The Arrow-Debreu theory was not originally put forward for the case of uncertainty, but an ingenious device introduced by Arrow, and further elaborated by Debreu, enabled the theory to be reinterpreted to cover the case of uncertainty about the availability of resources and about consumption and production possibilities...[Arrow's device permits one to incorporate] uncertainty about the environment into a Walrasian model of competitive equilibrium. The basic idea is that commodities are to be distinguished, not only by their physical characteristics and by the location and dates of their availability and/or use, but also by the environmental event in which they are made available and/or used. For example, ice cream made available (at a particular location on a particular date) if the weather is hot may be considered to be a different commodity from the same kind of ice cream made available (at the same location and date) if the weather is cold. We are thus led to consider a list of "commodities" that is greatly expanded by comparison with the corresponding case of certainty about the environment. The standard arguments of the theory of competitive equilibrium, applied to an economy with this expanded list of commodities, then require that we envisage a "price" for each commodity in the list, or, more precisely, a set of price ratios specifying the rate of exchange between each pair of commodities.
it will be useful to give a more precise account of the concepts of environment and event that I shall be employing. The description of the "physical world" is decomposed into three sets of variables: (1) decision variables, which are controlled (chosen) by economic agents; (2) environmental variables, which are not controlled by any economic agent; and (3) all other variables, which are completely determined (possibly jointly) by decisions and environmental variables. A state of the environment is a complete specification (history) of the environmental variables from the beginning to the end of the economic system in question. An event is a set of states...Granting that we cannot know the future with certainty, at any given date, there will be a family of elementary observable (knowable) events, which can be represented by a partition of the set of all possible states (histories) into a family of mutually exclusive subsets. It is natural to assume that the partitions corresponding to successive dates are successively finer, which represents the accumulation of information about the environment.
The definition of equilibrium requires that the agents have the access to the complete system of prices when choosing their plans. In effect, this requires that at the beginning of time all agents have available a (common) forecast of the equilibrium spot prices that will prevail at every future date and event. Radner (1970) "New Ideas in Pure Theory: Problems in the Theory of Markets under Uncertainty," 454-6 [HT M. Ali Khan]
The title of this post is an allusion to Radner's comment (454) that he had intended to submit his paper to a session called, "Old Ideas in Pure Theory." Radner treats so-called, Arrow or Contingent Commodities (recall) as a way to handle "uncertainty about the environment" within a Walrasian model of competitive or general equilibrium. Arrow's key (1953) insight is to connect such a commodity to every possible state (date and event) of the environment in advance. This commodity is a kind of insurance (or option). Below I return to this move as a way to handle uncertainty. One assumption that Arrow (and Debreu after him) relies on is that "at any one date each agent will have incomplete information about the state of the environment, but all the agents will have the same information." That is to say, Arrow had been working with a kind of representative agent model. One of Radner's contributions to this literature is to show how to represent "differences in information among the economic agents." (456)
Now, while extending the model, Radner also recognized several limitations on the Arrow-Debreu model. One line of limitation is that "the theory does not take account of at least three important institutional features of modern capitalist economies: money, the stock market, and active markets at every date." (457) And, in fact, Radner goes on to suggest quite elegantly that money, the stock market, and active markets are responses to uncertainty not just about the environment but also uncertainty about future prices. (457-8) To put the significance of this point poetically -- and, perhaps, go against Radner's more sober spirit -- here Radner shows how from within a pure (friction-less) model universe, the development of capitalist institutions that allow agents to manage uncertainty may be understood without formalizing this understanding. That's not the end of the matter because Radner then goes on to summarize his earlier results (now known as a Radner Equilibrium) what active (or "incomplete") markets may look like in his model universe (and so make the Arrow-Debreu far more dynamic):
Suppose that no market at any one date is complete in the Arrow-Debreu sense; i.e., at every date and for every commodity there will be some future dates and some events at those future dates for which it will not be possible to make current contracts for future delivery contingent on those events. In such a model, several types of "equilibrium" concept suggest themselves. First, we may think of a sequence of "momentary" equilibria in which the current market is cleared at each date. The prices at which the current market is cleared at any one date will depend upon (among other things) the expectations that the agents hold concerning prices in future markets (to be distinguished from future prices on the current market!). We can represent a given agent's expectations in a precise manner as a function (schedule) that indicates what the prices will be at a given future date in each elementary event at that date. This includes, in particular, the representation of future prices as random variables, if we admit that the uncertainty of the agent about future events can be scaled in terms of subjective probabilities [16].
Radner's reference ("[16]") is to Savage, who had indeed represented uncertainty about future events in terms of subjective probabilities, but the more fundamental trick -- the representation of future prices as random variables --, is (recall) due to Alchian (and flagged by Arrow), but by 1970 was being used more widely (recall Foucault on Becker 1962). So, in Radner's paper we find the three standard ways in which true Knightian uncertainty is displaced by three successor concepts in pure theory (or mathematical economics): (i) treat it as subjective probability (Savage); (ii) treat it as Randomness (Alchian); (iii) treat it as an Arrow commodity.* (It's possible that Radner is the first to use them jointly.) Elsewhere I have explained (for example here) why (i-iii) fail to do justice to Knightian uncertainty (and I add a note below).
Now, the resulting model has one interesting feature that is flagged by Radner: "since the equilibrium price expectations are self-fulfilling, the observation of the prices in any current market provides information about the true state of the environment (i.e., the specification of the values of particular prices defines an "event" in the set of possible states of the environment)." (459) To the best of my knowledge, Radner is the first formal modeler to note this species of performativity in a (model) market economy.
Some other time I explore what happens if one confuses this model for reality. I close with another observation. Radner had noted another limitation of the original Arrow-Debreu model:
If the Arrow-Debreu model is given a literal interpretation, then it clearly requires that the economic agents possess capabilities of imagination and calculation that exceed reality by many orders of magnitude. Related to this is the observation that the theory requires in principle a complete system of insurance and futures markets, which appears to be too complex, detailed, and refined to have practical significance. (457)
Without noticing it, perhaps, Radner here echoes one of Keynes's (1921) original reasons for embracing the reality of uncertainty. The cognitive capacities of economic agents are limited. The key word, in fact, is "imagination." After all, cheapening computer power is quickly diminishing the problem of "calculation." Rather, in the Arrow-Debreu universe we need to imagine future possible states exhaustively in advance. Elsewhere, I have complained about the modal commitment to finite, countable universes built into this model (which is supposed to model uncertainty); but even if we leave aside modal metaphysics, Radner is right to note that it is peculiar to think one can eliminate true uncertainty by relying on the imaginations of economic agents; if agents were imagining the unknown unknowns, they wouldn't be.
Recent Comments