Wiley, The London School of Economics and Political Science and The Suntory and Toyota International Centresfor Economics and Related Disciplinesare collaborating with JSTOR to digitize, preserve and extend access toEconomica.http:/www.jstor.orgThe Suntory and Toyota International Centres for Economics and RelatedThe Suntory and Toyota International Centres for Economics and Related DisciplinesDisciplinesRevealed Preference and the Utility Function Author(s): H. S. Houthakker Source: Economica,New Series, Vol. 17, No. 66 (May, 1950), pp. 159-174 Published by: on behalf of and WileyThe London School of Economics and Political ScienceTheSuntory and Toyota International Centres for Economics and Related Disciplines Stable URL: http:/www.jstor.org/stable/2549382 Accessed: 15-08-2014 21:33 UTCYour use of the JSTOR archive indicates your acceptance of the Terms i.e., as a substitute for the “ utility function “ and related formulations, it has since tended to becomie complementary to the latter ; in his Foundations Professor Samuelson uses it to express the empirical meaning of utility analysis, to which he apparently no longer objects. 1 p. A. Samuelson, “ A Note on the Pure Theory of Consumers Behaviour “, Economica, 1938, pp. 61-71, 353-354; “ The Empirical Implications of Utility Analysis “, Econometrica, 1938, pp. 344-356; Foundationzs of Economic Analysis (Cambridge, Mass., I947), Ch. V-VI; “Consumption Theory in Terms of Revealed Preference “, Econosnica, s94S, pp. 243-253. Cf. also I. M. D. Little, “A Reformulation of the Theory of Consumers Behaviour “, Oxford Economic Papers, I949, No. s, pp. 9o-99. 2 Samuelson, Foundations, p. III, note 14. For integrability in general, cf. R. G. D. Allen, Mathematical Analysis for Economists (London, 1938), pp- 42-2-4Z3, 513-517; H. T. Davis, The Theory of Economnetrics (Bloomington, Ind., 1947), pp. 77-81. 3 Samuelson, “ Pure Theory “, Economica, 1938, p. 71. 59 This content downloaded from 220.180.56.238 on Fri, 15 Aug 2014 21:33:54 UTC All use subject to JSTOR Terms and ConditionsECONOMICA Its fundamental concept is the relation which exists between a batch of goods Xa, bought by a consumer at prices Pa and income PaXa, and another batch Xb that might have been bought at the same income P“X“ and prices Pa, its cost PaXb being smaller than or equal to this income, but was not in fact bought. This relation is described by the formula Pa“Xb - PaXa and may be interpreted to mean that the consumer prefers Xa to Xb. Hence, by this com- parison of value sums, Xb is “ revealed to be inferior to“ Xa, and Xa is “revealed to be superior to“ XI. The expressions between quotation marks, the first of which is fundamental, can be described under the name “ revealed preference “. In order to erect a theory on the basis of this relation a rule must be introduced for drawing conclusions from it. For this purpose Professor Samuelson remarks that if the inequality stated really reveals the inferiority of Xb to Xa, then Xa must not also be revealed to be inferior to Xb, as would be the case if pbXa P“Xc; this means that if Xb is revealed to be inferior to X“ and X“ is revealed to be inferior to Xb, then the hypothesis does not rule out the possibility that X“ is revealed to be inferior to X“. This paradox, 1 Foundations, pp. I o, i 6. 2 H. Wold, “A Synthesis of Pure Demand Analysis II “, Skandinavisk Aktuarietidskrift, 1943, Pp. 227-228. To this penetrating exposition the present note owes much. 3 J. R. Hicks, Value and Capital (2nd edition, Oxford, 1946), p. 51. This quotation refers to a proposition still more restricted than Samuelsons hypothesis. This content downloaded from 220.180.56.238 on Fri, 15 Aug 2014 21:33:54 UTC All use subject to JSTOR Terms and Conditionswhich is borne out by numerical examples, proves that Samuelsons hypothesis does not tell everything about revealed preference. The general proposition we are looking for should prevent such anomalies; this can be achieved as follows. We consider a sequence of batches of goods Xt (t = o, I, 2, . . .T; T arbitrary but finite) each of which is bought at prices pt and an income PtXt. First we assume that each batch Xt is different from and revealed to be inferior to its prede- cessor in the sequence Xt-1. The sequence is then described by the formula Pt-i Xt pt-l Xt-l; t now runs from I to T. If a utility function exists, and the utility of Xt is called U(Xt), then the revealed inferiority we assumed implies that U(X) U(X1) U(X2) . U(XT); consequently X is superior to XT. We cannot demand that in these circumstances XT must be revealed to be inferior to X, for a batch may be inferior to another batch without being revealed to be inferior “; but our axiom should exclude the case that X is revealed to be inferior to XT. It must therefore state that for the sequence under consideration PTX2 U(X2) U(X3) so that X3 is still inferior to X. It is