Existence and equivalence of competitive and core allocations in large exchange economies with differential information

We study the equivalence of the core and competitive allocations in exchange economies with a continuum of traders and differential information. We show that if the economy is "irreducible", then a competitive equilibrium (in the sense of Radner) exists. Moreover, the set of competitive equilibrium allocations coincides with the "private core". We also show that the "weak fine core" of an economy coincides with the set of competitive allocations of an associated symmetric information economy in which the traders information is the "joint information" of all the traders in the original economy.