Core allocations and the dimension of the cone of efficiency price vectors (Q759626)

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 \(R'_ is\). k is ''large enough'' if \(k\geq r+1\), where r is the linear dimension of the cone P of the efficiency price vectors for x. Recall that it is always true that \(r\leq n\), 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.