P J Messe
email: pierre-jean.messe@univ-lemans.fr

Taxation of early retirement windows and delaying retirement: the French experience

Keywords: Delaying retirement, early retirement windows, job matching models, employment protection J23, J63, J65

This paper investigates the eect of the 2003 French pension reform on hiring, ring and employment rates among older workers. This reform increased the mandatory retirement age and simultaneously it set a tax levied on early retirement windows payed by rms to their older workers, to encourage them to leave their job early. We use a matching model with endogenous job destruction extended to account for a mandatory retirement age and we calibrate the model with data drawn from the French Labor Force Surveys for the years 2002 and 2003. We show that in the case of a high tax rate, delaying retirement raises job separation rates, which partially osets its positive eect on job nding rates. Consequently, the combination of an increase in the retirement age and a taxation on early retirement windows may have negative eects on the employment rate among older workers.

Introduction

Eective retirement ages had fallen signicantly over the last 30 years in most OECD countries. This pattern may lead to scal sustainability problems, especially for countries with Pay-As-You Go pension schemes, in which pensions of retired individuals are nanced through contributions of workers. The creation of early retirement schemes in the 1970's may be an explanation of the decline in employment among elderly people observed in Europe during the same period. [START_REF] Zaidmann | Les dispositifs de cessation d'activité : état des lieux et évolutions souhaitables[END_REF] has shown that in France these schemes led to a consensus between older workers, rms and government. This phenomenon may be due to two main reasons. First, early retirement schemes were partly nanced by the government so they could be treated by rms as a layo subsidy (Hutchens, 1999, Tuulia and[START_REF] Hakola | Not so voluntary retirement decisions? evidence from a pension reform[END_REF]. Second, the government encouraged early retirement to make more room for young workers in the labor market in a setting of high youth unemployment (Zaidman, 2000).

Since the employment rate among workers aged more than 55 in France was in the early 2000's one of the lower in the European Union (29.9% in France with respect to an European average of 37.8%1 ), the French government implemented important changes to constrain early retirement. These changes aimed simultaneously at restricting the access conditions to publicly subsidized early retirement schemes for workers and rms and at increasing the share of early retirement expenses charged to employers. However, in spite of the increase of early retirement costs for rms, employers continue to encourage their older workers to leave early their job, oering them generous nancial incentives called "early retirement windows".

We can discuss the motivations that lead employers to oer early retirement windows to their older workers rather than ring them. One potential explanation suggested by Amauger-Lattes and Desbarrats (2006) is that the legislation regarding separations among older workers implies a cumbersome and often costly procedure for employers. Studying more than 300 court rulings of the Court of Cassation during the period 1994-2004 regarding job separations for older workers, they highlight that in most cases, employers strike a mutual agreement with their older workers, oering them generous early retirement windows and reporting a "dismissal for serious misconduct", although it is not the case.

In the face of this widespread phenomenon, especially in the case of big rms, the French government set in 2003 a tax levied on the amount of early retirement windows payed by rms. The tax rate amounted to 23.85% in August 2003 and in August 2007 it raised to 50%. In addition, to deter rms from pushing their older workers into retirement too early, the 2003 reform also led to an increase in the mandatory retirement age. Initially, this age was 60, which means that when a worker reached 60 and if her insurance period was sucient to allow her to draw a full pension, an employer could push her into retirement paying her a low retirement allowance. In 2003, this age has been increased to 65 and since 2008 it is 70.

The goal of this paper is to investigate the eect of the combination of these two reforms on the hiring rate, the job destruction rate and the employment rate among older workers. Let us rst dene what job destruction means in this paper. As we allow employers to oer early retirement windows to their older workers to encourage them to leave their job and to avoid a layo, the denition of a separation (or job destruction) diers across the age group of workers. On the one hand, for middle-age workers, a separation results from a layo and the employer has to pay to the worker a severance pay, as dened in the French Employment Protection Legislation. On the other hand, for older workers close to retirement, a separation has not the same nature. Indeed, for this age group of workers, job destruction means that the employer encourages a worker to leave her job, before she reaches the mandatory retirement age, oering her early retirement windows.

In this setting, the tax set in 2003 and levied on the amount of early retirement windows payed by rms is a sort of age-dependent employment protection, given that it concerns separations for one specic age group of workers. In this paper, we put forward the idea that in the case of a high tax rate on early retirement windows, delaying retirement may raise the separation rate among the group of workers concerned by the tax. We refer to this eect as the "impatience eect". The idea is the following: as shown in the previous literature on age-dependent employment protection, especially on the Contribution Delalande in the French case2 [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF][START_REF] Behaghel | The perverse eects of partial employment protection reform : the case of french older workers[END_REF][START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF]Cheron et al., 2008), a high tax rate deters employers from laying their older workers o after a shock, even though their lled jobs imply negative prots for rms. Indeed, as long as their ring cost exceed the expected losses, employers prefer waiting until their older workers reach the mandatory retirement age. This is the well-known labor-hoarding eect of the tax. In this paper, we point out that in this setting, delaying retirement leads to an increase in expected losses incurred by rms, making them more impatient to get rid of their older workers. Consequently, an increase in the retirement age may encourage rms to oer generous early retirement windows to their older workers to force them to leave, rather than waiting until they reach the new mandatory retirement age.

Highlighting this so-called impatience eect, our paper gains new insights into the eect of an increase in the retirement age on job creation, job destruction and employment. The eect of an increase in the legal retirement age on employment among older workers has been already considered in the literature. Previous studies pointed out that delaying retirement may have a positive "horizon eect" on job creation, when labor is treated as a quasixed factor that implies xed costs [START_REF] Oi | Labor as a quasi-xed factor[END_REF]. These costs result from either a bilateral monopoly problem (Hutchens, 1986) or from an accumulation of specic human capital through training (cf [START_REF] Hashimoto | Firm-specic human capital as a shared investment[END_REF] and imply that rms are more reluctant to hire a worker close to the retirement age. Furthermore focussing on job destruction, [START_REF] Caroli | New technologies, organisation and age: rm-level evidence[END_REF] showed that there is an age-bias technological change, so employers are less likely to retain an older worker in the case of a shock on her job, if her employment duration is too short. Extending the model of [START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF] in order to account for the life cycle of the worker with a bounded retirement age, [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF] have drawn similar conclusions and argue that an increase in the retirement age may have a positive eect on hiring rates and may reduce ring rates by lengthening the employment duration of older workers.

In this respect, the impatience eect that we highlight in this paper may oset the horizon eect in the case of high age-dependent employment protection, leading therefore to a rise in separation rates among the age group concerned by the tax. While previous studies on age-dependent employment protection [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF][START_REF] Behaghel | The perverse eects of partial employment protection reform : the case of french older workers[END_REF][START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF]Cheron et al., 2008) showed theoretically and empirically that an age-specic ring tax may have a strong negative eect on job creation among the protected age group and also that it may raise separation rates among the previous cohort of workers not concerned by the tax, these works did not pay any attention to the eect of delaying retirement on job creation, job destruction and employment in the case of a high age-specic separation cost.

In this paper, we determine a critical value of the rate of the tax on early retirement windows, above which the impatience eect is higher than the horizon eect. If the tax rate is higher than this critical value, an increase in the mandatory retirement age raises job destruction among older workers.

In that case, the positive horizon eect of delaying retirement on job nding rates among the age group of workers concerned by the tax is attenuated. In addition, as an increase in the retirement age reduces transitions from unemployment to retirement, it exerts a negative eect on employment among older workers, if the horizon eect is not suciently strong. Consequently, the higher the tax rate on early retirement windows, the lower the horizon effect with respect to the impatience eect and the stronger the negative eect of postponing retirement on employment among the protected age group.

Calibrating our model using data drawn from the French Labor Force Survey for the years 2002-2003, we provide a numerical illustration of these ndings. We show that the change in the job separation rate for older workers after an increase in the retirement age strongly depends on the tax rate. Consequently, the eect of delaying retirement on employment among the protected age group is sensitive to the level of taxation of early retirement windows.

The remainder of the paper is structured as follows. In the next section, using the data drawn from the French Labor Force Survey for the period 2001-2009, we provide a brief presentation of the patterns in job destruction rates among older workers between 2001 and 2009, investigating to what extent separation rates have been aected by the 2003 and 2008 reforms. Then, using data from the Survey on Health Ageing and Retirement in Europe (SHARE), we show the incidence of early retirement windows in the job destruction among older workers in France. In section 3, we describe in detail the theoretical model, following the specication of [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF].

In section 4 we present our main theoretical ndings on the eect of an increase in the mandatory retirement age on hiring rates and ring rates of middle-age and older workers in a setting of partial employment protection. In the section 5, we describe our quantitative analysis based on the French Labor Force Surveys for the years 2002 and 2003 and we present our results. Section 6 concludes.

Early retirement in France from 2001 to 2009

As mentioned in the introduction, in France access conditions to publicly subsidized early retirement schemes have been severely restricted for workers and rms since the early 2000's. This policy has had two main eects: on the one hand, it has divided by 10 the number of yearly entries in such early retirement schemes. On the other hand, it has led to a sharp increase in the number of older unemployed without job-search requirements but who remain on unemployment benets until they reach the retirement age (DARES, 2010). As this so-called "unemployment tunnel" has been used by rms since the early 2000's as a new way to get rid of their older workers, unemployment spells can be treated as pre-retirement periods in the case of older workers. In addition, as stated by Amauger-Lattes and Desbarrats ( 2006), separations result in most cases from a mutual agreement between older workers and employers, for which rms accept to pay early retirement windows to encourage their workers to benet from the unemployment tunnel.

When investigating the eect of the 2003 reform on separation rates among older workers, we would expect that setting a tax on early retirement windows payed by rms would reduce separations, especially for workers close to the retirement age. Furthermore, as this reform led to an increase in the mandatory retirement age, the horizon eect would imply a stronger decrease in the transition rate from employment to unemployment or inactivity. However, using data from the French Labor Force Survey for the period 2002 to 2009, we observe the reverse story. Indeed, the gure 1 shows that the yearly separation rate among the workers aged 55-59 rose from 8.37% for the period between March 2002 and March 2003 to 11.1% for the period March 2003-March 2004. At the same time, the separation rate among the workers aged 50-54 displays a more slightly increase from 3.53% to 4.12%. Similarly, when examining the eect of the August 2007 increase in the tax rate from 23.85% to 50% along with an increase in the mandatory retirement age from 65 to 70 on separation rates among older workers, we also remark a sharp increase in transition rates out of employment among workers aged 55-59 from 9.21% for the period between March 2006 and March 2007 to 11% for the period March 2007-March 2008. At the same time, the job destruction rate among the workers 50-54 remained unchanged. These stylized facts leave some room for other theoretical explanations than the well-known horizon eect of an increase in the retirement age [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF] or the labor-hoarding eect of age-dependent employment protection [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF][START_REF] Behaghel | The perverse eects of partial employment protection reform : the case of french older workers[END_REF][START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF]Cheron et al., 2008).

However, for our theoretical framework to be relevant, we have to investigate the incidence of early retirement windows in the job destruction rate among older workers. While this pattern has been already considered in the Note: The yearly separation rate is the probability that a worker employed in March of year t is out of employment in March of year t+1 Source: French Labor Force Survey (2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009) American case3 , too few studies examined this rms behavior in the European case. Using data from the 1997 International Social Survey Program, [START_REF] Dorn | Voluntary and involuntary early retirement: An international analysis[END_REF] investigate the incidence of involuntary early retirement for 19 countries4 , asking retired respondents if they retired early "by choice" or "not by choice". Their analysis covers the early retirement of individuals aged between 45 and 64 who retired between 1983 and 1997. Providing some descriptive statistics, they show that in some European countries like Germany or Portugal, more than half of all retired respondents state that they retire early "not by choice". In France this proportion amounts to 41%, which is also very high.

To provide a decomposition of the dierent reasons that led older workers to leave their job, we use the wave 2006 of the Survey on Health Ageing and Retirement in Europe (SHARE). This data provides some information for eleven countries5 on the factors of early exit. We include in our sample only French retired respondents aged between 57 and 69 in 2006, who were not entitled to a public or private pension when they left their activity. We obtain a sample made up of 182 individuals. We remark in the table 1 that in France early exit stems from 2 main reasons. First, almost one third of the sample report that they left their job after they had been given early retirement windows from their employer. Second, around 30% of the individuals report that they left early their job due to their bad health status.

Even though the incidence of early retirement windows in separations among older workers may be overestimated owing to a justication bias, we observe that the fraction of individuals reporting an early exit due to nancial incentives oered by their employer is strongly higher than the fraction of individuals reporting an exit due to a simple layo. So the table 1 provides some empirical evidence of the specic nature of the separations regarding older workers, justifying therefore our theoretical framework. 3. The model

The economy

Following the specication of [START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF], we consider an economy in which rms produce one type of good using only one factor of production: the labor. For a sake of simplicity, we assume that a rm can not employ more than one worker and the number of jobs is endogenous. Following [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF]

H = +∞ 0 tη 2 e -η 2 t dt = 1 η 2
In the remainder of the paper we assimilate an increase in the mandatory retirement age to a decrease in η 2 .

We consider an economy "à la [START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF] with endogenous job creation and destruction. In this model, workers and rms with vacant jobs meet each other according to a matching function m(u i , v i ), that represents the number of matches as a function of the unemployment rate u i among the group C i and the vacancy rate v i targeted to job-seekers belonging to the group C i . Here, we assume that matching markets are segmented by age, so rms can target their job oer toward one particular age group. To test the robustness of our theoretical ndings, we will discuss in the end of the section 5 the implications of an alternative specication with only one matching market for both types of workers.

We also assume that the matching function is increasing, concave in each argument and linear homogeneous. Let θ i be the labor market tightness namely the number of vacancies per worker, so we can dene the Poisson arrival rate q(θ i ) of a match for an employer posting a vacancy targeted to job seekers belonging to the group C i .

q(θ i ) = m(u i , v i ) v i = m( 1 θ i , 1) i = 1, 2 (1)
Therefore q(θ i ) is a decreasing function of the tightness θ i . Furthermore we can dene the Poisson arrival rate p(θ i ) of a match for a job seeker belonging to the group C i :

p(θ i ) = m(u i , v i ) u i = θ i q(θ i ) i = 1, 2 (2)
So p(θ i ) is an increasing function of the tightness θ i . Consequently, θ i is an endogenous key variable to determine the job-nding rates of each age group of workers in our economy.

The rms' behaviour

In our model each rm has one job that can be either lled and producing or vacant and searching. As long as the job is vacant, rms pay a cost c of maintaining a vacancy. When the vacancy is matched with a worker, her idiosyncratic productivity y is drawn randomly from the xed distribution G( ) with ∈ [0, ]. The rm hires the worker if is higher than the productivity threshold c i . Consequently, c i is an other endogenous key variable to determine the hiring rate of each age group of workers.

Let J i v be the value to an employer of posting a vacancy targeted on workers belonging to the group C i . At steady-state, we obtain the following Bellman equations:

rJ i v = -c + q(θ i )[ 0 max{J i (x), 0}dG(x) -J i v ] i = 1, 2 (3)
where J i ( ) is the asset value of a job lled by a worker belonging to the age group C i with a productivity level . Under the free-entry condition, the ow value to the employer from opening a new vacancy is equal to zero at steady-state equilibrium. Therefore we get :

c i J i (x)dG(x) = c q(θ i ) i = 1, 2 (4)
This rst condition implies that the mean search cost must be equal to the value to an employer of a lled job. An increase in this value encourages therefore employers to open more vacancies. When a job is lled, a worker belonging to the age group C i starts producing an output y , where is the random component of the productivity, and he receives a productivity-contingent wage w i ( ). Then the job can be hit by an idiosyncratic shock at a Poisson arrival rate λ. In that case, a new random productivity level is drawn according to a cumulative distribution function G(x) and the employer has no other choice either to close down the job or to keep the worker. Existing lled jobs are destroyed if the productivity level falls below a productivity threshold d i . Consequently, a job occupied by a worker belonging to the age group C i may be destroyed at a Poisson arrival rate λG( d i ). We assume that λ does not dier across age groups so d i is a key endogenous variable to determine job destruction among each age group of workers.

In our model, we allow employers to oer early retirement windows to their older workers to encourage them to leave their job and to avoid a layo. As we mentioned in the introduction, this rm's behavior is the response of employers to the tightening of access conditions to publicly nanced early retirement schemes and an increasingly part of employers oer early retirement windows since the early 2000's. Consequently, in our model, we consider that the separation cost for an employer depends on the age of the worker. If a job occupied by a middle-age worker breaks up, her employer has to pay her a severance pay denoted by f 1 according to the French Employment Protection Legislation. However, if a job occupied by an older worker breaks up, before she reaches the mandatory retirement age, her employer has to oer to her early retirement windows denoted by f 2 . Reproducing the 2003 French pension reform, we set a tax levied on the amount of the early retirement window payed by the employer at a rate τ . However, once an older worker reaches the mandatory retirement age, the job automatically breaks up and the separation cost for the employer is only f r , that is a low retirement allowance.

In addition, recall that in our model a middle-age worker may switch to the next age group at a Poisson arrival rate η 1 . In that case, if her random component of productivity is lower than the productivity threshold d 2 , her job breaks up. So for each age group of workers C i (i ∈ 1, 2), the value to an employer of hiring a worker with a random component of productivity level equal to is dened by the following Bellman equations:

rJ 1 ( ) = y -w 1 ( ) + λ[ 0 max{J 1 (x), -f 1 }dG(x) -J 1 ( )] +η 1 [max{J 2 ( ), -f 2 (1 + τ )} -J 1 ( )] (5)
And:

rJ 2 ( ) = y -w 2 ( ) + λ[ 0 max{J 2 (x), -f 2 (1 + τ )}dG(x) -J 2 ( )] +η 2 [(J 2 v -f r ) -J 2 ( )] (6)

Rent-sharing rules

We assume that the wage is set to split the match surplus between the rm and the worker at all times and in xed proportions, as in the case of a standard Nash wage bargaining. The worker's share is β. In a setting of employment protection, we have to consider two rent-sharing rules. Indeed, when a worker is matched with a vacancy, no severance payment has to be paid if negotiation fails. However, following the standard model of [START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF], we assume that wages are renegotiated continuously so that the wage received by a worker accounts for the employment protection he will benet from in the case of a layo. So, we may dene a rst rentsharing rule when the worker is hired, that determines a potential wage w 0 i in the following way:

w 0 i ( ) = argmax{[W i ( ) -U i ] β [J i ( )] (1-β) } i = 1, 2 (7)
where W i ( ) is the ow value to a worker belonging to the age group C i from employment and U i is the ow value to an unemployed worker belonging to the age group C i . Solving the program (7), we get the following rst rent-sharing rule:

W i ( ) -U i = βS 0 i ( ) = β[J i ( ) + W i ( ) -U i ] (8)
where S 0 i ( ) is the match surplus from a job creation targeted to workers belonging to the age group C i . As the wage is assumed to be renegotiated immediately, we get the new maximization program: 9) we get the following rent-sharing rule:

w i ( ) = argmax{[W i ( )-U i -f i ] β [J i ( )+f i (1+τ i )] (1-β) } with τ 1 = 0 and τ 2 = τ (9) Solving (
W i ( ) -U i -f i = βS i ( ) = β[J i ( ) + τ i f i W i ( ) -U i ] (10)
We remark that a job lled by a middle-age worker breaks up if

J 1 ( ) ≤ -f 1 which implies W 1 ≤ U 1 + f 1 .
In a similar way, a job lled by an older worker breaks up if

J 2 ( ) ≤ -f 2 (1 + τ ), which implies W 2 ≤ U 2 + f 2 .
Let us rst dene the ow value from employment to a worker of the group C i . When he is hired and as long as her job is not hit by an idiosyncratic shock, she receives the productivity-contingent wage w i ( ). When a shock occurs at a Poisson arrival rate λ, the productivity level changes and the match may be dissolved. In that case the worker receives a payment from her employer f i (f 1 corresponding to a severance payment and f 2 corresponding to the amount of the early retirement window received). If she remains employed despite the shock, she receives a new wage w i ( ), which changes the value of her job W i ( ). Furthermore, a middle age-worker occupying a job with a random component of productivity level equal to may switch to the next age group at a Poisson arrival rate η 1 and if her job does not break up she benets from the discounted income ows an older worker derives from the same job. Similarly, an older worker may reach the mandatory retirement age at a Poisson arrival rate η 2 and then her job breaks up and she receives a retirement allowance f r . In that case, the worker is retired and she benets from a pension P discounted over an innity of time. So for each age group of workers, the ow value from employment to a worker satises the following Bellman equations:

rW 1 ( ) = w 1 ( ) + λ[ 0 max{W 1 (x), U 1 + f 1 }dG(x) -W 1 ( )] +η 1 [max{W 2 ( ), U 2 + f 2 } -W 1 ( )]
(11) and:

rW 2 ( ) = w 2 ( ) + λ[ 0 max{W 2 (x), U 2 + f 2 }dG(x) -W 2 ( )] +η 2 [(f r + U 3 ) -W 2 ( )] where rU 3 = P (12)
Furthermore, we determine the present ow value from unemployment to a worker belonging to the age group C i , denoted by U i by the following equation:

rU i = z i +p(θ i )[ 0 max{W 0 i (x), U i }dG(x)-U i ]+η i (U i+1 -U i ) i = 1, 2 (13)
where z i is the non-labor income received by an unemployed worker belonging to the age group C i . It is noteworthy that an older unemployed worker who retires does not receive any retirement allowance f r .

In the appendix (7.1), we determine the wage equations for each age group of workers:

w 1 ( ) = (1 -β)z 1 + β(y + cθ 1 -η 1 f 2 τ ) + f 1 (r + η 1 ) -η 1 f 2 (14) w 2 ( ) = βy + (1 -β)z 2 + βcθ 2 + (r + η 2 )[f 2 (1 + βτ )] -η 2 f r (15)
We observe rst that the wage received by a worker belonging to the group C i is an increasing function of the non-labor income z i and of the probability to be matched with a job p(θ i ), given that these variables raise the worker's threat point, allowing her to extract a higher share of the match surplus from wage bargaining. Furthermore, we remark that the wage of a middle-age worker decreases with the amount of ring costs f 2 and that the wage received by an older worker decreases with the amount of retirement allowance f r . These results are consistent with Lazear's ndings (1990) that show that the higher the employment protection of a worker the lower his wage has to be in the beginning of his career.

Job destruction and job creation at steady-state

As we already mentioned in the subsection (3.2), in the case of a shock on a job, the employer has no other choice either to retain the worker with the new value of , the random component of productivity drawn from a distribution G(x), or to close down the job. In the appendix 7.2, we determine two productivity thresholds, below which existing matches are dissolved.

y d 1 = z 1 + βc 1 -β θ 1 -λ d 1 S 1 (x)dG(x) -η 1 max{S 2 ( d 1 ), 0} + η 1 f 2 τ (16)
and:

y d 2 = z 2 + βc 1 -β θ 2 -λ d 2 S 2 (x)dG(x) -(r + η 2 )f 2 τ (17)
We observe that the productivity threshold is less than the opportunity cost of employment, composed of non-labor income z i and of the expected gain to search for a job. Indeed, the third term on the right-hand side of ( 16) and ( 17) represents the option value of retaining an existing match despite a shock. This labor-hoarding phenomenon is due to the fact that rms are faced with a positive cost of maintaining a vacancy and therefore they accept to incur a loss in anticipation of a future improvement in the value of the match's product 6 .

Regarding the eects on employment protection in the case of the workers belonging to the group C 1 , we draw the same conclusions as [START_REF] Lazear | Job security provisions and employment[END_REF]: any severance payment arrangement is neutral on the ring decision of rms through an optimal labor contract, in which a worker is willing to pay a fee when she signs the contract to buy the protection of her job. In a similar way, the amount of retirement allowance f r is also neutral on the ring decision of rms regarding the older workers.

Furthermore in the case of the older workers, we observe that the amount of early retirement window payed by the rm f 2 has a negative impact on the threshold productivity d 2 . This may be due to the fact that a third agent, the government, receives a part of this payment through a tax at a rate τ , which implies that the worker is not given the whole payment when he is red: in that case, following the Lazear's theory (1990), ring incentives are distorted7 . Indeed, as rms expect that ring older workers is more costly, they are more reluctant to close down their jobs. However, we also observe a threshold eect regarding the younger cohort of workers, in the sense that rms have interest to lay middle-age workers o before they switch to the next age group8 .

Regarding hiring rate of each age group of workers, we determine the productivity threshold c i below which the employer does not recruit an unemployed worker belonging to the group C i , so S 0 i ( c i ) = 0. Using ( 8) and (10), we deduce:

S i ( ) = S 0 i ( ) + f i τ i with τ 1 = 0 and τ 2 = τ (18)
Using the expression (18) we determine the productivity threshold c i as a function of d i :

c i = d i + (r + η i + λ)f i τ i i = {1, 2} (19)
Consequently, an increase in the ring costs f i reduces the hiring rate of the workers belonging to the group C i , only if the tax rate τ i is higher than 0. Furthermore, the free-entry condition (4) and the two rent-sharing rules (8) and (10) imply:

(1 -β) c i S 0 i (x)dG(x) = c q(θ i ) (20)
At steady-state equilibrium, θ 2 and d 2 solve the following equation system:

     c q(θ 2 ) = (1 -β) c 2 [ y(x-d 2 ) (r+η 2 +λ) -f 2 τ ]dG(x) y d 2 =z 2 + βc 1-β θ 2 - λ (r+η 2 +λ) d 2 [y(x -d 2 )]dG(x) -(r + η 2 )f 2 τ y c 2 = y d 2 + (r + η 2 + λ)f 2 τ
As the rst and the second equation describe respectively a downwardsloping and an upward-sloping curve, there exists one unique solution (θ 2 , d 2 ) to this problem.

Regarding the workers belonging to the age group C 1 , there may be two cases, depending on whether their job may break up when they are ageing or not. In the case 1, the worker keeps his job even though he switches to the next age group. It implies that the reservation productivity d 1 is higher than

d 2 .
Determining the match surplus in appendix 7.3, we show that ( ˜ d 1 , θ1 ) solves the following equation system:

   c q( θ1 ) = (1 -β) (r+η 1 +η 2 +λ) (r+η 1 +λ)(r+η 2 +λ) ˜ d 1 y(x -˜ d 1 )dG(x) y ˜ d 1 =z 1 + βc 1-β θ1 -λ(r+η 1 +η 2 +λ) (r+η 1 +λ)(r+η 2 +λ) ˜ d 1 y[x -˜ d 1 ]dG(x) -η 1 y ˜ d 1 -d 2 (r+η 2 +λ) + η 1 f 2 τ
In the case 2, the worker does not keep necessarily his job when he switches to the next age group. It implies that d 1 ≤ d 2 . Determining the match surplus in appendix 7.3 we show that ( ˆ d 1 , θ1 ) solves the following equation system:

     c q( θ1 ) = (1-β) (r+η 1 +λ) ˆ d 1 y(x -ˆ d 1 )dG(x) + (1 -β) η 1 (r+η 1 +λ)(r+η 2 +λ) d 2 [y(x -d 2 )]dG(x) y ˆ d 1 = z 1 + βc 1-β θ1 - λ (r+η 1 +λ) ˆ d 1 [y(x -ˆ d 1 )]dG(x) -λη 1 (r+η 1 +λ) d 2 [ y(x-d 2 ) (r+η 2 +λ) ]dG(x) + η 1 f 2 τ
In the appendix 7.4 we show that the equilibrium solves either the rst system or the second. Furthermore as in each system the two rst equations describe respectively a downward-sloping and an upward-sloping curve, we deduce that there exists one unique solution, that may be either the couple ( ˜ d 1 , θ1 ) or the couple ( ˆ d 1 , θ1 ).

Unemployment at steady-state equilibrium

We determine the equilibrium values of the unemployment rate u i and the vacancy rate v i among each group of workers, using two sets of steady-state conditions. The rst implies that, for each age group C i of workers, the labor force N i is constant so we get:

N 0 η 0 = N 1 η 1 = N 2 η 2 (21)
These two equations characterize N 1 and N 2 for exogenous values of N 0 , η 0 , η 1 and η 2 . The second condition implies that for each age group, the ow of workers out of unemployment equals the ow of workers back into unemployment:

λG( d i )(1 -u i )N i + N i-1 η i-1 u i-1 = p(θ i )u i N i [1 -G( c i )] + η i u i N i (22)
So combining ( 21) and ( 22), we determine the unemployment rate u i in the following way:

u i = λG( d i ) + u i-1 η i λG( d i ) + p(θ i )[1 -G( c i )] + η i (23)
As p(θ i ) is an increasing function of θ i and therefore of the vacancy rate v i , we nd the expression of the Beveridge curve9 for each generation of workers, that is an inverse relation between vacancy and unemployment rate. Furthermore, dening the labor market tightness at steady-state equilibrium, we nd an other relation between u i and v i such that v i = θ i u i . Therefore the equilibrium unemployment exists and is unique, at the intersection between the Beveridge Curve and the increasing curve whose equation is

v i = θ i u i .

The eect of the 2003 French pension reform on hiring and job separation rates by age group of workers

To study the eect of a tax on early retirement windows payed by rms combined with an increase in the mandatory retirement age on hiring and separation rates by age, we make some assumptions regarding the functional forms of the matching function and of the distribution of the component of the productivity levels. First, we assume that matching function is Cobb-Douglas such that:

m(u i , v i ) = u α i v 1-α i
where α is the elasticity of the matching function. Furthermore, we assume that i follows an uniform distribution in the interval [0, 1].

In this section, we study in a rst subsection the eect of an increase in the tax rate τ on hiring and job separation rates of older workers. Then, we examine the eect of an increase in the mandatory retirement age in a setting of a taxation of early retirement windows, to investigate the eect of the combination of both these reforms on transition rates among older workers. In a second subsection, we investigate the eect of an increase in the tax rate τ on job nding and separation rates among middle-age workers, then we examine the eect of a combination of this tax with an increase in the mandatory retirement age on these rates.

A qualitative analysis for older workers

Under the assumptions dened in the beginning of this section, we determine the job creation condition C 2 and the job destruction condition D 2 such that:

     C 2 (θ 2 , d 2 , τ, η 2 ) = (1-β)y 2(r+η 2 +λ) (1 -d 2 -(r+η 2 +λ)f 2 τ y ) 2 -cθ α 2 = 0 D 2 (θ 2 , d 2 , τ, η 2 ) = z 2 + βc (1-β) θ 2 - λy 2(r+η 2 +λ) (1 -d 2 ) 2 -(r + η 2 )f 2 τ -y d 2 = 0
Let C 2 i and D 2 i be respectively the partial derivatives of C 2 and D 2 with respect to their i-th argument. Dierentiating this equations system with respect to d 2 , we obtain:

     ∂ d 2 ∂τ = C 2 3 D 2 1 -D 2 3 C 2 1 D 2 2 C 2 1 -C 2 2 D 2 1 ∂ d 2 ∂η 2 = C 2 4 D 2 1 -D 2 4 C 2 1 D 2 2 C 2 1 -C 2 2 D 2 1
We show in the appendix (7.5) that an increase in τ reduces job separation rates among older workers, which is consistent with existing literature on agedependent employment protection [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF][START_REF] Behaghel | The perverse eects of partial employment protection reform : the case of french older workers[END_REF][START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF]Cheron et al., 2008).

Then, we examine the eect of a decrease in η 2 , that is an increase in the mandatory retirement age, on separation rates. In absence of taxation of early retirement windows, delaying retirement reduces unambiguously the separation rate among older workers through a labor-hoarding eect, as shown in previous studies [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF]. The underlying intuition is that an increase in the mandatory retirement age raises the option value of the employer to retain an existing match despite a shock, given that the employer expects a higher duration of this job.

The new theoretical result that we highlight in this paper is about the effect of an increase in the mandatory retirement age in a setting of a taxation of early retirement windows, as this is the case for the French 2003 and 2008 reforms. In that case, we show that delaying mandatory retirement age has two osetting eects on the productivity threshold d 2 . On the one hand, it reduces job destruction through the well-known labor-hoarding eect of the tax. On the other hand in the case of a high tax rate τ , an increase in the mandatory retirement age may encourage employers to dismiss their older workers, osetting therefore the dissuasive eect of the tax through a new eect that we refer to as the "impatience eect". The idea is the following: in the case of a high taxation of early retirement windows, employers have interest to retain their older workers, even though the present value of their job is negative after a productivity shock. Indeed, as long as the loss in prots does not exceed the separation costs due to the tax, employers prefer waiting until their worker reach the mandatory retirement age. In this setting, when the government raises the mandatory retirement age, the horizon along which rms will incur losses in prots is longer and employers could be more impatient to dismiss their older workers. In that goal, they will oer to them early retirement windows to force them to leave. This impatience eect will therefore raise job separations among older workers, osetting the initial labor-hoarding eect of the tax.

Consequently, the eect of postponing retirement on job separations among older workers is ambiguous and depends widely on the tax rate τ . Indeed the higher the tax rate, the more likely employers to accept to incur important loss in prots, waiting until their workers reach the mandatory retirement age and the stronger the impatience eect after an increase in the mandatory retirement age. So we dene a sucient condition under which the impatience eect is higher than the labor-hoarding eect, which implies that delaying retirement will raise separation rate among older workers. We represent this sucient condition with a critical value of the tax rate denoted by τ c , above which the impatience eect will dominate the labor-hoarding eect.

Proposition 1. For values of the tax rate τ suciently high such that τ >

τ c = 2λy(1-d 2 ) 2 f 2 [2(r+η 2 )] 2 , then ∂ d 2 /∂η 2 < 0.
Proof: Computing the partial derivative ∂ d 2 /∂η 2 , we deduce the following condition:

∂ d 2 ∂η 2 < 0 ⇔ D 2 4 < 0
In the appendix (7.5), we determine D 2 4 , and we nd that delaying retirement 20 may lead to raise the productivity threshold d 2 if:

τ > 2λy(1 -ε d 2 ) 2 [2(r + η 2 + λ)] 2 f 2 (24)
Consequently, as

τ c > 2λy(1-ε d 2 ) 2 [2(r+η 2 +λ)] 2 f 2 , it is a sucient condition under which ∂ d 2 /∂η 2 < 0.
First, we observe that the critical value τ c falls when the amount of the early retirement window f 2 is high. Indeed, this tax is levied on the whole amount of nancial incentives payed by employers to older workers. Consequently, the higher the amount of early retirement window the more likely an increase in mandatory retirement age will oset the labor-hoarding eect of the tax.

Second, the critical value τ c is increasing with λ, which means that the incidence of the impatience eect on the separation rate is reduced for high values of λ, that is when the persistence of idiosyncratic shocks is low. Indeed, in that case, employers expect a higher probability to benet from a future improvement in the value of the match's product and have therefore more interest to retain existing matches after a shock. As the labor-hoarding eect is reinforced through an increase in λ, the impatience eect is less likely to dominate.

Lastly, we remark that τ c is a decreasing function of the productivity threshold d 2 . So, in the setting of a high job separation rate of older workers, delaying retirement may lead to more dismissals among this age group of workers when early retirement windows are taxed.

Furthermore, a rise in the productivity threshold d 2 may lead to reduce the hiring rate of older workers, all other things being equal and may therefore attenuate and even oset the positive horizon eect due to an increase in the mandatory retirement age. So we have to determine to what extent a combination of a tax on early retirement windows and of an increase in the mandatory retirement age aects the tightness θ 2 . Dierentiating the job destruction condition (D 2 ) and the job creation condition (C 2 ) with respect to θ 2 we get:

     ∂θ 2 ∂τ = D 2 3 C 2 2 -C 2 3 D 2 2 D 2 2 C 2 1 -C 2 2 D 2 1 ∂θ 2 ∂η 2 = D 2 4 C 2 2 -C 2 4 D 2 2 D 2 2 C 2 1 -C 2 2 D 2 1 21
We show in the appendix (7.5) that an increase in the tax rate τ reduces θ 2 , which is consistent with previous theoretical and empirical ndings [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF][START_REF] Behaghel | The perverse eects of partial employment protection reform : the case of french older workers[END_REF][START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF]Cheron et al., 2008). Furthermore, in absence of age-dependent employment protection, delaying retirement exerts a positive horizon eect on job nding rates of older workers, as shown by [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF].

Once again, our paper diers from these previous works given that we address the case of an increase in the retirement age in a setting of high agedependent employment protection. As we have showed that for a high tax rate τ , such that τ ≥ τ c , this reform may increase d 2 through an impatience eect, it may reduce the present value to an employer of a job lled by an older worker, making rms more reluctant to hire older workers. Consequently, we highlight the fact that at some values of the tax rate τ an increase in the mandatory retirement age reduces job creation among older workers. So we determine a second critical value of the tax, denoted τ cc , above which delaying retirement may reduce the tightness in the labor market for older workers.

Proposition 2. For a high value of the tax rate τ , such that τ > τ cc where

τ cc = y{(1-ε d 2 )[(r+η 2 )+λε d 2 ]+(1-ε d 2 ) 2 } f 2 (r+η 2 +λ)[λ(1-ε d 2 )+(r+η 2 +λ)] , then ∂θ 2 /∂η 2 > 0.
Proof: see appendix (7.5)

In addition, recall that the job nding rate of older workers also depends on the productivity threshold c 2 = d 2 + (r + η 2 + λ)(f 2 τ /y). It is noteworthy that even though τ > τ c , which implies ∂ d 2 /∂η 2 < 0, c 2 does not necessarily rise after an increase in retirement age. Indeed, calculating the derivative of c 2 with respect to η 2 we get the following expression :

∂ c 2 ∂η 2 = ∂ d 2 ∂η 2 + (f 2 τ /y) (25)
This expression may be negative only if -

∂ d 2 ∂η 2 > (f 2 τ /y).
The right-hand side of this inequality is due to the fact that after an increase in retirement age, the expected duration of a job is higher, so an employer may be less reluctant to hire an unemployed worker aged 55-59 years.

Let Π 2 be the job nding rate among older workers, we dene Π 2 in the following way :

Π 2 = θ 1-α 2 [1 -d 2 -(r + η 2 + λ) f 2 τ y ]
Determining the partial derivative of Π 2 with respect to η 2 , we obtain the following expression :

∂Π 2 ∂η 2 = θ -α 2 [(1 -α) ∂θ 2 ∂η 2 (1 -c 2 ) -θ 2 ( ∂ d 2 ∂η 2 + f 2 τ y )]
Although this expression seems to be complicated, its sign depends strongly on the tax rate τ . It shows that contrarily to the ndings of [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF], delaying retirement may have ambiguous eect on job nding rates of older workers, if we account for dierent levels of age-dependent employment protection. In our case, we highlight three cases. In the rst case, the tax rate is so low that τ < τ c and an increase in the mandatory retirement age raises unambiguously the job nding rate of an unemployed older worker. In the second case, the tax rate belongs to the interval [τ c , τ cc ]. In this case, an increase in mandatory retirement age leads to a rise in d 2 but also in the tightness θ 2 . Consequently if the impatience eect is suciently high, then ∂Π 2 /∂η 2 > 0 which means that delaying mandatory retirement age reduces the job nding rate of an unemployed older worker. But it is the reverse story if the horizon eect dominates the impatience eect. Last but not least if the tax rate τ is higher that the critical value τ cc , then delaying retirement increases the productivity threshold d 2 and reduces simultaneously the tightness θ 2 , leading unambiguously to a fall in the job nding rate of an older job-seeker.

To summarize the new theoretical ndings that we highlight in this paper, we could say that there are some cases of high age-dependent employment protection, in which delaying retirement may increase separation rate and reduce hirings among older workers. In other words, the positive eect of delaying retirement may be undermined when such a reform is combined with a high level of taxation of separation cost regarding older workers.

A qualitative analysis for middle-age workers

As mentioned in the previous section, we have to distinguish two cases: the case 1, where max{S 2 ( d 1 ), 0} = S 2 and the case 2 where max{S 2 ( d 1 ), 0} = 0. In the case 1, the tightness θ 1 and the productivity threshold d 1 are dened at equilibrium by the following equations system:

     C 1 (θ 1 , d 1 , τ, η 2 , d 2 ) = -cθ α 1 + (1-β)(r+η 1 +η 2 +λ)y 2(r+η 1 +λ)(r+η 2 +λ) (1 -d 1 ) 2 D 1 (θ 1 , d 1 , τ, η 2 , d 2 ) = -y d 1 + z 1 + βcθ 1 (1-β) -λ(r+η 1 +η 2 +λ)y 2(r+η 1 +λ)(r+η 2 +λ) (1 -d 1 ) 2 -η 1 y d 1 -d 2 (r+η 2 +λ) + η 1 f 2 τ
And in the case 2, the tightness θ 1 and the productivity threshold d 1 are dened at equilibrium by the following equations system:

     C 1 (θ 1 , d 1 , τ, η 2 , d 2 ) = -cθ α 1 + (1-β)y 2(r+η 1 +λ) [1 -d 1 ] 2 + (1-β)η 1 y 2(r+η 1 +λ)(r+η 2 +λ) [1 -d 2 ] 2 D 1 (θ 1 , d 1 , τ, η 2 , d 2 ) = -y d 1 + z 1 + βcθ 1 (1-β) - λy[1-d 1 ] 2 2(r+η 1 +λ) - η 1 y 2(r+η 1 +λ)(r+η 2 +λ) [1 -d 2 ] 2 + η 1 f 2 τ
We observe that in the case 1, the job creation condition does not depend on d 2 , contrarily to the case 2. This dierence is noteworthy particularly when we study the eect of an increase in the tax rate τ on the productivity threshold d

1 . We show in the appendix (7.6) that in the case 1, an increase in τ raises d 1 if the following condition holds:

η 1 f 2 > - η 1 y (r + η 2 + λ) d d 2 dτ ( 26
)
The term on the left-hand side of the inequality represents the direct positive eect of an increase in τ on the ring rate of middle-age workers, already highlighted by [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF]. Indeed, if it is more costly for an employer to get rid of an older worker, he has interest to lay his worker o before he switches to next age group. The right-hand side of the inequality represents the indirect eect of an increase in τ . Indeed, given that an increase in the tax rate reduces job destruction among older workers, so it raises the present value to the employer of a job lled by a middle-age worker who will switch to the next age group. Consequently, in the case of a shock on a job, rms have interest to retain the existing match.

Furthermore, in the case 2, we obtain in the appendix (7.6) a sucient condition such that an increase in τ raises d 1 :

η 1 f 2 αcθ α-1 1 > - η 1 y (r + η 1 + λ)(r + η 2 + λ) [1 -d 2 ] d d 2 dτ [αcθ α-1 1 -βc] (27)
The term on the left-hand side of the inequality represents the direct eect and the term on the right-hand side corresponds to the indirect eect. Contrarily to the condition (26), the direct eect and the indirect eect have not necessarily opposite signs. Indeed, if the bargaining power of workers is too high, such that β > αθ α-1 1 , then the inequality ( 27) is necessarily true. The idea is the following. If β is too high, then the decrease in d 2 raises θ 1 , leading to a strong increase in wage and it may encourage employers to lay more middle-age workers o. However, if we assume that β is relatively low, such that β < αθ α-1 1 , then in the case 1 or in the case 2, an increase in τ leads to a rise in d 1 provided that the direct eect osets the indirect eect (through the fall in d

2 ). In addition, we show in the appendix (7.6) that in the case 1, the eect of an increase in the mandatory retirement age on d 1 depends on the sign of the following expression:

d d 1 dη 2 > 0 ⇔ 2η 1 (r + η 1 + λ)y [2(r + η 1 + λ)(r + η 2 + λ)] 2 (λαcθ α-1 1 -βc) +αcθ α-1 1 [η 1 y ( d 1 -d 2 ) (r + η 2 + λ) 2 + η 1 y r + η 2 + λ d d 2 dη 2 ] > 0 (28)
The rst term of this sum corresponds to the direct eect, that reduces job destruction among middle-age workers if β < λαθ α-1 1 . Indeed if the bargaining power is too high, then delaying mandatory retirement age may raise the job nding rate of middle-age workers, increasing therefore wages and encouraging employers to close down more jobs hit by a shock. However, if β is relatively low, a decrease in η 2 raises the option value of retaining an existing match in the case of a shock. This discussion around the role of β has been already considered in the paper of [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF].

However, as in this paper we address the case in which τ = 0, we exhibit a second term corresponding to the indirect eect that depends on the tax rate τ and may attenuate the direct eect. Indeed, we have previously shown that a decrease in η 2 raises job destruction among older workers if τ > τ c . In that case, a rm employing a middle-age worker expects her job is more likely to break up when she will switch to the next age group, which makes her employer more reluctant to retain her job in the case of a negative productivity shock.

It is the same story in the case 2. Indeed, we show in the appendix (7.6) that the eect of a decrease in η 2 on d 1 depends on the sign of the following expression:

2(r + η 1 + λ)η 1 y [2(r + η 1 + λ)(r + η 2 + λ)] 2 [1 -d 2 ] 2 [αcθ α-1 1 -βc] + η 1 y (r + η 1 + λ)(r + η 2 + λ) [1 -d 2 ][αcθ α-1 1 -βc] d d 2 dη 2 (29)
We nd again both well-known direct and novel indirect eects. Consequently, under the condition β < αθ α-1 1 , delaying retirement raises the option value to the employer to retain an existing match, but this direct eect may be attenuated if τ > τ c . Now, we investigate the eect of an increase in τ on the tightness θ 1 in the case 1. We show in the appendix (7.6) that an increase in τ reduces θ 1 under the following condition:

η 1 f 2 > - η 1 y r + η 2 + λ d d 2 dτ ( 30
)
This condition ensures that an increase in τ raises job destruction among middle-age workers. Consequently, as d 1 = c 1 , a rise in d 1 reduces the job nding rate of middle-age workers.

We draw similar conclusions in the case 2. Indeed in the appendix (7.6), we show that an increase in τ reduces θ 1 under the following condition:

-(1 -β)y (r + η 1 + λ) [1-d 1 ]η 1 f 2 + η 1 y (r + η 1 + λ)(r + η 2 + λ) [1-d 2 ][αcθ α-1 1 -βc] d d 2 dτ < 0
(31) So provided that β is suciently low, such that β < αθ α-1 1 , then the hiring rate of middle-age workers decreases with τ only if the direct eect through the increase in c 1 osets the indirect eect through the fall in d 2 . In addition, we have to determine the eect of an increase in the mandatory retirement age on the tightness θ 1 . In the appendix (7.6), we show that it has a positive direct eect on θ 1 . Nevertheless, a decrease in η 2 may lead to more separations among older workers and therefore to an increase in d 1 , if the tax rate τ is higher than τ c . In that case, this indirect eect raises c 1 and attenuates therefore the direct positive eect of an increase in the mandatory retirement age on θ 1 .

To sum up our ndings, we have shown that in the case of an high taxation of early retirement windows, an increase in the mandatory retirement age may encourage employers to close down more jobs hit by idiosyncratic shocks which attenuates the positive horizon eect on job creation. To investigate the magnitude of these eects and to study their impact on employment rate among both age groups of workers, we implement a numerical illustration.

A numerical illustration

5.1. The eects of an increase in the mandatory retirement age on hiring and separation rates for both age groups of workers For this numerical illustration, we consider two age groups of workers : the workers aged 55-59 years and the workers aged 50-54 years. Considering that one period is a year, we set initially η 1 = η 2 = 0.2, so the average duration of each age group equals 5 years and the mandatory retirement age is set to 60 before the 2003 French pension reform.

This numerical illustration aims at determining the magnitude of the eects of an increase in the mandatory retirement age raising from 60 to 65, namely a decrease in η 2 from 0.2 to 0.1, on the job nding and separation rates among each group of workers, for dierent values of the tax rate τ . The values chosen for this numerical illustration are reported in the table 2.

A rst set of parameters Φ 1 = {α, β, λ, r, z 1 , z 2 , f 1 , f 2 } is based on external information. Following the standard literature [START_REF] Mortensen | Unemployment responses to 'skillbiased' technology shocks: The role of labour market policy[END_REF][START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF]) the elasticity of the matching function α is set to the extensively-used value α = 0.5. The bargaining power of workers β is set to 0.5 so that the Hosios' condition (1990) holds. The annual interest rate r is set to 3%. The amounts of unemployment benets z 1 and z 2 are computed from the values of the average wages of each group of workers. Using empirical results of Aubert (2005) drawn from a rm-level survey called DADS (Déclaration Annuelle des Données Sociales) for the year 2001, we set the gross hourly wage of a worker aged 50-54 to 16 euros and the gross hourly wage of a worker aged 55-59 to 17 euros. We determine therefore the gross yearly average wage for each group of workers, considering a basis of 35 hours per week. We obtain w 1 = 29121 euros and w 2 = 30941 euros.

To determine the amounts of unemployment benets z 1 and z 2 , we have to set a replacement rate, that is a percentage of net earnings in work. We use the measure constructed by the OECD, that is an average of the net replacement rates for dierent categories of workers, dierent family situations and dierent durations of unemployment. According to the OECD, for the period 2002-2004, the French average net replacement rate is around 57%. In this respect, as typical-case calculations relate to a 40-year-old worker, we have also to account for the fact that workers aged more than 50 benet from more generous unemployment compensations. Following [START_REF] Hairault | Distance to retirement and older workers' employment: The case for delaying the retirement age[END_REF], we consider a 6% (11.5%) premium on unemployment benets received by workers aged 50-54 (55-59) relative to the previous age group.

Consequently, we end up with a replacement rate equal to 60.4% (67.4%) for workers aged 50-54 (55-59), so we set z 1 = 17589.08 euros and z 2 = 20854.23 euros.

We have also to calibrate age-dependent separation costs. In the case of a layo of a worker aged 50-54 years, the employer has to give him severance payment according to the French Employment Legislation. We have shown previously that this payment had no eect on the hiring or ring behavior of rms. Regarding the 55-59 years old, we assume that employers prefer dismissing a worker invoking the serious misconduct and paying him early retirement window f 2 as highlighted by Amauger- [START_REF] Amauger-Lattes | La rupture du contrat des 55-64 ans : tendances et ambiguités du droit[END_REF]. A little is known about this payment so we set it in a rst step to a yearly gross wage (f 2 = 30941 euros). Then, we will perform some sensitivity analysis to check whether our results are robust to dierent values of f 2 . Similarly, as we expect that our results are sensitive to the value of the Poisson arrival rate of idiosyncratic shock λ, we set rst this parameter to λ = 0.2 and in a second step, we will implement some sensitivity analysis with other values of λ.

Lastly, we choose to calibrate a second set of parameters Φ 2 = {y, c} to reproduce some stylized facts about hiring and separation rates among workers aged 55-59 years. According to the French Labor Force Survey for the period March 2002-March 2003, the job nding rate of an unemployed worker aged 55-59 equals 5.82%. In addition, the job separation rate for this cohort of workers equals 8.37%. Solving our model using these values, we obtain an average productivity of job y equal to 43677.31 and a cost of maintaining a vacancy c equal to 55793.35 euros. Using the French Labor Force Survey for the period March 2002-March 2003, we remark that these values allow us to match in a satisfying way the observed job nding and job destruction rates among the workers aged 50-54. Indeed, solving the equations system composed of the job creation and the job destruction condition for the workers aged 50-54 setting c = 55793.35 and y = 43677.31, we obtain a job nding rate equal to 20.02% and a job destruction rate equal to 3.63% for this group while the observed rates equal respectively 15.95% and 3.53% according to the French Labor Force Survey (cf gure 2). In addition, the expression of the steady-state unemployment rate (23) shows that the employment rate among the group C 1 of workers depends on the employment rate among the previous cohort of workers. According to the French Labor Force Survey, we set the employment rate among workers aged 45-49 years to 79.32%. From the steady-state expression (23), we compute employment rates and we see that the values obtained are close to those observed from the data. Indeed, we nd an employment rate equal to 82.2% among the 50-54 years old and equal to 65.11% among the workers aged 55-59, while the observed rates equal respectively 75.8% and 55.68%. From this numerical illustration, we seek to highlight the eect due to a tax on early retirement windows oered by rms, when the mandatory retirement age rises. First, we observe in the gure 3 that in absence of taxation of early retirement windows, our results regarding the eects of an increase in retirement age on job ows are consistent with those of [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF]. Indeed, for both age groups of workers, delaying retirement age exerts a strong positive eect on the job nding rates (+38.5% for workers aged 55-59 and +21% for workers aged 50-54) through the well-known horizon eect, and it reduces separation rates (-3% for workers aged 55-59 and -23% for workers aged 50-54) through a labor-hoarding eect, even though this eect is quite lower for the workers aged 55-59 than for workers aged 50-54.

All the interest of our numerical illustration lies on the way the eect of delaying retirement on job ows is aected when the level of age-dependent employment protection varies. Focussing rst on separation rates, we see that the labor-hoarding eect is attenuated when the tax rate τ rises, due to Lecture: Each graph represents the relative variation in job ows after a decrease in η 2 from 0.2 to 0.1 as a function of the tax rate τ .

the impatience eect that we highlighted throughout this paper. As regards the workers aged 50-54, this labor-hoarding remains still higher than the impatience eect, even for higher values of the tax rate τ . Consequently, for this age group of workers, the impatience eect has a low incidence on the quantitative results of delaying retirement on the separation rate. However, regarding the workers aged 55-59, we remark that for high values of the tax rate τ , delaying retirement may lead to raise the separation rate, rather than reducing it. For this age group of workers, the labor-hoarding eect is more than oset by the impatience eect, so the latter has a strong incidence on the predicted eects of an increase in the retirement age on the separation rate.

In addition, as regards the workers aged 50-54, the positive horizon effect due to an increase in the retirement age on the job nding rate is also attenuated when the tax rate τ rises. Indeed, as the labor-hoarding eect of delaying retirement on the separation rate among this age group of workers is decreasing with τ , the increase in the job nding rate, that depends negatively on the productivity threshold d 1 , is also lower for a high level of taxation of early retirement windows. Nevertheless, once again for this age group of workers, the horizon eect remains still positive even for high values of the tax rate so the impatience eect has a low incidence on the predicted eects of delaying retirement on job creation among workers aged 50-54. When turning to the job nding rates among workers aged 55-59, it can appear to be surprising to obtain inverted U-shaped curve. We would expect that the higher the impatience eect the lower is the horizon eect. However, recall that the eect of a decrease in η 2 on the job creation threshold c

2 is ambiguous and is dened by the expression (25). We observe that the higher the tax rate τ , the higher the second term of the derivative and therefore delaying retirement may reduce the threshold productivity c 2 while it raises the threshold d 2 . It could correspond to the increasing part of the curve. Then, when the impatience eect is too strong for high values of τ , leading to a substantial increase in d 2 with the retirement age, the rst term of the derivative in the expression (25) is higher than the second one, implying that c 2 is increasing with the retirement age, which attenuates the positive horizon eect. This could correspond to the decreasing part of the curve. Nevertheless, even for high values of the tax rate τ , the horizon eect remains still strongly positive. We conclude that the positive horizon eect due to an increase in the retirement age on the job nding rate of workers aged 55-59 years is very strong whatever the value of the tax rate τ . Lecture: Each graph represents the relative variation in steady-state employment rates after a decrease in η 2 from 0.2 to 0.1. as a function of the tax rate τ . employment rate among workers aged 55-59 years. In that case, even though the direct eect of a decrease in η 2 on employment dominates, it is strongly attenuated by the horizon eect and therefore, the decrease in employment rate after an increase in retirement age is quite low. However, for a tax rate τ equal to 50%, the impatience eect exerts a supplementary negative impact on the employment rate. Consequently, after an increase in retirement age, the employment rate falls by 5%. So in the case of a high taxation of early retirement windows, the decrease in the employment rate that results from an increase in retirement age is twice higher than in the case where early retirement windows are not taxed. We deduce therefore that the combination of a rise in the mandatory retirement age and of a taxation of the nancial incentives to retire payed by rms to their workers aged 55-59 years may have negative eects on employment among this age group of workers.

A sensitivity analysis

As for our numerical illustration, we set arbitrarily some parameters, especially the amount of early retirement windows f 2 and the arrival rate of idiosyncratic shocks λ, we have to check whether our results are robust to other values of these parameters. Recall that in the beginning of the subsection 4.1, we have determined a critical value of the tax rate τ c , above which the impatience eect is higher than the labor-hoarding eect, leading to a rise in the separation rate among workers aged 55-59 after an increase in the retirement age. It is worth noting that this critical value is a sucient condition but not necessary, which means that we could observe ∂ d 2 /∂η 2 < 0 even though τ < τ c . Nevertheless, studying the critical value of tax rate τ c provides some information on the probability that the impatience eect will dominate over the labor-hoarding eect. Indeed, the higher τ c and the lower this probability.

After some comparative statics in the subsection 4.1, we have shown that the incidence of the impatience eect on the predicted eects of delaying retirement on the separation rate among workers aged 55-59 should increase with the amount of early retirement windows f 2 and decrease with the arrival rate of idiosyncratic shocks λ. Considering dierent values of each of these two parameters, we can check whether our theoretical ndings are consistent with our numerical results.

We investigate rst the incidence of the impatience eect on job destruction among workers aged 55-59 for dierent values of f 2 and setting λ = 0.2. The gure 5 shows that in the benchmark case corresponding to the green circled curve, where f 2 is equal to one year of the average wage, the impatience eect starts to dominate over the labor-hoarding eect for a value of τ around 18%. Indeed, for higher values of τ , a decrease in η 2 from 0.2 to 0.1 leads to a rise in the separation rate in that case. However, for lower values of f 2 , for instance in the case where f 2 is only the half of one year of the average wage, corresponding to the red curve, the impatience eect starts to dominate for a value of τ around 35%. In a similar way, in the case where f 2 represents only one third of the average yearly wage, that corresponds to the dashed blue curve, the impatience eect is still lower than the labor-hoarding eect even for a value of τ equal to 50%. Consequently, these ndings show that the higher f 2 and the lower τ c , which is in line with our theoretical predictions.

Proceeding in a similar way for λ, setting f 2 = 30941, we remark in the gure 6 that while in the benchmark case (red curve) where λ = 0.2, the impatience eect starts to dominate for a value of τ around 18%, this value is strongly higher (around 40%) when we set λ = 0.4 (blue dashed curve). Once again, it implies that the incidence of the impatience eect is lower when λ is high, which is consistent with our theoretical results. Lecture: Each graph represents the relative variation in job ows after a decrease in η 2 from 0.2 to 0.1. as a function of the tax rate τ . The red curve corresponds to f 2 = 0.5 * 30941, the dashed blue curve corresponds to f 2 = 0.3 * 30941 and the circled green curve corresponds to f 2 = 30941 Lecture: Each graph represents the relative variation in job ows after a decrease in η 2 from 0.2 to 0.1. as a function of the tax rate τ . The red curve corresponds to λ = 0.2, the dashed blue curve corresponds to λ = 0.4

Regarding the robustness of our key theoretical result, that is an impatience eect in the case of taxation of early retirement windows that may lead to a rise in the separation rate among older workers after an increase in the retirement age, we could discuss some of the assumptions made throughout the paper. First, in our theoretical framework, we have considered segmented matching markets by age. However, it could be interesting to discuss how undirected search may aect our results. If we consider only one matching function for the two types of workers, it implies a change in the free-entry condition. Indeed, in the case of undirected search, vacancies and the two types of unemployed workers (according to the age group) meet each other according to a matching function denoted by h(u 1 + u 2 , v), that determines the number of hirings as a function of the number of unemployed of type i, with i ∈ {1, 2}, and of the number of vacancies v. In this case, θ represents the number of vacancies per job-seeker. The value to an employer of posting a vacancy, denoted by J v , is now dened by the following Bellman equation: 32) where γ represents the share of older unemployed workers, dened by ow equations ( 21) and ( 22). Consequently, the new free-entry condition is the following:

rJ v = -c+q(θ){γ[ ε 0 max{J 2 (x), 0}dG(x)-J v ]+(1-γ)[ ε 0 max{J 1 (x), 0}dG(x)-J v ]} (
c q(θ) = γ ε ε c 2 J 2 (x)dG(x) + (1 -γ) ε ε c 1 J 1 (x)dG(x) (33)
In the appendix 7.7, we determine the new wage equations and productivity thresholds in the case of undirected search. We nd that the destruction condition for older workers in the case of undirected search is the following:

yε d 2 = z 2 + βcθ (1 -β)γ -βp(θ) (1 -γ) (1 -β)γ ¯ ε c 1 J 1 (x)dG(x)-λ ε ε d 2 S 2 (x)dG(x)-(r+η 2 )f 2 τ
(34) With respect to the destruction condition for older workers in the case of segmented markets by age, considering only one matching function implies an additional eect, that corresponds to the third term on the right hand side of the equation (34). However, the last term on the right-hand side of the expression remains unchanged, so in the case of a high value of τ , a decrease in η 2 implies an impatience eect, even though matching markets are not segmented by age. In addition, the higher γ, the lower the incidence of the additional eect due to undirected search. As delaying retirement has a positive eect on the employment rate among middle-age workers, while regarding employment among older workers, the positive horizon eect is oset by a direct negative eect through η 2 , we would expect that delaying retirement may raise γ and therefore reduce the incidence of the additional eect due to undirected search on the impatience eect.

Furthermore, we could discuss briey what would be the consequence of a substitution between early retirement windows and the tax rate τ . It is true that if the substitution was perfect, a rise in tau would imply a severe reduction in the amount of early retirement windows payed to older workers, so that the whole separation costs due by the rm remains unchanged. In that case, there would be no room for an impatience eect. Nevertheless, our theoretical ndings still hold in the case of imperfect substitution. We could assume for instance that older workers constrain rms to oer a minimal lump-sum, given that separation results from a mutual agreement. In that case, a strong increase in tau combined with a minimum severance payment may lead to a rise in the separation costs for the rms, which may aect ring decision and leaves room for an impatience eect. The way this minimal amount of early retirement windows would be negotiated between older workers and their employers go beyond the scope of this paper. Nevertheless, as it is well-known that in France, job destruction among older workers result from mutual agreement between employers and employees, this issue could be a future premising eld of research.

Conclusion

In this paper, we studied the eects of postponing retirement in a setting of an age-specic employment protection on the hiring and separation rates of older workers and also on employment of the elderly. Reproducing the 2003 French pension reform, we set a tax levied on early retirement windows payed by rms to their older workers to dismiss them. We provided some theoretical ndings considering a matching model with endogenous destruction extended to account for a mandatory retirement age and in which we introduce agedependent separation costs.

We highlighted that in the case of a high tax rate, delaying retirement may raise separations among the targeted age group of workers through an impatience eect. Indeed, a high tax rate discourages rms from dismissing older workers paying them nancial incentives, so employers prefer waiting until their workers reach the mandatory retirement age. In this setting, delaying retirement forces employers to retain their workers for a longer time, and they could have interest to dismiss them before they reach the retirement age in spite of the cost induced by the tax. We pointed out that there exists a critical value of the tax rate above which the impatience eect osets the labor-hoarding eect of postponing retirement, leading to a rise of the separation rate of older workers after an increase in the mandatory retirement age. Calibrating our data using the French Labor Force Surveys for the years 2002 and 2003, we showed that this critical value is negatively correlated to the amount of early retirement windows but it is positively correlated to the Poisson arrival rate of idiosyncratic shocks. Nevertheless, for reasonable value of parameters, we have shown that the impatience eect may have a strong incidence on the predicted eects of delaying retirement on the separation rate among older workers.

In addition, the impatience eect that we highlighted in this paper may partially oset the positive impact of an increase in the retirement age on the hiring rate of these workers. Theoretically, we determined a second critical value of the tax rate above which delaying retirement reduces the hiring rate among the group of workers targeted by the tax. However, we remarked in our numerical illustration that the extent to which the tax rate inuences the eect of postponing retirement on the job nding rate is negligible for both age groups of workers.

However, the relative variation in the employment rate of the elderly after an increase in the mandatory retirement age strongly depends on the taxation level. Indeed, in the case of a high tax rate, delaying retirement may lead to more separations, which exerts a negative eect on the employment rate.

Similarly, regarding the previous cohort of workers, we have shown theoretically that the impatience eect may aect the impact of delaying retirement on the ring and the hiring rates but we found in our numerical illustration that the impatience eect has a lower incidence on the predicted eects of an increase in the retirement age on either job creation or job destruction. Consequently, the relative variation in the employment rate among this cohort of workers after an increase in the mandatory retirement age is not altered in a signicant way by a taxation of early retirement windows.

However, these results have to be considered with caution. Indeed, in our model we consider an exogenous and constant amount of early retirement windows, while we can expect that the amount oered by employers to their workers may depend on several factors as the level of the tax, implying therefore a substitution between early retirement windows and the tax, or the characteristics of this worker. Too few information are still available on early retirement windows in France, however since the 2008 reform, employers who oer such nancial incentives to their older workers have to indicate their names, their ages and the amount that they received prior to exit. Using this data, we could carry out an empirical study aiming at better understanding the factors that lead employers to oer such nancial incentives and the determinants of the amount oered. It could also allow us to better grasp the way separation costs are negotiated between employers and older employees, through a mutual agreement. We leave this issue for further investigation.

Substituting the Bellman equation ( 13) into this expression, we obtain:

-(1 -β)[p(θ 1 )[ 0 max{W 1 (x), U 1 }dG(x) -U 1 ] + z 1 ] = β -w 1 ( ) -(1 -β)η 1 f 2 -η 1 β[f 2 (1 + τ )] -λf 1 (38)
Combining the rent-sharing rule (8) and the free-entry condition (4) we get:

c i [W 1 (x) -U 1 ]dG(x) = β (1 -β) c q(θ 1 ) (39)
So, substituting this expression into (38), we deduce the following wage equation:

w 1 ( ) = (1 -β)z 1 + β(y + cθ 1 -η 1 f 2 τ ) + f 1 (r + η 1 ) -η 1 f 2 (40)
Then we determine the wage equations for the older workers belonging to the group C 2 . Substituting Bellman equations ( 6) and ( 12) into the expression (10), we get:

-(r + η 2 + λ)(1 -β)U 2 = β{y + λ d 2 S 2 (x)dG(x) + λ(U 2 -f 2 τ ) + η 2 P r } -w 2 ( ) -λ d 2 [W 2 (x) -U 2 -f 2 ]dG(x) -λ(U 2 + f 2 ) -η 2 ( P r + f r ) +(r + η 2 + λ)f 2 (1 + βτ ) (41)
The rent-sharing rule (10) implies:

λβ d 2 S 2 (x)dG(x) = λ d 2 [W 2 (x) -U 2 -f 2 ]dG(x)
Therefore we get:

-(r + η 2 + λ)(1 -β)U 2 = βy -w 2 ( ) -U 2 λ[1 -β] -(1 -β)η 2 P r -η 2 f r + (r + η 2 )[f 2 (1 + βτ )] (42)
So substituting this expression into (38) we get the following wage equation:

w 1 (ε) = (1-β)z 1 + βcθ (1 -γ) -βp(θ) γ (1 -γ) ε ε c 2 J 2 (x)dG(x)+β(yε-η 1 f 2 τ )+f 1 (r+η 1 )-η 1 f 2 (60)
In the same way, to determine the wage equation for older workers, the rent-sharing rule and the new free entry condition in the case of undirected search yield:

w 2 (ε) = (1-β)z 2 +βyε+ βcθ γ -βp(θ) (1 -γ) γ ε ε c 1 J 1 (x)dG(x)+(r+η 2 )[f 2 (1+βτ )]-η 2 f τ
(61) First we determine ε d 1 in the case of undirected search:

yε d 1 = z 1 + βcθ (1 -β)(1 -γ) -βp(θ) γ (1 -β)(1 -γ) ε ε c 2 J 2 (x)dG(x) -λ ε ε d 1 S 1 (x)dG(x) -η 1 max{S 2 (ε d 1 ), 0} + η 1 f 2 τ (62)
Then we determine ε d 2 in the case of undirected search:

yε d 2 = z 2 + βcθ (1 -β)γ -βp(θ) (1 -γ) (1 -β)γ ε ε c 1 J 1 (x)dG(x)-λ ε ε d 2 S 2 (x)dG(x)-(r+η 2 )f 2 τ (63)

Figure 1 :

 1 Figure 1: Yearly separation rates among older workers in France from 2002 to 2009

Figure 3 :

 3 Figure 3: The eect of a decrease in η 2 from 0.2 to 0.1 on job ows among each age group of workers for dierent values of the tax rate τ

Figure 4 :

 4 Figure 4: The eect of a decrease in η 2 from 0.2 to 0.1 on employment rates among each age group of workers

Figure

  Figure 5: A sensitivity analysis for f 2

Figure

  Figure 6: A sensitivity analysis for λ

Table 1 :

 1 Reasons for early retirement in France

	Reasons for early exit	Number of observations Frequency (in %)
	Early retirement windows	71	39.01
	Layo	10	5.49
	Own ill health	54	29.67
	Ill health of a relative	13	7.14
	Retire at same time as spouse	11	6.04
	To spend more time with family	13	7.14
	To enjoy life	10	5.49
	Total	182	100
	Note: Our sample contains all French retired individuals aged between 55 and 69 in 2006
	who were not entitled to a public or private pension when they left their activity
	Source: SHARE (waves 2004 and 2006)

, we consider two age groups of workers: the middle-age workers (group C 1 ) and the older workers (group C 2 ). A middle-age worker may switch to the next age group at a Poisson arrival rate η 1 . Similarly, an older worker may reach the mandatory retirement age at a Poisson arrival rate

  η 2 . In our model the Poisson arrival rate η 2 is a key parameter to determine the horizon of older workers denoted by H. We dene H in the following way:

Table 2

 2

	: Values of parameters
	Elasticity of the matching function	α	0.5
	Bargaining power	β	0.5
	Poisson arrival rate of shocks	λ	0.2
	Annual interest rate	r	0.03
	Workers aged between 50 and 54
	Unemployment benets
	(as a fraction of the net average wage)	z 1	60.4%
	Probability to switch to the next age group Workers aged between 55 and 59	η 1	0.2
	Unemployment benets
	(as a fraction of the net average wage)	z 2	67.4%
	Early retirement windows (as a fraction of the average yearly wage) f 2	100%
	Probability to reach the mandatory retirement age Calibrated values	η 2	0.2
	Cost of maintaining a vacancy	c 55793.35
	Average productivity	y 43677.31

Source: Eurostat

The Contribution Delalande is a tax payed by rms who re workers aged more than 50

See for instance[START_REF] Brown | Early retirement windows[END_REF]

Canada, Cyprus, Denmark, France, Germany, Great Britain, Hungary, Italy, Japan, the Netherlands, New Zealand, Norway, Poland, Portugal, Slovenia, Spain, Sweden, Switzerland and USA

Austria, Germany, Sweden, Netherlands, Spain, Italy, France, Denmark, Greece, Switzerland and Belgium

This type of labor-hoarding behavior has been well investigated by[START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF]

There may be several other cases in which severance payment would be non neutral on ring decision. For instance, if we had considered a wage posting case or if we had introduced risk aversion, the amount of early retirement windows would have also aected separation rates

For similar results, see[START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF],[START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF] and[START_REF] Chéron | Age-dependent employment protection[END_REF]

see notably[START_REF] Blanchard | The beveridge curve[END_REF]

Acknowledgements

I am grateful to Pascal Belan and François-Charles Wol for their guidance and support. Helpful comments have been received from Arnaud Chéron and François Langot. I have also benetted from comments by seminar participants at the Annual Conference of the European Society for Population Economics. I also thank two anonymous referees for their remarks and suggestions. I thank ANR for nancial assistance. Any remaining errors are mine.

Rate (in %)

Model Data

Lecture: The job nding rate is the fraction of workers aged 50-54 unemployed in March 2002 who nd a job in the next twelve months. The separation rate is the fraction of workers aged 50-54 employed in March 2002 who get into unemployment in the next twelve months. Source: French Labor Force Survey (2002Survey ( -2003) ) To sum up our numerical ndings, we show that in the case of taxation of early retirement windows, the impatience eect may have a strong incidence but only on the predicted eects of an increase in the mandatory retirement age on the separation rates among workers aged 55-59. This impatience eect could be therefore one explanation of the increase in the separation rate among workers aged 55-59 observed in France between the periods March 2002-2003and March 2003-2004, as mentioned in the section 2.

The eects of an increase in retirement age on employment rates among both age groups of workers

Using the expression (23) of the steady-state unemployment rate among the age group of workers C i , we deduce the eects of an increase in retirement age on employment rates and we investigate to what extent a taxation on early retirement windows alters this eect.

Focussing rst on the workers aged 50-54 years, we observe in the gure 4 that in the absence of taxation of early retirement windows, a decrease in η 2 from 0.2 to 0.1 leads to a rise in employment rate by around 3.8%. Even though the positive eect of delaying retirement on employment rate among this age group of workers is attenuated as the tax rate rises, it remains strongly positive even for high values of τ . So, we conclude that the impatience eect that we highlighted in this paper does not alter signicantly the impact of postponing retirement age on employment rate among workers aged 50-54.

However, when we investigate the eect of an increase in retirement age on employment rate among the workers aged 55-59 years, we see that the tax rate τ plays a more important role. Using the expression (23), we observe that a decrease in η 2 exerts a direct negative eect on employment rate among workers aged 55-59 years. Indeed, when the mandatory retirement age raises, it leads to reduce ows out of unemployment regarding job seekers aged 55-59 years. In addition, a decrease in η 2 may have indirect eects on the employment rate. First, it may have a positive eect on employment through an increase in the job nding rate of workers aged 55-59 years. But this horizon eect may be partially oset by the impatience eect for high values of the tax rate τ . Indeed, in this setting, delaying retirement age leads to a rise in job destruction among the workers aged 55-59 years, which aects negatively employment rate among this group of workers.

We see in the gure 4 that in the absence of a taxation of early retirement windows, a decrease in η 2 from 0.2 to 0.1 reduces by around 3% the 7. Appendix

The wage equations

Using the sharing rule (10) and rearranging terms, we obtain:

Let us rst dene the wage equation for the middle-age workers belonging to the age group C 1 . Bellman equations ( 5) and ( 11) imply:

Using the sharing-rule (10) we obtain:

And:

Therefore we get:

Substituting the Bellman equation ( 13) into this expression and rearranging terms, we obtain:

Using the sharing rule (8) and the free-entry condition (4), we deduce the following wage equation:

The productivity thresholds

We have to determine the productivity threshold d i , below which the rm closes down the job lled by a worker belonging to the group C i . First we dene d 1 . Substituting the wage equation ( 14) into the expression (5) we get:

Simplifying this expression we obtain:

Evaluating (45) at = d 1 gives the following productivity threshold:

We proceed in a similar way to determine d 2 , substituting the wage equation (15) into the expression (6) we get:

Simplifying this expression we obtain:

So evaluating (47) at = d 2 , we get the following job destruction condition:

Furthermore, rent-sharing rules ( 8) and ( 10) imply:

So using ( 49) we can deduce the productivity threshold c i , such that:

Match surpluses

Given that S 2 ( d 2 ) = 0, we get:

Furthermore, in the case where d 1 > d 2 , given that S 1 ( d 1 ) = 0, we obtain:

The expression (51) allows us to simplify this expression so we get:

In the case where d 1 ≤ d 2 , in a similar way we obtain:

7.4. The equilibrium for the middle-age workers

In the case 1, where max{S 2 ( d 1 ), 0} = S 2 ( d 1 ), an unique equilibrium ( ˜ d 1 , θ1 ) is dened by the following equation system:

And in the case 2, where max{S 2 ( d 1 ), 0} = 0, an unique equilibrium

) is dened by the following equation system:

So, we have to show that the equilibrium for the younger generation is either the couple ( ˜ d 1 , θ1 ) or the couple ( ˆ d 1 , θ1 ). In other words, we have to show that there can be neither 0 solutions nor 2 solutions to this problem. We borrow the proof of [START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF]. Indeed, if the problem had 0 solutions, it would imply :

So subtracting the two job creation conditions each other we would get:

Therefore, we deduce that θ1 > θ1 .

Furthermore, subtracting the two job destruction conditions each other we would get:

So we deduce that θ1 < θ1 . This result is not possible given the previous result, so it is the proof that there exists at least one solution to the problem. Furthermore, we proceed in a similar way to show that the problem does not admit two solutions. Indeed, if he admitted two solutions, it would imply:

In that case, the equation ( 55) would imply θ1 < θ1 and the equation ( 56) would imply θ1 > θ1 , so this case is absurd. Consequently, the problem admits one unique solution: either the couple

7.5. Eects of the tax rate and the retirement age on hiring and separation rates among older workers

We determine the partial derivatives of the job creation condition:

Then we determine the partial derivatives for the job destruction condi-tion:

We examine the eect of an increase in τ on d 2 :

< 0

Then we investigate the eect of a decrease in η 2 (namely an increase in the mandatory retirement age) on d 2 :

We have to study the sign of D 2 4 :

Consequently, a sucient condition such that D 2 4 < 0 may be expressed as follows:

Now, we determine the eect of an increase in τ on the tightness θ 2 :

Factorizing this expression by (1-β) (r+η 2 +λ) we deduce that its sign is the same as the sign of the following expression:

Simplifying and rearranging terms, we nd that this expression equals:

Then we determine the eect of an increase in the mandatory retirement age on θ 2 :

4 > 0, then dθ 2 dη 2 < 0. However, if D 2 4 < 0, then :

] we deduce that this expression has the same sign as the following expression:

The sign of this expression may be ambiguous given that at τ > τ c ,

We can dene a sucient condition such that ∂θ 2 ∂η 2 > 0:

7.6. Eects of the tax rate and the retirement age on hiring and separation rates among middle-age workers

We consider rst the case 1 where max{S 2 ( d 1 ), 0} = S 2 ( d 1 ). We determine the eect of an increase in τ and of a decrease in η 2 on the productivity threshold d 1 . Dierentiating this equations system we nd the two following expressions:

We determine the partial derivatives for the job creation condition:

Then we calculate the partial derivatives for the job destruction condition:

(r+η 2 +λ) 2 > 0

We determine then the eect of an increase in τ and of a decrease in η 2 on the tightness θ 1 . We obtain the following equations system:

Furthermore:

2 > 0 and D 1 5 C 1 2 < 0, so the direct eect implies that a decrease in η 2 raises θ 1 . The indirect eect through ∂ d 2 /∂η 2 may reinforce the direct eect if τ < τ c , but it may attenuate the direct eect if τ > τ c .

In a second step, we consider the case 2 where max{S 2 ( d 1 ), 0} = 0. We calculate the partial derivatives for the job creation condition:

We also calculate the partial derivatives for the job destruction condition:

We determine rst the eect of an increase in τ and of a decrease in η 2 on the productivity threshold d 1 . Dierentiating our equations system, we obtain the two following expressions:

We determine a sucient condition under which an increase in the tax rate τ raises d 1 :

Furthermore,

dη 2 has the same sign as the following expression:

Then, we determine the eect of an increase in τ and of a decrease in η 2 on the tightness θ 1 . We obtain the two following expressions:

Consequently, an increase in τ leads to a fall in the tightness θ 1 if the following condition holds:

(58) Furthermore, regarding the eect of a decrease in η 2 on θ 1 , we draw similar conclusions as in the case 1.

Wage equations and productivity thresholds in the case of undirected search

Let us start from the equation (39). When substituting the rent-sharing rule and the free-entry condition in the wage equation. Undirected search implies: