Greqam

Bruno Decreuse
email: decreuse@univmed.fr

Elvira Kazbakova
email: elvira@univmed.fr

Pierre Granier

Nicolas Gravel

André Zylberberg

ON THE SPIKE IN HAZARD RATES AT BENEFIT EXPIRATION: THE SIGNALLING HYPOTHESIS REVISITED

Keywords: Worker heterogeneity, Signalling, Hazard rate, Unemployment compensation, Moral hazard J.E.L. classi…cation: J64, J65

We revisit the signalling hypothesis, whereby potential employers use the duration of unemployment as a signal as to the productivity of applicants. We suggest that the quality of such a signal is very low when the unemployed receive unemployment bene…ts: individuals have good reasons to remain unemployed. Conversely, the signal becomes much more e¢ cient once bene…ts have elapsed: skilled workers should not stay unemployed in such cases. Therefore, the potential duration of unemployment bene…ts should drive employers'expectations and their recruitment practices. This mechanism can explain why hazards fall after bene…t expiration, and why hazards respond more to the potential duration of bene…ts than to replacement rates.

Introduction

This paper is a theoretical contribution to the literature on job search and unemployment insurance. We revisit the signalling hypothesis, whereby potential employers use the duration of unemployment as a signal as to the productivity of applicants. Managers are typically reluctant to interview the longterm unemployed because other managers interviewed and would have hired these workers if they had been productive 1 . We suggest that the quality of such a signal is very low when the unemployed receive unemployment bene…ts: individuals have good reasons to remain unemployed. Conversely, the signal becomes much more e¢ cient once bene…ts have elapsed: skilled workers should not stay unemployed in such cases.

Therefore, the potential duration of unemployment bene…ts should drive employers' expectations and their recruitment practices. This mechanism can explain why hazards fall after bene…t expiration, and why hazard rates respond more to the potential duration of bene…ts than to replacement rates.

Why is this important? Our paper is mostly theoretical. However, it can also be used to address two empirical puzzles that standard job search theory hardly explains.

On the one hand, hazard rates increase prior to the exhaustion date, and strongly decline afterwards.

Most of the unemployment compensation systems of the OECD countries deliver declining bene…ts with the unemployment spell. Bene…ts are proportional to the pre-unemployment wage for short unemployment spells, while they drop to a common standard determined by the public assistance system for longer durations. Since the late 1980s, a number of contributions have shown that the probability of leaving unemployment dramatically rises just prior to bene…ts lapse (see e.g. Mo¢ tt, 1985, [START_REF] Meyer | Unemployment insurance and unemployment spells[END_REF], and [START_REF] Katz | The impact of the potential duration of unemployment bene…ts on the duration of unemployment[END_REF], for the US, Ham and Rhea, 1987, for Canada, Carling et al, 1996, for Sweden, [START_REF] Joutard | Taux de sortie du chômage à l'approche de la …n des droits à l'indemnisation: une étude de périodes atypiques durant l'épisode de chômage[END_REF]Ruggiero, 1996, and[START_REF] Dormont | The e¤ects of the time pro…le of unemployment insurance bene…ts on the reemployment rate[END_REF], for France; Van Ours and Vodopivec, 2006, for Slovenia; see also Card et al, 2007, for a di¤erent perspective). However, hazard rates fall after they peak around the exhaustion date. The rise in hazard is predicted by the standard job search theory: reservation wages go down and search e¤orts go up as the exhaustion date becomes closer (see for instance Mortensen, 1977, 1986, and [START_REF] Van Den Berg | Non stationarity in job-search theory[END_REF]. However, the theory also predicts that hazard rates should stay constant afterwards, while they typically fall.

On the other hand, estimates show that the duration of unemployment positively responds to the various components of unemployment compensation generosity. However, it is more responsive to changes in potential duration than to changes in replacement rate. Well-known studies …nd positive and signi…cant e¤ects on unemployment duration from higher bene…ts (see for instance Narendranathan et al, 1985, Katz and[START_REF] Meyer | Unemployment insurance and unemployment spells[END_REF], Van den [START_REF] Van Den Berg | Non stationarity in job-search theory[END_REF], who obtain an elasticity of duration to bene…t typically lower than one). However, there is a wide dispersion in estimates, and several studies do not …nd any e¤ects (see Nickell, 1979, for the UK, [START_REF] Lynch | The youth labour market in the eighties: determinants of re-employment probabilities for young men and women[END_REF], for the US, [START_REF] Hujer | The analysis of labour market mobility using panel data[END_REF], for Germany, [START_REF] Groot | The e¤ects of bene…ts and duration dependence on re-employment probabilities[END_REF], for the Netherlands), or even a negative impact (see Jones, 1996, for Canada). Fewer studies examine the elasticity of average duration to potential duration. However, they conclude that this elasticity is indubitably positive (see [START_REF] Mo¢ Tt | Unemployment insurance and the distribution of unemployment spells[END_REF][START_REF] Katz | The impact of the potential duration of unemployment bene…ts on the duration of unemployment[END_REF][START_REF] Meyer | Unemployment insurance and unemployment spells[END_REF], for the US, Ham and Rea, 1987, for Canada, [START_REF] Ham | Unemployment and the social safety net during transitions to a market economy: evidence from the Czech and Slovak Republics[END_REF], for the Czech and Slovak Republics). This asymmetric response of unemployment duration to bene…t level and bene…t duration is nicely illustrated by [START_REF] Katz | The impact of the potential duration of unemployment bene…ts on the duration of unemployment[END_REF], also quoted in [START_REF] Atkinson | Unemployment compensation and labor market transitions: a critical review[END_REF]. They perform simulations from their estimates and show that a given UI expenditure cut achieved via reducing the length of entitlement has twice the e¤ect on unemployment duration of one coming via a cut in bene…t levels. This asymmetry is usually downplayed on the basis of its supposed inconsistency. If the potential duration of bene…ts plays a role, this must be so because the unemployed lose some income. As the magnitude of the loss is governed by replacement rates, replacement rates should also be part of the story 2 .

In this paper, equilibrium hazard rates result from the interplay between workers'job search strategies and employers' hiring strategies. We elaborate on [START_REF] Lockwood | Information externalities in the labour market and the duration of unemployment[END_REF], who examines the argument according to which the duration of unemployment conveys a signal on worker's ability, which leads employers to discriminate against the long-term unemployed. We argue that moral hazard e¤ects induced by unemployment compensation alter the value of the signal in a way that is consistent with the two empirical regularities highlighted previously.

A key aspect of our contribution relies on its ability to feature a realistic pattern of hazard rates as an equilibrium outcome of a model with worker heterogeneity, imperfect information, signalling, and moral hazard. Our model follows [START_REF] Lockwood | Information externalities in the labour market and the duration of unemployment[END_REF] with the noticeable exceptions that workers are allowed to set their search e¤ort and there is duration-dependent unemployment compensation. There are two types of workers, good and bad, and …rms are only willing to hire the good workers. At the beginning of unemployment episode, all workers are fully entitled to unemployment bene…ts. After a …xed interval of time, bene…ts fall to a lower level. Workers can set a low search e¤ort or a high one. At the time of interview, …rms have a positive probability of detecting a bad worker. This assumption has two implications. First, this ensures that exit rates are lower for the bad workers than for the good ones at given search e¤ort. Second, this drives employers' expectations on workers' type by unemployment duration. The bottom line argument is that workers who stayed unemployed for long have probably been interviewed elsewhere, and some other manager detected something wrong with the worker.

In this environment, agents have to select unemployment duration-contingent strategies that are mutually consistent in equilibrium. Workers set the pace of search e¤ort, while …rms set their hiring policy. The model may display di¤erent equilibrium con…gurations. We focus on one of them that is empirically relevant. We name it a baseline equilibrium 3 . A baseline equilibrium features three properties.

(i) Bad workers always choose a low search e¤ort. Unlike good workers, bad workers may be rejected by employers on the basis of the signal they send while interviewed. The marginal return to high search e¤ort is so low that they decide not to seek jobs with a high intensity. (ii) Good agents start seeking jobs with a low e¤ort, then set a high e¤ort prior to the potential duration of bene…ts. The opportunity cost of high search e¤ort goes down with unemployment duration. This leads the good workers to set a high search e¤ort as the exhaustion date becomes closer. (iii) Firms set a larger than the exhaustion date duration above which applicants always get rejected. The proportion of good workers among the applicants rapidly falls with duration around the exhaustion date. Firms adjust their beliefs accordingly and do not hire the long-term unemployed. We provide the set of necessary and su¢ cient conditions that leads to the existence and uniqueness of such a baseline equilibrium.

Then, we analyze how hazard rates respond to changes in the institutional environment. All the different components of unemployment insurance (potential duration and bene…t levels before and after the exhaustion date) originate moral hazard e¤ects and as such weaken the signal conveyed by unemployment duration. Therefore, both the duration at which good workers start searching for a job with a high e¤ort, and the cut-o¤ duration above which employers start rejecting applicants, are increasing in unemployment insurance generosity. However, the magnitude of the impact of each component depends on one key parameter: the return to search of a high e¤ort. When this parameter is large, good agents wait for a long time before setting a high search investment. In the non-frictional case where this parameter tends to in…nity, individuals set a high e¤ort once they have reached the exhaustion date and immediately get a job o¤er. Firms rationally expect they will only meet bad agents once the exhaustion date has elapsed and systematically reject all such applicants. More generally, the higher the return to search of a high e¤ort, the closer the cuto¤ duration to the exhaustion date. Interestingly, bene…t levels before and after the exhaustion date only marginally a¤ect this statement. This may explain why hazard rates respond substantially to the potential duration of bene…ts and not that much to bene…t levels.

Very few papers explain the decline in hazard after the exhaustion date. Two types of arguments have been put forward. First, seeking a job may require reducing the leisure time. [START_REF] Mortensen | Unemployment insurance and job search decisions[END_REF] shows that if leisure and consumption are substitutes, then a fall in bene…ts raises the opportunity cost of searching and hazard rates go down. Second, bene…ts can be used to improve job search e¢ ciency. This argument is due to [START_REF] Tannery | Search e¤ort and unemployment insurance reconsidered[END_REF]. In such a case, the opportunity cost of seeking a job is the marginal utility of consumption. When bene…ts go down, the marginal utility of consumption goes up and search spendings are reduced (see [START_REF] Ben-Horim | The e¤ect of unemployment insurance on unemployment duration[END_REF]Zuckerman, 1988, and[START_REF] Decreuse | On the time sequence of unemployment bene…ts when search is costly[END_REF].

Our paper is related to contributions that emphasize the role of employers'beliefs and hiring strategies to explain duration dependence in hazard rates. In [START_REF] Blanchard | Ranking, unemployment duration, and wages[END_REF], employers can meet several applicants at a time and marginally prefer workers with a short duration. Resulting hazard rates display negative duration dependence. In [START_REF] Coles | Retraining and long-term unemployment in a model of unlearning by not doing[END_REF], skills depreciate during the unemployment episode. Owing to recruitment costs, employers set a cut-o¤ duration very similar to Lockwood's and ours above which employers reject all applications. Our paper goes a step forward by highlighting the interaction between job search and recruitment strategies. It also argues that the design of unemployment compensation is a key variable a¤ecting the outcome of such an interaction.

The rest of the paper is organized as follows. Section 2 introduces our model and examines individual strategies. Section 3 is devoted to the analysis of equilibrium. Section 4 discusses some empirical implications. Section 5 concludes.

All proofs are gathered in the Appendix.

The model

This section introduces the main assumptions of the model, considers the microeconomic choices made by individuals and …rms, and characterizes the composition of unemployment by workers'type and duration.

Environment

We depart from [START_REF] Lockwood | Information externalities in the labour market and the duration of unemployment[END_REF] in three ways. First, there are unemployment bene…ts whose potential duration is …nite. Second, workers make search e¤orts. Third, the matching technology does not depend on the number of vacancies. 4We are interested in the steady-state of a continuous time economy populated by a continuum of …rms and workers. All agents have the same felicity function v which positively depends on consumption.

They discount time at rate > 0.

At each instant, a new cohort of individuals of total size n > 0 enters unemployment. This cohort is composed of 0 n good individuals, and (1 0 ) n bad individuals, 0 2 (0; 1). All these people are initially entitled to unemployment bene…ts. Workers only di¤er in their productivity, with y g > y b .

Workers can either be employed, unemployed, or non-participant. To ensure the existence of a steadystate number of unemployed, we assume that agents die/retire at rate n. When unemployed, agents have to seek a job, which means choosing a search e¤ort e. There are two levels of e¤ort: either e¤ort is high and e = h > 0, or e¤ort is low and e = 0. The cost of e¤ort is ce, c > 0. This implies that the cost of a low e¤ort is normalized to zero. The probability of contacting a vacant position in the time interval dt is

m (1 + e) dt.
Each …rm is endowed with a single job slot, which can either be …lled or vacant. Firms endowed with a vacant position must incur the cost > 0. Filled jobs produce either y g or y b , depending on the worker's type. There is a single wage w 2 (y b ; y g ), which is set exogenously. We discuss alternative assumptions on the wage setting in subsection 4.2. As a result of this wage, …rms do not want to hire bad workers who generate negative pro…ts. However, the worker's type is imperfectly observable. Firms receive a private signal on the worker's type at the time of interview. If the worker is good, the signal is good with probability g = 1. If the worker is bad, the signal is good with probability b = 2 (0; 1).

It is bad with the complementary probability 5 1

. Once a worker has found a job, he/she leaves the search market forever and enjoys the utility level W = v (w) =r.

The fact that employed workers leave the search market forever deserves further comments. We here follow Lockwood who argues that accounting for job loss would complicate the model in an unexpected way: …rms would condition their hiring strategies on cumulative durations of unemployment spells.

Although there is some empirical support for this phenomenon, it can be neglected in the …rst place to shed light on the mechanisms that are speci…c to this paper. 6The ‡ow number of matches depends only on the number and e¤orts of job-seekers. Let u denote the mass-number of active unemployed, and let e denote their average search e¤ort. The total number of matches is um (1 + e). Matching is random, which means that meetings are equiprobably distributed between the two sides of the markets, i.e. good and bad workers have the same probability of contacting a vacant job. The contact rate for a vacant position is m= , where = v=[u (1 + e)] is the labor market tightness.

The unemployment compensation scheme runs as follows. Unemployed workers receive unemployment bene…ts b (s) contingent on unemployment duration s. Let T be the potential duration of unemployment bene…ts. We have:

b (s) = ( b max > 0 if s T b min < b max else (1)
To ensure that agents do not refuse job o¤ers, b max < w.

Job-seeker behavior

In this sub-section, we examine the job-seeking behavior of unemployed individuals at given …rms'hiring policy.

Let U i (s) denote the value function of a type i unemployed whose unemployment duration is s. We have

rU i (s) = max e=h;0 fv (b (s)) + m (1 + e) i (s) [W U i (s)] ce + U 0 i (s)g ( )
where r = +n is the e¤ective discount rate, and i (s) 2 [0; 1] is the probability that the worker becomes hired once he contacted a vacancy. The reason why this probability is denoted i (s) will be made clear below. The hazard rate has three components: job availability, summarized by parameter m, worker' search e¤orts, captured by (1 + e), and …rms'hiring policy i (s).

Workers' search behavior depends on the probability of getting the job once interviewed. In the remaining, we solve the optimization problem ( ) for a particular hiring policy that will be the equilibrium one. This policy is de…ned by Assumption A1.

Assumption A1 Firms' hiring policy is given by

i (s) = ( i if s , with > T 0 else
According to this policy, …rms reject applicants whenever they receive a bad signal during the interview. This event occurs with probability 0 when the worker is good, and with probability 1 when the worker is bad. In addition, employers discriminate against the long-term unemployed: employers do not hire the workers who remain unemployed more time than the cut-o¤ duration . We restrict our attention to the empirically plausible case where T . In what follows, we will refer to this particular con…guration as the baseline equilibrium con…guration. We will provide additional restrictions later to ensure that such an equilibrium exists.

The optimization problem ( ) can be written as follows

rU i (s) = max e=h;0 fv (b (s)) + m (1 + e) i [W U i (s)] ce + U 0 i (s)g (2)
rU i ( ) = v (b min ) (3)
The resulting value function U i (s) is strictly decreasing for all s 2 [0; ].

Let e i (s) denote the optimal trajectory of e¤ort. This trajectory satis…es e i (s) = h if and only if

i m [W U i (s)] c (4)
This implies that either e i (s) = 0 for all s 0, or there exists a unique duration i 2 [0; ] such that e i (s) = h i¤ s i . The problem must be solved backward. In the remaining,

a i = i m (1 + h). Step 1. If i m [v (w) v (b min )] < rc, then e i (s) = 0 for all s 0. If not, go to Step 2.
Step 2. Solve the following Cauchy problem for all s

rx 1 i (s) = v (b min ) + a i W x 1 i (s) ch + x 10 i (s) (5)
rx 1 i ( ) = v (b min ) (6)
This yields

x 1 i (s) = v (b min ) + a i W ch r + a i h 1 e (r+ai)( s) i + v (b min ) r e (r+ai)( s) If i m W x 1 i (T ) < c, then i is such that i m W x 1 i ( i ) = c. If not, go to Step 3. Step 3. Solve the following Cauchy problem for all s T rx 2 i (s) = v (b max ) + a i W x 2 i (s) ch + x 20 i (s) (7)
x 2 i (T ) = x 1 i (T ) (8)
This yields

x 2 i (s) = v (b max ) + a i W ch r + a i h 1 e (r+ai)(T s) i +x 1 i (T ) e (r+ai)(T s) If i m W x 2 i (0) < c, then i is such that i m W x 2 i ( i ) = c. If not, e i (s) = h for all s .
Note that e b ( ) = h implies that e h ( ) = h. This property allows us to focus on the con…guration where e g ( ) = h and e b ( ) = 0.

Assumption A2 m [v (w) v (b min )] < rc Assumption A3 m W x 2 g (0) < c < m W x 1 g (T )
The following Proposition summarizes our results.

Proposition 1 Job-seekers'efforts

Let = T + ln 2 4 v (b max ) + c r+m m v (w) v (b max ) v (b min ) + m (1 + h) v(w) v(bmin)
r ch e (r+m(1+h))( T ) (ii) Good workers set e g (s) = h if s 2 [ ; ] and e g (s) = 0 else Assumption A2 ensures that bad workers always set the low level of e¤ort. Assumptions A3 guarantees that good workers set the high level of e¤ort before they have reached the potential duration of bene…ts T , but after some time spent in unemployment -that is 2 (0; T ). Note that Assumptions A1 to A3 are necessary and su¢ cient conditions.

Figure 1 depicts the resulting patterns of hazard rates.

[Insert Figure 1] The hazard rate of good workers features the typical spike prior to losing bene…t entitlement. The spike in hazard lasts until …rms discriminate against the long-term unemployed. At the same time, the hazard rate of bad workers is ‡at throughout the spell of unemployment.

We focus on this particular con…guration for simplicity. We discuss alternative equilibrium con…gurations and some empirical implications in Section 4.

The composition of unemployment

The previous subsection carefully examines job-seekers'behavior and resulting hazards. In this Subsection, we analyze the implications of such hazard rates on the distribution of unemployment spells, the distribution of type by unemployment duration, and the distribution of contact by type and unemployment duration.

Let u i (s; t) denote the size of the cohort of type-i unemployed whose unemployment duration is s as of time t. It evolves according to the following partial di¤erential equation:

@u i (s; t) @s + @u i (s; t) @t = [ i (s) m (1 + e i (s)) + n] u i (s; t) (10)
u i (0; t) = n i (0) (11)
The total size of the cohort of duration s unemployed is u (s; t) = u g (s; t) + u b (s; t). Finally, the number of job-seekers is U (t) = R 1 0 u (s; t) ds. Given that we only focus on a steady-state, @u i (s; t) =@t = 0 and the dependence vis-à-vis time t will be neglected.

Let (s) denote the pdf of the distribution of unemployment duration, while i (s) denotes the proportion of type-i unemployed conditional on duration s. By construction

(s) = u (s) U ( 12
) i (s) = u i (s) u g (s) + u b (s) (13)
Finally, let p i (s) denote the probability of contacting a type-i worker conditional on contacting a worker whose unemployment duration is s. Random matching implies that

p i (s) = m [1 + e i (s)] u i (s) m [1 + e g (s)] u g (s) + m [1 + e b (s)] u b (s) (14)
Proposition 2 Distribution of types and unemployment spells Under Assumptions A1 to A3,

g (s) = ( 0 0+(1 0)e m( 1
)s if s < 0 0+(1 0)e mh +m(1+h )s if s 2 [ ; ] (15)
p g (s) = ( 0 0+(1 0)e m( 1
)s if s < (1+h) 0 (1+h) 0+(1 0)e mh +m(1+h )s if s 2 [ ; ] (16)
(s) =

P i=g;b n i (0) exp R s 0 [m i (z) (1 + e i (z)) + n] dz R 1 0 P i=g;b n i (0) exp R x 0 [m i (z) (1 + e i (z)) + n] dz dx 8s 2 [0; 1) (17)
where , e g (s) and e b (s) are de…ned in Proposition 1

Figure 2 depicts the pattern of the proportion of good workers by unemployment duration.

[Insert Figure 2] As the hazard rate of good workers is always higher than the hazard rate of bad workers, the proportion of good workers falls with the duration of unemployment. Indeed,

0 g (s) = m [1 + e g (s) ] g (s) [1 g (s)] < 0 (18)
Note that the function g is continuous on [0; M], while its derivative is not continuous in , when good workers start searching harder. [Insert Figure 3]

The function p g is strictly decreasing on each interval where it is continuous. Indeed,

p 0 g (s) = m [1 + e g (s) ] p g (s) [1 p g (s)] < 0 (19)
However, the function p g jumps upwards in , as good workers suddenly become overrepresented among the applicants.

This discontinuity is important because the function p g shapes employers'beliefs on the composition of applicants by duration. The resulting hiring policy must be consistent with the one that has been postulated in Assumption A1. Typically, the existence of an equilibrium will require that lim s"

p g (s) > p g ( ).
The probability of contacting a good agent conditional on contacting an unemployed person of duration must be larger than the probability of contacting a good agent conditional on contacting an unemployed person of duration . Figure 3 has been drawn assuming that this restriction holds.

Finally, note that the pdf of the unemployment duration distribution has three properties: it is continuous on the support [0; ], it is strictly decreasing in s, re ‡ecting the fact that none can enter the distribution at some positive duration, and its derivative is discontinuous in s = , as good workers start …nding jobs at a faster rate.

Firm behavior

Let V > b denote the value of a vacant position, while i is the value of a job …lled by a worker of type i. We have7

V = + m E s max fE i [ i V j s] ; 0g (20)
r i = y i w (21)
Firms endowed with a vacant slot have to predict the type of workers they may meet. They do so on the basis of (i) the unemployment duration s that they observe, (ii) their expectation on the trajectory of individual search e¤orts, and (iii) the probability 2 (0; 1) of not detecting a bad worker at the time of interview.

Good workers do not send bad signals. Firms, therefore, can immediately detect a bad worker when they receive a bad signal. As b < V , …rms always reject such workers.

It follows that

E i [ i V j s] = Pr (signal is good) Pr (worker is good j signal is good) ( g V )
+ Pr (signal is good) Pr (worker is bad j signal is good)

( b V ) + Pr (signal is bad) 0
Firms'beliefs on workers'types obey the Bayes rule. Therefore, Pr (worker is good j signal is good) = Pr (worker is good \ signal is good) Pr (signal is good)

= p g (s) p g (s) + p b (s)
and Pr (worker is bad j signal is good) = Pr (worker is bad \ signal is good)

Pr (signal is good) = p b (s) p g (s) + p b (s)
For a particular …rm, we have

V = + m Z 1 0 (s) max f[p g (s) ( g V ) + (1 p g (s)) ( b V )] ; 0g ds (22)
Expected pro…ts depend on two distributions: the distribution of unemployment durations, and the distribution of workers'types conditional on contact and unemployment duration8 . Firm's hiring policy e i (s) follows.

Proposition 3 Firms'hiring strategy

The policy function e i (s) is such that

e i (s) = ( i if p g (s) b V g b (1 )V 0 else
A duration-s type-i worker that is interviewed is hired with probability i if the proportion of good workers among contacted workers is su¢ ciently large, and is not hired if not.

Finally, there is free entry of new …rms on the search market. This drives the value V of vacancy to zero.

Equilibrium time-dependence in hazard rates

This section considers the equilibrium of the model. We proceed in three steps. First, we study the existence and uniqueness of equilibrium. Second, we make comparative exercises and study how hazard rates respond to changes in the design of unemployment compensation. Finally, we discuss the impact of the exhaustion date on mean unemployment duration.

Equilibrium

In a symmetric equilibrium, all …rms have the same hiring policy i (s), all workers of the same type have the same job search behavior e i (s), agents maximize their gains, and expectations are compatible with equilibrium outcomes. We focus on the particular type of equilibrium that we have highlighted so far. Given that our framework may feature other types of equilibrium, we need to di¤erentiate this equilibrium from other types. In the sequel, we will call it the baseline equilibrium.

De…nition Baseline equilibrium

A baseline equilibrium is a vector ( ; ; ) and a set of four functions (e g ; e b ; ; p g ) such that (i) e b (s) = 0 for all s 0 and e g (s) = h i¤ s 2 [ ; ], where is de…ned in Proposition 1

(ii) and p g are de…ned in Proposition 2

(iii) For i = g; b, f i (s) = e f i (s) = i i¤ s 2 [0; ], where e f i (s) is de…ned in Proposition 3 (iv) V = 0 (v) 0 < < T <
(i) states that the postulated job-seeking behavior is optimal for both types of workers. (ii) recalls that the distribution of unemployment duration and the proportion of good workers by duration are implied by individual strategies. (iii) states that the postulated hiring policy is optimal for …rms. (iv)

is the free-entry condition. Finally, (v) makes clear that good workers set a high search intensity before bene…ts have elapsed and potential employers discriminate against the long-term unemployed after the loss of bene…t entitlement.

Proposition 4 Existence and uniqueness of a baseline equilibrium (i) In a baseline equilibrium, we have

= T + ln 2 4 v (b max ) + c r+m m v (w) v (b max ) v (b min ) + m (1 + h) v(w) v(bmin)
r ch e (r+m(1+h))( T ) 3 5

1 r+m(1+h) (JS) = h 1 + h + 1 m (1 + h ) ln y g w (w y b ) 0 (1 + h) 1 0 (HS) = m Z 0 (s) [p g (s) g + (1 p g (s)) b ] ds (

FE)

(ii) There may exist a baseline equilibrium (iii) If a baseline equilibrium exists, it is unique (i) The (JS) locus results from good workers'equilibrium job search strategy. The (HS) locus results from …rms' equilibrium hiring strategy. This strategy implies that p g ( ) = b = ( g b ), which yields (HS). Finally, tightness results from the free-entry condition. This yields (FE). Given that we assume that job search e¤orts do not create congestion e¤ects, tightness determination has no feed-back e¤ects on individual choices. Solving the equilibrium can be reduced to …nding a couple ( ; ) that satis…es (JS) and (HS).

(ii) We provide during the proof of Proposition 4 the set of necessary and su¢ cient conditions leading to the existence of a baseline equilibrium. These conditions are not particularly appealing. Indeed, we must check that (JS) and (HS) intersect at least once. We must also check that 0 < < T < . Finally, we must check that lim s! p g (s) p g ( ). This leads to four inequalities that de…ne the parameter space compatible with the existence of a baseline equilibrium. Of course, we show that this parameter space is nonempty.

(iii) The (JS) locus and the (HS) are both strictly increasing. On the one hand, a longer cut-o¤ duration raises the value of search, and good workers delay the moment at which they start searching with high intensity. On the other hand, an increase in raises the proportion of bad workers at all durations. In turn, this leads …rms to delaying the duration above which they reject all applications.

However, the slope of the (JS) curve is always lower than the slope of the (HS) curve. This establishes the uniqueness of equilibrium.

Note that there is a single equilibrium in our model, while there may be multiple equilibria in Lockwood's. This is so because Lockwood assumes that the matching rate m depends on the market tightness . This originates a feed-back e¤ect from job creation to the composition of unemployment. Namely, job pro…tability increases with the mean productivity of the job-seekers. But, tightness raises such a mean productivity. Job pro…tability may increase with tightness as a result, which explains multiple equilibria.

Abstracting from such feed-back e¤ects allows us to focus on the novelty of our paper: the interaction between workers'search and employers'hiring strategies.

Figure 4 depicts the equilibrium.

[Insert Figure 4] Proposition 4 allows us to interpret the duration-dependence in hazard rates observed in the data as an equilibrium outcome. This results from the interplay between job search and hiring strategies in an environment characterized by matching frictions, worker heterogeneity and asymmetric information on workers'types. Employers have no reasons to discriminate against the unemployed before the exhaustion date because good workers have strong reasons to stay unemployed. Things become di¤erent after the exhaustion date because good workers should manage to exit unemployment around the exhaustion date. This explains the fall in hazards after the exhaustion date: employers reject the applications of unemployed who have no reasons to stay unemployed unless they are of the bad type.

Changes in unemployment compensation scheme

In this Subsection, we examine the impacts of unemployment compensation on equilibrium hazard rates.

Proposition 5 Properties of the baseline equilibrium

Assume that there exists a baseline equilibrium. Then, (i) and are strictly increasing in b min , b max and T

(ii) as h tends to in…nity, and tend to T (i) shows that the various components of unemployment compensation generosity originate moral hazard e¤ects that are detrimental to search e¤orts. Confronted with a more generous scheme, good workers wait longer to make high search e¤orts and increases. However, such moral hazard e¤ects alter the signalling value of unemployment duration. The probability of recruiting a good worker increases at all durations. In turn, employers are less reluctant to hire the long-term unemployed and the cut-o¤ duration also increases. (ii) shows that and tend to T in the non-frictional case where h becomes arbitrarily large.

Indeed, good workers await the exhaustion date to set the high search e¤ort. They immediately exit unemployment and no good workers remain among the cohort of unemployed. Employers expect this and do not hire the workers who have overtaken the exhaustion date in unemployment.

According to (i), all the di¤erent components of unemployment insurance originate moral hazard e¤ects. But, this says nothing about the magnitude of the di¤erent e¤ects. (ii) tells us that the magnitude depends on parameter h. In the non-frictional case where h tends to in…nity, the potential duration of bene…ts governs equilibrium hazard rates, while replacement rates b max and b min have no impacts.

This provides a simple explanation to the fact that estimated hazard rates respond less to changes in bene…t levels than to changes in potential duration. If good workers can activate a su¢ ciently e¢cient technology, they massively exit the unemployment state around the exhaustion date. This drives employers'beliefs who reject all the applicants of a cohort whose unemployment duration is larger than

T .
Proposition 5 tells a general lesson. The design of unemployment insurance a¤ects the signalling value of unemployment duration. This should be taken into account by policy makers.

From bene…t expiration to mean unemployment duration

The model can explain why hazard rates mostly respond to the potential duration of bene…ts. However, the model does not necessarily predict that the average unemployment duration always increases with the exhaustion date. On the one hand, good workers respond to the exhaustion date by delaying high-intensity search. This tends to raise the mean duration. On the other hand, …rms are less discriminating against the long-term unemployed, which bene…ts the bad workers. This tends to lower the mean duration.

Formally, group-i speci…c mean duration is

s i = Z 1 0 e R s 0 (m i (z)(1+ei(z))+n)dz ds (23)
In the limit case where h tends to in…nity, we obtain

s g = 1 e (m+n)T m + n ( 24
)
s b = 1 e ( m+n)T m + n + e ( m+n)T n ( 25
)
Good workers'durations are truncated in the exhaustion date T , date at which they immediately exit the unemployment state. Their mean duration increases with T . Bad workers do not escape unemployment at the exhaustion date. Rather, they start being discriminated against afterwards. As a consequence, ds b =dT < 0, and their mean duration decreases with T .

Consider a worker who has just entered unemployment. This worker is good with probability 0 and bad with probability 1 0 . Therefore, the expected unemployment duration for such a worker is s = 0 s g + ( 10 ) s b . A marginal increase in the exhaustion date leads to ds dT = e (m+n)T h 0 e (1 )mT (1

0 ) i ( 26
)
The mean duration of unemployment increases with the exhaustion date whenever 0 e (1 )mT >

(1 0 ). The main reason why the potential duration of bene…ts has an ambiguous impact on mean unemployment duration is the fact that bad workers do not react to changes in unemployment compensation schemes. Unemployment insurance originates moral hazard e¤ects for the good, and not for the bad. In subsection 4.1, we discuss another equilibrium con…guration in which good and bad workers start seeking jobs with a high intensity. In this case, the mean duration increases with the exhaustion date.

Discussions

In this section, we discuss several aspects of our model. First, we consider two theoretical issues: the existence of alternative equilibrium con…gurations, and the case of endogenous wage. Then, we turn to empirical considerations: the model makes predictions on individual hazards that may be confronted to data, and we also make particular assumptions as to employers'information set that merit discussion.

Alternative equilibrium con…gurations

In this subsection, we explore the alternative equilibrium con…gurations that our model may feature. Doing so, we provide further motivations for our focus on a baseline equilibrium.

The panel of Figures 5 shows various combinations of hazard rates for good and bad agents. Each con…guration corresponds to a di¤erent type of equilibrium. The baseline equilibrium is depicted by Figure 5a to make the comparison easier.

[Insert panel of Figures 5] Figure 5b shows a Lockwood equilibrium, in which good and bad workers never seek jobs with a high intensity. This type of equilibrium may arise when m [v (w) v (b min )] < rc, which is not compatible with assumption A3. In such a case, good agents' marginal return to high search e¤ort is lower than marginal cost. A Lockwood equilibrium also arises when h tends to 0. In a Lockwood equilibrium, unemployment insurance design has no e¤ects on job search behavior, and, therefore, no impacts on employers'expectations and recruitment strategies.

Figure 5c depicts a credible set of hazard rates. Both good and bad workers start seeking jobs with high intensity before the exhaustion date. Such a pooling equilibrium may arise when assumption A2 does not hold. The threshold duration is b for bad workers, while it is g for good workers. Figure 5c assumes that g > b , but we may also have g < b . The striking feature is that g generally di¤ers from b . In a pooling equilibrium, unemployment compensation originates moral hazard e¤ects on all the individuals. However, given that good workers bene…t from a higher hazard rate at given search e¤ort, the main results featured by Proposition 5 should not be a¤ected. In particular, g , b , and would tend to the exhaustion date T as h would tend to in…nity.

Interestingly, this equilibrium con…guration may lead to non-ambiguous predictions concerning the impact of the exhaustion date on mean unemployment duration. Consider for instance the case where h tends to in…nity. In such a case, the mean durations of good and bad workers are

s g = 1 e (m+n)T m + n (27)
s b = 1 e ( m+n)T m + n (28)
The two durations are now increasing in T . As a result, the average mean duration increases with the exhaustion date.

We have decided not to focus on this equilibrium since it is associated with a technical di¢ culty that obscures the main message of the paper. When a given group of individuals start seeking jobs with a high intensity, this originates a discontinuity in the function p g that shapes the proportion of good workers among contacted individuals of a given unemployment duration. This function plays a key role as this drives employers'expectations on workers'types. This discontinuity gives birth to a parametric restriction to make sure that the proportion p g stays larger than the threshold implied by the zero-pro…t condition. With two groups of workers, the function p g is discontinuous in g (where it jumps upward)

and b (where it jumps downward). The former case is covered by the de…nition of a baseline equilibrium.

The latter case would be associated to a new parametric restriction.

Figures 5d to 5f show equilibrium con…gurations that are not empirically credible. In Figure 5d, the function p g decreases so rapidly that …rms discriminate against the unemployed before good agents seek jobs at a high intensity. Then, …rms hire the workers again, before turning down all the applicants once the …nal duration is reached. In Figures 5e, good workers start seeking jobs after the exhaustion date ( > T ), while …rms set a termination date shorter than the exhaustion date ( < T ) in Figure 5f. These equilibria imply that hazard rates do not peak at the exhaustion date.

Endogenous wage

The wage does not depend on unemployment duration. This may result from some minimum wage, or collective wage setting. However, one may discuss two alternative assumptions.

First, …rms may set the wage as in [START_REF] Lockwood | Information externalities in the labour market and the duration of unemployment[END_REF]. The monopsony wage would be set so as to guarantee that W g (s) = U g (s). Firms have no reasons to choose a wage that yields a utility level larger than the utility reached by a good unemployed person. Similarly, they would not set a lower wage, because good workers would refuse the jobs. The equilibrium wage decreases over the unemployment duration from 0 to the exhaustion date T . However, this would leave no search incentive to the good workers 9 .

They would never set the high search e¤ort as a result. Meanwhile, unemployment compensation would not alter workers'search incentives.

Second, there could be wage bargaining over match surplus without possible renegotiation. One of the players' types is unknown to the other player at the time of interview. One way to deal with this issue is to consider a simple process of take-it-or-leave-it o¤er where the player who initiates the o¤er is chosen randomly. For simplicity, we consider risk-neutral individuals. Assume that a worker whose unemployment duration is s meets an employer endowed with a vacancy. Suppose that the …rm makes the o¤er with probability 1 . The …rm won't o¤er less than U g (s). If the worker is good, he will refuse the o¤er. If the worker is bad, he will refuse the contract too. If he accepted, he would reveal that he is bad, leading the …rm to reject his application. Similarly, the …rm won't o¤er more than U g (s).

The worker makes the o¤er with probability . The …rm obtains 0 in that case, while the worker gets the whole pie. Due to type uncertainty, the worker obtains Y (s) = E i [Y i j s], with Y i = y i =r. As U g and Y fall over time, the mean wage falls with unemployment duration. The termination date is set so that Y ( ) = U g ( ). Good workers obtain no rents at the exhaustion date. This implies that they do not seek jobs with a high intensity around the exhaustion date. This may be compatible with the fact that hazard rates peak at the exhaustion date, but this requires additional parametric restrictions.

The …xed-wage assumption allows these additional di¢ culties to be neglected, without losing the main insights.

From theoretical to empirical hazards

The signalling argument relies on the coexistence of two subpopulations. They correspond to the divide between movers and stayers highlighted by the empirical literature on unobserved heterogeneity. In our paper as in Lockwood, the distinction between movers and stayers is an equilibrium outcome. A key feature of this approach is that ex-ante heterogeneity shapes employers' and workers' beliefs in a way that originates true duration dependence in hazard rates.

Hazard rates have three main properties in our model: (i) they are piecewise continuous, (ii) unobserved good workers bene…t from a higher exit rate at all durations than unobserved bad workers, (iii) good workers'and bad workers'hazards respond di¤erently to bene…t exhaustion. 9 There would be no more rents for bad workers. In equilibrium,

Wg (s) = W b (s) = Ug (s) = U b (s) = R 1 s e r(t s) b (t) dt.
(i) is consistent with piecewise constant hazard models introduced by [START_REF] Lancaster | The econometric analysis of transition data[END_REF]. In such models, the hazard is typically written as follows

(s j x) = 0 (s) l (x) (29)
where x is a vector of observed individual characteristics, and 0 is the baseline hazard. The duration axis is divided into M intervals with

0 (s) = 8 > > > > < > > > > : 1 if s 2 [0; 1 ] 2 if s 2 ( 1 ; 2 ] ::: M if s 2 ( M 1 ; 1) (30)
where k are constant and k are de…ne points in time, 0 < 1 < 2 < ::: < M 1 < 1.

In our model, workers have all the same observed characteristics. However, the analysis can be generalized to workers with heterogenous observed characteristics, with the assumption that search markets are segmented by observed characteristics. In the case of good workers, our model predicts three time intervals (M = 3), with 1 = , 2 = , 1 = m, 2 = m (1 + h) and 3 = 0.

(ii) Our model hinges on the fact that there are two groups of workers on each market segment, the good and the bad, that cannot be di¤erentiated by employers. These proportions may or may not di¤er between observable groups, yet the intuition suggests that the initial proportion of good workers should increase with skill level. This means that there is unobserved heterogeneity, and the econometrician should account for it.

(iii) There are several ways to model unobserved heterogeneity to capture the mover-stayer dichotomy. However, they are not all compatible with our model. The simplest way to deal with unobserved heterogeneity is to assume that there is an individual speci…c component " in hazard that is independent of both s and x. Formally,

(s j x; ") = 0 (s) l (x) " (31)
It is then usual to assume a simple functional form for the distribution of the error term, like the Gamma distribution, or, closer to our model, a discrete distribution (see [START_REF] Lancaster | Econometric methods for the duration of unemployment[END_REF].

However, this hazard function implies that the two subpopulations must experience similar qualitative patterns in baseline hazard rates. By contrast, our model predicts distinctive qualitative patterns across the two groups of workers. In other words, the individual component " should not be independent of sand probably of x as well. This requirement may be too strong to be compatible with identi…cation, yet it is an essential feature of our model.

Further evidence

Dormont et al (2006) provide another type of evidence. They estimate hazard rates by pre-unemployment earnings. They show that the hazard rate of formerly high-paid workers features a spike, while the hazard rate of formerly low-paid workers is fairly smooth around the exhaustion date. Of course, the distinction between low-paid and high-paid workers does not really …t with our model in which there is a single wage.

However, this suggests that ability to respond to bene…t exhaustion should be positively correlated to skill level.

In addition, our model provides a simple explanation to Dormont et al …nding. Suppose that there is a separate search market for each skill, and that formerly high-paid workers compete on the high-skill segment, while formerly low-paid workers compete on the low-skill segment. Suppose also that for one reason or another, estimation techniques used by Dormont et al fail to identify the two groups of workers in each segment. In such a case, they estimate the average hazard among each group of workers. Provided that the proportion of good workers increases with skill level, our model predicts that the mean average rate and the mean peak in such a rate should be higher among the formerly higher-paid workers than among the formerly lower-paid workers.

Employers'information set

Our model hinges on the assumption that employers can observe both one's unemployment duration and one's potential duration of bene…ts.

Several other papers assume that employers observe the unemployment duration (see e.g. [START_REF] Blanchard | Ranking, unemployment duration, and wages[END_REF]Diamond, 1994, Coles and[START_REF] Coles | Retraining and long-term unemployment in a model of unlearning by not doing[END_REF]. They rely on the fact that cvs implicitly display this information, or that employers should be able to obtain it during the interview. Oberholzer-Gee (2008) o¤ers empirical evidence consistent with this view. He sent applications to jobs advertised in a Swiss newspaper. Applicants only di¤ered with respect to employment status (employed, nonemployed) and unemployment duration (if unemployed). He shows that the longer the spell of nonemployment, the lower is the probability that …rms will invite a job applicant for an interview.

The potential duration of bene…ts may or may not be di¢ cult to observe. There are two main obstacles.

First, eligibility rules may be very complicated. They can vary with former job duration or demographic characteristics. Nevertheless, employers should always be able to evaluate the applicant's situation vis-à-vis the bene…t system using the information displayed by the applicant's cv (former job duration and current unemployment duration). In addition, there are certain regularities that depend on the type of advertised job that should be known by employers. For instance, a senior position should mostly attract relatively experienced workers who entered unemployment with full coverage and maximum potential duration 10 (or, similarly, workers bene…ting from a longer duration because of their age as in France, Germany or Sweden). Conversely, a very junior position should attract individuals who are mostly non-entitled to unemployment bene…ts.

Second, the legislation may be volatile. One interesting question concerns the impact of such volatility on employers'beliefs vis-à-vis the long-term unemployed. Intuitively, risk-averse workers facing institutional uncertainty should be more prompt to exit the unemployment state. Unless labor market skills and risk aversion are too negatively correlated, this e¤ect should be stronger for the good workers. Overall, the quality of the signal conveyed by unemployment duration should increase with institutional volatility, strengthening employers'discrimination vis-à-vis the long-term unemployed. The rigorous formalization of this argument is left for future work.

However, employers are likely to know whether one is covered by unemployment bene…ts or not.

Indeed, it is in the interest of covered workers to say and prove that they are covered. An unemployed worker who is no longer covered can easily be detected in such a case.

Conclusion

This paper is a theoretical contribution to the literature on job search and unemployment. We revisit the signalling hypothesis, whereby potential employers use the duration of unemployment as a signal on the productivity of applicants. We suggest that the quality of such a signal is very low when the unemployed get unemployment bene…ts: individuals have good reasons to stay unemployed. Conversely, the signal becomes much more e¢ cient once bene…ts have elapsed: skilled workers should not remain unemployed in such cases. Therefore, the potential duration of unemployment bene…ts should drive employers' expectations and their recruitment practices. This mechanism can explain why hazards fall after bene…t expiration, and why hazards respond more to the potential duration of bene…ts than to replacement rates.

Our paper can be extended in two directions. First, we plan to enrich agents'decision sets to account for non-trivial search and recruitment strategies. Such a model could be calibrated on country data and used to simulate policy changes. Second, beyond its focus on hazard rates, our paper tells that the design of unemployment compensation alters the signalling value of unemployment duration. This should be taken into account by policy makers. Policy implications such as the design of optimal unemployment insurance are on our research agenda.

and are jointly determined by (JS) and (HS). Then follows from (FE). Therefore, the solution can be reduced to …nding and that solve (JS) and (HS) and satisfy conditions (36) to (38).

We now provide necessary and su¢ cient conditions for the existence of a baseline equilibrium.

Lemma 1. Let The function P is strictly increasing, with P ( 1) = A 1 and P (1) = 1. It follows that there is a unique such that P ( ) = 0. This and associated given by (HS) is an equilibrium candidate. This candidate must satisfy constraints (36) to (38). First, > 0 i¤ P (0) < 0. Second, < T i¤ P (T ) > 0. Third, using (HS), one can see that > T is equivalent to > Z 1 . Therefore, > T i¤ P (Z 1 ) < 0. Fourth, lim

  Under Assumptions A1 to A3, (i) Bad workers set e b (s) = 0 for all s 0;

Figure 3

 3 Figure 3 depicts the pattern of the proportion of good agents by contact among individuals who have spent the duration s in unemployment.

Figure 4

 4 Figure 4 depicts these e¤ects of unemployment insurance. The components of unemployment insurance only a¤ect the (JS) locus that shifts rightward. The equilibrium moves along the (HS) locus that is positively sloped. and increase as a result.

A 1 = 1 0( 1 1 y g w (w y b ) 0 1 0

 11111 v (b max ) + c r + m m v (w) A 2 = v (b max ) v (b min ) A 3 = m (1 + h) v (w) v (b min ) r ch A 4 = ln y g w (w y b ) 0 + h)Let alsoa 1 = r + m (1 + h) a 2 = m (1 + h )Consider the function P : R ! R such thatP (x) = A 2 e a1T e a1x + A 3 (e a1x )There exists a baseline equilibrium i¤m [v (w) v (b min )] < rc (C1) P (max f0; Z 1 g) < 0 (C2) P (min fT; Z 2 g) > 0Proof. Note …rst that bad agents never set the high e¤ort i¤ condition (C1) holds. Using (HS), one can replace in (JS). After simple computations, one obtains P ( ) = 0

  s"p g (s) > b = ( g b ) is equivalent to < p 1 g ( b = ( g b )).In turn, this is equivalent to P (Z 2 ) > 0. Conditions C2 and C3 result from these four cases.

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :Fig. 6 :

 12346 Fig.1 : Hazard rates of good and bad workers -Baseline equilibrium

[START_REF] Oberholzer-Gee | Nonemployment stigma as rational herding: A …eld experiment[END_REF] displays direct evidence in favor of such a thesis. He conducted a survey among 766 managers in Switzerland who are responsible for hiring at administrative assistant level. Among the various reasons why they may prefer an employed worker to a

[START_REF] Tannery | Search e¤ort and unemployment insurance reconsidered[END_REF]-month nonemployed, they predominantly answer that they "prefer the candidate with a job because the unemployed applicant is probably not very productive. If she were productive, she would have been hired by another …rm."

For instance, the entitlement e¤ect put forward by[START_REF] Mortensen | Unemployment insurance and job search decisions[END_REF] can explain why bene…t levels may have a negative impact on average duration. However, the potential duration of bene…ts should also have a negative e¤ect in such a case.

We discuss alternative equilibrium con…gurations in subsection

4.1.

The latter assumption allows us to neglect the labor demand side of the model, while focusing on the novel aspects that we stress.

This is a reduced form of Lockwood, who assumes that …rms can choose whether to test workers prior to hiring them.The parameter i is then the probability of passing the test. Good workers are always successful and g = 1, while bad workers may fail and b 2 (0; 1).

One may wonder where workers entitled to unemployment bene…ts come from in a world without job destruction. We implicitly focus on a particular segment of the job market that is mostly reserved to experienced workers who already spent some time in their …rst job. The new cohort of workers is composed of those workers who lost or quit their …rst jobs.Alternative assumptions may be considered to preserve the general meaning of the model. They would come at the cost of losing its simplicity.

Equation[START_REF] Mortensen | Job search and labor market analysis[END_REF] assumes that the job is destroyed whenever the worker retires. This assumption is innocuous, because the value of a vacancy V is driven to 0 in equilibrium.

Note that unemployable workers participate in the search market. This results from the cost structure of search e¤orts:given zero cost for a low e¤ort, agents are marginally indi¤erent between searching for a job or not. Assuming an "-cost would su¢ ce to prevent non-employable workers from searching. In such a case, one would have to consider the distribution of unemployment duration conditional on the fact that i (s) > 0. Working of the model would not be a¤ected.

0 One may argue that signalling problems are less relevant for such workers who have a long work record. This perspective abstracts from human capital transferability problems. For instance, human capital accumulated during the previous job could either be purely speci…c (in such a case, the worker is bad) or fully transferable to a new job (in such a case the worker is good).

APPENDIX A Proof of Proposition 1

Given Assumption A1, the resolution of the maximization problem ( ) is given in the main text that precedes Proposition 1.

(i) results from Step 1 of the resolution and Assumption A2.

(ii) results from Step 2, Step 3, and Assumption A3.

By solving, Assumptions A1 to A3 are not only su¢ cient but also necessary conditions.

B Proof of Proposition 2

Assumption A1 to A4 imply that Proposition 1 holds. Therefore, e b (s) = 0 for all s 0 (32) and e g (s) =

Solving the Cauchy problem ( 10)- [START_REF] Joutard | Taux de sortie du chômage à l'approche de la …n des droits à l'indemnisation: une étude de périodes atypiques durant l'épisode de chômage[END_REF] leads to

One can use these di¤erent equations and the de…nitions of , g and p g given in the main text to show (i) to (iii).

C Proof of Proposition 3

The main text makes it clear that e i (s) = i i¤

The result follows.

D Proof of Proposition 4

(i) follows from the de…nition of a baseline equilibrium. The (JS) locus is implied by Proposition 1.

The (HS) locus results from Proposition 2, Proposition 3 and the free-entry assumption V = 0. From Proposition 3 and free entry,

From Proposition 2, the function p g is continuous and strictly decreasing on ( ; ]. It follows that =

. The computation leads to (HS). Finally, the (FE) locus results from imposing V = 0 in equation ( 22) that de…nes the value of a vacancy.

(ii) The de…nition of baseline equilibrium involves …nding a positive vector ( ; ; ) that solves (JS), (HS) and (FE), and satis…es

To conclude the proof, we show that the parameter space de…ned by conditions C1 to C3 is nonempty.

The parameters are set as follows: r = 5%, 0 = 0:45, m = 1:0, c = 0:31; = 0:3, b min = 0:3, b max = 0:6, T = :5, w = 1:0, y g = 1:2, y b = 0:8, v (x) = 1:0x 0:5 . Figure 6 depicts the resulting equilibrium and as h goes from 0 to 100.

[Insert Figure 6] (iii) Uniqueness is a by-product of the former proof.

E Proof of Proposition 5 (ii) As h tends to in…nity, the (HS) locus tends to the 45-degree line so that = , while the (JS) locus tends to the vertical line = T . Therefore, and tend to T .

Not to be published -Numerical simulations

We use Matlab to simulate the model and …nd equilibrium values for and as h varies from very small (h = 0:1) to very large (h = 100). The results are reported by Figure 6. The program is based on Appendix D, where we give necessary and su¢ cient conditions for the existence of a unique baseline equilibrium. The program …nds the roots of the function P (x), and checks the set of constraints (C1) to (C3). There are two curves, corresponding to and . These two curves are plain blue whenever all the constraints are satis…ed. They are dashed red whenever one of the constraints is not satis…ed.

It is di¢ cult to satisfy all the constraints for all possible values of h. In our search for an adequate parameterization, we proceeded as follows. First, set the wage w and the parameters of the bene…t system: potential duration T , high replacement rate b max , low replacement rate b min . We normalize w to 1 and broadly replicate the US system with T = 0:5, b max = 0:6, and b min = 0:3. Second, set preference parameters: e¤ective discount rate r, utility function v. We set r = 5%, and v (x) = x 1=2 . Then, set the contact rate per unit of search e¤ort m, and the probability that a bad worker sends a good signal . We set m = 1:0, and = 0:3. This implies that bad workers need on average three interviews to get hired when good workers need one. Then you can use the residual set of parameters to match the various constraints. The marginal search cost c governs the (JS) locus, while y g and y b the output levels, and 0 the initial proportion of good workers, a¤ect the (HS) locus. We normalize y g to 1:2 and y b to 0:8 so that the marginal productivity of good workers is 20% larger than their wage, while the marginal productivity of bad workers is 20% lower than their wage. Finally, we set c to 3:1 and 0 to 0:45 to match constraints (C1) to (C3). Once there are only two parameters to play with, this may take some time to …nd the correct combination -remember that we must check all the constraints for all possible values for h, that is from the case where high search e¤orts are as e¢ cient as low e¤orts to the Walrasian case where high search e¤orts instantaneously provide a job.

Here is the program that can be copied and pasted in the Matlab Editor.