Cones with semi-interior points and equilibrium

We study exchange economies in ordered normed spaces (X,‖⋅‖) where agents have possibly different consumption sets. We define the notion of semi-interior point of the positive cone X+ of X, a notion weaker than the one of interior point and we study the existence of equilibrium in the case where X+ has semi-interior points. In Section  4, we study the case where X+ has interior points and we prove a second welfare theorem and the existence of equilibrium. Subsequently we apply these results in the case where X+ has semi-interior points. In the case of semi-interior points the supporting price vectors are continuous with respect to a new norm ∣∣∣⋅∣∣∣ on X which is strongly related with the initial norm and the ordering, and in some sense can be considered as an extension of the norm adopted in classical equilibrium models. Many examples of cones in normed and Banach spaces with semi-interior points but with empty interior are provided, showing that this class of cones is a rich one. In the last section we apply our results to strongly reflexive cones.