Existence of Edgeworth and competitive equilibria and fuzzy cores in coalition production economies

Debreu and Scarf (Int Econ Rev 4:235-246, 1963) proved that for an exchange economy or a production economy with the same production set for all coalitions, under some standard assumptions, an Edgeworth equilibrium is a competitive equilibrium, and Florenzano (J Math Anal Appl 153:18-36, 1990) proved that for such an economy, any allocation in the fuzzy core is an Edgeworth equilibrium. These results are extended to coalition production economies where each coalition can have a different production set. In fact, we establish the coincidence of the fuzzy core, the set of Edgeworth equilibria, and the set of competitive equilibria in a coalition production economy under some standard assumptions. We then prove the existence of the fuzzy core in such a coalition production economy by using a fuzzy extension of Scarf's core existence theorem, thereby establishing the existence of Edgeworth equilibria and competitive equilibria in such economies.