On the Theory of Risk Aversion

Abstract

the existing theory by establishing the economic significance of the partial relative risk aversion function. Let u(t) be a utility function for wealth. The functions A(t) = -u(t)/u'(t) and R(t) -tu(t)/u'(t) are the Arrow-Pratt absolute and relative risk aversion functions. The importance of A(t) arises when considering an individual's aversion to risk as wealth is varied but the risk remains unchanged, while R(t) becomes relevant when wealth and the risk are changed in the same proportion. We shall demonstrate that the partial relative risk aversion function P(t; w) = -tu(t + w)/u'(t + w) is important when the risk is varied but wealth w remains fixed. In addition, we indicate the economic relationships between the functions A, R, and P; present some results about the behavior of P; and relate its behavior to that of A and R. The analysis in this paper is based on Pratt's risk premium which we feel is the only function which actually measures risk aversion for arbitrary risks.2 Our analysis differs from that of both Arrow and Pratt in that we do not use infinitesimal risks. Arrow and Pratt interpret A and R as local measures of absolute and relative risk aversion. The results of this paper show that the functions A, R, and P have a significance beyond their interpretation as local measures of risk aversion in that they determine the behavior of the risk premium in different comparative static contexts. In Section 2 basic concepts are briefly discussed. Section 3 contains our main results. In it the economic significance of A, R, and the new function P is established through their relationship with the risk premium, and we show how the behavior of these functions is relevant for the theory of risk aversion. Section 4 contains a comparison of our results with those of Arrow and Pratt, and indicates the usefulness of A, P, and R. for comparative static analysis of expected utility maximization models.