Univariate and multivariate comparisons of risk aversion: a new approach

Abstract

Introduction A generation ago, Kenneth Arrow (1965) and John Pratt (1964) independently developed a remarkable framework for carrying out comparisons of risk aversion. At the center of this framework lay the construction of risk aversion indices, measuring the degree of concavity of an agent's utility for wealth: The more concave the utility, the more risk averse the agent. [See Machina (1983) for an excellent account.] The Arrow–Pratt framework proved invaluable for comparative statics. For example, in order to show that increased riskiness of future income tends to stimulate present saving, it turned out to be necessary to make assumptions about the behavior of the consumer's indices of risk aversion. Generally speaking, those comparative statics results, which rest upon so-called third derivative conditions, turned out in many cases to be readily and intuitively interpretable in terms of the Arrow–Pratt indices. The foundation upon which Arrow and Pratt had built was expected utility theory: For agents who evaluate risky prospects by calculating their expected utilities, comparisons of risk aversion can always be stated in terms of the Arrow–Pratt indices (see, for example, Yaari 1969). However, for agents whose evaluation of risky prospects does not involve expected utility, the Arrow–Pratt indices are either undefined or else irrelevant. This is true, for example, for a states-of-nature theory of choice under risk. It is also true for the non-expected-utility theory proposed recently by the present author (Yaari 1985). I know, of course, that is is premature – indeed pretentious – to place the theory that I have proposed alongside expected utility theory and attempt to measure the one against the other.

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