The Economic Theory of Index Numbers

Ti-iIS paper attempts a synthesis of what might be called the classical economic theory of index numbers, developed by Staehle and others,' with the later work of Hicks2 and Samuelson.:3 The problem can be expressed in its simplest form as follows. A given individual consumer is considered in two situations, i and z, usually (but not necessarily) two points of time. Complete price-quantity data are available for the consumer in each of the situations, i.e., the prices of (n + i) commodities, Pl, P', Pl . . P.1p(), and the quantities purchased, ql, ql', q1 . . . ql(n), at situation i, and the corresponding prices and quantities denoted with a suffix z at situation 2. A definite preference map, subject to the usual conditions of convexity, etc., is assumed for the consumer in each situation, but the map for situation i may be different from that for situation 2. The consumer is assumed to have free choice. In particular, nothing will be said about a situation in which there is rationing. Can we define an index of the change in prices and an index of the change in the volume of consumption from situation i to situation z ? If so, can we obtain a measure of each index or, at least, limits for each index ? We can start with the problem of the price index. A perfectly definite index of price change can be defined for the consumer for situation i, which has reference only to the indifference level I, on which the consumer finds himself at this situation. The index, P12 (situation i), is the income which leaves the consumer on the indifference level I, when expended optimally at prices (P2), divided by the actual income expended at prices (Pl). Here the prices (P2) can be regarded as a hypothetical alternative to the prices (Pl) in situation i. There is no reference to the behaviour of the consumer in situation 2 ; the index is independent of the preference map of situation z and of the purchases (q2) made in this situation. We know what purchases (ql) the consumer makes in situation i ; to derive the index we should need to know the purchases, call them (q), he would make if the prices had been (P2) to maintain the same indifference level. We are not interested in what the consumer actually does at prices (p2) in situation 2. By definition, then: