Core allocations and the dimension of the cone of efficiency price vectors
Abstract Let ( R 1 ,…, R k ) be an arbitrary partition of the grand coalition in an atomless exchange economy with k “large enough.” We prove that an optimal allocation x belongs to the core if and only if x cannot be improved upon by any coalition that includes at least one of the R i 's. K is “large enough” if k ⩾ r + 1, where r is the linear dimension of the cone P of the efficiency price vectors for x . Recall that it is always true that r ⩽ n , when n is the number of commodities in the market, and that under differentiability and interiority r = 1; thus k can be chosen to be 2 (i.e., for any coalition R , an allocation x belongs to the core of the market if and only if x is not blocked by any coalition that either contains R or contains its complement).