The Arrow-Debreu Contingent Claims Model of Investment

Throughout the discussion of speculation and stability, we emphasized that uncertainty theorists now have a generally accepted framework for modeling choice under uncertainty. Economic theorists have chosen to model uncertainty as the revelation of a state of the world. Individuals in these models face investment and consumption decisions based on payoffs that vary across different states of the world.

This chapter examines the state-preference framework (Arrow-Debreu Theory) in detail. The Arrow-Debreu world has two versions: a state-contingent claims model and a securities version. After 1975, revised general equilibrium models began to incorporate future spot market prices into the definition of the state space. This change was brought about to remove speculative considerations identified in the literature. Yet the revised state definition introduces new problems of its own.

Following the publication of Arrow's seminal work, a large and complex literature on general equilibrium theory and contingent claims analysis evolved. The literature contains many optimality and non-optimality results spanning various extensions of the Arrow-Debreu model; it would be infeasible to attempt to review all of the works here. Fortunately, Radner (1982) summarized the key findings of the early literature.

Although some of the works discussed in this section were published after 1975, they all commonly assume that the state of the world described one or more joint events about the external environment. This early literature also accepted the equivalence of the contingent claims and securities version of Arrow's model without objection.

Theorists interpreted Arrow's results in different ways. A lemma circulated in the literature that with a complete set of contingent claim markets, all desired trading would take place in the prior trading round. In the absence of new information or a change in preferences or budget constraints, no one would want to retrade from their prior round position even if given the opportunity in sequential trading rounds. The subsequent trading rounds would be pointless.

In an earlier article, Radner (1968) indicated that this widely-circulating lemma only worked in one direction. If everyone believes future spot prices are inessential, they will be. However, if some individuals believe something new will change expected spot market prices, they can take positions in intermediate and sequential trading rounds that will force prices to depart from the prior trading round equilibrium. Ultimately, these individual positions may have to be reversed, but in the intermediate trading periods, the terms of trade may adversely affect the value of the prior trading round positions. In short, the traders can adopt paradoxical strategies that become self-fulfilling equilibria.

Radner (1968) extended the Arrow-Debreu model to include agents with differing information about the economy. He found that when information was restricted to the environment, the Arrow-Debreu contingent claims equilibrium can achieve an optimum (relative to a given structure of information). However, if agents receive information about the trading behavior of other market participants, then externalities arise. These externalities often distort preferences or otherwise diminish the optimality of the competitive equilibrium. In particular, the `set-up cost' of gathering information, which may be independent of the scale of production, introduces non-convexity into the production possibility set. And non-convexities, of course, violate the basic assumptions of the optimality theorems.

Radner's (1968) formal model dealt only with the case in which agents had fixed information structures. His informal remarks in that article, some of which are quoted in this chapter, went beyond that to suggest what might happen (and how Arrow-Debreu theory would have to be changed) if agents learned from prices and the actions of others.

Radner (1970) noted that the original Arrow-Debreu model assumes that all individuals have equal access to and the same information. Concerning information needed by market participants in the prior trading round(s) of the securities version of the Arrow-Debreu model, Radner observed

Although the second part of the price system might be interpreted as spot prices, it would be a mistake to think of the determination of the equilibrium values of these prices as being deferred in real time to the dates to which they refer. The definition of equilibrium requires that the agents have 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 price that will prevail at every future date and event. Radner (1970), p.456.

Radner's point about implied knowledge of spot market prices became the focus of the post-1975 Arrow-Debreu literature.

Radner (1982) identified a second line of criticism of Arrow-Debreu theory as inadequate treatment of money, the stock market, and active markets at every date. To correct these deficiencies Radner recommended that future extensions of the Arrow-Debreu model include 1) uncertainty about future prices as well as uncertainty about the environment; 2) a method for producers to compare net revenues at different dates and across states of the world; 3) consumers facing a sequence of budget constraints over time, rather than the single present net worth budget constraint of the Arrow-Debreu model; 4) speculation in future markets by storage, hedging, etc.; and 5) agents' attempts to forecast future prices based on information about both the environment and other market participants' behavior up to that point in time.

Radner's own work addressed some of these issues. Radner (1968) assumed that markets were complete but argued that some of these markets would be redundant and have no trading if agents' information structures were sufficiently different. Four years later, Radner (1972) provided a formal treatment of multiperiod incomplete markets, but agents were restricted from learning about the environment through prices. Finally, Radner (1979) studied what happens when agents are allowed to learn from prices, although he worked with a two-period model. These different information structures and corresponding equilibrium notions are clarified in Radner (1982).[6]

Another branch of the Arrow-Debreu literature questioned whether ex ante optimality or ex post optimality was the appropriate measure of efficiency.

As a practical matter, the achievement of an Arrow optimum is a normative dead end. After all, we are not so much interested in expectations as in results. Given an Arrow optimal distribution of contingent claims and supposing the occurrence of some event, we can then ask whether in that event the distribution of real goods resulting from the given distribution of contingent claims is a Pareto optimal distribution of real goods. If the answer is `no,' then it is comparatively small comfort to know that the economy had achieved an optimal allocation of risk bearing....the appropriate quality to seek is that there be no redistribution that will increase some trader's realized utility while decreasing no trader's realized utility. Such a situation will be termed an ex post Pareto optimum. Starr (1973), p.82.

For the pure exchange economy, Starr (1973) finds that Arrow's contingent claims equilibrium will be ex post Pareto optimal if and only if all of the market participants assign the same probability value to a given state s occurring. Starr refers to this property as `universally similar' beliefs.

For the case of production, Starr finds the Arrow-Debreu equilibrium will be ex post Pareto optimal under even more restrictive conditions. Market participants must have `universally similar' beliefs, and the prevailing contingent claim prices must be consistent with both universal similarity and profit-maximizing production. For both the pure exchange and the production economy, information about what state will occur is not particularly important for achieving ex post Pareto optimality in Starr's model. Pareto optimality results from the unanimity of traders' beliefs rather than their accuracy.[7]

Harris (1978) addressed the issues of (1) whether a decentralized resource allocation mechanism could be found such that ex ante choices result in an ex post optimal equilibrium, and (2) given an ex post efficient allocation, can an ex ante resource allocation mechanism be found to achieve that equilibrium solution? Recall that a Lindahl equilibrium achieves an efficient allocation of a public good by providing each individual with a specific price corresponding to the utility he receives from consuming that public good. Harris (1978) borrowed this concept to introduce a `Personalized Price Mechanism,' which turns out to be the product of the contingent claims market price times the individual's subjective probability for that state to occur. Thus, the personalized price of commodity c in state s for individual i is , using the notation of Section 2.1. `Compared to Lindahl prices, these `personal prices' are very special, since the relative prices of two goods to be delivered in the same state of the world are the same for all persons.' Harris (1978), p.430.

Harris starts by assuming (1) all states of nature are assigned positive probability by all consumers, (2) non-satiated consumers in all states of nature (follows from assumptions on concave, continuous, and strictly monotone utility functions), (3) additively-separable utility functions, and (4) a pure exchange economy. He then shows that his Personalized Price Mechanism will yield an ex post efficient allocation for a given state s, a `universally ex post efficient' allocation across every state, and an ex ante optimal allocation for each consumer's endowed probability beliefs. Conversely, by further assuming strictly positive consumption of goods and that all consumer utility functions are continuously differentiable, Harris shows a universally ex post efficient allocation can be achieved as the outcome of market trading with a Personalized Price Mechanism.

Grossman (1981) examined the nature of a rational expectations equilibrium (REE) in an Arrow-Debreu contingent claims economy with diverse information. A Walrasian equilibrium, in such an economy, will generally allocate resources differently than if each trader had access to all the information available in the market. Furthermore, traders will learn over time how market clearing prices relate to changes in underlying demand. Individuals will use this information to revise their demand schedules and want to retrade.

In the long run, prices will clear at a level at which no one desires to retrade. Grossman calls this latter solution a REE. In an economy with asymmetric information, the REE may yield an allocation that is identical to one in which all traders had full access to the information, but there is no guarantee. Grossman demonstrates that if the Arrow-Debreu markets are complete in the sense of spanning the entire range of the commodity-state space, and if traders have (1) additively separable, (2) non-satiable, (3) strictly concave, and (4) differentiable utility functions, then there exists a REE that is ex post Pareto optimal. Grossman (1981) characterized this finding as `a powerful extension of the fundamental theorem of welfare economics to economies with diverse information....However, the reader is cautioned that there may be multiple REE.' (p.555)

The distinction between (1) uncertainty and information about the environment, and (2) uncertainty and information about others' behavior or the outcome of as yet unperformed computations appears to be fundamental. The analyses of Arrow and Debreu deal with uncertainty about the environment. Radner (1968), p.32.

The revised state space incorporating future spot market prices creates new objections:

[T]here can be no uncertainty about 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. Arrow (1975), p.487.

The first objection is that treating prices as exogenous undermines the general equilibrium character of the Arrow-Debreu framework. If shocks to prices - rather than shifts in underlying demand and supply - are the focus of attention, then we are back in the realm of pre-modern partial equilibrium analysis. We should also note that Debreu's (1959) extension defined futures prices in terms of events, not vice versa.

Next, the problem identified by Nagatani, namely uncertainty over future spot market prices, is actually just one example of a class of potential dimensions of uncertainty affecting the Arrow-Debreu model. We call this class of problems `intrinsic uncertainty' in Chapter 4. Individuals in the Arrow-Debreu economy might reasonably also face uncertainty over possible (1) changes in preferences over time, (2) changes in beliefs stemming from new information, (3) the effects of `sunspots' on the equilibrium,[9] and (4) virtually any other object of uncertainty that individuals feel might influence other market participants. Complete contingent claims markets under such circumstances are impossible to create,[10] and all the individuals would likely never agree on how many relevant factors or variables must be accounted for in the contingent claims contracts. Anyone could dream up a new factor and say it is relevant.

Harris (1978) previously noted the problem with changing preferences in connection with the ex post optimality literature.

The conflict between ex post and ex ante Pareto efficiency of intertemporal resource allocation under uncertainty is an example of the problems caused by changing tastes. The problem has serious implications for making welfare judgments, as there may well be a divergence between ex ante choice and ex post preference. This (problem) casts doubt on the validity of the principle of consumer sovereignty as a means of evaluating resource allocations. Harris (1978), p.427.

A third problem relates to expanding the state space to eliminate uncertainty about changing preferences. Suppose arguendo that the state space also depicted consumer preferences as well. A moral hazard problem would then likely arise in that individuals would recognize their payoffs from alternative securities depend in part on their own preferences. Individuals who own securities paying off for a given value of their preferences would clearly benefit by changing their preferences to match that value. Similarly, they could cancel their liabilities by modifying preferences from those values of the state space that match the contingent securities they had sold.

Radner (1970) gives lack of information and moral hazard as two distinct reasons for the failure of some markets for contingent claims to exist. But in fact the latter is a special case of the former; if an insurance company could distinguish whether a fire was due to arson or not, it could pay in the latter case but not in the former. Thus moral hazard arises only because the insurance company cannot distinguish between two states of nature. Arrow (1970), p.463.

A fourth problem for the revised Arrow-Debreu economy is that individuals trading claims in a sequence of markets, and Debreu extended Arrow's word into a multiperiod model, would not know what state has been revealed until they witnessed the unfolding strategies of the other market participants. Prices in these sequential markets could follow any number of transient paths before arriving at the same final equilibrium value. Radner expressed this point as follows:

(Spot market prices) would depend, at a given date, on the evolution of the economy up to that date, including the evolution of the environment, both through direct observations of the environment...and indirectly through the decisions made up to that date...Unfortunately, in order correctly to infer something about the state of nature from the value of the new prices, an agent must in principle know the strategies used by other agents up to that date....In particular, an agent will no longer be able to assign a definite value to a strategy for given prices in the futures market. Radner (1968), p.35.

Other authors expressed this point somewhat differently:

A state of the world in this model is a complete specification of the physical environment and of spot market equilibrium prices as well, for all dates from the present to the end of the history of the economic system....[I]ndividuals will not know what state of the world has actually occurred until the history of the economic system is completed, hence there is no way that securities paying off on the basis of states of the world can be cashed in prior to that time, and hence no way that consumption plans can be implemented in the spot markets. It appears that incorporating spot market prices into the specification of states of the world leads to a restriction of the model to a two-period framework, today's security markets and tomorrow's spot markets and consumption. Burness, Cummings, and Quirk (1980), p.15.

Finally, from a theoretical viewpoint, the construction of the revised Arrow-Debreu economy - with subjective probabilities over possible spot market prices - drives an inappropriate nexus between Pareto optimality (a welfare concept) and particular institutions (that generate prices).

By including subjective probabilities as to equilibrium prices in the objective functions of consumers, and by using these objective functions in defining an ex ante optimum for the economy, the idea of an optimum has now become tied directly to a specific institution for allocating resources. How would one go about making a comparison between, say, a centrally planned allocation of resources and a competitive allocation with such a criterion? It seems clear that this is just an incorrect mixing of categories; from a descriptive or predictive point of view, beliefs of consumers as to equilibrium prices should be included in their objective functions, but from the point of view of welfare economics, they don't belong in the picture. Thus it seems that to the extent that future spot markets are to be active, the welfare results of the Arrow-Debreu model generally hold only because a flawed notion of ex ante optimality - one incorporating beliefs of consumers as to future spot prices - is employed. Burness, Cummings, and Quirk (1980), p.13.

cite as Michael A. S. Guth, "Arrow-Debreu Theory," Chapter 2 in Michael A. S. Guth, SPECULATIVE BEHAVIOR AND THE OPERATION OF COMPETITIVE MARKETS UNDER UNCERTAINTY, Avebury Ashgate Publishing, Aldorshot, England (1994), ISBN 1856289850.

Dr. Michael A. S. Guth, Ph.D., J.D. is a Professor of Financial Economics and Law for several universities with on-line degree programs and an attorney at law in Tennessee. He wrote his doctoral dissertation on topics in speculation theory, and this article on profitable destabilizing speculation is the first chapter in his published book, and one of his favorites.

In addition, Dr. Guth is a financial quant and former investment banker, having worked for Credit Suisse First Boston and Deutsche Bank in London and Frankfurt. He specializes in developing investment strategies and hedging techniques using derivatives. For five years, he consulted to the electric power and gas industry in the USA, even managing the Middle Office (financial risk control) function for two trading floors.

Dr. Guth has taught over 30 courses on-line at the undergraduate and graduate level on topics ranging from Managerial Economics to Strategic Management to Business Law. He can be reached through web page http://riskmgmt.biz/economist.htm