A Remark on the Core of an Atomless Economy

In an atomless economy any allocation that is not blocked by "small" coalitions is in the core. Hence, in such an economy, a competitive equilibrium is characterized by the blocking power of part of the coalitions which excludes all "big" coalitions. THE CORE of an economy consists of all the allocations that are not blocked by any coalition. In this note we prove that for any positive number 8, the core of an atomless economy coincides with the set of allocations that are not blocked by any coalition of measure less than s. This result implies that even if the "large" coalitions cannot be formed, any unblocked allocation is still in equilibrium with respect to some price system. In particular, the formation of the coalition of all traders or any "large" coalition is not needed to insure the Pareto-optimality of final allocation. We prove here directly that the core is equal to the set of allocations that are not blocked by small coalitions. The same result can also be obtained by proving that Aumann's [1], Vind's [5], or Hildenbrand's [2] equivalence theorems hold with the additional restriction on the measure of the blocking coalitions. In any case the proof is a simple application of Liapunov's convexity theorem [3 and 4] (for the statement of the theorem see also [2, Appendix, p. 451]).