Where Utility Functions Do Not Exist - A Note on Lexicographic Orders

There seems to be some amount of confusion in the finance text books regarding the conditions under which an individual’s preferences can be represented by a utility function. Fama and Miller, for example, assert that two axioms (comparability and transitivity) are sufficient to establish the existence of a utility function (when the set of alternatives is Rn). This is totally false: a real valued utility function need not exist even in the single good case (R1). One might hope that a vector valued utility might exist (with lexicographic ordering of the utility vector); but this is not the case. Indeed we cannot salvage the situation even by allowing the utility to be a vector in Rn) (i.e. to have an (countably) infinite number of components); only an uncountable number of real components can do the job. None of these results are new, but they do not seem to be sufficiently well known to researchers in finance. This may be because the original papers are mathematically forbidding or because they are scattered in sources somewhat removed form the mainstream finance literature. If that be so, this note should be of some help; some of our proofs and examples are new and hopefully more elementary (for example we avoid taking recourse to Sierpinski’s lemma).