price indexes that allow for substitution do not rise faster than the CPI (see Shapiro and Wilcox, 1997).

Two points are noteworthy in this context. First, even if one is committed to a COGI approach for the CPI, there is nothing to stop one from using a COLI for compensation purposes. Indeed, if Congress mandated that social security benefits be indexed to hold constant the living standards of social security recipients who have no other income, the COLI would certainly be the appropriate index for escalation (at least subject to the issues related to compensation; see below). In practice, this might mean retaining the Laspeyres approach for the CPI, while recognizing that CPI-based compensation is overgenerous, and compensating people by the growth in the CPI less some modest amount in recognition of substitution bias. More sophisticated compensation schemes could be implemented using superlative indexes, albeit with a lag (discussed further in Chapter 7).

One alternative to waiting for superlative indexes is to use other indexes that make a somewhat less exact allowance for substitution but that can be produced on the same schedule as the CPI. There are a number of possibilities. One is a constant-elasticity-of-substitution (CES) price index, suggested by Lloyd (1975) and recently evaluated by Shapiro and Wilcox (1997). A CES index starts from the price relatives for each good, the ratios of the price in the current period to the price in the base period. In the Laspeyres index, these relatives are averaged, using as weights the share of the budget devoted to each good. In a CES index, each price relative is raised to a power (for example, 0.5) before being weighted and added up. The final index is then obtained by raising this weighted sum to the power not of the exponent but of its reciprocal (see “Technical Note”). If the exponent is 0.5, one is weighting together the square roots of the price relatives and squaring the result. If the exponent is 1.0 (unity), one would be reproducing the Laspeyres; at the other extreme, with an exponent of zero, one would have the expenditure-weighted geometric mean of the price relatives. The smaller the exponent, the more goods are substitutable for one another. Indeed, if one subtracts the exponent from unity, the result is the measure known as the elasticity of substitution.

The constant elasticity of substitution index is exactly equal to the cost-of-living index number if preferences are homothetic and if all goods are equally substitutable for one another. In practice, historical data could be used to choose the exponent that brings the CES index as close as possible to some superlative index, such as Fisher’s ideal index. And although the assumptions of homotheticity and equal substitution are not realistic, such an index will nevertheless capture substitution bias in a way that the Laspeyres does not, and it will do so without requiring data on current purchases. Thus, if substitution is the main concern, the CES index has attractions as a basis for the CPI.

But there are also arguments against the use of a CES price index. Goods are



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