Econometrica: Jul, 1983, Volume 51, Issue 4

A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox

https://doi.org/0012-9682(198307)51:4<1065:AGOTQM>2.0.CO;2-Z
p. 1065-1092

Chew Soo Hong

The main result of this paper is a generalization of the quasilinear mean of Nagumo [29], Kolmogorov [26], and de Finetti [17]. We prove that the most general class of mean values, denoted by M"@?@?, satisfying Consistency with Certainty, Betweenness, Substitution-independence, Continuity, and Extension, is characterized by a continuous, nonvanishing weight function a? and a continuous, strictly monotone value-like function @?. The quasilinear mean M"@? results whenever the weight function @? is constant. Existence conditions and consistency conditions with first and higher degree stochastic dominance are derived and an extension of a well known inequality among quasilinear means, which is related to Pratt's [31] condition for comparative risk aversion, is obtained. Under the interpretation of mean value as a certainty equivalent for a lottery, the M"@?@? mean gives rise to a generalization of the expected utility hypothesis which has testable implications, one of which is the resolution of the Allias "paradox." The M"@?@? mean can also be used to model the equally-distributed-equivalent or representative income corresponding to an income distribution. This generates a family of relative and absolute inequality measures and a related family of weighted utilitarian social welfare functions.