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Article
May 2024

# Introducing Producer Price Index research series based on a geometric-mean formula

This article introduces Producer Price Index (PPI) research series calculated with a geometric-mean (geomean) formula and explains the formula’s methodological advantages. The article also compares official and geomean PPIs to illustrate how formula choice affects index values.

Most users of Producer Price Index (PPI) data know that PPIs measure the average changes over time in the selling prices received by domestic producers of goods and services. What is less commonly known, however, is how these changes are calculated—which formulas are used, what determines formula choice, and how that choice affects an index. This article explains the motivation behind the U.S. Bureau of Labor Statistics (BLS) PPI research series calculated with a geometric-mean (geomean) formula. In addition, the article presents comparisons between official and geomean PPIs, illustrating the effects of formula choice on index values. Important empirical insights are also introduced by comparing PPI and Consumer Price Index (CPI) data and methods.

### Index calculation and substitution

PPIs are calculated in two stages. First, price changes for items within a narrowly defined product group—for example, grapefruits—are combined to form an elementary-level index for that group. Then, elementary-level indexes for different groups are aggregated into broader measures; examples of such broader measures include the PPIs for citrus fruits, fresh fruits and melons, and, ultimately, headline final demand. The calculation of PPIs at each stage of aggregation is based on a modified Laspeyres formula.1 This formula uses an arithmetic mean of price changes between a base period and a comparison period (hereafter also referred to as the current period) by fixing the quantities of goods and services sold in the base period, although the mix of these goods and services may change between the two periods.2 For instance, in a hypothetical example for grapefruits, if 10,000 red grapefruits were sold in the base period and 12,000 red grapefruits were sold in the current period, BLS would measure the price of 10,000 red grapefruits in the current period and compare that price with the base-period price.

The calculation of elementary-level indexes for the PPI geomean research series uses a geomean formula rather than an arithmetic-mean (modified Laspeyres) formula.3 Instead of fixing quantities in the base period, the geomean formula fixes revenue shares in the base period. By fixing revenue shares, this formula yields a geomean index that captures substitution toward relatively less expensive goods and services. The formula is used only at the elementary (first) level of aggregation, assuming substitution only within an elementary index. If applied to the grapefruit example, the geomean formula would include substitution between pink and red grapefruits, but not between grapefruits and oranges. Further, if 60 percent of total grapefruit revenue in the base period was from red grapefruits, the formula would hold that share constant in the current period. Therefore, if the price of red grapefruits between the two periods were to fall relative to the price of pink grapefruits, keeping the revenue share of red grapefruits constant (at 60 percent) would have to assume that more red grapefruits were sold in the current period.

By capturing substitution toward relatively less expensive goods and services, geomean indexes tend to be smaller than or equal to their corresponding arithmetic-mean indexes (when the indexes are based on the same weights). For this reason, index levels for the PPI geomean research series are generally lower than index levels for the official PPI series. However, if all prices within an elementary-level index change at the same rate (or remain unchanged), the geomean and official versions of that index would be equal.

### Comparing the use of geometric means in consumer and producer price indexes

In 1999, BLS introduced a geomean formula for calculating most elementary-level CPI estimates. This shift largely aimed to capture consumer substitutions toward less expensive products when product prices change at varying rates. Because substitution considerations provide the primary rationale for using geometric means in the CPI, not all elementary-level CPIs are calculated with a geomean formula. Indexes for which substitution is less common, such as those for certain utilities, government fees, or medical care services, are still calculated with a modified Laspeyres formula.4

For consumers, the incentive to substitute toward less expensive products and services is straightforward—most consumers will choose to save money when faced with relative price changes. For producers, however, the direction of substitution is less clear because these market participants seek to maximize profits. Although many empirical studies have analyzed substitution by using consumer price data, fewer studies have examined substitution by using producer price data. However, research using PPI data for goods for the 2002–16 period has provided evidence of substitution toward relatively less expensive goods.5 This finding indicates that while producers may have an incentive to substitute toward higher priced goods and services, this incentive is likely outweighed by other factors (e.g., changes in demand).

### Mathematical considerations

Despite evidence that substitution in the PPI is toward less expensive goods, substitution is not the only, or even the primary, reason for using a geomean formula to calculate elementary-level PPIs. Specifically, the geomean formula satisfies certain mathematical properties that the modified Laspeyres formula does not.

First, the modified Laspeyres formula fails the time-reversal test, which posits that an index measurement should be independent of the period regarded as the base (reference) period. For example, if an index shows that a price level doubled from period 1 to period 2 (with period 1 being regarded as the base period and period 2 as the comparison period), it should also show that the price level fell by half from period 2 to period 1 (with period 2 now being regarded as the base period and period 1 as the comparison period). This condition is not satisfied by the modified Laspeyres formula, which always yields a higher measurement in the case without period reversal than in the case with period reversal—a difference that can be interpreted as an upward bias. The geomean formula passes the time-reversal test.

The modified Laspeyres formula also fails the transitivity (or circularity) test, which posits that, when chained together, indexes over adjacent intervals should equal their direct counterpart. For example, for periods 1, 2, and 3, the product of an index calculated from period 1 to period 2 and an index calculated from period 2 to period 3 should equal the value of an index calculated directly from period 1 to period 3. The failure of this test does not necessarily imply bias, but it is a user limitation. Unlike the modified Laspeyres formula, the geomean formula passes this test.

Another shortcoming of the modified Laspeyres formula is its susceptibility to “formula bias.” To the extent that prices fluctuate around a common mean (or are “mean reverting”), the modified Laspeyres formula used to calculate elementary-level PPIs would exhibit an upward bias, yielding index values higher than the true values.6 This bias occurs because items with temporarily low or sale prices in the base period will be overweighted in an index, but these prices will likely rise faster than the prices of other index items as they mean revert in the subsequent period. This price dynamic will result in a higher measure of inflation, a bias that can be alleviated with the use of a geomean formula.

Because of its superior mathematical properties, the geomean formula is used for all elementary-level indexes in the PPI geomean research series, not just for indexes for which substitution is most likely.

### Index results based on a geomean formula

Considering the aforementioned advantages of the geomean formula, BLS introduced the PPI geomean research series in April 2023. Although this section presents data only for selected final-demand and intermediate-demand indexes, data are also available for additional final-demand and intermediate-demand indexes; commodity indexes at the two-digit commodity classification level; and industry indexes, mostly at the three-digit North American Industry Classification System level.7

PPIs for final demand measure price changes for outputs sold for personal consumption, as capital investment, to government, and as exports. Chart 1 shows official and geomean PPIs for final demand for the 2011–22 period. As seen in the chart, over this 11-year period, the official PPI for final demand increased 31.87 percent, at an average annual rate of 2.55 percent, whereas the geomean PPI for final demand increased 24.17 percent, at an average annual rate of 1.99 percent. The 0.56-percentage-point difference between the two PPI growth rates is somewhat larger than that found for consumer prices in two previous studies (these studies found average differences of 0.25 percentage point and 0.47 percentage point).8

When the CPI program began using a geomean formula, researchers found that such use would reduce the CPI annual growth rate by about 0.2 percentage point per year.9 As already noted, the CPI program uses both a geomean formula and a modified Laspeyres formula to calculate its elementary-level indexes, with formula choice depending on the specific good or service of interest and the ease with which consumers make product substitutions. BLS publishes two research series that allow a closer comparison between CPI and PPI data. The first series, a CPI based on Laspeyres arithmetic means (R-CPI-L), uses a modified Laspeyres formula for all elementary-level indexes and, therefore, is comparable to the official PPI series. The second series, a CPI based on geometric means (R-CPI-G), uses a geomean formula for all elementary-level indexes and, therefore, is comparable to the PPI geomean research series. Over the 2011–22 period covered in chart 1, the R-CPI-L increased 37.12 percent, at an average annual rate of 2.91 percent, and the R-CPI-G increased 30.61 percent, at an average annual rate of 2.46 percent. The 0.45-percentage-point difference between the two CPI growth rates is smaller than that for the PPI, suggesting that formula choice has a greater impact on PPI values. To determine why this is the case, it is useful to look at a more detailed level of data.

Chart 2 shows official and geomean PPIs for final-demand goods and final-demand services, indicating that the differences between official and geomean growth rates are smaller for goods than for services. Between 2011 and 2022, the official PPI for final-demand goods rose 28.78 percent, at an average annual rate of 2.33 percent, and the geomean PPI for final-demand goods increased 25.31 percent, at an average annual rate of 2.07 percent. Over the same period, the official PPI for final-demand services increased 32.83 percent, at an average annual rate of 2.61 percent, and the geomean PPI for final-demand services increased 22.66 percent, at an average annual rate of 1.87 percent. The differences between the official and geomean average annual growth rates were 0.26 percentage point for goods and 0.74 percentage point for services.

The relatively larger difference between the official and geomean growth rates in the PPI services component is due to price dynamics for trade services. PPIs for trade services measure changes in price margins (differences between the selling and acquisition prices of goods).10 These margins reflect the value added by a wholesaler or a retailer for services such as marketing, storing, and making goods easily available for customer purchase. The margin prices for these services tend to be more volatile than the prices of other goods and services and are more dispersed within elementary-level indexes, resulting in larger differences between official and geomean PPIs.11

Chart 3 illustrates the stark differences in growth rates between the official and geomean PPIs for trade services. From 2011 to 2022, the official PPI for final-demand trade services rose 46.04 percent, at an average annual rate of 3.50 percent, the highest rate across all indexes shown previously. In contrast, the geomean PPI for final-demand trade services increased a more modest 22.82 percent, at an average annual rate of 1.89 percent. On the other hand, removing trade services from the official and geomean indexes for final demand leads to more similar growth patterns. Over the 2011–22 period, the official PPI for final demand less trade services rose 28.58 percent, at an average annual rate of 2.31 percent, and the geomean PPI for final demand less trade services increased 24.47 percent, at an average annual rate of 2.01 percent. The 0.30-percentage-point difference between these growth rates is noticeably smaller than the 0.45-percentage-point difference between the growth rates for the R-CPI-L and the R-CPI-G.

PPIs for intermediate demand measure price changes for outputs sold to businesses as inputs to production, excluding capital investment. Chart 4 shows official and geomean PPIs for processed goods for intermediate demand, unprocessed goods for intermediate demand, and services for intermediate demand. As seen in the chart, from 2011 to 2022, the differences in growth rates between the official and geomean PPIs for processed and unprocessed goods for intermediate demand were similar. For processed goods for intermediate demand, the official and geomean PPIs increased at average annual rates of 2.46 and 2.04 percent, respectively, for a difference of 0.42 percentage point. For unprocessed goods for intermediate demand, the official and geomean PPIs increased at average annual rates of 1.78 and 1.30 percent, respectively, for a slightly higher difference of 0.48 percentage point. Both differences are larger than the 0.26-percentage-point difference found for final-demand goods. In addition, compared with goods, services once again exhibit larger growth-rate differences. From 2011 to 2022, the official PPI for services for intermediate demand grew at an average annual rate of 2.96 percent, and the geomean PPI for services for intermediate demand grew at an average annual rate of 2.12 percent. As was the case for goods, the 0.84-percentage-point difference in growth rates for intermediate-demand services was greater than that for final-demand services.

### Conclusion

Currently, BLS uses an arithmetic-mean formula for calculating elementary-level indexes in the PPI. However, performing these calculations with a geomean formula presents mathematical advantages and can eliminate some sources of bias inherent in the arithmetic-mean formula. Although substitution toward relatively less expensive goods and services is not the primary factor motivating the use of a geomean formula, the evidence suggests that substitution in the PPI is toward less expensive goods, a pattern similar to that found for consumer prices.

As shown in this article, formula choice has a noticeable effect on index values. For the 2011–22 period, the use of a geomean formula at the elementary level would have lowered the PPI for final demand by 0.56 percentage point per year. The formula’s effect on service indexes, in particular those for trade services, is much larger than the effect on goods indexes. Between 2011 and 2022, the average annual growth rate for the official PPI series exceeded that for the PPI geomean research series by 0.26 percentage point for final-demand goods, 0.74 percentage point for final-demand services, and 1.61 percentage points for final-demand trade services. A similar pattern is observed for intermediate-demand indexes, with growth-rate differences being larger for services than for goods. In addition, for both goods and services, the magnitude of differences is greater for intermediate demand than for final demand.

Suggested citation:

Sara Stanley, "Introducing Producer Price Index research series based on a geometric-mean formula," Monthly Labor Review, U.S. Bureau of Labor Statistics, May 2024, https://doi.org/10.21916/mlr.2024.8

### Notes

1 In practice, the Producer Price Index (PPI) program uses a formula closer to a Young formula at the elementary level and a formula closer to a Lowe formula at higher levels of aggregation.

2 Price-index research often refers to weights rather than quantities. These weights are calculated with quantity data, whereby base-period quantities are multiplied by base-period prices to arrive at base-period revenue weights.

3 Specifically, the PPI program uses a geometric Young formula to calculate the PPI geomean research series.

4 Kenneth V. Dalton, John S. Greenlees, and Kenneth J. Stewart, “Incorporating a geometric mean formula into the CPI,” Monthly Labor Review, October 1998, https://www.bls.gov/opub/mlr/1998/10/art1full.pdf.

5 Jonathan C. Weinhagen, “Measuring the substitution effect in Producer Price Index goods data: 2002–16,” Monthly Labor Review, July 2020, https://doi.org/10.21916/mlr.2020.16.

6 Marshall B. Reinsdorf, “Formula bias and within-stratum substitution bias in the U.S. CPI,” The Review of Economics and Statistics, vol. 80, no. 2, May 1998, pp. 175–187, https://www.jstor.org/stable/2646630.

7 “BLS research series: producer price indexes using a geometric mean,” Producer Price Index (U.S. Bureau of Labor Statistics), https://www.bls.gov/ppi/research/bls-research-series-producer-price-indexes-using-a-geometric-mean.htm.

8 Michael J. Boskin, Ellen R. Dulberger, Robert J. Gordon, Zvi Griliches, and Dale Jorgenson, Toward a More Accurate Measure of the Cost of Living, final report to the Senate Finance Committee from the Advisory Commission to Study the Consumer Price Index (Washington, DC: Senate Finance Committee, December 4, 1996); and Marshall B. Reinsdorf and Brent R. Moulton, “The construction of basic components of cost-of-living indexes,” in Timothy F. Bresnahan and Robert J. Gordon, eds., The Economics of New Goods (Chicago, IL: University of Chicago Press, 1996), pp. 397–436, http://www.nber.org/chapters/c6073.

9 Dalton, Greenlees, and Stewart, “Incorporating a geometric mean formula into the CPI.”

10 Producer Price Index program staff, “Wholesale and retail producer price indexes: margin prices,” Beyond the Numbers: Prices & Spending, vol. 1, no. 8 (U.S. Bureau of Labor Statistics, August 2012), https://www.bls.gov/opub/btn/volume-1/wholesale-and-retail-producer-price-indexes-margin-prices.htm.

11 Robert S. Martin, Andy Sadler, Sara Stanley, William Thompson, and Jonathan Weinhagen, “The geometric Young formula for elementary aggregate producer price indexes,” Journal of Official Statistics, vol. 38, no. 1, March 2022, pp. 239–253, https://sciendo.com/article/10.2478/jos-2022-0011.