The Problem of Integrability in Utility Theory

Abstract

A chapter in the history of utility theory has now been brought to a close by Mr. Houthakker's2 important discussion of integrability. As far back as i886, G. B. Antonelli3 had noted that such a problem existed and seems to have given the correct integrability conditions . But to most economists this problem was first introduced by Irving Fisher's4 I892 study-perhaps the best of all doctoral dissertations in economics. Even here it is introduced pretty much as an afterthought so that it is no great wonder that Pareto's Manuale di economica politica5 (I906) should have neglected this topic-even though Pareto was clearly aware of Fisher's work and had discussed integrability in journal articles as far back as I893. Vito Volterra6 in his I906 review of the Manuale performed one of the few services professional mathematicians have ever rendered to economic. theory: he pointed out that when Pareto treats the case of three or more goods, his discussion of indifference directions is marred by the failure to recognise explicitly the integrability problem. Pareto admitted his mistake and discussed the problem a few