Some Stronger Measures of Risk Aversion in the Small and the Large with Applications

THE ARROW-PRATT MEASURES of risk aversion for von Neumann-Morgenstern utility functions have become workhorses for analyzing problems in the microeconomics of uncertainty. They have been used to characterize the qualitative properties of demand in insurance and asset markets, to examine the properties of risk taking in taxation models, and to study the interaction between risk and life-cycle savings problems to name just a few applications. Equally importantly, they have generated the linear risk tolerance class of utility functions which has provided canonical examples in such diverse areas as portfolio theory and the theory of teams. Despite these successes, there have been a number of areas for which the results have been weaker than hoped. It is natural to use the risk aversion measures to compare the behavior of individuals in risky choice situations. For example, consider the individual portfolio choice problem in a two asset world with a riskless asset and a risky asset. If individual A has a uniformly higher Arrow-Pratt coefficient of risk aversion than individual B, then B will always choose a portfolio combination with more wealth invested in the risky asset. But, suppose that both assets are risky. Now, there is no obvious sense in which the more risk averse individual can be said to hold a less risky portfolio, but it seems strange that such a simple alteration should destroy the analytics which support the basic intuition. Similarly, consider the basic insurance problem. If one individual, A, is more risk averse than another, B, in the Arrow-Pratt sense, it follows that A will pay a larger premium to insure against a random loss than will B. Typically, though, an individual evaluates partial rather than total insurance, that is, only some gambles can be insured against and others must be retained. In this case, even when the gambles which are retained are independent from those which are insured, it is no longer true that the individual whose Arrow-Pratt measure of risk aversion is higher will pay a larger insurance premium. The situation is no better when we consider comparative statics exercises for a single individual. Decreasing absolute risk aversion in the sense of Arrow and