Rational responses and rational conjectures

‘OURNAL OF ECONOMIC THEORY 36, 289-301 (1985)

Rational Responses and Rational conjectures

Jo& TRUJILLO*

Dt” de Teoria Econdmica, Far&ad de Econ6micas, Universidad Complutense de Madrid (Somosaguas), Madrid-23, Spain

Received September 17, 1982; revised December 21, 1984

This paper utilizes a pure-trade imperfect-competition model and discusses the rationality problem arising from the fact that agents have ?o make conjectures about the markets to set prices optimally. The objective is to show that there may be inefficient allocations with the agents behaving rationaly. In the same context Hahn (Rev. Econ. Stud. 45 (1978), 1-17) proposed a notion of rationality under which the existence results obtained were negative. This paper argues that Hahn’s is a strong notion of rationality that requires the mechanism to be incentive compatible. Thus, an alternative notion that refers exclusively to the rationality of tlze conjectures is proposed, and positive results on existence obtained. Joiouvna/ of

Economic Literature Classification Numbers: 021, 022. c: 1985 Academic Press, Inc.

Recently, there has been an increasing interest in developing non- Walrasian resource allocation mechanisms, specially those in which, at equilibrium, agents may be quantity constrained and the allocation is inefficient. One of the reasons for this interest, is that modefling macroeconomic phenomena with non-Walrasian mechanisms may be a way of accounting for situations, like persistent unemployment, that the Walrasian-based macroeconomics fails to explain.

General equilibrium-fixed price models (see D&en [ I] for a review) provide some insight into the problems of quantity constrained equilibria, but a convincing explanation for the price rigidity is lacking. presents a pure exchange model with endogenous prices that generates some of the results of the fixed-price literature.

Hahn proposes a model with price-setting behavior in which agents, due to their imperfect knowledge, have to make conjectures about market excess-demand functions in order to set prices optimally. The class of

* I am very grateful to Professor L. Hurwicz whose encouraging advice made possible this paper. I am also in debt with L. Corchbn, J. Silvestre, and especially J. Ruiz-Castillo for very helpful comments. Research support from NSF Grant SOC-7825734 and the March Foun- dation of Madrid during my stage at the University of Minnesota is gratefully acknowledged.

289 002%0531/85 $3.00

Copyright 0 1985 by Academic Press, Inc. All rights of reproductmn in any form reserved.

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admissible conjectures is restricted exogenously to those in which agents are willing to change the observed prices only in those markets in which they are quantity constrained. In particular, prices have to be raised (lowered) if the plan is to buy (sell) in excess of the constraint. An equilibrium relative to given admissible conjectures, is a list of quantity constraints and prices such that, when all agents have optimized relative to their constraints and conjectures, markets clear and no agent wishes to change prices. Under certain assumptions on the form of the conjectures, it is shown that there are non-Walrasian (inefficient) conjectural equilibria.

The possibility of explaining inefficiency in equilibrium models by introducing free parameters (exogenous constraints) like fixed prices, quantity constraints, imperfect information, or conjectures, is not critizable in itself. However, the persistence of such inefficiency requires a justification of the free parameter’s stability. This might be done, either with an exogenous theory consistent with the model, or through means of some endogenous notion of rationality.

Since exogenously fixed parameters otherwise are endogenous features of the model, it is reasonable to characterize equilibria as rational-or stable-by the absence of incentives for a change in the parameter’s value. An example of such characterization is Muth’s rational expectations hypothesis. In a dynamic model, (the distribution of) future prices can be taken as a free parameter. The rational expectations hypothesis characterizes those equilibrium paths at which there are no incentives to revise the expectations because they coincide with the true distributions. Another example is the notion of rationality proposed by Hahn [4] to justify exogenously given conjectures. Under his notion, which will be presented in detail below, Hahn shows that very mild assumptions on the economy are sufficient to preclude the existence of rational conjectural equilibria. This negative result suggests that further discussion on rational conjectures is worthwhile.

Our claim is that, if a characterization of equilibria pursues the justification of the free parameter-in Hahn’s case, the conjectures-one has to make sure that the absence of incentives refers to the free parameter and not to other aspects of the resource allocation mechanism. In this respect, we find that there are, at least, two different characterizations of conjectural equilibria: one in terms of the rationality of the response rules of the allocation mechanism; and another in terms of the rationality of the conjectures. In this paper, first we argue that Hahn’s characterization belongs to the former, and that the problem of the rationality of the response rules is akin to the problem of a mechanism incentive compatibility as defined in Hurwicz [S]. Second, we propose an alternative characterization of equilibria which is based on a notion of rational conjectures that, contrary to Hahn’s, leads to positive results.

RATIONAL CONJECTURES 291

To clarify the above distinction, we must review very briefly the notion of an incentive compatible mechanism given by Wurwicz [5]. The behavior of each agent in an allocation process can be summarized by the response function qi=fi($, e’), that gives the signal qi sent by the agent to the market as a function of the signal 4 received, and his characteristics ei. More compactly, we write g’(Q) ~f’(4, e’) for the response function of agent i, and denote g = (g’,..., g’,..., g”). The signal 4 = (ql ,..., 4”) is said to be an equilibrium signal if and only if the response process is stationary for each agent: that is, g’(q) = q’ for each i, where cl’ is the ith component of @. Given an equilibrium signal 4, the mechanism specifies through an outcome function h the corresponding equilibrium allocation h(q).

A resource allocation process (g, h) is said to be individually incentive compatible with respect to a set of admissible response rules 6’ for each agent, if and only if the equilibria of the process are Nash equilibria of a game in which the strategies are the response rules g’ E G’, and the payoffs the utilities derived from the corresponding equilibrium allocations.

If a resource allocation mechanism is individually incentive compatible we say that, at equilibrium, the response of each agent is “rational,” in sense that, for each i, given the response rules of the rest of the agents, alternative rule g’ in the admissible set G’ could give rise tn a preferred equilibrium allocation for agent i.

Hahn [4] defines a mechanism whose response rules can be represented by functions g’(q, 8) that take into account explicitly the ith agent conjecture ci about market excess-demand functions within an admissible class C’. Hahn proposes to characterize as rational those equilibria at which the responses of all agents are rational, without any restriction on the set of admissible responses. Any response which, given the response rules of the rest of the agents, produces a feasible outcome is considered admissible. Not surprisingly in view of the negative results obtained in the incentive- compatibility literature, Hahn [4], as we said above, establishes that given mild assumptions on the economy no such rational equilibrium exists.

Since the problem, as we see it, is to justify the conjectures without inter- fering with the question of rational responses, we make the followin proposal: to characterize as conjecture-rational those equilibria at whit the conjecture of each agent i is such that no alternative admissible conjecture reveals the existence of a feasible allocation (given the be of all agents and the conjectures of all but i) which is preferr Nash-equilibrium condition in the space of conjectures that we propose, is a guarantee for the stability of the conjectures, in the sense that there are no incentives for the agents to revise their conjectures in the admissible sets ci.

Under the proposed characterization we are able to obtain some positive results concerning the existence of inefficient equilibria with rational con-

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jectures. Hahn obtains negative results because his characterization is based on a very strong notion of rationality for the response process which amounts to require the mechanism to be incentive compatible.

The rest of the paper is organized as follows. Section I sets up the model which is basically in Hahn [4]. Section II contains a formal discussion of rational responses and rational conjectures, and introduces the notion of conjecture-rational equilibrium. In Section III we stablish that a Walrasian equilibrium is always conjecture rational and that there are non-Walrasian conjecture-rational equilibria. Finally, we establish that unsatisfactory equilibria, which are non-Walrasian equilibria at Walrasian equilibrium prices in Hahn’s terminology, while they may exist cannot be conjecture rational.

I. THE MODEL

A pure exchange economy e consists of a set of agents I of cardinality n. Each agent is described by a pair ei = (T’, U’), where T’ c R’ is a set oj possible net trades t’ and U’ is a utility function delined on T’. An allocation is a collection of net trades t = (t’,..., t”) such that xi t’= 0.

ASSUMPTION I. For each agent i E I: (a) T’ is closed, convex, bounded below and contains the origin of R’ in its interior; and (b) u’ is a continuous, real-valued, strictly monotonic, and strictly quasiconcave function.

In the Walrasian resource allocation process, the agents perceive a signal consisting of a vector of prices, and respond with proposed net trades which are the result of the maximization of utility taking prices parametrically.

We denote the Walrasian excess demand of an agent i by:

zi(p) = (tic T’and pt’< 0: Ui(ti) 3 Ui(t’), V? satisfyingpi’< 0).

A Walrasian equilibrium (WE) for an economy e is a price p* E R: and an allocation t* = (t*l,..., t*“) such that t*‘=z’(p*) for each iEl.

For an economy satisfying Assumption 1, the following two results are well known: for each i E 1, z’(p) is a single valued function at p E R:, , and a WE p* E R:, exists.

Next we present the allocation process which is basically in Hahn [4]. It differs from the Walrasian in two features: the signals received by the agents are not only prices but also quantity constraints, and these signals are not taken parametrically, since agents are allowed to be price setters.

DEFINITION 1. For each agent, a signal qi is a list qi= (p, bi, si), where


PER:, is a price vector, and b’ E R’ and si E R’ are vectors of quantity constraints. In particular bj (resp. ,$) is the maximum amount of commodity j that agent i perceives he can buy (sell) at the given prices pi Let Q = Rc+ x RiXRC.

When it is obvious from the context, a signal may be represented by (p, ti), indicating buying or selling constraints if tj is positive or negative, respectively.

Notice that a signal q’= (p, t’) is a point in the excess-demand function, of the market perceived by agent i.

The agent’s imperfect knowledge about the economy and the reaction of the other agents make him conjecture the shape of the market excess- demand functions they face. Next, we define what a conjecture is and restrict the set of admissible ones.

DEFINITION 2. A conjecture is a function ci: Q x T’ + R$, that is, c’(q’, ti) = ~7. It specifies the prices p at which a trade t’ can be accom- plished, given the information qi.

ASSUMPTIQN 2. For each ~EI:

(a) c’(q’, ti) is a continuous function of qi and t’.

(b) For each qi E Q, c’(q’, ti). t’ defines a function of t’ that is convex.

(c) The conjecture is a vector-valued function:

ci(qi, ti) = [c:(q;, t;,, c;(q;, t;, )...) c;(q(, ij)-J

and for each signal qi = (p, b’, si) and each commodity j either c$qj, tj) = pi for ail ti E R (competitive conjecture) or

cj(qj, tj) = pj if -si< $6 bj,

= an increasing function ?f tj otherwise.

(d) For each qiE Q, Ci(q’,, ti,) = pl.

We will denote by C’ the set of conjectures c’(.) that satisfy Assumption 2. Let C= C” x .. . x C”.

DEFINITION 3. The conjectured constraint of an agent i that perceives a signal qi E Q and conjectures ci(. ) E C’ is

B’(q’, c’)= {tie T’: c’(q’, to. t’<O).

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p q cl($,ti(qi ,cl))

p’ = c’(sji, t )

x

FIG. 1. The shaded area represents the agent’s conjectural constraint. p’ is the value of the conjecture for an arbitrary trade t’ when the signal is $= (D, bk, 3:). The trade t’($, cl) represents the utility maximizing choice. In this particular case, the choice is such that the value of the conjecture p coincides with the price contained in the signal 4’.

The constrained demand of agent i is:

t’(q), CT’)= (l%B’(qi, c’): ui(t’) > V(C), VEBi(‘)}.

Notice that even though the conjecture is made market by market, the demand is decided simultaneously in all I- 1 markets. Notice also, that the Walrasian demand is a special case in which ci( .) is the competitive conjecture. Figure 1 illustrates the conjectured constraint and demand of an agent.

DEFINITION 4. A conjectural equilibrium (CE) relative to CE C, is a signal q* = (q*‘,..., q*“) and an allocation t* = (t*‘,..., t*n), where q*‘E Q and t*‘c T”, such that:

and for each i E I.

FIG. 2. For 2 agents and 2 commodities the figure represents the Walrasian offer curves z1 and z’, and a conjectural equilibrium where both sides of the market are constrained.


That is, a CE relative to c is a signal q* and an allocation t* such that, for each agent, t*’ is the desired trade given q*j, and no agent wishes to change prices, given the conjecture c’( . ).

Figure 2, for a 2 x 2 economy illustrates a CE in which both sides of the market are constrained.

It is inmediate that a WE allocation is always a CE ailocation relative to all conjectures c E C. That is, a signal q* = (p*, t*), where p* and t* are the WE price and allocation, respectively, is a CE since the agents are unconstrained.

Remark. Notice that if all conjectures are differentiable, then the only CE are the WE because, by definition, at a conjectural equilibrium Ci(Pi> t*‘) = p* for each i, but if the conjecture is differentiable then the slope of the walrasian and the conjectured budget set coincide at the binding constraint point. Therefore ti($, ci) =?(p), at a GE if the conjecture is differentiable.

Thus, a necessary condition for the existence of a non-Walrasian CE is that the conjecture of at least one agent in one market be ~o~differentiab~e at the binding constraining point.

DEFINITION 5. The conjecture cJqj, r$, as a function of $, is kinked at qj, if it is nondifferentiable at ti = bj and/or ti = -si.

THEOREM 1 (Existence of non-Walrasian CE). Let (p*? t*) be the unique WE price and allocation of the economy, respectively. If the economy and the conjectures satisfy, respectively, Assumptions 1 and 2, and if for some agent i and some commodity j # 1:

(a) tT#O

(b) u’ is continuously differentiable in a neighborhood of Pi, avmd

(c) c$qj, .) is kinked at all qj= (p, tj), in a neighborhood of (p*, t*‘), then, there is a non- Walrasian CE relative to c.

ProoJ: Gale [2].’

’ Notice that our definition of kinked conjecture and Gale’s are not the same. But if I’ is the Walrasian demand of agent i at prices p *, fi > 0 and under our definition. c; is kinked at a?: q in a neighborhood of q* = (p*, t*‘), then, under our assumptions on the utility function. WP can assnre that there is always a 6 > 0 such that, if //q -q* j/ < 6, where q = (p; t’) and tj, = lk*’ for all k #i, 1 then t;(q, c’) < tf, which is the definition of kinked conjecture in Gale [z, p. 352.

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II. RATIONAL RESPONSES AND RATIONAL CONJECTURES

As we pointed out in the introduction, modeling with free parameters (like given conjectures) to produce inefficient equilibria, has to be taken only as a first step towards a more self-contained explanation in which the free parameter is, in some way, endogenized.

In this section, we give two alternative characterizations of the CE. The first one, which we will label Hahn-rational equilibria (HRE), was proposed by Hahn and is based on the rationality of the response functions. Hahn [4] establishes that, under very mild assumptions on the economy, no HRE exists. The second is the notion of conjecture-rational equilibria (CRE) which is based on a notion of rationality applied to the conjectures only. In order to clarify the discussion of both notions, we write explicitly the response function of our model.

Following the notation developed in the introduction, and applying it to our model, the response rule of an agent i is the function gi(qi, ci) = (p, t”), where t’ = t’(q’, c’) is the utility maximizing trade given q’ and ci, and ~5 = ci(qi, t’) is the value of the conjecture given qi and i’. Hence, according to Definition 4, a price and an allocation q* = (p*, t*) are a CE relative to c E C if for each agent i; g’(q*‘, ci) = q *L that is, all responses are stationary. , To motivate the definition of Hahn-rational equilibrium we first define what an i-equilibrium-feasible bundle is, and next what we understand as a rational response.

DEFINITION 6. A bundle t’~ T’ is i-equilibrium-feasible relative to the response rules gk( .) and conjectures ck of all the other agents k # i if and only if, there is a collection of signals q* = (q*‘,..., q*n), where q*k = (p*, t*k) for each k # i and q*i = (p*, P), such that gk(q*k, c”) = q*k for all k# i and (t”‘,..., P-l, t’, t*‘+‘,..., t*n) is an allocation. In other words, agent i considers a bundle t’ as i-equilibrium-feasible, if there is a signal that makes stationary the responses of all other agents, and the collection of trades t *k for all kfi p lus t’ is an allocation.

As an illustration notice that for agent i, the bundles t’ which are in the Walrasian excess demand function of the market-minus agent i, i.e., t’= -CkZiZk(p), PER:+, are i-equilibrium-feasible. On the other hand, other bundles might be i-equilibrium-feasible if, at least, a conjecture of an agent k# i is kinked, since such an agent may accept as stationary a signal at which he is quantity constrained.

DEFINITION 7. A signal qi= (p, t’) is a rational response for agent i if and only if, there is no bundle i’ E T’ satisfying Ui(?) > U(t’), which is i-equilibrium-feasible relative to the response rules and conjectures of the other agents k # i.


DEFINITION 8. A Hahn-rational equilibrium (I-IRE) is a CE at which t responses of all agents are rational.

This notion of rationality is closely related to the notion of an individually incentive-compatible mechanism delined in Hurwicz [5]. Notice, that an HRE is a Nash equilibrium, in a game in which the strategies are the response rules gi(. ), with no restriction imposed on the class admissible rules, and the payoffs are the utilities derived from t equilibrium allocation. Hurwicz [.5] characterizes the rules of a mechanism as incentive compatible, if the resulting equilibria are Nash equilibria oft game proposed above, but where the admissible strategies may be restr ted to a set of rules G’, for each agent. Consequently, the nonexistence of HRE is not surprising given the negative results of the incentive co patibility literature.

We think it is necessary to separate the incentive-compatibility problem-in Hahn’s, like in most other decentralized mechanisms-from that of the rationality of the conjectures or expectations, in the presence of uncertain knowledge. In the latter, one approaches the rationality of a single aspect of the response behavior of the agents: the selection of a conjecture within the admissible set; while in the former, one questions the rationality of the response behavior itself. The fact that a competitive agent may have incentives to become a monopolist, and the fact that a competitive agent should conjecture about the unknown in a rational manner, must be treated as separate problems.

Hence, we propose a characterization of CE based on the rationality of the conjectures, taking as given the response rules of all agents. To motivate the notion, we define first the concept of an i-conjecturable bundle.

DEFINITION 9. A bundle ?E T’ is i-conjecturable given the signal q’ an the response rule g’ if and only if there is an admissible conjecture ci E CI such that g’(q’, ci) = (~7, t’), for some Ip E R’++ . That is, a brindle is i-conjecturable given a signal if and only if the given rule g’ produces it as a response for some admissible conjecture.*

DEFINITION 10. Let qi = (p, t’) and g’(q’, ci) = (p*, P’). The conjecture C’E C’ is rational for agent i given the signal qi if, and only if, there is no i’ E r’ which is: i-conjecturable given qi and g’(. ), ~-equilibrium-fcasi~~e

’ It could seem more reasonable to call i-conjecturable given qi all those bundles that behmg to a conjectured budget set B’(q’, c’), for some adissible conjecture c’ E c’. Nevertheless, since we want the response rule gi(’ ) to be taken as given, we impose the extra restriction that the bundle has to be the result of the response process given the signal qi and an admissible conjecture.

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given gk( *) and ck for all k # i, and vi(?) > V(F). In other words, ci E c’ is rational for agent i given g’(q’, . ) if, and only if, there is no alternative conjecture C’ E c’ such that g’(q’, Ci) = (~7, ii), Vi(?) > Ui(t*‘), and P is i-equilibrium-feasible given g”( *) and ck for all k # i.

Notice that a conjecture c’ is rational given qi if all the information available, represented by g = (g’ . . . g”) and ck all k # i, is not able to reveal the existence of an alternative conjecture in the set c’ which is better than ci. Then, we could say that a rational conjecture is likely to persist, since, for the given economy, mechanism, and set C’, the agent has no incentive to conjecture otherwise.

DEFINITION 11. A conjecture-rational equilibrium (CRE), relative to an admissible set of conjectures, is a CE at which the conjectures of all agents are rational. Naturally, within a mechanism, the class of CRE depends on the exogenously given sets of admissible conjectures.

It is easy to verify that we impose less restrictive conditions than Hahn to further characterize the CE. While an HRE requires the responses of all agents to be rational, a CRE only requires the conjectures to be rational. Since the latter are parameters of the response functions, it may occur that the conjectures are rational while the responses are not. The inverse is not true, because an alternative conjecture gives rise to a new response.

III. RESULTS

Now we state our first positive result, establishing that a WE is always a CRE.

THEOREM 2 (A WE is a CRE). If q* = (p*, t*) is a WE then q* is a CRE, relative to the set of admissible conjectures C defined by Assumption 2.

Proof: q* is a CE relative to any collection of conjectures c E C. Due to the definition of Ci, i= l,..., n if a bundle t’ is i-conjecturable given q* then p*t’<O. But since q* is a WE, U’(t*‘) > Ui(ti) for all t’ such that p*t’< 0, i= l,..., n. Hence, there is no i-conjecturable bundle preferred to Pi by agent i= l,..., n. Therefore, the conjectures at a WE are always rational relative to the admissible set C, and consequently a WE is a CRE. Q.E.D.

It could be argued that Theorem 2 implies that by not requiring rational responses we are able to justify perfect competition for arbitrarily small economies. But notice that the theorem says that, given a particular response rule, Walrasian equilibrium allocations are sustainable by conjectures that can be considered as rational within an admissible set C. Hence,


the result depends both on the response mechanism and the admissible set of conjectures. Within these constraints, Theorem 2 is a justification of t competitive conjecture, when made in a Walrasian equilibrium, even for arbitrarily small economies. Nevertheless, this is no longer true if, for instance, we enlarge the set of admissible conjectures allowing the agents to conjecture new prices when unconstrained. We are not requiring our equilibria to satisfy all possible rationality aspects but, at least, those related to the given set of admissible conjectures. An interesting question is to characterize, for a given economy and response rules, the maximal set of admissible conjectures that makes a particular conjectural equilibrium, say a WE, a CRE.

Next we show the existence of non-Walrasian CRE.

THEOREM 3 (Existence of non-Walrasian CRE). If q* = (p*, t*) is the unique Walrusian equilibrium of an economy satisf$ng Assumption 1 and $

(a) for each agent ke Z, Uk( .) is twice continuously differentiable ouer an open neighborhood N, of t*k, and

(b) for some agent i E I and some commodity j # 1, tlFi # 0,

then, there is a non- Walrasian conjecture-rational equilibrium.

Pro05 For given conjectures ck, k = l,..., n such that cJ is kinked, Theorem 2 guarantees that there is a non-Walrasian equilibrium 4 = (0, 3). Assuming that for all k # i the conjectures ck are either competitive or differentiable (not kinked), the conjectured demand of all k z i is, at the equilibrium 4 = (d, t^), necessarily Walrasian (see remark to Definition 4). That is, tk(Qk, ck) = zk(b). From the proof of Theorem 2 trivially follows that such an equilibrium 4 can be found as close as desired to the Walrasian 9”.

As we mentioned above, due to Assumption 2 on the set of admissible conjectures, for a bundle tk to be k-conjecturable given Qk = (8, fk) necessarily $tk 60. Hence, since all agents k f i are unconstrained at 4 it follows that their respective conjectures ck are rational given gk, k # i. Therefore, the only thing left to prove is that ci is rational given @,

Define the function u’(p) z -&# i z”(p), where z”(p) is the Wa~rasia~ excess demand function of agent k. Since the conjectures of all k # i are competitive or not kinked, it follows that if t’ is i-eq~~~ibr~~rn-feasible then t’= u’(p) for some p.

On the other hand, since for t’ to be i-conjecturable given 4 = (B, 3’), necessarily fit’< 0, it follows that ci is not rational if and only if for some PER’ ++ , @u’(p) < 0 and v’(p) is strictly preferred to l*i. We show that for a price fi sufficiently close to p*, if flu’(p) d 0 then U(P) 3 u’(u’(p)).

Define the function d: T’ -+ R’ as the supporting hyperplane of the set of

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bundles at least as good as a given bundle t^‘. By the assumptions on the utility function, p”( .) is a well-defined single valued continuous function on T’.

At a WE p* it follows that p”(v’(p*)) = p*. Then, if p*v’(p) < 0 it follows that p(vi(p))(vi(p)- u’(p*))<O. By the continuity of the functions p”(.), and for @ sufficiently close to p*, if @z?(p) <O then p”(~‘(fi))(v~(p)- u’(b)) < 0. But, by definition of the supporting hyperplane b(. ), it follows that for @ sufficiently close to p *, if $0’(p) < 0 the v’(p) lies below the hyperplane p(z@)), and therefore vi(p) = f’ is preferred to v’(p). Hence, there is no i-feasible and conjecturable bundle preferred by agent i to t*’ given 4’ = (@, ?). Therefore ci is rational given 4’. Q.E.D.

Quantity constrained equilibria are sometimes identified with “wrong” prices (non-Walrasian equilibrium prices). Hahn [3] constructs a model in which the economy may settle at a non-Walrasian equilibrium allocation (in particular, at Dr&ze equilibrium, i.e., a fixed price equilibrium in which at most one side of each market is constrained) even if the prices are “right” (i.e., Walrasian). He labelled such equilibria “unsatisfactory.”

Fixed-price unsatisfactory equilibria have been studied by Silvestre [7] who shows, through some examples, that there may be a continuum of unsatisfactory equilibria when there are more than two agents on one side of the market. He also demonstrates that the closure of the set of unsatisfactory equilibria contains the WE.

On the other hand, it has been established by John [6] that a Dr&ze equilibrium is a conjectural equilibrium for some vector of admissible conjectures c = (cl,..., c”). It follows that there are conjectural equilibria at Walrasian prices. The next theorem shows that no unsatisfactory equilibrium can be conjecture-rational, for the given set of admissible conjectures.

THEOREM 4. Let p* be a Walrusian equilibrium price. If q = (p*, t] is a conjectural equilibrium and t‘ # t*, where t * is the Walrasian equilibrium allocation associated with p*, then q is not a conjecture-rational equilibrium.

Proof: The only thing to prove is that at least for an agent i, the conjecture c’ that generates the conjectural demand t’= t’(q’, 2) is not rational. For that matter, we show that the bundle t*’ is i-equilibrium-feasible, i-conjecturable given q, and strictly preferred to i’.

The bundle t*’ is i-equilibrium feasible independently of the conjectures of the other agents since it belongs to a Walrasian allocation t* = (t*l,..., t**). Moreover t” is i-conjecturable given qi, since if Cam is the competitive conjecture t”(q’, Cam) = z’(p*) = t*‘. Given Assumption 1 the fact that t*’ is strictly preferred to t” is obvious since at t” the agent is constrained. Therefore, the conjecture ci that generated ?= t’(q’, c’) is not rational. Q.E.D.

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CONCLUXONS

Since most likely real economies work inefficiently, it is interesting to build equilibrium models that reproduce that behavior. Wahn’s model of conjectures is interesting because it is capable of explaining both efficient and inefficient allocations. But his proposal to characterize equilibria as rational produces negative results because it is too strong. It requires to be rational, not the conjectures, but the response behavior of the agents. This requirement is akin to that of individual incentive campatibility which, as is well known, is a strong rationality property for decentralized mechanisms.

In this paper we proposed a notion of rational conjectures which is free of incentival aspects. We also established that inefficient equilibria with rational conjectures may exist. This result is important for macroeconomics, because it shows that there may be inefficiencies-like unemployment-which are due to the lack of Walrasian behavior. “‘Wrong prices” do not change because no agent foresees the possibility of making profits by setting new prices, and the economy is unable to reveal it.

REFERENCES

1. A. DRAZEK, Recent developments in macroeconomic, disequilibrium theory, Eco~omeriica 46 (1980), 2833306.

2. D. GALE. A note on conjectural equilibria, Rev. Econ. Stud. 45 (1978), 33-38. 3. F. HAHN, “Unsatisfactory Equilibria,” IMSS Technical Report No. 247, Stanford Univer-

sity. 1977. 4. F. HAHK, On non-Walrasian equilibria, Rev. Econ. Stud. 45 (1978), l-17. 5. L. HURWICZ, On informationally decentralized systems, in “Studies in Resource Allocation

Systems” (.I. Arrow and L. Hurwicz, Eds.), pp. 425459: Cambridge Univ. Press, New York, 1977.

6. R. JOHN, A remark on conjectural equilibria, DP 8009. CORE Universitt Catholique de Louvain, France, 1980.

7. J. SILVESTRE, Continua of Hahn unsatisfactory equilibria: Some examples, Econ. Lat. 5 (1980), 201-208.

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