Division of the Humanities and Social Sciences California Institute of Technology Pasadena, California 91125 When Is the Core Equivalence Theorem Valid?

In 1983 L. E. Jones exhibited a surprising example of a weakly Pareto optimal al­ location in a two consumer pure exchange economy that failed to be supported by prices. In this example the price space is not a vector lattice (Riesz space). Inspired by Jones' example, A. Mas-Colell and S. F. Richard proved that this pathological phenomenon cannot happen when the price space is a vector lattice. In particu­ lar, they established that (under certain conditions) in a pure exchange economy the lattice structure of the price space is sufficient to guarantee the supportability of weakly Pareto optimal allocations by prices-i.e., they showed that the second welfare theorem holds true in an exchange economy whose price space is a vector lattice. In addition, C. D. Aliprantis, D. J. Brown and 0. Burkinshaw have shown that when the price space of an exchange economy is a certain vector lattice, the Debreu-Scarf core equivalence theorem holds true, i.e., the sets of Walrasian equilib­ ria and Edgeworth equilibria coincide. (An Edgeworth equilibrium is an allocation that belongs to the core of every replica economy of the original economy.) In other words, the lattice structure of the price space is a sufficient condition for avoiding the pathological situation occuring in Jones' example. This work shows that the lattice structure of the price space is also a necessary condition. That is, "optimum" allocations in an exchange economy are supported by prices (if and) only if the price space is a vector lattice. Specifically, the following converse-type result of the Debreu-Scarf core equivalence theorem is established: If in a pure exchange economy every Edgeworth equilibrium is supported by prices, then the price space is neceBSarily a vector lattice. WHEN IS THE CORE EQUIVALENCE THEOREM VALID? Charalambos D. Aliprantis and Owen Burkinshaw* An answer to the question of the title is provided. It is shown that under some conditions the core equivalence theorem holds true (if and) only if the price space is a vector lattice.