A necessary and sufficient condition for the compactness of individually rational and feasible outcomes and the existence of an equilibrium

In an economy with arbitrary closed and convex consumption sets we show that Pageʹs (Journal of Economic Theory, 1987, 41, 392–404) condition of no unbounded arbitrage is necessary and sufficient for compactness of the set of individually rational and feasible allocations. In strictly reconcilable economies, in which at most one agent may have half-lines in his indifference surfaces, we show that Pageʹs condition is necessary and sufficient for compactness of the set of individually rational utility possibilities, for the existence of an equilibrium, for non-emptiness of the core, and for the existence of a Pareto-optimal allocation. These results allow one agent to be risk-neutral, not permitted by Wernerʹs (Econometrica, 1987, 55, 1403–1418) assumption of no half-lines in indifference surfaces. Our proofs that no unbounded arbitrage is necessary and sufficient for the existence of an equilibrium and for a Pareto-optimal allocation both rely on the result that non-emptiness of the core implies no unbounded arbitrage.