A cautionary tale for my undergraduate economics students
Reader Steven Kopits wonders why, in order to show the relative prominence of government spending, I don’t merely take the ratio of one real index to another real index. Specifically, he admonishes me:
I find this presentation confusing. … Is it not possible to present this data as a simple percent of GDP?
The answer is no. The reason is that real quantities (in the national income and product accounts) are measured using chain-weighted series. From Whelan (2000):
To address the problems with its fixed-weight measures, in 1996 BEA began calculating real GDP and all other published real aggregates according to a chain index formula. Specifically, BEA now calculates the growth rate of real aggregates according to the so-called “ideal” chain index popularized by Irving Fisher (1922):
The gross growth rate of the real aggregate at time t is calculated as a geometric average of the gross growth rates of two separate fixed-weight indexes, one a Paasche index (using
period t prices as weights) and the other a Laspeyres index (using period t − 1 prices as weights.)
The weights change over time, mitigating the problem with substitution bias. However, a consequence of measuring quantities using this approach is the loss of additivity. The sum of the chain-weighted components does not equal the corresponding chain-weighted aggregate. This point is highlighted in Figure 1.
Figure 1: Fixed investment (blue), and sum of equipment, structures, intellectual property, nonresidential investment (red), billions of Ch.09$. Source: BEA, 2013Q3 3rd release, and author’s calculations.
This problem occurs as long as relative prices change; the problem is more pronounced the greater the changes in relative prices.
Now, suppose one wanted to measure the importance of one component by taking a ratio — for instance equipment investment to total fixed investment. One could calculate the ratio of nominal equipment to nominal fixed investment. This, however, would not be informative regarding the quantities. Because of the non-additivity aspect of chain weighted measures, one can’t take the simple ratio of real variables.
However, one can compare the growth rates of the two series, for instance Z and W. Taking log ratios accomplishes this goal. Consider the difference in cumulative growth rates.
log(Zt/Z0) – log(Wt/W0) = log(Zt/Wt) – [log(Z0/W0)]
The right hand side of the expression is the log ratio of Z/W, minus a constant, in the [square brackets].
Now consider Figure 2, which shows the nominal ratio and the log ratio of real series.
Figure 2: Ratio of nominal equipment investment to nominal fixed investment in (blue, left scale), and log ratio of real equipment investment to real fixed investment, in Ch.09$ (red, right scale). Source: BEA, 2013Q3 3rd release, and author’s calculations.
The level of the right hand scale has no meaning, except that an upward trend indicates equipment investment grows faster than aggregate fixed investment. The difference between the initial value and the final is equal to the cumulative growth differential.
Note that the use of nominal magnitudes would completely obscure the trend increase in real equipment investment as compared to real fixed investment. On the other hand, merely taking the ratio of real equipment to real fixed investment would lead to an overstatement of the importance of equipment at the beginning of the sample, given the sum of the components exceeds the aggregate fixed investment by about 20%.
The bottom line: when discussing variables like GDP, and its components, understanding of their measurement in practice is useful.