A simple proof of the equivalence theorem for oligopolistic mixed markets

In a rather long and complicated way Shitovitz proved that in mixed markets with at least two large traders all of the same 'type', all core allocations are competitive. We provide a simple and short proof for this important result. Unlike Shitovitz, we do not prove directly this equivalence. Rather, we first show that the core of an oligopolistic market coincides with the core of the atomless market derived by splitting the atoms into a continuum of traders. We then apply Aumann's Equivalence Theorem to this atomless market, whose set of competitive equilibria coincides with that of the original mixed market.