# Box: Note on Calculating Output and Price Indexes

Estimation of most components of gross domestic product (GDP) consists of two broad computational stages: (1) Estimation of current-dollar values, and (2) separation of the current-dollar values into a price-change element and a quantity-change element./1/

In the first step, the current market values of spending for each component of GDP are determined from basic source data. That is, consumer spending on apples and oranges, on small appliances, on movie admissions, and on all of the other components of personal consumer expenditures are estimated using a variety of source data, such as retail sales data from the Bureau of the Census. These calculations are usually referred to as the "current-dollar" value of a component. Current-dollar values of all the GDP components always "add up" to current-dollar GDP.

Though many technical problems arise in computing current-dollar GDP and its components, it is conceptually straightforward: Current-dollar GDP is a measure of what is actually spent in the economy in a particular period. Measuring the change in current-dollar GDP is equally straightforward, conceptually, because it is, the actual change in spending that occurs in the economy between two periods.

In the second step, the period-to-period change in current- dollar GDP, or in the current-dollar value of a GDP component, is separated into a price-change element and a quantity-change element. For example, a 10-percent increase in expenditures on oranges could result from (1) a 10-percent increase in the number of oranges purchased with no change in the price of oranges, (2) a 10-percent increase in the price of oranges with no change in the number purchased, or (3) some combination of price and quantity increase totaling 10 percent. The quantity-change element in a GDP component, or in GDP itself, has in the past usually been referred to as the "constant-dollar" increase in the component, or sometimes as the change in the "real" component of GDP or in "real" GDP. Calculation of the quantity-change component is usually carried out by a process known as "deflation."/2/

Though measuring the change in current-dollar GDP is conceptually straightforward, partitioning the change into price- and quantity-change elements is not. This partitioning is an analytic step, because aggregate price change and aggregate quantity change cannot be observed directly in the economy. Instead, aggregate price and quantity changes must be calculated, and the calculation method is determined by analytic requirements.

In particular, it is important to recognize that real GDP is an analytic concept. Despite the name, real GDP is not "real" in the sense that it can, even in principle, be observed or collected directly, in the same sense that current-dollar GDP can in principle be observed or collected as the sum of actual spending on final goods and services in the economy. Quantities of apples and oranges can in principle be collected, but they cannot be added to obtain the total quantity of "fruit" output in the economy.

For this reason, real GDP must be computed by valuing the various components of GDP, using the prices of some period or periods. Real GDP is simply an index number—a computation, like the consumer price index or the price index for GDP, except that real GDP is an index number that measures quantities. Its computation cannot be determined by reference, or by analogy, to the methods used for the construction of current-dollar GDP.

In the past, measures of real GDP change were calculated by fixing the valuations of GDP components in some period (currently, the year 1987) and holding those valuations fixed over all years and quarters for which real GDP estimates are produced. This approach can be illustrated using a hypothetical two-commodity economy (exhibit 1) with total current-dollar spending of \$5.00 in year 1 and \$9.00 in year 2. If we take year 1 to be the "base" (or "weighting" or "valuation") period, then the prices in year 1 are used to value the quantities in both years and the changes in quantities from year 1 to year 2. This is shown in panel A. In the exhibit, the consumption of oranges fell in year 2 because the price of oranges rose rapidly, while the consumption of apples, whose price rose less rapidly, increased. With this calculation, the weighted-quantity-change measure for "fruit" increased by 20 percent.

There is no reason why year 1 must always be chosen as the weighting period. In the past, BEA has periodically shifted its weighting period—before December 1991, 1982 was used as the weighting year for measuring real GDP, and before December 1985, 1972 was the weighting year. Panel B shows what happens to the quantity measure if we shift the valuation, or weight year, to year 2.

If year 2 is used for valuation, the quantities in year 1 and in year 2 are calculated as before, but both sets of quantities are valued in year 2 prices, rather than year 1 prices. Using year 2 prices results in a 6-percent increase in quantities, substantially lower than the 20-percent increase that resulted from using year 1 prices.

This example illustrates a regularity that has often been observed in the calculation of real GDP. Moving the weighting period forward tends to reduce the quantity-change measure, because in general the quantities that have increased the most are those whose prices have increased, relatively, the least. To put it another way, the use of a more recent period of valuation tends to put a lower valuation on the quantities that have increased most rapidly. Thus, measuring the change in real GDP is subject to "weighting effects," because the measure is sensitive to the valuation period, the period chosen for the weights in the calculating formula.

Which calculation, panel A or panel B, is "correct"? There is no single answer to this question, because each year's prices are equally valid for valuing the changes in quantities. A common sense approach to the weighting problem is to take an average of the panel A and panel B calculations. Economic theory indicates that taking a geometric mean of the two measures is the preferred form of averaging. The geometric mean can be calculated by multiplying the panel A and panel B results together and then taking the square root—that is: $1.20 &215; 1.06= 1.13$. In the index number literature, this geometric average calculation of quantities is known as the "Fisher Ideal" index number.

BEA has adopted geometric averaging as the new method for calculating real GDP and for calculating measures of price change in GDP and its components. This method is presently employed in calculating the "chain-type annual-weighted" measures in NIPA tables 7.1–7.3 and 8.1.

Why is BEA changing its calculation method for real GDP? What are the advantages of the new calculating method over the old one? The main advantage of the old method is its simplicity: Only one set of valuations is necessary for calculating GDP for all periods. In the past, BEA has used one set of valuations (currently, those for 1987) to construct real GDP measures from the most recent period all the way back to 1929.

In addition, experience shows that the use of a single weighting period generally produces accurate measures of GDP as long as the periods being compared are close to the weighting period. The reason is that changes in relative valuations are usually small for periods close to the weighting period, so that "weighting" effects are also small.

The main disadvantage of using a single valuation period for calculating real GDP is that the measure becomes increasingly subject to "weighting effects" as the time between weighting, or valuation, period and the current period lengthens.

BEA's new method of calculating real GDP has another advantage. It permits shifting the valuations on a year-by-year basis, which means that long-term growth, past business cycles, and productivity are measured in the valuations that are appropriate to the period being studied. For example, in the present 1987-weight calculating method, change in output in both the 1980–81 recession and the 1974–75 recession is measured in 1987 prices. In the new method, output change in these recessions will be measured in the prices that prevailed at that time—that is, the 1981–82 recession will be measured in prices of the early 1980's, and the 1974–75 recession, in the prices of the mid-1970's. Experience has shown that applying a single, fixed valuation to historical time periods tends to statistically dampen economic recessions and recoveries and also distorts the picture of long-term economic growth. Cyclical fluctuations in the economy are best measured using valuations that are appropriate to the period being studied rather than valuations from some distant period.

1. There are a small number of exceptions to the description in the following sections, notably where extrapolators must be used because spending data are not available on a current basis. See "Annual Revision of the U.S. National Income and Product Accounts," SURVEY OF CURRENT BUSINESS 74 (July 1994): 26–27.

2. The quantity-change measure for GDP is probably the most widely used number from the NIPA's. For example, the first line of the monthly GDP press release reports the percentage change in real GDP.