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The Index-Number Theorem: A Geometrical Note
Published online by Cambridge University Press: 07 November 2014
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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.
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- Information
- Canadian Journal of Economics and Political Science/Revue canadienne de economiques et science politique , Volume 25 , Issue 2 , May 1959 , pp. 174 - 179
- Copyright
- Copyright © Canadian Political Science Association 1959
References
1 Hicks, J. R., A Revision of Demand Theory (Oxford, 1956), chap. XIX.Google Scholar
2 Marris, R. L., “Professor Hicks' Index Number Theorem,” Review of Economic Studies, XXV (1), no. 66, 10, 1957, 25–40.CrossRefGoogle Scholar See also his Economic Arithmetic (London, 1958), chap. IX.Google Scholar
3 In the sense defined implicitly above, of the collection of goods that could have been purchased in the second situation.
4 See below for further discussion.
5 In the limiting case of no substitution-effect (P 1 coinciding with P′1) the two index numbers would be equal.
6 Fulfilment of the index-number test might be the consequence either of consistency or of a situation-effect biased towards consumption of the goods whose relative prices are lower in the second situation.
7 For a discussion of this point in terms of statistical theory, see Marris, “Professor Hicks' Index Number Theorem,” and Economic Arithmetic.
8 To treat such adjustment as changes in tastes rather than part of the substitution-effect implies restricting the latter to a short-run phenomenon; this seems reasonable for index-number comparisons.
9 I am indebted to an unpublished paper by F. S. Chen for this illuminating way of expressing the point.
10 The alternative would be to eliminate the situation-effect by confining the index-number test to cost-indifferent collections such as P′1 and P 1 in Figure 1. The necessity of ensuring both that each individual compensation had been carried out and that no tastes had changed would seem to rule this alternative out as a practical possibility.