Frédéric Gavrel
email: frederic.gavrel@unicaen.fr

Participation, Recruitment Selection, and the Minimum Wage

Keywords: Search and matching, participation, heterogeneous workers, applicant ranking, efficiency, minimum wage. 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 state that participation is insufficient whatever workers' bargaining strengths. The reason for this is that, when holding a job, the marginal participant should receive the entire output.

As a consequence, introducing a (small) minimum wage raises participation, job creation, and employment. Therefore the aggregate income of the economy is enhanced.

Introduction

In this paper we study the efficiency of participation in a search-matching model with heterogeneous workers and recruitment selection. We also describe the effects of the minimum wage in this environment.

Labor theory generally accounts for participation choices by assuming that individuals give different values to inactivity. The usual search and matching model follows the same line [START_REF] Pissarides | Equilibrium Unemployment Theory[END_REF]. Individuals are homogenous when holding a job (they have the same productivity) whereas they are heterogeneous when inactive. According to this traditional view of participation, inactive individuals are "lazy". Recently [START_REF] Albrecht | Efficiency in a search and matching model with endogenous participation[END_REF] have introduced another approach to participation. Workers are assumed to be vertically differentiated. Since there are search frictions in the labor market only highly-skilled workers choose to join it. By contrast, low-skilled workers choose inactivity; the wages they expect from holding a job are not high enough to compensate for the risk of unemployment. According to this alternative view, individuals are inactive because they are insufficiently skilled.

Building on Albrecht et al., Gavrel (2011) finds that too many individuals participate in the labor market according to the social surplus criterion. The intuition behind this finding is that when deciding to join the market, lower-skilled workers are not aware of the (negative) impact of their choices on the average output of matches in the market. This result is obtained under what we consider to be a very strong assumption.

Recruitment is random; hence all workers face the same probability of finding a job, independently of their productivity. We instead presume that firms are selective.

They prefer good workers to bad ones. They then rank their applicants and hire the best. Under this assumption highly-skilled job-seekers have a greater chance of success than low-skilled ones. Hence the model that we develop does not contradict the observed facts. But this is not our main point. What is really of importance is the influence this has on conclusions regarding the efficiency of participation. In this respect our findings are very different. We state that when firms are selective, too few individuals participate in the labor market. The reason for this is that, contrary to [START_REF] Gavrel | On the efficiency of participation with vertically differentiated workers[END_REF], with recruitment selection the probability of a firm hiring a highskilled worker is no longer affected by the entry of lower-skilled ones. Consequently, any individual whose expected output compensates for her inactivity income should enter the labor market. As part of the output goes to firms, the private return to participation is unavoidably lower than the social one.

With respect to the efficiency of job creation, we state that firms tend to create too many vacancies. Similar to [START_REF] Gavrel | On the inefficiency of matching models of unemployment with heterogeneous workers and jobs when firms rank their applicants[END_REF], firms do not internalize the fact that an increase in labor demand lowers the expected productivity of their best applicants.

What could be done to improve the efficiency of the labor market? From a theoretical point a view a range of measures can be considered. We here show that the introduction of a mandatory minimum wage is an effective instrument in rendering the labor market more attractive. Intuitively, a minimum wage (provided it is not set at too high a level) increases labor supply and the unemployment rate. More surprisingly, labor demand increases. We explain how recruitment selection generates this unexpected result. Therefore, despite the unemployment rate increase, the aggregate income necessarily rises. Subsidies to participants also enhance the aggregate income.

Following Hosios and Pissarides, most papers dealing with the efficiency of the labor market retain the criterion of the social surplus (also referred to as the aggregate income). In retaining this criterion one implicitly assumes that incomes can be redistributed at no cost. Otherwise, an increase in the aggregate income is not necessarily Pareto-improving. As a consequence, different authors prefer to focus on the value of being unemployed. In our setting with heterogeneous workers, introducing a minimum wage appears to raise the expected income of all participants. This also results from recruitment selection. In the relevant literature we can distinguish between two subsets. First, some papers build on Pissarides (2000, Chap.8). 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. 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 1 . 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 improve aggregate income by acting as a substitute for an increase in workers' bargaining strengths. Although this empirical point is controversial, 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, according to the social surplus criterion, 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. In a well-known paper [START_REF] Blanchard | Ranking, unemployment duration, and wages[END_REF] account for the persistence of unemployment by assuming that firms rank their applicants according to their unemployment spells.

Along the same line of research, [START_REF] Moscarini | Unobserved heterogeneity and unemployment duration: a fallacy of composition[END_REF] shows that ranking of job applicants by ability gives rise to spurious duration dependence of unemployment when workers' heterogeneity is unobserved. 2 Moen (1999) states that recruitment selection is likely to create an over-education effect, while [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] argues that, where applicants are ranked, the technical skill bias can be seen as a consequence of a rise in unemployment. As mentioned above, using the circular matching model, [START_REF] Gavrel | On the inefficiency of matching models of unemployment with heterogeneous workers and jobs when firms rank their applicants[END_REF] shows that selection makes job creation excessive.

To summarize, our contribution is twofold. First, relative to [START_REF] Gavrel | On the efficiency of participation with vertically differentiated workers[END_REF] 3 , which is close to [START_REF] Albrecht | Efficiency in a search and matching model with endogenous participation[END_REF] 4 , we show that participation is unavoidably insufficient when firms rank their applicants. Second, relative to the usual search-matching model 1 In the absence of recruitment selection this condition is quite robust. See for instance Zenou (2009) in a spatial context.

2 Mocarini proposes a more general matching function with recruitment selection. For expositional simplicity, we restrict the analysis to urn-ball matching with applicant ranking. However, our results extend to Moscarini's specification.

3 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.

4 Contrary to [START_REF] Gavrel | On the efficiency of participation with vertically differentiated workers[END_REF], [START_REF] Albrecht | Efficiency in a search and matching model with endogenous participation[END_REF] study the efficiency of job creation when participation is set by individuals' decisions.

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 state that the effects of the introduction of a minimum wage are quite different with heterogeneous workers and recruitment selection. The paper is organized as follows. Section 2 presents our setting. Next we define a laissez-faire equilibrium and study its welfare properties (section 3). The results concerning the introduction of a minimum wage are stated in section 4 while section 5 provides an analysis of the impact of subsidies to participants. Section 6 concludes.

In Appendix F, we check that results extend to a dynamic setting. Appendix G considers posted wages à la [START_REF] Moen | Competitive search equilibrium[END_REF]. Unsurprisingly, job creation becomes efficient under competitive search. Conversely, participation remains insufficient. The reason is that firms cannot influence individuals' participation choices. We then surmise that this inefficiency is likely to hold in any wage-posting model.

Endogenous participation with applicant ranking 2.1 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 rate 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 migration-equilibrium equation

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 rate, q(., y), for a firm to hire a worker of productivity y7 . One can state the following preliminary result Lemma 1. With recruitment selection, the probability rate 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 rate q((v/n), y) is quite intuitive. As the second term,

f (y)n
v , is the probability rate 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 probabilities q(., y), we also deduce the probabilities of finding a job. It follows that

p(., y) = p((v/n), y) = exp - [1 -F (y)]n v (5) Integrating [p((v/n), 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)
3 Equilibrium and efficiency

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

Job creation and equilibrium

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.

Efficiency

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 of participation

Let us first study the 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 rate, q(., y) for a firm to hire a worker whose productivity, y, is greater than the cutoff y * , is not affected by a decrease in the cutoff.8 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.

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].

Partial efficiency of job creation

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 state the following proposition:

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). Differ- entiating 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 values).

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

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

Participation

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?[END_REF]. This amounts to examining the effect of an increase in the cutoff on the 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 9

Σ cp = 1 y * [p((v/n), y)β(y -z) -γ]f (y)dy ( 10
)
9 See Appendix B for detailed calculus.

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 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 stated the result below.

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

Job creation

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

As ∂Σ ∂y * < 0 (Proposition 1a), it follows that the first term of the derivative is negative whereas the second term is positive. 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 (constrained) over-optimal under the Hosios rule. The reason for this is twofold. First, holding the participation cutoff as a 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.

Introducing a minimum wage

Laissez-faire is inefficient. 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 wage10 . 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 state 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 (expected11 ) productivities, introducing a minimum wage tends to lower the (expected) profit of all matches,12 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.

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). We have shown that 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 state 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.

As mentioned in the introduction, 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 for by the minimum wage.

5 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 considered. We here study the implications of a lump-sum subsidy to participants paid for by a neutral tax13 . 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.14

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 condition for migration equilibrium is now written as This basic income clearly acts as a subsidy to job search, tending then to make participation more attractive. In Appendix E1, 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] Petrongolo | Looking into the black box: A survey of the matching function[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 condition for migration equilibrium becomes

z = -γ+p(., y * )w(y * )+[1-p(., y * )](z+b) = -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) as a constant, unemployment compensation also tends to lower the participation trigger y * . More workers enter the labor market because they receive the income b if they are not successful in finding a job. In Appendix E2, 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 state 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 E2.

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.

Final comments

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 the aggregate income.

In order to state 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 Portugal and Cardoso (2006) 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. Another line for further research.

Cambridge MA. 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

[ n v (1 -F (y) -1]Q -(Q -1)(1 -F (y * ) n v
The previous expression is a decreasing function of y.

As

< (>)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 ∂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 par-3.3.1), this shows that ∂Σ cp ∂y * < 0 and proves Proposition 2.

Appendix C: Proof of Proposition 3

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

J ≡ J((v/n), ŷ, y * , ŵ) = ŷ y * q((v/n), y)(y -ŵ)dy + (1 -β) 1 ŷ q((v/n), y)(y -z)dy
In the neighborhood of the laissez-faire equilibrium (Definition 2), the result is obtained

∂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) It results that ∂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 states that introducing a minimum raises the (average) unemployment rate of participating workers.

The employment rate, [1 -F (y * )]P (.) can be written as

[1 -F (y * )]P (.) = Q(.) v n
We know that introducing a minimum wage increases v and lowers θ. Consequently, Q(.) rises. This shows that the employment rate is enhanced, completing then the proof of Proposition 3.

p((v/n), y * )(y * -z -b) = γ -b β
Substitution into the job creation equation yields

-c + (1 -β)[1 -z -b - γ -b β - 1 y * p((v/n), y)dy] = 0 Or -c + (1 -β)[ β(1 -z) -γ β + 1 -β β b - 1 y * p((v/n), y)dy] = 0
Let us now state 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.

Efficiency

As the impact on job creation is indeterminate we are led to distinguish between the two cases. 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

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(.))z-(1-F (y * )γ
As the pair (v, y * ) satisfies ( 14), Σ is reduced to

Σ = (1 -β) v n Q(.)(b + z) + β v n Q(.)ȳ + (1 - v n Q(.))z -(1 -F (y * )γ Or Σ = (1 -β) v n Q(.)b + β 1 y * p((v/n), y)f (y)(y -z)dy + z -(1 -F (y * )γ
One 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 * )
From equation ( 15), we deduce that the previous derivative is close to zero in the neighborhood of laissez-faire. This proves that (small) benefits enhance the aggregate income.

jobs, s, does not depend on workers' productivities. For the sake of expositional simplicity, the welfare analysis is restricted to the case in which the time preference rate, r, is very close to zero. See Hosios (1990).

From the usual Bellman relations, we deduce the equilibrium equation for job creation -c + Notice that in this equation, the term S(y) = y-z+γ r+s+p(y) , which represents the surplus of a match with a worker of productivity y, should grow with y. Ranking workers by ability would not otherwise make sense. This consistency requirement imposes that 1 -βq(y)S(y) > 0 This existence condition is assumed to be satisfied henceforth. It comes from the fact that the ranking of applicants by ability is ordinal. See [START_REF] Moscarini | Unobserved heterogeneity and unemployment duration: a fallacy of composition[END_REF]. Combining the two latter equations yields

sg(z) + θq(z) = s + P (θ) 1 -F (y * ) f (z)
By integrating this equation on [y, 1], we obtain

s[1 -G(y)] + θ[1 -p(y)] = s + P (θ) 1 -F (y * ) [1 -F (y)] (18)
As ln p(y) = -1 -G(y) θ equation ( 18) can be rewritten as

p(y) + s ln p(y) = 1 - s + P (θ) θ 1 -F (y) 1 -F (y * ) (19) As θ = v/u and u = s(1 -F (y * ))n s + P (θ)
we also have

p(y) + s ln p(y) = 1 - s(1 -F (y))n v (20)
According to the above equation, the probabilities p(y) still grow with a job creation increase when taking into account the incidence of market tightness on the repartition function G(y). Likewise, they remain independent of the cutoff y * . Consequently their derivatives (with respect to y), q(y) = p(y) g(y) θ , are also independent of y * .

F.1.2 Employment shares and market tightness

In the following, we will need to know how the employment shares ρ(y) are affected by θ. To that end we make use of ( 19). We have

ρ(y) = q(y) Q(θ) = p(y)g(y) P (θ) From g(y) = 1 1 -F (y * ) s + P (θ) s + p(y) f (y)
we deduce

ρ(y) = 1 1 -F (y * ) p(y)(s + P (θ)) (s + p(y))P (θ) f (y)
Using F.1.1, we obtain that the derivative of ρ(y) with respect to θ has the same sign as

Γ = 1 p(y)(s + p(y)) ∂p(y) ∂θ - P ′ (θ) P (s + P ) Differentiating (19) gives ∂p(y) ∂θ = p(y) s + p(y) [ s θ 2 -Q ′ (θ)] 1 -F (y) 1 -F (y * ) Substitution into the expression Γ yields Γ = 1 -F (y) (s + p(y)) 2 [ s θ 2 -Q ′ (θ)] 1 1 -F (y * ) - P ′ (θ) P (s + P )
This shows that ∂ρ(y) ∂θ is a strictly decreasing function in y. As The proof is a straightforward extension of Appendix A. Notice that setting ϕ(y) = y

shows that an increase in market tightness still reduces the average output, meaning then that the externality which was highlighted in the static model (Proposition 1b) extends to the dynamic setting.

On the other hand, because a market tightness increase lowers S(y), setting ϕ(y) = S(y) shows that The last term of the derivative can be rewritten as

β 1 y * p(y) r + s + βp(y) ∂(q(y)S(y)) ∂θ = β 1 y * p(y) r + s + βp(y) q(y) ∂S(y) ∂θ dy +β 1 y * p(y) r + s + βp(y) S(y)Q ′ (θ)ρ(y)dy + β 1 y * Q p(y) r + s + βp(y) S(y) ∂ρ(y) ∂θ dy
As ∂S(y) ∂θ < 0, the first term of the right hand side (of the previous equation) is strictly negative. From Q ′ (θ) < 0, we deduce that the same holds for the second term. Consider the expression p(y) r+s+βp(y) S(y). This is a strictly increasing function of y. So, by setting ϕ(y) = p(y) r+s+βp(y) S(y), we obtain that the third term is also (strictly) negative (See F.1.2). (ii) We now turn to the derivative of H with respect to y * . As q(y)S(y) does not depend on y * for all y in ]y * , 1] (see equation ( 20)), this derivative reduces to

∂H(v, y * ) ∂y * = -q(y * )S(y * ) < 0 (23)
So, according to the job creation equation ( 16), an increase in participation (a decrease in y * ) stimulates job creation: v ′ (y * ) < 0.

F.2 Constrained (in)efficiency of participation

Let l(y) and u(y) denote the employment and unemployment levels of y-workers respectively. The social surplus per period is defined as follows18

Σ = 1 y * l(y)ydy + 1 y * u(y)(z -γ)dy + F (y * )nz -vc
Let l be the total employment level. As l(y) = q(y)l Q and sl = Qv, the job creation equation ( 16) can be written as -vc + we find that this direct derivative is zero in a laissez-faire equilibrium (equation ( 17)).

Let us now consider the derivative of Σ cp with respect to v. As p(y) grows with v, it results that l(y), rU (y) and, w(y) are increasing functions of v.

From F.1.3, we know that v ′ (y * ) < 0. Consequently, we obtain ∂Σ cp (v(y * ), y * ) ∂y * = ∂Σ cp (v(y * ), y * ) ∂v(y * ) v ′ (y * ) < 0

This states that participation is constrained suboptimal.

F.3 Introducing a minimum wage

In the presence of a binding minimum, the job creation equation is rewritten as 19 -c +

F.3.2 Impact on welfare

We claim that introducing a small minimum wage improves market efficiency21 . This result stems from the constrained inefficiency of participation.

As equation ( 24) is satisfied, the social surplus per period reduces to Σ =

z

  = -γ+p(., y * )[w(y * )+a]+[1-p(., y * )](z+a) = -γ+p(., y * )[z+a+β(y * -z)]+[1-p(., y * )](z+a). Or -(γ -a) + p(., y * )β(y * -z) = 0 (13)

  positive output ỹ (ỹ > y * ) such that ∂ρ((v/n), ỹ, y * ) ∂v = 0 and y

  Case 1. ∂v ∂b ≤ 0 In this case we use Propositions 1a,b. The derivative of Σ can be written as

  the cost at keeping a vacancy open.

  With respect to participation, the equation for migration equilibrium now isγ = βp(y * )S(y * )(17)In this dynamic setting, the distribution of productivities, F (y), among the total population remains exogenous, but the distribution among unemployed workers becomes endogenous. It is denoted by G(y) and defined on [y * , 1]. Its density is g(y). Below, we study how this endogenous distribution is affected by an increase in job creation or in participation.F.1 Preliminary resultsF.1.1 Flow equilibrium and hiring probabilitiesLet u denote the number of unemployed workers (searching for a job). The market tightness, θ, is then determined by the ratio v/u. For almost all z > y * , flow equilibrium imposes the condition thats[f (z)n -g(z)u] = p(z)g(z)uIntegrating the previous equation on [y * , 1] gives s[(1 -F (y * ))n -u] = P (θ)u with P (θ) = 1 y * p(y)g(y)dy = θ(1 -e (-1/θ) ) being the average probability of an unemployed worker finding a job.

  continuity that there exists some median output ỹ (1> ỹ > y * )This preliminary result is used to state that, for any (strictly) increasing function ϕ(y) defined on [y * , 1],

  with respect to θ (hence to v) as well as to y * .(i) We first show that the derivative ∂H ∂θ , which is negative, is bounded above. Integrating H by parts givesH = S(1) -p(y * )S(y * ) -1 y * p(y) r + s + βp(y)[1 -βq(y)S(y)]dy -p(y * )S(y * ) is a decreasing function of θ. On the other hand, developing the derivative of the integral yields βq(y)S(y)]dy < 0

  )[y -w(y)]dy = 0 with w(y) being the wage of y-workers. )(z -γ)dy + F (y * )nz As equation (16) is treated as a constraint, the social surplus is reduced to y)n -l(y)](z -γ)dy + F (y * )nzOr Σ cp = 1 y * l(y)[w(y) -z + γ]dy -γ[1 -F (y * )]n + nzAs usual, wages satisfyw(y) = rU (y) + β(y -rU (y))with U (y) being the asset value of unemployment:rU (y) = z -γ + p(y)βS(y) Σ cp is a function of the form Σ cp = Σ cp (v(y * ), y * )with v(y * ) being the implicit (decreasing) function derived from equation (16).We first show that the direct derivative ∂Σ cp (v,y * ) ∂y * is reduced to zero in a laissez-faire equilibrium. Noting that l(y) = p(y) s + p(y) f (y)n one can see that this derivative is proportional to γ -p(y * ) s + p(y * ) [w(y * ) -z + γ] Because w(y * ) -z + γ = rU (y * ) -z + γ + β(y * -rU (y * )) = β(s + p(y * ))S(y * )

19

  for migration equilibrium becomesγ = p(y * ) ŵ -z + γ r + s + p(y * )(25)F.3.1 Impacts on participation and job creationWe claim that a small minimum wage stimulates job creation and participation 20 . To that end, let J denote the following quantityAs in the static study, we examine the implications of introducing a binding minimum in the neighborhood of an equilibrium where all wages are bargained. As a consequence, we obtain (see paragraph F.1.3) Remember that ŷ is the productivity level such that the bargained wage coincides with the minimum wage, ŵ. 20 This result holds for any positive interest rate. s)( ŵ -z + γ) [r + s + p(y * )] 2 [

  )(-γ + z)dy + F (y * )nz Notice that in the previous equation, w(y) is the bargained wage (defined as in the constrained efficiency study, F.2) for y ≥ ŷ.Σ can be rewritten as followsΣ = ŷ y * l(y)( ŵ -z + γ)dy + 1 ŷ l(y)(w(y) -z + γ)dy + (1 -F (y * ))n(-γ + z) + F (y * )nzLet us consider the derivatives of Σ( ŵ, ŷ, v, y * ). As ŷ initially coincides with y * , we have ∂Σ( ŵ, ŷ, v, y * ) y) grows with v, l(y) and w(y) are increasing functions of v. Consequently ∂Σ( ŵ, ŷ, v, y * ) ∂v > 0On the other hand the derivative of Σ( ŵ, ŷ, v, y * ) with respect to y * is ∂Σ( ŵ, ŷ, v, y * ) ∂y * = -l(y * )( ŵ -z + γ) + f (y * )nγ

  Pissarides, C.A., 2001. Employment protection. Labour Economics 8 (2), 131-159.

	Portugal, P., Cardoso, A.R., 2006. Disentangling the minimum wage puzzle: An
	analysis of worker accessions and separations. Journal of the European Economic
	Association 4 (5), 988-1013.
	Zenou, Y., 2009. Search in cities. European Economic Review 53 (6), 607-624.

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

See Gavrel (2012) for a more detailed exposition.

Notice that the probability for a firm to receive several applications of the same type is zero.

With random recruitment, this probability rate is [(1 -e -1 θ ) f (y) 1-F (y * ) ].

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

Flinn assumes match specific productivities.

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

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.

Remember that η(θ) is the elasticity of Q(θ) with respect to θ in absolute values.

In this efficiency study, the interest rate is close to zero.

In this welfare study, the interest rate is very close to zero.

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

From (v/n)q((v/n), y) = p((v/n), y)f (y), we deduce

Using equation (2), we obtain

Appendix D: Proof of Proposition 4

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

From the job creation equation ( 12), we deduce

Differentiating Σ with respect to ŵ gives

From Proposition 3, we know that ∂v ∂ ŵ > 0. On the other hand we have ∂p((v/n),y) ∂v > 0.

From the condition for migration equilibrium (11), we deduce that introducing a minimum wage improves market efficiency.

Appendix E: Other public policies E.1 Basic income

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

Next, differentiating equation ( 13) gives

From the previous inequality, we deduce

This proves that introducing a basic income (a) lowers the cutoff y * .

E.2 Unemployment compensation Participation

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

we deduce that ( 14) can be rewritten as

On the other hand, the participation equation ( 15) gives

Appendix F: Extension to a dynamic setting

This appendix provides an extension of the main points of the analysis to a dynamic setting 15 . We first state that participation remains (constrained) suboptimal. Next we prove that introducing a minimum wage still raises participation and job creation.

Consequently, a small minimum wage improves market efficiency. For the sake of simplicity, time is discrete 16 . It is also assumed that the destruction probability of 15 Unless otherwise stated, the notation used is the same as in text.

16 Following [START_REF] Moen | Education, ranking and competition for jobs[END_REF] and [START_REF] Gavrel | On the inefficiency of matching models of unemployment with heterogeneous workers and jobs when firms rank their applicants[END_REF], switching to continuous time is straightforward.

From equation ( 25), we deduce that this derivative is close to zero when r tends to zero. Consequently, we obtain

Introducing a (small) minimum wage improves market efficiency.

Appendix G: Competitive search

The purpose of this appendix is to consider an alternative wage-setting mechanism, referred to as competitive search. This wage-setting mechanism was introduced by [START_REF] Moen | Competitive search equilibrium[END_REF] in an economy which is divided into a very large number of sub-markets.

In each sub-market, prior to search, firms post a wage that maximizes expected profits for a given expected income of workers. [START_REF] Moen | Competitive search equilibrium[END_REF] finds that job creation is efficient under these assumptions.

In the presence of vertically differentiated workers, firms are led to commit to wage functions that relate wages to productivities. In addition, there are as many constraints as productivity levels. The analysis is restricted to a symmetric equilibrium.

Furthermore, in what we regard as a first pass, participating workers do not know the distribution of y among the queues of applicants. This implies that firms cannot make use of wages to influence the distribution of y among their applicant pools.

The wage functions, w(y), defined on [y * , 1], are assumed to be affine.

Under the previous assumptions, we find that, despite recruitment selection, job creation becomes (partially) efficient. Conversely, participation remains (partially) insufficient. The reason for this is that firms are too small to exert any influence on individuals' participation decisions. We therefore surmise that the inefficiency of participation is likely to hold in any wage-posting model. These results are established as follows.

Consider one sub-market. In this sub-market, the wage function, w(y), and the market tightness, θ, are set by maximizing 22

22 The rates q(y) and p(y) are defined as in the text when replacing v/n with θ.

V ≡ -c + where the expected incomes of participants, Φ(y), are treated as exogenous.

As p(y) = q(y)(f (y)/θ), the indifference constraint implies

Market tightness θ appears to be a free maximum of V (θ). The first order condition for this maximum is This shows that job creation is (partially) efficient in this setting. See Proposition 1b.

As wage functions are (assumed to be) affine, participation is determined by γ = p(y * )(η + α)(y * -z)

Since η + α > 0, this shows that participation remains (partially) insufficient. See Proposition 1a.

23 Remember that α is the elasticity of (ȳ -z) with respect to v, hence to θ, in absolute values.