The index-number theorem is the theorem that, if two sets of price and quantity data on consumption are compared, the Laspeyres (base-weighted) index-number of the price or quantity of one collection as compared with the other will be greater than the Paasche index-number for the same comparison: the Laspeyres index will show a larger increase or a smaller decrease than the Paasche index, as the case may be. In symbols, the theorem is that
where p and q refer respectively to price and quantity, and the subscripts 0 and 1 to the base and the other collection of price and quantity data.
As developed by J. R. Hicks, the index-number theorem is presented as a deduction from, and an empirical test of, the “preference hypothesis” (that consumers choose according to a scale of preferences). But R. L. Marris has shown that the theorem is not necessarily contradicted by a change in tastes, so that it cannot be regarded as a test of the hypothesis of given preferences; rather, it is an inference from, and test of, the hypothesis that consumers' preferences are consistent. As such, it depends on an assumption, which must either be argued in general or proved in the particular case, about what consumption would have been had relative prices been the same in the two situations. The purpose of this paper is to explain the index-number theorem, and the assumptions on which it depends, by means of an application of the geometry of the familiar two-goods case.