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Economic News Release
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Research Issues Related to the Geometric Mean Formula for Elementary Indexes


The possibility of using the geometric mean formula to calculate the elementary (i.e., the lowest level of aggregation or the within-stratum level) indexes in the U.S. CPI was first raised by BLS researchers in the December 1993 issue of the Monthly Labor Review. Since then BLS researchers have continued to conduct research and have written a number of papers, but not all issues related to the geometric mean formula have been resolved. This note will briefly discuss the conceptual and empirical issues that arise in comparing the geometric mean formula to the CPI's current modified Laspeyres formula.

What does the geometric mean formula do?

The modified Laspeyres formula currently used by the CPI estimates the price each month of a fixed basket of goods and services. In contrast, the geometric mean estimates the price of a varying basket of goods and services. If all prices within the basket increase by the same amount, say 5 percent, then both the modified Laspeyres and the geometric mean will show the index increasing by 5 percent. The two formulas will give different results, however, if prices of items within the basket increase by different proportions.

For example, suppose that the sample market basket for lettuce in Boston consists of two items, a pound of iceberg lettuce and a pound of Romaine lettuce. If the price of iceberg lettuce increases from $1.00 to $1.50, while the price of Romaine lettuce remains equal to $1.00, then the price of the fixed market basket increases from $2.00 to $2.50, an increase of 25%. That is the price increase that would be reported by the current CPI formula.

The geometric mean formula, however, assumes that the market basket varies in a specific manner with the change in relative price between iceberg lettuce and Romaine lettuce. In particular, the geometric mean formula assumes that the quantities of the two types of lettuce that are purchased adjust so that relative expenditures on the two items remain constant. In our example, the market basket shifts to include roughly 20% more of the Romaine lettuce (now relatively less expensive) and 20% less of the iceberg lettuce (now relatively more expensive). The price of the market basket increases 22.5% under the geometric mean formula.

Low-level Consumer Substitution

Since consumers do respond to changes in relative prices by changing their consumption bundles, the conceptual cost-of-living index ought to incorporate those responses. However, the data collected in constructing the CPI do not provide enough information about shifts in quantities and expenditures to determine whether consumer substitution behavior at the lowest level more closely mimics the first, fixed market-basket scenario, or the second scenario in which quantities are adjusted to hold the share of expenditures on each item constant.

The issue, framed in terms of economic theory, has to do with the price elasticity of demand, or the closely related concept of consumer elasticity of substitution. Economists have shown that the geometric mean is the appropriate or "exact" cost-of-living index formula if the elasticity of substitution is equal to one, whereas the fixed basket formula is the appropriate formula if the elasticity of substitution is zero.1 Thus one important issue in comparing the formulas is determining the best approximation for the within-stratum elasticity of substitution.

Unfortunately, determining this value may be quite difficult for several reasons. First, the lowest level of aggregation is, by definition, the level at which the Consumer Expenditure Survey ceases to provide much information on levels of and changes in consumer expenditures. Thus the empirical information available for learning about substitution elasticities is quite limited.

Economists have suggested some possible rules for inferring elasticities in the absence of empirical data. George Stigler wrote, "The only general rule is that the elasticity of demand will be (numerically) greater, the better the substitutes for the commodity."2

A further complication is that relative price changes at this level can derive from many sources. In particular, we need to consider at least the following factors:

  1. Shifts in relative price between brands of items.
  2. Shifts in relative price between outlets.
  3. Shifts in relative prices between categories of items within the stratum. For example, both roasted coffee and instant coffee are within the CPI coffee stratum, even though empirical evidence shows that consumers do not substitute much between the two when their relative prices change.
  4. Shifts in relative prices between geographic areas. Several of the CPI strata are regional aggregates containing a sample of metropolitan or non-metropolitan urban areas located throughout a region. Also, some of the large urban areas cover a substantial geographical area that may cross state boundaries. Rents and prices that are subject to state regulation may be particularly affected by geographical differences.

A number of studies have suggested that the brand-level elasticity is usually quite large, typically around 1.5-2.0.3 How relevant this estimate is to the CPI will depend, however, on how much Factor 1 above contributes to the relative price change within the typical CPI stratum. The recent changes in CPI methodology that corrected the formula bias problem have removed the inappropriate weighting that had previously been applied to temporary price changes, such as one-time sales or promotions. Consequently, if most relative price changes between brands are temporary, then Factor 1 may not contribute much to the long-run variation in relative prices.

One can think of examples where Factor 3 or Factor 4 may be important and because of the lack of close substitutes for an item whose relative price has changed, e.g., insulin or local telephone service, one could conclude on a priori grounds that the relevant elasticity of substitution is much closer to zero than one. On the other hand, some consumer items, such as home computers, have shown sales revenue growth while prices have fallen, which would appear to be consistent with an elasticity greater than one. If the divergence between the geometric mean and Laspeyres index formulas tends to be large in strata where the elasticities of substitution are small, then the Laspeyres could provide the more accurate approximation to a cost-of-living index. Alternatively, if the strata with large divergences between the two indexes tend to have large elasticities of substitution, then the geometric mean index may provide the more accurate approximation to a cost-of-living index. Finally, it may be reasonable to consider the case where neither an assumption of an elasticity of zero nor an elasticity of one is universally appropriate, and different estimators might be used for different strata.

Research plans

BLS researchers have been at the forefront in studying the geometric mean and other issues related to the construction of the CPI. We expect the continuing BLS research to be able to address at least the following two questions:

  1. What is the decomposition of within-stratum price change among the four factors shown earlier?
  2. Using available data (e.g., the limited within-stratum data available from the Consumer Expenditures Survey, data from other non-BLS sources such as scanner data, surveys of published economic and marketing research) what can we learn about the magnitudes of the price elasticity of demand or the consumer elasticity of substitution at the below-stratum level?

1 These results are derived in a number of sources, for example, Robert A. Pollak, The Theory of the Cost-of-Living Index, Oxford University Press, 1989.
2 George J. Stigler, The Theory of Price, 3rd ed., Macmillan, 1966, p. 24.
3 Gerard J. Tellis, "The Price Elasticity of Selective Demand: A Meta-Analysis of Econometric Models of Sales," Journal of Marketing Research, November 1988, pp. 331-341.