Chain-Weight GDP Calculations:
I present a step-by-step process for computing the chain-weight GDP measures using price and quantity data. This process is useful because the equation that directly defines chain –weight GDP is difficult for students to understand, let alone use. If Y denotes real GDP, p denotes the vector of product prices, q denotes the vector of product quantities, and t denotes the time period, chain-weight real GDP can be expressed as the following:
Note that the first term under the square root is a Laspeyres quantity index (as it uses the previous period’s prices), and the second term is a Paasche quantity index (as it uses end of period prices). The BEA does not directly calculate chain-weight GDP using an equation like (1) , but the actual method employed by the BEA to calculate real GDP growth is also probably impractical at the intermediate level. The BEA first uses a chain-weight price index to deflate nominal figures (rather than calculating real figures directly from price and quantity data), and this price index is difficult to directly calculate. The process outlined below gives equivalent results to equation (1) and the BEA process, but is more intuitive. It is in fact the method suggested in numerical examples provided by the BEA (Landefeld and Parker 1995). Unfortunately, these steps do require repeated, tedious calculations. A spreadsheet can help to keep the calculations organized, increase accuracy, and make the assignment less time consuming.
The steps are outlined in the spreadsheet depicted in Figure 1. Column A labels the rows and columns B through F display five years of data. Rows 3-8 display price and quantity data for two goods
(it is straightforward to add more goods), where figures are chosen to depict the relative price for good 1 to be rising and its relative production falling.12 Row 9 calculates nominal GDP.
The first goal is to calculate the chain-weight growth rate of real GDP for each year. To accomplish this, it is first necessary to calculate the annual growth rate of real output using the current year’s prices as weights (row 11) and then the annual growth rate of real output using the previous year’s 7 prices as weights (row 12). These are growth rates associated with Paasche and Laspeyres indexes, respectively. The chain-weight real growth rate is then computed as the geometric mean of these two fixed-weight real growth rates (row 13). The next step is to choose a base year and calculate real GDP figures for each year by using the chain-weight real growth rates. In Figure 1, year 3 is chosen as the base year, so nominal GDP in year 3 is declared chain-weight real GDP. Real chain-weight GDP in year 4 is calculated by adding the chain-weight growth to year 3’s real GDP, and chain-weight GDP in year 5 is similarly calculated using its chain-weight growth rate and year 4’s real GDP value. Year 2’s real chainweight
GDP level is calculated by dividing year 3’s real GDP figure by year 3’s growth rate plus one, and year 1’s real GDP is calculated similarly. It is here that the term “chain-weight” shows its relevancy: year 5’s real GDP contains price information on years 5 and 4 (through the chain-weight growth rate calculation) and years 4 and 3 (though year 4’s calculation). Any given year’s real GDP figure contains a “chain” of prices back to the base year. Finally, the chain-weight GDP deflator (called the chain-weight price index) can be found by dividing the nominal GDP figure by the chain-weight real GDP figure (row 14). Note that because it uses chain-weight real GDP data, it also contains price information “chaining” to the base year. Chain-weight GDP data are presented as “chained base-year dollars”; however, they actually employ a weighted average of prices from any given year to the base year.
An interesting follow-up example is to calculate fixed-weight real GDP, real GDP growth, and the GDP deflator using the base-year prices, and then compare the results to the chain-weight results. If the figures are chosen carefully , the figures will exhibit the substitution bias.
Other rows have been added to calculate inflation rates implied by the fixed-weight and chain-weight deflator.
Reference: Miles B. Cahill (2003), “Theaching Chain-Weight Real GDP Measures“, The Journal of Economic Education, Vol 34, NO.3, pp. 224-234.