The Contingent Claims Model of Speculation

For the modern reader, the literature on profitable destabilizing speculation as a whole appears confusing and contradictory.  Many of the early works in the 1950s and 1960s employed models which simply failed to come to grips adequately with a complex phenomenon like speculation.  For example, speculation can only occur under uncertainty.  Yet many of the early works contained models with no randomness.  In a world of certainty, profits from purchases and sales at a known price illustrate arbitrage not speculation.

    Financial economists now have a general equilibrium, contingent claims model to illustrate speculation.  This model solves comparative static issues via standard economic reasoning.  The standard format provides individuals with production functions, preferences, time-distributed endowments, etc., and specifies the available market range.  Economists then solve for the general equilibrium and pose comparative static questions, e.g., what happens if preferences change?

    In the Hirshleifer (1975, 1976, 1977) model, investors buy Arrow-Debreu contingent claims knowing that they will have the opportunity to revise their portfolio after new information arrives and before the state of the world is revealed.  In presenting Hirshleifer's model, we benefit from hindsight.  A zero complementarity assumption in the original model led to several objections.  We will introduce the model without the assumption, show the first-order conditions, and then impose the assumption to simplify the expressions.

    Circa 1975 speculation theorists included, but were not limited to, two competing groups.  The first group maintained in the Keynes-Hicks tradition that different tolerances for risk largely explained who would hedge and who would speculate.  In the Keynes-Hicks view, futures markets serve the beneficial social role of transferring price risks from those more risk averse (hedgers) to those less risk averse (speculators).  The opposing group, led by the empirical writings of Holbrook Working, contended that different beliefs (subjective probability values) primarily determined who chose to speculate.  Working regarded speculation as the means by which market prices come to reflect the best information among market participants.

    Jack Hirshleifer (1975) initially slanted his work toward providing evidence for the Working view of speculation:  differences in beliefs account for speculation.  For example, his article illustrates how prices would change if unexpected information arrives and causes beliefs to change.  He also attempted to show that only a `belief deviant' individual would speculate.  But Hirshleifer's (1976, 1989) subsequent replies to comments expanded on the impact of other significant factors including the regime/completeness of markets; risk preferences; the nature of information and endowments; and, due to subsequent work by his son David Hirshleifer (1989a, 1989b, 1990, 1991), demand elasticities and transaction costs.

    Hirshleifer's valuable framework reflects the importance of (1) planning based on the arrival of new information, (2) multiple trading periods C so that traders have time to buy (sell) and resell (repurchase), (3) subjective probability distributions over shifts in underlying demand or supply;[1] and (4) market incompleteness.  As we describe in Sections 3.4 and 3.5, Hirshleifer's model represents a philosophy about looking at speculative activity:  identifying the factors most important to speculation, some of which are missing even from recent attempts to portray speculation.  Perhaps this chapter will rekindle interest his contingent claims model of speculation.  If so, the general theory of speculation will benefit.

3.2  Hirshleifer's (1975) Model Specification

Consider a two-commodity pure exchange economy with a risky commodity, Z, and a riskless commodity, N.  The total quantity of `riskless' commodity N is fixed and consequently does not depend on the state of the world.  The economy begins in period 0 and ends in period 2, at which time either state A or state B is revealed.  The total quantity of commodity Z will depend on the state of the world; states A and B can be thought of as a `good crop' and `bad crop' scenario, respectively.  Let some individual believe state A will occur with probability p and thus state B with probability 1-p.  This individual's endowment vector can then be represented as

                         ,                  (3.1)

and his expected utility as

                             .                     (3.2)

We can treat the riskless commodity, N, as a numeraire and set its price equal to 1:  .  The market regime is `semicomplete' in that state-contingent claims are available on asset Z  but not on asset N.  Hence, the available contingent claims have prices .  If state-contingent claims  and  were available on N as well, the market regime would be fully complete.

3.2.1.  The Noninformed Equilibrium

Consider now the following sequence of events.  First, the individual trades from his endowment, E, to a consumptive optimum, , based on his preferences, beliefs, and budget constraint.  After the trading has been completed, state A or state B is revealed.  The state contingent claims pay off according to which state of the world occurs.  Graphically, the sequence of events appears as

Endowment                Trading                      Consumptive                   Gamble E                               (period 0)               Gamble C ^*

Nature's Choice of                    Consumption

state A or state B                           (period 2)

(period 1)

    In this pure exchange economy, the individual faces the budget constraint

                            .                     (3.3)

When the expected utility function (3.2) is maximized with respect to  and  and subject to (3.3), the consumptive optimality conditions are

           and   .                              (3.4)

where    We can eliminate the  term from (3.4) by assuming the individual's utility function is additively-separable:    Recall this assumption appeared in numerous extensions of the Arrow-Debreu model reviewed in Chapter 2.  Additive separability implies the individual derives no complementarity from consuming the two goods.  If the two goods are food and gold, the zero complementarity assumption seems reason­able.  If the two goods are different kinds of food, then some complementarity effects would be expected.

    With zero complementarity, conditions (3.4) reduce to

                             and  .                     (3.5)

Conditions (3.5) together with the budget constraint (3.3) determine what quantity of commodities N and Z the individual will purchase.  But the right-hand side of the equations in (3.5) still contain the market prices, which any individual will take as given under the competitive assumptions of this model.

    The prevailing market prices cannot be determined from (3.5), because we do not know how many other market participants share this individual's endowments, preferences (utility function), and beliefs (probability assignments).  We have arrived at the fundamental issue confronting all asset pricing theory and the central theme of Chapter 5:  how can individual equilibrium conditions be translated into an aggregate, market-clearing equilibrium model?  Hirshleifer (1975) used the heuristic device of a `representative individual,' i.e., all the people in the market are so similar that any one individual's beliefs and preferences could be assumed to reflect the market's as well.  As Hirshleifer noted,

        In a world of pure exchange the equilibrium prices are then governed by the necessity of `sustaining' the representative endowment vector; since a representative individual can find no one to trade with in a closed economy, his endowment vector must be his consumptive optimum....Without a representative individual, equilibrium prices would of course reflect some average measure of the various individuals' differing probability beliefs and some average measure of their respective marginal utilities.  Hirshleifer (1975), p. 527.

If  denotes the market's probability assigned to state A occurring and   denotes the market's collective utility function, then market clearing prices can be determined from the optimality conditions

                             and  .                    (3.6)

    Assuming zero complementarity and the arrival of unexpected infor­mation that changes the representative probability beliefs from () to (), the new market clearing prices  and  will adjust according to the following ratios:

                          and  .                (3.7)

Note that (3.7) contains no marginal utility functions.  Hirshleifer observed that condition (3.7) will hold even if the `representative individual' assumption is relaxed.  In its place, he assumes individuals have homogeneous or `concordant beliefs.'  His logic then holds that if all of the individual probability assignments, p, equaled   before the new information and now equal , then condition (3.7) still holds.  Of course, while beliefs change, we must assume that preferences and endowments are held constant.

3.2.2  The Informed Equilibrium

Suppose the following sequence of events now occurs.  Individuals trade from their endowment position to a consumptive optimum, knowing that new information will arrive that will change probability beliefs.  After the information event, individuals can retrade their contingent claims. Following the second round of trading, the state of the world will be revealed as either state A or state B.  Traders will then consume their commodities.  Graphically, the new sequence of events prior to the state of the world being revealed is

Endowment                Prior Trading               Speculative

Gamble E                       Round                              Gamble T ^*

                                      (period 0)

Arrival of                        Posterior                 Consumptive

Information                Trading Round             Gamble C ^**

(period 1)                       (period 2)

    The complexity of the individual's decision problem has now increased substantially, because in selecting his prior trading consumptive optimum, he must not only take into account the prices that prevail in the market at that time, but also the prices he anticipates will prevail in the posterior trading round as well.  To simplify the analysis further, Hirshleifer assumes that the information that arrives will be decisive, so that every individual will know with certainty what the state of the world will be.  Thus in the posterior trading round, the probability assigned to either state A or state B will drop to zero.  Consequently, the prices of contingent claims for that state will drop to zero as well.  Thus the budget constraint in the prior trading round will contain both state A and state B contingent prices, whereas the posterior trading round budget constraint will only contain the actual state's prices.

    In the prior trading round, the individual faces the following dynamic optimization problem:

                                                       (3.8)

subject to

                                                               (3.9)

The single primed notation reflects the information that the state of the world will be A, and the double primed notation applies to state B.  The first two budget constraints indicate that the individual's wealth in period 2 (when the state is revealed to be either state A or B) must equal the value of his prior round trading position.  The third budget constraint shows that the value of the prior trading round (with superscript t) must equal the value of the individual's endowed wealth (with superscript e).

    One prominent feature emerging from this informed environment relates to prices in the uninformed environment:   and .  The prior round prices in the informed environment match the prevailing prices in the uninformed environment.  Hirshleifer (1976) shows that this key result does not depend on the zero complementarity assumption.  However, individuals must be able to compute future prices, a subject we will return to in Section 3.2.4.

3.2.3  Speculation Defined

Hirshleifer defined `speculation' as any portfolio position T ^* in the informed environment that differed from the consumptive optimum C ^* in the noninformed environment.  However, in a regime of incomplete markets, essentially everyone would speculate once the zero complementarity assumption is relaxed. 

    In earlier work, Hirshleifer and Rubinstein (1973) examined the operation of a single commodity market with a state-complete regime as well as for information event-state-complete markets.  Obviously, with a single commodity, no complementarity effects will complicate the first-order conditions.  For this model, Hirshleifer and Rubinstein defined a speculator as someone who plans for differential contingent consumptions over information events.

    Speculation in this earlier model does not depend on portfolio revision but does require a capital gains motive or plan.  If information unexpectedly arrived in the market, prices would normally adjust and some traders would revise their portfolios.  However, these transactions would not amount to speculation, since potential capital gains did not motivate any prior behavior.  The only individuals who met the nonspeculator definition had ratios of conditional posterior probabilities equal to the corresponding ratio of conditional posterior prices.  The nonspeculator definition in this earlier model thus depends only on beliefs and prices, not on resources or preferences.

    Similarly, Rubinstein (1975) found if all individuals have homogeneous beliefs in a single commodity contingent claims economy, then all would have `nonspeculative' beliefs.  In a heterogeneous beliefs version of the same economy, those individuals who happened to match the consensus beliefs would be `nonspeculators.'  These consensus beliefs are more complicated than just the average of individual beliefs, because additional information is imparted in computing the average.  Joseph Stiglitz has similarly encountered problems in defining speculation.

        Stiglitz (1970) also identifies speculation with portfolio revision....  Stiglitz introduces uncertainty by assuming the date t=1 price of long-term bonds is uncertain at date t=0.  Unfortunately, since Stiglitz admits no purely risky securities into his model, in an equilibrium context his economy is riskless and the injected uncertainty disappears together with the purpose of his model.  Rubinstein (1975), p.815n.

    Recall from Chapter 1 that a semantic controversy arose over how to define a `nonspeculator' in the speculation and stability literature.  If  `nonspeculators' could be defined, then everyone else left in the model would presumably be a speculator.  Friedman and Telser maintained that `nonspeculators' could not base their investment decisions on past prices or price trends.  Baumol and others countered that people would be irrational to ignore past prices.

    Against this backdrop, Hirshleifer (1975) introduced his contingent claims model of speculation, and the resolution appeared clear and simple.  Traders who utilize the informed environment to alter their choice of a consumption lottery from what it would have been in a noninformed environment are speculators.  Those who choose the same lottery in the two environments are nonspeculators.  Yet this result depended on the particular regime of incomplete markets selected and the zero complementarity assumption.  When the latter assumption was relaxed, then utility-deviant traders as well as belief-deviant traders were found to be speculators.  The only nonspeculators in the semicomplete market regime were those who exhibited zero complementarity in their preferences for goods.  As a result, the current generation of speculation theorists seem disillusioned with the general equilibrium contingent claims framework and favor instead noise trader, partial equilibrium, and so-called rational expectations models of speculation.

    Returning now to the Keynes-Hicks tradition that risk preferring individuals speculate, the Hirshleifer (1977) model offers some support for this position.  `[G]iven differences of belief so that speculation occurs, the extent of an individual's speculative commitments will vary with his risk-tolerance.'  But without trading rounds before and after the known arrival of information, traders will have no capital gains motive.  The new information can certainly change equilibrium beliefs.  Preferences and degrees of risk aversion normally will not change in a model just with the arrival of information; the theorist afterall has fixed the utility functions as a specification of the model.  These facts suggest that differences in beliefs have a greater impact than differences in risk aversion in defining who will speculate.

    Ironically, we have come full circle.  The extent to which speculators profitably destabilize markets remains as an empirical question.  The same can be said for whether differences in beliefs or risk aversion are more important determinants of speculation.

3.2.4.  Common Knowledge Beliefs and Price Computability

Even under the zero complementarity assumption, the relationship between prices and probability assignments will not be valid with merely a `concordant beliefs' assumption.  Although individual market traders may agree in their probability beliefs, they have no way of knowing a priori they agree.  Chapter 4 is devoted entirely to this topic, and Chapter 5 applies this topic to asset pricing theories.  Unless the traders' beliefs are both agreed and common knowledge, equation (3.7) will not hold once the representative individual assumption is relaxed.[2]  To say that traders' beliefs are common knowledge means that all the traders know their own and everyone else's beliefs, know everyone knows everyone else's beliefs, knows everyone knows everyone knows everyone else's beliefs, and so on.

    Hirshleifer (1976) was aware of this point when he stated `If all have concordant beliefs as to the likelihood of the possible states of the world, and all know this, there will be no problem in assigning probability weights.' (p.693, emphasis added)  However, we face an additional difficulty, which is precisely the same problem Nagatani (1975) identified for the Arrow-Debreu securities market in Chapter 2.  The future spot market prices to which probability measures are to be assigned cannot be computed from the information available in the prior trading round.[3]

    Hirshleifer avoided this difficulty by establishing a martingale process on prices:

        If everyone believes that the martingale proposition is valid, and acts accordingly, then (in a zero-complementarity world of concordant beliefs) the proposition will be valid!  But...equation (3.7) does not in general hold true if the zero-complementarity assumption is relaxed.  Absent independence in demand, if everyone believes in martingales, and acts accordingly, then everyone will then be wrong!  So without zero complementarity, the conditional posterior prices that are necessary to form a correct probability distribution for guiding decisions in the prior round of trading cannot simply be computed from publicly available data (the agreed probabilities) available in the prior round.  In general, detailed information as to endowments and utility functions would be needed to compute the conditional posterior prices properly, knowledge of which is needed to guide prior-round decisions properly.  (Hirshleifer 1976, p.693)

The other alternative is to assume that future spot market prices are common knowledge.  This assumption would limit the robustness of the martingale results, but it would enable individuals to solve the maximization problem set forth in (3.8) and (3.9).

3.2.5.  Zero Complementarity Revisited

Two of Hirshleifer's results in one application of his framework C speculation only by a belief-deviant individual and the martingale property of prices (and ) C hinge on the zero comple­men­tarity assumption.  One critic complained that zero complementarity was unrealistic for a consumptive lottery between a left and a right shoe or for different grades of wheat.  But clearly Hirsh­leifer's (1975) Semicomplete Market model intended to portray consumption among more varied commodities.

    Readers may be left with the unfortunate impression that most economists find the zero complementarity assumption objectionable.  Any theoretical model abstracts from reality.  Useful models include the most important elements of reality and exclude the rest.  For many commodities, zero complementarity is not only a close but a perfect approximation to reality. 

    David Hirshleifer (1991) uses the zero complementarity assumption to establish futures prices follow a martingale, i.e., the futures price exhibits no upward or downward bias.  However, futures prices do reflect seasonal variations with harvesting.  David Hirshleifer (1990) also shows that with positive (negative) complementarity, the futures prices is a downward (upward) biased predictor of the future spot price.  Furthermore when transactions costs are introduced, if demand is inelastic/elastic, the futures price is a downward/upward biased predictor of the future spot price.  However, David Hirshleifer's results on the effect of demand elasticity depend on the extent of consumer participation.  David Hirshleifer (1990) finds that nonparticipation by consumers causes the direction of the bias to correlate to the hedging positions of producers.  If consumers more actively trade futures, then the bias results could be reversed.  Zero complementarity tends to support the martingale property and offset bias effects in contingent claims or futures prices.

3.3  Market Incompleteness and Speculation

Commenting on Hirshleifer's article, Feiger (1976) noted that with complete contingent claims markets, no one has any desire to speculate.  With incomplete markets, everyone will form contracts with the intent of revising them.[4]  A reader may question why only belief-deviant individuals were found to speculate in Hirshleifer (1975), and whether the zero complementarity assumption is so strong that it negates the general proposition that everyone will speculate.  We will therefore employ another variant of Hirshleifer's (1977) framework to explore how speculation and incomplete markets relate. 

    This section will show that in a general equilibrium contingent claims model, utility maximizing individuals will generally speculate even if zero complementarity is assumed.  The particular form of the contingent claims model we employ resembles a stock market economy.  It turns out that the zero complementarity assumption only eliminated speculation for a so-called Semicomplete Market (with claims for  and ), which is the application covered in Section 3.2.  First we will define a Fully Complete Market, an Unconditional (incomplete) Market, and speculation.

    Again we have a pure exchange economy with riskless good N, risky good Z, and states A and B.  All individuals receive the same endowment for good N, but society's total endowment for good Z differs across states:   and

Definition 1:  A Fully Complete Market contains separate contingent claims for  and

    A fully complete regime only requires enough contingent claims to span the space of potential consumption vectors; separate state-commodity contracts shown in Definition 1 are sufficient but not necessary.

Definition 2 (Hirshleifer 1977):  An Unconditional Market contains claims to N and , where   represents a 1:1 entitlement package of

    The unconditional market regime corresponds to a claim for a numeraire, such as money, and a stock market security that entitles the purchaser of  to one share of Z for every possible state of the world.

Definition 3:  Speculation occurs in any market regime when individuals trade to intermediate consumption lotteries in an informed environment that differ from their consumption lotteries in a non-informed situation.

Proposition:  Assuming zero complementarity between N and Z, a utility-maximizing individual will speculate in an Unconditional Market regime.

Proof:  The Noninformed Environment.  Let some individual assign probability  to state A occurring and (1-) to state B.  The individual receives the same endowment for the riskless asset N in either state; moreover, the individual cannot trade claims for N conditioned on each state of the world so that .  The individual maximizes expected utility

subject to the budget constraints

where  represents the 1:1 sandwich price for  and the  superscripts e denote the individual's endowment.

    Let .  Feasi­­bility in the pure exchange economy requires  and   Because  = 0 by the zero complementarity assumption, the optimality conditions emerging from this expected utility maximizing problem reduce to

As a further consequence of zero complementarity, the individual's marginal rate of substitution for N across states will equal unity:

                                               = 1.                                     (3.10)

The Informed Environment.  The individual trades first to an intermediate contingent  claim  bundle,  denoted  with  superscript  t.  In  period  1,

conclusive information on the state emerges.  In period 2, the contingent claim market reopens, and the individual retrades to his final period 2 consumptive optimum.  The conclusive information collapses the 1:1 sandwich across states to simply a claim for Z in the known state.

    The individual now maximizes expected utility

                                         (3.11)

subject to the budget constraints

                                                                     (3.12)

                                                                    (3.13)

                                              (3.14)

 and  represent the implicit prices for good Z in states A and B, had such claims existed.  In particular, since  represents a 1:1 package of  ,   The primes denote the conditional prices and quantities that would prevail in each state:  one prime denotes state A, two primes denote state B.  Furthermore

    Let