Self-Dual Preferences

Abstract

Recent papers by Houthakker [3 and 4], Samuelson [8 and 9], Sir John Hicks [2], and others deal with the question of the existence of a nontrivial preference ordering which exhibits the mathematical properties in terms of both the direct and indirect utility functions. It is shown that homogeneity and separability are compatible with both the direct and indirect utility functions, but that direct and indirect additivity is consistent with only limited classes of utility functions. Samuelson has raised the question of whether there exists a nontrivial self-dual preference ordering which requires more stringent conditions than homogeneity and separability. By a self-dual preference is meant a preference ordering such that the direct utility function is identical with the corresponding indirect utility function. The purpose of this paper is to present a complete solution to the problem of self-duality. First, we elaborate on the'necessary and sufficient conditions for an exactly or strongly self-dual utility function (in Samuelson's sense). Then, using some well-known concepts of the continuous group theory of transformations, we study the case of weakly self-dual preference orderings and give a precise formulation of the concept of the same mathematical form. Some special classes of self-dual preferences are subjected to detailed analysis. RECENT PAPERS BY Houthakker [3 and 4], Samuelson [8 and 9], Sir John Hicks [2], Pollak [6], and Lau [5] deal with the question of the existence of a nontrivial preference ordering which exhibits the mathematical properties in terms of both the direct and indirect utility functions. Some of the mathematical properties investigated in these works are homogeneity, separability, and additivity. It is shown that homogeneity and separability are compatible with both the direct and indirect utility functions, but that direct and indirect additivity is consistent with only limited classes of utility functions ([2, 8, and 9]). Samuelson has raised the question of whether there exists a nontrivial self-dual preference ordering which may require more stringent conditions than homogeneity and separability [8]. By a self-dual preference is meant a preference ordering such that the direct utility function is identical with the corresponding indirect utility function [8].2 Although partial answers given by Houthakker [4], Pollak [6], and Russell [7] are very illuminating, the solution is far from complete. The purpose of this paper is to present a more complete solution to the problem of self-duality. First, we elaborate on the necessary and sufficient conditions for an exactly self-dual utility function (in Samuelson's sense). Then, using some well-known concepts in the theory of the continuous family of transformations, we study the case of weakly self-dual preference orderings and give a precise formulation of the concept of the same mathematical form. Some special classes of self-dual preference orderings which are convenient for empirical estimation are subjected to detailed analysis.