Preferences, Risk Neutrality, and Utility Functions

Abstract

A binary preference relation on a real vector space V with zero element 0 satisfying four (natural) axioms induces a utility function U = f u : V ! R where u: V ! R is linear as a map of vector spaces and f : R! R is weakly monotone. The weak ordering and non-triviality axioms are familiar, but the continuity axiom does not require topological assumptions. The fourth axiom (decomposition) is X Y ( ) 0) X ( ) Y (X, Y 2 V ). The utility function U : R ! R can be taken to be linear if and only if also satises the converse of the fourth axiom. When V is a real vector space of stochastic processes and 0 is the zero process, it is known that the four axioms imply the existence of discount factors and, with the converse, the linearity of an intra-period utility function. Thus preferences correspond to discounting and are not risk neutral only if the converse of the fourth axiom is not satised. One consequence is that, within the developed context, a utility function that is not risk neutral can be replaced by one that is risk neutral and still be consistent with the underlying preference relation. Previous axiomatizations of the discounted utility model by T. C. Koopmans and others unwittingly imply risk neutrality because the axiom sets include the four and the converse.

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