Alfredo Canelas
email: acanelas@fing.edu.uy

Jean R Roche
email: roche@iecn.u-nancy.fr

José Herskovits

SHAPE OPTIMIZATION FOR INVERSE ELECTROMAGNETIC CASTING PROBLEMS

In this paper we present an algorithm for inverse optimization problems concerning electromagnetic casting of molten metals. We are interested in locating suitable inductors around the molten metal so that the equilibrium shape be as near as possible to a desired target shape. A Simultaneous Analysis and Design (SAND) mathematical programming formulation is stated for the inverse problem. The resulting optimization problem is solved with FAIPA, a feasible directions interior-point algorithm.

INTRODUCTION

The industrial technique of electromagnetic casting allows for contactless heating, shaping and controlling of chemical aggressive, hot melts. The main advantage over the conventional crucible shape forming is that the liquid metal does not come into contact with the crucible wall, so there is no danger of contamination. This is very important in the preparation of very pure specimens in metallurgical experiments, as even small traces of impurities, such as carbon and sulphur, can affect the physical properties of the sample. Industrial applications are, for example, electromagnetic shaping of aluminum ingots using soft-contact confinement of the liquid metal, electromagnetic shaping of components of aeronautical engines made of superalloy materials (Ni,Ti, . . . ), control of the structure solidification, etc.

The direct problem is to determine the resulting liquid metal shape for a given external current distribution. The model considered here concerns a vertically falling molten metal column shaped by an externally applied magnetic field created by a set of inductors. In general, the direct problem can be solved either directly studying the equilibrium equation at the interface, or minimizing an appropriate energy functional, the main advantage of this last method being that the resulting shapes are then mechanically stable [START_REF] Moffat | Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. I. Fundamentals[END_REF][START_REF] Shercliff | Magnetic shaping of molten metal columns, Modèlisation mathématique et analyse numérique[END_REF][START_REF] Henrot | Existence of equilibria in electromagnetic casting[END_REF].

The inverse problem consists in determining the exterior field, and therefore the external currents, for which the liquid metal takes on a given desired shape. In the two-dimensional case, the inverse shaping problem consists in finding a distribution of inductors in order that the generated exterior field makes the horizontal cross-section of the molten metal attain a prescribed shape. This is a very important problem that one needs to solve in order to define a process of electromagnetic liquid metal forming. In addition, from a practical point of view, the magnetic field has to be created by a simple configuration of inductors.

The direct problem is inherently well posed, i.e. small variations in the applied currents cause small variations in the shape. The inverse problem is inherently ill posed: small variation of the liquid boundary may cause dramatic variations in the applied exterior field [START_REF] Henrot | Un problème inverse en formage de mètaux liquides[END_REF][START_REF] Felici | The inverse shaping problem[END_REF].

In a previous work we studied the inverse electromagnetic shaping problem considering the case where the inductors are made of single solid-core wires with a negligible area of the cross-section [START_REF] Canelas | The inverse electromagnetic shaping problem[END_REF]. Thus, the inductors were represented by points in the horizontal plane. In a second paper we considered the more realistic case where each inductor is a set of bundled insulated strands [START_REF] Canelas | Inductor shape optimization for electromag[END_REF].

In the present paper we introduce a regularized cost function in order to treat the numerical instability. The most important effect of the regularization term is to control the magnetic pressure in the boundary of the computed shape. Thus, this term ensures that the boundary of the liquid metal is smooth.

The model

Two different approaches are considered, the first one seeks for a set of inductors such that the distance between the equilibrium shape and the given target one is minimized. The second approach looks for a set of inductors such that a slack function related to the equilibrium relation on the boundary of the target shape is minimized.

A SAND mathematical programming method is stated for both inverse problem formulations. This method considers the position of the inductors, shape parameters and state variables as unknowns, see [START_REF] Haftka | Simultaneous nonlinear structural analysis and design[END_REF][START_REF] Canelas | Shape Optimization using the Boundary Element Method and a SAND Interior Point Algorithm for Constrained Optimization[END_REF]. The obtained optimization problems are solved employing FAIPA [START_REF] Herskovits | Feasible direction interiorpoint technique for nonlinear optimization[END_REF][START_REF] Herskovits | Mathematical programming models and algorithms for engineering design optimization[END_REF], a feasible directions interior point algorithm.

THE MATHEMATICAL MODEL The model

The simplified model of the electromagnetic shaping problem studied here concerns the case of a vertical column of liquid metal falling down into an electromagnetic field created by vertical inductors. We assume that the frequency of the imposed current is very high so that the magnetic field does not penetrate into the metal. In other words, we neglect the skin effect. Moreover, we assume that a stationary horizontal section is reached so that the 2-dimensional model is valid. The equilibrium of the system is ensured by the static balance on the surface of the metal between the surface tension and the electromagnetic forces. This problem and other similar ones have been considered by several authors, we refer the reader to the following papers for the physical analysis of the simplifying assumptions of the model: see [START_REF] Brancher | Étude de la déformation d'un liquide magnétique[END_REF][START_REF] Gagnoud | Le probléme de frontiére libre en lévitation électromagnétique[END_REF][START_REF] Henrot | Un problème inverse en formage de mètaux liquides[END_REF][START_REF] Moffat | Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. I. Fundamentals[END_REF][START_REF] Novruzi | Second Order Derivatives, Newton Method[END_REF][START_REF] Pierre | Computation of free surfaces in the electromagnetic shaping of liquid metals by optimization algorithms[END_REF][START_REF] Shercliff | Magnetic shaping of molten metal columns, Modèlisation mathématique et analyse numérique[END_REF].

We denote by Ω the exterior in the plane of the closed and simply connected domain ω occupied by the cross-section of the metal column. The exterior magnetic field can be found as the solution of the following boundary value problem:

∇×B = µ 0 J in Ω, (1)
∇ • B = 0 in Ω, (2)
B •ν = 0 on Γ, (3)
B(x) = O( x -1 ) as x → ∞ in Ω. ( 4
)
Here the fields J = (0, 0, j 0 ) and B = (B 1 , B 2 , 0) represent the mean square values of the current density vector and the total magnetic field, respectively. The constant µ 0 is the vacuum permeability, Figure 1. Direct electromagnetic casting problem.

ν the unit normal vector to the boundary Γ and • denotes the Euclidean norm. We assume that j 0 has compact support in Ω and satisfies:

Ω j 0 dx = 0. (5)
Since it is assumed that the molten metal is incompressible, we have the following condition:

ω dx = S 0 , (6)
where S 0 is given. On the other hand, the magnetic field produces a surface pressure that acts on the liquid metal changing its shape until the equilibrium is attained. The equilibrium is characterized by the following equation [START_REF] Pierre | Computation of free surfaces in the electromagnetic shaping of liquid metals by optimization algorithms[END_REF][START_REF] Pierre | Numerical simulation of tridimensional electromagnetic shaping of liquid metals[END_REF]:

1 2µ 0 B 2 + σC = p 0 on Γ, (7)
where C is the curvature of Γ seen from the metal, σ is the surface tension of the liquid and the constant p 0 is an unknown of the problem. Physically, p 0 represents the difference between the internal and external pressures.

In the direct problem the electric current density J is given and one needs to find the shape of ω that satisfies [START_REF] Canelas | The inverse electromagnetic shaping problem[END_REF] and such that the magnetic field B ω solution of (1)-(4) satisfies also the equilibrium equation [START_REF] Canelas | Inductor shape optimization for electromag[END_REF] for a real constant p 0 .

First order optimality conditions

Conditions (1)-( 5), with the function j 0 compactly supported in Ω, imply that there exists the flux function ϕ : Ω → R such that B = ( ∂ϕ ∂x2 , -∂ϕ ∂x1 , 0) with ϕ solution of:

-∆ϕ = µ 0 j 0 in Ω, ϕ = 0 on Γ, ϕ(x) = O(1) as x → ∞. (8)
The equilibrium equation [START_REF] Canelas | Inductor shape optimization for electromag[END_REF] in terms of the flux becomes:

1 2µ 0 ∂ϕ ∂ν 2 + σC = p 0 on Γ. ( 9
)
The direct problem, in terms of the flux, consists in looking for a domain ω such that the solution ϕ of (8) satisfies ( 9) for a real constant p 0 .

Variational formulation

Under suitable assumptions, the equilibrium configurations are given by the local stationary points with respect to the domain of the following total energy:

E(ω) = - 1 2µ 0 Ω ∂ϕ ω ∂ν 2 dx + σP (ω), (10)
subject to the equality constraint in the measure of ω:

ω dx = S 0 . ( 11
)
where in [START_REF] Herskovits | Feasible direction interiorpoint technique for nonlinear optimization[END_REF], we have been used the notation ϕ ω for the solution of (8), with the purpose of highlighting the dependence of the flux with respect to the domain ω. P (ω) is the perimeter of ω, i.e., the length of Γ = ∂ω when ∂ω is regular enough (for instance of class C 1 ):

P (ω) = Γ dγ, (12)
where dγ = length measure on Γ.

The variational formulation of the direct problem consists in finding the domain ω as a stationary point of the total energy [START_REF] Herskovits | Feasible direction interiorpoint technique for nonlinear optimization[END_REF], subject to the constraint [START_REF] Herskovits | Mathematical programming models and algorithms for engineering design optimization[END_REF]. As ϕ ω is solution of [START_REF] Haftka | Simultaneous nonlinear structural analysis and design[END_REF], to prove that this variational formulation is equivalent to the previous one it remains to show that the equilibrium relation is automatically ensured for all the stationary points.

First order optimality conditions

In this section we derive the necessary condition for a domain ω to be a stationary point of the total energy [START_REF] Herskovits | Feasible direction interiorpoint technique for nonlinear optimization[END_REF], subject to the constraint [START_REF] Herskovits | Mathematical programming models and algorithms for engineering design optimization[END_REF]. For that purpose we consider shape derivatives. Differentiation with respect to the domain is a classical issue, in this work we consider the point of view of F. Murat and J. Simon; see [START_REF] Allaire | Conception Optimale de Structures[END_REF][START_REF] Murat | Sur le contrôle par un domaine géométrique[END_REF][START_REF] Simon | Differentiation with respect to the domain in boundary value problems[END_REF].

Let V ∈ W 1,∞ (R 2 , R 2
) the set of the Lipschitz functions φ from R 2 to R 2 such that φ and ∇φ are uniformly bounded [START_REF] Allaire | Conception Optimale de Structures[END_REF]. Let ω be a bounded domain in R 2 of class C 2 . We consider a shape deformation given by the mapping Id + V , where Id is the identity mapping. Then, the deformed domain ω V is defined by ω

V = {x + V (x) | x ∈ ω}; see Fig. 2.
For every

V ∈ W 1,∞ (R 2 , R 2 ) the mapping Id + V is a diffeomorphism provided V W 1,∞ (R 2 ,R 2 ) < 1 [17].
Let O(ω) be the collection of images of ω considering all possible diffeomorphisms. If F is a scalar function defined in O(ω) we say that it is shape differentiable if the function

V → F (ω V ) is differentiable at V = 0 in the Banach space W 1,∞ (R 2 , R 2 ).
The derivative of F , defined in W 1,∞ (R 2 , R 2 ), is called shape gradient and is denoted by F (ω). It can be shown that the linear application V → F (ω)(V ) is determined by the normal component of V in the boundary of ω, see [START_REF] Allaire | Conception Optimale de Structures[END_REF], [START_REF] Novruzi | Structure of shape derivatives[END_REF] and [START_REF] Novruzi | Second Order Derivatives, Newton Method[END_REF] for a detailed description of the shape derivative structure.

Let L be the Lagrangian function defined in O(ω) × R by:

L(ω, p 0 ) = E(ω) -p 0 (m(ω) -S 0 ), (13)
Then, the first order optimality condition is the following:

L (ω, p 0 )(V ) = 0 ∀ V ∈ W 1,∞ (R 2 , R 2 ). ( 14
)
This kind of optimality conditions often appear in hydrodynamic problems and other fluid problems; let us refer for instance to reference [START_REF] Shercliff | Magnetic shaping of molten metal columns, Modèlisation mathématique et analyse numérique[END_REF] where a large class of liquid metal equilibria is considered. The next theorem shows how the term L (ω, p 0 )(V ) of ( 14) can be calculated.

Theorem

Let Ω be the complement of a compact set ω in R 2 with nonempty interior. Assume that

Γ = ∂ω = ∂Ω is of class C 2 . Let V be in W 1,∞ (R 2 , R 2 ) with compact support and V W 1,∞ (R 2 ,R 2 ) < 1. Let j 0 be a square integrable function from Ω into R with compact support in Ω.
Then, there exists a unique solution [START_REF] Kress | Linear integral equations[END_REF] and [START_REF] Henrot | Un problème inverse en formage de mètaux liquides[END_REF]) of:

ϕ ω V in C 1 (Ω V ) (see
-∆ϕ ω V = µ 0 j 0 in Ω V , ϕ ω V = 0 on ∂Ω V , ϕ ω V (x) = O(1) as x → ∞. ( 15
)
and the shape derivative of the lagrangian L is given by:

L (ω, p 0 )(V ) = Γ 1 2µ 0 ∂ϕ ω ∂ν 2 (V • ν) dγ + Γ (σC -p 0 ) (V • ν) dγ,
where ν is the unit normal to Γ oriented toward Ω, C is the curvature of Γ (seen from the metal) and ϕ ω the solution of [START_REF] Haftka | Simultaneous nonlinear structural analysis and design[END_REF]. See [START_REF] Henrot | Un problème inverse en formage de mètaux liquides[END_REF][START_REF] Pierre | Computation of free surfaces in the electromagnetic shaping of liquid metals by optimization algorithms[END_REF][START_REF] Novruzi | Second Order Derivatives, Newton Method[END_REF][START_REF] Allaire | Conception Optimale de Structures[END_REF].

THE INVERSE PROBLEM

The goal of the inverse problem is to find a distribution of current around the liquid metal column so that it attains a given shape. This topic was already studied and there are a few number of papers about the existence of such solutions. See [START_REF] Henrot | Un problème inverse en formage de mètaux liquides[END_REF][START_REF] Felici | The inverse shaping problem[END_REF].

It has been shown that the magnetic field B is the unique analytic extension of the field B •τ defined on the boundary Γ of the liquid metal (τ being the unit tangent vector to Γ) [START_REF] Henrot | Un problème inverse en formage de mètaux liquides[END_REF]. Furthermore, from ( 7) and ( 9), it is possible to show the following:

B •τ = p 0 -σC with = ±1 . ( 16
)
The constant p 0 must be such that:

p 0 ≥ max Γ σC . (17)
In [START_REF] Henrot | Un problème inverse en formage de mètaux liquides[END_REF], the authors proved that a solution for B exists only if Γ is an analytic curve, and the function B • τ is analytic. If p 0 > max Γ σC then is constant on Γ. So if we have p 0 = max Γ σC another restriction is imposed on Γ since change sign around points where B.τ = 0 (i.e. where C attaints it maximum), depending on the multiplicity order of these zero points: if the multiplicity order of a zero point is even, remains constant in its neighborhood. On the other hand if the multiplicity order of a zero point is odd, will change sign.

Hence by periodicity the number of singular points of odd order must be even. For example any curve which has its curvature attaining its maximum value at an odd number of points (at which B.τ has non-degenerate zeros), is in fact impossible to form.

In the present approach, given the target shape ω * , we look for a electric current density j 0 as the solution of the following optimization problem:

min j0 d(ω, ω * ), (18)
where d(ω, ω * ) is a "distance" between ω and ω * , see [START_REF] Allaire | Conception Optimale de Structures[END_REF]. The domain ω belongs to the set of admissible domains, i.e., ω ∈ O(ω * ), and satisfies the first order optimality condition [START_REF] Novruzi | Second Order Derivatives, Newton Method[END_REF] under the action of the electric current density j 0 . In addition, the magnetic field has to be created by a simple configuration of inductors. Hence we consider a distribution of the electric current density j 0 of the form:

j 0 = I m p=1 α p χ Θp , ( 19
)
where I is a given intensity of current, Θ p , with 1 ≤ p ≤ m, are inductors of circular crosssection, χ Θp are their characteristic functions, and α p are dimensionless coefficients. Then, the inverse problem consists in determining the position and radius of the circles Θ p . Thus, we have to solve a finite dimensional optimization problem with 3m project variables. Note that the expression [START_REF] Simon | Differentiation with respect to the domain in boundary value problems[END_REF] assumes that the electric current density is uniform on each region Θ p . Inductors made of bundled insulated strands allow the use of ( 19) as a good approximation, see [START_REF] Sullivan | Optimal Choice for Number of Strands in a Litz-Wire Transformer Winding[END_REF] and references therein.

To find an approximate solution of problem [START_REF] Murat | Sur le contrôle par un domaine géométrique[END_REF] we considers a domain deformation of ω * defined by the diffeomorphism Id + Z, with Z ∈ W 1,∞ (R 2 , R 2 ). Then denoting T Z the mapping:

T Z (x) = x + Z(x) , (20)
we define:

ω Z = T Z (ω * ) , Γ Z = T Z (Γ * ) .
The first formulation of the inverse problem that is considered here is:

min j0,Z Z 2 L 2 (Γ * ) + r ∂ϕ ω ∂ν 2 L 2 (Γ * ) , subject to: (21) ω Z is in equilibrium under j 0 .
where

∂ϕω ∂ν 2 L 2 (Γ * )
is a penalty term, and r is a positive parameter. The regularization term penalize solutions presenting a high magnetic pressure in the shape boundary. Hence the computed solutions to the inverse problem will give a magnetic field corresponding to a smooth boundary Γ * .

The equilibrium of ω Z is understood in the variational sense, i.e.,

Γ Z 1 2µ 0 ∂ϕ ω Z ∂ν 2 (V • ν) dγ + Γ Z (σC -p 0 ) (V • ν) dγ = 0 ∀ V in W 1,∞ (R 2 , R 2 ) . (22)
A second formulation of the inverse problem can be considered if we introduce a slack variable function p(x) : Γ * → R in the equilibrium equation. That is, we say that ω * is in equilibrium under j 0 and p if

Γ 1 2µ 0 ∂ϕ ω ∂ν 2 (V • ν) dγ + Γ (σC -p 0 + p) (V • ν) dγ = 0 ∀ V in W 1,∞ (R 2 , R 2 ) , (23)
where ϕ ω * is the solution of the state equation [START_REF] Haftka | Simultaneous nonlinear structural analysis and design[END_REF] in Ω * . Then, we obtain the following formulation of the problem:

min j0,p p 2 L 2 (Γ * ) + r ∂ϕ ω ∂ν 2 L 2 (Γ * )
, subject to: (24) ω * is in equilibrium under j 0 and p.

Note that in this formulation the shape of the liquid metal is no more an unknown of the optimization problem.

NUMERICAL METHOD The exterior Dirichlet problem

To solve [START_REF] Haftka | Simultaneous nonlinear structural analysis and design[END_REF] in the exterior domain Ω we consider a particular solution ϕ 1 of the differential equation given by:

ϕ 1 (x) = - µ 0 2π R 2 ln x -y j 0 (y) dy. ( 25
)
This function is a solution of the problem:

-∆ϕ 1 (x) = µ 0 j 0 in R 2 , ( 26
) ϕ 1 (x) = O(1) as x → ∞. (27)
Note that for the current density distribution defined by [START_REF] Simon | Differentiation with respect to the domain in boundary value problems[END_REF], the expression of ϕ 1 is

ϕ 1 (x) = - µ 0 I 2π m p=1 α p Θp ln x -y dy. (28)
The function ϕ 1 can be calculated as a sum of line integrals on the boundaries Γ p of domains Θ p . Consider the function w : R 2 × R 2 → R 2 defined as:

w(x, y) = (1/4)(1 -2 ln x -y )(x -y). ( 29
)
The divergence of w is ∇ y • w = ln x -y . Then, (28) becomes:

ϕ 1 (x) = - µ 0 I 2π m p=1 α p Γp w(x, y) • ν dγ. ( 30
)
The function ϕ can be computed as:

ϕ(x) = ξ(x) + ϕ 1 (x), (31)
where the function ξ is the solution of the following exterior problem:

-∆ξ(x) = 0 in Ω, ξ(x) = -ϕ 1 (x) on Γ, ξ(x) = O(1)
as x → ∞.

(

) 32
The SAND formulation of the inverse problems

Following Kress [START_REF] Kress | Linear integral equations[END_REF], an integral single layer representation of the solution of (32) is given by:

ξ(x) = - 1 2π Γ q(y) ln x -y dγ + c, (33)
where the constant c is the value at the infinity of ξ and the function q(y) ∈ H -1/2 (Γ) satisfies:

Γ q(y) dγ = 0. ( 34
)
It remains to impose the boundary conditions on Γ. Here, this is done with a weak formulation. Let a Γ (q, g) be the following elliptic bilinear form:

a Γ (q, g) = = - 1 2π Γ g(x) Γ q(y) ln x -y dγ dγ +c Γ g(x) dγ (35)
defined on H -1/2 (Γ) × H -1/2 (Γ). We look for a function q(y) ∈ H -1/2 (Γ) that satisfies (34) and:

a Γ (q, g) = - Γ ϕ 1 (x)g(x) dγ ∀ g ∈ H -1/2 (Γ).
(36) Equation ( 36) with ϕ 1 given by (30) will be used instead of [START_REF] Haftka | Simultaneous nonlinear structural analysis and design[END_REF]. Note that the unknown variables are now the function q and the scalar c.

Finally, ∂ϕ ∂ν in the equilibrium equation ( 22) can be computed as:

∂ϕ ∂ν = ∂ϕ 1 ∂ν + ∂ξ ∂ν , (37)
where the normal derivative of ϕ 1 is obtained from (30):

∂ϕ 1 ∂ν x (x) = - µ 0 I 2π m p=1 α p Γp ∂ ∂ν x (w(x, y) • ν) dγ.
(38) The following expression can be used for ξ:

∂ξ ∂ν x (x) = - 1 2π Γ q(y) ∂ ∂ν x ln x-y dγ+ 1 2 q(x), (39)
for all x ∈ Γ, where the integral of (39) is understood in the Cauchy principal value sense.

The SAND formulation of the inverse problems

A SAND formulation of the inverse problems ( 21) and ( 24) is employed here. In other words, the state variables, c and q are incorporated as unknowns of the optimization problem and the state and equilibrium equations are incorporated as equality constraints. The optimization problem of the formulation (21) becomes:

min j0,Z,c,q Z 2 L 2 (Γ * ) + r ∂ϕ ω ∂ν 2 L 2 (Γ * ) , (40)
subject to the area constraint:

ω Z dx = S 0 , (41)
the state equations:

a Γ Z (q, g) = - Γ Z ϕ 1 (x)g(x) dγ ∀ g ∈ H -1/2 (Γ Z ), (42)
Γ Z q(y) dγ = 0, (43)
and the equilibrium equation:

Γ Z 1 2µ 0 ∂ϕ ω ∂ν 2 (V •ν) dγ + Γ Z (σC -p 0 ) (V •ν) dγ = 0 ∀ V in C 1 (R 2 , R 2 ), (44)
where ϕ 1 , ϕ, and ξ are given by ( 25), (31) and (33).

The optimization problem of the formulation (24) becomes:

min j0,p,c,q p 2 L 2 (Γ * ) + r ∂ϕ ω ∂ν 2 L 2 (Γ * ) , (45)
subject to the state equations:

a Γ * (q, g) = - Γ * ϕ 1 (x)g(x) dγ ∀ g ∈ H -1/2 (Γ * ), (46)
Γ * q(y) dγ = 0, (47)
The numerical model and the equilibrium equation:

Γ * 1 2µ 0 ∂ϕ ω ∂ν 2 (V •ν) dγ + Γ * (σC -p 0 + p) (V •ν) dγ = 0 ∀ V in C 1 (R 2 , R 2 ), (48)
The numerical model Discretization of the domain We consider an approximation of the domain ω * defined by the piecewise linear closed boundary Γ h , i.e., Γ h is the union of the n linear finite elements j in R 2 , j ∈ {1, . . . , n}. The nodes of the boundary Γ h are denoted by x i .

A direction Ẑi ∈ R 2 is associated to each vertex x i of Γ h . We construct a continuous piecewise linear vector field

Z i from Γ h in R 2 such that Z i (x k ) = δ ik
Ẑi . The support of Z i is equal to the union of the finite elements for which x i is a node. The vector field Z of ( 20) is computed as:

Z(x) = n i=1 u i Z i (x), (49)
and the updated boundary Γ u is then given by:

Γ u = X | X = x + Z(x); u i ∈ R, x ∈ Γ h , (50)
where u T = (u 1 , . . . , u n ) ∈ R n is the vector of unknowns which determine the evolution of the boundary. This representation has the advantage of defining only one degree of freedom for each node. We denote by ω u the interior domain related to Γ u in order to show the dependence with respect to the vector u.

Inductors

We consider inductors with a transversal crosssection Θ p = {x ∈ R 2 : x -y p ≤ η}, with η a positive real unknown. The contribution of each inductor to the function ϕ 1 is calculated using (30). The entire set of shape parameters corresponding to the inductors is denoted by u p .

Exterior boundary value problem

For numerical calculations we consider a piecewise constant approximation q h (x) of q(x):

q h (x) = n j=1 q j e j (x), (51)
where e j (x) = 1 if x ∈ j and zero elsewhere.

Replacing the function g in ( 42) by e i , with i ∈ {1, . . . , n}, the weak formulation of the boundary value problem, given by equations ( 42) and (43), becomes:

A(u)q = b(u p , u), (52)
where the vector q T = (q 1 , . . . , q n , c) is in R n+1 , u is the vector of shape variables and u p is the vector that contains the shape parameters of the inductors. The coefficients a ij of the symmetric matrix A(u) are:

a ij (u) = - 1 2π i j ln x -y dγ dγ i, j ∈ {1, . . . , n}, (53)
a ij (u) = j dγ i = n + 1,
and j ∈ {1, . . . , n}, (54)
and the components b i of the vector b are:

b i (u p , u) = - i ϕ 1 (x) dγ i ∈ {1, . . . , n}, (55)
b i (u p , u) = 0 i = n + 1. (56)
For given vectors u and u p , the linear system (52) is symmetric and non-sparse. Numerical approximations of the element integrals of previous and later equations are computed by Gauss quadrature.

If q is the solution of the system (34), (36) and the piecewise constant approximation q h given by the solution of (52), then we have the following error bounds (see [START_REF] Nédélec | Approximation des équations intégrales en mécanique et en physique[END_REF]):

q -q h H -1/2 (Γ) ≤ C 1 h q H 1 (Γ) , (57)
and if ξ h is the approximation of (33) then

∂ξ ∂ν - ∂ξ h ∂ν H -1/2 (Γ) ≤ C 2 h q H 1 (Γ) . ( 58
)
The approximation of the normal derivative ∂ξ ∂ν at x l ∈ l is given by:

∂ξ h ∂ν (x l ) = - 1 2π n i=1 i =l q i K m=1 p m ∂ ln x l -x i (s m ) ∂ν
The numerical model

+ 1 2 q l , (59)
where x i (s m ) are the integration points and p m the weights of the Gauss quadrature formula. Thus, the computation of ∂ξ h ∂ν (x l ) needs O(n) floating point operations.

Equilibrium equation

Consider a direction V i ∈ R 2 associated to each vertex x i of Γ h and the continuous piecewise linear vector field

V i from Γ h in R 2 such that V i (x k ) = δ ik V i .
If we project the equation (44) in the finite dimensional space generated by V i , i ∈ {1, . . . , n}, the discrete version of the equilibrium is the following:

DE i (u p , u, q) = = Γu 1 2µ 0 ∂ϕ ω ∂ν 2 (V i •ν) dγ + Γu (-p 0 ) (V i •ν) dγ + σC i • V i , ( 60
)
where i ∈ {1, . . . , n} and C i is an approximation of the mean curvature at x i , given by:

C i = (x i -x i-1 ) x i -x i-1 - (x i+1 -x i ) x i+1 -x i . (61)
The gradient ∇ϕ is computed using (37), ( 38) and (39).

In the case of equation ( 48), we consider a piecewise linear function p h defined as:

p h (x) = n i=1 p i f i , (62)
where the function

f i satisfies f i (x k ) = δ ik .
Then, defining p T = (p 1 , . . . , p n ), the equilibrium equation is defined as:

DF i (u p , p, q) = = Γ * 1 2µ 0 ∂ϕ ω ∂ν 2 (V i •ν) dγ + Γ * (-p 0 + p h ) (V i •ν) dγ+ +σC i • V i . (63)
Figure 3. Geometric constraints.

Geometric constraints

In order to avoid the possibility of overlapping between the domains occupied by the liquid metal and the inductors, we consider the following inequality constraints:

ψ(x j ) ≤ ψ 0 , for all x j ∈ X, (64)
where X is a chosen set of points belonging to the boundary of the inductors. The real valued function ψ is zero in the interior of the liquid metal and negative in the exterior. Then, choosing a negative value for the parameter ψ 0 , (64) enforces the points x j to be in the exterior of the liquid metal as illustrated by Fig. 3.

The function ψ that we propose is defined as the solution of:

∆ψ(x) = 0 in Ω * , ψ(x) = 0 on Γ * , Γ * ∇ψ(x) • ν dγ = -1. (65)
In a similar way as the function ξ in Section , ψ can be calculated as:

ψ(x) = - 1 2π Γ q(y) ln x -y dγ + c, (66)
where q must satisfy:

Γ q(y) dγ = -1. (67)
As in Section , an approximated solution of q and c can be obtained solving a linear system similar to Example 1

(52). The numerical approximation of the function ψ is obtained employing (66). The value ψ 0 can be defined choosing a point in the exterior of the liquid metal and calculating the value of the function ψ at this point.

Defining h j (u p ) = ψ(x j (u p )) -ψ 0 , all the geometric constraints are expressed as:

h(u p ) ≤ 0. (68)
Note that the function ψ was defined for the fixed domain Ω * . Then, strictly speaking, the constraints (64) are suitable just for the the second formulation of the inverse problem. In the case of the first formulation, ψ should be defined for the changing domain Ω Z . In that case there is an extra computational cost associated to the computation of q and c each time that the domain Ω Z is updated. However, we have observed for all the examples considered that the use of ψ defined for the fixed domain Ω * is enough to prevent the overlapping between domains. The reason is that the domain occupied by the liquid metal keeps very close to the target shape all along the optimization process.

NUMERICAL EXAMPLES

We consider several examples to illustrate the behavior of the proposed formulations of the inverse problem. The shape and the surface S 0 of the target shape, the surface tension σ, the intensity I and the dimensionless coefficients α p are given.

For each example all the parameters, including the parameters ψ 0 of the geometric constraints, are the same for both formulations. The initial values of the state variables q and p 0 , the shape variable u of the first formulation and the pressure p of the second one are set equal to zero for all the examples.

For the solution of the optimization problems, the line search interior-point algorithm for nonlinear constrained optimization problems FAIPA was employed. For a given feasible point with respect to the inequality constraints, FAIPA defines a feasible and descent arc solving three linear systems of equations with the same coefficient matrix. Then, it performs a line search along this arc to define the next point. FAIPA makes subsequent iterations until a convergence criterion is satisfied.

At each iteration FAIPA needs the vector of partial derivatives with respect to all the design variables of the objective function and all the constraints. Herein, we have calculated exact derivatives for all the functions of the discretized model. For more details about FAIPA see [START_REF] Herskovits | Feasible direction interiorpoint technique for nonlinear optimization[END_REF][START_REF] Herskovits | Mathematical programming models and algorithms for engineering design optimization[END_REF][START_REF] Herskovits | A quasi-Newton interior point algorithm applied to constrained optimum design in computational fluid dynamics[END_REF].

For each example we plot the initial inductors, the target shape of the liquid metal, the inductors obtained by the optimization algorithm and the evolution of the objective function during the iterative process.

Example 1

The target shape of this example is the solution of the direct free-surface problem considering four concentrated intensities of value I = 0.1, with the sign given by Fig. 4, and an area S 0 equal to π; see Example 1a in [START_REF] Canelas | The inverse electromagnetic shaping problem[END_REF].

For the inverse problem we consider four inductors. The intensity I is equal to 0.1 and the surface tension σ is equal to 1.0 × 10 -4 . The dimensionless coefficients α p have absolute value equal to 4.0 with the sign given by Fig. 4. Solutions for the first formulation and three different values of the penalty parameter r are given by Fig. 5. For the second formulation, the solutions are given by Fig. 7. The evolution of the objective functions throughout the optimization process are shown in Figs. 6 and8.

Example 2

The target shape of this example is the bar of area S 0 equal to 7.86 depicted by Fig. 9. For the inverse problem we consider twelve inductors. The intensity I is equal to 0.1 and the surface tension σ is equal to 1.0×10 -4 . The dimensionless coefficients α p have absolute value equal to 4.0 with the sign given by Fig. 9. Solutions for the first formulation and three different values of the penalty parameter r are given by Fig. 10. For the second formulation, the solutions are given by Fig. 12.

Example 3

The target shape of this example is the bar of area S 0 equal to 4.99 depicted by Fig. 14. For the inverse problem we consider twelve inductors. The intensity I is equal to 0.1 and the surface tension σ is equal to 1.0×10 -4 . The dimensionless coefficients α p have absolute value equal to 4.0 with the sign given by Fig. penalty parameter r are given by Fig. 15. For the second formulation, the solutions are given by Fig. 17.

CONCLUSIONS

This paper deals with the shape design of the inductors used in the electromagnetic casting of molten metals. Two different approaches based on nonlinear optimization have been proposed in order to find the position and shape of suitable inductors. The first one minimizes the difference between the geometries of the best possible equilibrium domain and the target shape; the second minimizes a slack variable function related to the equilibrium equation on the target boundary.

It was also considered the addition of a regularizing term to the objective function, in order to penalize the solutions presenting a high magnetic field vector on the boundary of the liquid metal. This regularizing term have a "smoothing" action needed, because the computed magnetic field must correspond with an analytic Γ, if a solution B is to exist.

The finite dimensional optimization problems obtained after discretization were solved employing the line search interior-point algorithm FAIPA.

Some exhibited examples show that both formulations are effective to design suitable inductors.

Figure 2 .

 2 Figure 2. Domain perturbation.

14 .

 14 Solutions for the first formulation and three different values of the

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Example 1 -Target shape and initial configuration. (a) Equilibrium shape considering concentrated intensities. Solid line: equilibrium shape, pluses: positive currents, circles: negative currents. (b) Initial configuration. Circles with pluses: inductors of positive current, circles without pluses: inductors of negative current. Dash-dotted line: target shape. Thin solid line: geometric constraint.

Figure 7 .

 7 Figure 7. Example 1 -Solutions of the second formulation. (a) Solution for r = 0. (b) Solution for r = 0.0001. (c) Solution for r = 0.0005. Solid line: equilibrium shape, pluses: positive currents, circles: negative currents. (b) Initial configuration. Circles with pluses: inductors of positive current, circles without pluses: inductors of negative current. Dash-dotted line: target shape. Thin solid line: geometric constraint.

Figure 8 .

 8 Figure 8. Example 1 -Evolution of the objective function, second formulation.

Figure 9 .Figure 10 .Figure 11 .

 91011 Figure 9. Example 2 -Target shape and initial configuration. Circles with pluses: inductors of positive current, circles without pluses: inductors of negative current. Dash-dotted line: target shape.Thin solid line: geometric constraint.

Figure 12 .

 12 Figure 12. Example 2 -Solutions of the second formulation. (a) Solution for r = 0. (b) Solution for r = 0.00002. (c) Solution for r = 0.0001. Circles with pluses: inductors of positive current, circles without pluses: inductors of negative current. Dash-dotted line: target shape. Thick solid line: equilibrium shape. Thin solid line: geometric constraint.

Figure 13 .

 13 Figure 13. Example 1 -Evolution of the objective function, second formulation.

Figure 14 .Figure 15 .

 1415 Figure 14. Example 3 -Target shape and initial configuration. Circles with pluses: inductors of positive current, circles without pluses: inductors of negative current. Dash-dotted line: target shape.Thin solid line: geometric constraint.

Figure 16 .Figure 17 .

 1617 Figure 16. Example 1 -Evolution of the objective function, first formulation.

Figure 18 .

 18 Figure 18. Example 1 -Evolution of the objective function, second formulation.

ACKNOWLEDGEMENTS

The authors thank the Brazilian Research Councils CAPES, CNPq and Faperj for the financial support. The first author thanks the National Research and Innovation Agency (ANII) of Uruguay


Thomas Janssoone

Grégoire Chevreau

Lucile Vadcard

Pierre Mozer

Jocelyne Troccaz

Biopsym: a Learning Environment for Trans-Rectal Ultrasound Guided Prostate Biopsies

Keywords: Simulator, ultrasound, biopsy, educational content

This paper describes a learning environment for image-guided prostate biopsies in cancer diagnosis; it is based on an ultrasound probe simulator virtually exploring real datasets obtained from patients. The aim is to make the training of young physicians easier and faster with a tool that combines lectures, biopsy simulations and recommended exercises to master this medical gesture. It will particularly help acquiring the three-dimensional representation of the prostate needed for practicing biopsy sequences. The simulator uses a haptic feedback to compute the position of the virtual probe from three-dimensional (3D) ultrasound recorded data. This paper presents the current version of this learning environment.

Introduction

Prostate cancer is the most widespread cancer for men and the second cause of death after lung cancer in many countries. It can be suspected from a blood analysis with an abnormal rate of Prostate Specific Antigen (PSA) or from a suspicious digital rectal examination of this gland. However, the only way to confirm a prostate cancer is to detect one or several positives samples from the anatomo-pathological analysis of prostate biopsies. The simplest way to access to the prostate for biopsy is through the rectum (see Figure 1 left and middle).

Biopsy is performed through a trans-rectal way: the doctor introduces an ultrasound (US) probe into the rectum to visualize the prostate and thus directs the needle to the sites which must be sampled. A mechanical guide attached to the US probe directs the needle to the target. The procedure must follow a well defined protocol in order to take sufficient samples to affirm or cancel the presence of cancer. One very standard protocol is the 12 cores process (see figure 1 right), which must be carried out in a precise order to minimize the discomfort of the patient. The protocol is needed because cancer is generally not visible in the US images; though the samples have to be taken as regularly as possible over the prostate. Additional sites corresponding for example to suspicious zones located during a MRI examination can be added to the standard protocol. This gesture needs a good 3D representation of the prostate whilst standard US images are 2D. Building an accurate mental 3D representation, well coordinated with 2D US images, requires very good hand-eye coordination which may be difficult to acquire; as a consequence biopsies can quickly be poorly distributed.

The conventional teaching method for this practice is only based on companionship: the student learns by observing an expert doing the operation and then reproduces it. With this method, it may be very difficult to understand the intra-rectal gesture only by observation. The hand-eye coordination and the spatial localization may be hard to acquire and may need a long training, difficult to realize in a clinical environment. Moreover the evaluation of the performance of the trainee is purely qualitative.

The following sections present a new way to teach this gesture using an ultrasound-based prostate biopsy simulator integrated in a complete learning environment with exercises.

Material and Methods

In order to set up a new way to teach ultrasound guided prostate biopsies, a complete learning environment is being built. The user, most often a urologist resident, must be completely autonomous with the system and should be able to learn everything about prostate biopsy, from theoretical aspects like anatomy of this organ, to practice exercises with simulation of biopsy sequences on specific cases. This learning environment should also provide practical advices about what to do during the practice of this gesture. Such a system aims at proposing a new method to learn this gesture as realistically and completely as possible.

The Simulator

BiopSym simulates ultrasound biopsies of prostate. A first kernel of the system was developed in 2008 [START_REF] Sclaverano | BiopSym: a simulator for enhanced learning of ultrasound -guided prostate biopsy[END_REF]. Its principle is to use a force feedback device Sensable Omni Phantom (cf. Figure 2) to simulate the interaction of the probe with the anatomy of the patients. The stylus of this device represents the ultrasound probe used during this medical gesture. 3D real ultrasound images acquired on real patients are used to compute the visualization. In the new version which graphical user interface has been redesigned, evaluation of the trainee has been included; exercises and educational content have been added. The evaluation of the virtual biopsies is made thanks to the prostate segmentation [START_REF] Martin | MR prior based automatic segmentation of the prostate in trus images for mr/trus data fusion[END_REF]: the issue is to quantify the way the 12 zones of the prostate have been reached and which amount of tissue has been sampled inside the prostate (it is not rare to get samples out of the gland). The segmentation produces a mesh representing the organ and its bounding box; its edges are known. To locate all the sample sites, a simple decomposition of the box into 12 zones is done and comparisons between them and the coordinates of the needles in the simulator validate or not the sample.

Architecture

3D US volumes used in the application were acquired during real biopsy sessions at La Pitiè Salpètrière Hospital, Paris, France. Patient data (age, prostate size, PSA level, etc) are also recorded in a SQL database. BiopSym uses open source libraries to compute the simulation.

Users of this application will be physicians in urology so a significant care was taken to avoid common errors of medical software user interfaces. Lack of consistency and non intuitive design [START_REF] Noriyoshi | Human-Centered design in medical fields[END_REF], [START_REF] Batu Salman | Participatory icon design process for medical information system[END_REF], [START_REF] Flory | Nouvelles interfaces pour la représentation de l'information médicale[END_REF] are the most frequent criticisms made to medical software. These properties were considered in order to build user-friendly software with a "look & feel" design (see Figure 3).

Exercises

The list of exercises was described in a didactic survey based on an analysis of stakes and methods of urologists for prostate biopsies; it was made by a student in educational science of UPMF University working on the project. Its aim is both the acquisition of a good mental representation of the prostate during an echo-guided intervention and the improvement of the user theoretical knowledge. Exercises include: (1) a questionnaire to determine the probability of a cancer regarding patients' data (PSA level, prostate volume, age, etc), ( 2) image recognition like volume computation on US images (selection of diameters and height) or area localizations (selection of the bladder on a US volume for instance) and ( 3) several types of simulations with different constraints (volume of the prostate, position of the patient, target, link MRI/US, etc). During simulation, assistance can be provided to help the user; such assistance can consist in adding additional views (coronal, 3D, etc) that the user would not have in a standard biopsy procedure (see Figure 4). Evaluations of these exercises allow targeting the weakness of the user and specific exercises to correct them are recommended to him in order to improve his control of this gesture. Thus, the user will be totally autonomous with the application and he will be able to progress alone.

User feedbacks, integrated in the personal panel of the application, display the pedagogical path and evolution of the performances of the user [START_REF] Holzinger | Learning performance with interactive simulations in medical education: Lessons learned from results of learning complex physiological models with the HAEMO dynamics SIMulator[END_REF], [START_REF] Ebner | Successful Implementation of User-Centered Game Based Learning in Higher Educationan Example from Civil Engineering[END_REF]. He is given the possibility to see his timeline representing everything he has done with the application; he can also get details about each activity. For example, he is able to visualize the result of the sequence of biopsies he made two months ago and to watch the results on a 3D view. Charts representing the evolution of his score are also available.

Educational Content

The application also contains a complete lecture about prostate biopsy; it was prepared by a clinical expert. It can be decomposed in two parts:

-A slideshow where the expert reminds and explains the basic knowledge required to practice prostate biopsy: the anatomy of the prostate and its appearance in ultrasound images, its different components and the realisation of the gesture (see Figure 5). -Lectures about the practice of biopsy: the way it is realized, the checklist of all the details needed to make it successful and an explanation text from the "Association Française d'Urologie" (French Urologist Association) which explains it for the patient.

The educational part provides a theoretical support to the exercises and helps the user to acquire a solid background about this medical gesture. All the resources can be transferred to a USB pen for the trainee.

Results

BiopSym allows now the teaching of ultrasound guided prostate biopsy as pertinently and completely as possible. It provides a theoretical support about this gesture and above all a new way to learn this gesture with the simulations. The exercises and their recommendations will help the users to progress and allow them to be totally autonomous. The assistances will help them to get quickly a good mental representation of the prostate and so master faster the biopsy realization. The feedbacks about user performance allow him to visualize his progress and involve him in this practice. BiopSym evaluation with real medical student is scheduled for 2011.

Figure 1 .

 1 Figure 1. US-guided prostate biopsy. (Left) anatomy, (middle) typical ultrasound image, (right) 12-core protocol.

Figure 2 .

 2 Figure 2. (Left) US probe, (middle) Phantom modeling and (right) clipping planes.

Figure 3 .

 3 Figure 3. Screenshots of BiopSym.

Figure 4 .

 4 Figure 4. Virtual biopsy simulation with assistance.

Figure 5 .

 5 Figure 5. Examples of educational content.

Author for correspondance. Jocelyne.Troccaz@imag.fr -TIMC Lab, IN3S -Domaine de la Merci -School of Medicine -38706 La Tronche cedex -France


Nicolas Aubert
email: nicolas.aubert@univmed.fr.

André Lapied

Patrick Rousseau

EMPLOYEE OWNERSHIP AS A SIGNAL OF MANAGEMENT QUALITY September 2010

Keywords: Employee ownership, corporate governance, management entrenchment. JEL classification: G11, G32, G34, J33

The employees' decision to become shareholder of the company they work for can be a consequence of employers' matching contribution in company stock. From a behavioral perspective, employees would regard these contributions as an implicit investment advice made by their employer. This paper adopts another viewpoint. Since employee ownership can be used as an entrenchment mechanism, we suggest that employer's matching policy can be considered as an imperfect signal of management quality. This paper suggests that employee ownership can be used by managers to compensate their management skills to the market. It recommends that employee ownership policy should not be influenced by the managers.

Introduction

There are conflicting arguments regarding employee stock ownership in the academic literature. On one hand, employee ownership would enhance corporate performance through its incentive effects. Moreover, [START_REF] Weitzman | The Share Economy[END_REF] argued that profit sharing mechanism could solve stagflation. On another hand, it would result in poor corporate governance due to the potential collusion between employee owners and management. Indeed, Blasi (1988, p. xi) remarked that employee ownership could be regarded as a revolution or a rip-off. From his point, employee ownership could be regarded as a revolution when it improves corporate performance and lead to a better workplace's satisfaction. But it could lead to a rip-off when it is used as a management entrenchment mechanism.

These conflicting points lead us to investigate management's motivations to offer company stock to employees. [START_REF] Benartzi | Excessive extrapolation and the allocation of company stock to retirement accounts[END_REF] argues that employees tend to consider management matching contribution in company stock as an implicit investment advice. He calls this phenomenon the "endorsement effect" and regards it as primarily behavioral. This paper rather regards matching contribution in company stock as a signal of management quality. From this standpoint, by choosing the level of company stock granted to employees, management would signal their type to the market. Good managers would use employee ownership as an incentive mechanism while bad managers would use it as an entrenchment mechanism.

We emphasize three main results. Firstly, we show that bad managers wish to appear as good ones by using employee ownership as an imperfect signal of their management quality. Secondly, good managers must offer higher level of employee ownership to signal themselves because of the presence of bad managers. Due to this phenomenon, bad managers have an incentive to appear as good managers, but also, good managers have an incentive to appear as bad. Thirdly, introducing prior commitments by the managers could solve this problem. From a managerial perspective, this paper suggests that employee ownership can be used by managers to compensate their management skills to the market. It recommends that employee ownership policy should defined by the board of directors before the manager is hired.

This paper proposes a sequential game where a risk neutral manager grants company stock to his risk averse employee. The remainder of the paper is organized as follows. Section 2 analyzes the literature on employee ownership incentive effects and its implication for corporate governance. Section 3 presents the model set-up. Section 4 develops the conditions of the optimal contracting. Section 5 presents comparative statics analyses. Section 6 offers concluding remarks. All proofs are presented in the appendix.

Literature

A first body of literature on employee stock ownership focused on its positive effects on employees' behavior. These researches tested how employee ownership affects work attitudes such as implication, involvement, satisfaction, turnover and turnover intention. [START_REF] Klein | Employee stock ownership and employees attitudes: a test of three models[END_REF] identified three perspectives to explain the effects of employee stock ownership on employees' behavior: -intrinsic, -instrumental, -extrinsic. The first perspective is the intrinsic satisfaction. It states that employee ownership per se can increase employees' commitment to the organization and its satisfaction. Empirical results gave a little support to this model. The second perspective is the instrumental satisfaction. It states that the employee owner satisfaction and commitment come from the participation in decision-making.

Empirical tests assessed the relationship between employee ownership and perceived employees' influence in decision-making and linked these perceptions with employees' attitudes. The results of empirical studies support this model [START_REF] Long | The effects of employee ownership on organizational identification, employee job attitudes and organizational performance: a tentative framework and empirical findings[END_REF][START_REF] Long | Job attitudes and organizational performance under employee ownership[END_REF][START_REF] Klein | Employee stock ownership and employees attitudes: a test of three models[END_REF][START_REF] Klein | Correlates of employee satisfaction with stock ownership: who likes an ESOP most?[END_REF][START_REF] French | Employee ownership, work attitudes, and power relationships[END_REF]Buchko, 1992aBuchko, , 1992b[START_REF] Buchko | The effects of employee ownership on employee attitudes: an integrated causal model and path analysis[END_REF][START_REF] Pendleton | The perception and effects of share ownership: empirical evidence from employee buy-outs[END_REF][START_REF] Pendleton | Employee Ownership, Participation and Governance: a Study of ESOPs in the UK[END_REF]. The third perspective is the extrinsic satisfaction. It states that employee stock ownership is motivating when it is financially rewarding. Most of the empirical literature suggests that this model could explain employee owners' positive attitudes [START_REF] French | Employee perspectives on stock ownership: financial investment or mechanism of control?[END_REF][START_REF] Klein | Employee stock ownership and employees attitudes: a test of three models[END_REF][START_REF] Rosen | Employee Ownership in America: the Equity Solution[END_REF]Buchko 1992b[START_REF] Buchko | The effects of employee ownership on employee attitudes: an integrated causal model and path analysis[END_REF][START_REF] Gamble | ESOPs and employee attitudes: the importance of empowerment and financial value[END_REF]. Klein concluded this model is in line with the conclusions of the Principal-Agent Model. Since employee stock ownership is a way to motivate employees, it has effects on corporate performance. In the literature, performance is measured in terms of productivity and profitability. [START_REF] Kruse | Employee ownership, employee attitudes, and firm perfomance: a review of the evidence[END_REF] reviewed all the empirical tests of employee stock ownership on performance. To sum up all the findings available, [START_REF] Kruse | Research evidence on prevalence and effects of employee ownership[END_REF] states that empirical literature considers employee stock ownership as having either a positive or null effects on performance.

Another body of literature on employee stock ownership focused on its negative effects on corporate governance. Employee stock ownership is often regarded as a powerful antitakeover tool since it reduces the probability that a takeover arises [START_REF] Shivdasani | Board composition, ownership structure and hostile takeover[END_REF][START_REF] Beatty | The cash flow and informational effects of ESOP[END_REF]. From this standpoint, management would use employee ownership to put shares of the company in "friendly hands" [START_REF] Benartzi | The law and economics of company stock in 401(k) plans[END_REF]. The argument is that collusion between management and employee owners is possible. [START_REF] Pagano | Workers, managers, and corporate control[END_REF] states that: "managers and workers are natural allies against takeover threats" (p. 841). They conclude that: "managers offer long-term contracts to guard against raiders, and workers are willing to take action to protect their high wages" (p. 864). Employee ownership would serve as a management entrenchment mechanism. [START_REF] Gordon | ESOPs and corporate control[END_REF] argue that many employee ownership plans were established in the US during the late 1980s explicitly to defend against takeovers. When an employee stock ownership plan is implemented, event studies report negative reaction of the financial markets in line with the management entrenchment hypothesis [START_REF] Chang | ESOPs and shareholder wealth: an empirical investigation[END_REF][START_REF] Chang | Managerial vote ownership and shareholder wealth: evidence from ESOP[END_REF][START_REF] Conte | Financial returns of public ESOP companies: investors effecs vs. manager effects[END_REF]. As a takeover defence, employee ownership could even be more powerful than poison pills or golden parachutes [START_REF] Chaplinsky | The role of ESOPs in takeover contests[END_REF]. Poison pills and golden parachutes are less used when an employee stock ownership plans is implemented [START_REF] Park | Employee stock ownership plans, firm performance and monitoring by outside blockholders[END_REF]. [START_REF] Rauh | Own company stock in defined contribution pension plans: a takeover defense?[END_REF] confirms that employee ownership has a deterrence effect on takeover probabilities. He further claims that: "Strategic corporate control motives are, therefore, one significant reason managers encourage employees to hold company stock in their defined contribution pension accounts. As laws grant managers more insulation from market discipline, the incentive for management to encourage employees to hold company stock decreases." [START_REF] Brown | 401(k) matching contributions in company stock: costs and benefits for firms and workers[END_REF] find that offering matching contribution in company stock in 401(k) plans is more likely when they do not have multiple classes of stock which is an alternative mechanism to reduce takeover threats. According to them, employee ownership can then be interpreted as a mean to put shares in friendly hands to thwart takeovers.

According to the literature, employee ownership has therefore two sides: a bright one involving a better corporate performance and a dark side leading to management entrenchment. These arguments can be both regarded as management's motives to stimulate employee ownership. In several countries such as the US, management encourages employee ownership through matching contributions in company stock or discount on stock prize. Even if [START_REF] Brown | 401(k) matching contributions in company stock: costs and benefits for firms and workers[END_REF] show that firms tend to take into account the effect of their matching policy on participant retirement security, it is unclear whether workers take advantage of holding their company stock. Meulbroek (2005) and Ramaswamy (2003) estimate that the cost of investing in employer's stock is prohibitive. Since employee owners' saving strategy seems to be irrational, financial economists investigated behavioral biases (see [START_REF] Benartzi | The law and economics of company stock in 401(k) plans[END_REF] for a review). Along with other cognitive biases, [START_REF] Benartzi | Excessive extrapolation and the allocation of company stock to retirement accounts[END_REF] finds that: "when the employer's contributions are automatically directed to company stock, employees invest more of their own contributions in company stock (p. 1748)". According to him this phenomenon is consistent with an "endorsement effect: employees interpret the allocation of the employer's contributions as implicit investment advice (p. 1748)". Purcell (2003) confirms this effect using archival data by emphasizing that 401(k) concentration in company stock is significantly higher in companies where matching contributions are offered in company stock.

Our paper is an attempt to investigate an alternative explanation of employees' response to their employer's policy regarding company stock. We suggest that granting company stock to employees can be interpreted as an imperfect signal on management quality.

The model

Consider a model with three stages involving a risk neutral entrepreneur and a risk averse employee. This employee is representative of all company employees. We thus consider all employees to have homogeneous preferences. Figure 1 illustrates the time line.

t=0 t=1 t=2

The nature chooses the management's type k={G,B}: good (G) with a probability p 0 or bad (B) with a probability 1p 0 .

The manager defines the level of employee ownership c granted to the employees (c ≥ 0).

The employee observes the level of c and decides the level of effort implemented j={H,L}:

high (H) or low (L).

FIGURE 1: TIME LINE

At t = 0, the nature chooses the manager's quality. The type of the entrepreneur can either be good (G) or bad (B) with k={G,B}. The entrepreneur knows his type and the employee has an a priori probability distribution on it: P 0 , p 0 = P 0 (G), 1p 0 = P 0 (B), 0 < p 0 < 1.

At t = 1, the entrepreneur designs the compensation system aimed at motivating the employee to expand his desired level of effort. The compensation system we investigate only consists in employee ownership. We thus consider a setup where the entrepreneur has to decide whether or not to offer company stock to his employee and the amount of it. The entrepreneur's wealth W d is positive and totally invested in company stock. When employee ownership is offered to employee, it takes the form of a contribution in company stock that diminishes entrepreneur's wealth. Suppose c to be a proportion of W s , the positive initial employee's wealth. cW s is the money value of company stock granted to employee. Indeed, the money value of the contribution granted to the employee is very often commensurate with the amount invested by the employee. For instance, it takes the form of a discount or a matching contribution.

Degeorge et al. ( 2004) confirmed empirically that wealthier employees are more willing to take a firm exposure.

At t = 2, the employee observes the compensation system and chooses e, the level of effort that affects the stochastic part of the return on company stock r. The employee's expected utility is supposed to be additively separable between the gains and the disutility of effort. r is a random variable with a zero mean and a probability distribution f(r). r and k are supposed to . Therefore, the employee's effort is productive in the sense of first-order stochastic dominance. The effort is personally costly to the employee that endures ) (e  the disutility that is increasing with effort. Employee's initial wealth W s is totally invested in a risk free asset whose rate of return is r 0 . If employee ownership is granted, employee's wealth increases by an amount cW s of company stock.

The return of company stock is then

) 1 ( ,k j r   
for a level of effort j={H,L} and for different managers' types k={G,B}. According to managers' type and employees' effort, the conditional means of the conditional distributions of the return take the following values:

 L,B = ,  L,G =  + ,  H,B =  + ,  H,G =  +  + , with  > 0,  > 0,  > 0.
The entrepreneur's and employee's outcomes at the end of the game depend on the strategies they adopt. If, at some period of the game, one of the players decides to not participate, their expected utility at the end of the game is null. The agents' expected utilities at the end of the game depend on the manager' quality (G or B), on the level of employee ownership (c) and on the agent's level of effort (H or L). Let's denote

k j i V , ,
the manager expected utility if a contribution i={0,c} is granted, a level of effort j={H,L} is implemented according to his type k={G,B}. The risk neutral manager maximizes the expected value of his wealth V 0 ,j,k or V c,j,k depending on whether c is granted or not:

 V 0, j,k  W d (1 r   j,k ) f (r)dr  W d (1  j,k )    (1)  V c, j,k  (W d  cW s )(1 r   j,k ) f (r)dr  (W d  cW s )(1  j,k )    (2)
With c = 0 and c > 0, respectively, the expected utility of the employee is:

 U 0, j  u[W s (1 r 0 )] (e j ) ( 3
)  U c, j  {p 0 u[W s (1 r 0 )  cW s (1 r   j,G )]  (1 p 0 )u[W s (1 r 0 )  cW s (1 r   j,B )]} f (r)    dr (e j ) (4)
where u(.) is a risk averse von Neumann-Morgenstern utility function.

Derivation of the employee ownership contracts

Optimal contract

First, we characterize an equilibrium of the game without a minimum constraint on company' return. A minimum constraint is introduced in the next sub-section.

Assumption 1: The employee is risk averse and cautious4 .

 x : u'(x)  0,u''(x)  0,RRA(x)  x u''(x) u'(x)  1 (A1)
This hypothesis implies:

 x,[xu'(x)]' 0 (5)
Assumption 2: The employee's expected utility is strictly positive when company stock is not granted, and effort is low:

 U 0,L  u[W s (1 r 0 )] (e L )  0 (A2)
Assumption 3: The employee's expected utility is higher with employee ownership for a given level of effort:

 U c, j  {p 0 u[W s (1 r 0 )  cW s (1 r   j,G )]  (1 p 0 )u[W s (1 r 0 )  cW s (1 r   j,B )]}    f (r)dr (e j )  U 0, j  u[W s (1 r 0 )] (e j ),c  0,j  H,L (A3)
Lemma 1: The relation defined by

(c) = U c,H -U c,L = (e H ) -(e L
) has the following properties: ω(0) = 0, and '(c) > 0.

Proof: See the appendix.

Proposition 1: Without the constraint on the value V, under assumptions 1 to 3, for

  1 [(e H ) (e L )]  c k ,k  G,B there exists a unique  c*  (0,c k ],
where

 c k : W d W s  H ,k   L,k
1  H ,k  0, which is the Perfect Subgame Nash Equilibrium, for j = H, and k = {G, B}. Furthermore, c* is given by the following relation: . Indeed, the higher the amount of company stock, the lower the entrepreneur's wealth V c,i,j . The additional disutility bore by the employee due to a higher effort is exactly compensated by a supplementary utility of wealth. Figure 2 shows that the difference between the two levels of disutility of effort cannot be compensated above

k c . Beyond k c
, employee ownership becomes too costly to the entrepreneur.

Remark 1:

 c G  W d W s  1       c B  W d W s  1    (6)
The two threshold constraints differ according to the two types of managers who know their type G or B whereas the employee cannot observe it.

FIGURE 2: DETERMINATION OF THE OPTIMAL EMPLOYEE OWNERSHIP CONTRACTS

0 c* ω(c) V c,H V c*,H c ) ( ) ( L H e e    G c B c c G * c B * ω B (c) ω G (c) V c*,H,B V c*,H,G
Notes. Thick lines represent (i) entrepreneur's wealth V c,H with the optimal contract; (ii) the maximum thresholds of company stock granted G c and B c ; (iii) the difference of employee's utility wealth due to a high level of effort ω k (c). Dash line represents the disutility of effort bore by the employee.

Revelation of information

Now, we introduce the condition of a minimum return for the entrepreneur. The equilibrium decision is noted c e in the sequel.

If V < V m with V m > 0, the entrepreneur is fired out5 and his value is fixed to 0, except if the employee's ownership is greater than or equal to C m , C m > 0, or equivalently if c > c m , with

c m = C m /W s . In the sequel, W d > C m . (i) If (W d -c* W s )(1 +  H,G ) > (W d -c* W s )(1 +  H,B ) ≥ V m , (7)
then, c e = c*, j = H, and

V = V c*,H,k = (W d -c* W s )(1 +  H,k ), for k = {G, B}. (ii) If V m > (W d -c* W s )(1 +  H,G ) > (W d -c* W s )(1 +  H,B ), (8)
then, c e = Max(c*, c m ) j = H, and

V = V ce,H,k = (W d -c e W s )(1 +  H,k ), for k = {G, B}. (iii) If (W d -c* W s )(1 +  H,G ) ≥ V m > (W d -c* W s )(1 +  H,B ), (9)
(j) for k = G, then, c e = c*, j = H, and

V = V c*,H,G = (W d -c* W s )(1 +  H,G ), (jj) for k = B, then, c e = Max(c*, c m ) j = H, and V = V ce,H,B = (W d -c e W s )(1 +  H,B ).

Remark 2:

We have an increase of c for a given level of effort (H), and then a decrease of the shareholder value. If c*>c m , the manager chooses to give c*. When c e >c*, the manager's type is revealed to the employee.

The case (iii) corresponds to a separating equilibrium since G and B are identified.

Assumption 4: (W d -c* W s )(1 +  H,G ) ≥ V m > (W d -c* W s )(1 +  H,B ), (A4)
with c* characterized in Proposition 1.

Assumption 4 applies in the sequel.

The employee observes the value of c*, then, he can identify the type of the manager. In the case where c m > c*:

(j) If c e = c*, then k = G, (jj) If c e = c m , then, k = B.
The employee adapts his beliefs and then his behavior. Then, his expected utility depends now on the type k = {G, B} of the manager:

) ( ) ( )] 1 ( ) 1 ( [ , 0 , , j k j s s k j c e dr r f r cW r W u U             (10)
and the relation (.) becomes  k (c) = U c,H,k -U c,L,k + (e H ) -(e L ) since the employee is now able identify the manager's type. (j) for k = G,

(i) If (W d -c G * W s )(1 +  H,G ) ≥ V m , c G e = c G *, ( 11
) (ii) If (W d -c G * W s )(1 +  H,G ) < V m , , c G e = Max(c G *, c m ), (12)
In (ii), we underlines that the good manager's wealth can now be below the V m constraint whereas it was not possible with the contract described in the previous section.

(jj) for k = B,

(i) If (W d -c B * W s )(1 +  H,G ) ≥ V m , c B e = c B *, ( 13
) (ii) If (W d -c B * W s )(1 +  H,G ) < V m , , c B e = c m , (14)
where

 c G *  (0,c G ] and  c B *  (0,c G ]
are given by the following relation: , which is the Perfect Subgame Nash Equilibrium, for j = H, and k = {G, B}. Furthermore, c* is given by:

 (c*)  (e H ) (e L ) ( 15
)
Proof: See the appendix.

Remark 4: The revelation of information increases the contribution for k = G, and decreases the contribution for k = B, for a given level of effort H. These relations hold when the constraint k c and the revelation of the information have an effect. This conclusion does not invalidate the previous findings. Bad managers offer a higher contribution to avoid a layoff.

Because of the presence of bad managers, the level of the contribution increases. This phenomenon can be analyzed as an adverse selection problem where signaling is costly to the good managers only because of the presence of bad managers. The employee is always more demanding with good managers. This situation occurs because the disutilities of effort difference does not depend on manager's type. Furthermore, the equilibrium is reached when this difference is equal to the expected utilities difference. This latter difference is higher for good managers (see figure 2).

Cheating and commitments

Both managers' types have an incentive to cheat. The good and the bad managers can respectively offer the bad and the good contribution. In this situation, the equilibrium conditions do not hold any more. Let's examine this assertion. When

c G * < c m and (W d -c G * W s )(1 +  H,G ) ≥ V m >(W d -c B * W s )(1 +  H,G ),
-c G * W s )(1 +  H,G ) ≥ V m >(W d -c B * W s )(1 +  H,G ).
It follows that, in any case, B gives the signal G or G gives the signal B. Both managers' types have an incentive to choose the same behavior and the revelation is no longer possible. But this result opens the question of a second round problem: what is the optimal decision in the situation where it is possible to cheat? This question is submitted to an infinite regression phenomenon and no solution can appear without the introduction of necessarily arbitrary meta-believes.

Another way to solve the problem is to introduce commitments. The level of c can be set before the nature chose the manager's type. His expected value is then:

V 0,j = p 0 W d (1 +  j,G ) + (1 -p 0 )W d (1 +  j,B ) = W d [1 + p 0  j,G + (1 -p 0 )  j,B ], (16)
or:

V c,j = p 0 (W d -c W s ) (1 +  j,G ) + (1 -p 0 ) (W d -c W s ) (1 +  j,B ) = (W d -c W s ) [1 + p 0  j,G + (1 -p 0 )  j,B ]. ( 17
)
Proposition 3: Under assumptions A1 to A4, for

  G 1 [(e H ) (e L )]  c
there exists an unique c k e , which is the Perfect Subgame Nash Equilibrium, for j = H, and k = G, B, such that:

-In the case where c m > c*, Where c* is given by the following relation:

(i) If  p 0 (W d  c *W s )(1  H ,G )  (W d  c m W s )(1  ) , c k e = c*, ( 18
) (ii) If  p 0 (W d  c *W s )(1  H ,G )  (W d  c m W s )(1  ) , c k e = c m , (19)
 (c*)  (e H
) (e L ) , and

      0 , 1 p W W c s d      (20)
Proof: See the appendix.

Remark 5: According to proposition 3, a better compensation policy involving employee ownership should be designed before the manager type is known. From a managerial perspective, this means that the stockholders in charge of hiring and controlling the manager should not leave decisions regarding employee ownership policy to the manager's discretion.

Otherwise, our conclusions show that managers are likely to use employee ownership to compensate their type. But if such a recommendation applies, employee ownership does not allow separating good managers from bad managers any more.

Comparative statics

To illustrate how relational contract can be used, this section provides numerical results of the model. The calibration needs the specification of the distributions of company stock returns and the employee's utility function. We take the usual assumption of normality for the distribution and a negative exponential utility function. Given the conditions of the perfect Nash equilibrium in sub-game expounded in section 4.2 by the proposition 2 and the form of the negative exponential utility function, the analytical solution of c k * is given by

) ( ) ( ) ( ) ( ) ( * , * , * L H k k L k k H k k e e c F c F c       
, then:

 D L,k (c k * )  D H,k (c k * ) (e H ) (e L ), k = {G, B}, where: ]} 2 ) 1 ( ) 1 [( exp{ ) ( 2 2 , 0 ,    s k j s k j W ac c r aW c D      
, with j = {H, L}.

(21)

In Proposition 1, the equilibrium value c* is given by:

 (c*)  [ p 0 F H,G (c*)  (1 p 0 )F H ,B (c*)] [ p 0 F L,G (c*)  (1 p 0 )F L,B (c*)] (e H ) (e L ), ( 22
)  p 0 [F H,G (c*)  F L,G (c*)]  (1 p 0 )[F H,B (c*)  F L,B (c*)] (e H ) (e L ) ( 23
)  p 0 [D L,G (c*)  D H,G (c*)]  (1 p 0 )[D L,B (c*)  D H,B (c*)] (e H ) (e L
) . ( 24)

Proof: See the appendix.

Traditional comparative statics analysis makes it possible to emphasize several properties of the solution. Figure 3 displays the results of the simulations.

Figure 3 (a) displays the positive relationship between p 0 and c*. The higher the probability of having a good manager is, the higher the amount of company stock granted to the employee.

In an economy where good managers are more numerous, the employees request a higher . This result confirms and reinforces the previous conclusion of figure 3 (a) and remark 4. As the difference between bad and good managers increases, the employees become more demanding and seek to capture a higher proportion of company's wealth. In a situation where the employees think they face good managers, they therefore ask a higher amount of company stock. . The μ parameter is the company basic stock mean return. As for this chart, the more profitable is the firm, the more the employee wants to be compensated with company stock. We also notice a higher difference between * G c and c* than between * B c and c*. This result emphasizes that, for a given level of μ, the employees are more sensitive to the company's performance when the company is led by a good manager. . In fact, the employee's effort does not depend on the manager's type. It remains the same whatever the manager's type is. But, as the productivity of the effort increases, the amount of company stock diminishes. In order to obtain the same return associated with a high level of effort, the manager pays the employee less because the productivity increases.  . This latter effect is the more difficult to explain since it depends on the values of four different parameters i.e. α, μ, β, σ. The two former parameters are positively associated with the amount of company stock whereas the two latter are negatively associated with it (see figures 3(b) to 3(e)). As far as the increasing part of the curve, it is completely determined by the value of the parameter a i.e. the employee's absolute risk aversion. Indeed, when the simulation is run with a lower value of a (for instance 0,3), the increasing section of the curve disappears.

 G c ; 2 , 0 0  p ; 05 , 0 0  r 1 , 0   ; 5 , 0  m  ; 1  m  ; 5 , 7  e  ; 2 , 0  em  ; 5 , 0  s W ; 5  d W ; 6 , 0  a ; 05 , 0   ; 05 , 0   ; 1 , 0   .

Concluding remarks

In this paper, we set up a model which regards employees' compensation in company stock as an imperfect signal of management quality. This viewpoint significantly differs from the existing literature in behavioral finance. This body of literature assumes that employees invest their money in their firm because they consider their employer's contribution in company stock as implicit investment advice. However, employee ownership is often analyzed as an entrenchment mechanism by corporate governance literature.

This paper brings about three main conclusions. Firstly, we that employee ownership can be used by managers to compensate their actual management's skills. Secondly, we demonstrate that higher contributions in company stock as demanded to the good managers than to the bad. This situation is similar to an adverse selection problem where the presence of bad managers makes it costly to the good managers to signal themselves. But this phenomenon comes with another problem since it incentivizes both managers' type to hide their actual type. It is valuable for the bad managers to appear as good ones. Similarly, appearing as bad managers can also be profitable to good managers. To solve this problem, we introduce commitments which lead us to our third result. Our third main result has a normative thrust. In order to prevent managers from hiding their actual type, our model suggests that compensation mechanisms involving employee ownership should be defined before the manager's type is known. In other words, managers should not interfere with employee ownership policy. The comparative statics section highlights several main results.

First, in a situation where the employees think they face good managers, they therefore ask a higher amount of company stock. Second, for a given level of μ, the employees are more sensitive to the company's performance when the company is led by a good manager. Third, as the productivity of the effort increases, the amount of company stock granted to the employee decreases.

Appendix

Proof of Lemma 1:

 (c)  {p 0 u[W s (1 r 0 )  cW s (1 r   H ,G )]  (1 p 0 )u[W s (1 r 0 )  cW s (1 r   H ,B )]} f (r)    dr  {p 0 u[W s (1 r 0 )  cW s (1 r   L,G )]  (1 p 0 )u[W s (1 r 0 )  cW s (1 r   L,B )]} f (r)    dr (25)  (c)  {p 0 (u[W s (1 r 0 )  cW s (1 r      )]  u[W s (1 r 0 )  cW s (1 r    )])    (1 p 0 )(u[W s (1 r 0 )  cW s (1 r    )]  u[W s (1 r 0 )  cW s (1 r  )])} f (r)dr (26)
We have first ω(0) = 0. For the second property:

 '(c)  { p 0 (W s (1 r      )u'[W s (1 r 0 )  cW s (1 r      )]    W s (1 r    )u'[W s (1 r 0 )  cW s (1 r    )]) (1 p 0 )(W s (1 r    )u'[W s (1 r 0 )  cW s (1 r    )] W s (1 r  )u'[W s (1 r 0 )  cW s (1 r  )])} f (r)dr (27) Let  X : W s (1 r 0 )  cW s (1 r      ) Y : W s (1 r 0 )  cW s (1 r    ) Z : W s (1 r 0 )  cW s (1 r    ) T : W s (1 r 0 )  cW s (1 r  ) Replacing X, Y, Z, and T in equation (8) gives:  '(c)  {p 0 [ X  W s (1 r 0 ) c u'(X)     Y  W s (1 r 0 ) c u'(Y )] (1 p 0 )[ Z  W s (1 r 0 ) c u'(Z)  T  W s (1 r 0 ) c u'(T)]} f (r)dr (28)  '(c)  1 c {p 0 [Xu'(X)  Yu'(Y)]  (1 p 0 )[Zu'(Z)  Tu'(T)]    W s (1 r 0 )( p 0 [u'(Y)  u'(X)]  (1 p 0 )[u'(T)  u'(Z)])} f (r)dr (29)
Under hypothesis A1, and with:  r, X > Y, and Z > T, we have:

 r, u'(Y) > u'(X), u'(T) > u'(Z), Xu'(X) ≥ Yu'(Y), Zu'(Z) ≥ Tu'(T)
, and then '(c) > 0.

Proof of Proposition 1. A Perfect Subgame Nash Equilibrium (V c,H , U c,H ), with c > 0, must satisfy the following conditions:

The manager plays 0

 c if V c,H ≥ V 0,L and V c,H ≥ 0.
The employee chooses the level of effort k = H if U c,H ≥ U c,L and U c,H ≥ 0.

First, by assumption 2, when c = 0, the expected utility of employee is strictly positive with the level of effort j = L, and strictly greater than the value obtained with j = H, because

(e H ) > (e L ) > 0. Therefore, participation of the employee in the firm is optimal. Then, U c,H ≥ U c,L and U c,L ≥ U 0,L (by assumption 3), implies U c,H ≥ 0.

Second, the entrepreneur's exit cannot be an equilibrium because, V 0,L is always possible, as it is stated in the first point, and it is strictly positive. Then, V c,H ≥ V 0,L implies V c,H ≥ 0.

Third, with

 (c*)  (e H

) (e L ) , and the lemma 1, the condition

  1 [(e H ) (e L )]  c k ,k  G,B , implies  c*  c k . Furthermore,  c*  c k : W d W s  H ,k   L,k 1  H ,k  (W d  c *W s )(1  H ,k )  W d (1  L,k )  V c*,H ,k  V 0,L,k (30)
and then, we have:

V c*,H,k ≥ V 0,L,k , for any k. Fourth, U c,H ≥ U c,L is equivalent to  (c)  (e H
) (e L ) This characterize a positive Perfect Subgame Nash Equilibrium c*, with an equality in the preceding relation.

Proof of Lemma 2. The properties of the functions 

k (c) result of a straightforward adaptation of the proof of Lemma 1. Furthermore,   G (c)  {u[W s (1 r 0 )  cW s (1 r      )]  u[W s (1 r 0 )  cW s (1 r    )]} f (r)    dr (31)  B (c)  {u[W s (1 r 0 )  cW s (1 r    )]  u[W s (1 r 0 )  cW s (1 r  )]} f (r)    dr (32) Let  X : W s (1 r 0 )  cW s (1 r    ) Y : W s (1 r 0 )  cW s (1 r  ) a : cW s    G (c)  [u(X  a)  u(Y  a)] f (r)    dr   B (c)  [u(X)  u(Y )] f (r)    dr with X > Y.
As u(.) is a strictly concave function, for any r, and for any strictly positive a, Proof of proposition 3. First, the condition V c,H ≥ V 0,L is now:

u(X + a) -u(Y + a) < u(X) -u(Y),
 c  c : W d W s p 0 ( H ,G   L,G )  (1 p 0 )( H ,B   L,B ) 1 p 0  H ,G  (1 p 0 ) H ,B  W d W s  1  ,     p 0  (33)
Second, when c m > c*, for c = c*, the expected value of the entrepreneur is:

V c*,H = p 0 (W d -c* W s ) (1 +  H,G ),
and for c = c m , the expected value of the entrepreneur is:

 V c m , H  (W d  c m W s )(1  ) .
Then, if V c*,H ≥ V cm,H , the optimal choice is c*, and if V c*,H < V cm,H the optimal choice is c m .

Third, if c m ≤ c*, the optimal choice is c*.

These results are a direct adaptation of Proposition 1 and 2.

Application with a negative exponential utility function

This function is defined up to a positive linear transformation: u(w) =  v(w) + , and must satisfy the following normalization 6 : u(0) = 0. Then,  = , and, without lost of generality, we take  = 1. Therefore, in the sequel, the utility is 7 : u(w) = 1exp(-aw).

Let:

 F j,k (c) : u[W s (1 r 0 )  cW s (1 r   j,k )] f (r)dr    , ( 34
)
and we suppose that f(.) is a centered normal density with variance  2 . Then: (38)

 F j,k (c)

   (c*)  (e H ) (e L ) Proof: See the appendix.

Figure 2

 2 Figure 2 illustrates propositions 1 and 2. The manager selects the minimum level of c that insures a high level of effort at the intersection of ω(c) and ) ( ) ( L H e e   . Indeed, the

Lemma 2 :-

 2 The relations  k (c) have the following properties: ω k (0) = 0, and  k '(c) > 0.The solutions c k *, for k = {G, B}, of the equations:  k (c k *) (e H ) (e L ) are such that: c B * < c* < c G *.Proof: See the appendix.Remark 3: Irrespective of the k c constraint, a higher amount is required from the good manager. In the case where c m > c*, there exists a unique c k e , which is the Perfect Subgame Nash Equilibrium, for j = H, and k = {G, B} such that:

)

  (e L ) for k = G, B. -In the case where c m ≤ c*, there exists a unique ]

-

  In the case where c m ≤ c*: c k e = c*.

  contribution in company stock according to remark 4. If p 0 = 0, c*= * α parameter makes the difference between good and bad managers. Whereas it has no effect on * B c , it increases the value of the contribution * G c

Figure 3

 3 Figure 3 (c) displays the positive relationship between μ and c*, * G c and * B c. The μ parameter is the company basic stock mean return. As for this chart, the more profitable is the firm, the

Figure 3

 3 Figure 3 (d) displays the negative relationship between σ and c*, * G c and * B c . The σ parameter is the company's returns standard deviation. According to this figure, the riskier is the firm, the less the employee wants to be compensated with company stock. We also find the comparable difference between * G c and c* than between * B c and c* as in figure 3 (c). In figures 3 (c) and 3 (d), we cannot draw any conclusion from the relative positions of the lines c*, * G c and * B c . Although we notice that the distance between c* and * G c is higher than the one between c* and * B c , these differences are only determined by the value of the parameter p 0 used in the simulations (i.e. 0,2). Another value would have result in a different position on the chart. For instance, if p 0 = 0,5, the c* line lies exactly halfway through * G c and

Figure 3

 3 Figure 3 (f) shows the negative relationship between a and c*, * G c and * B c . The a parameter is the employee's absolute risk aversion. This parameter has the same negative effect on c*, * G c and * B cbecause the employee's risk aversion does not depend on the manager's type. As the employee's risk aversion increases, he becomes more reluctant to hold risky assets.

Figure 3

 3 Figure 3 (g) shows the curvilinear relationship between W s and c*, * G c and * B c . The W s parameter measures the employee's wealth. Considering assumption 3, on can distinguish two different effects: a wealth effect due to the value of W s and a scale effect related to the value of k j,

  (e), (f) and (g). FIGURE 3 (CONTINUED): RELATIONSHIPS BETWEEN THE OPTIMAL EMPLOYEE OWNERSHIP CONTRACTS AND THE VARIABLES OF THE MODEL

  and then,  G (c) <  B (c), for any positive c. With (c) = p 0  G (c) + (1p 0 )  B (c), and the preceding equation, we have: c B * < c* < c G *. Proof of Proposition 2. First, when c m ≤ c*, with the condition c B and c B * < c* < c G * (Lemma 2), without revelation, the equilibrium is obtained as in Proposition 1. Second, when c m > c*, c G * and c B * are defined as c* in Proposition 1, with  G (.) and  B (.) instead of (.). Their existence is satisfied with respect to Lemma 2 and the inequalities stated in the first point. (j) for k = G,(i) If (W dc G * W s )(1 +  H,G ) ≥ V m , c G e = c G *, (ii) If (W dc G * W s )(1 +  H,G ) < V m , , c G e = Max(c G *, c m ). (jj) for k = B, (i) If (W dc B * W s )(1 +  H,G ) ≥ V m , c B e = c B *, (ii) If (W dc B * W s )(1 +  H,G ) < V m , , c B e = Max(c B *, c m ) = c m .

  , where (x) is a normal density with mean m x and variance  by term of this last relation, for x = r, leads to: x = , m x =a c W s  2 ,and 6 Because U = 0 when the employee exert his exit option (see §1). 7 Assumption A1 restrict the domain studied to a w ≤ 1.D j,k (c)  exp{a[W s (1 r 0 )  cW s (1  j,k )k (c)  exp{aW s [(1 r 0 )  c(1  j,k )  ac 2 W s 

See Gollier (2008).

This minimum return condition is consistent with the argument of[START_REF] Jensen | The Market for Corporate Control: The Scientific Evidence[END_REF]. They argue that takeover target firms experience negative abnormal returns in the period prior to their acquisition.

Acknowledgements

We wish to thank Guillaume Garnotel for a helpful comment.