Translating Economic Essence of the Independence Axiom into a General Equilibrium Mathematical Condition

Satisfaction of either of the independence axiom, or its less stringent counterpart, `smoothness of utility functions' is necessary condition for robustness of applications of expected utility theory to modeling of recurring choice. This study arrives at a general equilibrium mathematical condition for functions that, simultaneously, satisfy each of the independence axiom or smoothness of utility functions, conditions. The mathematical condition shows functions that are not `well defined', equivalently, not `one-to-one' induce violations of the independence axiom. In stated respect, artifice of truncation of concave functions at first encounter of a zero derivative induces a function that is not well defined. While strictly convex functions are well defined, existence of an optimum requires artifice of restriction of domain for the function, as such, arrival at pre-specified utility, as opposed to risky expected utility. With focus on stock markets, in presence of satisfaction of ordering, continuity, and independence axioms, regardless, adoption of concave or strictly convex functions for modeling of expected utility embeds several reinforcing endogeneities, namely pre-specification of the optimum (i.e. supremum); dichotomization of expectations of positive returns from information; and for concave (respectively, strictly convex) functions only, dependence of expectations of positive returns on increase to risk aversion parameters of economic agents, as such, absence of any well defined representative agent (respectively, parameterization of stock markets by First Order Stochastic Dominance, as such, arrival at a contradiction to risky expected utility). Study findings reiterate importance of searches for new approaches to modeling of risky choice that subsists in general equilibrium