In a famous article [1], Arrow studied the allocation of risk-bearing in a competitive economy consisting of I individuals, C commodities and S states. Two schemes were considered. The first scheme was one equipped with S x C complete contingent commodity claims. These claims were assumed to be tradable in the market. The second scheme, in contrast, was equipped with S types of independent2 money claims and a set of C " spot " markets to be opened upon the occurrence of a particular state. Arrow then arrived at a remarkable conclusion (his Theorem 2) that the two schemes would lead to the same optimal allocation of risk-bearing. The " social significance" of the second scheme, he said, was that " it permits economizing on markets; only S+ C markets are needed to achieve the optimal allocation, instead of the SC markets ". I wish to argue, however, that the two schemes will not generally lead to the same allocation and consequently that the theorem is false.