Self-Fulfilling Mechanisms and Rational Expectations in Markets with a Continuum of Traders ∗ Giulio Codognato†and Sayantan Ghosal‡ April 2001

Abstract We extend an equivalence theorem proved by Forges and Minelli (1997) to a market game which does not require that a specific commodity is used as money. Under the assumption that the set of commodities is a net, we show an equivalence between the set of allocations generated by self-fulfilling mecahnisms and the set of rational expectations equilibrium allocations. We construct an example which shows that this equivalence breaks down when the net assumption fails to hold: in such cases, traders cannot obtain, through a self-fulfilling mechanism, the information revealed by rational expectations equilibrium prices. Journal of Economic Literature Classification Numbers: C72, D51. ∗

We would like to thank Enrico Minelli for some conversations which motivated the present paper. † Dipartimento di Scienze Economiche, Universit` a degli Studi di Udine, Via Tomadini 30, 33100 Udine, Italy, and SET, Universit` a degli Studi di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy. ‡ Department of Economics, University of Warwick, Coventry CV4 7AL, United Kingdom.

1

1

Introduction

In a recent paper, Forges and Minelli (1997) introduce the notion of selffulfilling mechanism to provide a strategic foundation of rational expectations equilibria. To this end, they use a market game, proposed by Shapley and Shubik (1977), which requires that a specific commodity plays the role of money. In order to guarantee that their self-fulfillig mechanism generates enough trade to support rational expectations equilibria, they need a joint assumption on traders’ endowments and preferences which assures that money is plentiful and well-liked (see also Dubey and Shapley (1994)). In this paper, we extend the work of Forges and Minelli (1997) on selffulfilling mechanisms to a more general market game which does not require the use of a specific commodity as money. More precisely, we extend to the case of differential information a game analysed by Codognato and Ghosal (2000) in exchange economies with an atomless continuum of traders (see also Sahi and Yao (1989) for an original version of the gane in exchange economies with a finite number of traders). We show an equivalence between the set of allocations generated by self-fulfilling mechanisms and the set of rational expectations equilibrium allocations. This result is achieved by extending to differential information economies the assumption that the set of commodities is a net, a joint restriction on traders’ endowments and preferences, introduced by Codognato and Ghosal (2000), which allows all traders to have boundary endowments and indifference curves which intersect the boundary of their consumption set. We would like to note that, as the market game we use is different, our proof of the equivalence theorem is not a straightforward generalization of the proof provided by Forges and Minelli (1997). We also show, by an example, that when the net assumption fails to hold, there are differential information exchange economies with fully revealing rational expectations equilibria which are supported by no self-fulfilling mechanism. The example we construct has the feature that traders cannot obtain, through a self-fulfilling mechanism, the information which is fully revealed by rational expectations equilibrium prices. Section 2 contains the model and the definitions. Section 3 states and proves the equivalence theorem and the example. Section 4 concludes.

2

2

Model and definitions

We consider the following differential information economy E. The space of traders is denoted by (T, T , λ). We take T = [0, 1], T the σ-algebra of Lebesgue measurable subsets of T and λ the Lebesgue measure on T . A null set of traders is a set of Lebesgue measure 0. Null sets of traders are systematically ignored throughout the paper. Thus, a statement asserted for “all” traders, or “each” trader, or “each” trader in a certain set, is to be understood to hold for all such traders except possibly for a null set of traders. l , the There are l commodities in the economy and the consumption set is R+ nonnegative orthant of the Euclidean space of dimension l. We shall denote l . Uncertainty is represented by a finite set by x = (x1 , . . . , xl ) a vector of R+ of states of nature denoted by S. All traders t ∈ T have the same probability distribution q over S, with q(s) > 0, for all s ∈ S. Given a partition P of S, we denote by P (s) the element of P containing s. Given two partitions, P and Q we denote by P ∨Q their coarsest common refinement. Given a partition P of S, a function f on S is P -measurable if P (s) = P (r) ⇒ f (s) = f (r), for all s, r ∈ S. A function f defined on S induces a partition of S, which we denote by Pf , such that Pf (s) = Pf (r) ⇔ f (s) = f (r), for all s, r ∈ S. The private information of each trader t ∈ T is described by a partition P t of S. If s in S occurs, each trader t ∈ T is informed of the element P t (s) in his partition W t P the coarsest common refinement P t which contains s. We denote by t∈T

of traders’ partitions. Given a function f defined on S, we denote Pft for P t ∨ Pf . We assume that there exists a finite partition {T1 , . . . , Th } of the set of traders T = [0, 1] such that, for each i = 1, . . . , h, Ti ∈ T and λ(Ti ) > 0; 0 and such that, for all t, t0 ∈ Ti , P t = P t , for each i = 1, . . . , h. Moreover, T t we assume that P (s) = {s}, for each s ∈ S. A commodity bundle is t∈T

l a vector in R+ , which specifies a quantity of each commodity in each state l , which assigns, in of nature. An assignment is a function x : S × T → R+ each state of nature, a commodity bundle to each trader. We assume that, for each s ∈ S, each coordinate of the function x(s, ·) is Lebesgue integrable. There is a fixed initial assignment w, which represents traders’endowments, satisfying the following assumption.

Assumption 1. w(·, t)Ris P t -measurable, for all t ∈ T ; w(s, t) > 0, for all s ∈ S and for all t ∈ T ; T w(s, t) dλ  0, for each s ∈ S. 3

R

R

An allocation is an assignment x for which T x(s, t) dλ = T w(s, t) dλ, for each s ∈ S. The preferences of each trader t ∈ T are described by a von l → R, satisfying the folNeumann-Morgenstern utility function ut : S × R+ lowing assumption. Assumption 2. For all t ∈ T , ut (s, ·) is continuous and strictly monotonic, for each s ∈ S.

l l denote Let L denote the set of commodities {1, . . . , l} and let R+j>0 ⊂ R+ l the set of nonnegative vectors in R+ whose jth component is strictly positive. For each s ∈ S and for each i ∈ L, consider the set Tis = {t ∈ T : wi (s, t) > 0}. Clearly, by Assumption 1, λ(Tis ) > 0. Given s ∈ S, we say that any two commodities i, j ∈ L stand in the relation C s if there is a measurable subset Tis0 of Tis with λ(Tis0 ) > 0 such that, for each trader t ∈ Tis0 , the l l l set {x ∈ R+ : u(s, x) = u(s, y)} ⊂ R+j>0 , for all y ∈ R+ . The following definition is an extension to a differential information economy of a notion introduced by Codognato and Ghosal (2000) to whom we refer for further details.

Definition 1. The set of commodities L is said to be a net if, for each s ∈ S, {hi, ji : iC s j} = 6 ∅. and the directed graph DLs (L, C s ) is strongly connected. We make the following assumption.

Assumption 3. The set of commodities L is a net. This assumption allows all traders to have, in each state of nature, boundary endowments and indifference curves which intersect the boundary of the consumption set. We refer to Codognato and Ghosal (2000) for a further l discussion of this assunption. Prices are a function p : S → R+ which asl sociates to each s ∈ S a price vector in R+ . Now, we are able to define a rational expectations equilibrium concept for the differential information economy E. Definition 2. A rational expectations equilibrium is a pair (p∗ , x∗ ) consisting of a price p∗ and an allocation x∗ such that (i) x∗ (·, t) is Ppt ∗ -measurable, for all t ∈ T ; (ii) for all s ∈ S and for all t ∈ T , X

t (s) r∈Pp ∗

q(r|Ppt ∗ (s))ut (r, x∗ (r, t)) ≥ 4

X

t (s) r∈Pp ∗

q(r|Ppt ∗ (s))ut (r, y),

l for all y ∈ {x ∈ R+ : p∗ (s)x ≤ p∗ (s)w(s, t)}. A rational expectations W t ∗ ∗ equilibrium (p , x ) is said to be fully revealing if Pp∗ = P . t∈T

We associate a game Γ(E) with the differential information economy E. More precisely, we extend the game analysed by Codognato and Ghosal (2000) l2 to the case of differential information. Let b ∈ R+ be a vector such that b = (b11 , b12 , . . . , bl−1l , bll ). A strategy function is a map B : S × T → l2 l2 l2 ), where P(R+ ) denotes the collection of all subsets of R+ P(R+ , such that, 2 l for each s ∈ S and for each t ∈ T , B(s, t) = {b ∈ R+ : bij ≥ 0, i, j = Pl bij ≤ wi (s, t), i = 1, . . . , l}. A strategy selection is a function 1, . . . , l; j=1 l2 b : S × T → R+ , such that b(s, t) ∈ B(s, t), for each s ∈ S and for each t t ∈ T ; b(·, t) is P -measurable, for all t ∈ T ; each coordinate of the function b(s, ·) is Lebesgue integrable, for all s ∈ S. For each s ∈ S and for each t ∈ T , bij (s, t), i, j = 1, . . . , l, is the amount of commodity i that trader t offers in exchange for commodity j in the state of nature s. Given a strategy ¯ the function which associates to each s ∈ S selection b, we denoteR by B(s) ¯ the matrix B(s) = ( T bij (s, t) dλ). Moreover, we denote by b \ b(s, t) a strategy selection obtained by replacing b(r, t) in b by b(s, t) ∈ B(s, t), for each r ∈ P t (s). Now, we are able to give the following definition (see Sahi and Yao (1989)). Definition 3. Given a strategy selection b, we say that a price vector p is market clearing in s ∈ S if p  0,

l X i=1

i¯

j

p bij (s) = p (

l X

¯ ji (s)), j = 1, . . . , l. b

(1)

i=1

By Lemma 1 in Sahi and Yao (1989), there is a unique, up to a scalar ¯ multiple, price vector p satisfying (1) if and only if B(s) is irreducible. Given a strategy selection B such that p is market clearing in s, consider the function determined as follows xj (s, t, b(s, t), p) = wj (s, t) −

l X i=1

bji (s, t) +

l X i=1

bij (s, t)

pi , pj

j = 1, . . . , l, for all t ∈ T . Given a strategy selection b, the final holdings of the traders are j

x (s, t) =

(

xj (s, t, b(s, t), p) if p is market clearing in s, wj (s, t) otherwise, 5

(2)

j = 1, . . . , l, for all s ∈ S, for all t ∈ T . It easy to verify that the assignment determined by (2) is an allocation. Codognato and Ghosal (2000) show that the nonuniqueness of market clearing prices induces an indeterminacy in traders’ payoffs for individual deviations. In order to overcome this indeterminacy problem, we must consider as possible Bayesian Nash equilibria only the strategy selections for which the aggregate bid matrix is irreducible in each state of nature. Denote by π(s, b) the function which associates to ¯ each s ∈ S and to each strategy selection b such that B(s) is irreducible, for all s ∈ S, the unique, up to a scalar multiple, market clearing price vector p satisfying (1). Now, we are able to define the equilibrium concept. ˆ ˆ such that B(s) ¯ Definition 4. A strategy selection b is irreducible, for all s ∈ S, is a Bayesian Nash equilibrium if, for all t ∈ T and for all s ∈ S, X

r∈P t (s)

ˆ t), π(r, b))) ˆ ≥ q(r|P t (s))ut (r, x(r, t, b(r,

r∈P t (s)

ˆ \ b(s, t)))), q(r|P t (s))ut (r, x(r, t, b(s, t), π(r, b

X

for all b(s, t) ∈ B(s, t).

Following Forges and Minelli (1997), we convert the Bayesian game Γ(E) into a game with communication in order to define an equilibrium concept which, like rational expectations, by taking into account the revelation of information, is robust to ex post regret. To this end, we introduce the foll2 lowing notions. A mechanism is a function µ : S × T → R+ , such that µ(s, t) ∈ B(s, t), for all s ∈ S and for all t ∈ T ; each coordinate of the function µ(s, ·) is Lebesgue integrable, for all s ∈ S. Given a mechanism ¯ M(s) the function which associates to each s ∈ S the maµ, we denote by R ¯ trix M(s) = ( T µij (s, t) dλ). A mechanism µ is said to be irreducible if ¯ M(s) is irreducible, for each s ∈ S. We say that an irreducible mechanism t µ is adapted to the price function π(s, µ) if µ(·, t) is Pπ (·,µ) -measurable, for all t ∈ T . Given a mechanism µ adapted to the price function π(·, µ), we denote by µ \ µ(s, t) the function obtained by replacing µ(r, t) in µ by t µ(s, t) ∈ B(s, t), for each r ∈ Pπ (·,µ) (s). A game with communication, which we denote by Γµ (E), is constructed by adding to the Bayesian game Γ(E) a mechanism µ adapted to the price function π(s, µ). As, in our model, there is a continuum of traders with the same information, an individual lie is of no 6

consequence. It follows that, as noticed by Forges and Minelli (1997), every mechanism µ is incentive compatible. Γµ (E) is played as follows. Nature chooses s in S according to q. Each trader t ∈ T is informed of P t (s). By the argument sketched above, we can proceed, in the description of the game, as if the mediator knew the true the state of nature s. Therefore, he is able to announce the price π(s, µ) to all traders. Given the public signal π(s, µ), t each trader t ∈ T refines his information to Pπ (·,µ) (s). Now, we are able to provide the definition of self-fulfilling mechanism for Γ(E). Definition 5. Given a game Γµ (E), the mechanism µ is self-fulfilling if, for all t ∈ T and for all s ∈ S, r∈P t

X

π (·,µ) (s)

r∈P t

X

π (·,µ) (s)

t q(r|Pπ (·,µ) (s))ut (r, x(r, t, µ(r, t), π(r, µ))) ≥ t q(r|Pπ (·,µ) (s))ut (r, x(r, t, µ(s, t), π(r, µ \ µ(s, t)))),

for all µ(s, t) ∈ B(s, t).

3

An equivalence theorem

Codognato and Ghosal (2000) prove an equivalence theorem `a la Aumann (see Aumann (1964)) between the set of Cournot-Nash equilibrium allocations and the set of Walras equilbrium allocations for the game they analyse. Here, considering an extension of the game to the case of differential information, we show an equivalence between the set of allocations generated by self-fulfilling mechanisms and the set of rational expectations equilibrium allocations. To this end, we state and prove the following theorem. Theorem. Under Assumptions 1, 2, 3, (i) if Γµ∗ (E) is a game such that the mechanism µ∗ is self-fulfilling, the pair (p∗ , x∗ ) such that p∗ (s) = π(s, µ∗ ), for all s ∈ S, and x∗ (s, t) = x(s, t, µ∗ (s, t), p∗ (s)), for all s ∈ S and for all t ∈ T , is a rational expectations equilibrium; (ii) if (p∗ , x∗ ) is a rational expectations equilibrium, there exists a game Γµ∗ (E) such that the mechanism µ∗ is self-fulfilling, p∗ = π(s, µ∗ ), for all s ∈ S, and x∗ (s, t) = x(s, t, µ∗ (s, t), p∗ (s)), for all s ∈ S and for all t ∈ T . 7

Proof. (i) Let Γµ∗ be a game such that the mechanism µ∗ is self-fulfilling and let p∗ (s) = π(s, µ∗ ), for all s ∈ S, and x∗ (s, t) = x(s, t, µ∗ (s, t), p∗ (s)), ¯ ∗ is irreducible, for each for all s ∈ S and for all t ∈ T . Since the matrix M s ∈ S, Lemma 1 in Sahi and Yao (1989) implies that p∗ (s)  0, for each s ∈ S. Moreover, the function x∗ (·, t) is Ppt ∗ -measurable since the mechanism µ∗ is adapted to the price function π(s, µ∗ ). Furthermore, it should be clear that, for all s ∈ S and for all t ∈ T , p∗ (s)x∗ (s, t) = p∗ (s)w(s, t). Now, let us show that, for all s ∈ S and for all t ∈ T , X

r∈P t

p∗ (s)

q(r|Ppt ∗ (s))ut (r, x∗ (r, t)) ≥

X

r∈P t

q(r|Ppt ∗ (s))ut (r, y),

p∗ (s)

l : p∗ (s)x ≤ p∗ (s)w(s, t)}. Suppose that this is not the for all y ∈ {x ∈ R+ case for a state of nature s ∈ S and a trader t ∈ T . Then, by Assumption 2, l : p(s)x = p(s)w(s, t)} such that there exists a bundle y ∈ {x ∈ R+

X

t (s) r∈Pp ∗

q(r|Ppt ∗ (s))ut (r, y) >

X

t (s) r∈Pp ∗

q(r|Ppt ∗ (s))ut (r, x∗ (r, t)).

By Lemma 5 in Codognato and Ghosal (2000), there exists β j ≥ 0, 1, such that Pl ∗j j j=1 p (s)w (s, t) j j y =β , j = 1 . . . , l. p∗j (s)

Pl

j=1

βj =

Let us put µij (s, t) = wi (s, t)β j , i, j = 1 . . . , l. By taking into account the fact that, as an immediate consequence of Definition 3, π(s, µ∗ ) = π(s, µ∗ \ t µ(s, t)), it is easy to verify that, for all r ∈ Pπ (·,µ∗ )(s), y j = xj (r, t, µ(s, t), π(µ∗ \ µ(s, t)), j = 1, . . . , l. But then, r∈P t

X

π (·,µ∗ ) (s)

r∈P t

X

t ∗ q(r|Pπ (·,µ∗ ) (s))ut (r, x(r, t, µ(s, t), π(r, µ \ µ(s, t)))) > t ∗ ∗ q(r|Pπ (·,µ∗ ) (s))ut (r, x(r, t, µ (s, t), π(r, µ ))),

π (·,µ∗ ) (s)

contradicting the fact that the mechanism µ∗ is self-fulfilling. (ii) Let (p∗ , x∗ ) be a rational expectations equilibrium. We start by noting that, by Assumption 2, p∗ (s)  0, for each s ∈ S, and p∗ (s)x∗ (s, t) = p∗ (s)w(s, t), for all s ∈ 8

S and for all t ∈ T . But then, by Lemma 5 in Codognato and Ghosal (2000), P for all s ∈ S and for all t ∈ T , there exist β ∗j (s, t) ≥ 0, lj=1 β ∗j (s, t) = 1, such that ∗j

∗j

x (s, t) = β (s, t)

Pl

j=1

p∗j (s)wj (s, t) , j = 1 . . . , l. p(s)∗j

l such that β ∗j (s, t) = β ∗j (s, t), j = Now, define a function β ∗ : S × T → R+ 1, . . . , l, for all s ∈ S and for all t ∈ T and, then, define a function µ∗ : l2 such that µ∗ij (s, t) = wi (s, t)β ∗j (s, t), i, j = 1, . . . , l, for all S × T → R+ s ∈ S and for all t ∈ T . It is easy to verify that each coordinate of the function µ∗ (s, ·) is Lebesgue integrable, for all s ∈ S, and this, together with the fact that µ∗ (s, t) ∈ B(s, t), for all s ∈ S and for all t ∈ T , implies that the function µ∗ is a mechanism. Moreover, it should be clear that µ∗ (·, t) is Ppt ∗ measurable, for all t ∈ T . Now, we want to show that the mechanism µ∗ is irreducible. Consider a state of nature s ∈ S. Let i, j ∈ L be two commodities which stand in the relation C s . Consider a trader t ∈ Tis0 . First, notice that p∗ (s)w(s, t) > 0 since, by Assumption 2, p∗ (s)  0. This, together with Assumption 2, implies that x∗ (s, t) > 0 and, thus, that x∗j (s, t) > 0. Now, R L L L ¯ = (µ ¯ ij ) such that µ ¯ ij = T s0 wi (s, t)β ∗j (t) dλ if iC s j, consider the matrix M i L ¯ ij ¯ Lij > 0 because, for each t ∈ Tis0 , wi (s, t) > 0 µ = 0 otherwise. If iC s j, µ ¯ ∗ (s), and, by the above argument, β ∗j (s, t) > 0. But then, the matrix M ¯ Lij , i, j = 1, . . . , l, by Assumption 3 and by the ¯ ∗ij (s) ≥ µ which is such that µ same argument used in the proof of Theorem 2 in Codognato and Ghosal (2000), is irreducible. Since, by Assumption 3, this holds for each s ∈ S, we can conclude that the mechanism µ∗ is irreducible. It is easy to verify that x∗ (s, t) = x(s, t, µ∗ (s, t), p∗ (s)), for all s ∈ S and for all t ∈ T . Moreover, as x∗ is an allocation, it follows that

Z

T

∗j

x (s, t) dλ =

Z

T

wj (s, t) dλ −

l X

¯ ∗ji (s) + µ

i=1

l X

¯ ∗ij (s) µ

i=1

p∗i (s) Z = wj (s, t) dλ, p∗j (s) T

j = 1, . . . , l, for all s ∈ S. But this implies that p∗j (s)(

l X i=1

¯ ∗ji (s)) = µ

l X

¯ ∗ij (s)p∗i (s), j = 1, . . . , l, µ

i=1

for all s ∈ S, and then, by (1), p∗ (s) = π(s, µ∗ ), for all s ∈ S. Now, consider the game Γµ∗ (E). It remains to show that the mechanism µ∗ is self-fulfilling. 9

Suppose, on the contrary, that, in a state of nature s ∈ S, there exist a trader t ∈ T and a vector µ(s, t) ∈ B(s, t) such that r∈P t

X

π (·,µ∗ ) (s)

r∈P t

X

∗ t q(r|Pπ (·,µ∗ ) (s))ut (r, x(r, t, µ(s, t), π(r, µ \ µ(s, t)))) > t ∗ ∗ q(r|Pπ (·,µ∗ ) (s))ut (r, x(r, t, µ (r, t), π(r, µ ))).

π (·,µ∗ ) (s)

Since, as an immediate conequence of Definition 3, π(s, µ∗ \ µ(s, t)) = π(s, µ∗ ) = p∗ (s), for all s ∈ S, the former inequality implies that X

t (s) r∈Pp ∗

X

t (s) r∈Pp ∗

q(r|Ppt ∗ (s))ut (r, x(r, t, µ(s, t), p∗ (r))) > q(r|Ppt ∗ (s))ut (r, x∗ (r, t)).

Moreover, it is easy to check that p∗ (s)x(s, t, µ(s, t), p∗ (s)) = p∗ (s)w(s, t). But then, the pair (p∗ , x∗ ) is not a rational expectations equilibrium, generating a contradiction. The following example shows that the net assumption is the weakest joint restriction on traders’ endowments and preferences which guarantees that that a mechanism associated with a rational expectations equilibrium is irreducible, while allowing traders to have boundary endowments and indifference curves which intersect the boundary of the consumption set. Therefore, the net assumption is crucial to obtain an equivalence between the set of allocations generated by self-fulfilling mechanisms and the set of rational expectations equilibrium allocations. The example concerns a differential information economy with linear utility functions. In this case, the preferences of the traders are such that the set of commodities is not a net and the example exhibits a fully revealing rational expectations equilibrium allocation which is generated by no self-fulfilling mechanism. Example. Consider a differential information economy E with T = [0, 1]; {T1 , T2 , T3 } such that λ(T1 ) = λ(T2 ) = λ(T3 ); L = {1, 2, 3}; S = {1, 2}; q(1) = q(2); P t = {{1}, {2}}, for all t ∈ T1 ; P t = {1, 2}, for all T ∈ T \ T1 ; w(s, t) = (1, 0, 0), for s = 1, 2, and for all t ∈ T1 ; w(s, t) = (0, 1, 0), for s = 1, 2 and for all t ∈ T2 ; w(s, t) = (0, 0, 1), for s = 1, 2 and for all t ∈ T3 ; 10

ut (s, x) = x1 +sx2 +x3 , for s = 1, 2 and for all t ∈ T1 ; ut (s, x) = sx1 +x2 +x3 , for s = 1, 2 and for all t ∈ T2 ; ut (s, x) = x1 + x2 + x3 , for s = 1, 2 and for all t ∈ T3 . Then, (i) the pair (p∗ , x∗ ) such that p∗ (1) = (1, 1, 1), p∗ (2) = (1 + α, 1 + α, 1); x∗ (1, t) = w(1, t), for all t ∈ T ; x∗ (2, t) = (0, 1, 0), for all t ∈ T1 ; x∗ (2, t) = (1, 0, 0), for all t ∈ T2 ; x∗ (2, t) = (0, 0, 1), for all t ∈ T3 ; is a fully revealing rational expectations equilibrium; (ii) there exists no game Γµ∗ (E) such that the mechanism µ∗ is self-fulfilling, p∗ = π(s, µ∗ ), for s = 1, 2, and x∗ (s, t) = x(s, t, µ∗ (s, t), p∗ (s)), for s = 1, 2 and for all t ∈ T .

Proof.(i) It is straightforward. (ii) Suppose that there exists a self-fulfilling mechanism µ∗ such that p∗ (s) = π(s, µ∗ ), for s = 1, 2, and x∗ (s, t) = x(s, t, µ∗ (s, t), p∗ (s)), for s = 1, 2 and for all t ∈ T . Since the mecha¯ ∗ (s) must be irreducible, for s = 1, 2. nism µ∗ is self-fulfilling, the matrix M On the other hand, the structure of the initial assignment implies that µ∗2j (s, t) = µ∗3j (s, t) = 0, j = 1, 2, 3, for s = 1, 2 and for all t ∈ T1 ; µ∗1j (s, t) = µ∗3j (s, t) = 0, j = 1, 2, 3, for s = 1, 2 and for all t ∈ T2 ; µ∗1j (s, t) = µ∗2j (s, t) = 0, j = 1, 2, 3, for s = 1, 2 and for all t ∈ T3 . Moreover, the fact that x∗ (s, t) = x(s, t, µ∗ (s, t), p∗ (s)), for s = 1, 2 and for all ∗ (1, t) = µ∗13 (1, t) = µ∗13 (2, t) = 0, for all t ∈ T1 ; t ∈ T , implies that µ12 ∗ ∗ ∗ µ21 (1, t) = µ23 (1, t) = µ23 (2, t) = 0, for all t ∈ T2 ; µ∗31 (s, t) = µ∗32 (s, t) = 0, ¯ ∗ (s) is such that for s = 1, 2 and for all t ∈ T3 . But then, the matrix M ∗ ∗ ¯ ∗31 (s) = µ ¯ 13 ¯ 23 ¯ ∗32 (s) = 0, for s = 1, 2. This, in turn, imµ (s) = µ (s) = µ plies, by applying to each state of nature the argument used in the proof of ¯ ∗ (s) is not Example 1 by Codognato and Ghosal (2000), that the matrix M irreducible, for s = 1, 2, generating a contradiction.

From this example, we should also realize the fact that an irreducible mechanism not only does not generate rational expectations equilibrium allocations, but also, by leaving traders with their initial assignment, prevent them from achieving the information which may be fully revealed by rational expectations equilibrium prices.

4

Conclusion

Under the assumption that the set of commodities is a net, we show an equivalence between the set of allocations generated by self-fulfilling mechanisms and the set of rational expectations equilibrium allocations. We construct an 11

example which demonstrates that the equivalence result breaks down when the net assumption fails to hold: in such cases, traders cannot obtain the information revealed by rational expectations equilibrium prices. This should motivate a further investigation in the direction of looking for less centralized ways of conveying information into prices.

References [1] Aumann, R.J. (1964), “Markets with a continuum of traders,” Econometrica 32, 39-50. [2] Codognato G., Ghosal S. (2000), “Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices,” Journal of Mathematical Economics 34, 39-53. [3] Dubey P., Shapley L.S. (1994), “Noncooperative general exchange with a continuum of traders: two models,” Journal of Mathematical Economics 23, 253-293. [4] Forges F., Minelli E. (1997), “Self-fulfilling machanisms and rational expectations,” Journal of Economic Theory 75, 388-406. [5] Sahi S., Yao S. (1989), “The noncooperative equilibria of a trading economy with complete markets and consistent prices,” Journal of Mathematical Economics 18, 325-346. [6] Shapley, L.S., Shubik M. (1977), “Trade using one commodity as a means of payment,” Journal of Political Economy 85, 937-968.

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