The Riesz-Kantorovich formula and general equilibrium theory

Let L be an ordered topological vector space with topological dual L′ and order dual L^~. Also, let f and g be two order-bounded linear functionals on L for which the supremum f∨g exists in L. We say that f∨g satisfies the Riesz-Kantorovich formula if for any 0≤ω∈L we havef∨g(ω)=sup0≤x≤ω [f(x)+g(ω-x)].This is always the case when L is a vector lattice and more generally when L has the Riesz Decomposition Property and its cone is generating. The formula has appeared as the crucial step in many recent proofs of the existence of equilibrium in economies with infinite dimensional commodity spaces. It has also been interpreted by the authors in terms of the revenue function of a discriminatory price auction for commodity bundles and has been used to extend the existence of equilibrium results in models beyond the vector lattice settings. This paper addresses the following open mathematical question: Is there an example of a pair of order-bounded linear functionals f and g for which the supremum f∨g exists but does not satisfy the Riesz-Kantorovich formula?We show that if f and g are continuous, then f∨g must satisfy the Riesz-Kantorovich formula when L has an order unit and has weakly compact order intervals. If in addition L is locally convex, f∨g exists in L^~ for any pair of continuous linear functionals f and g if and only if L has the Riesz Decomposition Property. In particular, if L^~ separates points in L and order intervals are σ(L,L^~)-compact, then the order dual L^~ is a vector lattice if and only if L has the Riesz Decomposition Property - that is, if and only if commodity bundles are perfectly divisible.