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

Sequential equilibrium without rational expectations of prices: a theorem of full existence

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

, among others, based on the rational expectations of prices.

Introduction

When agents'information is incomplete or asymmetric, the issue of how markets may reveal information is essential and, yet, debated. Quoting Ross [START_REF] Starr | The new palgrave …nance: a dictionary of economics[END_REF], "the theory with asymmetric information is not well understood at all. In short, the exact mechanism by which prices incorporate information is still a mystery and an attendant theory of volume is simply missing." A traditional response is given by the REE (rational expectations equilibrium) model by assuming, quoting [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF], that "agents have a 'model' or 'expectations' of how equilibrium prices are determined". Under this assumption, agents know the relationship between private information signals and equilibrium prices, along a so-called "forecast function".

Cornet-De Boisde¤re (2002) suggests an alternative approach, where agents' asymmetric information is represented by private information signals, which correctly inform each agent that tomorrow's state of nature will be in a subset of the state space. The latter paper extends the classical de…nitions of equilibrium, prices and no-arbitrage condition to asymmetric information. Generalizing [START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF], De [START_REF] De Boisde¤re | No-arbitrage equilibria with di¤erential information: an existence proof[END_REF] shows the existence of equilibrium on purely …nancial markets is characterized, in this setting, by that no-arbitrage condition. This existence result di¤ers from Radner's (1979) REE generic one. Finally, [START_REF] Cornet | Elimination of arbitrage states in asymmetric information models[END_REF] shows the above no-arbitrage condition may always be reached by agents, with no price model, from observing exchange opportunities on …nancial markets.

The above papers may picture the information transmission on actual markets and restore a full existence property of equilibrium. But they still retain [START_REF] Arrow | Le rôle des valeurs boursières pour la répartition la meilleure des risques[END_REF] and [START_REF] Radner | Existence of equilibrium plans, prices and price expectations in a sequence of markets[END_REF] rational expectation hypothesis (also called the conditional perfect foresight hypothesis), stating that agents know the map between 1 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) future realized states and equilibrium prices. In such a setting, the states of nature are exogenous and represent all individual ex ante uncertainty.

Yet, actual states typically encompass unobservable variables. [START_REF] Arrow | Le rôle des valeurs boursières pour la répartition la meilleure des risques[END_REF] acknowleges this by noticing that a complete market of exogenous state-contingent claims does not exist and should be replaced by state-contingent …nancial transfers.

In his setting, [START_REF] Kurz | Endogenous uncertainty in a general equilibrium model 34 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) with price contingent contracts[END_REF] notice, "agents need to know the maps from states at future dates to prices in the future and it is entirely unrealistic to assume that agents can …nd out what this sequence of maps is." Quoting [START_REF] Radner | Equilibrium under uncertainty[END_REF] himself, this condition "seems to require of the traders a capacity for imagination and computation far beyond what is realistic". So the question of the possibility and the way to discard rational expectations in the sequential equilibrium model. Radner's (1972-79) rational expectation assumptions would be justi…ed if agents knew all the primitives of the economy (endowments, preferences, etc...) and their relations to equilibrium prices, and if they had elected one common price anticipation in each state (amongst typically many possibilities and interests), with the common knowledge of game theory. Otherwise, the equilibrium outcome would typically di¤er from the standard sequential equilibrium. Such conditions are unrealistic.

Probably the …rst, best known and most radical escape to rational expectations was the temporary equilibrium model, introduced by J. Hicks and later developed by J.-M. Grandmont. It is traditionally presented as dichotomic from the sequential equilibrium model (see [START_REF] Grandmont | Temporary general equilibrium theory[END_REF]. At a temporary equilibrium, agents have exogenous anticipations, which need not be self-ful…lling. Current markets clear at agents' initial plans, which are typically revised, at each period, after observing realized prices and events. Equilibrium allocations need not clear on future spot markets, where agents may face bankruptcy, due to mistaken anticipations. This outcome explains why the temporary equilibrium did not thrive as the perfect foresight's, which lets agents coordinate across periods, on perfectly anticipated prices.

A less radical approach is referred to as bounded rationality. In this line of research, Kurz'(1994) rational belief equilibrium (RBE) allows agents to lack the "structural knowledge" of how equilibrium prices are determined. This unawareness may be due to uncertainty about the beliefs, characteristics and actions of other agents. It leads to an additional uncertainty on future variables, which Kurz calls "endogenous uncertainty", describes as the major cause of economic ‡uctuations, and shows to be consistent with heterogenous beliefs.

Bounded rationality models also serve to study learning processes with di¤erential information (alternative to the REE's), and the links between the information structure and equilibrium or core allocations. This is done, in particular, by [START_REF] Koutsougeras | Bounded rational learning in di¤erential information economies: core and value[END_REF], who emphasize "that the study of cooperative solution concepts (e.g., the core and the (Shapley) value) in di¤erential information economies appears to be a successful alternative to the traditional rational expectations equilibrium, because they provide sensible and reasonable outcomes in situations where any rational expectations equilibrium (REE) notion fails to do so."

The current paper departs from both perfect foresight and bounded rationality models, though it resumes endogenous uncertainty in de…ning the state space. Its asymmetric information concerns the probability assessments over future prices, but also the sets of possible states of nature and anticipations in each state. In our view, bounded rationality still demands inference and computational skills, as well as informations, which typically exceed agents'possibilities. In the real world, their beliefs, actions and characteristics are all private and their observations are limited. This restricts their reckonning capacities to a bare minimum and, consequently, their ability to construct any model, such as one of consistent beliefs. Kurz'RBE focusses on non-stationnary price solutions, so as to allow for heterogeneity of beliefs and dynamic ‡uctuations. An asymptotic limit to the probability distributions over price series is assumed to exist and to be approximated on the …nite observations that agents can make. Yet, with non-stationnary distributions, the asymptotic limits typically di¤er from their …nite proxies. This is one example of why we think bounded rationality is still too demanding from the layman's reckonning skills. The model we propose requires no structural knowledge, nor computation from agents.

Due to their private characteristics, agents face an incompressible uncertainty over the set of clearing market prices to expect, represented by a so-called and never empty "minimum uncertainty set". The set consists of all possible equilibrium prices along agents' private beliefs today and is consistent with [START_REF] Kurz | Endogenous uncertainty in a general equilibrium model 34 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) with price contingent contracts[END_REF] notice that price uncertainty and economic ‡uctuations are "primarily endogenous and internally propagated phenomena (...) generated by the actions and beliefs of the agents (...) and by their uncertainty about the actions of other agents".

That set (or a bigger one) might be inferred, we argue, by a tradehouse or a …nancial institution from observing and treating past data on long time series, rather than by consumers themselves. Yet, future equilibrium prices cannot be reckonned precisely by any agent or institution, because this would require to know every agent's beliefs and characteristics. Only a set of possible equilibrium prices could be assessed ex ante, or the minimum uncertainty set, but not the precise location of future prices within that set. Locating equilibrium prices obeys an uncertainty principle. The uncertainty over a set of anticipations is assessed by agents privately.

The current model's sequential equilibrium concept of "correct foresight equilib-rium" (CFE) is thus de…ned as De Boisde¤re's (2007), except for agents'forecasts, which need no longer be unique, but form sets containing the prices to prevail. The CFE, we argue, reconciles into one concept the sequential and temporary equilibria.

It is sequential, since anticipations are self-ful…lling ex post. It is also temporary since forecasts are exogenously given. Along our main Theorems, whether the …nancial stucture be nominal or real, and beliefs be symmetric or asymmetric, a CFE exists whenever agents'anticipation sets include the minimum uncertainty set.

In our view, this approach to information transmission and equilibrium pictures actual behaviours on markets. Endowed with no price model, unaware of the primitives of the economy, and with limited observational and reckonning capacities, consumers have exogenous anticipations and face endogenous uncertainty. They infer, …rst, the coarsest arbitrage-free re…nement of their initial anticipations from observing trade, along De Boisde¤re (2016). Whence reached, they have no means of further re…ning their anticipation sets. Then, market forces, driven by price and demand correspondences, lead to equilibrium.

The paper is organized as follows: Section 2 presents the model. Section 3 states the existence Theorem for purely …nancial markets. Section 4 proves this Theorem.

Section 5 shows the full existence of equilibria when assets are nominal , or real, or a mix of both. An Appendix proves technical Lemmas.

The basic model

We consider, throughout, a two-period economy, with private information signals, a consumption market and a …nancial market. The sets, I, S, L and J, respectively, of consumers, states of nature, goods and assets are all …nite. The …rst period is also referred to as t = 0 and the second, as t = 1. At t = 0, there is an uncertainty upon which state of nature, s 2 S, will prevail tomorrow. The non random state at t = 0 is denoted by s = 0 and, whenever S, we let 0 := f0g [ . Similarly, we denote by l = 0 the unit of account and let L 0 = f0g [ L.

Markets, information and beliefs

Agents consume and may exchange the same consumption goods, l 2 L, on the spot markets of each period. The generic i th agent's welfare is measured, ex post, by a utility index, u i : R L L + ! R + , over her consumptions at both dates.

At the …rst period (t = 0), each agent, i 2 I, receives a private information signal, S i S, about which states of the world may occur at t = 1. That is, she knows that no state, s 2 SnS i , will prevail tomorrow. Each set S i is assumed to contain the true state. Hence, the pooled information set, denoted by S := \ i2I S i , is non-empty and we let, w.l.o.g., S = [ i2I S i . Such a collection of #I …nite sets, whose intersection is non-empty, is called an information structure. Agents'information structure, (S i ),

is henceforth set as given and always referred to.

Agents are unaware of the primitives of the economy and of other agents'beliefs and actions. They fail to know how market prices are determined and face uncertainty over future spot prices. Thus, at t = 0, the generic i th agent elects a private set of anticipations, out of the price set, P := fp 2 R L ++ : kpk = 1g, in each state s 2 S i . We refer to := S P as the set of forecasts and denote by ! its generic element, and by B( ) its Borel -algebra. A forecast, ! := (s; p) 2 , is thus a pair of a random state, s 2 S, and a conditional spot price, p 2 P , expected in that state.

Remark 1 Strictly positive prices in P are related to strictly increasing preferences, as assumed below. For simplicity, but w.l.o.g., the set, P , normalizes all 6 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) agents'price expectations to one. In each state, this common value of one could be replaced by any other positive value without changing the model's properties.

We now de…ne anticipation structures and beliefs.

De…nition 1 An anticipation set is a closed subset of := S P . A collection of anticipation sets, i := [ s2Si fsg P i s , for each i 2 I, is an anticipation structure if:

(a) P i s 6 = ?, for every (i; s) 2 I S i , and \ i2I P i s 6 = ?, for every s 2 S. Let ( i ) be a given anticipation structure. An anticipation structure, ( 0 i ), which is smaller (for the inclusion relation) than ( i ), is called a re…nement of ( i ), and denoted by

( 0 i ) ( i ). It is said to be self-attainable if \ i2I 0 i = \ i2I i .
A belief is a probability distibution over ( ; B( )), whose support is an anticipation set. A collection of beliefs, ( i ), whose supports de…ne an anticipation structure, ( i ), is called a structure of beliefs, said to support ( i ) and denoted by ( i ) 2 ( i ) .

Only spot markets in states s 2 S 0 may open. We therefore restrict admissible commodity prices in states of s 2 S 0 to the set P := fp 2 R L ++ : kpk 6 1g P S , which is consistent with consumers'anticipations.

Agents may operate …nancial transfers across states in S 0 by exchanging, at t = 0, …nitely many assets, j 2 J, which pay o¤, at t = 1, conditionally on the realization of forecasts. According to Sections, these assets may be nominal (i.e., pay in cash) or real (i.e., pay in godds) or a mix of both. All assets'payo¤s de…ne a (S L 0 ) J return matrix, V , whose generic row across forecasts, ! 2 , is denoted

V (!) 2 R J .
We let V be the set of (S L 0 ) J matrices. Since payo¤s will face "trembles" in the …fth Section, for every n 2 N, we let

V n := fV 0 2 V : kV 0 V k 6 1=ng.
The generic payo¤ of an asset, j 2 J, in a state, s 2 S, is a bundle

v j s := (v jl s ) 2 R L 0 , of the quantities, v j0
s , of cash, and v jl s , of each good l 2 L, that the asset delivers if state s 2 S obtains. 2 We restrict asset prices to the set Q := fq 2 R J : kqk 6 1g w.l.o.g. and let P 0 := fp 2 R L ++ : kpk 6 1g Q be the set of …rst period prices. Along the forecast ! = (s; p := (p l )) 2 , the generic j th asset is a contract which promises to pay v j0 s + P l2L p l v jl s in cash if the forecast, !, obtains. Thus, at asset price, q 2 Q, agents may buy or sell unrestrictively portfolios of assets, z = (z j ) 2 R J , for q z units of account at t = 0, against the promise of delivery of a ‡ow, V (!) z, of conditional cash payo¤s across forecasts, ! 2 .

We now de…ne arbitrage-free anticipation structures.

De…nition 2 Given price q 2 Q, an anticipation structure, ( i ), is said to be qarbitrage-free if following Condition holds:

(a) @(i;z) 2 I R J : q z > 0 and V (!) z > 0, 8! 2 i , with one strict inequality.
An anticipation structure, ( i ), is said to be arbitrage-free if it is q-arbitrage-free for some price, q 2 Q, and we denote by AS their set. We denote by SB the set of structures of beliefs, which support an arbitrage-free anticipation structure, ( i ) 2 AS.

The agent' s behaviour and the concept of equilibrium

The generic i th agent receives an endowment, e i := (e is ) 2 R L S 0 i ++ , granting 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 it prevails. We let e := (e i ) 2 i2I R L S 0 i ++ be the bundle of endowments across agents. Since endowments will face "trembles" in the …fth Section, for every n 2 N, we

let E n := fe 0 2 i2I R L S 0 i + : ke 0 ek 6 1=ng and assume w.l.o.g. that E 1 i2I R L S 0 i ++ , henceforth considered as a …xed set.
Agents'forecasts are represented by an arbitrage-free anticipation structure, say ( i ) 2 AS, which is reached when they elect their strategies at t = 0, jointly with beliefs, ( i ) 2 ( i ) , along De…nition 1. The assumption that agents'forecasts are arbitrage-free is proved to be non restrictive in De Boisde¤re (2016), since they may always infer from markets a (unique coarse) self-attainable arbitrage-free re…nement of any anticipation structure. Then, the i th the agent's consumption set is that of continuous mappings, x :

0 i !R L + (where 0 i := f0g [ i ), denoted by X i := C ( 0 i ; R L + ).
Given the observed prices, ! 0 := (p 0 ; q) 2 P 0 , at t = 0, and her anticipation set, i , the generic i th agent's consumptions, x 2 X i , are mappings, relating s = 0 to a consumption decision, x !0 := x 0 2 R L + , at t = 0, and, continuously on i , every forecast, ! 2 i , to a consumption decision, x ! 2 R L + , at t = 1, which is conditional on the realization of the forecast !. Her budget set is de…ned as follows:

B i (! 0 ; i ) := f(x; z) 2 X i R J : p 0 (x 0 -e i0 )6 q z; p s (x ! -e is )6V (!) z; 8! := (s; p s ) 2 i g.
Given agents'structure of beliefs at the time of trading, ( i ) 2 ( i ) , each consumer, i 2 I, has preferences represented by the V.N.M. utility function:

x 2 X i 7 ! U i i (x) := R !2 i u i (x 0 ; x ! )d i (!).
The above economy, denoted

E ( i ) = f(I; S; L; J); V; (S i ); ( i ); ( i ); (e i ); (u i )g, retains
the small consumer price-taker hypothesis, by which no single agent may, alone, have a signi…cant impact on prices. It is called standard under the following Conditions:

Assumption A1 (strong survival): for each i 2 I; e i 2 R L S 0 i ++ ;
Assumption A2: for each i 2 I, u i is continuous, strictly concave and increasing:

[(x; y; x 0 ; y 0 ) 2 R 4L + ; (x; y) 6 (x 0 ; y 0 ); (x; y) 6 = (x 0 ; y 0 )] ) [u i (x 0 ; y 0 ) > u i (x; y)].
Strict concavity is retained to alleviate the proof of a selection amongst optimal strategies (see proof of Lemma 4). The consumer elects an optimal strategy in her budget set. This yields the following concept of sequential equilibrium:

(b) 8i 2 I; (x i ; z i ) 2 arg max (x;z)2Bi(!0; i ) U i i (x); (c) P i2I (x i0 e i0 ) = 0; (d) P i2I (x i!s e is ) = 0; 8s 2 S;
(e) P i2I z i = 0.

Under the above conditions, price p 2 P, and each forecast, ! s := (s; p s ) 2 , for s 2 S, are said to support equilibrium. A collection, fp; q; ( i ); ( i ); (x i ); (z i )g, which meets

Conditions (b)-(c)-(e)
is called a temporary equilibrium.

The model' s notations

For convenience, we summarize the model's notations in this single sub-Section: For every n 2 N, we let E n := fe 0 2 i2I R L S 0 i + : ke 0 ek 6 1=ng and assume w.l.o.g.

E ( i ) =
that E 1 i2I R L S 0 i
++ , henceforth considered as …xed.

We let s = 0 be the non-random state at t = 0 and denote S 0 := f0g [ S and S 0 i := f0g [ S i , for each i 2 I. We let l = 0 be the account unit and L 0 := f0g [ L.

Q := fq 2 R J : kqk 6 1g, P := fp := (p l ) 2 R L ++ : kpk = 1g, P := fp 2 R L ++ : kpk 6 1g P S and := S P are the sets, respectively, of asset prices, expected spot prices, market prices (for goods) and forecasts.

V is the set of all (S L 0 ) J matrices (V 2 V). For every n 2 N, we let This set is the "minimum uncertainty set", de…ned below. The following Theorems of Section 3 and 5 show that equilibrium exists, whenever agents'forecasts embed the latter set. This existence result holds whatever the anticipations and beliefs agents have, and the types of assets (nominal or real or a mix of both) they exchange. This full existence result is worth noticing, so it di¤ers from the generic ones of the classical sequential equilibrium models. It builds on a core Theorem 1.

V n := fV 0 2 V : kV 0 V k 6 1=ng.

Endogenous uncertainty and the existence of equilibrium

We start with a de…nition.

:= f! 2
: 9p := (p s ) 2 ; 9s 2 S; ! = (s; p s )g, which support an equilibrium, is called the minimum uncertainty set.

Lemma 1 Under Assumptions A1-A2, the following Assertions hold:

(i) 9 > 0 : R L + [ ; 1] L S , hence, S [ ; 1] L ;
(ii) the bound, , may be chosen independent of

V 2 V 1 and (e i ) 2 E 1 .
Proof See the Appendix.

Assumption A3 (correct foresight): for each i 2 I, the relation i holds, in which ( i ) 2 AS is the given anticipation structure of the economy, E ( i ) .

Theorem 1 Under Assumptions A1-A2-A3, an economy, E ( i ) , with purely …nancial markets admits an equilibrium (C.F.E.), for any structure of beliefs, ( i ) 2 SB.

Endogenous uncertainty and how to reach correct anticipations

Along Theorem 1, above, as long as agents have correct foresight (i.e., meet Assumption A3 ), a C.F.E. exists whatever their beliefs. Markets clear ex post at one self-ful…lling common anticipation. We now argue why the set of all equilibrium forecasts may be one of "minimum uncertainty" and how it could be assessed.

On the …rst issue, when today's beliefs are private, no equilibrium price should be ruled out a priori, given agents'unknown anticipations today. Theoretically, this set is of incompressible uncertainty. Practically, it would be so because no agent knows the beliefs and characteristics of other agents, nor has structural knowledge, along Kurz (1994). Past price series con…rm that erratic ‡uctuations may occur not only in periods of enhanced uncertainty. Yet, if no agent has structural knowledge and access to private data, how can this minimum uncertainty set, or a bigger set, be inferred ? The response may simply be empirical, that is, only require observations.

On this issue, the model speci…es normalized prices (extended by Remark 1). It is often possible to observe past prices and reckon their relative values, in a wide array of situations, or states, which typically replicate over time (hence, embed S).

Relative prices vary between observable upper and lower bounds.

Along a sensible assumption, markets are mostly at equilibrium and, with long enough series, all equilibrium forecasts would lie within the bounds of the series' convex hulls. 3 Such a statistical method and its iterative veri…cation across periods require no price model and need not be performed by consumers, but by a tradehouse or …nancial institution, having greater computational facilities. The applications to …nance they might infer are obvious. On consumer side, if agents should agree on a minimal span of price risk, they typically keep private their beliefs and have idiosyncratic anticipations, explaining their likely asymmetries. [START_REF] Cornet | Elimination of arbitrage states in asymmetric information models[END_REF] The existence proof

Hereafter, we set as given an arbitrary anticipation structure, ( i ) 2 AS, and beliefs, ( i ) 2 ( i ) , and assume that the economy, E ( i ) , meets Assumptions A1-A2-A3. In the following sub-Section 4.1, the …nancial structure is represented by an arbitrary payo¤ matrix V 2 V (which needs not be nominal), to present results that 3 e.g., if the future re ‡ects the past, if S is also a set of past states and, for every s 2 S, the past price serie, (p t s ) 2 (P ) Ts (where will serve in the following Section 5. In sub-Sections 4.2 and 4.3, the payo¤ matrix is restricted to be nominal.

The proof proceeds in three steps. Sub-Section 4.1 de…nes, via …nite partitions, a non-decreasing sequence, f( n i )g n2N , of …nite re…nements of ( i ), whose limit is dense in ( i ). Sub-Section 4.2 constructs a sequence of …nite auxiliary economies, which all admit equilibria along De Boisde¤re (2007). Sub-Section 4.3 derives a CFE of the economy E ( i ) from these auxiliary equilibria.

Finite partitions of agents'anticipation sets

Let (i; n) 2 I N be given. We de…ne an integer, K (i;n) 2 N, and a partition,

P n i = f k (i;n) g 16k6K (i;n) , of i , such that i ( k (i;n) ) > 0,
for each k 2 f1; :::; K (i;n) g.

In each set k (i;n) (for k 6 K (i;n) ), we select exactly one element, ! k (i;n) , to form the discrete sub-set,

n i := f! k (i;n) g 16k6K (i;n) , of i .
We de…ne mappings, n i :

n i ! R + , by n i (! k (i;n) ) = i ( k (i;n) ) and n i : i ! n i , by its restrictions, n i = k (i;n) (!) = ! k (i;n) , for each k 6 K (i;n) and every ! 2 k (i;n) .
And we henceforth assume that the Assertions of the following Lemma hold.

Lemma 2 For each (i; n) 2 I N, we may choose the above P n i , n i , n i , such that:

(i) n i n+1 i
and P n+1 i is …ner than P n i ;

(ii) [ n2N n i is everywhere dense in i ;

(iii) for every ! 2 i , ! = lim n!1 n i (!), and n i (!) converges uniformly to !;

(iv) there exist N 2 N, such that ( n i ) is arbitrage-free for every n > N . For simplicity, we henceforth assume that N = 1.

Proof See the Appendix, which provides one example of such sets and maps.

The auxiliary economies, E n

Given n 2 N, we de…ne an economy, E n =f(I; S; L; J); V; ( 0n i ); (e i ); (u n i )g, with same periods, sets of agents, goods and endowments as above. The realizable states and the generic i th agent's expectations are artefactual and de…ned as follows:

0n i := S [ n i
is the agent's information set, de…ning the information structure,

( 0n i ), of a formal state space, n := [ i2I 0n
i , whose set of realizable states is S.

In each state s 2 S, the i th agent has a perfect foresight of the spot price.

In each state (s; p) 2 n i , the i th agent is certain that price p 2 P will prevail.

By induction on n 2 N, we de…ne a sequence of equilibrium prices, (p n ; q n ) 2 P Q in the following way. For all prices, (p := (p s ); q) 2 P Q, we let the generic i th agent's consumption set, budget set, and utility function in the economy E n be:

X n i := R L 0n i +
, whose generic element is denoted by

x := [(x s ) s2S 0 ; (x ! ) !2 n i ];
B n i (p; q) := f (x; z) 2 X n i R J : p 0 (x 0 e i0 )6 q z; p s (x s e is )6 V (s; p s ) z; 8s 2 S and p (x ! e is )6 V (!) z; 8! := (s; p) 2 n i g;

and x 2 X n i 7 ! u n i (x) := 1 n#S X s2S u i (x 0 ; x s ) + (1 1 n ) X !2 n i u i (x 0 ; x ! ) n i (!).
Henceforth, the payo¤ matrix, V , is assumed to be nominal, so that V (s) := V (s; p), for every (s; p) 2 , only depends on s 2 S. The above economy, E n , is of the De Boisde¤re's (2007) type. Hence, from its Theorem 1 and proof, it admits an equilibrium, for every n 2 N, de…ned as follows:

De…nition 5 A collection of prices, (p; q) 2 P Q, and strategies, (x i ; z i ) 2 B n i (p; q), for each i 2 I, is an equilibrium of the economy E n , if the following Conditions hold:

P i2I (x is e is ) = 0; 8s 2 S 0 ; (c) P i2I z i = 0.
We set as given, for every n 2 N, such equilibria, C n := fp n ; q n ; (x n i ); (z n i )g, in each economy E n . From the proof of Theorem 1 in De Boisde¤re (2007), the elected equilibrium satis…es kp n 0 k + kq n k > 1, for each n 2 N, hence, kp 0 k + kq k > 1. Moreover, the sequence, fC n g := fn 2 N 7 ! C n g, meets the following properties:

Lemma 3 For each i 2 I, we let Z i := fz 2 R J : V (!) z = 0; 8! 2 i g, Z ?
i be its orthogonal complement and Z :=

P i2I Z i . Given fC n g, we let z n i = z on i + z ?n i be the decom- position of z n i on Z i Z ? i , for each (i; n) 2 I N.
The following Assertions hold:

(i) the price sequence f(p n ; q n )g may be assumed to converge to (p ; q ) 2 P Q, such that f(s; p s )g s2S (\ i2I i );

(ii) the sequences f(x n is ) s2S 0 g and f(z ?n i ) i2I g may be assumed to converge, say to (x is ) s2S 0 and (z ? i ) 2 R J I , such that P i2I (x is e is ) s2S 0 = 0 and

P i2I z ? i 2 Z;
(iii) there exists (z i ) 2 R J I , such that P i2I z i = 0 and (z i z ? i ) 2 Z i for every i 2 I.

Lemma 4 Let B i (!; z) = fx 2 R L + : p (x e is ) 6 V (!) zg, be given sets, for every z 2 R J and all ! := (s; p) 2 i . Along Lemma 3, the following Assertions hold for all i 2 I:

(i) the correspondence ! 2 i 7 ! arg max u i (x i0 ; x), for x 2 B i (!; z i ), is a continuous map, whose embedding, x i : ! 2 0 i 7 ! x i! , is a consumption, that is, x i 2 X i ; (ii) U i i (x i ) = lim n!1 u n i (x n i ).
Proof of the Lemmas See the Appendix.

An equilibrium of the initial economy

We now prove Theorem 1, via the following Claim. From the de…nition of C n , the relations p n 0 (x n i0 e i0 ) 6 q n z n i hold, for each (i; n) 2 I N, and yield p 0 (x i0 e i0 ) 6 q z i , for each i 2 I, in the limit (n ! 1). We let ! 0 := (p 0 ; q ). From Lemma 4-(i), the relations x i 2 X i and p s (x i! e is ) 6 V (!) z i also hold, for every i 2 I and every ! = (s; p s ) 2 i , and imply [(

x i ; z i )] i2I 2 i2I B i (! 0 ; i ).
Next, we assume, by contraposition, that C fails to meet Condition (b) of De…nition 3, that is, 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, moreover, assume that (x; z) 2 B i (! 0 ; i ) is such that:

(II) 9 ( ; M ) 2 R 2 ++ : x ! 2 [ ; M ] L ; 8! 2 i .
The existence of an upper bound to consumptions x ! (for ! 2 i ) results from the relation (x; z) 2 B i (! 0 ; i ), which implies a bound to …nancial transfers and from the fact that i is closed in S P . Moreover, for 2]0; 1] small enough, the strategy

(x ; z ) := ((1 )x + e i ; (1
)z) 2 B i (! 0 ; i ) meets both relations (I) and (II), from Assumption A1 and from the uniform continuity (on a compact set) of the mapping

( ; !) 2 [0; 1] i 7 ! (x ! , u i (x 0 ; x ! ))
. So, relations (II) may indeed be assumed.

From Lemmas 1-3,

p 2 R L + [ ; 1] L S .
Then, from the relations (I)-(II) and

(x; z) 2 B i (! 0 ; i ), the de…nition of i , Assumptions A1-A2 and uniform continuity arguments, we may also assume there exists 2 R ++ , such that: (III) p 0 (x 0 e i0 ) 6 q z and p s (x ! e is ) <

+ V (!) z, 8! := (s; p s ) 2 i .
From relations (I)-(II)-(III), we may also assume there exists 0 2]0; [, such that:

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

We recall from above that kp 0 k + kq k > 1. The above assertion is obvious, from relations (III), if p 0 (x 0 e i0 ) < q z. Assume that p 0 (x 0 e i0 ) = q z. If p 0 = 0, then, q 6 = 0, and relations (IV ) hold if we replace z by z q =N , for N 2 N big enough.

If p 0 6 = 0 and x 0 6 = 0, the desired assertion results from Assumption A1 and above.

Else, q z = p 0 e i0 < 0, and a slight change in portfolio insures relations (IV ).

From relations (IV ), the continuity of the scalar product and Lemmas 1-2-3, there exists N 1 2 N, such that, for every n > N 1 :

(V ) 8 > > > > > > < > > > > > > :
p n 0 (x 0 e i0 ) 6 q n z p n s (x (s;p s ) e is ) 6 V (s; p n s ) z; 8s 2 S p s (x ! e is ) 6 V (s; p s ) z; 8! := (s; p s ) 2 n i Along relations (V ), for each n > N 1 , we de…ne, in E n , the strategy (x n ; z) 2

B n i (p n ; q n ) by x n 0 := x 0 , x n s := x (s;p s ) , x n ! := x ! , for (s; !) 2 S n i
, and recall that:

U i i (x) := R !2 i u i (x 0 ; x ! )d i (!); u n i (x n ) := 1 n#S X s2S u i (x 0 ; x n s ) + (1 1 n ) X !2 n i u i (x 0 ; x ! ) n i (!).
Then, from above, from relation (II), Lemma 2, and the uniform continuity of

x 2 X i and u i on compact sets, there exists N 2 > N 1 such that:

(V I) jU i i (x)-u n i (x n )j < R !2 i ju i (x 0 ; x ! )-u i (x 0 ; x n i (!) )jd i (!) + " 4 < " 2 , for every n > N 2 .
From equilibrium conditions and Lemma 4-(ii), there exists N 3 > N 2 , such that:

(V II) u n i (x n ) 6 u n i (x n i ) < " 2 + U i i (x i ),
for every n > N 3 . Let n > N 3 be given. The above Conditions (I)-(V I)-(V II) yield, jointly:

U i i (x) < " 2 + u n i (x n ) 6 " 2 + u n i (x n i ) < " + U i i (x i ) < U i i (x).
This contradiction proves that C meets Condition (b) of De…nition 3, hence, from above, is a C.F.E. of the economy E ( i ) . This completes the proof of Theorem 1.

Theorem 1, above, holds for nominal asset structures when agents have correct foresight. We now examine existence for other …nancial and anticipation structures.

The existence theorems with arbitrary assets

In sub-Section 5.1, we show that temporary equilibria always exist, that is, for arbitrary beliefs and …nancial structures. In the following sub-Sections, we extend the above Theorem 1 to an economy with smooth preferences and arbitrary assets.

Temporay equilibria with arbitrary structures of payo¤s and beliefs

Theorem 2 Under Assumptions A1-A2, an economy, E ( i ) , with an arbitrary payo¤ matrix, V 2 V, admits a temporary equilibrium, for any structure of beliefs, ( i ) 2 SB.

Proof In the de…nition of auxiliary economies in Section 4, we may assume that the set of realizable states, S, is empty. This assumption is purely formal, artefactual.

Then, for each n 2 N, the economy, E n , is well de…ned, anticipation sets, ( n i ), are exogenous, whereas, for ! 0 := (p 0 ; q) 2 P 0 , the generic i th agent's budget set is:

B n i (! 0 ) := f(x; z) 2 X n i R J : p 0 (
x 0 e i0 )6 q z and p (x ! e is )6 V (!) z; 8! := (s; p) 2 n i g:

From De Boisde¤re's (2007) existence theorem and proof, for every n 2 N, the above economy, E n , admits a temporary equilibrium, de…ned as follows:

De…nition 6 A collection of prices, ! 0 := (p 0 ; q) 2 P 0 , such that k! 0 k > 1, and strategies, (x i ; z i ) 2 B n i (! 0 ), de…ned for each i 2 I, is an equilibrium of the economy E n , if the following Conditions hold:

(a) 8i 2 I; (x i ; z i ) 2 arg max (x;z)2B n i (!0) u n i (x); (b) P i2I (x i0 e i0 ) = 0; (c) P i2I z i = 0.
Indeed, all arguments of the proof of Theorem 1 in De Boisde¤re (2007) apply, mutatis mutandis, with the artefactual assumption that S is empty and yield an equilibrium along the above De…nition, say

C n := f! n 0 ; (x n i ); (z n i )g.
Similarly, all arguments of Lemmas 3 and 4 and Claim 1 above apply, mutatis mutandis, and yield a temporary equilibrium, C := f! 0 ; ( i ); ( i ); (x i ); (z i )g, along De…nition 3.

The economy with arbitrary assets and correct beliefs

We have to change the framework slightly, so as to be able to apply standard generic existence results of the litterature. We will conform to Du¢ e-Shafer's (1985) setting. Admissible commodity prices are now restricted to the new price set:

P := fp := (p l s ) 2 R L S 0 ++ : P (l;s)2L S 0 p l s = 1g. 4
The set Q = fq 2 R J : kqk 6 1g may still be retained for admissible asset prices (with a bound to be one w.l.o.g.). Indeed, asset prices may always be bounded via individual state prices -or price functions along De Boisde¤re's (2016). Since the de…nition of equilibrium is De…nition 3, above. The proof of its full existence builds on auxiliary …nite economies, which we now present.

5.2

The auxiliary economies, E n

( i )
We set as given (a generic) structure of beliefs, ( i ) 2 SB, whose supports de…ne a structure, ( i ) 2 AS. We construct, for all n 2 N, an auxiliary economy, denoted

E n ( i )
, resuming all Section 4 de…nitions and notations, in anything but budget sets.

For every tuple, (i; p; q; V 0 ) 2 I R L S 0 + Q V and endowment bundles, e 0 := (e 0 i ) 2

E 1 := f(e 0 i ) 2 i2I R L S 0 i + : k(e 0 i ) (e i )k 6
1g, the generic i th agent's budget set is now:

B n i (p; q; V 0 ; e 0 i ) := f (x; z) 2 X n i R J : p 0 (x 0 e 0 i0 )6 q z; p s (x s e 0 is )6 V 0 (s; p s ) z; 8s 2 S and p (x ! e is )6 V (!) z; 8! := (s; p) 2 n i g:
In the above budget sets, the payo¤ matrix and endowments may only di¤er from the original ones (i.e., V 2 V and (e i ) 2 E 1 ) in realizable states (i.e., s 2 S 0 ).

This restriction yields the following simple concept of auxiliary equilibrium:

De…nition 7 A collection of prices, (p; q) 2 R L S 0 ++ Q, payo¤ matrix, V 0 2 V, endow- ments, (e 0 i ) 2 E 1 ,
and strategies, (x i ; z i ) 2 B n i (p; q; V 0 ; e 0 i ), de…ned for each i 2 I, is an equilibrium of the economy E n ( i ) , if the following conditions hold:

(a) 8i 2 I; (x i ; z i ) 2 arg max (x;z)2B n i (p;q;V 0 ;e 0 i ) u n i (x); (b) P i2I (x is e 0 is ) = 0; 8s 2 S 0 ; (c) P i2I z i = 0; (d) p e 0 1 = 1.
We de…ne a related concept of pseudo-equilibrium, after introducing new sets:

we let G be the set of all full column rank S J matrices;

for every L 2 G, we denote by < L > its #J-dimensional span in R S ;

for every V 0 2 V and p := (p s ) 2 R L S 0 ++ , we let V 0 (p) be the S J matrix, whose generic row is V 0 (s; p s ) 2 R J (for s 2 S) and denote by < V 0 (p) > its span in R S ; for every triple (p; i; x) 2 R L S 0 ++ I X n i , we let p x 2 R 0n i be the vector, whose …rst components are the scalar products, p s x s , for each s 2 S, and subsequent components are the products, p ! x ! , for each ! := (s; p ! ) 2 n i ;

we let E n := R L 0n i I ++ be the sets of arbitrary endowment bundles, namely, for each agent, i 2 I, the conditional bundles, e 0 is 2 R L ++ , in each realizable state, s 2 S 0 , and

e 0 i! 2 R L ++ , in each idiosyncratic state, ! 2 n i ;
for every L := (L s ) s2S 2 G, and every i 2 I, we let [ L Vi ] be the 0n i J matrix whose …rst generic rows are the L s 2 R J , in each state s 2 S, and subsequent rows are the V (!) 2 R J , in each state ! 2 n i . We denote < L Vi > the matrix'span in R 0n i .

We now de…ne the following concept of pseudo-equilibrium in the economy E n ( i ) :

De…nition 8 A collection of prices, p := (p s ) 2 R L S 0 ++ , payo¤ matrices, L 2 G and V 0 2 V, endowments, e 0 := (e 0 i ) 2 E n , and an allocation, (x i ) 2 i2I X n i de…nes a pseudo-equilibrium of the economy E n ( i ) , if the following conditions hold:

(a) x 1 2 arg max u n i (x), for x 2 f x 2 X n 1 : p (x e 0 1 ) = 0 g; (b) for every i 2 Inf1g, x i 2 arg max u n i (x),
for x 2 f x 2 X n i :

P s2S 0 p s (x s e 0 is )+ P !2 n i p ! (x ! e 0 i! ) = 0 and p (x e 0 i ) 2 < L Vi > g; (c) < V 0 (p) > < L >; (d) P i2I (
x is e 0 is ) = 0; 8s 2 S 0 ;

(e) P i2I z i = 0;

(f ) p e 0 1 = 1.

Given (e 0 ; V 0 ) 2 E n V, we say that (p; L) is a pseudo-equilibrium, if there exists x 2 23 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) i2I X n i , such that (x; p; L) is a pseudo-equilibrium. We let E be the pseudo-equilibria manifold, that is, the set of collections, (p; L; e 0 ; V 0 ), such that (p; L) is a pseudoequilibrium. We de…ne the projection, : E ! E n V, by (p; L; e 0 ; V 0 ) := (e 0 ; V 0 ).

The above de…nitions extend Du¢ e-Shafer's (1985, pp. 288-289) to the economy

E n ( i )
. The following Claim states the full existence of pseudo-equilibria.

Claim 2 Given De…nition 8, the following Assertions hold:

(i) E is a smooth manifold without boundary of same dimension than (E);

(ii) is proper;

(iii) there exists a regular value (e ; V ) of , such that # 1 (e ; V ) = 1;

(iv)

1 (e 0 ; V 0 ) 6 = ?, for every (e 0 ; V 0 ) 2 E n V;

(v) the set of singular values of is closed and null. Assertion (iii) A price, p := (p s ) 2 R L S 0 ++ , and matrix, V 2 V, are set as given, such that V (p) 2 G, and we let L := V (p). The fact that there exist endowments, (e i ) 2

E n , which are optimal for each agent (meet Conditions (a)-(b) of De…nition 8) is obvious from Assumption A5 (align gradients with common and individual prices). That is, the elected price, p , and endowments, (e i ), yield a pseudo-equilibrium, (p ; L ), with no trade. This pseudo-equilibrium is Pareto optimal and a¤ordable at any price, hence, unique, whereas the value ((e i ); V ) is regular, as shown, mutatis mutandis, by Du¢ e-Shafer (see p. 296). Assertion (iv) From Assertion (iii) and mod. 2 degree theory, there is an odd number of pseudo-equilibria at any regular value of . The value (e 0 ; V 0 ) is regular from the De…nition, if 1 (e 0 ; V 0 ) = ?. Hence, 1 (e 0 ; V 0 ) 6 = ? for all (e 0 ; V 0 ) 2 E n V.

Assertion (v) is a standard application of Sard'theorem and demonstrated, mutatis mutandis, in Du¢ e-Shafer (p. 297), to which we refer the reader.

Remark 2 In De…nition 8, payo¤s and anticipations in all idiosyncratic states (! 2 n nS) were …xed, independently of n 2 N. Contrarily, the endowments, (e 0 i ) 2 E n , were allowed to vary in these idiosyncratic states. This ‡exibility was required to …nd a unique pseudo-equilibrium in Claim 2-(ii). Consequently, the generic set of regular values of in Claim 2-(v) was a sub-set of E n V. To simplify exposition, but w.l.o.g., we henceforth consider this set of regular values of to be a generic subset of E n V, where E n := f(e 0 i ) 2 E n : e 0 i! = e is ; 8i 2 I; 8! = (s; p) 2 n i g. That is, endowments are …xed at their original values, in all idiosyncratic states (! 2 n nS).

This simpli…cation is formal. It is unnecessary for proving Theorem 3, below, but it avoids heavy notations in de…ning auxiliary equilibria, and for proving Lemma 4.

Claim 2 and Remark 2 yield the following existence result.

Claim 3 For every n 2 N and every ( i ) 2 SB, there exist prices, (p

n ; q n ) 2 R L S 0 ++ Q, endowments, (e n i ) 2 E n , a payo¤ matrix, V n 2 V n , and strategies, (x n i ; z n i ) 2 B n i (p n ; q n ; V n ; e n i ),
for each i 2 I, which de…ne an equilibrium of the economy, E n ( i ) , along De…nition 7.

; V n ) 2 1 (e n ; V n ).
From the de…nitions of regularity and pseudo-equilibria, the relations

L n = V n (p n ) 2 G hold.
As standard (e.g., Du¢ e-Shafer, p. 289), the pseudo-equilibrium, (p n ; L n ; e n ; V n ), is equivalent to an equilibrium, fp n ; q n ; V n ; (e n i ); (x n i ); (z n i )g, along De…nition 7.

Henceforth, we set as given one equilibrium, C n

( i ) := fp n ; q n ; V n ; (e n i ); (x n i ); (z n i )g, in the economy, E n ( i )
, for each n 2 N. The sequences, f(s; p n s )g, for s 2 S, meet the lower bound condition of Lemma 1, as shown in the Appendix, and admit cluster points, whose set is denoted by ( i ) . The above structure of beliefs, ( i ) 2 SB, was generic.

We may proceed in the same way as above for all structures of beliefs, ( i ) 2 SB.

This leads to a well de…ned set, := [ ( i )2SB ( i ) , and to the following Assumption.

Assumption A7 : for every i 2 I, the relation i holds, in which ( i ) 2 AS is the given anticipation structure of the economy, E ( i ) .

We will show that is in fact a subset of . Before, we have to prove Theorem 3.

Theorem 3 Under Assumptions A1 to A7, the economy with arbitrary assets, E ( i ) , admits an equilibrium (C.F.E.), for any structure of beliefs, ( i ) 2 SB.

Proof First, we set …xed and given an arbitrary structure of beliefs, ( i ) 2 SB.

For each n 2 N, an equilibrium, C n ( i ) := fp n ; q n ; V n ; (e n i ); (x n i ); (z n i )g, is well de…ned from above. Under Assumptions A1 to A7, their sequence, fC n ( i ) g, meets the above Assertions of Lemma 3, upon replacing by , and of Lemma 4, as is. Both results are demonstrated in the Appendix. Then, Theorem 3 results from the following Claim 4.

Claim 4 The collection, fp ; q ; ( i ); ( i ); (x i );(z i )g, of prices, anticipation sets, beliefs, allocation and portfolios of Lemmas 3-4 is a CFE of the economy E ( i ) .

Proof The proof is identical to that of Claim 1, which we let the reader check. The only di¢ culty is for proving the relations (IV ) of sub-Section 4.3. From Lemma 1, Lemma 2-(iv) and Assumption A5, the sequence fkq n kg admits positive lower and upper bounds, for an appropriate choice of individual state prices in the auxiliary economies. Then, kq k > 0 and relations (IV ) in sub-Section 4.3 hold. The arguments of Claim 1, which all apply, lead to a price p 2 P (from De…nition 7-(d)), which is shown to be an equilibrium price. From Assumption A2, this implies p 2 P.

It follows from the above proof that any element in is an equilibrium forecast, that is, . Hence, Asssumption A7 can be replaced by A3, in Theorem 3.

Appendix

Lemma 1 Under Assumptions A1-A2, the following Assertions hold:

(i) 9 > 0 : R L + [ ; 1] L S , hence, S [ ; 1] L ;
(ii) the bound, , may be chosen independent of V 2 V 1 and (e i ) 2 E 1 .

Proof First, we introduce new notations and let, for every (i; s; x) Since all equilibrium prices belong to \ s2S P s , from the de…nition, it su¢ ces to prove that the following Lemmata 1 holds with an independent bound, .

27 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

Lemmata 1 The following Assertions hold:

(i) 8s 2 S, P s is a closed, hence, compact set;

(ii) 9 > 0 : 8(s; l) 2 S L, 8p := (p l s 0 ) 2 P s , p l s > .

Proof of Lemmata 1 Assertion (i) From the de…nition, for each (n; s) 2 N S the set P s contains p n . Let s 2 S and a converging sequence fp k g k2N of elements of P s be given. Its limit, p, is in P, a closed set. We may assume there exist (a same) j 2 I and a sequence, fx k g k2N := f(x k i )g k2N , of elements of A, converging to some x := (x i ) in the closure of A in i2I (R + [ f+1g) L S 0 i , such that, for each k 2 N, (p k ; j; x k ) satis…es the conditions of the de…nition of P s . From the de…nition of A, f(x k is 0 )g k2N , is bounded, hence, x s 0 := (x is 0 ) 2 R L I + is …nite, for each s 0 2 S 0 .

For every k 2 N, we let e We now show that there exists k 2 N, such that (p; j; e x k ) satis…es the conditions of the de…nition of P s (i:e:, p = lim p k 2 P s and P s is closed).

By contraposition, assume that, for each k 2 N, there exists y k 2 R L S 0 j + , such that y k s 0 = e x k js 0 , for each s 0 2 S 0 j nfsg, u j (x j0 ; y k s ) > u j (x j0 ; x js ) and p s (y k s x js ) < 0. Then, given k 2 N, we show the following relations:

(I) 8K > k, 9k 0 > K, u j (x k 0 j0 ; y k s ) > u j (x k 0 j0 ; x k 0 js ).

If not, one has u j (x k 0 j0 ; y k s ) 6 u j (x k 0 j0 ; x k 0 js ), for k 0 big enough, which implies, in the limit (k 0 ! 1), u j (x j0 ; y k s ) 6 u j (x j0 ; x js ), in contradiction with the above assumption that u j (x j0 ; y k s ) > u j (x j0 ; x js ). Hence, relations (I) hold. From the de…nition of the sequence fx k g k2N , relations (I) imply p k 0 s (y k s x k 0 js ) > 0, and, in the limit (k 0 !1), 28 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) p s (y k s x js ) > 0, in contradiction with the inequality, p s (y k s x js ) < 0, assumed above. This contradiction proves that p := lim p k 2 P s , hence, all P s are compact. Assertion (ii) Let (s; l) 2 S L and p := (p l s 0 ) 2 P s be given. Let e 2 R L have zero components but the l th , equal to 1. We prove that p l s = p s e > 0. Indeed, let Let E := (E l 0 s 0 ) 2 R L S 0 j + be de…ned by E l s = 1 and E l 0 s 0 = 0, for every (s 0 ; l 0 ) 6 = (s; l). Along Assumption A2, there exists n 2 N, such that y := (x n j + ( 11 n )E) satis…es u j (y 0 ; y s ) > u j (x j0 ; x js ), implying p s x js 6 p s y s = p s (x n js + (1 1 n )e) < p s x js + (1 1 n )p s e.

Hence, p l s = p s e > 0. The mapping ' (s;l) : P s ! R ++ , de…ned by ' (s;l) (p) := p s e is continuous and attains its minimum for some element p on the compact set P s , say (s;l) > 0. Then, Assertion (ii) holds for := min (s;l) , for (s; l) 2 S L.

Lemmata 1 proves the …rst part of Lemma 1. Since was chosen independent of V 2 V 1 and of (e i ) 2 E 1 , the second part of Lemma 1 also holds. Lemma 1 also holds for the economy of sub-Section 5.2 above. To see this, for every s 2 S, and every n 2 N, we replace in the de…nition of the above sets P s , the price set, P, of Section 2 by those of Section 5, namely, P n := fp 2 R L S 0 ++ : p e n 1 = 1g.

We let the reader check that Lemma 1 holds by the very same arguments as above (with a bound, , which does not depend on n 2 N).

Lemma 2 For each (i; n) 2 I N, we may choose the above P n i , n i , n i , such that:

(i) n i n+1 i
and P n+1 i is …ner than P n i ;

(ii) [ n2N n i is everywhere dense in i ;

3

  The core existence theorem With the model's endogenous uncertainty, only the set of possible equilibrium forecasts could be assessed. No agent or institution would know the true forcasts' location within that set, because this would require to know all agents' private beliefs, characteristics and actions.

  (a) 8i 2 I; (x i ; z i ) 2 arg max (x;z)2B n i (p;q) u n i (x); 15 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) (b)

  (a)-(c)-(d)-(e) of De…nition 3 of equilibrium, above. We now show that C meets Condition (b) of the same De…nition 3.

x k := (e x k i ) 2 A

 2 be de…ned by (ex k i0 ) := (x i0 ) 2 R L I + and (e x k is ) := (x is ) 2 R L I + and (e x k is 0 ) := (x k is 0 ), for each (i; s 0 ) 2 I S 0 i nfsg.Then, the relations p k s (x k js e e) > 0, which hold for every k 2 N, yield, in the limit, p s (e x k js e e) = p s (x js e e) > 0.

(we let x n j 2 R 1 n

 21 p; j; (x i )) 2 P s I A meet the conditions of the de…nition of P s . For every n > 1, )x js and x n js 0 := x js 0 for s 0 6 = s. It satis…es p s (x n js x js ) < 0 (since p s x js > p s e e > 0).

  De…nition 3 A collection of prices, p := (p s ) 2 P and q 2 Q, of an anticipation structure, ( i ) 2 AS, beliefs, ( i ) 2 ( i ) , and strategies, (x i ; z i ) 2 B i (! 0 ; i ), de…ned for each i 2 I (where ! 0 := (p 0 ; q)) is a sequential equilibrium of the economy, E (

i ) , or correct foresight equilibrium (C.F.E.), if the following Conditions hold: (a) 8i 2 I; 8s 2 S; ! s := (s; p s ) 2 i ;

  Let be the set of prices, p := (p s ) 2 P, which support the equilibrium of an economy, E ( i ) , for some arbitrary structure of beliefs, (

11 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) De…nition 4 i ) 2 SB. The set of forecasts,

  Ts 2 N) is large, then, iteratively, the set

	Ts X	tp	t s =k	Ts X	tp t
	t=1			t=1

f(s; ys) 2 S P : ys = s k; ( t) 2 R Ts + ; Ts X t=1 t = 1g, could easily be checked to always contain the self-ful…lling forecasts. 13 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

  The collection, fp ; q ; ( i ); ( i ); (x i );(z i )g, of prices, anticipation sets, beliefs, allocation and portfolios of Lemmas 3-4, de…nes a CFE of the economy E ( i ) .

16 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) Claim 1 Proof We let C := fp ; q ; ( i ); ( i ); (x i ); (z i )g be de…ned as in Claim 1. From Lemma 3, C meets Conditions

  Proof As we let the reader check, no argument in Du¢ e-Shafer (1985) is altered by the presence of the …xed set of unrealizable states, n nS, in which payo¤s are …xed exogenous, as are anticipations. Only the spans generated by payo¤s in realizable states (s 2 S) matter. No argument is altered by the presence of nominal payo¤s.

	Assertion (i) results, mutatis mutandis, from Du¢ e-Shafer's (1985) Section 4.
	Assertion (ii) results, mutatis mutandis, from Du¢ e-Shafer's Fact 10 (p 295).

  Proof Let n 2 N and ( i ) 2 SB be given. From Claim 2-(iii)-(v) there exist a regular value, (e n ; V n ) 2 E n V n , and a pseudo-equilibrium, (p n ; L n ; e n

25 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

  2 I S R L S 0 denote a consumption, s.t. u i (y 0 ; y s ) > u i (x 0 ; x s ) and y s 0 = x s 0 , 8s 0 2 S 0

					+	i	:
	e e 2 R L ++ have all components equal to = min e 0l is > 0 for (i; s; l; (e 0 i )) 2 I S 0 i L E 1 ;
	y i s x i nfsg;
	A := f(x i ) 2 i2I R L S 0 i +	:	P i2I x is =	P	i2I e is ; 8s 2 S 0 g;
	P s := fp 2 P : 9j 2 I; 9(x i ) 2 A; such that (y j s x

j ) ) (p s y s > p s x js > p s e e)g.

Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

if the asset, j 2 J, is nominal v jl s = 0, for every pair (s; l) 2 S L. If the asset is real, v j0 s = 0, for every s 2 S.

Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

We keep the same notations as in Section 2, so as to refer to the same De…nitions as above, with reference to the new sets. In particular, with Section

5's new de…nitions of the price and forecast sets, De…nitions 1, 2 and 3 may be kept as is. 20 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

anticipation structures are arbitrage-free, the asset price weighs the rows of payo¤s on every agent's forecasts, and can always be bounded uniformly. Moreover, Du¢ e-Shafer (1985) proceeds in the same way. It sets as given a state price vector (whose components are all ones, p. 295), instead of an upper bound to asset prices.

Consistently with the latter de…nition of the price set, P, the set of forecasts is now := S fp 2 R L ++ : kpk < 1g. An anticipation set is a closed subset of , and the structures of anticipations and beliefs are de…ned accordingly, along De…nition 1.

We assume that one agent, say i = 1, is fully informed upon the true states (S 1 = S) and true spot price that can prevail in any state and that she is endowed with exactly one unit of each good in any state, s 2 S 0 . Thus, for all price p 2 P, one has p e 1 = 1. The latter relation, p e 1 = 1, holds, at so called pseudo-equilibria in Du¢ e-Shafer (1985), and below. Under the above small consumer hypothesis, the latter assumption is an artefact of no cost, which will permit to normalize price anticipations on the unit simplex. The other characteristics of the current economy are the same as above, in Section 2, to which we add Du¢ e-Shafer's following smoothness assumptions, the Inada Conditions and an additional assumption on payo¤s, which will permit to bound the asset price norm uniformly from below:

Assumption A5: for each i 2 I, u i meets the Inada Conditions;

Assumption A6: there exists one asset, with non-negative payo¤s in all states, and having at least one positive payo¤ in one state s 2 S.

In the current economy, denoted by E ( i ) = f(I; S; L; J); V; (S i ); ( i ); ( i ); (e i ); (u i )g, Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

(iii) for every ! 2 i , ! = lim n!1 n i (!), and n i (!) converges uniformly to !;

(iv) there exist N 2 N, such that ( n i ) is arbitrage-free for every n > N . For simplicity, we henceforth assume that N = 1.

Proof Let i 2 I, n 2 N and K n := f1; :::; 2 n 1 g L be given (letting N start from n = 1).

From the de…nition, i := [ s2Si fsg P i s S P . For each pair (s; k := (k l )) 2 S i K n , we de…ne the (possibly empty) subset, (s;k)

), of i . To simplify notations, we let

and identify the latter set, f(s; k) 2 S i K n : i ( (s;k) (i;n) ) > 0g, to the subset, f1; :::; K (i;n) g, of N. Then, the partitions,

, and just one. We may always construct the sets,

, for every n 2 N. And we de…ne the mapping, n i , as in sub-Section 4.1. Then, Assertions

it is stationary. We let the reader check, from Assertion (ii) and the continuity of the scalar product that its limit is (Z i ). Hence, there exists

For simplicity, we assume costlessly that N = 1. Then, for every pair

Assume, by contraposition, that Assertion (iv) fails. Then, from De Boisde¤re's (2016) Claim 2, for every n 2 N, there exist n 0 > n and portfolios, (z

i . The sequence, f(z n 0 i )g, may be assumed to converge, say to (z i ) 2 Z. From the continuity of the scalar product, Assertion (ii) 30 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée) and above, the relations V (! i ) z i > 0 hold, for every (i; ! i ) 2 I i . The latter imply

from above, and from De Boisde¤re's (2016) Claim 2 jointly with the fact that ( i ) is arbitrage-free. This contradiction completes the proof.

Lemma 3 For each i 2 I, we let

i be its orthogonal complement and Z :=

The following Assertions hold:

(i) the price sequence f(p n ; q n )g may be assumed to converge to (p ; q ) 2 P Q, such that f(s; p s )g s2S (\ i2I i );

(ii) the sequences f(x n is ) s2S 0 g and f(z ?n i ) i2I g may be assumed to converge, say to

(iii) there exists (z i ) 2 R J I , such that

Proof Assertion (i) is obvious from the de…nitions, Lemma 1, the relations p n 2 (in sub-Section 4.2) for every n 2 N, Assumption A3 and compactness arguments.

Assertion (ii) The non-negativity and market clearance conditions over auxiliary equilibrium allocations imply that f(x n is ) s2S 0 g is bounded, hence, may be assumed to converge. The market clearance conditions of equilibrium,

which hold for each n 2 N, yield the limit:

By contraposition, assume that there exists an extracted sequence, f(z

To simplify, we assume w.l.o.g. that '(n) = n for every n 2 N, and we let := sup ke 0 k > 0, for e 0 := (e 0 i ) 2 E 1 . From the de…nition, for every n 2 N, the matrix V n of sub-Section 5.2 is identical to V in all rows except those of states s 2 S, that is,

Then, for every n 2 N, the de…nition of ( i ), the budget constraints and market clearing conditions of the equilibrium, C n , yield, in both sub-Sections 4.2 and 5.2: 31 Documents de travail du Centre d'Economie de la Sorbonne -2017.36R (Version révisée)

For every (i; n) 2 I N, let z 0n i := z ?n i kn . The bounded sequence f(z 0n i )g admits a cluster point, (z i ), such that k(z i )k = 1. The above relations and Lemma 2 yield:

, and

i , when passing to the limit.

The structure ( i ) 2 AS is arbitrage-free, along De…nition 2, above. The latter relations, imply z i 2 Z i \ Z ? i = f0g, from De Boisde¤re's (2016) Claim 2 and above, for each i 2 I. This contradicts the fact that k(z i )k = 1. It follows that the sequence, f(z ?n i )g, is bounded and may be assumed to converge, say to (z ? i ), and the above relations, (

, for all n 2 N, pass to the limit, that is, (

Assertion (iii) is obvious from the de…nitions and Assertion (ii).

Lemma 4 Let B i (!; z) = fx 2 R L + : p (x e is ) 6 V (!) zg, be given sets, for every z 2 R J and all ! := (s; p) 2 i . Along Lemma 3, the following Assertions hold for all i 2 I:

Proof Assertion (i) Let i 2 I be given. We denote simply

for each n 2 N, the equilibrium chosen in either sub-Sections 4.2 or 5.2.

To simplify notations, we henceforth let $ := (!; z) for every (!; z) 2 i R J , we let $ i := (!; z ? i ), for every (i; !) 2 I i and $ n i := ( n i (!); z ?n i ), for every (i; !; n) 2 I i N.

We recall that in sub-Section 5.2, the relation V n (!) = V (!) holds, and we notice that B i (!; z) = B i (!; z ? ), for every (n; !; z) 2 N n i R J , where z ? is the orthogonal projection of z on Z ? i := fz 2 R J : V (!) z = 0; 8! 2 i g ? , whereas B i (!;

For every (!; n) 2 i N, the fact that C n is an equilibrium of E n (or E n ( i ) ) and Assumption A2 imply:

Let R be the subset of i R J upon which the correspondence

for n 2 N, pass to the limit and yield a continuous map,

Assertion (ii) Let i 2 I be given and x i 2 X i be de…ned from above. By the same token (with same notations as above), we let

be de…ned continuous on its domain. The continuity of u i implies that of U i :

(x 0 ; $) 2 R L + R 7 ! u i (x 0 ; ' i (x 0 ; $)). Moreover, the relations (x i0 ; $ i ) = lim n!1 (x n i0 ; $ n i ), u i (x i0 ; x i! ) = U i (x i0 ; $ i ) and u i (x n i0 ; x n i n i (!) ) = U i (x n i0 ; $ n i ) hold, for every (!; n) 2 i N. Then, Lemma 2 and the uniform continuity of u i and U i on compact sets, yield:

(I) 8" > 0; 9N 2 N : 8n > N; 8! 2 i , j u i (x i0 ; x i! ) u i (x n i0 ; x n i n i (!) ) j < ".

Moreover, we recall the following de…nitions, for every n 2 N:

Then, Assertion (ii) results immediately from relations


Issam Mjejri

Manuel Gaudon

Giljoo Song

Christine Labrugère

Aline Rougier
email: aline.rougier@icmcb.cnrs.fr

Crystallized V 2 O 5 as Oxidized Phase for Unexpected Multi-Color Electrochromism in V 2 O 3 thick film

Keywords: Electrochromism materials, Vanadium oxides, Polyol synthesis, Doctor Blade, Optical properties

Our e-connected society is eager to develop devices with tunable colors. Electrochromic materials, able to modify their optical properties under an applied voltage, offer a smart solution. In the present study, we have successfully synthesized two vanadium oxide powders from a polyol mediated synthesis and powder suspensions were coated on glass/ITO substrates by doctor blading. The electrochemical and optical properties of the VxOy films are investigated by cyclic voltammetry (CV) coupled with insitu UV-Visible spectroscopy. Both V2O5 and V2O3 films exhibit reasonably good cycling stability, significant reflectance modulation, high optical contrast and good memory effects revealing the unknown EC properties of V2O3. The similar green ⇄ blue ⇄ orange reversible color changes for both vanadium oxides appears suitable for display application. Then, the evolution of the vanadium cation oxidation states and of the structure of V2O5 and V2O3 upon cycling are analyzed by ex-situ XPS and ex-situ XRD (at grazing incident angle). This work highlights a robust and novel scenario upon cycling, nearly the same whatever the raw film composition that shows, for each cycle, the crystallization of V2O5 upon oxidation, followed by amorphization upon reduction.

I.

INTRODUCTION

With the rapid development of clean renewable energy and increasing demands for multi-electrochromic devices, tremendous efforts have been pursued to find alternative materials that modulate several colors depending on redox potential with high performances and low cost. [START_REF] Li | Near-Infrared and Two-Wavelength Electrochromism Based on Nanocrystalline TiO2 Films Functionalized with Ruthenium-Amine Conjugated Complexes[END_REF][START_REF] Cai | Next-Generation Multifunctional Electrochromic Devices[END_REF][START_REF] Yang | Electrochromic Energy Storage Devices[END_REF][START_REF] Liu | Highly Stable and Flexible ITO-Free Electrochromic Films with Bi-functional Stacked MoO3/Ag/MoO3 Structures[END_REF][START_REF] Papaefthimiou | Advanced Electrochromic Devices Based on WO3 Thin Films[END_REF] For many years electrochromic (EC) materials have been of great academic and commercial interest and recognized as one of the key "green" technologies for sustainability and energy savings in building environments. [START_REF] Bucha | Review on Electrochromic Property for WO3 Thin Films Using Different Deposition Techniques[END_REF][START_REF] Granqvist | Oxide electrochromics: An Introduction to Devices and Materials[END_REF] Over the past decades, tungsten oxide (WO3), that switches to blue upon reduction, has been extensively studied due to its outstanding electrochromic properties. [START_REF] Dalavi | Efficient Electrochromic Performance of Nanoparticulate WO3 Thin Films[END_REF][START_REF] Chu | Novel WO3-Based Electrochromic Device for High Optical Modulation and Infrared Suppression[END_REF][START_REF] Danine | Room Temperature UV Treated WO3 Thin Films for Electrochromic Devices on Paper Substrate[END_REF] It is currently used in most of the commercial applications based on the modulation of transmittance including in particular smart windows. Aiming at enlarging the range of available colors well suitable for displays applications, intensive researches have been focusing on alternative materials, including polymers [START_REF] Beaujuge | The Donor-Acceptor Approach Allows a Black-to-Transmissive Switching Polymeric Electrochrome[END_REF] or more recently Metal Organic Framework. [START_REF] Mjejri | Double Sided Electrochromic Device Based on Metal-Organic Frameworks[END_REF] In the oxide family, V2O5 has come up as a promising electrochromic material for device application because of its multi-color reported in various electrolytes. [START_REF] Mjejri | Nano-Vanadium Pentoxide Films for Electrochromic Displays[END_REF][START_REF] Talledo | Electrochromic Vanadium-Pentoxide-Based Films: Structural, Electrochemical, and Optical Properties[END_REF][START_REF] Cogan | Optical Properties of Electrochromic Vanadium Pentoxide[END_REF][START_REF] Kovendhan | Prototype Electrochromic Device and Dye Sensitized Solar Cell using Spray Deposited Undoped and 'Li' Doped V2O5 Thin Film Electrodes[END_REF][START_REF] Lee | Improving the Durability of Ion Insertion Materials in a Liquid Electrolyte[END_REF] Besides, its abundance and layered structure facilitate the exchange of large amount of Li cations. [START_REF] Wang | Nanostructured Vanadium Oxide Electrodes for Enhanced Lithium-Ion Intercalation[END_REF][START_REF] Wang | Li + -Intercalation Electrochemical/Electrochromic Properties of Vanadium Pentoxide Films by Sol Electrophoretic Deposition[END_REF][START_REF] Yao | Revitalized Interest in Vanadium Pentoxide as Cathode Materials for Lithium-ion Batteries and Beyond[END_REF] Vanadium cation exists in V 2+ , V 3+ , V 4+ and V 5+ states in vanadium oxide compounds that are categorized as single and mixed oxides based on Magneli (VnO2n-1) and Wadsley (V2nO5n-2) series. [START_REF] Garcia-Lobato | Elucidation of the Electrochromic Mechanism of Nanostructured Iron Oxides Films[END_REF] Indeed, the multivalent nature of vanadium cation associated with distinct colors earns it the name of the Norse Goddess of beauty, Vanadis. Meanwhile, the coexistence of various oxidation numbers offers a drastic challenge to prepare with a stoichiometric target only one single phase, especially as vanadium oxide thin or thick films. The uses of vanadium dioxide: VO2 and vanadium sesquioxide: V2O3 are more popular than other VxOy forms due to their unique reversible metal-insulator phase transition (MIT). [START_REF] Gonçalves | Smart Optically Active VO2 Nanostructured Layers Applied in Roof-Type Ceramic Tiles for Energy Efficiency[END_REF][START_REF] Wu | Design of Vanadium Oxide Structures with Controllable Electrical Properties for Energy Applications[END_REF] V2O3 phase is known for its phase transition coupled with a magnetic disordering versus temperature. [START_REF] Wu | Design of Vanadium Oxide Structures with Controllable Electrical Properties for Energy Applications[END_REF][START_REF] Santulli | Synthesis and Characterization of V2O3 Nanorods[END_REF] In this work, it will be shown that, next to the magnetic properties, the V2O3 phase is characterized by unexpected electrochromic properties further proving the high diversity of the physico-chemical properties of vanadium oxides. On the contrary, due to its stability and great properties, V2O5 has been mentioned many times in literature as positive electrode for lithium/sodium batteries or optical batteries (electrochromic device). However, despite the diversity of papers, dealing with the electrochromic properties of V2O5 films, explanations on the observed preferential crystallographic orientation and the origin of the color changes remain a subject of debate. [START_REF] Quinzeni | Morphological and Electrochemical Properties of Nanocrystalline V2O5 Thin Films Deposited by Means of Radiofrequency Magnetron Sputtering[END_REF] Besides, V2O5 is one of the only materials that clearly shows both anodic and cathodic colorations as a result of the occurrence of various oxidation states associated with different colors. [START_REF] Li | Improved Electrochromic Performance in Inverse opal Vanadium Oxide Films[END_REF] Indeed, depending on the vanadium oxidation number, V2O5 based electrochromic films exhibit several colors, thus offering the possibility of obtaining multicolor displays.

Several synthetic ways such as microemulsion-mediated systems, arc discharge, laser-assisted catalysis growth, solution, vapor transport, and solvothermal and hydrothermal methods have been successfully explored to fabricate various kinds of nanostructured vanadium oxides and their derived compounds. [START_REF] Niu | Synthesis and Optical Property of Size-Tunable Vanadium Oxide Nano-Dandelions[END_REF][START_REF] Chu | Hydrothermal Synthesis of Vanadium Oxide Nanorods and their Electrochromic Performance[END_REF][START_REF] Kamalam | Solvothermal Synthesis and Characterization of Reduced Graphene Oxide/ Vanadium Pentoxide Hybrid Nanostructures[END_REF] In this paper, we use a simple method for the synthesis of various submicronic particles vanadium oxides with valence state controlled.

The electrochromic performance of vanadium oxide depends on several parameters including film thickness, morphology and porosity and the deposition method. Various physical and chemical techniques such as, thermal evaporation, electron beam evaporation, magnetron sputtering, sol-gel, electrochemical deposition and pulsed laser ablation have been used for the deposition of vanadium oxide films. [START_REF] Berezina | Vanadium Oxide Thin Films and Fibers Obtained by Acetylacetonate Sol-Gel Method[END_REF][START_REF] Kawashima | Investigation on a Source of Dominant Donor in Vanadium-Doped ZnO Films Grown by Reactive RF Magnetron Sputtering[END_REF][START_REF] Huotari | Pulsed Laser Deposited Nanostructured Vanadium Oxide Thin Films Characterized as Ammonia Sensors Actuator[END_REF] Among several deposition techniques, Doctor Blading presents the advantage of high versatility, low cost, reasonable deposition rates and simplicity in experimental parameters modulation.

In this work, based on our recent study, [START_REF] Mjejri | Low-Cost and Facile Synthesis of the Vanadium Oxides V2O3, VO2, and V2O5 and Their Magnetic, Thermochromic and Electrochromic Properties[END_REF] we report a robust synthetic methodology that provides access to stoichiometrycontrol of the final films (V2O5 or V2O3). The polyol route used to synthesize the pure vanadium oxides is low cost and easy transferable in large scale production.

In a second part, the electrochromic properties of the Doctor Blading V2O3 and V2O5 thick films are characterized by cyclic voltammetry in ionic liquid combined with in-situ optical measurements and ex-situ XRD after cycling, and compared. Several redox steps are observed and discussed, which gives rise to a variety of color transitions as a function of the applied voltage. Moreover, the optical properties of the vanadium oxide films assigned to the different oxidation-reduction steps upon electrochemical cycling are characterized by ex-situ X-ray photoelectron spectroscopy measurements.

Finally, consisting of the main discovery issued from these studies, the very surprising similarities of the electrochromic behaviors of V2O3 and V2O5 raw films will be analyzed and deeply discussed thanks to grazing incidence X-ray diffraction.

II. RESULTS AND DISCUSSION

II.1. Characterization of V2O3 and V2O5 powders

The XRD patterns of the V2O3 and V2O5 powders synthesized by the polyol process as described in the experimental section (IV) and supporting information, are shown in Figure 1. It is obvious that the crystalline phases for vanadium oxides are discriminatory depending on the gas nature used in the thermal treatment, either Ar/H2 for V2O3 or air for V2O5. No secondary phases are found in these patterns, and the sharpness of the diffraction peaks highlights the highly crystalline nature of the powders. Indeed, the diffraction pattern of the powder annealed under 95% Ar/5% H2 gas mixture (Figure 1a), shows the presence of narrow peaks, suggesting a material with high crystallinity. The d-spacing values of all diffraction peaks are identical to those of the rhombohedral crystalline phase V2O3 with (00-034-0187 JCPDS data file; R-3c space group). No peak of any other phase or impurity was detected from the XRD pattern indicating that V2O3 with high purity can be synthesized via polyol synthesis at 160 °C for 1 h followed by an annealing at 500 °C under 95% Ar/5% H2 gas mixture.

The X-ray pattern of the powder annealed under air (Figure 1b) shows that all the peaks are perfectly indexed to the orthorhombic V2O5 with Pmmn space group. All of the isolated phases are well-crystallized; nevertheless, despite the same temperatures and dwell times used for the annealing, the peak widths vary with the vanadium oxidation state: they increase as the valence state decreases.

To complete the structural studies, refinements using two models are compared: full-pattern matching using an isotropic peak profile function (Caglioti function) and full-pattern matching using an anisotropic peak profile function with a crystallographic direction corresponding to the (110) axis for both compounds. By comparison of the results obtained with the two full-pattern matching, the crystallite shapes are clearly anisotropic with an elongation along the (110) crystallographic axis. The use of an anisotropic model allows a significant decrease of the reliability factors Rp, Rwp, and Rexp. The unit cells and reliability factors of the anisotropic refinement for V2O3 and V2O5 are gathered in Table 1. The TEM images (Figure 3) show that the oxides consist of submicronic crystallites aggregated in packs. In good agreement with the narrower diffraction peak widths (Figure 1b), the V2O5 crystallites exhibit larger diameter. The average crystallite diameter can be roughly estimated as ∼100-150 nm for V2O5 and decreases to about 30 nm for V2O3.

II.2. From V2O3 and V2O5 powders to thick films

The SEM top-surface images of V2O3 and V2O5 films are reported in Figure 4a and4b, respectively. The morphology of the V2O3 film is homogeneous, without any large aggregates or agglomerates. The XRD pattern of the as deposited V2O3 film is similar to the one of the powder (Figure S1); thus the crystallite size and shape of the V2O3 powder appear not modified by the film deposition Doctor Blading process. To support this assertion, it can be added that the X-ray diffraction pattern of the V2O3 film can be refined using an isotropic peak profile function, leading to roughly the same parameters and peak profile as for the starting powder. On the contrary, a drastic evolution of the morphology from powder to film is observed for V2O5. It is obvious that the deposition method plays an important role in modifying the surface morphology of the vanadium oxide films. Indeed, the V2O5 film consists of platelets arranged parallel to the substrate (Figure 4b). For the V2O5 film, the X-ray diffraction pattern (Figure 4c) reveals that the d-spacing values of the diffraction peaks match with the ones of the orthorhombic crystalline phase and the ones of the ITO sub-coating. A clear preferential orientation is shown. Indeed, the main peaks located at 2θ = 20.2° and 40.0° corresponding to the (001) and (002) miller indexes, are of very large intensity, which indicates that the a-b plane of the film is parallel to the substrate (i.e. with c-axis as preferential orientation). From the X-ray pattern refinement using anisotropic peak profile function, the crystallite sizes along (001) and (hk0) directions are calculated of about 1.2 µm and 20 nm, respectively. The remarkable anisotropy of the V2O5 film's crystallites, may be due to surface dissolutionrecrystallization of the crystallites into the suspension before the Doctor-Blade coating. A phenomenon of surface dissolution-recrystallisation of the V2O5 particles into the asprepared inks for deposition involves that a change of stoichiometry between vanadium and oxygen must not be excluded, especially near the particles' surface.

II.3. Electrochromic properties of V2O3 and V2O5 films

The electrochemical stability is one of the key parameters to be taken into account when aiming at applications. Electrochemical properties of V2O3 and V2O5 films coated on ITO / glass substrate were evaluated for comparison by cyclic voltammograms (CV) followed by ex-situ optical measurements. As discussed below, the repetitive cycling of V2O3 and V2O5 thick film in ionic liquid (LiTFSI-EMITFSI) at a scan rate of 20 mV/s shows good cyclability and stability up to 500 cycles. It should be emphasized that to our knowledge such cyclability of V2O3 thick films has not been reported in the literature yet.

Figure 5a displays the 1 st , 5 th and 500 th CV of V2O3 film cycled in ITO/V2O3/LiTFSI-EMITFSI/Pt vs SCE cells started in oxidation. The small capacity of the first oxidation is rapidly followed by a reversible cycling. Upon reduction, the voltammograms show a large current wave, constituted of two main peaks located at 0.45 V and -0.47 V while in oxidation, two main peaks located at 0.51 V and 0.93 V are visible. The reversibility of the process is illustrated by a Coulombic efficiency defined as a Qred/Qox ratio keeping close to 100% (Figure S2a), while a slight increase in capacity is observed with cycling. The colors reached by the films at different voltages measured by diffuse reflectance in the range of 400 -800 nm are presented in Figure 5b. The V2O3 films switch reversibly in-between a reduced blue state (-0.9 V) and an oxidized-orange state (+1.9 V) associated with reflectance values less than 5% and of 35% at 600 nm, respectively, leading to an optical reflectance modulation, ΔR, of about 30%. The repetitive cycling of V2O5 films using similar conditions shows also good cyclability and stability in terms of capacity up to 500 cycles (Figure 6a). CVs of V2O5 film cycled in ITO/V2O5/LiTFSI-EMITFSI/Pt vs SCE cells exhibit a large redox current wave constituted of different convoluted peaks, during both the positive and cathodic scans, however slightly shifted to higher potential values as compared to the ones of V2O3. Furthermore, as for V2O3 films, and in agreement with a clear reversible process, the charges deduced from V2O5 cyclic voltammetric measurements in reduction and in oxidation are equal (Qred/Qox=99%) whatever the cycle number. Upon cycling the capacity significantly increases, of about 10 % in between the very first cycles and the 500 th cycle (Figure S2b). In respect to the optical behavior, the lithium insertion/ deinsertion process is accompanied by a clear three-step electrochromism, deep blue ⇄ green ⇄orange. The V2O5 films switch reversibly in-between a reduced blue state (-0.9 V) and an oxidized-orange state (+1.9 V) associated with reflectance values of about 5% and 55% at 600 nm, respectively, leading to an optical reflectance modulation, ΔR, of about 50% (Figure 6b). For the blue, reduced state (at -0.9 V), the calculated L*a*b* parameters are 31, -14, and -9, respectively, while for the orange, oxidized state (at +1.9 V) the L*a*b* parameters are 71, 12 and 48, respectively. The contrast ΔE* is of 74. Interestingly, if similar ΔE* contrast values, of about 75 to 80 % are recorded for both oxides, the higher electrochemical capacity associated with this orange to blue transition for the V2O3 thick film as compared to the V2O5 one emphasized its good EC properties. Commonly evaluated by the color efficiency, CE, for transmisttive systems (i.e. CE = 1/Q log (Tb/Tc) where Q is the electrochemical capacity, Tb and Tc the transmittance in the bleached and colored state). In the case of thick films, the modulation of ΔE* vs Q may be indeed considered as a criteria of merit in favor of V2O3. In addition, the evaluation of the switching times from chronoamperometry measurements confirms the slightly higher properties of V2O3. Both oxides exhibit switching times of few seconds in between orange (+ 1.9 V) to blue (-0.9 V) colors, namely 7s (oxidized orange) and 5s (reduced blue) for V2O3 and 9s (oxidized orange) 6s (reduced blue) for V2O5. The memory effect, namely persistence of the colored state after the voltage being cut, is a key characteristic of electrochromic materials. In this work, both oxides show good memory effect after long times (30 days). All the films in different states (blue and orange) for both oxides were exposed in air at room temperature, and the stability of the films was tested after 30 days with no electrical voltage applied. To illustrate this stability, the L*a*b parameters for blue-reduced state (-0.9V) and orange oxidized state (+1.9V) in initial state and after 30 days were determined and are summarized in Table S1 and Table S2. The analysis of the L*a*b parameters for both oxides at the initial state and after one month shows that there is no change in chromaticity parameters after 30 days. This demonstrates that V2O3 and V2O5 films possess great electrochromic color memory properties with excellent color stability at the oxidized/reduced state. Their good electrochromic color memory properties may be ascribed to the intrinsic stability and their porosity of Doctor Blading V2O3 and V2O5 films that help the electrolyte ions insertion/deinsertion into the film to stabilize the reduced and oxidized state by Coulomb force. The close EC behavior for both V2O3 and V2O5 films, exhibiting different morphologies, structures and compositions (i.e. different vanadium oxidation states) raises the question of the mechanism at the origin of the multichromism.

II.4. Mechanism

From what is generally accepted in V2O5, Li insertion and deinsertion takes place following the V2O5 + xLi + + xe -= LixV2O5 reaction. Thus upon reduction the simultaneous injection of electrons and Li + ions into the V2O5 film leads to a progressive reduction of pentavalent vanadium (V 5+ ) to its lower valence state (V 4+ ) and eventually to (V 3+ ) [START_REF] Tong | From Amorphous Macroporous Film to 3D Crystalline Nanorod Architecture: A New Approach to Obtain High-Performance V2O5 Electrochromism[END_REF] . In oxidation, the extraction of electrons and Li + ions from LixV2O5 film leads to the oxidation of lower valence vanadium (V 4+ or V 3+ ) to its pentavalent state (V 5+ ).

The number of exchanged electrons (x) involved during each coloration/bleaching process was estimated using the following equation: [START_REF] Mjejri | Nano-Vanadium Pentoxide Films for Electrochromic Displays[END_REF] Q = F*nV2O5*x, where Q is the electrochemical capacity, F = 96500 C, and n is number of moles, assuming a density of 75 %. The x values calculated from V2O5 films is about 1.1. The number of exchanged electrons remains much lower than to the expected one if V 5+ ions are fully reduced to V 4+ or even V 3+ ions.

The electrochemical behaviors of V2O5 and V2O3 films do not appear as different as it could be expected. The oxidation of V2O3 during the first half cycle can only be explained by oxygen anion sources. Nonetheless, the electrochromic behavior of V2O3, on the point of view of the change of colorations, is similar to other reports of other vanadium oxides with a threestep mechanism implying a blue to green color first change and then a green to orange color change (in oxidation scan).

The multi-step electrochromism from orange to green and green to blue, during the reduction of V2O5 films (with reverse phenomenon during oxidation) was previously observed by several authors. [START_REF] Tong | From Amorphous Macroporous Film to 3D Crystalline Nanorod Architecture: A New Approach to Obtain High-Performance V2O5 Electrochromism[END_REF][START_REF] Tong | Novel Morphology Changes From 3D Ordered Macroporous Structure to V2O5 Nanofiber Grassland and its Application in Electrochromism[END_REF][START_REF] Tong | Improved Electrochromic Performance and Lithium Diffusion Coefficient in Three-Dimensionally Ordered Macroporous V2O5 Films[END_REF][START_REF] Lu | Mechanism of Electrochemical Deposition and Coloration of Electrochromic V2O5 Nano Thin Films: an In Situ X-Ray Spectroscopy Study[END_REF] It can be associated, as we proposed here, to a double oxidation number change (V 5+ to V 4+ and then V 4+ to V 3+ ). In this way, Tong et al. [START_REF] Tong | From Amorphous Macroporous Film to 3D Crystalline Nanorod Architecture: A New Approach to Obtain High-Performance V2O5 Electrochromism[END_REF] recently ascribed the color change to olive green to the presence of V 3+ ions, associating the multiple colors in V2O5 to the presence of V 3+ /V 4+ and V 5+ . Such observations well agree with the color of V 3+ , V 4+ , and V 5+ in aqueous solutions: i.e., in an octahedral ligand field with aqua ligands. In the meantime, current investigations in our group on V2O5 thin films grown by RF sputtering confirm similar color changes of orange to green and green to blue upon cycling while only 1.5 Li are being exchanged with the single presence of V 5+ and V 4+ . In addition, similar multicolor electrochromism has been detected on amorphous VOx films, with x between 1.5 and 2. Nonetheless some authors [START_REF] Livage | Optical and Electrical Properties of Vanadium Oxides Synthesized From Alkoxides[END_REF][START_REF] Alsawafta | Improved Electrochromic Properties of Vanadium Pentoxide Nanorods Prepared by Thermal Treatment of Sol-Gel Dip-Coated Thin Films[END_REF][START_REF] Costa | Electrochromic Properties of Inkjet Printed Vanadium Oxide Gel on Flexible Polyethylene Terephthalate/Indium Tin Oxide Electrodes[END_REF] have shown that the occurrence of a single V 5+ /V 4+ redox couple, can explain the green intermediate coloration by a potential range where the vanadium ions with the two oxidation numbers coexist with a well-defined ratio. Also, the electrochemical capacities we have measured, close to 1 electron only per cycle, tend to show that (i) a redox process with only oxidation numbers involving a change of only one unit occurs, or (ii) only a fraction of about 25% of the film is submitted to the redox process. To study, in our case, the oxidation number really implied in the color changes observed in both V2O3 and V2O5 films, i.e to investigate what are the different amount mixtures of V 5+ , V 4+ , and V 3+ ions, during cycling, ex-situ XPS spectroscopy, for different applied voltages, was performed. The V2p3/2 core peak spectra of the film in different potential states corresponding to the distinct colorations: for the as-prepared films, the orange state (oxidized state after 50 cycles), the green (intermediate state after 50 cycles) and the blue one (reduced state after 50 cycles), reached during electrochromic recording were analyzed (Figures 7 and8), and the corresponding fitting results are shown in Table 2, for both V2O3 and V2O5 films. In the potential range from +1.9V to -0.9 V, only two V2p3/2 core peaks are identified. The orange coloration being associated with the occurrence of a very main peak near 517.5 eV, this peak can be without any ambiguities ascribed to the V 5+ ion. [START_REF] Chalker | Fabrication and Electrochemical Performance of Structured Mesoscale Open Shell V2O5 Networks[END_REF] The second peak centered on 516 eV is, based on literature on the previous VOx measurements, ascribed to the V 4+ ions. [START_REF] Xiong | Fabrication of Silver Vanadium Oxide and V2O5 Nanowires for Electrochromics[END_REF] But, this last attribution can be contested and possibly, this peak is the convolution of the signals characteristic of both V 3+ and V 4+ ions. Indeed, it can be noted that none occurrence of a distinguishable V 3+ peak for the V2O3 as-prepared film is surprising. On the other hand, and interestingly, V 4+ ions which are detected in the as-prepared film may be derived from the partial dissolution of the V2O5 film which was strongly suggested by the observation of a change of the crystallite morphology between the film's crystallites and the powder ones, as previously discussed. The existence of V 4+ in the as-prepared film also explains the observed "brownish" coloration of the as-prepared film in comparison with the brilliant orange color reached in oxidative regime. Anyway, it can be supposed that the oxidation numbers measured from XPS spectroscopy are not fully representative of the oxidation numbers in the film's bulk. Surface redox processes may occurr before the analysis. So, the attribution of the detected peak to the V 5+ and V 4+ ions as representative of our film composition cannot be made without uncertainties. Whatever, the difficulty to ascribe the different XPS core peaks to well-defined oxidation states, the evolution during cycling of the two different compositions studied and their comparison put into evidence remarkable similarities. The R=V 5+ /V 4+ ratios evolve quite similarly versus the applied potential for both V2O3 and V2O5 films. For a green coloration at -0.3 V, the V 5+ /V 4+ ratio is in favor of the V 5+ ion, but with a significant minor peak of V 4+ signal; whereas this ratio is roughly around the unit, i.e. showing an equilibrate partition between V 5+ and V 4+ ions, when the blue color is obtained at -0.9 V. For the return sweep, in oxidative regime at 1.9 V, leading to an orange color, it is found that the V 5+ ions are at the origin of nearly the whole XPS signal. From the ex-situ XPS analysis it seems that the multi-electrochromism showed in our VxOy films, whatever the starting chemical composition, is due to switches between the vanadium states +4 and +5. Especially, and without any ambiguities on this important suggestion, the oxidation of the vanadium oxide films, whatever their initial composition/structure, leads to a complete vanadium oxidation, it means that a pure V2O5 orange compound is reached.

To confirm the most significant result obtained by XPS, that is the complete oxidation of the vanadium species to oxidation state equal to 5 while an oxidative potential of 1.9 V is applied, starting from V2O5 as well as V2O3 film composition, X-ray analyses were performed (Figures 9 and10). The X-ray patterns were recorded with the same grazing incidence (2°) on both V2O3 and V2O5 films for the four states of coloration previously discussed: black or brown, respectively (as-deposited), green (-0.3V), blue (-0.9V) and orange (+ 1.9V). The measurements were realized after 50 cycles.

In Figure 9, all the peaks of the as-deposited V2O3 with black coloration are perfectly indexed to V2O3 phase without any preferential orientation (Figure 9a); the X-ray pattern for the film and the powder (Figure 1a) can be superimposed. For the two potentials corresponding to the green and the blue colors (Figure 9b and9c), one can note the absence of the characteristic peaks of any VOx phase: the film becomes amorphous; the peaks, which are observed, are due to the ITO sub-layer. In oxidative regime (+1.9 V) (Figure 9d), leading to an orange coloration, the additional peaks observed on the Xray pattern, besides the ITO's peaks, can be indexed as well crystallized orthorhombic V2O5 (SG: Pmmn).

In Figure 10, the X-ray patterns show that for as-deposited V2O5 with brownish coloration, all the peaks are perfectly indexed to V2O5 phase with a preferential orientation along the c-axis while for the reduction (to green and blue colors), only the peaks associated with ITO are observed. As previously, well-crystallized V2O5 is reformed upon oxidation.

These results are in agreement with the ones obtained by XPS; especially, V2O5 single phase in oxidation is confirmed. Indeed, whatever the starting composition, the X-ray pattern for the re-oxidized film after 50 cycles reveals that all peaks are identified to the vanadium pentoxide with high crystallinity. From ex-situ XRD analysis, it can be concluded that during cycling vanadium oxides pass through amorphous state during the reduction and that this state should be characterized by a mixture of V 4+ and V 5+ . During the return sweep, all the vanadium ions are fully oxidized into V 5+ state, in agreement with the orange color. More surprisingly, the complete reoxidation, at the end of each cycle, allows each time the recrystallization of V2O5. Finally, the kinetic of the phenomenon was studied on the as-prepared V2O3 film, comparing the X-ray pattern recorded for an applied voltage of +1.9 V, after 5 cycles and 50 cycles (Figure S3). This comparison shows that even after 5 cycles, V2O5 single phase is already obtained. However, the strong preferential orientation of the film drastically increases between the 5 th and the 50 th cycle. This observation confirms a reorganization of the atomic matter all along the cycling processes, and can justify the slight coulombic efficiency increase observed for the V2O3 starting film with the number of cycles.

III. CONCLUSION

In this study, we show a robust scenario that provides access to stoichiometry-controlled and which leads to: (i) the synthesis of two oxides V2O3 and V2O5 from a single polyol route, (ii) the demonstration for the first time of the electrochromic properties of V2O3, (iii) a "multi-color" electrochromic behavior very similar while one compares V2O3 and V2O5 (i.e. independently of the starting stoichiometry), which is associated with an unusual recrystallization process of V2O5 in oxidative regime. The latter represents a great step forward for the understanding of the electrochromic properties of vanadium oxides. Moreover, it is also a progress in solid state chemistry since a new synthesis method for obtaining well-crystallized V2O5 films, with a strong preferential orientation, can be proposed from the oxidation of any VOx films.

IV. EXPERIMENTAL SECTION

A detailed description of all the experimental procedures and techniques characterization is given in the supporting information (SI).

ASSOCIATED CONTENT

Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: It contains a detailed description of all the experimental procedures and techniques of characterization including the XRD of V2O3 powders and film, and ITO substrate and coulombic efficiency and its evolution during cycling for V2O3 and V2O5 films cycled in ITO/V2O3 (V2O5)/LiTFSI-EMITFSI/Pt vs SCE cell. We also determine the L*a*b* parameters in the blue-reduced state and orange-oxidized state of V2O3 and V2O5 at initial state and after 30 days and also the ex-situ XRD of V2O3 film before cycling, after 5 cycles and 50 cycles in oxidized state.

FIGURE 1 . 7 SEMFIGURE 2 .

 172 FIGURE 1. Full-pattern matching using an anisotropic peak profile of V2O3 (vanadium sesquioxide; R3̅ c space group) powders (a) and V2O5 (vanadium pentoxide; Pmmn space group) powders (b). TABLE 1. Unit-cells parameters and R reliability factors extracted from the pattern matching refinement of V2O3 and V2O5 powders. a (Å) b (Å) c (Å) R factors

FIGURE 3 .

 3 FIGURE 3. TEM Images of V2O3 (a, b) and V2O5 (c, d) powders.

FIGURE 4 .

 4 FIGURE 4. SEM images of V2O3 film (a) and V2O5 film (b); fullpattern matching using an anisotropic peak profile for V2O5 film (c) deposited by Doctor-Blade method.

FIGURE 5 .

 5 FIGURE 5. Cyclic voltammograms for V2O3 film cycled in ITO/V2O3/LiTFSI-EMITFSI/Pt vs SCE (a). ex-situ diffuse reflectance spectra of the V2O3 films at various potentials at 50 th cycle : black initial state (as-deposited), oxidized orange state (+1.9V), reduced green state (-0.3V) and more reduced blue state (-0.9V) (b). In CIE colorimetric space, the color is represented by three parameters, the luminance axis (L*) and two hue axes (a*) and (b*), which can be used to define and compare quantitatively the colors. The relative luminance (L*), the hue (a*) and (b*) values of V2O3 films in different states were calculated from the reflectance spectra. For the light blue-reduced state (at -0.9 V), the L*a*b* parameters are 26, -15, and -6, respectively, while for the bright orange-oxidized state (at +1.9 V) the L*a*b* parameters are 68, 11 and 56, respectively. The contrast ΔE* = [(L*2 -L*1)² + (a*2 -a*1)² + (b*2 -b*1)²] ½ is of 79.The repetitive cycling of V2O5 films using similar conditions shows also good cyclability and stability in terms of capacity up to 500 cycles (Figure6a). CVs of V2O5 film cycled in ITO/V2O5/LiTFSI-EMITFSI/Pt vs SCE cells exhibit a large redox current wave constituted of different convoluted peaks, during both the positive and cathodic scans, however slightly shifted to higher potential values as compared to the ones of

FIGURE 6 .

 6 FIGURE 6. Cyclic voltammograms for V2O5 film cycled in ITO/V2O5/LiTFSI-EMITFSI/Pt vs SCE (a). ex situ diffuse reflectance spectra of the V2O5 films at various potentials at 50 th cycle : initial state (as-deposited), oxidized orange state (+1.9V), reduced green state (-0.3V) and more reduced blue state (-0.9V) (b).

FIGURE 7 .

 7 FIGURE 7. ex-situ XPS of V2O3 film before cycling (a) and after 50 cycles at oxidized state (b) reduced green state (c) and reduced blue state (d).

FIGURE 8 .

 8 FIGURE 8. ex situ XPS of V2O5 film before cycling (a) and after 50 cycles at oxidized state (b) reduced green state (c) and reduced blue state (d).

FIGURE 9 .

 9 FIGURE 9. ex-situ Grazing incidence XRD of V2O3 film before cycling (a) and after 50 cycles at oxidized state (b) reduced green state (c) and reduced blue state (d).

FIGURE 10 .

 10 FIGURE 10. ex-situ Grazing Incidence XRD of V2O5 film before cycling (a) and after 50 cycles at oxidized state (b) reduced green state (c) and reduced blue state (d).

TABLE 2 .

 2 V 5+ and V 4+ quantification deduced from ex-situ XPS study after 50 cycles (100 % corresponds to normalization of the strongest contribution).

	V2O3 films	V 5+ (%)	V 4+ (%)	R=V 5+ /V 4+
	As-deposited (Black)	100	48	2.1
	Green color (-0.3V)	100	17	5.2
	Blue color (-0.9V)	87	100	0.9
	Orange color (+1.9V)	100	nearly 0	-
	V2O5 films	V 5+ (%)	V 4+ (%)	R=V 5+ /V 4+
	As-deposited (Brown)	100	9	11.1
	Green color (-0.3V)	100	32	3.1
	Blue color (-0.9V)	100	73	1.4
	Orange color (+1.9V)	100	7	13.8

AUTHOR INFORMATION

Corresponding Author *E-mail: aline.rougier@icmcb.cnrs.fr ORCID Issam Mjejri: 0000-0001-5313-3198 Manuel Gaudon : 0000-0002-6918-2004 Aline Rougier: 0000-0002-1340-734X

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.