Knowledge that Transforms

It is often true that a manager's opinions about events relevant to production are valued but are not fully known by others. This note suggests that in such circumstances there may be a problem with production. Consider a competitive equilibrium in a standard Arrow-Debreu model of an economy. In such an equilibrium production decisions are guided by prices and, in particular, by contingent commodity prices (which in fact may be implicit in stock market prices). Moreover, in such an equilibrium the managers of production processes play a strictly passive role since complete instructions for production are implicit in the criterion of profit maximization.' However, if the probabilistic beliefs of the managers are valued but are not fully known by the other agents in the economy, then it seems that these agents might well prefer to have the managers play an active role in making production decisions. In other words, it seems that profit maximization with respect to contingent commodity prices may encourage managers to act contrary to what would be the best wish of others, and consequently that the absence of markets in certain contingent commodities might not be undesirable.2 Our discussion of this issue will make reference to a simple example. An economy with many identical individuals and few identical managers uses seed to produce wheat which may be grown in two regions, A and B. Managers decide where to plant the seed. The wheat harvest is uncertainit is either positive or zero-depending on which of the two possible states of nature, a and ,B, occurs. This is described in Table 1, where si is the amount of seed planted in region i and f is the usual type of production function (f' > 0, f < 0). Let us suppose for simplicity that consumers alone determine prices in competitive equilibrium, that is, the few managers have only a negligible impact on the prices. Assume initially that consumers have fixed beliefs, independent of those which the managers might have. Specifically, assume that consumers believe the state a will occur with probability a. Then, since a competitive equilibrium in which there are markets for contingent wheat is Pareto efficient, it must in this case maximize expected utility of consumers. Consequently, if each consumer's endowment consists of one unit of seed and his von Neumann-Morgenstern concave utility function U(.) depends only on consumption of wheat, the problem solved by the market is to maximize expected utility:

The theory of the shrinking of the core of an exchange economy to the competitive equilibrium (or set of equilibria) when the number of participants increases is one of the most important and interesting contributions to general equilibrium theory in recent decades, and ought to become part of standard courses in economic theory. It is important to have an exposition of this idea which appears as a simple and natural extension of the tools of analysis familiar to most students of economics. The purpose of the present paper is to make an attempt at such an exposition along traditional calculus lines. The paper does not contain results which are new to specialists in the field. In the literature there are, of course, some expositions which point in the direction taken here, but I have not seen the approach spelled out in the way it is done in the sequel. (Some relevant references are given at the end of the paper.)