Lionel De
email: lionel.de.boisde¤re@wanadoo.fr

Equilibrium in incomplete markets with differential information: a basic model of generic existence

Keywords: sequential equilibrium, temporary equilibrium, perfect foresight, existence, rational expectations, …nancial markets, asymmetric information, arbitrage

The paper demonstrates the generic existence of general equilibria in incomplete …nancial markets with asymmetric information. The economy has two periods and an ex ante uncertainty over the state of nature to be revealed at the second period.

Securities pay o¤ in cash or commodities at the second period, conditionally on the state of nature to be revealed. They permit transfers across periods and states, which are typically insu¢ cient to span all state contingent claims to value, whatever the spot price to prevail. Under the standard smooth preference and perfect foresight assumptions, the paper shows that equilibria exist, except for a closed set of measure zero of securities and endowments. This theorem generalizes Du¢ e-Shafer's (1985) to arbitrary …nancial and information structures. The equilibrium prices are consistent with any collection of state prices and norm values on spot markets. This re…nement permits to extend to asymmetric information Cass'(1984) theorem that any collection of state prices supports an equilibrium on purely …nancial markets.

Introduction

This paper demonstrates the generic existence of equilibrium in incomplete …nancial markets with di¤erential information, in the context of a two-period pure exchange economy where uncertainty prevails at the …rst period over the state of nature to be revealed at the second period. Asymmetric information is represented by private …nite subsets of states of nature, which each agent is correctly informed to contain the true state of the second period. The scope of this speci…cation is discussed in Section 3, jointly with the information that …nancial markets may reveal.

Consumers exchange consumption goods on spot markets, and, unrestrictively, assets of any kind on typically incomplete …nancial markets. They are endowed with a bundle of goods in every state, with ordered smooth preferences over consumptions and a perfect foresight of future prices, along [START_REF] Radner | Existence of equilibrium plans, prices and price expectations in a sequence of markets[END_REF]. The paper's generic existence result generalizes a classical theorem of symmetric information with real assets due to Du¢ e-Shafer (1985). The current proof builds on a …xed arbitrary set of state prices. This device permits to extend to the current model other results of symmetric information, such as Cass ' (1984), stating that any collection of state prices supports an equilibrium on purely …nancial markets.

When assets pay o¤ in goods, equilibrium needs not exist, as shown by [START_REF] Hart | On the optimality of equilibrium when the market structure is incomplete[END_REF] in the symmetric information case. His example is based on the collapse of the span of assets'payo¤s, that occurs exceptionnaly at clearing prices. Du¢ e-Shafer (1985) shows that equilibrium with real assets exists, except for a closed set of measure zero of economies, parametrized by assets'payo¤s and agents'endowments.

The current model extends Du¢ e-Shafer's (1985) in three ways. First, it allows for asymmetric information amongst consumers. Second, its …nancial structure may cover any mix of nominal and real assets. Third, but not least, it normalizes to arbitrary values equilibrium prices on every spot market. Normalization to relevant values serves to study subsequently the existence of sequential equilibria when agents loose the perfect foresight of future prices. With no price map, agents typically face an endogenous uncertainty a la [START_REF] Kurz | On rational belief equilibria[END_REF]. To be self-ful…lling, their anticipations need therefore focus on sets of relevant values, as argued in Section 7.

The current paper drops [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF] rational expectations assumption. It prefers a learning process, presented in Section 3, where agents may infer information from markets with no price model. It is a step towards also replacing [START_REF] Radner | Existence of equilibrium plans, prices and price expectations in a sequence of markets[END_REF] perfect foresight assumption by a milder condition on anticipations, which remains consistent with the de…nition of sequential equilibrium (see Section 7).

The current proof uses standard di¤erential topology arguments, introduced by [START_REF] Debreu | Economies with a …nite set of equilibria[END_REF][START_REF] Debreu | Smooth preferences[END_REF] for the study of general equilibrium. It de…nes an auxiliary concept of "pseudo-equilibrium" with asymmetric information, shows its full existence from modulo 2 degree theory and derives the generic existence of equilibrium with asymmetric information from Sard's theorem and Grassmannians'properties.

The paper is organized as follows: Section 2 presents the model and the concepts of equilibrium and pseudo-equilibrium. Section 3 describes the information that markets reveal. Section 4 presents Grassmannians and their main properties. Section 5 derives from the latter properties the full existence of pseudo-equilibria. Section 6 proves the existence theorems. Section 7 concludes. An Appendix proves Lemmas. mix of both. They de…ne a …nancial structure, or (S L 0 ) J matrix, denoted by V . Thus, the expected payo¤s of a generic asset, j 2 J, in a state, s 2 S, are a bundle, v j (s) := (v l j (s)) 2 R L 0 , of the quantities, v 0 j (s), of cash, and v l j (s), of each good l 2 L, which one unit of the asset, j, promises to deliver if state s prevails.

The …nancial structure, V , identi…es (with same notation) to a map, V : S ! R J , relating every forecast of a state and spot price, ! := (s; p := (p l )) 2 S , to the row of all assets' payo¤s in cash, V (!) := (v 0 j (s) + P l2L p l v l j (s)) j2J 2 R J , delivered if both state s and price p obtain. Thus, when the asset price is q 2 R J , agents may buy or sell portfolios of assets, z = (z j ) 2 R J , for q z units of account at t = 0, against the promised delivery of a ‡ow, V (!) z, of conditional payo¤s across forecasts, ! 2 S .

Remark 1

The incompleteness of …nancial markets is a standard assumption.

It states that no agent can insure her risks on markets completely. That is, asset payo¤s (in realizable states) cannot span R S , which is written: #J < #S. The above condition is milder and allows markets to be complete (when #J = #S) to informed agents only. Asset payo¤s show that potentially complete markets to the informed, are incomplete to the uninformed: payo¤s can never span R Si if i 2 fj 2 I : S j 6 = Sg.

Consumers'behaviour and the concept of equilibrium

Each agent, i 2 I, receives an endowment, e i := (e is ), promising the commodity bundles, e i0 2 R L ++ at t = 0, and e is 2 R L ++ , in each state, s 2 S i , if this state prevails.

Her consumption set is

X i := R L S 0 i ++ .
Given the prices, p := (p s ) 2 P , for goods, and q 2 R J , for assets, the endowment, e i 2 R L S 0 i ++ , and the payo¤ matrix, V 2 R (S L 0 ) J , that she faces at the …rst period when she elects her strategy, her budget set is: 2 B i (p; q; e i ; V ) := f (x; z) 2 X i R J : p 0 (x 0 e i0 ) 6 q z and p s (x s e is ) 6 V (s; p s ) z; 8s 2 S; and p i s (x s e is ) 6 V (s; p i s ) z; 8s 2 S i nS g.

The budget set consists of consumption and investment plans, which are feasible at the agent's information and anticipations. Preferences are assumed to be ordered, represented by a utility function, u i : X i ! R, for each i 2 I. Given endowments and payo¤s, (e := (e i ); V ) 2 ( i2I X i ) R (S L 0 ) J , the above economy is denoted by

E (e;V ) = f(I; S; L; J); V; (S i ); (p i ); (e i ); (u i )g.
Its equilibrium concept is de…ned as follows:

De…nition 1 Given endowments, e := (e i ) 2 i2I X i , and a payo¤ matrix, V 2 R (S L 0 ) J , a collection of prices, (p; q) 2 P R J , and strategies, (x i ; z i ) 2 B i (p; q; e i ; V ), de…ned for each i 2 I, is an equilibrium of the economy, E (e;V ) , if the following Conditions hold:

(a) 8i 2 I; x i 2 arg max u i (x), for (x; z) 2 B i (p; q; e i ; V ); (b) P i2I (x is e is ) = 0, 8s 2 S 0 ; (c) P i2I z i = 0.
The economy is called standard if it meets the following classical Assumptions:

A1 8i 2 I; u i is C 1 ; A2 (Inada Conditions) 8(i; s; l; x := (x l s )) 2 I S 0 i L X i , @u i (x)=@x l s 2 R ++ , lim x l s !0 @u i (x)=@x l s = 1
(where x l s ! 0 stands for "x l s tends to zero while other components of x are …xed"), lim x l s !1 @u i (x)=@x l s = 0 (in which x l s ! 1 stands for "x l s tends to in…nity at other components of x …xed"),

lim x l s !0 u s i (x) = 0; A3 (di¤erentiably strictly convex preferences) 8i 2 I, 8x 2 X i , h T D 2 u i (x) h < 0, for every h 2 R L S 0 i nf0g, such that ru i (x) h = 0;
A4 at least one agent is fully informed and let i = 1 be such that S 1 = S.

Assumption A1 guarantees the smoothness of preferences and subsequent maps.

The above conditions permit to demonstrate the generic existence of equilibrium.

Before proceeding, we need introduce the model's notations. S is identi…ed to #S, to # (for S 0 ), L to #L, J to #J, whenever needed;

v := JS(L + 1) = dim R (S L 0 ) J ; v := (S J)J = dim R (S J) J ; e := P i2I LS 0 i ; l := (S + 1)(L 1) = dim P .
Second, the following notations are used for vectors, vector spaces and matrices:

for every agent, i 2 I, consumption, x := (x s ) 2 X i := R L S 0 i ++ , and subset

S 0 i , the R L extraction of x is denoted by x( ), that is, x( ) := (x s ) s2 2 R L ++ ;
for all i 2 I, p := (p s ) 2 P and x := (x s ) 2 X i , the vector p i x 2 R Si has for components the scalar products p s x s , for each s 2 S, and p i s x s , for all (s; p i s ) 2 i .

R

, for every …nite sets and , stands for the set of real matrices;

< V >, V (!) and V ( ") denote, for all …nite sets, , and " , element, ! 2 , and matrix, V 2 R , respectively, the span of the matrix'columns, in R , the matrix'! row, in R , and the extracted " submatrix of V ;

V p 2 R S J , for every V 2 R (S L 0 ) J and every p := (p s ) 2 P , denotes the matrix de…ned by V p (s) := V (s; p s ), for each s 2 S, and V p (!) := V (!), for each ! 2 ;

G := f V 2 R S J : rank V = #J g and G := f< V > : V 2 Gg
G := f V 2 G : rank V (S) = #J g and G := f< V > : V 2 Gg; G( ) is the R truncation of G 2 G
, and G ( ) := fG( ) : G 2 G g, for all S.

The concept of pseudo-equilibrium

The concept of pseudo-equilibrium is introduced to circumvent the fall in rank problem a la [START_REF] Hart | On the optimality of equilibrium when the market structure is incomplete[END_REF], which may occur and prevent equilibrium to exist when real assets are traded. In the de…nition, …nancial transfers belong to a …xed Jdimensional vector space, which includes the span of asset payo¤s valued at market prices. Hereafter, the pseudo-equilibrium is de…ned with reference to a given vector of state prices, 2 R S ++ . This will permit to extend Cass'(1984) result to the model.

The two concepts of equilibrium and pseudo-equilibrium coincide when assets are nominal or numeraire and it follows that equilibrium always exists with such assets (see Lemma 1 and Theorem 2, below). An asset, j 2 J, is said to be nominal if it pays in cash only, that is, v l j (s) = 0, for every pair, (s; l) 2 S L. It is numeraire if it pays in a bundle of goods, a 2 R L + nf0g, that is, v j (s) 2 a R, for every s 2 S.

In the general case, however, the concept of pseudo-equilibrium di¤ers and has little economic signi…cance. It is only used to prove the generic existence of equilibrium. Indeed, pseudo-equilibria exist in standard conditions (see Theorem 1, below), and generically coincide with equilibria (Lemma 1, hereafter, and Lemma 3, below).

De…nition 2 Let := ( s ) 2 R S ++ be given. The collection of a scalar, y 2 R ++ , matrix,

V 2 R (S L 0 ) J ,
for x 2 f x := (x s ) 2 X i : p 0 (x 0 e i0 )+ P s2S s p s (x s e is ) = 0 and p i (x e i ) 2 G(S i ) g; (c) < V p > G; (d) 8s 2 S 0 ; P i2I (x is e is ) = 0;
(e) p 0 e 10 + P s2S s p s e 1s = y.

Given (e := (e i );

V ) 2 R e ++ R (S L 0 ) J , we say that (y; p; G) 2 R ++ P G is a pseudo- -equilibrium, if there exists x 2 i2I X i , such that (x; y; p; G; e; V ) is a pseudo-- equilibrium along Conditions (a)
to (e), above. We let E be the pseudo--equilibrium manifold, or the set of collections, (y; p; G; e; V ) 2

R ++ P G R e ++ R (S L 0 ) J , such that (y; p; G) is a pseudo--equilibrium, given (e; V ) 2 R e ++ R (S L 0 ) J .
We de…ne the projection map,

: (y; p; G; e; V ) 2 E 7 ! (e; V ) 2 R e ++ R (S L 0 ) J .
We end with typical conditions, which turn pseudo-equilibria into equilibria:

Lemma 1 Given := ( s ) 2 R S ++ , let (x; y; p; G; e; V ) be a pseudo--equilibrium of a standard economy, E (e;V ) , and let q := P s2S s V p (s) be given. There exists portfolios, z 2 R J I , such that (p; q; x; z) de…nes an equilibrium of the economy E (e;V ) whenever one of the following Conditions holds:

(i) assets are nominal or numeraire;

(ii) rankV p (S) = J, i.e., G 2 G.
Proof See the Appendix.

Before studying Grassmanians, we recall what information markets may reveal.

The information which …nancial markets reveal

The paper's learning behaviour, presented in sub-Section 3.2, di¤ers from Radner's (1979) at the rational expectations equilibrium (REE), presented …rst.

Agents'inferences from observing prices in the REE model

Radner summarizes his 1979 paper on rational expectations equilibria as follows:

"When traders come to a market with di¤erent information about the items to be traded, the resulting market prices may reveal to some traders information originally available only to others. The possibility for such inferences rests upon traders having "models" or "expectations" of how equilibrium prices are related to initial information. This relationship is endogenous, which motivates the term "rational expectations equilibrium." This paper shows that, in a particular model of asset trading, if the number of alternative states of initial information is …nite then, generically, REE exist that reveal to all traders all their initial information." Radner establishes the generic existence of a fully revealing REE as follows: he shows that equilibria with symmetric pooled information exist and generically have distinct prices in each state of pooled information. In this approach, each term of the summary matters.

First, Radner's (1979) existence and inference results hold for "a particular model of asset trading". The author points out (p. 677): "Whether or not rational expectations equilibria exist generically in some fairly general model, is an open question". [START_REF] Jordan | Example of Nonexistence of Rational Expectations Equilibrium[END_REF] show that the REE model results are not general. Radner's (1979) outcomes build on a speci…c model with separable expected utility functions and no spot market of various goods, hence, no real asset markets. Moreover, the "number of alternative states of initial information" needs to be …nite, Radner adds, since prices are typically non-revealing if signal sets are "too large".

Second, reaching equilibrium requires that all traders had models "or "expectations" of how equilibrium prices are related to initial information" and used them to infer correct information. This assumption is seen as unrealistic (see Section 7). The study of information that markets reveal with no price model is therefore needed.

Agents'inferences from trade opportunities with no price model

The information that …nancial markets may reveal to agents, when they have no price model, is presented hereafter in a simpli…ed setting, where assets are nominal and the state space is …nite. These simpli…cations are not restrictive. The inference principle and properties, presented below, extend to arbitrary …nancial structures and in…nitely many states of nature, events or forecasts (see De Boisde¤re, 2016).

In the simplest two period model that we consider, any collection of information signals may be represented by a structure, (S i ), consisting of subsets of a …nite state space, S, along Section 2. Indeed, any information reduces tomorrow's set of possibilities, which may be represented by a state space. Given a payo¤ matrix, V , the pair [V; (S i )] is called the payo¤ and information structure, or structure. Assume, …rst, that an equilibrium exists when agents are endowed with the structure

[V; (S i )].
Then, the asset price reveals no information. Agents are price-takers, unaware of a relation between prices and signals and witness no speci…c volatility on …nancial markets in the absence of arbitrage. Along De Boisde¤re (2007), the existence of equilibrium is characterized by the following condition of absence of future arbitrage opportunity of the structure [V; (S i )], henceforth referred to as the AFAO Condition: 4 Grassmannians with asymmetric information Claim 1 Let G be a sub-vector space of R S . The following Assertions hold:

@(z i ) 2 R J I : P i2I z i = 0 and V (S i )z i > 0,
(i) (G 2 G ) , (9W 2 Z : G = f z 2 R S : W z = 0 g); (ii) (G 2 G ) , (9W 2 Z : G = f z 2 R S : W z = 0 g); (iii) G = f < W > : W 2 Z g.
Proof Claim 1 is immediate from the de…nitions. The proof is left to the reader.

The manifold, Z , is obviously open in the sense that the relation W 2 Z implies that W 0 2 R (S J) S belongs to Z in a neighbourhood of W for the Euclidean distance on R (S J) S . To be precise, we say that U R (S J) S is open in R (S J) S if it satis…es the following condition: 8W 2 U; 9" > 0; fW 0 2 R (S J) S : kW 0 W k < "g U . This de…nes a topology, , on R (S J) S , for which Z is closed, hence, compact.

Two elements, W and W 0 , of Z are said to be equivalent, which we denote by

W W 0 , if f z 2 R S : W z = 0 g = f z 2 R S : W 0 z = 0 g. From Claim 1, the Grassmannian,
G , identi…es to the set of equivalence classes, Z = , of Z . This permits to de…ne a topology, , on G Z = . We say that U is open in G for the topology , if

p 1 (U )
is open in Z for the topology, , where p : Z ! Z = is the identi…cation map.

Claim 2

The Grassmannian, G , is a compact set for the above topology, .

Proof Let fG k g k2N be a sequence of elements of G . From Claim 1, there exists a sequence, fW k g k2N , of elements of Z, such that G k = f z 2 R S : W k z = 0 g, for every k 2 N. Since Z is compact for the topology , the sequence fW k g k2N may be assu- W . Hence, there exists N 2 N, such that W k 2 p 1 (U ), and G k 2 U , for every k > N .

Thus, the sequence fG k g k2N converges to G 2 G , which proves that G is compact.

The other main properties of the Grassmannian

Let be the set of permutations between the elements s 2 S. For every 2 , we let P 2 R S S be the corresponding permutation matrix. The elements of S are ranked, so that the …rst or "upper" elements (for a matrix) are those of and the lower of S. For every V 2 R S J , P :V 2 R S J is obtained by permuting the matrix' rows along . From the de…nition of G, for every V 2 G, there exists 2 , which needs not be unique, such that the last J rows of P :V are linearly independent.

Thus, for each 2 , we let:

G := fV 2 R S J : P :V = 0 B B @ W V 1 C C A 2 R S J , with W 2 R (S J) J and rank V = Jg; G := f < V > : V 2 G g.
For each 2 , the generic vector space G 2 G , admits, from above, a unique matrix representation of the form P 1 :

0 B B @ (G) I 1 C C
A , where (G) 2 R (S J) J takes arbitrary values when G varies, and we let:

[ I j (G) ]
be the (S J) S matrix, whose …rst (S J) columns are those of the identiy matrix, I 2 R (S J) (S J) , followed by the columns of (G) 2 R (S J) J ; (v) G is a manifold without boundary;

K : P G R (S L 0 ) J ! R (S J) J
(vi) the map (p; V ) 2 P R (S L 0 ) J ! K (p; G; V ) 2 R (S J) J is C 1 ;

(vii) the sets Im K , G , G and G are manifolds of dimension v := (S J):J; the derivative

D V K (p; G; V ) has full rank, v .
Proof Throughout, 2 , is given and we typically assume, w.l.o.g., that = Id.

Assertion (i) results from the de…nitions.

Assertion (ii) Let G 2 G Id be given. The relation [ I j Id (G) ] 0 B B @ Id (G) I 1 C C A = 0 holds
from the de…nition. Let z 2 G be given. From above, there exists z 0 2 R J , such that

z = 0 B B @ Id (G) I 1 C C A z 0 . Hence, the relation [ I j Id (G) ] z = [ I j Id (G) ] 0 B B @ Id (G) I 1 C C A z 0 = 0
holds and the relation G fz 2 R S : [ I j Id (G) ] z = 0g follows.

Conversely, let z :=

0 B B @ z 1 z 2 1 C C A 2 R S be such that [ I j Id (G) ] z = 0, where z 1 2 R S J
and z 2 2 R J . From the above de…nitions, the relation [ I j Id (G) ] z = 0 is written:

z 1 = Id (G) z 2 , that is, z = 0 B B @ Id (G) I 1 C C A z 2 2 < 0 B B @ Id (G) I 1 C C A > = G.
Assertion (iii) From above, the map Id is one-to-one and onto. We show that

Id is bicontinuous, that is, Id : G 2 G Id 7 ! [ I j Id (G) ] 2 Z is bicontinuous. Let G 2 G Id , " 2 ]0; 1[, W := Id (G) and W n := fW 0 2 Z : kW 0 W k < "=ng be
given, for every n 2 N. From the de…nitions, the relation p(W 1 ) G Id holds and the reader will easily check that Id (p(W N )) W 1 , for N 2 N large enough (which we set as given). By construction, U := p(W N ), is a neighbourhood of G, such that

Id (U ) := Id (p(W N )) W 1 . Hence, Id is continuous at G 2 G Id and Id is continuous. Conversely, let W 2 Id (G Id ), G := 1 Id (W ) 2 G
Id and a neighbourhood, U , of G, be given. From the de…nition of , there exists " > 0, such that W := fW 0 2 Z :

kW 0 W k < "g p 1 (U ). Then, W := W \ Id (G Id ) is a neighbourhood of W in Id (G Id ), such that 1 Id (W) p(W ) U . Thus, 1
Id is continous at W , hence, continuous.

Assertions (iv) and (v) result from Assertions (i)-(iii) and the de…nition of G .

Assertion (vi) results from the de…nition of K .

Assertion (vii) Let 2 be given. From Assertion (iii) and above, G is homeomorphic to fP 1 :

0 B B @ W I 1 C C A , W 2 R (S J) J g, whose dimension is v := (S J):J.
Hence, from the de…nitions, Assertions (i) (iv) and above, G , G , G and Im K are all manifolds of dimension v . Let J S be the set of last J states and notice:

K Id (p; G; V ) := [ I j Id (G)]:V p = V p (SnJ ) + Id (G):V p (J ), for (p; G; V ) 2P G Id R (S L 0 ) J .
The derivatives of K Id (p; G; V ) with respect to payo¤s, for s 2 SnJ , are of the form of a (S J) (S J) block diagonal matrix, P , of diagonal elements:

the J J(L+1) matrices P (!) = 0 B B B B B B B B B B @
(1; p iT s ) 0 0 :: 0 0 (1; p iT s ) 0 :: 0 : : :

0 0 0 :: (1; p iT s ) 1 C C C C C C C C C C A
, for every

(i; ! := (s; p i s )) 2 I i ,
and J J(L+1) matrices

P (s) = 0 B B B B B B B B B B @
(1; p T s ) 0 0 :: 0 0 (1; p T s ) 0 :: 0 : : :

0 0 0 :: (1; p T s ) 1 C C C C C C C C C C A
, for every s 2 SnJ .

The matrix P , therefore, has rank (S J):J. It follows from above that the derivative D V K Id (p; G; V ) has maximal rank, v = (S J):J.

The pseudo-equilibrium manifold and existence theorem

We now de…ne agents' demands, characterize the pseudo-equilibrium manifold and prove the full existence of the pseudo-equilibrium. Throughout, := ( s ) 2 R S ++ and 0 := 1 are set as given and we let e x := (e x s ) := ( s :x s ), for every x := (x s ) 2 R L S 0 ++ .

The demand and excess demand correspondences

The …rst agent's (i = 1) demand is de…ned as: D 1 : (y; p) 2 R ++ P 7 ! D 1 (y; p) := arg max u 1 (x), for x 2 f x 2 X 1 : e p x = y g, where y > 0 is given. As classical results, in a standard economy, D 1 is a C 1 map, such that lim p!p kD 1 (y; p)k = +1, if p 2 @P nf0g.

Similarly, for each i 2 Inf1g, the agent's demand correspondence, D i : (p; G; e i ) 2

P G X i 7 ! arg max u i (x) for x 2 f x 2 X i : e p (x e i )(S 0 ) = 0 and p i (x e i ) 2 G(S i ) g, is a
continuous map in a standard economy, which is C 1 with respect to (p; e i ) 2 P X i .

Using Walras' law, we select one good, say l = 1. We recall that dim P = l := (S + 1)(L 1). For every i 2 I, and every consumption x i 2 X i , we denote by From above, Z is a same C 1 map, with partial derivative: D e 1 Z(y; p; W; (e i )) = I.

x i := (x is ) 2 R l ++ ,

The pseudo-equilibrium manifold' s characterization and properties

For each 2 , we now consider the maps: To show that 0 is regular, consider the derivative of H with respect to y, e 1 and V :

K : (p; G; V ) 2 P G R (S L 0 ) J 7 ! [ I j (G) ]:P :V p 2 R v , along
Claim 5 E is a submanifold of R l +1 ++ G R e ++ R v without boundary of dimen- sion e + v .
Hence, is a map between manifolds of the same dimension.

Proof From Claims 3 and 4 and the pre-image theorem, the pseudo--equilibrium

set, E = [ 2 H (0) 1 , is a boundaryless submanifold of R l +1 ++ G R e ++ R (S L 0 ) J of dimension (l + 1 + v + e + v ) (1 + l + v ) = e + v .
Claim 6 The map

: E ! R e ++ R (S L 0 )
J is smooth and proper, that is, the inverse image by of a compact set is compact.

Proof The map is smooth from the de…nitions. To show that it is proper, let

Y R e ++ R (S L 0 ) J be a compact set and f C k := (y k ; p k ; G k ; (e k i ); V k ) g k2N be a given sequence of elements of 1 (Y ). Since Y is compact, the sequence f(e k i ); V k
)g k2N may be assumed to converge, say to ((e i ); V ) 2 Y . From the above limit relation on the demand map D 1 and relation E = [ 2 H (0) 1 , the price sequence has a positive lower bound. Hence, f (y k ; p k ) g k2N is assumed to converge, say to (y; p) 2 R ++ P .

From Claim 2, the sequence f G k g k2N may be assumed to converge, say to G 2 G .

Let C := (y; p; G; (e i ); V ) := lim k!1 C k be given from above. From Claims 3-4 and the relation E = [ 2 H (0) 1 , there exists 2 , such that the relations H (C k ) = 0 hold, for k 2 N big enough. From Claim 4, the latter relations pass to the limit and yield: H (C) = 0 and, therefore,

lim k!1 C k = C 2 E . Thus, 1 (Y ) is compact.
Lemma 2 There is a regular value, ((e i ); V ), of , such that # 1 ((e i ); V ) = 1.

Proof See the Appendix.

The full existence of pseudo--equilibra follows from the above properties:

Theorem 1 For every 2 R S
++ and every collection of endowments and payo¤s, (e; V ) 2 R e ++ R (S L 0 ) J , a standard economy, E (e;V ) , admits a pseudo--equilibrium. Moreover, the set of regular values, R , of , is open and of null complement.

Proof As standard from modulo 2 degree theory, if f : X ! Y is a smooth proper map between two boundaryless manifolds of same dimension, Y being connected, the number, #f 1 (y), of elements x 2 X, such that y = f (x), is the same, mod. 2, for every regular value y 2 Y . In particular, if one regular value, y, of f , satis…es #f 1 (y) = 1, then, f 1 (y) is non-empty for every y 2 Y . Indeed, y 2 Y is regular by de…nition if f 1 (y) = ?. From Claims 5-6 and Lemma 2, the map, , meets all the desired condititions above, for X := E and Y := R e ++ R v . It follows that, for every pair (e; V ) 2 R e ++ R v , a standard economy, E (e;V ) , admits a pseudo--equilibrium. Since is proper, its sets of values, R c , is closed (i.e., R is open). From Sard's Theorem (see [START_REF] Milnor | Topology from the Di¤erentiable Viewpoint[END_REF], p. 10), R c is of zero measure. [START_REF] De Boisde¤re | Learning from arbitrage[END_REF] The existence of equilibrium 6.1 The full existence of equilibrium with nominal or numeraire assets

Nominal and numeraire assets are de…ned in Section 2. The de…nition of a numeraire di¤ers from Geanakoplos-Polemarchakis' (1986), where it is one good. From Lemma 1, pseudo-equilibria and equilibria coincide with such assets, which yields:

Theorem 2 Let := ( s ) 2 R S
++ and e := (e i ) 2 R e ++ be given, and let V 2 R (S L 0 ) J be the payo¤ matrix of either nominal, or numeraire, assets. In a standard economy, E (e;V ) , there exist prices, (p; q) 2 P R J , consumptions, x := (x i ) 2 R e ++ , and portfolios, z := (z i ) 2 R J I , such that q := P s2S s V p (s) and (p; q; x; z) de…nes an equilibrium.

Proof Theorem 2 is a direct consequence of Theorem 1 and Lemma 1, above.

The existence of equilibrium on arbitrage-free purely …nancial markets was proved in De Boisde¤re (2007) in a broader setting, where no agent needed to have ordered preferences or full information. When the economy is standard along Section 2, Theorem 2 shows the additional result that equilibria can be supported by any state prices, 2 R S ++ . With symmetric information, this result is known as Cass'(1984).

To extend Cass'theorem, De…nition 2 introduced so-called "pseudo--equilibria".

With nominal assets and symmetric information, it follows from Assumption A2, Lemma 1 and the no-arbitrage condition, that the budget constraints in [START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF] and in De…nition 2 are equivalent. Moreover, one agent (i = 1) has a single budget constraint, a la Debreu, while the other agents are constrained a la Radner, i.e., in every state of nature. This so-called "Cass trick" is the device, which both this paper and Cass'(1984) use to prove that any collection of state prices supports equilibria on purely …nancial markets. But the two papers'techniques of proof di¤er.

With standard assumptions and a slightly di¤erent …nancial structure, Geanakoplos-Polemarchakis (1986) also demonstrates the full existence of …nancial equilibria with numeraire assets and symmetric information, and shows their generic local uniqueness. The issue of supporting equilibrium state prices is not considered. Thus, Theorem 2 extends Cass'to a larger set of assets and information signals.

With reference to the REE model, Theorem 2 provides a stronger existence result.

As recalled in Section 3, the [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF] model has no spot markets, for goods, nor real asset markets. Whereas generic existence holds in the latter model, full existence holds from Theorem 2 in a broader setting, which includes spot markets.

The generic existence of equilibrium with arbitrary assets

The proof proceeds as follows: given := ( s ) 2 R S ++ , a pseudo--equilibrium of a standard economy, E (e;V ) , exists from Theorem 1 and coincides with equilibrium for an open set of null complement of payo¤s and endowments. The latter is supported 20 Documents de travail du Centre d'Economie de la Sorbonne -2021.08 by the state prices, , i.e., equilibrium prices, (p; q) 2 P R J , satisfy q = P s2S s V p (s).

Lemma 3 Given 2 R S ++ , there exists an open set, , of null complement in the set of regular values, R , of , such that, for every (e; V ) 2 , a standard economy, E (e;V ) , admits a pseudo--equilibrium, (x; y; p; G; e; V ), such that rankV p (S) = J.

Proof See the Appendix. Proof Theorem 3 results immediately from Theorem 1 and Lemmas 1 and 3.

Moreover, we notice from the proof of Lemmas 2 and 3 that equilibria in 1 (e; V ) are in odd number, and are continuous functions of (e; V ), for every (e; V ) 2 .

Concluding remarks

The current paper drops [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF] inferences, in an attempt to meet the main criticisms faced by REE models. This led existence results to be stronger and to apply to a broader setting with spot markets. Under close assumptions, the paper's results also extend classical theorems of symmetric information, such as Cass '(1984), or Du¢ e-Shafer's (1985). Yet, the paper's sequential equilibrium concept retains the perfect foresigth hypothesis. This restriction needs be explained.

The paper's additional existence results, in relation to the arbitrary choice of state prices, are presented in sub-Section 7.1. Sub-Sections 7.2 and 7.3 are devoted to a discussion of the perfect foresight restriction.

The existence of equilibrium for arbitrary state prices

On a purely …nancial market with symmetric information, [START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF] shows that any collection of state prices is consistent with the existence of a …nancial equilibrium, whether markets be complete or not. The current paper extends this result to other …nancial and information structures than Cass' (see Section 6).

It also proves the generic existence of equilibrium for arbitrary assets, signals and state prices. Beyond Cass'paper, the fact that the existence of equilibria, or pseudoequilibria, is consistent with arbitrary state prices and assets is little emphasized in the literature, as shown by the example of Geanakoplos-Polemarchakis' (1986) paper. The fact that state prices may be set as given is yet easily understood from the de…nition of equilibrium. In De…nition 1 above, exogenously given state prices, := ( s ) 2 R S ++ (using the model's notations) do not a¤ect any separate budget constraint of the second period. Moreover, the equilibrium asset price, when it exists, is q := P s2S s V p (s), where p := (p s ) s2S is the collection of spot prices. The variables, which adapt, are agents'consumptions and portfolios and the spot prices.

There is no need to let vary to reach equilibrium. It may be …xed.

In complete markets, the argument of this property follows immediately from the standard equivalence between the Arrow-Debreu and Radner equilibria. Indeed, let a consumption price, p = (p s ) 2 R S 0 L + , and an allocation, (x i ) 2 R S 0 LI + , de…ne an equilibrium of some Arrow-Debreu economy, which exists in standard conditions. 3 Let E be the equivalent Radner economy, with #S random states and Arrow-securities, whose payo¤s de…ne the S S identity matrix. Let state prices, := ( s ) 2 R S ++ , and 0 := 1, be given. From the equivalence between the Arrow-Debreu and Radner equilibria, the asset price, would require from traders a reckonning skill, which is seen as unrealistic.

On same grounds, the perfect foresight hypothesis, faces similar criticisms as [START_REF] Kurz | Endogenous uncertainty in a general equilibrium model with price contingent contracts[END_REF]: "agents need to know the maps from states at future dates sequential equilibrium, anticipations on realizable spot markets are endogenous and perfect. This setting is a step towards dropping both sides of rational expectations.

7.3 From perfect to correct foresight: revisiting sequential equilibrium

The paths of the classical models of temporary and sequential equilibrium diverge, as recalled by [START_REF] Grandmont | Temporary general equilibrium theory[END_REF]. The two concepts might yet be reconciled.

To see this, an equilibrium could be de…ned as "sequential" whenever agents make self-ful…lling -if not perfect -anticipations, at other conditions of equilibrium unchanged. That is, agents would anticipate sets of admissible prices, containing (but not necessarily limited to) future equilibrium prices, on each spot market.

Call the latter sequential equilibrium a "correct foresight" equilibrium (CFE). With symmetric information, classical sequential equilibria are CFE, and the two concepts coincide when agents anticipate one price on each spot market with certainty.

If markets clear and agents keep their plans unchanged, face no bankruptcy tomorrow and, yet, have no price model to refer to, their anticipations should typically be uncountable. Ex ante, a continuum of anticipations would not be ruled out completely by cautious rational agents. Call "forecast" a joint expectation, (s; p) 2 S , of a state of nature, s, and a spot price, p, in state s. With no price model, the generic i th agent's anticipation set, say i , consists in her plausible forecasts, given her idiosyncratic information and beliefs. Forecasts are exogenous and typically differ across agents. Thus, the CFE shares a characteristic of temporary equilibria:

anticipations are exogenously given and re ‡ect beliefs. Contrarily to the standard sequential equilibrium, they result from no calculation and require no price map.

So the argument that the CFE may reconcile sequential and temporary equilibria.

At a CFE, the collection of agents'anticipation sets, ( i ), replaces the previous conditions (ru i (e i ) = p i , for each i 2 I) and budget constraints. By construction, the allocation, (e i ), is Pareto-optimal in the following sense: "there is no other attainable allocation, which preserves agents'wealth in unrealizable states and strictly increases the utility of one agent without decreasing the utility of another agent". This follows from the optimality of e i , for every i 2 I, in the set B i (p ) := fx 2 X i : p i (x e i ) 6 0g.

Let C := (y; p := (p s ); G; (e i ); V ) 2 1 ((e i ); V ) be a pseudo--equilibrium, and

x := (x i ) be a related allocation. From Lemma 1, C de…nes an equilibrium for the allocation x and price q = P s2S s V p (s), and G = < V p > = G holds, by construction.

From De…nition 2 and the identity G = G , the relations u i (x i ) > u i (e i ) and p i s x is = p i s e is hold for every i 2 I and every s 2 S i nS. It follows from the Pareto-optimality of (e i ) that x = (e i ) and, hence, that # 1 ((e i ); V ) = 1.

The last part is to show that ((e i ); V ) is regular, relying on Lemmata 1:

Lemmata 1 For every i 2 Inf1g and every p 2 P , let D i (p) 2 R L S 0 ++ be the vector of the i th agent's demands in realizable states, that is, D i (p) = D i (p; G ; e i )(S 0 ). Let D 1 (p) = D 1 (y ; p) and Z(p) := P i2I D i (p). For all (i; p) 2 Inf1g P , the following holds:

4. 1 A

 1 characterization of the set G Introduced in sub-Section 2.3, the set of J-dimensional subspaces of R S , denoted by G , is called the Grassmannian. Claim 1 characterizes G by the following vector spaces, and leads to de…ne a topology, which makes that set compact. First, we let: 11 Documents de travail du Centre d'Economie de la Sorbonne -2021.08 Z := f W 2 R (S J) S : the rows of W are linearly independent g; Z := f W 2 R (S J) S : the rows of W are orthonormal g; Z := f W 2 R S J : the columns of W are orthonormal g.

  med to converge, say to W 2 Z. From Claim 1, G := f z 2 R S : W z = 0 g 2 G . Let U be an open set in G containing G. From the de…nition, p 1 (U ) is open and contains 12 Documents de travail du Centre d'Economie de la Sorbonne -2021.08

Claim 3

 3 be the map de…ned by K (p; G; V ) := [ I j (G) ]:P :V p . Let 2 and G 2 G be given. The following Assertions hold:(i) fG g 2 is an open cover of G; (ii) G = fz 2 R S : [ I j (G) ]:P z = 0g; (iii)is a homeomorphism; 13 Documents de travail du Centre d'Economie de la Sorbonne -2021.08 (iv) fG g 2 is an open cover of G ;

  the extracted vector of x i , which drops all consumptions in states s = 2 S 0 and in good l = 1. We denote similarly (with stars) the extracted demands in R l and de…ne the excess demand correspondence, Z : R l +1 ++ G R e ++ ! R l , namely: (y; p; G; e:=(e i )) 7 ! Z(y; p; G; e) := D 1 (y; p) + P i2Inf1g D i (p; G; e i ) P i2I e i .

DD:A

  (y;e 1 ;V ) H (y; p; G; e; V ) : = y h(y; p) = 1 D y D 1 (y; p) 0 D e 1 h(y; p) D e 1 Z(y; p; G; e) We need only show the above matrix has full rank, 1+l +v (for all (y; p; G; e; V )) or, from Claim 3, that the matrix A := 0 B B @ D y h(y; p) D y D 1 (y; p) D e 1 h(y; p) D e 1 Z(y; p; G; e) has rank l +1. The relations D e 1 h(y; p) = e p = D e 1 Z(y; p; G; e)e p and D y D 1 (y; p) >> 0 hold from the de…nitions and Assumption A2, whereas the relation D y D 1 (y; p) e p = 1 follows from di¤erentiating the condition, D 1 (y; p) e p = y, on demand. Hence, the matrices A and 0 B B @ D y D 1 (y; p) e p D y D 1 (y; p) e p D y D 1 (y; p) which D y D 1 (y; p) e p D y D 1 (y; p) e p < 0, have the same rank. The relation rankA = l + 1 follows.

  (i) e p T D p (D 1 (p)) = p T D p ( f D 1 (p)) = f D 1 (p) T ; (ii) h T D p ( f D 1 (p)) h < 0, 8h 2 R L S 0 nf0g, such that e h D 1 (p) = h f D 1 (p) = 0; (iii) e p T D p (D i (p )) = p T D p ( f D i (p ) = 0; (iv) D p ( f D i (p )), is negative semi-de…nite.Proof of Lemmata 1 Assertion(i) results from di¤erentiating the budget constraint, e p D 1 (p) = p f D 1 (p) = y , which holds from the de…nition of demands. Assertion (ii) For every p 2 P , let D(e p) := D 1 (p). From Assumptions A2-A3, it is standard that the map e p 7 ! D(e p) meets the relation, e h T D e p (D(e p)) e h < 0, for every h 2 R L S 0 nf0g, such that e h D(e p) = 0 (see, e.g., Du…e-Shafer, 1985, fact 4, p. 292).

  vector space, G 2 G , prices, p := (p s ) 2 P , endowments, e i := (e is ) 2 X i , and consumptions, x i := (x is ) 2 X i , de…ned for each i 2 I, is a pseudo--equilibrium of the economy, E ((e i );V ) , if the following Conditions hold:

(a) x 1 2 arg max u 1 (x), for x 2 f x := (x s ) 2 X 1 : p 0 (x 0 e 10 ) + P s2S s p s (x s e 1s ) = 0 g; 7 Documents de travail du Centre d'Economie de la Sorbonne -2021.08 (b) for every i 2 Inf1g, x i 2 arg max u i (x),

  At zero or su¢ ciently low prices, traders must infer that any state of their prior information sets, upon which they expected to achieve a "free lunch", is actually unrealizable. That is, agents, observing market, learn from arbitrage. This learning process with no price model (or price) features the trader's actual behaviour on …nancial markets. Call it the "no-arbitrage principle".If the structure, [V; (S i )], fails to be arbitrage-free at the outset, the no-arbitrage principle leads each agent, i 2 I, to infer, in …nitely many steps, a subset, S i nS

8i 2 I, with one strict inequality. Not all structures, [V; (S i )], meet the AFAO Condition at the outset. If not, asset pricing is impossible, since agents cannot agree on any price. Financial markets are prone to volatility. Arbitragists take advantage of it and of di¤erences in agents'assessments of portfolios. As long as the AFAO Condition fails, they may sell pro…tably to various buyers a bundle of clearing-market potfolios. However, as arbitragists compete to attract buyers, the prices at which they may sell such clear-ing portfolios fall to zero. i , of unrealizable states, such that [V; (S i )], is the coarsest arbitrage-free re…nement of [V; (S i )]: see Cornet-De Boisde¤re (2009), for …nite state spaces, and De Boisde¤re (2016), for arbitrary (possibly uncountable) state spaces. Once agents have inferred the re…nement (S i ), from the no-arbitrage principle, they cannot learn more information. Either from trade, since the AFAO Condition holds and no arbitrage-state remains, or from prices, since agents agree on prices and have no price model. Erratic market movements vanish. From De Boisde¤re (2007), an equilibrium would obtain, whose no-arbitrage price, for the re…nement (S i ), reveals no additional information.

Henceforth, agents are assumed to have reached the coarsest arbitrage-free re-…nements of their information signals, if needed, that is, [V; (S i )] is arbitrage-free.

  From Claim 3, the pseudo--equilibrium manifold, E , coincides with the union of inverse images, [ 2 H (0) 1 . Its properties therefore stem from the Claims below. Given 2 , the image 0 is a regular value of the map H , which is continuous, and class C 1 with respect to the (y; p; e; V ) derivatives.Proof Let 2 be given. From Claim 3, the proof that H is continuous and C 1

	Claim 4

Section 4;

H : (y; p; G; e; V ) 2 R ++ P G R e ++ R (S L 0 ) J 7 ! (h(y; p); Z(y; p; G; e); K (p; G; V )).

with to (y; p; e; V ) is standard and akin to Du¢ e-Shafer (1985, pp. 292-293).

  Theorem 3 Given := ( s ) 2 R S ++ , there exists an open set, R , of null complement in R e ++ R v , such that, for every (e; V ) 2 , a standard economy, E (e;V ) , admits an equilibrium supported by the state prices .

  2 R S ++ , and commodity price, p = (p s ) 2 R S 0 L + , where , for every s 2 S 0 , are joint equilibrium prices in the Radner economy, E. Thus, any collection of state prices supports an equilibrium if …nancial markets are complete. The same result holds in incomplete markets, from the Cass trick, if assets are nominal and agents have symmetric information, or from Theorem 2 above, when a broader class of assets and information signals is considered.

	p s s Such characteristics are typically private. If so, the construction of a forecast p s =
	function (relating unobservable signals to observable prices), would normally require
	agents to truthfully disclose their private information, in relation to the observed
	price, at every occurence of the joint signals. This requirement is more demanding
	than sharing information at just one period. Alternatively, if all individual charac-
	teristics were commonly known by every agent, inferring a price function from them

7.2 The two sides of rational expectations

In the literature, the rational expectations'assumption refers either to Radner's (1979) treatment of asymmetric information, or to the Arrow (1953) -

[START_REF] Radner | Existence of equilibrium plans, prices and price expectations in a sequence of markets[END_REF]

perfect foresight hypothesis, which characterizes the classical sequential equilibrium.

As seen in Section 3, in

[START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF]

, equilibrium prices are typically distinct in agents'joint information signals. This outcome holds in a speci…c model and does not prove, by itself, the generic existence result. The latter requires that agents be aware of a map relating signals to equilibrium prices, so as to typically infer joint signals from prices. This map is endogenous. It depends on all agents'characteristics.

Documents de travail du Centre d'Economie de la Sorbonne -2021.08

Documents de travail du Centre d'Economie de la Sorbonne -2021.08

As in Du¢ e-Shafer(1985), our generic existence proof could not avoid the arti…cial interior consumptions at equilibrium. Doing without would require to drop the standard modulo 2 degree argument in the existence proof, which is central.

Documents de travail du Centre d'Economie de la Sorbonne -2021.08

We recall that agents have the same information, S = S, and face no …nancial market, in the Arrow-Debreu economy, but a complete market of contingent goods.

Documents de travail du Centre d'Economie de la Sorbonne -2021.08

to prices in the future and it is entirely unrealistic to assume that agents can …nd out what this sequence of maps is." [START_REF] Radner | Equilibrium under uncertainty[END_REF] himself admits it "seems to require of the traders a capacity for imagination and computation far beyond what is realistic".

Yet, the classical concept of sequential equilibrium relies on perfect foresight.

Probably the …rst, best known and most radical escape to the assumption comes from the temporary equilibrium framework, introduced by J. Hicks and developed by J.-M. Grandmont later. At a temporary equilibrium, agents have exogenous anticipations and their demands clear on all current markets. Agents typically revise their beliefs and their plans ex post, at each period, after the realized state of nature and the actual spot prices (if uncorrectly anticipated) are observed. Moreover, temporary equilibrium allocations need no longer clear tomorrow on spot markets, where consumers may also face bankruptcy, due to mistaken anticipations.

A less radical approach is referred to as bounded rationality. In this line of research, Kurz'(1994) equilibrium allows agents to lack the "structural knowledge" of how equilibrium prices obtain. This unawareness may be due to uncertainty about the beliefs, characteristics and actions of other agents. Agents'behaviour consists in ruling out anticipations, which observation contradicts. It typically leads to an additional uncertainty over future prices, which Kurz calls "endogenous uncertainty", describes as the major cause of economic ‡uctuations, and shows to be consistent with heterogenous beliefs. Bounded rationality may require of agents less awareness than perfect foresight to compute a price map, but requires no less reckonning skill.

The current paper stems from a dissatisfaction with both rationality assumptions, but presents a hybrid model. Borrowing to the temporary equilibrium, anticipations are exogenous in unrealizable states, SnS. Borrowing to the standard Sections'information structure, (S i ). Anticipations play the role of random states. So, dropping perfect foresight makes the study of agents' inferences, and of the existence of equilibrium, more complex. It implies dealing with in…nite dimensional models. On existence issues, conditions need be studied, which make exogenous anticipations self-ful…lling. Keeping perfect foresight, in a …rst step, is easy to explain.

First, perfect foresight permits to consider a …nite economy, where exposition and proofs are simpler. In a …nite setting, for instance, the no-arbitrage principle of Section 3 relies on simple induction arguments [START_REF] Cornet | Elimination of arbitrage states in asymmetric information models[END_REF]. This inference behaviour leads to the same results when agents' anticipation sets are in…nite. But the proof relies on in…nite dimensional topolgy (De Boisde¤re, 2016).

Second, it is standard to prove a property in a …nite economy, …rst, in order to extend it to the in…nite setting. The typical method applies an asymptotic argument to a sequence of …nite economies with the desired property, which approximate the in…nite economy more and more closely. This method can be used to study correct foresight equilibra, building on existence results of …nite economies with perfect foresight (see De [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF][START_REF] De Boisde¤re | Price revelation and existence of …nancial equilibrium with incomplete markets and private beliefs[END_REF]. So the usefulness of the above theorems.

Finally, assuming perfect foresight circumvents a di¢ culty in …nding alternative conditions, which make anticipations self-ful…lling ex post. To our best knowledge, no such alternative exists in the literature. Yet, CFE prices, which depend on agents' private, possibly changing, beliefs, describe a "minimum uncertainty set". Without structural knowledge, agents cannot infer which particular price of that limited set will prevail tomorrow (any of which can prevail). Yet, agents'anticipations, if they embed the minimum uncertainty set, are always self-ful…lling, along De Boisde¤re (2015). The issue is a topic of research by itself and goes beyond the paper's scope.

So the focus on …nite anticipation sets, which led to assume perfect foresight.

Appendix

Lemma 1 Given := ( s ) 2 R S ++ , let (x; y; p; G; e; V ) be a pseudo--equilibrium of a standard economy, E (e;V ) , and let q := P s2S s V p (s) be given. There exists portfolios, z 2 R J I , such that (p; q; x; z) de…nes an equilibrium of the economy E (e;V ) whenever one of the following Conditions holds:

(i) assets are nominal or numeraire;

be given. Let (x := (x i ); y; p; G; e; V ) be a pseudo--equilibrium of a standard economy, E (e;V ) , and denote q :=

Assume that V is the payo¤ matrix of either nominal or numeraire assets, or that the condition rankV p (S) = J holds. If assets are numeraire, there exists a vector, a 2 R L + nf0g, and matrix, W 2 R S J , such that V (s; p) = p a W (s), for every forecast (s; p) 2 S (hence, p a > 0). If assets are nominal, there exists a …xed matrix, W 2 R S J , such that V p 0 = W , for every p 0 2 P . In both cases, the matrix W has full rank (redundant assets are eliminated).

From the de…nitions and above, the relation

or (ii) of Lemma 1, and implies that

From above, the collection, C := (p; q; x; z), is such that (x i ; z i ) 2 B i (p; q; e i ; V ), for every i 2 I, and it meets Conditions (b)-(c) of De…nition 1 of equilibrium. Let i 2 Inf1g

be given. From Assumption A2, the budget set B i (p; q; e i ; V ) may be replaced by

i (p; q; e i ; V ) := f (x; z) 2 X i R J : p 0 (x 0 e i0 ) = q z and p i (x e i ) = V p (S i )z g in De…nition 1 at no cost. From the de…nition of q and above, the i th agent's pseudo-equilibrium budget set coincides with

Since x i is optimal in B i , the strategy (x i ; z i ) is optimal in B i (p; q; e i ; V ) from above.

By similar arguments, we show that (x 1 ; z 1 ) is optimal in B 1 (p; q; e 1 ; V ). Thus, C also meets Condition (a) of De…nition 1 and is an equilibrium.

Lemma 2 There is a regular value, ((e i ); V ), of , such that # 1 ((e i ); V ) = 1.

Proof We order the set S := [ S, so that its "last" J elements be in S. We denote by s j , for j 2 J, these last states, and by J := fs j g j2J S their set. We set as given 0 := 1 and := ( s ) 2 R S ++ and let e x := (e x s ) := ( s x s ) s2S 0 , for every x := (x s ) 2 R S 0 .

We let a := (0; 1; 0; :::; 0) 2 R L 0 be the payo¤ bundle made of one unit of the …rst good and V 2 R (S L 0 ) J be the numeraire asset matrix de…ned (for each j 2 J) by the payo¤s v j (s) = a, if s = s j , and v j (s) = 0 otherwise (N.B. choosing real assets is not restrictive, since a could be replaced by a 0 := (1; 0; :::; 0) 2 R L 0 costlessly). We let p := (p l s ) 2 P , such that p 1 s = 1 2 for all s 2 J , G := < V p > and q := P s2S s V p (s) be given. Then, V p (J ) = 1 2 I is half the identity matrix and V p (SnJ ) = 0.

From Assumption A2, we may choose agents' endowments, (e i ) 2 i2I X i , such that, for each i 2 I, the relation

holds, where p i s = s p s , for every s 2 S 0 , and p i (S i nS) = p i (if S i 6 = S). A standard economy, E ((e i );V ) , obtains.

The relations e 1 2 D 1 (y ; p ) and e i 2 D i (p ; G ; e i ) hold, for each i 2 Inf1g, and C := (y := p e 1 ; p ; G ; (e i ); V ) is a pseudo--equilibrium and de…nes a tradeless equilibrium, from Lemma 1. Indeed, the allocation (e i ) is interior, meets gradients' Then, Assertion (ii) follows from the relation h T D p ( f D 1 (p)) h = e h T D e p (D(e p)) e h, which holds from the de…nitions, for every h 2 R L S 0 and every p 2 P . Assertion (iii) Let i 2 Inf1g be given. For every p 2 P , the …rst budget constraint in the i th agent's demand, D i (p; G ; e i ), is written:

for every p 2 P . Assertion (iv) follows as a standard corollary.

To complete the proof of Lemma 2, following Du¢ e-Shafer (1985, pp. 296-297),

we let E := Id (G ) = 0 and E := Id (G), for every G 2 G Id , and de…ne the maps:

Showing that C is a regular point of , is equivalent to showing that the derivative, D H (y ; p ; E ), has full rank, 1 + l + v . The relations K (p; E ) = 0, for every p 2 P , hence, D p K (p ; E ) = 0 hold from the de…nitions. Moreover, rank D E K (p ; E ) = v := (S J)J. Indeed, V p (J ) := 1 2 I and E is a (S J) J matrix. From the de…nitions, the relation K (p ; E) = 1 2 E holds for every E 2 Id (G Id ) and the derivative, D E K (p ; E ), has rank (S J)J = v , no matter the value of E . Therefore, it su¢ ces to show that D p Z (p ; E ), has full rank l . We may write D p Z (p ; E ) as a LS 0 (L 1)S 0 real matrix. If it fails to have full rank, there exists h 2 R (L 1) S 0 nf0g, such that D p Z (p ; E ) h = 0. Let h 2 R L S 0 be the vector whose R (L 1) S 0 extraction is h and whose components in good l = 1 are all zeros. Let

and, hence:

From Lemmata 1, the above relation, f D 1 (p ) h = 0, implies the following ones:

This contradiction proves that D p Z (p ; E ) and, from above, D H (y ; p ; E ) have full rank, i.e., ((e i ); V ) is a regular value of . The proof of Lemma 2 is complete.

Lemma 3 Given 2 R S ++ , there exists an open set, , of null complement in the set of regular values, R , of , such that, for every (e; V ) 2 , a standard economy, E (e;V ) , admits a pseudo--equilibrium, (x; y; p; G; e; V ), such that rankV p (S) = J.

Proof Given 2 R S ++ , let (e 0 ; V 0 ) 2 R be a regular value of , which exists from Lemma 2. From Claims 3 and 4, Theorem 1, the implicit function theorem and the de…nition of a regular value, there exists a pseudo-equilibrium, C 0 := (y 0 ; p 0 ; G 0 ; e 0 ; V 0 ) 2

is di¤erentiable and zero-valued, from the de…nition and Claims 3 and 4 (we can identify 1

Id (E) to E for the derivations). From above, D H = 0 holds, which implies:

The latter equation states that the rows of D (e 1 ;V ) H(C ) are linear combinations of those of D (e 1 ;V ) (f 1 ; f 2 )(e; V ). From the de…nitions and Claim 4, D (e 1 ;V ) H(C ) has full rank, with l + (S J):J independent rows, and so does D (f 1 ; f 2 )(e; V ). Thus, rank Df 1 (e; V ) = l holds for every (e; V ) 2 U . Consider now maps and sets as follows:

: (e; V ) 2 U 7 ! (f 1 (e; V ); V ) 2 P R (S L 0 ) J ;

: (p; V ) 2 P R (S L 0 ) J 7 ! V p 2 R S J ;

Q := ++ R v , such that, for every (e; V ) 2 , a standard economy, E (e;V ) , has a pseudo--equilibrium, (x; y; p; G; e; V ), such that rankV p (S) = J.


Sheng Yuan

Sébastien Linas

Catherine Journet

Philippe Steyer

Vincent Garnier

Guillaume Bonnefont

Arnaud Brioude

Bérangère Toury
email: toury@univ-lyon1.fr

Pure & crystallized 2D Boron Nitride sheets synthesized via a novel process coupling both PDCs and SPS methods

Within the context of emergent researches linked to graphene, it is well known that h-BN nanosheets (BNNSs), also referred as 2D BN, are considered as the best candidate for replacing SiO 2 as dielectric support or capping layers for graphene. As a consequence, the development of a novel alternative source for highly crystallized h-BN crystals, suitable for a further exfoliation, is a prime scientific issue. This paper proposes a promising approach to synthesize pure and well-crystallized h-BN flakes, which can be easily exfoliated into BNNSs. This new accessible production process represents a relevant alternative source of supply in response to the increasing need of high quality BNNSs. The synthesis strategy to prepare pure h-BN is based on a unique combination of the Polymer Derived Ceramics (PDCs) route with the Spark Plasma Sintering (SPS) process. Through a multi-scale chemical and structural investigation, it is clearly shown that obtained flakes are large (up to 30 μm), defect-free and well crystallized, which are key-characteristics for a subsequent exfoliation into relevant BNNSs.

Thanks to their remarkable potential for electronic applications, 2D-nanomaterials related researches are currently booming. Especially, hexagonal boron nitride (h-BN) is a key material in particular within the context of emergent researches linked to graphene. Indeed, it is well known that BNNSs have been considered as one of the best candidates for replacing SiO 2 as dielectric support or capping layers for graphene [START_REF] Lee | Electron tunneling through atomically flat and ultrathin hexagonal boron nitride[END_REF] . The most important reason is linked to SiO 2 surface imperfections (roughness, charged traps …), which can significantly limit the carriers' mobility within graphene's atomic sheets, then affecting their electronic transport properties. Since lattice parameters of BNNSs perfectly match the ones of graphene, this drawback should be overcome [START_REF] Golberg | Boron Nitride Nanotubes and Nanosheets[END_REF] . Recent works [START_REF] Britnell | Field-Effect Tunneling Transistor Based on Vertical Graphene Heterostructures[END_REF][START_REF] Mehr | Vertical Graphene Base Transistor[END_REF] even present the possibility of building multi-layered hetero-structures in which h-BN and exfoliated graphene flakes are alternatively stacked to form vertically-oriented graphene-based transistors [START_REF] Liu | In-plane heterostructures of graphene and hexagonal boron nitride with controlled domain sizes[END_REF][START_REF] Song | Large scale growth and characterization of atomic hexagonal boron nitride layers[END_REF] .

To succeed in these domains of applications, the development of an accessible resource of pure and highly crystallized h-BN crystals remains challenging. Today, only two main sources of hexagonal boron nitride (h-BN) crystals are commonly available for the production of BNNSs after a mechanical or chemical cleavage. Firstly, commercial sources [START_REF] Clere | Method for making high thermal diffusivity boron nitride powders[END_REF][START_REF] Lin | Composition comprising exfoliated boron nitride and method for forming such compositions[END_REF] , generally obtained from an oxygen-containing boron compound reacting with a nitrogen-containing source, which are most often characterized by a high level of defects with relatively small crystalline areas (from less than 2 μ m up to 10 μ m). Secondly, h-BN-crystals can also be found using the High Pressure High Temperature (HPHT) method adopted at the National Institute for Materials Science (NIMS Japan) [START_REF] Watanabe | Direct-bandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal[END_REF][START_REF] Kubota | Deep Ultraviolet Light-Emitting Hexagonal Boron Nitride Synthesized at Atmospheric Pressure[END_REF] . This process allows the cleavage of relatively large (~100 μ m) and thin (several nm) BN samples with an atomically flat surface and low defects density. However, widespread of this latter resource is hindered by its severe production conditions, e.g. high temperature (up to 2100 °C), high pressure (5.5 GPa) and treatments time (80 h). Those specificities do not allow, up to now (to our knowledge), the reproducibility of that route by other group.

In order to reach large and highly crystallized h-BN sheets, two strategies can be considered. The first one consists in a direct deposition of the required BNNSs on a substrate, while the second one is based on the generation of BNNSs by exfoliation of large h-BN single crystals.

Considering the first approach, literature reports CVD and CVD-derivatives [START_REF] Guo | Fabrication of large area hexagonal boron nitride thin films for bendable capacitors[END_REF][START_REF] Kim | Exploring carrier transport phenomena in a CVD-assembled graphene FET on hexagonal boron nitride[END_REF][START_REF] Liu | Direct Growth of Graphene/Hexagonal Boron Nitride Stacked Layers[END_REF][START_REF] Gibb | Atomic Resolution Imaging of Grain Boundary Defects in Monolayer Chemical Vapor Deposition-Grown Hexagonal Boron Nitride[END_REF][START_REF] Ismach | Toward the Controlled Synthesis of Hexagonal Boron Nitride Films[END_REF] syntheses of BN nanolayers deposited on different substrates using boron and nitrogen based precursors [START_REF] Corso | Boron Nitride Nanomesh[END_REF][START_REF] Simonson | A vibrational study of borazine adsorbed on Pt(111) and Au(111) surfaces[END_REF][START_REF] Preobrajenski | Monolayer h-BN on lattice-mismatched metal surfaces: On the formation of the nanomesh[END_REF][START_REF] Sutter | Chemical Vapor Deposition and Etching of High-Quality Monolayer Hexagonal Boron Nitride Films[END_REF] . However, adopting such gaseous-based methods, it is difficult to control the defects formation and to tune the number of layers. The second approach takes advantage of the weak h-BN inter-planar force to get, by exfoliation, a two-dimensional few layers graphene-like structure (often referred as BN nanosheets). Exfoliation is well-documented in the literature and can be performed by mechanical [START_REF] Corso | Boron Nitride Nanomesh[END_REF][START_REF] Novoselov | Two-dimensional atomic crystals[END_REF][START_REF] Li | Large-scale mechanical peeling of boron nitride nanosheets by low-energy ball milling[END_REF][START_REF] Pacilé | The two-dimensional phase of boron nitride: Few-atomic-layer sheets and suspended membranes[END_REF][START_REF] Meyer | Selective Sputtering and Atomic Resolution Imaging of Atomically Thin Boron Nitride Membranes[END_REF][START_REF] Gorbachev | Hunting for Monolayer Boron Nitride: Optical and Raman Signatures[END_REF] or chemical [START_REF] Han | Structure of chemically derived mono-and few-atomic-layer boron nitride sheets[END_REF][START_REF] Warner | Atomic Resolution Imaging and Topography of Boron Nitride Sheets Produced by Chemical Exfoliation[END_REF][START_REF] Zhi | Large-Scale Fabrication of Boron Nitride Nanosheets and Their Utilization in Polymeric Composites with Improved Thermal and Mechanical Properties[END_REF][START_REF] Lin | Exfoliated Hexagonal Boron Nitride Nanosheets[END_REF][START_REF] Yurdakul | Nanoscopic characterization of two-dimensional (2D) boron nitride nanosheets (BNNSs) produced by microfluidization[END_REF][START_REF] Li | Exfoliation of Hexagonal Boron Nitride by Molten Hydroxides[END_REF][START_REF] Chen | Vortex fluidic exfoliation of graphite and boron nitride[END_REF][START_REF] Yi | Hydrodynamics-assisted scalable production of boron nitride nanosheets and their application in improving oxygenatom erosion resistance of polymeric composites[END_REF][START_REF] Jan | Enhancing the mechanical properties of BN nanosheet-polymer composites by uniaxial drawing[END_REF][START_REF] Yi | Size-selected boron nitride nanosheets as oxygen-atom corrosion resistant fillers[END_REF][START_REF] Liu | Boron nitride nanosheets with controlled size and thickness for enhancing mechanical properties and atomic oxygen erosion resistance[END_REF] methods. As a consequence, this indirect way for generating BNNSs by exfoliation from high quality h-BN crystals seems more relevant, but actually suffers from a lack of pure h-BN sources of supply, as discussed above.

Keeping this issue in mind, we propose an innovative alternative experimental procedure, which can be widely applied to produce important amount of h-BN flakes with large-scale (> 10 μ m) and high purity based on an original combination of two advanced techniques: the Polymer Derived Ceramics (PDCs) route, and the Spark Plasma Sintering (SPS) process.

In a first step, the PDCs route is involved in the preparation of very pure preceramic polymer from a molecular resource. Among all available precursors for producing h-BN from the PDCs route, recent works have shown that polyborazylene (PBN) was the best candidate [START_REF] Kusari | Formation of nanostructured, nanocrystalline boron nitride microparticles with diatom-derived 3-D shapes[END_REF][START_REF] Zhong | Design of carbon fiber reinforced boron nitride matrix composites by vacuum-assisted polyborazylene transfer molding and pyrolysis[END_REF][START_REF] Termoss | Preparation of boron nitride-based coatings on metallic substrates via infrared irradiation of dip-coated polyborazylene[END_REF][START_REF] Li | Preparation of Polyborazylene-Derived Bulk Boron Nitride with Tunable Properties by Warm-Pressing and Pressureless Pyrolysis[END_REF][START_REF] Yuan | New potential for preparation of performing h-BN coatings via polymer pyrolysis in RTA furnace[END_REF] , due to its easy and reproducible synthesis, combined with its relatively high ceramic yield and high purity. Moreover, the B/N ratio within the polymer is ideal to get further stoichiometric h-BN and there is no contaminant except hydrogen atoms that are easily removed during the ceramization step. In other words, the hexagonal arrangement of h-BN is already present within the molecular precursor's structure in the borazinic ring. Further developments have shown that an incorporation of lithium nitride as crystallization promoter into the polymeric precursor noticeably improves, from outstandingly low temperatures (1000 °C), both the crystallization rate and the crystallinity level [START_REF] Yuan | Low-Temperature Synthesis of Highly Crystallized Hexagonal Boron Nitride Sheets with Li3N as Additive Agent[END_REF] . Recently, we have also demonstrated that, without any sintering step, a simple and classical resistive thermal treatment (<1200 °C) of this additivated preceramic polymer led to a highly crystallized h-BN, prone to be easily exfoliated to h-BN graphene like few layers [START_REF] Yuan | Synthesis of hexagonal boron nitride graphene-like few layers[END_REF] . In this previous study, the obtained BNNSs displayed size of couple of micrometers, which is not enough to envisage the above-mentioned applications.

The preceramisation process gives rise to an amorphous BN powder. With the purpose to favour crystallisation and increase crystals size, powder is then sintered by SPS [START_REF] Bernard | Ordered mesoporous polymer-derived ceramics and their processing into hierarchically porous boron nitride and silicoboron carbonitride monoliths[END_REF] . Traditionally, SPS is used to get very dense material. Compared to conventional sintering techniques, near-theoretical density is indeed achieved at lower sintering temperature and for shorter treatment durations [START_REF] Munir | Electric Current Activation of Sintering: A Review of the Pulsed Electric Current Sintering Process[END_REF] . Even if the SPS process is most often applied for a densification purpose, it is known that its parameters [START_REF] Aman | A Screening Design Approach for the Understanding of Spark Plasma Sintering Parameters: A Case of Translucent Polycrystalline Undoped Alumina[END_REF] (current pulse, pressure, heating rate) may also influence nucleation-growth mechanisms [START_REF] Omori | Sintering, consolidation, reaction and crystal growth by the spark plasma system (SPS)[END_REF] . It is reasonable to assume that the intense pulse current lines generated during the SPS process may modify the crystallization step promoting the crystals growth. In that direction, the synergy between PDCs and SPS is a promising and accessible strategy to get pure, large h-BN flakes. The targeted BNNSs could then result in a subsequent exfoliation step.

Results and Discussion

Bulk sample. Liquid PBN is first mixed with Li 3 N micro powder (5 wt. %) and preceramized at 650 °C prior to be sintered in the SPS apparatus. After 1 h under optimized experimental conditions (90 MPa pressure, 1800 °C, constant current of 770 A, 10 ms pulses ON and 5 ms OFF, vacuum in the SPS chamber of 0.15 mbar) a white pellet of 1 cm in diameter is obtained (Fig. 1a inset). A preliminary characterization of the chemical bonding within the specimen is investigated by X-ray photoelectron spectroscopy (XPS). Two analyses are performed on the same sample before and after an Ar + sputtering of the surface, leading to 1 μ m depth abrasion. Both general surveys appear on Fig. 2. Photoelectron peaks from B1s, N1s, O1s and C1s are clearly recognized in the spectrum recorded on the raw sample, giving an elemental chemical composition of 42.5; 43.6; 2.4 and 11.5%at., respectively. The binding energies of the B1s and N1s, at 190.7 eV and 398.6 eV, respectively, are consistent with reported XPS data for BN [START_REF] Moulder | Handbook of X-ray Photoelectron Spectroscopy[END_REF] . Both peaks are perfectly symmetric, confirming that B and N atoms are exclusively involved in BN bonds [START_REF] Saitoh | Preparation and characterization of boron rich boron nitride microcrystals by hot filament assisted chemical vapor deposition[END_REF][START_REF] Patibandla | Chemical Vapor Deposition of Boron Nitride[END_REF] . Moreover, the carbon peak at 285.0 eV, attributed to a pollution source, is very common with XPS measurements. The asymmetric shape of the C1s peak, typical of carbon pollution, still emphasizes this interpretation [START_REF] Vincent | Microtexture and structure of boron nitride fibres by transmission electron microscopy, X-ray diffraction, photoelectron spectroscopy and Raman scattering[END_REF] . In the same way presence of the slight amount of oxygen (signal at 532.7 eV) would be linked to an oxygen impurity probably resulting from residual water traces or from oxygen adsorbed at the sample surface. However, this first analysis gives an elemental B/N ratio of 0,97, i.e. very close to the one of stoichiometric BN. The second analysis performed after abrasion (4 keV Ar + sputtering), allows confirming that carbon and oxygen resulted from extrinsic pollution since they are almost absent in the sample (< 0.5%at. for O1s and 1.3%at. for C1s). After etching, determination of the B/N ratio is more complicated since a preferential sputtering of nitrogen atoms occurs, leaving behind a boron-rich surface (54.6%at. for B1s vs 44.1%at. for N1s) [START_REF] Guimon | XPS study of BN thin films deposited by CVD on SiC plane substrates[END_REF] .

The as-sintered sample is then characterized adopting a multiscale approach (Scanning Electron Microscopy (SEM), Raman spectroscopy and X-Ray Diffraction). Figure 1a presents a representative SEM image recorded on the raw sample. It clearly shows very large and homogenous BN domains (up to 30 μ m) well staked into differentiated flakes. If the pristine bulk pellet is clearly polycrystalline, we will show that each flake is a single crystal. Furthermore, it can be highlighted that BN flakes have a preferential orientation perpendicularly to the pellet surface e.g. aligned in the compressed direction (Fig. 1b). Figure 1c presents the Raman spectrum recorded on the raw sample. It is well known that Raman peak position and Full Width at Half Maximum (FWHM) are both parameters relative to crystallinity level of the sample. We can observe a well-defined single peak at 1366 ± 1 cm -1 , which is characteristic of the E 2g vibration mode of h-BN crystal [START_REF] Yuan | New potential for preparation of performing h-BN coatings via polymer pyrolysis in RTA furnace[END_REF] . This value is very close to the theoretical one reported for h-BN bulk material at 1367 cm -1 52 . The slight difference with the bulk reference material is due to residual inter and/or intragranular stresses developed into the pellet and may result from the high energetic sintering process. This peak is symmetric with a FWHM of 7.7 cm -1 . To the best of our knowledge, such a value is comparable to the one reported for the best h-BN single crystals obtained by HPHT [START_REF] Watanabe | Direct-bandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal[END_REF][START_REF] Kubota | Deep Ultraviolet Light-Emitting Hexagonal Boron Nitride Synthesized at Atmospheric Pressure[END_REF][START_REF] Clubine | Synthesis and characterization of bulk single crystal hexagonal boron nitride from metal solvents[END_REF][START_REF] Watanabe | Ultraviolet luminescence spectra of boron nitride single crystals grown under high pressure and high temperature[END_REF] . This result is correlated with a very low defect density and a very large crystallite size. Figure 1d presents two XRD patterns performed on the raw sample. The upper spectrum is an X-Ray analysis of the pellet surface, and the other one is an X-ray analysis of the pellet cross-section. Both patterns exhibit thin, intense and well-separated peaks indicating a good structural ordering. All peaks have been assigned to expected crystallographic planes from an h-BN crystal according to the JCPDS file number 01-073-2095 (Table 1). The comparison of both patterns clearly shows a drastic decrease of the intensity of all peaks except the {001} ones pointing out a preferential orientation of the h-BN crystallites along the a axis (characteristic of the covalent B-N bond) parallel to the applied load.

In summary, large micrometers-crystallized domains (up to 30 μ m) grew in the sintering load direction, thus validating our method strategy based on the coupling of PDCs and SPS methods.

Boron Nitride NanoSheets (BNNSs). Mechanical exfoliation. For a finer scale investigation, the h-BN raw pellet was scrubbed and mechanically exfoliated by the tape method, leading to 10 to 200 nm thick pieces with side size up to 30 μ m (Fig. 3b). These BNNSs were deposited on a Si substrate, covered by a 80 nm SiO 2 oxide layer in order to be characterized by Raman spectroscopy. Two spectra were recorded on two different BNNSs of the same sample differentiated by their contrast under optical microscope (Fig. 3a,b). Both spectra, shown in Fig. 3c, display the single peak of h-BN located at 1366 cm -1 with different intensities in relation to different BNNSs thicknesses. By AFM (Fig. 3c, inset), their thicknesses were measured at 13 nm and 114 nm. This result shows that mechanical exfoliation of h-BN leads to micrometric BNNSs of thicknesses down to few nanometers. Furthermore, the high crystalline quality of BNNS flakes is also confirmed by TEM (Fig. 4). Indeed, diffraction pattern associated with 5-30 μ m BNNSs exhibited clearly hexagonal bright dots, which are representative of single crystals.

Chemical exfoliation. Finally, in order to perform a deeper structural investigation, single crystal flakes can be easily isolated into thinner BNNS by an appropriate ultrasonic exfoliation treatment in ethanol. We have studied the crystal and BNNSs size distribution, respectively before and after the chemical exfoliation, from SEM and optical images (see Supplementary Fig. S1 online). It is worth mentioning that such a size could still be increased by optimizing the exfoliation process. Figure 5 presents a TEM image recorded on an isolated BNNS measuring 10 to 15 μ m. After exfoliation, their thickness slows down to several nanometers, which is approaching the targeted structure of graphene-like two-dimensional sheets. Then, the flatness of the BNNSs has been investigated by AFM (see Supplementary Fig. S2 online). In a typical study, the RMS (Root Mean Square) roughness has been determined to 0.7 nm. This value is similar to the one measured for standard silica. Even if we did not observe a lower value as it was shown in the work of Dean et al. [START_REF] Dean | Boron nitride substrates for high-quality graphene electronics[END_REF] we reasonably think that the flatness of our samples would be compatible with graphene deposition as substrate or to produce van der Waals heterostructures. Figure 6a shows homogeneous crystalline areas with Moiré patterns on extended zones confirming the high level of crystallinity. The selective area electron diffraction (SAED) pattern (Fig. 6b, inset), obtained from the high resolution image (Fig. 6b), shows six bright spots hexagonally distributed demonstrating the achievement of h-BN single crystals. This corresponding SAED pattern can be interpreted as the AA' atomic stacking in the c-direction. Atomic layers can also be stacked in a more complex turbostratic structure, as demonstrated by the hexagonal-shaped Moiré patterns obtained on different samples area (Fig. 6c-e). Misorientations between h-BN layers similar to graphene can explain those Moiré patterns, which reveal themselves by multiple spots in corresponding Fast Fourier Transform images (Fig. 6c-e inset) [START_REF] Zan | Advances in Graphene Science[END_REF] . Those images show two sets of hexagonal spots rotated with angles of 12°, 21° and 25° respectively. By modelling these misorientations with two BN monolayers, it is possible to match quite well corresponding FFT images (Fig. 6f-h). This relative rotation could be due to the stain induced by rolled structure, as already imaged for rotational stacking fault in few-layer graphene sheets [START_REF] Warner | Direct Imaging of Rotational Stacking Faults in Few Layer Graphene[END_REF] .

(h,k,l) 2θ (°) d (Å) R. I. * (%) 2θ (°) d (Å) R.I. (%) 2θ (°) d (Å) R.I. (%) (002
In order to get further information about the chemical composition, analysis by electron energy loss spectroscopy (EELS) was directly performed on thin BNNSs (Fig. 7). Only boron and nitrogen K-edges are recorded while no traces of lithium, carbon or oxygen is detected. This supplementary chemical analysis corroborates previous XPS results concluding on the excellent purity of samples: bulk, as well as BNNSs.

To summarize, the novelty of the method comes from the association of two synthesis processes: the PDCs route which is known to lead to high purity ceramic when a pure preceramic polymer is used and the SPS one that allows reaching high crystallinity to the final ceramic. The beneficial effect of SPS is obvious when comparing characteristics of the present BNNSs with the best ones synthesized by a single PDCs route [START_REF] Yuan | Synthesis of hexagonal boron nitride graphene-like few layers[END_REF] . For instance the crystalline quality is first significantly improved as shown by the FWHM Raman peaks measurement that decreased from 40 cm -1 for PDCs BNNSs to 7.7 cm -1 for the PDCs-SPS'BNNSs. Moreover, we demonstrated that applying an external current (by SPS) led to preferentially orientated flakes with much larger domains (up to 30 μ m) than the ones obtained previously (1-2 μ m). On the other hand, even if the NIMS laboratory may bring to the scientific community larger crystals (500 μ m), their experimental procedure is more energy and time consuming. Indeed, it can be underlined that to reach GPa pressure conditions, a specific and dedicated instrument (diamond anvil) is required, which limits the entire size of the sample and is not easy to access. Summarized comparative data are gathered as Supplementary Table S2 online.

Finally, the quality of our samples may open new perspectives for many applications as graphene substrate or to produce van der Waals heterostructures.

Conclusions

To conclude, we have demonstrated that a large amount of self-standing h-BN few-layers can be obtained by exfoliation of highly crystallized h-BN bulk material, the latter being prepared using a relevant dual synthesis method involving the PDCs route and the SPS process. This original method offers a new alternative way to generate high quality h-BN crystals as new source of supply for BBNSs.

Methods

A dispersion of lithium nitride (Li 3 N, Aldrich) micro powders (5 wt%) in liquid polyborazilene (PBN), was prepared following a procedure described elsewhere [START_REF] Yuan | Synthesis of hexagonal boron nitride graphene-like few layers[END_REF] . This dispersion was slowly heated in a Schlenk tube under argon from room temperature to 200 °C and held at 200 °C for 1 h leading to a solid-state polymer avoiding further oligomer evaporation. The obtained white powder (82% mass yield) was then placed into an alumina crucible to be heated under N 2 at 650 °C (1 °C.min -1 ), and kept at this final temperature for 1 h.

In a gloves box, the SPS graphite die covered with a papyex ® was filled with the preceramic powder and then transferred into the SPS chamber (HP D 25, FCT System) in-between the two punches and compressed at room temperature under uniaxial load until 90 MPa. SPS heating (10 ms pulses ON and 5 ms OFF) was then conducted up to 1800 °C keeping the normal load constant. During the 1 h dwell temperature the current was 770A and the vacuum in the SPS chamber was 0.15 mbar. The annealing process was held at this maximum temperature to complete the BN ceramization reaction and the densification process. Removing the protective papyex, a white pellet (300 mg) was finally obtained. SEM observations were carried out at very low accelerating voltage (200 V) using a SUPRA Zeiss microscope. Raman spectra were acquired in a Horiba Jobin-Yvon ARAMIS spectrometer equipped with a 1800 lines/nm grating, using an excitation wavelength of 532 nm and a x50 objective (the main Raman band of the silica peak is located at 520.1 cm -1 ). XPS measurements were done using a PHI Quantera SXM instrument equipped with a 180 hemispherical electron energy analyzer and a monochromatized Al Kα (1486.6 eV) source operated at 15 kV and 4 mA. Analyses were performed on the raw sample and after a 1 μ m abrasion using a 4 keV Ar+ sputtering. XRD patterns were recorded with a Phillips PW 1830/40 diffractometer, with a CuK radiation source. BNNSs are then prepared by chemical (in ethanol, ultrasonication: 1 min at 25 W, Hielscher UP400S and centrifugation) or physical (tape) exfoliation. A droplet of the solution was simply deposited onto a 300 mesh holey carbon copper grid in order to perform TEM observations (JEOL 2100F and 2010F microscopes operating at 200 kV). EELS spectra were recorded using a JEOL JEM-ARM200F Cold FEG, operating at 200 kV.

Figure 1 .

 1 Figure 1. (a) SEM image showing stacked layers with micrometric sizes, inset: 1 cm-diameter pellet; (b) SEM image showing the pellet cross section and BNNSs aligned perpendicularly to the surface pellet; (c) Raman scattering spectrum recorded on the raw sample and (d) XRD patterns obtained with the raw sample (upper) analysis of the pellet surface and (bottom) analysis of the pellet cross-section.

Figure 2 .

 2 Figure 2. (a) XPS general surveys obtained on the raw sample (1) and after 1-μ m abrasion (2); (b) B1s peak and the corresponding deconvolution recorded on the raw sample; (c) N1s peak and the corresponding deconvolution recorded on the raw sample; (d) C1s peak and the corresponding deconvolution recorded on the raw sample and (e) O1s peak and the corresponding deconvolution recorded on the raw sample.

( a )

 a Data from the JCPDS 01-073-2095 (b) Data from XRD pattern recorded (⊥) (c) Data from XRD pattern recorded (∥)

  ) as reported for h-BN in the JCPDS file number 01-073-2095; (b) obtained from the pattern recorded on the raw sample for an X-Ray analysis of the pellet surface and (c) obtained from the pattern recorded on the raw sample for an X-Ray analysis of the pellet cross section. * Relative intensity normalized by the (002) diffraction peak.

Figure 3 .

 3 Figure 3. (a,b) optical images of BNNSs with different thicknesses deposited on Si0 2 surface and (c) Raman spectra recorded on the corresponding BNNSs, inset: AFM image obtained on the thicker sample.

Figure 4 .

 4 Figure 4. TEM BF-image showing a large crystal and its corresponding diffraction pattern (inset).

Figure 5 .

 5 Figure 5. TEM BF-image showing a 10 μm BNNS and its corresponding diffraction pattern (inset).

Figure 6 .

 6 Figure 6. (a) BF-TEM micrograph of large h-BN sheets (10 μ m) with Moiré patterns. Three misorientated single crystals were delimitated with dashed line; (b) typical hexagonal atomic arrangement of h-BN few layers, inset: corresponding SAED pattern; (c-e) particular atomic arrangement exhibiting hexagonal shaped Moiré patterns with angles misorientation of 12°, 21° and 25°, insets: corresponding FFT images; (f-h) modeling of two h-BN layers rotated of corresponding angles.

Figure 7 .

 7 Figure 7. (a) Wide range EELS spectrum obtained on a BNNS, inset: high resolution spectra of B and N K-edge; (b) high resolution spectrum acquired around the K-edge of Li; (c) high resolution spectrum acquired around the K-edge of C and (d) high resolution spectrum acquired around the K-edge of O.

Table 1 . XRD data (a

 1

	)	26.75	3.33	100	26.77	3.33	100	26.56	3.353	100
	(100)	41.61	2.17	14	41.60	2.17	60	41.57	2.17	2.2
	(101)	43.87	2.06	4.5	43.88	2.06	17	43.12	2.1	1
	(102)	50.16	1.82	12	50.15	1.82	23	50.09	1.82	1
	(004)	55.11	1.67	5	55.10	1.67	6	54.64	1.68	2.7
	(103)	59.54	1.55	1	59.52	1.55	2	59.43	1.55	0.1
	(104)	71.35	1.32	2.5	71.32	1.32	3	71.31	1.32	0.1
	(110)	75.94	1.25	4	75.93	1.25	20	75.87	1.25	1
	(112)	82.19	1.17	6	82.15	1.17	16	82.08	1.17	0.1

Scientific RepoRts | 6:20388 | DOI: 10.1038/srep20388 www.nature.com/scientificreports

Scientific RepoRts | 6:20388 | DOI: 10.1038/srep20388

Acknowledgements

Authors gratefully acknowledge the ANR for its financial support through the project 14-CE08-0018 «GoBN», the CTμ (Centre Technologique des Microstructures) of the Université Lyon 1 and CLYM (Centre Lyonnais de Microscopie) for access to TEMs.

Author Contributions

S.Y. and B.T. produced the PBN by PDCs and conducted the measurements by XRD, V.G. and G.B. conducted the SPS process. C.J., A.B. and P.S. performed the SEM and TEM characterization. S.L. characterized the samples by AFM and Raman spectroscopy. All authors contributed to write the paper.

Additional Information

Supplementary information accompanies this paper at http://www.nature.com/srep Competing financial interests: The authors declare no competing financial interests.