The author proves several versions of the second welfare theorem for exchange economies with non convex preferences. One theorem asserts that, given a Pareto optimum f, one can find income transfers and a Walrasian quasiequilibrium g s uch that all but k agents are indifferent between f and g, where k is the number of commodities. Another theorem shows that, with probabil ity one in a particular formulation of a random sequence of economies , every Pareto optimum is close to a Walrasian equilibrium with incom e transfers.