Messe Pierre-Jean

Bia Michela

Leombruni Roberto

Young-in Old-out: a new evaluation

Keywords: exit age, youth employment, propensity score, matching model, on-the-job search JEL Classi…cation : J23, J62, J63

In this paper we examine the consequences of a decrease in exits of older workers on youth employment, allowing for job-to-job ‡ows. We distinguish between young workers with no experience and experienced workers and we consider jobs with di¤erent experience requirements. Using a theoretical matching model with heterogeneous jobs and workers, we point out that a fall in the exit rate of older workers increases job-to-job ‡ows, which makes …rms more reluctant to open jobs for young workers. Then, we provide some empirical evidence for the Italian case.

Introduction

Over the last three decades, most European countries experienced an ageing of their population and a fall in the labor force participation of older workers. In the face of steadily increasing dependency rates, some countries raised their legal retirement age to balance the budget of their pay-as-you-go retirement schemes. However, the e¢ ciency of such a policy option depends strongly on its impact on the hiring rate and employment among every age group of workers and particularly the young workers.

At …rst sight, we can think that an increase in legal retirement age, encouraging …rms to maintain their ageing workforce, could slowdown the dynamics of new hirings, if …rms were expected to squeeze out their older workers and to replace them by new young workers. However, this simplistic idea, that led to the creation of early retirement schemes, has been severely criticized over the past ten years. Following the idea of [START_REF] Bozio | Réformes des retraites : estimations sur données françaises[END_REF] we may argue that young and older workers may be imperfectly substitutable in the labor market. Consequently, once an older worker retires, his job can not be …lled by a young worker. We can discuss about what makes these two cohorts of workers imperfectly substitutable. [START_REF] Cadiou | Evolutions démographiques et marché du travail : des liens complexes parfois contradictoires[END_REF] highlight the fact that the low job …nding rates among young workers and among older workers are not of the same nature. While the low hiring rate of senior workers is due to the fact that they are close to the retirement age 1 , the low job …nding rate for young workers stems from their lack of experience. Consequently, they can not …ll a job requiring some professional experience to replace an older worker retired.

However, even though the two sorts of jobs for the young and the old generation of workers di¤er greatly in the set of skill and experience requirements, a reduction in exits of older workers after an increase in the retirement age may have a negative impact on the hiring rate of young workers through a vacancy-chain e¤ect [START_REF] Contini | Gross ‡ows vs. net ‡ows in the labor market: What is there to be learned?[END_REF]. The idea is the following: if we consider within a local labor market 2 , that a vacancy opens up when an older worker leaves his job, there is a non-negative probability that it will be …lled by an employed job-seeker (attracted by better opportunities or higher wage), who accumulated a su¢ cient level of experience to replace the older worker and whose job requires less professional experience. Therefore, after this job-to job ‡ow, a new vacancy opens up and may be …lled by another young employed job-seeker, involving again job-to-job ‡ows, that may result …nally in hiring a young worker from out of the local labor market. So, holding job creation constant and considering a vacancy-chain pattern, retaining older workers in the labor force leads to fewer complex vacant jobs, i.e. vacant jobs requiring some professional experience, which reduces job to job ‡ows and decreases the number of vacant jobs requiring no experience. As a result, job …nding rates of young workers are negatively a¤ected.

For this idea to be relevant, two conditions are needed. First, the economy has to be characterised by a large share of job-to-job transitions in the total number of separations. Second, job creation has to be uncorrelated with a variation in the exit rates of older workers. Regarding the …rst condition, recent empirical studies highlight the major role of job-to-job ‡ows to explain the total labour market turnover. In the case of the US labor market, Nagypal (2003) reports that job to job transitions account for more than 50% of the total separations of workers with a college degree and for more than 30% of separations of workers without a college degree. Regarding the European case, Theodossiou and Zangelis (2007) report cross-country evidence on job to job transitions for 6 European countries and they …nd that job to job ‡ows account for up to 17% of total separations in the di¤erent European economies considered. So it appears that workers in Europe are less mobile than in the US labor market. Nevertheless, in Italy, [START_REF] Contini | Young in, old out: nuovi pattern di mobilita nella economia italiana[END_REF] show that job to job transitions are about 25% of total separations. The Italian labor market can therefore be characterized by a large share of job to job transitions. The second condition requires job creation to be exogenous. As argued by [START_REF] Layard | Unemployment: macroeconomic performance and the labour market[END_REF], it seems unrealistic and they refer to this idea as a lump-of-labor fallacy. Indeed, job creation is expected to be a¤ected by a change in the exit rate of older workers, so we can not view the number of jobs in the economy as a …xed stock. One …rst goal of this paper is to predict the e¤ect of delaying retirement on youth employment using the vacancychain setting and allowing for endogenous job creation. Using a theoretical matching model "à la" Mortensen-Pissarides (1994), [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF] already pointed out that there exists a positive relation between the distance from retirement age and the hiring rate. This result implies that postponing retirement has a positive horizon e¤ect on the hiring rate of each group of workers, including the young workers. Nevertheless, while it is clear that the distance from retirement strongly a¤ects job …nding rates of older workers, it is less straightforward that it plays a signi…cative role in the determination of job opportunities for young workers.

Our theoretical contribution is to investigate the e¤ect of a decrease in the exit rate of older workers on the hiring rate among young workers, using a matching model with heterogeneous jobs and workers 3 and considering two types of workers, workers with experience and young workers with no experience and two types of jobs, jobs requiring experience and jobs requiring no experience. Through this theoretical model, we represent an overlapping labour markets economy in the sense of [START_REF] Sattinger | Overlapping labour markets[END_REF], where new young entrants into a local labor market can not be matched with complex jobs requiring experience. However, as we allow for job-to-job ‡ows, some experienced workers may be mismatched with jobs requiring no experience. These workers are therefore looking for a complex job due to better wage o¤ers. In this framework we show that a decrease in the exit rate of older workers leads to an intensi…cation of job-to-job ‡ows from simple to complex jobs and reduces the ‡ow value of a simple job to an employer. Consequently, it reduces the creation of simple jobs and exerts a negative e¤ect on the job …nding rate of young workers with no experience.

Then, in the second part of the paper, we provide some evidence of negative e¤ects of retaining older workers in the labor force on the hiring rate of young workers at a local labor market level, examining the Italian case given that in this country job to job ‡ows account for a large share of total worker ‡ows. While the relation between elderly labor force participation and youth employment has been widely studied at a country-level [START_REF] Bozio | Labor force participation by the elderly and employment of the young: the case of france[END_REF], Jousten et al., 2009), too few empirical studies have been carried out in a microeconomic perspective. However, as worker and job ‡ows are strongly a¤ected by …rm-speci…c variables [START_REF] Davis | Gross job ‡ows[END_REF], it is interesting to investigate the e¤ect of a slowdown in exits of older workers on the hiring rate of young workers at the …rm-level, controling for all regressors that may a¤ect job creation.

However, such a study raises an empirical issue due to the endogeneity of the variation in exit rates of older workers. To address this issue, Portugal et al. (2009) investigate the e¤ect of a decrease in the exit rate of elderly, due to an increase in the legal retirement age for women, on worker ‡ows of di¤erent age groups, using detailed matched employer-employee Portuguese data. Their empirical methodology lies on a quasi-experiment given the fact that only women have been a¤ected by the retirement reform. Using a di¤erence-in-di¤erence matching method [START_REF] Heckman | Matching as an econometric evaluation estimator : Evidence from evaluating a job-training programme[END_REF], they show that …rms employing women treated by the reform hire one to two fewer workers for each senior retained after the increase in legal retirement age and this decrease in the hiring rate is particularly strong for younger workers.

Our paper is close to the spirit of their study. Indeed, our goal is to examine the e¤ect of a fall in the exit rate of older workers on the hiring rate of young workers, using Italian matched employer-employee data from the Worker Histories Italian Panel and considering the period 1988-1989. Nevertheless, our paper di¤ers from the study of Portugal et al. for two reasons: …rst, we do not consider …rms but rather local labor markets, using a synthetic-…rm approach to account for job-to-job ‡ows, that represent the starting point of the vacancy-chain theory. Second we do not exploit a quasiexperiment. We examine the period 1988-1989 given the fact that after 1990, the Italian labor market has been severely a¤ected by changes in employment legislation [START_REF] Kugler | E¤ects of employment protection on worker and job ‡ows : Evidence from the 1990 italian reform[END_REF], then it has been hit by a severe recession and after 1997, job creation has been strongly in ‡uenced by the introduction of temporary contracts and of other atypical contracts by the Treu reform, making the Italian labor market more ‡exible [START_REF] Sciulli | Making the italian labor market more ‡exible: An evaluation of the treu reform[END_REF]. So, we choose to study the period 1988-1989 to be able to isolate the e¤ect on hirings of young workers due to a fall in exits of older workers from e¤ects due to other institutional e¤ects or to economic ‡uctuations. However, we are faced with a selection bias, in the sense that local labor markets with di¤erent exit rates of elderly may also di¤er in their characteristics likely to a¤ect the accession rate of young workers.

In the empirical part of the paper, we proceed in two steps. In a …rst step, we view a fall in the exit rate of older workers as a binary treatment and we examine the average e¤ect of the treatment on the treated, using a di¤erence-in-di¤erence estimator to provide some preliminary results that are not robust to the selection bias. In a second step, we view a fall in the exit rate of older workers as a continuous treatment 4 , and we control for the selection bias given observables using propensity score matching methods, generalized in a setting with a continuous treatment [START_REF] Hirano | The propensity score with continuous treatments[END_REF]. After correcting for the selection bias we …nd that the higher the fall in exits of older workers, the less hirings of young workers.

The remainder of the paper is structured in the following way: in section 2, we develop our theoretical model and we present our theoretical …ndings. Then, in section 3, we present our empirical study, describing the type of data used, explaining our methodology and presenting our empirical results. Section 4 concludes.

The theoretical model: A labor market with heterogeneous jobs and workers

Following the wide literature about skilled and unskilled workers, we consider a theoretical model with two sorts of workers and two kinds of jobs. But in the contrary of the standard literature, we distinguish workers according to their professional experience and not according to their educational level. To allow for job-to-job ‡ows, we focus on a local labor market where all …rms belong to the same sector of activity, are located in the same area and whose jobs require the same educational level. In our model, we consider the young workers, that we denote by ne as subscript, who entry into the labor market with no professional experience and the workers with experience, denoted by e as subscript. We assume that a number 0 of young workers exogenously arrive in the labor market as unemployed. They can only apply for jobs requiring no experience. In the remainder of the paper, we denote this type of jobs by S (simple job) as upscript. In our model, once a young is hired, he accumulates professional experience and can reach a su¢ cient experience level at a Poisson arrival rate to apply for complex jobs, denoted by the upscript C. Workers have interest to apply for complex jobs to have better wage opportunities, given that in our model, the wage received by a worker occupying a simple job is assumed to be the minimum wage 5 .

Both types of jobs may be broken up in the case of an idiosyncratic shock, according to a Poisson process with an arrival rate . In that case, a worker who occupied the job becomes unemployed but keeps his experience level. Indeed, whatever the unemployment duration of a worker, his working experience remains the same over time. Furthermore, as we introduce onthe-job search, workers with experience may always have interest to search for a simple job, implying a cross-skill equilibrium contrarily to the paper of Albrecht and Vroman (2002) 6 .

Following the Behaghel's speci…cation (2007), we consider a guillotine retirement age, introducing a …nite horizon in the career of worker 7 . We consider that a worker can retire as soon as he accumulates a su¢ cient experience level to apply for complex jobs. Let the Poisson arrival rate at which an experienced worker retires.The horizon H of an experienced worker is de…ned in the following way:

H = Z 1 0 t e t dt = 1 (1)
Consequently, a decrease in lengthens the career of an experienced worker. In the remainder of the paper, we assimilate a decrease in to an increase in the retirement age.

Our speci…cation is twofold: …rst, it allows us to reproduce the life cycle of a worker, who arrives in the labor market with no experience and who has to accumulate a su¢ cient level of experience occupying simple jobs to apply for complex jobs. Second, we build an economy with overlapping labor markets [START_REF] Sattinger | Overlapping labour markets[END_REF] in which young workers can not be matched with jobs requiring experience. Through this theoretical framework, we introduce an imperfect substitution between a worker occupying a complex job who retires and a young worker. Our aim is to study the e¤ect of a decrease in the exit rate of older workers, that is , on hirings of young workers.

In our model we assume that search is undirected, so workers have perfect information about the type of new vacancies but employers can not target their hirings to a special type of workers 8 . In our economy, vacancies of type S and the two types of workers (with or without experience) meet each other according to a matching function denoted by h S (u ne + u e ; v S ), that determines the number of hirings for this kind of jobs as a function of the number u i of unemployed of type i, with i 2 fne; eg , and of the number of vacancies v S of type S. We assume that the matching function is increasing, concave and linear homogenous.

An employer opening a S-type vacancy may …ll his job according to a Poisson process at an arrival rate h S ( 1 S ; 1) = q( S ). q( S ) is decreasing with the tightness on the simple job market, denoted by S , equal to the number of S-type vacancies over the number of job-seekers. We deduce therefore that an unemployed worker (whatever his type) may be matched with a simple job according to a Poisson process at an arrival rate p( S ), such that p( S ) = S q( S ) . p( S ) is therefore increasing with S .

In a similar way, vacancies of type C meet job seekers with experience according to a matching function denoted by h C (u e +o e ; v C ), that determines the number of hirings for this kind of jobs as a function of the number u e of unemployed workers with experience, of the number o e of employed job seekers and of the number v C of C-type vacancies. Consequently, a C-type vacancy may be matched with a job seeker according to a Poisson process at an arrival rate h C ( 1 C ; 1) = q( C ). This arrival rate is decreasing with the tightness in the complex jobs market C , equal to the number of Ctype vacancies over the number of job seekers with experience. We deduce therefore that an experienced job-seeker may …nd a complex job according to a Poisson process at an arrival rate p( C ) = C q( C ) increasing with C .

The …gure 1 represents the worker ‡ows in our model. The dashed arrow indicates job-to-job ‡ows. Flow equations at steady-state allow us to determine the labor force structure at the local labor market level.

We …nd the six following steady-state ‡ow equations:

(u e + o e )p( C ) = E C ( + ) (2)

Experienced workers employed in complex jobs

Unemployed

(E C + o e ) = u e [ + p( S ) + p( C )] (3)
u e p( S ) + E S ne = o e [ + + p( C )] (4)
u ne p( S ) = E S ne ( + ) (5)
E S ne + 0 = u ne p( S ) (6)
0 = (u e + o e + E C e ) (7)
where u e and o e represent respectively the number of experienced jobseekers unemployed and employed, E S ne and E e refer respectively to the number of workers with no experience occupying a simple job and to the number of experienced workers occupying a complex job and 0 represent the number of young workers who arrive in the labor market.

Combining equations ( 5) and ( 6) we obtain the expression of the number of unemployed workers with no experience:

u ne = 0 ( + ) p( S ) (8)
Furthermore, combining the equations ( 3) and ( 7) we deduce the expression of unemployment for experienced workers:

u e = 0 [ + p( S ) + p( C ) + ] (9)
The equation ( 8) is quite intuitive. The youth unemployment increases with the number of new entrants into the labor market and decreases with their probability to …nd a simple job denoted by p( S ). Our theoretical model allows us to endogenize this probability as a function of the tightness S , depending on the simple-job creation behaviour of employers. As this behaviour may be strongly in ‡uenced by the horizon of workers, our approach allows us to investigate theoretically the channels through which a decrease in the exit rate has an impact on hirings of young workers with no experience. In the next subsection, we represent the job creation decision of …rms.

The …rms'behaviour

In our model, for a sake of simplicity, each …rm can not open more than one job. As long as the job is vacant, the …rm pays a search cost c i that depends on the type of job opened (i 2 fS; Cg). When a …rm decides to open a S-type vacancy, it may be …lled by a young worker with no experience according to a Poisson process at an arrival rate q( S ), where is the share of unemployed workers with no experience over the whole unemployment, namely = u ne =(u ne + u e ). In a similar way, a S-type vacancy may be …lled by an experienced job seeker according to a Poisson process at an arrival rate ( 1)q( S ). At the steady-state equilibrium, the ‡ow value V S to an employer from opening a S-type vacancy satis…es the following Bellman equation:

rV S = c S + q( S )[ (J S ne V S ) + (1 )(J S e V S )] (10)
where J S i is the ‡ow value to a …rm of a simple job …lled by a worker of type i, with i 2 fne; eg.

Furthermore, the ‡ow value V C to a …rm which opens a C-type vacancy is de…ned by the following Bellman equation :

rV C = c C + q( C )(J C V C ) (11)
where J C is the ‡ow value to an employer of a …lled complex job. We assume that …rms are free to entry the market. Consequently, at steady-state equilibrium all pro…ts from opening a new vacancy are exhausted for the two sorts of jobs, which implies the following free-entry condition :

V C = V S = 0 (12)
In the submarket of simple jobs, a …rm opening a S-type job vacancy may either recruit a unexperienced unemployed worker or an experienced one. Let J S i be the present ‡ow value from hiring a worker of type i, with i 2 fne; eg. We assume that both types of workers have the same productivity on the same type of job. However, an experienced worker can quit his job to …nd a complex job at a Poisson arrival rate p( C ) and can also retire at a Poisson arrival rate , contrarily to workers with no experience.

Borrowing the idea of [START_REF] Belan | Subsidizing low-skilled jobs in a dual labor market[END_REF], we assume that each worker occupying a simple job receives a minimum wage w. Consequently, the present values of expected pro…t to a …rm from a simple job …lled either with a worker with no experience or with an experienced worker satisfy the following Bellman equations:

rJ S ne = y S w + (V S J S ne ) + (J S e J S ne ) (13)
And :

rJ S e = y S w + [ + + p( C )](V S J S e ) (14)
Furthermore, when an experienced job seeker …nds a complex job, he starts producing an output y C > y S and he receives a wage w C > w. This part aims at providing a theoretical intuition, so we focus on the wageposting equilibrium. His job may break up for two reasons. First, it is hit by an idiosyncratic shock at a Poisson arrival rate and the worker gets into unemployment. Second, the worker retires at an arrival rate . So, the ‡ow value to a …rm from a …lled complex job satis…es the following Bellman equation:

rJ C = y C w C + ( + )(V C J C ) (15)

Wage posting equilibrium

We determine the steady-state equilibrium. Combining the Bellman equation [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF] and the free-entry condition we obtain the following expression:

J S ne + (1 )J S e = c S q( S ) (16)
Substituting the equations ( 13) and ( 14) into the expression ( 16) we get the following relation between S et C :

(r + + ) [(y S w) + [y S w] [r + + p( C ) + ] ] + (1 )(y S w) [r + + p( C ) + ] = c S q( S ) (17)
Furthermore, substituting the Bellman equation [START_REF] Davis | Gross job ‡ows[END_REF] in the expression (11) and using the free-entry condition, we obtain the following expression:

c C q( C ) = (y C w C ) (r + + ) (18)
To determine the e¤ect of a decrease in on job creation for each type of job, we use speci…c functional form to describe the two matching processes. We assume as in the standard literature that each matching function is Cobb-Douglas with an elasticity equal to = 0; 5, such that:

h S (u ne + u e ; v S ) = (u ne + u e ) 1 2 (v S ) 1 2 (19) h C (u e + o e ; v C ) = (u e + o e ) 1 2 (v C ) 1 2 (20)
Substituting ( 20) into [START_REF] Heckman | Matching as an econometric evaluation estimator : Evidence from evaluating a job-training programme[END_REF], we obtain the following expression:

( C ) = (y C w C ) c C (r + + ) 2 (21)
Consequently, through our theoretical model, we can write the tightness for the submarket of complex jobs as a function of only exogenous parameters. This expression shows us clearly that a decrease in the exit rate among older workers denoted by fosters the creation of complex jobs. Our result is in accordance with …ndings of [START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF] that highlight a positive horizon e¤ect of an increase in the retirement age on job creation. However, in this paper, we investigate the e¤ect of a decrease in the exit rate among older workers on hirings of young workers. We mention previously that young workers can not apply for complex jobs so we have to determine the e¤ect of a decrease in on the creation of simple jobs, that is on S . Substituting ( 21) into [START_REF] Gautier | Unemployment and search externalities in a model with heterogeneous jobs and workers[END_REF], we …nd that at steady-state equilibrium, S satis…es the following equation:

L( S ; ) = (r + + ) [(y S w) + [y S w] [r + + h (y C w C ) c C (r+ + ) i + ] ] + (1 )(y S w) [r + + h (y C w C ) c C (r+ + ) i + ] c S q( S ) = 0 (22)
In the wage posting equilibrium, we …nd two main o¤setting e¤ects of a decrease in the exit rate among older workers denoted by on the tightness for the submarket of simple jobs S . On the one hand, a decrease in raises the horizon of experienced workers and therefore increases the average duration of simple jobs, which encourages employers to open more simple jobs. On the other hand, a decrease in has a negative e¤ect on S through job-to-job ‡ows. This is the key e¤ect that we highlight in our model. Indeed, given that the creation of complex jobs is fostered by a decrease in , the probability for an experimented worker occupying a simple job to be matched with a complex job is higher. Consequently, the average duration of simple jobs decreases and …rms are more reluctant to open such jobs, reducing hiring of young workers.

Nevertheless, recall that the share of unemployed workers without experience denoted by is given by our ‡ow equations at steady-state and therefore it is a function of both S and C . Indeed, combining equations ( 8) and ( 9), we write in the following way:

= ( + ) [ + p( S ) + p( C ) + ] [ + p( S ) + p( C ) + ]( + ) + p( S )
We see that the expression of is quite complicated and depends both on S and C . So it is not so straightforward to examine the sign of the derivative of S with respect to .

Proposition 1 In the wage posting equilibrium, a decrease in reduces S ,i.e @ S @ > 0 only if

y C w C > c C (r + + ) 2 Proof. See Appendix
We have shown theoretically that a decrease in the exit rate among older workers within a local labor market may reduce the tightness S and therefore it has a negative e¤ect on the hiring rate p( S ) of young unemployed workers, provided that the ‡ow value to an employer of a …lled complex job is su¢ ciently high. Thus, when this condition holds, the horizon e¤ect is o¤set by the indirect e¤ect due to job-to-job ‡ows. Given the expression of u ne we deduce that a fall in the exit rate raises unemployment among young workers with no experience.

3 The empirical study

The data

For our empirical study, we use matched employer-employee data from the Worker Histories Italian Panel (WHIP) a database built by the University of Turin in cooperation with the Italian Social Security Administration (INPS). The original data set collects social security forms of a 1/90 random sample of employees every year. It provides accurate information about the employment spells of each worker. The data also includes also longitudinal records for …rms employing the randomly selected workers in the sample. However, the data from the WHIP does not provide information about all individuals who work in a …rm, so we can not observe the distribution of individual characteristics for each …rm of the sample. To adress this issue, we carry out our study at the local labor market level, given that each observation unit is made up of a group of …rms belonging to the same sector of activity, located in the same area and whose jobs require the same educational level.

We can discuss our synthetic-…rm approach. Indeed, it leads us to underestimate the variance of each …rm characteristic between …rms given that each observation unit is a group of …rms. To test the relevance of this approach, Contini et al. (2008) investigate the wage dynamics in the province of Veneto, confronting the results obtained through the synthetic-…rm method with those obtained through an analysis at the …rm-level. They conclude that both approaches lead to similar results when the number of …rms in each local labor market is quite small (between 50 and 95 …rms per observation unit). Consequently, there is a trade-o¤ when choosing the agregation level between having the …nest grid and having a su¢ cient number of …rms in each observation unit.

Furthermore, studying the mobility of workers at a synthetic-…rm-level allows us to account for the job-to-job ‡ows that account for the major part of the worker ‡ows observed in Italy. Using data from the WHIP, [START_REF] Leombruni | Eppure si muoveva già. una breve storia della mobilità dei lavoratori in italia[END_REF] show that job-to-job ‡ows in the Italian labor market are observed within a sector, and within a geographical area. Consequently, we choose to agregate our data by province and by sector of activity, and we end up with a sample of 517 synthetic …rms. Studying the mobility of workers at a local labor market level, allows us to account for the vacancy-chain pattern and we can expect that exits of older workers may imply hirings of young workers, holding job creation constant.

Our empirical analysis aims at investigating the e¤ect of a decrease in the exit rate among older workers on the hiring rate of the young, during the period 1988-1989. Here, the exit rate of older workers is de…ned as the number of exits of older workers observed in a year over the stock of older workers employed in the start of the year. In addition, the hiring rate of young workers is de…ned as the number of hirings of young workers observed in a year over the stock of young workers employed in the end of this year. We choose this period given that since 1990, employment protection legislation has been severely changed and has had a strong e¤ect on the Italian labour market [START_REF] Kugler | E¤ects of employment protection on worker and job ‡ows : Evidence from the 1990 italian reform[END_REF]. Moreover, since 1993, the Italian labour market has been strongly a¤ected by the economic recession. At last, since 1997, the Treu reform liberalized the use of temporary contracts and introduced new atypical contracts, involving a strong e¤ect on Italian worker ‡ows [START_REF] Sciulli | Making the italian labor market more ‡exible: An evaluation of the treu reform[END_REF].

Consequently, we study the period 1988-1989 to isolate the e¤ect on youth employment due to a slowdown in the exit rate among older workers from other e¤ects due to institutional changes or to economic ‡uctuations. However, even during the period 1988-1989, job creation among young workers has been increased by the introduction of CFL (Contratto Formazione e Lavoro) 9 contracts [START_REF] Tattara | Evaluating the italian training on the job contract (c ‡)[END_REF]. To control for this e¤ect, we include in our set of regressors the share of CFL contracts observed in each local labour market.

As mentioned in the introduction, we carry out our empirical study in two steps. In the …rst step, we consider a binary variable equal to one if the observation unit experienced a reduction in the exit rate of workers aged 50 or more between 1988 and 1989 and we study the e¤ect of this dummy on the hiring rate of young workers aged 30 or less in 1989. In a second step, we include in our sample only the treated observations and we examine to what extent a fall in the exit rate among older workers a¤ect hirings of young workers. In this step, we view the absolute value of the decrease in the exit rate among older workers between 1988 and 1989 as the continuous treatment. For some descriptive statistics, we group our treated observations into two categories: the group 1 is made up of synthetic …rms, in which the value of the continuous treatment is lower than the median level, that is 13.5% and the group 2 is composed of synthetic …rms, in which the value of the treatment exceeds 13.5%. We observe in the table 1 that the average hiring rate of young workers is higher in the group 1 (around 54,58%) than in the group 2 (around 51,57%). So a strong slowdown in the exit rate among older workers may reduce new hirings of young workers. To control for potential composition e¤ects likely to in ‡uence the hiring rate of young workers or the exit rate of older workers observed within a local labor market, we use the set of information provided by our panel matched employer-employee data. We include …rst the age structure of each synthetic …rms controling for the share of younger workers and of older workers in each local labor market. We add also the share of women to test for gender-speci…c e¤ects. We also introduce the average ratio between the average wage of young workers and the average wage of older workers in each synthetic …rm. Indeed, this ratio may have a strong e¤ect on the substitutability between young and older workers. We include also the share of CFL contracts signed in each local labor market, following empirical results supporting evidence of the strong positive impact of these contracts on youth employment [START_REF] Tattara | Evaluating the italian training on the job contract (c ‡)[END_REF].

Furthermore, empirical study of [START_REF] Contini | Job creation and job destruction[END_REF] show that the age or the size of the …rm a¤ect the workers'mobility. Consequently, in our set of regressors, we include the share of young …rms (less than 8 years) and the share of small …rms (less than 9 workers) in each local labor market. To control for a potential e¤ect due to economic trends in each synthetic …rm, we add in our set of explantory variables the unemployment rate and a dummy equal to one if the local labor market is declining 10 . We provide some descriptive statistics in the table 2. it . However, the estimation of this di¤erence raises an issue of missing data. Indeed, at time t an observation i is either treated or not treated and we can not observe simultaneously the value of the dependent variable for one observation i in the two states D = 0 and D = 1.

To adress this issue, we can determine the average treatment e¤ect on the treated group using a di¤erence-in-di¤erence estimator denoted by DD de…ned in the following way:

DD = fE(Y it jD = 1) E(Y it jD = 0)g fE(Y it 1 jD = 1) E(Y it 1 jD = 0)g
The idea is to estimate the gap between the hiring rates of young workers observed respectively in treated and in control synthetic …rms at time t 1 and then to evaluate to what extent this gap has been a¤ected at time t. However, this method holds only if the time invariance assumption is checked. This assumption implies that the hiring rate of young workers for the control group of observations would have evolved between t 1 and t in the same fashion that the hiring rate for the treated group of observations, in the event that they had not been exposed to the treatment. The time invariance assumption may be formulated in the following way:

E(Y 0 it Y 0 it 1 jD = 0) = E(Y 0 it Y 0 it 1 jD = 1)
This assumption is particularly strong, given that treated synthetic …rms may di¤er widely of not treated synthetic …rms regarding some characteristics likely to a¤ect hirings of young workers (a known issue that we refer to as the Ashenfelter's dip, 1978). To attenuate this bias, our strategy consists in including a set of regressors when computing the di¤erence-in-di¤erence estimator, de…ning the following model:

Y it = X it + D + t + DD D t + c i + " it with " it N (0; 2 ) ( 23
)
where X it is a vector composed of a set of regressors observed in a synthetic …rm i at time t, c i is an individual-speci…c e¤ect, that allows us to control for unobserved heterogeneity and " it is the error term of our regression. In our model, we include a calendar e¤ect adding a binary variable t equal to one if the time t corresponds to the period when we consider our treatment, that is the year 1989. We also introduce a binary variable denoted by D equal to one if the synthetic …rm i experienced a decrease in the exit rate among older workers between 1988 and 1989. Our aim is to estimate the parameter DD to determine the e¤ect of the interaction between the variables D and t on the hiring rate of young workers. In this regression, the parameter DD is the di¤erence-in-di¤erence estimator.

Results

We present our results in the table 3. We observe …rst that the Pearson's coe¢ cient is equal to 0.3952. Consequently, our random-e¤ect model is more relevant than a pooled-OLS regression, given that it allows to control for unobserved heterogeneity. We also see that the hiring rate of young workers at time t is strongly correlated with a lot of regressors like the job declining dummy, the share of young or small …rms, the share of young workers or the share of CFL signed. It implies that the time invariance assumption is not a priori checked and that we need to control for the characteristics of each local labour market to examine the e¤ect of the treatment on the treated group. We …nd that the di¤erence-in-di¤erence estimator DD is not signi…cant.

We can discuss the relevance of our results given that treated synthetic …rms are de…nitely not comparable with the control group. For instance, implementing a simple t-test of di¤erence in means between these two groups for the job declining dummy, we …nd in the table 4 a t-stat equal to 3:3114, quite higher in absolute value than 1:96. So the probability for the treated synthetic …rms to experience job destruction in 1988 is quite higher than this probability for synthetic …rms of the control group. It may be explained by the fact that declining sector tend to reduce their hirings, maintaining their ageing workforce. Consequently, we can not draw clear conclusions of a simple di¤erence-in di¤erence model due to the heterogeneity of the two groups regarding their covariates. Furthermore, in this subsection, we viewed the decrease in the exit rate among older workers as a binary treatment. It could be interesting to estimate the e¤ect of this decrease on the hirings of young workers, viewing this decrease as a continuous treatment. On the one hand, this strategy forces us to keep in our sample only the treated observations. On the other hand, this approach allows us to separate observations with a low level of treatment from those with a high level of treatment and to see how the hiring rate of young workers varies according to the level of the treatment.

Consequently, we use in the next subsection a propensity score matching procedure and we implement it to a continuous treatment, that is the decrease in the exit rate among older workers between 1988 and 1989 in absolute value.

Generalized propensity score matching method 3.3.1 Identi…cation strategy

As mentioned previously, our strategy does not consist in regressing the hiring rate of young workers on absolute value of the decrease in the exit rate among older workers. Indeed, even though we include in our model all the information provided by our database, we are in the face of a selection bias, in the sense that our explanatory variable of interest may be a¤ected by a set of observables, likely to be correlated with the error term of our regression. To control for the selection bias through observables, we use a propensity score matching procedure generalized to a continuous treatment. As in the binary treatment context, propensity score methods in a setting with continuous treatment rely heavily on the key assumption that adjusting for pre-treatment di¤erences solve the problem of drawing causal inference. Formally, we make the weak unconfoundedness assumption, introduced by [START_REF] Hirano | The propensity score with continuous treatments[END_REF], that requires that the treatment assignment mechanism is conditional independent of each potential outcome given the pre-treatment variables.

Let us consider a set of N synthetic …rms characterized by a vector X it of observables, by a number T i equal to the absolute value of the decrease in the exit rate among older workers between 1988 and 1989 and by a number Y i ( ) equal to the hiring rate of young workers, associated with the treatment level T i = . T i is our continuous treatment and Y i is our dependent variable. The unconfoundedness assumption may be formulated in the following way:

Y i ( ) ? T i jX i
Given unconfoundedness, we can apply matching methods based on the Generalized Propensity Score (GPS) with continuous treatment introduced by Imbens and [START_REF] Hirano | The propensity score with continuous treatments[END_REF]. The GPS is de…ned as the conditional density of the actual treatment given the observed covariates. Formally, let r( ; x) = f T jX ( jx) be the conditional density of the treatment given the covariates. Then, the GPS is R i = r( i ; X i ). The GPS is a balancing score, that is, within strata with the same value of r( ; x), the probability that T = t does not depend on the value of X. In combination with the weak unconfoundedness assumption, this balancing property implies that:

f T ( jr( ; X i ); Y i ( )) = f T ( jr( ; X i ))
. As a result, the GPS can be used to eliminate any biases associated with di¤erences in the covariates. Formally, if the assignment to the treatment is weakly unconfounded, given pre-treatment variables X i , then we can de…ne in a second step a speci…cation regressing the conditional mean of our dependent variable on the value of the treatment T i and on the value of the generalized propensity score R i . In our analysis, we choose a simple linear model de…ned in the following way:

E(Y jT = ; R = r) = b 0 + b 1 T + b 2 R + b 3 T R ( 24
)
where and r are speci…c values of respectively the treatment and the propensity score.

At last, we have to determine the average causality e¤ect or dose-response function to evaluate the e¤ect of several values of the treatment on the hiring rate of young workers. The dose-response function denoted by ( ) may be formulated in the following way:

( ) = E[E(Y ( )jr( ; X)] (25)
In other words we have to estimate ^ ( ) such that:

^ ( ) = 1 N N X i=1 b0 + b1 + b2 R + b3 R (26)
To obtain the dose-response function, we have to calculate ^ ( ) for different values of treatment, given that for each value , standard errors are computed through bootstrapping methods.

Results

The …rst step consists in matching our observations calculating the Generalized Propensity Score (GPS in the remainder of the paper). The GPS methods are implemented in our application using the gpscore, dose-response STATA package [START_REF] Bia | A stata package for the estimation of the dose-response function through adjustment for the generalized propensity score[END_REF]. A key assumption in the STATA implemented version of the GPS methods is the normality of the treatment variable conditional on the pre-treatment covariates. In our study we assume that the Box -Cox transformation of the treatment has a Normal Prior to evaluate the dose-response function, we have to ensure that the balancing property is checked. In the table 5, we report the values of the tstatistic when we carry out a test of di¤erence in means for each explanatory variable and for di¤erent treatment levels after having adjusted by the GPS. The idea is to test for each group of observations, de…ned by a speci…c treatment interval, whether the observables are similar with those of synthetic …rms of the other groups, after matching. We use the bold case values to indicate that the t-stat is higher than 1.96. In that case, balancing property does not hold. We remark in the table 5 that no t-stat values are higher than 1.96. We can conclude that our GPS matching procedure allows us to check the balancing property. This function describes to what extent a decrease in the exit rate among older workers has an e¤ect on the hiring rate of young workers. Given that the treatment is equal to the absolute value of the decrease, a high treatment level corresponds to a strong fall in exits of older workers. We observe in the …gure 2 that for a treatment level that ranges from 0.1 to 0.4, the treatment a¤ects slightly the hiring rate of young workers. However, for a dose of treatment higher than 0.4, we remark that a slowdown in the exits of older workers has a negative e¤ect on hirings of young workers.

Thus we show empirically that a strong decrease in exits of older workers reduces sharply hirings of young workers at a local labor market level, which provides some empirical evidence of our theoretical results. Nevertheless, we have to take these empirical …ndings with caution. Indeed, we matched our observations given a set of observables which allows us to check the balancing property only for the variables that we include. Therefore we did not correct for a selection bias on unobservables.

Furthermore, recall that our study has been carried out through a synthetic-…rm approach. It implies that our observations has been built grouping in a same cell …rms belonging to the same sector of activity and located in the same province. Our approach is therefore strongly in ‡uenced by the aggregation level chosen to build each synthetic …rm.

Concluding remarks

In this paper, we investigated the e¤ect of delaying retirement age on hirings of young workers. Our starting point was to consider an imperfect substitutability between young and older workers. We explained the vacancy-chain pattern, through which an exit of an older worker may involve a new hiring of a young worker, even though these two types of workers are not perfectly substitutable for an employer. This vacancy-chain results from job-to-job ‡ows observed within a local labor market composed of …rms belonging to the same sector of activity, located in the same area and whose jobs require the same educational level.

In the …rst part of the paper, we highlighted the theoretical channels through which exits of older workers a¤ect hirings of young workers within a local labor market. In this part, we represented a vacancy-chain in a matching model with heterogeneous jobs and workers, considering that workers are either experimented or not and that workers with no experience (i.e. the young workers) can not apply for a complex job requiring experience. In this model, to represent a vacancy-chain, we allowed for job-to-job ‡ows from simple jobs to complex jobs, but only if workers have a su¢ cient level of professional experience to apply for complex jobs.

We pointed out two main e¤ects of a fall in the exit rate among older workers on the hiring rate of young workers. First, this decrease lengthens the horizon of an experienced worker and encourages job creation for both types of jobs. Second, the increase in complex job creation raises the probability for an experienced employed job seeker to …nd a complex job. Therefore, it reduces the average duration of a simple job making …rms more reluctant to open simple jobs and reducing hirings of young workers with no experience. We show that the net e¤ect of a decrease in the exit rate among older workers on the hirings of young workers is negative if the ‡ow value to an employer of a …lled complex job is su¢ ciently high.

Then, in the second part of the paper, we have studied empirically the e¤ect of a slowdown in the exit rate among older workers on the hiring rate of young workers within a local labor market. We use matched employeremployee data drawn from the Worker Histories Italian Panel. We considered a sample of synthetic …rms, each of these being composed of …rms belonging to the same sector and located in the same area. We grouped …rst our observations into two categories: synthetic …rms that experienced a decrease in the exit rate among older workers between 1988 and 1989 and those that did not experience such a decrease during this period. Using a di¤erence-in-di¤erence approach and controling for characteristics of each synthetic …rm, we showed that a slowdown in the exits of older workers does not exert a signi…cative e¤ect on hirings of young workers.

Nevertheless, we pointed out that some variables di¤er greatly among the two groups and therefore a simple di¤erence-in-di¤erence analysis has no sense, because it implies a strong selection bias. Consequently, keeping in our sample only synthetic …rms that experienced a decrease in the exit rate among older workers between 1988 and 1989 and matching observations each other adjusting by the generalized propensity score to correct for the selection bias on observables, we found that a strong decrease in the exit rate among older workers reduces signi…catively the hiring rate of young workers within a local labor market. However, we have to take our results with caution given that our matching method does not allow us to correct for a selection bias on unobservables. Furthermore, our …ndings may depend on the aggregation level chosen to build our synthetic …rms. To test the robustness of our results, we shoud reproduce this study on very big …rms assimilating job-to-job ‡ows to promotions within the …rm.

One extension of this paper may come to mind. We could allow for both types of worker mobility, within a local labor market, through job-to-job ‡ows and within a …rm, through internal promotions. This speci…cation could allow us to better grasp channels through which a decrease in the exit rate among older workers a¤ects the hiring rate among young workers. We leave this idea for further investigation. We observe that:

p( C ) + @p( C ) @ = (y C w C ) c C (r + + ) (y C w C ) c C (r + + ) 2
Substituting this expression into the previous one, we obtain: @( 1 ) @ = p( S ) ( + ) < 0

In addition, given that < 1 + (r+ ) , we deduce that: Consequently, @ ln(E(J S )=@ > 0 only if

[ @p( C ) @ + 1] < 0 , (y C w C ) > c C (r + + ) 2
If this condition holds, it ensures that L 2 > 0 and therefore @ S =@ > 0.

Figure 1 :

 1 Figure 1: Worker ‡ows in a local labor market

Figure 2 :

 2 Figure 2: Dose-response function of the treatment on the hiring rate of young workers

  ln[E(J S )] = ln[ (y S w)[p( C ) + ] + (r + + )(y S w)] ln[r + + p( C ) + ] ln(r + + )So, di¤erentiating this expression with respect to , we obtain:@ ln E(J S ) @ = [ @p( C ) @ + 1] + @ @ [p( C ) + ] [p( C ) + ] + (r + + ) [ @p( C ) @ + 1] [r + + p( C ) + ] = [ @p( C ) @ + 1][ 1 [p( C ) + ] + (r+ + ) 1 p( C ) + + r + ] + @ @ [p( C ) + ] [p( C ) + ] + (r + + )Recall that:1 = 1 + p( S ) [ + p( S ) + p( C ) + ]( + )Di¤erentiating this expression with respect to , we obtain:@( 1 ) @ = p( S ) ( + ) ( + p( S ) + p( C ) + + [1 + @p( C ) @ ]) ( [ + p( S ) + p( C ) + ]) 2

( 2 +

 2 p( S ) + + (y C w C ) c C (r+ + ) [1 (r+ + ) ]) ( [ + p( S ) + p( C ) + ]) 2

[ 1 [

 1 p( C ) + ] + (r+ + ) 1 p( C ) + + r + ] < 0

Table 1 :

 1 Average hiring rate among workers aged 30 or less for di¤erent values of the continuous treatment.

		Group 1 Group 2 Total
	Average hiring rate
	among young workers	54.58% 51.57% 53.07%
	Number of observations	117	117	234

Note :

The group 1 is composed of synthetic …rms that experienced a decrease in the exit rate among older workers between 1988 and 1989 lower than 13.5% in absolute value. The group 2 is composed of synthetic …rms that experienced a decrease in the exit rate among older workers between 1988 and 1989 higher than 13.5% in absolute value. Source :

WHIP (waves 1988WHIP (waves -1989) )

Table 2 :

 2 Descriptive statistics for the years 1988 and 1989Let Y D it , the hiring rate of young workers observed at time t in a local labor market i in the state D, with D 2 f0; 1g and t 2 f1988; 1989g. The binary variable D is equal to one if the synthetic …rm experienced a decrease in the exit rate among older workers between 1988 and 1989. In that case, the synthetic …rm is viewed as a treated observation. We have to determine the e¤ect of the treatment on our dependent variable Y it , in other words we have to estimate the di¤erence Y 1

	Variable	Year 1988 Year 1989
	Hiring rate of young workers	50.37%	49.34%
	Exit rate of older workers	21.62%	20.5%
	Share of young workers	38.81%	38.91%
	Share of older workers	14.89%	14.94%
	Share of women	24.18%	24.6%
	Share of CFL contracts	22.4%	21.35%
	Share of young …rms (less than 8 years)	33.96%	32.92%
	Share of small …rms
	(less than 9 employees)	40%	40.1%
	Ratio between the average wage of
	youngsters and that of older workers	85.63%	86.97%
	Unemployment rate	11.11%	10.84%
	Dummy equal to one if the synthetic …rms
	is declining	23.21%	26.31%
	Number of observations	517	517
	Source : WHIP (waves 1988 and 1989)
	3.2 A di¤erence-in-di¤erence approach

Table 3 :

 3 A di¤erence-in-di¤erence estimation of the e¤ect of a decrease in the exit rate among older workers on the hiring rate among youngsters on the period[1988][1989]

	Variable			Coe¢ cient Standard error
	Binary variable t = 1
	if t = 1989			-0.013	0.013
	Binary variable D = 1 if the
	synthetic …rm is treated			0.024	0.016
	Interaction term D t			0.007	0.019
	Share of young workers			0.3634	0.108
	Share of older workers			0.266	0.14
	Share of women			-0.088	0.046
	Share of CFL contracts			0.273	0.05
	Share of young …rms (less than 8 years)	0.173	0.08
	Share of small …rms
	(less than 9 employees)			0,2	0,058
	Ratio between the average wage
	of young and that of older workers		0.07	0.037
	Unemployment rate			0.156	0.128
	Dummy equal to one if the synthetic …rms
	is declining			-0.089	0.013
	R 2			0.3091
	Number of observations			1034
	Pearson's coe¢ cient			0.3952
	Note : Coe¢ cients have been estimated through a random-e¤ect model. Standard errors
	are corrected from the heteroscedasticity bias through the method of White.
	Signi…cativity thresholds : : 10%	: 5%	: 1%
	Source : WHIP (waves 1988 and 1989)

Table 4 :

 4 Di¤erence in means of the job declining dummy in 1988 between the treated and the control group

	Group	Observations Mean	Standard error
	Control	283	0.177	0.023
	Treated	234	0.299	0.029
	di¤erence		-0.0122 t-stat = -3.3114

Source : WHIP wave 1988

Table 5 :

 5 Testing di¤erence between means for di¤erent treatment interval after adjustment by the GPS

	Variables for the year 1988		t-stat for di¤erent
			treatment intervals
		[0;0.05] [0.05;0.15] [0.15;0.3] [0.3;1]
	Share of young workers	-0.18	0.34	-1.11	0.781
	Share of older workers	0.369	-0.38	0.524	-0.918
	Share of women	-1.15	1.26	0.01	-0.791
	Share of CFL contracts	-0.389	0.78	-0.103	0.039
	Share of young …rms (less than 8 years)	-0.11	-0.243	-0.42	0.577
	Share of small …rms
	(less than 9 employees)	0.625	-0.705	-0.235	0.438
	Ratio between the average wage of
	young and that of older workers	0.364	-1.073	0.494	0.592
	Unemployment rate	0.941	-0.77	0.148	-0.252
	Dummy equal to one if the synthetic …rm
	is declining	-0.218	0.287	0.205	-0.304
	Hiring rate of young workers	-0.011	-0.451	0.448	-0.505
	Note : the treatment corresponds to the absolute value of the decrease in the exit rate
	among older workers between 1988 and 1989 for each synthetic …rm.
	Bold case values indicate that the t-statistic is higher than 1.96 in absolute value.
	Source : WHIP (waves 1988 and 1989)

See also[START_REF] Chéron | Job creation and job destruction over the life cycle: The older workers in the spotlight[END_REF] for this question

A local labour market is a group of …rms belonging to the same sector of activity, located in the same area and whose jobs require the same educational level

For similar speci…cation see also[START_REF] Dolado | On-the-Job Search in a Matching Model with Heterogeneous Jobs and Workers*[END_REF][START_REF] Gautier | Unemployment and search externalities in a model with heterogeneous jobs and workers[END_REF][START_REF] Mortensen | Unemployment responses to 'skill-biased' technology shocks: The role of labour market policy[END_REF], Albrecht and Vroman, 1999

Actually, we consider the absolute value of the decrease in the exit rate of older workers. Consequently, an high value of the treatment means that the observation unit experienced a severe fall in the exit rate of older workers

See also[START_REF] Belan | Subsidizing low-skilled jobs in a dual labor market[END_REF] for a similar speci…cation

Contrarily to their paper, we de…ne the skill level of a worker by his experience level

The implications of a …nite horizon in a matching model on job creation and job destruction have been already investigated by[START_REF] Cheron | Job creation, job destruction and the life cycle[END_REF][START_REF] Behaghel | La protection de l'emploi des travailleurs Âgés en france : une étude de la contribution delalande[END_REF]

However, in the case of complex jobs, employers are sure to be matched with only workers with experience, given that youngsters can not apply for this type of jobs.

The CFL is a …xed-term contract with a maximum duration of 24 months not renewable. This contract was introduced in 1984 to easy the entrance of youngsters aged between 15 and 29 into the labour market, providing …scal bene…ts to …rms hiring young people.

It means that in this synthetic …rm, the number of jobs in the end of the year is lower that the number of jobs observed in the beginning of the year

distribution, given the covariates. Formally, let BoxCox(T i ) denote the Box-Cox transformation of the treatment variable, we de…ne BoxCox(T i ) in the following way:

Using the Shapiro-Wilk test, we check that BoxCox(T i )jX i N ( 0 + 1 X i ; 2 v ), where 0 , 1 and 2 v are parameters to be estimated through a maximum likelihood procedure. So for each observation, the generalized propensity score is de…ned in the following way:

Then, according to the distribution of the values of the treatment, that is the absolute value of the decrease in the exit rate among older workers between 1988 and 1989, we divide our sample into di¤erent groups. In our study, we choose to break down our treatment into four intervals : [0;0.05], [0.05;0.15], [0.15;0.3] and [0.3;1]. Therefore, at each time t, we divide our sample into quartiles according to the value of the treatment.

In each of these groups, we calculate the GPS for a treatment level T i corresponding to the median treatment level for each group. We can determine the GPS of each observation i belonging to the group j, with j = f1; 2; 3; 4g in the following way:

where T j is the median treatment level for each group j. According to the distribution of the GPS in each of these groups, we divide them into K blocks. Here, we consider 5 blocks. For each block k j , with k = f1; 2; 3; 4; 5g, composed of synthetic …rms presenting a speci…c treatment level corresponding to their group j and a speci…c GPS corresponding to their block k, we compare the values of observables with those of observations belonging to the block k j 00 avec j 0 6 = j, namely synthetic …rms with a similar GPS but a di¤erent treatment level.

Appendix

Rearranging terms of L( S ; ), we obtain the following equation:

Di¤erentiating L( S ; ) we get:

where L i is the partial derivative of L( S ; ) with respect to its ith element, with i 2 f1; 2g. We determine

We have to determine the sign of @ =@ S . We remark that:

As the second term on the right-hand side of the equation is increasing with S , we deduce that is decreasing with S . Consequently, L 1 < 0.

Furthermore, let E(J S ) be the the esperance of the expected income ‡ows to an employer from a …lled simple job such that:

where p( C ) = (y C w C ) c C (r+ + ) . Consequently, L 2 has the same sign as @E(J S )=@ . We consider the logarithm of the expression to simplify calculation so:

TEPP