for some matrix A = [amn] (which must be symmetric and have a single positive eigenvalue), then the Fisher ideal index (8) is exact in the sense that if we calculated the cost function associated with (29) and used it to calculate the COLIs (18), (19) or (24), we would obtain (8) (Byushgens, 1925). If the matrix A has an inverse B, say, the cost function associated with (29) takes the form
(If A is not invertible, (30) will still lead to the Fisher ideal index.) The demand functions associated with equation (30) can be written in the form
The remarkable thing about this result is not that it is possible to find a cost function and a set of demand functions that justify a given price index, but the fact that the result is so general. Although preferences (29) are homothetic—and indeed we can see directly from (30) that the cost function is the product of utility and a function of prices, or from (31) that the shares of the budget pnqn/x are independent of x—the matrices A and B are not specified, except that they must be symmetric and have a single nonnegative eigenvalue, a requirement that comes from the general theory of consumer demand and guarantees, among other things, that demand curves slope down. As a result, and always subject to homotheticity, the demand functions (31) allow the consumer to respond to price changes in a general way; the price elasticities of demand from (31) are unrestricted, except by the general restrictions of consumer theory. The Fisher ideal index is therefore exact for a set of preferences and demand functions that do not restrict substitution behavior in ways beyond that required for the theory. It therefore permits a way of computing a general cost-of-living index without having to estimate the demand functions.
Diewert (1976) extended and generalized these results. A particular specification of preferences, or of the cost function, is said to be a second-order flexible functional form if the utility (or cost) function can provide a second-order approximation to an arbitrary utility (or cost) function. A superlative price index is then one that is exact for some second-order flexible functional form for either the cost or utility function but with preferences restricted to be homothetic. Diewert showed that the utility and cost functions (29) and (30) are flexible for homothetic preferences, so that the Fisher ideal index is an example of a superlative index.