Constant, Increasing and Decreasing Risk Aversion with Many Commodities

Abstract

The Arrow (1971)-Pratt (1964) theory of risk aversion has achieved important successes in the study of economic responses to risk. This theory applies to a special class of cases in which the von Neumann-Morgenstern expected utility hypothesis is used to explain these responses. Under this hypothesis, individuals who maximize the expected utility of different utility functions will in general make different choices in the face of uncertainty. For the purpose of predicting which of two different utility functions will lead to locally more risk averse economic behaviour, the essential feature of these functions is their associated Arrow-Pratt absolute risk aversion measures. In fact, the local comparisons can be extended to global comparisons if the Arrow-Pratt absolute measures are uniformly comparable. The Arrow-Pratt measure of absolute risk aversion and a closely related measure of relative risk aversion also play crucial roles in the investigation of changes, induced by wealth increases, in the risk-taking behaviour of a single expected utility maximizing individual. This study has resulted in a satisfactory theory of increasing, decreasing and constant absolute and relative risk aversion. The special cases to which the Arrow-Pratt theory is directly applicable are those in which the utility functions in question are functions of a single variable, e.g. wealth. Unfortunately, the models required to explain many economic decisions employ a version of the expected utility hypothesis in which the argument of the utility function is a multidimensional vector. One example arises in the study of the consumption-savings decision. Another occurs in the analysis of the combined consumption-savings, portfolio decision. A final example is obtained in the investigation of demand for goods of uncertain quality. The papers of Kihlstrom-Mirman (1974) and Diamond-Stiglitz (1974) have suggested a partial generalization of the Arrow-Pratt theory of absolute risk aversion. This