We study economies that involve both small and large traders as well as the choice of a public project. Within this framework, we establish two sufficient conditions under which the set of competitive allocations coincides with the core. Our first core equivalence result holds under the assumption that there is a countably infinite set of large traders similar to each other. The second result, independent of the number of large traders, requires the existence of a coalition of small traders with the same characteristics of the large traders. Finally, we show how the generalized Aubin approach to cooperation may dispense with both conditions.