VNM utility and risk aversion tendency with multiple commodities: A theoretical study

Expected utility function forms the mathematical foundation of game theory, finance and management science. Rooted in the expected utility enigma, this paper examines the Von Neumann-Morgenstern (VNM) utility function and risk aversion tendency of an individual in the situation of multiple commodities. In order to establish a solid utility foundation for risk decision-making in a multivariate context, the traditional expected utility theory is reexamined, and the satisfactory expected utility theorem and VNM utility theorem are obtained by introducing the intermediate axiom and a new concept, the P-continuity of risk preferences. It is demonstrated that under the independence and intermediate axioms, individuals have continuous VNM utility functions if and only if their risk preferences are P-continuous. To identify the VNM function among various utility functions of an individual, the Arrow-Pratt risk premium and risk aversion coefficient are extended successfully from a single variable case to a multiple variable situation. It is shown that the two extended indices are all measures of risk aversion for an individual. Particularly, the risk aversion tendency, as an extension of the Arrow-Pratt coefficient of relative risk aversion, characterizes all individuals in uncertain environments. Finally, it is established that a VNM function is homogeneous with order $k>0$ if and only if the individual has a constant risk aversion tendency $\theta<1$. This knowledge provides significant convenience in identifying the VNM utility function of an individual.