Job Creation and Job Destruction over the Life Cycle: The Older Workers in the Spotlight

DISCUSSION PAPER SERIES IZA DP No. 2597 Job Creation and Job Destrution over the Life Cyle: The Older Workers in the Spotlight Jean-Olivier Hairault Arnaud Chéron François Langot February 2007 Forshungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor

Job Creation and Job Destrution over the Life Cyle: The Older Workers in the Spotlight Jean-Olivier Hairault EUREQua, University of Paris I, CEPREMAP and IZA Arnaud Chéron GAINS, University of Le Mans and EDHEC François Langot GAINS, University of Le Mans, CEPREMAP and EUREQua Disussion Paper No. 2597 February 2007 IZA P.O. Box 7240 53072 Bonn Germany Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: iza@iza.org Any opinions expressed here are those of the author(s) and not those of the institute. Researh disseminated by IZA may inlude views on poliy, but the institute itself takes no institutional poliy positions. The Institute for the Study of Labor (IZA) in Bonn is a loal and virtual international researh enter and a plae of ommuniation between siene, politis and business. IZA is an independent nonprofit ompany supported by Deutshe Post World Net. The enter is assoiated with the University of Bonn and offers a stimulating researh environment through its researh networks, researh support, and visitors and dotoral programs. IZA engages in (i) original and internationally ompetitive researh in all fields of labor eonomis, (ii) development of poliy onepts, and (iii) dissemination of researh results and onepts to the interested publi. IZA Disussion Papers often represent preliminary work and are irulated to enourage disussion. Citation of suh a paper should aount for its provisional harater. A revised version may be available diretly from the author.

IZA Disussion Paper No. 2597 February 2007 ABSTRACT Job Creation and Job Destrution over the Life Cyle: The Older Workers in the Spotlight This paper extends the job reation - job destrution approah to the labor market to take into aount the life-yle of workers. Forward looking deisions about hiring and firing depend on the time over whih to reoup adjustment osts. The equilibrium is typially featured by inreasing (dereasing) firing (hiring) rates with age, and a hump-shaped age-dynamis of employment. The empirial plausibility of the model is assessed by inorporating existing age-speifi labor market poliies in Frane. Finally we show that the age-dynamis of employment is optimal when the Hosios ondition holds and we design the optimal agepattern for employment poliies when this ondition does not apply. JEL Classifiation: J22, J26, H55 Keywords: job searh, mathing, life yle Corresponding author: Jean-Olivier Hairault EUREQua University of Paris I 106-112, Boulevard de l'hôpital 75647 Paris Cedex 13 Frane E-mail: Jean-Olivier.Hairault@univ-paris1.fr

1 Introdution It is now well known that the low employment rate of older workers aounts for half of the European employment gap (see OECD [2006]). The longrun unemployment inidene is 50% higher for older workers (see Farber [1997] and Mahin and Manning [1999]). A rst strand of the empirial literature emphasizes the negative role played by labor market institutions (spei insurane programs) on the job-searh deisions of older unemployed workers (see for instane Blöndal and Sarpetta [1998]). A seond strand gives greater importane to skill obsolesene, arguing that older workers suer from a biased tehnologial progress. Under wage stikiness, this gives rms inentives to send older workers into early retirement (see Crépon, Deniau and Perrez-Duarte [2002] and and Aubert, Caroli and Roger [2006]). 1 However, something is missing in this whole piture. Figure 1 shows that the fall in the employment rate of older workers is steeper when the retirement age gets loser, whatever the ountry onsidered. Two ountry groups emerged very learly in the mid-nineties: those with high employment rates for workers aged 55-59 (Canada, Great Britain, Japan, the United States and Sweden) and those whih experiene a huge derease in employment rates at these ages, around 25 points with respet to the 50-54 age group (Belgium, Frane, Italy and the Netherlands). As doumented by Gruber and Wise [1999], the seond group of ountries is haraterized by an eetive retirement age of 60 (versus 65 in the rst group). However, there is no reason to believe that these ountries are more sensitive to ongoing tehnologial progress. In this paper, it is rst argued that the proximity of the retirement age is the primary ause of the derease in the employment rate of older workers. We study the diret inuene of impending retirement on both job reation and job destrution, and abstrat it from the labor produtivity dimension. 2 Sine Oi [1962], labor is indeed viewed as a quasi-xed input 1This point has already been put forward by Lazear [1979], from a theoretial standpoint. 2See Bartel and Siherman [1993] and Friedberg [2003] for an investigation of the relation between the impending retirement and the labor produtivity. 2

Figure 1: Employment rates from age 30 to 64 for OECD Countries 100 Jap 90 80 Swe US UK Can Spa Ger Ita Net Fra Bel 70 employment rate 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 Ranking of eetive retirement age (from the highest to the lowest) Soure: OECD data for 1995 (authors' alulation). In eah ountry, eah bar refers to employment rates of the age groups : 30-49 (rst bar on the left), 50-54, 55-59 and 60-64 (last bar on the right) fator so that the hiring proess is ostly, leading rms to implement laborhoarding strategies. An important ontribution of Mortensen and Pissarides [1994] (MP hereafter) has been to provide theoretial foundations for these mehanisms in an overall theory of equilibrium unemployment. In that ontext, endogenous hirings and separations depend on the expeted duration of jobs. But beause workers live forever, workers' age plays no role on MP's labor market equilibrium. Surprisingly enough, the impat of the life yle of workers in this extensively-used framework has not been yet addressed 3. From this point of view, our paper lls a gap. Beause the horizon of older workers is shorter, we show that rms and workers invest less in job-searh 3The proximity of retirement has been srutinized by Seater [1977] and Lungqvist and Sargent [2005], but only to explain the low searh intensity of unemployed older workers. Bettendorf and Broer [2003] examine the age-dynamis of a labor market equilibrium with mathing fritions, but rings are exogenous and there exists a perfet insurane assumption against unemployment and death risks. 3

and labor-hoarding ativities at the end of the life yle. This implies that the hiring (ring) rate dereases (inreases) with the age of the worker. This approah ould help to better understand the eonomi rationale underlying the disrimination against older workers in the proess of hirings and rings. Furthermore, sine all new entrants (youngest workers) are unemployed, we show that the overall equilibrium age-dynamis of employment is humpshaped. Lastly, the rst part of this paper also revisits the positive impat of onventional labor market poliy tools in our life yle setting. Importantly, we nd the assumption of innite-lived agents understates (overstates) the potential employment gains (osts) related to ring taxes (unemployment benets). Beyond its theoretial interest, we believe that this approah is able to deliver realisti empirial preditions. More partiularly, we assess the ability of the model to mimi the older workers' experiene, for the Frenh eonomy. This leads to taking into aount the existing age-dependent poliies that have been put in plae in order to ompensate for the fall in the older workers' employment (spei employment protetion and assistane programs). Simulation results show that our theoretial framework is able to repliate the main features of the Frenh labor market. The distane from retirement appears as the key variable, muh more than labor market institutions, to explain higher (lower) ring (hiring) rates for older workers. These empirial results onrm our basi intuition: the retirement age is the entral fator to take into onsideration in any analysis of older worker employment rates. On the other hand, the employment rate of younger workers is more onventionally governed by labor market institutions, beause the inuene of the retirement age vanishes as the distane from retirement inreases. The last step of our quantitative investigation emphasizes that rms' behaviors play a primary role in aounting for the derease of employment at the end of the life yle, that is, more than that of workers. If the low employment rate of older workers an be explained by the short horizon reated by impending retirement, the next issue is then to examine the soial optimality of suh outomes. Before engineering any poliy devies to prevent rms from disriminating against older workers, it is neessary to study the soial optimality of suh behaviors. By maximizing 4

the soial welfare, we show that older workers ome rst (last) in the ring (hiring) proess. Without any other distortions than the mathing proess, it is optimal to disriminate against older workers due to their impending retirement. The deentralized equilibrium even oinides with the rst best outome when the Hosios ondition holds. However, what should be the age prole of ring osts and hiring subsidies when the Hosios ondition no longer holds or unemployment benets exist? As the eet of any distortion depends on the expeted life-time of the math, we show that age onstitutes the ornerstone of any optimal labor market poliies, and in a way whih is sometimes quite opposed to the sense of urrent OECD legislation. The rst setion presents the benhmark model and the age-dynamis properties of the equilibrium. The seond setion addresses the issue of older workers' employment, by examining the impat of old-spei labor market poliies and assessing the empirial performane of the model, based on the Frenh experiene. The last setion examines the rst best alloation and establishes the optimal age-prole for employment poliies. 2 How Do Job Creation and Job Destrution Vary over the Life Cyle? Let us onsider an eonomy à la Mortensen - Pissarides [1994]. Labor market fritions imply that there is a ostly delay in the proess of lling vaanies, and endogenous job destrutions losely interat with job reations. Wages are determined by a spei sharing rule of the rent generated by a job. The latter an be interpreted as the result of a bargaining between workers and employers. At this stage, no other fritions or ineienies are introdued. 4 Contrary to the large literature following Mortensen - Pissarides [1994], we onsider a life yle setting haraterized by a deterministi age at whih workers exit the labor market. Firms are free to target their hirings by age: direted-searh by age is tehnologially possible and legally authorized. This 4Unemployed workers job-searh eort is disarded at this point for larity of exposition. We present in Appendix A an extended version of the model with endogenous job-searh eort for unemployed people. It allows us to assess the robustness of our results. 5

means that we are onsidering at this stage a laissez-faire eonomy. 5 2.1 Benhmark Model Desription 2.1.1 Worker Flows We onsider a disrete time model and assume that at eah period the older worker generation retiring from the labor market is replaed by a younger worker generation of the same size (normalized to unity) so that there is no labor fore growth in the eonomy. We denote i the worker's age and T the exogenous age at whih workers exit the labor market: they are both perfetly known by employers. There is no other heterogeneity aross workers. The eonomy is at steady-state, and we do not allow for any aggregate unertainty. We assume that eah worker of the new generation enters the labor market as unemployed. Job reation takes plae when a rm and a worker meet. Firms are small and eah has one job. The ows of newly reated jobs result from a mathing funtion the inputs of whih are vaanies and unemployed workers. The destrution ows derive from idiosynrati produtivity shoks that hit the jobs at random. One a shok arrives, the rm has no hoie but either to ontinue prodution or to destroy the job. Then, for age i (2, T 1), employed workers are faed with layos when their job beomes unprotable. At the beginning of eah period, a job produtivity ɛ is drawn in the general distribution G(ɛ) with ɛ [0, 1]. The rms deide to lose down any jobs whose produtivity is below an (endogenous) produtivity threshold (produtivity reservation) denoted R i. Let u i be the unemployment rate and v i the vaany rate of age i. For any age, we assume that there are mathing funtions that give the number of hirings as a funtion of the number of vaanies and the number of unemployed workers, M(v i, u i ), where M is inreasing and onave in both its arguments, and with onstant returns-to-sale. Let θ i = v i /u i denote the 5If we onsider ex-ante undireted searh but ex-post math-spei heterogeneity, it adds some omplexities but our main results should hold. Typially, the reservation produtivity allowing the job to start will inrease with a worker's age, hene reduing the probability of being employed. In our equilibrium the latter result will hold due to the derease in the number of vaanies targeted at older workers. 6

tightness of the labor market of age i. It is then straightforward to dene the probability of lling a vaany as q(θ i ) M(u i,v i ) v i and the probability for unemployed workers to meet a vaany as p(θ i ) M(u i,v i ) u i. At the beginning of their age i, the realization of the produtivity level on eah job is revealed. Workers hired when they were i 1 years old (at the end of the period) are now produtive. Workers whose produtivity is below the reservation produtivity R i are laid o. For any age i, the ow from employment to unemployment is then equal to G(R i )(1 u i 1 ). The other workers who remain employed (1 G(R i ))(1 u i 1 ) an renegotiate their wage. The dynamis by age of unemployment are then given by: u i = u i 1 (1 p(θ i 1 )) + G(R i )(1 u i 1 ) i (2, T 1) (1) for a given initial ondition u 1 = 1. The overall unemployment rate u is then 1 i=1 dened by u =PT u i T 1. 2.1.2 The Hiring Deision Any rm is free to open a job vaany and engage in hiring. denotes the ow ost of reruiting a worker and β [0, 1] the disount fator. Let V i be the expeted value of a vaant job direted to a worker of age i: V i = + β [q(θ i )J i+1 (1) + (1 q(θ i ))V i ] where J i (ɛ) is the expeted value of a lled job by a worker of age i with idiosynrati produtivity ɛ. Following Mortensen and Pissarides, we assume that new jobs start at the highest produtivity level, ɛ = 1. As J T (1) = 0, no rms searh for workers of age T 1, that is θ T 1 = 0. The zero-prot ondition V i = 0 i (1, T 2) allows us to determine the labor market tightness for eah age θ i from the following ondition: βj i+1 (1) = q(θ i ) As 1/q(θ i ) is the expeted duration of a vaany direted to a worker of age i, the market tightness is suh that the expeted and disounted job value is equal to the expeted ost of hiring a worker of age i. 7 (2)

2.1.3 The Firing Deision For a bargained wage w i (ɛ), the expeted value J i (ɛ) of a lled job by a worker of age i is dened by: 1 J i (ɛ) = ɛ w i (ɛ) + β J i+1 (x)dg(x) + βg(r i+1 ) max{v i } i [1, T 1] R i i+1 (3) A rst thing to note is that with probability G(R i+1 ) the job is destroyed and the rm an freely hoose to diret its vaant job to workers of any age. It is also worth emphasizing that the deterministi exit at age T leads to an exogenous job destrution, whatever the produtivity realization: J T (ɛ) = 0 ɛ. The (endogenous) job destrution rule J i (ɛ) < 0 leads to a reservation produtivity R i dened by J i (R i ) = 0 i [2, T 1]: 1 R i = w i (R i ) β J i+1 (x)dg(x) βg(r i+1 ) max{v i } i [2, T 1] R i (4) i+1 The higher the wage, the higher the reservation produtivity, and hene the higher the job destrution ows. On the other hand, the higher the option value of lled jobs (expeted gains in the future), the weaker the job destrutions. Beause the job value vanishes at the end of the working life, labor hoarding of older workers is less protable. It is again worth determining the terminal age ondition: R T 1 = w T 1 (R T 1 ). 2.1.4 Wage Bargaining The rent assoiated with a job is divided between the employer and the worker aording to a wage rule. Following the most ommon speiation, wages are determined by the Nash solution to a bargaining problem 6. Values of employed (on a job of produtivity ɛ) and unemployed workers 6Reently, this wage setting rule has been somewhat disputed (See e.g. Shimer [2005] and Hall [2005]). We leave for future researh the exploration of alternative wage rules. 8

of any age i, i < T, are respetively given by: [ 1 ] W i (ɛ) = w i (ɛ) + β W i+1 (x)dg(x) + G(R i+1 )U i+1 R i+1 U i = b + β [p(θ i )W i+1 (1) + (1 p(θ i ))U i+1 ] (6) with b 0 denoting the opportunity ost of employment. 7 For a given bargaining power of the workers, onsidered as onstant aross ages, the global surplus generated by a job, S i J i (ɛ)+w i (ɛ) U i, is divided aording to the following sharing rule: (5) W i (ɛ) U i = γ [J i (ɛ) + W i (ɛ) U i ] (7) As in MP, a ruial impliation of this rule is that the job destrution is optimal not only from the rm's point of view but also from that of the worker. J i (R i ) = 0 indeed entails W i (R i ) = U i. Aording to this Nash bargaining solution, we then derive the following expression for the wage (see Appendix C for details): w i (ɛ) = (1 γ)b + γ (ɛ + θ i ) i [1, T 1] (8) This is a traditional wage equation, exept that age matters through the market tightness. If this latter diminishes along the life yle, the age prole of wages is dereasing. This ould ounterat the inentives for rms to re older workers. 8 2.2 The Laissez-Faire Equilibrium The main objetive of this setion is to haraterize the life yle pattern of hirings and rings. For didati reasons, we rst rely exlusively on the rm behavior, without onsidering wage retroations. Wages are assumed 7We assume that W T = U T so that the soial seurity provisions do not aet the wage bargaining and the labor market equilibrium. 8From an empirial standpoint, suh a derease in wages along the life yle is a shortoming. A similar result holds also in MP, where wages are expeted to derease after hiring. As stated in Appendix A, it ould be easily overome by allowing for general human apital aumulation. If the worker's reservation wage and the produtivity grow at the same rate over the life yle, only the disount fator is hanged at the equilibrium (see Appendix A). 9

to be xed at the reservation wage level b. This wage posting ase ould be rationalized by a bargaining power for workers equal to 0 (γ = 0 in (8)). Then, we will turn to our benhmark labor market equilibrium when it allows for wages adjustments over the life yle. The introdution of endogenous job-searh eort and its impliations on equilibrium job reations and job destrutions will be also examined. Lastly, we will show that the age prole of employment rates is typially hump-shaped. 2.2.1 The Wage Posting Equilibrium If wages are equal to b, the ring poliy, dened by R i, is independent of the hiring one. Proposition 1. If γ = 0, a labor market equilibrium with wage posting exists and it is haraterized by a sequene {R i, θ i } solving: q(θ i = β(1 R ) i+1 ) (JC P artialeq ) R i = b β 1 R i+1 [1 G(x)]dx (JD P artialeq ) with terminal onditions R T 1 = b and θ T 1 = 0. Proof. See Appendix D.1. It is then possible to derive the age prole of hirings and rings along the life yle. Property 1. R i+1 R i and θ i+1 θ i Proof. See appendix D.2. Older workers are more vulnerable to idiosynrati shoks. A shortened horizon relative to younger workers make them more exposed to rings. Otherwise stated, this reets that labor-hoarding dereases with worker's age. In turn, it reates a downward pressure on the hirings of older workers. It reinfores the derease in hirings at the end of the working yle due to a shortened horizon whih makes vaanies unprotable. i. 10

2.2.2 The Equilibrium with Wage Bargaining If wages are bargained aording to the equation (8), the ring poliy depends now on the market tightness. The wage derease over the life yle is then likely to oset the diret eet of the shortening horizon on rings. This ould put into question the dereasing age prole of rings, hene of hirings. Proposition 2. A labor market equilibrium with wage bargaining exists and it is haraterized by a sequene {R i, θ i } solving: q(θ i = β(1 γ)(1 R ) i+1 ) (JC) ( ) γ R i = b + (JD) 1 γ θ i β 1 R i+1 [1 G(x)]dx with terminal onditions R T 1 = b and θ T 1 = 0. Proof. See Appendix D.3. Corollary 1. Let be M(v, u) = v ψ u 1 ψ with 0 < ψ < 1, and G(ɛ) = ɛ, ɛ [0, 1], with b 1 2b/β, 9 the labor market equilibrium with wage bargaining an be summarized by a sequene {R i } T 1 i=2 solving: R i = b + with terminal ondition R T 1 = b. Proof. Straightforward. ( ) [ ] 1 γ β(1 γ) 1 ψ (1 R i+1 ) 1 γ β 2 (1 R i+1) 2 (9) The sequene of R i is no longer neessarily monotoni. If the wage dereases suiently at the end of working life beause of the weakness of the market tightness, then rms might be less inlined to re these older workers. 10 The following property and orollary state restritions, implying that this indiret eet of age through wages does not dominate the diret impat of age on labor-hoarding and ring. 9From (9) it is straightforward to see that b 1 2b/β is a suient ondition for an interior solution to exist (R i 0 i). 10It is worth emphasizing that this inentive to keep older workers still remains when there is human apital aumulation that allows wages to grow over the life yle at the same rate as produtivity (see Appendix A). 11

Property 2. γ if 1 2γ [ β(1 γ) 1 ψ [ β(1 γ) ] ψ 1 ψ for ψ 1/2 ] ψ 1 ψ (1 b) 2ψ 1 1 ψ for ψ 1/2 then the labor market equilibrium veries R i+1 R i and θ i+1 θ i Proof. See Appendix D.4. Figure 2 gives the age dynamis of the labor market. This gure shows dr that if i dr i+1 0 and b R i, i, the model generates a monotonous inreasing sequene of R i. While this result is not ambiguous in the ase of an exogenous wage, a parameter restrition is required when the wage derives from a Nash bargaining. The value of reruiting osts () is entral for understanding this R(i) Figure 2: Equilibrium dynamis of R i i. R* R* R(T-2) R(T-1) R(i+1) result. It determines how age inuenes the vaany rate. The higher the reruiting ost, the less sensitive labor market tightness to age, the steeper the age prole of wages. If is suiently high, the wage eet annot 12

ounterat the horizon eet on the reservation produtivity: the age-prole of the ring rate is inreasing. Corollary 2. If ψ = 1/2 the ondition βγ(1 γ)2 ensures that the labor market equilibrium veries R i+1 R i and θ i+1 θ i i. Proof. Straightforward from Property 2 with ψ = 1/2. Lastly, it should be emphasized that if Property 2 (hene Corollary 2) are not satised, the age dynamis of the labor market is osillatory 11. In the light of empirial fats on employment, hiring and ring rates by age, we rule out suh solutions of the labor market equilibrium. As it will be stated below, this is also onsistent with realisti alibrations of the model. 2.2.3 The Role of the Endogenous Job-Searh Eort For didati reasons, until now we have negleted the inuene of the life yle hypothesis on workers' job-searh eort. Making the latter endogenous would atually reinfore the derease in the employment rate at the end of working life. As the retirement age gets loser, the return on job-searh investments dereases beause the horizon (the expeted job duration) over whih they an reoup their investment is redued. This point an easily be stated by onsidering the following unemployed problem to dene job-searh intensity: 12 { } U i = max b e2 i e i 2 + β [e ip(θ i )W i+1 (1) + (1 e i p(θ i ))U i+1 ] where the labor market tightness is now dened by θ i v i /[e i u i )] with a mathing funtion M(v i, e i u i ) where e i is the average job-searh eort of workers of age i. The optimal deision rule shows that it is in the older unemployed workers' interest to redue their job-searh intensity, sine the disounted sum of surplus related to employment is dereasing with age: 11Proof available upon request. 12See the Appendix A for a detailed desription of the model with a more general speiation of preferenes. 13

e i = βp(θ i ) [W i+1 (1) U i+1 ] This provides an addition to existing fores that lead job reation to derease with age. This point an be highlighted by examining the equilibrium properties with job-searh eort. Proposition 3. A labor market equilibrium with wage bargaining and endogenous job-searh eort exists and it is haraterized by a sequene {R i, θ i } solving: q(θ i = β(1 γ)(1 R ) i+1 ) (JC eff ) R i = b + 1 2 ( γ 1 γ θ i ) 2 1 β R i+1 [1 G(x)]dx (JD eff ) with terminal onditions R T 1 = b and θ T 1 = 0. Proof. See Appendix D.5. Property 3. Let be M(v, eu) = v ψ (eu) 1 ψ with 0 < ψ < 1, and G(ɛ) = ɛ, ( ) 2ψ ɛ [0, 1], if 1 β γ2 β(1 γ) 1 ψ, then the labor market equilibrium with 1 ψ wage bargaining and endogenous job-searh eort veries R i+1 R i and θ i+1 θ i i. Proof. See Appendix D.6. Corollary 3. If ψ = 1/2 the ondition βγ(1 γ) 2β ensures that the labor market equilibrium veries R i+1 R i and θ i+1 θ i i. Proof. Straightforward from Property 3 with ψ = 1/2. This orollary shows that the restrition on to ensure the existene of an equilibrium with R i+1 R i is weaker than that imposed by orollary 2. This point demonstrates the positive role of job-searh eort in explaining the inrease (derease) of rings (hirings) with age. 2.2.4 The Age Prole of the Employment Rate The age prole of hirings and rings has been reursively determined from terminal onditions. On the other hand, the age prole of unemployment u i 14

(or employment n i = 1 u i ) depends on the arbitrary initial ondition u 1. This explains why it is ambiguous: u i G(R i+1 ) G(R i+1 ) + p(θ i ) n i+1 n i i Figure 3: Equilibrium dynamis of {R i, n i } Property 4. For u 1 = 1, there exists a threshold age T so that n i n i 1 i T and n i n i 1 i T. Proof. See Appendix D.7. In the ase where all the new entrants are unemployed, high vaany rates and low ring rates at the beginning of the working life yle make the employment rate inreasing with age until the age T. Until this threshold age, this inrease in employment rate is simply the result of a queue phenomenon. From T on, the employment rate evolution by age mimis the age prole of rings and hirings. The age heterogeneity aross workers then leads to a low employment rate for older workers. The overall age-dynamis of employment is thus hump-shaped, as found in OECD data. 15

2.2.5 The Age Prole of Wages Sine the seminal empirial work of Miner [1962], it is well-known that the wage inreases with age and delines at the end of the life yle. This stylized fat an be explained in our model simply by inluding general human apital aumulation as shown in Appendix A. The wage inreases with the aumulation of human apital, but the derease of the labor market tightness slows down this growth. At the end of the life yle, the large derease of the tightness an dominate the dynami of human apital aumulation. Property 5. Let ψ = 1/2, if > 2β(1 γ) 2, then w i+1 (R i+1 ) > w i (R i ), i T. Proof. See Appendix D.8. The disrimination against the older workers leads also to another phenomenon: at the end of the life yle, rms hoard their workers less (the reservation produtivity inreases with the age of the worker). Only the more produtive remain at work. Property 5 illustrates this result. Aordingly, the average wage an inrease with age due to a omposition eet. 2.2.6 Distane from Retirement instead of Age An important parameter of the model is the retirement age. Only the distane between the urrent age and the retirement age matters aording to a horizon eet. On the ontrary, the biologial age does not matter in itself. Property 6. For two retirement ages, T and T + N, we have R T 1 i = R T +N 1 i and θ T 1 i = θ T +N 1 i, i. Proof. Proposition 2 and Corollary 1 learly show that for all T, we have the same terminal ondition: for two retirement ages, T and T + N, we have R T 1 = R T +N 1 = b and θ T 1 = θ T +N 1 = 0. Then, using bakward indution, the equations (JC) and (JD) show that the R T 1 i = R T +N 1 i and θ T 1 i = θ T +N 1 i, i. This explains why some ountries experiene a drop in their employment rate as soon as the age of 55 (see gure 1 in the introdution): the 55-59 years-old workers are lose to their retirement age. The model is thus 16

able to provide some foundations to the observed OECD employment rate dierenes, by relying on a pure horizon eet. 2.3 Labor Market Poliy Revisited in a Life Cyle Setting In a laissez-faire eonomy, the labor hoarding strategy is less valuable for older workers. This setion revisits the impat of ring taxes and unemployment benets in a nite horizon ontext. It analyzes to what extent labor market poliy tools either reinfore or ounterat this primary mehanism at work. We rst emphasize that a onstant ring tax implies a higher derease in rings in our eonomy than in an innite-lived agents' model à la MP. The short horizon of older workers implies that rms an esape from the tax by waiting for the pending retirement age. Seondly, the unemployment bene- t system has a lower negative impat on the employment of older workers. Indeed, the horizon during whih the rms must bear higher labor osts is shorter for older workers. Overall, these results highlight that onventional labor market instruments introdue a bias in favor of the older worker employment due to a horizon eet. We will also show that the global impat on employment of these poliies is more favorable in our life-yle setting. 2.3.1 The Equilibrium with Unemployment Benets and Firing Costs Let us denote z the unemployment benet naned by a non-distortionary tax. F denotes a ring ost whih refers to the impliit osts in mandated employment protetion legislation and in experiene-rated unemployment insurane taxes. Furthermore, we onsider a two-tier wage struture in line with Mortensen and Pissarides [1999] and Pissarides [2000]. Proposition 4. A labor market equilibrium with wage bargaining exists and it is haraterized by a sequene {R i, θ i } solving: q(θ i ) = β(1 γ)(1 R i+1 F ) R i = b + z (1 β)f + γ 1 γ θ i β 17 1 R i+1 [1 G(x)] dx

with terminal onditions R T 1 = b + z F and θ T 1 = 0. Proof. See Appendix B and onsider F i = F i+1 = F, z i = z as well as H i = 0. In a MP type of eonomy, T, and the produtivity threshold and labor market tightness jump on stationary values that we denote, R and θ. 13 Corollary 4. If T, the labor market equilibrium is haraterized by {R, θ} solving: q(θ) = β(1 γ)(1 R F ) R = b + z (1 β)f + γ 1 γ θ β Proof. Straightforward from Proposition 4. 1 R [1 G(x)] dx Corollary 5. Let be M(v, u) = v ψ u 1 ψ and G(ɛ) = ɛ, when ψ = 1/2 the ondition βγ(1 γ)2 ensures that the labor market equilibrium with poliy veries R i+1 R i and θ i+1 θ i i. Proof. Similar to proof of Corollary 2. 2.3.2 The Impat of Firing Costs Revisited We rst re-examine the impat of the ring tax in our life yle setting. Property 7. If Corollary 5 is satised, the labor market equilibrium is haraterized by, i: 0 dr df > dr i df 0 dθ i+1 df Proof. See Appendix D.9 13When F = 0, R is equivalent to R on gure 2. > dr i+1 df > dθ i df > dθ df 18

Property 7 emphasizes that the older workers benet more from the employment protetion, and, by onsequene, that the ring tax redues more job destrutions than in an innite horizon eonomy. 14 At the end of the working yle, introduing a ring tax inreases the present ring ost without any future onsequenes on the job value as the worker will be retired in the next period. On the other hand, in an innite horizon, the present ring ost inreases in the same proportion as in our life-yle model, but the job value dereases, as the rm rationally expet the future ost of the ring tax. In some sense, retirement allows rms to esape from the ring tax, leading them to more labor hoarding for older workers. By bakward indution, rings of younger workers are also redued sine expeted durations of jobs inrease. This explains why the global eet of a ring tax on job destrutions is higher in our life-yle eonomy. Similarly, these longer expeted durations for jobs also translate into an inrease in labor market tightness. This ounterats the diret negative impat of the ring ost on job reation and this explains why 0 θ i F i > θ This suggests that evaluating employment protetion in an innite-lived agents ontext underestimates the potential positive impat on employment. Overall, however the eet of a ring tax in our life yle setting remains ambiguous, sine job reation is still negatively aeted by the level of the ring osts. 2.3.3 The Impat of Unemployment Benets Revisited We now analyze the impat of unemployment benets. Property 8. If Corollary 5 is satised, the labor market equilibrium is haraterized by, i: F. dr dz dr i dz > dr i+1 dz > 0 dθ dz dθ i dz 14Ultimately, when γ = 0 and β 1, dr df 1). 19 < dθ i+1 dz = 0 whereas dr i df < 0 < 0 (for instane dr T 1 df =

Proof. See Appendix D.9 In our life yle setting, unemployment benets imply higher distortions for younger workers than for older workers. Labor hoarding strategies are indeed diretly related to the sum of labor osts until retirement. The expeted ost of unemployment benets is thus higher for younger workers. This also explains why the unemployment benet level has a larger global impat on job destrution in a MP type of eonomy than in our life yle setting. In turn, the labor market tightness is less aeted in our setup. 3 The Older Workers in the Spotlight It is in the interest of rms to dierentiate their hiring and ring deisions by age in a life yle setting. It potentially delivers an original explanation for the observed derease in the employment rate at the end of the working yle. This setion is preisely foused on older workers' employment. Beyond their proximity to retirement, the labor market of older workers is also haraterized by spei poliies whih have been designed to ompensate for age disrimination. Our life-yle model allows us to study the theoretial impat of the age-dependent poliies. They ould explain a large part of the employment apart from the retirement age. In a seond step, we propose a quantitative analysis to measure to what extent the distane from retirement is indeed the main feature of older workers' employment, based on the Frenh experiene. 3.1 Older Worker Poliies and Labor Market Equilibrium Faed with the low employment rate of older workers, governments have put in plae dierent poliies to ounterat this trend or to ompensate for it by spei assistane programs for older workers. These poliies are ertainly ruial to understanding the dierent fores at work when onsidering the end of the working life yle. Our approah provides a theoretial framework to analyze their onsequenes by age on hirings and rings. 20

It is possible to distinguish two types of labor market poliy oriented toward older workers. The rst type is designed to protet them. Some poliies for older workers are indeed haraterized by (i) higher ring taxes (ii) higher "unemployment" benets through spei assistane programs. In some ountries, (e.g. Belgium, Finland, Frane, Japan, Korea, Norway), it is indeed more ostly for rms to lay o older workers beause of longer notie periods or higher severane pay. Another important feature onerning older workers is that there exist spei inativity and disability programs in most European ountries. They provide generous substitution inomes for people eligible for these programs until retirement: this leads to an inrease in the non-employment inomes. Moreover, these workers must be onsidered as inative (the job-searh osts are higher than for the younger) as the benets are not onditional on a job-searh ativity. The seond type of poliy aims at inreasing the likelihood for older workers to nd a job. In the UK, rms reeive a subsidy if they hire an older worker. In the US, the objetive of the anti-age disrimination law is to give the same employment opportunities to all individuals, whatever their age. 3.1.1 The Equilibrium with Age-dependent Labor Market Poliies We now let z i be the age-dependent unemployment benet naned by a non-distortionary tax. 15 F i is the tax that the rm must pay when it res a worker of age i, and we introdue H i as the hiring subsidy that the rm gets when it hires a worker of age i. The equilibrium alloation with wage bargaining and endogenous searh eort is now featured by (see Appendix B for derivation details): Proposition 5. For given sequenes of poliy instruments {H i, F i, z i }, a labor market equilibrium with wage bargaining and endogenous searh eort 15We assume that unemployed and employed workers fae an age-spei lump sum tax, X i, so that X i = z i u i i. 21

exists and it is haraterized by a sequene {R i, θ i } solving: q(θ i = β(1 γ)(1 R ) i+1 + H i+1 F i+1 ) (JC pol ) R i = b + z i + 1 2 ( γ 1 γ θ i ) 2 [ ] 1 β R i+1 [1 G(x)] dx F i+1 F i (JD pol ) with terminal onditions R T 1 = b + z T 1 F T 1 and θ T 1 = 0. Proof. See Appendix B. 3.1.2 The impat of age-dependent labor market poliies From the system (JC pol )-(JD pol ), it an easily be shown that age-spei poliies have an impat on the targeted population, but also on younger agents due to the forward-looking behaviors of rms. The impat of age-inreasing unemployment benets. The unemployment benet (z i ) is found to exert a onventional upward pressure on wages. This leads to an inrease in the produtivity threshold R i : the number of job destrutions rises and the hiring rate dereases. To get more insights into the role of the age prole of unemployment benets, let us examine the impat of an additional unemployment inome for a senior. For simpliity, let us now onsider that z T 1 = z and z i = 0 i [2, T 2]. Assume also H i = F i = 0 i and γ = 0. From proposition 5, it is straightforward to see that: R T 2 = b β 1 R T 1 = b + z T 1 ; R T 1 [1 G(x)]dx ; q(θ T 2 ) = β(1 b z T 1) q(θ T 3 ) = β(1 R T 2) As ould be expeted, this rst shows that the greater generosity of assistane programs at the end of the working life inreases rings and dereases hirings of older workers ( R T 1 z T 1 > 0 and θ T 2 z T 1 > 0). But it is also important to notie that the value of labor-hoarding for younger workers is also redued ( βr1 R [1 G(x)]dx βr1 T 1 R [1 G(x)]dx z T 1 = T 1 R T 1 R T 1 z T 1 < 0). This leads to higher (lower) ring (hiring) rate for workers who are not yet eligible for the generous unemployment benet system ( R T 2 z T 1 > 0 and θ T 3 z T 1 < 0). 22

Unambiguously, the age-inreasing non-employment inome implies a rise in the non-employment rate of all workers (see equation (1)). Nevertheless, the ring probability of older workers is higher than that of the younger workers beause the expeted eets of this poliy are smaller than its instantaneous impat. The impat of the age-inreasing ring tax. Whereas F i tends, as expeted, to push down R i by inreasing the urrent ost of ring, F i+1 inreases R i by reduing the value of labor-hoarding, i.e. the expeted future gain assoiated with the job (see the term in brakets, equation JD pol ). This suggests that job destrution is ruially related to the age prole of the employment protetion, F i+1 F i. More severe employment legislations protet workers who already have a job, but at the expense of those without a job. This point has already been stated theoretially by Mortensen and Pissarides [2000]. To get further insights into the role of the age prole of ring taxes, let us, for instane, examine the impat of a poliy introduing an additional tax when laying o older workers. For simpliity, onsider F T 1 > 0 and F i = 0 i [2, T 2]. Without loss of generality, we assume again that the bargaining power of the workers is equal to zero (γ = 0). Assume also H i = 0 and z i = z, i. From proposition 5, it is straightforward to see that: R T 2 = b + z + βf T 1 β R T 1 = b + z F T 1 ; 1 R T 1 [1 G(x)]dx ; = β(1 b z) q(θ T 2 ) q(θ T 3 ) = β(1 R T 2) This learly shows that for workers of age T 1, the introdution of the tax redues rings ( R T 1 F T 1 < 0) while the number of newly-hired workers who enter to employment at age T 1 is unhanged ( θ T 2 F T 1 = 0): unambiguously, the ring tax implies an inrease in the employment rate of the oldest workers (see equation (1)). On the ontrary, for workers of age T 2, it appears that the ring probability inreases and simultaneously the hiring one dereases ( R T 2 F T 1 > 0 and θ T 3 F T 1 θ T 3 R T 2 R T 2 F T 1 < 0). The expeted ring tax in T 1 indeed redues the value of job ontinuation when the worker is T 2 years old on the labor market (less labor-hoarding). The assoiated 23

inrease in produtivity thresholds translates into lower expeted duration of jobs, hene lower hirings: the employment rate of workers of age T 2 falls unambiguously with F T 1. Overall, our framework illustrates that the aggregate employment impat of age-dependent employment protetion is not determined a priori. The impat of the age-inreasing hiring subsidy. In the equations (JC pol ) and (JD pol ), the hiring subsidies have a diret impat only on the job reation rule, unlike the ring taxes whih aet both (JC pol ) and (JD pol ). Let us, for instane, examine the impat of a poliy introduing an additional subsidy when hiring older workers. For simpliity, onsider H T 1 > 0 and H i = 0 i [2, T 2]. We assume that the bargaining power of the workers is equal to zero (γ = 0), and F i = 0 and z i = z, i. From Proposition 5, it is straightforward to see that: R T 2 = b + z β 1 R T 1 = b + z ; R T 1 [1 G(x)]dx ; q(θ T 2 ) = β(1 b z + H T 1) q(θ T 3 ) = β(1 R T 2) It learly appears that when the bargaining power of the workers is equal to zero, the hiring subsidy targeted for the age T 1 workers has an impat only on this age group, through an inrease in their hiring rate. This poliy unambiguously inreases the employment rate of the older workers, without introduing any distortions for the younger workers. Nevertheless, this result is not general. Indeed, when the bargaining power of the workers is stritly positive, the higher exit rate from unemployment tomorrow pushes up the wage today and then the produtivity threshold. This leads to an inrease (derease) of the ring (hiring) rate for the younger workers. This adjustment of the wage ould be negleted in the analysis of the unemployment benet and the ring tax beause it amplies the diret eets of the institutional hanges. This is no longer the ase for the hiring subsidies. 24

3.1.3 The Inuene of Legislation Prohibiting Age-Disrimination The seond issue of this setion is about legislation prohibiting age disrimination. This legislation in the US dates bak to the 60's (Age Disrimination in Employment At in 1967, and subsequent amendments). In 2000, the European Union Counil Diretive also required all 15 EU ountries to introdue legislation prohibiting diret and indiret disrimination at work on the grounds of age. To extend our benhmark eonomy we should notie that when age disrimination is eiently prohibited, the job-searh is undireted and there is only one labor market tightness, θ. However, there still exists distint equilibrium wages sine produtivity shoks are job-spei. Lastly, under the assumption one job - one rm, it not possible to examine the question of the age-disrimination legislation related to rings. Let onsider that searh eort is onstant, and denote by V the value of a vaant position, value funtions now solve: [ ] T 2 u i V = + β q(θ) u J i+1(1) + (1 q(θ))v i=1 [ 1 J i (ɛ) = ɛ w i (ɛ) + β G(R i+1 )V + W i (ɛ) = w i (ɛ) + β [ G(R i+1 )U i+1 + ] J i+1 (x)dg(x) R i+1 ] W i+1 (x)dg(x) R i+1 1 (10) (11) (12) U i = b + z + β [p(θ)w i+1 (1) + (1 p(θ))u i+1 ] (13) The free entry ondition, V = 0, now implies: T 2 q(θ) = β i=1 u i u J i+1(1) whih means that the average reruiting ost equals the rm's expeted gain related to the math aording to the age of the worker she meets. With probability q(θ) the vaant job is lled, and with probability u i /u it is lled with a worker of age i. Proposition 6. Let G(ɛ) = ɛ, ɛ [0, 1], a labor market equilibrium with wage bargaining and age disrimination legislation exists and it is harater- 25

ized by a sequene {R i, θ, u i } solving: q(θ) T 2 = β(1 γ) i=1 u i u (1 R i+1) R i = b + z + γp(θ)β(1 R i+1 ) β 2 (1 R i+1) 2 u i = G(R i )(1 u i 1 ) + (1 p(θ))u i 1 with boundary onditions R T 1 = b + z and u 1 = 1. Proof. See Appendix C.3 for the equilibrium derivation. If we assume for simpliity that the bargaining power of the workers is equal to zero (γ = 0), Proposition 6 implies that the dynami of the produtivity threshold is the same as if there were no anti-disrimination law. Then, this law has no impat on ring rates by age in this ase. Turning to the hiring proess, if we assume that θ = θ T 2, with θ T 2 obtained in the model with disrimination, then θ > θ T 1 sine θ T 2 > θ T 1. Hene, unambiguously, the employment rate of the older workers inreases with the anti-disrimination law. More generally, suh a law imposes the same job reation rate whatever the worker's age, on the basis of the average expeted gain. As a onsequene, it is favorable to older workers' reruitment as regards the laissez-faire eonomy. Labor market tightness for older workers is indeed inreased by the age disrimination legislation. But, in turn, sine wages are positively related with labor market tightness, equilibrium wages and destrution rates of older workers are also inreased. The overall impat on employment is thus theoretially undetermined, but one might expet that the diret impat on hirings is less than oset by the indiret impat on rings, that is, this legislation inreases the employment rate of older workers. 3.2 A Quantitative Investigation: the Frenh Older Worker Experiene The objetive of this setion is twofold. On the one hand, we aim at showing that the model is able to aount for the observed age-dynamis of job reation and job destrutions ows. On the other hand, we verify that the 26

feedbak eet of the retirement age is ruial in the explanation of the derease in the employment rate of older workers, relative to existing labor market institutions that are traditionally favored in any model aiming at explaining labor market ows. We ompletely disard the entry-stage proess of the labor market and so the younger worker ows, typially fousing on employment age heterogeneity after the age of 30 in our empirial investigations. We hoose to evaluate the model on Frenh data (males). Firstly, the retirement age is known without any unertainty. All men exit the labor fore when they reah their full pension age and they are not allowed to ontinue working after drawing their pension. Until reently, more than 90% of men retired at 60. All these features make Frane very lose to our framework, whih leaves aside any questions related to heterogeneity and unertainty about the exit date from the labor fore. Frane is also haraterized by the existene of spei and generous assistane programs for older workers. Workers are eligible for these programs in the ase of layos, onditional on being aged more than 57. They then reeive a generous inome until retirement. Another important feature of the Frenh labor market is the spei ring tax that rms inur in the ase of laying o older workers (Delalande Tax). 3.2.1 Calibration and Quantitative Assessment The empirial performane of the model is now evaluated on Frenh data (male workers) by simulating the equilibrium with age poliies (JC pol )-(JD pol ). The alibration is based on the period prior to 1993, that is before the Soial Seurity reform whih has introdued more heterogeneity in the retirement age. Similarly, we disard modiations to the layo tax sheme that have been implemented from 1993 on. We onsider ows between employment and non-employment as older workers are mostly entitled to spei assistane programs whih are slightly more generous than unemployment benets. 27

Calibration. Speiations of funtional forms for the distribution of idiosynrati shoks and the mathing funtion are rst required. As is usually assumed, we onsider an uniform distribution and a Cobb-Douglas mathing funtion. More preisely, we set: G(ɛ) = ɛ ɛ [0, 1] ln(p(θ i )) = ψθ i 0 < ψ < 1 A rst set of parameters Φ 1 = {β, ψ, γ, F, z, F (old), z(old)} is then based on external information, where F (resp. z) is the ring tax (resp. unemployment benets) for workers less than 55 (resp. 57) years old, and F (old) (resp. (zold)) refers to older worker-spei labor market poliies for workers more than 55 (resp. 57) years old. The disount fator β equals 0.96, whih yields an annual interest rate of 4%. The elastiity of the mathing funtion is set to the extensivelyused value ψ = 0.5 (see, among others, Mortensen and Pissarides [2000]). The bargaining power of the workers is xed at γ = 0.3, whih is the mean of admissible values estimated on Frenh mirodata (see Abowd and Allain [1996]). Overall ring osts represent in Frane 15 months of average wage earnings (see Abowd and Kramarz [2003]). No evidene exists for Frane on the deomposition of these osts into severane transfers to workers and administrative osts. Garibaldi and Violante [2002] provide suh a breadown for Italy, whose employment protetion OECD indiator is reported to be very lose to Frane: a third of ring osts are related to administrative osts. Applying this value for the overall ring ost in Frane leads the layo tax to represent in our eonomy 5 months of the average annual wage earnings, whih is obtained by setting F = 0.2. For workers between 55 and 60, we also inlude the additional administrative ost targeted at older workers (the Delalande tax). This additional omponent represents 25% of the annual wage earnings (Frenh law until July 1992). This leads to F (old) = 0.32, whih implies that the overall ring tax for older workers is 8 months of the average annual wage earnings. Unemployment benets are set to reprodue an average net replaement ratio of 55% (Martin [1996]) for workers less than 57 years old: this implies z = 0.275. For workers over 57, we add a premium 28

of 10% whih reets the fat that they are exempt from the derease in the replaement ratio throughout the unemployment spell. 16 Lastly, and due to a lak of information on the value of leisure and the ow of reruiting osts, we hoose to alibrate a seond set of parameters Φ 2 = {b, } in order to reprodue stylized fats on the employment rate in Frane in 1993. Aordingly, the model will be assessed on its ability to aount for job ows in and out of non-employment rather than on the repliation of the age dynamis of the employment stok variable. The values of b and are set suh that the model mathes the observed average employment rate for 30-44 and 55-59 year old male workers, denoted respetively n (30 44) and n (55 59). This implies b = 0.18 and = 0.46. Model Assessment. Our alibration strategy hene implies that by denition the model mathes the employment rate for the two age groups, 30-44 and 55-59. The employment rate of workers aged between 45 and 55 is also well repliated. The key issue about the empirial relevane of our model then relies on its ability to aount for job reation and job destrution ows. The model ould repliate the age-dynamis of the stok variable (employment), but understates or overstates by ten both hirings and rings. Figure 4 shows entry and exit rates for three age groups, 30-44, 45-54 and 55-59, so that our assessment is based on six moments. Interestingly, this gure shows that our model mathes quite well the age-pattern of ring and hiring rates over the life yle. However, it must be aknowledged that the levels are not perfetly reprodued. More partiularly, the ring and hiring rates of the age group 45-55 are overestimated. Yet, we onsider that this simple model works surprisingly well to generate the age prole of hirings and rings over the life yle. 3.2.2 Identifying the mehanisms at work What are the main driving fores behind the age-dynamis generated by our model? First, it is of primary interest to identify the relative importane of the age poliies and the horizon eet in this result. It is often argued that 16The rst year of the replaement rate is approximated around 0.65 (Martin [1996]). This implies z(old) = 0.3025. 29

Figure 4: The age-dynamis of the labor market 0.9 0.8 Employment Rate by Age Group Model Data 0.7 0.6 0.5 Rate 0.4 0.3 0.2 0.1 0 30 30 44 45 54 55 59 Annual Hiring Rate by Age Group 25 20 Rate in % 15 10 5 0 14 30 44 45 54 55 59 Annual Firing Rate by Age Group 12 10 Rate in % 8 6 4 2 0 30 44 45 54 55 59 Data = Frenh Labor Fore Survey (males, averages over 1990-1993) Hiring rate = outows from non-employment / non-employment Firing rate = inows into non-employment / employment 30

when generosity of unemployment benets is the primary ause of low employment (see Lunqjvist and Sargent [2005]). Aordingly, one ould expet that higher UB for workers aged 57 or more in Frane is the key feature at the origin of the deline in the older worker employment rate. We will show that the intrinsi horizon eet atually is dominant. Seondly, it is ruial to identify the respetive ontributions of the supply and demand for older workers. Whih side of the labor market is the most sensitive to the horizon eet? It will be revealed that the demand side seems dominant in explaining the derease in the employment rate when the retirement age omes loser. Disentangling the horizon eet and the age poliies. The generosity of assistane programs after 55 ould explain the deline in the employment rate for these workers, but also that of the 45-54 age-groups by a feedbak eet. Even if this last eet is inherent to our framework, this explanation would be a simple transposition to our life yle model of a well-identied and lassi mehanism. In order to identify the relative ontribution of the horizon eet, we onsider along our benhmark alibration the ase where the unemployment benet and/or the ring tax are onstant throughout the working life yle. Table 1 ("without age poliies") gives a quantitative measure of the derease of employment when both poliies are the same for all ages. It appears that there still exists a large derease in the employment rate after 55. Consistently, the hiring and ring rates are deeply aeted by the proximity of the retirement age. Without any age poliies, the derease in the 55-59 employment rate (with resp. to 45-54) is equal to 25.8 points. These results learly show that the horizon eet largely dominates the dynamis of the employment rate at the end of the life yle. By omparing the rst and the seond lines, it appears that the inreasing Table 1: Poliies versus Horizon Eet Employment Hiring Firing Age 30-44 45-54 55-59 30-44 45-54 55-59 30-44 45-54 55-59 Without Age Poliies 89.6 86.2 60.4 27.9 24.9 11.0 3.1 4.9 17.3 Without Age Firing Cost 89.6 85.2 54.2 27.6 22.0 7.3 3.3 7.3 20.3 Benhmark 89.4 82.5 58.4 27.9 22.0 8.2 3.1 6.5 12.7 31

Figure 5: Age-dynamis of the labor market with and without age-spei poliies e i p(θ i ) 0.35 0.3 0.25 0.2 0.15 Annual Hiring Rate with z(old) without z(old) G(R i ) 0.35 0.3 0.25 0.2 0.15 Annual Firing Rate with z(old) without z(old) 0.1 0.05 0 40 45 50 55 60 Age (i) 0.1 0.05 0 40 45 50 55 60 Age(i) 0.35 0.3 Annual Hiring Rate with F(old) without F(old). 0.35 0.3 Annual Firing Rate e i p(θ i ) 0.25 0.2 0.15 G(R i ) 0.25 0.2 0.15 with F(old) without F(old). 0.1 0.1 0.05 0 40 45 50 55 60 Age (i) 1 0.8 Employment Rate 0.05 0 40 45 50 55 60 Age(i) 1 0.8 Employment Rate 0.6 0.6 n i 0.4 with z(old) without z(old) n i 0.4 with F(old) without F(old). 0.2 0.2 0 20 30 40 50 60 Age (i) 0 20 30 40 50 60 Age (i) 32

prole of UB after 57 is responsible for a derease of 6 points for the 55-59 employment rate, and also of 1 point for the 45-54 age-groups. Higher ring osts after 55 lead to better employment protetion for the 55-59 years-old, who benet from an inrease in the employment rate of 4.2 points due to this legislation. However, this latter negatively aets the employment rate for workers aged 45-54 whih dereases by 2.7 points. The net eet on employment rates is thus lose to zero (see also gure 5). These results are onsistent with a reent eonometrial evaluation of the Delalande Tax (Behaghel, Crépon and Sédillot [2005]). The respetive role of the rms and the workers. If the horizon eet seems to play an important role at the end of the working life, both rms and workers are potentially responsible for this outome. It is possible to identify their respetive roles by onsidering eonomies with a onstant jobsearh eort for workers. More preisely, we ompare two eonomies, with and without an endogenous job-searh eort at eah age. In this latter ase, the onstant job-searh eort is set to its average endogenous value over the 30-44 age range. Table 2: Workers versus Firms Employment Hiring Firing 30-44 45-54 55-59 30-44 45-54 55-59 30-44 45-54 55-59 Benhmark 89.4 82.5 58.4 27.9 22.0 8.2 3.1 6.5 12.7 Benhmark e i = e 89.4 84.9 72.6 27.9 26.2 15.9 3.1 5.8 8.9 Without Age Poliies 89.6 86.2 60.4 27.9 24.9 11.0 3.1 4.9 17.3 Without Age poliies, e i = e 89.6 87.4 68.9 27.9 27.2 18.0 3.1 4.5 15.1 We rst onsider the benhmark eonomy (with age poliies) in these two ases (the rst two lines of Table 2). By omparing these two eonomies, we see that the employment rate of 55-59 years old workers is redued by 12.8 points (with resp. to 45-54), instead of 24.1 points when searh eort is endogenous. This suggests that more than 50% of the derease in the employment rate is related to the rms' behavior (demand side). Moreover, the intrinsi role of the workers' behavior (supply side) is ertainly overemphasized, as the unemployment benets are partiularly generous at the end of the working life. The relative ontribution of the labor demand ould 33

be even greater when restriting our attention to the horizon eet. We now onsider the two eonomies without age poliies (the last two lines) with and without endogenous searh eorts. The ontribution of the rms' behavior goes up to more than 70%. 4 Eient Job Creation and Job Destrution over the Life Cyle Our model preisely highlights eonomi rationales sustaining age disrimination: there is less time over whih to reoup reation osts and temporary negative produtivity shoks for older workers. A laissez-faire equilibrium is then typially featured by job reation (destrution) rates dereasing (inreasing) with age. Hene, there is age disrimination against older workers at the equilibrium. In some ountries, poliies have been implemented to sustain the demand for older workers. If the previous setion has questioned their eieny, their optimality is still an open question. This setion is devoted to the analysis of this issue. 4.1 On the Optimality of Age Disrimination We rst wonder to what extent the laissez-faire equilibrium is optimal. We preisely show that the Hosios ondition leads to equilibrium eieny. In this partiular ase, the disrimination against the older workers is optimal. The intuition of the result is simple: a entral planner maximising soial welfare will alloate fewer resoures to the hiring and the labor-hoarding of older workers, as the same phenomenon as in the deentralized equilibrium is at work: a short horizon makes less eient any investments in older worker employment. In line with the analysis of Pissarides [2000] with innite-lived agents, we derive the optimal steady-state alloation by maximizing the sum of disounted output ows net of reruiting osts for a new entrant generation. This is done over the life yle of workers 17. In addition, sine generations 17It is possible to show that it is equivalent to maximizing the expeted gain of unemployed workers. 34

are independent of eah other, doing this maximization for eah one would lead to the same result. The planner's problem is stated as: under the onstraints: T 1 max β (y i i + {R i } T 1 i=2,{θ i,e i } T 1 i=1 i=1 where y i is the average output. [ b 1 ] ) 2 e2 i u i θ i u i e i (14) u i+1 = G(R i+1 )(1 u i ) + u i (1 e i p(θ i )) (15) y i+1 = u i e i p(θ i )1 + (1 u i ) 1 R i+1 ɛdg(ɛ) (16) Proposition 7. Let η(θ i ) = θ i q (θ i )/q(θ i ) and G(ɛ) = ɛ, the eient alloation exists and it is haraterized by a sequene {Ri, θi } solving: = β (1 η(θ q(θi ) i )) ( 1 Ri+1 ( ) Ri = b + 1 η(θ 2 i ) 1 2 1 η(θi )θ i β ) R i+1 with terminal onditions R T 1 = b and θ T 1 = 0. Proof. See Appendix D.10. (JC ) [1 G(x)] dx (JD ) Property 9. Let η(θi ) = 1 ψ i and G(ɛ) = ɛ, if 1 β(1 ψ) ( ) βψ 2ψ 1 ψ, then the eient alloation veries Ri+1 Ri and θi+1 θi i. Proof. Let substitute 1 γ by ψ in proof of property 3, and the proof is straightforward. These results therefore suggest that it is soially eient to disriminate against older workers by providing them with a lower probability of hiring and a higher probability of ring. Property 10. Let η(θ i ) = 1 ψ, if γ = 1 ψ then R i = R i et θ i = θ i i. Proof. The proof is straightforward by substituting ψ by 1 γ in Proposition 7 and by omparing with Proposition 2. 35

As in Mortensen and Pissarides [1994], the equilibrium is in general not optimal. This is only in the ase of the Hosios ondition (Property 10). It is easy to show that a ompetitive job-searh equilibrium à la Moen [1997] is also able to generate the soially optimal math surplus sharing rule haraterized by the Hosios ondition (proof available upon request). Our life yle eonomy indeed does not introdue any additional soure of externalities. 4.2 Optimal Age-Dependent Firing Costs and Hiring Subsidies It is well-known that the existene of unemployment benets distorts the laisser-faire equilibrium and legitimates the implementation of ring ost and hiring subsidy poliies (Mortensen and Pissarides (1999)). We show in this setion that the rst best alloation an also be reovered in our life yle setting by suh poliies, but only on ondition that they are age-dependent. Overall, we show that the age prole of hiring subsidies and ring taxes is ruially related to the value of the worker's bargaining power and the level of unemployment benets. The intuition of these results is the following: Firstly, high unemployment benets inrease the labor ost and then redue the inentives for rms to hire workers. Nevertheless the redution of the labor demand is not the same at eah age. Indeed, for an older worker, the rm only pays the tax introdued by the UB during a short period. On the other hand, for a younger worker, the rm antiipates paying this tax for a longer period. In this ase the omparative advantage of the younger worker beomes a handiap. In order to restore the eient alloation, ring taxes and hiring subsidies typially must derease with age. Seondly, if the bargaining power of the workers is too low relatively to its optimal value, rms over-invest in the proess of hiring the younger workers. Beause of ongestion eets, the rotation osts are higher than their optimal value. In order to restore the rst-best alloation, ring taxes and hiring subsidies typially must inrease with age. 36

By omparing the equilibrium alloation with the rst best one, it is possible to feature the rst best poliies. Proposition 8. Let η(θ i ) = θ i q (θ i )/q(θ i ) = 1 ψ, assume a given sequene for unemployment benets {z i }, the optimal labor market poliy is a sequene {Hi, Fi } solving: H i+1 = F i+1 + F i = z i + βf i+1 + [ ] γ (1 ψ) (1 γ)ψ [ ( γ 1 γ βq(θ i ) (17) ) 2 ( ) ] 2 1 ψ ψ 1 2 (θ i ) 2 (18) with boundary onditions HT 1 = F T 1 = z T 1, and where θi is given by the solution of the dynamial system (JC )-(JD ). Proof. The proof is straightforward by allowing H i and F i in (JC pol ) and (JD pol ) to be onsistent with (JC ) and (JD ). To understand the poliy impliations of this proposition, we disentangle the role played by eah distortion, either related to unemployment benets or searh externalities. The two following orollaries deal suessively with these two soures of distortions and their respetive impliations on poliy. Corollary 6. Let η(θ i ) = θ i q (θ i )/q(θ i ) = 1 ψ, assume γ = 1 ψ and take z i = z i as given, the age dynamis of hiring subsidies and ring taxes is haraterized by F i F i+1 z and H i H i+1 z. Proof. The proof is straightforward by onsidering γ = 1 ψ in Proposition 8. Assuming γ = 1 ψ (job-searh externalities are internalized), we are fousing on the poliy impliations of the distortions related to unemployment ompensations. Why does the ring tax derease with worker's age? Let us rst onsider a job with a worker of age T 1. Correting for an exessive wage implies F T 1 = z. With a worker of age T 2, not only z but also F T 1 must be internalized: both z, by inreasing the wage, and F T 1 by reduing the 37

value of labor-hoarding are found to inrease R i. Aordingly, F T 2 > F T 1. By bakward indution, it thus appears that Fi = z T 1 i j=0 β j : the ring tax internalizes the sum of disounted unemployment benets until the exit from the labor market. 18 Hiring subsidies are then introdued to avoid the distortion indued by termination osts, H i = F i i. If we allow for some exogenous heterogeneity in unemployment benets, that is z i z i+1, we have Fi = z i + βfi+1. Then, if z i z i+1, it is straightforward to see that the optimal age prole of ring taxes is still dereasing with age, Fi Fi+1. On the ontrary, if z i z i+1 it an be the ase that the optimal employment protetion for older workers who benet from high unemployment ompensations is higher than for the younger. Corollary 7. Assume z i = 0 i and Proposition 7 is satised, the age dependene of hiring subsidies and ring taxes is haraterized by: if γ > 1 ψ, then H i if γ < 1 ψ, then H i Proof. Imposing θi+1 θi is straightforward. > H i+1 0 and F i < H i+1 0 and F i > F i+1 0. < F i+1 0. from Proposition 7 into proposition 8, the proof If γ > 1 ψ, the worker's bargaining power is higher than its eient value. This implies that equilibrium wages are higher than required by the optimum. Consequently, there are not enough vaanies at the equilibrium. Hiring subsidies have to be introdued in order to be onsistent with θ i = θi. But at the same time, the large value of γ together with hiring subsidies are γ responsible for an exessive rate of job destrution: 1 γ θ i (from (JD pol )) > 1 ψ ψ θ i (from (JD )). This requires a positive tax on rings. Until now, the same results would have been obtained in a Mortensen-Pissarides eonomy with innite life horizon. Our additional point is that the size of the distortions related to γ 1 ψ is dereasing with a worker's age. This is due to θ i θ i+1, whih indiates 18If it was assumed that agents have an innite life horizon on the labor market, as in Mortensen-Pissarides (T ), it would be straightforward to see that F i = z/(1 β) i. 38

that the wage inidene of γ is as smaller the older the worker. Ultimately, even if γ > 1 ψ, we have F T 1 = H T 1 = 0 (for z = 0). Consistently, when γ > 1 ψ, we nd at optimal to redue the size of employment protetion and the amount of hiring subsidies as a worker's age inreases. In turn, when γ < 1 ψ, equilibrium wages are not high enough so that it is optimal to tax hirings and simultaneously enourage rings. For the same reason as before, distortions being lower for older workers, hiring taxes and ring subsidies are optimally inreasing with a worker's age. 19 Overall, the age dynamis of ring taxes and hiring subsidies depend both on the value of unemployment benets and on worker bargaining power. In partiular, even though γ < 1 ψ, it an be the ase that F i F i+1 if the value of z is high enough to have equilibrium wages higher than their eient value. In other words, higher unemployment benets make a dereasing prole of hiring subsidies and ring taxes by age more likely. On the ontrary, if γ and z are low enough, the dynamis are reversed. Interestingly, this an easily be formally stated by onsidering the ase β 1. Corollary 8. Let η(θ i ) = θ i q (θ i )/q(θ i ) = 1 ψ, assume β 1, γ < 1 ψ, z i = z i and property 9 is satised, then z z is a suient ondition for F i > F i+1 z and H i > H i+1 z, z ẑ is a suient ondition for F i < F i+1 z and H i < H i+1 z. where ẑ = [ ( 1 ψ ψ ) 2 ( γ 1 γ Proof. See Appendix D.11. 20 ) 2 ] 2 2 [ ψ(1 b) ] 2 1 ψ and z = [ ( 1 ψ ψ ) 2 ( These results should provide important insights on the optimal age-pattern of employment protetion and hiring subsidies in OECD ountries. In Anglosaxon ountries with low unemployment benets, the shape of the ring 19Lastly, it is worth emphasizing that if we had assigned an eieny motive to unemployment benets by removing redistributive onsiderations, it is straightforward to see from Proposition 8 that for F i = 0 i and γ > 1 ψ, zi zi+1 0, whereas if γ < 1 ψ, zi z i+1 20If 0. γ < 1 ψ and z [ẑ, z], the age dynamis of H i and F i are typially non-monotonous (rst inreasing and then dereasing). 39 γ 1 γ ) 2 ] 2 2 [ ψ ] 2 1 ψ.

tax should be inreasing, whereas the revers should hold in European ountries suh as in Frane. 4.3 The Role of Legislation Prohibiting Age Disrimination As emphasized above, some OECD ountries have more or less reently adopted a law to prohibit age disrimination. Despite its eetiveness being still ontroversial, it is of primary interest to examine whether this labor market poliy is welfare-improving or not. Interestingly, our answer is again that it depends on the level of unemployment benets. Of ourse, this law annot be a rst best poliy 21. However, in a seond best world, where job-searh externalities are not internalized or unemployment benets are paid to unemployed workers, prohibiting age disrimination an be optimal. Figure 6: Welfare gain from age disrimination legislation 0.04 0.03 Welfare gap (Wnd Wd) 0.02 0.01 0 0.01 0.02 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Unemployment benefits (z) It is quite intuitive that the optimality of the law prohibiting age disrimination depends on the level of unemployment benets. Indeed, when 21This an be easily demonstrated. The program of the soial planner when age disrimination is prohibited inorporates a supplementary onstraint, θ i = θ. 40

unemployment benets are high, wages are higher than required by the eieny, so that employment rates are too low. The legislation is then ounterprodutive, sine it gives more weight to less protable workers (the older ones). On the other hand, when employment is too high (too muh rotation), the legislation whih redues the expeted gain from hirings is now welfare improving. Sine no analytial results an be derived to state this point, we run some simulations of the dynami system dened by Proposition 6 (the same parameters as detailed in the alibration of the Frenh eonomy). The results are illustrated in Figure 6. 22 This suggests that when unemployment benets are high, adopting a legislation prohibiting age-disrimination is welfare-degrading. This result alls into question the adoption of this legislation in some European eonomies suh as Frane. 5 Conlusion This paper puts the emphasis on a life yle view of the labor market, both for understanding supply and demand harateristis, and for implementing welfare-improving poliies. We rst inorporate life-yle features into the job reation - job destrution approah to the labor market. The equilibrium is typially featured by inreasing (dereasing) ring (hiring) rates with age, and an hump-shaped age-dynamis of employment. In that ontext, we nd that ring taxes and unemployment benets introdue a bias in favor of older workers. 22To ompare equilibrium welfare with and without direted job-searh we use the following denitions, respetively: W d = W nd = T 1 β i ( yi d + bu d i θi d u d ) i i=1 T 1 β i ( yi nd i=1 + bu nd i θ nd u d ) i where subsripts d and nd stand for the equilibrium with age disrimination (benhmark) and without age disrimination, respetively. In order to apture only the impat of a onstant hiring rate implied by the age disrimination law on the labor market dynamis, we assume that the job-searh eort is onstant over the life-yle. 41

The empirial plausibility of the model is assessed on Frenh data by inorporating existing age-spei labor market poliies. We show the primary role played by the retirement date in aounting for the observed fall in the older worker employment rate. This result relies neither on retirement programs nor on delines in produtivity. It simply refers to the expeted distane from retirement, that determines the expeted duration of jobs and, in turn, both rms and workers' forward-looking strategies at the end of the working life. Lastly, we emphasize that the optimal age prole of older worker poliies should sharply dier among ountries, aording to dierenes in unemployment benet institutions. While in a US-type eonomy hiring subsidies and ring taxes should be more favorable to older workers, the reverse holds true in European ountries with high unemployment ompensation. This learly alls into question the optimality of higher employment protetion for older workers and age disrimination legislation adopted in some European ountries. Several interesting aspets of a life yle approah to the labor market have not been addressed in this paper. Our benhmark ould also provide interesting insights to explain younger workers' employment from a quantitative standpoint. But it would imply modeling the speiities of the labor market at the rst stages of the life yle, for instane the imperfet information on the produtive harateristis of younger workers. This line of researh has, for instane, been examined by Pries and Rogerson [2005] in an innite-lived agents ontext. Finally, the long run impat of the inreasing weight of older workers in the labor fore ould be also addressed in our framework. All these topis are left for further researh. Referenes [1] J. Abowd and L. Allain, Compensation Struture and Produt Market Competition, Annales d'eonomie et Statistiques, 41/42 (1996), 207-217. 42

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A Extended Model with Endogenous Searh Effort and Human Capital Aumulation This setion desribes an extended version of our benhmark model that allows both for endogenous job-searh eort and general human apital aumulation. Our objetive is twofold: It is rst to examine the robustness of our results with respet to the introdution of job-searh eort. It is then to show that a basi inorporation of human apital aumulation reoniles the model with the hump-shaped prole of wages over the life yle. The mathing funtion that gives the number of hirings, M(v i, e i u i ), now inludes the average job-searh eort e i. Aordingly, labor market tightness is dened as θ i = v i /[e i u i ], so that the probability for unemployed workers to M(v meet a vaany is e i p(θ i ) = e i,e i u i ) i e i u i and the probability for a rm to ll a vaany is q(θ i ) = M(v i,e i u i ) v i. Let us onsider some general preferenes for the opportunity ost of employment and the searh eort, φ(b i, e i ), 23 the value of the unemployed position writes as: U i = max e i 0 {φ(b i, e i ) + β [e i p(θ i )W i+1 (1) + (1 e i p(θ i ))U i+1 ]} with φ 1 > 0, φ 2 < 0 and φ 1 < 0, φ 2 > 0. The optimal job-searh eort deision rule then solves: φ 2(b i, e i ) = βp(θ i ) [W i+1 (1) U i+1 ] (19) General human apital aumulation over the life yle is for further simpliity assumed to ontinuously inrease with age, whatever the worker's 23In the main text we onsider a simpler speiation, φ(b i, e i ) = b i e2 i 2. 46

status on the labor market (either employed or unemployed). 24 Firms' value funtions are now dened by: V i = i + β [q(θ i )J i+1 (1) + (1 q(θ i ))V i ] J i (ɛ) = h i ɛ w i (ɛ) + β 1 R i+1 J i+1 (x)dg(x) + βg(r i+1 ) max i {V i } i [1, T 1] where h i reets upon the produtivity impat of human apital for a worker of age i. i is the reruiting ost whih eventually depends on the worker's age due to dierent levels in human apital levels (see hereafter). In that ontext, the wage bargaining (see Appendix C for more details) solves: w i (ɛ) = (1 γ)φ(b i, e i ) + γ (h i ɛ + i θ i e i ) (20) Before turning to the equilibrium denition, one must provide some assumptions on laws of motion for human apital, reruiting osts and opportunity osts of employment. In a way onsistent with an innite-lived agents mathing model with growth (see Pissarides [2000]), we onsider a deterministi trend on human apital, and that reruiting osts and workers' reservation wage are indexed to this trend: h i+1 = (1 + µ)h i i = h i φ(b i, e i ) = h i φ(b, e i ) with µ 0, a given initial ondition h 1 = 1 and some positive parameters, b 0. 24A useful extension of this model would be to allow for some depreiation in human apital skill with periods of unemployment. Currently, our goal is primarily to show that human apital aumulation provides an opposite fore to the age-horizon on the age dynamis of wages. 47

Proposition 9. A labor market equilibrium with wage bargaining, general human apital aumulation and endogenous searh eort exists and it is haraterized by a sequene {R i, θ i } solving: q(θ i = β(1 + µ)(1 γ)(1 R ) i+1 ) (JC eff ) ( ) γ R i = φ (b, Φ(θ i ))) + 1 γ θ i e i β(1 + µ) 1 R i+1 [1 G(x)]dx (JD eff ) with terminal onditions θ T 1 = 0 and R T 1 = φ(b, 0), and where Φ(θ i ) φ 1 2 ( γ θ 1 γ i). Proof. Similar to the proof of Proposition 2, but by making in addition use of the fat that φ 2(b i, e i ) = βp(θ i ) [W i+1 (1) U i+1 ] = γ i θ 1 γ i whih implies e i = φ 1 2 ( b, γ 1 γ θ i ) Φ(θ i ). If µ = 0, it is then possible to derive the Corollary 3 in the main text (see appendix D.6). This orollary demonstrates the positive role of the jobsearh eort in explaining the inrease (derease) of rings (hirings) with age Ṫurning to the inuene of human apital aumulation on equilibrium wages, let us remove the endogenous searh eort. The equilibrium wage bargaining is: w i (ɛ) = h i [(1 γ)b + γ (ɛ + θ i )] Sine h i is inreasing with age whereas θ i θ i+1, aording to the value of µ, the age dynamis prole of wages is typially hump-shaped. B Benhmark Model with Labor Market Poliy Let us denote H i a lump sum paid to the employer when a new worker of age i is hired, F i the ring ost and z i the unemployment benets. We follow MP by onsidering that the wage struture that arises as a Nash bargaining solution has two tiers. The rst tier wage reets the fat that hiring subsidy is diretly relevant to the deision to aept a math and that the possibility of inurring ring osts in the future aets the value the employer plaes 48

on the math. In turn, the seond tier wage applies when ring osts are diretly relevant to a ontinuation deision. Let the subsript i = 0 index the initial wage and the value of a job under the terms of the two-tier ontrat, rms' value funtions solve: V i = + βq(θ i ) [ ] Ji+1 0 + H i+1 + β(1 q(θi ))V i 1 [ ] Ji 0 = 1 wi 0 + β J i+1 (x)dg(x) + βg(r i+1 ) max{v i } F i+1 R i i+1 1 [ ] J i (ɛ) = ɛ w i (ɛ) + β J i+1 (x)dg(x) + βg(r i+1 ) max{v i } F i+1 R i i+1 The optimal ring deision rule now solves: J i (R i ) = F i Adding the free entry ondition, V i = 0, it emerges that labor market tightness and produtivity threshold are derived from the following two equations: = β [ ] Ji+1 0 + H i+1 q(θ i ) [ 1 ] R i = w(r i ) F i β J i+1 (x)dg(x) G(R i+1 )F i+1 R i+1 (21) (22) Let us now examine the derivation of the two-tier wage struture. The latter is haraterized by the following two sharing rules (as a result of Nash bargaining): W i (1) U i = γ [ ] Ji 0 + H i + W i (1) U i w 0 i (23) W i (ɛ) U i = γ w i (ɛ) (24) [ J i (ɛ) (max i {V i } F i ) + W i (ɛ) U i ] so that the equations for the initial and subsequent wage bargaining are (see Appendix C.2 for details): wi 0 = (1 γ)(b + z i ) + γ (1 + θ i + H i βf i+1 ) (25) w i (ɛ) = (1 γ)(b + z i ) + γ (ɛ + θ i + F i βf i+1 ) (26) 49

The derivation of (JC pol ) and (JD pol ) is then straightforward by notiing that J i+1(ɛ) = 1 γ and J i (R i ) = F i implies that J i (ɛ) = (1 γ)(1 R i ) F i, and similarly making use of the fat that J 0 i = J 0 i J i (R i ) F i. C Wage Equations Derivations C.1 Wage Bargaining in a Laissez-Faire Equilibrium Let rst onsider the derivation of the wage equation in a Laissez-Faire equilibrium with both endogenous job-searh eort and human apital aumulation. The sharing rule (7) an take the form: (1 γ)u i = γ [J i (ɛ) + W i (ɛ)] W i (ɛ) (27) >From value funtions (3),(5) and (6), it turns out to be that: γ [J i (ɛ) + W i (ɛ)] W i (ɛ) = γh i ɛ w i (ɛ) + γβ Similarly, γβ 1 β 1 1 R i+1 W i+1 (x)dg(x) R i+1 [J i+1 (x) + W i+1 (x)] dg(x) (1 γ)βg(r i+1 )U i+1 (28) [J i+1 (x) + W i+1 (x)] dg(x) = γβ [J i+1 (x) + W i+1 (x) U i+1 ] dg(x) R i+1 R i+1 +γβ[1 G(R i+1 )]U i+1 β 1 Sine (7) holds for eah age: γβ 1 1 1 W i+1 (x)dg(x) = β [W i+1 (x) U i+1 ] dg(x) R i+1 R i+1 +β[1 G(R i+1 )]U i+1 R i+1 [J i+1 (x) + W i+1 (x) U i+1 ] dg(x) = β so that (28) an be written as: 50 1 R i+1 [W i+1 (x) U i+1 ] dg(x)

γ [J i (ɛ) + W i (ɛ)] W i (ɛ) = γh i ɛ w i (ɛ) (1 γ)βu i+1 (29) In turn, the unemployed value (6) is: U i = φ(b i, e i ) + β [e i p(θ i ) (W i+1 (1) U i+1 ) + U i+1 ] [ ] γ = φ(b, e i ) + β e i p(θ i ) 1 γ J i+1(1) + U i+1 From the free entry, it derives: J i+1 (1) = i v i βm(e i u i, v i ) i q(θ i ) so that sine p(θ i ) = M(e i u i, v i )/[e i u i ] and e i = e i in equilibrium, we have: (1 γ)u i = (1 γ)φ(b i, e i ) + γ i e i θ i + (1 γ)βu i+1 (30) Let substitute for (29) and (30) in (27), we nd: (1 γ)φ(b i, e i ) γ i e i θ i (1 γ)βu i+1 = γh i ɛ w i (ɛ) β(1 γ)u i+1 This lastly leads to the following wage equation: w i (ɛ) = h i [(1 γ)φ(b, e i ) + γ (ɛ + e i θ i )] Abstrating from the endogenous job-searh eort and human apital aumulation, it is straightforward to get the equation (8). C.2 Wage Bargaining with Labor Market Poliy For further simpliity, let us assume h i+1 = h i = 1. Under this assumption, we derive the two-tier wage struture. The sharing rule (23) rst an be written as: γh i (1 γ)u i = γ [ J 0 i + W i (1) ] W i (1) (31) 51

Following the same derivation strategy as for the ase without poliy, we nd that: γ [ J 0 i + W i (1) ] W i (1) = γ1 w 0 i (1 γ)βu i+1 γβf i+1 (32) (1 γ)u i = (1 γ)φ(b, e i ) + γe i θ i + (1 γ)βu i+1 (33) Substituting out for (32) and (33) in (31), it emerges that the initial wage solves: w 0 i = (1 γ)φ(b, e i ) + γ (1 + e i θ i + H i βf i+1 ) Similarly, the sharing rule (24) an be written as: γf i (1 γ)u i = γ [J i (ɛ) + W i (ɛ)] W i (ɛ) (34) This in turn leads to the following wage equation: w i (ɛ) = (1 γ)φ(b, e i ) + γ (ɛ + e i θ i + F i βf i+1 ) C.3 Wage Bargaining with Age-Disrimination Legislation From the equation (10)-(11)-(12)-(13) and the free-entry ondition, it is possible to show that the total surplus related to the job math, S i (ɛ) J i (ɛ) V + W i (ɛ) U i solves S i (ɛ) = ɛ b z βp(θ)(w i+1 (1) U i+1 ) + β 1 R i+1 S i+1 (x)dg(x) In addition, sine the sharing rule an be stated both as γs i (ɛ) = W i (ɛ) U i and (1 γ)s i (ɛ) = J i (ɛ), it emerges that: J i (ɛ) = (1 γ)[ɛ b z βp(θ)(w i+1 (1) U i+1 )]+β 1 R i+1 J i+1 (x)dg(x) (35) This entails J i(ɛ) = 1 γ, hene J i (ɛ) = (1 γ)(ɛ R i ) and W i (ɛ) U i = γ(ɛ R i ). It is then straightforward to derive from the sharing rule the following wage equation: w i (ɛ) = γɛ + (1 γ)(b + z) + γ(1 γ)βp(θ)(1 R i+1 ) 52

In turn, the job destrution rule solves S i (R i ) = 0, R i = b + z + βγp(θ)(1 R i+1 ) β 1 R i+1 [x R i+1 ]dg(x) D Proofs of propositions, properties and orollaries D.1 Proof of proposition 1 For γ = 0, the dierentiation of (3) with respet to ɛ, implies that J i(ɛ) = 1 i. Sine J i (R i ) = 0, the value of a lled job veries J i (ɛ) = ɛ R i. The equation (2) an be written as: q(θ i ) = β(1 R i+1) This gives immediately (JC P artialeq ). Sine by integrating by parts 1 R i+1 J i+1 (x)dg(x) = 1 R i+1 J i+1(x)[1 G(x)]dx = 1 R i+1 [1 G(x)]dx and V i = 0 i (free entry ondition), it is straightforward to see that the equation (4) veries (JD P artialeq ). D.2 Proof of property 1 The proof is straightforward. Making use of (JD P artialeq ), we indeed obtain: R T 1 = b R T 2 = b β R T 3 = b β... 1 R T 1 [1 G(x)]dx 1 R T 2 [1 G(x)]dx R T 1 R T 2 As an be seen in (JC P artialeq ), θ i depends negatively on R i+1, and it turns out that θ i+1 θ i i. D.3 Proof of proposition 2 By dierentiating (3) with respet to ɛ, it emerges that J i(ɛ) = 1 γ i. Sine J i (R i ) = 0, the value of a lled job veries J i (ɛ) = (1 γ) (ɛ R i ). 53

The equation (2) an then be written in order to determine the sequene of θ i as an expression of R i (equation (JC)). Moreover, by ombining the wage equation (8) and the equation (4), and integrating by parts as in proof of proposition 1, one gets the equation (JD) desribing the age prole of R i. D.4 Proof of property 2 dr If i dr i+1 0 [ondition (C1)] and R i b [ondition (C2)] the solution to the dynamial equation (9) neessarily veries R i+1 R i. Given the denition of (JC) and q (θ i ) 0, it then omes that θ i+1 θ i. Making use of (9), we have: ( dr i γ 0 1 dr i+1 1 ψ [ β(1 γ) R i b 0 1 2γ ) [ β(1 γ) ] ψ 1 ψ (1 Ri+1 ) 2ψ 1 1 ψ (36) ] ψ 1 ψ (1 Ri+1 ) 2ψ 1 1 ψ (37) If ψ 1/2, evaluating (36) for R i+1 = 0 is suient to insure that both onditions (C1) and (C2) hold simultaneously. On the ontrary, if ψ 1/2, evaluating (37) for max{r i } = b implies that (C1) and (C2) hold. D.5 Proof of proposition 3 Similar to proof of proposition 2, but by making in addition use of the fat that: e i = βp(θ i ) [W i+1 (1) U i+1 ] = γ 1 γ θ i (38) Sine the wage solves in that ase the equation (20) for φ(b i, e i ) = b e2 i 2, ombine this wage equation with (38) and plug into (4) we get (JD eff ). Proof of (JC eff ) is straightforward. 54

D.6 Proof of property 3 Following the same proedure as for the proof of Property 2, we now nd that onditions (C1) and (C2) are (from Proposition 3): dr i dr i+1 0 1 β γ2 1 ψ ( β(1 γ) R i b 0 1 βγ 2 ( β(1 γ) ) 2ψ 1 ψ (1 Ri+1 ) 2ψ 1 ψ (39) ) 2ψ 1 ψ (1 Ri+1 ) 2ψ 1 ψ (40) It is thus obvious that the ondition (39) is more stringent than ondition (40) whatever the value of ψ. Evaluating the former ondition with R i+1 = 0, one gets the suient ondition reported in the main text. D.7 Proof of property 4 Let us denote Ψ(R i+1, θ i ) = G(R i+1) G(R i+1 )+p(θ i ). By denition, Ψ(R i+1,θ i ) R i+1 > 0 and Ψ(R i+1,θ i ) θ i < 0. For θ i+1 θ i and R i+1 R i (property 2), it thus appears that Ψ(R i+1, θ i ) Ψ(R i+2, θ i+1 ) i. Let us rst reason by ontradition by assuming u 1 < Ψ(R 2, θ 1 ). Sine Ψ(R i+1, θ i ) Ψ(R i+2, θ i+1 ) this neessarily implies u 2 < u 1. In turn sine Ψ(R 2, θ 1 ) < Ψ(R 3, θ 2 ), we have u 3 < u 2, and Ψ(R 3, θ 2 ) < Ψ(R 4, θ 3 )... This shows that in that ase we would have n i+1 n i i. On the ontrary, sine by denition u 1 = 1 > Ψ(R 2, θ 1 ), we have that n 2 > n 1. Ψ(R i+1, θ i ) < Ψ(R i+2, θ i+1 ) i then insures that there exists an age T whih veries u T = G(R T +1 ) G(R T +1 )+p(θ T ), and so that n T +1 n T. D.8 Proof of property 5 The equation (8) gives the wage at age i for a produtivity level ɛ. Then the wage gap by age, at the level of the produtivity reservation by age is given by the dierene of the equation (8) for age i + 1 and age i, for the produtivity R i+1, R i : w i+1 (R i+1 ) w i (R i ) = γ[(r i+1 R i ) + (θ i+1 θ i )] 55

Let substitute out from R i+1 (R i ) as a funtion of θ i+1 (θ i ) aording to proposition 2 under the parameter restrition ψ = 1/2 and G(ɛ) = ɛ, we nd: w i+1 (R i+1 ) w i (R i ) = [ 1 1 γ ] (θ 2β(1 γ) 2 i+1 θ i ) Then w i+1 (R i+1 ) w i (R i ) > 0 if 1 < 0 beause θ 2β(1 γ) 2 i+1 θ i < 0. D.9 Proofs of properties 7 and 8 From proposition 4 and orollary 4, let onsider M(v, u) = v ψ u 1 ψ with ψ = 1/2 and G(ɛ) = ɛ, it omes: 2γ(1 γ)β2 dr i = dz [1 β + [ +βdr i+1 1 R i+1 dr T 1 = dz df 2γ(1 γ)β2 dr = dz [1 β + [ +βdr 1 R Let examine the impat of ring ost, we have: ] (1 R i+1 F ) df 2βγ(1 γ) (1 R i+1 F ) ] (1 R F ) ] 2βγ(1 γ) (1 R F ) df ] dr T 1 = 1 df dr df = 1 β + 2γ(1 γ)β2 (1 R F ) 1 β + 2γ(1 γ)β2 (1 R F ) + βr > 1 so that we have dr T 1 df β = 1 and γ = 0, dr df = 0 < dr i < dr 2 df df dr df 0. It an be notied that when Similarly, the inidene of unemployment benets is derived from: dr T 1 dz dr dz = 1 = 1 1 β(1 R) + 2γ(1 γ)β2 (1 R F ) 56 > 1 from orollary 5.

so that we have 0 dr T 1 dz < dr i < dr 2 dz dz dr dz. Lastly, from orollary 5, it is straightforward to see that: dθ i df dθ df dθ i dz dθ dz Taking into onsideration dr i df θ θ i, it omes that dθ i dθ df df = β(1 γ) (1 ψ) θψ i = β(1 γ) (1 ψ) θψ β(1 γ) = (1 ψ) θψ i β(1 γ) = (1 ψ) dr df and dθ i dz [ 1 + dr ] i+1 df [ 1 + dr ] df dr i+1 dz θψ dr dz 0 and 0 dr i dz dz. dθ dr dz, as well as D.10 Proof of proposition 7 Let us denote λ i and µ i the Lagrange multiplier assoiated with onstraints (15) and (16), optimal deision rules with respet to R i+1, θ i and u i, y i are respetively given by: λ i = µ i R i+1 β i = (µ i λ i ) p (θ i ) β i (e i + θ i ) = p(θ i ) (µ i λ i ) ( λ i 1 = β i b 1 ) 2 e2 i θ i e i + λ i [1 e i p(θ i ) G(R i+1 )] 1 ] +µ i [e i p(θ i ) ɛdg(ɛ) R i+1 µ i = β i+1 It is then possible to derive the following expression of the produtivity threshold: R i = b 1 2 e2 i θ i e i β 1 R i+1 ɛdg(ɛ) + e ip(θ i ) p (θ i ) + βr i+1(1 G(R i+1 )) 57

[ ] Let us remark that θ i + e i = p(θ i )/p (θ i ) implies e i = p(θi ) θ p (θ i ) i θ i η(θ i ) 1 η(θ i ) (sine p(θ i )/p (θ i ) = β 1 R i+1 (ɛ R i+1 )dg(ɛ) = 1 θ i 1 η(θ i )) and βr i+1 (1 G(R i+1 )) β 1 R i+1 ɛdg(ɛ) = R i+1 [1 G(x)]dx, one gets (JD ). The derivation of (JC ) is straightforward from the optimality onditions of the planner's problem. D.11 Proof of orollary 8 From proposition 8, we have that the optimal age prole of rings solve for β 1: F i F i+1 = [ ( ) 2 γ 1 γ Aordingly, from (JC ) sine θ i [ ] 1 ψ 1 ψ ( ) ] 2 1 ψ 1 ψ 2 (θ i) 2 z [ ( ) 2 ( 1 ψ i, z ψ insures that F i > F i+1. In turn, sine for i [1, T 2], θ i emerges that z [ ( 1 ψ ψ ) 2 ( γ 1 γ ) 2 ] 2 2 [ ψ(1 b) ] 2 1 ψ γ 1 γ [ ψ(1 b) ) 2 ] 2 2 = ] 1 1 ψ, it insures that F i < F i+1. [ ψ ] 2 1 ψ 58