Equity, Arrow's Conditions, and Rawls' Difference Principle

Abstract

An Arrow social welfare function was designed not to incorporate any interpersonal comparisons. But some notions of equity rest on interpersonal comparisons. It is shown that a generalized social welfare function, incorporating interpersonal comparisons, can satisfy modifications of the Arrow conditions, and also a strong version of an equity axiom due to Sen. One such generalized social welfare function is the lexicographic form of Rawls' ARRow (1) INVESTIGATED the problem of how to amalgamate the personal welfare orderings of the members of a society into a social welfare ordering. His approach was deliberately designed to avoid making any kind of interpersonal comparison. He was then able to show that such an approach must fail as long as one insists on certain other apparently appropriate conditions. It would therefore seem that an obvious way around Arrow's impossibility theorem is to make interpersonal comparisons and to use them in the construc- tion of a social ordering. Moreover, some considerations of equity which many people would think relevant for making social choices are specifically excluded by Arrow's approach. This paper shows how, if interpersonal comparisons are made in a certain way, one can construct a social welfare ordering by a method which satisfies suitably modified forms of Arrow's 1963 conditions. Moreover-as is just as well, given that the interpersonal comparisons are deliberately based on a notion of equity- it is also possible to satisfy an extra condition, which is a kind of equity axiom. The lexicographic extension of Rawls' difference principle, or maximin rule, satisfies all these conditions. In addition, it is the only rule or principle which satisfies a condition which underlies Suppes' grading principle, together with these condi- tions. Section 2 presents preliminary definitions and notation, and shows how some considerations of equity are excluded by Arrow's approach to social choice. Section 3 shows how these considerations of equity may be represented by ordinal interpersonal comparisons of the kind discussed in Sen (6), how they are related to an equity axiom due to Sen (7), and how Sen's equity axiom may be generalized. Section 4 defines generalized social welfare functions (GSWF's) and shows how Arrow's conditions can be modified to apply to GSWF's. Section 5 'This is an expanded and subsequently revised version of a paper presented to the European