On a Theorem of Arrow: Comment'

The paper of K. Nagatani [1] will be useful in correcting a possible misunderstanding in reading my article on the allocation of risk-bearing [2]. I asserted (Section 3) that " any optimal allocation of risk-bearing can be achieved by . . . a competitive system involving securities payable in money " (emphasis added). The italicized word, " can " is somewhat ambiguous. As Nagatani notes, my construction requires each economic agent to be aware of what commodity prices, ps, will prevail for each possible state of nature. My construction, therefore, is a contingent-market version of the old idea of perfect foresight. With uncertainty present, it is not required that the future be known in order to achieve optimal allocation, only that commodity prices be known conditional on the state of nature; such knowledge is needed to give meaning to conditional contracts payable in money. Nagatani suggests that since the ex post commodity markets do not exist at the time of writing the contracts for money claims, no individual can in fact know what prices those markets will lead to. Hence, each individual will form a probability distribution for prices holding in each state of nature before trading in money claims, and the outcome will not be the same as it would be if contingent commodity contracts were traded directly. Let me suggest two interpretations which would justify my position. One is an equilibrium interpretation of ex post commodity prices. Clearly, if the correct ex post prices are believed in at the time the money claims markets are operating, they will in fact be achieved if the appropriate state of nature occurs; that is the way they are arrived at in my construction. This does not explain the process by which they are discovered on the market. One story which accomplishes this is to assume that the world consists of a succession of identical lotteries. In each, the allocation takes place in two stages, first a money claims market and then, when the state is revealed, a system of commodity markets. Which state actually occurs in each drawing is a random event with known probabilities. After a sufficiently long period, the commodity prices conditional on each possible state will become known, and the construction given by me will be the equilibrium outcome of this process (I have not investigated its stability). An alternative interpretation is that the definition of a state of the world includes a statement of the prices that will prevail. The money claims, then, are payable conditional on the occurrence of specified possible price vectors. In fact, many contracts in the real world which have some contingent features in them are payable in amounts which are functions of the ex post prices, the returns on common stocks for example. The second interpretation obviously eliminates a good many difficulties; there can be no uncertainty about the prices that will prevail in a given state if those prices are made part of the very definition of the state. But it must be admitted that there are some difficulties with this interpretation. Implicitly, at least, the uncertainties in the model are exogenous to the economic system; but prices are endogenous to it, and this might complicate our understanding of the model.