statistical offices. Econometric modeling is often controversial because it relies heavily on judgment, and the assumptions needed to justify a given inference can often be challenged. This would make it difficult for the BLS to defend a CPI whose construction was crucially dependent on this sort of subjective work.

There is a less controversial approach that holds great promise for calculating good approximations to cost-of-living indexes. This uses what are known as superlative price indexes, which are better approximations to a COLI than the basket price indexes and can be calculated without knowing demand functions. Consider a concrete example. Using the “test” approach to price index construction, Fisher (1922) recommended what is known as Fisher’s ideal index, which is a geometric mean (the square root of the product) of the Paasche and the Laspeyres indexes. Although Fisher’s index was not derived from cost-of-living considerations, a natural question is whether it has a cost-of-living interpretation. This would be the case if there were consumer demand functions that led back to the Fisher index. This turns out to be true, as was demonstrated by the Russian mathematician Byushgens and economist Konüs in the 1920s (see Konüs and Byushgens, 1926). Indeed, the demand functions that do the trick are relatively general; although they are homothetic—which, as we have seen, is a considerable disadvantage—and have a specific functional form, they leave a large number of parameters unspecified. Subject to homotheticity, these parameters can be chosen to match any pattern of consumer substitution that is consistent with the theory. In consequence, the Fisher ideal index can be interpreted as a cost-of-living index without being specific about exactly how consumers substitute in response to changes in relative prices. Apart from the homotheticity (see below), this result comes close to squaring the circle. The statistical agency does not need to make potentially controversial estimates of demand functions. Instead, it can use the two basket price indexes, Paasche and Laspeyres, to calculate another index, the Fisher ideal index, that does what neither basket index can do by itself, namely, capture substitution behavior in a relatively general way.

In his work on superlative indexes, Diewert (1976) extended these results in important ways. First, he went beyond the Fisher ideal index and defined a whole class of superlative indexes whose members, like the Fisher ideal index, are capable of capturing general substitution responses. All of these, like the Fisher index, can be calculated from the same information that goes into basket price indexes—reference and comparison period prices and quantities. They also all require information on comparison baskets so that, like the Paasche index, they can only be produced as quickly as quantity data can be collected.

Diewert also addressed the homotheticity issue. He showed that when demand functions are not homothetic, so that there are different cost-of-living index numbers at different levels of living—and this is the relevant case in practice— superlative indexes can be interpreted as cost-of-living indexes for some level of living intermediate between those of the reference and current periods. If, over the interval of comparison, changes in the level of living are not very important



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