Formal Education Versus Learning-by-Doing

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1 DISCUSSION PAPER SERIES IZA DP No Foral Education Versus Learning-by-Doing Frédéric Gavrel Isabelle Lebon Thérèse Rebière July 2014 Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor

2 Foral Education Versus Learning-by-Doing Frédéric Gavrel CREM-CNRS, University of Caen and TEPP-CNRS Isabelle Lebon CREM-CNRS, University of Caen and TEPP-CNRS Thérèse Rebière LIRSA-CNAM and IZA Discussion Paper No July 2014 IZA P.O. Box Bonn Gerany Phone: Fax: E-ail: Any opinions expressed here are those of the author(s) and not those of IZA. Research published in this series ay include views on policy, but the institute itself takes no institutional policy positions. The IZA research network is coitted to the IZA Guiding Principles of Research Integrity. The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of counication between science, politics and business. IZA is an independent nonprofit organization supported by Deutsche Post Foundation. The center is associated with the University of Bonn and offers a stiulating research environent through its international network, workshops and conferences, data service, project support, research visits and doctoral progra. IZA engages in (i) original and internationally copetitive research in all fields of labor econoics, (ii) developent of policy concepts, and (iii) disseination of research results and concepts to the interested public. IZA Discussion Papers often represent preliinary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version ay be available directly fro the author.

3 IZA Discussion Paper No July 2014 ABSTRACT Foral Education Versus Learning-by-Doing * The efficiency of educational choices is studied in a search-atching odel where individuals face a tradeoff: acquiring foral education or learning while on the job. When their education effort is successful, newcoers directly obtain a high-skill job; otherwise, they begin with a low-skill job, learn-by-doing and then search while on-the-job for a high-skill job. Low-skill firs suffer fro hold-up behavior by high-skill firs. The low-skill sector is insufficiently attractive and individuals devote too uch effort to foral education. A selffinancing tax and subsidy policy restores arket efficiency. JEL Classification: H21, I20, J21, J64, J68 Keywords: foral education, learning-by-doing, arket efficiency, on-the-job search, search uneployent Corresponding author: Thérèse Rebière LIRSA-CNAM Departent of Econoics 40 rue des Jeûneurs Case 1D2P Paris France E-ail: therese.rebiere@cna.fr * We would like to thank Björn Brügeann, Gary Fields, and Bernd Fitzenberger for their helpful coents on earlier versions of that work. The usual caveat applies.

4 1 Introduction Foral education is not the only way to acquire skills that give workers the opportunity of gaining a good job. Learning-by-doing in a low-skill job and then searching (while onthe-job) for a high-skill job is another way of reaching the sae goal. Do workers choose the right aount of foral education when faced with this trade-off? If not, what kind of public policy should be ipleented? Although huan capital is generally easured by the aount of foral education, any skills are best learned on-the-job through participating in the production process. Consequently, training here also deterines workers productivity. In the absence of learning-by-doing, workers would always hold a job equivalent to their education level, without any prospect of iproveent. Using US data, Andersson et al. (2005) find that 15% to 20% of workers with a high school diploa or less had escaped low-wage eployent after nine years. This result sees to indicate that there is a stepping-stone effect toward better paid jobs (Connolly and Gotschalk 2001) which is ore likely to occur when workers voluntarily change jobs (Sicheran and Galor 1990, Gotschalk 2001, Holzer 2004), and even ore likely when eployent is gained at a higher-wage fir (Andersson et al. 2003). Such a springboard effect is deonstrated for Gerany by Knabe and Plu (2013) who state that the rate of transition to a high-paid job conditional on first accepting a low-paid job is particularly significant for low-skilled workers. Likewise, epirical evidence for France shows that, since the 1980s, upward professional obility has iproved, especially for low-skilled workers: 29% of blue collar workers in 1998 had experienced upward obility between 1998 and 2003, against 19% between 1980 and We can think of a secretary becoing an executive secretary, or an unskilled worker in a routine occupation becoing skilled in a lower technical occupation in sectors such as anufacturing, industrial crafts, construction, and warehousing and transport; and then transitioning fro lower to interediate technical occupations, such as technician to forean or supervisor (Monso 2006). Most of those occupations could also be entered by obtaining a diploa in the relevant field of copetence. This evidence is consistent with our fraework, in which educated workers gain a well-paid job directly, whereas workers with a lower level of education have to train theselves on-the-job before gaining a better-paid job. The fact reains that, during the past few decades, ore and ore individuals have chosen to reinforce their effort in foral education (see for instance Machin (1996), Aceoglu (2002), Mincer (1994, 2003), and Moscarini and Vella (2008)). Did these private educational choices lead to an efficient outcoe? The purpose of this paper is to shed soe light on this issue. We argue that individuals tend to put too uch ephasis on foral education, copared with training in the workplace. The reason for this does not involve educational decisions theselves. This distortion originates in the fact that firs with high-skill jobs underestiate the social cost of filling their vacancies with workers previously eployed in low-skill jobs in which they have practised learning-by-doing. Firs create too any high-skill jobs. In response to this hold-up behavior, job creation is 2

5 suboptial in the low-skill sub-arket. As a result, high-skill jobs are too appealing, and individuals ake too great an effort to acquire foral education. This creates a proble requiring governent involveent. To assess the consistency of our arguent we use a two-sector search-atching odel in which workers have a finite life expectancy (Moen and Rosén 2004, Gavrel et al. 2010). In contrast to these previous papers, our odel assues that workers can becoe skilled via foral education. Before entering the labor arket, new workers decide on the aount of effort to devote to foral education. If they succeed in acquiring the required skills, they directly join the pool of applicants for good jobs. If they fail, they have to search for a low-skill job, and then begin to learn while on-the-job. When the learning-by-doing process coes to its end, workers are endowed with the sae skills as (forally) educated workers (following Arrow (1962)). They then can join the pool of applicants for good jobs. First, we describe a (decentralized) stationary equilibriu of the labor arket and its efficiency conditions. Assuing that firs internalize the well-known congestion effect (Hosios 1990, Pissarides 2000), high-skill job creation appears to be too high; whereas low-skill job creation, as well as individuals educational choices, are constrained efficient. In other words, they are optial for the equilibriu value of the tightness of the highskill sub-arket. This eans that inefficiency derives entirely fro an excessive creation of high-skill vacancies. Next, we copare the decentralized equilibriu with a social optiu. The results validate the consistency of our arguent: low skill jobs are too few in nuber, and individuals put too uch ephasis upon foral education. Second, we show that a Tax and Subsidy Policy (TSP) can decentralize the social optiu. Taxes ust be levied on (filled) good jobs. They ensure that perceived hiring costs coincide with social costs. However, these taxes distort low-skill job creation as well as educational choices. In order to restore arket efficiency, these taxes ust be dedicated to the funding of two kinds of copensatory transfer. One is allocated to firs in the low-skill sub-arket when they lose workers leaving the for a better job. The other is a reward that workers receive if their foral education is successfully copleted. Rewarding graduates is necessary since the taxes which have to be levied on high-skill jobs excessively reduce the surplus for a atch with such jobs, hence the returns to foral education for workers. Econoists have been interested in the efficiency of huan capital investent for a long tie. A controversial issue, going back to Pigou (1912), concerns governental involveent that seeks to enhance skills. Since firs would not have an interest in investing in workers skills because of the risk that their experienced workers ight quit for external opportunities, governent subsidies appear to be a necessary easure for the iproveent of training and of schooling. By contrast, Becker (1964) pointed out that the solution for huan capital inefficiency ay be iproved loan arkets rather than governent regulation and training subsidies. A copetitive labor arket iplies that workers are the only ones who have an incentive to invest in their general training, bearing the cost theselves, either directly or by taking a wage cut. Hence, the appro- 3

6 priate aount of investent for an efficient arket would be undertaken unless workers are credit-constrained. More recently, labor theory has re-exained the issue of educational choices in the presence of arket iperfections. Our paper is a contribution to this literature. Aceoglu and Pischke (1999) argue that search frictions allow us to account for eployerprovided on-the-job training, since firs are able to recover such investent in huan capital. Moen (1999) studies the efficiency of educational choices. Firs rank their applicants and hire the best, while workers use foral education to copete for jobs. Here, the education effort can be too intensive. The sae result is presented by Charlot and Decreuse (2007), in which workers self-select their educational choices (see also Charlot and Decreuse (2005)). Workers of low ability place too uch ephasis on the value of a higher foral education for gaining a job, even though education can be costly. Such inefficient behavior leads the authors to suggest that educational subsidies be prohibited. The paper is organized as follows: Section 2 outlines the analytical fraework. We define a labor arket decentralized (stationary) equilibriu in section 3. Section 4 studies arket efficiency and states two ain results: a decentralized equilibriu is constrained efficient in ters of low-skill job creation and educational choices but inefficient in ters of high-skill job creation; the laissez-faire situation is inefficient. In section 5, we outline a self-financed fiscal policy which rewards educational success and leads to a social optiu. In section 6, we calibrate the odel with US epirical evidence. Finally, section 7 contains soe concluding coents. 2 Analytical fraework The econoy consists of two types of agents: workers and firs. Firs are infinitely-lived whereas workers have a finite life expectancy of 1/. Tie is continuous and paraeter easures the workers labor arket exit rate. Each worker who leaves the arket is replaced with a newcoer. The easure of the total labor force is constant and noralized to one. All agents are risk-neutral and discount future payoffs at rate r (r 0). The labor arket is segented into two interacting sub-arkets (sectors arranged into a hierarchy). Sector 2 offers low-skill jobs, while sector 1 offers high-skill jobs. Workers decide on their effort in foral education e when entering the econoy. If their effort is successful (which occurs with the probability π), workers will enter the pool of applicants for high-skill jobs (high-skill uneployent); whereas workers with unsuccessful effort will enter the pool of applicants for low-skill jobs (see figure 1). The probability π is assued to be an increasing and concave function π(e) of the education effort e (π (.) > 0, π (.) < 0). Workers with low-skill jobs will therefore have to engage in a learning-by-doing process in order to becoe skilled enough to be eployable in a high-skill fir. The expected duration of this process is denoted by (1/λ). Workers thus acquire the required skills at a Poisson rate λ. When the learning period coes to an end, workers engage in an on-thejob search process, hoping to get a high-skill job. The incentive to look for a high-skill job is the wage differential between sectors. 4

7 Figure 1: Workers flows Sector 2 eployent Sector 2 uneployent Low-skill workers Trained Workers Foral Education Sector 1 uneployent Sector 1 eployent When entering the labor arket firs choose the sub-arket i (i = 1, 2) in which they will operate. They then create a single job in their chosen sub-arket. Frictions exist that prevent the instantaneous atching of jobs with workers. Firs thus have to pay a cost, c, in order to keep their vacancy open. When atched with a worker, jobs yield output y 1 in sector 1, ŷ 2 in sector 2 when workers are trained, and y 2 when workers are novice (with y 1 > ŷ 2 > y 2 ). Wages are negotiated. Workers have a bargaining power of β and firs have a bargaining power of (1 β). Sector 1 offers the wage w 1 ; whereas sector 2 offers the wage w 2 when workers are untrained, and the wage ŵ 2 when workers have learned by doing. Job creation results fro the usual assuption of free entry in both sectors. Market frictions in sector-i are suarized in a constant-returns atching function that defines the arrival rate of workers to job vacancies q i (θ i ) with q i(θ i ) < 0. The arrival rate of job offers to searching workers is p i = θ i q i with p i(θ i ) > 0 where θ i is the sub-arket tightness. 2.1 High-skill jobs Asset values In sub-arket 1, the lifetie utility of an eployed worker, called W 1, satisfies: (r + )(W 1 U 1 ) = w 1 (r + )U 1 (1) where U 1 is the lifetie utility of a high-skilled worker when uneployed. We have: (r + )U 1 = d + p 1 (W 1 U 1 ) (2) with d being the value of leisure. Regarding sector-1 firs, the value of a filled job, called J 1, verifies: 5

8 (r + )(J 1 V 1 ) = y 1 w 1 (r + )V 1 (3) where V 1 is the asset value of a sector-1 fir whose job is vacant. We have: Private surplus and arket tightness rv 1 = c + q 1 (J 1 V 1 ) (4) Firs are unlikely to coit to wages. Following Shier (2006), we assue that when an on-the-job seeker eets a fir, she ust reject her current job and bargain with the other eployer with no possibility of reverting to her old job. Therefore, the worker s threat point is always uneployent rather than the value of her previous job. This assuption follows Mortensen (2003, p.99) who pointed out that aking counteroffers is not the nor in any labor arkets. The outside option of a worker in sector 1 cannot be the lifetie utility of an eployed worker in sector 2 who chooses to keep her low-skill job, but uneployent in sector 1 1. When a worker and a fir eet and agree to for a atch, the private surplus S 1 = [W 1 U 1 ] + [J 1 V 1 ] of this atch is shared between the worker and the fir according to their bargaining power. Fro equations (1) and (3), we deduce that the (private) surplus of a atch in sub-arket 1, satisfies: (r + )S 1 = y 1 (r + )(U 1 + V 1 ) As the wage w 1 stes fro static Nash bargaining, we have: βs 1 = [W 1 U 1 ] (5) As already entioned, in both sub-arkets job creation results fro the assuption of free-entry (V 1 = 0). We thus have: (r + + βp 1 (θ 1 ))S 1 = y 1 d (6) Consequently, by using (4), the arket tightness θ 1 is deterined by the following equilibriu equation: c + q 1 (θ 1 )(1 β)s 1 = 0 (7) This equilibriu equation is equivalent to the reduced for of the basic atching odel (Pissarides (2000)). Hence, an increase in paraeters c, β, d, r and lowers the arket tightness θ 1, whereas an increase in the output y 1 stiulates job creation in this subarket. 1 Other outside options are nevertheless studied in section

9 2.2 Low-skill jobs Asset values When the training period coes to an end, the output of a worker in a low-skill job rises fro y 2 to ŷ 2 and the worker begins to search (while on the job) for a high-skill vacancy. Her outside opportunities are defined by the lifetie utility of an uneployed worker in sub-arket 1 (utility U 1 ). As Nash bargaining is static, the wage jups fro w 2 to ŵ 2. It eans that we first need to define the asset values associated with a atch between a low-skill job and a trained worker (hereafter referred to as an on-the-job seeker). So let Ŵ 2 be the lifetie utility of such a worker. Using (5), one can show that this asset value satisfies: (r + + p 1 )(Ŵ2 U 1 ) = ŵ 2 + p 1 βs 1 (r + )U 1 (8) Regarding sector 2 firs, the value of a low-skill job when atched with an on-the-job seeker, called Ĵ2, verifies: (r + + p 1 )(Ĵ2 V 2 ) = ŷ 2 ŵ 2 rv 2 (9) where V 2 is the value of a sector 2 vacancy. Fro equations (8) and (9), we derive the (private) surplus of a atch of a sector 2 fir with an on-the-job seeker, Ŝ2. Knowing that Ŝ2 = [Ŵ2 U 1 ]+[Ĵ2 V 2 ], the private surplus Ŝ 2 satisfies: (r + + p 1 )Ŝ2 = ŷ 2 + p 1 βs 1 (r + )U 1 rv 2 (10) Under the assuption of free-entry (V 2 = 0), the substitution of (2) into (10) yields: (r + + p 1 (θ 1 ))Ŝ2 = ŷ 2 d (11) We can now define the asset values associated with a atch between a sector 2 fir and a newcoer. As Nash bargaining iplies that: Ŵ 2 U 1 = βŝ2, we obtain the result that the lifetie utility of an unskilled worker when holding a sector 2 job, W 2, satisfies: (r + + λ)(w 2 U 2 ) = w 2 + λβŝ2 + λu 1 (r + + λ)u 2 (12) where U 2 is the value of uneployent in this sub-arket. We have: (r + )U 2 = d + p 2 (W 2 U 2 ) (13) On the firs side, the value of a job when held by a newcoer verifies: 7

10 rj 2 = y 2 w 2 (J 2 V 2 ) + λ(ĵ2 J 2 ) (14) Under the assuptions of free-entry (V 2 = 0) and Nash bargaining, the latter equation can be rewritten as: (r + + λ)j 2 = y 2 w 2 + λ(1 β)ŝ2 (15) Private surplus and arket tightness The private surplus of an untrained worker atched with a sector 2 fir is such that S 2 = [W 2 U 2 ] + [J 2 V 2 ]. Fro equations (12) and (15), we deduce S 2 as a function of Ŝ 2 : (r + + λ)s 2 = y 2 + λŝ2 + λu 1 (r + + λ)u 2 (16) Finally, by using (2) and (13), one can see that equation (16) can be rewritten as follows: (r + + λ)(r + + βp 2 (θ 2 )) S 2 = y 2 + λŝ2 d + λ r + r + βp 1(θ 1 )S 1 (17) According to equation (17), the tightness of sub-arket 2 is a function of the tightness of sub-arket 1 via the ter βp 1 S 1. Equilibriu in sector 2 thus depends on the equilibriu in sector 1. This results fro the fact that workers asset values in sector 2 depend on workers asset values in sector 1. This one-way interdependence will play a crucial role in the efficiency study. As a result, the assuption of free-entry deterines the arket tightness θ 2 by the following equilibriu equation: c + q 2 (1 β)s 2 = 0 (18) where the cost of keeping a vacancy open, c, is assued to be the sae in both subarkets. 2.3 Educational choices When entering the econoy, a new worker decides on the aount of effort to devote to foral education. Her effort, denoted by e, deterines the probability π of becoing a high-skilled worker. If she succeeds, she enters the pool of applicants for high-skill jobs; if she fails, she ust search for a low-skill job and ust begin a learning-by-doing process after finding one. Reeber that the probability π is an increasing and concave function π(e) of effort e such that π (.) > 0, π (.) < 0. Education effort is then obtained by axiizing the following objective: We obtain the following first order condition: ED e + π(e)u 1 + (1 π(e))u 2 (19) 8

11 π (e)(u 1 U 2 ) 1 = 0 (20) For obvious reasons, the effort e increases with the difference (U 1 U 2 ). Fro the concavity of function π(.) we deduce that the second order condition is satisfied. Using equations (2) and (13), we can rewrite the optiality condition as follows: π (e)β(p 1 S 1 p 2 S 2 ) (r + ) = 0 (21) Education effort is an increasing function of the private surplus S 1, whereas it is a decreasing function of the private surplus S 2. In other words, workers would have an incentive to increase (reduce) their education effort if the gain fro holding a high-skill (low-skill) job increases. 3 Equilibriu 3.1 Definition In su, a labor arket equilibriu can be defined as follows: Definition 1. A labor arket equilibriu is a set of values (S 1, θ 1, Ŝ2, S 2, θ 2, e ) which jointly satisfy equations (6), (7), (11), (17), (18) and (21). Fro arket tightness and probability π, the eployent and uneployent levels in both sub-arkets can be deduced by using the conditions for flow-equilibriu. 3.2 Eployent and uneployent levels In a steady state, eployent and uneployent levels are deduced fro the flowequilibriu conditions. In sub-arket 1, high-skill uneployent u 1 and high-skill eployent l 1 are obtained fro the following equations: π = ( + p 1 )u 1 (22) l 1 = p 1 (u 1 + ˆl 2 ) (23) where ˆl 2 is the nuber of on-the-job seekers (i.e. the level of high-skill eployent in sub-arket 2). In sub-arket 2, low-skill uneployent u 2, low-skill eployent l 2 and high-skill eployent ˆl 2 are derived fro the following conditions: (1 π) = ( + p 2 )u 2 (24) l 2 + λl 2 = p 2 u 2 (25) ( + p 1 )ˆl 2 = λl 2 (26) 9

12 With v i denoting vacant jobs in the labor sub-arket i, the sub-arket tightness of sector 1 is given by θ 1 = v 1 /(u 1 + ˆl 2 ) and the sub-arket tightness of sector 2 is given by θ 2 = v 2 /u 2. Fro these flow-equilibriu conditions, we derive the ipacts of variables θ 1, θ 2 and π on all eployent and uneployent levels. Table 1 reports these partial derivatives. The variable η i (i = 1, 2) denotes the elasticity of rate q i with respect to arket tightness θ i (in absolute value). Table 1: Partial derivatives of eployent and uneployent levels u 2 l 2 θ θ 2 (1 π)q 2(1 η 2 ) 2 (1 π)q 2 (1 η 2 ) (+p 2 ) 2 (+λ)(+p 2 ) 2 π +p 2 p 2 (+λ)(+p 2 ) ˆl 2 u 1 l 1 θ 1 λl 2q 1 (1 η 1 ) (+p 1 ) 2 πq 1(1 η 1 ) (+p 1 ) 2 (u 1 +ˆl 2 )q 1 (1 η 1 ) +p 1 θ 2 π λ +p 1 l 2 θ 2 0 λp 2 (+p 1 )(+λ)(+p 2 ) λp 1 l 2 (+p 1 ) θ 2 p 1 +p 1 +p 1 + p 1 ˆl 2 π The tightness in sub-arket 1 is independent of that in sub-arket 2. However owing to the interactions between the two sub-arkets, high-skill eployent depends on the transition rates in sectors 1 and 2. Therefore high-skill eployent depends on job creation in the low-skill sub-arket. 4 Efficiency We now study the welfare properties of a decentralized equilibriu (Definition 1). As in Gavrel et al. (2010), firs do not internalize the social cost of hiring a high-skilled worker coing fro the low-skill sector. The creation of high-skill jobs thus appears to be too high. Due to this hold-up phenoenon, job creation is suboptial in the low-skill sub-arket. As a consequence, educational choices are inefficient; workers devote too uch effort to foral education. Siilar to Hosios (1990) and Pissarides (2000), let us consider a social planner who is only subject to search frictions and can redistribute incoe at no cost. In this case, the efficiency criterion is the social surplus. For the sake of expositional siplicity, the interest 10

13 rate r is assued to be equal to zero. This assuption allows us to copare steady states according to the social surplus per period. Denoted by Σ, the social surplus per head and per period is given by: Σ = l 1 y 1 + l 2 y 2 + ˆl 2 ŷ 2 + (u 1 + u 2 )d θ 1 (u 1 + ˆl 2 )c θ 2 u 2 c e (27) Notice that in (27) the last ter, e, easures the cost of foral education, per period. In what follows, for ethodological reasons, we will assue that the usual Hosios condition holds true in both sub-arkets, that is: η 1 = η 2 = β The atching functions are therefore Cobb-Douglas. 4.1 High-skill job creation Let us first study the efficiency of high-skill job creation. Using Table 1, one can show that the derivative of the surplus Σ with respect to θ 1 has the sae sign as: HS (1 η 1 )q 1 [y 1 ( )] ˆl2 u 1 + ˆl ŷ 2 + u 1 2 u 1 + ˆl d ( + η 1 p 1 )c (28) 2 Under the Hosios condition and with r = 0, the decentralized equilibriu in sub-arket 1 can be rewritten (see equations (6) and (7)): As ŷ 2 > d, we have: (1 η 1 )q 1 (y 1 d) ( + η 1 p 1 )c = 0 ˆl 2 u 1 + ˆl 2 ŷ 2 + u 1 u 1 + ˆl 2 d > d This iplies HS < 0 in a decentralized equilibriu. Sector 1 firs thus create too any vacancies. This results eans that θ S 1 < θ 1, where θ S 1 denotes the social optiu value of the tightness of sub-arket 1. When the relevant outside option for sector 1 workers (coing fro sector 2) is the asset value of high-skill uneployent, the wage bargaining of on-the-job seekers with sector 1 firs yields an inefficiency that is not solved by the Hosios rule. The intuition behind this inefficiency result is that, with static Nash bargaining, sector 1 firs underestiate the (social) opportunity cost of a atch realized with a worker coing fro sector 2. This cost is given by the output ŷ 2 which is higher than the value of leisure, d. As a consequence, high-skill job creation is the ore inefficient, the larger the share of on-the-job seekers in the pool of applicants for high-skill jobs. In other words, sector 2 firs suffer fro hold-up behavior fro sector 1 firs. This distortion clearly depends on the anner in which one defines the outside option for skilled workers. We coe back to this issue in section 4.4 where we study the consequences of alternative threat points. 11

14 4.2 Low-skill job creation One can show that the derivative of the social surplus Σ with respect to the arket tightness θ 2 has the sae sign as (see A): λ LS (1 η 2 )q 2 ( + p 1 ) [p 1(y 1 d) θ 1 c] [ +(1 η 2 )q 2 y 2 d + λ(ŷ ] 2 d) ( + λ)( + η 2p 2 ) c (29) + p 1 We shall state that LS is equal to zero in a decentralized equilibriu (Definition 1). For r = 0, cobining equilibriu equations (11) (17) and (18) gives: [ λ (1 β)q 2 βp 1S 1 + (1 β)q 2 y 2 d + λ(ŷ ] 2 d) + p 1 Let us consider the first ter of the previous equation: (1 β)q 2 λ βp 1S 1 Fro the equilibriu equation (7), we deduce: βp 1 S 1 = p 1 S 1 θ 1 c ( + λ)( + βp 2) c = 0 (30) With Nash bargaining, part of the surplus goes to the firs. However, under the assuption of free entry, profits are dedicated to the funding of vacancy costs. On the other hand, fro the definition of private surplus S 1, it follows that: We then obtain: βp 1 S 1 = y 1 d S 1 p 1 S 1 θ 1 c = y 1 d S 1 S 1 = y 1 d + θ 1 c + p 1 Consequently, the first ter of the equilibriu equation for tightness θ 2 can be rewritten as: λ (1 β)q 2 βp λ 1S 1 = (1 β)q 2 ( + p 1 ) [p 1(y 1 d) θ 1 c] Under the Hosios condition, this proves that the derivative of the social surplus with respect to the sub-arket tightness θ 2 is zero in a decentralized equilibriu (hence for an inefficient decentralized equilibriu value of θ 1 ). The decentralized equilibriu value of the tightness of sub-arket 2, θ 2, does not coincide with its social optiu value, θ S 2. The value of θ 2 indeed depends on the tightness 12

15 of sub-arket 1, θ 1. We have shown in the previous section that θ 1 takes a value which is higher than its optial value θ S 1. Thus θ 2 differs fro θ S 2 even if LS = 0 in a decentralized equilibriu. We copare those two values in section 4.4. LS = 0 can however be read as a "constrained" efficient result of sub-arket 2 in the sense of Moen and Rosén (2004): job creation is socially optial in sub-arket 2 if the social planner chooses the sae tightness as the one deterined in sub-arket 1 in a decentralized equilibriu, θ 1. At first glance, this (constrained) efficiency result ight look surprising as, via onthe-job search, high-skill eployent depends positively on low-skill job creation (see Table 1). In fact, an increase in high-skill eployent raises the output of sector 1 and it also generates an increase in vacancy costs. What we have shown is that sector 2 firs correctly evaluate and internalize these consequences through the ter βp 1 S 1. It is worth noting that this result is driven by the assuption of free entry which ensures that sector 2 firs correctly internalize the vacancy costs generated by the transition of their trained workers toward high-skill eployent. 4.3 Educational choices One can check that for a nil interest rate, the derivative of the social surplus with respect to the education effort has the sae sign as: E [ ] [ λp 2 p1 (y 1 d) 1 θ ] 1c ( + λ)( + p 2 ) + p 1 + p 1 [ p 2 (y 2 d + λŝ2) + λ ] θ 2 c + λ + p 2 + p 2 π (e) (31) Here also we shall state that E is equal to zero in a decentralized equilibriu. In other words, the education effort appears to be constrained efficient in the sae sense as θ 2 is. In order to verify the previous stateent, let us first consider the quantity βp 1 S 1 (see B for detailed calculus). Fro the definition of the private surplus S 1 (see equation (6)) and fro the equilibriu equation (7), we obtain (for r = 0): p 1 (y 1 d) θ 1 c = βp 1 S 1 (32) + p 1 + p 1 Substitution of equation (32) into equation (31) then yields: E = βp 1 S 1 [ p 2 + λ + p 2 ( y 2 d + λŝ2 + λβp ) 1S 1 + λ ] θ 2 c + p 2 π (e) (33) Let us now consider the quantity βp 2 S 2 (see B for detailed calculus). Fro the definition of the private surplus S 2 (see equation (17)) and by using equation (18), we have (for r = 0): p 2 + p 2 [ y 2 d + λŝ2 + λβp ] 1S 1 + λ θ 2 c = + λ + p 2 βp 2S 2 (34) 13

16 Substitution of (34) into (33) finally yields: E = βp 1 S 1 βp 2 S 2 π (e) This shows that in a decentralized equilibriu, the derivative of the social surplus with respect to the education effort is nil (see equilibriu equation (21)). We can write e = e C. 2 For the sae reason given in the previous section, e does not coincide with the social optiu e S. It is worth noting that the constrained efficiency of educational choices also results fro the condition of free entry. Because profits go to the funding of job creation, this condition akes individuals correctly internalize the social return to education. 4.4 Social optiu and decentralized equilibriu: coparison and discussion We now exaine how the decentralized equilibriu is located relative to the social optiu. Next, we provide a discussion of our welfare results by considering alternative the threat points of trained workers in wage bargaining with high-skill firs Market efficiency A social optiu can be defined as follows: Definition 2. A social optiu is a set of values (θ S 1, θ S 2, e S ) which jointly satisfy HS = LS = E = 0. The following proposition suarizes our (constrained) efficiency results: Proposition 1. A decentralized labor arket equilibriu is constrained efficient in ters of low-skill job creation (θ 2 = θ C 2 ) and education effort (e = e C ), but inefficient in ters of high-skill job creation (θ 1 > θ S 1 ). The decentralized equilibriu does not coincide with a social optiu. It is worth noting that the efficiency of the tightness of sub-arket 2 only holds for the (decentralized) equilibriu value of the arket tightness θ 1. One can verify that in the absence of on-the-job search, a decentralized equilibriu would coincide with a social optiu (under the Hosios condition). The constrained efficiency results enable us to explain why the decentralized equilibriu (Definition 1) is not a social optiu (Definition 2). Let us underline how the decentralized equilibriu is located relative to the social optiu. We already know that job creation is beyond its optial level in sub-arket 1 (θ S 1 < θ 1): arket tightness θ 1 is too high. What can be said about low-skill job creation and the education effort of newcoers? Under the Hosios condition (η 1 = η 2 = β), we state the following proposition: 2 Notice that the Hosios condition was not used in stating this point. Furtherore this result reains true whatever the workers bargaining strength β is. 14

17 Proposition 2. Relative to a social optiu, low-skill job creation (θ2) is too low in a decentralized equilibriu (θ2 S > θ2), and individuals education effort (e ) is too high (e S < e ). Proof: See C. As already shown, low-skill job creation is not optial as θ1 is too high copared to its social optiu value θ1 S. Since a significant level of high-skill job creation facilitates transitions of trained workers toward high-skill jobs, sector 2 firs create less jobs than required at the social optiu: θ2(= θ2 C ) < θs S. Siilarly, a significant nuber of sector 1 vacancies encourages newcoers to display a significant education effort (which is higher than the social optial level) so that they apply to high-skill jobs directly. We thus have e (= e C ) > e S Discussion In order to reach a ore precise understanding of our welfare analysis, let us consider a (very) hypothetical arket structure in which the decentralized equilibriu would be efficient. First, siilar to Albrecht et al. (2006) as well as Gautier and Wolthoff (2009), suppose that, in the bargaining with a high-skill fir, the threat point of trained workers, Û 1, is directly deduced fro their productivity on a low-skill job. rû1 = ŷ 2 Û1 Expressed in words, when a trained worker initially atched with a low-skill fir encounters a high-skill fir, the high-skill fir (Bertrand) copetes for the sae worker with the low-skill fir. Consequently, the high-skill fir will win and pay a wage beyond the productivity level of the worker at the low-skill fir. This clearly requires that sector 1 firs can coit to wages. In this case, the private surplus of a atch with a trained worker in sector 1 will obviously reflect its social costs and benefits. Job creation will be efficient in sector 1. Is this sufficient to ensure the efficiency of a decentralized equilibriu? Consideration of the efficiency of job creation in sub-arket 2 and of educational choices shows that this is not the case. The reason for this is that two different levels of wages would prevail in sector 1: high wages for trained on-the-job seekers having a threat point equal to Û1, and lower wages for high-skill uneployed having a threat point based on U 1. The profitability of high-skill firs, and high-skill job creation, would therefore depend on the probability of recruiting one type of worker or the other. Since low-skill firs do not integrate the existence of lower-wage sector 1 workers, they underestiate the profitability of high-skill firs. This distortion iplies that the gain a trained worker expects fro a atch with a high-skill fir no longer reflects the vacancy costs that her transition to sector 1 generates. As a result, low-skill job creation is too high copared with its social optial value 3. 3 See additional inforation which can be found in an unpublished Appendix available fro the authors. 15

18 As shown in D, with this threat point, arket efficiency also requires the segentation of the high-skill sub-arket according to the applicants type (educated or trained). In the high-skill graduate sector, firs only hire workers who have successfully copleted their foral education; whereas in the trainees high-skill sector, firs only recruit trained workers coing fro sub-arket 2. Siilar to Sattinger (2006), segenting sub-arket 1 eliinates this distortion, since each segent of the sub-arket is coposed of workers with identical wages 4. As a result low-skill firs correctly evaluate the social costs and benefits of the departure of their trained workers. Segenting the high-skill sector restores efficiency, but this arket structure is not rational. As Sattinger would put it, the highskill sub-arket is "irreducible". On the other hand, in confority with a scenario soeties used in the literature (Cahuc et al. (2006)), the workers threat could be deduced fro the lifetie utility of a low-skilled worker (Ŵ2). Under this assuption, high-skill job creation would no longer be efficient since the wage of trained sector 2 workers is lower than their productivity. Highskill firs would not internalize the entire loss of low-skill firs whose eployees quit. They would therefore create too any vacancies copared to what is socially efficient. This scenario is presented in the unpublished Appendix. Our discussion led us to consider different threat points of trained workers hired in sector 1. Assuing that a trained worker could bargain over her wage according to the low-skill fir productivity (ŷ 2 ) or even according to her lifetie utility while eployed in a low-skill job (Ŵ2), she would obtain a higher wage than that of a worker who successfully copleted her foral education. Beyond the arguent presented by Shier (2006) and Mortensen (2003), this result allows us to justify our choice in ters of a unique threat point equal to U 1 for all workers in the high-skill sector. It is indeed not reasonable to suppose that workers who failed in their foral education would obtain higher wages than educated workers. 5 Optial public policy The laissez-faire situation is not an optiu. What then should a governent do? We now present a self-financed Taxes and Subsidies Policy (TSP) leading to a social optiu. The sae assuptions as above have been adopted. The interest rate is equal to zero and the Hosios condition holds on both sub-arkets. Which public policy shall be ipleented? Fro the previous analysis of inefficiency sources, we could think that the decentralized equilibriu would be efficient if firs in sub-arket 1 could offer a direct transfer to firs in sub-arket 2 in order to copensate for the loss high-skill firs ipose on low-skill firs when their trained eployees leave for sub-arket 1. This relationship between firs, which would have to be iposed by the social planner, does not however restore arket efficiency (the proof is available in point. 4 Note that this distortion does not exist in our odel as all high-skilled workers have the sae threat 16

19 the unpublished Appendix available fro the authors). Such a transfer policy raises a proble siilar to the one we identified when the trained workers threat point is deduced fro their productivity on a low-skill job. Job creation becoes excessive in sector 2. As a direct transfer between firs cannot restore labor arket efficiency, the next section focuses on a ore coplex public policy. 5.1 Taxing sector 1 As previously highlighted, job creation is too high in sub-arket 1. The governent can decentralize the social optiu by ipleenting an appropriate fiscal policy. We now prove that in order to restore efficiency, a tax τ could be levied in sub-arket 1. Thus, the value of a filled job J 1 (see equation (3)) now depends on τ: (r + )(J 1 V 1 ) = y 1 τ w 1 (r + )V 1 (35) By coparison between (7) and the optial condition (28), we obtain the result that the tax would restore sub-arket efficiency if it is equal to: τ = ˆl 2 u 1 + ˆl 2 ŷ 2 + u 1 u 1 + ˆl 2 d d = ˆl 2 ˆl 2 + u 1 (ŷ 2 d) > 0 (36) Let α be the share, in sub-arket 1 eployent, of workers coing fro sub-arket 2: The tax can therefore be written as: α = ˆl 2 ˆl 2 + u 1 τ = α(ŷ 2 d) and S 1 is now given by: S 1 = y 1 (αŷ 2 + (1 α)d) + βp 1 Therefore, sector-1 equilibriu (7) becoes: 0 = c + q 1 (1 β) y 1 (αŷ 2 + (1 α)d) + βp 1 (37) Equation (37) coincides with the optiality condition in sector 1 (28). With a tax τ, high-skill job creation becoes efficient. In short the Pigovian tax τ akes sector 1 firs internalize the real cost of hiring a worker coing fro sector 2. However, ipleenting this tax does not only restore efficiency in sector 1, it also odifies efficiency results for sector 2 and for educational choices: job creation in sector 2 is no longer efficient and the sae holds for the education effort. 17

20 These distortions lead to the tax τ being dedicated to the funding of two copensatory transfers. The first, denoted by σ q is allocated to sector 2 firs when a worker quits her low-skill job. The transfer σ q is given by: σ q = τ (38) The second transfer, denoted by σ e, is allocated to (entrant) workers whose education effort e is successful. The transfer σ e is given by: σ e = p 1 + p 1 σ q (39) Before showing that these transfers offset the distortions created by the tax τ, we need to verify that the policy is self-financing. As there are π workers whose effort e is successful and p 1ˆl2 quit, the governent s expenditures are equal to: πσ e + p 1ˆl2 σ q = Fro equations (22), (23), and (38), we deduce that: ( ) π + + p ˆl 2 p 1 σ q 1 ( ) π + + p ˆl 2 p 1 σ q = (u 1 + ˆl 2 )p 1 σ q = l 1 σ q = l 1 τ 1 As the governent s receipts are given by (l 1 τ) per period, this shows that the governent s balanced budget constraint is satisfied for this self-financed TSP. 5.2 Subsidizing Sector 2 By restoring efficiency in sector 1 one has reduced job creation in sector 2 in excess of the efficiency level. The reduction of the value of a sector 1 job does reduce the future opportunities of sector 2 eployees. It thus decreases the value associated with their low-skill job. As a consequence, the private surplus associated with a filled sector 2 job is reduced. In order to restore efficiency in the overall labor arket, we propose to subsidize sector 2 firs whose workers leave for sector 1. With the copensatory transfer σ q (see equation (38)), the private surplus S 2 now satisfies (for r = 0): ( + λ)( + βp 2 ) S 2 = y 2 d + λ ŷ2 d + λ p 1 σ q + λ + p 1 + p 1 βp 1S 1 On the other hand, using equation (37) one can see that, with the tax, the quantity (βp 1 S 1 ) is now given by: Cobining the two previous equations yields: βp 1 S 1 = p 1(y 1 d) θ 1 c + p 1 p 1 + p 1 σ q (40) ( + λ)( + βp 2 ) S 2 = y 2 d + λ ŷ2 d + λ p 1 (y 1 d) θ 1 c + p 1 + p 1 18

21 Substituting S 2 into equation (18) shows that transfer σ q enables the restoration of the efficiency of low-skill vacancy creation (see equation (29) for β = η 2 ). 5.3 Rewarding educational success With the reward σ e defined by equation (39), the private optiality condition (20) has to be rewritten as follows (for r = 0): or ( π (e) U 1 + p ) 1 σ q U p 1 ( π (e) βp 1 S 1 + p ) 1 σ q βp 2 S 2 = + p 1 Using equation (40), the previous equation can be rewritten as: ( p1 + p 1 (y 1 d) or ( p1 + p 1 (y 1 d) θ 1 c p 1 σ q + p ) 1 σ q βp 2 S 2 + p 1 + p 1 + p 1 ) θ 1 c βp 2 S 2 + p 1 π (e) = 0 As the efficiency of θ 2 is restored (despite the tax), βp 2 S 2 reains equal to: [ ( p 2 y 2 d + λŝ2 + λβp ) 1S 1 + λ ] θ 2 c + λ + p 2 + p 2 Therefore the social optiality condition (33) holds true. π (e) = 0 At first glance, the idea of rewarding educational success ight look counterintuitive, since one could point out that, according to Proposition 2, education effort is lower in a social optiu than in a decentralized equilibriu. The reason for this is that without subsidies, private educational choices are no longer efficient for the optiu value of subarket tightness θ1 S (coputed with the tax τ). In the absence of a reward, the return to education (the opportunity to get a "good" job) would be too weak, thus leading to a reduction in foral education below its optial level. The reward copensates for this effect. The following proposition suarizes the above results: Proposition 3. With the TSP (τ; σ e ; σ q ) the decentralized equilibriu is a social optiu (θ 1(T SP ) = θs 1, θ 2(T SP ) = θs 2, e (T SP ) = es ). 19

22 6 Calibration In this section we verify that the previous TSP restores arket efficiency through a calibration of the odel using U.S. labor arket evidence. We therefore assue that the Hosios condition holds and that the discount rate is nil (r = 0). We consider that the low-skill group is constituted of individuals whose educational attainent is a high-school diploa with no college attendance, or less. The high-skill group is therefore coposed of individuals with soe college or associate degree, plus all college graduates. 6.1 Baseline scenario The uneployent rates presented in the theoretical odel have to be understood as the uneployent of young workers only, since the odel does not copute any job separation rate. For this reason we atch the uneployent rates of the calibration with that of educated and non-educated young workers aged 16 to 24 years old for 2006, given by the U.S. Bureau of Labor Statistics (BLS). The rates are coputed as the youth uneployed population divided by the total labor force 5. We obtain fro the sae source a share of educated workers in the total labor force π of about 42%. The probability of leaving peranently the labor arket (dying) is fixed at = which represents about 42.7 years in the labor force (Ljungqvist and Sargent 1998). Equations (22) and (24) can then be used to retrieve the average onthly job finding rates p 1 and p 2. Table 2 gives the fixed values of variables and paraeters and table 3 gives the inferred values. Table 2: Paraeters and variables whose value is fixed Nae Description Value Source β worker s bargaining power 0.5 Pissarides (2000) η 1 elasticity sub-arket Pissarides (2000) η 2 elasticity sub-arket Pissarides (2000), Hosios (1990) peranent exist rate Ljungqvist and Sargent (1998) M 1 atching paraeter Hagedorn et al. (2010) M 2 atching paraeter Hagedorn et al. (2010) y 1 high-skill productivity 100 noralized y 2 low-skill prod. (untrained) Hagedorn et al. (2010) d non-arket output Hall and Milgro (2008), see text π share of educated in LF BLS, year 2006 u 1 high-skill youth uneployent BLS, year 2006 u 2 low-skill youth uneployent BLS, year 2006 AHL low/high-skill transition share 0.20 Andersson et al. (2005) 5 That is to say - educated (for u 1 ) or non-educated (for u 2 ) - youth uneployent rates ties the share of the youth labor force in overall labor force. 20

23 As a standard feature, bargaining power β and the elasticities η 1 and η 2 are assued to be equal to 0.5 (the Hosios condition holds). This choice of calibration will allow us to observe the ipact on the labor arket structure of a TSP as proposed in the theoretical part of this work. High-skill productivity y 1 is noralized to 100. We assue the atching functions to be Cobb-Douglas with atching paraeters M 1 and M 2, so that h 1 = M 1 v η 1 1 (u 1 + ˆl 2 ) 1 η 1 and h 2 = M 2 v η 2 2 u 1 η 2 2 are the atching function of respectively the high-skill sub-arket and the low-skill sub-arket. The atching paraeters values are set according to Hagedorn et al. (2010) who present the calibration of a search-atching odel with the sae two subgroups (high-skill and low-skill individuals) using data fro the CPS between 1979 and We can then deduce the tightness of each sub-arket θ 1 and θ 2 as well and the probability that a fir fills a job q 1 and q 2. Hagedorn et al. (2010) give a arginal product ratio of between skilled and unskilled workers. We use this ratio to retrieve the value for y 2. According to Hall and Milgro (2008) non-arket activities, which includes the value of leisure, represent a fraction 0.71 of the average productivity. In order to iniize the presence of potential uneployent benefits, which are not taken into account in our odeling, we chose to fix the value of d as a proportion of y 2. Table 3: Paraeters and variables whose value is inferred Nae Description Value Source ŷ 2 low-skill prod. (trained) 85 see text p 1 high-skill job transition rate equation (22) p 2 low-skill job transition rate equation (24) θ 1 tightness sector atching function definition θ 2 tightness sector atching function definition ˆl 1 low-skilled leaving for a high-skill job see text l 2 low-skill eploy. (untrained) equations (23), (25), (26), (41) ˆl 2 low-skill eploy. (trained) equations (23), (25), (26), (41) l 1 high-skill eployent equations (23), (25), (26), (41) λ training transition rate equations (23), (25), (26), (41) c 1 vacancy cost sub-arket E+2 equation (7) c 2 vacancy cost sub-arket E+2 equation (18) π (e) arginal success rate of education equation (21) e education effort E+4 see text The probability λ of upgrading the skills of low-skilled workers is obtained knowing that the share of overall low-skilled workers in the labor force ust be equal to the following equation: (1 π) = u 2 + l 2 + ˆl 2 + ˆl 1 (41) 21

24 Where ˆl 1 is the share of low-skilled workers who succeed in obtaining a high-skill job. According to Andersson et al. (2005) between 15% and 20% of workers with a low level of education had escaped low wage eployent after nine years. We use this proportion to estiate ˆl 1 = AHL (1 π) where AHL = 0.2 is the proportion of low-skilled workers escaping fro low-skill jobs. We then deduce the training transition rate λ and the eployent levels l 2, ˆl 2, and l 1 fro equations (23), (25), (26), and (41). Regarding the education technology, we retain the specification π(e) = e φ. Knowing e the baseline value for π and π (e) enables us to retrieve the paraeter φ and the value of education effort e. The reaining paraeters are the cost of a vacancy, and the trained workers productivity ŷ 2. In order to ensure an equilibriu in each sub-arket, we assue fro now on the existence of two different vacancy costs depending on the sub-arket, so that c i is the cost of vacancy in sector i. Finally, we arbitrarily fix a value for productivity ŷ 2 of 0.85 ties the high-skill output (sensitivity test could be perfored on the basis of this paraeter but such a test goes beyond the scope of this work). This choice ensures a progressiveness in the workers productivity (y 2 < ŷ 2 < y 1 ). A trained worker fro sub-arket 2 will indeed search on-the-job for a high-skill job rather than choosing highskill uneployent (this iplies ŷ 2 < d), and low-skill wages will reain below high-skill wages even for trained workers (this iplies ŷ 2 < y 1 ). Using the preceding values in equations (7) and (18) gives the values for c i. 6.2 The Tax and Subsidy Policy (TSP) The TSP requires the iposition of a tax τ on the output of high-skill firs, a subsidy σ q to low-skill firs whose workers quit for a high-skill job, and a reward σ e for educational success. Table 4 synthesizes the values of variables at the baseline and after the TSP has been ipleented. As expected, the decentralized equilibriu is not a social optiu since HS < 0. The tax iposed on high-skill firs that is required to restore a social optiu creates a distortion which is restored by the ipleentation of a subsidy to low-skill firs, whose workers quit for high-skill jobs, in addition with a reward for educational success. The calibration of the policy that would restore the social optiu gives us the opportunity of studying the consequences of such a policy for different variables in the econoy. We can appraise the iportance of the tax and the subsidies required to restore the efficiency based on US epirical evidence. The tax that should be ipleented represents about 10.9% of high-skill firs output, and provokes a reduction in the surplus (S 1 ) of about 9.1%. The reason why such a tax is necessary to lower the over-creation of high-skill vacancies and to restore efficiency in sub-arket 1 lies in the strong differential between d (35.77) and ŷ 2 (85) in our baseline scenario. Since the product of the tax is split aong a sall nuber of agents, the subsidies distributed are iportant. This creates a strong increase in the surplus of low-skill firs whose eployees are trained (Ŝ2). The increase of this surplus is alost copensated by the reduction of the high-skill surplus (sall rise of S 2 ), hence job creation rises only in 22