There are many other superlative indexes, for example, the Törnqvist index defined by
which is exact for the translog cost function, in which the logarithm of costs is a quadratic form in the logarithms of prices. The Walsh price index (9) is exact for a utility function that is a quadratic form in the square roots of the quantities; it too is therefore a superlative index. Diewert (1978) shows that these three superlative price indexes approximate one another to the second order around any given price-quantity combination, so that the choice between them is unlikely to matter much in practice.
The Fisher ideal index is computed from both the Paasche and Laspeyres, and thus requires information on both base period and current baskets. The (logarithm of the) Törnqvist index (31) is a weighted average of logarithmic price relatives, with weights that are the average of current and base period patterns of demand. Indeed, superlative indexes always require both current and base period quantity information. Intuitively, their ability to capture the substitution effects of prices has to be based on observation of the effects of the price change, which requires data on demand both before and after the change.
The analysis so far has been entirely within the framework of homothetic preferences, something that is unattractive in practice. It is possible to accommodate nonhomotheticity at the price of interpreting the superlative index as the cost-of-living index for some specific intermediate level of living. For example, Diewert (1976:122) showed that the Törnqvist price index is exact at the level of utility that is the geometric mean of the utility in periods 0 and t.
The analysis of the passage from individual to aggregate indexes is essentially identical to the same analysis for the basket price indexes in the second section of these notes. Nevertheless, it is worth defining Pollak’s (1980, 1981) social cost-of-living index which is the ratio of the aggregate cost of obtaining the base levels of living at current prices to the aggregate cost of obtaining the base levels of living at the base period prices. Hence, adding superscripts h to denote individual households
where ch(uh,p) is the cost function of household h—note that there is no requirement that different households have the same preferences—and u0h is the label