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The numerical results of a calculation based on a basket of goods can get a little messy. The simplified example in [link] 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 [link] are shown in [link] . 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
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.
Total Spending | Index Number | Inflation Rate Since Previous Period | |
---|---|---|---|
Period 1 | $100 | $\begin{array}{rcl}\frac{\text{100}}{\text{1.07}}& \text{=}& \text{93.4}\end{array}$ | |
Period 2 | $106.50 | $\begin{array}{rcl}\frac{\text{106.50}}{\text{1.07}}& \text{=}& \text{99.5}\end{array}$ | $\begin{array}{rclll}\frac{\left(\text{99.5\u201393.4}\right)}{\text{93.4}}& \text{=}& \text{0.065}& \text{=}& \text{6.5\%}\end{array}$ |
Period 3 | $107 | $\begin{array}{rcl}\frac{\text{107}}{\text{1.07}}& \text{=}& \text{100.0}\end{array}$ | $\begin{array}{rclll}\frac{\text{100\u201399.5}}{\text{99.5}}& \text{=}& \text{0.005}& \text{=}& \text{0.5\%}\end{array}$ |
Period 4 | $117.50 | $\begin{array}{rcl}\frac{\text{117.50}}{\text{1.07}}& \text{=}& \text{109.8}\end{array}$ | $\begin{array}{rclll}\frac{\text{109.8\u2013100}}{\text{100}}& \text{=}& \text{0.098}& \text{=}& \text{9.8\%}\end{array}$ |
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.
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