending was $100, was also chosen as the base year, and given an index number of 100. At a glance, you can see that the index numbers would now exactly match the dollar figures, the inflation rate in the first period would be 6.5%, and so on.

Now that we see how indexes work to track inflation, the next module will show us how the cost of living is measured.

**Note:** Watch this video from the cartoon *Duck Tales* to view a mini-lesson on inflation.