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
Chew Soo HongThe main result of this paper is a generalization of the quasilinear mean of Nagumo , Kolmogorov , and de Finetti . 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  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.
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