On the non-emptiness of the fuzzy core

The seminal contribution of Debreu and Scarf (Int Econ Rev 4:235–246, 1963) connects the two concepts of core and competitive equilibrium in exchange economies. In effect, their core-equilibrium equivalence result states that, when the set of economic agents is replicated, the set of core allocations of the replica economy shrinks to the set of competitive allocations. Florenzano (J Math Anal Appl 153:18–36, 1990) defines the fuzzy core as the set of allocations which cannot be blocked by any coalition with an arbitrary rate of participation and then shows the asymptotic limit of cores of replica economies coincides with the fuzzy core. In this note, we provide an elementary proof of the non-emptiness of the fuzzy core for an exchange economy. Hence, in motivation, our result is similar to the contribution of Vohra (On Scarf’s theorem on the non-emptiness of the core: a direct proof through Kakutani’s fixed point theorem. Brown University Working Paper, 1987) and Shapley and Vohra (Econ Theory 1:108–116, 1991) for the core. Unlike the classical Debreu–Scarf limit theorem (Debreu and Scarf in Int Econ Rev 4:235–246, 1963) and its numerous extensions our result does not require any asymptotic intersection—or limit—of the set of core allocations of replica economies.