Demand functions and reflexivity

In the theory of ordered spaces and in microeconomic theory two important notions, the notion of the base for a cone which is defined by a continuous linear functional and the notion of the budget set are equivalent. In economic theory the maximization of the preference relation of a consumer on any budget set defines the demand correspondence which at any price vector indicates the preferred vectors of goods and this is one of the fundamental notions of this theory. Contrary to the finite-dimensional economies, in the infinite-dimensional ones, the existence of the demand correspondence is not ensured. In this article we show that in reflexive spaces (and in some other classes of Banach spaces), there are only two classes of closed cones, i.e. cones whose any budget set is bounded and cones whose any budget set is unbounded. Based on this dichotomy result, we prove that in the first category of these cones the demand correspondence exists and that it is upper hemicontinuous. We prove also a characterization of reflexive spaces based on the existence of the demand correspondences.