Risk Neutrality and Ordered Vector Spaces
James C. Alexander
Case Western Reserve University - Department of Mathematics and Cognitive Science
Matthew J. Sobel
Case Western Reserve University - Department of Operations
May 10, 2006
Case Western Reserve Technical Report No. TM-808A
The following result clarifies when preferences over time and under risk correspond to discounting and are not risk neutral. If a binary relation on a real vector space V satisfies four axioms, then there is a utility function U=fu from V to R where u from V to R is linear as a map of vector spaces and f from R to R is continuous and weakly monotone. Three axioms are familiar: weak ordering, continuity, and non-triviality. The fourth axiom is X-Y is weakly preferred to 0 (where 0 is the zero element in V) implies X is weakly preferred to Y (X and Y in V). The function f from R to R is linear if and only if the binary relation also satisfies 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 the linearity of an intra-period utility function. So preferences correspond to discounting and are not risk neutral only if the converse of the fourth axiom is not satisfied.
Number of Pages in PDF File: 10
Keywords: risk neutrality, ordering, discounting, vector space, preference
JEL Classification: D81, D99
Date posted: April 20, 2006
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