A Theory of Risk Aversion without the Independence Axiom

I study preferences defined on the set of real valued random variables as a model of economic behavior under uncertainty. It is well-known that under the Independence Axiom, the utility functional has an expected utility representation. However, the Independence Aiom is often found contradictory to empirical evidences. The purpose of this paper is to study risk averse utility functions without assuming the Independence Axiom. The major difference between the approach in this paper and that in the literature is I take the point of that prference are defined on set of random variables, in stead of on probability distribution functions. This approach gives simple characterizations of risk aversion, which cannot be expressed when preference is viewd as defined on probability distribution functions. The second advantage is that the differentiability property of utility function studied in this paper does not rely on the assumption that the random variable is bounded (which has to be assumed if one require the utility function is Frechet differentiable in probability distributions). Considering the importance of the tools developed in continuous time asset pricing theory where asset prices are driven by diffusion process, which is clearly not bounded, this approach looks promising in applying nonexpected utility analysis to asset pricing theories. The first part of the paper studies the relation between convexity of preference and risk aversion. When utility function does not have an expected utility representation, equivalence between convexity and risk aversion breaks down. I showed that under appropriate continuity conditions, risk aversion can be characterized by a simple condition that is weaker than convexity, which I call equal-distribution convexity, that is a preference is risk averse iff convex combinations of random variables with the same distribution are preferred to the random variables themselves. Differential properties of risk averse utility functionals are studied. A representation theorem for the form of the Frechet derivative of continuously differentiable utility functionals is given. Characterization of monotonicity and risk aversion in terms of the Frechet derivative of utility functionals are given. I also provide a criteria of comparing individual's attitude toward risk by the properties of the Frechet differential of the utility functions. This criteria, when applied to expected utility, reduces to the usual Arrow-Platt measure of absolute risk aversion. The last part compares the notion of Machina (1982)'s differentiability and the notion of differentiability proposed in this paper