The Second Welfare Theorem of Classical Welfare Economics

We extend the Second Fundamental Theorem of Welfare Economics in several directions. For pure exchange economies, we drop all insatiability requirements on preferences. For economies with production, we use a concept of directional optimality to provide necessary and sufficient conditions for a given allocation to be competitive. This enables us to show, for example, that not all consumers need to be locally nonsatiated, if the economy is "connected." (An example due to Stanley Reiter shows that such extra conditions are unavoidable.) We use weak assumptions on feasibility sets, allowing, but not requiring, short sales and a very general form of disposability. We do not require that preferences be reflexive, transitive, total, or negatively transitive; and we replace full continuity of preferences by a semicontinuity condition for strict preferences. This provides decentralization results extending some of Arrow's original results [1], as well as those in Arrow and Hahn [2, Theorem 4, pp. 93-94] Debreu [6, Theorem 6.4, p. 95], [4, p. 281], and elsewhere