Cyclical Upgrading of Labor and Unemployment Differences across Skill Groups

Transcription

1 Cyclical Upgrading of Labor and Unemploymen Differences across Skill Groups Andri Chassamboulli Ocober 1, 2007 Absrac This paper examines he labor marke cyclical dynamics in a search and maching model which allows for job finding raes o vary endogenously across skill groups. In he economy I examine skilled individuals can perform boh skilled and unskilled jobs, whereas unskilled individuals can only perform unskilled jobs. The possibiliy of onhe-job search induces skilled workers o ake ransiorily jobs below heir skill level, hereby influencing he employmen prospecs of lower skill groups. As he relaive profiabiliy of skilled and unskilled jobs changes over he business cycle, firms respond accordingly by adjusing he skill mix of vacancies, hereby influencing he chances skilled and unskilled workers find jobs unevenly. The model highlighs he imporance of a verical ype of skill mismach ha akes he form of workers downgrading o lower job levels o escape unemploymen and upgrading by on he job search, in explaining why ypically skilled unemploymen lower and less responsive o business cycles. A he same ime, he model is consisen wih well esablished evidence ha he qualiy of job-worker maches and job-o-job ransiion raes are procyclical. Deparmen of Economics, Universiy of Cyprus ( andricha@ucy.ac.cy). I am grealy indebed o John Haliwanger, Jeff Smih, John Shea, Michael Pries for he invaluable suppor and advice and Chrisopher Pissarides for exensive commens on an earlier version of he paper. 1

2 1 Inroducion Average unemploymen raes hide dramaic differences across skill groups. As shown in Figure 1, he unemploymen rae falls subsanially wih educaion. Moreover, i is obvious o he naked eye ha he lower he level of educaion, he higher he rise in he unemploymen rae in downurns. The quesion ha follows is why ypically he burden of unemploymen falls more heavily on he lower educaed. Several empirical sudies parly explain his fac by showing ha while he enry rae ino unemploymen is lower, he exi rae from unemploymen is higher a higher educaional aainmen. 1 Wih regard o cyclical changes in unemploymen raes, Hall (2005) and Shimer (2005a) shif he aenion in job finding raes. Boh argue ha while job separaion raes are almos acyclical, job finding raes are highly procyclical, suggesing ha differences in he cyclical behavior of unemploymen raes across skill groups are mainly due o differences in he cyclical behavior of job finding raes. There are good heoreical grounds as o why enry ino unemploymen declines wih educaion. Skilled workers are more likely o accumulae firm specific capial, making i more difficul for firms o make hem redundan. 2 However, hiring skilled individuals is ypically more cosly, herefore i is less apparen why should he job finding raes of skilled individuals be relaively high and less responsive o business cycles. One argumen is ha a he prevailing wages, he relaive demand for skilled as opposed o unskilled workers remains sufficienly large. Bu as also emphasized in Nickell (1979), i may be he case ha skilled workers manage o keep heir employmen raes relaively high, by aking jobs below heir skill level while searching for more suiable employmen insead of searching while unemployed. 3 Naurally, skilled workers qualify for a wider range of job 1 For evidence ha joblessness falls more heavily on he lower skill groups, see e.g., Topel (1993), van Ours and Ridder (1995), Manacorda and Perongolo (1999), and Juhn e al. (2002). For evidence ha job finding raes rise wih educaion, see e.g., Topel (1984), Beach and Kaliski (1987), and Perongolo (2001), and for evidence ha separaion raes decrease wih educaion, see e.g., Nickell (1979), Royaly (1998), Polsky (1999), Fallick and Fleischman (2001), and Nagypal (2004). 2 The heoreical relaion beween urnover and firm specific capial is emphasized in Jovanovic (1979). 3 There is a variey of reasons skilled workers may be hired for low-skill posiions. These include imperfecions in he maching process and produciviy gains. Similarly, here are several reasons skilled workers may accep ransiorily low-skill job offers. As Blau and Robins (1990), and Belzil (1996) show, on-he-job search is more efficien han unemployed search. The former also find ha high unemploymen and fewer job offers in recessions induce searchers o rejec fewer job offers. Moreover, sigma effecs associaed wih long unemploymen spells may induce workers o be less selecive in maching; see for insance, Vishwanah (1989). 2

3 ypes, hus are relaively more capable of finding ransiory employmen as opposed o remaining unemployed unil a suiable job offer comes along. In his paper I develop a search and maching model o invesigae he imporance of on-he-job search a he higher end of he skill disribuion, in explaining he observed differences in he cyclical responsiveness of unemploymen raes across skill groups. In he economy I examine firms open vacancies for eiher high-produciviy jobs, which have high skill requiremens, or low-produciviy jobs, wih lower skill requiremens. High-skill workers are bes suied for high-skill jobs, bu hey also qualify for low-skill jobs, whereas, low-skill workers qualify only for low-skill jobs. To capure maching imperfecions, I assume ha workers canno ex-ane idenify he ypes of vacancies, hus canno arge only he jobs hey are suied for. Consequenly, he job finding rae of low-skill workers is proporional o he share of low-skill vacancies in he oal number of vacancies posed. However, he skill mix of vacancies does no influence high-skill workers job finding raes. This is because he possibiliy of on-he-job search induces hem o accep he low-skill jobs hey encouner, and coninue searching while employed for high-skill jobs. Cyclical changes in he skill composiion of new employmen opporuniies affec he job finding raes of he wo ypes of workers unevenly, causing he skill composiion of job seekers o change. In urn, cyclical changes in he composiion of job seekers affec he effecive maching raes of firms wih low- and high-skill vacancies unevenly, causing he skill composiion of he vacancies opened o change, resuling in a circle of ineracions. In his framework, when high-skill workers accep ransiorily low-skill jobs, hey influence he profis of low-skill jobs in wo counervailing ways. On he one hand, since hey are likely o abandon low-skill jobs sooner, by crowding ou low-skill workers from low-skill posiions, hey lower he profis of low-skill jobs, and discourage firms from opening low-skill vacancies. On he oher hand, when high-skill workers join he queues for low-skill jobs, firms wih low-skill vacancies can fill heir vacancies faser, hus face lower he recruimen coss. This in urn encourages firms o open more low-skill vacancies. Wheher he chances low-skill workers find jobs improve or worsen when high-skill workers accep ransiorily low-skill jobs, depends on which of he wo effecs dominaes. The cyclical paern in he maching behavior of skilled workers his paper refers o, which akes he form of downgrading o lower job levels o escape unemploymen and upgrading by on-he-job search, is suppored by several empirical observaions. Evidence by Bowlus (1995) and Davis e al. (1996), among ohers, ha jobs creaed in recessions are of lower duraion and offer lower wages han jobs creaed in booms, sugges ha he qualiy 3

4 of job-worker maches falls in periods of high unemploymen. Moreover, he imporance and he procyclical behavior of job-o-job ransiion raes has been emphasized in a number of empirical sudies, bu recenly, Nagypal (2004) documens in addiion ha boh he share of separaions accouned by job-o-job flows, and he share of quis ha lead o a direc ransiion ino a new job rises wih educaion. 4 The finding of Pissarides and Wadsworh (1994) ha he propensiy o search on he job is higher among he more educaed also suppors he view ha more educaed individuals are prone o search while employed as opposed o unemployed. Finally, here is direc evidence of over-educaion phenomena a high levels of educaional aainmen. As documened in Hecker (1992, 1995), since he early 1980 s, beween 17 and 18 percen of college graduaes in he U.S. were employed in jobs ha do no require a degree. 5 I calibrae he model o he U.S. labor marke, assuming ha he high-skill ype refers o college graduaes and he low-skill ype o individuals wih less han college educaion. The calibraion accouns for he much lower separaion rae of college graduaes and much higher produciviy gains associaed wih hiring college graduaes, as refleced in he observed large wage premium for college graduaes. Sill, he simulaions reveal ha he so much lower unemploymen rae of college graduaes can be explained only if a subsanial fracion of hem is being underemployed in jobs ha require less han college educaion. The calibraion yields ha on average, 17.8% of college graduaes are over-educaed, well in line wih he empirical graduae over-educaion measures repored above. I also invesigae he possibiliy ha eliminaing maching imperfecions would yield lower unemploymen raes for college graduaes, simply because he marke for college graduaes is igher due o larger produciviy gains associaed wih skilled posiions. I simulae he model assuming ha workers can arge only he jobs hey are bes suied for (i.e., search is direced). However, o mach he observed unemploymen differences, he model wih direced search requires unrealisically high wage premiums for college graduaes. Unless he naure of maching or wage seing differs significanly across he wo secors, his also suggess ha employmen a higher educaional aainmen remains high, due o emporary over-educaion and upgrading by on-he-job search. 4 Similarly, Polsky (1999) shows ha he negaive correlaion beween educaional aainmen and separaions becomes less apparen when accouning also for quis insead of only layoffs. 5 Graduae over-educaion measures of he same range can also be found for many European counries. For insance, Green e al. (1999) find ha jus over 20% of graduaes in he UK are genuinely over-educaed for heir jobs, and Oliver and Raymond (2003) show ha in 1998 he proporion of over-educaed college graduaes in Spain was 21%. 4

5 In addiion, wih 17.8% on average of college graduaes being over-educaed, he cyclical behavior of unemploymen raes in he model maches quie well he observed differences in he cyclical behavior of unemploymen raes beween college graduaes and individuals wih less han college educaion. In he model he burden of recessions falls more heavily on he lower educaed for wo reasons. Firs, firms respond o a fall in aggregae produciviy by shifing he skill mix of vacancies owards he more producive ype, which is high-skill in he model. Consequenly, boh ypes of workers suffer reducions in heir chances of finding jobs, as firms open fewer vacancies per job seeker, bu low-skill workers are hur he mos. Second, a higher number of high-skill unemployed resors o emporary employmen in low-skill jobs in recessions, insead of remaining unemployed, while upgrades o high-skill jobs happen more frequenly in booms, when job finding raes rise. Afer esablishing ha ransiory over-educaion is crucial in explaining why skilled unemploymen remains relaive low a all saes of he business cycle, he quesion I ask is how his behavior affecs he chances low-skill workers find jobs, and he economy overall. I find ha i acually improves he chances lower educaed workers find jobs. In he calibraed model, he negaive crowding ou effec ha lowers he average qualiy of low-skill jobs is small relaive o he posiive impac of lower recruimen coss for firms wih low-skill vacancies. Moreover, when high-skill workers accep low-skill jobs, he pool of poenial hires is higher a boh segmens of he labor marke. No only firms wih low-skill vacancies benefi from high-skill unemployed joining he queues for low-skill jobs, bu also firms wih high-skill vacancies benefi from he presence of on-he-job searchers. The resuling higher recruimen aciviy a boh segmens of he labor marke, mainains a higher incenive for firms o open vacancies in boh secors. Hence, boh high- and low-skill employmen is higher when high-skill workers accep ransiorily low-skill jobs. In addiion, high-skill employmen is higher no only because of he on-he-job searchers, bu also because he number of suiably mached high-skill workers is higher. Finally, I find ha boh secors exhibi more cyclical employmen growh when high-skill workers accep ransiorily lowskill jobs, bu especially he high-skill secor. This is because wih on-he-job searchers, poenial hires for firms wih high-skill vacancies capure a higher share in he pool of job seekers in booms, when here are fewer unemployed job seekers, and vice versa in recessions. Alhough his paper is no he firs o accoun for he asymmeric naure of maching, i is he firs o explore he cyclical dynamics of he labor marke in a model ha allows for job finding raes o vary endogenously across skill groups. Albrech and Vroman (2002), Gauier (2002), and Dolado e. al. (2004), also assume ha skilled workers can perform 5

6 boh skilled and unskilled jobs, whereas unskilled workers can perform only unskilled jobs. By incorporaing asymmeric maching, hese sudies link over-educaion phenomena o skill-biased echnological shocks and he crowding ou of workers a he lower segmen of he labor marke. Their focus, however, is o explain long-run uneven developmens in he unemploymen raes of differen skill groups. Therefore, by focusing only on seady saes, lile aenion has been paid o he cyclical implicaions of ransiory over-educaion and upgrading by on-he-job search. By incorporaing job heerogeneiy and on-he-job he cyclical upgrading of labor has been emphasized in Barlevy (2002) and more recenly in Krause and Lubik (2006). The view formalized in Barlevy is ha recessions impede he reallocaion of workers from less producive o more producive jobs, because firms pos fewer vacancies per job seeker. Similarly, Krause and Lubik highligh he role of on-he-job search in explaining cyclical changes in he composiion of new employmen opporuniies, and observed worker mobiliy paerns. The model in Krause and Lubik does no feaure worker heerogeneiy, while Barlevy s model accouns for worker heerogeneiy, bu as he auhor argues, in order o make he role of aggregae shocks more ransparen, employs a symmeric framework, in which maching raes are equal across skill caegories. Hence, exising models offer a characerizaion of he cyclical behavior of worker flows only in erms of average or represenaive values and overlook he observed salien differences across skill groups. More imporanly, hey overlook he across-skill exernaliies ha arise, when workers of differen skill compee for he same ypes of jobs. The res of he paper is organized as follows. Secion 2 oulines he dynamic model in which aggregae produciviy flucuaes over ime. Secion 3 defines a seady sae equilibrium and uses analyic resuls o provide a more rigorous inuiion for he resuls of he dynamic model ha follow. In secion 4, I calibrae and numerically solve he dynamic model oulined in secion 2, and discuss is qualiaive and quaniaive implicaions. Secion 4 also sudies he consequences of ransiory over-educaion on low-skill employabiliy and is implicaions for he cyclical dynamics of he labor marke overall. In secion 5, I examine he properies of he model in which search is direced. Finally, secion 6 concludes. 2 The Model 2.1 The Labor Marke The labor force is composed by wo ypes of risk neural workers: a fracion δ is low-skill (l) and he remaining (1 δ) is high-skill (h). Similarly, vacancies can be eiher high-skill 6

7 (h) or low-skill (l), bu he mix is deermined endogenously. High-skill workers can perform boh ypes of jobs, whereas low-skill workers can only perform low-skill jobs. Accordingly, a low-skill worker can be eiher employed and producing in a low-skill job or unemployed and searching, while a high-skill worker can be in any if he following hree saes: employed and producing in a high-skill job, unemployed and searching, and employed and producing in a low-skill job, bu simulaneously searching for a high-skill job. I label a worker in he laer sae as over-qualified job seeker. Each firm has a mos one job, which can be eiher vacan and searching for candidaes or filled and producing. The mass of each ype of vacancy is deermined endogenously by a free-enry condiion. The exogenous componen of job desrucion follows a Poisson process wih arrival rae s i, where i = (h, l), and is assumed o be specific o he ype of worker. Whenever a mach is desroyed he job becomes vacan and bears a mainenance cos c l, specific o is ype. Wages are chosen o divide he surplus of he mach beween he worker and he firm in fixed proporions, in line wih Nash bargaining. Wih γ being he workers bargaining power, a share γ of he surplus goes o he worker, while he res 1 γ goes o he firm. When unemployed he worker enjoys a produciviy flow b i, which can be inerpreed as he opporuniy cos of working Mach Produciviies The produciviy of each job-worker mach is assumed o be he produc of a sochasic aggregae componen y, and a mach specific componen α ij, when a worker of ype i = (h, l) is mached wih a job of ype j = (h, l). The aggregae componen is assumed o follow a discree-sae Markov process. The vecor of possible aggregae produciviy realizaions is given by ȳ and he elemens of he ransiion marix Π are given by π nm = prob{y +1 = ȳ m \ y = ȳ n }. The condiion ha ensures a mach is formed in equilibrium is simply ha he produciviy of he mach is higher han he worker s opporuniy cos of working, i.e., yα ij > b i. This condiion ensures ha he surplus of he job is posiive, and he wage i offers is higher han he opporuniy cos of working. Since low-skill workers do no have he minimum required skills o perform high-skill jobs, he underlying assumpion is ha 6 Since here is no governmen or any form of axaion in he model, I avoid naming b i as unemploymen benefi. In realiy, he unemploymen benefi is only one of he facors ha deermine b i. A variey of addiional facors could influence a worker s opporuniy cos of working, including he value aribued o leisure, spousal income, and he value of home producion. 7

8 yα lh b l 0, which implies ha when filled by a low-skill worker, high-skill jobs generae losses insead of a surplus. I assume ha high-skill workers are bes suied for high-skill jobs and are herefore more producive when mached wih high- insead of low-skill jobs. This implies ha yα hh b h > yα hl b h and ensures ha over-qualified workers receive lower wages han suiably mached workers, because he former generae lower surplus. Hence, under his assumpion, over-qualified workers have a naural incenive o search on he job for highskill jobs. Moreover, I assume for convenience ha he rae a which workers mee high-skill vacancies is he same regardless of wheher he worker is employed or no. Consequenly, as long as yα hl > b h, so ha he surplus of a low-skill job filled by an over-qualified worker is posiive, i is opimal for unemployed high-skill workers o accep ransiorily low-skill jobs, since hey reain heir chances of finding a high-skill job by coninuing heir search while employed. Finally, I assume ha he ne produciviy of a suiably mached low-skill worker is a leas as high as he ne produciviy of an over-qualified worker. Tha is, yα ll b l yα hl b h. This is o ensure wihou any addiional resricions in he parameer space ha firms wih low-skill vacancies are beer off hiring low-skill insead of over-qualified workers. Since he laer are more likely o qui, low-skill posiions always generae higher profis when filled by suiable insead of over-qualified workers. Noe ha his assumpion does no necessarily imply ha over-qualified workers are less producive han suiably mached low-skill workers. Even if α hl > α ll, his condiion is sill saisfied when high-skill workers opporuniy cos of working is sufficienly higher han ha of low-skill workers Maching and Timing Firms and workers mee each oher via a maching echnology m(ν, z ), where ν = ν h + ν l is he number of high- and low-skill vacancies, and z = u h + u l + e hl (1 s h ) is he number of job seekers; u h and u l denoe he number of high- and low-skill unemployed, and e hl (1 s h ) he number of over-qualified workers who survive separaion, and hus coninue searching on he job. The funcion m (, ) is sricly increasing in is argumens, and exhibis consan reurns o scale. workers mee vacancies as m(θ ), where θ = This allows me o wrie he flow rae a which ν h +νl u h +ul +ehl (1 s h) capures he degree of labor 7 In secion 4 where I calibrae he model, I choose o le he daa guide me on my choice of parameer values for α hh α hl, α ll, b h and b h, insead of imposing ex-ane hypoheical resricions. Neverheless, he calibraed values conform o hese assumpions. 8

9 marke ighness. I assume ha workers canno ex-ane disinguish he vacancy ype, and hus canno direc heir search owards a specific ype of vacancy. I make his assumpion o capure he noion ha maching is imperfec: workers do no always arrive a he jobs hey are bes suied for. Consequenly, low-skill workers encouner low-skill vacancies wih probabiliy per uni of ime ha is proporional o he fracion of low-skill vacancies. Similarly, highskill workers encouner low- and high-skill vacancies wih a probabiliy per uni of ime ha is proporional o he fracion of low- and high-skill vacancies, respecively. Assuming ha η = νl ν l +νh, he effecive maching rae of low-skill workers is η m(θ ), while over-qualified workers relocae ino high-skill jobs a rae (1 η )m(θ ). Unemployed high-skill workers accep boh high- and low-skill jobs, hus heir effecive maching rae is m(θ ). The iming wihin a period is as follows. A he beginning of he period, he realizaion of aggregae produciviy is revealed and agens produce. Afer agens produce, some of he exising maches are exogenously desroyed. Subsequenly, firms pos vacancies o ensure zero profis. Finally, search akes place. Based on he maching raes specified above, some over-qualified workers qui o high-skill jobs, while some unemployed workers find jobs. { } Leing e = e hh, e hl, e ll be he disribuion of employed workers across ypes of maches a he beginning of period, he disribuion of employed workers a he beginning of period + 1 is given by: [ e ll +1 = e ll (1 s l ) + η m(θ ) e hh +1 = e hh (1 s h ) + (1 η )m(θ ) e hl +1 = e hl (1 s h ) + η m(θ ) ] δ e ll (1 s l ) [ 1 δ e hh (1 s h ) [ 1 δ (e hl + e hh )(1 s h ) (1 η )m(θ )e hl (1 s h ) (1) I now urn o he effecive maching raes of firms. The rae a which a firm mees a job seeker of any ype is equal o q(θ ) = m(1, 1 θ ), which is decreasing in θ and exhibis he sandard properies: lim q(θ ) θ 0 = limθ q(θ ) θ =, and limq(θ ) θ = limθ q(θ ) θ 0 ] ] = 0. Lowskill vacancies mach only wih unemployed job seekers. This is because an over-qualified worker has no incenive o change employer unless he new employer offers a high-skill job. Accordingly, someimes firms wih low-skill vacancies mee over-qualified workers who refuse o mach. I follows ha low-skill vacancies mach wih low-skill workers a rae ψ ϕ q(θ ), and wih high-skill workers a rae ψ (1 ϕ ) q(θ ), where ϕ = u l +uh u l +uh +ehl (1 s h). ul u l +uh and ψ = Likewise, employers wih high-skill vacancies do no hire he low-skill 9

10 workers hey mee. Consequenly, high-skill vacancies mach only wih eiher over-qualified or unemployed high-skill workers, and hus, heir effecive maching rae is (1 ψ ϕ ) q(θ ). 2.2 Value Funcions To describe he value funcions I adop he following noaion. For he worker, U i is he value of being unemployed, and W ij is he value of a vacancy, and J ij is he value of being employed. For he firm, V j is he value of a filled job. In all cases, i denoes he ype of worker and j he ype of job. Moreover, in wha follows β = 1 1+r is he discoun facor Workers A low-skill worker s value of being unemployed saisfies [ ] U l = b l + βe η m (θ ) W+1 ll + (1 η m(θ )) U+1 l (2) The inerpreaion is sraighforward. The value of being unemployed is equal o he payoff of being unemployed in he curren period, b l, plus he presen value of he expeced payoff nex period. The laer is given by he probabiliy he worker finds a low-skill job, η m(θ ), imes he value of having a low-skill job nex period, W+1 ll, plus he probabiliy he worker remains unemployed, (1 η m(θ )), imes he value of being unemployed in he nex period, U l +1. The expecaions operaor E depends on he ransiion marix of aggregae produciviy Π, and he ransiion equaions described in (1). The res of he value funcions ake a similar form. Given ha high-skill workers accep boh ypes of jobs, he value of unemploymen o a high-skill worker saisfies U h = b h + βe [ m(θ )[η W hl +1 + (1 η )W hh +1 ] + (1 m(θ ))U h +1 ] (3) The values of being suiably mached saisfy W hh W ll [ ] = w hh + βe s h U+1 h + (1 s h )W+1 hh [ ] = w ll + βe s l U+1 l + (1 s l )W+1 ll while he value of being over-qualified is given by W hl = w hl + βe [ ] s h U+1 h + (1 s h)w+1 hl + (1 s h)(1 η )m(θ )[W+1 hh W +1 hl ] (4) (5) (6) where w ij denoes he wage rae in each case. The value of being over-qualified incorporaes in addiion he expeced gain from on-he-job search. This is given by he las erm in he 10

11 bracke, which is inerpreed as follows: given ha he mach survives job desrucion wih a probabiliy (1 s h ), he worker mees a high-skill vacancy wih a probabiliy (1 η )m(θ ), and obains a surplus [W+1 hh W +1 hl ] from swiching jobs Firms For he firms, he asse values of filling low- and high-skill vacancies wih suiable workers are given, respecively, by J hh J ll [ ] = yα hh w hh + βe s h V+1 h + (1 s h )J+1 hh [ ] = yα ll w ll + βe s l V+1 l + (1 s l )J+1 ll (7) (8) The value of filling a low-skill vacancy wih an over-qualified worker is given by J hl = yα hl w hl + βe [ s h V l +1 + (1 s h)j hl +1 (1 s h)(1 η )m(θ )[J hl +1 V l +1 ] ] I incorporaes in addiion he loss due o endogenous quis. This is capured by he las erm in he bracke. If he job is no exogenously desroyed, he worker coninues searching on he job, and quis wih a probabiliy (1 η )m(θ ), in which case he job becomes vacan. Finally, he values of opening high- and low-skill vacancies are given by (9) [ ] V h = c h + βq(θ )E (1 ψ ϕ )J+1 hh + ψ ϕ V+1 h V l = c l + βq(θ )E [ ψ ϕ J ll +1 + ψ (1 ϕ )J hl +1 + (1 ψ )V l +1 ] (10) (11) Surpluses Given ha he worker and he firm share he surplus in fixed proporions wih γ being he worker s share, he wage w ij saisfies he following Nash bargaining condiions W ij U j = γs ij J ij V i = (1 γ)s ij. (12) where S ij denoes he surplus of he mach, defined as S ij W ij + J ij U i V j (13) Subsiuing he value funcions ogeher wih he Nash bargaining condiions in (12), ino he surplus expression above yields S ll = yα ll b l + βe [(1 s l )S ll +1 η m(θ )γs ll +1] (14) 11

12 S hh = yα hh b h + βe [ S hl (1 s h )S hh +1 (1 η )m(θ )γs hh +1 η m(θ )γs hl +1 = yα hl b h + βe (1 s h)s+1 hl (1 η )m(θ )γs+1 hh η m(θ )γs hl (1 s h )(1 η )m(θ )[S+1 hl γshh +1 ] +1 ] (15) (16) The surplus of a low-skill job filled by a low-skill worker, S ll, akes he sandard form. The erm ouside of he bracke gives he mach produciviy ne of he opporuniy cos of working. The firs erm in he bracke gives he surplus given ha he mach survives o he nex period, and he second erm in he bracke he loss o he worker for giving up searching for a job while unemployed. Once employed he worker gives up he opporuniy o mach wih a low-skill vacancy wih a probabiliy η m(θ ) and gain a share γ of he resuling surplus S+1 ll. Therefore, he value of his opporuniy is subraced from he surplus. As regards he surplus of a high-skill job, S hh, he only difference is ha when suiably mached, high-skill workers give up searching for boh high- and low-skill jobs. Consequenly, he addiional erm (1 η )m(θ )γs+1 hh, which reflecs he value of he opporuniy o mach wih a high-skill vacancy is also subraced from he surplus. The surplus of a low-skill job filled by an over-qualified worker, S hl, akes also ino accoun he cos of endogenous quis, which is given by he las erm in he bracke. Given ha he mach survives o he nex period, wih a probabiliy (1 η )m (θ ), he over-qualified worker quis o a high-skill job, in which case he worker obains a share γ of S+1 hh, bu Shl +1 is los. I is easy o verify jus by looking a he surplus expressions above, ha as in he sandard model, an increase in he meeing rae m(θ ) lowers he surpluses of all jobs. This is because a higher meeing rae raises workers value of searching while unemployed. As firms need o compensae he workers hey hire for giving up searching, his in urn lowers he surplus of filled jobs. I is also sraighforward o verify ha as long as S hh +1 Shl +1, so ha low-skill jobs offer lower wages o high-skill workers han high-skill jobs, upgrading he skill composiion of vacancies (i.e., lowering η ), raises he surplus of low-skill jobs,, bu lowers he surplus of high-skill jobs, S hh. The inuiion is similar; when high-skill S ll vacancies are relaively more abundan, high-skill workers can more easily find high-skill jobs, which offer higher wages, and avoid emporary employmen in low-skill jobs, while lowskill workers have more difficuly finding jobs. This in urn, raises he value of searching while unemployed for high-skill workers, and lowers i for low-skill workers. The surpluses of high- and low-skill jobs change accordingly. The impac of a fall in η on S hl is more cumbersome o deermine. On he one hand, wih high-skill vacancies relaively more abundan, over-qualified workers can more easily 12

13 upgrade o high-skill jobs. Hence, as endogenous quis are more likely, he surplus declines. On he oher hand, he value of he forgone opporuniy o mach wih a low-skill vacancy while unemployed, which is subraced from he surplus, is lower when η is lower. The overall impac on he surplus depends on which of he wo effecs dominaes, making i difficul o esablish i analyically. 2.3 Equilibrium Given free enry, V i = 0 should be saisfied in equilibrium. Therefore, E V i = 0 mus also hold in equilibrium. Applying hese condiions o (10) and (11) ogeher wih he Nash bargaining condiions in (12) yields he following free-enry condiions for low- and high-skill vacancies, respecively [ ] (1 γ)βe ψ ϕ S+1 ll + ψ (1 ϕ )S+1 hl = c l (1 γ)βe [(1 ψ ϕ )S hh +1] = q(θ ) c h q(θ ) The free-enry condiions are such ha in equilibrium he expeced profi from filling a vacancy (lef hand side) is equal o he coss of keeping he vacancy unfilled (righ hand side), and implicily define θ and η. More formally, he equilibrium is given by a vecor {θ, η} ha for each realizaion of aggregae sae, y, and disribuion of employmen, e = { e hh, e hl, e ll }, saisfies he following: (i) he hree ypes of maches are formed volunarily, i.e., yα hh > b h, yα ll > b l, and yα lh > b h ; (ii) he wo free enry condiions in (17) and (18) are saisfied so ha he values of mainaining low- and high-skill vacancies are zero; and (iii) he sae variables e hh, e hl, and e ll are deermined by he se of flow equaions in (1). Wih he characerizaion of he equilibrium I complee he descripion of he model. Before digging deeper ino he model a few words are in line regarding he properies of he equilibrium. Firs, noice ha uniform changes in he expeced profis of boh ypes of vacancies require offseing changes in marke ighens, θ, while unequal changes in he expeced profis of high- and low-skill vacancies require adjusmens in he equilibrium value of η (i.e., adjusmens in he skill mix of vacancies) o keep he values of boh ypes of vacancies equal o zero. To esablish analyically he inuiive noion ha firms respond by shifing he vacancy mix owards high-skill vacancies when he expeced surplus of highskill jobs increases relaive o he expeced surplus of low-skill jobs, one has o prove ha η lowers he expeced surplus of low-skill vacancies, as capured by he lef-hand-side of (17). However, his is no an easy ask. As menioned earlier, alhough an increase in η lowers (17) (18) 13

14 S ll, he impac on S hl can go eiher way. The illusraive seady-sae exercise ha follows specifies parameer resricions, which ensure ha an increase in η lowers S hl. Moreover, he simulaions of he calibraed dynamic model ha follow, confirm his resul. Observe also ha unlike he sandard model, shifs in he skill composiion of job seekers affec he wo secors unevenly, hus alering he skill composiion of vacancies opened. A reducion in he fracion of unemployed job seekers, ψ, (or equivalenly, an increase in he fracion of over-qualified job seekers) lowers he expeced surplus of low-skill jobs, while i raises he expeced surplus of high-skill jobs. Assuming ha he equilibrium value of η declines when he expeced surplus of low-skill jobs declines relaive o he expeced surplus of high-skill jobs, i follows ha an increase in he fracion of over-qualified job seekers induces firms o open relaively more high-skill vacancies, making i more difficul for low-skill workers o find suiable jobs. Moreover, when S+1 ll Shl +1 > 0, i can be easily verified by rearranging erms in (17) ha an increase in he fracion of high-skill job seekers (i.e., a reducion in ψ φ ), lowers he expeced surplus of low-skill vacancies, bu raises he expeced surplus of high-skill vacancies. Hence, when low-skill jobs generae a higher surplus when filled by low-skill insead of over-qualified workers, a rise in he fracion of high-skill job seekers discourages firms from opening low-skill vacancies. 3 Seady Sae In his secion, I firs solve for a unique seady sae equilibrium and hen, I illusrae he impac of a permanen decline in aggregae produciviy y on marke ighness θ and skill composiion of vacancies as capured by η. The proofs of he resuls presened in his secion are given in he Appendix. The purpose of his analyic exercise is o provide a more rigorous inuiion for he resuls of he numerical analysis ha follow. Evidenly, his exercise is limied, because i does no provide insighs ino he dynamic associaed wih shocks. The ask of characerizing he dynamic responses of variables o emporary shocks is aken up in subsequen secions. To keep calculaions racable, I consider he case s h = s l = s, b h = b l = b, and α hl = α ll. This choice of parameers ensures ha over-qualified workers have an incenive o search on he job, because high-skill jobs offer beer wages (i.e., S hh S hl ), and ha firms wih low-skill vacancies are beer off hiring low- as opposed o high-skill workers (i.e., S ll S hl ). Assuming coninuous ime, he seady sae free enry condiions along which he value 14

15 of opening a vacancy is equal o zero, are given by he se of equaions below. (1 γ)[ψϕs ll + ψ(1 ϕ)s hl ] = c l q(θ) (1 γ)(1 ψϕ)s hh = c h q(θ) (19) (20) where S ll = S hl = S hh = yα ll b (r + s + γηm(θ)) (21) yα ll b (r + s + γηm(θ) + (1 η)m(θ)) (22) (yα hh b) (r + s + γ(1 η)m(θ)) γηm(θ)shl (23) Sufficien parameer resricions o ensure he seady sae equilibrium is unique are: i) (yα [ ] ll b) δ (yα hh b) (1 δ) + 2γ γ+1 c l c h ; ii) γ 1 2 and δ 1 2 ; iii) (yα ll b) (yα hh b) γ. The firs condiion ensures ha he fracion of low-skill job seekers, ψφ, decreases when θ increases, and is sufficien o esablish ha he value of low-skill vacancies declines wih θ. Condiions ii) and iii) ensure ha a higher η increases he surplus of low-skill vacancies (lef-hand-side of (19)), bu lowers he surplus of high-skill vacancies (lef-hand-side of (20)). Therefore, if for some exogenous reason he surplus of high-skill jobs increases relaive o he surplus of low-skill jobs, η mus decline, for he free-enry condiions o be saisfied in equilibrium. Under hese condiions, he free-enry condiions (19) and (20) have opposie slopes in he [η, θ] plane, and he equilibrium is characerized by he inersecion of he wo loci as shown in Figure 2. A reducion in y lowers he surpluses of boh ypes of jobs. Therefore, boh loci shif down in response o a fall in y, and he equilibrium value of θ declines. Inuiively, when aggregae produciviy is low, each job is proporionally less producive, hus firms pos fewer vacancies per job seeker. The impac on η depends on which of he wo ypes of jobs is hur he mos. In oher words, i depends on which of he wo loci shifs down by more. To deermine his, I firs ake he raio of he low-skill free-enry condiion o he high-skill free-enry condiion, and hen evaluae is derivaive wih respec o y. The raio is given by, S ll S hl ψϕ ψ(1 ϕ) + (1 ψϕ) Shh (1 ψϕ) S hh = c l (24) c h Afer subsiuing for he surpluses given in expressions (21) o (23), he derivaive of his raio wih respec o y is given by, [ ] R y = bλ 3 λ 1 (ψϕλ 3 + ψ(1 ϕ)λ 2 ) (1 ψϕ)λ 2 [(yα hh b)λ 3 γηm(θ)(yα ll b)] 2 (α hh α ll ) (25) 15

16 where λ 1 = [δ(s + (1 η)m(θ)) + (1 δ)(s + ηm(θ))], λ 2 = (r + s + γηm(θ) + (1 η)m(θ)), and λ 3 = (r + s + γ(1 η)m(θ)). As long as (a hh a ll ) 0, he above derivaive is posiive. This implies ha a reducion in aggregae produciviy has a sronger negaive impac on he value of low-skill vacancies. Therefore, he free-enry condiion for low-skill vacancies shifs down relaively more so ha boh η and θ decline, as illusraed in Figure 3. The reason low-skill jobs are hur he mos is simply ha he ne produciviy of low-skill jobs, (yα ll b), is lower han he ne produciviy of high-skill jobs, (yα hh b). As Shimer (2005b), Hagedorn and Manovskii (2007), and Pries (2007) also poin ou, he surplus of a mach is more sensiive o changes in aggregae produciviy when he produciviy of he mach ne of he opporuniy cos of employmen is small. In he presen model, a lower values of y, he percenage gap beween he produciviy of he job and he opporuniy cos of employmen declines more for low- han for high-skill jobs, pushing he relaive surplus of high-skill jobs up. 8 Consequenly, he burden of a permanen reducion in aggregae produciviy falls more heavily on low-skill workers. The reducion in θ implies ha high-skill workers also have more difficuly finding vacancies, because m(θ) declines. However, in addiion o he reducion in m(θ), low-skill workers bear he reducion in η. Hence, hey suffer a higher reducion in heir maching rae relaively o high-skill workers, implying a relaively higher increase in low-skill unemploymen in recessions. A conclusion regarding he impac of a fall in y on he number of over-qualified highskill workers canno be reached based on his analyic resul alone. A fall in y implies ha high-skill workers encouner low-skill vacancies less frequenly, as ηm(θ) declines. However, if he rise in high-skill unemploymen due o he fall in m(θ) is sufficienly high, hen he number of over-qualified workers may sill rise. For now i is enough o noe ha aggregae shocks have uneven consequences on he wo ypes of workers. Firms face he choice of which ype of vacancy o open and how many vacancies o open. As he relaive surplus of he wo ypes of jobs changes when aggregae produciviy falls, firms respond no only by lowering he number of vacancies 8 I is imporan o poin ou ha his resul is no sensiive o he assumpion he aggregae produciviy shock is muliplicaive. An addiive aggregae produciviy shock (i.e., y + a ij insead of yα ij) would imply an even higher increase in he ne produciviy of high-skill jobs relaive o he ne produciviy of low-skill jobs. Moreover, his resul is no sensiive o he assumpion ha b is he same for boh ypes of workers. Assuming ha high-skill workers generae b h while unemployed and low-skill workers generae b l while unemployed, he same resul would sill hold as long as he ne produciviy of high-skill jobs (yα hh b h ) is greaer han he ne produciviy of low-skill jobs (yα ll b l ). 16

17 posed per job seeker, as in he sandard model, bu also by adjusing he skill mix of vacancies owards he relaively more profiable ype, which is high-skill vacancies in he model. Given ha high-skill workers qualify for boh ypes of jobs, and search on he job is manageable, high-skill employabiliy is less vulnerable o recessions. On he conrary, low-skill workers who qualify only for low-skill jobs are subjec o unfavorable shifs in he vacancy mix in recessions. 4 Analysis of he Dynamic Model I now proceed wih characerizing he dynamic version of he model oulined in secion 2. I firs describe he calibraion of he model, and I subsequenly simulae he model and describe he dynamic evoluion of key variables: high- and low-skill exi raes from unemploymen, job-o-job ransiion rae, over-qualificaion rae, and high- and low-skill unemploymen raes. The calibraion of he model is summarized in Table Calibraion I consider he high-skill ype as represening workers who hold a leas a college degree. I herefore se he proporion of high-skill workers o δ = 0.25, which based on he March CPS Annual Demographic Survey Files for he period from 1964 o 2003, equals he average proporion of U.S. labor force ha holds a college degree or more. I choose he model period o be one quarer and herefore se he discoun rae o r= For he maching funcion I make he sandard choices. I assume a Cobb Douglas funcional form so ha m = z a v 1 a, and choose an elasiciy parameer a = 0.4, which lies a he lower range of esimaes repored in Perongolo and Pissarides (2001). I also make a sandard choice for he worker s bargaining power. I assume ha workers and firms spli he surplus equally, i.e., γ = 0.5. Following he lieraure, I selec values for he separaion raes, s h and s l, which are higher han he empirical measures of ransiion raes from employmen o unemploymen, o ake ino accoun workers who exi he labor force, bu whose behavior is similar o hose couned as unemployed. 9 Blanchard and Diamond (1990) show ha in he U.S., he 9 Since Clark and Summers (1979) i became eminen ha he disincion beween he pool of unemployed and he pools of hose ou of he labor force is fuzzy, wih many workers going back an forh beween he wo saes. 17

18 wan-a-job pool in he sock of hose no in he labor force is roughly equal o he sock of unemployed. Moreover, hey documen ha only half of he average flow ino employmen comes from unemploymen, wih he oher half coming from people classified as no in he labor force, signifying ha ou of he labor force job seekers also ake par in maching. Assuming ha all people classified as ou of he labor force paricipae in he maching process ses an upper bound o he value of he separaion rae, which can be compued by adding ogeher he flows from employmen o unemploymen and ou of he labor force. A lower bound can be compued by looking only a flows o unemploymen, assuming ha only hose classified as unemployed search for jobs. To calculae hese upper and lower bounds, I use he monhly esimaes of ransiion raes from employmen o unemploymen and ou of he labor force, for college and non-college graduaes, repored in Nagypal (2004). Afer convering he monhly esimaes ino quarerly frequencies, I find ha s h should lie in he range [ ] and s l in he range [ ]. 10 I chose o se s h = 0.03 and s l = 0.07, which pus more weigh on low-skill separaions, and resuls in an average separaion rae in he model of 0.06, which is line wih CPS esimaes of Hall (2005), when roughly half of he flows from employmen o ou of he labor force are flows ino a job seeking sae. Noe ha he calibraed separaion raes do no accoun for job-o-job ransiions. For high-skill workers job-o-job ransiions in he form of upgrading o higher job levels, are endogenous in he model. For low-skill workers hey are no. Bu since he focus of he analysis is unemploymen differences across skill-groups, I choose no o include job-o-job flows, because workers who direcly move ino a new job are no accouned as unemployed. For he parameer values for job creaion coss I consruc an upper bound as follows. According o Hamermesh (1993), in 1990 average recruimen and raining coss in he U.S. represen abou on-sixh of average annual labor earnings. Moreover, he job creaion coss canno be oo large relaive o aggregae oupu in he model. The sandard upper bound in he lieraure is 5% of oupu devoed in job creaion aciviies. Based on hese 10 As far as I know, esimaes for he U.S. of flows from employmen o unemploymen and ou of he labor force for differen educaional groups, can only be found in Fallick and Fleischman (2001) who uses he basic monhly CPS survey from February 1994 o December (2000), and Nagypal (2004) who expands he period o January Using average employmen shares by educaion, from he March CPS Annual Demographic Survey files, and Nagypal s esimaes, I firs calculaed he monhly separaion raes for college graduaes and workers wihou a college degree. Then, by couning pahs in a probabiliy ree, I derived he quarerly raes as: s m[(1 f m) 2 + f ms m] + (1 s m)[s m(1 s m) + s m(1 f m)], were s m is he monhly separaion rae and f m he monhly job finding probabiliy. I am graeful o Bruce Fallick for providing me wih he CPS esimaes of monhly job finding raes by educaion. 18

19 wo observaions, I se c l = 0.13 and c h = 0.22, which are roughly equal o one hird of quarerly low- and high-skill wages, respecively, when he laer are suiably mached wih high-skill jobs. Wih hese values, he simulaed average vacancy coss in he model are less han 5% of simulaed oupu. I nex urn o he calibraion of high- and low-skill produciviies, α hh and α ll, he produciviy of over-qualified high-skill workers, α hl, and he opporuniy coss of working, b h and b l. These parameers are seleced o mach saisics from he simulaed daa o empirical measures of, i) wage differences beween college educaed and non-college educaed workers, ii) wages differences beween over-educaed and correcly mached workers, ii) average job finding rae, and iv) unemploymen raes of workers wih college and less han college educaion. To mach hese saisics, I se α ll = 0.4, α hh = 0.68, and α hl = 0.6. The values for he opporuniy coss of working are se o b h = 0.52, and b l = 0.28, which are less han he simulaed average high- and low-skill wages, respecively. Below I discuss my choice of he relevan arges. I begin wih my choice of arge for he wage difference beween workers wih college and less han college educaion. Based on he March CPS, Auor e al. (2008) find ha he college-plus o high school log wage premium (i.e., he average log wage raio of college o high school graduaes) ranges form 0.4 o 0.65 in he period from 1963 o This implies an average log wage premium of approximaely The low-skill group in he model is no resriced o high-school graduaes only; i also includes workers wih some college educaion and workers wih less han high-school educaion. However, he average share of employed workers who have no compleed college educaion, bu have some college educaion is no more han Therefore, I consider an average log wage premium of 0.5 as a fair arge. My choice of parameer value for he produciviy of over-educaed college graduaes was guided by evidence on wage differenials beween over-educaed and correcly mached workers. For he U.S., Sicherman (1991) finds ha over-educaed workers earn more han heir co-workers who are no over-educaed, bu less han similar workers wih he same level of schooling ha work in jobs ha require heir acual level of schooling (i.e., correcly allocaed workers). In paricular, he wage rae of over-educaed workers is on average 5% lower han ha of correcly allocaed workers. Considering his as a lower bound, I choose he value of α hl ha implies ha he wage of an over-educaed college graduae is 10% lower han he wage of a suiably mached college graduae. 11 Esimaes in he same range can also be found in Wheeler (2005). 19

20 The average job finding rae I choose o arge, incorporaes ou of he labor force job seekers, in line wih my choice of separaion raes. I make use of he Hall (2005) esimae ha incorporaes his group ino he group unemployed. Hall ook advanage of he expanded unemploymen rae series, available from he BLS saring in 1994, which includes hose classified as discourage workers who wan a job bu believe a job is unavailable for several reasons, and hose marginally aached o he labor force, who indicae a likelihood of reurning o he labor force in he near fuure. The series was approximaed for earlier years by regressing he expanded series o he sandard unemploymen rae for he years 1994 hrough 2004 and using he fied value for he years before. Afer exending he expanded unemploymen rae series o earlier years, he job finding rae was calculaed as he raio of new hires o he number of job seekers, as measured by he expanded unemploymen rae series. For he period from 1964 o 2003 he esimaed monhly job finding rae averages o 0.28, which works ou o an average quarerly job finding rae of abou 0.6. Consisen wih my choice of separaion and job finding raes, he argeed unemploymen raes are higher han he official empirical measures, o ake ino accoun workers classified as ou of he labor force who paricipae in he maching process. Unforunaely, he series of marginally aached or discouraged workers in he BLS is no available by educaion. Therefore, he mehodology of Hall, of impuing he expanded unemploymen rae series for earlier years using he years afer, canno implemened o consruc expanded unemploymen rae series by educaion. Insead, guided by he Blanchard and Diamond (1990) finding ha he wan-a-job group is roughly equal o he number of unemployed, I approximae he expanded unemploymen raes as 2u u +l, were u is he number of unemployed and l is he size of he labor force. Based on he March CPS Annual Demographic Survey files from 1964 o 2003, his calculaion yields an average unemploymen rae of for college graduaes, and for workers wih less college educaion. The resuling average unemploymen rae of 0.10 in he model is consisen wih he average expanded unemploymen rae in Hall (2005). Finally, I urn o he calibraion of he aggregae produciviy process. I approximae hrough a 9-sae Markov chain he quarerly deviaions from a linear rend of he U.S. GDP for he period from 1964 o The esimaed auocorrelaion coefficien of he sandard AR(1) model is and he sandard error of he innovaion is Hall (2005) and Shimer (2005b) show ha he sandard model, along he lines of Morensen and Pissarides (1994), can explain he magniude of cyclical changes in unemploymen only by assuming implausibly large produciviy shocks. The reason is ha for reasonable calibra- 20