Introduction

The literature on index numbers is so vast that we can cover only a small fraction of it in this chapter. Frisch (1936) distinguishes three approaches to index number theory: (i) ‘statistical’ approaches, (ii) the test approach, and (iii) the functional approach, which Wold (1953, p. 135) calls the preference field approach and Samuelson and Swamy (1974, p. 573) call the economic theory of index numbers. We shall mainly cover the essentials of the third approach. In the following two sections, we define the different index number concepts that have been suggested in the literature and develop various numerical bounds. Then in section 4, we briefly survey some of the other approaches to index number theory. In section 5, we relate various functional forms for utility or production functions to various index number formulae. In section 6, we develop the link between ‘flexible’ functional forms and ‘superlative’ index number formulae. The final section offers a few historical notes and some comments on some related topics such as the measurement of consumer surplus and the Divisia index.

Price indexes and the Konüs cost of living index

We assume that a consumer is maximizing a utility function F(x) subject to the expenditure constraint where x ≡ (x1, …, xN)T ≥ 0N is a non-negative vector of commodity rentals, p ≡ (p1, …, PN)T ≫ 0N is a positive vector of commodity prices and y > 0 is expenditure on the N commodities.