A tangent cone analysis of smooth preferences on a topological vector space

A theory of smooth preferences on a locally convex, topological vector space is developed by characterizing the existence of a unique supporting hyperplane to a convex set at a given point. The results are expressed in terms of the tangent cone. A new concept of properness, called strict properness, is also proposed to characterize the existence of a strictly supporting hyperplane to a convex set at a given point. We say that strict preferences are properly smooth at a given point provided that they are smooth, strictly proper, and the secant cone has a non-empty interior. Proper smoothness is broadly consistent with Gâteaux-differentiable utility even when the preference domain has an empty interior. Yet proper smoothness also allows the possibility of incomplete or intransitive preferences. This concept has immediate applications to optimization and equilibrium theory. For example, we demonstrate a version of the Second Welfare Theorem for agents with properly smooth preferences.