# The Experimental CPI using Geometric Means (CPI-U-XG)

I. Overview

Effective April 10, 1997, the Bureau of Labor Statistics (BLS) is releasing an experimental Consumer Price Index for All Urban Consumers that uses a geometric mean formula to combine individual price quotations at the lower level of aggregation while keeping the current Laspeyres arithmetic mean formula for higher level aggregation. This experimental index, called the experimental CPI using geometric means, or CPI-U-XG, is an additional index and does not replace the official CPI. The CPI-U-XG indexes are available from December 1990 through the present, and BLS currently is evaluating the full or partial adoption of the geometric mean formula in the official CPI.

The official CPI uses a modified version of the Laspeyres formula at both levels of aggregation. In a Laspeyres index, relative price change R from the base period (time 0) to comparison time t is measured by comparing the sum of the weighted prices in the comparison period with the sum of the weighted prices in the base period. Algebraically, the formula is:

where Pt,i is the price of the i-th item in comparison period t, P0,i is the price in the base period 0,
Q0,0i is the quantity consumed in the base period 0, and indicates the summation across items. Since price times quantity equals expenditure, the denominator is the total amount of money people spent on items at time 0, and the numerator is the total amount of money they would have spent, had they chosen to buy the exact same items in the exact same quantities they did at time 0.

With geometric means, price change R is measured as

where S0,iis the base-period expenditure share associated with each item in the sample, and indicates the product of the relative price changes across items. The relationship to the Laspeyres index can be seen by rewriting the Laspeyres formula as an arithmetic mean of price relatives, also weighted by base-period expenditure shares:

For a given set of expenditure weights and prices, the geometric mean measure of average price change RGwill lie below the Laspeyres index RL(unless all prices change in the same proportion, in which case both formulas yield the same answer). For example, suppose there are two equally-weighted items sampled within an individual item category, a pound of iceberg lettuce and a pound of Romaine lettuce. Assume that both prices are \$1.00 in time period 0, but that the price of Romaine lettuce increases to \$1.50 in time period t:

 price relative base period expenditures relative expenditure share Q0,i P0,i P t,i (P t,i /P 0,i ) (Q 0,i *P 0,i ) in time 0 (S 0,i ) Iceberg lettuce 1 \$1.00 \$1.00 1.0 \$1.00 0.5 Romaine lettuce 1 \$1.00 \$1.50 1.5 \$1.00 0.5

In the official Laspeyres CPI, the expenditure for the fixed marketbasket would increase from \$2.00 in the base period to \$2.50 in month t, as [(\$1.00*1 + \$1.50*1)]/[(\$1.00*1 + \$1.00*1)] = \$2.50/\$2.00 = 1.25, representing a price increase of 25 percent. Using geometric means, the price change would be 22.5 percent, as = (1.00/1.00) 0.5 *(1.50/1.00) 0.5 = 1.0*1.225 = 1.225. If both types of lettuce had increased from \$1.00 to \$1.50, then both formulas would have shown a 50 percent increase.

From a theoretical standpoint, the potential advantage of the geometric mean index would be as a closer approximation to a cost-of-living index. A cost-of-living index measures the change in the cost of purchasing goods and services that will yield a constant level of satisfaction to consumers. As a fixed-weight index, the Laspeyres formula can be viewed as an upper bound to a true cost-of-living index because it does not reflect the fact that consumers can and do change spending patterns as relative prices change. If consumers respond to changes in relative prices in such a way that relative expenditure shares remain constant, an index using geometric means would equal the cost-of-living index 1. In the example above, the geometric mean estimate equals the change in the cost-of-living index if the purchase of 1.225 pounds of iceberg lettuce and 0.816 pounds of Romaine lettuce in period t (so that the same dollar amount is spent on each) yields the same satisfaction as the one pound of each type purchased in period 0. Within any CPI item category, however, the precise relationships among a Laspeyres index, an index using geometric means, and a true cost-of-living index are not known.

Section II of this announcement reviews the concepts and construction of the official CPI in more detail. Section III summarizes the concepts and construction of the experimental CPI-U-XG. Section IV discusses the limitations of comparing the official CPI to the CPI­U-XG and introduces a comparison Laspeyres index called the CPI-U-XL. Section V presents the results of the CPI-U-XG and compares those results to the CPI-U-XL from December 1990 to the present. Finally, section VI outlines future plans by the BLS for the evaluation of geometric means.

II. How the official (Laspeyres) CPI is currently constructed

To compute the CPI, the universe of consumer goods and services is stratified into 9,108 basic indexes (207 item strata times 44 geographic strata) 2. A two-tiered weighting system is used to calculate the CPI. The first tier of weights are used during the calculation of the basic indexes to weight the individual price quotations together. These weights are derived from the Point-of-Purchase Survey (POPS). Basic indexes are updated each month by the Laspeyres-based price relative:

where P is the price of an item and Q b is the quantity consumed during the base period corresponding to the POPS survey. Only expenditures are collected during the outlet and item selection process; however, base period quantities (Q b ) are not directly available. To handle this, the base period quantities are rewritten in terms of base period expenditures X, as in

where is the estimated "base period price" of the i-th sampled item. The base period price is estimated, not collected, because the base period price is typically not available when the new item samples are introduced into the CPI.

Each month in the official CPI, these expenditure estimates are updated by the price relatives calculated for each of the 9,108 basic cells using the Laspeyres framework. It is at this initial or basic level where the CPI-U-XG uses geometric means instead of the Laspeyres arithmetic means in the estimation of price relatives.

At the second or aggregate tier, base period expenditure weights for each of the 9,108 basic indexes of the CPI are derived from the Consumer Expenditure Survey. The current market basket represents expenditures drawn from the Consumer Expenditure Survey during 1982-84; these weights were introduced into the CPI in December 1986. In both the official CPI as well as the CPI-U-XG, basic indexes are aggregated across basic item-area categories using the Laspeyres method (of arithmetic means) and not geometric means.

III. Geometric Means and the CPI-U-XG

As noted above, the Laspeyres approach currently used in the CPI can be interpreted as an upper bound to a true cost-of-living index in that it does not account for changes in consumption induced by changes in relative prices. Item substitution occurs at both the basic level (e.g., price-induced substitution may occur within an individual item category such as apples) and at the aggregate level (e.g., between chicken and beef).

Historically, analysis of the effect of consumer substitution in the CPI has focused on substitution at the second or aggregate level. Most recently, Aizcorbe, Cage and Jackman (1996) and Shapiro and Wilcox (1996), for example, have estimated the substitution effect at the aggregate level by constructing superlative Fisher or Tornqvist indexes, which use current as well as base period Consumer Expenditure Survey weights 3.

At the basic index level, the calculation of a CPI using geometric means is a potential mechanism for reflecting consumer substitution behavior and thereby eliminating what has been termed "lower-level substitution bias" in the Laspeyres CPI 4. Superlative indexes cannot be constructed at the basic level because only base-period expenditure information is available corresponding to the individual prices being aggregated. For the same reason, the geometric mean's implicit assumption of constant expenditure shares is not testable, but in many markets it is likely to be more plausible than the assumption of fixed consumption quantities. The possibility of using the geometric mean formula to calculate basic indexes in the CPI was first raised by BLS researchers in 1993 5. To assess the potential impact of using geometric means, Moulton (1993) recalculated most non-shelter basic indexes from June 1992 through June 1993 utilizing geometric means instead of arithmetic means. Later estimates by Moulton and Smedley (1995), which include the shelter component of the CPI, cover the period from June 1992 through December 1994. Differences between the two indexes were most significant for food, apparel, and entertainment items.

The earlier estimates by BLS researchers were not of production-grade quality in that some item categories were not included in the estimates, and in that the methods of estimation differed slightly from those used for the official CPI. The BLS now has created the near production-grade quality CPI-U-XG using geometric means to calculate basic indexes from December 1990 to the present. 6 As noted earlier, to distinguish these indexes from earlier estimates by BLS researchers, we call these near production-quality indexes the experimental CPI using geometric means, or CPI-U-XG.

IV. Comparability of series

It is important to note that there are several factors that seriously reduce the meaningfulness of directly comparing the official CPI with the CPI-U-XG, especially before 1996. In particular, the CPI-U-XG incorporates changes other than that associated with the geometric mean formula. These additional factors can be characterized as "bounding" and "methodological" effects. The geometric mean estimator is quite sensitive to extremely large price decreases (the geometric mean is undefined when any price equals zero, as can occur in the CPI in rare cases when formerly priced items are offered without charge). Unlike the official CPI, therefore, the CPI-U­XG has special "bounding" rules for handling extremely large percentage price changes. Specifically, within most commodity and service categories, monthly prices are bounded from above at 20 times the item's base period price, and from below at 5 percent of the base price, to avoid extremely large percentage price changes in the index. Similarly, any rents less than \$20 are reset to \$20 in the CPI-U-XG.

In addition, there are significant "methodological" effects present in the CPI-U-XG. BLS began the process of constructing the CPI-U-XG in early 1996. Hence, CPI-U-XG indexes from December 1990 through 1995 use the estimation methods in place as of early 1996. In other words, any methodological changes made between 1990 and 1995 potentially affect the CPI-U-XG over the entire 1990-1995 period. These same changes affect the official CPI only as they were implemented. For example, in January 1995, improvements were made in the imputation of owners' equivalent rents to eliminate an upward bias in that estimator; at the same time, a six-month chain estimator was incorporated to estimate rents (Bureau of Labor Statistics, 1994b). While these owners' equivalent rent and residential rent methodological changes affect the CPI-U-XG over the entire period from 1990 forward, these same changes affect the official CPI from January 1995 forward.

In order to better isolate the impact of using a Laspeyres arithmetic means estimator versus a geometric means estimator in the calculation of basic indexes, BLS created a Test Laspeyres index, the CPI-U-XL, which incorporates the same bounding rules described above for the CPI-U-XG. In addition, both the CPI-U-XL and CPI-U-XG use the calculation methods in effect at the beginning of 1996. That is, methodological changes are introduced into use for the CPI-U-XL and the CPI-U-XG at the same time 7. For these reasons, we will focus on the differences between the CPI-U-XL and the CPI-U-XG.

V. Results

From December 1990 through February 1997, the CPI-U-XG rose 16.2 percent, which is equivalent to an annual growth rate of 2.46 percent. During that same time, the CPI-U-XL rose 18.6 percent, which is equivalent to an annual growth rate of 2.80 percent, for an annualized difference of 0.34 percent. Tables 1 and 2 give index levels from December 1990 through February 1997 for the CPI-U-XG and CPI-U-XL, respectively, for All Items, each of the 7 major groups, and the categories for food, energy, and all items less food and energy. Table 3 displays the differences between the geometric mean and Test Laspeyres indexes by year for the major CPI item groups. As in previous BLS research, the differences between the two indexes prior to 1995 were greatest for food, apparel, and entertainment items 8. The relatively small differences in the housing component reflect, in part, the incorporation in the CPI-U-XL throughout the study period of the January 1995 changes to the shelter component of the official CPI-U 9.

It is important to note that the difference in the rate of increase between the CPI-U-XG and the CPI-U-XL (as well as the official CPI) has fallen since January 1995, when several methodological improvements to the official CPI were made. For example, in January 1995, an improvement was made in the estimation of quantity weights for newly initiated items within the CPI food-at-home component (Bureau of Labor Statistics, 1994a). That reduced the rate of growth in the CPI-U-XL (as well as the official CPI) indexes from January 1995 forward, but had no effect on the CPI-U-XG, which is weighted not by quantities but by expenditure shares. From December 1990 through December 1994, the annualized difference between the CPI-U-XL and CPI-U-XG was approximately 0.37 percent, but from December 1994 through February 1997, this difference fell to 0.28 percent. For the food-at-home category, the annualized difference between the CPI-U-XL and CPI-U-XG fell from approximately 0.88 percent (Dec. 1990 to Dec. 1994) to 0.26 percent (Dec. 1994 to Feb. 1997).

In June 1996, BLS extended the food-at-home improvement in the estimation of quantity weights to other commodities and services; in July 1996, a functional form bias occurring at item substitution for all commodities and services was eliminated (Bureau of Labor Statistics, 1996a and 1996b). The differences between the CPI-U-XG and the CPI-U-XL (and official CPI) are expected to decline further as these methodological changes reduce the rate of growth in the CPI-U-XL (and official) indexes without affecting the CPI-U-XG.

Graph 1 shows the difference in the 12-month changes for the CPI All Items index for the CPI-U-XL and the CPI-U-XG. For comparison purposes, this graph also shows the difference between the official CPI and the CPI-U-XG over the same time frame. This graph shows that changes in the official CPI over time have reduced the difference between the official CPI and the CPI-U-XG.

Graph 2 shows the same differences for the food-at-home component of the CPI. In this instance, the running 12-month changes between the official CPI and the CPI-U-XL are virtually indistinguishable. The effect of the January 1995 functional form correction is very clear, as differences between the Laspeyres indexes (both the official and the CPI-U-XL) and the CPI-U-XG decline significantly.

As demonstrated in Graph 3, the case of housing contrasts with that of food at home. The Test Laspeyres index for housing exceeds the Geometric Mean index by a relatively small and consistent amount over the study period. The various changes since 1990 in the CPI treatment of residential rent, owners' equivalent rent, and lodging while out of town are visible in Graph 3 as differences between the official CPI series and the CPI-U-XL for housing 10.

Finally, graphs 4 through 11 show, for the CPI-U-XL and CPI-U-XG, the running 12-month percent changes for All Items and each of the seven major groups.

VI. Summary and future BLS plans for geometric means

While the official CPI reflects no consumer substitution behavior, the CPI-U-XG reflects an assumed, but in many cases reasonable, degree of substitution within individual item categories. Methodological improvements made to the official CPI since January 1995 have reduced the differences between it and the CPI-U-XG. Nevertheless, the BLS estimates that an index using geometric means for all basic indexes would increase approximately one-quarter of one percent a year less rapidly than the official CPI, given the current environment of relatively modest inflation. Partial adoption of the geometric mean formula would be expected to have a downward impact of between zero and one-quarter of one percent per year, depending on how many, and which, indexes continue to be based on the Laspeyres approach.

The BLS is now evaluating the adoption of a geometric mean formula as the official CPI, and the CPI-U-XG will be used to generate further research. Scanner data, studies of substitutions between brands, and other information will be used to assess elasticity of demand as the relative prices of items within individual item categories change. The experiences of other countries using geometric means also will be considered.

By the end of 1997, BLS will announce the findings of its research, including its determination of which CPI basic indexes are best calculated with the geometric mean formula and when the implementation of any change will take place. The likely date for that implementation is with the release of January 1999 CPI data.

Footnotes

1 In mathematical terms, the geometric mean index equals the cost-of-living index if all elasticities of substitution in consumption between items equal minus one. The Laspeyres index exceeds the cost-of-living index unless all the elasticities of substitution equal zero.

2 We will use the term "basic indexes" for the sake of consistency. Such indexes have also been called subindexes, elementary aggregates, basic item-area component indexes, low-level indexes, first-level indexes, item-area indexes, stratum indexes, etc.

3 Under certain conditions, superlative indexes can be shown to be closer approximations than Laspeyres indexes to true cost-of-living indexes (see Diewert, 1987). However, they can be produced only with a significant time lag (i.e., current period expenditures are not typically available in real time for most items).

4 See, for example, Reinsdorf (1994), Advisory Commission to Study the Consumer Price Index (1996) and Moulton (1996).

5 Many countries employ the geometric mean formula within some or all index categories, although differences in sample item selection procedures limit the formula’s general interpretation as a cost-of-living approximation.

6 The CPI-U-XG is not considered a production-grade index largely because the quote weights within the shelter component are based not on the theoretically appropriate expenditures but on quantities. Base-period expenditure data for geometric mean weighting will not be available until January 1999, when the new CPI housing sample is introduced. In addition, due to a flaw in computing the weights within the commodity and service components of the CPI-U-XG, the basic geometric mean indexes were inefficiently estimated prior to the index for January 1997.

7 Even though the Test Laspeyres index (CPI-U-XL) uses early 1996 calculation methods retroactively to December 1990, some improvements made between 1990 and 1996 are not shown in the CPI-U-XL from its inception, because certain data are not available. For example, an improvement made in January 1995 to the estimation of quantity weights included the use of "overlap samples," or the collection of two samples during the same (overlap) time period. Since these overlap samples did not exist before January 1995, the effect of this improvement is not incorporated into the CPI-U-XL until January 1995, when it was incorporated into the official CPI. Since this January 1995 improvement affects the estimation of quantity weights, the CPI-U-XG, which is weighted by expenditures, is not affected by this change. The table below identifies important methodological changes that have occurred since 1990 in which the impact of the change is different for the official CPI, the CPI-U-XL, and the CPI-U-XG.

### Methodological changes between 1990 and 1996 affecting the official Laspeyres, test Laspeyres, and experimental CPI using geometric means at different times.

 Methodological change Official CPI-U affected starting CPI-U-XL affected starting CPI-U-XG affected starting Improvements to residential rent and owners’ equivalent rent January 1995 December 1990 December 1990 Improved estimates of quantity weights for food-at-home categories January 1995 January 1995 has no effect Improved estimates of quantity weights for other commodities and services June 1996 June 1996 has no effect Improved estimates of quantity weights for item substitutions July 1996 July 1996 has no effect bounding N/A December 1990 December 1990

8 Baskin and Leaver (1996) have estimated variances for the CPI-U-XG, CPI-U-XL, and official CPI for the major shelter components, and concluded that differences in measures of cumulative index change for the two test series are statistically significant.

9 The unavoidable reliance on quantity weights rather than the theoretically appropriate expenditure weights in constructing the CPI-U-XG housing indexes also affect their movements relative to the corresponding CPI-U-XL series. Rent changes for less expensive housing units have a greater influence in the CPI-U-XG. In addition, 1991 and 1995 changes in the treatment of landlord-provided furniture had a greater effect on the shelter components of the CPI-U-XG than the corresponding components of the CPI-U-XL.

10 The differences between the CPI-U-XL and official CPI-U housing series reflect not only the earlier incorporation of methodological changes in the shelter component of the CPI-U-XL, but also the special computational procedures used during 1991 and 1992 in the CPI-U’s lodging while out of town index to compensate for small sample sizes in that component. These procedures were not replicated in the CPI-U-XL.

References

Advisory Commission to Study the Consumer Price Index, Toward a More Accurate Measure of the Cost of Living. Final Report to the Senate Finance Committee, December 4, 1996.

Aizcorbe, Ana M., Cage, Robert A. and Jackman, Patrick C., "Commodity substitution bias in Laspeyres Indexes: Analysis Using CPI Source Data for 1982-1994," presented at the Western Economic Association International Conference in San Francisco, July 1996.

Baskin, R.M., and Leaver, S. G., "Estimating the Sampling Variance for Alternative Forms of the Consumer Price Index, "Proceedings of the Survey Research Methods Section, 1996, American Statistical Association, to appear.

Bureau of Labor Statistics, "Improving CPI Sample Rotation Procedures," CPI Detailed Report, October 1994, pp.7-8 (cited as Bureau of Labor Statistics, 1994a).

Bureau of Labor Statistics, "Improvements in estimating the shelter indexes in the CPI", Consumer Price Index Detailed Report, October 1994, pp.5-6 (cited as Bureau of Labor Statistics, 1994b).

Bureau of Labor Statistics, "Extending the Improvement in CPI Sample Rotation Procedures," CPI Detailed Report, June 1996, pp.9-10 (cited as Bureau of Labor Statistics, 1996a).

Bureau of Labor Statistics, "Improving CPI item substitution procedures," CPI Detailed Report, July 1996, pp.8-9 (cited as Bureau of Labor Statistics, 1996b).

Diewert, W. Erwin, "Index numbers," The New Palgrave Dictionary of Economics, Vol. 2, The Stockton Press, New York, 1987, pp. 767-780.

Moulton, Brent, "Basic components of the CPI: Estimation of price changes," Monthly Labor Review, December 1993, pp. 13-24.

Moulton, Brent, "Bias in the Consumer Price Index: What is the Evidence?" Journal of Economic Perspectives, Fall 1996, pp. 159-177.

Moulton, Brent R., and Smedley, Karin E., "A Comparison of Estimators for Elementary Aggregates of the CPI," presented at the Western Economic Association International Conference, San Diego, July 7, 1995.

Reinsdorf, Marshall, "Price dispersion, seller substitution, and the CPI," Bureau of Labor Statistics Working Paper 252, March 1994 (revised version of paper presented at Statistics Canada's Price Measurement Advisory Committee, Ottawa, Ontario, Canada, May 1993).

Shapiro, Matthew D., and Wilcox, David W., "Alternative Strategies for Aggregating Prices in the CPI," paper prepared for the conference on Measuring Inflation and Real Growth, Federal Reserve Bank of St. Louis, October 16-17, 1996.