Rational expectations in the aggregate.

Haltiwanger, John C. ; Waldman, Michael

RATIONAL EXPECTATIONS IN THE AGGREGATE

I. INTRODUCTION

One of the major recent innovations in economic theory is the emergence of the rational expectations hypothesis, the hypothesis that expectations of agents tend to be consistent with the predictions of the relevant economic theory. This paper considers the relationship between the way rational expectations is typically employed in practice and the argument frequently put forth to justify its use.

Rational expectations has typically meant what we will refer to as standard rational expectations: the expectation of each agent taken separately is by itself consistent with the predictions of the relevant theory. This, however, is different from the argument frequently put forth by proponents of the rational expectations hypothesis to justify its use. This argument is that on an aggregate level expectations should be consistent with the predictions of the relevant theory. This justification recently in the works of Kantor [1979], Maddaock and Carter [1982], and Hoover [1984]; it was first expressed by Muth [1961, 316]:

The hypothesis can be rephrased a little more precisely as follows: that expectations of firms (or, more generally, the subjective probability distribution of outcomes) tend to be distributed, for the same information set, about the predictions of the theory (or the "objective" probability distributions of outcomes).

Underlying the above argument is a belief that if expectations are rational in the aggregate, then expectational deviations across agents will tend to cancel out. The statement of this belief also appeared in Muth [1961, 321]:

...Allowing for cross-sectional differences in expectations is a simple matter, because their aggregate effect is negligible as long as the deviation from the rational forecast for an individual firm is not strongly correlated with those of the others...

Charles Schultze [1985, 10] expressed the same notion in his 1984 presidential address to the American Economic Association: (1)

In a word of auction markets, the fact that forecasts of individual agents are widely distributed around the "true" mean is for most purposes irrelevant...

This paper formally investigates the relationship between standard rational expectations and what occurs when expectations are rational only in the aggregate, i.e., what will be referred to here as aggregate rational expectation. (2) The goal is two fold. First it is to show that the above view is overly simple. It is generally not the case that an aggregate rational expectations world can e accurately modeled using a standard rational expectations assumption. The second objective is to consider environments where standard and aggregate rational expectations equilibria differ, and investigate what factors affect the size of the difference.

These issues are examined by analyzing a model wherein agents choose between alternative activities. A crucial factor in determining the relationsip between standard and aggregate rational expectations equilibria is the nature of the interaction among agents. This interaction is characterized as being either of two types. First, activities can exhibit congestion, i.e., the larger is the total number of agents who choose to participate in a given activity, the lower is the incentive for agent i to choose that activity. Examples of situations which exhibit congestion are the problem of agents choosing between different roads which lead to the same final destination, and market decisions such as the problem of carrer choice or the problem of firms deciding where to locate. Second, the activities can exhibit synergism, i.e., the larger is the total number of agents who choose to participate in a given activity, the higher is the incentive for agent i to choose that activity. An example of a situation which exhibits synergism is the problem faced by consumers in choosing a computer hardware system; the larger is the number of individuals who purchase a particular system, the greater will be the subsequent availability of computer peripherals and software for that system. This model is considered because of its generality; many common models are actually special cases of it, and that will be demonstrated. (3)

Only under very special conditions do standard rational expectations and aggregate rational expectations yield equivalent results. The difference between the two equilibria is larger when: (i) the divergence in expectations under aggregate rational expectations is increased; (ii) in a world which exhibits congestion, the severity of the congestion is decreased; (iii) in a worls which exhibits synergism, the severity of the synergism is increased; and (iv) the activities exhibit synergism rather than congestion.

These results to our own earlier work on the robustness of rational expectations equilibria. Haltiwanager and Waldman [1985] considered an environment in which agents vary in terms of their ability to form expectations; but in contrast to the present paper, no aggregate rational expectations assumption was imposed. Rather, that paper looked at an environment in which there are two types of agents. Agents termed "sophisticated" satisfied a standard rational expectations assumption, while those referred to as "naive" all had the same incorrect set of expectations. (4) The issue addressed was, given an environment in which agents vary in this manner, is it the sophisticated agents or the naive agents who are disproportionately important in the resulting equilibrium? If the environment exhibits congestion, then the sophisticated agents turn out to be disproportionately important, while with synergism the naive agents dominate. These two papers are complementary. Although thoy employ different tests to determine whether standard rational expectations equilibria are robust, they reach quite similar conclusions. In both, standard rational expectations equilibria tend to be robust in environments of congestion but not synergism.

Section Ii sets forth a model agents choose between alternative activities. Section III analyzes the model, with special attention paid to the factors which affect the sizes of the different between standard and aggregate rational expectations equilibria. Section IV presents two special cases of the genereal model: (i) a model of career choice, and (ii) a variant of the macroeconomic trading externality model of Diamond [1982]. In addition to showing the general applicability of the model, these examples demonstrate a number of real-world implications of the approach. Section V presents concluding remarks.

II. THE MODEL

The analysis is conducted within the context of a simple model wherein agents choose whether or not to participate in a given activity (note: the model is formally equivalent to a model where agents choose between two different activities). It is assumed that the choice concerning participation is made prior to the realization of the returns to participation, that it is irreversible, and that it is made simultaneously by all agents in the population. The benefit to participating for a given agent is given by B([pi]), where [pi] is the proportion of agents who choose to participate, i.e., [pi][Epsilon](0.1). (5) If B'[is less than]([is greater than]) 0, then the activity congestion (synergism). Agents are parametrized by variables ([x.sub.i], [e.sub.i]). The variable [x.sub.i] determines agent i's cost of engaging in the activity, C([x.sub.i]), where [x.sub.i] is uniformly distributed on [0,1]. (6) [x.sub.i] can be interpreted either as a random draw of costs or as representing a characteristic of agents that is distributed uniformly across the population. The variable e.sub.i is agent's i's error in the expectation of the benefit B([pi]). Under standard rational expressions e.sub.i is identically zero for all agents. (7) Under aggregate rational expectations, the distributions of e.sub.i.]s is described by a density functional f(e.sub.i.), where f(.) is continously differentiable and positive in the interval [-E,E], and equals zero elsewhere. In this case, it is also assumed that x.sub.i and e.sub.i are independently distributed. Note our specification states that the size of the largest positive expectational error is given by E, while the most negatice is -E. Further, since aggregate rational expectations means there is no aggregate bias, [Mathematical Expression Omitted].

Agent i chooses to participate if

C([x.sub.i] [is less than or equal to] B([pi]) + [e.sub.i]. (1)

Both B and C are assumed continuously differentiable and satisfy

C(0) = 0 [is less than] B(0), (2)

B(1) [is less than] C(1), (3)

C'(z) [is greater than] for all z [Epsilon] [0,1], (4)

B'(z) - C'(z) [is less than] 0 for all z [Episilon] [0,1], (5)

and

2E [is less than] C(1). (6)

Conditions (2) and (3) insure an interior solution for [pi]. Condition (4) stipulates that agents with higher values for [x.sub.i] face higher costs of participating. Condition (5) insures that the equilibrium is unique. Condition (6) simply states that individual biases in expectations are small relative to the variation in participation costs levels in the population. This last assumption reduces the number of cases that need to be analyzed.

Under standard rational expectations, the equilibrium participation rate [pi.sup.S] is such that all agents with [x.sub.i] [is less than or equal to] [pi.sup.S] participate, and [pi.sup.S] satisfies B([pi.sup.S]) = C([pi.sup.S]). Under aggregate rational expectations, the equilibrium participation rate [pi.sup.A] satisfies (8)

[Mathematical Expression Omitted]

III. ANALYSIS

This section analyzes the model developed above by first comparing standard and aggregate rational expectations in terms of the proportion of agents who choose to participate. Second, the ramifications of varying the divergence of expectations under aggregate rational expectations are explored. The third topic is the effects of varying the severity of the interaction among agents. The first proposition considers the relationship between the standard and aggregate rational expression equilibria. All proofs are relegated to an appendix.

PROPOSITION 1.

If E [is less than or greater than] B([pi.sup.S]) [is less than or greater than] C(1) - E and C" = 0, then [pi.sup.S] = [pi.sup.A]. (1.i)

If E [is greater than] B([pi.sup.S]) [C(1) - E [is less than] B([pi.sup.S])] and C" = 0, then [pi.sup.S] [is less than] ([is greater than]) [pi.sup.A]. (1.ii)

If C" [is greater than] ([is less than]) 0 and E [is less than or equal to] B([pi.sup.S]) [B([pi.sup.S]) [is less than or equal to] C(1) - E], then [pi.sup.S] [is greater than]

Part (1.i) of proposition 1 states that, given two restrictions on the model standard and aggregate rational expectations result in the same participation rates. (9) The first restriction is that the benefit from participating under standard rational expectations, i.e., B([pi.sup.S]), is further than E from the extreme values of C. The second restriction is that the cost function is linear. Parts (1.ii) and (1.iii) of proposition 1 state that standard and aggregate rational expectations do not in general yield equivalent results. In particular, (1,ii) and (1.iii) identify situations for which there is a systematic difference beteen the two equilibria.

The intuition underlying these results is as follows. Under aggregate rational expectations there is a set of agents who choose to participate because they overvalue the true benefit to participating. These agents are called optimistic participants. Similarly, there is a set of agents who do not participate because they undervalue the true benefit; they are called pessimistic nonparticipants. If under aggregate rational expectations the proportion of optimistic participants identically equals the proportion of pessimistic nonparticipants, then the equilibrium participation rate will be independent of the type of expectations assumed. As a general rule, however, there is no guarantee that these two groups will be equal. On the one hand, there could be a truncation problem. This is what underlies (1.ii). For example, suppose E [is greater than] B([pi.sup.S]) and C" = 0. In this situation the proportion of pessimistic nonparticipants will be relatively small because the range of participation cost levels from which these agents are drawn is truncated. Hence, more agents incorrectly participate than incorrectly do not participate, which in turn causes [pi.sup.A] [is greater than] [pi.sup.S]. On the other hand, there could be a problem due to nonlinearities in the cost of participating. This is what underlies (1.iii). To understand this point suppose C" [is greater than] 0 and E [is less than or equal to] B([[pi].sup.5]). Optimistic participants are drawn from agents with relatively high values for [X.sub.i], while, given C" [is greater than] 0, high values for [x.sub.i] are associated with increasingly higher marginal costs. This tends to discourage incorrect participation which in tern yields [[pi].sup.S] [is greater than] [[pi].sup.A].

Proposition 1 identifies conditions sufficient to guarantee that standard and aggregate rational expectations yield equivalent results, and identifies cases in which there is a systematic difference between the two equilibria. It should be clear from the proposition and the above discussion that the two equilibria will in general differ. This is because, unless the special conditions stipulated in (1) hold, only by chance will the proportion who incorrectly participate be exactly offset by those who incorrectly do noy participate. (10)

Next, consider a situation in which standard and aggregate rational expectations do not yield equivalent results. How does an increase in the divergence in expectations under aggregate rational expectations affect the size of the difference between the two equilibria?

PROPOSITION 2. Let g([e.sub.i]) be a density function defined on the interval [-E, E] which is the result of a mean-preserving spread of f([e.sub.i]), and let F(a) = [[integral].sup.a/.sub.-E]f([e.sub.i])[de.sub.i] and G(a) = [[integral].sup.a/.sub.-Eg([e.sub.i])[de.sub.i]. Also, suppose F(0) = G(0) and F(a) [is not equal to] G(a) for all a [is not equal to] 0 and -E [is less than] a [is less than] E. If (2.i), (2.ii), or (2.iii) holds, then this mean-preserving spread causes ~[[pi].sup.S] - [[pi].sup.A]~ to increase.

C" = 0 and either E [is greater than] B([[pi].sup.S]) or B([[pi].sup.S]) [is greater than] C(1) - E; (2.i)

C" [is greater than] 0 and E [is less than or equal to] B([[pi].sup.S]); (2.ii)

C" [is greater than] 0 and B([[pi].sup.S]) [is less than or equal to] C(1) - E. (2.iii)

Conditions (2.i), (2.ii) and (2.iii) are the conditions identified in proposition 1 for which there is a systematic difference between standard and aggregate rational expectations equilibria. Proposition 2 therefore states that, given an environment where the standard and aggregate equilibria systematically differ, the size of the difference is positively related to the divergence in expectations under aggregate rational expectations. That is, in terms of participation rates, a mean-preserving spread of the expectations distribution tends to drive the aggregate equilibrium away from the standard equilibrium. The intuition for this result is best understood by considering a particular example. Suppose C" = 0 and E [is greater than] B([[pi].sup.S]) so that [[pi].sup.S] [is less than] [pi.sup.A]]. In this situation [[pi].sup.S] [is less than] [[pi].sup.A] because the proportion of optimistic participants is larger than the proportion of pessimistic nonparticipants due to a truncation problem. In particular, benefits are sufficiently low that the range of participation cost levels from which pessimistic nonparticipants are drawn is truncated. In this situation, a mean-preserving spread of the expectations distribution exacerbates the truncation problem which has the effect of decreasing the proportion of pessimistic nonparticipants. This is in turn causes [[pi].sup.A] to increase, with the final result being that the aggregate equilibrium is driven away from the standard equilibrium.

Finally, again consider a situation in which standard and aggregate rational expectations do not yield equivalent results. How is the size of the difference between the standard equilibrium and the aggregate equilibrium affected by an increase in the severity of the interaction among agents? Define B(z) as a "normalized increasing synergistic transformation" of B(.), where B'(z) [is greater than] B'(z) for all z [epsilon] [0,1], and [[pi].sup.S] is independent of whether the benefits to participating are given by B(.) or B(.). In other words, a normalized increasing synergistic transformation of B(.) is one which increases the severity of the synergism (or equivalently decreases the congestion), but leaves the participation rate under standard rational expectations unchanged. (11)

PROPOSITION 3. If (2.i), (2.ii) or (2.iii) holds, then a normalized increasing synergistic transformation of B(.) causes ~[[pi].sup.S] - [[pi].sup.A]~ to increase.

Proposition 3 shows that if the standard and aggregate rational expectations equilibria systematically differ, then the size of the difference is positively (negatively) related to the severity of the synergistic (congestion) effects. For example, if in a synergistic environment one were to increase the severity of the synergistic effects, but at the same time leave the standard rational expectations equilibrium unchanged, then the participation rate under aggregate rational expectations would be driven away from the standard equilibrium. Note further, an immediate implication of this proposition is that the size of the difference between the two equilibria tends to be larger under synergism than under congestion.

The intuition underlying these results is as follows. Suppose that conditions are such that the proportion of optimistic participants is smaller than the proportion of pessimistic nonparticipants, i.e., [[pi].sup.S] [is greater than] [[pi].sup.A. Consider what happens when there is a normalized increase in the degree of synergism in this situation. Given that it is normalized, [[pi].sup.S] remains unchanged. However, since under aggregate rational expectations the pessimistic nonparticipants are larger in this situation, the higher degree of synergism reduces the return to participation for each agent so that this overly pessimistic behavior is reinforced and [[pi].sup.A] decreases. Now consider what happens when there is a normalized increase in the degree of congestion starting from a situation where [[pi].sup.S] [is greater than] [[pi].sup.A. Under aggregate rational expectations the pessimistic nonparticipants are again larger, but now the higher degree of congestion increases the return to participation for each agent so that there is a tendency for the overly pessimistic behavior to be offset. Hence, an increase in the degree of congestion causes [[pi].sup.A] to increase in this situation. Overall, then with an increase in synergism the behavior of the group whose erroneous beliefs dominate tends to be reinforced, while with an increase in congestion the behavior of the group whose erroneous beliefs dominate tends to be offset.

IV. APPLICATIONS

Many common models are special cases of section III's general model. The analysis of two special cases will both demonstrate the general applicability of the model, and yield additional economic insights.

Application 1: Career Choice

Assume there is a continuum of risk-neutral workers. Each worker must decide whether to choose career j or career k. Prior to participating in a career workers must make an investment in human capital in order to acquire the skills required for the chosen career. For the time period under consideration each worker makes an irreversible choice concerning this decision, and all workers make this choice simultaneously and prior to the realization of the return to each career.

Each worker is a price taker. If worker i chooses career j then the net return is given by [W.sub.j] - [C.sub.j.(x.sub.j]), where [W.sub.j] is the wage earned in careeer j, [C.sub.j.(x.sub.j]) is the cost of acquiring the skills for career j, and [C.sup.1/.sub.j] [is greater than] 0. For career k, the net return is given by [W.sub.k] - [C.sub.k.(1-X.sub.i]), [C.sub.1/.sub.k] [is greater than] 0. The term [x.sub.i] reflects heterogeneity across workers in terms of comparative advantage in acquiring skills for career j have by construction a relatively high marginal cost of skills. That is, workers with a relatively low marginal cost of acquiring the acquiring the skills for career k. The distribution of [x.sub.i]'s across workers is described by a uniform density function over the unit interval.

The total number of workers choosing careers j and k is [N.sub.j] and [N.sub.k] respectively. The wage associted with each career is given by the labor demand equations [W.sub.j] = [f.sub.j]([N.sub.k]) and [W.sub.k] = [f.sub.k]([N.sub.k]), where [f'.sub.j] [is less than] 0 and [f'.sub.k] [is less than] 0. The latter imply that in the terminology of the general analysis this is a model which exhibits congestion. Demand and cost conditions are assumed to be such that, first, all workers choose a career, and second, both careers have positive participation rates.

As in the general analysis, there are two assumptions concerning expectations. The standard rational expectations assumption is that each worker taken individually has correct expectations concerning the resulting value for [W.sub.j] - [W.sub.k]. In contrast, under aggregate rational expectations worker i's expectation concerning [W.sub.j] - [W.sub.k] equals [W.sub.j] - [W.sub.k] + [e.sub.i], where the distribution of [e.sub.i]'s across workers has the properties which characterized te analogous distribution in the general analysis above.

There is a one-to-one correspondence between this model and the one found in the general analysis above. (12) Hence, as in the general analysis, the standard and the aggregate rational expectations equilibria may not be the same because of either a truncation problem or nonlinearities in the cost functions. Of more interest are the following corollaries of propositions 2 and 3. In what follows, the statement that the two equilibria are systematically different means that the analogues of either (2.i), (2.ii) or (2.iii) of proposition 2 hold.

COROLLARY 1. Given a situation where the two equilibria are systematically different, an increase in the dispersion of expectations under aggregate rational expectations will increase ~[N.sup.S./.sub.J] - [N.sup.A./.sub.J]~, ~[N.sup.S/sub.k] - [N.sup.A/sub.k~., ~[W.sup.S/sub.j] - [W.sup.A/sub.j]~ and ~[W.sup.S/sub.k] - [W.sup.A/sub.k]~.

COROLLARY 2. Given a situation where the two equilibria are systematically different, the more inelastic the demand for labor in each sector (i.e., the greater the degree of congestion), the smaller are ~[N.sup.S/.sub.j] - [N.sup.A/.sub.j]~ and ~[N.sup.S/.sub.k] - [N.sup.A/.sub.k~, and the greater are ~[W.sup.S/.sub.j] - [W.sup.A/.sub.j]~ and ~[W.sup.S/.sub.k] - [W.sup.A/.sub.k]~.

These two corollaries yield a number of insights concerning the implications of aggregate rational expectations within a typical labor market context. First, the dispersion of expectations is an important determinant of the difference between predicted wages and allocation of labor under aggregate relative to standard rational expectations. Second, labor demand elasticities are important for determining whether the impact of changes in the dispersion of expectations will be greater on predicted wages or the predicted allocation of labor. This highlights an insight that is not readily apparent from the general analysis. That is, suppose a standard rational expectations assumption is employed to model a labor market situation similar to that considered here, when in fact the situation under consideration is better characterized by aggregate rational expectations. The analysis in this section suggests that if labor demands are inelastic, then the standard rational expectations assumption will yield relatively good predictions concerning the allocation of labor across sectors and relatively poor ones concerning the distribution of wages across sectors. However, with elastic demands the results are reversed--poor predictions on the allocation of labor and good predictions about the distribution of wages. Hence, there is a trade-off between the accuracy of price and quantity predictions, where the relative accuracies depend upon the elasticities of the demand curves under consideration.

Application 2: Trading Externalities and Aggregate Output

This application considers a situation characterized by synergism--in particular, a macroeconomic model where synergism is present because the technology of exchange exhibits positive trading externalities. The model is closely related to one presented in Diamond [1982, 886-87]. (13) Workers and firms are not distinguished. Rather, there is a continuum of agents who must decide whether or not to undertake a production project. If agent i decides to undertake a project, then he produces y units of output at a cost c([x.sub.i]). The heterogeneity in costs across agents captures the idea that, prior to deciding whether or not to produce, each agent i draws a production project from the distribution of projects. The distribution of [x.sub.k]'s in the population is described by a uniform density function over the unit interval.

The key restriction on behavior is that each individual cannot consume what he himself produces, but must rather trade his own output for that which is produced by others. This assumption reflects the advantage that specialized production and trade have over self-sufficiency. Let Y be the aggregate output level. The probability of making a trade is given by p(Y), where the assumption p' [is greater than] 0 captures the trading externality. Untraded output is assumed to be wasted. Further, agents must decide whether or not to produce prior to the realization of p(Y), and it is over this probability that agents form expectations. Finally, agents are assumed to have standard or aggregate rational expectations, where each type is specified in exactly the same manner as in the general analysis above.

Agent i will undertake his production project if it has positive expected value. This implies agent i will (will not) undertake his production project when

[p(Y} _ [e.sub.i]] y [is greater than] ([is less than]) c([x.sub.i]). (8)

Further, the analysis is restricted to parameterizations for which there is a unique value for aggregate output. (14)

The correspondence between the current model and the one found in the general analysis allows for a series of results that follow immediately from the propositions in the general analysis. As in the general analysis, the standard and aggregate rational expectations equilibria may differ because of either a truncation problem or nonlinearities in the cost of undertaking a project. Of more interest are the following two corollaries that follow immediately from propositions 2 and 3.

COROLLARY 3. Given a situation where the two equilibria are systematically different, an increase in the dispersion of expectations under aggregate rational expectations will result in an increase in ~Y.sup.S - Y.sub.A~.

COROLLARY 4. Given a situation where the two equilibria are systematically different, an increase in the severity of the trading externality will result in an increase in ~Y.sup.S - Y.sub.A~.

Corollaries 3 and 4 indicate some of the factors which can affect the size of the aggregate output difference between standard and aggregate rational expectations equilibria. Of particular interest is the result concerning dispersion of expectations. A number of empirical studies have found that dispersion of expectations concerning inflation is an important factor in the determination of aggregate output. (15) In particular, increases in the dispersion of expectations have in general been found to lead to decrease in aggregate output. Although the expectations here do not concer inflation, corollary 3 states that in this model an increase in the dispersion of expectations can yield a similar result. That is, even constraining expectations to be rational in the aggregate, the analysis states that an increase in the dispersion of expectations can depress aggregate output. (16) This suggests that theoretical work employing an aggregate rational expectations assumption may prove fruitful in the explanation of this empirical observation.

One final comment concerns the relationship between the results derived here and those of Diamond. Diamond deals solely with a standard rational expectations assumption. Further, in order for his model to be consistent with fluctuations in aggregate output, he assumes that the trading externality is sufficiently severe that multiple equilibria exist. Our analysis suggests that the existence of multiple equilibria may not be necessary for trading externalities to be important for explaining fluctuations in aggregate output. Rather, under an aggregate rational expectations assumption, changes in the dispersion of expectations yield interesting fluctuations in aggregate output even when multiple equilibria are ruled out. (17)

V. CONCLUSION

In practice rational expectations has typically meant that the expectation of each agent taken separately is by itself consistent with the predictions of the relevant economic theory, i.e., what is called here standard rational expectations. This differs, however, from the argument frequently put forth by proponents of the rational expectations assumption to justify its use. That argument is that on an aggregate level it would be surprising if expectations were inconsistent with the predictions of the relevant theory. The employment of the stronger assumption of standard rational expectations is then justified by the argument that, if expectations were rational in the aggregate, then expectational deviations across agents would tend to cancel out. This paper formally investigates the relationship between standard and what is called here aggregate retional expectations.

The first finding is that the above argument is incorrect, i.e., only under very special conditions do standard rational expectations and aggregate rational expectations yield equivalent results. The remaining findings concern the factors which determine the size of the difference. The size of the difference will be larger (i) the larger is the divergence in expectations under aggregate rational expectations, and (ii) the more synergistic is the environment. This last result s of particular interest because of its relationship to our own earlier work on the rebustness of rational expectations equilibria (Haltiwanger and Waldman [1985]). That work shows that in a world characterized by congestion, standard rational expectations equilibria are relatively robust to the introduction of agents who make systematic errors. However, in a world characterized by synergism, the introduction of such agents can have a dramatic effect. In other words, although the two papers employ different criteria to judge whether standard rational expectations equilibria are robust, they reach the same conclusion: standard rational expectations equilibria are robust in environments of congestion but not synergism.

This paper has compared standard rational expectations and aggregate rational expectations where the overall distribution of expectations under the two regimes differ, although both regimes exhibit aggregate unbiasedness. In particular, aggregate rational expectations was characterized by disperse expectations while standard rational expectations lacked any dispersion. Given that the two regimes potentially generate significantly different predictions, it is of interest to investigate the ramifications of alternative sources of dispersion. In the formal analysis above the source of the dispersion is not of much importance since it is assumed (see footnote 5) that agents cannot obtai or infer any information on the beliefs of other agents prior to making decsions. However, a possible issue for future research might be to relax this assumption. It would then be of interest to compare alternative expectations regimes where the distribution of expectations across regimes is the same, both satisfy aggregate unbiasedness, but the source of the heterogeneity in expectations is differenct. For example, it would be of interest to compare a regime with dispersion generated from agents facing different information sets to one with dispersion generated from systematic errors. Our conjecture s that even in this type of setting there might be significant differences between equilibria across the regimes. If dispersion is due to differences in information, then to the extent possible when making decisions individuals will attempt to incorporate the information held by other agents. For example, some decisions may be delayed until the information held by other agents can be inferred from market prices. On the other hand, if dispersion is due to systematic errors, then agent swould probably not attempt to infer the opinions of others before making their own decisions. (18) Hence, even though the initial distribution of expectations would be the same, the two environments can be expected to work quite differently. Further, based upon the above analysis, the differences are apt to be larger if synergism is present since synergism provides incentives for systematic errors to be reinforced.

APPENDIX

Before proceeding to the proofs, it is helpful to note that (7) can also be written as

[Mathematical Expression Omitted]

where [theta].sup.A.=max(E,C(x.sub.i.)-B([pi].sup.A.)) and [phi].sup.A.=max(-E,C(x.sub.i.)-B ([pi].sup.A)). Using (A.1) and reversing the order of integration, [pi].sup.A can be written as

[Mathematical Expression Omitted]

where M.sup.A.= min[C(1)-B([pi].sup.A.),E], N.sup.A.=max[-E, -B([pi].sup.A.)] and x.sup.A satisfies C(x.sup.A.)-B([pi].sup.A.)-e.sub.i.=0. Further, it is helpful to define a function, H(z), that satisfies

[Mathematical Expression Omitted]

where [M.sup.z]=min[C(1)-B(z),E], [N.sup.z]=max[-E,-B(z)] and [x.sup.z] satisfies C([X.sup.z])-B(z)-[e.sub.i]=0. Assumptions (2) - (6) imply that H' [is greater than]0, and in turn H(z) [is greater than]0 for z [is greater than] [pi.sup.A], H(z)=0 for z= [pi.sup.A], and H(z) [is less than] 0 for z [is less than] [pi.sup.A]. We can now proceed to the proofs.

Proof of Proposition 1. (i) Letting [pi].sup.A.=[pi].sup.S in equation (A.1) under the restrictions E [is less than or equal to] B([pi].sup.S.) [is less than or equal to] C(1)-E and C" = 0 shows that [pi].sup.A.=[pi].sup.S is a solution to equaltion (A.1). Since there is a unique solution to (A.1), this implies that [pi].sup.A.=[pi].sup.S..

(ii) Consider the case E>B([pi].sup.S.). Suppose [pi].sup.S.[is less than or equal to] [pi].sup.A.. This implies H([pi].sup.S.)[is less than or equal to]0. Further, the assumption that C"=0 together with the other assumptions yields

[Mathematical Expression Omitted]

where x.sup.S.=[B([pi].sup.S.)+e.sub.i.]/C'. Using the assumption that [Mathematical Expression Omitted] and that by construction x.sup.S.=[pi].sup.S for e.sub.i.=0 allows us to rewrite (A.4) as

[Mathematical Expression Omitted]

However, (A.5) implies H([pi].sup.S.) <0, i.e., a contractiction. Thus, [pi].sup.A.> [pi].sup.S.. The case B([pi].sup.S.)>C(1)-E follows in a similar fashion.

(iii) Consider the case C">0. Suppose [pi].sup.S.[is less than or equal to][pi].sup.A.. This implies H([pi].sup.S..)[is less than or equal to]0. However, under the assumptions in this case H([pi].sup.S.) can be written as

[Mathematical Expression Omitted]

where x.sup.S satisfies C(x.sup.S.)-e.sub.i.=0. Let x.sup.S.=x.sup.S when e.sub.i.=0 and define [k.up.S.=x.sup.S.+(x.sup.S./B([pi].sup.S.))e.sup.i.. Using these definitions and given the that [pi].sup.S.=x.sup.S., (A.6) can be rewritten as

[Mathematical Expression Omitted]

Further, since C">0, k.sup.S.[is less than or equal to] x.sup.S where the ineuality is strict except for e.sub.i.=0. Hence, (A.7) yields

[Mathematical Expression Omitted]

Using the definition of k.sup.S and the assumption that [Mathematical Expression Omitted] yields that the right-hand side of (A.8) equals zero. This implies H([pi].sup.S.)>0 which is a contradiction. Hence, [pi].sup.S.>[pi.sup.A.. The case C" <0 follows in a similar fashion.

Proof of Proposition 2. Given the assumptions of g.(e.sub.e.) and f.(e.sub.i.),

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted]

In what follows, let H.sub.g.(z) be defined analogously to H(z) but with the distribution g.(e.sub.i.) substituted for f(e.sub.i.). Similarly, let [pi].sup.A.sub.g be aggregate rational expectations equilibrium when e.sub.i has the distribution g(e.sub.i.).

First, consider (i) under the assumption E > B([pi].sup.S.). Proposition 1 yields [pi].sup.A.>[pi].sup.S and [pi].sup.A.sub.g > [pi].sup.S.. Thus, we need to prove that [pi].sup.S.sub.g.>[pi].sup.A.. Using the restrictions on f(.) and g(.), H.sub.g.([pi.sup.A.)-H([pi].sup.A.) can be written as

[Mathematical Expression Omitted]

Given (A.9), (A.10) and F(0)=G(0), (A.11) implies H.sub.g.([pi].sup.A.)-H([pi].sup.A.)<0. Since H([pi].sup.A.)=0, this implies H.sub.g.([pi].sup.A.)<0. Since H.sub.g.([pi].sup.A.sub.g.)=0 and G'.sub.j.>0, this in turn yields [pi].sup.A.sub.g.>[pi].sup.A.. Thus, (i) under the assumption E > B([pi].sup.S.), ~ [pi].sup.S.-[pi].sup.A.~ is increasing as a result of a mean-preserving spread of f(.). The other cases (i.e., (i) under the assumption B([pi].sup.S.)> C(1)-E, (ii) and (iii)) follow in a similar manner.

Proof of Proposition 3. Let H(z) have the same formulation as H(z) but with B(.) substituted for B(z). Also, let [pi].sup.A be the aggregate rational expectations equilibrium when the benefit to participating is given by B(.). Since B([pi].sup.S.)=B([pi].sup.S.), this implies H([pi].sup.S.)=H([pi].sup.S.). Further, since B' > B', H(z)> ([is less than])H(z) if z [is less than] ([is greater than]) [pi].sup.S.].

First, consider (i) under the assumption E>B([pi].sup.S.) or (iii). By proposition 1, this implies [pi].sup.A.>[pi].sup.A and [pi].sup.A.>[pi].sup.S.. This means we want to prove [pi].sup.A.>[pi].sup.A.. Suppose [pi].sup.A.[is less than or equal to] [pi].sup.A.. This implies H([pi].sup.A.)<H([pi].sup.A.)[is less than or equal to] H([pi].sup.A.) where the latter inequality is due to H' > 0. Yet, this yields a contradiction since by construction H([pi].sup.A.)=H([pi].sup.A.)=0. Thus, given either (i) under the assumption E = B([pi].sup.S.) or (iii), a normalized increasing synergistic transformation of B(.) causes ~ [pi].sup.S.-[pi].sup.A.~ to increase. The other cases (i.e., (i) under the assumption B([pi].sup.S.) > C(1)-E and (ii)) follow in a similar manner.

(1) Schultze questions the validity of Muth's claim for the analysis of environments with implicit contracts. From this perspective our analysis can be partially interpreted as saying that, even in the absence of implicit contracts, Muth's calim is incorrect.

(2) Aggregate rational expectations does not necessarily mean that agents are not behaving "rationally" under some broad definition of the term, but rather that individuals do not satisfy the standard criterion for rational expectations that appears in the literature.

(3) The concepts referred to as congestiona dn synergism have appeared elsewhere in the literature under different names. What is referred to as congestion here has elsewhere been referred to as decreasing returns (see Hirshleifer [1982; 1985] and Schelling [1978]) and strategic substitutes (see Bulow et al. [1985] and Cooper and John [1988]). Similarly, what is called synergism has been referred to as increasing returns (Hirshleifer, Schelling), strategic complements (Bulow et al., Cooper and John), and network externalities (see Farell and Saloner [1985; 1986], and Katz and Shapiro [1986]).

(4) Other papers which consider this type of heterogeneity include Conlisk [1980], Akerlof and Yellen [1985a, b], Russell and Thaler [1985], and Haltiwanger and Waldman [1989]. See Kahneman, Slovic and Tversky [1982] for evidence that expectational errors of agents do tend to be conrelated across individuals.

(5) It is assumed that there is not means by which agents can obtain or infer the beliefs of others prior to making the participation decision. For example, there is no allowance for a futures market on the benefits to participating in an activity. Such a market would clearly act as an information aggregator. For most of the real world examples that have been mentioned, the lack of an organized futures market is in accordance with empirical observation. For a discussion of which factors determine the existence of futures markets and how the presence or absence of futures markets influence expectations formation, see Russell and Thaler [1985].

(6) The interpretation of B(.) and C(.) when treating this as a model of the choice between two activities is slightly different. Suppose the two activities are X and Y. Then B can be interpreted as the net benefit of choosing X and C as the net cost of choosing X, where [pi] would be the proportion of agents who choose X and 1-[pi] the proportion of agents who choose Y.

(7) In our model, standard rational expectations implies perfect foresight. However, the results here easily generalize to a specification in which the benefit from participating is stochastic, agents are risk neutral, and under standard rational expectations each agent has unbiased expectations about the expected benefit.

(8) In evaluating (7) it is important to recall that by assumption f(.) is identically zero for e.sub.i outside the range [-E, E]. Hence, for x.sub.i such that C(x.sub.i.)-B([pi].sup.A.) <-E the inner integral in (7) is equal to one, while for x.sub.2 such that C(x.sub.i.)-B([pi].sup.A.)>E the inner integral is equal is equal to zero.

(9) Note that, even given these two restrictions, standard and aggregate rational expectations do not yield equivalent results regarding social welfare. Under aggregate rational expectations individual mistakes are being made. This implies that even when the proportion of agents participating is independent of the type of expectations assumed, social welfare is lower under aggregate rational expectations because agents do not sort themselves efficiently among activities.

One might at first think that this logic should lead to a general conclusion that social welfare is ower under aggregate rational expectations than under standard rational expections. This is incorrect. When the externalities being considered are technological (as opposed to pecuniary), standard rational expectations does not typically yield a first best results with regard to the proportion of agents participating. Combining this with the result that the aggregate rational expectations participation rate can be different than the standard rational expectations participation rate, the aggregate rational expectations participation rate may actually be closer to the first best. The subsequent result is that for some cases the social welfare ranking is actually higher under aggregate than under standard rational expectations.

(10) Note than even in the absence of any truncation problems, it is not necessary for C" to be everywhere the same sign for the two equilibria to be different. Rather, it can be demonstrated that it is only necessary that C" is not everywhere equal to zero for there to be, in general, a difference between the two equilibria.

(11) This transformation is derived by a rotation of the B function with the standard rational expectations equilibrium as the origin. Note further that the transformation is restricted such that B satisfies equations (2), (3), and (5).

(12) To be precise, let B(N.sub.j.)=W.sub.j.-W.sub.k=f.sub.j.(N.sub.j.)-f.sub.k.(1-N.sub.j.) and C(x.sub.j.)=C.sub.j.(x.sub.i.)-C.sub.k.(1-x.sub.i.). Given the assumptions made, these B and C function satisfy all of the properties of the B and C functions in the general analysis above.

(13) There is a growing literature which investigates the ramifications of synergism in macroeconomic settings. See Cooper and John [1988] for a survey of this recent literature.

(14) That the severity of the trading externality is restrictecd so that a unique equilibrium exists represents a significant departure from Diamond. Further discussion of this difference is provided below.

(15) See, for example, Mullineaux [1980] and Lovi and Makin [1980]. These empirical studies are an outgrowth of a discussion in Friedman [1977] concerning the influence of an increase in the dispersion of expectations on aggregate economic activity.

(16) This will be true if conditions are such that Y.sup.S.> Y.sup.A.. By proposition 1, a reasonable condition that will yeld this result is C" >0.

(17) See also Haltiwanger and Waldman [1989].

(18) This is especially true given the propensity of individuals to overestimate their own abilities, i.e., underestimate their own likelihood of making mistakes. See Kahneman, Slovic, and Tversky [1982] for evidence concerning the overestimation of abilities.

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