A Schur Concave Characterization of Risk Aversion for Non-expected Utility Preferences
Abstract This paper extends Machina′s ( Econometrica 50 (1982), 277-323) characterization of risk aversion for Frechet differentiable non-expected utility preferences to the class of continuous non-expected utility preferences without imposing any differentiability requirement. The necessary and sufficient condition for risk aversion is derived in terms of the Schur concavity of the preference functional when evaluated on finite lotteries with equal probabilities. The latter is characterized by its marginal-rate-of-substitution between a high-income state and a low-income state being not less than unity. Correspondingly, the more risk averse the preference ordering, the greater is its marginal-rate-of-substitution between a high-income state and a low-income state. Finally, we apply our results to characterize individual as well as comparative risk aversion for the class of Gâteaux differentiable preferences in terms of the concavity of the Gâteaux derivative or local utility function. This extends Machina′s Frechet-based local expected utility analysis to the larger class of Gâteaux differentiable preferences. Journal of Economic Literature Classification Number: D81.