Matching, sorting and wages

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Contents lists available atScienceDirect

Review

of

Economic

Dynamics

www.elsevier.com/locate/red

Matching,

sorting

and

wages

✩

Jeremy Lise

a

,

b

,

Costas Meghir

c

,

d

,

b

,

Jean-Marc Robin

e

,

a a_University_College_London,_United_Kingdom

b_IFS,_United_Kingdom c_Yale_University,_United_States d_NBER,_United_States e_Sciences_Po,_Paris,_France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received8January2015

Receivedinrevisedform7November2015 Availableonline2December2015 JELclassiﬁcation: J3 J64 J65 Keywords: Sorting Mismatch Search-matching Wagedynamics

Wedevelopanempiricalsearch-matchingmodelwhichissuitableforanalyzingthewage, employment and welfare impact of regulation in a labor market with heterogeneous workersandjobs.Toachievethiswedevelopanequilibriummodelofwagedetermination andemploymentwhichextendsthecurrentliteratureonequilibriumwagedetermination withmatchingandprovidesabridgebetweensomeofthemostprominentmacromodels and microeconometric research. The model incorporates productivity shocks, long-term contracts, on-the-job search and counter-offers. Importantly, the model allows for the possibility ofassortative matchingbetweenworkers and jobs due tocomplementarities betweenworkerandjobcharacteristics.Weusethemodeltoestimatethepotentialgain from optimal regulationand weconsider the potential gains and redistributiveimpacts fromoptimalunemploymentbeneﬁtpolicy.Hereoptimalpolicyisdeﬁnedasthatwhich maximizestotaloutputandhomeproduction,accountingforthevariousconstraintsthat arise from search frictions. The model is estimated on the NLSY using the method of moments.

1. Introduction

Labormarketimperfectionsmayjustifylabormarketinterventions.Withinacompetitiveframeworkregulationwouldbe welfarereducing,asitwouldtypicallyreduceemploymentandincreaseinsiders’wages.Bycontrast,anyfriction constrain-ingtheallocationofworkerstojobsinevitablyallowssomeagentstoappropriateagreatershareoftherentthanacentral plannerwoulddeemﬁt.Indeed,ifthereareimportantcomplementaritiesinproduction,mismatchmayproducesubstantial welfarelossesrelativetotheﬁrstbestoraconstrainedplanner.

✩ _This_paper_has_also_circulated_under_the_title_“Mismatch,_Sorting_and_Wage_Dynamics”._We_have_beneﬁtted_greatly_from_comments_and_questions_from

DaleMortensen,JoeAltonji,ThibautLamadon,GuiseppeMoscarini,JohnMoore,BentleyMcLeod,RobShimer,twoanonymousrefereesandtheeditorof thisjournalaswellasseminarparticipantsatBerkeley,Chicago,Columbia,Georgetown,Harvard,Indiana,MinneapolisFed,NYU,Northwestern,Princeton, Stanford,Wisconsin,UBC,SimonFraserUniversity,IIES,StockholmSchoolofEconomics,Cergy-Pontoise, Edinburgh,York,Southampton,Bristol,CEMFI, Oslo,Oxford,Tilburg,EUI,Uppsala,QueenMary,CarlosIII,SMU,CentralBankofTurkey,Koc,SITE,NBERsummermeeting,SEDmeetings,CowlesSummer Conference,Crest-ThemaSaM,andSOLEmeetingsinNY.CostasMeghirthankstheESRCforfundingundertheProfessorialFellowshipRES-051-27-0204 andtheCowlesfoundationandISPSatYale.Jean-MarcRobingratefullyacknowledgesﬁnancialsupportfromtheEconomicandSocialResearchCouncil throughtheESRCCentreforMicrodataMethodsandPracticegrantRES-589-28-0001,andfromtheEuropeanResearchCouncil(ERC)grant ERC-2010-AdG-269693-WASP.Theauthorsareresponsibleforallerrorsorinterpretations.

E-mailaddresses:[email protected](J. Lise),[email protected](C. Meghir),[email protected](J.-M. Robin). http://dx.doi.org/10.1016/j.red.2015.11.004

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Wedevelopasearch-matchingmodelinwhichworkerswithdifferentabilitiesareassignedtodifferenttasks.Ourmodel combines elements fromkey papers in the equilibriumsearch literature. Thus we allow for endogenousjob destruction becauseofproductivityshocks,drawingfromtheseminalpaperof

Mortensen and Pissarides (1994)

.Tothisframeworkwe introducetwosidedheterogeneitywithpotentialcomplementaritiesbetweenjobandworkerproductivityfollowing

Shimer

and Smith (2000).In thiswaywe caninvestigatesortinginthelabormarket, oneofourmotivatinginterests.Toaccount forjob-to-jobtransitionsandtobetterexplainwagegrowthweallowforon-the-jobsearch,whichisnotpresentin

Shimer

and Smith (2000).Wagedeterminationisdrawnfrom

Postel-Vinay and Robin (2002)

,

Dey and Flinn (2005)

and

Cahuc et al.

(2006).Inotherwordswecombinebargainingoverthesurplus(forworkersoutofunemploymentormovingtoanewjob) withBertrandcompetitionwhenapoachingﬁrmisinvolved.

In this way,our model provides a bridge betweenan essentially theoretical literature on allocation of heterogeneous workerstojobs,1 _and_the _large_{microeconometric}_literature_on_job_mobility_and_wage_dynamics.2 _The _framework_that_we

develop allowsforfrictionsandinefficienciesinthelabormarket.Thefrictionsareduetothetime ittakestolocatejobs, which meansthat individualswillspendtime searchingforajob whileunemployedandifworkingthey are likelytobe mismatched (iftherearecomplementaritiesinproduction).Onemayarguethatsuch frictionsareinevitable;nevertheless itisimportanttounderstandthewelfarelossthattheycauserelativetothebenchmarkofafrictionlesseconomy,because thisprovidesa measureofhowdominantsearchfrictionsareindeterminingeconomicoutcomes.3 Moreimportantly,our modelcanquantifythewelfarelossfrominefficienciesthatcanpotentiallybe addressedby labormarketregulation:first, the number of job seekers cause congestionmaking it harder forothers to find jobs –this is a standard externality in modelswithendogenousarrivalrates.Beyondthatthepotentialcomplementaritiesbetweenworkerandfirmproductivities, the possibilityofon-the-job search andthelackof commitmentonthe workerside allows forthepossibilityof another inefficiency: workersandjobsaresometimeswillingtoformmatcheswhoseflowoutputislowerthanthecombinedcost ofavacancyandthelostout-of-workbenefit.Ontheonehandthejob,withitslocalmonopsonypowermanagestoextract sufficient surplustomakeit worth hiringtheworker; ontheother handtheworker preferstheresultingcurrentloss to theincreasedflowofincomewhenout ofworkbecausebeinginajobprovidesabetteroutsideoffertonegotiateawage once an alternativeoffer arrives.These featuresmay beimportantin thelabor market andourmodelcan quantifytheir importanceforwelfare.

Weestimateourmodelusingpaneldataonworkersandusetheestimatestoquantifyhowmuchsortingthereiswith respect to unobserved characteristics. A ﬁrst set of studies interested in thisquestion followAbowd et al. (1999, AKM)

andassessthedegreeofassortativematchinginthelabor marketbycalculatingthecorrelationbetweenworker andfirm fixed effectsestimatedbyapanel-data regressionofindividualwagesonworkers’ andemployers’indexes.4 Theytypically find non-significant or negative correlations. Andrews et al. (2008) show that thisestimated correlation iscontaminated by a spurious statisticalbias andproposea bias-correctedestimator. Using GermanIAB data,they findthat thenegative OLS estimateisturnedintoa positivenumberafterapplyingthebiascorrection (0

.

23 insteadofsomewhereintherange

[−

0

.

19

,

−

0

.

15

]

).

However,

Eeckhout and Kircher (2011)

stronglyargueagainstthepossibilityofidentifyingassortativematchinginlabor marketsbythisapproach,bias-correctedornot.Thisisbecausethesurplusofamatchisnotamonotonicfunctionofworker and firm characteristicsingeneral. Depending onthe distribution ofmatches aroundthe optimal, Beckerianallocation,a positive ora negative AKMcorrelation canbe estimatedirrespective ofthesignof thecorrelation betweenworkers’and firms’ true unobserved characteristics in the population of active matches. In the previous working paper we also find that theAKM correlation is misleading and demonstratethat positive sorting withrespect to unobserved characteristics may inducea negative correlation in theworker andfirm effects estimatedon a panel ofwages. Lopes de Melo (2009),

Hagedorn et al. (2012),

Bagger and Lentz (2014)

reachsimilarconclusionswithdifferentmodels.

Notethat thisargumentinvalidating astructuralinterpretation ofAKMisimplicitin

Gautier and

Teulings (2006),who estimatearegressionmodeloflogwagesonaquadraticfunctionofworkerandemployertypes.Thesetypesarecalculated by projecting logwagesseparately onworkers’ andemployers’observed characteristics.GautierandTeulings’sestimation rests on various parametric restrictions, but nevertheless convincingly claims 1) that the wage equation is nonlinear in workers’andemployers’types,and2)thattheyarepositivelycorrelated.

1

SeealsoShi (2001),TeulingsandGautier (2004),Moscarini (2005)andGautieretal. (2010)foralternativeapproachestotwo-sidedmatchingmodels withoutandwithon-the-jobsearch.

2 _See_amongst_others_{MaCurdy (1982)}_,_Altonji_and_{Shakotko (1987)}_,_Abowd_and_{Card (1989)}_,_{Topel (1991)}_,_Topel_and_{Ward (1992)}_,_Meghir_and_Pistaferri (2004),AltonjiandWilliams (2005),Guvenen (2007),BonhommeandRobin (2009),Guvenen (2009),Lowetal. (2010),Lise (2013),Altonjietal. (2013).

3 _See_also_Teulings_and_{Gautier (2004)}_,_Gautier_and_{Teulings (2015)}_for_an_approach_to_measuring_the_welfare_cost_of_search_frictions._Their_approach

differsboththeoreticallyandempiricallyfromours:Theirproductionfunctionisaquadraticinthedistanceoftheworkerandjobcharacteristics–itisnot estimated.Therearenoshockstoﬁrmproductivity.Therelativeeﬃciencyofon-the-jobsearchissetatdifferentvaluesbutnotestimatedfromthedata. Finally,wagesareapproximatedwithobservedandunobserveddeterminantsassumedtobeorthogonal.

4 _See_Goux_and_{Maurin (1999)}_and_Abowd_et_{al. (2009)}_who_present_results_for_French_and_U.S._matched_{employer–employee}_data,_and_Gruetter_and Lalive (2009)forAustriandata.

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Identiﬁcationofcomplementarityandsorting,giventhatweuseapanelofworkers’wagesandlabormarkettransitions extractedfromtheNLSY,5_is_also_hard_to_prove_or_disprove_{theoretically.}_However,_we_argue_that_sorting_can_be_seen_in_the

waywageandemployment mobilityvary asafunction ofthelength oftime spent workingfollowing an unemployment spell. Due to search frictions, the cohort of workers entering the labor market following a spell out of work will start offmismatched,butthroughon-the-job searchthey willbecome betterandbettermatched withtime spent inthelabor market.Ifthetendencytosortinequilibriumisstrongthiswillresultinwagesspreadingout asworkerssortthemselves. MonteCarlosimulationsfortheSimulatedMethodofMomentestimatorthatwehaveimplementedhereseemtosuggest thatthesetofmomentswematchdoidentifythedegreeofsorting.WeﬁndthattheNLSYdataarebestﬁttedbyourmodel assumingnocomplementarityandzerosortingforworkerswithhigh-schooleducationorlessandpositivecomplementarity andsortingforcollege-graduates.6

Finally,ourmodeloffersanempiricalframeworkforunderstandingemploymentandwagedeterminationinthepresence offirm–workercomplementarities,searchfrictionsandproductivityshocks.Asaresultitoffersawayforevaluatingthe ex-tenttowhichregulationmaybewelfareimprovingandcanevaluatetheimpactofspecificpoliciessuchasunemployment benefit.Ina searchframework withmatchcomplementaritiesunemploymentbenefitcanhaveambiguouseffectson em-ploymentandtotaloutput.Ontheonehand,itallowsworkerstobemorepickyandformbettermatches,whichcomesat thecostoflongerunemploymentspellsandhigherunemployment.Ontheotherhand,thefactthathigherqualitymatches willbe formedmayinducefirmstocreatemorejobs, increasing thecontactrateandpotentially reducingunemployment duration.Ourframeworkallowsthiseffecttobequantified(seealso

Acemoglu and Shimer, 2000

).And,inaddition,itallows ustoanalyzetheeffectofsuchpoliciesonthedistributionofwelfarethusshowingwhopaysandwhobeneﬁtsfromsuch apolicyinthisnon-competitiveenvironment.

Wefindthatthedegreetowhichlabormarketinterventionsarejustifieddependsonwhetherwearelookingatthelow orhighskilledmarkets.Ourfindingthatthemarketforunskilledlabor(highschoolgraduatesorless)ischaracterizedbyan extremelylowdegreeofcomplementarityimpliesthatmismatchisnotverycostly.Ifwereallocatedtheemployedworkers inthisgroupoptimallythedifferencein(steadystate)welfarewouldbe1.1percent.Ontheother hand,amongthecollege graduates,optimallyreallocatingtheemployedworkersproduces adifference in(steadystate)welfareof6.8percent.We alsofindthatsearch frictionsaresignificant.Ifwedothesameexperimentwithfullemploymentthewelfarechangesare 7.8and19.6forthehighschoolandcollegegroupsrespectively.Theinteractionofsearchfrictionsandthecostofmismatch (the degreeofcomplementarity)differbetweenthetwo labormarkets, implyingthata socialplannerwhoisconstrained bythesefrictioncouldattainawelfareincrease of2percentforthehighschoolgroupbutonly0.7percentforthecollege group,largelybytradingoffthelevelofemploymentagainstthecostofcreatingvacancies.Finally,wefindthatifwelimit theplannertoanoptimalunemploymentbenefitprogram,thiscangoalongwaytorealizingthepotential welfaregains forthehighschoolgroup.However,itisineffectualforthecollegeeducatedgroupastheresultingemploymentdistortions outweighthegainsfromimprovedmatchquality.

Thepaperproceedsasfollows.Section2describesthemodel,Section3andSection4presenttheestimationprocedure. Sections5 and 6describethedataandthechoiceofmoments.Section7presentstheresultsofestimation,analysestheﬁt anddiscussesestimationofthedegree ofsorting.Section 8presentsthewelfareanalysis, thecostofsearchfrictions, and analysesindetailtheoptimalunemploymentbeneﬁtpolicy.Section9concludes.

2. Themodel

Webuildamodelofindividualemploymentandwagedynamics,withheterogeneousworkersandjobsandwith produc-tivecomplementaritiesatthematchlevel.Thismodeldrawsfrom

Mortensen and Pissarides (1994)

,asfarastheprocessof matchcreationanddestructionisconcerned,andfrom

Postel-Vinay and Robin (2002)

,

Dey and Flinn (2005)

and

Cahuc et al.

(2006)inordertoincorporateon-the-jobsearchintheMortensen–Pissaridesmodel.Inaddition,wedrawfrom

Postel-Vinay

and Turon (2010)therenegotiationmechanismforwagesfollowingﬁrm-levelproductivityshocks.Thewagedynamics fol-low fromthe process ofsearch andmatching andfromﬁrm-level productivityshocks,butentirely abstractfromhuman capital accumulationand idiosyncraticability shocks.While important, incorporatinghuman capital accumulationwould complicatemattersbeyondthescopeofthispaper.7

2.1. Workers, jobs and matches

Inthiseconomy,thereisaﬁxedmeasureofinﬁnitelylivedindividualsthatisnormalizedtoone.Theydifferfromeach otheraccordingtoability,andabilitydifferencesarepermanent,continuouslydistributedacrossworkers,andobservableby

5 _We_preferred_to_stick_to_standard_worker_panel_data_for_two_reasons._First,_matched_{employer–employee}_data_are_less_universal_and_not_always_easily

accessibletoresearchers.Second,value-addedperworker(usedbyCahucetal.,2006; Baggeretal.,2014; BaggerandLentz,2014)doesnotmeasurewell thelaborproductivityofasinglejobisolatedfromtheotherjobsinaﬁrm.

6 _There_may_be_other_reasons_for_increasing_inequality_over_time_for_a_cohort,_including_permanent_shocks_to_productivity_as_in_Meghir_and_Pistaferri (2004)andevenheterogeneousaccumulationofhumancapitalasinGuvenen (2007).Thesealternativeexplanationshavenotbeenexploredhereandwe believethatIdentiﬁcationmayrequirericherinformation,suchasmatchedemployer–employeedata.

7 _See_Bagger_et_{al. (2014)}_for_a_recent_estimation_of_a_search_model_with_human_capital_and_shocks_to_worker_ability_rendered_tractable_by_assuming

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allagents,butnotbytheresearcher.Let

x

∈ [

x

,

x

]

denoteworkerabilityandlet

(

x

)

beitsPDF.Wealsodenoteby

u

(

x

)

the (endogenous)numberof

x among

theunemployed,with

U

=

x

x u

(

x

)

dx.

Workers are matchedpairwise tojobsortasks.Jobsarecharacterized bya technologicalfactor,saylabor productivity, denoted y, whichisalsocontinuousandobservablebyallagents,butnot theresearcher.Thereisnomatch-speciﬁctype; matchspeciﬁcityresultsfromthecombinationofaworkertype

x with

ajobtype y; thesetogetherwiththecurrentwage paid to the worker will form the relevant state space for the worker and the firm value. Jobs are subject to persistent idiosyncraticproductivityshocks,whichcantriggerseparationsandwagerenegotiations.So,ifan employeequits,thejob continues to existandis available foranother match. Theseidiosyncraticproductivity shocksmay reflectchanges inthe product market (shifts indemand) or events such aschanges in work practices orinvestments (which are not modeled here).Specifically,weassumethat

y ﬂuctuates

accordingtoajumpprocess:shocksarriveatarate

δ

,andanewproductivity level yisdrawnfromadistributionwithPDF

γ

(

y

)

on

[

y

,

y

]

.Thusthesizeof

δ

determinesthepersistenceoftheshocks.

The numberofavailabletype- y jobsintheeconomy isn

(

y

)

,andthe totalnumberofjobsis

N

=

y

yn

(

y

)

d y. Jobsare eithervacantormatched.Thenumberofvacanttype- y postsis

v

(

y

)

andtheresultingtotalnumberofvacanciesisdenoted by

V

=

v

(

y

)

d y.Bothdistributionsareendogenousduetoafreeentrymechanismthatwillbedescribedlater.

A matchbetweena worker of type x and a job of productivity y produces a ﬂow ofoutput f

(

x

,

y

)

. We willspecify thisfunctiontoallowforthepossibilitythat

x and

y are complementaryinproduction,implyingthatsortingwillincrease total output.However, we wishto determinethisempirically, asitisimportantboth forunderstandingthe labormarket and forevaluating the potential effectsof regulation. Matches can end both endogenously,as we characterize later, and exogenously.Wedenoteby

ξ

therateofexogenousjobdestructions,deﬁnedasaneventthatisunrelatedtoeitherworker orjobcharacteristics.

Equilibrium willresultinajointdistributionof

x and

y, thedistributionofmatches.Wedenotethisby

h

(

x

,

y

)

,which satisﬁesthebalanceequations:

h

(

x

,

y

)

d y

+

u

(

x

)

= (

x

),

(1)

and

h

(

x

,

y

)

dx

+

v

(

y

)

=

n

(

y

).

(2)

2.2. Meetings and match formation

We relate the aggregate number ofmeetingsbetweenvacancies andsearchingworkers througha standard aggregate matchingfunction

M

(

·,·)

.Thistakesasinputsthetotalnumberofvacancies

V and

thetotalamountofeffectivejobseekers

U

+

s

(

1

−

U

)

,where

s is

therelativesearchintensityofemployedworkersvis-a-visunemployed.Thematchingfunctionis assumedtobeincreasinginbothargumentsandexhibitconstantreturnstoscale.

Forthepurposeofexpositionitisusefultodeﬁne

κ

=

M

(

U

+

s

(

1

−

U

),

V

)

[U

+

s

(

1

−

U

)

] V

,

whichsummarizestheeffectofmarkettightnessinasinglevariable.Inastationaryequilibrium

κ

isconstant,butitisnot invariant to policy,anditis importanttoallow it tochangewhen evaluating interventionsorcounterfactualregulations. Notethatwhile

M

(

·,·)

governstheaggregatenumberofmeetings,whetherornotameetingtranslatesintoamatchwillbe determinedbythedecisionsofindividualworkersandﬁrms.

The matchingparameter

κ

allowsustocalculateall relevantmeeting rates.Theinstantaneous rateatwhichan unem-ployed worker meetsa vacancyoftype y is κV

·

v(y)

V

=

κ

v

(

y

)

.The instantaneousprobability foranyvacancy tomakea contactwithanunemployedworkeroftype

x is κ

u

(

x

)

.Employedworkersarecontactedbyjobsoftype y with instanta-neousprobability

s

κ

v

(

y

)

.Finally,therateatwhichvacanciesarecontactedbyaworker

x employed

atajob y is s

κ

h

(

x

,

y

)

. Individualsandjobsareriskneutralandweassumeeﬃciency,inthesensethatanymatchwherethesurplusispositive will be formed when the worker and the job meet. Under these conditions we can characterize the set of equilibrium matchesandtheirsurplusseparatelyfromthesharingofthesurplusbetweenworkersandjobs.

Let W0

(

x

)

denotethe presentvalue ofunemploymentfor aworker withcharacteristic x. This willreﬂect theﬂow of income when out of work (or home production) and the expected present value of income that will arise following a successful jobmatch. Similarly

0

(

y

)

denotesthe presentvalue toa job ofposting avacancy arisingfromthe expected revenues of employing a suitable worker net of expected posting costs. Let also W1

(

w

,

x

,

y

)

(respectively

1

(

w

,

x

,

y

)

) denote the present value of a wagecontract w for a worker x employed at a job y (respectively the ﬁrm’sdiscounted proﬁt).

Thesurplusofan

(

x

,

y

)

matchisdeﬁnedby

(5)

Forapair(x

,

y)

a matchisfeasibleandsustainableifthe surplusisnonnegative, S

(

x

,

y

)

≥

0.Wefocusonequilibriasuch thatforall

x there

isatleastone y such that

S

(

x

,

y

)

≥

0.

2.3. Wages

Differentwagesarenegotiatedwhenleavingunemployment,uponpoaching,orafterashocktothejobproductivity.

2.3.1. Wage negotiation with unemployed workers

Thewageforaworker transitingfromunemploymentis w

= φ

0

(

x

,

y

)

andweassume thatitissettosplitthesurplus accordingtoNashbargainingwithworker’sbargainingparameter

β

8_:

W1

(φ

0

,

x

,

y

)

−

W0

(

x

)

= β

S

(

x

,

y

).

(4)

A simpleassumption wouldbe to set

β

to zero.However thismay leadto the counterfactualimplication that forsome matches initial wages are negative implying workers pay to obtain a high value job with the potential of future wage increases.Althoughthisisnotimplausible(unpaidinternships)thisisneverobservedinoursurveydata.Byallowing

β

to bedeterminedbythedataweavoidthisproblemandallowgreaterﬂexibilityinﬁttingthefacts.

2.3.2. Poaching

Wagescan onlybe renegotiatedwheneitherside hasan interest toseparate ifthey donot obtainan improvedoffer, assuming thatthematchremains viableforbothparties.The eventsthatcantrigger renegotiationoccurwhena suitable outside offer ismade, orwhen a productivity shockchanges the value of thesurplus suﬃciently. We consider ﬁrst the impactofanoutsideoffer.

Weassumethatincumbentemployersrespondtooutsideoffers:anegotiationgameisthenplayedbetweentheworker andboth jobsasin Dey and Flinn (2005) andCahuc et al. (2006). Ifa worker x, currentlypaired to a job y such that

S

(

x

,

y

)

≥

0,ﬁndsanalternativejob ysuchthat S

(

x

,

y

)

≥

S

(

x

,

y

)

,theworkermovestothealternativejob.Thisisbecause the poachingﬁrm can always paymorethan the currentonecan match. Alternatively,if thealternative job y produces lesssurplusthanthe currentjob,butmorethan theworker’s shareofthesurplusatthe currentjob,i.e. W1

−

W0

(

x

)

<

S

(

x

,

y

)

<

S

(

x

,

y

)

(where W1 denotesthepresentvalueofthecurrentcontract), thentheworkerusestheoutsideofferto negotiate upher wage.Lastly,if S

(

x

,

y

)

≤

W1

−

W0

(

x

)

,theworker hasnothingtogain fromthecompetitionbetween y and

y

becauseshecannotmakeacrediblethreattoleave,andthewagedoesnotchange.

Ineitherone ofthe firsttwocases,theworker endsupinthe highersurplusmatch. Iftheworkerchanges firms,she usesthesurplusatthepreviousmatchastheoutsideoptionwhenbargaining.Ifthesurplusatthecurrentmatchexceeds thesurplusatthepoachingfirmweassumetheincumbentfirmmakesatakeitorleaveitofferandthereisnobargaining. Assume S

(

x

,

y

)

≥

S

(

x

,

y

)

.The bargainedwagewhen switchingﬁrms (from y to y) in thiscaseis w

= φ

1

(

x

,

y

,

y

)

such thattheworker obtainstheentiresurplusoftheincumbentjobplus ashareoftheincremental surplusbetweenthetwo jobs,i.e.

W1

(φ

1

,

x

,

y

)

−

W0

(

x

)

=

S

(

x

,

y

)

+ β

S

(

x

,

y

)

−

S

(

x

,

y

)

= β

S

(

x

,

y

)

+ (

1

− β)

S

(

x

,

y

).

(5)

Theshareofthe increasedsurplus

β

accruing totheworker willbe determinedempirically.Ifthesurplusinthe current matchisgreaterthan withthe poachingﬁrm, theincumbentﬁrmretainstheworker by offeringawage w

= φ

2

(

x

,

y

,

y

)

thatdeliverstotheworkerthetotalsurplusatthepoachingﬁrm,precludingbarraging,i.e.,

W1

(φ

2

,

x

,

y

)

−

W0

(

x

)

=

S

(

x

,

y

).

(6)

Inour approachthere isan asymmetry betweenworkers andfirmsbecausethe latterdo notsearch when thejob is filled.Asaresulttheydonotfireworkerswhentheyfindanalternativeworker whowouldleadtoalargertotalsurplus, nordotheyforcewagesdownwhenanalternativeworkerisfoundwhosepaywouldimplyanincreasedshareforthefirm. Wedecidedtoimposethisasymmetrybecauseinmanyinstitutionalcontextsitishardforthefirmtoreplaceworkersin thisway. Moreover,we suspect that even when allowed firmswouldbe reluctant todo so inpractice.We do,however, allowfirmstofireaworkerwhenthecurrentsurplusbecomesnegativeandthenimmediatelysearchforareplacement.

2.3.3. Productivity shocks

Anotherpotential sourceofrenegotiation iswhena productivityshockchanges y to y thus alteringthevalue ofthe surplus.If yissuchthat S

(

x

,

y

)

<

0,thematchisendogenouslydestroyed:theworkerbecomesunemployedandthejob willpostavacancy.

8 _In_what_follows_we_will_distinguish_between_wages_obtained_when_moving_from_unemployment_φ

0,whenswitchingjobsφ1andwhentheﬁrmsretains

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Supposenowthat

S

(

x

,

y

)

≥

0.Thevalueofthecurrentwagecontract w becomes W1

(

w

,

x

,

y

)

.Futurepaynegotiations whether dueto productivity shocks orcompetition withoutside offers will be affected by the newvalue ofthe match. However, thecurrent wagemay ormaynot change. Ifthe wage w is such that theworker is still obtaining at leastas much asher outsideoption,withouttakingmore thanthe newsurplus, 0

≤

W1

(

w

,

x

,

y

)

−

W0

(

x

)

≤

S

(

x

,

y

)

,neither the worker nor thejob has acredible threat to force renegotiation:both are better offwiththe currentwage w being paid to theworkerthan walkingawayfromthematchtounemploymentandtoavacancyrespectively. Inthiscasethere will be no renegotiation. If, however, W1

(

w

,

x

,

y

)

−

W0

(

x

)

<

0 or W1

(

w

,

x

,

y

)

−

W0

(

x

)

>

S

(

x

,

y

)

(with S

(

x

,

y

)

≥

0) then renegotiationwilltakeplacebecauseawagecanbefoundthatkeepsthematchviableandeachpartnerbetter offwithin thematchrelativetounemploymentfortheworkerandavacancyforthejob.

Todeﬁnehowtherenegotiationtakesplaceandwhatisthepossibleoutcomeweuseasetupsimilartothatconsidered by

MacLeod and Malcomson (1993)

andPostel-Vinay and Turon (2010).Thenewwagecontract issuchthat itmovesthe currentwagethesmallestamountnecessarytoputitbackinthebargainingset.Thus,ifattheoldcontract

W

1

(

w

,

x

,

y

)

−

W0

(

x

)

<

0,anewwage

w

= ψ

0

(

x

,

y

)

isnegotiatedsuchthat

W1

(ψ

0

,

x

,

y

)

−

W0

(

x

)

=

0

,

(7)

which just satisﬁesthe worker’s participationconstraint. Ifat the new y, W1

(

w

,

x

,

y

)

−

W0

(

x

)

>

S

(

x

,

y

)

, a newwage

w

= ψ

1

(

x

,

y

)

isnegotiatedsuchthattheﬁrm’sparticipationconstraintisjustbinding:

W1

(ψ

1

,

x

,

y

)

−

W0

(

x

)

=

S

(

x

,

y

).

(8)

Notethat afirm hiringanunemployedworker offersthevalue ofunemploymentplusa shareofthe surplus,whereas afterapositiveshocktothesurplusvaluethereisnoNashbargaining.Thismayseembothadhocandinelegant.However, considertheothersituationwhenthematchsurplusfallsbelowtheworkersurplusatcurrentwage.UnderNashbargaining the workerwouldget thesameshareofthenewsurplusasintheother triggering situation,wipingout anywagegains due tooutsideoffers. Thatdoesnot soundright. Intuitively,the worker shouldhave morebargainingpower inone case thanintheother.Ourassumption(zerobargainingpowerwhentheworkervaluesfallsbelowthevalueofunemployment; fullbargainingpowerwhenthefirmprofitfallsbelowthevalueofavacancy)mayseemabitextreme,butitismotivated by theidea thatthe worker’sbargainingpower shoulddepend onwhetherrenegotiationiswantedby theworker orthe employer,anditensuresthatthevaluefunctionoftheworkerismonotonicallyincreasinginthewage.

Furthermore,wagesrespondtojobspecificproductivityshocks,butnotalwaysinanobviousdirection.Separationsand paychangesmayhappenfollowingbothgoodshocksthatincreasethevalueofproductivityy and badshocksthatdecrease it.Itisallaboutmismatch:whatmattersiswhathappenstotheoverallsurplus.Apositiveproductivityshock,forexample, can imply that the quality ofthe matchbecomes worse and the surplus declines, since the outside option of the firm has changedandit maybe worthwhileto separate fromthecurrentworker andposta vacancyto findabetter worker. Conversely anegative productivityshockcanimprovethesurplusifthismeans thejobtype isnowclosertotheoptimal onesoughtbytheworker.Ashockthatreducesthesurpluscanstillleadtoawageincreasetocompensatetheworkerwho isnowmatchedwithajobwithfewerfutureprospectsofwageincreases.Thuswhatreallymattersasfarastheviabilityof thematchandthepossibleoptionsforrenegotiationiswhetherashockimprovesorworsensaparticularmatch,measured bywhetheritleadstoanincreaseoradecrease,respectively,ofthesurplus.

2.4. Value functions

The next step insolving the modelis to characterizethe value functionsof workersandjobs, which havebeenkept implicituptonow.Thesedeﬁnethedecisionrulesforeachagent.Weproceedbyassumingthattimeiscontinuous.

2.4.1. Unemployed workers

Unemployed workers are always assumed to be available for work at a suitable wage rate. While unemployed they receive income or money-metric utility (home production) depending on their ability x and denoted by b

(

x

)

. Thus the presentvalueofunemploymenttoaworkeroftype

x is W

0

(

x

)

,whichsatisﬁestheoptionvalueequation

rW0

(

x

)

=

b

(

x

)

+

κ

β

S

(

x

,

y

)

+v

(

y

)

d y

,

(9)

where thesubjectivediscount rateisdenotedby

r and

we deﬁne ingeneral

a

+

=

max

(

a

,

0

)

.Thustheintegral represents theexpectedvalueofthesurplusoffeasiblematchesgiventheworkerdrawsfromthedistributionofvacantjobs

v

(

y

)

.She contactsajoboftype

y at

arate

κv

(

y

)

andthematchisconsummatedifthesurplus

S

(

x

,

y

)

isnon-negative,inwhichcase shegetsashare

β

ofthesurplus.

2.4.2. Vacant jobs

Using

(3)

,

(4)

,and

(5)

,thepresentvalue ofproﬁtsforanunmatchedjob meetingaworkerwithhumancapital

x from

unemploymentis

(7)

where 1

− β

representsthe proportion of the surplusretained by the ﬁrm and

0

(

y

)

is the value of a vacancy. A job meeting aworker whois alreadyemployedwill havetopaymore toattracther.The value ofajob withproductivity y,

meetinganemployedworkerinalowersurplusmatchwithproductivity yis

1

(φ

1

(

x

,

y

,

y

),

x

,

y

)

−

0

(

y

)

= (

1

− β)

S

(

x

,

y

)

−

S

(

x

,

y

)

.

Basedontheseconsiderations,thepresentvalueofavacancyforajobwithproductivity

y is

r

0

(

y

)

= −

c

+ δ

0

(

y

)

−

0

(

y

)

γ

(

y

)

d y

+

κ

(

1

− β)

S

(

x

,

y

)

+u

(

x

)

dx

+

s

κ

(

1

− β)

S

(

x

,

y

)

−

S

(

x

,

y

)

+h

(

x

,

y

)

dx d y

.

(10) The c is a per-period cost of keeping a vacancy open. The second termreﬂects the impact of a change in productivity from y to y,assuming that productivityshocks continue to accrue atrate

δ

when the job isvacant andassuming that the distribution of productivity shocks is

γ

(

y

)

. The third term is the expected gain from matching with a previously unemployedworker. Thefourthtermisthe expectedgain frompoaching aworker whoisalreadymatched withanother job.Thenotation

[·]

+ensuresthatintegrationisoverallpossible

x and

ythatincreasethecurrentsurplus.

Asformatches,apostisdestroyed if

0

(

y

)

<

0.However,we onlyconsiderstationaryequilibriasuch thatpostsexist forever,rulingoutjobcreationanddestructionaswehaveruledoutthebirthanddeathofworkers.Aswediscussbelow ourfreeentryconditionisthatthelowestproductivityjobinthesupportmakeszeroproﬁts.Assumingthattheproduction functionincreasesin y, in

Appendix A

weshowthat

₀

(

y

)

>

0.So

0

(

y

)

>

0 forall y

∈ [

y

,

y

]

if

0

(

y

)

>

0.Theresulting equilibriumdistributionofjobsis

n

(

y

)

=

N

γ

(

y

),

(11)

andisthusexogenousup tothenumberofjobs

N.

Wedothisforsimplicitysothat wedonot havetoendogenizeboth thenumberofjobsandtheproﬁtabilitythresholdattheequilibrium.

2.4.3. Employed workers

In orderto derive the wagerates we need to deﬁne the value of ajob toa worker W1

(

w

,

x

,

y

)

. Thisis the present value totheworkerofa wagecontract w for a feasiblematch

(

x

,

y

)

(if S

(

x

,

y

)

<

0 thereisnomatchandnowage).Any wagecontract

w delivers

w in theﬁrstunitoftime.Thecontinuationvaluedependsoncompetingevents:thejobmaybe terminatedwithprobability

ξ

;thejobmaybehitbyaproductivityshockwithprobability

δ

;theworkermaybepoached withprobability

s

κ

V .

Giventheabove discussion,theﬂow valueofworkingatwagerate w in afeasible match

(

x

,

y

)

isdetermined bythe followingBellmanequation,

[r

+ δ + ξ +

s

κ

v

(

A

(

w

,

x

,

y

))

] [W1

(

w

,

x

,

y

)

−

W0

(

x

)

]

=

w

−

b

(

x

)

−

κ

β

[

S

(

x

,

y

)

]

+v

(

y

)

d y

+ δ

min

S

(

x

,

y

),

W1

(

w

,

x

,

y

)

−

W0

(

x

)

₊

γ

(

y

)

d y

+

s

κ

A(w,x,y)

min

S

(

x

,

y

),

S

(

x

,

y

)

+ β

S

(

x

,

y

)

−

S

(

x

,

y

)

+

v

(

y

)

d y

,

(12) where

A

(

w

,

x

,

y

)

=

y

:

W1

(

w

,

x

,

y

)

−

W0

(

x

) <

S

(

x

,

y

)

isthesetofjobsthatcanleadtoawagechange(eitherbymovingorrenegotiation)andv

(

A

)

=

_Av

(

y

)

d y,foranyset A.

Ontherighthandsidetheﬁrsttermisthewagenetoftheﬂowvalueofunemployment(allinparentheses).Thesecond termistheexpectedexcessvaluetotheworker ofa productivityshock(timestheprobabilitythatitoccurs): inthiscase theworkereitherendsupwiththeentirenewsurplusorthenewvalueorindeednothingifthematchisnolongerfeasible (see equations

(7)

and

(8)

).The third lineistheexpectedexcess value followingan outsideofferasinequation

(5)

.The integralisoveralloffersthatcanimprovethevalue(whethertheworkermovesornot).

2.4.4. The match output and joint surplus

Havingdeﬁnedthenon-employmentvalue fortheworker,thevacancyvalue fortheﬁrm,thevalueofacontract

w to

theworker(withasimilarexpressionforthevaluetotheﬁrm),wecannowcalculatethesurplusvalue S

(

x

,

y

)

deﬁnedby equation (3).Weshow in

Appendix A

that thematchsurplus S

(

x

,

y

)

isindependentofthe currentwagecontractandis deﬁnedbytheﬁxedpointinthefollowingequation,

(8)

(

r

+ ξ + δ)

S

(

x

,

y

)

=

f

(

x

,

y

)

−

b

(

x

)

+

c

−

κ

β

S

(

x

,

y

)

+v

(

y

)

d y

−

κ

(

1

− β)

S

(

x

,

y

)

+u

(

x

)

dx

−

s

κ

(

1

− β)

S

(

x

,

y

)

−

S

(

x

,

y

)

+h

(

x

,

y

)

dxd y

+

s

κ

β

S

(

x

,

y

)

−

S

(

x

,

y

)

+v

(

y

)

d y

+ δ

S

(

x

,

y

)

+

γ

(

y

)

d y

.

(13) Notethatthesurplusofan

(

x

,

y

)

matchneverdependsonthewage.ThisfollowsfromtheBertrandcompetitionbetween theincumbentandthepoachingﬁrmthatisinducedbyon-the-jobsearchanddisconnectsthepoachedemployee’soutside optionfromboththevalueofunemploymentandhercurrentwagecontract.ThustheParetopossibilitysetforthevalueof the workerandthejobisconvexinall cases,implyingthattheconditionsforaNashbargain aresatisﬁed.Thiscontrasts with Shimer (2006) where jobs do not respond to outside offers and where the actual value of the wage determines employmentdurationinaparticularjob.Thisfeaturealsohasacomputationaladvantagesincetheequilibriumdistribution ofmatchescanbedeterminedwithoutsimultaneouslycomputingthewageratesforworkers.

2.5. Steady-state equilibrium

Theexogenouscomponentsofthemodelarethedistributionofability

(

x

)

,theformofthematchingfunction

M

(

·,

·)

as well asthearrivalrateofshocks

δ

,thetransitionprobability

γ

(

y

)

,thejobdestructionrate

ξ

,therelativesearchintensity ofemployedworkers

s,

thediscountrate

r,

thevalueofhomeproduction(orleisure)

b

(

x

)

,thecostofpostingavacancy

c,

bargainingpower

β

,andtheproductionfunction f

(

x

,

y

)

.

Inequilibriumallagentsfollowtheir optimalstrategy. Thedistributionofmatches

h

(

x

,

y

)

,isdeterminedbythesteady state ﬂow equation, andthenumberof jobsisdetermined by afree-entry mechanismthat wemake explicitbelow. The unemploymentdistribution

u

(

x

)

andthedistributionofvacancies

v

(

y

)

followfromthebalanceequations(1)and

(2)

.

2.5.1. Steady-state ﬂow equation

In a stationary equilibrium ﬂows in and out of any worker stock must balance each other out. Let us consider the stocks ofexistingmatches oftype

(

x

,

y

)

.Theycanbe destroyedforanumberofreasons.First,there isexogenousmatch destruction at a rate

ξ

; second, with probability

δ

,the job componentof matchproductivity changes to some value y

differentfromy, andtheworkermaymovetounemploymentormaykeepthejobforminganewmatch

(

x

,

y

)

;third,the worker mayreceiveanofferandquittoahighersurplus

(

x

,

y

)

match.Ontheinﬂowside,new

(

x

,

y

)

matchesareformed whensome unemployedoremployedworkersoftype

x match

withvacantjobs y, orwhen

(

x

,

y

)

matchesarehitwitha productivityshockandexogenouslychangefrom

(

x

,

y

)

to

(

x

,

y

)

.

Equatinginﬂowstooutﬂows,wehaveforall

(

x

,

y

)

suchthatthematchisfeasible(i.e.

S

(

x

,

y

)

≥

0),

ξ

+ δ +

s

κ

v

B

(

x

,

y

)

h

(

x

,

y

)

=

[u

(

x

)

+

sh

(

x

,

B

(

x

,

y

))

]

κ

v

(

y

)

+ δ

γ

(

y

)

h

(

x

,

y

)

d y

,

(14) where

B

(

x

,

y

)

= {

y

:

0

≤

S

(

x

,

y

) <

S

(

x

,

y

)

},

B

(

x

,

y

)

= {

y

:

S

(

x

,

y

)

≥

S

(

x

,

y

)

},

arethesetsofjobsthatwouldimply(

B

)ornotimply(

B

)animprovementofthematchsurplusfor

x.

Thus,

s

κ

v

B(

x

,

y

)

istheprobabilityofreceivinganalternativeoffer,whenemployed,fromajobthatbeatsthecurrentone.

Thisequationdeﬁnesthesteady-stateequilibrium,togetherwiththeaccountingequations

(1)

and

(2)

.

2.5.2. Free entry

At this stage the only parameterthat is not yet determined is thenumber of jobs N, aswe only consider equilibria withinﬁnitelylivedjobs.Hence,

n

(

y

)

=

N

γ

(

y

)

.Jobcreationisone-shoteventandanumber

N of

jobsiscreatedaslongas

0

(

y

)

remainspositive.9

Attheequilibrium,

0

(

y

)

=

0.Hence, c

= δ

0

(

y

)

γ

(

y

)

d y

+

κ

(

1

− β)

S

(

x

,

y

)

+u

(

x

)

dx

+

s

κ

(

1

− β)

S

(

x

,

y

)

−

S

(

x

,

y

)

₊ h

(

x

,

y

)

dx d y

.

(15)

9 _We_implicitly_assume_that

(9)

Thisconditionwilldeterminethenumberofpotentialjobs

N in

theeconomy.Thisisacomplicatedequationas

N enters κ

via V

=

N

−

L

−

U and U , and

κ

conditionsallotherequilibriumvariables.

Appendix B provides a simple iterativealgorithm that uses these equations and that of the surplus to compute the equilibriumobjects.

3. Modelspeciﬁcation

Wecomplete thepresentationofthemodelwithits parametricspeciﬁcation. Thecentral elementofthe modelisthe productionfunction,whichdrivesthepotentialgainsfromsorting.Here wespecifythistobe aCESintheindividualand ﬁrmcharacteristic.

f

(

x

,

y

)

=

A

α

xρ

+ (

1

−

α

)

yρ

1ρ

_.

Thisallowsforvariousdegreesofcomplementaritydepending ontheestimatedvalue of

ρ

,includingtheextremecaseof linearity,in whichcasethere are noproductivity gainsfromsorting.The question of nonparametricidentiﬁcation ofthe degreeofcomplementarityisaddressedinanotherpaper(Lamadon et al., 2014).Followingthedatadescriptionweoffera heuristicargumentonthehowtheparametersareidentiﬁedhere.

Akeyelementofthemodelisthedistributionofworkerandﬁrmtypes.Wechoosetheﬂexible,yetparsimoniousBeta distribution,eachwithsupport

x

=

y

=

0 and x

=

y

=

1.Theparametersofthedistributionof

x (a

x,

b

x) andthose ofthe distributionof y (ay,

b

y)areestimatedtogetherwiththeremainingparametersofthemodel.When

x and

y are unobserv-able,it isnot possibleto separate the productionfunctionfrom thedistributions. Inother words we canreparameterize the modelfor x and y to haveuniformdistributions andchange theway thesetwo characteristicsenter the production function.Nevertheless,thecomplementarityparameterisidentiﬁedinanycase.

The ﬁnal componentofthe modelis thematching function.This regulatesthe numberof matchesdepending on the numberofvacanciesandthenumberofunemployedandisimportantforevaluatingcounterfactualpolicies.However, with-outﬂuctuationsin

U and V it

isaconstant.Indeedthematchingfunctioncanonlybeidentiﬁedasaresultofﬂuctuations inthe aggregate vacancies andunemploymentandconsequentlycannot be estimatedwithin ourenvironment, whereno aggregatequantityisallowedtochange.Wethustakethematchingfunctionfromtheliterature,forthepurposesof simu-latingcounterfactuals.Weuseanelasticityof0

.

5 andthefunctionalform

M

(

U

+

s

(

1

−

U

) ,

V

)

=

η

[U

+

s

(

1

−

U

)

] V

,

where the search intensity for the unemployed has been normalized to one (see, Blanchard and Diamond, 1991 and

Petrongolo and Pissarides, 2001);the search intensity of the employed s as well asthe overall level of matches

η

are estimatedfromthedatatogetherwiththeremainingparametersofthemodel.

3.1. Measurement error

Inourmodelwages arestochastic becauseofthetype ofﬁrm thatan individual mayencounter, theoutsideoffersor the layoffsthat mayoccur andthe productivityshocks.There will alsobe unexplained variation dueto her productivity characteristic

x.

Inthedataanadditionalsourceofvariationismeasurementerrorthatweneedtoaccountfor,soasnotto biastheothersourcesofvariation.

We use the monthly records of wages in the NLSY. Thus, while it may be reasonable to assume that measurement errorisindependentfromone yearto thenextit maynotbe sowithinthe year,asall recordsarereportedatthesame interviewwithrecall.Havingexperimentedwithanumberofalternatives,includingacommonequi-correlatedcomponent acrossallmonths,wesettledonameasurementerrorstructurethatiscommonwithinyearandindependentacrossyears. Thevarianceofmeasurementerror,whichisassumedtobelognormal,isestimatedalongsidetheotherparameters ofthe model.

4. Estimationmethod

Toestimate the modelwe usethe SimulatedMethod ofMoments (SMM).10 Toimplement theapproach we calculate a vectorofmoments fromthedatam

N

=

1_N

N

i=1mi,where forexample,m

N maybethe meanwageforthoseone year out of unemploymentor wage growth t periods following a job move, etc. Counterparts to these moments, deﬁned as

mM_S

(θ )

=

1_S

S

_s₌₁mM_s

(θ )

,are thenconstructed basedon S simulated careersfromthe model,givenaparametervector

θ

. Theprocedurethenistoiterateontheparameterssoastobringthemomentsconstructedfromthesimulateddataasclose aspossibletotheonesestimatedfromthedata.Speciﬁcallyweﬁndavaluefor

θ

tomaximize

LN

(θ )

= −

1 2

mN

−

mMS

(θ )

T

W_N−1

mN

−

mMS

(θ )

,

(10)

where

WN istakento be thediagonal ofthe estimatedcovariancematrix of

mN.To compute thevariance of

m

i in the actualsampleweusethebootstrap.

Thesimulatedmomentsarenotnecessarilyasmoothfunctionoftheparameters,althoughtheywouldbecomesoasthe numberofsimulationsincreasedtoinfinity.However,withanyfiniteandrelativelysmallnumberofsimulationsderivative basedmethodsarenotappropriateforfindingtheminimum.Wethususeamethoddevelopedby

Chernozhukov and Hong

(2003), which doesnot require derivatives ofthe criterion function. Theyconstruct a Markov chainthat converges to a stationary process ofwhich theergodic distributionhas amode that is asymptotically equivalent to theSMM estimator.

Appendix Cdescribesthisprocedureindetail.

5. Thedata

Weusethe1979to2002wavesoftheNationalLongitudinalSurveyofYouth1979(NLSY).TheNLSYconsistsof12,686 individualswhowere14to21yearsofageasofJanuary,1979.Itcontainsanationallyrepresentativecorerandomsample, aswellasanover-sampleofblack,Hispanic,themilitary,andpoorwhiteindividuals.Forouranalysis,wekeeponlywhite malesfromthecoresample.

We only includedata for individuals once they have completed their education. We also drop individualswho have servedinthemilitary,andfollowworkersuptothepointofanon-employmentspellof36monthsorlonger.Weconsider theseworkers tohaveleft thelabor force. Wesubdivide thedata intotwo educationgroups:highschool degreeorless, andcollegegraduate.Themodelisestimatedseparatelyonthesesubgroups.

Individualsareinterviewedonceayearandprovideretrospectiveinformationontheirlabormarkettransitionsandtheir earnings. Fromthisweconstructhistoriesatamonthlyfrequencyaggregatingthedataasfollows:we deﬁnea workeras employed in agivenweek ifheworkedmorethan 35hours inthe week.We deﬁne aworker’s employment statusina monthastheactivityhewasengagedinforthemajorityofthemonth,treatingunemploymentspellsoftwoweeksorless asjob-to-jobtransitions.Aftersampleselection,weareleftwithanunbalancedpanelof2125 individuals(446,747person months).

Weremoveaggregategrowthfromwagesbasedonaveragewagegrowthinyears10to20fromcompletionofeducation. At thispoint themainsourceofwagegrowthisduetoaggregateproductivity.Havingremovedthisconstantgrowthrate fromwagesweassumethattheremaininggrowthisattributabletogainsfromjobsearch.11

Finally,we trimthedatatoremoveavery smallnumberofoutlierswhencalculatingwagechanges.When calculating monthtomonthwagechanges,weexcludeobservationswherethewagefallsbymorethanhalforincreasesbymorethan afactorofﬁve.

6. Choiceofmomentsandidentiﬁcation

The model will be estimated based on worker level data recording transitions between jobs andbetween work and unemploymentaswellwagesovertime.Wehavenoinformationonthefirmsideitself(suchasforexampleproductivity). As such identification is challenging. Lamadon et al. (2014) discuss the formal non-parametric identification of a model similar tothis,albeitwithoutproductivityshocksandusingmatchedemployeremployeedata.Otherinsightfulworkwith moreformal discussionofidentificationis

Hagedorn et al. (2012)

,who analyzeaversion ofthemodelwithouton-the-job search,and

Bagger and Lentz (2014)

,wheresortingresultsfromheterogeneoussearchstrategies.Thekeyresultisthatthe complementaritiesintheworker–jobmatchcanbeidentifiednonparametricallyintheircontext,usingjob-sideinformation on productivity and job duration. Here we presenta heuristic description of the identification argument based on the simplerdataatourdisposalandusingaspecificparametricformforthematchproductionfunction.

Transitions in andout of work andbetweenjobs play a key role in parameters controlling labor market mobility. In particular

η

(matchingeﬃciency)and

s (relative

searchintensity)thathelpdeterminethearrivalrateofoffersareidentiﬁed by exitfromunemploymentandby mobilitybetweenjobs. Exitfromemploymentisgovernedbytwo componentsofthe model:productivityshockstothejobandtheexogenousdestructionrate.However, productivityshocksarealsorelatedto job-to-jobtransitionsandwagegrowthandcannotbesettoﬁtperfectlytheexitratefromemployment.Theremainder is capturedbytheexogenousmatchdestructionrate

ξ

.

Asnotedearlier,thesupportofthedistributionofunobservedproductivity

x and

y have bothbeennormalizedto(0

,

1) and each have beenassumed to followa beta distribution withparameters (ax, bx) and (ay,by) respectively; these are directlylinkedtothecrosssectionaldistributionofwages.Throughitseffectonproductivityshocksandjobmobility, y is

alsolinked tothevariabilityofwagegrowthwithin andbetweenjobs. Hencetheidentiﬁcationoftheweightof y in the match productionfunction,

α

, isdriven by the variancesof wagegrowth,conditional on the varioustypes oftransition. Similarlythewithinandbetween-jobvarianceofwagegrowthisinformativeabouttherateofarrivalofproductivityshocks

δ

because(togetherwitharrivalofjoboffers)suchshockscanleadtorenegotiationofwages.

Centraltothe modelisthe complementarityparameter,whichdetermines theamountofsorting.Key toidentiﬁcation with the type of data we have at our disposal is thefact that sorting, together with search frictions –identiﬁed using

11 _Our_model_does_not_allow_for_human_capital_{accumulation.}_Any_such_growth_in_the_data_is_accounted_for_by_improved_job_matches_through_job_search.

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theinformationontransitions,control thewaycross-sectionalwageinequalityincreasesforaparticularcohortofworkers asthey sortinto more suitable jobsfollowing entryintothe labor market. Tocapturethis we simulatethe evolutionof wagesandemploymentofanentrycohort–agroupofindividualsdrawnfrom

(

x

)

andallstartingoutunemployed.Under the assumptions of our model these are comparableto an NLSY entry cohort starting its labor market career following completion of education. Because of search frictions, thiscohort of workers will start off mismatched, but if there are productioncomplementaritiestheprocessofon-the-jobsearchwillresultinjobreallocationimprovingsorting;thisinturn willleadtoan increaseinwageinequality.Inaddition,ifsortingisanimportantfeatureofthelabormarket,theaverage wagegainstochangingjobswilldeclinewithtimeinthelabormarket,asthecohortofworkersmovetowardtheirideal matches.Similarly,thevarianceofwagechangesshoulddeclineastherearefeweroffersthatimprovematchessubstantially andworkersmakefewer transitions.Alternatively,ifthereare noproductioncomplementarities(or thesearevery weak) thensortingwillnotbeafeatureoftheequilibrium,andwageinequalitywillnotincreasewithtimeforourcohortasthere isnotendencyfortheeconomyto reallocatethemacrossjobs. Thedegreeofcomplementarity willaffectthevarianceof wages

within jobs

(aswellasbetween)becauseitaffectsthedegreetowhichafirmcanandwishestorespondtooutside offers.Forthesamereasonsasthosedescribed abovetheoffersthatwillbecapableofinducingwagegrowthwilldecline infrequencywithtime inthelabormarket.Thusthedegreeofcomplementarityinproductioncanbeidentifiedusingthe profileofwagegrowthanditsvariancebetweenjobsaswellaswithinjobs.12

Changingthedistribution of y and its shocksaffectstransitions bothbetweenjobsandinandout ofunemployment. In addition, changing the distribution of x will affect the variance of wage growth in a very speciﬁc way, governed by the structure ofpay settingandwill alsoaffect transitionsin andout of work. Thus the distributionof theseobjects is intimatelylinked toobservedtransitions,which willlimitthe extenttowhich wecan explainthe varianceofwages and theirgrowth.Wethusidentifythevarianceofmeasurementerrorinwagesfromthevarianceofwagesthattheeconomic modelisunabletoreproduce.

Theremainderoftheparametersareidentiﬁedasfollows.Thebargainingparameter

β

isprimarilyidentiﬁedbywages followinga job change,becauseit regulates theextent ofa job-to-jobwageincrease. However, asweshow belowmany modelgenerated moments depend onthis parameter.The parameter b in b

(

x

)

=

f

(

x

,

b

)

, whichreflects the incomeflow whileout ofwork, isidentifiedby thestarting wageinnewjobsfollowingunemployment, because(for example)a very highvaluewillnotbeconsistentwithlowinitialpay.Theflowcostofvacancyposting

c governs

theproﬁtabilityofposting vacancies,andisidentiﬁedbymatchingthevacancytounemploymentratio,whichisobservedfrommacroeconomicdata.

Identiﬁcationofthemodelinpracticerequiresacarefulchoiceofmomentsthatwillbesensitivetotheparameterswe need toestimate. In particular, tosummarizethe employment dynamics,we use thelong-run employment rateandthe transitions betweenemployment statesand betweenjobs, calculated onyears 16–20. To summarizewagedynamics, we includethelevelof(log)wagesandtheir crosssectionalvariance aswellaswagegrowthanditsvariancebothwithinand betweenjobs.Eachofthesemomentsiscalculatedseparately byyearinthelaborforce(year sinceleavingschool forthe cohort).Allthemomentsweuse,theirvaluesinintervalsofﬁveyearsandthevalueproducedbythemodelareshownin

Table 4.Inaddition,wetarget themeanvacancytounemploymentrate(basedonthemeanandstandard deviationfrom

Hagedorn and Manovskii, 2008).

Estimationandpracticalidentiﬁcationreliesonthemomentspredictedby themodelbeingsensitivetochangesinthe parameters:amodel-predictedmomentwhichissensitivetoaparticularparameterishelpfulinidentifyingit.In

Tables 6

and 7weshowthe elasticityofeachofthemoments weusewithrespect toeachoftheparameters(separatelyforeach educationgroup).Thusforexamplethecomplementarityparameter

ρ

hasanimportanteffectonmomentstodowiththe distributionofwagesandwagegrowth.Thesameistruefortheparametersrelatedtothedistributionofx and y (ax,

b

x,

ay,

b

y).Howevertheseparametersalsochangejobfindingandjobswitchingrates.Bycontrastthevarianceofmeasurement errorleavestransitionratescompletely unaffected.Examinationof thistablealso indicatesthe ourchoice ofmomentsis sufficientforidentifyingallparameters.Animportantlessontobedrawnfromthistableisthat,withsomeexceptions,itis difficulttoassociatetheidentificationofanyoneparameterwithasinglefeatureofthedata.However,acompleteanalysis ofidentificationwouldrequireananalysisoftherankofthismatrixofderivatives,whichisbeyondthescopeofthispaper.

7. Estimationresults

7.1. The ﬁt of the moments

Summaries of the ﬁt of the targeted moments by the model are presented in Table 4. As seen from the table the transitionsratesareﬁttedremarkablywell.

In Figs. 1 and 2 we summarize the ﬁt of the model for wages, wage growthand the corresponding variances, both overallandbytypeoftransitionforthelowerandhighereducationindividualsrespectively.Themodelgenerallyﬁtsthese

12 _Our_model_can_explain_the_increase_in_the_cross_sectional_variance_within_a_cohort_by_sorting._However,_in_principle_other_modeling_choices,_not_explored

here,couldalsoexplaintherise,suchaspermanentshockstoindividualproductivityorheterogeneousaccumulationofhumancapital.Nevertheless,as wepointoutsortinghasimplicationsnotonlyforthegrowthinthecrosssectionalvarianceofthewagesofacohortbutalsoforthewaythevarianceof wagegrowthevolves.Identiﬁcationinsucharichermodelfromworkerleveldataisanopenquestion.

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Notes:w isthelogwage,wtisoveralllogwagegrowth,wE Etiswithinjoblogwagegrowth,wJ t isbetweenjoblogwagegrowth;var

denotesvariance.Thesolidlinedenotesthemomentsaspredictedbythemodel,thedashedlineistheequivalentdatamomentandtheshadedarea correspondstoapoint-wise95%conﬁdenceinterval.

Fig. 1. Fit to wage moments: High school or less.

Notes:w isthelogwage,wtisoveralllogwagegrowth,wE Etiswithinjoblogwagegrowth,wJ t isbetweenjoblogwagegrowth;vardenotes

variance.Thesolidlinedenotesthemomentsaspredictedbythemodel,thedashedlineistheequivalentdatamomentandtheshadedarea cor-responds toapoint-wise95%conﬁdenceinterval.

Fig. 2. Fit to wage moments: College graduate.

patternsverywellandcertainlycapturesthequalitativefeaturesofthedata.13Anywagegrowthgeneratedbythemodelis duetothejobsearchprocessandreﬂectsmobilitytowardsbetterjobsand(forthehighereducatedpeople)improvements in sorting.Forthe lowereducated peopleaswe shallseethereare veryfew complementarities;howeverastheworkers movetohighersurplusjobs(becauseofimprovedﬁrmproductivity)andbyreceivingoutsideofferstheycanimprovetheir wage.Thisprocessofoutsideoffersisresponsiblefortheobservedwagegrowthforthelowskilled.

13 _This_comparison_does_not_reﬂect_estimation_error_and_hence_provides_a_narrower_conﬁdence_interval_for_evaluating_the_difference_between_the_model