MBS 93-41
A Schur-Concave Characterization of Risk Aversion for Non-Expected
Utility Preferences
S. H. Chew, M. H. Mao
For expected utility preferences, Rothschild and Stiglitz (Journal
of Economic Theory, 1970) showed that aversion to mean-preserving increases
in risk is equivalent to two additional definitions of risk aversion based
respectively on the behavior of the decision maker's risk premium and on
his demand for risky asset. This result has been extended in Machina (Econometrica,
1982a) for non-expected utility preferences which are Frechet differentiable
with respect to the L-metric. He showed that aversion to mean-preserving
increases in risk is equivalent to two strengthened definitions of risk
aversion based on the decision maker's conditional risk premium as well
as his conditional demand for risky asset. This paper further extends Machina's
characterization of risk aversion to 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.