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.

        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.

2.5  Price Uncertainty in the Contingent Claims Economy

    George Feiger examined the impact of the scope of markets on speculative behavior in the contingent claims economy.  Commenting on Radner's (1968) paradoxical trading hypothesis, Feiger (1976) essentially restated Nagatani's objection to the Arrow-Debreu securities economy. 

        The resolution of the paradox represents both the strength and the weakness of the concept of complete markets.  For its arises only when the markets are incomplete in that one cannot insure against future spot market prices.  Thus, logical incompleteness makes it necessary to suppose that the contingencies allowed in the prior trading include all future spot prices.  If this is acceptable, then speculation will never take place under complete markets.  Feiger (1976), p.680.

Hirshleifer's (1976) analysis of Feiger's point with respect to a multiperiod contingent claims model of speculation led him to conclude the following:

        My contention was that price uncertainty was an endogenous, not an exogenous consideration, that the probability distribution of prices was the resultant of an underlying stochastic variability of the quantity magnitudes.  But my contention, to be strictly valid, rested upon the computability of price conditional upon the requisite beliefs or information about quantity.  I emphasized certain special cases (e.g., concordant beliefs) where computability was clearly possible....[W]ith less special and more `realistic' assumptions, computability is attenuated.  In the extreme we can imagine computability entirely replaced by free-floating beliefs; more or less `wild fancies' about prices then indeed become autonomous determinants of behavior and equilibrium.  Feiger (1976) suggests that the remedy, in principle, would be to provide prior contingent contracts for any commodity conditional not only upon state and message but also upon posterior price.  This would grotesquely enlarge dimensionality of the decision problem, unfortunately.  Hirshleifer (1976), pp. 695-696.

So a contemporaneous exchange between Feiger and Hirshleifer - on the contingent claims model - parallels the conclusions of the exchange between Nagatani and Arrow on the securities model.  Yet look how the definition of `complete markets' has expanded.

    In the original Arrow article, a complete market would require   contingent claims.  With Feiger's comment, the revised complete market would contain  markets, where  depicts a continuum of possible spot market prices in a given state of nature.  Finally, for an intertemporal investment economy with information events, a complete market must contain  markets, where  represents the set of possible messages that could be received.  Even for the case of conclusive information, so that , market completeness we would still require  contingent claims markets in advance of the information.  The previous problems identified in the last section - reduction to partial equilibrium analysis, uncertainty over changing preferences, moral hazard, collapse to two period model - resurface in the revised state space definition in the contingent claims setting as well.

2.6  The Stock Market Efficiency Literature

In demonstrating the ex ante optimality of the competitive equilibrium with production, Debreu introduced the notion of maximizing the value of a firm's shares and explained its equivalence to profit maximization for a world of complete markets:

        Given a price system p and a production , the profit of the j^th producer is .  Considering the price system as a datum, the j^th producer tries to maximize his profit in his production set.  For this he needs neither an appraisal (conscious or unconscious) of the likelihoods of the various events, nor an attitude toward risk.  His behavior amounts to maximizing the value of the stock outstanding of the j^th corporation.  In other words, the j^th corporation announces a production plan ; as a result, its share has a determined value on the stock market; it chooses its plan so as to maximize the value of its share.  Debreu (1959), p.100.

    Real financial markets obviously contain incomplete contingent claim markets.  A share in a firm entitles the owner to receive a proportion of the firm's earnings across every state of the world.  A share of a firm is a so-called `unconditional' contingent claims contract.  Incomplete markets imply potential risks cannot be effectively hedged with prevailing financial instruments; and consequently, the competitive equilibrium allocation would be expected to be sub-optimal.  To accommodate this shortfall, Diamond (1967) introduced the notion of `constrained optimality':  Pareto optimality relative to the set of allocations that can be achieved through existing market structures.  In a one-good, two-period economy, Diamond showed that a competitive equilibrium will be a constrained Pareto optimum.

    Diamond contrasted the competitive allocation resulting from firms maximizing their market value with the allocation by the government under restricted conditions:  any taxes, subsidies, or other form of redistribution could not be state-dependent.  Each individual received a payoff from the profits of the firm according to the number of shares he owned.  Assuming constant returns to scale (so that the ratio of output in any two states is independent of scale) and each firm believes that its market value is proportional to its production scale, then firms operating to maximize their market values achieved a constrained Pareto optimum.

    However, Hart (1975) showed that Diamond's result holds with little generality; with two or more goods, or three or more periods, the stock market economy fails to achieve even constrained optimality.  Furthermore, an equilibrium need not exist.  For particular values of the exogenous parameters, Hart's generalized stock market model contained multiple equilibria, so that a given equilibrium could be Pareto dominated by another.  Starrett (1973) also found that transactions costs in Diamond's model could prevent efficient allocations.

    Hart's multiple equilibria result might be called a `structural inefficiency' in the stock market.  Stiglitz (1972) had also found structural inefficiencies in stock market allocations using the Capital Asset Pricing Model.  In Stiglitz (1982), he focused on `marginal inefficiencies':  the private market incorrectly decides how to invest at the margin and would do so even if there were a unique equilibrium.  Hart (1975) coined the phrase `strong optimality' to define an allocation with no marginal inefficiencies.  Like Stiglitz, Hart found an equilibrium may be strongly sub-optimal even if it is unique.

    Stiglitz summarizes his marginal inefficiency argument as follows:

        With a complete set of risks markets, we know we wish to equalize the marginal rates of substitution between any two states for all individuals.  With an incomplete set of markets, we cannot do this, but we may be able to have a more `efficient' distribution of risks (come closer to equalizing, on average, the marginal rates of substitution) if we change the price distribution (and thus the `profit distribution') associated with the risky asset.  The government recognizes that it can change this price distribution by altering the allocation of investment and the ownership of shares in the different assets.  The market ignores this effect.  Stiglitz (1982), p.242.

In essence, Stiglitz argues that while individual firms exhibit price taking behavior and perceive that their profits remain constant across any two states of nature even if they increase output, profits and prices for industries as a whole do change.  A central planner can take advantage of this `price distribution effect' to rearrange ownership of shares and the productive capital available to firms for expansion and thus create a Pareto superior allocation.

    In Stiglitz's (1982) model, individuals attempt to maximize the payoff from holding a portfolio of stocks; he shows the portfolio holdings will generally be inefficient.  All of the firms in his model produce a single good, and all have constant returns to scale.  A related model by Loong and Zeckhauser (1983) examined the effects of pecuniary externalities in incomplete contingent claims markets.

        Since the firms in (Stiglitz's (1982) model) have constant returns to scale, his example can be interpreted to refer to individuals who decide what to produce themselves, rather than individuals who invest in stock.  This makes Stiglitz's example directly comparable to ours.  The essential remaining difference is that we are explicitly concerned with individuals who have a choice of alternative technologies for producing the same good, and we show that under conditions of uncertainty they will often make inefficient choices.  Loong and Zeckhauser (1983), p.173.

Using a two-period, two-good model with two representative classes of individuals, Loong and Zeckhauser show that individuals will generally undertake inefficient and sometimes overly risky production decisions - even when compared against the constrained Pareto optimum concept of Diamond.  The source of the inefficiency rests with (1) technological externalities in the production of goods and (2) differences in marginal rates of substitution for the two goods between the two classes of individuals.  Bankruptcy introduces another form of non-convexity in production that can create optimality problems for a stock market economy.

2.7  Conclusions

The pioneering contributions of Arrow and Debreu have forever changed the way economic theorists formulate uncertainty models.  After more than forty years of scrutiny and extensions, their general equilibrium framework and approach continues to be the starting point for new theories on the operation of competitive markets under uncertainty.

    While the general approach has been widely endorsed, the substantive results of the Arrow-Debreu have been sharply restricted.  Following the exchange between Arrow and Nagatani, the optimality results may be restated as follows.  The competitive equilibrium of the two-period contingent claims economy achieves an ex ante Pareto optimum, and every contingent claims competitive equilibrium can be achieved, roughly speaking, with an appropriate redistribution of resources.  The two-period securities model and multiperiod versions of either the contingent claims or the securities models achieve optimality only in the flawed sense identified in Section 2.4.

    The optimality results of the original models were derived under certain idealizing assumptions about individual preferences, production, the information available to all market participants, and the scope of markets.  Grossman, Starr, Harris, and others, relaxed some of these simplifying assumptions and developed alternative optimality results based on variations of the original model.  Other authors such as Radner, Nagatani, and Feiger raised fundamental questions about the validity of the optimality results even under the restrictive conditions of the Arrow-Debreu world.  These authors uncovered various internal inconsistencies in the treatment of uncertainty faced by individual agents in these models.

    It is surprising how little the economics profession has understood these latter set of inconsistency results.  In university after university, Arrow-Debreu theory continues to be taught as a perfectly consistent paradigm for uncertainty economics.  We also find continued erroneous assertions about no trading in multiperiod contingent claims models, when these assertions apply to a two-period Arrow-Debreu model.

    For example, economists will assert that if contracts contingent on each state of nature were allowed for each commodity, speculation would not occur.  They will argue Arrow, among others, has shown that no one could gain from spot market trading if all traders had been previously able to trade in complete markets.  Moreover, they would boast that no one would trade even if inconclusive information arrived in the economy.[11]  The Milgrom and Stokey (1982) No Trading Theorem is a rational expectations variant of this same complete markets concept.

    Yet we know from Radner's work on sequential equilibria where traders do not have the same information that `complete' markets will not necessarily preclude subsequent rounds of trading.

        Suppose, however, that new markets were introduced at later dates; would there be any incentive to trade in these new markets?  In general there would, because the equilibrium prices in such markets would convey additional information beyond that contemplated in the original structure of information....The introduction of the spot markets brings with it the need for economic agents to be concerned not only with uncertainty about the environment, but also with uncertainty about other agents' strategies....Therefore the announcement of (spot prices) at the beginning of date T would typically provide each agent with (additional) information....to the extent that he could guess the strategies (or acts) of the other agents.  Radner (1968), p.35, p.55. 

    At present an open question remains as to the extent of speculation in a fully complete and perfect contingent claims regime.  Many economic theorists would argue that no speculation will take place.  It is unclear how many of these theorists are aware of the dimensions of uncertainty (future spot market prices, changing preferences, changing beliefs) inherent in the Arrow-Debreu model with the passage of time.[12]  Aside from these risks, individuals may still attempt to capture capital gains from others who do not share their probability beliefs over states of nature.  From the work of Harris (1978) and Starr (1973), we would expect that only multiperiod models where traders share the same beliefs would exhibit no subsequent trading rounds, but even these possible results would have to be limited by a ceteris paribus argument (e.g., holding preferences constant).  We will explore these issues in Chapter 3.

    This discussion of trader information, objects of speculation, and the scope of markets leads naturally to the development of the Hirshleifer contingent claims model of speculation.  This general equilibrium model is the focus of the next chapter.

Notes

1. Arrow (1951) provided a rigorous proof of the connection between competitive equilibria and Pareto optima.  Gerard Debreu, independently of K. J. Arrow (1951), introduced convex analysis methods into welfare theory.  See Gerard Debreu (1951), in particular Section 6, and W. Hildenbrand (1983), in particular Section 2.

2. Gerard Debreu (1959) extended Arrow's pure exchange model in several important ways, hence the name `Arrow-Debreu' to describe the contingent claims and securities economies.  Debreu added production and multiple periods, and his proofs demonstrated the importance of convexity, preordering or ranking of the consumption set, and other key assumptions of the model.  The complete citation for the 1972 Nobel Memorial Prize in Economics to J. R. Hicks and K. J. Arrow is `for their pioneering contributions to general economic equilibrium theory and welfare theory.'  The complete citation for the 1983 Nobel award to G. Debreu is `for having incorporated new analytical methods into economic theory and for his rigorous reformulation of the theory of general equilibrium.'

3. Aumann (1987) expresses the position that every object of uncertainty should be defined as part of the state space.

        The term `state of the world' implies a definite specification of all parameters that may be the object of uncertainty on the part of any player...Conditional on a given , everybody knows everything; but in general, nobody knows which is really the true .  Taking the `atoms' of  to represent all aspects of uncertainty on the part of any player - including uncertainty about the uncertainty of other players - by means of the partitions.  Aumann (1987), p.6.

4. Debreu (1959) subsumes the  contingent claims into his definition of a commodity under uncertainty:

        One is thus led to define a commodity in this new context by its physical characteristics, its location, and its event (or vertex of the event tree; this vertex defining implicitly the date of the commodity).  A contract for delivery of wheat between two agents takes, for example, the form:  the first agent shall deliver to the second agent, who shall accept delivery, five thousand bushels of what of a specified type at location s, at event .  If  does not obtain, no delivery takes place....Therefore the definition of an uncertain commodity may require here several events (and several locations)....An agent who buys a bushel of No. 2 Red Winter Wheat in Chicago at date t in any event buys in facts as many commodities as there are events at t.  Debreu (1959), pp.99-100.

5. Arrow's (1964) article is entitled `The Role of Securities in the Optimal Allocation of Risk-Bearing,' and emphasis seems to have been placed on the securities version of his model.

6. Most of these ideas are contained in Radner's (1967) French article.

7. Starr (1973), p.94.

8. Many authors subsequently developed revised general equilibrium models, in which prices are defined as part of the state of the world, without citing Nagatani's (1975) comment as the impetus for this salient change in the established Arrow-Debreu framework.

9. See Cass and Shell (1983).

10.    Loong and Zeckhauser (1982) note the nonexistence problem for contingent claims markets as follows:

        If there were some measure of performance for market failures that was in some way equivalent to a batting average, contingent claims markets might well lead the league.  That is, the ratio of non-established contingent claims markets to all desirable contingent claims markets is high in relation to, say, the ratio of public goods relative to all goods, or goods generating nontrivial externalities relative to private goods.  (p.171). 

        Chapter 6 contains several nonexistence propositions for fully `complete' markets.

11.    Salant (1976), p.674.

12.    In private correspondence with me in 1994, Radner emphasized that in a multiperiod model if (1) all traders have the same information at each date and in each event, (2) traders observe all `payoff relevant' events when they occur, and (3) markets completely span the elementary events that traders observe, then in an equilibrium of the prior round of Arrow-Debreu contingent claims economy, no one will want to reopen trading at subsequent dates.  Radner's carefully worded assumptions eliminate future spot market price uncertainty as well as uncertainty over changing beliefs or preferences, which are `payoff relevant.'  Radner's Assumption (2) also implies that no firm has uncertainty about the profitability of any given production plan.

References

Arrow, Kenneth J., (1951), `An Extension of the Basic Theorem of Classical Welfare Economics,' in Neyman, J., editor, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (Berkeley:  University of California Press), pp. 507-532.

Arrow, Kenneth J., (1964), `The Role of Securities in the Optimal Alloca­tion of Risk-Bearing,' Review of Economic Studies, Vol. 31, pp. 91-96.

Arrow, Kenneth J., (1970), untitled comment, American Economic Review, Vol. 60, pp. 462-463.

Arrow, Kenneth J., (1975), `On a Theorem of Arrow:  Comment,'        Review of Economic Studies, Vol. 42, pp. 487-488.

Aumann, Robert J., (1987), `Correlated Equilibrium as an Expression of Bayesian Rationality,' Econometrica, Vol. 55:1, pp. 1-18.

Burness, Stuart, Ronald Cummings, and James Quirk, (1980), `Speculative Behavior and the Operation of Competitive Markets Under Uncertainty,' Staff Paper 80-11, Department of Economics, Montana State University, Bozeman, Montana.

Cass, David and Karl Shell, (1983), `Do Sunspots Matter?,' Journal of          Political Economy, Vol. 91:2, pp. 193-227.

Coutinho, Paulo C., (1986), `Non-Optimality of Rational Expectations Equilibrium:  the Complete Markets Case,' Review of Economic Studies, 53, pp. 883-884.

Debreu, Gerard, (1951), Econometrica, Vol. 19, pp. 273-292.

Debreu, Gerard, (1959), Theory of Value, (New York:  John Wiley & Sons, Inc.), Chapter 7.

Dreze, J., (1970), `Market Allocation Under Uncertainty,' European Economic Review, Vol. 2, pp. 133-165.

Diamond, Peter A., (1967), `The Role of a Stock Market in a General Equilibrium Model with Technological Uncertainty,' American Economic Review, Vol. 57, pp. 759-776.

Feiger, George, (1976), `What is Speculation?,' Quarterly Journal of  Economics, Vol. 90, pp. 677-687.

Grossman, Sanford J., (1977), `The Existence of Futures Markets, Noisy Rational Expectations, and Informational Externalities,' Review of Economic Studies, Vol. 64, pp. 431-449.

Grossman, Sanford J., (1981), `An Introduction to the Theory of Rational Expectations Under Asymmetric Information,' Review of Economic Studies, Vol. 48, pp. 541-559.

Hammond, Peter J., (1976) `Ex Ante and Ex Post Welfare Optimality Under Uncertainty,' Essex University Discussion Paper No. 83.

Harris, Richard, (1978), `Ex-Post Efficiency and Resource Allocation Under Uncertainty,' Review of Economic Studies, Vol. 45, pp. 427-436.

Hart, Oliver D., (1975), `On the Optimality of Equilibrium When the Market Structure Is Incomplete,' Journal of Economic Theory, Vol. 11, pp. 418-443.

Hildenbrand, Werner, (1983), `Introduction,' in Gerard Debreu, Mathematical Economics, Cambridge University Press, pp. 1 -29.

Hirshleifer, Jack, (1976), `Reply to Comments,' Quarterly Journal of Economics, Vol. 90, pp. 689-96.

Loong, Lee Hsien, and Richard Zeckhauser, (1982), `Pecuniary Externalities Do Matter When Contingent Claims Markets Are Incomplete,' Quarterly Journal of Economics, Vol. 97, pp. 171-180.

Milgrom, Paul, and Nancy Stokey, (1982), `Information, Trade, and Common Knowledge,' Journal of Economic Theory, Vol. 26, pp. 17-27.

Nagatani, Keizo, (1975), `On a Theorem of Arrow,' Review of Economic Studies, Vol. 42, pp. 483-485.

Radner, Roy, (1967), `Equilibre des Marchés à Terme et au Comptant en Cas d'Incertitude,' Cahiers d'Econometrie, Vol. 9, pp. 30-47.

Radner, Roy, (1968), `Competitive Equilibrium Under Uncertainty,' Economet­r­ica, Vol. 36, pp. 31-58.

Radner, Roy, (1970), `Problems in the Theory of Markets Under Uncertainty,' American Economic Review, Vol. 60, pp. 454-460.

Radner, Roy, (1972), `Existence of Equilibrium in Plans, Prices, and Price Expectations in a Sequence of Markets,' Econometrica, Vol. 40:2, pp. 289-303.

Radner, Roy, (1979), `Rational Expectations Equilibrium:  Generic Existence and the Information Revealed by Price,' Econometrica, Vol. 47:3, pp. 655-678.

Radner, Roy, (1982), `Equilibrium Under Uncertainty,' in Arrow, K. J., and M. D. Intrilligator, eds., Handbook of Mathematical Economics, Vol. II, Chapter 20, (North Holland, Inc.).

Salant, S., (1976), `Hirshleifer on Speculation,' Quarterly Journal of Economics, Vol. 90, pp. 667-675.

Starr, Ross M., (1973), `Optimal Production and Allocation Under Uncertainty,' Quarterly Journal of Economics, Vol. 87, pp. 81-95.

Starrett, David, (1973), `Inefficiency and the Demand for Money in a Sequence Economy,' Review of Economic Studies, Vol. 40, pp. 437-448.

Stiglitz, Joseph E., (1972), `On the Optimality of the Stock Market Allocation of Investment,' Quarterly Journal of Economics, Vol. 86, pp.25-60.

Stiglitz, Joseph E., (1982), `The Inefficiency of the Stock Market Equilibrium,' Review of Economic Studies, Vol. 49, pp. 241-261.

Comment

I am happy to take up Michael Guth's invitation to comment on this chapter, a beautifully clear survey of state-preference theory.

    As background for the issue to be addressed, let me point to an informational paradox in standard micro theory.  Individuals supposedly optimize their consumption choices in the light of known commodity prices.  Yet these prices are themselves endogenous variables determined by the aggregate of such individual decisions, hence unknowable to traders at the time the decisions are made.  So prices are known, yet unknown!

    In our textbooks we deal with the seeming contradiction on two levels.  First, we say the traditional intersection of supply and demand generates a `static' equilibrium C or, in another terminology, a Nash solution.  Such an equilibrium is non-constructive, meaning we do not attempt to examine the process by which it is achieved.  It is an equilibrium in the sense that everyone is making a best response to circumstances; given objective conditions and other traders' choices, no one wants to modify his/her choices.  Second, if a constructive solution is wanted, we have to concern ourselves with `dynamics.'  In various versions we talk about auctioneers, recontract assumptions, cobwebs, optimal search theorems, etc.  For certain problems it is absolutely necessary to deal with dynamics.  Yet on the whole we have a well-justified confidence that the simpler static analysis very often successfully predicts economic behavior.  I will only add that the exciting field of experimental economics has shed considerable light upon the market institutions and dynamic protocols needed to support our textbook static solutions.

    Turning now to state-preference theory, I shall deal separately with Arrow's two models:  (1) the `state-claim markets' version, and (2) the `security markets' version.

    State-claim markets:  As pointed out in the chapter, Arrow's first version is a straightforward generalization of standard theory.  Instead of individuals choosing over C commodities subject to prices , they now choose over a space of  contingent commodities subject to prices .  Lurking within this generalization is the same paradox as before:  prices are assumed known, yet unknown.  But, I would claim, if we are looking only for `static' solutions, there is no more paradox than in standard micro theory.  The only issue is how successful this extension of standard static theory will be in describing the real world.

    Some theorists have, however, been disturbed by the following question.  Suppose markets were to re-open after determination of the state.  There would be  markets in the first round and C in the second.  But then beliefs about prices in the second round C let us denote these prices  C would affect first-round choices.  Hence, it has been suggested the possible second-round prices should themselves be among the contingencies priced in the opening round.  Thus, first-round markets would  generate price-quantity solutions over a space of

 tradable entities.

    I have been unsympathetic to this extension.  No so much because of unrealism, though I'm unaware of any real-world phenomenon that even approximates such tradable entities.  My main objection is to the discordant mixture of static and dynamic considerations.  Static analysis provides only a best response equilibrium.  Subjective preferences and beliefs, about re-opening of markets or whatever, are already incorporated into individuals' supply-demand offers in the original market.  At equilibrium no one is motivated to change his position, and nothing more needs to be said so long as we remain in the domain of statics.  To go further we must explicitly model the dynamics:  how are prior expectations formed, how are they modified by one's own experience, how does one go about making optimal inferences from the observed behavior of others, and so forth.  Expanding the space of tradable entities is not needed for statics, and provides no aid for dealing with dynamics.

    Securities markets:  Arrow's second version allows for two stages.  In the first round, S prices for claims to income in each state are determined; in the second round, after everyone knows which state has come about, C commodity prices are determined.  In this model only  rather than the previous  markets are required.

    Since second-round trading is essential to the model here, beliefs about posterior prices seemingly pose a more serious problem.  In the prior round when the state-prices  are determined, the second round commodity prices  are strictly unknown C rather than `known-yet-unknown' as in our familiar paradox.  In fact, not only are these   prices invisible in the first round, but only the one subset of them conditional upon the actually realized state s will ever become visible.

    Nevertheless, so long as we remain within the context of static equilibrium, a slight modification resolves the problem.  If a 2-round set of prices C  in the initial round and  in the succeeding round C is proposed ex ante to all individuals, and an associated set of allocations is arrived at such that no one wishes to make further changes, we have an equilibrium.  (Subject to a number of technical conditions not essentially different from those needed even for standard micro theory.)  These 2-round prices are known-yet-unknown in exactly the same sense as before.  And in particular, the posterior prices are nothing but the `rational expectations' prices arrived at in another branch of economic literature.  (There may be more than one such equilibrium, but that also is a possibility in standard micro theory.)  So once again, expansion of the space of tradable entities to include second-round prices as contingencies is not needed for a static equilibrium, and is unhelpful if we need to deal with dynamics.

    Returning to the static equilibrium, the important question is:  `What real world inferences can be drawn from it, and are those inferences actually observed?'  And indeed, until convincing evidence is obtained we are entitled to remain somewhat skeptical about rational expectations predictions.

    The only additional point I want to emphasize here is that, to apply the theory, we do not need to solve it in full generality.  More or less rough approximations are all that can ever be achieved.  In standard theory the argument against protective tariffs, with which the vast majority of economists certainly concur, rests upon such a rough-approximation theoretical development.  In a similar spirit we can use approximate solutions from state-preference theory to cast light upon many issues of practical concern.  For example:  (1) Under what circumstances are futures markets for a particular good likely to come into existence, and when not?  (Note that a regime of futures and spot markets for commodities corresponds to a  rather than to Arrow's   pattern.  Thus, state-beliefs have to somehow be translated into unconditional supply-demand offers for particular goods.)  (2) As a related question, given such a  regime, to what extent do individuals' speculative/hedging choices depend upon factors like differential risk aversion versus differential beliefs?  (This question is to be addressed in the chapter following.)  And finally, going somewhat farther afield, various rough-approximation versions of state-preference theory have been shown to cast light upon broader topics like (3) security prices (the CAPM model), (4) firms' balance sheet choices between debt versus equity, and (5) the demand for financial liquidity.

Jack Hirshleifer

Department of Economics

University of California, Los Angeles