Monotonicity along rays and consumer duality with nonconvex preferences

The goal of the paper is to prove a duality relation between the direct and indirect utility function without any reference to convexity of preferences, nor quasiconcavity of the utility function. To reach this result we allow for the pricing system to be sublinear, rather than linear. The relevance of this assumption has been illustrated both in Consumer Theory, to keep account of the presence of intermediation or of bundling, and in Mathematical Finance. The main tool is a nonlinear separation theory, which uses sublinear functionals to separate points from radiant or coradiant sets. This yields a characterization of the class of functions for which the duality can be proved, namely those whose upper level sets are evenly coradiant. Such functions are nondecreasing along each rays emanating from the origin, a very weak requirement of nonsatiation of preferences, and satisfy a further technical requirement. We underline that this further requirement is always satisfied if u is upper semicontinuous hence, in particular, if u is continuous or differentiable. The conditions that we obtain are necessary and sufficient and consequently they offer the minimal assumption under which a utility function coincides with the dual of the indirect utility.