The Riesz-Kantorovich formula and general equilibrium theory

Let L be an ordered topological vector space with topological dual L' and o rder dual L-similar to. Also, let f and g be two order-bounded linear funct ionals on L for which the supremum f V g exists in L. We say that f V g sat isfies the Riesz-Kantorovich formula if for any 0 less than or equal to ome ga is an element of L we have f V g(omega) = sup [f(x) + g(omega - x)]. 0 less than or equal to x less th an or equal to omega 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 form ula 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 t o extend the existence of equilibrium results in models beyond the vector l attice settings. This paper addresses the following open mathematical quest ion: Is there an example of a pair of order-bounded linear functionals f and g f or which the supremum f V g exists but does not satisfy the Riesz-Kantorovi ch formula? We show that if f and g are continuous, then f V g must satisfy the Riesz-K antorovich formula when L has an order unit and has weakly compact order in tervals. If in addition L is locally convex, f V g exists in L-similar to f or any pair of continuous linear functionals f and g if and only if L has t he Riesz Decomposition Property. In particular, if L-similar to separates p oints in L and order intervals are sigma(L,L-similar to)-compact, then the order dual L-similar to is a vector lattice if and only if L has the Riesz Decomposition Property - that is, if and only if commodity bundles are perf ectly divisible. (C) 2000 Elsevier Science S.A. All rights reserved.