ABSTRACT
This chapter contains a critical review of the theory of “exact” and “superlative” index numbers that is still dominating the field of economic index numbers. An index number is said to be “exact” for a function if it is identically equal to the ratio of numerical values of that function at any pair of points taken into comparison. The first use of the concept of “exact” index numbers appeared in the discovery of Byushgens (1925) that Irving Fisher’s ‘Ideal Index’ is an exact formula when demand is governed by a homogeneous quadratic utility (see also Konüs and Byushgens, 1926). This concept became more widely known with Schultz (1939) and its discussion in the Foundations of Paul A. Samuelson (1947, p. 155). In the introduction to the enlarged edition, Samuelson (1983, p. xx) later wrote: “Thirty-five years after that [revealed preference] analysis appeared there has been but one major advance in index number theory-namelyW. E. Diewert’s formalizing concept of a ‘superlative index number’, which is a formula based upon two periods (pj,qj) data that will be exactly correct as an ordinal indicator of utility for some specified family of indifference contours.(Only a few different ‘superlative’ formulas are known; perhaps the set of simple superlative formulas is a limited set.)”2
Diewert (1976, p. 117) used Irving Fisher’s (1922, p. 247) terminology of “superlative” index numbers to define index numbers that are exact for functions that provide a second-order differential approximation to an unknown true function. In another paper, Diewert (1978) pointed out that these index numbers approximate each other up to the second-order at the point where the compared
these two points do not vary very much3. Quoting Diewert (2004, p. 450) words, “Diewert (1978, p. 888) showed that the three superlative index number formulas listed approximate each other to the second order around any point where the two price vectors [in the calculation of the price index], p0 and p1,are equal and where the two quantity vectors [in the calculation of the quantity index], q0
and q1, are equal. He concluded that ’all superlative indices closely approximate each other’(Diewert, 1978, p. 884)”. This definition can be contrasted with that given by Irving Fisher (1922, pp. 244-48), who had singled out eleven index number formulas out of 134 examined formulas and called them “superlative” because, in his numerical example, they performed very closely to the “ideal” geometric mean of Laspeyres and Paasche indexes4. He claimed that all these superlative formulas correspond to combinations of the Laspeyres and Paasche including the direct and implicit Walsh index numbers, one combination of these last two formulas, and a couple of combinations of direct and implicit Törnqvist-type index numbers. Apart from the Fisher “ideal” and the implicit Walsh indexes, Diewert’s superlative index numbers differ widely in nature and, potentially, in numerical values from those defined superlative by Fisher (1922)5.
