Information

Despite these difficulties the Walrasian model, perfected by Arrow and Debreu, remains “the benchmark model” of modern economics. However, it is not obvious where pre­cisely the difficulty revealed by the SMD results comes from.

Given this, it is worth noting an important result due to Jordan (1982), which is the key to understanding the problem posed by the SMD results. What Jordan shows is that if you are interested in designing a mechanism to achieve an efficient allocation of resources in an economy, then the competitive mechanism uses less information than any other mechanism. What is more, there is no other mechanism that uses so little information.

To explain this result, which it is worth dwelling upon, consider the following very simple case: a simple barter economy in which there are n individuals each owning a bundle of l goods. We wish to design a mechanism to allocate these goods effi­ciently, that is, which achieves a Pareto optimum and which, at equilibrium, uses as little information as possible.

With each economy the messages that will be required are the vector of prices and the vector of trades each agent wishes to make. For each economy then we need a vector of l - 1 prices (the last one can be dropped since we can normalize prices) and for each of the n individuals a vector of “excess demands” or “net trades”. Since each individual satisfies Walras’s law (what he or she purchases must have the same value as what he or she sells) we only need to know l - 1 of these quantities. Furthermore, since we are considering an equilibrium the aggregate excess demand or net trade must be zero for each commodity, therefore we can drop one of the individuals. Thus we have 1∙ (l - 1) + (n - 1) ∙ (l - 1) “pieces of information”, that is, the messages we use must contain n(l - 1) pieces of information or, put more technically, “the dimension of the message space” of this mechanism is n(l - 1).

The remarkable fact is that any other mechanism satisfying certain minimal require­ments can only achieve an efficient outcome by using more information than this. In other words, the use of a price mechanism limits dramatically the amount of information required to achieve a desirable result.

All of this seems very positive and the idea that the Walrasian equilibrium is infor­mationally efficient is very appealing. However, the very same analysis shows why the SMD results are so destructive. Suppose that we come back to the idea of adjusting from an out-of-equilibrium position to an equilibrium. That is, suppose we take the stability issue seriously and, instead of just trying to solve a system of simultaneous equations, we wish to study how equilibrium is achieved. Even if we accept that all the individuals in the economy observe the same prices for all goods, we have to be able to specify the mechanism that adjusts prices towards equilibrium values.

The first suggestion to solve this problem was that the price adjustment process, which we have typically considered, was inadequate and by modifying it we might be able to get out of the dilemma posed by the SMD results. Yet, what became immediately clear after the innovative work of Smale (1976) was that stability could only be achieved at the price of an enormous increase in the amount of information.

Smale’s global Newton method is an extension of the standard tatonnement process for finding the equilibria of an aggregate excess demand function. However, unfortunately it uses a great deal of information. Without entering into technical details, let me just mention that what is needed is a knowledge of all the partial derivatives of the aggregate excess demand function and this greatly increases the size of the messages necessary to make the mechanism function. Worse, despite this increase in the information required, although the process leads to equilibria from a large set of starting prices, it still does not guarantee convergence from any arbitrary starting point. An additional problem is with the economic content of the process. While the original tatonnement process has a very natural interpretation, despite valiant efforts by some economists, this is not the case for the Newton method. Furthermore, all the alternative adjustment processes in order to solve the stability problem that have been constructed to date have no natural economic interpretation. The informational problem posed by the Newton problem turns out to be unavoidable, for what Saari and Simon (1978) showed was that any process that would always take the economy from any starting prices to an equilibrium would use an infinite amount of information. Thus, while at equilibrium the competitive process is extremely informationally parsimonious, as soon as we consider the Walrasian idea of an economy moving from an out-of-equilibrium position to an equilibrium, the amount of informa­tion necessarily explodes. This shows where one of the fundamental problems exposed by SMD comes from.