Lionel De
email: lionel.de-boisde¤re@univ-paris1.fr

On the existence of financial equilibrium when beliefs are private

Keywords: sequential equilibrium, temporary equilibrium, perfect foresight, existence, rational expectations, nancial markets, asymmetric information, arbitrage. JEL Classication: D52

, stem from the perfect foresight assumption.

Introduction

The traditional approach to sequential nancial equilibrium relies on Radners (1972)(1973)(1974)(1975)(1976)(1977)(1978)(1979) classical, but restrictive, assumptions that agents have the so-called rational expectations of private information signals, and perfect foresight of future prices. Along the former assumption, agents are endowed, quoting Radner, with a model of how equilibrium prices are determined and (possibly) infer private information of other agents from comparing actual prices and price expectations with theoretical values at a price revealing equilibrium. Along the latter, agents anticipate with certainty exactly one price for each commodity (or asset) in each prospective state, which turns out to be the true price if that state prevails. Both assumptions presume much of agents inference capacities. Both assumptions lead to classical cases of inexistence of equilibrium, as shown by [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF], [START_REF] Hart | On the Optimality of Equilibrium when the Market Structure is Incomplete[END_REF], Momi (2000), [START_REF] Busch | Robust nonexistence of equilibria with incomplete markets[END_REF], among others. Building on our earlier papers, we show hereafter that the relevance and properties of the sequential equilibrium model can jointly be improved, if we drop these standard assumptions.

In a rst model [START_REF] Cornet | Arbitrage and price revelation with asymmetric information and incomplete markets[END_REF], dropping rational expectations only, we provided the basic tools, concepts and properties for an arbitrage theory, embedding jointly the symmetric and asymmetric information settings. In this model, we showed in [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF], standard existence problems of asymmetric information vanished, namely, a nancial equilibrium with nominal assets existed, not only generically -as in Radners (1979) rational expectations model -but under the very same no-arbitrage condition, with symmetric or asymmetric information, namely under the generalized no-arbitrage condition introduced in [START_REF] Cornet | Arbitrage and price revelation with asymmetric information and incomplete markets[END_REF]. This result was consistent with and (partially) extended [START_REF] Cass | Competitive equilibrium with incomplete nancial markets[END_REF] standard existence theorem to the asymmetric information setting. In a second model [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF], dropping both the rational expectation and perfect foresight hypothesis, we extended the above model of asymmetric information to one with all kinds of assets, where agents could forecast future prices with some private uncertainty. This new model embeds both Cornet-de Boisde¤res (2002) and (2009), and their main results, as particular application cases, but is innite dimensional and more general. It introduces the basic tools and properties for an arbitrage theory, when agents have asymmetric information and private idiosyncratic price expectations. These concepts and properties generalize Cornet-de Boisde¤res (2002[START_REF] Cornet | Elimination of arbitrage states in asymmetric information models[END_REF]. In particular, the innite model displays what information current prices may reveal about the future, to agents having no clue of how equilibrium prices are determined, hence, being prone to uncertainty between (typically) uncountable forecasts. The model shows that no-arbitrage prices, observed on markets, always convey enough information to free markets from arbitrage, after a nite number of inference steps. Then, agents updated beliefs are said to be revealed by prices.

Formally, the latter model is a two-period pure exchange economy, where agents, possibly asymmetrically informed, face an exogenous uncertainty, represented bynitely many random states of nature, exchange consumption goods on spot markets, and (nominal or real) securities on nancial markets, so as to transfer wealth across periods and states. At the rst period, besides the above exogenous uncertainty, agents may face an endogenous uncertainty on the future price, in each state they expect. That is, consumers have private sets of price anticipations, distributed along idiosyncratic probability laws, called beliefs. This uncertainty on prices is said to be endogenous, because it may a¤ect and is focussed on the endogenous variables.

The latter model of [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF] is dealt with throughout this paper. Its equilibrium notion, or correct foresight equilibrium (C.F.E.), is reached when all agents anticipate the true price as a possible outcome, and elect optimal strategies, which clear on all markets, ex post. This equilibrium is, indeed, a sequential one, i.e., di¤ers from the temporary equilibrium notion, introduced by [START_REF] Hicks | Value and Capital[END_REF] and developed by [START_REF] Grandmont | Temporary general equilibrium theory[END_REF][START_REF] Grandmont | Temporary general equilibrium theory[END_REF], [START_REF] Green | Temporary general equilibrium in a sequential trading model with spot and futures transactions[END_REF], [START_REF] Hammond | Overlapping expectations and Harts condition for equilibrium in a securities model[END_REF], [START_REF] Balasko | Temporary nancial equilibrium[END_REF], among others, in which agents need not anticipate prices correctly at the outset, and may need revise their plans and beliefs, ex post. Building on the arbitrage theory developed in [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF], the purpose of the paper is to study the existence conditions of the C.F.E., and to show this concept may bring a response to classical existence problems, following, not only Radners (1979) rational expectations equilibrium -which we had shown in [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF] already, but also [START_REF] Hart | On the Optimality of Equilibrium when the Market Structure is Incomplete[END_REF], [START_REF] Momi | Non-existence of equilibrium in an incomplete stock market economy[END_REF], [START_REF] Busch | Robust nonexistence of equilibria with incomplete markets[END_REF], in particular. We prove that a C.F.E. exists whenever agents anticipations embed a so-called minimum uncertainty set, presented hereafter. If required, agents beliefs at the C.F.E. may be revealed (in the above sense) by the equilibrium price itself.

The paper is organized as follows: we present the model, in Section 2, the minimum uncertainty set and existence Theorems, in Section 3, and the Theorems proof, in Section 4. An Appendix proves technical Lemmas.

The basic model

In this Section, we recall the framework and results of the model we had introduced in [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF], to which we refer the reader for more details. This model describes a pure-exchange economy with two periods (t 2 f0; 1g), a commodity market and a nancial market. At t = 0, agents may be asymmetrically informed and uncertain of future prices; and they are also uncertain of the state of nature, which will randomly prevail tomorrow. The sets of agents, I := f1; :::; mg, commodities, L := f1; :::; Lg, states of nature, S, and nancial assets, J := f1; :::; Jg, are all nite.

The models notations

Throughout, we denote by 1 the scalar product and k:k the Euclidean norm on an Euclidean space and by B(K) the Borel sigma-algebra of a topological space, K.

We let s = 0 be the non-random state at t = 0 and S := f0g [ S. For all set 6 S and tuple (s; l; x; x ; y; y ) 2 62L2R 6 2R 6 2R L6 2R L6 , we denote by: F x s 2 R, y s 2 R L the scalar and vector, indexed by s 2 6, of x, y, respectively; F y l s the l th component of y s 2 R L ;

F x 6 x and y 6 y (respectively, x << x and y << y ) the relations x s 6 x s and y l s 6 y l s (resp., x s < x s and y l s < y l s ) for each (l; s) 2 f1; :::; Lg26; F x < x (resp., y < y ) the joint relations x 6 x , x 6 = x (resp., y 6 y , y 6 = y );

F R L6 + = fx 2 R L6 : x > 0g and R 6 + := fx 2 R 6 : x > 0g, R L6 ++ := fx 2 R L6 : x >> 0g and R 6 ++ := fx 2 R 6 : x >> 0g, F M 0 := f(p 0 ; q) 2 R L + 2R J : kp 0 k + kqk = 1g; F M s := f(s; p s ) : p s 2 R L + ; kp s k = 1g
, whenever s 2 S, and M := [ s2S M s .

The commodity and asset markets

The L consumption goods, l 2 L, may be exchanged by consumers, on the spot markets of both periods. In each state, s 2 S, an expectation of a spot price, p 2 R L + , or the spot price, p, in state s itself, are denoted by the pair ! s := (s; p) 2 S 2 R L + , and normalized, at little cost, to the above set M s .

Each agent, i 2 I, is granted an endowment, e i := (e is ) 2 R LS + , which secures her the commodity bundle, e i0 2 R L + at t = 0, and e is 2 R L + , in each state s 2 S, if this state prevails at t = 1. To simplify notations, we will also denote e i! := e is , for every triple (i; s; !) 2 I 2 S 0 2 M s . Ex post, the generic i th agents welfare is measured by a continuous utility index, u i : R 2L + ! R + , over her consumptions at both dates.

The nancial market permits limited transfers across periods and states, via J assets, or securities, j 2 J := f1; :::; Jg, which are exchanged at t = 0 and pay o¤, in commodities and/or in units of account, at t = 1. For any spot price, or expectation, ! 2 M, the cash payo¤s, v j (!) 2 R, of all assets, j 2 f1; :::; Jg, conditional on the occurence of price !, dene a row vector, V (!) = (v j (!)) 2 R J . By denition and from the continuity of the scalar product, the mapping ! 2 M 7 ! V (!) is continuous.

The nancial structure may be incomplete, namely, the span,

< (V (! s )) s2S > := f(V (! s ) 1 z) s2S : z 2 R J g may have lower rank (for all prices (! s ) 2 5 s2S M s ) than #S.
From above, assets provide no insurance against endogenous uncertainty.

Agents can take unrestrained positions (positive, if purchased; negative, if sold), in each security, which are the components of a portfolio, z 2 R J . Given an asset price, q 2 R J , a portofolio, z 2 R J , is thus a contract, which costs q 1 z units of account at t = 0, and promises to pay V (!) 1 z units tomorrow, for each spot price ! 2 M, if ! obtains. Similarly, we normalize rst period prices, ! 0 := (p 0 ; q), to the set M 0 .

Information and beliefs

Consistently with [START_REF] Cornet | Arbitrage and price revelation with asymmetric information and incomplete markets[END_REF], each agent receives a private information signal, S i S, during the rst period, which informs her that the true state, i.e., that which will prevail at t = 1, will be in S i . Henceforth, the collection, (S i ), of all agents signals is set as given and we let 5 := \ m i=1 S i . Agents are correctly informed, in the sense that no state of S n 5 will prevail. They form and may update their private anticipations of future spot prices in each state they expect, and these anticipations are distributed along idiosyncratic probability laws, called beliefs. Formally, we recall from [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF]: Denition 1 For all probability , on (M; B(M)), and pair (! := (s; p); ") 2 M2R ++ , we let B(!; ") := f(s 0 ; p 0 ) 2 M : kp 0 0pk+js 0 0sj < "g and P () := f! 2 M : (B(!; ")) > 0; 8" > 0g be a compact set, whose elements are called anticipations, expectations or forecasts.

A probability, , on (M; B(M)), is called a belief if the relation P () S 2 R L ++ holds (hence, " := inf (l;(s;ps))2L2P () p l s > 0). We denote by B the set of all beliefs. A belief, 0 2 B, is said to rene 2 B, and we denote it by 0 , if P (0) P ().

Two beliefs, (; 0) 2 B 2 , are said to be equivalent, and we denote it by 0 , if A collection of beliefs, ( i ) 2 B m , is called a structure (of beliefs), and we denote it by ( i ) 2 SB, if the following condition holds:

P (0) = P (),
(a) \ m i=1 P ( i ) 6 = ?, i.e., the common anticipation set is non-empty. Similarly, a collection of classes, ( o i ) 2 CB m , is called a class structure (of beliefs), and we denote it by

( o i ) 2 CSB, if \ m i=1 P ( o i ) 6 = ?. Let (( i ); ( 0 i )) 2 SB 2 , ( o
i ) ( 0 i ), if both relations ( i ) ( 0 i ) and ( 0 i ) ( i ) hold. A renement, ( 3
i ) 2 SB, of ( i ) 2 SB, is said to be self-attainable if the following Condition holds:

(b) \ m i=1 P ( 3 i ) = \ m i=1 P ( i ),
i.e., the common anticipation set is left unchanged. The notions of renement and self-attainable renement are dened alike on CSB.

Remark 1 We notice that a class of beliefs identies to a set of anticipations

(not yet ordered by a probability distribution, i.e., by a belief) and a class structure identies to a collection of anticipations, some of which are common to all agents. Though it is not required for the arbitrage theory we present in Section 2, and, hence, not stated as a condition in the above Denition, we will restrict, in the subsequent Sections 3 and 4, agents beliefs, ( i ) 2 SB, to be consistent with the information signals they receive during the rst period, that is, restrict beliefs to satisfy P ( i ) \ M s = ?, for every pair (i; s) 2 I 2 SnS i . This rationality assumption may also be made in the following sub-Section 2.4, but it is not required, formally.

Consumers behavior and the notion of equilibrium

In Section 3, we recall from [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF] how agents may rene their anticipations from observing markets in this model. Hereafter, we assume that agents implement their decisions after having reached their (nal) structure of beliefs, ( i ) 2 SB, and observed the market price, ! 0 := (p 0 ; q) 2 M 0 , at t = 0, which are set as given and referred to throughout. We also assume that, by the time agents trade, markets have eliminated redundant assets, in the sense that no non-zero portfolio, z 2 R J nf0g, is left, which yields an agent (say, i 2 I) no payo¤ at all, that is, V (!) 1 z = 0, for every

! 2 P ( i ).
Then, the generic i th agents consumption set,

X( i ) := C (P 0 ( i ); R L + ) , is the set of continuous mappings from P 0 ( i ) := f0g [ P ( i ) to R L + . A consumption,
x 2 X( i ), is, thus, a mapping, relating s = 0 to a (xed) consumption decision,

x 0 := x ! 2 R L
+ , at t = 0, and, continuously in ! 2 P ( i ), every anticipation, ! := (s; p) 2 P ( i ), of the spot price to a random consumption decision, x ! 2 R L + , at t = 1, which is conditional on the joint occurence of state s and spot price p at t = 1.

Each agent i 2 I elects and implements a consumption and investment decision, or strategy, (x; z) 2 X( i ) 2 R J , that she can a¤ord on markets, given her endowment, e i 2 R LS + , and her expectation set, P ( i ). This denes her budget set as follows:

B i (! 0 ; i ) := f(x; z) 2 X( i )2R J : p 0 1(x 0 0e i0 ) 6 0q1z; p s 1(x ! 0e i! ) 6 V (!)1z; 8! := (s; p s ) 2 P ( i )g . An allocation, (x i ) 2 X[( i )] := 5 m i=1 X( i )
, is a collection of consumptions across consumers. We dene the following set of attainable allocations:

A((! s ); ( i )) := f(x i ) 2 X[( i )] : P m i=1 (x i0 0e i0 ) = 0; P m i=1 (x i!s 0e i!s ) = 0; 8s 2 5; s:t: ! s 2 \ m i=1 P ( i )g ,
for every price collection, (! s ) := (! s ) s25 2 5 s25 M s . Each agent i 2 I has preferences represented by the V.N.M. utility function:

u i i : x 2 X( i ) 7 ! u i i (x) := R !2P (i) u i (x 0 ; x ! )d i (!).
The generic i th agent elects a strategy, which maximises her utility function in the buget set, i.e., a strategy of the set

B 3 i (! 0 ; i ) := arg max (x;z)2Bi(!;i) u i i (x).
The above economy is denoted by E. Its equilibrium concept is dened as follows:

Denition 2 A collection of prices, (! s ) 2 5 s25 M s , beliefs, ( i ) 2 SB, and strategies,

(x i ; z i ) 2 B i (! 0 ; i ), dened for each i 2 I, is a correct foresight equilibrium (C.F.E.),
or a sequential equilibrium (respectively, a temporary equilibrium) of the economy

E, if the following Conditions (a)-(b)-(c)-(d) (resp., Conditions (b)-(c)-(d)) hold: (a) 8s 2 5, ! s 2 \ m i=1 P ( i ); (b) 8i 2 I; (x i ; z i ) 2 B 3 i (! 0 ; i ) := arg max (x;z)2Bi(!;i) u i i (x); (c) (x i ) 2 A((! s ); ( i )); (d) P m i=1 z i = 0.
Under above conditions, ( i ) 2 SB, or (! s ) 2 5 s25 M s , are said to support equilibrium.

The models no-arbitrage prices and the information they reveal

We recall from [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF] the denitions of arbitrage-free prices, beliefs, and structures. , and a price, q 2 R J , be given. The couples,

(V; o ) or (V;
), are said to to be q-arbitrage-free (hence, arbitrage-free), or q to be a no-arbitrage price of (V; o ), or (V; ), if the following equivalent Conditions hold:

(a) there is no portfolio z 2 R J , such that 0q 1 z > 0 and V (!) 1 z > 0 for every

! 2 P () = P ( o )
, with at least one strict inequality;

(b) there exists a continuous mapping

: P () ! R ++ , such that q = R !2P () (!)V (!)d(!).
We let

Q[V; o ] be the set of no-arbitrage prices of (V; o ) (or (V; )) and Q c [V; ( o i )] := \ m i=1 Q[V; o i ] be the set of common no-arbitrage prices of [V; ( o i )]. The class structure [V; ( o i )] is said to be arbitrage-free (resp., q-arbitrage-free) if Q c [V; ( o i )] 6 = ? (resp., if q 2 Q c [V; ( o i )]
).We say that q is a no-arbitrage price (resp., a self-attainable no-

arbitrage price) of [V; ( o i )]
if there exists a renement (resp., a self-attainable renement), (

o 3 i ) ( o i ), such that q 2 Q c [V; ( o 3
i )], and we denote their set by Q[V;

( o i )],
which is non-empty. All above denitions and notations can be stated, equivalently, in terms of any representative structure, ( i ) 2 5 m i=1 o i . We then refer to

Q c [V; ( i )] := Q c [V; ( o i )] and Q[V; ( i )] := Q[V; ( o i )]
as, respectively, the sets of common no-arbitrage prices, and no-arbitrage prices, of the structure [V; ( i )]. When no confusion arises, the reference to V is dropped in the above denitions and notations.

We now summarize, into the following Claim 1, the main results proved in [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF].

Claim 1 Let ( o i )
2 CSB and a price, q 2 R J , be given. The following assertions hold:

(i) the class structure ( o i ) (or a structure, ( i ) 2 5 m i=1 o i )
) is arbitrage-free if and only if there exist no portfolios (z i ) 2 (R J ) I , such that

P m i=1 z i = 0 and V (! i ) 1 z i > 0 for every couple (i; ! i ) 2 I 2 P ( i ) = I 2 P ( o i
), with at least one strict inequality;

(ii) there exists a unique coarsest arbitrage-free renement of

( o i ), denoted by o 5[V; ( o i )] or o 5[( o i )]. It is self-attainable. It is equal to ( o i ) if and only if ( o i
) is arbitrage-free;

(iii) for each i 2 I, there exists a set, o 5(V; o i ; q)2 ? [ CB, also denoted by o 5( o i ; q), and said to be revealed by price q, which is empty, if and only if o i has no q-arbitragefree renement, and, otherwise, is the coarsest q-arbitrage-free renement of o i ;

(iv) along assertion (iii), above, and Denition 1, the following relations hold:

( q 2 Q[( o i )],
i.e., q is a no-arbitrage price) , (

i ; q)) 2 CSB) , ( ( o 5( o i ; q)) ( o i )) ( o 5( o
, (

i ; q)) is the coarsest q-arbitrage-free renement of ( o i )). ( o 5( o
When such conditions hold, the renement,

( o 5( o i ; q))
, is said to be revealed by price q. A renement is said to be price revealable if it can be revealed by some price.

(v) ? 6 = fq 2 R J : o 5[( o i )] = ( o 5( o i ; q))g = Q c [V; o 5[( o i )]], that is, the coarsest renement, o 5[( o i )],
is revealed by any of its common no-arbitrage prices;

(vi) ( o 5( o i ; q)) ( o i ) is self-attainable if and only if q 2 Q[( o i )] is self-attainable.
We now introduce and discuss a notion of minimum uncertainty on expectations, when agents form their anticipations privately, and state our existence Theorem.

3 An uncertainty principle and the existence of equilibrium

The existence Theorem

When anticipations are private, there exists a set of minimum uncertainty on future prices, any element of which is an equilibrium price for some structure.

Denition 4 Let an economy, E, as described in Section 2, be given and let A be its set of sequential equilibria (i.e., of C.F.E.). The minimum uncertainty set, 1, of this economy is the subset of [ s25 M s , whose elements support a C.F.E., that is:

In Section 4, we prove that 1 is non-empty and that a continuum of equilibria exists in standard conditions. Given this, the set 1 is typically uncountable. In the economy, E, agents information and beliefs are private. So, each agent sees other agents beliefs as (possibly arbitrary) elements of the set B. We retain the standard small consumer, price-taker, hypothesis, which states that no single agents belief, or strategy, may have a signicative impact on equilibrium prices. The economy, E, is said to be standard if, in addition to that hypothesis, it meets the Conditions:

F Assumption A1: 8i 2 I; e i >> 0;

F Assumption A2: 8i 2 I, u i is class C 1 , strictly concave, strictly increasing.

Along Theorem 1, in any standard economy, E, the minimum uncertainty set, (i) 9" > 0 : 8p := (s; ) 2 1, 8l 2 L, l > ";

(ii) 1 6 = ?.

Consistently, with the above Assertions

(i)-(ii), if the relation 1 \ m i=1 P ( o i ) holds,
that is, if every agents anticipations embed 1, then, the following Assertions hold: (iv) if a self-attainable renement, (

(iii) if the class structure, [V; ( o i )],
o 3 i ), of ( o i )
, is arbitrage-free (and such a renement exists), every representative renement,

( 3 i ) 2 5 m i=1 o 3 i , supports a C.F.E. (v) if ( o 3
i ) 2 CSB is a self-attainable price-revealable renement of ( We let the reader check, as standard from Assumption A2, that if

( 3 i ) 2 SB and ! 0 := (p 0 ; q) 2 M 0 support a C.F.E., then, q 2 Q c [V; ( 3 i )].
Consequently, if that supporting structure,

( 3 i ) 2 SB
, is price-revealable, then, it is revealed by the equilibrium price, i.e., the C.F.E. is price-revealed. Given this, Assertion (v) of Theorem 1 is a Corollary of Assertion (iv) and only Assertions (i)-(ii)-(iii) need be proved.

Before discussing the Theorem Condition, 1 \ m i=1 P ( o i ), we prove Assertion (i).

Proof of Assertion (i) Let A and 1 be the sets of Denition 4, and s 2 5 and p := (s; ) 2 1 be given. From the denition, there exist prices, (! s ) 2 5 s25 M s , beliefs,

( i ) 2 SB, and strategies, [(x i ; z i )] 2 5 m i=1 B i (! 0 ; i ), such that C : = ((! s ); ( i ); [(x i ; z i )]) 2
A and p = ! s . Let e := max (s;l)2S 2L P m i=1 e l is > 0 and C := sup @ui @y l (x; y)= @ui @y l (x; y), for (i; (x; y); (l; l 0 )) 2 I2[0; e] 2L 2L 2 , be given. Then, for each s 2 5 0 , the relations (x is ) > 0 and P m i=1 (x is 0e is ) = 0 hold, from Condition (c) of Denition 2 (applied to C), and imply x is 2 [0; e] L , for each i 2 I. Moreover, the relations C 2 R ++ and p 2 S2R L ++ are standard from Assumption A2 and Condition (b) of Denition 2 (on C). Let (l; l 0 ) 2 L 2 be given. We show that l l 6 C. Otherwise, it is standard, from Assumptions A1-A2 and Conditions (b)-(c) of Denition 2 (applied to C), that there exist i 2 I and x 2 X( i ), identical to x i in every image, x ! (for ! 2 f0g [ P ( i )), but two,

x l p := x l ip + E l and x l p := x l ip 0 E l (for E 2 R ++ small enough), such that (x; z i ) 2 B i (! 0 ; i ) and u i i (x) > u i i (x i ),
which contradicts the fact that C meets the Condition (b) of Denition 2. We let the reader check the joint relations >> 0, kk = 1 and l l 6 C, which hold, from above, for each (l; l 0 ) 2 L 2 , imply l > " := 1 C p L , for every l 2 L. 3

Referring the reader to [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF] for the models renement of information process, we now discuss the Theorems condition, namely, the uncertainty principle it embeds.

Discussing the Theorems Condition

The Theorems Condition, 1 \ m i=1 P ( o i ), is consistent with the models assumption of price-takers agents seeing other consumers beliefs as arbitrary: then, 1 is seen as the set of all possible equilibrium prices. It is also consistent with the no-arbitrage condition with asymmetric beliefs, i.e., with anticipations precluding arbitrage (see below). From Theorem 1, the Condition is su¢cient for the existence of a C.F.E. It may also be a necessary condition, if beliefs are unpredictable and erratic enough to let any price in 1 become a possible outcome. This situation may arise in times of enhanced uncertainty and volatility. Then, the equilibrium is, typically, non-cooperative. Contrarily, if agents co-operate and share rened beliefs, they might reach a nest sequential equilibrium, which exists from Theorem 2. Proof We assume that Theorem 1 holds and, non restrictively, that [V;

( o i )], is such that P ( o i ) = 1,
for each i 2 I (hence, arbitrage-free). Let R ( o i) 6 = ? be the set of symmetric renements of and1

( o i ), which satisfy Condition (i) of Theorem 2. Let f( o k i )g k2K 2 R K ( o i) be a chain, 1 k := P ( o k i ), for every (i; k) 2 I 2 K,
3 := \ k2K 1 k .
As an intersection of a chain of compact sets, 1 3 is a compact set, hence, nonempty. Let ( o 0 i ) ( o i ) be the symmetric renement dened by P ( o 0 i ) = 1 3 , for every i 2 I. We let the reader check by contraposition (as tedious and too long here), from similar arguments as for Lemmas 3-4 (see the Appendix), that ( An emprical justication of the Theorems Condition could be that relative prices may be observed and analysed, on long time series, for virtually all types of uncertainty, beliefs, price volatility, speculation, or stability, and past states of nature.

o 0 i ) 2 R ( o i) , whereas ( o 0 i ) (
From the joint set of relative prices so observed, the set, 1, of all normalized equilibrium prices, or a bigger set, might be inferred. Below, we hint at another possible explanation, which is theoretical and could be addressed in a subsequent paper.

It is important to recall that agents have no price model in the economy E.

Their anticipations start from an initial class structure, ( o i ) 2 CSB, which may be updated to a renement,

( 3 i ) ( o i
), when they trade. With no price model, agents can only infer from markets the information that arbitrage opportunities reveal. As shown in [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF], from assets payo¤s, such unsophisticated agents may always rene their anticipations, in nitely many steps, up to a coarse, self-attainable, arbitragefree renement, (

o 3 i ) = ( o i ).
Our conjecture is that a similar renement process as that of [START_REF] De Boisde¤re | Arbitrage and price revelation with private beliefs[END_REF], through trade, exists on spot markets, in which 1 is a xed set of no-arbitrage prices, that cannot be ruled out. Since arbitrage is limited by nite endowments, agents may also forecast positive prices outside 1 at equilibrium. Once inferred from arbitrage as above, the renement, ( The generic i th agents set and strategy are dened ex ante, with reference to ex ante conditions, and a xed expectation set, P ( o 3 i ). So, only her ex ante span of payo¤s matters, < V; )) i2I . Along Remark 3, we need only prove assertions (ii) and (iii) of Theorem 1. The proofs principle is to construct a sequence of auxiliary economies, with nite expectation sets, rening and tending to the initial expectations sets, (P i ) i2I . Each nite economy admits an equilibrium along Theorem 1 of [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF], which we set as given. Then, from the sequence of nite dimensional equilibria, we derive an equilibrium of the initial economy, E. To that aim, we introduce auxiliary sets.

o 3 i > := f! 2 P ( o 3 i ) 7 ! V (!) 1 z : z 2 R J g,

Auxiliary sets

We divide P i , for each i 2 I, in ever ner partitions, and let for each n 2 N:

K n := fk n := (k 1 n ; :::; k L n ) 2 (N \ [0; 2 n 0 1]) L g;
P (i;s;kn) := P i \ (fsg25 l2f1;:::

;Lg ] k l n 2 n ; k l n +1
2 n ]), for every (s; k n := (k Lemma 1 There exists N 2 N 3 , such that the following Assertion holds:

(i) 8( o 3 i ) ( o i ), (G N i P ( o 3 i ), 8i 2 I) ) (Q c [V; ( o 3 i )] 6 = ?).
Proof see the Appendix.

3

For every integer n > N along Lemma 1, and any element 2 ]0; 1], hereafter set as given, we consider an auxiliary economy, E n , which admits an equilibrium, C n .

Auxiliary economies, E n

We dene N N := N n f0; 1; :::; N 0 1g, along Lemma 1, and set as given, for each , presented hereafter.

We now let n 2 N N be given and derive from the set, G n i , of sub-Section 4.1, and prices, (! n01 s ) 2 5 s25 M s , assumed to be dened at the last induction step, an auxiliary economy, E n , referred to as the (n; )-economy, which is of the type described in [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF]. Namely, it is a pure exchange economy, with two period (t 2 f0; 1g), 2 Non restrictively (up to a shift in the upper boundary of P i;s;kn ), we assume that each g n i;s;kn P i;s;kn is in the interior of P i;s;kn $ = ?, to insure that i (P i;s;kn ) > 0.

16 Documents de Travail du Centre d'Economie de la Sorbonne -2012.55 m agents, having incomplete information, and exchanging L goods and J nominal assets, under uncertainty (at t = 0) about which state of a nite state space, S n , will prevail at t = 1. Referring to [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF], and to the above notations and denitions in the economy E, the generic (n; )-economys characteristics are as follows:

F The information structure is the collection, (S n i ), of sets

S n i := 5 [ e S n
i (and we let S 0n

i := 5 0 [ e S n
i ), such that e S n i := fig2G n i is dened for each i 2 I. The pooled information set (of the states which may prevail at t = 1) is, hence, 5 = \ i2I S n i . For each i 2 I, the set e S n i := fig2G n i consists of purely formal states, none of which will prevail. The state space of the (n; )-economy is

S n = [ i2I S n i . F The S n 2J payo¤ matrix, V n := (V n (s n ))
, is dened, with reference to the payo¤ mapping, V , of the economy E, by V n (s) := V ((s; p n01 s )), for each s 2 5, and

V n (s n ) := V (!),
for each s n := (i; !) 2 S n . The payo¤ matrix V n is purely nominal.

F In each formal state, s n := (i; (s; p s )) 2 e

S n i , the generic agent i 2 I is certain that price p s 2 R L ++ , and only that price, can prevail on the s n -spot market.

F In each realizable state, s 2 5, the generic agent i 2 I has perfect foresight, i.e., anticipates with certainty the true price, say

p n s 2 R L ++ (or ! n s := (s; p n s ) 2 M s ).
F The generic i th agents endowment,

e n i := (e n is n ) 2 R LS n i
++ , is dened (with reference to e i in E) by e n is := e is , for each s 2 5 0 , and e n is n := e is , for each s n := (i; (s; p s )) 2 e S n i .

F For every collection of the true market prices, ! n 0 := (p n 0 ; q n ) 2 M 0 , at t = 0, and ! n s := (s; p n s ) 2 M s , for all s 2 5, at t = 1, the generic i th agent has the following consumption set, X n i , budget set, B n i (S n i ; (! n s )), and utility function, u n i :

X n i := R LS n i + ; B n i (S n i ; (! n s )) := {(x; z) 2 X n i 2R J : 2 6 6 4
p n 0 1(x 0 0e i0 ) 6 0q n 1z and p n s 1(x s 0e is ) 6 V n (s)1z; 8s 2 5

p s 1(x s n 0e is ) 6 V n (s n )1z;
u n i : x 7 ! P s n 2S n i n i (s n )u i (x 0 ; x s n ),
where n i (s n ) := i (P (i;s;kn) ), for every (s;

k n ; s n ) 2 S i 2 K n 2 (fig 2 [G n i \ P (i;s;kn) ])
, is the probability of the set P (i;s;kn) 6 = ?, along the belief i , and n i (s) := , for each s 2 5.

Theorem 1 of [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF] and Lemma 1 above yield Lemma 2.

Lemma 2 The generic (n; )-economy admits an equilibrium, namely a collection of prices, ! n 0 := (p n 0 ; q n ) 2 M 0 , at t = 0, and ! n s := (s; p n s ) 2 M s , in each state s 2 5, and strategies, (

x n i ; z n i ) 2 B n i (S n i ; (! n s ))
, dened for each i 2 I, such that:

(i) 8i 2 I; (x n i ; z n i ) 2 arg max (x;z)2B n i (S n i ;(! n s )) u n i (x); (ii) 8s 2 5 0 ; P m i=1 (x n is 0 e is ) = 0; (iii) P m i=1 z n i = 0.
Moreover, the equilibrium prices and allocations satisfy the following Assertions:

(iv) 8(n; i; s) 2 N N 2 I 2 5 0 , x n is 2 [0; e] L
, where e := max (s;l)2S 2L P m i=1 e l is ;

(v) 9" 2]0; 1] : p nl s > ", 8(n; s; l) 2 N N 2 5 2 L.

Proof see the Appendix.

3

Along Lemma 2, we set as given an equilibrium of the (n; )-economy, namely:

C n := (! n 0 := (p n 0 ; q n ); (! n s ); [(x n i ; z n i )]) 2 M 0 2 5 s25 M s 2 5 m i=1 B n i (S n i ; (! n s )),
which is always referred to. The equilibrium prices, (! n s ) 2 5 s25 M s , permit to pursue the induction and dene the (n+1; )-economy in the same way as above, hence, the auxiliary economies and equilibria at all ranks. These meet the following Lemma.

Lemma 3 For the above sequence, fC n g, of equilibria, it may be assumed to exist:

(i) ! 0 := (p 0 ; q ) = lim n!1 ! n 0 2 M 0 and ! s := (s; p s ) = lim n!1 ! n s 2 M s , for each s 2 5; (ii) (x is ) := lim n!1 (x n is ) i2I 2 R Lm , such that P i2I (x
is 0 e is ) = 0, for each s 2 5 0 ;

18 Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

(iii) (z i ) = lim n!1 (z n i ) i2I 2 R Jm , such that P m i=1 z i = 0.
Moreover, we dene, for each i 2 I, the following sets and mappings:

G 1 i := [ n2N G n i = lim n!1 % G n i P i ; for each n 2 N, the mapping, ! 2 P i 7 ! =HC n i (!) 2 G n i , from the relations (!; =HC n i (!)) 2 P 2 (i;s;kn) , which hold (for every ! 2 P i ) for some (s; k n ) 2 S i 2K n ; from Assertion (i) and Lemma 2-(v), the belief, i := 1 1+#5 ( i + P s25 E s ),
where E s is (for each s 2 5) the Diracs measure of ! s ; the support of i 2 B, denoted by

P i := P ( i ) = P i [ f! s g s25 ; for all (! := (s; p s ); z) 2 M2R J , the set B i (!; z) := fx 2 R L + : p s 1(x 0 e is ) 6 V (!)1zg.
Then, the following Assertions hold, for each i 2 I:

(iv) G n i G n+1 i , 8n 2 N, G 1 i = P i and f=HC n i (!)g n2N converges to ! uniformly on P i ; (v) 8s 2 5, fx is g = arg max x2Bi(! s ;z i ) u i (x i0 ; x), along Assertion (ii); we let x i! s := x is ; (vi) the correspondence ! 2 P i 7 ! arg max x2Bi(!;z i ) u i (x i0 ; x) is a continuous mapping, denoted by ! 7 ! x i! . The mapping, x i : ! 2 f0g [ P i 7 ! x i! ,

dened from Assertions

(ii) and (v) and above, is a consumption plan, henceforth referred to as

x i 2 X( i ); (vii) u i i (x i ) = 1 1+#5 lim n!1 u n i (x n i ) 2 R + .
Proof see the Appendix. Let i 2 I be given. From the denition of C n , the relations p n 0 1(x n i0 -e i0 ) 6 0q n 1z n i and

p n s 1(x n is -e is ) 6 V ((s; p n01 s
))1z n i hold, for each (n; s) 2 N N 25, which yield in the limit (from the continuity of the scalar product):

p 0 1(x i0 -e i0 ) 6 0q 1z i and p s 1(x is -e is ) 6 V (! s )1z
i , for each s 2 5. The relations p s 1(x i! -e is ) 6 V (!)1z i hold, for all (s; ! := (s; p s )) 2 S i 2 P i , from Lemma 3-(v)-(vi). This implies, from Lemma 3-(vi):

[(x i ; z i )] 2 5 m i=1 B i (! 0 ; i ).
Assume, by contraposition, that C fails to meet Denition 2-(b), then, there exist

i 2 I, (x; z) 2 B i (! 0 ;
i ) and " 2 R ++ , such that:

(I) " + u i i (x i ) < u i i (x).
We may assume that there exists E 2 R ++ , such that:

(II) x l ! > E, for every (!; l) 2 f0g [ P i 2 L.
If not, for every B 2 [0; 1], we dene the strategy (x B ; z B ) := ((1 0 B)x

+ Be i ; (1 0 B)z),
which belongs to B i (! 0 ;

i ), a convex set. From Assumption A1, the strategy (x B ; z B ) meets relations (II) whenever B > 0. Moreover, from relation (I) and the uniform continuity of (B; !) 2 [0; 1] 2 P i 7 ! u i (x B 0 ; x B ! ) on a compact set (which holds from Assumption A2 and the relation x 2 X( i )), the strategy (x B ; z B ) also meets relation (I), for every B > 0, small enough. So, we may assume relations (II).

We let the reader check, as immediate from the relations (x; z) 2 B i (! 0 ;

i ) and (III) p 0 1(x 0 0 e i0 ) 6 D 0 q 1z and p s 1(x ! 0 e is ) 6 D + V (!)1z, 8! := (s; p s ) 2 P i .

From (III), the continuity of the scalar product (hence, of ! 7 ! V (!)) and Lemma 3-(i)-(iii)-(iv), there exists N 1 2 N N , such that, for every n > N 1 :

(IV ) )) be the strategy dened by x n 0 := x 0 , x n s := x ! s , for every s 2 5, and

x n s n := x ! , for every s n := (i; !) 2 e

S n i , and recall that:

F u i i (x) := 1 1+#5 R !2Pi u i (x 0 ; x ! )du i i (!) + 1+#5 P s25 u i (x 0 ; x ! s ); F u n i (x n ) := P s n 2S n i n5 u i (x 0 ; x s n ) n i (s n ) + P s25 u i (x 0 ; x ! s ).
Then, from above, Lemma 3-(i)-(iv) and the uniform continuity of x 2 X( i ) and u i on compact sets, there exists N 2 > N 1 such that (with C := (1+#5)):

(V ) jCu i i (x)-u n i (x n )j 6 R !2Pi ju i (x 0 ; x ! )-u i (x 0 ; x =HC n i (!) )jdu i i (!) < " 2 , for every n > N 2 .
From equilibrium conditions and Lemma 3-(vii), there exists N 3 > N 2 , such that:

(V I) u n i (x n ) 6 u n i (x n i ) < " 2 + Cu i i (x i ), for every n > N 3 (with C := (1+#5)).
Let n > N 3 be given. The above Conditions (I)-(V )-(V I) yield, jointly:

Cu i i (x) < " 2 + u n i (x n ) < " + Cu i i (x i ) < Cu i i (x).
This contradiction proves that C meets Condition (b) of Denition

) := fx 2 R L + : p s 1(x0e is ) 6 V (!)1zg
, for all (! := (s; p s ); z) 2 P i 2R J . Then, whenever 1 \ m i=1 P i along Denition 4, the following Assertions hold for each i 2 I:

(i) for each s 2 5 0 , it may be assumed to exist prices, ! 3

s = lim k!1 ! k s 2 M s , such that f! 3 s g s25 \ m i=1 P i , and consumptions, x 3 is = lim k!1 x k is , such that P i2I (x 3 is 0e is ) = 0;
(ii) it may be assumed to exist portfolios,

z 3 i = lim k!1 z k i , such that P m i=1 z 3 i = 0; (iii) 8s 2 5, fx 3 is g = arg max x2Bi(! 3 s ;z 3 i ) u i (x 3 i0 ; x) along Assertion (i); we let x 3 i! 3 s := x 3 is ; (iv) the correspondence ! 2 P i 7 ! arg max x2Bi(!;z 3 i ) u i (x 3 i0 ; x) is a continuous mapping, denoted by ! 7 ! x 3 i! . The mapping x 3 i : ! 2 f0g [ P i 7 ! x 3 i! , dened from Assertions (i)-(iii)
and above, is a consumption plan, referred to as

x 3 i 2 X( i ); (v) for every x 2 X( i ), u i i (x) = lim k!1 u k i (x) 2 R + and u i i (x 3 i ) = lim k!1 u k i (x k i ) 2 R + .
Proof see the Appendix.

3

Claim 3 Whenever 1 \ m i=1 P i , the collection of prices, (!

s ) = lim k!1 (! k s ), beliefs, 3
, allocation, (x 3 i ), and portfolios, (z

3 i ) = lim k!1 (z k i ), of Lemma 4, is a C.F.E.
Proof The proof is similar to that of Claim 2. We assume that 1 \ m i=1 P i and let

C 3 := ((! 3 s ); ( i ); [(x 3 i ; z 3
i )]) be dened from Lemma 4, whose notations will be used throughout. Given (i; k) 2 I 2 N, the relations f! k s g s25 1 \ m i=1 P i hold from Claim 2, and imply that P ( k i ) = P i , hence, B i (!

(I) " + u i i (x 3 i ) < u i i (x).
From relation (x; z) 2 B i (! 3 0 ; i ), Lemma 4-(i) and Assumptions A1-A2, the relation:

(II) p 3
0 1(x 0 -e i0 ) 6 D 0 q 3 1z, for some D 2 R ++ , may also be assumed.

From (II), Lemma 4-(i), continuity arguments and the identity of B i (! 3 0 ; i ) and

B i (! k 0 ; k i
) on all second period budget constraints, there exists K 2 N, such that:

(III) (x; z) 2 B i (! k 0 ; i ) = B i (! k 0 ; k i ), for every k > K.
Relations (I)-(III), Lemma 4-(v) and the fact that C k is a C.F.E., yield:

(IV ) u i i (x) < " 2 + u k i (x) 6 " 2 + u k i (x k i ) < " + u i i (x 3 i ) < u i i (x), for k > K big enough.
This contradiction proves that

Appendix: proof of the Lemmas

Lemma 1 There exists N 2 N 3 , such that the following Assertion holds:

(i) 8( o 3 i ) ( o i ), (G N i P ( o 3 i ), 8i 2 I) ) (Q c [V; ( o 3 i )] 6 = ?).
Proof Let the arbitrage-free class structure, [V; ( and sequences, f(G n i )g n2N , be dened as in Section 4. For each (i; n) 2 I2N, we consider the vector space Z n i := fz 2 R J : V (!) 1 z = 0; 8! 2 G n i g and its orthogonal, Z n? i , and, similarly, Z 3 i := fz 2 R J : V (!) 1 z = 0; 8! 2 P i g and Z 3? i . We show, rst, that, for denoted by ! 7 ! x i! . The mapping, x i : ! 2 f0g [ P i 7 ! x i! , dened from Assertions (ii) and (v) and above, is a consumption plan, henceforth referred to as x i 2 X( i );

(vii)

u i i (x i ) = 1 1+#5 lim n!1 u n i (x n i ) 2 R + .
Proof Assertions (i)-(ii) result from Lemma 2-(iv) and compactness arguments. 3

Assertion (iii) For all (i; n) 2 I2N N , we let Z 3 i := fz 2 R J : V (!) 1 z = 0; 8! 2 P i g and recall from the proof of Lemma 1 that Z 3 i = fz 2 R J : V (!) 1 z = 0; 8! 2 G n i g. We show that the portfolio sequence f(z n i ) i2I g is bounded in R Jm . Indeed, let E := max i2I ke i k. The denition of fC n g n2N N yields, from budget constraints and clearance conditions:

(I) [ P m i=1 z n i = 0 and V (! i )1z n i > 0E, 8(i; ! i ) 2 I 2 G n i ],
for every n 2 N N .

Assume, by contradiction, f(z n i )g is unbounded, i.e., there exists an extracted sequence, f(z '(n) i )g, such that n < k(z '(n) i )k 6 n+1, for all n 2 N N . From (I), the portfolios (z 3n i ) := 1 n (z

'(n) i
) meet, for all n 2 N N , the relations 1<k(z 3n i )k61+ 1 n and:

(II) P m i=1 z n i = 0 and V (! i )1z 3n i > 0 E n , 8(i; ! i ) 2 I 2 G n i .

From (II), the density of G 1 i in P i , scalar product continuity and above, the sequence f(z 3n i )g may be assumed to converge, say to (z i ), such that k(z i )k = 1 and:

(III) P m i=1 z i = 0 and V (! i )1z i > 0, 8(i; ! i ) 2 I2P i .

Relations (III) and the fact that [V; ( o i )] is arbitrage-free imply, from Claim 1,

(z i ) 2 5 m i=1 Z 3
i and, from the elimination of redundant assets (see sub-Section 2.4),

(z i ) = 0, which contradicts the fact that k(z i )k = 1. Hence, the sequence f(z n i )g is bounded and may be assumed to converge, say to (z i ) 2 R Jm . Then, the relation P m i=1 z i = 0 results asymptotically from clearance conditions in Lemma 2-(iv).

Assertion (v) Let (i; s) 2 I25 be given. For every (n; ! := (s; p s ); ! 0 ; z) 2 N N 2M s 2M s 2R J , we consider the following (possibly empty) sets:

B i (!; z) := fy 2 R L + : p s 1(y0e is ) 6 V (!)1zg and B 0 i (!; ! 0 ; z) := fy 2 R L + : p s 1(y0e is ) 6 V (! 0 )1zg.

For each n > N , the fact that C n is a (n; )-equilibrium implies, from Lemma 2:

( From Lemma 2-(ii)-(iii)-(iv), the relation (!; x i0 ; z i ) = lim n!1 (=HC n i (!); x n i0 ; z n i ) holds, whereas, from Assumption A2 and ( [START_REF] Debreu | Theory of Value[END_REF], p. 19), the correspondence (x; !; z) 2 R L + 2P i 2R J 7 ! arg max y2Bi(!;z) u i (x; y) is a continuous mapping, since u i and B i are continuous. Hence, passing to the limit into relations (I) yields a continuous mapping, ! 2 P i 7 ! x i! := arg max y2Bi(!;z i ) u i (x i0 ; y), which, from Lemma 3-(v) and above, is embedded into a continuous mapping, x i : ! 2 f0g [ P i 7 ! x i! , i.e., x i 2 X( i ).

3 Assertion (vii) Let i 2 I and x i 2 X( i ) be given, along . Let ' i : (x; !; z) 2 R L + 2P i 2R J 7 ! arg max y2Bi(!;z) u i (x; y) be dened on its domain. By the same

1

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5

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

: 2 2 o,

 22 and we let o := f 2 B : g be their (common) equivalence class. We denote by CB := f o Bg the set of classes, forming a partition, of B, and by P ( o ) the expectation support of any class o 2 CB, namely, the set of anticipations, P ( o ) := P (), which is common to all beliefs

Denition 3

 3 Let a class structure of payo¤s and beliefs, [V; ( o i )], a class of beliefs, o 2 CB, a representative belief, 2 o

  o i )], of payo¤s and beliefs be given. Let 1 be the minimum uncertainty set of the economy E, along Denition 4. Then, the following Assertions hold:

  is arbitrage-free, every representative structure of beliefs, ( i ) 2 5 m i=1 o i , supports a C.F.E.;

3 i ) 2 5 m i=1 o 3 i 3 i

 333 o i ) (and such a renement exists), then, every representative renement, ( , supports a C.F.E., which is price-revealed, i.e., revealed by the equilibrium price itself. 11 Documents de Travail du Centre d'Economie de la Sorbonne -2012.55 Remark 3 Assertion (iv) of Theorem 1 is a direct Corollary of assertion (iii), from replacing the structure [V; (o i )] by the arbitrage-free structure [V; ( o )].

Theorem 2 3 i ) 2 5 m i=1 o 3 i

 233 Let a standard economy, E, and a class structure of payo¤s and beliefs,[V; ( o i )], be given, such that 1 \ m i=1 P ( o i ).Then, there exists a nest symmetric resuch that:(i) every representative renement, ( , supports a C.F.E.

i 3 13

 3 ) holds, for every k 2 K, by construction. Then, from Zorns Lemma, R ( o i) admits a minimal element, that is, the desired nest renement of Theorem 2. Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

  which (typically) is consistent with uncountably many beliefs (those cannot be inferred). If ( o i ), is arbitrage-free, anticipations are kept unchanged, that is, (

3 i ) 2 5 m i=1 o 3 i 3 i ) = P ( o 3 i 3 i ) 2 5 m i=1 o 3 i 3 i

 3333333 cannot be narrowed down from observing prices at t = 0. Indeed, along Theorem 1, all possible equilibrium prices at t = 0 are supported by renements, ( , having the same expectation supports, P ( ), for each i 2 I. So, spot prices at t = 0 are non-informative. This outcome is consistent with agents having no price model.The model leaves room for individual di¤erences, but is also consistent with agents sharing information (along, e.g., Theorem 2). In all cases, agents nal beliefs, say ( 3 i ) 2 SB, belong to same classes of beliefs (( ), but remain private. This privacy, agents xed expectations sets, (P ( o )), and the Theorems Condition restore existence. Indeed, there can be no fall in rank problem a la[START_REF] Hart | On the Optimality of Equilibrium when the Market Structure is Incomplete[END_REF].

3 i

 3 which is xed independently of any (equilibrium) price, p 2 1 P ( o ), to the di¤erence of Hart. 4 The existence proof Throughout, we set as given a standard economy, E, an arbitrage-free class structure, [V; ( o i )], a supporting structure of beliefs, ( i )

s 2 5 ,

 5 an arbitrary spot price, ! N 01 s := (s; p N 01 s ) 2 M s . Then, we dene, by induction on n 2 N N , a sequence of prices, f(! n s )g 2 (5 s25 M s ) N N , which are, for each n 2 N N , the second period equilibrium prices of the economy E n

; 17 Documents

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3 4. 3 Claim 2

 332 An equilibrium of the initial economyWe now prove Assertion (ii) of Theorem 1, via the following Claim. The collection of prices, (!s ) = lim n!1 (! n s ), beliefs, ( i ), allocation, (x i ),and portfolios, (zi ) = lim n!1 (z n i ), ofLemma 3, is a C.F.E. of the economy E. Proof Let C := ((! s ); ( i ); [(x i ; z i )]) be dened from Claim 2 and use the notations of Lemma 3. From Lemma 3-(ii)-(iii)-(v)-(vi), C meets Conditions (c)-(d) of the above Denition 2 of equilibrium. From the denition, f! s g s25 \ m i=1 P ( i ), so C meets Condition (a) of Denition 2. To prove that C is a C.F.E., it su¢ces to show that C satises the relation [(x i ; z i )] 2 5 m i=1 B i (! 0 ; i ) and Condition (b) of Denition 2.

i 2 B

 2 (and the denition of a belief), from Lemma 3-(i), the relations (I) 0 (II), Assumption A2, and the same continuity arguments as above (and the continuity of the scalar product), that we may also assume there exists D 2 R ++ , such that: 20 Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

C 3 meets

 3 Denition 2-(b), i.e., is a C.F.E. From Claim 2, Remark 3, the arbitrary choice of the arbitrage-free class structure, [V; ( o i )], and representative structure, ( i ) 2 5 m i=1 o i , the proof of Theorem 1 is complete.3

  for n = 0, we select one g 0 (i;s;0) 2P (i;s;0) , for all s 2 S i , and let G 0 : s 2 S i ; k n01 2 K n01 g, we let, every (s; k n ) 2 S i 2 K n , such that P (i;s;kn) 6 = ?, P (i;s;kn) : s 2 S i ; k n 2 K n g, and, by induction, a non-decreasing dense sequence, fG n i g n2N , of subsets of P i , with a good property:

	i := fg 0 (i;s;0) : s 2 S i g;
	F for n 2 N 3 arbitrary, given G n01 i (i;s;kn0) 2 P (i;s;kn0) g n := fg n01 (i;s;kn) 8 > > : i \ P (i;s;kn) = ? be set f ixed in P (i;s;kn) ; if 2 G n01 > > < be equal to g n01 (i;s;kn0) , if there exists (k n01 ; g n01 (i;s;kn0) ) 2 K n01 2G n01 i	\ P (i;s;kn)	2
	This yields a set, G n i := fg n (i;s;kn) 2
	1 n ; :::; k L

n )) 2 S i 2K n .

For each (i; s; n; k n ) 2 I2S i 2N2K n , such that P (i;s;kn) 6 = ?, we select g n (i;s;kn) 2 P (i;s;kn) uniquely, and dene a set,

G n i := fg n (i;s ;k n ) : s 0 2 S i ; k 0 n 2 K n ; P (i;s ;k n ) 6 = ?g,

as follows: 15 Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

F

  Claim 3, below, completes the proof of Theorem 1 via the following Lemma. the related C.F.E., C k , of Claim 2, and we dene the set, B i (!; z

	Lemma 4 For each (i; k) 2 I2N, we let k := 1 2 k , denote simply u k i := u	i	i	k	and
	by C k = ((! k s ); ( k i ); [(x k i ; z k i )])

2, hence, from above, that C is a C.F.E. The sets A, of C.F.E., and 1, of minimum uncertainty, of Dention 4, which contains f! s g s25 , are non-empty, i.e., Theorem 1-(ii) holds. 3 21 Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

  3 0 ; i ) and B i (! k proved if we show that C 3 meets Condition (b) of Denition 2. By contraposition, assume this is not the case, i.e., there exists (i; (x; z); ") 2 I2B i (! 3 0 ; i )2R ++ , such that:

	0 ; k i ) may only di¤er by
	one budget constraint at t = 0. From Lemma 4, C 3 meets Conditions (a)-(c)-(d) of
	Denition 2. Moreover, for every (i; k) 2 I 2 N, the relations p k 0 1(x k i0 -e i0 ) 6 0q k 1z k i hold,
	from Claim 2, and, passing to the limit, yield p 3 0 1(x 3 i0 -e i0 ) 6 0q 3 1z 3 i , which implies,
	from Lemma 4-(iv) and above: (x 3

i ; z 3 i ) 2 B i (!

3 0 ; i ), for each i 2 I. Thus, Claim 3 will 22 Documents de Travail du Centre d'Economie de la Sorbonne -2012.55 be

  As a standard application of Berges Theorem (see, e.g.,[START_REF] Debreu | Theory of Value[END_REF], p. 19), the corre-spondence (x; !; ! 0 ; z) 2 R L + 2M s 2M s 2R J 7 ! arg max y2B i (!;! ;z) u i (x; y),which is actually a mapping (from Assumption A2 ), is continuous at (x holds from Lemma 2-(i)-(ii)-(iii). Hence, the relations (I) pass to that limit and yield: Let i 2 I be given. For every (!; n) 2 P i 2N N , the fact that C n is a

	I)	(! n01 s	; ! n s ) 2 M 2 s and x n is 2 arg max y2B	i (! n s ;! n0 s	;z n i ) u i (x n i0 ; y).
		fx i!

i0 ; ! s ; ! s ; z i ), since u i and B 0 i are continuous. Moreover, the relation (x i0 ; x is ; ! s

; z i ) = lim n!1 (x n i0 ; x n is ; ! n s ; z n i ) s g := fx is g = arg max y2Bi(! s ;z i ) u i (x i0 ; y). 3 Assertion (vi)

(n; )-equilibrium and Assumption A2 yield:

(I) fx n i=HC n i (!) g = arg max y2Bi(=HC n i (!);z n i ) u i (x n i0 ; y).

Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

Documents de Travail du Centre d'Economie de la Sorbonne -2012.55

= f! s

[ s25 M s : 9((!

s ); ( 3 i ); [(x 3 i ; z 3 i )]) 2 A; ! s = ! 3 s 2 M s g.

each i 2 I, there exists N i 2 N, such that Z n i = Z 3 i , for every n > N i . Indeed, since fG n i g n2N is non-decreasing, fZ n i g n2N is non increasing in R J , hence, stationary, that is, there exists N i 2 N, such that Z n i = Z Ni i , for every n > N i . From the denition,

i is dense in P i , we easily show, by contraposition, that

holds for every (i; ! i ) 2 I 2 P Nn i , with one strict inequality. The sequence, f(z n i )g n>N o , may be assumed to converge in a compact set, say to (z 3 i ) 2 Z. From the continuity of the scalar product and the fact that, for each i 2 I, lim n!1 G n i = [ n2N G n i is dense in P i , the above relations on f(z n i )g n>N o , imply, in the limit, that V (! i ) 1 z 3 i > 0 holds, for every (i; ! i ) 2 I 2 P i , with one strict inequality, since (z 3 i ) 2 Z. This contradicts the fact that [V; o i ] is arbitrage-free. This contradiction proves Lemma 1.

3 Lemma 2 The generic (n; )-economy admits an equilibrium, namely a collection of prices, ! n 0 := (p n 0 ; q n ) 2 M 0 , at t = 0, and ! n s := (s; p n s ) 2 M s , in each state s 2 5, and strategies, (

, dened for each i 2 I, such that:

Moreover, the equilibrium prices and allocations satisfy the following Assertions:

, where e := max (s;l)2S 2L P m i=1 e l is ;

(v) 9" 2]0; 1] :

Proof Let n 2 N N be given. From Lemma 1 and the fact that 5 is a set of common states for all agents, the structure [V n ; (S n i )] is arbitrage-free, along [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF], on a purely nancial market. Moreover, the (n; )-economy is, formally, one of the type presented in [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF] and, from above, admits an equilibrium along Denition 3 and Theorem 1 of [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF] and its proof, more precisely (up to a slight change in notations), it admits a collection of prices, ! n Lemma 3 For the above sequence, fC n g, of equilibria, it may be assumed to exist:

Moreover, we dene, for each i 2 I, the following sets and mappings:

i;s;kn) , which hold (for every ! 2 P i ) for some (s; k n ) 2 S i 2K n ; from Assertion (i) and Lemma 2-(v), the belief,

where E s is (for each s 2 5) the Diracs measure of ! s ; the support of i 2 B, denoted by

Then, the following Assertions hold, for each i 2 I: are continuous mappings and, moreover, the relations

Then, the uniform continuity of u i and U i on compact sets, and Lemma 3-(ii)-(iii)-(iv) yield:

Moreover, with C := (1+#5), we recall the following denitions, for every n 2 N N :

Then, Lemma 4-(vii) results immediately from relations (I)-(II)-(III) above.

3

Proof of Lemma 4 It is similar to that of Lemma 3, hence, left to the reader. 3