Arrow-Debreu Theory

Throughout the discussion of speculation and stability in Chapter 1, 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.

2.1   Review of Arrow's (1964) Contingent Claims and Securities Models

Any introduction to modern uncertainty theory in economics must begin by recalling two fundamental theorems from welfare economics.  The first states that assuming no externalities or non-convexities in production or consumption, then every competitive equilibrium is a Pareto optimum.  The second theorem holds that given an appropriate redistribution of resources, then, roughly speaking, every Pareto optimal allocation can be achieved as a competitive equilibrium.[1]  These theorems applied to an economy with no randomness or uncertainty.

    The contingent claims model enabled economists to extend the relationship between competitive equilibria and Pareto optimality to an economy operating under uncertainty.  So powerful a modeling tool was this conception that it provided the general framework for uncertainty theorists for the next forty years, spawned a new field of study known as contingent claims analysis in finance, and contributed to the award of Nobel prizes to two of its creators.[2]

    Arrow (1953, reprinted 1964) introduced uncertainty into the standard pure-exchange economy via a random variable designating the `state of nature.'  The Arrow-Debreu literature alternatively refers to this variable as a `state of the world.'  The first part of Arrow's article proposed a market with I individuals, C commodities, and S possible states of nature.  Prior to the realization of a state, individuals could buy and sell contingent claims, denoted , which entitled the owner to one unit of commodity c if state of nature s occurred.  In the earliest formulations of the model, the states represented physical conditions of the environment, e.g., rain or shine.  But the state variable soon became broadly interpreted to represent other exogenous forms of uncertainty a trader might face.[3]

    The basic feasibility constraint for the economy requires that the sum of the contingent claims equals the total stock of a commodity in that state of the world:

                                                                          (2.1)

With a complete set of contingent claims, i.e., one claim for each commodity in each possible state of nature, Arrow noted that the competitive economy operating under uncertainty was isomorphic to the standard pure exchange economy, with a couple exceptions.

    First, the total number of trading instruments had increased by a factor of S.  Whereas the standard pure exchange economy had C goods being traded, the contingent claims economy had  instruments.[4]  This point was relatively innocuous.  The second difference was that instead of maximizing utility from consumption across the range of C commodities, each individual i would maximize expected utility given by the product of his utility from consuming each commodity c and his subjective probability () for state s.  Arrow assumed each individual had preferences represented by the following utility function:

                         ,

where  is assumed to be a non-decreasing, concave function.  This assumption in turn implies that  is non-decreasing and quasi-concave. Expected utility maximization seemed like an innoc­uous change.  However, the change led to some controversy over what objects of uncertainty agents should have subjective probabilities.  This controversy continues to the present day.

    The sequence of events in Arrow's model begins with trading in the

 contingent claims.  Then a competitive equilibrium is achieved, trading stops, the state of nature is subsequently revealed, and only those contingent claims for the realized state  are executed.  Given the quasi-concavity of   and a set of positive weights , a central planner could maximize   subject to (2.1) and arrive at the optimal allocation  .  With a set of money incomes  for individual i and taking prices   for a claim to one unit of commodity c in state s as given, if the individual chooses the quantities  that maximize  subject to

                                            ,

then the chosen quantities  would be the optimal allocation .  Thus, the competitive equilibrium in the contingent claims market achieves a Pareto optimal allocation.  This result is Arrow's (1964) Theorem 1.

    In the second part of his article, Arrow formulated a securities market version of the contingent claims model, which introduced paper claims to money.  Each security s pays one dollar if state s occurs and zero otherwise.  Arrow assumes that there exist precisely S securities, whose S-dimensional payoff vectors are thus linearly independent.

    In the securities version of Arrow's model, individuals first purchase securities before the state has been revealed.  After the realization of a state, the individuals cash in their money claims and purchase commodities in a spot market.  With a complete market, individuals would allegedly need access to only S + C markets (down from   contingent claims in the first part of Arrow's article) to achieve the same Pareto optimal competitive equilibrium.

    To prove the equivalence of the contingent claims and securities market allocations, we shall follow the steps restated by Nagatani (1975), which are more fully developed than those in the original article.  The feasibility constraint in the securities model imposes a double condition on the income, Y_i, available to individual i.  Let  denote the price for security s,  the future spot market price for commodity c in state s, and   the quantity of commodity c purchased by individual i in state s.  Then technical feasibility requires

                                                ,                                        (2.2)

and

                                            .                                     (2.3)

The question is how are the prices  and  determined.

    Arrow chose the security prices to meet the condition

                                                 .                                         (2.4)

The values of  are precisely the contingent claims prices from the first part of Arrow's model.  Thus Arrow asserts that individuals facing these prices have the same range of alternatives in the market and, consequently, will acquire the same claims as in the first part of his article.  Each individual i will purchase the quantity

                                                                                      (2.5)

of security s.  The  must satisfy (2.2) and (2.3) above.  Furthermore, the total income in the economy must equal the sum of the individuals' income:   for all s.  After substituting this expression into (2.3), summing across individuals i, and combining it with (2.4); the security prices can be expressed as

                                            .                                     (2.6)

Arrow uses (2.6) and (2.4) to define  and , respectively.  As Nagatani (1975) and Radner (1970) pointed out, if the  were known to individuals in the Arrow-Debreu economy, then indeed there would exist a set of prices () such that the sequential securities trading followed by spot market trading would yield the same allocation .  This equivalent allocation result is Arrow's (1964) Theorem 2, perhaps the more important of the two theorems in his article.[5]

2.2  The `Early' Arrow-Debreu Literature, 1955 - 1975

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)

    Coutinho (1986) provided the complementary analysis to Grossman (1981):  he illustrated a REE under the same assumptions as Grossman (1981) that reflects only a portion of the available information and another REE that could be ex post Pareto dominated.  Since the REE plays an important role in subsequent chapters of this book and since Coutinho employs a particularly straight-forward Arrow-Debreu contingent claims model, we will retrace the key features of his examples.

    Consider a two-period contingent claims model with two possible states of the world, s₁ and s₂.  The market contains two consumers with identical preferences, which can be described by a von Neumann-Morgenstern utility function.  The endowments for consumers 1 and 2 across the two states of the world are e₁ = (0,1) and e₂ = (1,0), respectively.  Thus, if the state turns out to be s₁, consumer 1 has 0 units of the good and consumer 2 has 1 unit.

    Each consumer is informed about the result of one of two coin tosses.  Consumers know that if both coins come up heads or both come up tails, then the state will be s₁.  If one coin comes up heads and the other tails, then the state will be s₂; however, each consumer only knows the value of his own coin toss, not the other's toss.  After receiving this information, the consumers trade in a complete set of Arrow-Debreu contingent claims markets.  Let y_i denote the information signal received by consumer i.

    Coutinho (1986) defines a REE for this economy as a triple of vectors

 such that

with  and subject to  and    for i = 1, 2, and J = 1, 2.  To prove the existence of an equilibrium, Coutinho sets  which implies the consumers have Cobb-Douglas utility functions.  `It is a well-known result that an Arrow-Debreu economy where consumers have Cobb-Douglas utility functions has a unique equilibrium.'  (Coutinho (1986), p.884)

    The price vector p(y) = (1,1), for all y, coupled with demand functions x₁ = (1/2, 1/2) and x₂ = (1/2, 1/2), constitute an REE.  The price vector conveys no additional information about which of the two states is more likely, hence each consumer will continue to assign probability  for J, i = 1, 2.  In this equilibrium, each consumer insures against the uncertain outcome by allocating half of his endowment to each state of nature.  The equilibrium price vector reveals no information even though the economy as a whole has a conclusive signal about the state.  Given the price vector p(y) = (1,1), no consumer has any incentive to retrade if subsequent contingent claims trading markets were opened to him.  Thus Coutinho has illustrated a REE that imperfectly reveals information in the economy.

        According to Grossman (1977), this economy must have also a fully revealing equilibrium.  Indeed, consider now the following case:  p = (1,0) if , and p = (0,1) if   This is a fully revealing equilibrium price system since  =  and demand equals supply at each state of nature.  Coutinho (1986), p.884.

    Next, Coutinho illustrates a REE that is ex post Pareto dominated by another equilibrium.  First, let us assume the same structure in the economy as in the previous example, but with one exception.  Consumers can now chose a production technology that will allow them to select their endowment vector from the convex hull of (0,1), (1,0), and (1/2, 1/2).

    Facing the price vector (1,1), each consumer will chose an endowment that splits his income and risk across the two possible states:  (1/2, 1/2).  This strategy maximizes expected utility and yields an equilibrium.  However, a central planner could choose the production technology more efficiently.  If both coin tosses yielded the same result, the central planner could choose the production technology that leads to endowment vector  = (1,0) for i = 1, 2.  Similarly, if the coin tosses yield opposite results, the central planner could select technology leading to endowments  = (0,1) for both consumers.  In both instances, the central planner's allocation clearly dominates the competitive equilibrium allocation (1/2, 1/2).

    Coutinho has this to say about the optimality of these equilibria:

        It is interesting to note that while the non-revealing REE is ex post Pareto inferior to the fully revealing REE, it is ex ante Pareto superior to the fully revealing REE (in fact the non-revealing REE is ex ante Pareto optimal).  This is because ex ante (with) the realization of information, the non-revealing REE price vector p = (1,1) allocates income in a way that consumers fully insure each other.  Coutinho (1986), p.884.

    The appropriateness of either the ex ante or ex post optimality concept remains an open issue for discussion.  In addition to the works surveyed here, others who have contributed to this discussion include Dreze (1970) and Hammond (1976), who introduced the notion of ex post social welfare optimality.

    Articles in this literature that employed additively-separable utility functions follow the Arrow tradition of the Arrow-Debreu literature.  Debreu (1959), by contrast, specifies only the preference preorder of the  55 consumer, not his utility function.  `This preference preordering, reflects the tastes of the consumer for goods and services (including, in particular, their spatial and temporal specifications), his personal appraisal of the likelihoods of the various events, and his attitudes toward risk.'  Debreu (1959, p.101).  Endowed with this preference preordering and his wealth, an individual takes prices as given and chooses consumption bundles that optimize his preferences.

    As we shall see in Chapter 3, critics have focused upon the additively-separable utility function, which implies zero complementarity between the commodities, as a major objection to the contingent claims models of speculation.  Yet this assumption has significantly simplified the resulting first-order conditions.  Although it has yet to be explored in the speculation theory models, the preference preordering adopted by Debreu (1959) may yield a solution to the current modeling handicap.

2.3  The Revised Arrow-Debreu State Space

Extensions and comments on Arrow and Debreu's work in the early literature left the basic equilibrium solution and the equivalence theorem intact.  However, the situation changed when Nagatani (1975) raised a fundamental question about how individual traders could know future spot market prices in the securities version of the Arrow-Debreu model.  Recall that Arrow chose the security prices to meet the condition

                                                 .                                         (2.4)

As outside observers, we know that the  should equal the contingent claims market prices in the first part of Arrow's article.  Yet individuals trading in the securities version cannot be expected to know prices in a market that does not even exist.

    No mechanism in the securities version of Arrow's model transmits information about the value of  to these individuals trading securities in the prior round.  The lack of knowledge about  in turn means that individuals face uncertainty over what prices will prevail in the future spot markets once a given state of the world has been realized; without knowing , the individuals cannot compute  from (2.4).  This dimension of uncertainty creates risk and a source for speculation.  Absent securities that payoff according to the price in a given state of the world, individuals might seek inefficient allocations to offset real or perceived risks from price uncertainty.

    As Nagatani (1975) noted, individuals trading in the commodity claims version of Arrow's model reveal more information than they do in the securities market version.  In the former case each individual i reveals the whole SC vector , whereas in the securities version, he reveals only the S vector .  It is not surprising then that the securities market in turn contains less information to be inferred by any individual.  An agent will know how much income he will have in any state s, but he will not know at the time he purchases securities how much of commodity c he can afford to purchase in that state.

    Thus individual i would try to solve the following decision problem:

                                                      (2.7)

subject to income constraint .  In expression (2.7), E is an expectation operator taken over the subjective probability distribution of prices, , given state s, and  is individual i's demand as a function of the spot market prices he faces and his income.  Let  denote the solution to (2.7).  Nagatani stated the crux of the problem succinctly:

        For the given social endowments , these individual solutions would generally lead to the same allocation  if and only if  for all i, s.  But generally ...(and) some indi­vid­uals will have more money and others less relative to the allocation under perfect information.  Nagatani (1975), pp. 484-485.

    In a footnote Nagatani pointed out that for a Cobb-Douglas utility function, the proportion of income spent on any good is independent of prices.  For this one special case, .  But, in general, specula­tive considerations about future spot market prices will lead to sub-optimal trading allocations in the Arrow-Debreu securities market.

    Arrow's (1975) response offered two possible solutions to this dilemma.  First, the Arrow-Debreu world could be considered a succession of identical lotteries.  After a sufficiently long period of time, individuals would become familiar with what commodity prices prevail in a particular state of the world.  The mechanics of this identical lottery process appear cumbersome.  Every state of the world would have to be randomly selected multiple times so that the individuals could compare what prices had previously prevailed when that state occurred.  Even then individuals might face uncertainty over whether preferences or other features of the market have changed over time.

    Arrow's second, alternative response became the norm in the post-1975 Arrow-Debreu literature:  prices were defined as part of the state of the world.[8]  In fact, the state of the world evolved to a complete description of every conceivable dimension of uncertainty in the market.  But redefining the state space represented a major departure in the original formulation of the model.  As Radner observed

        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.

In the early Arrow-Debreu literature, the state of the world had only described the physical environment.

2.4  Problems with the Revised Version of Arrow's Model

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.