Nicolas Pinel

Cédric Le Bastard

Limei Liu

Christophe Bourlier

Yide Wang

Rough thin pavement thickness estimation by GPR

Keywords: Radar scattering, Electromagnetic scattering by rough surfaces, Ground wave propagation, Nondestructive testing, Delay estimation

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INTRODUCTION

Ground penetrating radar (GPR) is a useful means of media sounding, which is widely used at centimeter-scale wavelengths in road surfaces evaluation [START_REF] Dérobert | Step-frequency radar applied on thin road layers[END_REF][START_REF] Lee | A novel, compact, low-cost, impulse ground-penetrating radar for nondestructive evaluation of pavements[END_REF][START_REF] Bastard | Thin-pavement thickness estimation using GPR with high-resolution and superresolution methods[END_REF]. In this context, the roadway is usually considered as made up of perfectly flat stratified interfaces. Then, the vertical structure of the roadway is deduced from radar echo detection and amplitudes estimation. Echo detection provides the time-delay estimation (TDE) associated with each interface, and amplitude estimation is used to retrieve the wave speed within each layer. The case of small pavement thicknesses was studied in a recent paper [START_REF] Bastard | Thin-pavement thickness estimation using GPR with high-resolution and superresolution methods[END_REF].

In general, classical methods of pavement thickness estimation assume flat interfaces for the pavement. Even if this first approximation has sense and is rather realistic, to our knowledge the validity domain of this approximation, and its influence on the electromagnetic backscattering and on the GPR process have not been studied in details. This is the scope of this work. Thus, in this paper, the surface roughness of the pavement is taken into account in the electromagnetic backscattering modeling and in the GPR thickness estimation process, and compared with the case of neglecting the roughness of the pavement.

First, the amplitudes of the first two echoes from the rough thin pavement are calculated with a rigorous electromagnetic method, namely the PILE method [START_REF] Déchamps | Fast numerical method for electromagnetic scattering by rough layered interfaces: Propagationinside-layer expansion method[END_REF]. The frequency behavior of the echoes is then presented in the considered frequency band, f ∈ [1.0; 3.0] GHz, comparatively to the echoes obtained for flat interfaces. Finally, the influence of the pavement roughness on the thickness estimation is investigated by using the Maximum Likelihood Method.

ECHO AMPLITUDES: FREQUENCY BEHAVIOR

In this section, the frequency behavior of the first two backscattered echoes s 1 and s 2 of a rough pavement is presented. To calculate the echoes within the frequency band f ∈ [1.0; 3.0] GHz, the PILE (Propagation Inside Layer Expansion) method [START_REF] Déchamps | Fast numerical method for electromagnetic scattering by rough layered interfaces: Propagationinside-layer expansion method[END_REF] is used. It is a Method-of-Moments based method which is able to compute rigorously each echo reflected by a flat or a rough layer.

Simulation parameters

The pavement under study is an Ultra Thin Asphalt Surfacing (UTAS) of thickness H = 20 mm [START_REF]AFNOR standard NFP 98-137[END_REF], overlying a rolling band of same general composition. It is assumed that the UTAS and the rolling band can be assimilated to homogeneous media at the frequency band under study f ∈ [1.0; 3.0] GHz [START_REF] Bastard | Thin-pavement thickness estimation using GPR with high-resolution and superresolution methods[END_REF][START_REF] Spagnolini | Permittivity measurements of multilayered media with monostatic pulse radar[END_REF][START_REF] Gentili | Electromagnetic inversion in monostatic ground penetrating radar: TEM horn calibration and application[END_REF]. Their relative permittivities typically range between 4 and 8 [START_REF] Adous | EM characterization of bituminous concretes in a large frequency bandwidth: First results[END_REF], and their conductivities between 10 -3 and 10 -2 S/m [START_REF] Fauchard | Utilisation de Radars très hautes fréquences : Application à l'auscultation non destructive des chaussées[END_REF]. For the simulations, their relative permittivities are taken as ǫ r2 = 5 and ǫ r3 = 8, respectively, and their conductivities as σ 2 = 5 × 10 -3 S/m and σ 3 = 10 -2 S/m, respectively. The two rough interfaces Σ A and Σ B are assumed to be described by a Gaussian height probability density function (pdf), and an exponential auto-correlation function [START_REF] Li | Low grazing incidence millimeter-wave scattering models and measurements for various road surfaces[END_REF][START_REF] Koudogbo | Numerical and experimental validations of IEM for bistatic scattering from natural and manmade rough surfaces[END_REF]. For the upper interface Σ A , the root mean square (rms) height σ hA is of the order of 1 mm, and the correlation length L cA of the order of 5 -10 mm. For the lower interface Σ B , the rms height σ hB and the correlation length L cB are a bit greater. For the simulations, the chosen parameters are σ hA = 0.8 mm, L cA = 10.0 mm, σ hB = 1.6 mm, and L cB = 30.0 mm. In practice, the two rough surfaces are only very weakly correlated, so that it can be assumed here that they are statistically uncorrelated.

Concerning the incident wave, a monochromatic incident wave of TM polarization (also called vertical polarization) is considered, with normal incidence onto the pavement, θ i = 0. The emitter antenna is assumed to be in the far-field zone of the ground, so that the incident wave is assumed to be plane. The typical width of the central zone illuminated by the emitter antenna is of the order of 100 -200 mm [START_REF] Liu | Modélisation et Expérimentation Radar Impulsionnel et á Sauts de Fréquence Pour lAuscultation de Milieux Stratifiés du Génie Civil[END_REF]. Then, for the simulations of the numerical method, surfaces of length L = 600 mm will be considered, illuminated by a Thorsos beam of attenuation parameter g = L/6. A normal incident wave (with incidence angle θ i = 0) is taken, and the first two orders of the reflected echoes by the rough layer s 1 and s 2 are calculated under the PILE method. Then, to determine the frequency behavior of the received echoes [START_REF] Bastard | Thin-pavement thickness estimation using GPR with high-resolution and superresolution methods[END_REF]) in the backscattering direction θ s = θ i = 0, it is necessary to run the numerical simulation scenario at different frequencies f within the considered bandwidth, f ∈ [1.0; 3.0] GHz.

s k ≡ s k (f ) (see equation (2) of
The numerical process is described as follows. The rough layer, with two independent rough surfaces, is generated by a Monte-Carlo process (the two rough surfaces being generated from independent processes), for which the calculation of the backscattered signals s 1 and s 2 is led with the PILE. In order to study the variability of the echo amplitudes s 1 and s 2 , several independent Monte-Carlo processes are generated. Thus, it is possible to estimate the standard deviations of the echo amplitudes, and even a profile of their calculated probability density functions if a significant number of realizations is led (typically, of the order of 10000 [START_REF] Dusséaux | Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method[END_REF]). Indeed, for a practical scenario, the illuminated surface area is of the order of 100 -200 mm, which is not large in comparison with the two surface correlation lengths l cA = 10.0 mm and l cB = 30.0 mm. This implies that the received echo amplitudes depend on the location of the pavement where the measurement is made. As a consequence, in order to study the variability of the received echo amplitudes, a significant number of realizations must be generated.

To compute the numerical results, at least 1000 independent realizations of a Monte-Carlo process are generated, in order to simulate the variability of the received echoes. For the simulation of the numerical method, the two rough interfaces are sampled with a sampling step ∆x = λ 2 /10, with λ 2 the wavelength inside the inner medium Ω 2 .

Numerical results

First, numerical simulations are led at a fixed radar frequency f , in the middle of the radar band under study, i.e. f = 2.0 GHz. In order to study the probability density function (pdf) of the echoes s 1 and s 2 , the number of realizations of the Monte-Carlo process is taken as 10000. A comparison is also made between the case with rough interfaces and the case with flat interfaces. Numerical results of the simulated pdfs are plotted in Fig. 1 for the first echo s 1 , and in Fig. 2 for the second echo s 2 . In each figure, the pdf of the real part, the imaginary part, the modulus, and the phase (in degrees) of the echo are represented. The mean value is plotted in red dashed vertical line, and the mean value plus and minus the standard deviation are plotted in purple dashed vertical line. Then, the pdf is compared with a Gaussian pdf having the same mean value and standard deviation in full red line. A comparison is also made with the case of flat interfaces (whose pdf is a Dirac delta function) in green vertical line.

Concerning the first echo s 1 , the imaginary part and the phase do not highlight a significant difference between the flat case and the mean value of the rough case. Moreover, the dispersion around this mean value remains low: for instance, in the phase distribution, the RMS phase is of the order of 1 degree. By contrast, a relatively small but though significant difference occurs in the real part and in the modulus. As expected, the (upper) surface roughness induces a decrease of the echo (real part or modulus) comparatively to the flat case. However, this decrease remains small owing to the small electromagnetic roughness of the surface at this typical frequency. Moreover, the dispersion around this mean value remains low. The general shape of the pdf resembles a Gaussian for the imaginary part and the phase. For the real part and the modulus, the shape differs only slightly from a Gaussian, and is assimilated to a Gaussian as well in first approximation.

The same qualitative observations can be made for the second echo s 2 . In this case, the relative differences between the flat case and the mean value of the rough case are higher, owing to the larger electromagnetic roughness of the layer.

Second, in what follows the frequency behavior of the echoes s 1 and s 2 is investigated in the whole range of the two radar frequency bands under study, i.e. for f ∈ [0.5; 3.0] GHz, for which 1000 Monte-Carlo processes were used. Fig. 3 presents the frequency behavior of the real part of the first two echoes s 1 and s 2 . The flat case is plotted in green full line, the mean value of the rough case in red circled dashed line, the mean value plus or minus twice the standard deviation of the rough case in magenta circled dash-dot line, 4. RRMSE variations on the two estimated time delays T1 and T2 , as well as on the layer thickness Ĥ, vs. the SNR and one realization of the rough case in blue dotted line with plus signs. The results highlight that as the radar frequency increases, the amplitudes of the backscattered echoes s 1 and s 2 decrease, because the layer (electromagnetic) roughness increases relatively to the wavelength. Moreover, for the lower frequencies f ≈ 1 GHz, it can be seen that the difference with the flat case is relatively weak and could be neglected. On the contrary, for the higher frequencies f ≈ 3 GHz, the relative difference with the flat case is significant and cannot be neglected any more, as it exceeds 10 percent for instance for s 2 . Thus, significant differences appear in the backscattered echoes s 1 and s 2 between the rough and flat cases, in particular for the higher frequencies.

Then, let us have a look at the consequences of these differences in the electromagnetic backscattering on the thickness estimation by GPR, with the Maximum Likelihood Method (MLM).

THICKNESS ESTIMATION BY GPR

The process to determine the time delays of the first two echoes is explained in details in [START_REF] Bastard | Thin-pavement thickness estimation using GPR with high-resolution and superresolution methods[END_REF]. To perform time delay estimation (TDE), the MLM is used. An additive complex Gaussian white noise is considered to model the measurement uncertainties and the noise in the instruments. The radar pulse is a ricker pulse, defined as the second derivative of a Gaussian pulse. The data vector is made of 5 samples within the 2 GHz frequency bandwidth (see Fig. 3). The scenario under study is the same as described in the previous section. Thus, the data (i.e., the echo amplitudes s 1 and s 2 ) used to determine the time delays correspond to the realization plotted in blue dotted line with plus signs in Fig. 3. Fig. 4 represents the relative root mean square error (RRMSE) variations on the two estimated time delays T1 and T2 , as well as on the layer thickness Ĥ, vs. the signal-to-noise ratio (SNR), for the frequency band f ∈ [1.0; 3.0] GHz. First, for both flat and rough cases, it can be seen that the RRMSE decreases with increasing SNR. A difference between the flat and rough cases is observable in T1 for SNR higher than 40 dB, in T2 for SNR higher than 25 dB, and in Ĥ for SNR higher than 25 dB. As a consequence, taking the roughness of the surfaces into account makes it possible to increase the performances of the algorithm for moderate to high SNR. Thus, in the context of high SNR, it is important to take the roughness into account in the data modeling to obtain very low RRMSE, and this modeling allows in this case an even better precision of the thickness estimation. On the other hand, for low SNR and/or for a first estimate of the pavement thickness, these results confirm that taking the surface roughness into account is not necessary: this phenomenon can be neglected in this other context, as usually done in many previous studies.

CONCLUSION

In conclusion, taking the surface roughness into account in the pavement thickness estimation by standard GPR of bandwidth of the order of 2 GHz allows us to quantify the classical approximation which considers flat interfaces. Thus, for typical pavements encountered in practice, like here B = [1.0; 3.0] GHz in Fig. 4, the difference between the rough and the flat cases in the backscattered echoes amplitudes is significant for the higher frequencies of the bandwidth. Then, the influence of this difference in the GPR estimation process is significant for moderate to high SNR (typically, 25 dB for the thickness). As a consequence, taking the roughness into account in the data model is of interest, and this modeling allows a better precision in the thickness estimation process.

Fig. 1 .Fig. 2 .

 12 Fig.1. Probability density functions (pdf) of the first echo s 1 (real part, imaginary part, modulus and phase) obtained from 10000 realizations, at a radar frequency f = 2 GHz. The mean value is plotted in red dashed vertical line, and the mean value plus and minus the standard deviation are plotted in purple dashed vertical line. Then, the pdf is compared with a Gaussian pdf having the same mean value and standard deviation. Comparison is also made with the flat case in green vertical line.

Fig. 3 .

 3 Fig. 3. Frequency behavior of the real part of the first two echoes s 1 and s 2

2 ,

 2 Fig. 4. RRMSE variations on the two estimated time delays T1 and T2 , as well as on the layer thickness Ĥ, vs. the SNR


Zaier Aouani
email: aouani@gmail.com

Bernard Cornet
email: cornet@ku.edu

Existence of financial equilibria with restricted participation

Keywords: Restricted participation, portfolio constraints, financial exchange economy, equilibrium, existence, arbitrage-free asset prices JEL: C62, D52, D53

.

Introduction

Since the seminal paper by [START_REF] Radner | Existence of equilibrium of plans, prices, and price expectations[END_REF] proving the existence of equilibria in a financial exchange economy with bounded portfolio sets, and the non-existence issue raised by [START_REF] Hart | On the optimality of equilibrium when the market structure is incomplete[END_REF], [START_REF] Duffie | Stochastic equilibria with incomplete financial markets[END_REF]Shafer (1985, 1986) showed a generic existence result with real assets. An extensive body of literature built upon their argument, see e.g. [START_REF] Geanakopolos | Solving systems of simultaneous equations in economics[END_REF], [START_REF] Hirsch | A geometric approach to a class of equilibrium existence theorems[END_REF], [START_REF] Husseini | Existence of equilibrium with incomplete markets[END_REF] and Bich andCornet (2004, 2009). Subsequently, the literature on the existence problem paid particular attention to incomplete asset markets with only nominal assets or only numéraire assets; this was considered either in the case of unconstrained agents' portfolio holdings, e.g. [START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF][START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF], [START_REF] Werner | Equilibrium in economies with incomplete financial markets[END_REF], or [START_REF] Duffie | Stochastic equilibria with incomplete financial markets[END_REF] for nominal assets and [START_REF] Geanakoplos | Uncertainty, information and communication: Essays in honor of Kenneth J. Arrow[END_REF] for numéraire assets, as well as when agents' participation to financial markets might be restricted.

With restricted participation, in addition to the budget constraint, each agent i faces exogenous portfolio constraints z i ∈ Z i ⊂ R J , where J denotes the (finite) number of assets in the economy. The presence of such portfolio constraints is a natural cause of market incompleteness and allows to capture a wide range of imperfections in the financial markets, such as short selling constraints, collateral requirements, and more generally institutional constraints. [START_REF] Elsinger | Arbitrage and optimal portfolio choice with financial constraints[END_REF] give an extensive discussion of these institutional constraints and how to model them in a general financial framework. The existence problem had recently a growing interest since the first work by [START_REF] Siconolfi | Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets[END_REF], and [START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF][START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF]. Linear equality constraints are considered by [START_REF] Balasko | The structure of financial equilibrium with exogenous yields: The case of restricted participation[END_REF] with nominal assets, and by [START_REF] Polemarchakis | Generic existence of competitive equilibria with restricted participation[END_REF] with real assets. More recently, the case of portfolio sets Z i which are closed, convex subsets containing zero as in [START_REF] Siconolfi | Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets[END_REF] is considered by [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF] for real assets and by [START_REF] Martins-Da-Rocha | Equilibria in exchange economies with financial constraints: Beyond the Cass-trick[END_REF], [START_REF] Hahn | Constrained asset markets[END_REF], and [START_REF] Cornet | Arbitrage and equilibrium with portfolio constraints[END_REF] in the nominal case.

This paper considers a two-date stochastic model (t = 0 and t = 1) of a financial exchange economy with finitely many states of nature, one of which is revealed at t = 1. There is a market for finitely many physical goods at every state today or tomorrow and financial transfers across time and across states are allowed by means of finitely many assets. There are finitely many agents with non-ordered preferences and portfolio constraints described by closed, convex subsets containing zero. Our contribution is twofold. First, when financial assets are nominal or numéraire, we provide a general existence result of equilibria (Theorem 1). Apart from standard assumptions on the consumption side (preferences and endowments), we assume that portfolio restrictions are defined by linear inequality constraints. This extends the framework of linear equality constraints considered by [START_REF] Balasko | The structure of financial equilibrium with exogenous yields: The case of restricted participation[END_REF], and the standard model of unconstrained portfolios. Our existence result generalizes previous work by [START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF], [START_REF] Werner | Equilibrium in economies with incomplete financial markets[END_REF], [START_REF] Duffie | Stochastic equilibria with incomplete financial markets[END_REF], and [START_REF] Geanakoplos | Uncertainty, information and communication: Essays in honor of Kenneth J. Arrow[END_REF].

Our second contribution provides an auxiliary result (Theorem 2) which is the key tool in the proof of Theorem 1. In this auxiliary result, we make an additional "nonredundancy-type" (or "reduced form") assumption (F3 in the text) on the financial side. In the case of nominal assets and no restrictions on portfolio trades (Z i = R J for all i), Assumption F3 is equivalent to the fact that the payoff matrix V has no redundant assets, that is, rankV = J or equivalently ker V = {0}. In this case there is a priori no loss of generality in assuming that there are no redundant assets, otherwise, by deleting the redundant columns we obtain a "reduced" financial economy, whose equilibria yield equilibria in the original one. However, as mentioned in [START_REF] Balasko | The structure of financial equilibrium with exogenous yields: The case of restricted participation[END_REF], one significant source of restricted participation is financial intermediation which typically involves redundancy. So there is no a priori grounds for the standard Full Rank Assumption in the presence of restricted participation, which therefore will be superseded by Assumption F3.

In the case of linear equality portfolio constraints (i.e., the Z i 's are vector spaces), [START_REF] Balasko | The structure of financial equilibrium with exogenous yields: The case of restricted participation[END_REF] show how to transform the agents' financial opportunities to obtain a "reduced" financial economy in which each agent's portfolio choice is a subspace having the same dimension as the wealth space it generates, that is, Z i ∩ ker V = {0} for all i [START_REF] Siconolfi | Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets[END_REF]); moreover, every equilibrium in the "reduced" economy leads to an equilibrium in the original economy. In this paper, we extend the analysis to the case of linear inequality constraints with nominal or numéraire assets. We show how to "reduce" the financial structure to obtain a new financial structure satisfying Assumption F3, a weaker condition than Siconolfi (1989)'s, keeping the correspondence between the equilibria; moreover, Assumption F3 coincides with Siconolfi's when the Z i 's are linear subspaces. Finally, we mention the companion papers Aouani and Cornet (2008a,b) and the paper by [START_REF] Hahn | Constrained asset markets[END_REF], which study the more general case of closed convex portfolio sets.

The paper is organized as follows. In Section 2, we describe the financial exchange economy, we state our main existence result (Theorem 1) in the case of nominal or numéraire assets, and we state the auxiliary result (Theorem 2) under the additional Assumption F3. This section also provides examples under which Assumption F3 and the Financial Survival Assumption are satisfied. In Section 2.6, the proof of our main result (Theorem 1) is given as a consequence of Theorem 2 by "reducing" the initial economy into a new economy satisfying Assumption F3 whose equilibria yield equilibria in the original one. Finally, we also discuss the relationship with the existence results in the literature by [START_REF] Radner | Existence of equilibrium of plans, prices, and price expectations[END_REF], [START_REF] Siconolfi | Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets[END_REF], [START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF][START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF], [START_REF] Werner | Equilibrium in economies with incomplete financial markets[END_REF], [START_REF] Duffie | Stochastic equilibria with incomplete financial markets[END_REF] and [START_REF] Geanakoplos | Uncertainty, information and communication: Essays in honor of Kenneth J. Arrow[END_REF]. Section 3 is devoted to the proof of the auxiliary result (Theorem 2). The Appendix gathers the proofs of some lemmata used in the proofs of Theorems 1 and 2.

The model and the main result

The stochastic financial exchange economy

1 The stochastic model considers two dates: t = 0 (today) and t = 1 (tomorrow). At the second date, there is a nonempty finite set S := {1, . . . , S } of states of nature, one of which prevails at time t = 1 and is only known at time t = 1. For convenience, s = 0 denotes the state of the world (known with certainty) at period 0 and we let S = {0} ∪ S = {0, 1, . . . , S }. At each state, today and tomorrow, there is a spot market for a positive number of divisible physical goods and we assume that the goods are perishable, i.e., each good does not last more than one period. In this model a commodity is a couple (h, s), specifying the physical good h = 1, . . . , and the state 0, 1, . . . , S at which it is available. Thus the commodity space is R L , where L = (1 + S ). An element x (resp. p) in R L is called a consumption (resp. a price) and we will use the notation x = (x(s)) s∈ S ∈ R L , where x(s) = (x 1 (s), . . . , x (s)) ∈ R , denotes the spot consumption at node s ∈ S. 1 We shall use hereafter the following notations. If I is a finite set, whose number of elements is I, the space R I (identified to the space R I of functions x : I → R whenever necessary) is endowed with the scalar product x • y := I i=1 x i y i , and we denote by x := √ x • x the Euclidean norm, B I (x, r) := {y ∈ R I : yx ≤ r}, the closed ball centered at x ∈ R I of radius r > 0. For x = (x i ) and y = (y i ) in R I , the notation x ≥ y (resp. x > y, x y) means that, for every i, x i ≥ y i (resp. x ≥ y and x y, resp.

x i > y i ) and we let R I + = {x ∈ R I : x ≥ 0}, R I ++ = {x ∈ R I : x 0}. Let X ⊂ R I ,
the span of X is the linear subspace of R I , denoted < X >, which is the set of all the K-linear combinations K k=1 α k x k of vectors x k ∈ X for every integer K, and we denote by int X, cl X, respectively, the interior and the closure of X. Consider a I × J-matrix A with I rows and J columns, with entries A j i (i ∈ I, j ∈ J), we denote by A i the i-th row of A (hence a row vector, i.e., a (1 × J)-matrix, often identified to a vector in R J when there is no risk of confusion) and A j denotes the j-th column of A (hence a column vector, i.e., a I × 1-matrix, which may similarly be identified to a vector in R I ). If there is no risk of confusion, we will use the same notation for the I × J-matrix A and the associated linear mapping A : R J → R I . We shall denote by ker A := {x ∈ R J : Ax = 0} the kernel of A, by Im A := {Ax : x ∈ R J } the image of A, and by rank A the rank of the matrix A, that is, the dimension of Im A. We also denote ker A by {A = 0} and we let {A ≥ 0} := {x ∈ R J : Ax ≥ 0}.

In the exchange economy, there is nonempty finite set I := {1, . . . , I} of consumers. Each consumer i ∈ I is endowed with a consumption set X i ⊂ R L , a preference correspondence P i , from k∈I X k to X i , and an endowment vector e i ∈ R L . The set X i is the set of her possible consumptions, and for x ∈ i∈I X i , P i (x) is the set of consumption plans in X i which are strictly preferred to x i by consumer i, given the consumption plans (x i ) i i of the other agents. Finally e i = (e i (s)) S lists the state endowment e i (s) across states, with e i (0) being known with certainty and e i (s) (s 0) being available only if state s prevails at t = 1. The exchange economy can thus be summarized by E = (X i , P i , e i ) i∈I .

The financial structure consists of a finite set J := {1, . . . , J} of assets. An asset j is a contract which is issued at t = 0 and promises to deliver the financial payoff V j s (p) at state s ∈ S of period t = 1 if state s prevails (for a given commodity price p ∈ R L ). So, the payoff of asset j across tomorrow states is described by the mapping p → V j (p) := (V j s (p)) s∈S ∈ R S . The financial structure is described by the payoff matrix mapping V : p → V(p), where V(p) is the S × J-matrix, whose columns are the payoffs V j (p) (1, . . . , J) of the J assets. We denote by z = (z j ) ∈ R J , the portfolio of some consumer and we will use the standard convention: if z j > 0 (resp. z j < 0), then z j (resp. |z j |) represents the quantity of asset j bought (resp. sold) at period 0. We assume that portfolios may be constrained, that is, each agent i is endowed with a portfolio set Z i ⊂ R J which describes the portfolios that are possible for her. If some agent i ∈ I has no constraints on her portfolio choices, then Z i = R J . Throughout this paper we consider portfolio sets that are closed, convex and contain zero for every agent, a framework general enough to cover most of the constraints considered in the literature (see [START_REF] Elsinger | Arbitrage and optimal portfolio choice with financial constraints[END_REF]). The financial structure can be summarized by the list F := (V, (Z i ) i∈I ) and the stochastic financial exchange economy (E, F ) can thus be described by the list (E, F ) := (X i , P i , e i ) i∈I ; (V, (Z i ) i∈I ) .

Equilibria and absence of arbitrage opportunities

Given commodity and asset prices (p, q) ∈ R L × R J , the budget set of consumer i is2

B i F (p, q) = (x i , z i ) ∈ X i × Z i : p(0) • x i (0) + q • z i ≤ p(0) • e i (0) p(s) • x i (s) ≤ p(s) • e i (s) + V s (p) • z i , ∀s ∈ S = {(x i , z i ) ∈ X i × Z i : p (x i -e i ) ≤ W(p, q)z i },
where W(p, q) denotes the total payoff matrix, that is, the (1 + S ) × J-matrix -q V(p) .

We now introduce the standard equilibrium notion in this model.

Definition 1. An equilibrium of the economy (E, F ) is a list p, q, x, z ∈ R L × R J × (R L ) I × (R J ) I such that (i) for every i, ( xi , zi ) maximizes the preference P i under the budget constraint, that is

( xi , zi ) ∈ B i F ( p, q) and B i F ( p, q) ∩ (P i ( x) × Z i ) = ∅,
(ii) [Market Clearing] i∈I xi = i∈I e i and i∈I zi = 0.

We make the following standard assumptions C1-C6 on the consumption side. We denote by A(E) the set of attainable allocations of the economy, that is,

A(E) = {(x i ) i∈ ∈ i∈I X i : i∈I x i = i∈I e i }.
Consumption Assumption C For every i ∈ I and for every x = (x i ) i∈I ∈ i X i C1 Consumption Sets: X i is a closed, convex, bounded below subset of R L ;

C2 Continuity: The correspondence P i , from k∈I X k to X i , is lower semicontinuous 3 with open values in X i (for the relative topology of X i );

C3 Convexity: P i (x) is convex;

C4 Irreflexivity:

x i P i (x); C5 Local Non-Satiation LNS: (a) ∀x ∈ A(E), ∀s ∈ S , ∃x i (s) ∈ R , (x i (s), x i (-s)) ∈ P i (x), 4 (b) ∀y i ∈ P i (x)], (x i , y i ] ⊂ P i (x); C6 Consumption Survival CS: e i ∈ intX i .
We note that these assumptions are standard in a model with nonordered preferences; the assumptions on P i are satisfied in particular when agents' preferences are represented by utility functions that are continuous, strongly monotonic, and quasi-concave. We now recall that equilibrium asset prices preclude arbitrage opportunities under the above Non-Satiation Assumption. We denote by AZ the asymptotic cone 5 of a nonempty set Z ⊂ R J .

Proposition 1. Assume LNS and the convexity of the portfolio sets Z i (i ∈ I). If ( p, q, x, z) is an equilibrium of the economy (E, F ), then q is arbitrage-free at p, in the sense that

W( p, q)( i AZ i ) ∩ R S + = {0}.
We denote by Q( p) the set of arbitrage-free asset prices q at p ∈ R L .

3 Let Φ be a correspondence from X to Y, that is, Φ is a mapping from X to 2 Y . Then Φ is said to be lower semicontinuous (l.s.c.) at

x o ∈ X, if for every open set V ⊂ Y such that Φ(x o ) ∩ V ∅, there exists an open neighborhood U of x o in X such that Φ(x) ∩ V ∅ for all every x ∈ U.
The correspondence Φ is said to be l.s.c. if it is l.s.c.at every point of X. Finally, we denote by

G(Φ) := {(x, y) ∈ X × Y : y ∈ Φ(x)} the graph of Φ.
4 Given x i ∈ X i and s ∈ S, we denote x i (-s) := (x i (s )) s s . 5 The asymptotic cone of a nonempty subset Z of R J is the set AZ := {lim n λ n z n : (λ n ) n ↓ 0 and z n ∈ Z for all n}. As a consequence from the definition, one has A(clZ) = AZ and we refer to [START_REF] Debreu | Theory of Value[END_REF] for a general reference. When Z is additionally assumed to be convex, then AZ = 0 + (clZ), where 0 [START_REF] Rockafellar | Convex Analysis[END_REF]). When Z is convex, the inclusion 0 + (Z) ⊂ AZ holds but may be strict when Z is not closed.

+ (C) := {ζ ∈ R J : ζ + C ⊂ C} is the recession cone of the convex set C ⊂ R J (see
Proof. Suppose that for some i ∈ I, there exists a portfolio ζ i ∈ AZ i such that W( p, q)ζ i > 0, namely [W( q)ζ i ](s) ≥ 0, for every s ∈ S, with at least one strict inequality, say for s ∈ S. From assumption LNS, there exists

x i ∈ P i ( x) such that x i (-s) = xi (-s). For t > 0 large enough, p (x i -e i ) ≤ W( p, q)(z i + t ζ i ). Since zi + t ζ i ∈ Z i , we get (x i , zi + t ζ i ) ∈ B i F ( p, q) but since x i ∈ P i ( x), this contradicts the optimality of ( xi , zi ) in B i F ( p, q).

Nominal and numéraire assets

If the financial structure F is nominal, the matrix V(p) of financial payoffs does not depend on the commodities price vector p and is denoted R.

A numéraire asset is defined as follows. Let us choose a commodity bundle ν ∈ R , a typical example being ν = (0, . . . , 0, 1), when the -th good is chosen as numéraire. A numéraire asset j is a real asset which delivers the commodity bundle

A j s = R j s ν ∈ R at state s of date t = 1 if this state s prevails. Thus the payoff at state s is (V ν ) j s (p) = (p(s) • ν)R j s for the commodity price p = (p(s)) ∈ R L .
For a numéraire financial structure, i.e., all the assets are numéraire assets (for the same commodity bundle ν), we denote R the S × J-matrix with entries R j s and, for p ∈ R L , we denote V ν (p) the associated S × J-payoff-matrix, which has for entries (V ν )

j s (p) = (p(s) • ν)R j s , i.e. V ν (p) =                      p(1) • ν 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 p(S ) • ν                      R.
In the nominal case, the set Q(p) of arbitrage-free prices, that is, the set of asset prices q satisfying

-q R ( i AZ i ) ∩ R S + = {0} (2.1)
does not depend on the price p, hence is simply denoted Q R . In the numéraire case, under the Desirability Assumption (made in FN0) below, if ( p, q, x, z) is an equilibrium, then p(s) • ν > 0 for all s ∈ S (see the proof of Lemma 3 in Appendix 4.3) and we notice that, if p(s)

• ν > 0 for all s ∈ S , then Q( p) = Q R
as defined above by (2.1). Thus, every equilibrium asset price q belongs to Q R (by Proposition 1) in the nominal case and in the numéraire case.

Given the financial structure F = V, (Z i ) i∈I , we denote Z F =< ∪ i∈I Z i > the linear space where financial activity takes place. We say that Z ⊂ R J is a convex polyhedral set if it can be defined by finitely many linear inequalities, i.e., Z := {z ∈ R J : Bz ≥ b} for some K × J-matrix B and some b ∈ R K . Clearly this takes into account the case of a set defined by finitely many linear inequalities and/or equality constraints, since each equality constraint can be equivalently replaced by two inequality constraints. We can now present the general assumptions on the financial side:

FN Financial Assumption in the Nominal-Numéraire case:

FN0: The financial structure F is either (i) nominal, i.e., V(p) = R is independent of p, or (ii) numéraire, i.e., V(p) = V ν (p)
, for every agent i, the correspondence P i has an open graph and the commodity bundle ν ∈ R is desirable at every state s ∈ S, i.e., for all x ∈ A(E), for all t > 0, (x i (s) + tν, x i (-s)) ∈ P i (x);

FN1: For all i ∈ I, Z i is a convex polyhedral set, 0 ∈ Z i , and V : R L → R S ×J is continuous;

FN2: Financial Survival ∀i ∈ I, ∀q ∈ clQ R ∩ Z F , q 0, ∃ζ i ∈ Z i , q • ζ i < 0.
Assumption FN0 is standard; note however that the open graph assumption and the desirability assumption are slightly stronger than the corresponding ones in C, but these assumptions on P i are still satisfied when ν ∈ R + \{0} and agents' preferences are represented by utility functions that are continuous, strongly monotonic, and quasi-concave. Assumption FN1 considers the class of convex polyhedral portfolio sets of particular interest for economic applications; in particular it takes into account the case of linear equality constraints (portfolio sets which are linear spaces) as considered by [START_REF] Balasko | The structure of financial equilibrium with exogenous yields: The case of restricted participation[END_REF]. Sufficient conditions for Assumption FN2 to hold are provided in Section 2.4.

Existence of equilibria in the nominal and numéraire case

We now state our main existence result.

Theorem 1. [Nominal or numéraire assets] Let the economy (E, F ) satisfy assumptions C and FN, then it admits an equilibrium ( p, q, x, z) such that || p(0

)|| + || q|| = 1 and || p(s)|| = 1 for s ∈ S.
The proof of Theorem 1 is given in Section 2.6 as a consequence of an auxiliary result (Theorem 2) stated in the next section. We now give some consequences of Theorem 1. The following Corollary 2 and 3 allow to extend to the case of consumers with nonordered preferences the existence results of [START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF], [START_REF] Duffie | Stochastic equilibria with incomplete financial markets[END_REF][START_REF] Werner | Equilibrium in economies with incomplete financial markets[END_REF] in the nominal case and [START_REF] Geanakoplos | Uncertainty, information and communication: Essays in honor of Kenneth J. Arrow[END_REF] in the numéraire case.

Corollary 1. The financial exchange economy (E, F ) admits an equilibrium under Assumptions C, FN0, FN1 if -0 ∈ intZ i for all i.

Corollary 2. The financial exchange economy (E, F ) admits an equilibrium under Assumption C if -F consists of nominal assets;

-Z i = R J for all i.

Corollary 3. The financial exchange economy (E, F ) admits an equilibrium under Assumption C if -F consists of numéraire assets and satisfies FN0 (ii);

-Z i = R J for all i.
The proof of the above corollaries is a consequence of Theorem 1 and of the following proposition, which gives examples under which the Financial Survival Assumption FN2 is satisfied.

Proposition 2. Let F = V, (Z i ) i be a financial structure, then Assumption FN2 (and also Assumption F2 defined hereafter) is satisfied if, either (a) or (b) holds (a) For every i ∈ I, one of the following conditions holds:

(i) Z i = R J (unrestricted participation), (ii) Z i = -z i + R J + ,
for some z i ∈ R J ++ (bounds on short sales for every asset), (iii)

Z i = [-z 1 i + R J 1 + ] × R J 2 , for some z 1 i ∈ R J 1
++ , with J = J 1 + J 2 (bounds on short sales for some assets), (iv) Z i is the closed ball {z ∈ R J : z ≤ r} for some r > 0 (bounded portfolios)6 , (v) 0 ∈ intZ i ;

(b) For every i ∈ I, Z i = {0} (pure spot markets).

Proof. Part (a). First notice that each condition (i), (ii), (iii) or (iv) implies (v). Thus we need only show Assumption FN2 holds under Condition (v). Let q ∈ [clQ R ∩ Z F ] \ {0}, and let i ∈ I, then choose ζ i = -εq/ q . Then, for ε > 0 small enough,

ζ i ∈ B(0, ε) ⊂ Z i (since 0 ∈ intZ i by Condition (v)) and q • ζ i = -ε q < 0.
Part (b). We now prove that Assumption FN2 holds in the case of pure spot markets. Notice that

Z F = {0} and [clQ R ∩ Z F ] \ {0} = ∅, hence FN2 holds.

An auxilary existence result

In this section, we state an auxiliary existence result that will be the key tool to prove Theorem 1 in Section 2.6. This result, also of interest for itself, does not assume neither that the financial structure is nominal or numéraire, nor that the portfolio sets are convex polyhedra. It assumes however the additional Assumption F3 that is presented below. We posit the following new set of assumptions on the financial structure.

Financial Assumption F:

F0: The set A F (p) := i∈I (AZ i ∩{V(p) ≥ 0}) does not depend on p (hence is simply denoted A F hereafter); F1: For every i ∈ I, Z i is closed, convex, 0 ∈ Z i , and V : R L → R S ×J is continuous; F2: Financial Survival ∀i ∈ I, ∀p ∈ R L ,p(0) = 0, ∀q ∈ clQ(p) ∩ Z F , q 0, ∃ζ i ∈ Z i , q • ζ i < 0; F3: The set A F is pointed, i.e., A F ∩ -A F = {0}.
Assumptions F0 and F1 weaken the previous assumptions FN0 and FN1, respectively, and F2 adapts the previous Survival Assumption FN2 to the more general case considered here. Sufficient conditions for Assumptions F0 and F3 to hold are given at the end of this section.

We now state our second existence result.

Theorem 2. Let the economy (E, F ) satisfy assumptions C and F, then it admits an equilibrium ( p, q, x, z)

such that || p(0)|| + || q|| = 1 and || p(s)|| = 1 for s ∈ S.
The proof of Theorem 2 is given in Section 3.

We can now give two consequences of Theorem 2 which extend the results by [START_REF] Radner | Existence of equilibrium of plans, prices, and price expectations[END_REF] and the framework of [START_REF] Siconolfi | Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets[END_REF] to the case of agents with nonordered preferences. Note however that the Survival Assumption F2 is not directly comparable to the one used by [START_REF] Siconolfi | Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets[END_REF] and it is the purpose of further work to show the relationship between his existence result and Theorem 2.

Corollary 4. The economy (E, F ) admits an equilibrium under assumptions C, F1 in each of the following cases:

(i) [START_REF] Radner | Existence of equilibrium of plans, prices, and price expectations[END_REF]) For every i ∈ I, Z i is the closed ball B J (0, r i ), for some r i > 0.

(ii) [START_REF] Siconolfi | Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets[END_REF]) F consists of nominal assets, F2 holds and AZ i ∩ ker R = {0} for every i ∈ I.

The proof of the corollary is a consequence of Theorem 2, Proposition 2, and Remarks 1, and 2, which give sufficient conditions for Assumptions F2, F0, and F3 to hold.

Remark 1. Assumption F0 is satisfied when -the financial structure is nominal, i.e., V(p) does not depend on p;

-the set {V(p) ≥ 0} does not depend on p; -for every i, Z i is bounded [START_REF] Radner | Existence of equilibrium of plans, prices, and price expectations[END_REF]) since AZ i = {0}; -the set clQ(p) does not depend on p (under Assumption F3, by Lemma 4 below); -the case of numéraire assets is treated in Section 2.6.

Remark 2. Assumption F3 is satisfied when -ker V = {0} (No redundant assets); -for every i, Z i is bounded [START_REF] Radner | Existence of equilibrium of plans, prices, and price expectations[END_REF]); -for every i, ker V ∩ AZ i = {0} [START_REF] Siconolfi | Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets[END_REF]).

Moreover, Assumption F3 is equivalent to the assumption that V has no redundant assets when there are no restrictions on portfolio trades (Z i = R J for all i). In the case of nominal assets, it coincides with Siconolfi (1989)'s assumption when the Z i 's are linear subspaces.

Assumption F3 is crucial in the proof of Theorem 2. Together with F0, it enables us to look for equilibrium portfolios in an appropriate bounded set, creating thereby a suitable environment for the application of a fixed point theorem. In this regard, Assumption F3 and Siconolfi (1989)'s assumption serve the same purpose, but the weaker Assumption F3 is needed if we want to treat the case of linear inequalities instead of linear equalities. See Section 2.6 for more detail.

Remark 3. We can choose the equilibrium asset price q to be in Q( p) ∩ Z F . Indeed, let ( p, q, x, z) be an equilibrium of (E, F ) and let q * = proj Z F q then ( p, q * , x, z) is an equilibrium of (E, F ) since for every i ∈ I, and for every z i ∈ Z i , one has q * • z i = q • z i . Moreover, q * ∈ Q( p) (under LNS by Proposition 1).

Proof of Theorem 1

The proof of Theorem 1 is given hereafter, first in the case of nominal assets and then in the case of numéraire assets. Consider an economy (E, F ) satisfying Assumptions C and FN.

In the case of nominal assets with no restrictions on portfolio trades, there is a priori no loss of generality in assuming that the payoff matrix V has no redundant assets, that is, rankV = J or equivalently ker V = {0}. Indeed, otherwise by deleting the redundant columns we obtain a new financial economy whose equilibria yield equilibria in the original one. Hereafter, we show how to transform the financial structure to obtain a reduced financial structure, i.e., satisfying Assumption F3, which supersedes the standard assumption of no redundant assets, while keeping the correspondence between the equilibria. Existence of equilibrium then follows from Theorem 2. This analysis coincides with the one in [START_REF] Balasko | The structure of financial equilibrium with exogenous yields: The case of restricted participation[END_REF] when portfolio sets are defined by linear equality constraints and assets are nominal (since the transformed financial structure they obtain satisfies Siconolfi (1989)'s assumption which is equivalent to F3 in the framework of linear equality constraints), and extends it to the case of linear inequality constraints with nominal and numéraire assets.

We modify the financial structure F by considering the reduced financial structure F π which has the same payoff matrix as F and the portfolio sets πZ i (i ∈ I) where π is the orthogonal projection mapping7 on the orthogonal space to L F := A F ∩ -A F . We recall that Z F :=< ∪ i Z i >, Z F π :=< ∪ i πZ i > and the definition of F π can be summarized by

F π = V(p), (πZ i ) i , where π = proj (L F ) ⊥ , L F := A F ∩ -A F , and A F := i∈I ( AZ i ∩ {V(p) ≥ 0}) ⊂ Z F .

The nominal case

This section considers the case of a financial structure F = (V, (Z i ) i ) with nominal assets satisfying the Assumption FN, and in fact the more general case of a financial structure satisfying Assumptions F0, FN1, and F2. This more general framework will be needed in Section 2.6.2 to treat the case of numéraire assets.

Step 1. The first step is summarized by Lemma 1, the proof of which is given in Appendix 4.1.

Lemma 1. If F = (V, (Z i ) i ) satisfies Assumptions F0, FN1, and F2, then F π satisfies Assumptions F0-F2, together with Assumption F3, that is, A F π ∩ -A F π = {0}.
Step 2. From Theorem 2 and Step 1, the economy (E, F π ) admits an equilibrium ( p, q, x, z).

Step 3. There exists an equilibrium (p * , q * , x * , z * ) of (E, F ) as a consequence of the following lemma, the proof of which is given in the Appendix 4.2.

Lemma 2. Assume LNS, and let ( p, q, x, z) be an equilibrium of (E, F π ). Then there exists z * ∈ i Z i such that ( p, π q, x, z * ) is an equilibrium of (E, F ).

To end the proof of Theorem 1 we need to check that the equilibrium found in Lemma 2 can be chosen so that || p(0)|| + ||π q|| = 1 and || p(s)|| = 1 for s ∈ S. Recall that from Remark 3 we can choose the equilibrium asset price vector in (E, F π ), that is q, to be in Q F π ( p) ∩ Z F π ⊂ Z F π ⊂ Imπ. Hence π q = q and || p(0)|| + ||π q|| = || p(0)|| + || q|| = 1. The fact || p(s)|| = 1 for s ∈ S is immediate since we are not changing the equilibrium commodity prices when we go from an equilibrium in (E, F π ) to an equilibrium in (E, F ).

The numéraire case

Consider a financial economy with numéraire assets (E, F ) satisfying Assumptions FN (the part (ii) with numéraire assets). The proof of Theorem 1 consists in applying the Steps 1-3 of the previous section to a modified financial economy (E, F ε ) (for ε > 0 small enough), chosen so that (i) it satisfies Assumptions F0, FN1, and F2 and (ii) the equilibria of (E, F ε ) are also equilibria of the original financial economy (E, F ).

Step 0. We define the modified financial structure F ε = (V ε , (Z i ) i ) for ε > 0, by taking the same portfolio sets Z i as for F and defining the modified payoff matrix V ε , by

V ε (p) =                      max{ε, p(1) • ν} 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 max{ε, p(S ) • ν}                      R.
The financial exchange economy (E, F ε ) satisfies Assumptions F0, FN1, and F2 whenever the economy (E, F ) satisfies Assumptions FN0-FN2. Indeed, {V ε (p) ≥ 0} = {R ≥ 0} for every p ∈ R L , hence V ε satisfies Assumption F0. Assumptions F1 and FN1 are obviously satisfied, and assumption F2 is a consequence of FN2 and the fact that Q F ε (p) = Q R for every p. The relationship between the equilibria of the economies (E, F ε ) and (E, F ) is then given by the following lemma, the proof of which is given in Appendix 4.3. Lemma 3. For ε > 0 small enough, every equilibrium ( p, q, x, z) of (E, F ε ) such that || p(s)|| = 1 for s ∈ S is an equilibrium of the economy (E, F ).

Proof of Theorem 2

We first state a lemma summarizing several properties that will be used hereafter. We recall that

A F (p) := i∈I (AZ i ∩ {V(p) ≥ 0}) ⊂ Z F .
b) clQ(p) ∩ Z F o = -A F (p) + (Z F ) ⊥ . Hence A F (p) ⊂ Z F is closed.
(c) If we additionally assume F0, then for all v = (v i ) i ∈ (R S ) I the set K v is bounded, where

K v := (z 1 , • • • , z I , p) ∈ ( i Z i ) × B L (0, 1) : ∀i, V(p)z i ≥ v i , - i∈I z i ∈ A F .
The proof of Lemma 4 is given in the Appendix 4.4.

Truncating the economy

We denote by X i the projection of the set of attainable allocations A(E) on X i . Since A(E) is bounded (by Assumption C1), the sets X i are also bounded for every i ∈ I. Consequently, one can choose r 1 > 0 large enough such that X i ⊂ intB L (0, r 1 ) for every i ∈ I,

For i ∈ I, let v i = (v i (s)) ∈ R S , where v i (s) = -1 + min{p(s) • (x i (s) -e i (s)) : p(s) ∈ B (0, 1), x i ∈ B L (0, r 1 )} (s ∈ S), (3.1)
which is well defined from the compactness of the closed balls B (0, 1) and B L (0, r 1 ). We denote by Z i the projection of K v on Z i and the sets Z i are bounded for every i ∈ I, since K v is bounded (by Lemma 4). Consequently, one can choose r 2 > 0 large enough such that Z i ⊂ intB J (0, r 2 ) for every i ∈ I.

We let r = (r 1 , r 2 ) and for every i ∈ I,

X r i = X i ∩ B L (0, r 1 ), P r i (x) = P i (x) ∩ intB L (0, r 1 ), Z r i = Z i ∩ B J (0, r 2 ).
We define the truncated financial economy (E r , F r ), which has X r i , for consumption sets, P r i , for preference correspondences, Z r i for portfolio sets. The endowments of consumers and the payoff matrix are the same as for the economy (E, F ). To summarize, we let (E r , F r ) := X r i , P r i , e i i∈I , V, (Z r i ) i∈I .

The reaction correspondences and the fixed-point argument

We define the set of admissible prices for commodities and assets

Π = {(p, q) ∈ R L × R J : ||p(0)|| + q ≤ 1, ∀s ∈ S, p(s) ≤ 1, and q ∈ clQ(p) ∩ Z F },
and, following [START_REF] Bergstrom | How to discard "free disposability" -at no cost[END_REF], the "modified" budget sets of consumer i, for (p, q) ∈ Π: where ε(p, q) = (ε s (p, q)) s∈ S ∈ [0, 1] 1+S is defined, for (p, q) ∈ Π, by

B rε i (p, q) = {(x i , z i ) ∈ X r i × Z r i : p (x i -e i ) ≤ W(p, q)z i + ε(p, q)}, Brε i (p, q) = {(x i , z i ) ∈ X r i × Z r i : p (x i -e i ) W(p, q)z i + ε(p, q)}, 12
ε s (p, q) =          1 -p(0) -q for s = 0 1 -p(s)
for s ∈ S.

Claim 3.1. For all (p, q) ∈ Π, Brε i (p, q) ∅ and B rε i (p, q) = cl Brε i (p, q).

Proof. We first notice that e i ∈ intX r i for every i ∈ I; indeed, this is a consequence of the facts that e i ∈ X i ⊂ X i ∩ intB L (0, r 1 ) and e i ∈ intX i (Survival Assumption C6). Let (p, q) ∈ Π. Since e i ∈ intX r i , for t > 0 small enough, x r i := e itp ∈ X r i . Hence, p (x r ie i ) = -tp p ≤ 0, and

p(0) • x r i (0) -e i (0) -ε 0 (p, q) = -t p(0) 2 -1 + p(0) + q < 0 if [p(0) 0 or q = 0], p(s) • x r i (s) -e i (s) -ε s (p, q) = -t p(s) 2 -1 + p(s) < 0 for all s = 1, . . . , S .
Consequently, if p(0) 0 or q = 0, then (x r i , 0) ∈ Brε i (p, q). Assume now that p(0) = 0 and q 0, then there exists z i ∈ Z i such that q • z i < 0 (by Assumption F2). We can choose λ > 0 small enough (λ z i ≤ r 2 ) so that z r i := λz i ∈ Z r i := Z i ∩ B(0, r 2 ) (since z r i ∈ B(0, r 2 ), and z r i ∈ Z i recalling that z i ∈ Z i , 0 ∈ Z i and Z i is convex) and q • z r i < 0. Again, from above, we can choose λ > 0 small enough so that p(s)

• x r i (s) -e i (s) -ε s (p, q) < V s (p) • z r i for all s ∈ S. Thus, (x r i , z r i ) ∈ Brε i (p, q) ∅. We now prove the equality B rε i (p, q) = cl Brε i (p, q). The first inclusion cl Brε i (p, q) ⊂ B rε i (p, q) is im- mediate. Conversely, let (x i , z i ) ∈ B rε i (p, q) and let ( xi , zi ) ∈ Brε i (p, q) ∅, then (x i , z i ) = lim t→0 (x t i , z t i ) ∈ cl Brε i (p, q) taking (x t i , z t i ) := (1 -t)(x i , z i ) + t( xi , zi ) ∈ Brε i (p, q
) for all t ∈ [0, 1]. We now introduce an additional agent and, as in (Gale andMas-Colell (1975, 1979)), we set the following reaction correspondences defined on

Π × i∈I X r i × Z r i Φ i (p, q, x, z) =          B rε i (p, q) if (x i , z i ) B rε i (p, q) Brε i (p, q) ∩ (P r i (x) × Z r i ) if (x i , z i ) ∈ B rε i (p, q), Φ 0 (p, q, x, z) = {(p , q ) ∈ Π : (p -p) • i∈I (x i -e i ) + (q -q) • i∈I z i > 0}.
The main properties of the correspondences Φ i are stated in the following lemma, the proof of which is given in Appendix 4.5. Lemma 5. For all i ∈ {0} ∪ I, the correspondence Φ i is lower semicontinuous with convex values on Π × i∈I X r i × Z r i .

The proof of Theorem 2 relies on the following theorem due to [START_REF] Gale | An equilibrium existence Theorem for a general model without ordered preferences[END_REF].

Theorem 3. Let I o be a finite set, let C i (i ∈ I o ) be a nonempty, compact, convex subset of some Euclidean space, let C = i∈I 0 C i and let Φ i (i ∈ I o ) be a correspondence from C to C i , which is lower semicontinuous and convex-valued. Then, there exists c * = (c * i ) i ∈ C such that, for every i We apply Theorem 3 to the sets

∈ I 0 [either c * i ∈ Φ i (c * ) or Φ i (c * ) = ∅].
I o = {0} ∪ I, C 0 = Π, C i = X r i × Z r i (i ∈ I)
, and to the correspondences Φ i (i ∈ I o ) defined above. We check that the assumptions of Theorem 3 are fulfilled. The set Π is convex since the set clQ(p) ∩ Z F is convex and does not depend on p (by Assumption F0 and Lemma 4). Moreover Π is obviously compact, and nonempty (since (0, 0) ∈ Π). For every i ∈ I, the set X r i × Z r i is clearly compact, convex, and nonempty (since it contains (e i , 0)). Finally, for every i ∈ I o , the correspondence Φ i is lower semicontinuous and convex-valued by Lemma 5.

It follows from Theorem 3 that there exists ( p, q, x, z) ∈ Π × i∈I (X r i × Z r i ) such that for all i ∈ I, either Φ i ( p, q, x, z) = ∅ or ( xi , zi ) ∈ Φ i ( p, q, x, z), and for i = 0, either Φ 0 ( p, q, x, z) = ∅ or ( p, q) ∈ Φ 0 ( p, q, x, z).

Remark that, by construction, ( p, q) Φ 0 ( p, q, x, z), hence Φ 0 ( p, q, x, z) = ∅ or equivalently

p • i∈I ( xi -e i ) + q • i∈I zi ≤ p • i∈I ( xi -e i ) + q • i∈I zi for all (p, q) ∈ Π. (3.2)
Moreover, for every i ∈ I, ( xi , zi ) ∈ B rε i ( p, q); indeed if ( xi , zi ) B rε i ( p, q), then ( xi , zi ) Φ i ( p, q, x, z) = B rε i ( p, q) and Φ i ( p, q, x, z) ∅ (since it contains (e i , 0)), a contradiction. Consequently, for all i ∈ I,

either Brε i ( p, q) ∩ P r i ( x) × Z r i = ∅ or ( xi , zi ) ∈ Brε i ( p, q) ∩ (P r i ( x) × Z r i
) and the second condition cannot hold from the irreflexivity of P r i (by Assumption C). Thus we can conclude that for all i ∈ I, ( xi , zi ) ∈ B rε i ( p, q) and Brε i ( p, q) ∩ P r i ( x) × Z r i = ∅.

(3.3)

Checking the market clearing conditions

Since the Market Clearing Condition i∈I zi = 0 may not be satisfied by the portfolios z = (z i ) i , the purpose of the next claim is to define new portfolios zi (i ∈ I) that will satisfy the Market Clearing Condition i∈I zi = 0 and will also be admissible, i.e., zi ∈ Z i ⊂ Z r i for all i.

Claim 3.2. (a) For every i, V( p)z i ≥ v i , i∈I zi ∈ -A F , and q • i∈I zi = 0.

We let zi = zi + ζ i , for some

ζ i ∈ AZ i ∩ {V( p) ≥ 0} (i ∈ I) such that i∈I zi = -i∈I ζ i . . (b) Then i∈I zi = 0, for every i, zi ∈ Z i ⊂ Z r i , q • zi = q • zi , V( p) zi ≥ V( p)z i and ( xi , zi ) ∈ B rε i ( p, q). Proof. Part (a). First, V( p)z i ≥ v i follows from the definition of v i (in (3.1)) and ( xi , zi ) ∈ B rε i ( p, q) (by (3.3)).
Second, we show that i∈I zi ∈ clQ( p) ∩ Z F o . If this does not hold, then there exists q ∈ clQ( p) ∩ Z F such that q • ( i∈I zi ) > 0. Without any loss of generality, we can assume that q ∈ B J (0, 1). From (3.2) (taking (p, q) ∈ Π defined by p(0) = 0, p(s) = p(s) for s 0 and q = q ) we have

0 < q • i∈I zi ≤ p(0) • i∈I ( xi (0) -e i (0)) + q • i∈I zi . (3.4)
Since ( xi , zi ) ∈ B rε i ( p, q) (by (3.3)) we deduce that p(0) • xi (0)e i (0) + q • zi ≤ ε 0 ( p, q) := 1p(0)q for all i ∈ I.

Summing up over i we get p(0)

• i∈I xi (0) -e i (0) + q • i∈I zi ≤ (1 -p(0) -q )I,
which together with the above inequality (3.4) implies that p(0) + q < 1. Hence there exists α > 1 such that α p(0) + α q < 1 and α q ∈ clQ( p) ∩ Z F (since the latter set is a cone). Consequently, from (3.2), (taking (p, q) ∈ Π defined by p(0) = α p(0), p(s) = p(s) for s 0 and q = α q) we deduce that:

α p(0) • i∈I xi (0) -e i (0) + α q • i∈I zi ≤ p(0) • i∈I xi (0) -e i (0) + q • i∈I zi .
Dividing by p(0) • i∈I xi (0)e i (0) + q • i∈I zi > 0 (by inequality (3.4)), we get α ≤ 1, which contradicts that α > 1.

Third, we show that i∈I zi ∈ -A F . From above, we have

i∈I zi ∈ clQ( p) ∩ Z F o = -A F + (Z F ) ⊥ (by Lemma 4). Thus, there exists a ∈ A F such that i∈I zi + a ∈ (Z F ) ⊥ but i∈I zi + a ∈ Z F (since a ∈ A F ⊂ Z F and zi ∈ Z i ⊂ Z F for each i ∈ I). Hence i∈I zi + a ∈ (Z F ) ⊥ ∩ Z F = {0}. Therefore, i∈I zi = -a ∈ -A F .
Finally, we show that q• i∈I zi = 0. We have q• i∈I zi ≤ 0 since q ∈ clQ( p)∩Z F and i∈I zi ∈ clQ( p)∩Z F o (from above). Taking (p, q) = ( p, 0) ∈ Π (3.2) we deduce that 0 ≤ q • i∈I zi . Hence, q • i∈I zi = 0.

Part (b). The equality i∈I zi = 0 is straightforward and, for all i, V(

p) zi ≥ V( p)z i (since zi -zi = ζ i ∈ {V( p) ≥ 0}). To show that, for all i, zi ∈ Z i ⊂ Z r i , it is sufficient to prove that ( z, p) ∈ K v = (z 1 , • • • , z I , p) ∈ ( i Z i ) × B L (0, 1) : ∀i, V(p)z i ≥ v i , - i∈I z i ∈ A F .
Indeed, for all i, zi = zi

+ ζ i ∈ Z i (since zi ∈ Z r i ⊂ Z i and ζ i ∈ AZ i ). Moreover, V( p) zi ≥ V( p)z i ≥ v i by Part (a). Finally, i∈I zi = 0 ∈ A F . This ends the proof that ( z, p) ∈ K v . We now show that q • ( zi -zi ) = q • ζ i = 0 for every i ∈ I. Indeed, first -q • ζ i ≤ 0 for every i since q ∈ clQ( p) ∩ Z F = -A F + (Z F ) ⊥ o
(by Lemma 4) and

ζ i ∈ AZ i ∩ {V( p) ≥ 0} ⊂ A F . Recalling that q • i∈I ζ i = -q • i∈I zi = 0 from Part (a), we deduce that q • ζ i = 0 for every i ∈ I.
Finally, for all i, ( xi , zi ) ∈ B rε i ( p, q) since ( xi , zi ) ∈ B rε i ( p, q) (by 3.3)) and, from above W( p, q)z i ≤ W( p, q) zi .

We now show that the Market Clearing Condition holds for the commodity markets.

Claim 3.3. i∈I xi = i∈I e i .
Proof. We first prove that the equality holds at state s = 0. If i∈I xi (0) i∈I e i (0), we deduce from (3.2), (taking (p, q) ∈ Π defined by p(0) = i∈I ( xi (0)e i (0))/ i∈I ( xi (0)e i (0)) , p(s) = p(s) for s 0 and

q = 0) that 0 < i∈I ( xi (0) -e i (0)) ≤ p(0) • i∈I xi (0) -e i (0) + q • i∈I zi ,
and in the exact same way as for inequality (3.4) in the proof of Claim 3.2 we obtain a contradiction.

We now prove that the equality holds for all state s 0. Suppose that, for some s 0, i∈I xi (s) i∈I e i (s). From (3.2), we deduce that p(s) = 1, hence ε s ( p, q) = 1p(s) = 0, and

0 < p(s) • i∈I xi (s) -e i (s) .
Since ( xi , zi ) ∈ B rε i ( p, q) (by Claim 3.2), and ε s ( p, q) = 0, we have p(s) • xi (s)e i (s) ≤ V s ( p) • zi for all i ∈ I, where V s ( p) denotes the s-th row of the matrix V( p). Summing up over i, and using the fact that i∈I zi = 0 (by Claim 3.2) we get p(s) • i∈I xi (s)e i (s) ≤ i∈I V s ( p) • zi = 0, a contradiction with the above strict inequality.

3.4. The list ( x, z, p, q) is an equilibrium of (E, F )

We prepare the proof with the following claims.

Claim 3.4. B rε i ( p, q) ∩ P r i ( x) × Z r i = ∅.

Proof. From (3.3), one has Brε i ( p, q) ∩ P r i ( x) × Z r i = ∅. Since P r i has open values and since B rε i ( p, q) = cl Brε i ( p, q) (by Claim 3.1 since ( p, q) ∈ Π), we deduce that B rε i ( p, q) ∩ P r i ( x) × Z r i = ∅.

Claim 3.5. ε( p, q) = 0, that is, || p(0)|| + || q|| = 1, and || p(s)|| = 1 for every s ∈ S.

Proof. From Claim 3.2, ( xi , zi ) ∈ B rε i ( p, q) for each i ∈ I, hence p ( xie i ) ≤ W( p, q) zi + ε( p, q). We now claim that the budget inequality is binding, that is p ( xie i ) = W( p, q) zi + ε( p, q) for all i ∈ I.

(3.5) Indeed, if it is not true then there exists s ∈ S such that p(s) • ( xi (s)e i (s)) < W s ( p, q) • zi + ε s ( p, q). From the Local Nonsatiation LNS, there exists x n i (s) → xi (s) such that x n i := (x n i (s), xi (-s)) ⊂ P r i ( x) for all n. Then, it is possible to choose n large enough so that (x n i , zi ) ∈ B rε i ( p, q), which together with x n i ∈ P r i ( x) contradicts the fact that B rε i ( p, q) ∩ P r i ( x) × Z r i = ∅ (by Claim 3.4). Summing up over i the equalities (3.5) and using the facts that i∈I zi = 0 (Claim 3.2) and i∈I xi = i∈I e i (Claim 3.3), we get Iε( p, q) = 0. Hence ε( p, q) = 0.

We now show that ( x, z, p, q) is an equilibrium of (E, F ). Since ε( p, q) = 0 (Claim 3.5) for every i, B rε i ( p, q) = B i F r ( p, q) and from Claims 3.2, 3.3, and 3.4, we deduce that ( x, z, p, q) is an equilibrium of (E r , F r ). To show that it is an equilibrium of (E, F ), we only have to prove that

B i F ( p, q) ∩ (P i ( x) × Z i ) = ∅ for every i ∈ I.
Assume that it is not true, then for some i, there exists (x i , z i ) ∈ B i F ( p, q) ∩ (P i ( x) × Z i ). From the choice of of r = (r 1 , r 2 ) in the definition of the truncated economy, for all i, xi ∈ X i ⊂ intB L (0, r 1 ) (since x is an attainable allocation) and zi ∈ Z i ⊂ intB J (0, r 2 ) (by Claim 3.2). Thus, for t > 0 small enough, 16 Document de Travail du Centre d'Economie de la Sorbonne -2009.63

(x t i , z t i ) := ( xi + t(x i -xi ), zi + t(z i -zi )) ∈ X r i × Z r i , hence (x t i , z t i ) ∈ B i F r ( p, q) (since (x i , z i ) ∈ B i F ( p, q
) and ( xi , zi ) ∈ B i F r ( p, q)). On the other hand, from the Local Nonsatiation Assumption LNS, for every t ∈ (0, 1], xi + t(x ixi ) ∈ P i ( x). Hence (x t i , z t i ) ∈ B i F r ( p, q) ∩ (P r i ( x) × Z r i ) ∅, in contradiction with the fact it is empty by Claim 3.4 .

Appendix

We modify the financial structure F by considering the reduced financial structure F π which has the same payoff matrix as F and the portfolio sets πZ i (i ∈ I) where π is the orthogonal projection mapping on the orthogonal space to L F := A F ∩ -A F . We recall that Z F :=< ∪ i Z i >, Z F π :=< ∪ i πZ i > and the definition of F π can be summarized by

F π = V(p), (πZ i ) i , where π = proj (L F ) ⊥ , L F := A F ∩ -A F , and A F := i∈I ( AZ i ∩ {V(p) ≥ 0}) ⊂ Z F ,
In the following, we will use extensively the following properties8 of the linear mapping π: for all (p, q, z) ∈

R L × R J × R J , q • πz = πq • z, V(p)πz = V(p)z, hence W(p, q)πz = W(p, πq)z.
(4.1)

Proof of Lemma 1

We prepare the proof with two claims.

Claim 4.1.

Q F π (p) ∩ Z(F π ) ⊂ Q F (p) ∩ Z F for every p ∈ R L .
Proof. We first claim that

Q F π (p) ∩ Z(F π ) ⊂ Q F π (p) ∩ Imπ ⊂ Q F (p) for every p ∈ R L .
The first inclusion is a consequence of the fact that Z(F π ) ⊂ Imπ. We prove the second inclusion by contradiction. Assume that there is some q ∈ Q F π (p) ∩ Imπ such that q Q F (p). Then there exists i ∈ I and [START_REF] Rockafellar | Convex Analysis[END_REF]) and by (4.1) (since q ∈ Imπ implies q = πq), W(p, q)(πζ i ) = W(p, q)ζ i > 0, which contradicts the fact that q ∈ Q F π (p). This ends the proof of the two inclusions.

ζ i ∈ AZ i such that W(p, q)ζ i > 0. But πζ i ∈ π(AZ i ) ⊂ A(πZ i ) (from
We end the proof of Claim 4.1 by showing that Z(F π ) ⊂ Z F . Indeed, let y ∈ Z(F π ), then y = πz for some z ∈ Z F and y

= πz = πz -z + z ∈ ker π + Z F ⊂ Z F since ker π = L F ⊂ Z F . Claim 4.2. i∈I πZ i ∩ {V(p) ≥ v i } ⊂ i∈I Z i ∩ {V(p) ≥ v i } for all v = (v i ) i∈I ∈ (R S ) I and all p ∈ R L .
Proof. The result is a consequence of the two following inclusions

i∈I πZ i ∩ {V(p) ≥ v i } ⊂ ker π + i∈I Z i ∩ {V(p) ≥ v i } ⊂ i∈I Z i ∩ {V(p) ≥ v i } .
To prove the first inclusion, let y = i∈I y i such that y i = πz i for some z i ∈ Z i and V(p)y i ≥ v i . Then y = i∈I y i = i∈I (πz iz i ) + i∈I z i ∈ ker π + i∈I Z i ∩ {V(p) ≥ v i } since πz iz i ∈ ker π and V(p)z i = V(p)πz i = V(p)y i ≥ v i , from the properties (4.1) of the mapping π.

To prove the second inclusion, from the definition of the asymptotic cone in the convex case it suffices to show that ker π

⊂ A i∈I (Z i ∩{V(p) ≥ v i }) and that the convex set i∈I (Z i ∩{V(p) ≥ v i }) is closed. From Rockafellar (1970), if the sets C k (k ∈ K) are polyhedral convex sets in R n , then k∈K C k is also a polyhedral convex set (hence is closed) and k∈K AC k = A( k∈K C k ). Since Z i (i ∈ I) is a polyhedral convex set (by Assumption FN1), the set Z i ∩ {V(p) ≥ v i )} is also a polyhedral convex set, hence i∈I (Z i ∩ {V(p) ≥ v i }) is a polyhedral set and it is closed. Recalling that ker π = L F ⊂ A F , from the previously mentioned result we get ker π = L F ⊂ A F = i∈I AZ i ∩ {V(p) ≥ 0} = i∈I A(Z i ∩ {V(p) ≥ v i }) = A i∈I (Z i ∩ {V(p) ≥ v i }) . (4.2)
We are now ready to give the proof of Lemma 1.

0. F π satisfies F0. The financial structure F satisfies F0, that is, the set A F := i∈I (AZ i ∩ {V(p) ≥ 0}) does not depend on p. Clearly, F π will also satisfy F0 if we show that A F π := i∈I (AπZ i ∩ {V(p) ≥ 0}) = π( A F ). We recall that, for all i, AπZ i = πAZ i since Z i is a polyhedral convex set (see [START_REF] Rockafellar | Convex Analysis[END_REF]). To prove the first inclusion, let y ∈ A F π , then y = i∈I y i for some y i such that y i = πz i for some z i ∈ AZ i and V(p)y i ≥ 0. But V(p)z i = V(p)πz i = V(p)y i ≥ 0 by (4.1), hence y ∈ π(A F ). Conversely, let y ∈ π( A F ), then y = π( i∈I z i ) for some z i ∈ AZ i ∩ {V(p) ≥ 0}). But πz i ∈ πAZ i = AπZ i and V(p)πz i = V(p)z i ≥ 0 from the properties (4.1) of the mapping π. Thus y = i∈I πz i ∈ A F π .

1. F π satisfies F1 and FN1. Indeed, πZ i is a polyhedral convex set, since Z i is a polyhedral convex set and π is linear (see for example [START_REF] Rockafellar | Convex Analysis[END_REF]) and πZ i contains 0 since Z i contains 0.

F

π satisfies F2. Let q ∈ clQ F π (p) ∩ Z(F π )\{0} and recall that clQ F π (p) ∩ Z(F π )\{0} ⊂ clQ F (p) ∩ Z F \{0}
by Claim 4.1. Since F satisfies F2, for all i ∈ I, there exists z i ∈ Z i such that q • z i < 0. But πz i ∈ πZ i and q • πz i = q • z i < 0 (from the properties (4.1) of the mapping π since q ∈ Z(F π ) ⊂ Imπ), thus F π satisfies F2.

3. F π satisfies F3. Taking v i = 0 for every i in Claim 4.2, and in (4.2), we deduce that

A F π = A i∈I πZ i ∩ {V(p) ≥ 0} ⊂ A i∈I Z i ∩ {V(p) ≥ 0} = A F . Consequently, L F π ⊂ L F . Moreover, L F π ⊂ A F π ⊂ i∈I πZ i ∩ {V(p) ≥ 0} ⊂ Imπ and L F = ker π (from the definition of the mapping π). Consequently L F π ⊂ L F ∩ Imπ = ker π ∩ Imπ = {0}.
This ends the proof of Lemma 1.

p n (s) • e 1 (s). Passing to the limit, when n → ∞, we get p(s) • x 1 (s) ≥ p(s) • x 1 (s) ≥ p(s) • e 1 (s), hence p(s) ∈ P s .

Claim 4.4. There exists ε > 0 such that p(s) • ν ≥ ε p(s) for all p ∈ P and all s ∈ S.

Proof. We start by proving that, for every p(s) ∈ P s ∩ S (0, 1), p(s) • ν > 0. Indeed, if p(s) ∈ P s , there exist i s ∈ I, say i s = 1, and x ∈ A(E) such that (x 1 (s), x 1 (-s)) ∈ P 1 (x) implies p(s) • x 1 (s) ≥ p(s) • x 1 (s) ≥ p(s) • e 1 (s). Since e 1 ∈ intX 1 (by the Survival Assumption in C) and ||p(s)|| = 1 (since p(s) ∈ S (0, 1)), there exists y 1 ∈ X 1 such that p(s)•x 1 (s) ≥ p(s)•e 1 (s) > p(s)•y 1 (s). But, for every t > 0, x 1 (s)+tν, x 1 (-s) ∈ P 1 (x) (from the Desirability Assumption in FN). Moreover, since P 1 (x) is open (by Assumption C), it is possible to choose x 1 (s) such that x 1 (s) + tν, x 1 (-s) ∈ P 1 (x) and p(s) • x 1 (s) > p(s) • x 1 (s)9 . Since p(s) ∈ P s we deduce that

p(s) • [x 1 (s) + tν] ≥ p(s) • x 1 (s), that is tp(s) • ν ≥ p(s) • x 1 (s) -p(s) • x 1 (s) > 0.
Hence we have proved that, for every p(s) ∈ P s ∩ S (0, 1), there exists ε p > 0 such that p(s) • ν > ε p . This implies that P s ∩ S (0, 1) ⊂ {p(s) ∈ S (0, 1) : p(s) • ν > ε p }.

Since P s is closed (by Claim 4.3) the set P s ∩ S (0, 1) is compact and there exist finitely many prices p 1 , . . . , p r such that

P s ∩ S (0, 1) ⊂ r k=1 {p(s) ∈ S (0, 1) : p(s) • ν > ε p k }.
We let ε(s) := min{ε p 1 , . . . , ε p r } and ε := min{ε(s) : s ∈ S}. Then, for every p(s) ∈ P s \{0}, p(s)/||p(s)|| ∈ P s (noticing that P s is a cone) hence (p(s)/||p(s)||) • ν > ε(s) > 0 and, for all p ∈ P and all s ∈ S, p(s) • ν ≥ ε||p(s)||.

Proof of Lemma 3. We choose ε > 0 as in Claim 4.4 and we show that every equilibrium ( p, q, x, z) of (E, F ε ) such that || p(s)|| = 1 for s ∈ S, is also an equilibrium of (E, F ). First we claim that p ∈ P, that is p(s) ∈ P s for every s ∈ S. From the Financial Market Clearing Condition i∈I zi = 0 we deduce that V ε s ( p) • ( i∈I zi ) = 0, hence there exists i s ∈ I, say i = 1, such that V ε s ( p) • z1 ≥ 0. We prove that p(s) ∈ P s by showing that it satisfies the condition defining P s with i = 1 and x = ( xi ) i ∈ A(E). Indeed, let x 1 := (x 1 (s), x1 (-s)) ∈ P 1 ( x). First we have p(s) • x 1 (s) ≥ p(s) • x1 (s). Otherwise p(s) • (x 1 (s)e 1 (s)) < p(s) • ( x1 (s)e 1 (s)) ≤ V ε s ( p) • z1 .

Consequently, (x 1 , z1 ) ∈ B 1 F ε ( p, q) ∩ [P 1 ( x) × Z 1 ], which contradicts the fact that ( p, q, x, z) is an equilibrium of (E, F ε ). Second we have p(s) • x1 (s) ≥ p(s) • e 1 (s). Otherwise p(s) • ( x1 (s)e 1 ) < 0. Hence for t > 0 small enough p(s) • ( x1 (s) + tν)e 1 < 0 ≤ V ε s ( p) • z1 .

Step 5. We are ready to conclude. First we note that for two closed convex cones M and N of R Part (c). Assume K v is not bounded. Then there exist sequences (p n ) n ⊂ B L (0, 1), (z n i ) n ⊂ Z i (i ∈ I) such that for all n and all i, V(p n )z

Proof of Lemma 5

First, we claim that, for all i ∈ I, the correspondence B rε i from Π to X r i × Z r i is lower semicontinuous, has a closed graph, and has closed convex values. Indeed, from Claim 3.1, B rε i is the closure of Brε i on Π. We then notice that Brε

i has an open graph, hence is lower semicontinuous. Consequently, B rε i , which is the closure of a lower semicontinuous correspondence, is lower semicontinuous. The fact that B rε i has a closed graph and closed convex values is immediate.

The correspondence Φ 0 has clearly an open graph thus it is lower semicontinuous and one easily checks that it has convex values. Let i 0, ( p, q, x, z) ∈ Π × k∈I X r k × Z r k , and let V be an open set in X r i × Z r i such that Φ i ( p, q, x, z) ∩ V ∅. We distinguish two cases:

1. ( xi , zi ) B rε i ( p, q). Then ( p, q, xi , zi ) G(B rε i ) and since B rε i has a closed graph there exists an open neighborhood W of ( p, q, xi , zi ) in Π × (X r i × Z r i ) such that W ∩ G(B rε i ) = ∅. On the other hand, V is open in X r i × Z r i and satisfies B rε i ( p, q) ∩ V ∅ (because Φ i ( p, q, x, z) ∩ V ∅ and Φ i ( p, q, x, z) ⊂ B rε i ( p, q)). Since B rε i is lower semicontinuous we conclude that there exists an open neighborhood O of ( p, q) in Π such that for every (p, q) ∈ O, B rε i (p, q) ∩ V ∅.

Then U := [W × ( k i X r k × Z r k )] ∩ [O × ( k X r k × Z r k )
] is an open neighborhood of ( p, q, x, z) and for every (p, q, x, z) ∈ U one has (p, q, x i , z i ) ∈ W ∩ [O × (X r i × Z r i )], hence Φ i (p, q, x, z) ∩ V = B rε i (p, q) ∩ V ∅. 2. ( xi , zi ) ∈ B rε i ( p, q). Note that Brε i ∩ (P r i × Z r i ) is lower semicontinuous since Brε i has an open graph and P r i × Z r i is lower semicontinuous. Since Brε i ( p, q) ∩ (P r i ( x) × Z r i ) ∩ V = Φ i ( p, q, x, z) ∩ V ∅, we conclude that there exists an open neighborhood T of ( p, q, x, z) in Π × k∈I X r k × Z r k such that for every (p, q, x, z) ∈ T one has Brε i (p, q) ∩ (P r i (x) × Z r i ) ∩ V ∅. Consequently, B rε i (p, q) ∩ V ∅ (because Brε i (p, q) ⊂ B rε i (p, q)) and Φ i (p, q, x, z) ∩ V ∅ for every (p, q, x, z) ∈ T . This ends the proof of the lower semicontinuity of Φ i .

11

  Document de Travail du Centre d'Economie de la Sorbonne -2009.63 Lemma 4. Under Assumptions F1 and F3, for every p ∈ R L : (a) The set Q(p) is a convex cone with vertex 0.

(

  Document de Travail du Centre d'Economie de la Sorbonne -2009.63

13

  Document de Travail du Centre d'Economie de la Sorbonne -2009.63

  J , one has (M ∩ N) o = cl(M o + N o ), hence from the above steps one gets clQ(p) ∩ Z F o = cl (clQ(p)) o + Z o F = cl Q(p) o + (Z F ) ⊥ = cl -A + (Z F ) ⊥ = -A + (Z F ) ⊥ .

  n i ≥ v i , -i∈I z n i ∈ A F ,and i∈I ||z n i || -→ Passing to a subsequence if necessary, we can assume that (p n ) n converges to some p ∈ B L (0, 1) and that, for each i, the bounded sequence (z n i / k∈I ||z n k ||) n converges to some ζ i . The vector ζ i belongs to AZ i since z n i ∈ Z i for every n and 1/ k∈I ||z n k || -→ n→∞ 0. Moreover, for every n, we haveV(p n )(z n i / k∈I ||z n k ||) ≥ v i / k∈I ||z n k ||. Passing to the limit, we obtain V(p)ζ i ≥ 0 (since V is continuous). Recalling that, for every n, -i∈I z n i ∈ A F , we get -i∈I (z n i / k∈I ||z n k ||) -→ n→∞ -i∈I ζ i ∈ A F since A F is closed by Part (b).Recalling that for each i,ζ i ∈ AZ i ∩ {V(p) ≥ 0} ⊂ A F we conclude that i∈I ζ i ∈ A F ∩ -A F = {0} (by Assumption F3). Hence ζ i ∈ A F for each i and i∈I ζ i = 0, which implies that ζ 1 =i 1 ζ i ∈ A F ∩ -A F = {0},and similarly ζ i = 0 for every i. But z n i / k∈I ||z n k || -→

	n→∞	ζ i , hence 1 = i∈I	||z n i || k∈I ||z n k || -→

n→∞

+∞.

n→∞ i∈I ||ζ i || = 0, a contradiction.

October 14, 2009 Document de Travail du Centre d'Economie de la Sorbonne -2009.63

Document de Travail du Centre d'Economie de la Sorbonne -2009.63

For every p = (p(s)), x = (x(s)) in R L , we denote by p x the vector p(s) • x(s) s∈ S. 4 Document de Travail du Centre d'Economie de la Sorbonne -2009.63

The result is true for any norm • of R J and the set Z i will be polyhedral if we take for example the norm z ∞ := max{|z j | : j = 1 . . . , J}. 8 Document de Travail du Centre d'Economie de la Sorbonne -2009.63

When L is a subset of R J , we define the polar set of L by L o := {z ∈ R J : z • ξ ≤ 0 for all ξ ∈ L} and the orthogonal set to L byL ⊥ := {z ∈ R J : z • ξ = 0 for all ξ ∈ L}.When L is a linear space and ϕ ∈ R J , we denote by proj L ϕ (resp. proj L ⊥ ϕ) the orthogonal projection of ϕ on L (resp. on L ⊥ ), that is, the unique α ∈ L (resp. β ∈ L ⊥ ) such that ϕα ∈ L ⊥ (resp. ϕβ ∈ L).

The first equality comes from the fact that πq • πz = πq • z, since πq ∈ Imπ and z -πz ∈ ker π = (Imπ) ⊥ since π is an orthogonal projection mapping; then by symmetry q • πz = πq • πz = πq • z. The second one holds since z -πz ∈ ker π = L F and L F := A F ∩ -A F ⊂ {V(p) ≥ 0} ∩ -{V(p) ≥ 0} = ker V(p).

Indeed, since P 1 (x) is open, there exists λ ∈ (0, 1] such that λ x 1 (s)+ tν, x 1 (-s) + (1 -λ) y 1 (s) + tν, x 1 (-s) = λx 1 (s) + (1λ)y 1 (s) + tν, x 1 (-s)) ∈ P 1 (x), then choose x 1 (s) = λx 1 (s) + (1 -λ)y 1 (s), then one has p(s) • x 1 (s) < p(s) • x 1 (s).20 Document de Travail du Centre d'Economie de la Sorbonne -2009.63

A finite collection {C i , i ∈ I} of nonempty convex cones in R n is positively semi-independent if c i ∈ C i , for all i ∈ I and i∈I c i = 0, imply that for all i ∈ I, c i = 0.

Proof of Lemma 2

Let ( p, q, x, z) be an equilibrium of the economy (E, F π ). We first claim that there exists z * i ∈ Z i (i ∈ I) such that V( p)z * i = V( p)z i for all i and z * i = 0. Indeed, by Claim 4.2, taking v i := V( p)z i for all i, one gets

Hence 0 = i∈I z * i for some z * i ∈ Z i such that V( p)z * i ≥ V( p)z i for every i. But i∈I V( p)(z * izi ) = 0 since i∈I (z * izi ) = 0 and we conclude that V( p)(z * izi ) = 0 for every i. This ends the proof of the claim.

We will show that ( p, π q, x, z * ) is an equilibrium of (E, F ). We first prove that ( xi ,

for all i and it only remains to show that π q • z * i = q • zi for all i. But, with the Market Clearing Condition z * i = 0 proved in the above claim, it suffices to show that π q • z * i ≥ q • zi for all i. Indeed, suppose that for some i one has -π q • z * i >q • zi . But this implies that W( p, q)πz * i > W( p, q)z i , with πz * i ∈ πZ i ; indeed, from above and the properties (4.1) of the mapping π, one hasq

In other words, the i-th agent has an arbitrage opportunity in the financial economy (E, F π ), which is impossible when ( p, q, x, z) is an equilibrium of (E, F π ) under Assumption C (see, for example, [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF]).

We end the proof by showing that

q) and W( p, π q)z i = W( p, q)πz i , from the properties (4.1) of the mapping π. But (x i , πz i ) ∈ B i F π ( p, q), together with (x i , πz i ) ∈ P i ( x) × πZ i contradicts the fact that ( p, q, x, z) is an equilibrium of (E, F π ).

Proof of Lemma 3

We prepare the proof of Lemma 3 with two claims. We let S (0, 1) := {y ∈ R : ||y|| = 1} and for every s ∈ S

Claim 4.3. The set P := Π s∈S P s is closed.

Proof. For every s ∈ S , let (p n (s)) n be a sequence in P s such that p n (s) → p(s). For every n ∈ N, the property defining P s is satisfied for some agent i n s ∈ I. Since there is a finite number of agents, by eventually considering a subsequence, we can assume there exists some agent i s ∈ I (independent of n), say i s = 1, such that, for all n, there exists x n ∈ A(E) and (x 1 (s), x n (-s))

• e 1 (s). Since A(E) is compact (by Assumption C), without any loss of generality, we can assume that x n → x ∈ A(E).

We show that p(s) ∈ P s . Indeed, let i = 1, let x = lim n→∞ x n as defined above, and let (x 1 (s), x 1 (-s)) ∈ P 1 (x). By Assumption FN0, P 1 has an open graph, hence for n large enough, (x 1 (s), x n 1 (-s)) ∈ P 1 (x n ) since (x 1 (s), x n 1 (-s)) → (x 1 (s), x 1 (-s)) and x n → x. From above, we deduce that p n (s)

Consequently, from the desirability assumption, we get x1 (s) + tν, x1 (-s) , z1 ∈ B 1 F ε ( p, q) ∩ [P 1 ( x) × Z 1 ], which contradicts the fact that ( p, q, x, z) is an equilibrium in (E, F ε ). This ends the proof that p ∈ P.

Since p ∈ P, and || p(s)|| = 1 for s ∈ S, we deduce that p(s) • ν ≥ ε|| p(s)|| = ε, by Claim 4.4. Consequently, V ε ( p) = V( p), for every i ∈ I, B i F ε ( p, q) = B i F ( p, q) and clearly ( p, q, x, z) is an equilibrium for (E, F ).

Proof of Lemma 4

Part (a). Let p ∈ R L . The set Q(p) is obviously a cone, and we now show that it is convex by contradiction. Suppose that there exist q 1 , q 2 in Q(p), α ∈ (0, 1) such that αq 1 + (1 -α)q 2 Q(p). Then there exists i ∈ I and there exists ζ

In the first case, we conclude that either -q 1 • ζ > 0 or -q 2 • ζ > 0 which, together with V(p)ζ ≥ 0, implies that W(p, q 1 )ζ > 0 or W(p, q 1 )ζ > 0, contradicting the fact that q 1 and q 2 are both in Q(p). Similarly, in the second case, we conclude that either -q 1 • ζ ≥ 0 or -q 2 • ζ ≥ 0 which, together with V(p)ζ > 0, contradicts the fact that q 1 and q 2 are both in Q(p).

Part (b).

Step 1. For every p, the set [START_REF] Debreu | Theory of Value[END_REF], it suffices to show that the convex cones AZ i ∩ {V(p) ≥ 0} are positively semi-independent. 10 Indeed, let ζ i ∈ AZ i ∩ {V(p) ≥ 0} for each i such that i∈I ζ i = 0. Then

Hence ζ 1 = 0 and similarly, ζ i = 0 for every i.

Step 2. -int(A o ) ⊂ Q(p). Suppose there exists q ∈ -int(A o ) and q Q(p). Then there exist i ∈ I and ζ ∈ AZ i such that W(p, q)ζ > 0. Thus ζ ∈ AZ i ∩ {V(p) ≥ 0} \ {0} ⊂ A\{0}. Recalling that A is a pointed closed convex cone (by assumption F3 and Step 1), from [START_REF] Rockafellar | Convex Analysis[END_REF], we have Step 4. The set -A + (Z F ) ⊥ is closed. Indeed, it suffices to show that the closed convex cones -A (which is closed by Step1) and (Z F ) ⊥ (a vector space, hence closed) are positively independent. This is clearly the case since -A ⊂ Z F .

The convexity of the values of Φ i is a consequence of the convexity of Brε i (p, q), B rε i (p, q), Z r i and the convexity of the values of P r i , by assumptions C and F1.