Screening on the job: should temporary jobs be subsidized? Julien Albertini

∗

Xavier Fairise∗ Florent Fremigacci∗

†

February 2009 Preliminary version, work in progress Abstract

The difficulty to allocate the right workers to the right jobs is an important source of market frictions. With the expansion of atypical jobs in the mid-1980’s, the idea that screening and flexibility could be complementary motivations arose. The purpose of this paper is threefold : (i) First, we investigate the screening effect of temporary jobs in terms of transitions to regular employment (ii) Then, we test alternative subsidy schemes and analyze which ones are efficient (iii) Finally, we approach the question empirically. We extend the framework of Pries and Rogerson (2005) to allow firms to hire workers on temporary jobs or in permanent jobs without screening. Screening takes the form of a learning process where both the employer and the employee infer the match quality during a temporary job. The probability a short-time contract become a permanent one is endogenous.

Keywords: Fixed-duration contract, Subsidised temporary jobs, Active labor market policies, Screening, Unemployment, Matching model.

JEL Classification: H29; J23; J38; J41; J64 ∗ EPEE et TEPP (FR CNRS no 3126), University of Evry, Bd. François Mitterand, 91025 Cedex, France. † CREST, 15 Bd Gabriel Peri, 92245 Malakoff Cedex, France. Corresponding author. E-mail addresses: [email protected] (J. Albertini), [email protected] (X. Fairise), [email protected] (F. Fremigacci).

1

1

Introduction

The difficulty to allocate the right workers to the right jobs is an important source of market frictions. Usually, the quality of a match is not completely known ex-ante but must be experienced (see Jovanovic (1979) and Jovanovic (1984)). With the expansion of atypical jobs in the mid-1980’s, the idea that screening and flexibility could be complementary motivations arose. Indeed, it is often argued that employers use temporary jobs as a way of screening workers before hiring them into regular jobs and to adjust their employment to changes in demand. An important aspect of the use of temporary contracts is their pattern of promotion of effective “stepping stones” to permanent employment (into regular contracts of indefinite duration). On the one hand, temporary jobs could be used to reduce employers uncertainty about the employability of job applicants when firing costs are high. In the same time, it also allows the unemployed to build up their work experience, prevent skill atrophy and signal their willingness to work. On the other hand, temporary employment can crowd-out job-search activities and create a lock-in effect, thus increasing the duration until a full-time job is found. Moreover, in a segmented labor market, individuals going through part-time jobs risk being caught in a temporary unemployment trap and lower their future chances of entering regular employment. The screening process may lead to prolonged probationary periods. In such a context, temporary workers will face recurrent periods of unemployment and temporary work, increasing the precariousness. Therefore, the desirability of atypical contract and their effects on labor market outcomes are ambiguous. The efficiency of such a policy depend whether flexible contracts are “stepping stones” or dead ends. But it also depends on the workers behavior and on the firms willingness to spend greater resources on screening workers. Empirical studies of the effectiveness of temporary employment in providing a better access to regular employment have not succeeded in giving unambiguous results. While Booth, Francesconi, and Frank (2002) for the UK, Lechner, Pfeiffer, pengler, and Almus (2000) for Germany found evidence of a positive treatment effect of temporary jobs; Autor and Houseman (2005) for the US, Amuedo-Dorantes (2002) for Spain, and Böheim and Weber (2006) for Germany reached the opposite conclusion. On the theoretical side, Bentolila and Saint-Paul (1992) underline that fixedterm contracts increase the size of the employment response to aggregate shocks, while decreasing its persistence when firing cost are low. Cahuc and Postel-Vinay (2002) show that the use of temporary jobs intensifies job flows and may increases unemployment and reduces efficiency when they are combined with a stringent employment protection. Blanchard and Landier (2002) obtained similar results in a simple matching model. They confirm their results using individual data from France on young workers and show that temporary jobs have no substantial effects 2

on unemployment durations. Neugart and Storrie (2006) find that temporary work agencies may increase regular employment. Despite the extensive literature on the consequences of flexible labor contracts, the effect of the screening process and the probationary period as been neglected so far. If temporary jobs increase the unemployed opportunities for a long lasting insertion in the labor market, then the question on the optimality of subventions to such jobs naturally arise. One of the problems faced by governments trying to implement a subsidized employment program is to design the measure in such way that it still encourages the unemployed to take such jobs without, in the meanwhile, promoting "dead end" jobs (Calmfors (1994)). Employment subsidy is an instrument of active labor market policies commonly used in many OECD but subsidy schemes exhibit wide differences among countries. In France, “reduced activity” implies that employed workers may receive a wage subsidy which correspond to a part of his benefits if working less than a given number of hours. In Switzerland, employed workers who accept jobs paying less than the unemployment benefits receive a wage subsidy. However in Canada, “The Targeted Wage Subsidy Program” enables employers to hire individuals who face barriers to employment by offering temporary wage subsidies. The principle of such a policy remains the same. It is a way to increase employment opportunities of the unemployed by providing a means of integration. It aims at improving chances to get unemployed a regular jobs through fixed-time contracts and increasing work experience. Then, the questions we ask are whether we should give a financial incentive to get unemployed back to work on temporary jobs and what kind of subsidy scheme is (the most) efficient. To our knowledge the only study that evaluate the effect of subsidised temporary jobs is the one of Gerfin, Lechner, and Steiger (2005). Using individual data from administrative records in Switzerland, they show that wage subsidies may get unemployed back to work. However, they do not test alternative schemes. They are no theoretical framework that study temporary employment subsidies. The purpose of this paper is threefold : (i) First, we investigate the screening effect of temporary jobs in terms of transitions to regular employment (ii) Then, we evaluate the most common form of wage subsidy implemented in many OECD countries. We test alternative subsidy schemes: hiring subsidies, permanent or not, with possibly for additional allowance for employers and/or employees if the short-term contract become a permanent one. (iii) Finally, we approach the question empirically. We extend the framework of Pries and Rogerson (2005) to allow firms to hire workers on temporary jobs or in permanent jobs without screening. Screening takes the form of a learning process where both the employer and the employee infer the match quality during a short-time contract. The probability a short time contract become a permanent one is endogenous. The reminder of this paper is organized as follows. The next section is devoted to the description of the model . The equilibrium and the subsidy schemes are defined in section 3. Section 4 is devoted to simulation exercises. Results are 3

discussed in Section 5. Section 6 finally concludes.

2

The model

We extend the model of Pries and Rogerson (2005) that use a simple learning process in the spirit of Jovanovic (1979). We allow firm to hire an unemployed worker in a “screening contract” or in a regular contract directly. It’s a convenient way to represent a dual labor market where temporary jobs and permanent jobs coexist. Employers use temporary jobs as a way of screening workers before hiring them into permanent jobs. Then, a temporary worker can switch to a regular job if its productivity is revealed to be good. Separations occur at no cost either exogenously or when the true quality of the match is revealed to be bad. Our model includes a Non-Walrasian labor market with search and matching frictions as in Mortensen and Pissarides (1999). Wages are determined through a Nash bargaining process.

2.1

Worker and entrepreneur preferences

Following Pries and Rogerson (2005) we assume there is a continuum of identical workers and entrepreneurs whose preferences are defined as follows : Workers Entrepreneurs

P∞ t=1

P∞ t=1

β t (Ct − aNt ) β t (Ct − kv Vt − kx Xt )

β is the discount factor and Ct is consumption. a represents labor disutility and Nt is the working time, with Nt = 1 if the worker is employed and Nt = 0 otherwise. Concerning the entrepreneur, Vt vacancies are posted and Xt employment positions are created. To post a vacancy and to create an employment position respectively incur costs kv and kx .

2.2

The matching process

Labor market flows are governed by a matching process. The number of matches M is given by a matching function M = M (U, V ), with V the vacancies and U the number of unemployed workers. The matching function satisfies the usual assumptions, it is increasing, concave and homogeneous of degree 1. Let θ = V /U be the labor market tightness, a vacancy is filled with probability q(θ) = M/V and an unemployed worker finds a job with probability p(θ) = θq(θ) = M/U . During the matching process, these probabilities are taken as given by the workers and entrepreneurs. Note that the properties of the matching function imply that q and p satisfy q 0 (θ) < 0 and p0 (θ) = q(θ) + θq 0 (θ) > 0. 4

2.3

Match quality information

We follow Pries (2004) and Pries and Rogerson (2005)’s modeling. The true match quality is unknown when the worker and employer meet. Worker and employer get informations about the match quality through a screening process. The observed match output is defined as follows : y = y+ε y is the match true quality and ε is a white noise with mean zero. There exists two types of matches : good matches with y = y g and bad matches with y = y b < y g . When a match is formed, its quality does not change through time. However, the quality of the match may not be instantaneously revealed, a sort of learning process may occur. It should be stressed that production is observed at the end of a period, after the wage contract being signed. Consider now a worker and employer meet. At the meeting date, a common signal is received by the worker and employer. It is the probability π the match be a good one, this probability being drawn from a distribution H(π). Worker and employer ought to decide to continue their relationship. Continuing the relationship incurs a cost c paid by the entrepreneur. Thereafter, at the beginning of the period, the match quality may be revealed with probability γ. It is assumed there is a fraction of matches for which the quality is known without a learning process. In such a case, the worker is hired on long-term contract. Conversely, if the match quality is not initially revealed and if the match is formed, an output y = y + ε is observed at the end of each period as long as the relationship continues. The match quality may be inferred from a learning process. The white noise ε is drawn from a uniform distribution whose support is [−ω, ω], with ω > 0. The observed value of output may reveal the match quality. Let g b , if the realized value of output lies between y g − ω and assume that ω > y −y 2 b y + ω, the match quality cannot be inferred. Conversely, if y ∈ [y b − ω, y g − ω[ (y ∈]y b + ω, y g + ω]), the match quality is bad (good). The probability α the g −y b match quality be observed can easily be determined, one has α = y 2ω .

2.4

Workers and employers behavior

Suppose a worker and firm meet. Let Jen (π) the value of a newly matched entrepreneur at the meeting date and before the quality of the match be possibly revealed and let Je (π) be the value of an already matched entrepreneur. Je (π) applies for new matches, after the match quality be revealed or not, and thereafter if the match continues. It represent a temporary job because the true quality is unknown and the relation can be severed at any time. Finally, the value of an unfilled employment position is denoted by Ju . These values satisfy : 5

Jen (π)

=

Je (π) = + Ju =

½ ¾ £ ¤ max Ju , γ πJe (1) + (1 − π)Ju + (1 − γ)Je (π) − c (1) ½ £ max Ju , πy g + (1 − π)y b − w(π) + β(1 − λ) α(πJe (1) + (1 − π)Ju ) ¾ ¤ (2) (1 − α)Je (π) ¸ · Z n −kv + β q Je (π)dH(π) + (1 − q)Ju (3)

Equation (1) says the value of a newly matched entrepreneur Jen (π) is equal to the expected gain of the match less the hiring cost c. Equation (2) states the value of an already matched entrepreneur is equal to the present expected gain of the match plus the present value of the expected futur gains. Equation (2) take into account the fact that wage contracts are signed before the output level be revealed. When the productivity is revealed to be good with probability (1 − λ)απ, the temporary job become a regular one: Je (1). The screening process stop but separations may occur exogenously. Finally, equation (3) says the value of a vacant job is equal to present value of the expected gains less the vacancy posting cost kv . Consider now workers and let Ven (π) denotes the value of a newly matched worker before the quality of the match be possibly revealed and let Ve (π) the value of an already matched worker. As in the case of an entrepreneur, Ve (π) applies for new matches, after the match quality be or not revealed, and thereafter if the match continues. Finally, the value of an unemployed worker writes Vu . Ven (π)

=

Ve (π) = + Vu =

½ ¾ £ ¤ max Vu , γ πVe (1) + (1 − π)Vu + (1 − π)Ve (π) ½ £ max Vu , w(π) − a + β(1 − λ) α(πVe (1) + (1 − π)Vu ) ¾ ¤ (1 − α)Ve (π) + βλVu · Z ¸ n β p Ve (π)dH(π) + (1 − p)Vu

(4) (5) (6) (7)

According to (4), the value of a newly matched worker Ven (π) is equal to the expected gain of the match. Equation (5) states the value of an already matched worker is equal to the present expected gain of the match (the current wage less the cost a) plus the present value of the expected futur gains. Finally, equation (7) says the value of an unemployed worker is equal to the present value of its expected gains. 6

2.5

Equilibrium

Wages are determined according to a Nash bargaining process. Let ν be the bargaining power of workers and S(π) = Je (π) − Ju + Ve (π) − Vu the total surplus of a job characterized by the probability π. The bargaining process leads to the following total surplus sharing between workers and entrepreneurs : Ve (π) − Vu = νS(π) Je (π) − Ju = (1 − ν)S(π)

(8) (9)

When a match is formed, two continuation decisions must be taken. Firstly, just after meeting and the probability π be drawn, then entrepreneur and the worker must decide if the continue their relationship. Continuation decision is taken if the probability the match be a good one π is greater or equal to a threshold π. Secondly, if the match quality is not initially revealed, the entrepreneur and worker has to decide, once again, to continue or not their relationship. Continuation decision is taken if π is greater or equal to a new threshold π. Observe that if c = 0, the threshold π is equal to 0, we thus conjecture that π < π. To characterize the equilibrium, we follow Pries and Rogerson (2005) approach. The equations system determining the equilibrium can be reduced to a three equations system whose unknown are π, π and θ = V /U . Consider the definition of total surplus, equations (2), (3), (5) and (7) may be rewritten as follows : ½ £ ¤ S(π) = max πy g + (1 − π)y b − a + β(1 − λ) απS(1) + (1 − α)S(π) ¾ − (1 − β)Vu − (1 − β(1 − λ))Ju , 0 (10) As their is an employment position creation cost, the free entry condition writes Ju = kx . Using equations (5), (7) and (8), one gets : Z 1 Z 1 (1 − β)Vu = βpγνS(1) πdH(π) + βp(1 − γ)ν S(π)dH(π) π

π

Similarly, using equations (2), (3) and (9), the following expression is obtained : Z

1

πdH(π)

(1 − β)Ju = −kv + βqγ(1 − ν)S(1) π

Z

1

+ βq(1 − γ)(1 − ν)

S(π)dH(π) − βqc(1 − H(π)) π

7

(11)

From the two above equations, one easily deduces that : (1 − β)Vu =

¤ νp £ (1 − β)Ju + kv + βqc(1 − H(π)) (1 − ν)q

(12)

Using the above equation to eliminate Vu from (10) and using the free entry condition, one gets : ½ £ ¤ S(π) = max πy g + (1 − π)y b − a + β(1 − λ) απS(1) + (1 − α)S(π) νp − (1 − β(1 − λ))kx − [(1 − β)kx + kv + βqc(1 − H(π))], 0 (1 − ν)q

¾ (13)

Consider now equations (1) and (4) at the threshold π. Knowing that Jen (π) = = 0, it is easily deduced that :

Ven (π)

πS(1) = c

(14)

The threshold π is such that the expected gain of the match is just equal to the cost c. Consider now equation (13) evaluate at π, it satisfies S(π) = 0. One gets : 0 = πy g + (1 − π)y b − a + β(1 − λ)απS(1) − (1 − β(1 − λ))kx νp [(1 − β)kx + kv + βqc(1 − H(π))] − (1 − ν)q

(15)

Substituting in equation (13) provides an expression of the total surplus S(π) for values of π greater or equal to π :

S(π) =

(y g − y b ) + β(1 − λ)αS(1) (π − π) ≡ ψ(π)(π − π) 1 − β(1 − λ)(1 − α)

(16)

We easily deduce that :

S(1) =

yg − yb (1 − π) ≡ ϕ(π) = ψ(π)(1 − π) 1 − β(1 − λ)(1 − απ) g

(17) b

αβ(1−λ)(y −y ) One easily checks that ϕ(π) > 0, ψ(π) > 0, ψ 0 (π) = − (1−β(1−λ)(1−απ)) 2 < 0,

ϕ0 (π) = −πψ(π) + (1 − π)ψ 0 (π) < 0 and πψ 0 (π) + ψ(π) = 8

(y g −y b )(1−β(1−λ)) (1−β(1−λ)(1−απ))2

> 0.

Using the two above equations, conditions (11), (14) and (15) may be rewitten as follows : πϕ(π) = c

(18)

0 = y b − a + (1 − β(1 − λ)(1 − α))πψ(π) − (1 − β(1 − λ))kx ν θ[(1 − β)kx + kv + βq(θ)c(1 − H(π))] − (1 − ν) Z 1 kv + (1 − β)kx = βq(θ)γ(1 − ν)ϕ(π) πdH(π)

(19)

π

Z

1

+βq(θ)(1 − γ)(1 − ν)ψ(π)

(π − π)dH(π) − βq(θ)c(1 − H(π)) (20) π

The above three equations system allows to determine the equilibrium value of π, π and θ = V /U . To complete the determination of the equilibrium, we have to determine labor market flows. Let define : • Eg : the number of goods quality matches; • En : the number of unknown quality matches; • Ev : the number of vacant employment positions previously created. It is also useful to define two functions X0 (π) and X1 (π) to denote the optimal decision rules. X0 (π) denotes the optimal decision rule for new matches and before the match quality be revealed or not. If π ≥ π, the relationship continues and X0 (π) = 1, otherwise, if π < π, the relationship is severed and X0 (π) = 0. X1 (π) denote the optimal decision rule for new matches if the match quality is not initially revealed and, thereafter, if the match continues, as long as the match quality is not revealed. Knowing that θ = V /U , π and π are determined by equations (18) — (20), steady state labor market equilibrium flows characterized by Eg , En , Ev and V are obtained by solving the following system of equations : λEg = (1 − λ)αEn E[π|X1 (π) = 1] + V qγE[π|X0 (π) = 1]

(21)

En (λ + (1 − λ)α) = V q(1 − γ)E[X0 (π)]

(22)

©

ª qγE[π|X0 (π) = 1] + q(1 − γ)E[X1 (π)] Ev = (1 − λ)α(1 − E[π|X1 (π) = 1])En © ª +(V − Ev ) 1 − qγE[π|X0 (π) = 1] + q(1 − γ)E[X1 (π)] (23) λ(Eg + En ) = V − Ev

(24)

9

R1

with E(π|X0 (π) = 1) = R1

πdH(π) Rπ1 π dH(π)

π πdH(π) R1 , π dH(π)

E(X0 (π)) = 1 − H(π), E(π|X1 (π) = 1) =

and E(X0 (π)) = 1 − H(π).

Equations (21)—(23) respectively describes steady state outflows and inflows from good quality matches, unknown quality matches and vacant employment position previously created. Equation (24) says at the steady state, the number of vacant employment positions newly created is equal to the total number of destroyed employment positions.

2.6

Uniqueness of the equilibrium for small values of c

Some analytical results are difficult to obtain about the existence of the steady state equilibrium. However, some interesting results may be obtained for small values of the cost c. To begin, suppose c = 0. From equation (18), it immediately Optimal match formation

6

c=0

V /U

c>0

c=0

c>0

-

¾

(V /U )c=0

(V /U )c>0

¾

-

Free entry

-

(π)c>0 (π)c=0

π

Figure 1: Equilibrium values of π and V /U . follows that π = 0. If a worker and an entrepreneur meet, as there is no cost, it is worthwhile to wait for the match quality be revealed or not, consequently, π = 0. Furthermore, it can be hoped that π will increase and become positive as c increases. The following result provides some properties about the equilibrium for small values of the cost c. 10

Result 1 If c = 0, then, π = 0 and, if it exists, the solution in θ and π of the system formed by equations (19) and (20) is unique. Furthermore, the derivatives of θ, π and π with respect to c satisfy ∂π < 0, ∂θ < 0 and ∂π > 0. ∂c ∂c ∂c Proof See appendix. Result 1 states that for c = 0, if it exists, the equilibrium is unique. Furthermore, result 1 says how the equilibrium is perturbed if c increases. It follows that, by continuity, for small values of c, the equilibrium still exists and is unique. The effect of an increase in c on the equilibrium is represented in figure 1. This argument will enable us to provide some analytical results concerning labor market policies measures for small values of c.

3 3.1

Labor market policies Subsidized temporary jobs: theoretical approach

We first explore the effect of a wage subsidy targeted on temporary jobs and earned by employee as in France and in Switzerland. In a second step ,we explore various subsidy schemes: hiring subsidies, permanent or not, with possibly for additional allowance for employers and/or employees if the short-term contract become a permanent one. We also analyze the effects when the employer receive the subsidy. The income of a temporary employed worker is composed of a wage w(π) and a subsidy ςw . He enjoys the wage subsidy as long as he remains on a screening job. When the match quality is revealed to be good (π = 1) the wage subsidy disappears. When an unemployed worker is hired on a temporary job (π < 1) he obtains at the meet time a subsidy ςH .

References Amuedo-Dorantes, C. (2002): “Work transitions into and out of involuntary temporary employment in a segmented market: Evidence from Spain,” Industrial and Labor Relations Review, 53, 309–325. Autor, D., and S. Houseman (2005): “Temporary Agency Employment as a Way out of Poverty?,” Working Paper No. 05-123. Bentolila, S., and G. Saint-Paul (1992): “The macroeconomic impact of flexible labor contracts, with an application to Spain,” European Economic Review, 36, 1013–1053.

11

Blanchard, O., and A. Landier (2002): “The Perverse Effects of Partial Labour Market Reforms: Fixed Duration Contracts in France?,” The Economic Journal, 112, F214–F244. Böheim, R., and A. Weber (2006): “The effects of marginal employment on subsequent labour market outcomes,” Economics working papers 2006-12. Booth, A., M. Francesconi, and J. Frank (2002): “Temporary Jobs: Stepping Stones Or Dead Ends?,” Economic Journal, 112, F189–F213. Cahuc, P., and F. Postel-Vinay (2002): “Temporary jobs, Employment Protection, and Labor Market Performance?,” Labour Economics, 9, 61–91. Calmfors, L. (1994): “What Can Sweden Learn from the European Unemployment Experiences?,” Swedish Economic Policy Review, 1-2. Gerfin, M., M. Lechner, and H. Steiger (2005): “Does Subsidised Temporary Employment Get the Unemployed Back to Work? An Econometric Analysis of Two Different Schemes,” Labour Economics, 12, 807–835. Jovanovic, B. (1979): “Job matching and the theory of turnover,” Journal of Political Economy, 87, 972–990. (1984): “Matching, Turnover, and Unemployment,” Journal of Political Economy, 92, 108–1220. Lechner, M., F. Pfeiffer, H. pengler, and M. Almus (2000): “The Impact of Non-profit Temping Agencies on Individual Labour Market Success in the West German State of Rhineland-Palatinate,” Discussion Paper No. 00-02. Mortensen, D., and C. Pissarides (1999): “New developments in models of search in the labor market,” in Handbook of Labor Economics, vol. 3, chap. 2, pp. 2567–2627. Elsevier Science, New York. Neugart, M., and D. Storrie (2006): “The Emergence of Temporary Work Agencies,” Oxford Economic Papers, Vol. 58, No. 1, pp. 137-156. Pries, M. (2004): “Persistence of unemployment fluctuations: a model of recurring job loss,” Review of Economic Studies, 71, 193–215. Pries, M., and R. Rogerson (2005): “Hiring policies, labor market institutions, and labor market flows,” Journal of Political Economy, 113, 811–839.

12

A

Proof of result 1

To begin, consider equations (19) and (20). Having eliminated π thanks to equation (18), we define g(π, θ; c) and f (π, θ; c) such that : g(π, θ; c) ≡ y b − a + (1 − β(1 − λ)(1 − α))πψ(π) − (1 − β(1 − λ))kx · µ ¶¸ ν c − θ (1 − β)kx + kv + βq(θ)c(1 − H ) =0 (1 − ν) ϕ(π) Z 1 f (π, θ; c) ≡ −kv − (1 − β)kx + βq(θ)γ(1 − ν)ϕ(π) πdH(π) Z

(25)

c ϕ(π)

1

+βq(θ)(1 − γ)(1 − ν)ψ(π) π

µ µ ¶¶ c =0 (π − π)dH(π) − βq(θ)c 1 − H ϕ(π)

(26)

Computing the derivatives of f and g with respect to π, θ and c and evaluating them at c = 0 provides : g1 (π, θ; 0) g2 (π, θ; 0) g3 (π, θ; 0)

(1 − β(1 − λ)(1 − α))(ψ(π) + πψ 0 (π) > 0 ν = − ((1 − β)kx + kv ) < 0 1−ν ν θβq(θ) < 0 = − 1−ν =

Z f1 (π, θ; 0)

=

1

0

βγq(θ)(1 − ν)ϕ (π)

πdH(π) 0

+

0

Z

1

βq(θ)(1 − γ)(1 − ν)ψ (π)

(π − π)dH(π) π

− f2 (π, θ; 0)

=

f3 (π, θ; 0)

=

βq(θ)(1 − γ)(1 − ν)ψ(π)(1 − H(π)) < 0 q 0 (θ) ((1 − β)kx + kv ) < 0 q(θ) −βq(θ) < 0

The signs of the above derivatives immediately follow from the signs of the derivatives of ϕ, ψ and q. That is ψ 0 (π) < 0, πψ 0 (π) + ψ(π) > 0, ϕ0 (π) < 0 and q 0 (θ) < 0.

Uniqueness Suppose c = 0, we easily deduce the slope of the optimal match formation equation (equation (25)) and the slope of the free entry condition (equation (26)) are respectively positive and négative (−g1 (π, θ; 0)/g2 (π, θ; 0) > 0 and −f1 (π, θ; 0)/f2 (π, θ; 0) < 0). It follows the solution of the system formed equations (25) and (26), if it exists, is unique (see figure 1).

Sign of the derivatives with respect to c The derivatives of θ and π with respect to c for c = 0 write :

13

∂θ ∂c

=

∂π ∂c

=

−f1 (π, θ; 0)g3 (π, θ; 0) + g1 (π, θ; 0)f3 (π, θ; 0) <0 f1 (π, θ; 0)g2 (π, θ; 0) − g1 (π, θ; 0)f2 (π, θ; 0)

−f3 (π, θ; 0)g2 (π, θ; 0) + g3 (π, θ; 0)f2 (π, θ; 0) f1 (π, θ; 0)g2 (π, θ; 0) − g1 (π, θ; 0)f2 (π, θ; 0) ν ((1 − β)kx + kv )β(q(θ) + θq 0 (θ)) <0 = − 1−ν f1 (π, θ; 0)g2 (π, θ; 0) − g1 (π, θ; 0)f2 (π, θ; 0)

Finally, from equation (18), it is easy to deduce that if c = 0,

14

∂π ∂c

=

1 ϕ(π)

> 0.