Massimo Pica Ciamarra

Patrick Richard

Matthias Schröter

Brian P Tighe

Statistical mechanics for static granular media: open questions

The theoretical description of granular materials, or assemblies of macroscopic particles, is a formidable task. Not only are granular materials out of thermal equilibrium, but they are also characterized by dissipative interactions and by static friction. Following a suggestion by S. F. Edwards, researchers have investigated the possible existence of a statistical mechanics of static granular systems, which would permit the description of macroscopic properties of mechanically stable granular assemblies from just a few parameters. The formulation and the validity of such an approach is still a matter of debate. This "emerging area" focuses on three important questions concerning such a statistical mechanics approach. First, we consider how the phase space of interest is affected by the requirement of mechanical stability. Second, we explore how the intensive parameters analogous to temperature can be determined from experimental or numerical data. Finally, we contrast different ways to express the granular counterpart to the classical Hamiltonian, known as the volume function.

Introduction

Describing the properties and the behaviour of granular materials is a problem of great interest, as granular materials are widely used in industrial processes and found in a number of natural phenomena. How can we characterize the disordered structure of a granular pack? Can we deduce from such a characterization the response (evolution of the packing fraction, of the force network. . . ) of a granular material submitted to mechanical perturbations or stresses? There is not yet a theoretical framework that permits such a description. From the theoretical point of view, the challenge is that of describing a disordered out-of-equilibrium many-particle system, as granular particles are so heavy that their thermal motion is negligible. The problem is further complicated by dissipation and static friction in the interparticle interaction, both of which arise due to the macroscopic size of the particles.

More than 20 years ago, S.F. Edwards and co-workers proposed that the methods of statistical mechanics could be extended to dense disordered granular assemblies. 1,2 Such a statistical mechanics of granular matter, if valid, would permit granular systems to be described with just a few parameters, and would represent a breakthrough in the field of granular media. Edwards' original suggestion has been the subject of intense research, both theoretically and experimentally. The statistical mechanics proposed by Edwards is specifically geared towards static granular materials, where 'static' indicates that the phase-space of interest is restricted to mechanically stable configurations.

Granular media are often static. Grains under gravity naturally settle down in a mechanically stable state, which is a local energy minimum in the absence of driving.

Stable states in mechanical equilibrium can also be prepared without gravity by imposing a confining stress. In either case, grain scale deformations under typical conditions are extremely small; hence one might expect that elastic energy plays a marginal role in static granular systems, and no role at all in the limit of hard spheres in the absence of gravity. Motivated by this, Edwards [START_REF] Edwards | Granular matter : an interdisciplinary approach[END_REF] suggested that in a granular statistical mechanics, the conventional Hamiltonian should be replaced with a volume function W that expresses the volume of the system as a function of the degrees of freedom of the system q. In strict analogy with equilibrium statistical mechanics, one assumes that all microstates (mechanically stable states) of given volume occur with the same probability. One then derives the partition function of the system,

Z(χ) = exp(-W (q)/χ) Θ(q) dq,
where χ is termed compactivity, a Lagrange parameter analogous to the temperature, and Θ(q) imposes the condition of mechanical stability, including a finite bulk and shear modulus. [START_REF] Dagois-Bohy | [END_REF] At a given compactivity χ, the equiprobability condition leads to the minimization of the granular free energy F(φ ) = W -χS, where we have introduced the granular entropy S(W ) = log (Ω(W )), Ω(W ) being the number of mechanically stable states of volume W . The inverse compactivity χ -1 = ∂ S/∂V is therefore related to the volume dependence of the entropy. If one considers not only the number of ways the grains can occupy the volume V but also the number of ways the grains support the boundary stresses σ , the partition function becomes Z(χ, α i j ) = exp(-W (q)/χ -Γ i j (q)/α i j ) Θ(q) dq,

where Γ i j and α i j are the virial and the tensorial angoricity, which reduce to scalars Γ and α for systems loaded isotropically. Similarly to the compactivity, the angoricity is related to the pressure dependence of the entropy S(σ ), which is now a measure of the number of mechanically stable states supporting a given stress σ . From this partition function, as for conventional statistical mechanics, many quantities can be obtainedin principle. As we shall show, doing so remains challenging.

In the following three sections we will discuss recent developments and open questions regarding three key ingredients of eqn (1): first, how does the requirement of mechanical stability, represented by Θ(q), shape the explored phase space? Second, how can the intensive parameters χ and α be determined from experimental or numerical data? And finally, what is the right way to compute the equivalent to the classical Hamiltonian, W ?

2 The phase space of mechanically stable configurations and its exploration

The function Θ(q) limits the integration in eqn 1 to the mechanically stable configurations of grains * . This raises two questions: what is the range in volume fraction φ

(the fraction of the total volume occupied by grains) and confining pressure for which mechanically stable configurations exist? And how does the system explore this phase space?

The condition of mechanical stability

For a better understanding of the role of Θ it is helpful to contrast the Edwards' approach with the hard sphere gas. In the latter we only request that the particles do not overlap. Consequently, the phase space volume of configurations where any two particles touch is of measure zero. In contrast, imposing mechanical stability requires sufficient permanent contacts to constrain all rotational and translational degrees of freedom -there can be no zero energy ("floppy") modes. [START_REF] Van Hecke | [END_REF] Absent geometric degeneracies such as crystallinity, mechanically stable frictional sphere packings must have an average coordination number Z ≥ 4, the so-called frictional isostatic contact number. 6,7 A second important reference point is the jamming paradigm for frictionless soft particulate systems such as foams, colloids and emulsions. [START_REF] Van Hecke | [END_REF] Mechanically stable packings of frictionless spherical particles exist only above the so-called point J with φ J ≈ 0.64. In frictionless packings both the packing fraction and the contact number display scaling with distance to point J, and both in turn are controlled by the pressureunjamming corresponds to zero pressure. The presence of nonzero surface friction µ renders granular materials qualitatively distinct from foams, emulsions, etc.. Friction allows for tangential contact forces, which permit frictional packings to support loads in ways unavailable to frictionless materials. 8 For various confining pressures, there are mechanically stable packings over a range of volume fractions. 9 Moreover, even for a given combination of φ and p, the average contact number can depend on the * In the following we will exclusively discuss spherical grains preparation history. 10 The empirical fact that mechanically stable packings of spheres can only be prepared in a finite range of φ was first established by Scott 11 and Bernal. 12 The apparent absence of crystallinity earned the upper and lower bounds the names Random Loose Packing (RLP) and Random Close Packing (RCP), respectively. From a statistical mechanics perspective, one expects that these two bounds should be related to a configurational entropy S corresponding to the logarithm of the phase space volume of all mechanically stable configurations at a given volume fraction compatible with the mechanical boundary conditions. It is then natural to ask what features characterize S(φ RLP ) and S(φ RCP ). We shall consider RLP first.

One possible interpretation of RLP is that below φ RLP the entropy is zero. † This is suggested by the observation from numerics that loose packings approach the isostatic value Z = 4 in the large friction limit. Provided lowering φ lowers Z -an assumption supported by numerics -then S must be zero below loose packing. By considering the fraction of "fully mobilized" contacts, which saturate the Coulomb inequality, Shundyak et al. 13 showed that packings with finite friction approach an "isostatic line" connecting frictionless and frictional isostatic states (Fig. 1a). In this sense even packings at finite µ are isostatic, at least in numerics. By the same reasoning as above, one would then expect S to vanish at RLP for any value of µ.

An alternative viewpoint holds that RLP corresponds to a maximum in S, 14,15 as outlined in Fig. 1 b. This explanation is supported by the observation that the experimental and numerical protocols used to build RLP packings are all gentle deposition protocols, which aim to avoid disturbing the already existing structure . 14,16,17 If S(φ RLP ) is a maximum, either S drops discontinuously to zero below φ RLP , or there is another, lower packing fraction where S goes to zero. Pica Ciamarra et al. termed this Random Very Loose Packing (RVLP), 14 and showed that states below RLP can be prepared numerically with carefully chosen initial conditions (Fig. 1b). If RVLP exists, one must explain why it is not observed in experiments. One such explanation is dynamical. During preparation the system explores the much larger phase space of Figure 1: Numerical results on frictional disc packings. (a) Taking into account the number of fully mobilized contacts (where the tangential force is just at the coulomb threshold) slowly prepared packing approach the isostatic limit for vanishing pressure. From Shundyak et al. 13 (b) The lower bound of the configurational entropy of twodimensional disc packings is below the RLP value (≈ 0.77) for the same system. From Pica Ciamarra et al. 14 non-overlapping hard sphere states before it settles down in a static configuration. The probability ending up in a particular configuration will depend on the phase space volume of the basin of attraction leading to this state. The apparent absence of very loose packings in the experiment could be simply due to their very small basins of attraction.

This in turn leads to the question of ergodicity, which we revisit below.

We now return to Random Close Packing. Similar to loose packing, RCP is primarily a phenomenological fact: multiple protocols, including tapping 18 and flow pulses, 19 slow down asymtotically when φ approaches 0.64. Under other protocols, like cyclic shear, no peculiar changes in dynamical properties occur at φ ≃ 0.64. 20 However, all protocols that cross RCP observe the onset of some crystalline order. While these facts seem uncontroversial, their interpretation is not. It has been proposed that RCP is a first order phase transition 21,22 between the amorphous and the crystalline state; this would imply that S is still finite at RCP. An alternative viewpoint holds that it is the endpoint of the amorphous branch of the hard sphere fluid, 23 corresponding to a vanishing S at RCP. Yet another possibility is that RCP corresponds to a local minima of the curve S versus φ ; the physical mechanism behind RCP is then explained by geometrical properties like the maximum frequency of polytetrahedral structures. 24 Clearly more work is needed.

Exploring the phase space

Besides establishing the range of φ where disordered mechanically stable states exist, we also need to consider how the system explores these states. A basic assumption of classical statistical mechanics is that all configurations are realized with equal probability. As pointed out above, due to their dissipative and macroscopic nature, granular materials cannot explore their phase space via thermal fluctuations. An external, macroscopic perturbation like shearing or tapping will always be needed to move the system from one mechanical stable configuration to another. Consequently, we must allow for the possibility, indeed likelihood, that the probability of visiting a microstate depends on the chosen protocol and preparation history. There is ample experimental evidence for such protocol dependence. For example, the force chains in a granular pile differ if it is poured from a single point source or through a wide sieve. 25 For at least two experimental driving protocols -shaking 18,26 and flow pulses 19 the existence of history-independent steady states has been established. While this does not guarantee equiprobability of the microstates, it at least assures the existence of an effective heat bath with time-independent probabilites. Moreover, numerical simulations of frictional particles driven via flow pulses 27 are compatible with an equivalence between time and ensemble averages, suggesting that this protocol allows to uniformly sample the phase space.

The only direct experimental test (with matching numerics) of the equiprobability hypothesis involved a complete enumeration of all mechanically stable configurations of 7 frictionless discs in a two dimensional box, 28 generated by tapping or slow compression. Here it was found that the probabilities of creating individual configurations differed by orders of magnitude. This can again be understood by looking at the dynamic part of the evolution towards these static state: the large probability differences of these states reflects a large size disparity in the volumes of their respective basins of attraction (assuming the preparation protocol adds no bias). However, the phase space volume of mechanically stable configurations of frictional particles is much larger than its frictionless counterpart. This means that the frictionless system will have passed through many states that would have been mechanically stable in the presence of fric-tion. Unfortunately, no results on the probability distribution of frictional particles are available.

On a more general note, it is not clear how much a statistical mechanics of granular media will require the equiprobability assumption. One could imagine, e.g., a partition function including a protocol-dependent degeneracy function, 29 which accounts for the different probabilities. Also it has been shown that equiprobability is not a necessary condition for the existence of intensive thermodynamic parameters like compactivity. 30 Again, more work is needed.

Measuring intensive thermodynamic parameters

Any canonical ensemble is characterized by one or more intensive thermodynamic parameters, e.g. the inverse temperature β = 1/k B T in equilibrium statistical mechanics.

In the usual thermodynamics the temperature is measured introducing a thermometer, a calibrated probe that acquires the temperature of the system of interest when the two equilibrate at constant energy, i.e. in the microcanonical ensemble. In the simplest form of a granular statistical mechanics, the counterpart to β is a scalar χ conjugate to the packing's volume V , termed "compactivity," or a scalar α, the "angoricity," conjugate to the product of pressure and volume Γ = pV . Γ is an extensive stress-related quantity known variously as the virial, force moment, or extensive pressure. How to measure these intensive parameters in the granular case is not so obvious, because there is no natural dynamical process by which two systems in contact can exchange volume (or force moment) and thus come to equilibrium, i.e. there is not a microcanonical ensemble. The analogy between compactivity (or angoricity) and the temperature must therefore be regarded as an assumption of Edwards' theory.

It is natural to ask how to measure these intensive parameters, and which aspects of Edwards's theory -if any -are tested by these measurements. This is the goal of the present section which focuses mainly on the "overlapping histograms" method and discusses some aspects of the Edwards' theory that remain to be tested. In the discussion below, there is no intrinsic difference between measurements of χ and α; the reader can freely exchange the conjugate pairs (χ,V ) and (α, Γ).

Several methods exist to measure intensive parameters like compactivity. Among them are the "specific heat" method, 26 which consists in measuring the compactivity from the fluctuations around the mean volume of a granular packing, just as the specific heat of a thermal system can be determined from its energy fluctuations. This method has been used widely for granular packings undergoing compaction. 18,19,26,31 However, the interpretation of the measured quantity as a granular analog of temperature relies on the assumptions built into the Edwards' approach; hence the method cannot test their validity. The compactivity can also be determined from the probability distribution of a the volume of cells tessellating the granular packing. 32 Here, we will focus on a third way: the "overlapping histograms method," proposed by Dean and Lefèvre 33 and applied to granular systems by Henkes et al. 34 and McNamara et al. 35 It has the advantage of allowing one to probe system size dependence from one ensemble of packings. Moreover it does not require detailed knowledge of the entropy and is a relevant test of the theory. We describe the method as a Gedankenexperiment, though of course ref. 34 and 35 have implemented it in numerics. Imagine sampling mgrain clusters from a packing of N ≫ m grains, labeled packing 1. The force moment Γ on each cluster can be measured and in this way the histogram or frequency distribution P

m (Γ) can be built up. If the system is statistically homogeneous and if it is entropy maximizing, then packing 1 can be seen, in the framework of Edwards' theory, as the analog of a thermal bath characterized by an intensive parameter α 1 . All other parameters P m can potentially depend on, e.g. packing fraction, friction coefficient, particle shape, etc. and will be summarized in x. In this case the distribution we have sampled is the stationary probability distribution P

(1)

m (Γ) = ρ m (Γ; x, α 1 ): ρ m (Γ; x, α) = Ω m (Γ; x) e -αΓ Z(α) , with Z = Ω m (Γ; x) e -αΓ dΓ. ( 2
)
where Z is the partition function, which normalizes ρ m . Eqn (2) lacks predictive power because we do not know Ω m (Γ; x), the density of states of m-grain clusters. Continuing the thought experiment, we prepare a second packing and label it 2. Care is taken to hold the parameters x fixed, but we admit the possibility that in constructing packing 2 it equilibrates in a state characterized by an intensive parameter α 2 different from packing 1 ‡ . Of course naïvely one expects that α -1 will bear some relation to the mean pressure in the bath Γ bath /N. As before we collect m-grain clusters and sample the frequency distribution P

(2)

m (Γ).
With malice aforethought we construct the ratio r m (Γ, Γ ′ ) from the sampled distributions:

r m (Γ, Γ ′ ) := ln P (1)
m (Γ)P ( 2)

m (Γ ′ ) P (1) m (Γ ′ )P (2) m (Γ)
.

(3)

If the system is entropy maximizing, eqn (2) holds and

r m (Γ, Γ ′ ) = -(α 1 -α 2 )(Γ -Γ ′ ) . (4)
The unknown density of states Ω m has dropped out, so that the ratio is linear in Γ -Γ ′ , with decay constant given by α 1 -α 2 .

A method for determining α (or χ) thus emerges: one constructs a family of packings at fixed x, sample distributions of Γ (or V ) on clusters of grains, and compares them pairwise via eqn (3). Obviously the method requires adequate statistics to make a comparison. It is also necessary to prepare enough packings so that for each packing i there is another j such that the histograms P (i)

m and P ( j)
m have sufficient overlap to construct the ratio r m -hence the name of the method. Note that the intensive parameter can only be determined up to an offset, which can be fixed assuming that RLP has infinite compactivity. The aforementioned Gendankenexperiments had been applied to 2D numerical simulations by Henkes et al. 34 and to 3D experiments and numerical simulations by McNamara et al. 35 An example, extracted from the former work, is given in Fig. 2.

By verifying the linearity of the ratio of two overlapping histograms, the present method has been used to successfully test Edwards' theory. 34,35 However, new exper- ‡ In fact equilibrating a new packing may produce fluctuations in some parameters, e.g. packing fraction (in an stress ensemble) or pressure (in a volume ensemble). One hopes that Ω m depends on these parameters sufficiently weakly that eqn (4) can still be used.

Expressing the partition function

The interest in the statistical mechanics of granular media originates from the possibility of predicting the properties of granular assemblies by solving the partition function, eqn 1. Such a task can only be accomplished if one is able to express in terms of the degrees of freedom q the partition function, namely the projector operator Θ(q) = 0, 1 and the volume function W (q). While this is easily done in lattice models, where one can solve the partition function using standard statistical mechanics tools, 36 how to express Θ and W on the continuum in a manageable form is not obvious.

We restrict the discussion in this section to hard sphere states. Hard spheres have no intrinsic stress scale and hence the magnitude of the confining pressure is arbitrary, analogous to temperature in thermal hard spheres. A treatment of deformable particles must keep track of the imposed stress as well as the imposed volume, as in the partition function of eqn (1). There is a growing literature focusing on stress-based ensemble approaches to granular statistical mechanics, see ref. 8 and 37 for details.

The function Θ weights mechanically stable states. As discussed above, there are similarities to the hard-sphere Hamiltonian H hs , which gives zero statistical weight to all states with particle overlaps. By writing W ms = -log Θ and assuming equiprobability, one may relate Θ to a volume function W ms = 0, and ∞ for mechanically stable and unstable states, respectively. However, while H hs can be expressed as a sum of two body contributions, W ms results from an N body interaction, N being the number of grains of the system. Formally, linear stability can be imposed in conservative systems by demanding that the system's Hessian have a positive definite spectrum (excluding rigid body modes); 4 extensions to dissipative systems are also possible. 38,39 Practically, however, such constraints are too complicated to include in the partition function. Some authors approximate them with global constraints on the connectivity 40 or moments of the stress. [41][42][43] Since the volume is an extensive quantity, the general approach to estimate W consists in tessellating the whole medium into N non-overlapping elements and associate to them a volume function W i (which can be seen, for example, as the free volume available per grain) such that W = ∑ i W i . Blumenfeld and collaborators proposed a way to determine the volume function W i . First introduced for 2D systems, 44,45 this method consists in partitioning the system into voids defined from the grain contacts, and the voids into quadrilaterals defined from the centers of the grains and the contact points (Fig. 3). This method can be generalized to 3D granular systems and also to cellular media. The description has several advantages: (i) it allows a quantitative description of the structure at any arbitrary point within the material; (ii) the shape tensor is a constitutive property; and (iii) all the basic elements are octahedra, allowing a unique tensorial description. One of the drawbacks is that this approach may illtessellate non-convex voids 46 and so overestimate their volumes. The importance of these non-convex voids, and the amount by which they overestimate volume remain a matter of debate. 47 Another drawback is that, while formally elegant, the volume function is a complicated object which is difficult to handle. Indeed, this volume function has never been used in conjunction with an estimation of the projector Θ to solve the partition function, and to predict properties of granular assemblies. As an alternative approach, a natural way to partition space is the well known Voronoï tessellation, 48 which is purely geometric and not tied to the connectivity of the contact network. 49 From this tessellation the theoretical derivation of the volume function is not straightforward, and no analytical formula exists. An approximation of this volume function has been introduced by Song et al. , 40 which expressed W i as a function of just one parameter, the local coordination number z i :

W i /V i = 2 √ 3/z i ,
where V i is the volume of a grain. Here z i refers to geometric contacts, i.e. it includes contacts that bear no force. Song et al. predict φ RCP ≈ 6/(6 + 2 √ 3) = 0.634. We stress that the calculation is not exact, but an analytical expression of an approximate value.

Song et al. 40 The validity of this theory is controversial. While there is reasonable agreement with some experiments 50 and simulations, 40 recent X-ray tomography experiments display significant differences with the prediction W i /V I = 2 √ 3/z i . For a given packing fraction, the difference between the number of contacts measured and the theoretical prediction can be as much as 15%. 51 Further, simulations show that φ RCP decreases monotonically on increasing the dimensionality;; 52 the approach by Song et al. does not reproduce this dependence. Finally, all packings in the tomographic experiments were hyperstatic (i.e. grains possess a number of contacts in excess of that which would uniquely determine the contact forces from mechanical equilibrium), contrary to the assumption that Θ(q) restricts packings to isostatic states. 40 This suggests that the discrepancy between the prediction and the observed behaviors originates from the identification of the isostatic condition with the ensemble of jammed configurations, i.e. the approximation used to deal with the projector operator Θ. In addition, since this approach resorts on an approximation of the volume function that neglects the contact network, it cannot be used to investigate the geometrical and the mechanical properties of granular systems.

More work is clearly need in this direction. The objective is to find approximations for Θ and W that are simple enough to allow analytical and/or numerical calculations, and which also give access to the calculations of properties of granular systems one could be interested in, such as the fabric tensor, fixed by the contact network.

Conclusion

The statistical mechanics of static granular systems is a theoretical effort towards the description of many particle systems out of thermal equilibrium. Here we have shown that there are a number of experimental and conceptual challenges that stand in the way of a predictive theoretical formalism. From the experimental viewpoint, the main problem is that of devising protocols that can uniformly sample the phase space of interest.

From the theoretical viewpoint, apart from the difficulties related to measuring the intensive parameters and to identifying the bounds on volume fraction, the main open problem is to express the volume fraction and the projector on the mechanically stable states in terms of the particles coordinates. In particular, the condition of mechanical stability, which imposes that the forces and the torques acting on each grain sum to zero, is expected to introduce correlations which have been neglected so far. A refined theory would find applications not only in granular media but also in other quenched, amorphous materials. Indeed, methods similar to those described here have also been applied to structural 53 and spin glasses, 54,55 to polymer and other elastic networks, 56 and to crumpling of sheets of paper. [57][58][59][60]

Figure 2 :

 2 Figure 2: The method of overlapping histograms. (a) P m (Γ) of m = 8-grain clusters sampled from packings with N = 4096 grains. The symbols correspond to runs with different packing fractions. (b) The ratio r m of eqn 3. (c) Distributions of (a) rescaled according to the values of α extracted from (b). From Phys. Rev. Lett. 99, 038002; Copyright (2007) by the American Physical Society.

Figure 3 :

 3 Figure 3: The geometric construction around grain v in 2D. The vectors r q connect contact points anticlockwise around each grain (and clockwise around each cell c). The vectors R q connect from grain centroids to centroids of neighboring cells. (a) The quadrilateral, or quadron, associated with grain v and cell c is shown shaded. (b) The sum of the areas of all the quadrons around grain v constitute the area associated with this grain. The sum of all these areas is the area of the entire system (reprinted with permission from J. Chem. Phys. B 113, 3982. Copyright 2009 American Chemical Society).

Figure 4 :

 4 Figure4: Phase diagram of jamming obtained from the theory exposed in ref.40 All disordered packings lie within the yellow triangle demarcated by the RCP line, RLP line and granular line. Iso-compativity lines are in color.

† States below RLP might still exist, but their number would have to grow subextensively with volume.

Acknowledgments

The authors acknowledge many interesting discussions with the participants of the workshop "Statistical mechanics of static granular media" held in 2009 at the Lorentz Center in Leiden. M.P.C. and B.P.T. gratefully acknowledge financial support from MIUR-FIRB RBFR081IUK and the Netherlands Organization for Scientific Research (NWO). P.R. thanks the Region Bretagne (CREATE grant: Sampleo).


Philippe Belley
email: pbelley@k-state.edu

Nathalie Havet
email: havet@gate.cnrs.fr

Guy Lacroix
email: guy.lacroix@ecn.ulaval.ca

Wage Growth and Job Mobility in the Early Career: Testing a Statistical Discrimination Model of the Gender Wage Gap

Keywords: Gender wage gap, job transitions, tenure, returns to mobility, experience JEL codes: J16, J71, J41

The paper focuses on the early career patterns of young male and female workers. It investigates potential dynamic links between statistical discrimination, mobility, tenure and wage profiles. The model assumes that it is more costly for an employer to assess female workers' productivity and that the noise/signal ratio tapers off more rapidly for male workers. These two assumptions yield numerous theoretical predictions pertaining to gender wage gaps. These predictions are tested using data from the 1979 cohort of the National Longitudinal Survey of Youth. As predicted by our statistical discrimination model, we find that men and women have the same wage at the start of their career, but that female wages grow at a slower rate, creating a gender wage gap. Also consistent with our model, we find that mean wages are higher for workers who keep their job, while wage growth is stronger for workers who change job.

Introduction

Economic analysis has made numerous contributions to our understanding of the causes of occupational segregation and the existence of earnings disparities between men and women on the labor market. 1 In particular, several theories of discrimination have attempted to explain why two groups with identical average productivity are paid different average wages. They have attributed differences to either employers' prejudices, i.e. taste discrimination [START_REF] Becker | The Economics of Discrimination[END_REF]], or imperfect information, i.e. statistical discrimination [START_REF] Phelps | The statistical theory of racism and sexism[END_REF]; [START_REF] Arrow | The theory of discrimination[END_REF]]. However, all of these theories in their earliest and simplest version are not supported by empirical evidence. Models built on taste discrimination predict the elimination of the male/female differential in the long run, whereas their persistency has been observed for decades in most OECD countries. Models based on statistical discrimination have proved deceptive since they were unable to generate a gender/racial gap in mean wages.

Consequently, second generation models have been developed to go beyond the strict opposition between discrimination and productivity differentials. For example, recent models of taste discrimination integrate henceforth the dynamic of job-search costs [START_REF] Flabbi | Gender discrimination estimation in a search model with matching and bargaining[END_REF]; [START_REF] Bowlus | Discrimination and skill differences in a equilibrium search model[END_REF]; [START_REF] Rosen | Search, bargaining and employer discrimination[END_REF]; [START_REF] Black | Discrimination in a equilibrium search model[END_REF]; [START_REF] Sattinger | Search and discrimination[END_REF]; [START_REF] Sasaki | An equilibrium search model with coworker discrimination[END_REF]; [START_REF] Sulis | Gender wage differentials in Italy: A structural estimation approach[END_REF]] and obtain more interesting conclusions in the long term. Nevertheless, these models are always little helpful in understanding gender discrimination as they fail to explain the root of the prejudices. In contrast to taste-based theories of discrimination, statistical discrimination theories derive group inequality without assuming gender animus or preference bias against members of a targeted group. Models of statistical discrimination are more appealing because they suggest gender discrimination may be a rational response by firms to imperfect information on individual productivity. While all models of statistical discrimination share that feature, there exist important differences which suggest different explanations for group inequality. Two principal paths in this literature have been followed. The first one, based on [START_REF] Arrow | The theory of discrimination[END_REF], examines how asymmetric beliefs (true or not) about different groups can influence hiring decisions or wages. Recent models of this kind combine dynamic job assignment models with human capital assumptions. They explain wage differentials as well as promotions [START_REF] Lazear | Male-female wage differentials in job ladders[END_REF]; [START_REF] Barron | Gender differences in training, capital, and wages[END_REF]] and their long term persistency [START_REF] Coate | Will affirmative-action policies eliminate negative stereotypes ?[END_REF]; [START_REF] Moro | General equilibrium model of statistical discrimination[END_REF]; [START_REF] Moro | The effect of statistical discrimination on black-white wage differentials: Estimating a model with multiple equilibrium[END_REF]; [START_REF] Fryer | Belief flipping in a dynamic model of statistical discrimination[END_REF]].

The second path followed in the field of statistical discrimination originates directly from the pioneer work of [START_REF] Phelps | The statistical theory of racism and sexism[END_REF]. It is based on the notion that employers are unable to precisely know the productivity of each employee insofar as the signals available to them (recruitment tests, diplomas, etc.) are less reliable for women than for men. Recent work has thus extended Phelps's seminal contribution by introducing human capital investment decisions [START_REF] Lundberg | Private discrimination and social intervention in competitive labor market[END_REF]; [START_REF] Havet | Why do women's wages increase so slowly throughout their career? A dynamic model of statistical discrimination[END_REF]] or by accounting for job matching [START_REF] Rothschild | A model of employment outcomes illustrating the effect of the structure of information on the level and distribution of income[END_REF]; [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF]]. In both cases, it turns out wage gaps arise endogenously.

This paper focuses on the early career patterns of young male and female workers. In order to achieve a better understanding of the overall gender wage gap, it seems to be of crucial importance to understand the factors contributing to the gender differential in early-career wage growth. Indeed, we empirically observe that the gender wage gap is fairly small at the entry into the labour market, but after a few years a considerable gender wage gap emerges [START_REF] Loprest | Gender differences in wage growth and job mobility[END_REF]; [START_REF] Manning | The gender gap in early-career wage growth[END_REF]; [START_REF] Napari | Gender differences in early-career wage growth[END_REF]; Del [START_REF] Bono | Job mobility and the gender wage gap in Italy[END_REF]]. Consequently, investigating the determinants of gender wage gap in the early career could help understand the growing male-female wage gap over the course of the worker's career. Moreover, focusing on career starts is particularly interesting since some of the common explanations for gender wage gap are irrelevant at this stage. For example, male and female workers differ fairly little in terms of career interruptions and thus gender wage gap cannot be convincingly explained by gender differences in promotion opportunities (glass ceillings) given the short durations of jobs and the small experience of young workers. By contrast, models of job mobility, which point to the heterogeneity in the quality of employee-employer matches [START_REF] Burdett | A theory of employee job search and quit rates[END_REF]; [START_REF] Jovanovic | Job matching and the theory of turnover[END_REF]] could be relevant. There is evidence that mobility plays a less important role in terms of wage growth for young women: not only are young women less likely to quit a job, but they also seem to receive lower returns to mobility than young men [START_REF] Simpson | Starting even? job mobility and the wage gap between young single males and females[END_REF]; [START_REF] Light | Panel estimates of male and female job turnover behavior: can female nonquitters be identified?[END_REF]; [START_REF] Loprest | Gender differences in wage growth and job mobility[END_REF]].

Therefore, our paper seeks to illustrate potential links between statistical discrimination, mobility, tenure and wage profiles. We use the statistical discrimination model proposed by [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF] to explain racial wage gaps as our starting point. Thus a worker's productivity is assumed to depend on the quality of the job match. It is further assumed that it is more costly for an employer to assess female workers' productivity. Finally, the model allows productivity to become less noisy with tenure. As in [START_REF] Altonji | Employer learning and statistical discrimination[END_REF] and [START_REF] Altonji | Employer learning, statistical discrimination and occupational attainment[END_REF], we assume that employers learn about workers' characteristics and productivity over time. However, unlike [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF], we assume that female workers' productivity remains noisy with tenure, while male workers' noise/signal ratio is assumed to be zero in the second period of their two-period lives. [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF] conjectured that even if asymmetries and informational imperfections were only transient, they could nevertheless generate permanent wage differentials between racial groups. By assuming away perfect productivity revelation for women, our model does indeed show that gender wage gaps may appear within the first years of the working lives and may be permanent. Furthermore, the model provides a simple framework within which gender differences in terms of tenure, experience and mobility can be better understood.

The model generates a series of predictions that we test against U.S. data from the 1979 cohort of the National Longitudinal Survey of Youth. It turns out most theoretical predictions are supported by the data. The paper is organized as follows. Section 2 describes the structure of the model and its basic assumptions. Section 3 presents the wage profiles that characterize the equilibrium and emphasizes theoretical implications with respect to gender differences in mobility. Finally, Section 4 presents the empirical results and Section 5 concludes the paper.

The Structure of the Model

Our approach is based on the dynamic statistical discrimination model setup by [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF], which incorporates the notion of job-matching. In particular, like Oettinger, we assume that the productivity of any job match is imperfectly observed ex ante, that the initial productivity signal is noisier for the "minority" group (i.e. women in the present paper) and that additional information about the productivity of the match is revealed with tenure on the job. Our contribution is to extend Oettinger's framework by allowing for the possibility that match productivity is not revealed completely for women as job tenure accumulates and consequently female workers' productivity remains less reliable. It turns out that this simple generalization substantially expands the set of empirical predictions with respect to male-female differences in starting wages, wage levels for experienced workers and wage growth.

In our model, the agents are competitive firms who negotiate compensation with employees one-on-one and offer each a wage equal to his or her expected productivity, conditional upon all available information, and income-maximizing workers who take mobility decisions based on their expected wage schedules. Before we describe the agent's objective functions and determine the equilibrium, it is essential to precise the informational context in which firms make their wage decisions.

Employees are assumed to work for two periods (t = 1, 2) and maximize expected compensation over their working lives. At the beginning of each period t, a worker receives exactly one job offer. The true productivity of an employee in the job offered at period t, µ ti , is a random variable whose distribution is known and identical for men and women:

µ ti ∼ N μi , σ 2 µi .
Individuals' productivity depends on the quality of their job match. Moreover, for each individual, the true quality of the new job offers received in the two periods, µ 1i and µ 2i , are independent draws from the underlying match productivity distribution. This latter assumption, standard in [START_REF] Jovanovic | Job matching and the theory of turnover[END_REF] ensures that employees' history is irrelevant to the evaluation of his/her productivity in any newly formed match.

The quality of the match is vulnerable to informational imperfections on both sides of the market. Productivity is ex ante unknown in any potential, but untested, match. Before the match actually occurs, employers and employees alike do not know precisely what the exact productivity will be. It can only be ascertained by observing employee i in the job offered at t. More precisely, during the first period all workers will be employed in the job they were offered at the beginning of the period. Let the observed productivity of worker i be given by:

s j 1i = µ 1i + ε j 1i , where ε j 1i ∼ N 0, σ 2 ε j , j ∈ {f , m} , (1)
and where superscripts f and m stand for female and male workers, respectively. We assume that µ 1i and ε 1i are not correlated. At the start of the second period, the worker must decide whether to stay on the job or move to a new job. If the new offer is accepted, both parties will observe the productivity in the new match with error, as in the first job, i.e.

s j 2i = µ 2i + ε j 2i , where ε j 2i ∼ N 0, σ 2 ε j , j ∈ {f , m} . (2)
If the worker stays in the first-period job, his/her true productivity remains µ 1i , since our model assumes away investment in human capital. On the other hand, the two parties will better assess the true productivity, µ 1i . For stayers we may write observed second-period productivity as:

s j 1i = µ 1i + v j i , where v j i ∼ N 0, σ 2 v j and σ 2 v j < σ 2 ε j , j ∈ {f , m}. (3)
As in [START_REF] Phelps | The statistical theory of racism and sexism[END_REF], gender differences occur essentially through the quality of the productivity signal, i.e. σ 2 ε f > σ 2 ε m . We further assume that a gap remains irrespective of tenure. This assumption departs from [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF] who assumed that the noise/signal ratio vanished after the first period. In fact we assume this to be the case for men, but not for women, i.e.

0 = σ 2 v m < σ 2 v f .
The gender difference in the individual productivity uncertainty has often been justified on cultural grounds [START_REF] Cornell | Culture, information, and screening discrimination[END_REF]; [START_REF] Lazear | Culture and language[END_REF]]. For example, [START_REF] Henley | Body Politics : Power, Sex and Nonverbal Communication[END_REF] and [START_REF] Lang | A language theory of discrimination[END_REF] stress that men and women differ in their communication style and therefore the evaluation process is subject to greater "error" if the candidate is a woman, given that employers and managers are disproportionately men. In the same spirit, [START_REF] Montgomery | Social networks and labour market outcomes : Towards an economic analysis[END_REF] shows formally that information about the least represented group in the top-level jobs is usually of lower quality. Recently, [START_REF] Altonji | Employer learning, statistical discrimination and occupational attainment[END_REF] has argued that unemployment is likely to interact with early statistical discrimination to deter employers' learning. To the extent women typically suffer from higher unemployment then men, they may be less able to convey their true productivity. For all these reasons it seems legitimate to assume that gender differences in the measurement of individual productivity arises not only at the time of hire, but also during employment.

Formally we assume that firms are competitive and risk-neutral, negotiating employee compensation on an individual basis. Employers will offer wages equal to individual expected productivity due to the binding zero expected profits in both periods. In the first period, wage profiles can thus be written as:

w j 1i = E µ 1i s j 1i , j ∈ {f , m}. (4)
Likewise, second-period wage contracts are determined by individuals' productivity signal, i.e. s j 1i if individual i remains on the job, and s j 2i otherwise. We thus have w j 2i = E(µ 1i |s j 1i ) for "stayers" and w j 2i = E(µ 2i |s j 2i ) for "movers". A worker will choose to change jobs if, and only if, the expected wage in the second-period job offer exceeds the expected wage in the current job, that is if μj

2i ≡ E µ 2i s j 2i > μj 1i ≡ E µ 1i s j 1i .
Wages in the second period can thus be written

w j 2i = μj 1i , if μj 1i = E µ 1i s j 1i ≥ μj 2i = E µ 2i s j 2i (stayer), μj 2i , if μj 1i < μj 2i (mover).
(5)

Note that the productivity of a male stayer will be perfectly observed. His compensation will be w m 2i = μm 1i = µ 1i and the condition to remain on his original job is μm 2i ≤ μ1i . Naturally these conditions do not apply to female workers. We will now examine the equilibrium solution and analyze how gender wage gaps may arise.

Equilibrium Wage Profiles

Equilibrium is determined by the optimization behavior of employers and employees. Firms, which are in a competitive environment, maximize profits by proposing wages, which reflect expected productivity conditional on the individual signal and the group membership. Workers take mobility decisions that maximize their expected lifetime earnings. We will characterize wage profiles in the two periods before drawing conclusions about the returns of mobility, tenure, and experience.

First-Period Wages

For the first period, our analytical framework is identical to the initial statistical discrimination model developed by [START_REF] Phelps | The statistical theory of racism and sexism[END_REF] and [START_REF] Aigner | Statistical theories of discrimination in labor market[END_REF]. We obtain the standard result according to which wage contracts are a weighted average of mean productivity (μ) and of the individual signal, s j 1i :

w j 1i = E(µ 1i |s j 1i ) = (1 -ρ 2 j )μ + ρ 2 j s j 1i , j ∈ {f , m}, (6)
where

ρ 2 j = σ 2 µ / σ 2 µ + σ 2 ε j . 2
The weight ρ 2 j can be interpreted as a measure of the quality of the signal. Thus the greater the reliability of the signal, the more employers will individualize wage rates. Clearly, given the assumption that women's signals are less reliable,3 employers will discriminate -rationallybetween men and women by offering them different wages. When setting the starting wage of women, they will tend to emphasize the average characteristics of the group over individual performance in order to guard against possible measurement errors. Consequently, men and women with the same productivity signals, s 1 , will receive different compensations. Women with a strong initial signal will receive a lower pay than their male counterparts, and conversely for a weak productivity signal. The wage profile offered to women during the first period is thus less steep than that offered to men, and women's compensation is more clustered around 2 To show this, observe that µ 1i and s j 1i are normal bivariates with correlation coefficient ρ

2 j = σ 2 µ / `σ2 µ + σ 2 ε j
´. The result follows from computing the conditional expectation.

mean productivity, μ. Men's wages will in fact have a higher variance ρ 2 m σ 2 µ than women's ρ 2 f σ 2 µ . Yet, men and women will receive the same wage rate upon entry into the labor market. Indeed, expected pay in the first period is invariant with respect to the reliability of the signals

E w j 1i = μ, j ∈ {f , m} , ∀ρ 2 j . (7)
Thus first period mean wages are equal to mean productivity, which we assume identical across gender.

Second-Period Wages

Second period wage profiles depend on mobility behavior. As shown previously, stayers' wage rates are characterized by

w j 2i = μj 1i = E µ 1i s j 1i
and those of the movers by

w j 2i = μj 2i = E µ 2i s j 2i .
A worker continues with the same job only if μj 1i ≥ μj 2i and conversely changes jobs if μj 1i < μj 2i . Because mobility is endogenous, non-random selection between movers and stayers must be accounted for when characterizing mean compensation. Thus,

• the average second-period wage is given by:

E μj 1i μj 1i -μj 2i ≥ 0 = μ + δ 2 j δ 2 j + ρ 2 j 2σ 2 µ π 1/2 , j ∈ {f , m} (8)
for stayers.

• and

E μj 2i μj 2i -μj 1i > 0 = μ + ρ 2 j δ 2 j + ρ 2 j 2σ 2 µ π 1/2 , j ∈ {f , m} (9)
for movers, where δ 2 j = σ 2 µ /(σ 2 µ + σ 2 v j ). For stayers, δ 2 j is a measure of the quality of the signal similar to ρ 2 j in the first period.4 Moreover, since the productivity revelation mechanism is perfect for men (σ 2 v m = 0) but imperfect for women (σ 2 v f > 0), we have 1 = δ 2 m > δ 2 f . Note that the mean conditional wage rate in the second period is equal to the worker's mean productivity (μ), adjusted for the quality of the signal. The expected wage of a mover is lower than that of a stayer ρ 2 j < δ 2 j because σ 2 v < σ 2 ε .

Our model generates positive returns to work experience and tenure. At the beginning of the second period, a mover has one period of experience as an asset, but no tenure, whereas a stayer has both one period of experience and one period of tenure. Thus movers' mean wage differential between the first and the second period characterizes the average return to experience, while the average return to tenure is given by the second-period mean wage differential between stayers and movers. Average returns to experience and tenure are thus given respectively by:

ρ 2 j δ 2 j + ρ 2 j 2σ 2 µ π 1/2 and δ 2 j -ρ 2 j δ 2 j + ρ 2 j 2σ 2 µ π 1/2
.

The positive return to tenure captures the fact that the signal is less noisy in the second period.

The unconditional second-period mean wage of group j can be derived from equations ( 8) and ( 9):

E(w j 2i ) = Pr μj 1i ≥ μj 2i E μj 1i μj 1i -μj 2i ≥ 0 + Pr μj 2i > μj 1i E μj 2i μj 2i -μj 1i < 0 = μ + (δ 2 j + ρ 2 j )σ 2 µ 2π 1/2 , j ∈ {f , m} . (10)
Thus on average workers earn more in the second period because they self-select into the best possible match. Unlike first-period wage rate, second-period wages increase with the reliability of the signals, δ 2 j and ρ 2 j . The better they are, the more profitable the selection process is likely to be on average. Indeed, mistakes such as changing jobs that prove to be a worse match, or foregoing a job change that would have been profitable can be better avoided when productivity signals are more precise.

In this context, male workers should benefit more from mobility. In the second period they should on average receive higher wages than their female co-workers. Our model thus predicts that even if there is no gender wage gap at entry into the labor market, it will appear as careers unfold.

Wages and Mobility

We now consider between-period wage changes. The expected wage change for stayers is given by

E μj 1i -μj 1i μj 1i -μj 2i ≥ 0 while that of movers is given by E μj 2i -μj 1i μj 2i -μj 1i > 0 .
It can easily be shown that

E μj 1i -μj 1i μj 1i -μj 2i ≥ 0 = δ 2 j 1 -ρ 2 j δ 2 j + ρ 2 j 2σ 2 µ π 1/2 , ( 11
) E μj 2i -μj 1i μj 2i -μj 1i > 0 = ρ 2 j 1 + δ 2 j δ 2 j + ρ 2 j 2σ 2 µ π 1/2 . ( 12
)
From equations ( 11) and ( 12) it is clear that the expected wage change is positive for both stayers and movers. This result is not surprising since mobility is endogenous.

If σ 2 ε < σ 2
µ -a reasonable assumption-movers will clearly experience greater wage increases than stayers on average. Indeed wage changes for stayers solely reflect corrections to productivity measurement errors. Conversely, wage changes are essentially attributable to productivity changes in the case of movers.

In summary, our model yields many unambiguous theoretical predictions that can be empirically tested. For both sexes we find that:

1. wage profiles are increasing, on average; 2. experience and tenure show positive returns; 3. movers' mean wage is lower than that of stayers. But 4. their wage growth is greater (assuming that σ 2 ε < σ 2 µ ).

As for the male-female wage gap, several results emerge:

1. for identical productivity signals, employers offer compensations that differ across gender; 2. upon entry into the labor market, men and women earn the same wage on average; 3. however, a gender wage gap emerges in the initial years of their working lives. Some of these predictions are similar to those derived by [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF]. In fact, the equilibrium described by [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF] is a special case of our model in which δ 2 j = 1, ∀j. However, this assumption is not innocuous since the productivity revelation mechanism plays an important role in the determination of the second period wage rate. Moreover, our generalization complicates the analysis with respect to differences in the return to mobility and tenure, and changes a number of conclusions. For instance, unlike [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF], we cannot assert that women should always have higher returns to tenure than men because the reliability of the initial signals ρ 2 j and the precision of the revelation mechanism δ 2 j act in opposite directions. Likewise, the impact of δ 2 j on movers' mean wage increase is ambiguous.

Male-Female Gap in the Return to Mobility

The analysis of gender differences in terms of return to job mobility and tenure is slightly more complex. However, we will show that the sign of these differences not only depends on the male-female gap in the reliability of the initial signals, but also on the magnitude of the variances of the shocks σ 2 ε f , σ 2 v f relative to the variance of the productivity σ 2 µ .

To ease the derivation of the results let k ∈ ]0, 1] be such that

σ 2 ε m = kσ 2 ε f , α = σ 2 ε f σ 2 µ
, and

β = σ 2 v f σ 2 µ
. We can rewrite the conditions pertaining to the gender differences in job mobility and tenure in terms of k, α and β. Thus for the average wage of male stayers to be higher to that of female stayers, it is necessary and sufficient according to equation (8) that:

5 1 1 + ρ 2 m ≥ δ 2 f δ 2 f + ρ 2 f , or equivalently that k ≥ k A = α -β (3 + α + β) α[1 + β (3 + α + β)] . (13)
By the same reasoning, we can derive the following predictions based on equations ( 8),( 9),( 11) and ( 12):

6 k B = -3(1 + β) + p 9(1 + β) 2 + 4(1 + α)(1 + β)(2 + α + β) -8(1 + β) 2 2α(1 + β) , k C = 3α + p 9α 2 + 8[(1 + α)(1 + β)(2 + α + β) -α 2 ] 2[(1 + α)(1 + β)(2 + α + β) -α 2 ] , k E = (α -β)[3α -3β + p 9(α -β) 2 + 8[(1 + α)(1 + β)(2 + α + β) -(α -β) 2 ] 2α[(1 + α)(1 + β)(2 + α + β) -(α -β) 2 ]
.

The derivation of k B , k C and k E is available from the authors.

4. The condition for the male movers' wage growth to exceed that of female movers always obtains; 5. Men's return to tenure will be higher than women's if k E ≤ k ≤ 1.

Ranking these various threshold values of k would allow us to characterize a limited number of baseline cases. The complexity of k A , k B , k C , and k E is such that we must turn to numerical simulation. However, if we make the reasonable assumption that the residual variances (σ 2 ε f , σ 2 v f ) are much smaller than the variance of productivity (σ 2 µ ), then α will be comprised in the interval [0, 1], and β in [0, α] due to the manner in which productivity gets less noisy with job tenure7 . It can be shown that

(k A -k E ), (k E -k C ), (k E -k B )
are always negative irrespective of α and β, while (k B -k C ) can be both positive or negative. Consequently only six baseline cases need be examined. Our model's predictions are summarized in Table 1. Contrary to [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF], our results depend on the discrepancy in the reliability of men's and women's signals. In [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF], productivity revelation is perfect, i.e. σ 2 v f = 0, implying β = 0. This assumption has important repercussions for the threshold values. In fact, for β = 0 we find that k A = k B = k C = k E = 1. Consequently, whatever the value of k ∈]0, 1], we find that k ≤ k A . Furthermore, we can show that the predictions in [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF] correspond to the first column of Table 1. Recall that his model yielded a positive gap in men's returns to mobility, and that tenure was more highly valued by women.

≤ k ≤ k A k A ≤ k ≤ k E k E ≤ k ≤ k B k B ≤ k ≤ k C k C ≤ k ≤ k B k C ≤ k ≤ 1
An empirical study based on wage equations will allow us to distinguish between the differences in the returns of job mobility and tenure for men and women. We can then establish whether the data are consistent with any of our theoretical predictions.

Data and Empirical Analysis

The Sample

In this section, we test the unambiguous predictions of our statistical discrimination model and focus on the theoretical ambiguities surrounding differences in the returns to job mobility and tenure. In particular, the analysis will attempt to shed some light on how the wage gap evolves in the earliest stages of work and on the importance of job mobility in the process.

To test the theory, we use data drawn from the 1979 cohort of the National Longitudinal Survey of Youth (NLSY). This survey follows a cohort representative of young Americans aged 14 to 21 in 1979. It provides extensive longitudinal data on educational achievements, wages, and work experience from 1979 to 2008. In this study we focus on white individuals from the cross-sectional sample of the NLSY and exclude the supplemental sample of economically disadvantaged whites.

Our aim is to follow the evolution of wages at the beginning of young adults' career. Since our model does not include labor supply, we follow [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF] by focusing on an individual's first observable transition from weak to strong labor force attachment. Respondents are considered to be strongly attached to the labor force during a specific survey year if they worked at least half of all weeks since the last survey, 8 and if they worked at least an average of 30 hours during each of those weeks. The onset of a respondent's career is the first of three consecutive survey years of strong labor force attachment that follow one year of weak labor force attachment. 9 We then follow each respondent's wage evolution for up to ten years following that first transition into strong labor force attachment, keeping in our sample only wage observations preceded by a year of strong labor force attachment. 10 Consistent with our requirement of measuring wages every year, we do not consider NLSY data collected after 1994 since surveys were then conducted every two years.

Wage is measured at the end of each year of strong labor force attachment. 11 We use hourly 8 The number of weeks between each survey is not necessarily 52 weeks, so we restrict our sample to respondents-survey years with 45 to 59 weeks since the last survey.

9 Respondents who are observed to have strong labor force attachment in survey year 1979 are excluded from the sample.

10 We also experimented with a sample that excluded individuals who experienced one year of weak labor force attachment after their first three years of strong attachment. This results in a much smaller sample, larger standard errors, but qualitatively similar results.

11 Ideally we would have identified the respondent's "first job" and the wage associated with that job.

wage for a job held at the time of the survey. If a respondent holds more than one job at the time of the survey, we use the hourly wage for the job at which the respondent worked most hours. Wages are measured in 2004 dollars and included in the sample if they are greater than $1 and less than $60.

Educational achievements is measured using years of completed schooling. We focus on individuals who have completed at least 13 years of education at the end of their career's first year. The rationale is that our model assumes male and female workers to have the same level of commitment to the labor market. This assumption is more likely to hold in a sample of individuals who have invested in post-secondary education. The sample of individuals with no post-secondary education may display greater variation in their attachment to the labor force, and the mechanisms behind this variation likely differ between men and women given that women's educational and labor supply choices are affected by their fertility decisions.

Although fertility decisions also affect female workers with post-secondary decisions, it can be argued that higher household earnings and stronger investments in human capital of more educated couples reduce the scope for fertility decisions affecting female labor supply.

The model has strong predictions about the impact of work experience, job tenure, and job changes on wages. Work experience is a variable that counts the number of year of strong labor force attachment since the first transition into strong attachment. Job tenure measures the number of consecutive weeks at a given job. To identify a job change, we compare the week when the currently held job started, to the week when the previous survey interview took place. If the currently held job started after the previous survey, a respondent is considered to have changed jobs.

Our model makes it clear that job changes are triggered if workers receive large shocks to their productivity signals, or if they get good productivity draws with another employer. These productivity draws and signal shocks are observed by the worker, but not by the econometrician, and are incorporated in the error term of our wage equation. This error term is therefore correlated with any explanatory variable describing job changes or job tenure. These variables are instrumented using lagged job changes, lagged job tenure, and lagged annual hours of However, the NLSY provides no question or variable referring to an individual's first job. Many respondents hold jobs while enrolled in school, but these may not be jobs associated with their eventual careers. Other respondents may be out of school and hold jobs, but are not strongly attached to the labor market. This makes it hard to identify the first job associated with their eventual career and we opted instead to look at hourly wage for jobs held at the end of each year of strong labor force attachment.

work. We used lagged values because our model assumes that productivity draws and signal shocks are not correlated across periods. The first two instruments should be correlated with current job changes and tenure through other factors not correlated with the worker's own wage, such as shocks to household structure, or a stronger preference for job mobility. Lagged hours of work should identify job changes triggered by adjustment on the labor supply margin, rather than wages.

To proxy ability, our regressions include the score percentile of the Armed Force Qualification Test (AFQT).12 This variable is meant to capture the fact that employers may partly observe the worker's ability and adjust compensation accordingly. We also include a dummy variable that controls for whether the job is covered by a collective bargaining agreement, acknowledging the fact that other external mechanisms may determine wages. In the same vein we include dummy variables for occupation and industry to reflect how work in these different professions and firms are rewarded in the product market.13

Our sample consists of 505 women, and 474 men with at least some college whom we observe a year following their first transition into strong labor force attachment. Table 2 provides summary statistics on some of the variables used in the empirical analysis. It shows the distribution of the year during which workers made their transition into strong labor force attachment. For 90% of the sample, this first transition took place in years 1980 to 1988. On average these workers made their first transition into the labor force at the age of 23, which is consistent with the fact that they completed on average 15 years of education. Almost 57% of the sample had completed 4 years or more of college by the time of their first transition.

At the end of the first year of strong labor force attachment, women and men earn an average hourly wage rate of $12 and $13, respectively. Two years later, their respective average hourly wage rate has increased to $14 and $16. As many as 72% of men and 67% for women change job at the end of their first year of employment. But mobility decreases as their careers unfold, down to about 30% for men and women by the end of the third year. By that time, 14% of male and female workers are covered by a collective agreement.

Labor supply during that first year of strong labor force attachment is quite similar for men and women. Indeed, men worked on average 1,950 hours over 46 weeks, while women worked 1,800 hours over 45 weeks. Two years on, labor supply is slightly higher for workers of both genders.

After their first year of strong attachment, the majority of women are either professionals (36%) or clerical workers (30%). Two years on, both occupations are still the most prevalent at 43% and 26%, respectively. Early on, men are more likely to work as professionals (37%) or service workers (11%) and managers (11%). Two years on, they are mostly professionals (38%), managers (13%), and sales workers (11%).

The two most prevalent industries for women are professional and related services (39%) and in wholesale or retail trade (21%). Two years on, these two industries represent respectively 38% and 19% of all jobs. Male workers work mostly in wholesale and retail trade, manufacturing and professional services industry (20% each). Two years on, the professional service industry represents 19% while manufacturing's share is higher (24%).

Estimation Results

We test our model's predictions using reduced-form log-wage regressions. Our sample includes all individuals that have completed at least three years of strong labor force attachment following one year of weak labor attachment. We follow their hourly wage over those first three years of strong labor force attachment, but we also add to our sample hourly wages observed after a year of strong labor force attachment over the next seven years. Our sample of individual-wage observations contains real wages (in 2004 dollars) from 1980 to 1994.

The Gender Wage Gap

The main results concerning the gender wage gap are presented in Table 3. All results presented in Table 3 are OLS estimates. The first column contains results for a model where we only control for gender, age, years of education, AFQT, and a dummy variable for collective bargaining. 14 This column shows that female wages are about 8.3% lower than that of men, and that this gender wage gap is statistically significant. Since there is no control for work experience, age is associated with strong wage growth. As expected, years of education, AFQT and collective bargaining are associated with higher wages.

Models 2 to 4 gradually add more variables to the model and allows us to study how that affects the gender wage gap. In Model 2 we add a variable for work experience, which is measured by years of strong labor force attachment. As expected years of experience are associated with strong wage growth: 4.2% for each year of strong labor force attachment.

Even after adding years of experience, we still find that women earn on average 8.3% less per hour of work.

In Model 3 we add an interaction term between work experience and the gender dummy variable. Remember that the model predicts that male and female workers have the same starting wage, and that the lower quality of productivity signaling for women implies that female wage should grow at a slower rate. Consistent with this, we find that the gender wage gap is reduced to less than 2% and is not statistically significant. Moreover, the parameter on the interaction between gender and work experience also reveals that wage growth is weaker for women: one year of strong labor force attachment leads to a 3.4% wage growth, compared to 4.9% for men, and this difference is statistically significant.15

Other mechanisms could generate weaker wage growth for women. If most women in our sample expect to have children and therefore make fewer on-the-job human capital investments, we should observe weaker wage growth for women. In Model 4, we include variables that control for marital status and the presence of children at home. The dummy variable for being married and its interaction with gender indicate that being married is associated with higher wages, but much less so for women. The parameter on a dummy variable for the presence of at least one child in the household indicates that having children is associated with lower wages. 16 Including these household related variables show that the gap in wage growth from strong labor force attachment decreases, but remains statistically significant at 1%, compared to 1.5% in Model 3. It suggests that part of the weaker female wage growth may be explained by different on-the-job human capital investments.

Job Changes and Job Tenure

Our model also has predictions regarding the comparative wages of workers who keep the same job (stayers) and those who change job (movers). We test these predictions in Table 4. The first column reports the results of Model 3 from Table 3 where it was found that controlling for years of experience resulted in a statistically insignificant gender wage gap. Models 5 to 7 are 2SLS estimates. As explained in Section 4.1, job changes, and by construction job tenure, are endogenous variables. We instrument them with their own lagged values, as well as lagged hours of work.

Model 5 of Table 4 adds a dummy variable indicating whether the individual changed job since the last survey interview. We also include an interaction between gender and job change. Our results show that movers have on average lower wages, as predicted by our model (see Section 3.3). Moreover, the mover wage penalty is greater for women, indicating that the mean wage of movers is greater for men. Note also that the coefficient on gender is now positive and statistically significant, indicating that, among stayers, women have a higher mean wage. Stronger mean wage for women among stayers, and stronger mean wage for men among movers is consistent with Case 1 of Table 1.

The coefficients associated with job changes in Model 5 of Table 4 are quite large. It is very likely that these job changes are not all voluntary. Individuals who, in the high unemployment rates of the early 1980s, suffered from layoffs, but remained in strong labor force attachment (a requirement of our sample selection), may have done so by taking on a job (or multiple jobs) that offered lower wages. Our results suggest that this might have been more severe for women. Compounding all of this is the fact that we use lagged job change variables as instruments. These may tend to identify the negative impact of job losses over multiple years on the wages of workers who nevertheless remain in strong labor force attachment. However, it should be kept in mind that these are log hourly wage regressions, so that measured in actual dollars, the impact of job changes represent small amounts. 17

found both the child dummy variable and its interaction with gender to be negative, but not statistically significant. 17 For example, the intercept of Model 5 in Table 4 implies a base hourly wage of $4.58, relative to an Model 6 includes a linear and quadratic term in job tenure weeks, as well as interactions between these terms and gender. 18 Tenure coefficients indicate that wages grow with tenure, as predicted by our model. Moreover, this growth is significantly stronger for women, which is also consistent with Case 1 of Table 1 which predicts larger returns to tenure for women. These coefficients on job tenure imply that 52 weeks with the same employer lead to 4.1% wage growth for men, compared to 7.5% for women.

Model 6 in Table 4 also suggests that women suffer from a marginally statistically significant wage penalty of approximately 5%. This is probably due to the fact that workers with very low tenure include those who changed jobs. These movers, and especially female movers, suffer from severe wage penalties as shown in Model 5, and the assumed quadratic impact of job tenure is not flexible enough to allow for this nonlinear impact of tenure. The gender dummy variable may therefore be picking up the larger penalty of job changes for women. With Model 7 in Table 4 we study the joint impact of job changes and tenure. We still find that workers suffer from changing jobs, although the effect is not statistically significant for men (this result remains consistent with Case 1 of Table 1). Tenure is still associated with statistically significant wage growth, but we find no stronger returns to tenure for women, contrary to Model 6. 19 Finally, we find that female stayers have higher average wage, which is consistent with Model 5 and Case 1 of Table 1.

Wage Growth and Job Changes

A clear prediction of the model from Section 3.3 is that although movers' mean wage is lower than that of stayers, their wage growth is stronger. To test this prediction, as well as predictions about the mean wage gain across genders of stayers and movers (last two rows of Table 1), we study wage growth in Table 5. The dependent variable is the difference between the current log hourly wage and the log wage recorded at the previous survey. All regressions presented in Table 5 control for collective bargaining coverage, occupation and industry.20 All average wage of $16.68 in that regression sample. The coefficients for job change variables imply that the base hourly wage of movers is around $3.52 for men and $1.78 for women.

regressions also control for age, years of schooling, ability as measured by the AFQT score percentile, as well as a dummy variable for gender. Model 8 shows that, consistent with our previous findings in Table 3, women have weaker wage growth, although the coefficient is not precisely estimated. However, once we control for hours of work in Model 9, the impact of gender is marginally statistically significant and in favor of men. Hours of work can be seen as controls for the amount of on-the-job human capital investment, and are associated with statistically significant stronger wage growth. Comparing the gender gap in wage growth from Model 8 to 9 indicates that women can make up for their noisier productivity signals through on-the-job investments.

One clear prediction of Section 3.3 is that movers should experience stronger wage growth. Model 9 tests this prediction by adding a dummy variable indicating whether the worker holds a job he was not holding at the previous interview. As previously explained, this variable is endogenous. Because we are now estimating wage growth equations, where lagged hourly wage are part of the dependent variable, we instrument job change variables with second order lags of job changes and hours worked. The coefficient on job change is positive and statistically significant, which is consistent with the prediction of our model that movers should have stronger wage growth relative to stayers. Model 11 includes an interaction between job change and gender to test among movers and stayers which gender has stronger mean wage gains. The results are inconclusive as the parameter estimate is not statistically different from zero.

Implications for Signal Quality and Productivity Revelation Across Genders

Our empirical results support Case 1 of Table 1. This has implications for the comparison across genders of signal quality in a new job, and the productivity revelation for workers who do not change job. These matters because they are the source of wage growth and job mobility in our model.

Assuming that productivity revelation is perfect for men, Case 1 entails that signal quality in a new job is good for men. For women signal quality is bad but productivity revelation is good. 21 This implies that only women with very low productivity or bad post-hiring noise do not keep leaving those who remained uncovered as the base case.

21 Remember from section 3.4 that Case 1 implies k < k A , where k decreases as the signal quality increases for men, while k A increases as the signal quality for women weakens, and as the revelation mechanism for women improves. their current job. This tends to remove the extreme left part of the signal distribution among female stayers, increasing their observed average wage. The average wage of male stayers also increases but to a smaller extent because employers do not revise their expectations about their productivity so much as they do for women. Contrary to women, the majority of male movers are not drawn only from extreme left tail of the signal distribution because of their lower noise/signal ratio. It follows that male workers move only if they get a much better draw (productivity and noise wise) at a new job, implying that among movers, men have the largest average wage, whereas women have a higher average wage then men among stayers. Estimates from Model 5 of Table 4 lend support to these conjectures: women stayers have higher wage (coefficient on dummy for gender is positive and statistically significant) while male movers earn on average more than female stayers (the interaction between gender and job change dummy variables is negative and statistically significant).

Wage gains from staying at the same job depends solely on the improvement in signalling stemming from holding on to a job. The conditions that lead to Case 1 implies that this improvement is stronger for women, with the results that returns to tenure are stronger for women as found in Model 6 of Table 4. This also implies that wage growth among stayers should be stronger for women, but Models 10 and 11 in Table 5 are inconclusive on this issue. The two models are also inconclusive with regards to the predicted stronger movers wage growth among men. Wage gains for movers depend on the quality of signalling at new jobs but also on the revelation mechanism, which are both stronger for men.

Conclusion

In this paper we investigate the gender wage gap using a two-period model based on the theories of matching and statistical discrimination. Simply by assuming that women's true productivity is more costly to measure, and that the noisiness of women's signal tapers off less rapidly than men's, it is possible to generate a series of theoretical predictions about wage gaps. These pertain to the relation between wage gaps and mobility, tenure and experience. To our knowledge, only three other papers [START_REF] Oettinger | Statistical discrimination and the early career evolution of the black-white wage gap[END_REF]; [START_REF] Neumark | Wage differentials by race and sex : The roles of taste discrimination and labor market information[END_REF]; [START_REF] Altonji | Employer learning and statistical discrimination[END_REF]] have empirically tested the validity of the theory of statistical discrimination within a similar framework.

The theoretical predictions are tested using U.S. data from the 1979 cohort of the National Longitudinal Survey of Youth. The data supports most of these predictions. We find that, though men and women with post-secondary education earn identical wages upon entry into the labor market, a substantial gap emerges in men's favor in the next few years. Other clear predictions of the model are supported: work experience and tenure are associated with higher wages. Average wages are higher among workers who do not change job (stayers), while job changers (movers) experience stronger wage growth.

The theoretical model has ambiguous predictions about gender-related mean wage among stayers, mean wage among movers, returns to tenure, mean wage gains among stayers, and mean wage gains of movers. However, the model provides enough structure to limit its predictions to six possible cases, which we present in Table 1. Although our results are inconclusive with regards to mean wages gains, we find evidence that supports Case 1 whereby women have stronger mean wages among stayers and stronger returns to tenure, while men have stronger mean wages among movers. Case 1 implies that employers are better at inferring the productivity of new male employees, while finding it much more difficult to do so with female employees. However, Case 1 also implies that employers eventually get a precise estimate of female workers' productivity with tenure.

Table 1 :

 1 Male-female differences in the return to job mobility and tenure

	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
	0

For a literature review, see[START_REF] Cain | The economic analysis of labor market discrimination : A survey[END_REF] and, more recently,[START_REF] Altonji | Race and gender in the labor market[END_REF] and[START_REF] Havet | Ecarts salariaux et disparités professionnelles entre sexes : développements théoriques et validité empirique[END_REF] and for statistical discrimination, see[START_REF] Fang | Theories of statistical discrimination and affirmative action: A survey[END_REF].

The assumptionσ 2 ε m < σ 2 ε f implies ρ 2 f < ρ 2 m .

Their wage profile isw j 2i = (1 -δ 2 j )μ + δ 2 j s j 1i

Recall that Oettinger (1996)'s model implicitly assumes that: σ 2 v f = 0 and thus that β = 0 in our framework. His model thus boils down to assuming that k = 1, i.e. that women's productivity signal is no more noisy than men's signal. Consequently his model precludes male stayers to have a higher mean average wage.

The expressions for k B , k C , and k E are respectively:

σ 2 ε f ≥ σ 2 v implies that β ≤ α.

The AFQT score aggregates results for tests on word knowledge, paragraph comprehension, mathematics knowledge, and arithmetic reasoning. The AFQT is a subset of the Armed Services Vocational Aptitude Battery administered to NLSY respondents in 1980. At that time, age varied from 15 to 22 among respondents, it is known that performance on the AFQT test is positively correlated with age at time of the test. We therefore use AFQT test scores adjusted for age at the time of the test.

Previous research has shown that occupations and industries are coded with error in the NLSY. We ignore any industry change that is not associated with a job change. Occupation is allowed to change without any job change. However, we ignore any occupation "cycling" within the same job: if an occupation change is observed within the same job, it is ignored if the occupation at this same job is observed to have changed again at the next survey interview.

Although we do not present the results here, we also include in all models a set of dummy variables to account for occupation and industry.

These results are consistent with[START_REF] Loprest | Gender differences in wage growth and job mobility[END_REF] who uses the NLSY79 to find that women have weaker wage growth than men. Although she also considers a sample of workers with consecutive years of strong labor force attachment, her sample contains workers who did not go to college. If we consider workers of all educational levels we still find that returns to experience are lower for women, however this difference is not statistically different once marital and parental status are accounted for, variables that are not considered in[START_REF] Loprest | Gender differences in wage growth and job mobility[END_REF].

We also tried adding an interaction term between gender and the child dummy variable to Model 4. We

The quadratic term in tenure is divided by 10,000 for the sake of presentation. This should be kept in mind when interpreting the value of its coefficient.

We also experimented with linear as well as quadratic terms in tenure (as in Model 6) but all were imprecisely estimated. We therefore opted for the more parsimonious Model 7 with no quadratic term.

We include dummy variables for all possible state transitions. For example, collective bargaining includes a dummy variable for individuals who stayed covered by a collective agreement, a dummy variable for individuals who lost their coverage, and a dummy variable for individuals who gained collective bargaining agreement,

white men and women who are observed transitioning from one year of weak labor force attachment (LFA) to three consecutive years of strong LFA, and who had completed at least one year of college by the end of their first year of strong LFA. The definition of strong LFA is provided in section 4.1. The entry year is the survey year during which the individual is observed having completed their first year of strong LFA.