Joël Puibasset
email: puibasset@cnrs-orleans.fr

Elastic Compliance and Stiffness Matrix of the FCC Lennard-Jones Thin Films: Influence of Thickness and Temperature

The fcc Lennard-Jones crystal is used as a generic model of solid to study the elastic properties of thin films as a function of thickness and temperature. The Monte Carlo algorithm is used to calculate the average deformations along the axes in the isostress-isothermal ensemble that mimics a real uniaxial loading experiment. The four independent parameters (tetragonal symmetry without shear) have been calculated for film thicknesses ranging from 4 to 12 atomic layers, and for five reduced temperatures between 0 and 0.5 /kB, where  is the energetic parameter of the Lennard Jones potential and kB is Boltzmann's constant. These parameters (Poisson's ratio and moduli) give the compliance matrix, which is inverted to get the stiffness coefficients. It is shown that the three Poisson's ratios exhibit a good linearity with the inverse of the film thickness, while this is not the case for the moduli and the compliance coefficients.

Remarkably, the stiffness coefficients do exhibit a good linearity with the inverse of the film thickness, including the limiting value of infinite thickness (bulk solid) obtained by applying periodic boundary conditions in all directions. This linearity suggests to interpret the results in terms of a bulk+surface decomposition. However, the surface stiffness matrix deduced from the slopes has nonzero components along the out-of-plane direction, an unexpected observation in the framework of the surface stress theory.

I Introduction

Solids are not rigid systems: their deformation under loading [START_REF] Landau | Theory of Elasticity[END_REF] can play an important role in various situations. In particular, in the context of fluid adsorption in solids, it has long been observed that the presence of a condensable gas in a porous material induces its deformation. [START_REF] Meehan | The Expansion of Charcoal on Sorption of Carbon Dioxide[END_REF][START_REF] Bangham | The Expansion of Charcoal Accompanying Sorption of Gases and Vapours[END_REF] Obviously, soft or deformable systems exhibit large deformations which are due to and reciprocally influence adsorption. [START_REF] Dušek | Responsive Gels: Volume Transitions I[END_REF][START_REF] Scherer | Deformation of Aerogels During Characterization[END_REF][START_REF] Ogieglo | In Situ Ellipsometry Studies on Swelling of Thin Polymer Films: A Review[END_REF][START_REF] Kulasinski | Poroelastic Model for Adsorption-Induced Deformation of Biopolymers Obtained from Molecular Simulations[END_REF] But in most cases, adsorbents are solid bodies, and the deformations are small (elastic regime). [START_REF] Scherer | Dilatation of Porous Glass[END_REF][START_REF] Yates | The Expansion of Porous Glass on the Adsorption of Non-Polar Gases[END_REF][START_REF] Haines | Length Changes of Activated Carbon Rods Caused by Adsorption of Vapors[END_REF][START_REF] Balzer | Deformation of Microporous Carbon During Adsorption of Nitrogen, Argon, Carbon Dioxide, and Water Studied by in Situ Dilatometry[END_REF][START_REF] Fomkin | Adsorption-Induced Deformation of Adsorbents[END_REF] The theory of poroelasticity has been developed in this context to study the adsorption-induced deformation of systems like soils or reservoirs, where it can play an important role. [START_REF] Biot | General Theory of Three-Dimensional Consolidation[END_REF][14][START_REF] Coasne | Poroelastic Theory Applied to the Adsorption-Induced Deformation of Vitreous Silica[END_REF][START_REF] Brochard | Poromechanics of Microporous Media[END_REF][START_REF] Gor | Adsorption-Induced Deformation of Nanoporous Materials-a Review[END_REF][START_REF] Vandamme | Adsorption and Strain: The CO2-Induced Swelling of Coal[END_REF][START_REF] Vandamme | Coupling between Adsorption and Mechanics (and Vice Versa)[END_REF] The smaller the pores, the larger the capillary forces that induce the deformations: it is thus important to have good models of the mechanical properties of nanometric systems in order to be able to predict their deformation during adsorption.

Conversely, the deformation of a saturated nanoporous medium along the main desorption curve is also a powerful tool to measure the elastic constants of nanometric systems, in complement to direct measurements. [START_REF] Dolino | Adsorption Strains in Porous Silicon[END_REF][START_REF] Boissiere | Porosity and Mechanical Properties of Mesoporous Thin Films Assessed by Environmental Ellipsometric Porosimetry[END_REF][START_REF] Herman | Deformation of Silica Aerogel During Fluid Adsorption[END_REF][START_REF] Dourdain | Determination of the Elastic Modulus of Mesoporous Silica Thin Films by X-Ray Reflectivity via the Capillary Condensation of Water[END_REF][START_REF] Prass | Capillarity-Driven Deformation of Ordered Nanoporous Silica[END_REF][START_REF] Schoen | Pore-Lattice Deformations in Ordered Mesoporous Matrices: Experimental Studies and Theoretical Analysis[END_REF][START_REF] Sharifi | Humidity-Driven Deformation of Ordered Mesoporous Silica Films[END_REF][START_REF] Grosman | Adsorption-Induced Strain of a Nanoscale Silicon Honeycomb[END_REF][START_REF] Liu | The Pore-Load Modulus of Ordered Nanoporous Materials with Surface Effects[END_REF][START_REF] Rolley | Using Capillary Forces to Determine the Elastic Properties of Mesoporous Materials[END_REF][START_REF] Zhang | Determining the Effects of Surface Elasticity and Surface Stress by Measuring the Shifts of Resonant Frequencies[END_REF][START_REF] Fu | Determination of Young's Modulus and Poisson's Ratio of Thin Films by X-Ray Methods[END_REF][START_REF] Hoogeboom-Pot | Nondestructive Measurement of the Evolution of Layer-Specific Mechanical Properties in Sub-10 nm Bilayer Films[END_REF] The knowledge of the fluid-solid interactions has also proven to be important due to specific effects. [START_REF] Puibasset | Adsorption-Induced Deformation of a Nanoporous Material: Influence of the Fluid-Adsorbent Interaction and Surface Freezing on the Pore-Load Modulus Measurement[END_REF] Understanding the mechanical properties of nanosystems is also important in the context of nanoporous solids, in particular regarding the adsorption/desorption hysteresis that appears for a sufficiently low temperature. Many explanations have been proposed: the pore morphology which influences the adsorption and desorption mechanism, the topological effect of the porous network, the disorder created by the solid adsorbent, etc. [START_REF] Cohan | Sorption Hysteresis and the Vapor Pressure of Concave Surfaces[END_REF][START_REF] Everett | A General Approach to Hysteresis[END_REF][START_REF] Everett | A General Approach to Hysteresis. Part2: Development of the Domain Theory[END_REF][START_REF] Mason | The Effect of Pore Space Connectivity on the Hysteresis of Capillary Condensation in Adsorption Desorption Isotherms[END_REF][START_REF] Evans | Capillary Condensation and Adsorption in Cylindrical and Slit-Like Pores[END_REF][START_REF] Kierlik | Phase Diagrams of Single-Component Fluids in Disordered Porous Materials: Predictions from Integral-Equation Theory[END_REF][START_REF] Kierlik | Equilibrium and out-of-Equilibrium (Hysteretic) Behavior of Fluids in Disordered Porous Materials : Theoretical Predictions[END_REF][START_REF] Kierlik | Adsorption Hysteresis and Capillary Condensation in Disordered Porous Solids: A Density Functional Study[END_REF][START_REF] Puibasset | Adsorption/Desorption Hysteresis of Simple Fluids Confined in Realistic Heterogeneous Silica Mesopores of Micrometric Length: A New Analysis Exploiting a Multiscale Monte Carlo Approach[END_REF][START_REF] Bonnet | Evidence for a Disorder-Driven Phase Transition in the Condensation of 4 He in Aerogels[END_REF][START_REF] Puibasset | Monte Carlo Multiscale Simulation Study of Argon Adsorption/Desorption Hysteresis in Mesoporous Heterogeneous Tubular Pores Like MCM-41 or Oxidized Porous Silicon[END_REF][START_REF] Puibasset | Counting Metastable States within the Adsorption/Desorption Hysteresis Loop: A Molecular Simulation Study of Confinement in Heterogeneous Pores[END_REF][START_REF] Puibasset | Numerical Characterization of the Density of Metastable States within the Hysteresis Loop in Disordered Systems[END_REF][START_REF] Puibasset | Stability Intervals of Metastable States in Hysteretic Systems[END_REF][START_REF] Puibasset | Fluid Adsorption in Linear Pores: A Molecular Simulation Study of the Influence of Heterogeneities on the Hysteresis Loop and the Distribution of Metastable States[END_REF][START_REF] Bonnet | On Condensation and Evaporation Mechanisms in Disordered Porous Materials[END_REF] The observation of correlations between the adsorption/desorption hysteresis and the elastic deformations [START_REF] Amberg | A Study of Adsorption Hysteresis by Means of Length Changes of a Rod of Porous Glass[END_REF][START_REF] Quinn | The Hysteresis Loop in Adsorption Isotherms on Porous Vycor Glass and Associated Dimensional Changes of the Adsorbent[END_REF][START_REF] Fomkin | Adsorption of Gases, Vapors and Liquids by Microporous Adsorbents[END_REF][START_REF] Günther | Novel Insights into Nanopore Deformation Caused by Capillary Condensation[END_REF] has been the starting point of recent studies which suggest that the free energy variations due to the solid deformations could affect the isotherm, and explain the shape of the hysteresis curve, in particular the sharp desorption branch generally associated with collective or avalanche effects. 54- 57 Very recently, the development of nanometric devices (nanoelectromechanical systems, NEMS) [START_REF] Ekinci | Nanoelectromechanical Systems[END_REF] has been the starting point of many studies, in particular on silicon nanocantilevers. [START_REF] Li | Ultrathin Single-Crystalline-Silicon Cantilever Resonators: Fabrication Technology and Significant Specimen Size Effect on Young's Modulus[END_REF][START_REF] Park | Molecular Dynamics Study on Size-Dependent Elastic Properties of Silicon Nanocantilevers[END_REF][START_REF] Gordon | Size Effects in Mechanical Deformation and Fracture of Cantilevered Silicon Nanowires[END_REF][START_REF] Sadeghian | On the Size-Dependent Elasticity of Silicon Nanocantilevers: Impact of Defects[END_REF] It is thus desirable to have accurate models of the elastic properties of nanometric systems, and many studies have focused on the size dependence of the mechanical properties of nanometric devices.

The effect of thickness is generally attributed to the surface stress induced by the free surfaces. 63- 72 A clear understanding of experimental data is still lacking, and theoretical approaches have been developed for a long time. [START_REF] Domb | The Zero Point Energy and Θ Crystals[END_REF][START_REF] Horton | Ideal Rare-Gas Crystals[END_REF][START_REF] Klein | Comparison of Classical Monte Carlo Experiments with Improved Self-Consistent Phonon Theory: Thermodynamic Properties of Solid Xe[END_REF][START_REF] De Wette | Lattice Dynamics, Thermal Expansion and Specific Heat of a Lennard-Jones Solid in the Quasi-Harmonic Approximation[END_REF][START_REF] Westera | Cell-Cluster Expansion for an Anharmonic Solid[END_REF][START_REF] Mohazzabi | Simple Classical Calculation of Thermal Expansion for Rare-Gas Solids[END_REF][START_REF] Tretiakov | Quick and Accurate Estimation of the Elastic Constants Using the Minimum Image Method[END_REF][START_REF] Phuong | Mechanical Properties of Metallic Thin Films: Theoretical Approach[END_REF] The nanometric size being also well-suited for atomistic studies, many algorithms have been proposed to track the problem by molecular simulations in various statistical ensembles. [START_REF] Squire | Isothermal Elastic Constants for Argon. Theory and Monte Carlo Calculations[END_REF][START_REF] Sprik | Second-Order Elastic Constants for the Lennard-Jones Solid[END_REF][START_REF] Quesnel | Elastic Compliances and Stiffnesses of the FCC Lennard-Jones Solid[END_REF][START_REF] Ray | Fluctuations and Thermodynamic Properties of Anisotropic Solids[END_REF][START_REF] Ray | Elastic Constants and Statistical Ensembles in Molecular Dynamics[END_REF][START_REF] Miyazaki | Calculation of Mechanical Properties of Solids Using Molecular Dynamics Method[END_REF][START_REF] Van Workum | Mechanical Properties of Solids and Nanoscopic Structures via Molecular Simulations[END_REF][START_REF] Shenoy | Atomistic Calculations of Elastic Properties of Metallic FCC Crystal Surfaces[END_REF][START_REF] Clavier | Computation of Elastic Constants of Solids Using Molecular Simulation: Comparison of Constant Volume and Constant Pressure Ensemble Methods[END_REF] These molecular simulations are useful for accurate predictions since in principle any well-suited interatomic potential may be used to describe the interactions.

On the other hand, non-specific mechanical properties of solid may be studied using simple pairwise additive interactions like the Lennard Jones potential. [START_REF] Cowley | Some Monte Carlo Calculations for the Lennard-Jones Solid[END_REF][START_REF] Li | Fluctuations and Thermodynamic Response Functions in a Lennard-Jones Solid[END_REF][START_REF] Quesnel | The Poisson Ratio for an FCC Lennard-Jones Solid[END_REF][START_REF] Iwaki | Molecular Dynamics Study on Stress-Strain in Very Thin Film : Size and Location of Region for Defining Stress and Strain[END_REF][START_REF] Heinz | Accurate Simulation of Surfaces and Interfaces of Face-Centered Cubic Metals Using[END_REF] These simple models show that surface relaxations occur over few atomic layers: the effects are thus enhanced in ultrathin films or multilayers. [START_REF] Cammarata | Surface and Interface Stress Effects in Thin Films[END_REF][START_REF] Miller | Size-Dependent Elastic Properties of Nanosized Structural Elements[END_REF][START_REF] Fedorchenko | Thickness Dependence of Nanofilm Elastic Modulus[END_REF][START_REF] Hamilton | Theories of Surface Elasticity for Nanoscale Objects[END_REF][START_REF] Zhou | Size-Dependent Elastic Properties of Thin Films: Surface Anisotropy and Surface Bonding[END_REF][START_REF] Streitz | Surface-Stress Effects on Elastic Properties. I. Thin Metal Films[END_REF][START_REF] Lu | Thin Plate Theory Including Surface Effects[END_REF][START_REF] Chang | The Theoretical Model of FCC Ultrathin Film[END_REF][100][101][102][103][104][105] The simple decomposition of the elastic properties in a bulk + surface contribution, and the expected linear dependence with the inverse system size, remain controversial. [START_REF] Wolf | Surface-Stress-Induced Structure and Elastic Behavior of Thin Films[END_REF][START_REF] Miller | Size-Dependent Elastic Properties of Nanosized Structural Elements[END_REF][START_REF] Shenoy | Atomistic Calculations of Elastic Properties of Metallic FCC Crystal Surfaces[END_REF][START_REF] Streitz | Surface-Stress Effects on Elastic Properties. I. Thin Metal Films[END_REF]106,107 Furthermore, the effect of temperature is not so frequently studied. [START_REF] Quesnel | Elastic Compliances and Stiffnesses of the FCC Lennard-Jones Solid[END_REF]108 The following study thus focuses on a simple Lennard-Jones thin film model to determine the effect of both the film thickness and the temperature on its elastic properties. We present the model and methods, in particular the two numerical deformation experiments used to calculate the elastic constants of the thin film. The results are then given for various film thicknesses and temperatures, and discussed in terms of surface stress.

II Model and methods

We investigate the elastic properties of atomic thin films, in particular the influence of temperature and film thickness. The model is the Lennard-Jones (12-6) crystal that catches the main features of elastic solids. The approach can be easily extended to any solid of interest described with more accurate potentials. The procedure used to determine the elastic constants follows the experimental route. In this computational method the elastic constants are defined as the initial slope (linear regime) of stress-strain curves given by the molecular simulations. [START_REF] Sprik | Second-Order Elastic Constants for the Lennard-Jones Solid[END_REF][START_REF] Quesnel | Elastic Compliances and Stiffnesses of the FCC Lennard-Jones Solid[END_REF] The calculations are limited to deformations without shear, and the statistical averages are performed in the framework of the Monte Carlo simulations, well-suited to monitor strain fluctuations in the constant pressure ensemble. [START_REF] Clavier | Computation of Elastic Constants of Solids Using Molecular Simulation: Comparison of Constant Volume and Constant Pressure Ensemble Methods[END_REF] We have first determined the bulk elastic constants as a function of temperature. Then, the system is cut along one crystallographic plane and the corresponding periodic boundary conditions are removed, in order to obtain a film with two free surfaces. The film thickness is varied gradually by removing the external layers, and the film elastic constants are determined for each thickness.

The reduced symmetry of the film compared with that of the bulk solid has consequences on the number of elastic constants to be determined and on the method to measure these parameters.

II_1 Stiffness and compliance matrices

The stress tensor is the derivative of the free energy of the solid with respect to the strain tensor components. In the limit of small deformations, the stress tensor is linear with the strain tensor (Hooke's law): the elastic stiffness matrix C is defined as the fourth rank tensor that relates stress to strain. The Voigt notation will be used to simplify the equations. The generalized Hooke's law then writes: 1

   6 1 j ij i j C   (1) with xx   
The stiffness matrix is symmetric. Furthermore, the intrinsic symmetry of the solid reduces the number of independent coefficients in the matrix. Choosing [100], [010] and [001] as the axes, the cubic symmetry of the bulk fcc crystal imposes that only three independent elastic constants are expected (C11 = C22 = C33, C12 = C13 = C23, and C44 = C55 = C66). For a film of infinite extension in directions x and y, and cut from the bulk crystal along parallel planes perpendicular to the z axis (see Figure 1), the tetragonal symmetry implies that the compliance matrix has only six distinct coefficients. Since there is no shear stress in the mechanical tests that we have performed, the shear components will be discarded. The stress-strain relation then reduces to:

                              
      C C C C C C C C C (2)
with only four independent coefficients. The stiffness coefficients Cij may be obtained directly from imposed deformations applied to the system along the three axes, and measuring the corresponding components of the stress tensor (finite-strain method). Conversely, constant pressure or stress may be applied to the system: the average deformation in each direction then gives the compliance constants. When expressed in terms of Young's modulus (E) and Poisson's ratio (), the compliance matrix of the film writes: The two x and y axes parallel to the film being equivalent, 12 21    , as expected from the symmetry of the matrix. On the other hand, 13 31    , but the symmetry of the matrix imposes that:

            E E E E E E E E E (3)
1 13 3 31 E E     (4)
We thus have 5 parameters and a relation, which gives only four independent parameters as for the stiffness matrix (eq 2). In the case of the bulk solid,    , reducing the number of independent parameters to two.

II_2 Imposed stress method.

The coefficients of the compliance matrix are obtained by applying uniaxial loading and measuring the average deformation with respect to the unloaded system. This procedure allows to take into account the internal stress and strain in the solid due to the long range interactions and the solid/vacuum interfaces, as well as the thermal expansion at finite temperature due to the anharmonicity of the interactions.

For the bulk crystal, the three axes being equivalent, the two independent parameters are obtained in a single simulation: the stress is applied along the x direction ( 1  , 0, 0), and the measured deformation along the same axis gives Young's modulus

1 1 1 /   E
, while the measured deformation along one of the two other equivalent axes gives the Poisson's ratio For the thin film, at least two simulations are required (see Figure 2). (i) In the first one, a uniaxial stress is applied along the x axis, parallel to the film, ( 1  , 0, 0). The measured deformations along the x, y and z directions are all informative, and give respectively the Young's modulus

1 3 1 2 12 / /          .
1 1 1 /   E
, and the two Poisson's ratio

1 2 12 /     and 1 3 13 /      . (ii)
The second simulation consists in applying the uniaxial stress along the z direction, perpendicular to the surface, (0, 0, 3  ). The measured deformation along the same axis gives the second Young's modulus

3 3 3 /    E
. In principle, at this point, all parameters are determined.

However, it will prove useful to calculate independently the third Poisson's ratio 31

 by measuring the deformation along one of the two equivalent directions parallel to the film:

3 1 31 /    
. This will allow to cross-check for the consistency of the five independent measurements through the symmetry relation (eq 4). We thus introduce the quantity

13 3 31 1    E E 
which is expected to be equal to 1 for a symmetric matrix.

II_3 Atomic Model

The numerical bulk samples were prepared by positioning atoms on the site of an fcc regular lattice. The initial lattice parameter is chosen so that nearest neighbor atoms are at a distance corresponding to the van der Waals diameter. For the bulk measurements, two simulation box sizes have been chosen, equal to 6 and 7 lattice units along each direction (the axes are chosen parallel to the crystallographic directions). The number of atoms is respectively 864 and 1372.

Periodic boundary conditions are applied in the three directions. The two systems are relaxed at zero temperature and zero pressure (Monte Carlo run in the isostress-isothermal ensemble). These relaxed systems are then used to calculate the bulk elastic constants as a function of temperature.

The thin films are issued from the largest bulk box. A gap is introduced in the z-direction so that the opposite faces of the film do not interact through the periodic boundary conditions (see Figure 1). The film thickness is varied by removing atomic planes on both sides. Five thicknesses are considered, equal to 2, 3, 4, 5 and 6 lattice units (corresponding to 4, 6, 8, 10 and 12 atomic planes).

All atoms interact with the Lennard-Jones potential:

                      6 12 4 ) ( r r r V LJ     (5)
where  represents the binding energy between atoms and  their van der Waals diameter. A cutoff radius of  4 is used, and a shift of the potential at the cutoff radius is applied to avoid discontinuities. For comparison, Quesnel et al. [START_REF] Quesnel | Elastic Compliances and Stiffnesses of the FCC Lennard-Jones Solid[END_REF][START_REF] Quesnel | The Poisson Ratio for an FCC Lennard-Jones Solid[END_REF] have used the same potential with a cutoff radius of  3 . Small differences are expected, because of the importance of the long range contributions to the interactions, as shown by Tretiakov and Wojciechowski. [START_REF] Tretiakov | Quick and Accurate Estimation of the Elastic Constants Using the Minimum Image Method[END_REF] Their work shows that using  4 instead of  3 significantly improves the accuracy, the discrepancy with the full potential including the long range corrections being less than few percent. The Lennard-Jones parameters will be used as fundamental units in which all other quantities can be expressed (reduced units): distances are given in units of  , temperature is given in units of

II_4 Simulation method

The calculations of the elastic constants are performed at five temperatures between 0 and 0.5 B / k  . The zero-temperature calculations are done by minimization of the energy of the system, while the finite temperature calculations are done with the Monte Carlo algorithm to sample the system phase space. Clavier and collaborators have shown that Monte Carlo is the best approach for constant pressure runs. [START_REF] Clavier | Computation of Elastic Constants of Solids Using Molecular Simulation: Comparison of Constant Volume and Constant Pressure Ensemble Methods[END_REF] Since the imposed thermodynamic parameters are the temperature and the stress or pressure, we use the isostress-isothermal ensemble. The Monte Carlo trials consist in particle displacements to thermalize the system, and homothetic dilatations or contractions along the three space directions to equilibrate the system with the three components of the external stress i  . Different configurations have to be considered for the bulk and the film calculations.

For bulk calculations, the simulation box is free to deform in all directions independently. The stress is uniaxial, i.e. a finite pressure is applied in one direction (chosen to be x), while zero pressure is applied in the y and z directions. The solid deformation in a given direction is given directly by averaging the fluctuating dimension of the simulation box along that direction. As an example, Figure 3 For film calculations, the simulation box is free to deform only in the two directions parallel to the film (x and y), while the box size is fixed along the z direction: this is necessary to avoid the shrinkage of the simulation box along z. The stress is applied in directions x and y using the conventional isostress algorithm, and the corresponding solid deformations are given by the box size. One has to be careful in choosing the value of the stress parameters for the simulation code.

In the Monte Carlo algorithm, the stress is applied to the whole lateral faces of the simulation box

(for example z y L L 
for the ( 1  , 0 0) uniaxial loading). But in a real experiment, it is applied only to the (smaller) lateral surface of the film: z L is replaced by the film thickness. The value of the stress parameter is thus corrected so that the total force applied laterally on the film is the same in the simulation and in the real experiment. This applies only for the stresses in the x and y directions. In the z direction, the pressure is applied directly and uniformly to the atoms located on the outermost atomic planes. The force acting on each atom equals

N L L y x / 3 
where N is the number of atoms in each plane. Note that the deformation parallel to the film is so small that the forces (proportional to y x L L ) are essentially constant during the simulation. The deformation of the solid along z is monitored by measuring the average position z± of the outermost planes: the film thickness is then given by z+-z-. It is important to mention here that this approach is expected to apply in our case consisting of a fcc crystal, but may not apply in general. In particular one has to be cautious if the solid is disordered, presents a rough surface, or if interactions are long range.

A more general approach in such cases would be more appropriate, such as applying the stress via a frozen solid on top of the simulated film.

The Gibbs' surface: Some subtleties have to be carefully considered in the case of the film. In a real experiment, the pressure would be applied to the film by another solid, or a fluid. The thermodynamic analysis of the energy received by the system during a mechanical test requires the introduction of the Gibbs' dividing surface situated at mid-distance between the film and the solid or the fluid used by the experimentalist to apply the pressure. If the film deforms ( 3

 ) the 3   z y x H L L
where z H is the distance between the two Gibbs' surfaces. In our simulations, the forces are applied directly to the center of the outermost atoms, and the corresponding work received by the solid is We thus have to apply either the correction factor z z h H / to the stress or its inverse to the strain.

Without this correction the compliance matrix is not symmetric, the discrepancy increasing for very thin films. In our simulations, the measured value of the parameter

13 3 31 1    E E 
introduced to measure this symmetry decreases from 0.9 for the thickest film down to 0.75 for the thinnest one!

On the other hand, with the Gibbs' surface correction, the  parameter oscillates between 0.996 and 1.02: the symmetry of the matrix is well verified.

Conversely, the Gibbs' surface may as well be defined so that alpha precisely equals 1. As a matter of fact, from a thermodynamic point of view, the Gibbs' surface should be chosen so that the macroscopic description of the system distinguishes a contribution of the volume and a contribution of the surface. The position of the surface should then ensure the symmetry of the compliance tensor, and thus alpha = 1.

Improving accuracy:

The errors on the average values of deformation can be evaluated from the standard block analysis. 109 The typical relative error that can be reached is within 4×10 -5 for the lateral dimensions of the solid, for 2×10 [START_REF] Ogieglo | In Situ Ellipsometry Studies on Swelling of Thin Polymer Films: A Review[END_REF] Monte Carlo steps per atom. The situation may be improved if one takes into account the natural symmetry of the system. For instance, in the bulk case (x-loading) or for the thin film (z-loading), one expects identical box deformations (in average) in the y and z directions for the bulk, or the x and y directions for the film. The method then consists in imposing the geometrical constraint y L = z L (for the bulk case) during the course of the simulation. The fluctuating value of

y L = z L is displayed in Figure 3-b.
As can be seen the average value of y L is not affected, but the fluctuations are smaller (the geometrical constraint stiffens the system). The same Monte Carlo run then results is more accurate results.

The corresponding relative error is now within 2×10 -5 . This procedure will be used each time it is possible: for the bulk calculations (uniaxial loading) and for the film calculations during the zloading experiment (Figure 2).

III Results and Discussion

III_1 bulk Lennard-Jones crystal

Zero temperature calculations: The first calculations have been performed at zero temperature for the bulk solid for the two system sizes. The aim was to determine the best choice for the stress amplitude in order to be in the linear regime of small deformations. Figure 4 E is the limiting value of this quantity for infinitesimal deformations. As can be seen, this quantity is not constant for large deformations (>10 -3 ), but is essentially constant for small deformations (<10 -3 ), as expected in the linear regime. At zero temperature, there is no limitation is choosing the deformation as small as possible except that due to the finite numerical precision: we get is varied between 0.1 and 0.5. The system size now fluctuates, and the strain-stress curves are obtained from the average values of the deformation. As previously, the calculations have been performed on several decades in order to find the best compromise between linearity (obtained for the smallest deformations) and numerical accuracy (obtained for the largest deformations). The error bars have been deduced from the standard block analysis. In practice, the higher the temperature, the larger the fluctuations. At T* = 0.1, the fluctuations are already so large that deformations smaller than 10 -5 cannot be measured, i.e. the error bars on the average deformation exceeds its value. For larger, measurable deformations, the corresponding effective Young's modulus

3 3 3 /    E is
given in Figure 5 as a function of the applied stress, and for all temperatures. As can be seen, the lowest stress values (below 10 -2 ) are now inappropriate at finite temperature because of the large uncertainty due to natural fluctuations of the system. On the other side, for a reduced stress larger than 0.1, the effective Young's modulus shows small variations: the system is entering the nonlinear regime of deformations. The best range of stress values is between 10 -2 and 10 -1 , corresponding to deformations around 10 -3 . The calculated elastic constants for the bulk crystal as a function of temperature are given in Table 1 and Figure 6. The data exhibit a global decrease of the modulus with increasing temperature, and a constant Poisson's ratio within errors, in agreement with the work of different authors. [START_REF] Tretiakov | Quick and Accurate Estimation of the Elastic Constants Using the Minimum Image Method[END_REF][START_REF] Quesnel | Elastic Compliances and Stiffnesses of the FCC Lennard-Jones Solid[END_REF][START_REF] Clavier | Computation of Elastic Constants of Solids Using Molecular Simulation: Comparison of Constant Volume and Constant Pressure Ensemble Methods[END_REF] Note that despite small error bars, the Young's modulus data look scattered around the expected linear behavior: this most probably reveals systematic errors missed by the block analysis. In particular, large time scale fluctuations inaccessible to our simulations could contribute to the results.

III_2 Thin film

Zero temperature calculations. The two loading experiments (Figure 2) are performed on the five film thicknesses (4 to 12 layers). As for the bulk, the loading stress is chosen very small to be in the linear regime of deformations (no limitation due to thermal fluctuations). The results for the five (redundant) elastic quantities are shown in Figure 7. In all cases the data are shown versus the inverse film thickness expressed in terms of the number of planes. The bulk results, obtained with periodic boundary conditions are reported as if the number of planes was infinite.

All curves are monotonous, showing smooth variations, with bulk results in agreement with the extrapolated evolution of the elastic constants to the infinite system. In particular, the anisotropy, revealed in the films by the differences between the three Poisson's ratio and the two Young's moduli, reduces when the film thickness increases, and is compatible with its vanishing for the infinite thickness. The consistency between the five mechanical constants is excellent whatever the system size: the parameter ) /( ) (

13 3 31 1    E E 
(see inset in Figure 7) is compatible, within 2%, with the value of 1, expected for a perfectly symmetric matrix.

The evolution of the three Poisson's ratio is almost linear with the inverse system size, even for the thinnest films. Departure from linearity is the largest for 13  but never exceeds 5%. An interesting point is that their evolution with the film thickness is compatible with an essentially constant average value:   3

/ 13 31 21     
is independent of the film thickness. On the other hand, the evolution of the Young's moduli departs from a linear behavior for the smallest thicknesses.

This non-linearity is also observed if one draws the compliance matrix coefficients (Eq 3):

1 21 1 / , / 1 E E   etc.
The general trend is a softening of the film when its thickness decreases, with amplification for the ultrathin films. versus the inverse of its thickness given by the number of atomic layers. The lines for the Young's moduli are guides to the eye; the straight lines for the Poisson's ratio are issued from the bulk value (inverse thickness equal to zero) and best fit the data for finite thicknesses. Inset:

evolution of the parameter ) /( ) ( 13 3 31 1    E E 
versus the film thickness.

The stiffness matrix has been calculated by inversion of the numerical compliance matrix, without making any assumption on the symmetry. The five independent constants, C11, C33, C12, C13 and C31 are shown in Figure 8 as a function of the inverse system thickness given by the number of layers. The symmetry of the stiffness matrix is well verified since C13=C31 within 5%. As previously, the anisotropy of the mechanical properties for the film appears through the differences between the two diagonal terms (C11 and C33) and the two off-diagonal terms (C12 and C13 = C31). As previously, the bulk values agree well with the expected extrapolation of the film constants for infinite thickness. The dependence with the inverse system size is close to linearity on the whole range of thicknesses for all stiffness constants. This is at odd with what was observed for the Young's moduli, which exhibited a poorly linear behavior with the inverse system size. versus the inverse of its thickness given by the number of atomic layers (eq 3). The straight lines are guides to the eye.

Finite temperature calculations.

The temperature of the film has been varied between 0.1 and 0.5 in reduced units by increments of 0.1. The thermal agitation of the surface atoms increases significantly with temperature. At T* = 0.4 and above, the thermal agitation is so large that the surface atoms are able to overcome the energy barrier that normally maintains them at their relative position in the crystal. The consequence is atomic migration on the external surface, with the formation of vacancies and adatoms. Since the film deforms irreversibly under loading (plasticity), the elastic constants become meaningless at high temperatures. Even when the structure of the film remains stable, the thermal fluctuations are large, and it is difficult to reach a good accuracy. In particular, at T* = 0.3 and for the largest values of the stress used for the loadings, the film deformations are accompanied by atomic displacements. It was thus necessary to use smaller values of the stress, resulting in a lack of accuracy. For T* = 0.1 and 0.2 the accuracy is comparable to that for the bulk solid. The straight lines drawn in the figures are guides to the eye passing through the bulk value (inverse thickness equal to zero) and best fitting the data for finite thicknesses.

The results for the Poisson's ratio are given in Figure 9. As previously, the data are plotted as a function of the inverse film thickness given by the number of atomic layers. As can be seen, for all temperatures, the Poisson's ratios follow an essentially linear behavior with the inverse of the film thickness. Note that this linearity extends remarkably well even for the thinnest film. The largest departure from linearity never exceeds 5% whatever the temperature. Here again the evolution of the average value of the Poisson ratios   3 / 13 31 21

    
with the film thickness is compatible with a constant.

The Poisson ratios prove to be barely dependent on temperature. For instance, the largest variations in the bulk values never exceed 3%, and those for the thinnest film remain below 10% for 21 and 31 and below 15% for 13. We however note the following trend: the differences between the three Poisson's ratios tend to decrease when T increases. This is easy to observe for 31 and 13. The anisotropy of the thin film is thus attenuated by increasing the temperature. The lines are guides to the eye.

The reduced Young's moduli obtained at different temperatures are given in Figure 10. For all temperatures, the general trend is a softening of the film when the thickness decreases, the dependence being non-linear with the inverse of the thickness. The data also show that the higher the temperature the lower the moduli, whatever the film thickness. In the in-plane directions, the decrease of E1 reaches 20% between T* = 0 and 0.3 for all thicknesses, while in the direction perpendicular to the film, the reduction of E3 reaches 40% for the thinnest film (4 layers) between T* = 0 and 0.3. The Young modulus in the out-of-plane direction is thus more affected by the temperature, probably in connection with the highest mobility of surface atoms. The Young moduli combined with the three Poisson's ratio give the compliance matrix (eq 3) that has been inverted to get the stiffness matrix (eq 2) for each temperature and film thickness.

The five constants C11, C33, C12, C13 and C31 are given in Figure 11. As expected from symmetry considerations of the matrix, the C13 and C31 constants are almost superimposed: the ratio between both remains equal to 1 within 2.5%. For all temperatures, the stiffness decreases for thinner films. As can be seen the stiffness constants exhibit a linear dependence with the inverse film thickness, which was not the case for the compliance matrix (see the discussion below). We also note an increase of the anisotropy (differences between the elastic constants) when the film thickness decreases.

A clear picture of the evolution of the elastic constants with the temperature does not emerge.

The general trend is a decrease of the stiffness parameters when temperature increases, following essentially the behavior observed for the bulk solid. The evolution of the slope of the curves is weak below T* = 0.2. At T* = 0.3 the slope become significantly more negative for all elastic constants, showing a drastic decrease of the thin film stiffness. A careful analysis of the data shows however a non-monotonous dependence of the elastic constants with temperature, as was observed for the bulk crystal. A possible source of errors could be due to the high sensitivity of the compliance-to-stiffness inversion process to the original errors in the data. It has been tried to perform an inversion taking into account the symmetry constraint (C13 = C31), but no significant improvement was observed. The errors in the data are most probably significantly larger than estimated. In particular, the Monte Carlo sampling may miss or overestimate some vibration modes in the simulation box associated with collective movements. 83

III_3 Discussion

The most striking behavior shown by the data is the linear dependence of the stiffness constants with the inverse film thickness, from zero up to the highest temperature considered in this study (T* = 0.3) above which the film loses its integrity under loading. The discrepancy never exceeds 5% at zero temperature (all thicknesses) or for thicknesses larger or equal to 6 layers (all temperatures), and remains below 15% in the worst case (four layers, high temperature).

The linear behavior of the elastic matrix versus the inverse system size suggests to interpret the evolution of the stiffness as due to a surface stress term. This can be seen by using the ansatz that any nanosystem = bulk + surface [START_REF] Zhang | Determining the Effects of Surface Elasticity and Surface Stress by Measuring the Shifts of Resonant Frequencies[END_REF][START_REF] Müller | Elastic Effects on Surface Physics[END_REF][START_REF] Shenoy | Atomistic Calculations of Elastic Properties of Metallic FCC Crystal Surfaces[END_REF] and writing the total force that applies laterally to an idealized thin film made of a bulk-like core and an effective surface stress. It has been shown that generally any elastic property D of such system would deviates from its bulk value b D according to [START_REF] Miller | Size-Dependent Elastic Properties of Nanosized Structural Elements[END_REF] 28

h h d D D s b   (3)
where h  is the thickness of the nanostructure, s h corresponds to the length that sets the scale below which the surface elastic effects are significant, d depends on the geometry, and s dh somehow relates to the surface elastic constants of the system. Applying this equation to the stiffness matrix, and replacing the thickness by the number of atomic planes, one gets:

N c C C s ij b ij ij   . ( 4
)
The s ij C may be interpreted as the surface stiffness constants associated with the first atomic layer, while the b ij C are the bulk values. These surface constants can be deduced from the best fit of the data shown in Figure 11 by extracting the slopes. The results are given in Table 2. This surface matrix is symmetric, with negative values except for s C 12 . The presence of the surfaces softens the thin film. An interesting point is that this surface matrix has non-negligible components in the z direction, perpendicular to the surface. This is actually unexpected in the usual formalism of surface stiffness. A possible interpretation is that the surface stress model applies only when the film thickness is large compared to the range of relaxation of the outermost layers. In our case, the film is so thin that the two external surfaces are not independent: the elastic constants in the direction perpendicular to the surface exhibit dependence with the film thickness. Another explanation could be found in the work of Chang et al. [START_REF] Chang | The Theoretical Model of FCC Ultrathin Film[END_REF] who have proposed a lattice model where only the total strain energy density of an ultrathin film is taken into account, without any consideration of surface effect. They have observed a size dependence of the elastic constants, in particular for the out-of-plane stiffness coefficient C13. This indicates that the surface effects might not be the only source of size dependence of the elastic constants of

IV Conclusion

The simple Lennard-Jones crystal has been considered to model the elastic properties of thin films as a function of temperature and film thickness. The method consists in performing mechanical loadings and measuring the corresponding deformations. The anisotropy of the films requires at least four independent measurements (shear is not considered in this study). The calculations are performed in the isotherm-isostress ensemble by Monte Carlo simulations. The Young's moduli and Poisson's ratio appearing in the compliance matrix have been calculated between 0 and 0.5 in reduced units, for the bulk crystal and for the film of thickness 4 to 12 atomic layers. The bulk moduli decrease with temperature, while the Poisson's ratio remains essentially constant. For the films, one observes a softening of the solid when the thickness diminishes, in particular in the in-plane directions. The evolution with temperature shows a global softening with increasing T, the effect being the largest in the out-of-plane direction. At T*

Figure 1 :

 1 Figure 1: Schematic representation of the thin film showing the axes parallel to the

Figure 2 :

 2 Figure 2: Schematic representation of the x-and z-loading experiments. The uniaxial stress is

k

  is Boltzmann's constant, and pressure and moduli are given in units of 3 /  . For instance, the reduced dimension of the initial lattice unit is  2 . Numerical values in SI units for quantities calculated with this potential can easily be obtained from the Lennard-Jones parameters tabulated for a number of materials.

Figure 3

 3 Figure 3: a) Spontaneous fluctuations of the lateral dimension y L of the bulk sample containing

3 

 3 -z-is the distance between the outermost atomic planes. The difference between z H and z h is typically one van der Waals diameter: the algorithm thus underestimates the energy transferred to the solid, and thus produces incorrect deformations. Another way to see the same problem is to notice that the deformation has to be calculated based on the variations of z H and not z h , which is equivalent to measuring it in the core of the solid. For thin films, monitoring the total thickness z h improves the statistics.

Figure 4 : 3 3 3 E

 433 Figure 4: Main circles: strain-stress data at zero temperature during the uniaxial loading along

3 ,

 3 that has been varied over four decades. The behavior is very close to linearity. The effective modulus in the inset (in reduced units). By definition, the Young's modulus 3

.

  's ratio is deduced from the deformation in the direction perpendicular to the uniaxial stress: 364. These results are comparable with those obtained by Quesnel et al.83,92 who have used a slightly different cutoff (3) and the constant-rate deformation method ( value in the first 0.2% strain).In order to evaluate the system size effects, the same calculations have been done in the larger cubic box comprising seven unit cells in each direction. The measured Young's modulus at zero temperature is 's ratio remains equal to 0.364. The system size effects are small (less than 2%). This has to be correlated with the fact that the simulation box is at least twice as large as the cutoff radius of the potential. These values are in good agreement with the work of Tretiakov and Wojciechowski 79 who take into account the long range Finite temperature calculations: The reduced temperature T* =  / B T k

3 Figure 6 :

 36 Figure 5: Effective reduced modulus 3 3 / *   as a function of the reduced stress * 3  , for the

Figure 7 :

 7 Figure 7: Poisson's ratio and Young's moduli at zero temperature for the fcc Lennard-Jones film,

31 Figure 8 :

 318 Figure 8: Stiffness matrix coefficients (eq 2) at zero temperature for the fcc Lennard-Jones film,

Figure 9 :

 9 Figure 9: Poisson's ratio at different reduced temperatures (given in the panels) for the fcc

3 Figure 10 :

 310 Figure 10: Young's moduli at different reduced temperatures (given in the panels) for the fcc

3 Figure 11 :

 311 Figure 11: Stiffness matrix coefficients (Eq 2) at different reduced temperatures (given in the

Table 1 :

 1 Reduced Young's modulus and Poisson's ratio for the bulk fcc Lennard-Jones crystal at different reduced temperatures (see text).

	k B T/	  	
	0	60.5	0.364
	0.1	62.2	0.354
	0.2	53.0	0.354
	0.3	49.0	0.368
	0.5	31.1	0.371

Table 2 :

 2 29ultrathin films, yet the fundamental mechanisms are not fully understood. Further investigations are necessary to elucidate this point. Reduced stiffness coefficients for the bulk and surface contributions (see eq 4) for an fcc Lennard-Jones thin film, at different reduced temperatures.

	T*	C b ii *	C b ij *	C s 11 *	C s 33 *	C s 21 *	C s 31 *
	0	103.7	59.4	-71.8	-118	+21.6	-83.6
	0.1	101.6	55.7	-81.6	-159	+20.4	-94.8
	0.2	86.5	47.4	-11.2	-110	+63.6	-39.6
	0.3	85.8	49.9	-132	-192	-58	-136

(109) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987.

Acknowledgement

The author acknowledges financial support from ANR-17-CE30-002-04.

under loading.

The coefficients of the stiffness matrix exhibit a linear dependence with the inverse of the system size. This has been analyzed in the framework of a bulk+surface contribution, and the corresponding surface stiffness has been determined from the corresponding slopes. However, the out-of-plane coefficients are non-zero, which is not compatible with the surface stress formalism.

The bulk+surface decomposition is thus not justified in the case of very thin films, essentially because the surfaces are so close that they influence each other (the distance associated to surface relaxations exceeds the film thickness). However, the dependence of the stiffness matrix with the inverse thickness is remarkably linear and deserves further studies.


Frédéric Gavrel
email: frederic.gavrel@unicaen.fr.

Participation, Recruitment Selection, and the Minimum Wage

Keywords: Search and matching, heterogeneous workers, applicant ranking, public policies, efficiency. JEL Classification numbers: D8, J6

This paper reexamines the efficiency of participation with heterogeneous workers in a search-matching model with bargained wages and free entry. Assuming that firms hire their best applicants, we show that participation is always too low. The reason for this is a hold-up phenomenon: To be active, a worker must pay the entire search cost whereas part of the gain from this investment goes to the firm. As a consequence, introducing a (small) minimum wage raises participation, job creation, and employment. Therefore, net aggregate income of the economy is increased.

creates a positive effect on all participating workers.

What could be done to improve the efficiency of the labor market? We show that the introduction of a mandatory minimum wage is an effective instrument in correcting this market failure. Intuitively, a minimum wage (provided it is not too high) increases participation. More surprisingly, labor demand as well as the employment rate increase. Again, the entry of lower-skilled workers tends to raise firms' expected profits leading to the creation of additional vacancies. Therefore, despite the unemployment rate increase, (net) aggregate income necessarily rises. Furthermore, introducing a minimum wage appears to raise the expected income of all participants. Subsidies to participants (like unemployment compensation or a negative income tax) also improve efficiency.

In the relevant literature we can distinguish between two subsets. First, some papers build on [START_REF] Pissarides | Equilibrium Unemployment Theory[END_REF] 1 . In Pissarides' model, job creation and individuals' participation decisions are efficient if and only if the elasticity of the matching function with respect to unemployment (usually denoted by η) coincides with workers' bargaining strengths (usually denoted by β): the famous Hosios condition. Conversely, if the bargaining power of workers is not high enough (if β is lower than η), participation will be insufficient. In this case, the minimum wage could raise aggregate income by acting as a substitute for an increase in workers' bargaining strengths. Empirically, however, it seems that the bargaining power of workers is too high. For instance, [START_REF] Flinn | The minimum wage and labor market outcomes[END_REF] finds β = 0.4 and η < 0.4. By contrast, assuming vertically differentiated workers and recruitment selection, we show that minimum wage efficiency no longer depends on workers' bargaining strengths.

The other subset includes papers which attempt to derive the matching function from first principles. As [START_REF] Petrongolo | Looking into the Black Box: A Survey of the Matching Function[END_REF] put it, this literature seeks to throw some light into the black box. Many of these articles are, like the present one, based on the urn-ball model. Some authors assume recruitment selection. 2To summarize, our contributions are twofold. First, relative to [START_REF] Gavrel | On the Efficiency of Participation with Vertically Differentiated Workers[END_REF] 3 we show that participation is unavoidably insufficient when firms rank their applicants.

Second, relative to the usual search-matching model with endogenous participation [START_REF] Flinn | Minimum Wages Effects on Labor Market Outcomes under Search, Matching, and Endogenous Contact Rates[END_REF][START_REF] Flinn | The minimum wage and labor market outcomes[END_REF], we find that a (small) minimum wage necessarily increases employment and improves market efficiency.

The paper is organized as follows. Section II presents our setting. Next we define a laissez-faire equilibrium and study its welfare properties (section III). The results concerning the introduction of a minimum wage are exposed in section IV while section V provides an analysis of the impact of subsidies to participants.4 Section VI concludes.

II. Endogenous participation with applicant ranking

Market structure

Apart from the matching process, the market structure used here is very close to [START_REF] Albrecht | Efficiency in a Search and Matching Model with Endogenous Participation[END_REF] and [START_REF] Gavrel | On the Efficiency of Participation with Vertically Differentiated Workers[END_REF]. We study the efficiency of participation in the following simplified static environment with risk-neutral agents5 .

The population (participants and non-participants) has a positive measure n. When holding a job, workers' productivities y are distributed according to the (strictly) increasing c.d.f. F (y) of support [0, 1]. The density F (y) is denoted by f (y) (f (y) > 0). When inactive all individuals earn the same positive income z. For expositional simplicity, it is assumed that z does not depend on y, but the analysis extends to the case in which z grows with y as long as the increase in domestic output is lower than the increase in market output. Expressed in words, individuals need to hold a job to fully take advantage of their skills.

Participating workers must pay some positive search cost, γ, and find a job with probability p(., y). They are then unemployed with probability [1 -p(., y)]. In this case, they all enjoy the same domestic income z as non-participants . All firms are identical and freely enter the market. Each active firm creates a single vacancy incurring some positive cost c to that end.

When matched with a worker of productivity y (an event which occurs with probability density q(., y)), a job generates the (private) surplus (y -z). This surplus is divided between the two parties according to their bargaining strengths. The wage, w(y), of an employed worker is then given by

w(y) = z + β(y -z) (1)
with β being workers' bargaining strengths (0 < β < 1).

Other arguments of functions p(., y) and q(., y) will be made explicit when we turn to the matching process.

Participation choices

Ignoring corner points, one can surmise that high-skilled workers (workers whose productivities are higher that some cutoff y * ) will choose to join the labor market, whereas low-skilled individuals (individuals whose productivities are lower than the cutoff y * ) will prefer to be inactive. This cutoff is thus determined by the following indifference condition

z = -γ + p(., y * )w(y * ) + [1 -p(., y * )]z Or -γ + p(., y * )β(y * -z) = 0 (2)
In the next paragraph, we shall show that in conformity with intuition, the probability p(., y) increases with the output y. It follows that

-γ + p(., y)w(y) + [1 -p(., y)]z > (<)z ⇐⇒ y > (<)y *
Consequently, labor supply has a measure equal to [1 -F (y * )]n while the set of nonparticipants has a measure equal to F (y * )n. Active firms offer a set of jobs of measure v. Hence the tightness of the labor market, θ, is given by the ratio

θ = v [1 -F (y * )]n
As already noted, the main innovative feature of our paper involves the matching process. Contrary to [START_REF] Albrecht | Efficiency in a Search and Matching Model with Endogenous Participation[END_REF] and [START_REF] Gavrel | On the Efficiency of Participation with Vertically Differentiated Workers[END_REF], (active) firms' recruitment is not random. As in [START_REF] Moen | Education, Ranking and Competition for Jobs[END_REF] and [START_REF] Gavrel | Technical Skill Bias as a Response of Firms to Unemployment. A Matching Model with Applicant Ranking and Endogenous Skill Requirements[END_REF][START_REF] Gavrel | On the Inefficiency of Matching Models of Unemployment with Heterogeneous Workers and Jobs when Firms Rank their Applicants[END_REF], they rank their applicants and select the best. As a consequence, workers do not face the same probability of finding a job. In accordance with empirical evidence, we find that high-skilled job seekers have a greater chance of success.

Matching process

We now provide a brief account of the modeling of meetings in the labor market. 6Applying the usual urn-ball model, we assume that each job seeker draws one firm at random. In general, firms will have several applicants of differing productivity. We assume that they have full knowledge of the sample of their applicants. They then recruit the best one.

Let us consider the probability density, q(., y), for a firm to hire a worker of productivity y. One can prove the following preliminary result Lemma 1. With recruitment selection, the probability density q(., y) (y

* ≤ y ≤ 1) is q(., y) = q((v/n), y) = exp - [1 -F (y)]n v f (y)n v (3)
Proof. The proof follows the same line as [START_REF] Gavrel | On the Inefficiency of Matching Models of Unemployment with Heterogeneous Workers and Jobs when Firms Rank their Applicants[END_REF].

The above expression for the density q((v/n), y) is quite intuitive. As the second term, f (y)n/v, is the probability density for a firm to meet at least one applicant of productivity y, the exponential term, exp (-[1 -F (y)]n/v), represents the probability of (a firm) not meeting any applicant of productivity higher than y. Because a firm will not hire a worker of productivity y if another applicant is more skilled the expression for q((v/n), y) follows.

Integrating q((v/n), y) on the range [y * , 1] gives the probability of filling a vacancy, Q((v/n), y * ). We get

Q((v/n), y * ) = 1 -exp - [1 -F (y * )]n v (4)
From the density q(., y), we also deduce the probability of finding a job, p(., y) . It follows that

p(., y) = p((v/n), y) = exp - [1 -F (y)]n v (5)
Integrating [p(., y)f (y)/(1 -F (y * ))] on the range [y * , 1] gives the average probability of finding a job, P ((v/n), y * ). We obtain

P ((v/n), y * ) = θQ((v/n), y * )
It is worth noting that neither q((v/n), y) nor p((v/n), y) depend on the cutoff y * . This point will have important implications in what follows.

ρ((v/n), y, y * ) will denote the density of the output y among employed workers (i.e.

the set of occupied jobs). This density is defined as follows

ρ((v/n), y, y * ) ≡ q((v/n), y) Q((v/n), y * ) (6)
Next, from the density ρ((v/n), y, y * ), we deduce the expected output, ȳ, of a job when matched with a worker ȳ =

1 y * ρ((v/n), y, y * )ydy (7)

III. Equilibrium and efficiency

We now define a labor market equilibrium, and will then study its welfare properties.

Equilibrium and social optimum

As indicated above, firms freely enter the labor market. This means that the profit expected from a vacancy should be equal to the cost of its creation (c). Job creation is then formally deduced from the following equation

-c + (1 -β) 1 y * q((v/n), y)(y -z)dy = 0 (8)
Consequently, a labor market equilibrium can be defined as follows Definition 1. A laissez-faire equilibrium is a pair (y * , v) which jointly satisfies equations(2) and (8).

From the equilibrium pair (y * , v), one can deduce the equilibrium values of all other endogenous variables.

Our welfare criterion is the social surplus (also referred to as the aggregate income).

Denoted by Σ, the social surplus per head is defined as

Σ = (v/n)Q((v/n), y * )ȳ + [1 -Q((v/n), y * )(v/n)]z -[1 -F (y * )]γ -(v/n)c (9)
In the previous expression, the quantity

(v/n)Q((v/n), y * ) measures the employment rate. It is also equal to P ((v/n), y * )[1 -F (y * )].
A social optimum is then determined as follows Definition 2. A social optimum is a pair (y * , v) which maximizes the social surplus

Σ.
In what follows, we will restrict the welfare analysis to the neighborhood of a laissezfaire equilibrium. We first study the impact of participation (job creation) on the social surplus for a given value of job creation (participation). We will refer to our results as partial efficiency properties. Partial efficiency should be clearly distinguished from constrained efficiency, which is the second step of our investigation.

Partial efficiency

Let us first study the partial efficiency of the output cutoff y * . Differentiating Σ with respect to y * gives

∂Σ ∂y * = - v n q((v/n), y * )(y * -z) + f (y * )γ From v n q((v/n), y * ) = p((v/n), y * )f (y * )
we deduce that this derivative has the same sign as

Y ≡ γ -p((v/n), y * )(y * -z)
We obtain the following proposition Proposition 1a. In a laissez-faire equilibrium, the cutoff y * is partially inefficient.

Participation [1 -F (y * )]n is necessarily too low.

Proof. The expression Y can be rewritten as follows

Y = γ -βp((v/n), y * )(y * -z) -(1 -β)p((v/n), y * )(y * -z)
Using equation ( 2), we obtain

Y = -(1 -β)p((v/n), y * )(y * -z) < 0
This proves that in the neighborhood of a laissez-faire equilibrium the cutoff y * is too high whatever the value of job creation v is.

Q.E.D.

Proposition 1a contradicts the results of [START_REF] Gavrel | On the Efficiency of Participation with Vertically Differentiated Workers[END_REF]. In this paper, where firms' recruitment is random, participation is excessive. This contradiction requires interpretation. The intuition behind it is that, with random recruitment, a decrease in the cutoff lowers the average output per filled job. As [START_REF] Albrecht | Efficiency in a Search and Matching Model with Endogenous Participation[END_REF] put it, the marginal participant reduces the average productivity of matches in the market.

On the contrary, when firms rank their applicants, the probability density, q(., y), for a firm to hire a worker whose productivity y lies on the range ]y * , 1] is not affected by a decrease in the cutoff 7 . Consequently, any participant of productivity y raises the aggregate income by the amount: -γ + p((v/n), y)(y -z).

In a competitive environment, all individuals whose productivities, y, are higher than the value of inactivity, z, should participate in the market. In this context, the productivity of the marginal participant should be equal to z. With matching frictions and recruitment selection, all individuals whose expected output (p(., y)y + (1p(., y))z -γ) is higher than z should join the labor market. In this case, the expected output of the marginal participant should be equal to the value of inactivity.

Because productivities are higher than wages (β < 1), the private return to participation (-γ +p((v/n), y)β(y -z)) is lower than the social return. This clearly shows why the productivity of the marginal participant is too high with recruitment selection: the core of this inefficiency is a hold-up problem. To enter the market, a worker must pay the entire search cost whereas part of the gain from this investment goes to the firm.

This interpretation also explains why there is no value for the bargaining strengths of workers which could make the laissez-faire equilibrium coincide with a social optimum. When β tends to 1, the entire output goes to workers. The cutoff y * becomes efficient, but job creation is reduced to zero.

It is worth noting that this insufficiency of participation does not stem from urn-ball matching by itself, but from recruitment selection. With random recruitment, the urn-ball matching function is H = v(1 -e -1/θ ), which is no more than a particular specification of the usual CRTS function. Thus, the results would be the same as in [START_REF] Gavrel | On the Efficiency of Participation with Vertically Differentiated Workers[END_REF].

Regarding the partial efficiency of job creation we build on [START_REF] Gavrel | On the Inefficiency of Matching Models of Unemployment with Heterogeneous Workers and Jobs when Firms Rank their Applicants[END_REF] to prove the following proposition

7 With random recruitment, this density is: (1 -e -1/θ )f (y)/(1 -F (y * )).
Proposition 1b. (i) An increase in job creation (v) lowers the expected output (ȳ).(ii)

Consequently, in the neighborhood of laissez-faire, job creation is partially inefficient under the Hosios rule. Firms create too many jobs.

Proof. Statement (i) is proved in Appendix A. Let us prove statement (ii). Differentiating Σ with respect to v gives

n ∂Σ ∂v = Q(.)(1 -η(.))(ȳ -z) -c + vQ(.) ∂ ȳ(.) ∂v
with η(.) being the elasticity of the probability Q(.) with respect to v (in absolute value).

Let α(.) denote the elasticity of (ȳ -z) with respect to v (in absolute value).

Since ∂ ȳ/∂v < 0, the derivative of Σ can be rewritten as

∂Σ ∂v = 1 n [Q(.)(1 -η(.) -α(.))(ȳ -z) -c]
On the other hand, equation ( 8) is equivalent to

Q(.)(1 -β)(ȳ -z) -c = 0
Applying the Hosios rule (β = η), we then obtain

∂Σ ∂v = - 1 n Q(.)α(.)(ȳ -z) < 0
This proves that job creation is too high in the neighborhood of an equilibrium, whatever the value of the cut off y * is.

Q.E.D.
The intuition behind this result is that, relative to the basic matching model, job creation acquires a new congestion effect in the presence of recruitment selection.

As market tightness increases, the number of applications that a firm can expect to receive falls and accordingly the expected maximum output (i.e. the expected value of the n th order statistic) among its applicants is reduced. In other words, when the market becomes tighter, firms are compelled to hire lower-skilled workers. This lowers the average output ȳ. See Appendix A. Since firms do not not internalize this negative externality, job creation is excessive.

It is worth noting that, with recruitment selection, an increase in the number of vacancies exerts the same influence as a decrease in the share of skilled workers with random recruitment.

Constrained efficiency

Proposition 1a suggests that any policy measure that tends to raise participation is likely to improve the efficiency of the labor market. On the other hand, a participation increase will affect firms' entry. Hence to assess the value of this conjecture it is convenient to study the constrained efficiency of the cutoff y * . See [START_REF] Moen | Does Poaching Distort Training? Review of Economic[END_REF]. This amounts to examining the effect of an increase in the cutoff on aggregate income when job creation is set by free-entry (equation ( 8)).

As equation ( 8) is treated as a constraint, the social surplus, Σ cp , can be rewritten as follows 8

Σ cp = 1 y * [p((v/n), y)β(y -z) -γ]f (y)dy (10)
On the other hand, according to equation (8) job creation (v) is an implicit function, v(.), of the participation cutoff y * . One can see that the derivative v (y * ) is negative.

The reason for this is that with recruitment selection a participation increase raises the expected profits of vacancies, prompting firms to enter the market. Differentiating 8 See Appendix B for detailed calculus.

Σ cp with respect to y * gives

∂Σ cp ∂y * = -f (y * )[p((v/n), y * )β(y * -z) -γ] + 1 y * ∂p((v/n), y) ∂v β(y -z)f (y)dyv (y * )
Using equation ( 2) and noting that ∂p((v/n), y) ∂v > 0 the following results

∂Σ cp ∂y * = 1 y * ∂p((v/n), y) ∂v β(y -z)f (y)dyv (y * ) < 0
So the participation cutoff appears to be too high. We have then proved the result below.

Proposition 2. In a laissez-faire equilibrium, participation is constrained suboptimal.

The intuition behind Proposition 2 is as follows. A participation increase tends to raise the value of vacancies. As free entry is imposed, this stimulates job creation, and hence creates a positive effect (an indirect externality) on all participating workers.

Symmetrically, as in [START_REF] Albrecht | Efficiency in a Search and Matching Model with Endogenous Participation[END_REF], we can study the constrained efficiency of job creation (v). This amounts to appraising the effect of an increase in v on the social surplus when the participation cutoff is set by individuals' choices, equation

(2). From (2) we deduce the following derivative of y *

∂y * ∂v = - (1 -F (y * ))(y * -z) vf (y * ) < 0
As an increase in job creation raises the probability of getting a job for all workers, lower-skilled workers find it profitable to enter the market. In other words, the productivity of the marginal participant decreases. This indirect effect of job creation is similar to Albrecht et al.

We then obtain the following expression for the impact on the social surplus

∂Σ cv ∂v = ∂Σ ∂v + ∂Σ ∂y * ∂y * ∂v
Under the Hosios rule, in the neighborhood of laissez-faire, this derivative can be rewritten as follows (see Proposition 1b)

∂Σ cv ∂v = ∂Σ ∂y * ∂y * ∂v - 1 n Q(.)α(.)(ȳ -z)
with α(.) being the elasticity of (ȳ -z) with respect to v (in absolute value).

As ∂Σ/∂y * < 0 (Proposition 1a), it follows that the first term of the derivative is positive whereas the second term is negative. The constrained efficiency of job creation appears to be indeterminate. This contrasts with [START_REF] Albrecht | Efficiency in a Search and Matching Model with Endogenous Participation[END_REF] who obtain the result that job creation is excessive under the Hosios rule. The reason for this is twofold. First, holding the participation cutoff constant, the expected output no longer depends on job creation when recruitment is random (as in Albrecht et al.).

Second , attracting lower-skilled workers improves market efficiency under recruitment selection, whereas this tends to depress efficiency under random recruitment. See our comment on Proposition 1a.

IV. Introducing a minimum wage

Participation is insufficient. What should be done to deal with this market failure?

In this section we provide a rationale for adopting a mandatory minimum wage.

We show that introducing a minimum wage improves the efficiency of the economy (measured by the aggregate income). This improvement of efficiency does not require workers' bargaining strengths to be low. It holds for any value of the surplus shares. Moreover a small minimum also appears to raise the expected incomes of all participants.

We first study the effects of introducing a minimum wage on participation, labor demand and unemployment. This study is interesting by itself as the vertical differentiation of workers may be a better assumption than match specific productivities in assessing the merits of the minimum wage9 . Surprisingly, we find that job creation is stimulated. We explain how recruitment selection causes this unexpected result.

The labor market in the presence of a binding minimum wage

In the presence of a binding minimum wage, ŵ, low-ability workers (workers whose productivities are lower that some trigger ŷ) now earn that minimum, whereas the wages of high-ability workers (workers whose productivities are greater that ŷ) are still subject to bargaining. Formally, we have

w(ŷ) = z + β(ŷ -z) = ŵ and y * ≤ y ≤ ŷ ⇐⇒ w(y) = ŵ ŷ ≤ y ≤ 1 ⇐⇒ w(y) = z + β(y -z).
Notice that the first equation determines the trigger ŷ.

The presence of a mandatory wage has two consequences. First, the equation which determines the cutoff y * is now written as follows

z = -γ + p((v/n), y * ) ŵ + (1 -p((v/n), y * ))z Or γ = p((v/n), y * )( ŵ -z) (11)
For a given value of job creation v, an increase in the minimum wage ŵ lowers the cutoff y * , consequently increasing the participation level. We will show that this intuitive result holds true when job creation is endogenous.

Next, when matched with a worker whose productivity lies on the range [y * , ŷ], a job generates the profit (y -ŵ). Otherwise the profit remains equal to [(1 -β)(y -z)].

As a consequence, the job creation equation must be rewritten as

-c + ŷ y * q((v/n), y)(y -ŵ)dy + (1 -β) 1 ŷ q((v/n), y)(y -z)dy = 0 (12)
This leads to the following definition Definition 3. In the presence of a binding minimum wage, a labor market equilibrium is a pair (y * , v) which jointly satisfies equations (11) and (12).

In what follows we examine the implications of introducing a binding minimum in the neighborhood of an equilibrium where all wages are bargained. In other words, this amounts to studying the effects of a (small) increase in the minimum wage in a situation where the legal minimum ŵ is initially equal to the bargained wage:

z + β(y * -z).
This also means that in our benchmark, the cutoff y * coincides with the trigger ŷ.

Effects of a minimum wage increase

With respect to the effects of introducing a small but binding minimum wage ( ŵ becomes higher than [z + β(y * -z)]), we first show the following proposition Proposition 3. Introducing a minimum wage lowers the cutoff (y * ) and raises job creation (v). Despite the increase in labor demand, the unemployment rate (1 -P )

rises but the employment rate (P (1 -F (y * ))) is enhanced.

Proof. See Appendix C.

At first glance this increase in labor demand might look surprising. The intuition behind it is very simple. Introducing a (small) minimum wage makes lower-skilled workers (infra-marginal workers) join the market. Because their productivities remain higher than the wages they earn when holding a job, this tends to raise the expected profits of an offered job. On the other hand the wage increase lowers the profits of matches with incumbent workers on the minimum. In the neighborhood of laissez-faire this negative effect on expected profits is very small. Job creation is then stimulated. Conversely, implementing a minimum wage lowers market tightness, consequently increasing the unemployment rate. However, employment rises, so that fewer workers are jobless, whether inactive or unemployed.

It is worth noting that with vertically differentiated workers and ranking of candidates, the minimum wage effects are quite different from the usual search and matching setting [START_REF] Flinn | Minimum Wages Effects on Labor Market Outcomes under Search, Matching, and Endogenous Contact Rates[END_REF][START_REF] Flinn | The minimum wage and labor market outcomes[END_REF]. When all workers have the same (expected 10 ) productivities, introducing a minimum wage tends to lower the (expected) profit of all matches 11 , leading then to a reduction of job creation. On the contrary, in our model, a minimum wage only affects the profit of matches with low-skilled workers.

10 Flinn assumes match specific productivities.

11 Independently of the increase in the value of being unemployed that occurs in a dynamic setting.

Minimum wage and market efficiency

We can now study the effect of implementing a minimum wage on the efficiency of the labor market. Differentiating the social surplus Σ with respect to the minimum wage ŵ yields

∂Σ ∂ ŵ = ∂Σ ∂v ∂v ∂ ŵ + ∂Σ ∂y * ∂y * ∂ ŵ
We have seen that the cutoff y * decreases with a minimum wage increase (Proposition 3). This increase in participation tends to improve market efficiency (Proposition 1a). On the other hand job creation is enhanced (Proposition 3). With applicant ranking this increase in labor demand tends to depress market efficiency under the Hosios rule (Proposition 1b). So, a priori, the impact on the social surplus would be indeterminate. However, when resulting from the introduction of a minimum wage, the stimulus to job creation follows from the decrease in the participation cutoff, which improves the expected income of vacancies. This is the reason why we can prove the result below.

Proposition 4. Introducing a small but binding minimum wage improves the efficiency of the labor market.

Proof. Proposition 4 results from the constrained inefficiency of participation (Proposition 2). See Appendix D.

Proposition 4 contradicts [START_REF] Gavrel | On the Efficiency of Participation with Vertically Differentiated Workers[END_REF] where the efficiency of the labor market is improved by subsidizing non-participants (i.e. making inactivity more attractive to workers). The reason for this is that in this paper (as well as in [START_REF] Albrecht | Efficiency in a Search and Matching Model with Endogenous Participation[END_REF] random recruitment creates a mixing effect which tends to lower the social surplus when low-ability workers join the market (see the comment on Proposition 1b). On the contrary, with applicant ranking, the presence of bad workers in the market does not affect the probability for a firm to hire a good one. That only enhances the probability of filling the vacancies in a less productive but efficient manner. Proposition 4 also contrasts the usual search-matching model in which minimum wage efficiency critically depends on workers' bargaining strengths.

Different authors prefer to focus on the value of being unemployed in lieu of the social surplus. In our static setting with heterogeneous workers this amounts to examining how the expected incomes of all individuals are affected by the minimum wage. Let us first consider the individuals whom the minimum wage prompts to enter the market.

The expected incomes of these new participants obviously rise; otherwise they would remain inactive. Let us now turn to the workers who already participated in the market. As labor demand rises, all incumbent workers have a higher probability of finding a job. See equation ( 5). Because workers earn the minimum wage only if it is binding, we obtain the following corollary to Proposition 3:

Result. A small minimum wage raises the expected incomes of all participating workers.

As introducing a minimum wage lowers market tightness (Proposition 3), this result might look counterintuitive. The intuition behind this point is that the entry of lower-skilled workers does not directly affect the probabilities with which incumbent workers get a job. New entrants find it more difficult to obtain a job (relative to incumbent) but this is compensated by the minimum wage.

V. Other public policies

Basic income

Introducing a minimum wage is a simple policy measure. Nevertheless, as French (2012) points out in his review of [START_REF] Flinn | The minimum wage and labor market outcomes[END_REF], other policy measures could be con-sidered. We here study the implications of a lump-sum subsidy to participants paid for by a neutral tax12 . See [START_REF] Holmlund | Unemployment Insurance in Theory and Practice[END_REF]. This income, denoted by a, is received by all participants whether they find a job or not. Although non-participants do not profit from it, this subsidy resembles a basic income.13

In the presence of such a basic income, the surplus of a match remains equal to (y-z).

Consequently, the equilibrium equation for job creation is left unchanged.

On the other hand, the equilibrium equation for participation becomes

z = -γ + p(., y * )[z + a + β(y * -z)] + [1 -p(., y * )](z + a) Or -(γ -a) + p(., y * )β(y * -z) = 0 (13)
This basic income clearly acts as a subsidy to job search, tending then to make participation more attractive. In Appendix E, we show that an increase in the subsidy a necessarily lowers the cutoff y * . So, as a consequence of Proposition 2, implementing a (small) basic income improves market efficiency.

Unemployment compensation

As workers are assumed to be risk-neutral, unemployment insurance does not really make sense in our setup. See [START_REF] Pissarides | Employment Protection[END_REF]. Nevertheless unemployment compensation can be used as a means of attracting more individuals in the labor market.

Contrary to a basic income, introducing unemployment benefits (denoted by b) changes the expression for the (private) surplus of a match. As a consequence, the job creation equation must be rewritten as

-c + (1 -β) 1 y * q((v/n), y)(y -z -b)dy = 0 (14)
On the other hand, the participation cutoff is now derived from the following equation

z = -γ + p(., y * )[z + b + β(y * -z -b)] + [1 -p(., y * )](z + b) Or -(γ -b) + p(., y * )β(y * -z -b) = 0 (15)
Holding job creation (v) constant, unemployment compensation also tends to lower the participation threshold y * . More workers enter the labor market because they receive the income b if they are not successful in finding a job. In Appendix F, we show that unemployment compensation actually increases participation when job creation is endogenous. But, due to the increase in all wages, the impact on job creation is indeterminate, even in the neighborhood of laissez-faire. Despite of this, we can prove that introducing (small) unemployment benefits improves market efficiency. To that end we make use of previous (partial) inefficiency results (Proposition 1a,b) when job creation is reduced. In the reverse case the proof is similar to that of Proposition 2.

See Appendix F.

Which policy is better for welfare? To answer this question, numerical simulations would indicate the extent to which these different measures can raise the aggregate income (outside the neighborhood of laissez-faire). We can however mention two arguments which can be made in favor of minimum wages. First, unlike subsidies to workers, introducing a minimum wage does not require the collection of taxes, consequently creating administrative costs. Second, and more importantly, to be efficient, subsidies to participants imply that their job search is monitored. This will generate a dead weight loss.

VI. Conclusion

Using a matching model with vertically differentiated workers, we show that in the presence of recruitment selection, participation in the labor market is insufficient.

Insofar it is not too large, a minimum wage improves the efficiency of the economy by rendering activity more attractive to low-skilled workers. This result holds despite an increase in unemployment, implying then that the effects of minimum wages on unemployment are not what really matters. Subsidies to participants also enhance net aggregate income. An online appendix extends the analysis to a dynamic model.

Although the distribution of productivities among unemployed workers becomes endogenous, results hold true in this setting. Participation remains (constrained) insufficient and introducing a minimum wage still improves market efficiency. This appendix also shows that directed search restores market efficiency.

In order to derive analytical insights, the study was restricted to the neighborhood of laissez-faire. A calibration of the dynamic model would allow one to go beyond and to compare the efficiency of different policy measures. Numerical analysis would also provide an assessment of the optimum minimum wage. This a topic for further work.

From an empirical perspective these analytical insights also make possible an alternative interpretation of the results from [START_REF] Card | Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania[END_REF] case study.

According to these authors, the observed employment increase in the fast-food industry is likely to result either from a higher inflow of workers into this industry, or from a lower outflow. See [START_REF] Portugal | Disentangling the Minimum Wage Puzzle: An Analysis of Worker Accessions and Separations[END_REF] for empirical findings that coincide with this explanation. In an earlier paper, we show that a minimum wage can induce an employment increase in the low-skill sector by prompting employees to reduce the intensity of their (on-the-job) search for better-paid jobs. A larger proportion of vacancies is filled. However, due to the reduction of job creation, the (possible) employment growth in low-skill industries is accompanied by a macroeconomic fall in employment. This would imply that one cannot infer macroeconomic predictions from sectoral results. See [START_REF] Gavrel | Minimum wage, On-the-job Search and Employment: On the Sectoral and Aggregate Equilibrium Effect of the Mandatory Minimum Wage[END_REF]. The present paper clearly weakens this pessimistic view of minimum wages. By giving firms the opportunity of filling their vacancies with new participants, a mandatory wage instead stimulates job creation when employers are selective.

To conclude, we would like to emphasize that both (perfect) applicant ranking and random recruitment are strong assumptions. In the real world firms carry out a costbenefit analysis when deciding on their recruitment process. In fact their selectiveness should be modeled as an endogenous variable that is likely to depend on market tightness as well as on the distribution of workers' productivities. We leave a more realistic modeling of alternative recruitment strategies for future research.

∂ρ((v/n), ỹ, y * ) ∂v = 0 and

y < (>)ỹ ⇐⇒ ∂ρ((v/n), y, y * ) ∂v > (<)0 This implies that, ∀y ∈ [y * , ỹ[ ∪ ]ỹ, 1] ∂ρ((v/n), y, y * ) ∂v y < ∂ρ((v/n), y, y * ) ∂v ỹ
Consequently we obtain

∂ ȳ ∂v < 1 y * ∂ρ((v/n), y, y * ) ∂v ỹdy = 0
This proves that job creation lowers the expected output.

Σ cp = (v/n)Q((v/n), y * )β(ȳ -z) -[1 -F (y * )]γ + z Or Σ cp = 1 y * [(v/n)q((v/n), y)β(y -z) -f (y)γ]dy + z From (v/n)q((v/n), y) = p((v/n), y)f (y), we deduce ∂Σ cp ∂y * = 1 y * ∂p((v/n), y) ∂y * β(y -z)f (y)dy -[p((v/n), y * )β(y * -z) -γ]f (y * )
Using equation ( 2), we obtain

∂Σ cp ∂y * = 1 y * ∂p((v/n), y) ∂y * β(y -z)f (y)dy
On the other hand, equation ( 8) implies that ∀y ∈

[y * , 1] ∂p((v/n), y) ∂y * = ∂p((v/n), y) ∂v v (y * ) < 0
As v (y * ) < 0 (see the text), this shows that ∂Σ cp /∂y * < 0 and proves Proposition 2.

∂J(.) ∂y

* = -(y * -ŵ)q((v/n), y * ) = -(1 -β)(y * -z)q((v/n), y * ) < 0 ∂J(.) ∂ ŷ = 0 ∂J(.) ∂ ŵ = - ŷ y * q((v/n), y)dy = 0
Let us now turn to the effect of job creation v on J(.). Let H denote

H ≡ 1 y * q((v/n), y)(y -z)dy
Integrating H by parts gives

H = 1 -z -p((v/n), y * )(y * -z) - 1 y * p((v/n), y)dy
In the neighborhood of laissez-faire, the derivative of J with respect to v is (very

) close to (1 -β) ∂H ∂v From ∂p((v/n), y) ∂v > 0 we deduce ∂H ∂v = - ∂p((v/n), y * ) ∂v (y * -z) - 1 y * ∂p((v/n), y) ∂v dy < - ∂p((v/n), y * ) ∂v (y * -z) ∂J(.) ∂v < -(1 -β) ∂p((v/n), y * ) ∂v (y * -z) < 0 Now, differentiating equation (12) yields -(1 -β)(y * -z)q((v/n), y * )dy * + ∂J(.) ∂v dv = 0
From the two previous relations, we deduce

0 > dv dy * > - q((v/n), y * ) ∂p((v/n),y * ) ∂v . As ∂p((v/n), y * ) ∂v = p((v/n), y * )[1 -F (y * )] n v 2 and q((v/n), y * ) = p((v/n), y * )f (y * ) n v we obtain dv dy * > - vf (y * ) 1 -F (y * ) Next, differentiating equation (11) gives 1 + n v [f (y * ) + 1 -F (y * ) v dv dy * ]( ŵ -z) ∂y * ∂ ŵ = 0
From the previous inequality, we get

[f (y * ) + 1 -F (y * ) v dv dy * ]( ŵ -z) = 1 -F (y * ) v [ vf (y * ) 1 -F (y * ) + dv dy * ]( ŵ -z) > 0
This proves that introducing a minimum wage lowers the cutoff y * . We already know that a decrease in y * is associated with an increase in v (dv/dy * < 0). Therefore, introducing a minimum wage raises job creation.

From the same equation ( 11), we deduce

∂θ ∂ ŵ = - θ 2 ŵ -z < 0
This shows that introducing a minimum raises the (average) unemployment rate of participating workers.

The employment rate, [1 -F (y * )]P (.), can be written as From Propositions 1a-b, we know that, in the neighborhood of laissez-faire, ∂Σ/∂v is negative under the Hosios condition, while ∂Σ/∂y * is always (strictly) negative. As ∂y * /∂b < 0, we obtain ∂Σ/∂b > 0. This proves that, in this first case, introducing (small) unemployment benefits improves market efficiency under the Hosios condition.

Case 2. ∂v/∂b > 0

In this case, we note that Σ can be written as From equation ( 15), we deduce that the previous derivative is close to zero in the neighborhood of laissez-faire. This proves that (small) benefits increase the aggregate income.

Σ = v n [-c+(1-β)Q(.)(ȳ-z-b)]+(1-β) v n Q(.)(b+z)+β v n Q(.)ȳ+(1- v n Q(.

[ 1 -

 1 F (y * )]P (.) = Q(.) v nWe know that introducing a minimum wage increases v and lowers θ. Consequently, Q(.) rises. This shows that the employment rate is enhanced, completing then(v/n), y)(z+β(y-z))]dy+[1-Q((v/n), y * )(v/n)]z-[1-F (y * )]γ From the job creation equation (12)(v/n), y)(z + β(y -z))dy] +[1 -Q((v/n), y * )(v/n)]z -[1 -F (y * )]γ Differentiating Σ with respect to ŵ gives ((v/n), y * )( ŵ -z) -γ]f (y * ) ∂y * ∂ ŵFrom Proposition 3, we know that ∂v/∂ ŵ > 0. On the other hand we have ∂p((v/n), y)/∂v > 0. From the indifference condition (11), we deduce that introducing a minimum wage improves market efficiency.1 + βp((v/n), y * ){1 + (y * -z)(1 -F (y * )prove that unemployment compensation (b) lowers the participation cutoff (y * ). Suppose that the derivative ∂y * /∂b is positive. From the job creation equation (as previously rewritten) we deduce ∂v/∂b > 0. Conversely the participation equation (15) imposes that ∂v/∂b < 0. This contradiction proves that unemployment compensation enhances participation.EfficiencyAs the impact on job creation is indeterminate we are led to distinguish between the two cases.Case 1. ∂v/∂b ≤ 0In this case we use Propositions 1a,b. The derivative of Σ can be written as

  ))z-(1-F (y * )γAs the pair (v, y * ) satisfies (14), Σ is reduced toΣ v/n), y)f (y)(y -z)dy + z -(1 -F (y * )γOne can check that Σ is an increasing function in b as in v. Differentiating Σ with respect to y * gives∂Σ ∂y * = [γ -b -βp((v/n), y * )(y * -z -b) + (1 -p((v/n), y * ))b]f (y * )

See for instance[START_REF] Lehmann | Optimal Income Taxation with Endogenous Participation and Search Unemployment[END_REF] for a study of optimal income taxation with endogenous participation.

See for instance[START_REF] Blanchard | Ranking, Unemployment Duration, and Wages[END_REF],[START_REF] Moscarini | Unobserved Heterogeneity and Unemployment Duration: A Fallacy of Composition[END_REF],[START_REF] Moen | Education, Ranking and Competition for Jobs[END_REF],[START_REF] Gavrel | Technical Skill Bias as a Response of Firms to Unemployment. A Matching Model with Applicant Ranking and Endogenous Skill Requirements[END_REF][START_REF] Gavrel | On the Inefficiency of Matching Models of Unemployment with Heterogeneous Workers and Jobs when Firms Rank their Applicants[END_REF].

Relative to[START_REF] Gavrel | On the Inefficiency of Matching Models of Unemployment with Heterogeneous Workers and Jobs when Firms Rank their Applicants[END_REF], who assumes recruitment selection, we extend the analysis to vertically differentiated workers and endogenous participation.

An online appendix proves that results extend to a dynamic setting. It is also shown that directed search restores market efficiency.

This static setting helps us present our results with clarity. See the online appendix for the extension to a dynamic model.

See Gavrel (2012) for a more detailed exposition.

Indeed, practical relevance requires that lower-skilled workers are on the minimum. Such a prediction is not compatible with horizontally differentiated workers.

One can also consider the case in which subsidies are financed by a tax on job creation. Not surprisingly, results are ambiguous.

This lump-sum subsidy also resembles certain elements of a tax credit conditional on employment, like EITC in US or WTC in UK.

Appendix A: Proof of Proposition 1b, Statement (i) We here prove that an increase in v reduces the expected output ȳ

One can show that the derivative of ρ((v/n), y, y * ) with respect to v has the same sign as the following expression

The previous expression is a decreasing function of y.

As

Appendix B: Constrained efficiency of participation

The aggregate income, Σ, can be rewritten as

As equation ( 8) is treated as a constraint, we obtain the following expression for the constrained (social) surplus

Appendix C: Proof of Proposition 3

Let J denote the expected profit of an offered vacancy (gross of job opening costs)

In the neighborhood of the laissez-faire equilibrium (Definition 2), the result is obtained Appendix D: Proof of Proposition 4

In the presence of a binding minimum wage, the aggregate income can be rewritten as

Appendix E: Basic income

Following the same reasoning as in the proof of Proposition 3 (Appendix C), we obtain

Next, differentiating equation (13) gives

Appendix F: Unemployment compensation

Participation

Let us consider the equilibrium equation for job creation ( 14). From

we deduce that ( 14) can be rewritten as