K Nafkha

E Richalot

S Mengué

O Picon

Plane wave spectra estimation in 2D reverberation chamber

Keywords: reverberation chamber, directions-of-arrival, MUSIC, mode stirring, FDTD I

In this article, we study the angular plane waves expansion of the standing electric field within a reverberation chamber. For this purpose, we adapted a high resolution Root-MUSIC based method to the context of the reverberation chamber by using a decorrelation procedure to estimate the angles of arrival and complex amplitudes of the planes waves. Using FDTD simulations to get the standing electric field cartographies, we demonstrate, at low frequencies, that directions of arrival are slightly affected by the stirrer rotation whereas the wave amplitudes are strongly perturbed. This result indicates that stirrer works as a plane wave powers tuner.

Introduction

Reverberation chambers (RC) constitute a performing tool for radiated emissions and immunity measurements. They are electrically large multimoded cavities that use either electronic [START_REF] Hill | Electronic mode stirring for reverberating chamber[END_REF] or mechanical [START_REF] Corona | Reverberating chambers as sources of stochastic electromagnetic fields[END_REF] stirring to reproduce, on the equipment under test (EUT), an average homogeneous and isotropic field. While fields within an RC are often described through their statistical distributions, it has been shown that they can be expanded on their angular plane wave spectra [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF] [START_REF] Moglie | FDTD analysis of plane wave superposition to simulate susceptibility tests in reverberation chambers[END_REF].

As plane wave distributions are often used to model indoor and urban propagation channels, it is of great interest to use a similar representation in an RC in order to relate its performance to practical multipath environments where devices under test are used.

The estimation of the directions of arrival (DOA) of wave fronts on an array of sensors is a common problem for many fields including radar, sonar [START_REF] Iwata | Application of the multiple signal classification (MUSIC) method for one-pulse burst-echo Doppler sonar data[END_REF], radio astronomy, and mobile communications [START_REF] Fuhl | High-Resolution 3-D Direction-of-Arrival Determination for Urban Mobile Radio[END_REF]. Fourier analysis is the simplest technique to determine the plane wave angular distribution in space; however, it suffers from a poor angular resolution for a limited observation surface (as in an RC), generates secondary lobes that disturb the result interpretation and necessitates a high number of sensors.

Other algorithms have been proposed to increase the angular resolution over conventional Fourier analysis. Super resolution techniques are advanced approaches possessing theoretically an infinite asymptotic resolving power: two sources can be theoretically resolved as closed and as weak as they can be. Limitations of resolving power come only from limited observation time. These techniques include multiple signal classification (MUSIC) [START_REF] Schmidt | Multiple Emitter Location and Signal Parameter Estimation[END_REF] [START_REF] Laghmardi | A Joint Space-Time Estimation Algorithm For Wideband Signals With High Resolution Capabilities[END_REF]), Root-MUSIC [START_REF] Barabell | Improving the resolution performance of eigenstructure-based direction finding algorithm[END_REF] [START_REF] Hwang | Direction of Arrival Estimation using a Root-MUSIC Algorithm[END_REF] and estimation of signal parameters via rotational invariance techniques (ESPRIT) [START_REF] Roy | ESPRIT-Estimation of Signal Parameters Via Rotational Invariance Techniques[END_REF]. We choose Root-MUSIC, one of the most powerful approaches, that returns the estimated discrete angular spectrum. This algorithm provides asymptotically unbiased estimates of the directions of arrival.

Super resolution techniques are based on eigen-decomposition of the autocorrelation matrix generated by a received signal vector. As this decomposition is possible only with a full rank matrix, the plane waves impinging on the array must be not fully correlated. We will see that, in an RC, received signals are highly correlated. Therefore, we perform a matrix preprocessing that restores the matrix rank.

The algorithm used to estimate the plane wave spectrum within the working volume of a reverberation chamber is presented in Section II. This algorithm is independent of the way signals on sensors are obtained, using simulations or measurements. Only 2D-FDTD simulation results are used in this paper. After its validation for the canonical case of an empty rectangular cavity, our method is applied to a reverberation chamber equipped with a fixed stirrer and leads to very accurate results (Section III). By rotating the stirrer, we then analyze the plane wave spectrum variation in the RC at frequencies below the lowest useable frequency (LUF). The results obtained show that, whereas directions of arrival stay approximately constant, the plane wave powers are highly affected by the stirrer rotation (Section IV). We conclude that, at low frequencies, the stirrer principally works as tuner of plane wave powers.

II.

Plane wave estimation algorithm MUSIC goniometer processes the signals received on a sensor array to compute the parameters of multiple incident wavefronts, which are the directions of arrival, the amplitudes and the phases of the waves. Root-MUSIC consists of a modification of this algorithm that improves the resolution performance, and is only applicable with an equally spaced sensor array.

II.1. The received signal

Let us consider a linear equally spaced array of N identical omnidirectional sensors constituting the receiving antenna (Figure 1). P plane wave fronts impinge on this array from directions θ i , iЄ{1,..,P}, defined from the array normal. These plane waves, generally considered as coming from P far-field point sources, are supposed to be narrowband and uncorrelated. The signal received by a sensor is a linear combination of the P plane wave contributions with added noise. Thus, the N-dimensional vector X(t m ) of the signals received

by each sensor at time t m , also called time observation, can be written as:

X(t m ) = A S(t m ) + B(t m ) (1)
where S represents the complex amplitudes of the incident wave fronts (we take the phase reference point at sensor 1), and B describes the noise at the sensors. The N*P rectangular matrix A is the directional matrix whose columns are the P directional vectors.

Each directional vector a(θ i ) is obtained by illuminating the N-sensors linear array by a unit wave plane of direction of arrival θ i , iЄ{1,..,P}. For sensors along the y-axis with interelement spacing d (Figure 1) and λ the plane wave wavelength, the directional vector is given by : a(θ i ) = (1 e j2πd.sin(θi)/λ e j4πd.sin(θi)/λ … e j(N-1).2πd.sin(θi)/λ ) T .

II.2. The Root-MUSIC algorithm

The N*N sample covariance matrix of the X vector is:

( ) ( ) ( ) ( ) ( ) 1 1 .
.

M H H xx m m m R E X t X t X t X t M ∆ = = = ∑ (2)
where (.) H denotes the Hermitian transpose, E(.) the expectation operator and M the number of observed samples.

Under the basic assumption that the incident signals and the noise are uncorrelated, R xx is a full rank matrix which can be written:

2 . . H xx R A S A = + Σ (3)
where S 2 = E(S(t).S(t) H ) is a diagonal matrix containing the plane wave powers and Σ = E(B(t).B(t) H ). In the following, the noise is assumed to be gaussian and spatially white so that Σ=σ 2 I, where I is the unit matrix.

If the number P of incident waves is less than the number of sensors N, then the minimum eigenvalue of A.S 2 .A H is zero, which occurs (N-P) times. According to Eq. ( 3) and to the eigenvector orthogonality property, the eigenvectors of R xx corresponding to the highest eigenvalues form a basis of the signal subspace, whereas the (N-P) others form a basis of the noise subspace. As a consequence, signal and noise subspaces are orthogonal. Therefore, the product between an eigenvector of the noise-subspace and a vector a(θ) is null for all direction θ corresponding to an incident wave. This orthogonality is used in MUSIC by defining a pseudo-spectrum P(θ) as:

( ) ( ) ( ) θ θ θ a V V a P H N N H 1 = (4)
where V N = [e P+1 , e P+2 , … e N ] is the matrix of noise-subspace eigenvectors. This function is maximal when θ corresponds to a signal incidence direction.

An alternative to the tracing of the MUSIC pseudo-spectrum consists of determining the poles of this function. This is the strategy adopted in Root-MUSIC [START_REF] Barabell | Improving the resolution performance of eigenstructure-based direction finding algorithm[END_REF].

The denominator of P(θ) can be written:

( ) ( ) ( ) 2 sin( ) 2 sin( ) 2 sin( ) 1 1 1 1 1 1 . . d d d N N N j k j n j l kn l k n l N D e G e g e π θ π θ π θ λ λ λ θ - - - - - = = =-+ = = ∑∑ ∑ (5)
where

G = V N. V N H
, and g l is the sum of G-coefficients along the l th diagonal.

If we define the polynomial:

( )

1 1 N l l l N D z g z - - =-+ = ∑ (6),
then the research of the directions of arrival is tantamount to one of the roots of D(z) on the unit circle [START_REF] Castanié | Spectral Analysis Parametric and Non-Parametric Digital Methods[END_REF]. The Root-MUSIC algorithm is known to offer better resolution performance and to require a lower signal-to-noise ratio than MUSIC itself.

II.3. Plane wave decorrelation procedure

The presented algorithm in II.2 has been developed in the case of several independent time observations X(t m ). We will show that an appropriate pre-processing makes it possible to use Root-MUSIC in the case of a single time observation.

In our study of the reverberation chamber, the signal is analysed at a resonant frequency, so that we only have a single snapshot of the data. It implies that the phase differences between the incident plane waves are fixed, so that the signals are seen as highly correlated. Moreover, an RC is classically excited by a single antenna, so that the plane waves are all generated by the same source and not P uncorrelated sources as required for the application of MUSIC.

When the sources (or plane waves) are highly correlated, the rank of R xx matrix is no longer the number P of directions of arrival, but the number of groups of coherent sources that are uncorrelated with each other [START_REF] Grenier | Effective correlation factor after translational & rotational invariances processing: spatial smoothing & DEESE methods[END_REF]. As a result, the rank of R xx matrix is in our case equal to one [START_REF] Gabriel | Spectral analysis and adaptive array superresolution techniques[END_REF] as all plane waves are fully correlated, and the estimation of the P directions of arrival is impossible using the Root-MUSIC algorithm.

To overcome this problem, a spatial smoothing is applied to the data vector [START_REF] Shan | On Spatial Smoothing for Direction-of-Arrival Estimation of Coherent Signals[END_REF]. This operation guaranties the nonsingularity of the covariance matrix. For an array of identical sensors, the duality between time and space can be used. To obtain several samples, we take a subarray of Q adjacent sensors out of the whole array, from one edge of the array, and write the measured field values into a Q-dimensional vector X 1 . The second subarray is obtained by shifting the first one by one sensor. The corresponding values are stored into X 2 .

This subarray translation introduces a phase difference between the observation vectors X 1 and X 2 , and is equivalent to a temporal shift. The input vectors X i obtained from the (N-Q+1) subarrays constitute the observed samples used to calculate the covariance matrix R xx as the average of the elementary covariance matrices of each X i vector, using Eq. (2). It has been shown by [START_REF] Shan | On Spatial Smoothing for Direction-of-Arrival Estimation of Coherent Signals[END_REF] that if the number of subarrays is greater than or equal to the number of signals, then the modified covariance matrix is non-singular; thus Root-MUSIC can be successfully apply to the smoothed covariance. In the studied examples, we will take subarrays having the recommended length N/2.

Thus the use of translated subarrays to average the covariance matrix compensates the absence of time variation with the frequential signal. To improve the correlation matrix properties and to enhance the method accuracy, a second treatment is performed to the covariance matrix after spatial smoothing. As we will see, this method artificially increases the number of sensors, so that the fall of the sensor number due to the use of subarrays is partially compensated.

When signals are completely uncorrelated, the correlation matrix is Toeplitz. We perform a second processing to reduce the coefficients variation on R xx -matrix diagonals [START_REF] Jansson | Analysis of forward-only and forward-backward sample covariances[END_REF][START_REF] Phaisal-Atsawasenee | Improved angular resolution of beampsace MUSIC for finding directions of coherent sources[END_REF]. It consists of averaging the previous covariance matrix and the one obtained with the same sensors but ranged in decreasing order, from N to 1. We notice here R F the first covariance matrix, and R B the second one. The average covariance matrix R FB is given by:

R FB = (R F + J R F H J)/2 (7),
where J is an N*N exchange matrix given by

              = 0 0 1 0 1 0 1 0 0 L L M O M M L J (8).
This averaging process leads to a hermitian and centro-symmetric matrix.

II.4. Plane waves number estimation

Root-MUSIC algorithm is a parametric method requiring prior knowledge of the number of plane waves to separate. In the canonical case of an empty parallelepipedic reverberation chamber (without stirrer), the number of plane waves is known analytically, but after the stirrer introduction, this number in no longer known theoretically. To overcome this problem, we refer to the information theory criterion of Rissanen and Schwartz [START_REF] Barron | The Minimum Description Length Principe in Coding and Modeling[END_REF]) (Wax & Kailath, 1985) [START_REF] Athanasios | On the Behavior of Information Theoretic Criteria for Model Order Selection[END_REF]. According to this criterion, the number of plane waves is the k value which minimizes the MDL (Minimum Description Length) quantity defined as:

( ) M k N k k N M k MDL k N N k i i N k i i log . 2 . . 2 1 . 1 log . ) ( 1 1 - +                     - - = - + = + = ∑ ∏ λ λ (9)
for k varying from 1 to N-1. N is the number of sensors, M the number of time observations and

N i i ≤ ≤ 1 , λ are R xx -matrix eigenvalues.
Because of the decorrelation procedure, we have N/2 sensors by subarray and the parameter k varies from 1 to N/2-1. The number of time observations is M = N/2-1.

II.5. Plane wave complex amplitudes

After the determination of the number of waves using the MDL criterion, the use of the previously presented Root-MUSIC algorithm associated to the decorrelation procedure leads to directions of arrival. However, for a line of sensors parallel to y-axis, only k y component of wave vectors can be determined, and the angles of incidence are extracted using the equation k y =k 0 .sinθ, where k 0 is the free space propagation constant. One k y value corresponds to two directions of arrival, θ and (π-θ), and both possible directions are considered while evaluating wave amplitudes. To resolve this ambiguity on symmetric directions of arrival, a second line of sensors parallel to the first one is used.

By neglecting the noise, Eq. (1) becomes X = A S, where (NxP) rectangular matrix A is calculated analytically using the found directions of arrival. X is a 2N-dimensional vector containing the observations on the two sensor arrays. A H A matrix is non singular because of the orthogonality of a(θ i ) vectors, so by a simple matrix inversion we can find the plane waves complex amplitudes vector S:

S = (A H A) -1 A H X (10)
Thus, amplitudes and phases of the received plane waves are known without any ambiguity.

It has to be noticed that, as standing waves are generated within the cavity, if θ i is a DOA and A i the associated complex amplitude, then (θ i -π) is also a DOA and the associated amplitude is the conjugate of A i :

( ) ( ) y k x k j y k x k j y x y x e A e A + + - + . .
* is real at every point (x,y). These two waves correspond to forward and backward travelling waves on the same propagation axis.

This study could be extended to the 3 D case of electromagnetic field distribution, where two orthogonal lines of sensors should be used to estimate correctly the elevation and azimuthal angles [START_REF] Harabi | Estimation of 2-D Direction of Arrival with an Extended Correlation Matrix[END_REF], however a particular attention should be taken to the orientation of sensors (antennas oriented along each axis) for electric field polarization consideration.

In this paper, we validate plane wave extraction technique in a 2D case in order to avoid considering the electric field polarization and for computation time reduction.

III.

Simulations results

III.1. 2D FDTD simulations

In order to reduce the computation time, simulations are performed in this paper with a 2D

RC. The simulated structure presents a cylindrical symmetry along z-axis. The fields are independent of z axis, so that only TM modes are excited with an electric field along z-axis. A 2D-FDTD tool [START_REF] Taflove | Computational electrodynamics, the finite difference time domain method[END_REF] is used to calculate the field distribution within the 2D reverberation chamber.

The simulated chambers, with and without stirrer, are of dimensions a=3.105m along x-axis and b=2.475m along y-axis as shown in Figure 1. The classical formula 6×f 0 (f 0 = empty cavity cut-off frequency) [START_REF] Mitra | Statistical Simulations and Measurements Inside a Microwave Reverberation Chamber[END_REF] for the LUF of chamber leads to 465MHz. A fine mesh of 414×330 rectangular Yee's cells is used. Chamber walls as well as the stirrer are taken to be perfectly conducting, so that the boundary condition is that electric field tangential components are equal to zero on their surfaces. The cavity excitation is performed by 10 punctual sources placed along a wall. This choice of numerous sources aims for a correct excitation of all the modes, whereas with a single punctual source a very weak amplitude is observed for a mode presenting a node near the excitation point. To obtain the frequential spectrum, a Dirac excitation imposes firstly on each of the sources a unit amplitude on E z component at t=0. Once the studied resonant frequency is chosen, the field cartography is obtained with a sinusoid excitation at the chosen frequency to avoid the excitation of adjacent modes. We can notice that, even if 10 punctual sources are used, the excitations are not independent, so that the correlation problem mentioned in Section II.3 remains unchanged and the decorrelation procedure is used.

Once electric field cartography is determined, the complex electric field is known on each cell of the mesh; so that each cell can be considered as an electric field sensor. To extract the directions of arrival, we use E z -field real values from a line of 330 cells parallel to y-axis as an observation vector. Spatial smoothing applied to decorrelate the signals on the 330 sensors of the array is performed by using subarrays of 165 sensors.

As already seen in Section II.5, the extraction of DOA from a single line of sensors does not allow distinguishing θ and (π-θ). To resolve this ambiguity, another line of sensors, parallel to and close to the first one, is used. The distance between the two sensor lines is 5×d, where d refers to intersensor distance which is equal to FDTD-cell dimension of 7.5 cm. The two lines are placed far away from the stirrer at x 1 = 3m and x 2 =3.0375m from the origin as presented in Figure 1.

The working volume of a reverberation chamber is the particular volume where the field is statistically isotropic and uniform when the RC is well stirred (beyond the LUF). It implies that plane waves have no preferred propagation direction, field polarisation nor phase. This means that, in this volume where the device under test is placed, propagation direction angles, polarisation angles and phases are uniformly distributed. In our 2D case, we define the working surface as being distant of λ LUF /3 from the cavity walls as well as from the stirrer (Figure 1), where λ LUF is the wavelength in free space at the LUF. We focus on the field properties on this particular surface of dimensions 1.778m along x-axis and 2.055m along yaxis.

Once the spectrum is estimated from 2D-FDTD simulation results, we measure its accuracy by reconstructing the electric field cartography on the working surface from the estimated directions of arrival, amplitudes and phases. This reconstructed cartography is compared to the one originally obtained from FDTD simulation, and the absolute difference between both electric fields is calculated on the working surface. We will use as an indicator the relative error obtained by normalizing this absolute difference by the average amplitude of the initial electric field on the whole working surface.

III.2. Plane wave spectrum in the empty RC

To validate our approach, an empty rectangular cavity is examined, as the fields are known analytically in this case. We consider a cavity of dimensions a along x-axis and b along yaxis, with a coordinate system origin placed at its corner. The TM mn0 modes, which satisfy the boundary conditions, have an E z -field of the following form:

            = b y n a x m E E z mn z π π sin sin 0 (11)
with a corresponding resonance frequency:

2 2 2       +       = b n a m c f mn (12)
where c is the light speed in vacuum.

By expanding the sine terms on exponentials, the resonant mode can be written as a superposition of four plane waves.

        + - - - =       + -       + -       -       + y b n x a m j y b n x a m j y b n x a m j y b n x a m j z mn z e e e e E E π π π π π π π π 4 0 (13)
The plane wave vectors are of the form k r = (k x , k y ) T = (±mπ/a, ±nπ/b) T , and the angles of incidence defined from x-axis are given by θ = arctan(k y /k x ) and θ = arctan(k y /k x ) + π.

To validate our method, we examine the TM 680 mode of theoretical resonance frequency 564.88MHz. Following the previous analytic equations, one finds four plane waves of directions of arrival θ 1 =59.13°, θ 2 = -59.13°, θ 3 =120.87°, θ 4 = -120.87°, of the same amplitude equal to 0.25× E z 0 .

The simulation of the empty rectangular cavity using the FDTD method leads to a resonance frequency of 564.97MHz, and the electric field cartography is normalized to have a maximal E z amplitude equal to one (E z 0 = 1 Volt/meter). The number of plane waves, theoretically equal to 4 for an empty cavity, is overestimated by MDL criterion as 20 planes waves are estimated; this overestimation is due to numerical noise that disturbs the waves number estimation process.

The estimated DOA corresponding to the two highest eigenvalues are θ 1 = 59.06°, θ 2 = -59.06°, θ 3 = 120.94°, θ 4 = -120.94° with the same amplitudes of 0.2493. The relative errors on directions of arrival and amplitudes are respectively about 0.1% and 0.3%. The amplitudes of the 16 other plane waves are very low, with a ratio between their amplitude and the one of the first modes below -41dB. This confirms that these modes can be considered as noise. The precision of our approach is also evaluated by comparing the initial and reconstructed cartographies on the working surface. The mean relative error, calculated as the average relative error on the whole working surface, is less than 1.5 %.

FDTD simulations as well as the DOA extraction method have been validated in this simple case where the angular spectrum is known analytically. With the study of an RC equipped with a fixed stirrer, we will now validate our plane wave spectrum estimation technique in a more realistic cavity.

III.3. Plane wave spectrum in the RC equipped with a fixed stirrer

The reverberation chamber is simulated with on oblique stirrer, rotated to 45° from the axes (Figure 1). We examine the field at the resonance frequency of 327.5MHz. The same lines of sensors as in the previous case are used to estimate the angular spectrum. After computing the smoothed bidirective covariance matrix, we use the MDL criterion which estimates 20 plane waves. The amplitudes of the 20 plane waves found by our algorithm are less disparate than for the empty cavity: the ratio between the maximal and the minimal amplitudes is of 2.7. It seems to indicate the MDL criterion doesn't overestimate the number of waves in this case.

As the solution is not analytically known with a stirrer, the accuracy of the spectrum evaluation can only be evaluated by the comparison between the initial and the reconstructed cartographies of E z -field. As the mean relative error found is less than 3%, the method is considered as validated and can be used to study the variation of the plane wave spectrum due to the stirrer rotation.

IV. Study of the stirrer rotating effect

Two fundamental parameters of the RC operation are deduced from FDTD simulations: the frequency spectrum which is the Fourier transform of the impulse response and the electromagnetic field cartography at each resonant frequency. By rotating the stirrer around its axis by steps of 2.5° from 0° to 180° (and not 360° for symmetry considerations), it is possible to follow the variation of resonant frequencies on a stirrer rotation and to deduce from it E z -field cartographies for each stirrer position.

We choose to follow the variation of a mode of resonant frequencies inferior than the LUF ([323MHz-328 MHz], see Figure 2). Its frequency excursion is about 5MHz. In order to explain the influence this frequency variation can have on the cavity response, we also present the resonant frequency variation of the two adjacent modes. Their resonant frequency excursions are about 7 MHz. Although the relative resonant frequency variations are small (about 2 %), the related influence on the field properties is important. Thus a given frequency, for example 327.3MHz, can correspond to the resonant frequency of two different modes for different stirrer positions (see Figure 2), and these two modes will present very different field distribution. Moreover, this effect increases with the frequency due to the increase of the resonant mode density. The choice of a low resonant frequency in the paper is justified by the low density of modes and by the low density of plane waves on angular spectrum that permits an easier analysis than at high frequencies.

For each stirrer position, once the resonant frequency and the associated E z -cartography are determined, we use the presented plane wave estimation algorithm to estimate all the plane wave parameters (DOA, complex amplitude).

For a better legibility, we focus on the DOA comprised between 0° and 90°. Five plane waves are found on this interval upon 20 waves in total. We first of all examine the plane wave DOA as a function of the stirrer position (Figure 3). We observe that directions of arrival are slightly affected by the stirrer rotation. The whole couples of DOA and associated amplitude, found on a complete stirrer rotation, are represented in a polar diagram (Figure 4), where the angle indicates the DOA and the radius the plane wave power. In agreement with the remark drawn with Figure 3, we can notice that DOA are grouped together around 5 principal directions. On the contrary, a large variation of wave powers is observed, with a variation of the wave amplitudes that reaches more than 45dB. We conclude that, below the LUF, the stirrer can be used as plane wave power tuner.

V. Conclusion

This paper presents a method to estimate the plane wave spectrum within a reverberation chamber. In our frequential study of the cavity, Root-MUSIC method cannot be directly applied because of the high correlation between wave fronts. Particular attention is paid to this problem, and a pre-processing is applied to decorrelate the received signals. The study of the canonical case of the empty cavity shows that directions of arrival, amplitudes and phases of the plane waves are obtained very precisely with relative errors below 0.3%. The accuracy of this method is also demonstrated for an RC equipped with a stirrer, with a mean difference between the cartography issued from FDTD and the one reconstructed from the extracted wave parameters below 3% at about 565MHz.

With this method we attempt to contribute to a better understanding of the stirring process within an RC. Thus a study of the stirrer rotation effect on plane wave DOA and powers is presented. For the studied frequency band, situated below the LUF, the stirrer rotation generates a large variation of wave powers whereas its effect on DOA distribution is very low.

Figure 1 .

 1 Figure 1. Geometry of the 2D RC studied

Figure 4 .

 4 Figure 4. Polar diagram of plane wave directions of arrival and powers on a stirrer rotation


Elena L Del Mercato
email: elena.delmercato@univ-paris1.fr

Vincenzo Platino
email: vin-cenzo.platino@gmail.com.

Vincent Iehlé

Michael Magill

Antoine Mandel

Martine Quinzii

Tarun Sabarwal

Ross Starr

Emily Tanimura

We

On the regularity of smooth production economies with externalities: Competitive equilibrium à la

Keywords: JEL classification: C62, D51, D62 externalities, private ownership economies, competitive equilibrium à la Nash, regular economies

We consider a general equilibrium model of a private ownership economy with consumption and production externalities. The choices of all agents (households and firms) may affect utility functions and production technologies. The allocation of a competitive equilibrium is a Nash equilibrium. We provide an example showing that, under standard assumptions, competitive equilibria are indeterminate in an open set of the household's endowments. We introduce the model with firms' endowments, in the spirit of Geanakoplos, Magill, Quinzii and Drèze (1990). In our model, firms' endowments impact the technologies and the marginal productivities of the other firms. We then prove that these economies are generically regular in the space of endowments of households and firms.

Introduction

The Arrow-Debreu model of general equilibrium has been extended to economies with consumption and production externalities. For such extensions, one has to choose an equilibrium notion. From a normative point of view, markets can be extended in order to obtain the two Fundamental Theorems of Welfare Economics also for such economies, i.e., a perfect internalization of the externalities. This is the idea sketched by [START_REF] Arrow | The organization of economic activity: Issues pertinent to the choice of market versus non-market allocation[END_REF] and first analyzed by [START_REF] Laffont | Decentralization with externalities[END_REF]. They enlarge the choice sets of the agents and introduce personalized prices that agents face on markets for rights on the consumption and production of any other agent in the economy. Recent contributions by [START_REF] Magill | A Theory of the Stakeholder Corporation[END_REF], and [START_REF] Crès | Production externalities: Internalization by voting[END_REF], explore instead corporate governance policies that induce firms to internalize the externalities by maximizing some "social criterion" in order to increase the welfare of stakeholders or shareholders, depending on their respective contribution. Their analysis then focuses on legal systems that allow these goals to be achieved.

On the other hand, the positive theory of competitive equilibrium leads to a definition of equilibrium that combines Arrow-Debreu with Nash, that is, agents (households and firms) maximize their goals by taking as given both the commodity prices and the choices of every other agent in the economy. At equilibrium, agents' optimal choices are mutually consistent and markets clear. This is the notion given in [START_REF] Arrow | General competitive analysis[END_REF], [START_REF] Laffont | Fundamentals of public economics[END_REF]. This notion includes as a special case the classical equilibrium definition without externalities. With such an equilibrium notion, agents cannot choose the consumption and production of other agents, i.e., externalities cannot be internalized, and competitive markets may prevent equilibrium allocations from being Pareto optimal.

We consider a private ownership economy with a finite number of commodities, households and firms. Utility and transformation functions may be affected by the consumption and production activities of all other agents. We take the conventional non-cooperative view of market equilibrium. Our purpose is to provide the genericity of regular economies. We recall that an economy is regular if it has a finite (odd) number of equilibria and every equilibrium locally depends in a differentiable manner on the parameters describing the economy. Therefore, the equilibria of a regular economy are locally unique and persistent under small perturbations of the economy. Furthermore, at a regular economy, it is possible to perform classical comparative statics, see [START_REF] Smale | Global analysis and economics[END_REF], [START_REF] Mas-Colell | The theory of general economic equilibrium. A differentiable approach[END_REF], [START_REF] Balasko | Foundations of the general equilibrium theory[END_REF]. Quite a few works deal with the generic regularity of economies with externalities, see [START_REF] Crès | Symmetric smooth consumption externalities[END_REF], [START_REF] Bonnisseau | Regular economies with non-ordered preferences[END_REF], [START_REF] Kung | Voluntary contributions to multiple public goods in a production economy with widespread externalities[END_REF], [START_REF] Mandel | An index formula for production economies with externalities[END_REF], [START_REF] Bonnisseau | Externalities, consumption constraints and regular economies[END_REF], and [START_REF] Balasko | Wealth concerns and equilibrium[END_REF]. However, with the exception of [START_REF] Mandel | An index formula for production economies with externalities[END_REF], all these authors focus on economies with externalities only on the consumer side. 2Apart from the intrinsic interest in a regularity result, regular economies are also important for the study of Pareto improving policies in terms of taxes and subsidies. In the presence of other sources of market failures, such as incomplete financial markets and public goods, there is a well established methodology for analyzing these policies, see [START_REF] Geanakoplos | Existence, regularity and constrained suboptimality of competitive allocations when the asset market is incomplete[END_REF], [START_REF] Geanakoplos | Generic inefficiency of stock market equilibrium when markets are incomplete[END_REF], [START_REF] Citanna | Constrained suboptimality in incomplete markets: A general approach and two applications[END_REF], [START_REF] Citanna | The taxation of trades in assets[END_REF], Villanacci andZenginobuz (2006, 2012). Such a methodology applies to the set of regular economies, since it requires equilibria to be differentiable maps of the fundamentals. Therefore, our contribution provides a solid foundation for the analysis of these kinds of policies in the presence of externalities, in the spirit of [START_REF] Greenwald | Externalities in economies with imperfect information and incomplete markets[END_REF] and [START_REF] Geanakoplos | Pareto improving taxes, Jour-29 Documents de travail du Centre d'Economie de la Sorbonne -2015.44R[END_REF]. 3We make basic assumptions on utility and transformation functions that are standard in "smooth" equilibrium models without externalities. These assumptions guarantee the non-emptiness and the compactness of the set of equilibria. 4 However, even in the simpler case of consumption externalities, they are not sufficient to establish classical generic regularity. For economies with consumption externalities, establishing generic regularity requires the introduction of an additional assumption on the second order effects of externalities on individual utility functions, see [START_REF] Bonnisseau | Externalities, consumption constraints and regular economies[END_REF]. 5 However, the analogous assumption on the effects of production externalities on transformation functions is not going to work. We provide an example of a private ownership economy with one household and two firms where, despite well behaved second order external effects, equilibria are indeterminate in an open set of the household's endowments (and the indeterminacy is payoff relevant).

In order to overcome indeterminacy, we incorporate firms' endowments in the general model, in the spirit of [START_REF] Geanakoplos | Generic inefficiency of stock market equilibrium when markets are incomplete[END_REF]. 6 Firms' endowments consist of amounts of commodities held initially by the firms. In our model, firms' endowments have an impact on the production sets of the other firms. Consequently, perturbing these endowments affects the first and the second order effects of production externalities on transformation functions, thereby allowing us to establish generic regularity. 7 Our main theorem (Theorem 16) states that competitive equilibria are determinate in an open and full measure subset of the endowments of households and firms. Most of the classical properties of the equilibrium manifold then hold true.

In order to prove Theorem 16, we adapt Smale's approach to economies with consumption and production externalities. Smale's approach has been used for other economic environments, see for instance [START_REF] Crès | Symmetric smooth consumption externalities[END_REF], [START_REF] Cass | Generic regularity of competitive equilibria with restricted participation[END_REF], [START_REF] Villanacci | Existence and regularity of equilibria in a general equilibrium model with private provision of a public good[END_REF], and [START_REF] Bonnisseau | Externalities, consumption constraints and regular economies[END_REF]. This approach is an alternative to the aggregate excess demand-excess supply approach. Notice that in our economic environment, the aggregate excess demand-excess supply approach is problematic. This is because the individual demands and supplies are interdependent, making it difficult to define the aggregate excess demand-excess supply out of equilibrium.

The paper is organized as follows. In Section 2, we introduce the classical model, basic assumptions and definitions. In Subsection 2.3, we discuss the difficulties of establishing classical generic regularity. In Section 3, we provide our example of indeterminacy and we analyze the second order external effects in the example. Section 4 is devoted to our regularity result. We present our model with firms' endowments and we prove that these economies are regular in an open and full measure subset of the endowments of households and firms. All the lemmas are proved in Section 5. In Section 6, one finds classical results 5 In the absence of this assumption, these authors provide an example where equilibria are indeterminate for all initial endowments. 6 In Geanakoplos, [START_REF] Geanakoplos | Generic inefficiency of stock market equilibrium when markets are incomplete[END_REF], there are no direct externalities in preferences and production sets. However, their model exhibits pecuniary externalities arising from the incompleteness of financial markets. 7 We do not rely on logarithmic or quadratic perturbations of the payoff functions that are commonly used to establish generic regularity in non-cooperative games, see for instance [START_REF] Harsanyi | Oddness of the number of equilibrium points: A new proof[END_REF], [START_REF] Ritzberger | The theory of normal form games from the differentiable viewpoint[END_REF][START_REF] Van Damme | Strategic equilibrium[END_REF].

from differential topology used in our analysis and the Jacobian matrix of the equilibrium function.

The general model

We consider a private ownership economy. There is a finite number of commodities labeled by the superscript c ∈ C := {1, . . . , C}. The commodity space is R C . There are a finite number of firms labeled by the subscript j ∈ J := {1, . . . , J} and a finite number of households labeled by the subscript h ∈ H := {1, . . . , H}.

The production plan of firm j is y j := (y 1 j , .., y c j , .., y C j ). As usual, if y c j > 0 then commodity c is produced as an output, if y j < 0 then commodity is used as an input. The production plan of firms other than j is y -j := (y f ) f =j , and let y := (y j ) j∈J . The consumption of household h is x h := (x 1 h , .., x c h , .., x C h ). The consumption of households other than h is x -h := (x k ) k =h , and let x := (x h ) h∈H .

The production set of firm j is described by a transformation function t j . 8 The main innovation of this paper is that the transformation function t j may depend on the production and consumption activities of all other agents. That is, t j describes both the technology of firm j and the way in which its technology is affected by the activities of the other agents. More precisely, for a given externality (y -j , x), the production set of the firm j is

Y j (y -j , x) := y j ∈ R C : t j (y j , y -j , x) ≤ 0 where t j is a function from R C × R C(J-1) × R CH ++ to R. Let t := (t j ) j∈J .
The preferences of household h are described by a utility function,

u h : (x h , x -h , y) ∈ R C ++ × R C(H-1) + × R CJ -→ u h (x h , x -h , y) ∈ R u h (x h , x -h , y) is the utility level of household h associated with (x h , x -h , y).
That is, u h also describes the way in which the preferences of household h are affected by the activities of the other agents. Let u := (u h ) h∈H .

The endowment of household h is e h := (e for every firm j ∈ J . A private ownership economy is E := ((u, e, s), t).

The price of one unit of commodity c is p c ∈ R ++ , and let p := (p 1 , .., p c , .., p C ). Given w = (w 1 , .., w c , .., w C ) ∈ R C , denote w \ := (w 1 , .., w c , .., w C-1 ) ∈ R C-1 .

Basic assumptions

In this subsection, we make the following set of assumptions to establish the non-emptiness and the compactness of the equilibrium set.

Assumption 1 For all j ∈ J ,

(1) The function t j is a C 2 function.

(2) For every (y

-j , x) ∈ R C(J-1) × R CH ++ , t j (0, y -j , x) = 0. (3) For every (y -j , x) ∈ R C(J-1) × R CH
++ , the function t j (•, y -j , x) is differentiably strictly quasi-convex, i.e., for all y j ∈ R C , D 2 y j t j (y j , y -j , x) is positive definite on Ker D y j t j (y j , y -j , x).9 (4) For every (y -j , x) ∈ R C(J-1) ×R CH ++ , D y j t j (y j , y -j , x) 0 for all y j ∈ R C .

Fixing the externalities, the assumptions on t j are standard in smooth general equilibrium models. Points 1 and 4 of Assumption 1 imply that the production set is a C 2 manifold of dimension C and its boundary is a C 2 manifold of dimension C -1. Point 2 of Assumption 1 states that inaction is possible. Consequently, using standard arguments from profit maximization, the wealth of household h associated with his endowment e h ∈ R C ++ and his profit shares is strictly positive for every price p ∈ R C ++ . Thus, one deduces the non-emptiness of the interior of the individual budget constraint. Point 3 of Assumption 1 implies that the production set is strictly convex. Furthermore, if the profit maximization problem has a solution then it is unique, because the function t j (•, y -j , x) is continuous and strictly quasi-convex. We remark that t j is not required to be quasi-convex with respect to all the variables, i.e., we do not require the production set to be convex with respect to the externalities. Point 4 of Assumption 1 implies that the function t j (•, y -j , x) is strictly increasing and so the production set satisfies the classical "free disposal" property.

Remark 2 Our analysis holds true if some commodities are not involved in the technological process of firm j. In this case, for every firm j, one defines the set C j of all the commodities c ∈ C that are involved in the technological process of firm j, where C j denotes the cardinality of the set C j with 2 ≤ C j ≤ C. The production plan of firm j is then defined as y j := (y c j ) c∈C j ∈ R C j and the transformation function

t j is a function from R C j × f =j R C f × R CH ++ to R.
In this case, all the assumptions on the transformation functions are written just replacing

R C × R C(J-1) × R CH ++ by R C j × f =j R C f × R CH ++ .
Furthermore, in the definition of a competitive equilibrium, one also adapts the market clearing condition for every commodity c by considering only firms that use commodity c in their technological process. That is, for every commodity c, the sum over j ∈ J is replaced by the sum over j ∈ J (c) := {j ∈ J : c ∈ C j }.

Given (x, y) ∈ R CH ++ × R CJ , the set of the production plans that are consistent with the externality (x, y) is defined by

Y (x, y) := {y ∈ R CJ : t j (y j , y -j , x) ≤ 0, ∀ j ∈ J } (1)
The next assumption provides a boundedness condition on all the sets of feasible production plans that are consistent with the externalities. 10

Assumption 3 (Uniform Boundedness) Given r ∈ R C ++ , there exists a bounded set C(r) ⊆ R CJ such that for every

(x, y) ∈ R CH ++ × R CJ , Y (x, y) ∩ {y ∈ R CJ : j∈J y j + r 0} ⊆ C(r)
The following lemma is an immediate consequence of Assumption 3.

Lemma 4 Given r ∈ R C ++ , there exists a bounded set K(r) ⊆ R CH ++ × R CJ such that for every (x, y) ∈ R CH ++ × R CJ , the following set is included in K(r). A(x, y; r) := {(x , y ) ∈ R CH ++ × R CJ : y ∈ Y (x, y) and h∈H x h - j∈J y j ≤ r}
10 Assumption 3 is analogous to several conditions used to establish the existence of an equilibrium with externalities in production sets, that is, the boundedness condition given in [START_REF] Arrow | General competitive analysis[END_REF], page 134 of Section 2 in Chapter 6; Assumption UB (Uniform Boundedness) in [START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF]; Assumption P(3) in [START_REF] Mandel | An index formula for production economies with externalities[END_REF]; Assumption 3 in del [START_REF] Del Mercato | Private ownership economy with externalities and existence of competitive equilibria: A differentiable approach[END_REF].

It is well known that the boundedness of the set of feasible allocations is a crucial condition for proving the existence of an equilibrium. However, Assumption 3 is a stronger version of the standard boundedness condition used for economies without externalities, because it guarantees that the set of feasible allocations A(x, y; r) is uniformly bounded with respect to the externalities. It means that the bounded set K(r) that includes the set of feasible allocations is independent of the externality effects. In particular, it implies the boundedness of the set of feasible allocations that are mutually consistent, i.e., the set

F(r) = {(x, y) ∈ R CH ++ ×R CJ : t j (y j , y -j , x) ≤ 0, ∀ j ∈ J and h∈H x h - j∈J y j ≤ r}.
Notice that in order to prove the existence of an equilibrium it would not be sufficient to assume only the boundedness of the set F(r). 11

Assumption 5 For all h ∈ H,

(1) The function u h is continuous in its domain and C 2 in the interior of its domain.

(2) For every

(x -h , y) ∈ R C(H-1) ++ × R CJ , the function u h (•, x -h , y) is diffe- rentiably strictly increasing, i.e., D x h u h (x h , x -h , y) 0 for all x h ∈ R C ++ . (3) For every (x -h , y) ∈ R C(H-1) ++ × R CJ , the function u h (•, x -h , y) is diffe- rentiably strictly quasi-concave, i.e., for all x h ∈ R C ++ , D 2 x h u h (x h , x -h , y) is negative definite on Ker D x h u h (x h , x -h , y). (4) For every (x -h , y) ∈ R C(H-1) + × R CJ and for every u ∈ Im u h (•, x -h , y), cl R C {x h ∈ R C ++ : u h (x h , x -h , y) ≥ u} ⊆ R C

++

Fixing the externalities, the assumptions on u h are standard in smooth general equilibrium models. Point 3 of Assumption 5 implies that the upper contour sets are strictly convex. Consequently, if the utility maximization problem has a solution then it is unique. Notice that u h is not required to be quasi-concave with respect to all the variables, i.e., we do not require preferences to be convex with respect to the externalities. Point 4 of Assumption 5 implies the classical Boundary Condition (BC), i.e., the closure of the upper counter sets is included in R C ++ .

Notice that in Points 1 and 4 of Assumption 5, we allow for consumption externalities x -h on the boundary of the set R

C(H-1) ++

in order to handle the 11 In Chapter 6 of [START_REF] Arrow | General competitive analysis[END_REF], the authors have recognized the need to assume the boundedness of a wider set of feasible production allocations than the ones that are mutually consistent, in order to extend their existence proof to the case of externalities. In [START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF], Assumption UB is needed to find the cube to compactify the economy in order to use fixed point arguments. Assumption P(3) in [START_REF] Mandel | An index formula for production economies with externalities[END_REF] or Assumption 3 in del Mercato and Platino ( 2015) are used to show that the set of feasible allocations is bounded once externalities move along a homotopy arc.

behavior of u h as consumption externalities approach the boundary. Points 1 and 4 of Assumption 5 imply that BC is still valid whenever consumption externalities converge to zero for some commodities. 12 This property is used to prove properness properties of the equilibrium set (see Step 2 of the proof of Lemma 17 in Section 5). If u h does not satisfy BC whenever consumption externalities converge to zero for some commodities, we provide below an alternative assumption on u h from which one still gets Step 2 in the proof of Lemma 17, while maintaining Points 1 and 4 of Assumption 5 only for consumption externalities in R

C(H-1) ++

. 13

(5) There exists δ > 0 such that for every

(x -h , y) ∈ R C(H-1) ++ × R CJ and for every (x h , x h ) ∈ R 2C ++ , if u h (x h , x -h , y) > u h (x h , x -h , y), then u h (x h , x -h + δ1, y) ≥ u h (x h , x -h + δ1, y) where 1 := (1, . . . , 1) ∈ R C ++ . 14

Competitive equilibrium and equilibrium function

In this subsection, we provide the definition of competitive equilibrium and the notion of equilibrium function.

We use commodity C as the "numeraire good". Given

p \ ∈ R C-1 ++ , let p := (p \ , 1) ∈ R C ++ . Definition 6 (Competitive equilibrium) (x * , y * , p * \ ) ∈ R CH ++ ×R CJ ×R C-1
++ is a competitive equilibrium for the economy E if for all j ∈ J , y * j solves the following problem

max y j ∈R C p * • y j subject to t j (y j , y * -j , x * ) ≤ 0
(2)

12 A simple example of utility function that satisfies this property is given by any additively separable function

u h (x h , x -h ) = u h (x h ) + v h (x -h ) where u h is defined on R C
++ and satisfies the classical BC, and v h is defined on R C(H-1) +

. 13 For example, consider two commodities and the utility function

u h (x 1 h , x 2 h , x 1 k ) := x 1 h x 2 h x 1 k where (x 1 h , x 2 h ) ∈ R 2 ++ and x 1 k ∈ R + .
This function does not satisfy BC whenever the externality x 1 k converges to zero, but it satisfies Point 5. 14 A simple example of utility function that satisfies Point 5 is given by any multiplicatively separable function

u h (x h , x -h ) = u h (x h )m h (x -h ) where u h is defined on R C ++ and m h (x -h ) > 0 for every x -h ∈ R C(H-1) ++ .
for all h ∈ H, x * h solves the following problem

max x h ∈R C ++ u h (x h , x * -h , y * ) subject to p * • x h ≤ p * • (e h + j∈J s jh y * j ) (3)
and (x * , y * ) satisfies market clearing conditions, that is

h∈H x * h = h∈H e h + j∈J y * j (4)
The proof of the following proposition is standard, because in problems ( 2) and ( 3) each agent takes as given both the price and the choices of every other agent in the economy.

Proposition 7

(1) From Assumption 1, if y * j is a solution to problem (2), then it is unique and it is completely characterized by KKT conditions. 15(2) From Point 2 of Assumption 1 and Assumption 5, there exists a unique solution x * h to problem (3) and it is completely characterized by KKT conditions.

(3) As usual, from Point 2 of Assumption 5, household h's budget constraint holds with an equality. Thus, at equilibrium, due to the Walras law, the market clearing condition for commodity C is "redundant". So, one replaces condition (4) with

h∈H x * \ h = h∈H e \ h + j∈J y * \ j . Let Ξ := (R C ++ × R ++ ) H × (R C × R ++ ) J × R C-1
++ be the set of endogenous variables with generic element ξ := (x, λ, y, α, p \ ) := ((x h , λ h ) h∈H , (y j , α j ) j∈J , p \ ) where λ h denotes the Lagrange multiplier associated with household h's budget constraint, and α j denotes the Lagrange multiplier associated with firm j's technological constraint. We describe the competitive equilibria associated with the economy E using the equilibrium function

F E : Ξ → R dim Ξ , F E (ξ) := ((F h.1 E (ξ) , F h.2 E (ξ)) h∈H , (F j.1 E (ξ) , F j.2 E (ξ)) j∈J , F M E (ξ)) (5)
where

F h.1 E (ξ) := D x h u h (x h , x -h , y) -λ h p, F h.2 E (ξ) := -p • (x h -e h - j∈J s jh y j ), F j.1 E (ξ) := p -α j D y j t j (y j , y -j , x), F j.2 E (ξ) := -t j (y j , y -j , x), and F M E (ξ) := h∈H x \ h - j∈J y \ j - h∈H e \ h .
The vector ξ * = (x * , λ * , y * , α * , p * \ ) ∈ Ξ is an extended equilibrium for the economy E if and only if F E (ξ * ) = 0. We simply call ξ * an equilibrium.

Theorem 8 (Existence and compactness) The equilibrium set F -1 E (0) is non-empty and compact.

del Mercato and Platino (2015) provide a proof of Theorem 8.

Regular economies

We recall below the formal notion of regular economy.

Definition 9 (Regular economy) E is a regular economy if F E is a C 1 function and 0 is a regular value of F , i.e., for every

ξ * ∈ F -1 E (0), the diffe- rential mapping D ξ F E (ξ * ) is onto.
Using Smale's approach, the definition of regular economy becomes a very intelligible notion. The fact that the Jacobian matrix D ξ F E (ξ * ) is non-singular simply means that the linear approximation at ξ * of the non-linear equilibrium system F E (ξ) = 0 has a unique solution. Then, applying the Implicit Function Theorem, around ξ * , the solution of the equilibrium system is a differentiable mapping of the parameters describing the economy. Furthermore, if the equilibrium set F -1 E (0) is non-empty and compact, as a consequence of the Regular Value Theorem (Corollary 22 in Subsection 6.1), one easily deduces that a regular economy has a finite number of equilibria.

In the presence of externalities, the possibility of equilibrium indeterminacy cannot be excluded by making standard assumptions. Indeed, the equilibrium notion given in Definition 6 has the characteristics described in what follows. All the agents take as given both the price and the choice of every other agent in the economy. Given the price and the choices of the other agents, the individual optimal solutions are completely determined. But, for a given price, the equilibrium allocation ((x * h ) h∈H , (y * j ) j∈J ) is a Nash equilibrium, and the problem is that, under standard assumptions, one may get indeterminacy in Nash equilibrium. We illustrate the reason below.

Consider the equilibrium function F E defined in (5). The economy E remains fixed, thus we omit the subscript E. The price p \ is fixed. Consider all the equilibrium equations except the L -1 market clearing conditions by defining the following function G. G(q) := ((F h.1 (q, p \ ), F h.2 (q, p \ )) h∈H , (F j.1 (q, p \ ), F j.2 (q, p \ )) j∈J )

where q := (x, λ, y, α) = ((x h , λ h ) h∈H , (y j , α j ) j∈J ), so that we write ξ as (q, p \ ).

Every q * = ((x * h , λ * h ) h∈H , (y * j , α * j ) j∈J ) such that G(q * ) = 0 provides the individual demands and supplies ((x * h ) h∈H , (y * j ) j∈J ) that are mutually consistent at the price p \ .

Consider the Jacobian matrix of that system, that is, D q G(q * ) := D q ((F h.1 (q * , p \ ), F h.2 (q * , p \ )) h∈H , (F j.1 (q * , p \ ), F j.2 (q * , p \ )) j∈J ) (6) In the absence of externalities, the Jacobian matrix D q G(q * ) is non-singular. 16This is because the transformation and utility functions are respectively differentiably strictly quasi-convex and strictly quasi-concave in the individual choice. Then, the Implicit Function Theorem implies that, locally, the individual demands and supplies are a C 1 mapping of the price.

It turns out that, in the presence of externalities, under standard assumptions, the Jacobian matrix D q G(q * ) is not necessarily non-singular. Consequently, the individual demands and supplies ((x * h ) h∈H , (y * j ) j∈J ) that are mutually consistent may not be a differentiable mapping of the price. The matrix D q G(q * ) may fail to be nonsingular due to the presence of some of the following effects:

(1) the second order external effects on utility functions arising from the derivatives of F h.1 (q, p \ ) with respect to (x -h , y), (2) the second order external effects on transformation functions arising from the derivatives of F j.1 (q, p \ ) with respect to (y -j , x), (3) the first order external effects on transformation functions arising from the derivatives of F j.2 (q, p \ ) with respect to (y -j , x).

In economies without externalities, all these effects are equal to zero.

One might believe that if the utility and transformation functions are respectively differentiably strictly quasi-convex and strictly quasi-concave with respect to all the variables (i.e., individual choice and externalities), then the matrix D q G(q * ) is non-singular. This belief is wrong, since the matrix D q G(q * ) does not actually involve the whole Hessian matrix of the utility and transformation functions. The matrix D q G(q * ) involves only a partial block of rows of those Hessian matrices. This is because the first order effects of externalities on utility and transformation functions do not appear in the first order conditions associated with the individual maximization problems, i.e., in the system G(q) = 0.

In the case of pure exchange economies with externalities, Bonnisseau and del Mercato (2010) have introduced a specific assumption on the second order external effects on utility and possibility functions.17 We adapt their assumption on utility functions to our framework.

Assumption 10 (Bonnisseau and del Mercato, 2010)

Let (x, y) ∈ R CH ++ × R CJ such that the gradients (D x h u h (x h , x -h , y)) h∈H are positively collinear. Let v ∈ R CH such that h∈H v h = 0 and v h ∈ Ker D x h u h (x h , x -h , y) for every h ∈ H. Then, v h k∈H D 2 x k x h u h (x h , x -h , y)(v k ) < 0 whenever v h = 0.
We refer to [START_REF] Bonnisseau | Externalities, consumption constraints and regular economies[END_REF] for the interpretation of this assumption and an example of utility functions that satisfy this assumption.

Notice that Assumption 10 does not require the Hessian matrix of u h (with respect to all the variables) to be positive definite on Ker Du h (x h , x -h , y).

That is, u h is not required to be differentiably strictly quasi-concave with respect to all the variables. We face positive forms that may induce to think of strict quasi-concavity, but actually in Assumption 10 one takes into account only a partial block of rows of the Hessian matrix of u h . Assumption 10 is in the same spirit as the assumption of diagonally strict concavity introduced in Rosen (1965) on a weighted sum of the payoff functions of the agents. However, compared with the condition of Theorem 6 of [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF], Assumption 10 is easier to read, it does not involve any vector of weights and it focuses only on directions (v h ) h∈H that sum to zero where v h is orthogonal to the gradient

D x h u h (x h , x -h , y).
Under Assumption 10, the classical result of generic regularity holds true if there are no externalities in the production sets.

In the presence of external effects on the production side, an assumption on the transformation functions analogous to Assumption 10 is not sufficient for establishing generic regularity in the space of households' endowments.18 This is shown in the next section by means of an example.

3 Infinitely many competitive equilibria: An example

In the previous section, we have explained why standard arguments for the genericity of regular economies may not apply to economies with externalities.

In this section, we provide an example of a private ownership economy with one household and production externalities between two firms where equilibria are indeterminate in an open set of the household's endowments. In the example, we get infinitely many equilibria because there are infinitely many mutually consistent supplies. Importantly, this allocations indeterminacy translates into a "price relevant indeterminacy". That is, one gets infinitely many equilibrium prices, and consequently the indeterminacy has an impact on the welfare of the economy.

Example. There are two commodities. There is one household, x = (x 1 , x 2 ) denotes the consumption of the household, e = (e 1 , e 2 ) is his initial endowment and the utility function is given by u(x 1 , x 2 ) = x 1 x 2 . Therefore, there are no externalities on the consumption side and then Assumption 10 is obviously satisfied.

There are two firms, the production plan of firm j is y j = (y 1 j , y 2 j ). Without loss of generality, for simplicity of exposition, the subscript f denotes the subscript -j, so that the production plan of the firm other than j is y f = (y 1 f , y 2 f ). Both firms use commodity 2 to produce commodity 1.

The production technology of firm j is affected by the production plan of the other firm in the following way. Given y f , the production set of firm j is

Y j (y f ) = {y j ∈ R 2 : y 2 j ≤ 0 and y 1 j ≤ f j (y 2 j , y f )}
where the production function f j is defined by

f j (y 2 j , y f ) := 2φ(y f ) -y 2 j with φ(y f ) :=            y 1 f 2 -y 2 f if y 2 f < -ε 2 y 1 f 2ε if y 2 f ∈ [-ε 2 , 0]
where 0 < ε < 1. The production function f j is not completely smooth with respect to the externality. 19 But, it goes in the essence of the problem.

For every firm j = 1, 2, Assumption 1 is satisfied for every externality y f = (y 1 f , y 2 f ) with y 1 f > 0 and y 2 f < -ε 2 . Therefore, in what follows,

(1) we focus on equilibria where the amounts of output are strictly positive and the amounts of input are strictly lower than -ε 2 ,

(2) φ(y f ) is then given by y 1 f 2 -y 2 f according to the definition above.

The price of commodity 2 is normalized to 1. By Definition 6, we have that (x * , y * 1 , y * 2 , (p * , 1)) is a competitive equilibrium if for every j = 1, 2, y * j solves max

y 1 j >0, y 2 j <-ε 2 p * y 1 j + y 2 j subject to y 1 j -2φ(y * f ) -y 2 j ≤ 0 x * solves the following problem max x∈R 2 ++ x 1 x 2 subject to p * x 1 + x 2 ≤ p * e 1 + e 2 + 2 j=1 (p * y * 1 j + y * 2 j ) (7)
and markets clear.

For each firm j = 1, 2, let α j be the Lagrange multiplier associated with firm j's technological constraint. The KKT conditions associated with firm j's maximization problem are given by

         p * = α j , 1 = α j φ(y * f ) -y 2 j y 1 j = 2φ(y * f ) -y 2 j
Consequently, one gets the following equilibrium equations for every j = 1, 2,

y * 2 j = -(p * ) 2 [φ(y * f )] 2 and y * 1 j = 2p * [φ(y * f )] 2
and one easily deduces that, at equilibrium,

y * 1 1 = y * 1 2 = - 2y * 2 2 p * and y * 2 1 = y * 2 2 for any y * 2 2 < -ε 2 (8)
Thus, at equilibrium, the aggregate profit is given by π * := -2y * 2 2 and the optimal solution of the household is given by

x * 1 = 1 2p * (p * e 1 + e 2 + π * ) and x * 2 = p * x * 1 (9)
Using the market clearing condition for commodity 2, the equilibrium price is

p * = e 2 + 6y * 2 2 e 1 (10)
Notice that p * > 0 if and only if y * 2 2 > -e 2 6 . Therefore, using (8), ( 9) and ( 10), any bundle

(x * , y * 1 , y * 2 , (p * , 1)) with y * 2 2 ∈ - e 2 6
, -ε 2 is a competitive equilibrium. Thus, for all initial endowments that belong to the open set {e = (e 1 , e 2 ) ∈ R 2 ++ : e 2 > 6ε 2 } we get infinitely many equilibria parametrized by y * 2 2 .

We now show that the economy of the example exhibits well behaved second order external effects. For this purpose, we provide below the condition on the transformation functions analogous to Assumption 10.

Let (x, y) ∈ R CH ++ × R CJ such that t j (y j , y -j , x) = 0 for every j ∈ J and the gradients (D y j t j (y j , y -j , x)) j∈J are positively collinear. Let z ∈ R CJ such that j∈J z j = 0 and z j ∈ Ker D y j t j (y j , y -j , x) for every j ∈ J . Then,

z j f ∈J D 2
y f y j t j (y j , y -j , x)(z f ) > 0 whenever z j = 0.

If there are no external effects, this condition is satisfied because t j is differentiably strictly quasi-convex in y j . Therefore, the sign of the inequality is strictly positive, whereas in Assumption 10 the analogous sign is strictly negative since u h is differentiably strictly quasi-concave in x h .

Consider the previous example. For every firm j, the transformation function is given by t j (y j , y f ) = y 1 j -2φ(y f ) -y 2 j . As above, we focus on production plans for which φ(y f ) is given by

y 1 f 2 -y 2 f . Take z = (z 1 , z 2 ) ∈ R 4 such that z 1 + z 2 = 0 (11)
and z j ∈ Ker D y j t j (y j , y f ). Then, one gets

z 1 j = - φ(y f ) -y 2 j z 2 j ( 12
)
We compute below the two matrices involved in the condition above,

D 2 y j t j (y j , y f ) =      0 0 0 φ(y f ) 2(-y 2 j ) -y 2 j      , D 2 y f y j t j (y j , y f ) =      0 0 φ(y f ) y 1 f -y 2 j φ(y f ) 2(-y 2 f ) -y 2 j     
Using ( 11) and ( 12), it is an easy matter to compute z j D 2 y j t j (y j , y f )(z j ) + z j D 2 y f y j t j (y j , y f )(z f ) which is given by

  1 (-y 2 j ) + 1 -y 2 j -y 2 f - 1 (-y 2 f )   φ(y f ) 2 (-y 2 j ) (z 2 j ) 2 (13)
Since z j = 0, from (12) we have that z 2 j = 0 . Since the gradients (D y j t j (y j , y f )) j=1,2 are positively collinear, one gets y 1 j = y 1 f . Then y 2 j = y 2 f , because t j (y j , y f ) = 0 for every j = 1, 2. Therefore, the quantity in (13) becomes which is strictly positive.

The general model with firms' endowments

In order to establish generic regularity, we incorporate firms' endowments in the model of Section 2. Every firm j is endowed with an exogenously given vector of commodities η j := (η 1 j , .., η c j , .., η C j ), where η c j denotes the amount of commodity c held initially by firm j. Let η := (η j ) j∈J ∈ R CJ . Notice that in the revised version accepted for publication in Economic Theory (forthcoming 2017), we consider only positive firms' endowments. However, our results also apply for negative firms' endowments, under a suitable survival condition given in Assumption 12 below. As for the interpretation of negative endowments, we refer to Subsection 1.3.2 in [START_REF] Arrow | Existence of an equilibrium for a competitive economy[END_REF]. That is, we may extend η c j to include all debts payable in terms of commodity c. Debts owed to firm j are positive, debts owed by firm j are negative. 20For every j, the total production plan y j introduced in the market by firm j is given by

y j := y j + η j (14)
where y j is the production decision of firm j according to its technology and the externality (y -j , x), that is, y j ∈ Y j (y -j , x). Notice that in Y j (y -j , x), the production externality is the total production plan y -j = y -j + η -j introduced in the market by firms other than j, and not the production decision y -j of firms other than j. That is, in our model firm j's decision y j is constrained by its technology and its technology is affected by the total production plans introduced in the market by the other firms. This seems reasonable in many economic applications.

Remark 11 One could consider a different model where the production externality is the production decision y -j of firms other than j, i.e., y j ∈ Y j (y -j , x).

In this case, despite the presence of firms' endowments, one obtains the same Jacobian matrix D q G(q * ) as in ( 6) of Subsection 2.3. In other words, firms' endowments do not impact the technologies of the other firms. Therefore, perturbing the endowments η does not affect firms' supplies. Moreover, as regards market clearing conditions, perturbing the endowments η is completely redundant, i.e., it has exactly the same effect as perturbing the endowments e, because household h's budget constraint and market clearing conditions are given by p

• (x h -e h - j∈J s jh (y j + η j )) = 0 and h∈H x \ h - j∈J (y j + η j ) \ - h∈H e \ h = 0.
Consequently, this way of introducing firms' endowments does not lead to the result of generic regularity.

Given the price p ∈ R C ++ and the externality (y -j , x) ∈ R C(J-1) × R CH ++ , the optimal production decision of firm j solves the following problem max y j ∈R C p • y j subject to t j (y j , y -j , x) ≤ 0

We write this problem in terms of the total production plan y j introduced in the market by firm j, that is, max y j ∈R C p • y j subject to t j (y j -η j , y -j , x) ≤ 0 ( 15) Problem ( 15) is equivalent to the previous maximization problem, because y j = y j -η j by ( 14) and firm j takes as given its endowment η j . Therefore, introducing firms' endowments entails a displacement of the argument of the transformation function t j .

The return from firm j generated by the price p, the production decision y j and the endowment η j is given by p • (y j + η j ). We write this return in terms of the total production plan introduced in the market by firm j, that is, p • y j .

According to the behavior of each firm j, we are led to introduce the following notation:

t j (y j , y -j , x; η j ) := t j (y j -η j , y -j , x)

Notice that the function t j (y j , y -j , x; η j ) satisfies all the assumptions given in Assumption 1 except Point 2. However, y j = η j acts for t j (y j , y -j , x; η j ) as y j = 0 acts for t j (y j , y -j , x). Indeed, Point 2 of Assumption 1 implies that t j (η j , y -j , x; η j ) = 0, and then using standard arguments for profit maximization, one gets p • y * j ≥ p • η j whenever y * j solves problem (15). The individual wealth of household h associated with his endowment and his shares on the returns from the firms is given by p • (e h + j∈J s jh y * j ). Consequently, the individual wealth of household h is greater than p • (e h + j∈J s jh η j ) which is strictly positive if e h + j∈J s jh η j 0. Then, in the following assumption we focus on the set of endowments (e, η) for which e h + j∈J s jh η j 0 for every h ∈ H, from which one deduces the non-emptiness of the interior of all the individual budget constraints.

Assumption 12 (Survival Assumption) (e, η) belongs to the following set.

E := {(e, η) ∈ R CH ++ × R CJ : e h + j∈J s jh η j 0, ∀ h ∈ H}
We assume below that Assumption 3 (Uniform Boundedness) is satisfied in some open subset of endowments of households and firms. The following assumption is used to prove non-emptiness and properness properties of the equilibrium set, see the proof of Lemma 15 and Step 1 of the proof of Lemma 17 in Section 5.

Assumption 13 There exists an open set Λ ⊆ E such that Assumption 3 is satisfied for every r ≥ j∈H e h + j∈J η j with (e, η) ∈ Λ.

Remark 14 In the setting of Remark 2, for every firm j the initial endowment η j belongs to the set R C j . In this case, in Assumptions 12 and 13, R CJ is replaced by j∈J R C j , and for every commodity c ∈ C the sum over j ∈ J is replaced by the sum over j ∈ J (c).

From now on,

(1) t = (t j ) j∈J is fixed and satisfies Assumption 1 and Assumption 13,

(2) u = (u h ) h∈H is fixed and satisfies Assumptions 5 and 10,

(3) a private ownership economy is completely parametrized by the endowments of households and firms (e, η) in the open set Λ given in Assumption 13, (4) we simply denote F e,η the equilibrium function associated with an economy (e, η) ∈ Λ.

Competitive equilibrium and equilibrium function. The notions of competitive equilibrium and equilibrium function given in Subsection 2.2 are adapted to the economy (e, η) according to the behavior of firm j given in (15) and the notations introduced in ( 14) and ( 16). More precisely, in Definition 6, one replaces t j (y j , y * -j , x * ) with t j (y j , y * -j , x * ; η j ). Consequently, the equilibrium function F e,η coincides with the equilibrium function defined in (5), except for the first order conditions associated with problem (15) which are replaced by

F j.1
e,η (ξ) := p -α j D y j t j (y j , y -j , x; η j ) and F j.2 e,η (ξ) := -t j (y j , y -j , x; η j ) projection (Lemma 17). 4) 0 is a regular value of the equilibrium function (Lemma 18). Consequently, the equilibrium set is a sub-manifold of the same dimension as the space of the parameters describing the economy. 5) The local differentiability of the associated equilibrium selection map (Proposition 19).

Proofs

In this section, we prove all the lemmas stated in Section 4.

Proof of Lemma 15. The compactness of the equilibrium set is a direct consequence of Lemma 17. For the existence of an equilibrium it suffices to adapt the proof of Lemma 10 given in del Mercato and Platino (2015) by replacing the transformation functions t j (y j , y -j , x) with the functions t j (y j , y -j , x; η j ).

Notice that the functions t j (y j , y -j , x; η j ) satisfy all the assumptions given in Assumption 1 except Point 2. Whenever Point 2 of Assumption 1 is invoked in the proof of Lemma 10 of del Mercato and Platino (2015), the production plan y j = 0 is replaced by y j = η j which acts for t j (y j , y -j , x; η j ) as y j = 0 acts for t j (y j , y -j , x). Furthermore, whenever Lemma 4 is invoked in the proof of Lemma 10 of del Mercato and Platino (2015), the bounded set K(r) given by Lemma 4 is replaced by the set K( r) + η where r := r + Proof of Lemma 17. We show that any sequence (ξ ν , e ν , η ν ) ν∈N ⊆ F -1 (0), up to a subsequence, converges to an element of F -1 (0), knowing that the sequence Π(ξ ν , e ν , η ν ) ν∈N = (e ν , η ν ) ν∈N ⊆ Λ converges to some (e * , η * ) ∈ Λ. We recall that ξ ν = (x ν , λ ν , y ν , α ν , p ν\ ).

Step 1. Up to a subsequence, (x ν , y ν ) ν∈N converges to (x * , y * ) ∈ R CH + × R CJ .

For every j ∈ J and ν ∈ N, define the following production plan

y ν j := y ν j -η ν j ( 17
)
F j.2 (ξ ν , e ν , η ν ) = 0 and the definition given in ( 16) imply t j (y ν j , y ν -j , x ν ) = 0. Then, y ν = (y ν j ) j∈J belongs to the set Y (y ν , x ν ) defined in (1). Summing Consequently, Assumption 13 implies that the sequence (x ν , y ν ) ν∈N is included in the bounded set K(r) given by Lemma 4, because (x ν , y ν ) belongs to the set A(x ν , y ν ; r) for every ν ∈ N. Thus, the sequence (x ν , y ν ) ν∈N is included in the compact set cl K(r) and, up to a subsequence, it converges to some (x * , y * ) ∈ R CH + × R CJ . Finally, from ( 17) it follows that the sequence (y ν ) ν∈N converges to y * := y * + η * which completes the proof of the step.

Step 2. The consumption allocation x * is strictly positive, i.e. x * 0. Define the bundle e * h := e * h + j∈J s jh η * j which is strictly positive by Assumption 12.

We show that for every h ∈ H, x * h belongs to the closure of the following set

{x h ∈ R C ++ : u h (x h , x * -h , y * ) ≥ u h ( e * h , x * -h , y * )} (18)
which is included in R C ++ by Point 4 of Assumption 5. Thus, x * h 0.

By F h.1 (ξ ν , e ν , η ν ) = F h.2 (ξ ν , e ν , η ν ) = 0 and KKT sufficient conditions, x ν h solves the following problem for every ν ∈ N.

max x h ∈R C ++ u h (x h , x ν -h , y ν ) subject to p ν • x h ≤ p ν • e ν h + p ν • j∈J s jh y ν j
We claim that the point e ν h := e ν h + j∈J s jh η ν j belongs to the budget constraint of the problem above. Notice that e ν h 0 by Assumption 12. By F j.1 (ξ ν , e ν , η ν ) = F j.2 (ξ ν , e ν , η ν ) = 0 and KKT sufficient conditions, y ν j solves the following problem for every ν ∈ N. max y j ∈R C p ν • y j subject to t j (y j , y ν -j , x ν ; η ν j ) ≤ 0

Point 2 of Assumption 1 and the definition given in ( 16) imply t j (η ν j , y ν -j , x ν ; η ν j ) = 0, and so p ν •y ν j ≥ p ν •η ν j for every j. Thus, one gets p ν •(e ν h + j∈J s jh y ν j ) ≥ p ν • e ν h which completes the proof of the claim. Therefore, for every ν ∈ N

u h (x ν h , x ν -h , y ν ) ≥ u h ( e ν h , x ν -h , y ν )
By Point 2 of Assumption 5, for every ε > 0 we have that u h (x ν h +ε1, x ν -h , y ν ) > u h ( e ν h , x ν -h , y ν ) where 1 := (1, . . . , 1) ∈ R C ++ . So, taking the limit for ν → +∞ and using the continuity of u h , one gets

u h (x * h + ε1, x * -h , y * ) ≥ u h ( e * h , x * -h , y * )
since ( e ν h ) ν∈N converges to e * h . Thus, for every ε > 0 the point (x * h +ε1) belongs to the set defined in (18), which implies that x * h belongs to the closure of this set.

Step 3. Up to a subsequence, (λ ν , p ν \ ) ν∈N converges to some (λ * , p * \ ) ∈ R H ++ × R C-1 ++ . By F h.1 (ξ ν , e ν , η ν ) = 0, fixing commodity C, for every ν ∈ N we have λ ν h = D x C h u h (x ν h , x ν -h , y ν ). Taking the limit over ν, by Points 1 and 2 of Assumption 5, we get λ *

h := D x C h u h (x * h , x * -h , y * ) > 0.
By F h.1 (ξ ν , e ν , η ν ) = 0, for all commodity c = C and for all ν ∈ N we

have p ν c = D x c h u h (x ν h , x ν -h , y ν ) λ ν h
. Taking the limit over ν, by Points 1 and 2 of Assumption 5, we get p * c :=

D x c h u h (x * h , x * -h , y * ) λ * h > 0. Therefore, p * \ 0.
Step 4. Up to a subsequence, (α ν ) ν∈N converges to some α * ∈ R J ++ . For every firm j, fix a commodity c(j) ∈ C. By F j.1 (ξ ν , e ν , η ν ) = 0, for every ν ∈ N we have that α

ν j = p ν c(j) D y c(j) j t j (y ν j , y ν -j , x ν ; η ν j )
which is strictly positive by Point 4 of Assumption 1. Taking the limit, by Points 1 and 4 of Assumption 1, we get

α * j := p * c(j) D y c(j) j t j (y * j , y * -j , x * ; η * j ) > 0.
Proof of Lemma 18. We show that for each (ξ * , e * , η * ) ∈ F -1 (0), the Jacobian matrix D ξ,e,η F (ξ * , e * , η * ) has full row rank. It is enough to prove that ∆p \ = 0. Thus, the above system becomes

                           (1) h∈H ∆x h D 2 x k x h u h (x * h , x * -h , y * ) = 0, ∀ k ∈ H (2) -∆x h • p * = 0, ∀ h ∈ H (3) h∈H ∆x h D 2 y f x h u h (x * h , x * -h , y * ) = 0, ∀ f ∈ J (6) - h∈H λ * h ∆x \ h = 0 (19) F h.1 (ξ * , e
v h λ * h D 2 x k x h u h (x * h , x * -h , y * )(v k ) = 0 for every k ∈ H. Summing up k ∈ H, we obtain h∈H v h λ * h k∈H D 2 x k x h u h (x * h , x * -h , y * )(v k ) = 0. F h.1 (ξ * , e * , η * ) = 0
implies that the gradients (D x h u h (x * h , x * -h , y * )) h∈H are positively proportional. So, by (20) all the conditions of Assumption 10 are satisfied, and then v h = 0 for each h ∈ H since λ * h > 0. Thus, we get ∆x h = 0 for all h ∈ H, and consequently ∆ = 0 which completes the proof.

Appendix

Regular values and transversality

The theory of general equilibrium from a differentiable view is based on results from differential topology. First, we remind the definition of a regular value. Second, we summarize the results used in our analysis. These results, as well as generalizations on these issues, can be found for instance in [START_REF] Guillemin | Differential topology[END_REF], [START_REF] Mas-Colell | The theory of general economic equilibrium. A differentiable approach[END_REF] and [START_REF] Villanacci | Differential topology and general equilibrium with complete and incomplete markets[END_REF].

Definition 20 Let M , N be C r manifolds of dimensions m and n, respectively. Let f : M → N be a C r function with r ≥ 1. An element y ∈ N is a regular value for f if for every x * ∈ f -1 (y), the differential mapping Df (x * ) is onto.

Theorem 21 (Regular Value Theorem) Let M , N be C r manifolds of dimensions m and n, respectively. Let f : M → N be a C r function with r ≥ 1. If y ∈ N is a regular value for f , then

(1) if m < n, f -1 (y) = ∅, (2) if m ≥ n, either f -1 (y) = ∅, or f -1 (y) is an (m -n)-dimensional sub- manifold of M .
Corollary 22 Let M , N be C r manifolds of the same dimension. Let f : M → N be a C r function with r ≥ 1. Let y ∈ N a regular value for f such that f -1 (y) is non-empty and compact. Then, f -1 (y) is a finite subset of M .

The following results is a consequence of Sard's Theorem for manifolds.

Theorem 23 (Transversality Theorem) Let M , Ω and N be C r manifolds of dimensions m, p and n, respectively. Let f : M × Ω → N be a C r function, assume r > max{m -n, 0}. If y ∈ N is a regular value for f , then there exists a full measure subset Ω * of Ω such that for any ω ∈ Ω * , y ∈ N is a regular value for f ω , where

f ω : ξ ∈ M → f ω (ξ) := f (ξ, ω) ∈ N
Definition 24 Let (X, d) and (Y, d ) be two metric spaces. A function π : X → Y is proper if it is continuous and one among the following conditions holds true.

(1) π is closed and π

-1 (y) is compact for each y ∈ Y , (2) if K is a compact subset of Y , then π -1 (K) is a compact subset of X, (3) if (x n ) n∈N is a sequence in X such that (π(x n )) n∈N converges in Y , then (x n ) n∈N has a converging subsequence in X.
The conditions above are equivalent.

Jacobian matrix of the equilibrium function

The

16

  Documents de travail du Centre d'Economie de la Sorbonne -2015.44R (Version révisée)

  η) which is bounded by Assumption 13. Finally, in order to complete the adaptation of the proof of Lemma 10 of del Mercato and Platino (2015), one also adapts in the following way the proof of Proposition 15 in del Mercato and Platino (2015). Namely, in Step 1.2 of Proposition 15, one replaces the point e h (τ ν ) with the point τ ν (e h + j∈J s jh η j ) + (1 -τ ν ) x h , and in Step 2.2, one replaces the point e h (τ ν ) with the point e h + j∈J s jh η j .

F

  h.2 (ξ ν , e ν , η ν ) = 0 over h, from F M (ξ ν , e ν , η ν ) = 0 one gets ν ∈ N. Now, for every commodity c consider the following compact set Q c := { r c := max Q c and r := (r c ) c∈C From the equality above, ≤ r for all ν ∈ N, and r ≥ h∈H e * h + j∈J η * j by definition.

  computation of D ξ,e,η F (ξ * , e * , η * ) is described below. Vectors are treated as row matrices. The symbol "T " means transpose. 0 denotes the zero vector. With innocuous abuse of notation, the dimension of 0 is C or C -1 depending on the dimension of the respective block of columns. 0 denotes the zero matrix. With innocuous abuse of notation, the size of 0 is C × C or (C -1) × (C -1) depending on the size of the respective block of rows and columns. I := [I C-1 |0 T ] (C-1)×C where I C-1 denotes the identity matrix of size (C -1).

  ) h∈H . The total resources are r := h∈H e h ∈ R C ++ . The share of firm j owned by household h is s jh ∈ [0, 1], and let s := (s jh ) j∈J ,h∈H . As usual,

	1 h , .., e c h , .., e C h ) ∈ R C ++ , and let e :=
	(e h

h∈H s jh = 1

  ) = 0 and equation (2) in system (19) imply that ∆x h ∈ Ker D x h u h (x * h , x *-h , y * ) for every h ∈ H. Now, for every h ∈ H define v h := λ * h ∆x h . Thus, the vector (x * h , v h ) h∈H satisfies the following conditions.

* , η * (v h ) h∈H ∈ h∈H Ker D x h u h (x * h , x * -h , y * ) and h∈H v h = 0 (20)

where the last equality comes from equation (6) in system (19). Multiplying by v k both sides of equation (1) in system (

19

) and using the definition of v h , one gets h∈H

Documents de travail du Centre d'Economie de la Sorbonne -2015.44R (Version révisée)

[START_REF] Mandel | An index formula for production economies with externalities[END_REF] makes use of the classical regularity result by assuming that the demands and supplies that are mutually consistent are differentiable and their differential is onto. Actually, the purpose of his paper is not the study of regularity, but merely the existence of an equilibrium via the degree theory for correspondences.

 3 The Pareto improving analysis of[START_REF] Greenwald | Externalities in economies with imperfect information and incomplete markets[END_REF] mainly focuses on economies with incomplete markets and imperfect information. However, in Section I, the authors consider a general equilibrium model of a private ownership economy with consumption and production externalities. They assume that the aggregate excess demand-supply function is differentiable and its differential is onto. Furthermore, the issue of the existence of such Pareto improving policies is not addressed for the general model provided in Section I. In[START_REF] Geanakoplos | Pareto improving taxes, Jour-29 Documents de travail du Centre d'Economie de la Sorbonne -2015.44R[END_REF], the Pareto improving analysis deals with economies with consumption externalities

only.4 Under these assumptions, the set of competitive equilibria is non-empty and compact. The existence of a competitive equilibrium is demonstrated byArrow and Hahn (Chapter 6, 1971),[START_REF] Laffont | Effets externes et théorie économique. Monographie du Séminaire d'Econométrie[END_REF] del Mercato and[START_REF] Del Mercato | Private ownership economy with externalities and existence of competitive equilibria: A differentiable approach[END_REF].

This is a convenient way to represent a production set using an inequality on a function called the transformation function. In the case of a single-output technology, the production set is commonly described by a production function. The transformation function is the counterpart of the production function in the case of production processes which involve several outputs, see for instance[START_REF] Mas-Colell | Microeconomic theory[END_REF] and[START_REF] Villanacci | Differential topology and general equilibrium with complete and incomplete markets[END_REF]. 5 Documents de travail du Centre d'Economie de la Sorbonne -2015.44R (Version révisée)

Let v and v be two vectors in R n , v • v denotes the scalar product of v and v . Let A be a real matrix with m rows and n columns, and B be a real matrix with n rows and l columns, AB denotes the matrix product of A and B. Without loss of generality, vectors are treated as row matrices and A denotes both the matrix and the following linear mapping A : v ∈ R n → A(v) := Av T ∈ R[m] where v T denotes the transpose of v and R[m] := {w T : w ∈ R m }. When m = 1, A(v) coincides with the scalar product A • v, treating A and v as vectors in R n .

"KKT conditions" means Karush-Kuhn-Tucker conditions. 10 Documents de travail du Centre d'Economie de la Sorbonne -2015.44R (Version révisée)

Then, the matrix D ξ F E (ξ * ) is trivially non-singular for almost all initial endowments, and one gets the generic regularity in the space of households endowments.

 17 The possibility functions represent general consumption sets with externalities.12 Documents de travail du Centre d'Economie de la Sorbonne -2015.44R (Version révisée)

Further research is open to find some kind of auxiliary assumption on the first order effects of externalities on transformation functions. In Section 4, we choose another approach.13 Documents de travail du Centre d'Economie de la Sorbonne -2015.44R (Version révisée)

In order to get a smooth approximation, one might approximate the function φ around -ε 2 by a polynomial function.

If η j is negative, then p•η j represents a cost for firm j, i.e., a penalty. For instance, in the literature of game theory, adding penalties to the payoff functions of the agents has been often used to restore the generic regularity of Nash equlibria, see for instance van[START_REF] Van Damme | Strategic equilibrium[END_REF].

See Definition 24 in Subsection 6.1.

The proof of Lemma 10 of del Mercato and Platino (2015) have been adapted to our model in the proof of Lemma 15. 21 Documents de travail du Centre d'Economie de la Sorbonne -2015.44R (Version révisée)

Under the previous assumptions one gets the result analogous to Theorem 8. The proof of the following lemma is given in Section 5.

Lemma 15 For every (e, η) ∈ Λ, the equilibrium set F -1 e,η (0) is non-empty and compact.

The regularity result

In this subsection, we provide and we prove our main theorem.

Theorem 16 The set Λ * of regular economies (e, η) is an open and full measure subset of Λ.

In order to prove this theorem, we introduce the following notations and we provide two auxiliary lemmas, namely Lemmas 17 and 18. Both lemmas are proved in Section 5.

Point 1 of Assumption 1 and Point 1 of Assumption 5 imply that the equilibrium function F e,η is C 1 everywhere. By Definition 9, the economy (e, η) is regular if

and denote Π the restriction to F -1 (0) of the projection of Ξ × Λ onto Λ, i.e. Π : (ξ, e, η) ∈ F -1 (0) → Π(ξ, e, η) := (e, η) ∈ Λ

We can now write the set Λ * given in Theorem 16 as

So, in order to prove Theorem 16, it is enough to show that Π(G) is a closed set in Λ and Π(G) is of measure zero.

We first claim that Π(G) is a closed set in Λ. From Point 1 of Assumptions 1 and 5, F and D ξ F are continuous on Ξ × Λ. The set G is characterized by the fact that the determinant of all the square submatrices of D ξ F (ξ, e, η) of dimension dim Ξ is equal to zero. Since the determinant is a continuous function and D ξ F is continuous on F -1 (0), the set G is closed in F -1 (0). Thus, Π(G) is closed since the projection Π is proper. 21 The properness of the projection Π is provided in the following lemma.

Lemma 17 The projection Π : F -1 (0) → Λ is a proper function.

To complete the proof of Theorem 16, we claim that Π(G) has measure zero in Λ. The result follows by Lemma 18 given below and a consequence of Sard's Theorem (Theorem 23 in Subsection 6.1). Indeed, Lemma 18 and Theorem 23 imply that there exists a full measure subset Ω of Λ such that for each (e, η) ∈ Ω and for each ξ * such that F (ξ * , e, η) = 0, rank D ξ F (ξ * , e, η) = dim Ξ. Now, let (e, η) ∈ Π(G), then there exists ξ ∈ Ξ such that F (ξ, e, η) = 0 and rank D ξ F (ξ, e, η) < dim Ξ. So, (e, η) /

∈ Ω. This prove that Π(G) is included in the complementary of Ω, that is in Ω C := Λ \ Ω. Since Ω C has measure zero, so too does Π(G). Thus, the set of regular economies Λ * is a full measure subset of Λ since Ω ⊆ Λ * , which completes the proof of Theorem 16.

Lemma 18 0 is a regular value for F .

0) holds for (ξ , e , η ) ∈ U i × I if and only if ξ = g i (e , η ).

We conclude this section by remarking that most of the classical properties of the equilibrium manifold and the natural projection are established for the model studied in this section. 1) For any given economy, the equilibrium set is non-empty and compact (Lemma 15). 2) For any given economy, the topological degree modulo 2 of the equilibrium function is equal to 1 (Lemma 10 in [START_REF] Del Mercato | Private ownership economy with externalities and existence of competitive equilibria: A differentiable approach[END_REF]. 22 Consequently, one gets that, at every regular economy, the number of equilibria is odd. 3) The properness of the natural ∆D ξ,e,η F (ξ * , e * , η * ) = 0 implies ∆ = 0, where

The computation of D ξ,e,η F (ξ * , e * , η * ) is described in Subsection 6.2 and the system ∆D ξ,e,η F (ξ * , e * , η * ) = 0 is written in detail below.

We remind that

(7) -α * j ∆y j D 2 η j y j t j (y * j , y * -j , x * ; η * j ) -∆α j D η j t j (y * j , y * -j , x * ; η * j ) = 0, ∀ j ∈ J

Using the definition given in ( 16), we have that D η j t j (y * j , y * -j , x * ; η * j ) = -D y j t j (y * j , y * -j , x * ; η * j ) and D 2 η j y j t j (y * j , y * -j , x * ; η * j ) = -D 2 y j t j (y * j , y * -j , x * ; η * j )

Then, for every j ∈ J equation ( 7) becomes α * j ∆y j D 2 y j t j (y * j , y * -j , x * ; η * j ) + ∆α j D y j t j (y * j , y * -j , x * ; η * j ) = 0 Multiplying the equation above by ∆y j and using equation (4), since α * j > 0 one gets ∆y j D 2 y j t j (y * j , y * -j , x * ; η * j )(∆y j ) = 0. So, equation (4) and Point 3 of Assumption 1 imply ∆y j = 0. Then, the equation above implies that ∆α j D y j t j (y * j , y * -j , x * ; η * j ) = 0, and so ∆α j = 0 by Point 4 of Assumption 1. Thus, (∆y j , ∆α j ) = 0 for every j ∈ J . Since p * C = 1, from equation ( 5) one gets ∆λ h = 0 for all h ∈ H, and so eral equilibrium model with private provision of public goods, Review of Economic Design 10, 249-271. Villanacci, A., Zenginobuz, U., 2012. Subscription equilibrium with production: Non-neutrality and constrained suboptimality, Journal of Economic Theory 147(2), 407-425.