1 Introduction

In this chapter we ask a simple question: What does rational behavior on the part of asset market participants imply about equilibrium asset prices? “Rationality” has a variety of meanings in economic thought, so to answer this question we must first be clear about what we intend by “rational behavior.” Here we take “rationality” to mean that investor preferences satisfy the Savage [14] axioms or some modern refinement of them. Thus, the investor is a subjective expected utility (SEU) maximizer. The existence of an SEU representation implies that the investor is a Bayesian, but it does not otherwise restrict the investor's beliefs about future prices. In particular, without ancillary assumptions on prior beliefs, it does not imply that the investor has correct beliefs, or even that he will eventually learn the truth.

The hypothesis of rationality can be strengthened by ancillary assumptions on the nature of beliefs. One such belief restriction is the requirement that investors' beliefs are conditional forecast distributions from prior beliefs over a (p. 50 ) class of models which includes the “true” model. A sharper belief restriction is the rational expectations hypothesis, that investors are certain of the true model. In other words, investors' beliefs are correct. In terms of prior beliefs, the first assumption has the support of the prior belief distribution containing the true model. The second assumption has the support of the prior belief distribution containing only the true model. In other words, the true model receives prior probability 1.

Here we examine the implications of this hierarchy of rationality hypotheses for equilibrium asset prices. The economy we analyze has investors who live forever, have stochastic endowments of a single consumption good, and trade in each period a dynamically complete set of Arrow securities. We do not explicitly consider richer sets of securities, but even with the weakest of our rationality hypotheses security prices are arbitrage proof, so more complex assets can be priced by arbitrage from the prices of the Arrow securities.

A belief-based learning rule is a map from partial histories of states into a forecast distribution on the next state. We show that any belief-based learning rule is consistent with Bayesian updating, and thus is consistent with subjective expected utility maximization. Consequently, any map from partial histories into prices of the Arrow securities is consistent with rational behavior of the first type. In this market model, restrictions on asset prices cannot come from the hypothesis of SEU maximization.

Belief restrictions, on the other hand, do have power. For example, if investors beliefs are generated by a prior on a set of models for the economy that includes the true model, then investors will learn the true model (if it is identified) and, under some assumptions, prices will converge to their rational expectations equilibrium values. If we further strengthen the rationality hypothesis to require correct beliefs from the outset, then prices will always be at their rational expectations equilibrium values.

It is important for these rational expectations equilibrium conclusions that belief restrictions as well as SEU rationality are required of all investors. This is surely problematic. How is it that all investors know the truth or even place positive probability on it? Where does this knowledge or prior restriction come from? It cannot be derived from learning, as the rationality model with a prior restriction is itself supposed to be a model of the learning process. What happens if, more realistically, we assume that some, but not all, investors know the truth or are able to learn it? This requires an analysis of an economy with heterogeneous investors.

In an economy in which investors have heterogeneous beliefs or heterogeneous learning rules, structure on long-run prices arises from the forces of market selection. It has long been assumed that those with better beliefs will make better decisions, driving out those with worse beliefs, and thus determining long run asset prices correctly. This argument is usually attributed to Alchian [1] and Friedman [11], and to Cootner [7] and Fama [10] for its application to financial markets. More generally the idea is that rational (in a strong sense) decisionmakers (p. 51 ) will drive out irrational decisionmakers. Thus, according to this argument, long-run asset prices will be correct. But until recently there was little formal investigation of this conjecture.

Delong, Shleifer, Summers, and Waldman [8, 9] provide one of the first formal analyses of wealth flows between rational and irrational traders. They argue that irrationally overconfident noise traders can come to dominate an asset market in which prices are set exogenously; a claim that contradicts Alchian's and Friedman's intuition. In Blume and Easley [5] we addressed the same issue in a general equilibrium model. We showed that if savings rates are equal across investors, general equilibrium wealth dynamics need not lead to investors making portfolio choices as if they were subjective expected utility maximizers with correct beliefs. We did not study the emergence of fully intertemporal expected utility maximization, nor did we say much about the emergence of beliefs. Sandroni [13] addressed the latter question. He built economies with intertemporal expected utility maximizers and studied the emergence of rational expectations. He showed in a Lucas trees economy that, controlling for discount factors, only investors with rational expectations, or those whose forecasts merge with rational expectations forecasts, survive. He also showed that even if no such investors are present, no investor whose forecasts are persistently wrong survives in the presence of a learner. In Blume and Easley [4] we showed that whenever markets are complete, and investors have a common discount factor, investors with correct beliefs drive out those with incorrect beliefs and thus drive prices to their rational expectations equilibrium values. Here we apply the analysis of Sandroni [13] and Blume and Easley [4] to show how selection works in the Arrow securities economy.

In the following sections we describe a simple model of an economy with Arrow securities (section 2), show that subjective expected utility maximizers are Bayesians (section 3), consider other non-SEU behavior (section 4), show the implications of various definitions of rationality for asset prices (section 5), and show how wealth dynamics work (section 6).

2 A Simple Model

We consider the implications of SEU rationality and various belief restrictions for asset pricing in a simple infinite-horizon general equilibrium model. Time is discrete and is indexed by t ∈ {0,1, …, T} with T ≤ ∞. At each date t ≥ 1, a state from the finite set of states {1, …, S} is realized. A Path of states is denoted σ = (σ₁, σ₂ …), where each σ_t ∈ S. The set of paths is denoted Σ and its product sigma-field is denoted Ϝ. States evolve according to a “true” probability p on (Σ,Ϝ)

All random variables dated t are assumed to be date-t; measurable; that is, their value depends only on the realization of states through date t. Formally, Ϝ_t is the σ-field of events measurable at date t, and each such random variable (p. 52 ) is assumed to be Ϝ_t-measurable. For a given path a1 σ,σ_t is the state at date t and σ^t = (σ₁, …, σ_t) is the partial history through date t. Let Ή denote the set of all partial histories. Let 0 denote the empty partial history.

There is a single, non-storable consumption good available at each date. Also at each date, S Arrow [3] securities are available. One unit of security 5 available at date t pays one unit of the consumption good at date t + 1 if and only if state s occurs at date t + l.¹ The price of the consumption good is one at each date. The prices of Arrow securities are random variables denoted as q_t = (q^l_t, …, q^s_t).

There are I investors, indexed by i. Investors have stochastic endowments of the good. The endowment stream for investor i is given by the random variable eⁱ = (eⁱ₀, eⁱ₁ …), where eⁱ_t ∈ R₊₊ is investor i's endowment of the good at date t. A consumption plan for investor i is a random variable denoted cⁱ = (cⁱ_o, cⁱ₁ …), where cⁱ_t ∈ R₊₊ is i's consumption of the good at date t. The set of consumption plans is denoted C.

Investors have preferences over entire consumption plans. Investor i's preference order over plans is denoted by ≸ⁱ. Under standard assumptions (see Savage [14] or Anscombe and Aumann [2]), these preferences have a subjective expected utility (SEU) representation.

Definition 1. Preferences ≸ have a subjective expected utility representation if there exists a payoff function U: C → R and subjective beliefs, a probability p on Σ, such that c ≸ c′ if and only if

E_p U (c) ≥ E_pU(c′).

We assume that investors' utility functions are time separable and permit geometric discounting, that is, there is a discount factor β and a function u: c ↦ R such that

This representation provides a utility function for consumption u, a discount factor β and, most importantly for our purposes, beliefs p over paths. The representation places no restrictions on beliefs other than the obvious requirement that they are a probability on (Σ, Ϝ). One important special case is that of beliefs generated by iid forecasts. If trader i believes that all the σ_t are iid draws from a common distribution then p, then pⁱ is the corresponding distribution on infinite sequences. In this case, the marginal probability on σ^t is pⁱ_t(σ) = ∏^t_{τ = 1} P(σ_τ).

(p. 53 ) The subjective expected utility representation also allows for investors who are uncertain about the process on states and who learn about it as Bayesians. In fact, it allows for no other learning rules. To formalize this claim we first need to define conditional preferences and Bayes' rule.

3 Bayesian Learning

If a preference order ≸ satisfies the Savage axioms, then there is a well-defined notion of conditional preference order. We say that one consumption plan c is at least as good as another consumption plan cʹ given some set of paths A, c ≸ _A cʹ, if the ranking between these two plans depends only on how they behave on A.

For c and cʹ in C, and A ∈ Ϝ, define

Definition 2. Consumption plan c is at least as good as cʹ given A, c ≸_A cʹ, if for all plans cʺ ∈ C, c_Acʺ ≸ cʹ_Acʺ.

Given that the path will be in A, only how plans compare on A matters to a subjective expected utility maximizer. If the individual's beliefs over paths are pⁱ then only beliefs conditional on A matter when A is known to occur. These conditional beliefs are given by the Bayes rule.

Definition 3. For A ∈ Ϝ with p(A) 〉 0 and any B ∈ Ϝ define _PA by

First we recall the elementary fact that SEU maximization implies Bayesian updating.² If a decisionmaker ranks consumption plans with beliefs p then, given knowledge of A, he ranks consumption plans using conditional beliefs p_A.

Theorem 1. Suppose ≸ has an SEU representation (U, p). Then for any A ∈ F with p(A) 〉 0 and c, cʹ ε C, c ≸ _A cʹ if and only if E_PA U(c) ≥ E_PA U(cʹ), where _PA is p conditioned on A.

Proof Choose any cʺ ∈ C. The following are all equivalent:

(p. 54 ) p(A) E_p {U(c)|A} + p(A^c) E_p {U(cʺ)|A^c} ≥ p(A^c) E_p {U(cʹ)|A} + p(A^c) E_p{U(cʺ)|A^c}

p(A) E_p {U(c)|A} ≥ p(A)E_p {U(cʹ)|A}

E_PA U(c) ≥ E_PA U(cʹ) ■.

4 Contingent Decision Problems

In asset markets investors make a sequence of decisions: a choice at the outset, a choice after observing the state at date 1, a choice after observing the states at dates 1 and 2, and so forth. We call these contingent decision problems. Each contingency has its own preference order with respect to which choice, given that contingency is maximal.

Formally, let Σσ^t denote the set of all paths with initial partial history σ^t, and let F_σ^t denote the restriction of F to subsets of Σ_σ^t.

The elements of the decision problem D_σ^t for contingency σ^t are the set of all consumption plans C_σ^t = {(c_t+1, …): c ε C} and on this set a preference order ≥^σt.

Definition 4. A contingent preference structure is a collection D = {(Dσ^t, ≸σ^t): σ^t ∈ H}. Combining the framework with appropriate budget constraints defines contingent decision problems from which asset demand can be derived.

One source of contingent preference structures is conditional SEU preferences. We know from Theorem 1 that SEU preferences imply Bayesian learning, so we call the decision framework they induce Bayes.

Definition 5. A contingent preference structure D is Bayes if there is a payoff function U and a belief p on Σ such that ≸σ^t is represented by (U, pσ^t) where each pσ^t is defined from p by Bayes' rule.

The SEU framework imposes a coherency over preferences in these different decision problems. If σ^t is a partial history, and σ^t+k is an extension of that partial history, then preferences in the decision problem after having observed partial histories σ^t and σ^t+k each have an SEU representation. The payoff functions are the same in both representations, and the beliefs at σ^t+k are the conditional probability of future events from the probability distribution representing beliefs at σ^t given the additional observations σ_t+1…, σ_t+k. This, of course, is a consequence of Theorem 1. This works both ways. The coherency condition just described is necessary as well as sufficient for a contingent preference structure to be Bayes. The following claim is trivially true but it makes an important point, that when conditional preferences are SEU, Bayesian behavior is nothing more nor less than the path-consistency of preferences.

(p. 55 ) Theorem 2. Let D be a contingent preference structure such that each ≸σ^t has an SEU representation. Then D is Bayes if and only if for each partial history σ^t and extension σ^t+k, the conditional preference order defined from ⪰^σt given σ^t+k, denoted , is identical to the preference order ⪰^σt+k.

Proof That every collection of conditional preference orders derived from SEU preferences has this property is obvious. Going the other way, let (U, p) represent the preference order ⪰⁰, and consider any expected utility representation (Uʹ,pʹ) for ⪰^σt. Since ⪰⁰_σ^t and ⪰^σt are identical, the uniqueness of the SEU representation implies that pʹ = pσ^t and that there exist numbers α and β 〉 0 such that Uʹ = α + βU. So (U, p_σ^t) represents ⪰^σt. Thus D is Bayes. ■

It is often argued that this Bayesian coherence imposes a significant restriction on beliefs, and therefore on behavior. A common “remedy” is to suppose that preferences after different partial histories have SEU representations with identical payoff functions, but to allow arbitrary belief evolution. Specifically, consider a decisionmaker who has initial beliefs p₁ on states at time 1. Suppose that the individual's beliefs on states at time 2, conditional on the realization of the time 1 state σ₁, are given by a learning rule P₂(σ₁.). Similarly, for each partial history σ^t the individual's beliefs on states at time t + 1 are given by a learning rule p_t+1(σ^t,·). An individual who follows this procedure uses a belief-based learning rule.

Definition 6. A belief-based learning rule is a probability p₁ on S and a sequence of F_t-measurable functions {p_t+1(σ^t,)}, fort ≥ 1, from partial histories into probabilities on S. A contingent preference structure D has a belief-based expected utility (BBEU) representation if there exists a U: C → R and a belief-based learning rule {p_t}∞t=1 such that ⪰^σt is represented by (U, p_t+1(σ^t)).

So many papers on bounded rationality employ this remedy that we resist the opportunity to single out one or two to pick on. But our point is that this is no remedy at all. If a contingent preference structure has a BBEU representation, then it is Bayes.

Theorem 3. If {p_t}^∞_t=1 is a belief-based learning rule, then there is a subjective belief p on Σ such that

1. 1. For all A ⊂ S, p₁(A) = p(σ₁ ε A), and

2. 2. For all partial histories σ^t and A ⊂ S, P_t+1 (σ^t, A) = p(σ_t+1 ε A\σ^t).

In other words, any belief-based learning rule is realizable as a collection of conditional expectations from a subjective belief distribution p.

(p. 56 ) Proof The functions p_t are conditional probabilities, and p₁ is a probability distribution. The initial beliefs p₁ and the conditional distributions P₂, … PT−1 can be integrated to generate a marginal distribution qT on partial histories σ^T such that the marginal distribution of σ^T with respect to any q_T+k is just qT. The Kolmogorov extension theorem then states there is a distribution p on Σ with the given marginals qT.

Any BBEU representation is Bayes. Thus, requiring an individual to be a Bayesian places no restrictions on his sequence of one-period forecasts. Restrictions on these forecasts are typically obtained by placing restrictions on the set of models for the stochastic process that the individual considers and by restricting his prior on the model set. It is worth emphasizing that even if we can observe an individual's entire sequence of one period ahead forecasts, observations that contradict Bayesian behavior, and thus a subjective expected utility representation, are not possible unless the observer has some prior knowledge about the individual's beliefs.

The usual method of employing this false remedy is to pose a problem with a space of models and a prior distribution (or class of prior distributions) on that space. Then one rejects Bayesian updating in favor of some other updating rule. Of course such learning rules can lead to conclusions different from Bayesian updating, but what is being rejected is not Bayesian behavior but the choice of the model space. This is a belief restriction, not a rationality hypothesis. The only implication of SEU rationality is coherence. We explore coherence violations in the next section.

5 Non-Subjective Expected Utility Motivated Behavior Preferences

Beliefs, and the maximization of expected utility using those beliefs, are obviously important for asset pricing. These beliefs could come from preferences over random consumption streams as in the subjective expected utility setting. Alternatively, they could be arbitrary beliefs of investors who are not operationally subjective expected utility maximizers. In this section we consider more general, or seeming more general, forms of behavior and ask if these behaviors are different from those generated by subjective expected utility maximizers. We show that any investor who correctly anticipates his future beliefs acts as if he is a Bayesian, although perhaps one with rather odd beliefs. Such investors are indistinguishable from subjective expected utility maximizers. Equilibria in economies populated with these investors thus cannot be differentiated from those arising in economies populated by subjective expected utility maximizers.

As we have already shown that the hypothesis that investors are subjective expected utility maximizers places no restrictions on learning rules, this claim is (p. 57 ) perhaps not surprising. However, the hypothesis that investors are subjective expected utility maximizers does impose consistency restrictions on plans. These restrictions arise from the hypothesis that the investor knows his preferences. This, in turn, implies that he acts as if he knows his belief-based learning rule and thus correctly anticipates how his beliefs will evolve along any path. Alternatively, a investor who does not correctly anticipate his future beliefs may not act according to his plans and thus may not behave as a subjective expected utility maximizer. This is the reason for the qualification that the behavior of any investor who correctly anticipates his future beliefs is not distinguishable from that of a subjective expected utility maximizer.

To see these claims most clearly we consider a simple three-period, two-state model, S = (s₁, s₂). Let w₀ be the present discounted value at t = 0 of the investor's endowment stream. At date 0 the investor has to allocate this wealth between consumption and purchases of date 0 Arrow securities. At date 1 she receives the proceeds from the Arrow securities that pay off in the realized state and she re-allocates between consumption and new Arrow security purchases. Finally, at date 2, she consumes the proceeds of the Arrow securities that pay off at date 2.

At date 0 the investor has beliefs pⁱ on sample paths ((σ₁, σ₂). These beliefs induce a probability pⁱ₁ on states at date 1 and two conditional probabilities at date 1 on states at date 2. Denote these conditional probabilities by pⁱ₂(s₁,·) if state s₁ occurs at date 1, and pⁱ₂(s₂,·) if state s₂ occurs at date 1.

Because markets are complete, the investor's sequential decision problem can be collapsed into a static problem in which she chooses consumption plans subject to a single budget constraint. Let qσt+1t (σt) be the price of consumption in state σt+1 given σt.

Let the plans that solve this decision problem be denoted {c_o, c₁ (σ₁), c₂ (σ₁, σ₂)}.

When period 1 arrives, this investor will carry out her plans as long as her beliefs at that point are as predicted. A subjective expected utility maximizer always correctly predicts the evolution of her beliefs along each path and thus always carries out her plans. We also want to allow for the possibility that the investor's beliefs are not as predicted. The investor's actual beliefs about states at date t = 2, given σ₁, are denoted by r σ₁ (σ₂). We say that an investor is naive if she does not always correctly anticipate how her beliefs will evolve; otherwise she is sophisticated.

(p. 58 ) Definition 7. A investor is naïve if for some { σ₁, σ₂}, P₁ (σ₁, σ₂) ≠ r_σ1 (σ₂) Otherwise she is sophisticated.

Investors who are subjective expected utility maximizers are sophisticated. They are Bayesians who act as if they understand how their beliefs will evolve. But an investor does not need to be a Bayesian to be sophisticated. For example, an investor who uses maximum likelihood to estimate a parameter for an iid distribution on states and who correctly anticipates how the maximum likelihood estimate will change, given any partial history is sophisticated. However, a maximum likelihood estimator who does not anticipate how her estimate will change over time is naïve.

The decision problem for an investor at time 1 if state σ₁ has occurred is

Let the decisions that solve this problem be denoted {d₁ (σ ₁), d₂ (σ₁, σ₂)}.

Definition 8. The investor is plan consistent if for each (σ₁, σ₂)

c₁ (σ₁) = d₁ (σ₁)

c₂ (σ_l, σ₂) = d₂ (σ₁, σ₂).

Subjective expected utility maximizers are obviously sophisticated investors and are plan consistent. In fact, any sophisticated investor solves a decision problem that can be recast as if the investor was a Bayesian who solves problem P₀. These investors are operationally Bayesian and are indistinguishable from subjective expected utility maximizers. They are thus plan consistent.

Definition 9. An investor is observationally equivalent to a subjective expected utility maximizer if there are beliefs pⁱ on paths, a discount factor β_i and a utility function uⁱ which generate her plans as a solution to problem P_o.

Theorem 4. In the framework of this section:

1. 1. All subjective expected utility maximizers are sophisticated investors.

2. 2. All sophisticated investors are plan consistent.

3. 3. All sophisticated investors are observationally equivalent to subjective expected utility maximizers.

4. 4. Naïve investors need not be plan consistent.

(p. 59 ) Proof

1. 1. Obvious.

2. 2. Rewrite the investor's objective function in P₀ as

   .

3. 3. Let p₀ (σ₁) = r_σ1 (σ₁) Let P₁ (σ₁, σ₂) = rσ₁ (σ₂). The sophisticated investor has the same plan as would a Bayesian with beliefs p(σ₁, σ₂) = P₀ (σ₁)p₁(σ_l, σ₂).

4. 4. To prove this we provide an example of a naïve, plan inconsistent investor. Suppose that all asset prices are 1 and that u(c) = γ⁻¹ c^γ. Let g = 1/(1 – γ).

The investor is naïve and a maximum likelihood learner. At date 0 her forecast for period 1 is (r, 1 − r). At date 1 her conditional forecasts are r_S1(·) = (r, 1 − r) and r_S2(·) = (p, 1 − p) ≠ (r, 1 − r).

At date 0 she plans in each state to allocate

to the state s₁ security. At date 1 she in fact chooses

Since her planned and actual date 1-state s₂ investment allocations differ, her planned and actual date 2 consumptions must differ on the event σ₁ = s₁

6 Equilibrium Prices

Because subjective expected utility maximization places no restrictions on beliefs, it places few restrictions on equilibrium prices. To see this in the simplest setting suppose that there is only one investor, i.e., (I = 1), whose endowment is constant over time and states, e_t (σ₁) = e ∈ R₊ for all σ^t and t. Let p¹_t+1 (σ_t, •) be the investor's date t forecast of the conditional probability on states at date t + 1. If the investor's discount factor is β₁ then it is easy to see that the following prices support the investor's endowment and so form an equilibrium. An equilibrium price of Arrow security s given partial history σ^t is

(p. 60 ) The prices do not depend on the investor's utility function or on the amount of the endowment.³ This is true in a multiple investor economy also as long as the investors have common beliefs and a common discount factor.

Theorem 5. Consider an economy with I investors with common beliefs p¹ and a common discount factor β. Suppose that each investor's endowment is constant over time and states, i.e., for each i there is a eⁱ ∈ R₊ such that eⁱ_t(σ^t) = eⁱ for all σ^t and t. Then, for each s ∈ S and partial history σ^t the equilibrium price of Arrow security s given partial history σ^t is

q^s_t (σ^t) = βp¹_t+1 (σ^t, s). (4)

Proof Because markets are dynamically complete, it is sufficient to consider the present value of investors' endowments and their complete market demands. At prices q^s_t (σ^t) = βp^l_t+1 (σ^ts) the present value of investor i's endowment is eⁱ/(l – β). Calculation shows that, regardless of the investor's utility function, at a date 0 optimum he saves fraction (5 of this value and invests a fraction of this savings in each Arrow security equal to the probability of its associated state. Thus, at date 0 he consumes eⁱ units of the good. This is clearly an equilibrium at date 0. The date 1 present value of his wealth will be el/(l – β) no matter what state occurs at date 1. So he again consumes eⁱ units of the good and we have an equilibrium at date 1. Repeating this argument shows that we have an equilibrium at each date. ■

We know from Theorem 3 that the beliefs are arbitrary. So equilibrium Arrow security prices are arbitrary. Of course there are restrictions on prices of securities that can be represented as bundles, over time or over states, of Arrow securities. Although these redundant securities are not present in our model, we could easily include them and price them by arbitrage. Inclusion of such securities would lead to falsifiable restrictions on security prices. But note that rejecting these restrictions would do far more than reject Bayesian learning—it would reject any decision theory in which individuals recognize and take advantage of arbitrage opportunities.

To obtain restrictions on asset prices in the simple economy of Theorem 5 we would need to have restrictions on investors' beliefs. For example, if we assume that these beliefs are correct, pⁱ = p for all i, then prices will also be “correct.” Or we could assume that all investors are learning about the true model. If the true model p is in the set of models they consider and if it is identified, they will learn it and prices will converge to “correct” prices.⁴ But how is it that all investors come to know the truth or to have it in their model set? What happens if some (p. 61 ) investors know the truth, have rational expectations, and others do not? Will the rational investors drive out the incorrect ones and force prices to converge to their “correct” values? These questions are addressed in the next section.

7 Wealth Dynamics

If investors have differing beliefs and or differing discount factors then they may make differing investment choices. Both the level of savings and the portfolio rule that investors choose may be affected by these individual factors. This will cause wealth to flow between the investors over time. In this section we analyze these wealth flows and their implications for asset prices. The analysis in this section is an application of the results in Sandroni [13] and Blume and Easley [4].

In order to do this analysis we make the following assumptions. First we assume that investors are strictly risk averse and that marginal utility converges to infinity as consumption converges to 0.

A. 1. The payoff functions u_i are C¹ functions from R₊ to R₊ which are strictly concave, strictly monotonic, and satisfy an Inada condition at 0.

We next assume that the aggregate endowment is uniformly bounded from above and away from 0:

A. 2. ∞ 〉 F = sup_t, σ ∑_i eⁱ_t(σ) ≥ inf_t,σ ∑_i eⁱ_t (σ) = f 〉 0.

Finally, we assume that investors believe to be possible anything which is possible.

A. 3 For all investors i, all dates t and all paths σ, p_t (σ) 〉 0 and pⁱ_t (σ) 〉 0.

The economy has a full set of Arrow securities so by the First Welfare Theorem any competitive equilibrium allocation is a Pareto optimal allocation. Further, because each investor has a strictly positive endowment, she must receive a strictly positive utility in any competitive equilibrium. So the relevant Pareto optimal allocations are those in which every investor has strictly positive utility. Our approach is to characterize this set of Pareto optimal allocations. Any property that holds for all of them must hold for any competitive equilibrium.⁵

Theses Pareto optimal allocations maximize a weighted sum of utilities with strictly positive welfare weights for each investor. If c^* = (c^1*, …, c^i*) is such a Pareto optimal allocation, then there is a vector of welfare weights (λ¹, …, λⁱ) » 0 such that c^* solves the problem

(p. 62 )

where e_t = ∑_i eⁱ_t.

From the first-order conditions for problem (5) it can be shown that for any investors i and j there is a constant K_ij such that for any path σ and for all t,

The left side of eq. (8) is not a marginal rate of substitution. It is the ratio of marginal utilities for two investors along the path σ. This ratio conveys information about the investors' consumption and wealth. To see this, note that because the aggregate endowment, and thus consumption, is bounded away from 0 and from above, our conditions on utility functions imply that:

So we can study the limit behavior of i's consumption, and thus his wealth and affect on prices, by studying the right side of eq. (8). The right side does not involve utility functions so we immediately have the result that, within the class of preferences that we consider, attitudes toward risk are irrelevant for long survival. All that matters are discount factors and beliefs.

8 Conclusion

The hypothesis of rational investor behavior has few implications for asset prices. The power of rationality lies in the ancillary assumptions of various restrictions on beliefs. Rational expectations impose strong conditions on asset prices. Belief restrictions guaranteeing that rational expectations are learnable imply strong conditions on asset prices in the long run. We believe that these assumptions on traders' beliefs are unreasonable, but that most mispricings of assets with reference to the rational expectations asset pricing formula are consistent with Bayesian rationality. We agree that behavioral models can provide convenient representations of preferences which highlight particular properties of asset price behavior that differ from conventional asset pricing predictions. But it is wrong to assert that the source of these differences must lie in the rejection of SEU preferences.

Assuming that some investors may have more accurate beliefs than others is more plausible than rational expectations assumptions. We have reported on some restrictions on asset prices that can be derived from economic models with heterogeneous beliefs. In the short run, investors with incorrect expectations can influence asset prices. But in the long run they may lose out to those with better expectations who choose better portfolio rules. Expectations also affect savings rates, and investors with incorrect expectations can be induced to over-save, so the conjecture that they are driven out is far from obvious. We show that if markets are dynamically complete and investors have a common discount factor, then in fact the market is dominated in the long run by those with correct expectations. In the long run, asset prices converge to their rational expectations values.

The assumptions that markets are complete and that investors have a common discount factor are both important for this selection result. In Blume and Easley [4] we provide an analysis of the tradeoff between the size of discount factors and the distance of beliefs from the truth. Investors with high discount factors and incorrect beliefs can drive out those with correct beliefs and lower discount factors, and this will cause even long-run asset prices to be incorrect. More importantly, we also show that if markets are not dynamically complete (p. 65 ) then, even when discount factors are common, investors with incorrect beliefs can drive out those with correct beliefs.