# On the notion of equilibrium when agents use mis-specified models

Jean-Marc Tallon

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CAHIERS ECO & MATHS

94.26

ON THE NOTION OF EQUILIBRIUM

WHEN AGENTS USE MIS-SPECIFIED MODELS

Jean-Marc TALLON

MAD
# On the notion of equilibrium when agents use mis-specified models

Jean-Marc Tallon*

First draft: February 1994

This draft: March 1994

## Abstract

This paper deals with equilibrium analysis in models where agents have mis-specified models of the economy. It is assumed, in a pure exchange two-period economy, that households do not necessarily agree on the list of possible states of nature. Each will thus express a demand based on a particular model of the economy, that is internally consistent (i.e., in which the rational expectations assumption is made). An equilibrium in this setup is defined to be a temporary equilibrium at which the observed price is consistent with all agents' models and clears markets. Thus, at an equilibrium, agents cannot infer from the observed price that their model is mis-specified. Examples illustrate this equilibrium concept.

## Résumé

Nous analysons dans cet article un modèle dans lequel les agents croient en des modèles mal spécifiés de l'économie. On suppose, dans le cadre d'une économie d'échange à deux périodes, que les agents ne sont pas nécessairement d'accord sur la liste des états de la nature. Chaque consommateur exprime une demande basée sur son propre modèle de l'économie, modèle cohérent et dans lequel l'hypothèse d'anticipations rationnelles est faite. Un équilibre de ce modèle est un équilibre temporaire tel que le prix aujourd'hui est compatible avec le modèle de chaque agent. A l'équilibre, les agents ne peuvent pas inférer du système de prix qu'ils ont un mauvais modèle de l'économie. Des exemples illustrent ce concept d'équilibre.

# 1 Introduction

In traditional representations of uncertainty in general equilibrium theory (e.g., Arrow [1953], Debreu [1959]), the notion of state of nature is essential. A state of nature is a complete list of

<footnote>* M.A.D.-C.N.R.S. Address : M.A.D., Université Paris I, 90 rue de Tolbiac, 75634 Paris Cedex 13, France. e-mail: jmtall@frmop22.cnusc.fr. I would like to thank F. Collard, J.P. Drugeon and F. Portier as well as participants at the MAD seminar, for useful comments.</footnote>
relevant data describing the economy at a point in time. At any moment, states must be mutually exclusive and exhaustive. In other words, agents are assumed to be able to construct a list of all “possible futures”. A point that is probably less emphasized is that in equilibrium analysis, all agents are assumed to build the same list of possible futures. This assumption is crucial for most studies of multi-period models. The concept of a rational expectations equilibrium (see Radner [1982]) for example presupposes the existence of an agreement on the possible states as well as on the explicit descriptions of the data in each state. How one can define a (sensible) equilibrium concept if this agreement is assumed away is the main concern of this paper. The motivation for abandoning this traditional assumption is twofold. First, it allows to study situations in which agents are surprised by an event. Second, relaxing it seems to go in the direction of recent developments in the theory of decision under uncertainty.

More precisely, in most models of asymmetric (or incomplete) information, agents might have different partitions over the set of states of nature, but they typically agree on zero-probability events. This assumption is rarely discussed, but from a technical viewpoint, whereas it is an hypothesis that needs to be made for the consistency of the model, as pointed out by Allen [1986] :

> "If agents disagree about the definition of subsets of states of the world which occur with probability zero, rational expectations equilibrium need not be well defined."
> (Allen [1986] p.4).

In this literature, one of the main results is that if the state space is finite the equilibrium price reveals all the private information to the market, i.e., equilibria are typically fully revealing. Thus, the price vector, and consequently “the market mechanism”, is efficient in diffusing private information in the economy. However, as previously mentioned, this result presupposes some exogenous coordination among agents, for they are assumed to agree on zero-probability events. In particular, in the case of a finite state space, agents are never truly surprised by the occurrence of an event (or of a particular price) since every possibility was expected with some positive probability. This absence of surprises could in itself be considered as a “bad implication” of the model, for it is clear that agents do sometimes get surprised by the occurrence of unforeseen events.

May be more convincingly, this construction which essentially rests on the foundations laid by Savage, has been questioned in the theory of decision under uncertainty<fnref n="1" />. Indeed, theories such as the one developed by Lipman [1993] or Gilboa and Schmeidler [1992]a basically assert the existence, at least in people's mind, of "impossible possible states". These states are logically impossible but agents believe they are in fact possible. Gilboa and Schmeidler [1992]b also point out the difficulties of an analysis in terms of states of the world:

<footnote n="1">¹See for instance Karni and Schmeidler [1991] for a survey.</footnote>
"In many decision problems under uncertainty, states of the world are neither naturally given, nor can they be simply formulated." (Gilboa and Schmeidler [1992]b p.1).

As pointed out by Machina [1987] for instance, these developments are not all directly relevant for the study of general equilibrium theory (in particular they deal only with one decision maker and static choices). However, the important message for general equilibrium theory is, I believe, that defining a complete list of states of nature is not such an easy task<fnref n="2" />.

If Gilboa and Schmeidler's claim<fnref n="3" /> is true at an individual's level, it seems even more convincing when comparing agents. If a state cannot be easily formulated by an agent alone, there is little chance that another agent will formulate the same one in the same situation. Thus I take the assumption that agents do not anticipate the same states to be plausible in a decentralized framework. There is no reason that different agents would reach the same conclusions as to what (exogenous) events could occur in the future (reflecting the co-existence of different conflicting theories at a point in time). Thus, this coordination, assumed in most of the literature on rational expectations equilibrium, is assumed away in this paper, which will be mainly concerned with the definition of a notion of equilibrium in this setup.

More specifically, it is assumed that agents do not necessarily agree on all possible unfoldings of the future or admissible states of the world, and therefore concieve of different “models of the economy”. The issue is then that of agents’ coordination on models of the economy. The basic idea is that it is possible to define a (temporary) equilibrium concept in such a way that it is the first-period component of different rational expectations equilibria, the latter corresponding to different (mis-) specifications of the exogenous future. Therefore, an agent, seeing a particular price in the first period cannot infer that he has a mis-specified model of the economy in mind when making his decisions.

For instance, I might have a “consistent model of the future” in mind when making my decisions today, say it might rain or be sunny tomorrow, while you believe in the occurence of some other states, say, that it might only be sunny or foggy (the overlap of agents’ expectations is not necessary as will become clear below). Therefore, we will take decisions today based on mis-specified models of the economy, even though these models might be intrinsically consistent. By consistent I mean here that each agent has a fully specified model in mind, that is, a complete description of what he thinks are possible futures, as well as corresponding equilibrium prices<fnref n="4" />.

<footnote n="2">²This remark is not new for it essentially is at the heart of any analysis in terms of sunspots (see e.g. Cass and Shell [1983]). Indeed, a sunspot precisely reflects the idea that agents are unable to assess what are the relevant exogenous variables they need to know to understand the model.</footnote>

<footnote n="3">³Gilboa and Schmeidler [1992]b go even further and question the very relevance of expected utility theory in many circumstances: “It seems that in many situations of choice under uncertainty, the very language of expected utility is inappropriate”. They propose an alternative theory of decision under uncertainty based on reasoning on the basis of cases previously observed.</footnote>

<footnote n="4">⁴At this stage, the model an agent has in mind can be basically anything, and the associated “equili-</footnote>
If the equilibrium price today is coherent with both models, then there is no incentive on either side to revise one's view of the future. Thus, at this (temporary) equilibrium, each agent wrongly believes that his model of the economy is the right one. This in turn implies that, comes the next period, agents might be surprised either by the realisation of the state or by the value of equilibrium prices (this even if the state that occurs was one expected by the agent, since they based their price expectations on a wrong model of the economy). This seems a natural way to introduce the notion of surprise in an economic model. Indeed, agents (research institutes) use a great deal of effort trying to make accurate forecast, which are almost always (at least in times of large uncertainty) proven wrong. One could argue that this is the case because they use mis-specified models, and are unable to contemplate all possible futures.

The equilibrium concept that one should use in this setup is not that clear. Notice first that it is not possible to define an intertemporal equilibrium in this setup, unless one explicitly deals with the issue of bankruptcy, for agents do not agree on the future<fnref n="5" />. As a consequence, the concept we will look at qualifies as a temporary equilibrium in that markets today only are required to clear. Therefore, we are led to define an “individually consistent temporary equilibrium” (or I.C.T.E.) to be a price vector today and (possibly different across agents’) expectations of prices tomorrow, such that markets today clear and, for each agent, today’s price vector together with his price expectations is a rational expectations equilibrium of his model. Thus, even though it is a temporary equilibrium, it entails restrictions on price expectations, a point that will be discussed more extensively after the presentation of the model. Indeed, the equilibrium concept defined has special characteristics that distinguish it from usual temporary equilibria. Such a temporary equilibrium is illustrated and is shown to exist by way of examples. At this equilibrium, agents cannot infer that the model they have in mind is not the true one, or that other agents are using different models.

That examples of existence of I.C.T.E. can be found should of course not be taken as a general existence result. Indeed, one obtains existence only if certain constraints on agents' endowments expectations are met, as will be clear from examples 1 and 2. Thus, a consistent temporary equilibrium will not, in general, exist. This raises the question of what would an agent do if he observed a price inconsistent with his model of the economy. A possible answer to that question is that he will be led to revise his model of the economy<fnref n="6" />. An alternative approach would be to stick to the usual temporary equilibrium analysis as much as possible, and to define expectations

<footnote>prices need not come from the solution of a rational expectations equilibrium. For example, one could probably develop a theory with other consistency requirements between the agent's (exogenous) model and its prediction of endogenous variables. However, in order to adopt a simple modeling, we will focus on the case where this link is the solving of a rational expectations problem.</footnote>

<footnote n="5">⁵See Tallon [1993] for a model where agents have different support for their expectations of possible states, and where an intertemporal (rational expectations) equilibrium with bankruptcy is defined.</footnote>

<footnote n="6">⁶Another answer would be that agents will not revise their model and that the temporary equilibrium will entail some rationing.</footnote>
functions satisfying some restrictions, that will be exposed below. This would permit to define
today's demand as a function of prices and endowments today alone. This approach also suffers
from conceptual difficulties as will be clear below.

The type of models revision that occurs if there is no prices coordinating agents' actions is
of the following sort. If today's prices are inconsistent with my model of the economy (say, that
it might rain or be sunny tomorrow), then I'll try to come up with a model (say, it might be
sunny, raining or snowing) that would yield the observed prices as an equilibrium price vector.
Clearly, this "revision" is of the eductive type (in the spirit of Guesnerie [1992]), and does
not take place in time. To take this into account, one is then led to define another concept of
temporary equilibrium, similar to the I.C.T.E., except that it endogenizes agents' expectations
about possible futures, i.e., it endogenizes the model an agent believes in. This equilibrium will be
called a (individually consistent) temporary equilibrium in price, probabilities and endowments
expectations.

These equilibria can themselves be of two types. The first type is called “coordinating”, and
corresponds to equilibria at which all agents have the same vision of the future (whether it is the
“true” one or not). The second, on the contrary, consists of equilibria at which different agents
believe (without being proven wrong in the first period) in different models.

The possibility that the first-period equilibrium price of a particular rational expectations e-
quilibrium could be obtained as the temporary equilibrium of a model where agents mis-represent
the economy (while behaving rationally in their own setup) is interesting for several reasons.
First, it allows agents to make mistakes in their forecast, while acting “rationally” (or at least
less arbitrarily than in traditional temporary equilibrium models). Two types of forecasting mis-
takes can be made. The first one consists of having omitted to forecast the state that actually
occured. The second one lies in the fact that agents are using mis-specified models to forecast
endogenous variables. Thus, comes the second period and a state that was expected, there is no
reason that the observed price will be the same as the one that was anticipated on the basis of
the “wrong” model of the economy. This points out that the price system does not coordinate,
or even “worse”, wrongly coordinates agents’ actions and beliefs in models encompassing pos-
sible mis-specifications. Second, as pointed out by Blume and Easley [1984] “a theory of market
evolution which requires all traders to have full knowledge of the structure of the economy is at
least imperfect, if not misguided”. The purpose of this paper is not to propose a fully specified
model of “market evolution” when traders have imperfect knowledge of the economy, but only
to suggest a possible representation of such an imperfect knowledge, that is different from what
is usually found in the literature<fnref n="7" />.

An obvious shortcoming of the paper is that it does not deal with any type of learning. Indeed,

<footnote n="7">⁷However, see e.g. Nyarko [1991] and references therein for examples where agents have mis-specified models of the economy.</footnote>
what happens in the next period is, strictly speaking, outside of the model. In a trivial sense, however, one could obtain convergence to the “true model” if the environment is stationary, for there is no information asymmetry and agents observe the state in each period<fnref n="8" />. It is nevertheless not clear how agents would revise their probability distribution upon seeing an event to which they had assigned a zero probability. This points out the need for a better understanding of the formation of agents’ expectations for both endogenous and exogenous variables.

Section 2 presents a formal model while section 3 is devoted to examples. Section 4 contains concluding comments.

## 2 The model

To keep the model developed here as simple as possible, the equilibrium concept will be looked at in a two-period pure exchange economy. There are *C* goods ($c = 1, ..., C$) and *H* agents ($h = 1, ..., H$). Each agent believes that the possible states of nature tomorrow (that specify utility functions and endowments for all agents in the economy) form a set<fnref n="9" /> $S_h$. Thus, each agent has a different view of what the future is made of if we assume that these sets are different across households.

A typical state is denoted $s$, prices in that state $p(s)$, and consumption by household $h$, $x_h(s)$. Today will be referred to as spot 0. An agent can transfer wealth across spots by holding $I$ assets ($i = 1, ..., I$), with return matrix $Y$ and prices $q$. A row $y(s)$ of this matrix specifies the returns of the different assets in a given state $s$. In order to be consistent with the structure of the model, the returns of an asset must be specified in such a way that an individual $h$ does not learn from the matrix that some other agents believe in the occurrence of states not in $S_h$. As a consequence, assets' returns should be given in the following way: asset $i$ yields $y^i(s)$ in state $s \in \bigcap_h S_h$ and $\bar{y}^i$ in states not in $\bigcap_h S_h$. This simply means that financial contracts specify returns state by state for commonly expected states, and include a clause "if none of these states occurs, the asset pays off a certain amount." Indeed, if the matrix is specified like this, agents do not learn from the matrix any information about other agents' models<fnref n="10" />. This would imply that the rank of this matrix cannot exceed $\bigcap_h S_h + 1$, and therefore that agents are likely to face incomplete markets. Observe that, in this formulation, assets could be either nominal or real, depending on the form taken by the returns $y^i(s)$. The specific form of these assets is not

<footnote n="8">⁸However, see Kurz [1990] for an argument that different rational beliefs (i.e., that cannot be proven wrong just by observing the data realisation) can coexist, even in a stationary environment.</footnote>

<footnote n="9">⁹$S_h$ will denote both the set and its cardinality, the context making it clear which meaning is intended.</footnote>

<footnote n="10">¹⁰An alternative interpretation would be to say that agents agree on the denomination of the states, but do not agree on what the data are in a state. For example, we agree that it can rain or snow tomorrow, but I might think that endowments are high if it snows and low if it rains, while you believe the reverse. In that case assets could have returns depending on the label of the state, without revealing to an agent that other agents believe that the state "rain" actually means something different for them.</footnote>
important for the rest of the analysis, and will be explicitly specified in the examples.

Endowments at spot 0 are known and denoted $e_h(0)$. Endowments in state $s \in \cup_h S_h$ are equal to $e_h(s)$ for household $h$. Utility functions $u_h$ are assumed to be separable across states, and only in order to keep notation to a minimum, will be taken to be equal across states. There is no uncertainty on utility functions, all the uncertainty in the economy resting on endowments and probabilities. We make the admittedly questionable assumption (especially with regard to the motivation given for partially "abandoning" the standard theory of choice under uncertainty) that agents maximize their expected utility<fnref n="11" />. Therefore, each agent has a utility function $u_h: \mathbb{R}_+^C \times \mathbb{R}_+^C \rightarrow \mathbb{R}$, that is assumed to be strictly concave. The overall utility is consequently given by $v_h(x_h) = \sum_{s \in S_h} \pi_h(s)u_h(x_h(0), x_h(s))$, where $\pi_h(s)$ is the (subjective) probability that $h$ places on the occurrence of state $s$. Observe that we make the assumption, at this stage, that an agent is confident in his model, i.e., does not realize that he might have not foreseen some contingencies<fnref n="12" />.

It is now possible to state the maximisation problem a typical agent solves, referred to as $P_h(h)$:

$$ \begin{array}{ll} \max_{x_h, b_h} & v_h(x_h) \\ \text{s.t.} & \begin{cases} p(0)(x_h(0) - e_h(0)) = -qb_h & P_h(h) \\ p(s)(x_h(s) - e_h(s)) = y(s)b_h & \text{for all } s \in S_h \end{cases} \end{array} $$

In writing this program, it is assumed that household $h$ is able to expect the right equilibrium prices had his model been the correct model of the economy, that all agents use. Therefore, the equilibrium prices of the economy according to $h$ are denoted $(p_h(0), p'_h)$ where $p'_h = (p_h(s))_{s \in S_h}$.

More precisely $(p_h(0), p'_h)$ is an equilibrium price vector of the model where each agent $j$ solves the program $P_h(j)$:

$$ \begin{array}{ll} \max_{x_j, b_j} & v_j^h(x_j) \\ \text{s.t.} & \begin{cases} p(0)(x_j(0) - e_j(0)) = -qb_j & P_h(j) \\ p(s)(x_j(s) - e_j(s)) = y(s)b_j & \text{for all } s \in S_h \end{cases} \end{array} $$

that is, where agent $j$ uses $h$'s model of the economy. Observe that in the program $P_h(j)$, the objective function is indexed by $h$. More precisely, one has $v_j^h(x_j) = \sum_{s \in S_h} \pi_h(s)u_j(x_j)$, that is, $h$ assigns his probability assessment to $j$'s utility. How an agent is able to compute such an equilibrium price vector is an issue that will not be approached, as in usual models of rational expectations. However, one of the two rationale usually advanced to justify that assumption, namely that agents know (have learned?) the true equilibrium price function, cannot

<footnote n="11">¹¹Thus, even if we agree with Gilboa and Schmeidler [1992]b that defining states of nature is an incredibly difficult task, we do not reject the associated hypothesis that agents maximize an expected utility function. However, the only assumption we truly need concerns the separability of the utility function across states.</footnote>

<footnote n="12">¹²See Kreps [1990] for a discussion of the case where agents know they do not foresee all possible contingencies.</footnote>
be considered here since the knowledge of this function would reveal some information to agents. Therefore, one is left with the second “justification”, that agents act “as if” they were able to compute the equilibrium price of their model, even though this requires that they know all the relevant information and have incredible computations ability. There is no real coordination issues at this point since these computations are done at an individual level<fnref n="13" />.

One can then define an individually consistent temporary equilibrium, to be a price vector such that markets clear in the first period and it is compatible with all agents' different models. Denote the solution to $P_h(j)$ by $x_j^h$ and $b_j^h$. The formal definition is then:

**Definition 1**

A price vector $(p(0), q, (p'_1, ..., p'_H))$ and an allocation $(x_j^h, b_j^h)_{h,j=1,...,H}$ are a individually consistent temporary equilibrium if:

* (i) Given $(p(0), q, (p'_1, ..., p'_H))$, $(x_j^h, b_j^h)$ is a solution to $P_h(j)$ for all $h$ and all $j$,

* (ii) $\sum_j (x_j^h(0) - e_j(0)) = 0$, $\sum_j (x_j^h(s) - e_j(s)) = 0$ and $\sum_j b_j^h = 0$ for all $h$ and all $s \in S_h$,

* (iii) $\sum_h (x_h^h(0) - e_h(0)) = 0$ and $\sum_h b_h^h = 0$.

Condition (i) ensures that agents maximize their utility, conditional to the model they expect. Condition (ii) says that the model each agent has in mind is consistent in the sense that the observed price is the first-period component of a rational expectations equilibrium of that model. Indeed, for the prices expected by agent $h$, markets clear at all spot (today and tomorrow) in $h$'s model. Finally, condition (iii) expresses the fact that markets clear today. Thus, the first-period demand, based on the observation of $(p(0), q)$ and on individual (rational) expectations is equal to total endowment. Observe that this equilibrium concept entails restrictions on which price expectations are “allowable”, since they must come from the (rational expectations) solution of a fully specified, coherent model. These restrictions therefore come from expected equilibrium conditions, the difference with standard intertemporal equilibrium being that those conditions are agent specific, and are consequently not necessarily fulfilled in the second period.

If at equilibrium all agents' expectations are the same (i.e., $p'_h = p'_{h'}$ for all $h, h'$, which in particular implies that $S_h = S_{h'}$), the equilibrium will be said to be coordinating, while it is uncoordinating if price expectations are different across households. Observe that it is possible to obtain equilibria at which coordination is effective, but on the basis of the wrong model (this is a point made in example 3).

The equilibrium thus defined (whether it is coordinating or not) is “non-informative” in the sense that agents cannot infer from first-period prices that the model they are using to compute their demand is potentially mis-specified. This is the reason why it is called “individually

<footnote n="13">¹³See Guesnerie [1993] for more on this problem.</footnote>
consistent”, since the observed price is consistent with agents’ different models, and does not
inform them that they actually disagree about the structure of the economy.

Observe that agents can never detect “collective mistakes” in the first period: suppose e-
verybody has the same model of the economy ($S_h = S$ for all $h$); then, the equilibrium is
coordinating, but everybody might be wrong in the sense that everybody might have “forgot-
ten” a possible state. However, if the state that occurs in the second period was expected by
everybody the equilibrium price vector corresponds to what was expected.

That such an equilibrium does not exist for all possible collections of $S_h$ is clear and in itself
not very interesting. On the other hand, this definition reduces to the standard definition of
a rational expectations equilibrium if all agents a priori agree on the states tomorrow, i.e., if
$S_h = S$ for all $h$, and those are the “true” states. Therefore, in that case, there is always one
structure of expectations such that an equilibrium exists. This leads to the question of knowing
what the non-existence of an equilibrium means.

Clearly, if no consistency constraints were imposed upon agents’ models, a temporary equili-
brium would always exist in this setup<fnref n="14" />. Therefore, the source of non-existence here is the fact
that agents might learn from today’s prices that their model is wrong. One could then argue
that what is missing here is a theory of how agents revise their model in view of this (a priori
impossible according to their computations) price<fnref n="15" />.

The following definition simply acknowledges the fact that agents can indeed revise their
models. More precisely, it seems interesting to endogenize the model an agent believes in. In
that case, expectations for exogenous variables (i.e., the list of all agents’ endowments and
the probability of occurence of a state) by an agent is determined at equilibrium. If one let
$e(s_h) \in \mathbb{R}_{++}^{CH}$ be the endowment vector expected by $h$ in state $s_h$ and $\pi_h(s)$ the probability
attached by $h$ to the occurence of $s$, one can define a consistent equilibrium in price, probabilities,
and endowments expectations.

**Definition 2**

$[p(0), q, p'_1, ..., p'_H, x, b, ([e(s_h), \pi_h(s_h)]_{s_h \in S_h})_{h=1,...,H}]$ is a consistent equilibrium in price, probabilities, and endowments expectations if:

Given $([e(s_h), \pi_h(s_h)]_{s_h \in S_h})_{h=1,...,H}$, $(p(0), q, p'_1, ..., p'_H, x, b)$ is an I.C.T.E. of that model.

Now, this equilibrium always exists, since it is enough to assume that agents all expect the
same states to obtain existence (one is back to the standard rational expectations framework).
This equilibrium could also be of two types, coordinating or not, depending on agents’ expec-
tations at equilibrium. It is worth emphasizing that the “revision” agents make of their model

<footnote n="14">¹⁴See Tallon [1992] for a proof of existence of temporary equilibrium in a model where assets are nominal.</footnote>

<footnote n="15">¹⁵Alternatively, if agents are stubborn and stick to their model, one would be led to change the definition of an equilibrium to encompass the possibility of quantity rationing.</footnote>
does not take place in time, but is of the eductive type. Finally, it is implicit in this definition that agents are confident in their new model if it is consistent with observed prices. This is a fairly strong assumption, for one could claim that agents, once proven wrong (the equilibrium price is not consistent with their first model<fnref n="16" />), would behave more carefully in their assessment (for example would have a probability distribution over all the consistent models). In contrast, we assume that agents remain confident (arrogant?) that their new model is the “true” one.

An alternative approach, alluded to in the introduction, rests on the following observation. Unlike in standard temporary equilibrium analysis, agents' demand at spot 0 is not defined for prices that are not first period components of the rational expectations equilibrium of the agent's model. Thus, the fact that demand is not defined for all $p(0)$ can be seen as the source of non-existence (it is of course related to the consistency requirement discussed above). Thus, in order to adopt a more traditional approach, one is led to define the set of models and price expectations, such that the observed price today together with a price expectation is a rational expectations of the model. More formally, let:

$$E_{e(0)}(p(0)) = \{ ((e(s_h), \pi_h(s_h))_{s_h \in S_h}, p'_h) \mid (p(0), p'_h) \text{ is a R.E.E. of the model } (e(0), ((e(s_h), \pi_h(s_h))_{s_h \in S_h})) \}$$

It is the set of models and expectations that admit $p(0)$ as the first period component of a rational expectations equilibrium, given endowments<fnref n="17" /> $e(0)$. If this correspondence was non empty for all $p(0)$, and continuous, then one could define an expectation function linking expectations continuously to $p(0)$ and $e(0)$ (alternatively $e_h(0)$). Then, one would be back in the standard temporary equilibrium framework. An equilibrium exists and given that we imposed agents' expectations to lie in $E_{e(0)}(p(0))$ it is a consistent equilibrium. This approach will be illustrated in example 1. Note however the difficulties there are to actually characterize this correspondence. An analogy with the questions raised by the Sonnenschein conjecture could be drawn, since what is at stake here is the “creation” of an economy that admits a particular price as an equilibrium (see Mas-Colell [1977]).

# 3 Examples

In this section, the proposed equilibrium concept is illustrated by way of simple examples. The first and second examples are illustrative of the equilibrium concept, and the kind of inferen-

<footnote n="16">¹⁶Interestingly enough, Gilboa and Schmeidler [1993] make the point that “two economic theorists may very often interpret given data differently, each contending it is consistent with one’s own theory. Furthermore, when the contested theories are formulated in different languages, it may be hard to obtain agreement on what it is that the two have observed.” We do not deal with this possibility here, and assume that agents “speak the same (neo-classical) language” and agree on what is observed and what consequences it has on one’s theory.</footnote>

<footnote n="17">¹⁷Alternatively, one could define a correspondence $E_{e_h(0)}(p(0))$ which would endogenize the expectations by agent $h$ of the others' endowments today.</footnote>
ce that lies behind definition 2, in models where agents do expect different states. Example 3 shows how “collective mistakes” (i.e., agents all expecting the same wrong model) lead to study the effect of optimisim or pessimism on the equilibrium allocation. The next three examples are in a sense less specific, and provide some partial results for the existence of coordina- ting/uncoordinating individually consistent equilibria. The example that follows proposes a gra- phical illustration of some of the mechanisms at work, while the last one is meant to relate the equilibrium concept to the common prior assumption.

**Example 1:** There are two households, one good and one financial asset, inside money. Each household expects one state, $s_1$ for household one and $s_2$ for household two. Their preferences are identical, equal to $u_h(x(0), x(s)) = x(0)x(s)$. The model agent 1 has in mind is the following:

Agent 1 solves the program:

$$ \begin{array}{ll} \max_{x_1, b_1} & x_1(0)x_1(s_1) \\ \text{s.t.} & \left\{ \begin{array}{l} p(0)(x_1(0) - e_1(0)) = -qb_1 \\ p(s_1)(x_1(s_1) - e_1(s_1)) = b_1 \end{array} \right. \end{array} $$

while agent 1 thinks agent 2 solves:

$$ \begin{array}{ll} \max_{x_2, b_2} & x_2(0)x_2(s_1) \\ \text{s.t.} & \left\{ \begin{array}{l} p(0)(x_2(0) - e_2(0)) = -qb_2 \\ p(s_1)(x_2(s_1) - e_2(s_1)) = b_2 \end{array} \right. \end{array} $$

Normalizing $q = 1$, the solution to these problems yields the following demand functions for goods:

$$ x_1^1(0) = \frac{p(0)e_1(0) + p(s_1)e_1(s_1)}{2p(0)} \quad \text{and} \quad x_1^1(s_1) = \frac{p(0)e_1(0) + p(s_1)e_1(s_1)}{2p(s_1)} $$

$$ x_2^1(0) = \frac{p(0)e_2(0) + p(s_1)e_2(s_1)}{2p(0)} \quad \text{and} \quad x_2^1(s_1) = \frac{p(0)e_2(0) + p(s_1)e_2(s_1)}{2p(s_1)} $$

The (rational expectations) equilibrium prices of this (mis-specified) model must therefore satisfy:

$$ \frac{p_1(s_1)}{p_1(0)} = \frac{e_1(0) + e_2(0)}{e_1(s_1) + e_2(s_1)} $$

The same analysis can be done for 2's model, yielding equilibrium prices satisfying the following relation:

$$ \frac{p_2(s_2)}{p_2(0)} = \frac{e_1(0) + e_2(0)}{e_1(s_2) + e_2(s_2)} $$

One now has to write the equilibrium condition on today's market given that the demands come from mis-specified models:

$$ x_1^1(0) + x_2^2(0) = e_1(0) + e_2(0) $$
where the demands are evaluated respectively at $(p_1(0), p_1(s_1))$ and $(p_2(0), p_2(s_2))$, yielding the condition:

$$ \frac{e_1(s_1)}{e_1(s_1) + e_2(s_1)} + \frac{e_2(s_2)}{e_1(s_2) + e_2(s_2)} = 1 \quad (*) $$

Therefore, if the models agents had in mind verify the above condition, there exists, for that economy, a I.C.T.E.. Thus, in an economy where household 1 believes with certainty that a particular state will occur, while household 2 believes another one will occur, there is a first-period price vector that clears markets today and confort them in their opinion if the expected states verify equation (*). It is clear that if both agents were to expect the same “true” state (i.e., $(e_1(s_1), e_2(s_1)) = (e_1(s_2), e_2(s_2))$), the equilibrium price observed today is the first-period component of the rational expectations equilibrium of the model.

It is worth pointing out that any price vector today is “rationalizable” in the sense that for any price $p(0)$, there exist states $s_1$ and $s_2$ and prices $p_1(s_1)$ and $p_2(s_2)$ such that $(p(0), p_1(s_1), p_2(s_2))$ is a I.C.T.E.. Indeed, these prices and states can be constructed so as to satisfy:

$$ p(0)(e_1(0) + e_2(0)) = p_1(s_1)(e_1(s_1) + e_2(s_1)) $$

$$ p(0)(e_1(0) + e_2(0)) = p_2(s_2)(e_1(s_2) + e_2(s_2)) $$

This remark has some consequences as to what an agent might expect if he “observes”<fnref n="18" /> a price $p(0)$ today inconsistent with his model of the economy. The agent admits that he made a mistake in his forecast and revises his model so that the above equation is satisfied (i.e., he will redefine his expectations $(e_1(s_1), e_2(s_1))$ so that $p(0)(e_1(0) + e_2(0)) = p_1(s_1)(e_1(s_1) + e_2(s_1))$). There is then no reason that he adopts the right model of the economy as a consequence of having “observed” a price incompatible with his first model. This approach is the one implicitly followed when a consistent equilibrium in prices and expectations was defined. Indeed, it amounts to endogenize endowments expectations so that the previous condition is met at equilibrium.

Finally, one could have started with expectations functions for both agents. Let $\phi_h(e_h(0), e_{h'}(0), p(0)) = (e_h(s_h), e_{h'}(s_h), p(s_h))$ be $h$'s expectations function. Notice that these expectations functions are of a somewhat special type, since they assume that the support of an agent's expectations depends on what is presently observed, an assumption often ruled out in the temporary equilibrium literature. Indeed, in the case under consideration, agents upon observing $(e_h(0), e_{h'}(0), p(0))$ place probability one on the occurrence of $\phi_h(e_h(0), e_{h'}(0), p(0))$ tomorrow.

Suppose that:

$$ \phi_1(e_1(0), e_2(0), p(0)) = (e_1(0), e_2(0), p(0)) \quad \text{and} $$

$$ \phi_2(e_2(0), e_1(0), p(0)) = (2e_2(0), 2e_1(0), \frac{1}{2}p(0)) $$

Then, it is clear that with such expectations functions, any price today constitutes an I.C.T.E..

<footnote n="18">¹⁸The expectations revision process we are considering here is made before any exchange takes place. It concerns changes of expectations upon hearing a price from the auctioneer that is incompatible with one's model of the economy.</footnote>
**Example 2:** This example points out that the simple structure of the above example is by no means necessary, and agents could well expect commonly some states. Suppose for instance that both agents place a probability 1/2 on state s occuring, and 1/2 on "something else" (state s₁ for the first household, s₂ for the second). Markets are assumed to be "complete" (there is a bond paying one unit of account if s is realized and zero if not, as well as another one paying zero if s occurs and one if not). Agents have the same utility function $u_h(x(0), x(1)) = x(0)^{1/2}x(1)^{1/2}$. Normalizing the price of the bonds to be equal to one, one can (after computations similar to the one in example 1) characterize the I.C.T.E. system:

$$p(0)(e_1(0) + e_2(0)) - p_1(s)(e_1(s) + e_2(s)) - p_1(s_1)(e_1(s_1) + e_2(s_1)) = 0$$

$$p(0)(e_1(0) + e_2(0)) - p_1(s) \left( 1 + 2 \frac{p_1(s)}{p_1(s_1)} \right) (e_1(s) + e_2(s)) + p_1(s_1)(e_1(s_1) + e_2(s_1)) = 0$$

$$p(0)(e_1(0) + e_2(0)) - p_2(s)(e_1(s) + e_2(s)) - p_2(s_2)(e_1(s_2) + e_2(s_2)) = 0$$

$$p(0)(e_1(0) + e_2(0)) - p_2(s) \left( 1 + 2 \frac{p_2(s)}{p_2(s_2)} \right) (e_1(s) + e_2(s)) + p_2(s_2)(e_1(s_2) + e_2(s_2)) = 0$$

$$p(0)(e_1(0) + e_2(0)) - p_1(s)e_1(s) - p_2(s)e_2(s) - p_1(s_1)e_1(s_1) - p_2(s_2)e_2(s_2) = 0$$

The first two equations represent equilibrium equations for agent 1's model (by Walras law, the market clearing equation in state s₁ is always verified). The next two equations state equilibrium conditions for agent 2's model, while the last equation is the market clearing condition in the first period of the "true" model. This system being linear homogeneous in prices, there are only four unknowns (say the prices $p_1(s)$, $p_1(s_1)$, $p_2(s)$, $p_2(s_2)$ relative to $p(0)$), but five equations. Thus, this system has a solution only if a certain constraint (whose specific form is of little interest) on endowments is met. Therefore, if this constraint is met, there is an I.C.T.E.. Said differently, this system defines the prices and endowments expectations yielding a consistent equilibrium.

**Comments on examples 1 and 2** In light of these examples, one might say that existence of an I.C.T.E. is in some sense non generic, since it entails restrictions on the fundamentals expected by households. First, it should be noted that imposing that agents expect the same things to happen in the second period (i.e., the rational expectations framework in case there is actually only one possible state) is even more specific than the restrictions imposed in order to obtain an I.C.T.E.. Example 1 illustrates this point, for the case in which agents expect the same states is one (among many) possibility such that an equilibrium exists. Second, and in some sense more fundamentally, agents' expectations about fundamentals (states) are somehow endogenous in this model. Indeed, as can be seen in example 1, if an agent observes a price today such that it is not compatible with his model, he can be led to revise his expectations about the fundamentals tomorrow, so that the observed price corresponds to that new model.
Therefore, one could make the case that the working of the model makes the “non-generic” case plausible, or, alternatively, that the right equilibrium concept to look at is the one defined as the equilibrium in prices, probabilities and endowments expectations, where the latter are endogenous, since such an equilibrium always exists.

**Example 3:** This example makes explicit the role of expectations on the equilibrium allocation (and utilities). Indeed, it is possible to speak about optimism vs pessimism in that type of models. Imagine an economy in which agents expect two states, one of them being a priori impossible, while the other one is the one that will occur. In much the same way as in Wakker [1990], optimism could be defined as agents expecting “good things” (i.e., a good configuration of exogenous variables) in the “impossible state”, whereas pessimism would consist of agents expecting “bad things” in that state.

Therefore, the purpose of this example is to compare two economies. In the first one, both agents expect the same two states ($s$ and $\alpha$), while in the second economy, both agents also expect two states ($s$, the same as in the first economy, and $\beta$), one of which being different from the first economy. Observe that the logic of the example is slightly different from the previous ones, since agents do agree on the states in these two economies. Suppose furthermore that the “true” economy is the one where there is actually only one possibility for the state (i.e., full certainty), and that this state is $s$. The equilibrium we look at is therefore of the coordinating type, at which agents coordinate their expectations on the wrong model. An example shows that agents’ ex post utilities are higher in the optimistic economy, say the first one, where agents expect “good things” (high endowments) to happen in $\alpha$, than in the pessimistic economy, the second one, where agents expect “bad things” (low endowments) to happen in $\beta$.

The structure of the model is the same as in example 2, except that there is only one real asset (paying one unit of the good in each state). Preferences are the same as in example 2, and endowments in the first economy are given by :

$$ (e_1(0), e_1(\alpha), e_1(s)) = (1, 4, 4) \text{ and } (e_2(0), e_2(\alpha), e_2(s)) = (1, 4, 2) $$

while in the second economy, one has:

$$ (e_1(0), e_1(\beta), e_1(s)) = (1, 1, 4) \text{ and } (e_2(0), e_2(\beta), e_2(s)) = (1, 2, 2). $$

It is easy to compute equilibrium in both economies, and then to compute ex post utilities, i.e., the utility of an agent given the realisation of the state. Since the realized state is $s$, utility is $x_h(0)^{(1/2)}x_h(s)^{(1/2)}$, which is equal, in the optimistic economy, to 2.0061 for agent 1 and 1.4476 for agent 2, while these values are respectively 2 and 1.4434 in the pessimistic economy. Therefore, agents are better off being optimistic than being pessimistic<fnref n="19" />. The mechanisms at

<footnote n="19">¹⁹This example is of course very particular, and models could be easily built that would yield different</footnote>
work are relatively easy to understand. Households' expectations in the bad states drive them to transfer too little wealth intertemporally, while they lead them, in the optimistic case, to a better intertemporal allocation. This incompleteness of the market is not really crucial here, for had markets been complete, agents would have transfered some wealth from the first period to the impossible state, therefore "wasting" some ressources.

Observe that the comparison is made on the basis of the utility derived from consumption in both periods since we specified which state occured and that this state was expected by both agents<fnref n="20" />. This result is reminiscent of an analysis in terms of sunspot. However, the argument is, in the present model, different for it is the expectation of an impossible rather than irrelevant state (one that cannot occur rather than one whose fundamentals are the same as in another state) that alter the equilibrium allocation. This gives a rationale for the usual behavior of public authorities, trying to convince people that everything will be for the best tomorrow. Then, even though prophecies are not self-fulfilling (in the sense that the state that is realized is independent of such announcements), agents are better off believing in those claims than being overly pessimistic<fnref n="21" />. Furthermore, such an effect would still be present had markets been complete, unlike the sunspot case<fnref n="22" />.

**Example 4:** This example is merely a restatement of a result in Pietra and Siconolfi [1993]. Consider a standard two-period pure exchange economy (with many goods and agents), with uncertainty in the second period and financial markets on which assets are traded. This general framework could be reinterpreted in our terms as an economy in which all agents agree on the "true" model. Then, if assets are of the "real" type, it is well known that rational expectations equilibria are locally unique. Therefore, this means that the consistent temporary equilibrium (which is nothing but the first period component of the rational expectations equilibrium) is coordinating. Indeed, an agent when observing $(p(0), q)$ today is bound to make the same price forecast as other agents.

This result is not so obvious when assets pay off in units of account and markets are incomplete, for the equilibrium allocation is indeterminate. One could then conjecture that several price vectors tomorrow could be equilibrium vectors given first-period prices. However, Pietra and Siconolfi [1993] show that even in the incomplete market case, if there are enough goods compared to states, the temporary equilibrium is typically coordinating: to an equilibrium price

<footnote>results, e.g. that being pessimistic is "better" for the economy. However, this example is only meant to give the flavor of what this equilibrium concept enables one to look at.</footnote>

<footnote n="20">²⁰Looking at the two-period allocation without having to deal with bankruptcy is permitted by the fact that the state that occured was commonly expected. Furthermore, since both agents have the same model in mind, the equilibrium price is equal to the expected price.</footnote>

<footnote n="21">²¹In the present example, if both agents "knew the true model", that is expected only the occurrence of $s$, then utility levels are 2.02 and 1.4434 for agent 1 and 2 respectively. Therefore, agent 1 would be better off if everybody agreed on the true model, while agent 2 prefers the optimistic case.</footnote>

<footnote n="22">²²See Cass and Shell [1983]</footnote>
today corresponds only one equilibrium price vector tomorrow.

This result is however tempered by the condition that states be not too numerous: if agents concieve of complex futures, then it is possible, even in the rational expectations framework that equilibrium prices be not coordinating. Since nothing fixes a priori the number of states expected by an agent in our framework, this last case of uncoordinating equilibrium (even though everybody expects the same future) is possible.

**Example 5:** This example shows, rather trivially, that agents can never infer the "true model" (exogenous states) when observing a particular price today. Let $(p^*(0), q^*)$ be a consistent temporary equilibrium of the economy where agents expect endowments $e_h(s)$, $h = 1, ..., H$, and let agent $h$'s expectation in state $s$ be $p_h^*(s)$. Then, it is also an equilibrium of the model where agent $h$ expects $e_h(s) + v$ and agent $h'$ expects $e_{h'}(s) - v$, where $v \in \mathbb{R}^C$ is such that $p_h^*(s)v = p_{h'}^*(s)v = 0$. Of course, this new model gives the same equilibrium allocation, and this example seems trivial. However, it could be the case that this new model has also other equilibria. Also, one might argue (but this is outside of the model) that, in a "dynamic" version of the model, agents' learning would not have the same properties in both models, even though they yield the same first-period temporary equilibrium. Therefore, even though these models are observationally equivalent in the first period, they might well have different properties.

**Example 6:** This example deals with an economy where there is no first-period consumption, and where assets pay off in units of account. Then, any economy that fits the "informational" structure imposed by the return matrix admits the same individually consistent temporary equilibrium price vector. This claim is based on the observation made by Cass [1984] in a model with nominal assets, that any asset price vector that satisfies a no-arbitrage condition can be embedded in a rational expectations equilibrium<fnref n="23" />.

More precisely, let $Y$ be an $(S \times I)$ matrix ($S \ge I$), such that the first $S-1$ rows correspond to states commonly expected (i.e., states in $\bigcap_h S_h$, as defined in section 2). This matrix is consistent with agents commonly expecting the states that correspond to the first $S-1$ rows, individually expecting some states in which assets pay according to what is specified in the last row of $Y$, and not being able to infer that other agents might expect different states.

$$ \text{“commonly expected states”} \quad \begin{cases} \vdots \\ \vdots \\ \vdots \end{cases} \quad \begin{bmatrix} \dots & y(1) & \dots \\ \vdots & \ddots & \vdots \\ \dots & y(S-1) & \dots \\ \vdots & \vdots & \ddots \end{bmatrix} = Y \\ \text{“none of the above occured”} \quad \begin{cases} \dots \\ \dots \\ \dots \end{cases} \quad \begin{bmatrix} y(1) \\ \vdots \\ y(S) \end{bmatrix} $$

<footnote n="23">²³See Rahi [1993] for an application of this observation to the case of asymmetric information, to find non-revealing equilibria.</footnote>
The no-arbitrage condition can be written as follows:

$$Q = \left\{ q \in \mathbb{R}^I \mid \exists \lambda \in \mathbb{R}_{++}^S \text{ such that } q = \sum_{s=1}^S \lambda(s)y(s) \right\}$$

Now, let $q \in Q$. Then there exist $p(s)$, for $s = 1, ..., S$, such that $(q, p(s))$ is a rational expectations equilibrium of the model where agents expect the “right” $S - 1$ states, and anything they want for the last one. Therefore, $q$ is the first-period component of rational expectations equilibria corresponding to many different models.

There is no need to restrict oneself to models with agents expecting the same number of states, for the last state “defined” by the return matrix is actually of the form “none of the above occured”. This could correspond to different states according to different agents. To see that the set of no-arbitrage prices is identical to the one defined above when one explicitly writes all the states expected by an agent, observe that if there exists $\lambda \in \mathbb{R}_{++}^S$ such that $q = \sum_{s=1}^S \lambda(s)y(s)$, then there exists $\tilde{\lambda} \in \mathbb{R}_{++}^{S_h}$ such that $q = \sum_{s=1}^{S_h} \tilde{\lambda}(s)y(s)$, where $y(s)$ for $s \in S_h \setminus S$ is equal to $y(S)$. This corresponds to the fact that assets pay the same return in all those states that are not commonly expected. Indeed, it is enough to take $\tilde{\lambda}(s) = \lambda(s)$ for $s = 1, ..., S - 1$ and $\tilde{\lambda}(s) = \lambda(S) / (S_h - (S - 1))$ for $s = S, ..., S_h$.

Therefore, for all economies whose structure is compatible with the asset returns matrix, there exist non-informative consistent temporary equilibria. Furthermore, if the no-arbitrage condition does not yield only one equilibrium price (i.e., if markets are incomplete, in the sense that $S > I$), these equilibria are indeterminate.

**Example 7:** In this example, we provide a graphical illustration of the equilibrium concept at hand. Consider a two agents, single good, static economy, where a signal $s$, publicly observable, can take the value $\alpha$ or the value $\beta$. Agents agree on the fact that this signal is perfectly correlated with the fundamentals of the economy. Let $e_h(s)$, for $h = 1, 2$ and $s = \alpha, \beta$ be the true endowment vector in each state. Suppose that agent 1 expects that 2's endowment if $s$ occurs is $\omega_2(s)$, while agent 2 believes that 1's endowment if $s$ occurs is $\omega_1(s)$. Suppose furthermore that agents make mistakes when expecting the other's endowment, so that $\omega_h(s) \neq e_h(s)$. Agents also have to guess the other's utility function. Finally, it is assumed that agents can, previous to the realisation of the state, exchange Arrow securities, written on each state (markets are then complete).

In the figures, the “true” Edgeworth box is the one at the top and on the left, while agent 1 believes it is the bottom one and agent 2 believes it is the one on the right. These three Edgeworth boxes have different height and width, for each household does not anticipate correctly what the other agent's endowment is. For instance, in figure 1, agent 1 expects agent 2 to get more in state $\alpha$ and less in state $\beta$ than he actually does (the bottom box is wider but less high than
the true one). On the contrary, agent 2 believes that 1 gets more in state $\beta$ than he actually does (the top right box is higher than the true one).

Each agent takes decisions on the basis of the expected good price and the observed bond price. The latter (which remains implicit in the figures) is given, since markets are complete.

In figure 1, agent 1 expects the equilibrium price to be $p_1^*$, while agent 2 believes it will be $p_2^*$. As a consequence they take the decisions shown, which are not consistent, that is to say, that do not clear the bonds' market.

One can then wonder what happens. Both agents are led to revise their model (that is, change the height and width of their “own” Edgeworth box) until their decisions (still based upon misspecified models) be consistent, as shown in figure 2. In that case, they have changed their expectations about the other’s endowments (and preferences) and they both expect the equilibrium prices to be $p^*$. This price will actually be the one observed, and it does clear the bonds’ market.

Observe that in the case of such a simple economy, the consistent equilibrium allocation is optimal. This need not be the case if, for example, agents were making mistakes about their own endowments on top of “poorly” forecasting the other’s. If agents wrongly expected both their and the other’s endowment (this is the case in figure 3, where $e^h$ denotes the endowment point expected by household $h$, while the true one is $e$), then it is possible to get that they both anticipate different equilibrium prices, market for bonds clears but the situation is suboptimal, since if agents had known the true model, they would have reached a situation where both are better off.

**Example 8:** While we have purposely presented so far “extreme” examples, in the sense that either agents disagreed on the states, or completely agreed on the states and their probability of occurrence, this last example shows that an intermediate situation is also possible. Thus, we make the assumption that the state space is common knowledge but that agents have different priors on it. We show in an excessively simple way that even in this case, it is possible that agents do not realize their disagreement about the economy.

Suppose there are two agents and two states, $\alpha$ and $\beta$. There exists a complete set of financial securities. Endowments satisfy the following relation:

$$ (e_1(0), e_1(\alpha), e_1(\beta)) = (e_2(0), e_2(\beta), e_2(\alpha)) $$

Agents have the same utility function, and their prior on the states, given by $\pi_h(s)$, $s = \alpha, \beta$ and $h = 1, 2$, satisfy the following restriction (together with $\pi_h(\alpha) + \pi_h(\beta) = 1$):

$$ \pi_1(\alpha) = \pi_2(\beta) \text{ and } \pi_1(\beta) = \pi_2(\alpha) $$

Therefore, the situation is symmetric between the two agents. Each agent thinks he will have
$e_1(\alpha)$ with probability $\pi_1(\alpha)$, while the other has $e_1(\beta)$. Consequently, the symmetry of the construction gives the following set of equations:

$$ \left\{ \begin{array}{l} x_1^1(\alpha) = x_2^2(\beta) \\ x_1^1(\beta) = x_2^2(\alpha) \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{l} x_2^1(\alpha) = x_1^2(\beta) \\ x_2^1(\beta) = x_1^2(\alpha) \end{array} \right. $$

It is then straightforward to see that equilibrium in agent 1's model is equivalent to equilibrium in agent 2's model (note that by Walras' law, one can omit equilibrium on spot 0 markets):

$$ \left\{ \begin{array}{l} x_1^1(\alpha) + x_2^1(\alpha) = e_1(\alpha) + e_2(\alpha) \\ x_1^1(\beta) + x_2^1(\beta) = e_1(\beta) + e_2(\beta) \end{array} \right. \iff \left\{ \begin{array}{l} x_1^2(\alpha) + x_2^2(\alpha) = e_1(\alpha) + e_2(\alpha) \\ x_1^2(\beta) + x_2^2(\beta) = e_1(\beta) + e_2(\beta) \end{array} \right. $$

Therefore, whenever 1's model is consistent, 2's model is consistent as well. Thus, the framework proposed can apply to a model where the state space (that does not encompass agents' beliefs) is common knowledge, but on which agents have different priors.

# 4 Concluding comments

We have shown in this paper that it is possible that an equilibrium could arise, that does not reveal the fact that agents took their decisions on the basis of different mis-specified models. Thus, the main conclusion reached is that the price system does not act as a good coordination mechanism when agents have mis-specified models of the economy in mind. More precisely several cases arise. First, it is possible that a consistent equilibrium exists, that does not reveal to agents that they have a mis-specified model of the economy. Second, there may be no price that would coordinate agents' actions in the first period given their initial models. They are led to revise<fnref n="24" /> these models, but the equilibrium thus achieved need not be one where everybody has the same true model in mind.

Another point of interest lies in the fact that this temporary equilibrium concept entails restrictions on agents' expectations, that are less arbitrary than in standard temporary equilibrium models.

There are several obvious shortcomings to this analysis. First, it does not propose a way to consider the intertemporal allocation that would emerge in this model. In itself, this is not too damaging, for it is indeed possible to define an intertemporal equilibrium concept, entailing bankruptcy in the second period, in this framework. Second, sequences of temporary equilibria are not studied. In order to study such a sequence one would have to specify a bankruptcy rule (as mentioned this should not be too demanding) and, more importantly, one would have to

<footnote n="24">²⁴Once more, it is important to stress that this revision is totally implicit in the model, and does not take place in time. Furthermore, there is no law (such as Bayes' rule for instance) according to which agents make their revision.</footnote>
define a process according to which agents revise their expectations, a learning rule<sup>25</sup>. This rule
would have to bear on expectations for exogenous variables (since agents are assumed to be
able to hold rational expectations within their model). This points out that this rule cannot
be exclusively based on economic arguments. Indeed, it is clear that allowing for any type of
expectations (as done in the text) is not very satisfactory. Even though I believe that agents
do get surprised, and do have different theories about how the economy evolves<sup>26</sup>, there is also
no doubt that there is some coordination among agents' expectations. How this coordination is
achieved deserves some studying, which would probably underline the role of institutions and
customs in this coordination process.

Finally, although I have avoided to talk about rationality in the text, it seems clear that
agents' behavior, as specified here, is at the border of bounded rationality. On the one hand they
are assumed to be able to perform incredibly complicated computations (i.e., they can compute
a rational expectations equilibrium of their model), but on the other hand, they are unable to
expect all relevant worlds, and to recognize their ignorance. There is a difference though between
computational abilities and what could be called "imaginative or creative abilities", which could
provide a rationale for such a behavior. One could also claim that the assumption that agents'
model of how prices are determined is of the rational expectations type is not necessary to
much of the discussion, and look into other assumptions. Adopting a more abstract approach,
let $f_h$ denote $h$'s model of the economy. Agent $h$ thus believes that today's equilibrium price
is determined according to $p(0) = f_h(p_h^e(1))$ where $p_h^e(1)$ is the expected price vector for (all
states in) period 1 by agent $h$. The true model is denoted by $f$, i.e., the price today is given
by $p(0) = f(p_1^e(1), \dots, p_H^e(1))$. An equilibrium would then be a collection of models ($f_1, \dots, f_H$
and a price vector today $p(0)$ such that it clears today's markets, $p(0) = f_h(p_h^e(1))$ for all $h$
and $p(0) = f(p_1^e(1), \dots, p_H^e(1))$. The definition requires some consistency between agents' models
$f_h$ and the true model $f$, but does not a priori require that agents' expectations be based on
the rational expectations solution of a well defined equilibrium model. Certainly, this point also
deserves some further studying.

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<footnote n="25"><sup>25</sup>Alternatively, one might try to apply Gilboa and Schmeidler [1992]b model of case-based decision theory, where the “learning” comes from the accumulation of observed cases. In that case, an agent would begin his life expecting only one state, and as time goes by, would “learn uncertainty”, i.e., he will learn that an act he thought certain actually bears some uncertainty.</footnote>

<footnote n="26"><sup>26</sup>This goes in the same direction as Kurz [1990], who claims that “(...) the existence of diversity of beliefs among intelligent and equally informed agents [is] an empirical observation which requires an explanation”.</footnote>
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<figure><img src="image_2.png" /><figcaption>Figure 2 : Consistent Edgeworth boxes</figcaption></figure>

<figure><img src="image_3.png" /></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3 : An example of sub-optimal consistent equilibrium</figcaption></figure>