M od el
ingM atc hed J ob -W orker F l
ow s
Dale T .M ortensen
Departm ent ofE c onomic s
Northw esternU niversity
E vanston,IL 60 2 0 8
d -m ortensen@nw u.ed u
J une 19 9 9 and revised Novemb er 19 99
A bstract:W hat can one infer about labor market ° ows from matched employer-employeepaneldata? T hepurposeofthis paperis tosketch possible answers tothis question.A generalbutsimplelabormarketequilibrium model ofhireandseparation° ows is developedinthepaper.T hemodelembodies the hypothesis thatworkerproductivity di®ers across employers and thatworker andemployer° ows re° ectresponses tothesedi®erences inalabormarketchar-acterized by friction.In themodeled market,each agentacts optimallytaking as given thewageo®erdistributionand markettightness and thesein turn are determined by theircollectiveaction.T heexistenceofalabormarketequilib-rium is establishedundertwodi®erentwagedeterminationmodels:rentsharing andwageposting.A demonstrationthatmarket° owparameters,searchandre-cruitinge®ortfunctions,andtheequilibrium wagedistributioncanbeestimated with matched job-work° owdatais theprincipalcontribution ofthepaper.
Keywords: Job Flows, W orkerFlows, L aborM arketEquilibrium, W age D istribution
1
In
trod uc tion
W hatcanoneinferabouttheroleofjobandworker° owsfrom paneldataonthe laborforcedynamicsofindividualestablishmentsmatchedtoworkeridenti¯ers? P reliminary answers tothis question are raised in this paper.Essentially the ideaproposedhereistouseasynthesisofthetheoreticalconstructsfoundinthe recentliteratureonthe`°ows approach'toequilibrium labormarketanalysis as developed in P issarides (1 990 ), M ortensen and P issarides (1 994), B ertola and Caballero(1 994), B urdettand M ortensen (1 998),and B ontemps etal.(1 998) as a toolforanalyzing the recently available matched employer-workerpanel datadescribed byA bowd and Kramerz (1 999).
A simple frameworkforthe analysis ofthe relationships between hire, lay-o®,and quit° ows and theirdeterminants designed toexplain theevolution of establishmentsize is presented in the paperand setwithin a synthesis ofthe searchand matchingequilibrium frameworks.FollowingtheargumentofIdson and O i (1 999), the explanation forlaborforce size di®erences studied in the paperis heterogeneity in employerproductivity.H eterogeneity ofthis form is alsoneeded tounderstand the behaviorofjob ° ows as documented by D avis, H altiwangerand Schuh (1 996). A n implied positive correlation between the wagepaid and plantsizeforboth abilateralbargainingrentsharingmodelof wagedeterminationas assumedbyP issarides (1 990 )andadynamicmonopsony modelin which each employerposts awageo®eras in B urdettand M ortensen (1 998)is an implication of the hypothesis.1 W ithin the framework studied,
more productive establishments hire at greaterrates, pay better, experience lowerturnoverrates,and,as aconsequence,growtoalargersizeinbothcases.
T hebehavioralrelationships ofthemodel,appropriatelyaggregated,deter-mine the distributions ofwage o®ers as wellas markettightness and the level ofemploymentundereitherwagedeterminationhypothesis.B ut,markettight-ness, unemployment, and the wage o®erdistribution provide the information thatagents usetomake search and recruitingdecisions.A proofthataratio-nalexpectations marketequilibrium exists forboth therentsharingand wage postingmodels ofwagedetermination is the principaltheoreticalresultofthe paper.A demonstrationthattheequilibrium behaviorrelationships andaggre-gates canbeestimatedusingmatchedavailablematchedjob-worker° owdatais themoreoriginaland interestingcontribution.In sum,thegoalofthepaperis thedevelopmentofatheoreticalmarketequilibrium approachfortheempirical analysis ofmatched employer-employeedata.
Insection2,adynamictheoryofanestablishment'slaborforcesizeiscreated bymergingthetheoryofoptimalworkersearche®ortwiththetheoryofvacancy creation.T he resultis a characterization ofestablishmentlaborforce size,n; as a simple `birth-death'M arkov process characterized by a hiring frequency, 1T he existence ofthe relationship isd oc um ented b y B row nand M ed o® (19 89) for the U S.
B ontem pset al. (199 8) and Ab ow d et al. (19 9 9) d em onstrate that the relationship exists inFrance, B lanch°ow er et al.(19 96) provid esevid ence for the relationship inE ngland , and Albraek (199 8) and B ingley and W estergaard -Nielsen(19 9 6) show that plant size e®ec tson w agesare ofsim ilar m agnitud e inDenm ark.
h;and separation rate,d.In thespecialcaseofconstantmarginalproductper workerbothareindependentofnandtheinvariantdistributionforthis process isP oissonwithmeanequaltoh=d:T heexpectedfrequencieswithwhichaworker receiveso®ersandanemployerreceivesjobapplicantsaswellasthedistribution ofwageo®ersareregardedas givenbyeachindividualagentbutaredetermined by theircollective activities.A n equilibrium is a setofoptimalagentactions thatgenerate the market° owparameters and wage o®erdistribution thatall theagents expect.Equilibrium is de¯ned and existenceproofs areprovided in section 3 forboth the rentsharing and wage posting cases. A s noted above, the model implies that an employer's labor force size is a known stochastic process.A recursivemaximum likelihoodestimationstrategybasedonthis fact is devised in section 4 ofthepaper.U nderthe strategy,search and recruiting levels foreachemployertypeandtheequilibrium distributionofwageo®ers are identi¯ed.
2
Lab or Forc e Dyn
am ic s
2 .
1
Searc h Strategy an
d R eservationW age
W orker's choose acceptance and search strategies to maximize the expected present value of their own future income stream. L etr¸0 represent the common discount rate and letF (w)represent the fraction of vacancies that o®erwageworless..T he worker's maximalvalue ofa match with thatpays wagewsolves rW (w)= max s¸0 ½ w+ ±(U ¡W ) +¸sRmaxhW (we);W (w)i¡W (w)]dF (we)¡cw(s) ¾ (1 ) where¸sis theP oissonrateatwhichoutsideo®ers arrivegivensearchintensity s,cw(s)denotes thecostsearch e®ort,±is theexogenous job destruction rate, andU represents thevalueofunemployment.T herelationship re° ects thefact thatthe workeronly quits when an outside o®erhas a highervalue than the currentjob and thefactthatworker's expected presentvalueoffutureincome falls toU whendestructionoccurs.Similarly,thevalueofunemploymentsolves
rU = max s¸0 ½ b+ ¸s Z [maxhW (we);Ui¡U]dF (w)¡cw(s) ¾ (2) wherebrepresents incomewhen unemployed.G iven thattheo®erdistribution has aboundedsupport,theexistenceanduniqueness ofaincreasingvaluefunc-tionW (w)and an associated value ofunemploymentU is consequence ofthe fact that both can be represented as ¯xed points of contraction maps. (See M ortensen and P issarides (1 999b).)
O necan rewritetheB ellman equation(1 )as rW (w)= max s¸0 ( w¡c(s)+ ¸s Zw w [W (we)¡W (w)]dF (we)¡±[W (w)¡U] )
wherewdenotestheuppersupportofthewageo®erdistribution.Consequently, W 0(w)= 1 r+±+ ¸s(w)[1 ¡F (w)] bytheenvelopetheorem,where s(w)= argmax s¸0 ( ¸s Zw w [W (we)¡W (w)]dF (we)¡cw(s) ) is theoptimalchoiceofsearche®ort. Integration byparts implies
s(w) = max s¸0 ( ¸s Zw w [W 0(we)[1 ¡F (we)]dwe¡cw(s) ) (3) = max s¸0 ( ¸s Zw w [1 ¡F (we)]dwe r+ ±+ ¸s(we)[1¡F (we)]¡cw(s) ) :
H ence, optimalsearch e®ortis the unique particularsolution to the ordinary di®erentialequation
c00w(s(w))s0(w)= ¡
¸[1 ¡F (w)] r+ ±+ ¸s(w)[1 ¡F (w)] consistentwith the boundaryconditionc0
w(s(w))= 0:A s an implication,s(w) is decreasingand di®erentiableifF (w)is continuous.
W henunemployed,thetypicalworkeraccepts ano®eronlyifitis nosmaller thanthereservationwage.A sthereservationwageequates thevalueofemploy-menttothevalueofunemployedsearch,denotedbyU ;acceptancerequires that w¸R where the reservation wage,R ;solvesU =W (R):G iven thata wage o®erarrives atrate¸sand thatwage is arandom drawfrom the distribution ofo®ersF ,U solves theB ellman equation
rU = max s¸0 ½ b¡cw(s)+ ¸s Z [maxhW (we);Ui¡U]dF (w) ¾ max s¸0 ( b¡cw(s)+ ¸s Zw R [W (we)¡W (R)]dF (w) )
wherebis the unemploymentbene¯tforgone when employed andW (R)=U is the de¯nition ofthe reservation wage. H ence, equations (1 )and (3)imply that the search intensity of an unemployed workeris the same as that of a workeremployed atthereservation wageprovided thatthereservation wageis theunemploymentbene¯t,
s0 =s(R) (4)
which in turn implies thatthereservation wageis equaltothe unemployment bene¯tifthelowestwageo®ered is theunemploymentbene¯t,i.e.,
2 .
2
R ec ruitin
gStrategy an
d W age O ®er
In this subsection,atheoryofrecruitingbehavioralongthelines developed by P issarides (1 990 )andB ertolaandCaballero(1 994)is usedtoderiveanoptimal establishmenthire ° ow. Each employerrecruits new workers by posting job vacancies,anumberdenoted byv.2
Employers di®erin productive e±ciency.L etprepresentthe productivity ofajob-workermatch and¹(p)representthe measure ofemployers with pro-ductivity equalto orless thanp:A llworkers are perfect substitutes and all havethesameoutsideoptiontoemploymentcharacterizedbyacommonreser-vation wageR =bas established above.Forthe moment, letthe pair(p;w) characterizeand employerwherewrepresents herwagero®er.
L et´0s0 and´1s(w)denote the frequencies with which unemployed
work-ers and employed workerrespectively apply fora vacancy.A s allworkwork-ers are identical,allwages o®eredinthemarketbyparticipatingemployermustbeac-ceptabletounemployedworkers.H owever,becauseanemployedworkeraccepts onlywheno®eredahigherpayingjob,therateatwhichavacancyo®eringwage wismatchedwithsomeworkeris´0s0+´1 Rw ws(z)dG(z)whereG(w)represents thefractionofworkers earnworless,i.e.,thewagedistribution.Eachemployer posts thenumberofvacancies thatmaximizes theexpectedpresentvalueofthe ¯rm's futurecash° owattributabletorecruitingactivity.Formally,thenumber ofvacancies posted solves
¼(p;w)= max v¸0 ½ v[´0s0 + ´1 Zw w s(z)dG(z)]J(p;w)¡cf(v) ¾ (6) whereJ(p;w)denotes theexpected presentvalueofan employed workergiven the employer's productivitypand wagewandcf(v)is an increasing, twice di®erentiable,and strictlyconvexcostofrecruitingfunction.
B ecauseanemployedworkerquits whenahigheroutsideo®eris located,the employer's valueofacontinuingmatch solves theB ellman equation
rJ(p;w)=p¡w¡±J(p)¡¸s(w)[1 ¡F (w)]J(p): Equivalently,
J(p;w)= p¡w
r+ ±+ ¸s(w)[1 ¡F (w)]: (7) A ftersubstitutionforthevalueofa¯lledjobfrom (7),equation(6)impliesthat theoptimalnumberofvacancies posted byan employerofproductivitypis
v(p;w)= argmax v¸0 ( Ã [´0s0 + ´1 Rw ws(z)dG(z)] r+ ±+ ¸s(w)[1¡F (w)] ! (p¡w)v¡cf(v) ) : (8) 2Vac ancieshere are b etter m easured b y anind ic ator ofrec ruitinge®ort rather thana c ount
T hesolution tothesu±cient¯rstordercondition c0f(v(p;w))= [´0s0 + ´1 Rw ws(z)dG(z)](p¡w) r+ ±+¸s(w)[1 ¡F (w)] : (9) Inalabormarketcharacterizedbyfriction,thewagerateo®eredbyanyem-ployeris essentiallyindeterminate.3 FollowingP issarides (1 990 )amongothers,
onemightsupposethatthewageis theoutcomeofasimplebilateralbargaining problem.Forexample,supposethatwhenanunemployedworkerandemployer meet, the workermakes a wage demand with probability¯ and the employer accepts orrejects.Conversely,theemployersets thewagewith complementary probability 1 ¡¯ and the workeraccepts orrejects.Ifnegotiation ends after the¯rstround,theexpectedoutcomeyields thesimplelinearrentsharingrule
w(p)=¯p+ (1 ¡¯)R (1 0 ) because each agent will demand all the rent when he or she has the power to setthe wage. N ote thatthe recruiting e®ortunderthe rentsharing wage rulev(p;w(p));increases with employerproductivitypfrom (9)given thatthe employer's expected shareofmatchrent¯ is afraction.
Instead of sharing the rent, B urdett and M ortensen (1 998)assume that employers postthewagetheypayallemployees onceand foralland thateach workeronlyhavethepowertoacceptorreject.B ecausetheemployersets the wageas wellas vacanciestomaximizepro¯tfrom recruitingactivities asde¯ned in equation (6),thewage postingruleis de¯ned by
w(p)= argmax w¸R ( Ã [´0s0 + ´1 Rw ws(z)dG(z)] r+ ±+ ¸s(w)[1 ¡F (w)] ! (p¡w) ) : (1 1 ) B ecausethe¯rstterm,theproductoftheacceptanceprobabilitydividedbythe rateatwhichmatchrentis discounted,is increasinginthewageo®er,thewage rule is wellde¯ned. B e implication, recruiting e®ortv(p;w(p))also increases with employerproductivityfrom equation (9)and theenvelopetheorem under wageposting.
Finally,itis usefultonotethattheproductivity ofthemarginalemployer, thesmallestwageo®eredandthereservationwageallequaltheunemployment bene¯tbunder either wage setting rule if employers exist with productivity slightlyhigherthan thebene¯t.Formally,
w=p=R =bif¹(b+ ")> 0 forallsmall"> 0: (1 2) T his result is implied by (8), (5), and the wage rule in each case. N amely, becausev(p;w)> (=)0 asp > (=)win both cases,v(p;w(p))= 0 =) p= w(p)=w=R =bunderboth the rentsharing rules, equation (1 0 )and the wage posting rule (1 1 ). T he quali¯cation is needed in the rent sharing case becausep=p0 if¹(p0)= 0 forsomep0 > band,therefore,w=w(p0)> R > b.
A lthoughw=R =balways in thecaseofwageposting,thequali¯cation itis stillnecessaryforp=b:
2 .
3 Lab or Forc e Siz e
T hatemployerproductivity,wageand laborsizeareallpositivelycorrelated is a welldocumented stylized fact.In this section we demonstrate thatpositive associationsbetweenanytwoofthesethreevariableisimpliedundereitherwage determination hypothesis.
B ecause in any su±ciently smalltimeintervaleitheraworkeris hired,one separates orneithereventoccurs,laborforcesizeis aspecialM arkov chain on theintegers knownas abirth-deathprocess.Formally,anemployer's laborforce sizefn(t)gevolves as astochasticprocess characterizedbythe`birth'frequency
h(p;w)= [´0s0 +´1
Zw w
s(z)dG(z)]v(p;w) (1 3) and a`death'rateperemployed worker
d(w)=±+ ¸s(w)[1 ¡F (w)]: (1 4) Itiswellknownthedistributionofthefuturevalueofn(t)convergestoaP oisson with mean Efng=n(p)=h(p;w(p)) d(w(p)) = [´0 + ´1 Rw ws(z)dG(z)]v(p;w(p)) ±+ ¸s(w)[1¡F (w(p))] : (1 5) att! 1 forany initialvalues (See, Feller(1 968)). B ecause the acceptance probability increases withw, the quit rate decreases withw;and recruiting e®ortv(p;w(p))increase withpunder either wage determination hypothesis implythatthelaborforcesizen(w)increases withbothproductivitypandthe wagew(p).O fcourse,thepositiveassociationis inducedbythefactthatmore productiveemployers both pay ahigherwageand employmoreworkers in the longrun.
Forfuturereference,itusefultoderivethelaws ofmotionfortheprobability thatthelaborforcesizewilltakeon anyparticularintegervaluein thefuture. T he argumentis based on the factthateitheraworkeris hired,aworkersep-arates, orneitherhappens in any su±ciently shortperiod oftime.A ny other changeinsizewouldinvolvesomecombinationofindependenthireandsepara-tionevents,theprobabilityofwhichvanishes relativetothelengthoftheperiod as theperiod length itselfconverges tozero.
Indeed,theprobabilityofatransitionfrom alaborforcesizen¡1 tooneof ninanysu±cientlyshorttimeintervaloflength¢ is approximateequaltoh¢ ; the probability oftransition from sizen+ 1 tonis approximatelyd(n+ 1 )¢ and theprobabilityofmakingatransition from anyotherlaborforcesizeton is oforder¢ 2 orhigher.H ence, the probability thatthe laborforce is ofsize
natdatet+ ¢ ;denotedaspn(t+ ¢ )conditionalon thedistribution overnat timetsatis¯es
pn(t+ ¢ ) = pn¡1(t)h¢ + pn+ 1(t)d(n+ 1 )¢
foralln¸1 and
p0(t+ ¢ )=p1(t)d¢ + [1¡h¢ ]p0(t)+ o(¢ )
whereo(¢ )=¢ ! 0 as ¢ ! 0:In continuous time the distribution evolves accordingtothelaws ofmotion _ pn= lim ¢! 0 ½pn(t+ ¢ ) ¡pn(t) ¢ ¾ =hpn¡1 + (n+ 1 )dpn+ 1 ¡(h+ nd)pn (1 6) forn¸1 and _ p0 =dp1 ¡hp0: (1 7)
T he invariantdistribution is the steady state solution tothese di®erential equations.T hesteadystateconditions imply
p1 = µ h d ¶ p0; and p2= (h+ d)p1 ¡hp0 2d = 1 2 µ h d ¶2 p0: H ence,byinduction pn = (h+ (n¡1 )d)pn¡1 ¡hpn¡2 nd = 1 n µh d ¶ pn¡1 = 1 n! µh d ¶n p0 forn= 1;2::: Finally,because 1 = 1 X 0 pn=p0 1 X 0 1 n! µh d ¶n =p0e h d theinvariantdistributionoflaborforcesizegivenhiringandseparationrates is theP oisson distribution with meanh=d,i.e.,
pn(h;d)=e ¡(h=d) n! µh d ¶n : (1 8)
3 Lab or M arket E quil
ib rium
3.
1
T he M atc hin
gP roc ess
T hevarious meetingrateparameters,¸;´0 and´1;areinterrelatedthroughthe
berepresentedbyamatchingfunctionthattransforms matchinginputs,search andrecruitinge®ort,intoanaggregatemeeting° ow.W emadetwoassumptions thatsimplifytheanalysis thatfollows:(1 )T hesearche®ortofunemployedand employed workers are perfectsubstitutes forone another. (2)T he matching function exhibits constantreturns.B oth oftheseassumptions arestandard in thesearch and matchingequilibrium literature.(SeeM ortensen and P issarides (1 999a,1 999b).)
T hemarketmeetingrateis thevalueofthelinearly homogenous matching functionM (V ;S)where
V =
Zp p
v(p;w(p))d¹(p) (1 9) represents theaggregatenumberofvacancies posted,
S =us0 + (1 ¡u)
Zw w
s(w)dG(w): (20 ) U nderthe usualassumption ofrandom matchingatrates proportionaltoown search(recruiting)e®ort,themeetingfrequencies perunitsearche®ortandper vacancyrespectivelyare ¸ = M (V ;S) S =M µV S;1 ¶ =m(µ) s0´0 = M (V ;S) V ³u S ´ = µm(µ) µ ¶ ³s0u S ´ (21 ) s(w)´1 = M (V ;S) V µ (1 ¡u)s(w) S ¶ = µ m(µ) µ ¶ µ (1¡u)s(w) S ¶
wheremarkettightnessis representedbytheratioofrecruitingtosearche®ort,
µ´VS =
Rw
wv(p;w(p))d¹(p) us0 + (1 ¡u)Rwws(w)dG(w)
: (22)
3.
2
Stead y State U nem pl
oym en
t an
d W age Distrib ution
B ecauseallo®ers areacceptabletoan unemployed worker,theunemployment ratethatequates ° owin and outofunemployment,thesteadystate,solves
u 1¡u= ± ¸s0 = ± s0m(µ) :
from (4)and (21 ).B ysubstitution foruin to(22),µsolves µh±s0 + m(µ)R w ws(w)dG(w) i ±+ s0m(µ) = Zw w v(p;w(p))d¹(p) (23)
given the search e®ortand vacancy function.B ecausem(µ)is increasing and concave, the left side is monotone increasing inµ and is zero if and only if µ = 0:H ence, there is a unique solution forµ given any search e®ort and vacancyfunction pair(s(w);v(p;w)).
A sthesteadystatemeasureofworkers earningworless,G(w);balances the ° owofworkers intoand outofthecategory,thesteadystatewagedistribution function solves ±G(w)+ ¸[1 ¡F (w)] Zw R s(z)dG(w)=¸s(R)F (w)u (1 ¡u) =±F (w) (24) wheretheleftsideaccounts forthefractionofemploymentis thecategorywho arelaid o®plus thefraction whoquittotakehigherpayingjobs and theright sideis the° owintothecategoryfrom unemploymentexpressed as afractionof employment.Finally,thewageo®erdistribution,thefractionofvacancies that o®erwageworless,is inducedbythevacancyfunctionv(p;w)andthemeasure of¯rms thatpaywagew(p)orless:Indeed,
F (w(p))= Rw(p) w v(p;we (pe))d¹(pe) Rw wv(p;we (pe))d¹(ep) : (25)
D e¯nition1 G iven a vacancy functionv(p);and a search intensity function
s(w);theassociatedlabormarketsteadystateis thetightness parameterµ;o®er distributionF (w);andwage distributionG(w)de¯nedbyequations (23),(24), and(25).
3.
3 E xistence
In asteadystateequilibrium,each workersearches and each employerrecruits atoptimalrates takingas giventhelabormarketsteadystateinducedbytheir collectiveactions.
D e¯nition2 A steady state rent sharing equilibrium is a pair of functions
(s(w);v(p))and the associated labor marketsteady state thatsatisfy the op-timalityconditions,equations (2)and(8),given thatthe wage is determinedby the linearrentsharingrule (9).
If search and recruiting e®ort are both essential inputs in the matching process, i.e.,M (0;S)=M (V ;0 )= 0 forallS and V ;then a trivialno-trade equilibrium always exists.Since the application rate pervacancy is´0 + ´1 = M (V ;0 )=V = 0 when noworkers search and because the o®erarrivalrate per unitofsearch e®ortis¸=M (0;S)=S = 0 when noemployerrecruits,notrade in the sense thatV =S = 0 is an equilibrium because every individualagent oneithersideofthemarkethas noincentivetomakethee®ortto¯ndamatch. B elowwedemonstratethatanon-trivialequilibrium alsoexists ifalittlesearch e®ortis costless.Fortechnicalreason,wealsoneedtoassumeanuniform upper bound on vacancies.
A ssumption1 T hecostofsearchcw(s)is strictlyincreasing,strictlyconvex, andtwicedi®erentiableon[s;1)forsomearbitrarilysmallnumbers > 0 andcw(s)= cw0(s)= 0 foralls2[0;s]:
A ssumption2 T hecostofrecruitingcf(v)is strictlyincreasing,strictlycon-vex, and twice di®erentiable on [0;v]forsome arbitrarily large number v < 1,cf(0 )=c0f(0 )= 0 andcf(v)=1 forv > v:
T heargumentfortheexistenceofanequilibrium isbasedonthefactthatthe equilibrium conditions de¯neamapping,T ;from thespaceofrealvaluedvector function pairsf(w)= (s(w);v(w))de¯ned on thecompactinterval[R ;w],call itz;toitselfwhereR =bundertheassumptionthattherearepotentialentrant whowould pay less than thereservation wage,i.e.,¹(b)> 0 .B y construction, any ¯xed pointofthe map is an equilibrium.T o deriveT, note thatforany f2z thesteadystatelabormarketequations (23)-(25)uniquelydeterminea vector(µ;F )2 <+ £¢ ;where¢ is thespaceofdistribution functions.L etT1
denote this transformation.T he optimality conditions, equations (2)and (8), generateauniquef2zforanychoiceof(µ;F )2 <+£¢ .L etT2representthis
map.T osum up then,atransformationT :z ! z exists,whereT f=T1T2f
and any¯xed pointf=T fis arentsharingequilibrium.
In the caseofameasureofemployers o®eringwageworless,¹(w), which is continuous,z is a subset of the space of continuous bounded real valued function pairs de¯ned on [R ;w];with the sup norm,underA ssumptions 1 and 2.Formally,equations (2)and(21 )implythat
c0w(s(w))=m(µ)
Zw w
[1 ¡F (we)]dwe
r+ ±+ m(µ)s(we)[1 ¡F (we)] (26) foranyµ.If¹(w)is continuous thenF (w)is also continuous given any con-tinuous functionv:[R ;w]! <+ from equation (24).H ence,thesolutions(w)
totheequation aboveis acontinuous map from theinterval[R ;w]to[s;s(R)]: Furthermore,s(w)! sasm(µ)! 0 and s(R) · c0¡f1 Ã m(µ) Zw R [1 ¡F (we)]dwe r+ ±+ m(µ)s(we)[1 ¡F (we)] ! (27) · s´c0¡f1 µm(µ)(w ¡R) r+ ± ¶
whereµ is a uniform upperbound on possible equilibrium values of market tightness implied by A ssumptions 1 and 2 and derived below.O fcourse, the ¯rstinequalityis impliedbythefactthatµ·µandthatm(µ)is increasingand thesecondbythefactthatF (w)2[0;1 ],s(w)¸0 .Insum,s(w)isacontinuous bounded decreasingfunction ifa uniform upperbound on markettightnessµ exists.
W eproceedtoshowthataboundonmarkettightness exists underA ssump-tions 1 and2.First,notethatthemarketequassump-tions of(21 )implythatthe¯rst
ordercondition forandinteriorsolution,equation (9),can bewritten as c0f(v(p:w(p))) = µm(µ) µ ¶2 4±+ m(µ) Rw(p) w s(z)dG(z) ±+ m(µ)Rwws(w)dG(w) 3 5 (28) £ · p ¡w(p) r+±+ s(w(p))m(µ)[1 ¡F (w(p))] ¸
by equation (8). T he solution is unique, continuous and increasing inpand v(R)= 0 forany continuous distribution functionF (w), a condition already established.O fcourse,forsu±cientlysmallvalues ofµ,v(p)is atthecornerv speci¯ed in A ssumption 2.
G iven that(26)and A ssumption 1 imply thats(w)is uniformly bounded belowbysand that(28)and A ssumption 2 implyv(x)is uniformly bounded abovebyv;
µ´v[¹(p)s¡¹(p)] (29) represents a uniform upper bound onµ by equation (22)given anyf 2 z. Consequently,wehavecompleted the demonstration thatT :z ! z wherez is asubsetofthespaceofcontinuous boundedrealvaluedfunctionpairs de¯ned on [R ;w]:
T heorem 3 Ifthe measure ofemployers ofproductivityporless,¹(p);is con-tinuous and¹(b)> 0, ifthe di®erence between the uppersupportofthe pro-ductivitydistribution,x,andtheunemploymentbene¯t,b;is positive and¯nite, and if the o®er arrivalrate per unitof search e®ort,m(µ);is non-negative, increasing,continuous andconcave,then a non-trivialrentsharingequilibrium exists underA ssumptions 1 an 2.
P roof.SeetheA ppendix.
3.
4
AnE xam pl
e
P issarides'(1 990 )theoryofequilibrium unemploymentis alimitingexampleof thegeneralframework.Speci¯cally,his modelcorresponds tothe case ofonly one employertype and a constantposting costpervacancy.Itis wellknown thatauniquenon-trivialequilibrium exists inthis specialcase.
T ocharacterize the equilibrium,¯rstnote thatthe lack ofwage dispersion implies nosearch whileemployed,i.e.,s(w)= 0 forw> R from equation (26) becausew=wis theonlywageo®ered.A s aconsequence,
W (w)¡U = w¡rU r+ ± R = rU = max s¸0 fb¡c(s)+ sm(µ)[W (w)¡U]g J(p) = p¡w r+ ±
from equation (1 ), equation (3), the de¯nition of the reservation wageU = W (R);andequation(7)wherep=pistheproductivitycommontoallemployer. H ence,P issarides'assumption ofN ash bargainingovermatch surplus,i.e.,
w= argmax w n (W (w)¡U)¯¡J(x;w)1¡¯¢o; is theequivalenttothelinearrentsharingrulede¯ned in equation (9).4 Equilibrium markettightnessisdeterminedbyequations(28)and(9),rewrit-ten hereas theP issarides (1 990 )`freeentry'condition
aµ m(µ)= p¡w(p) r+± = (1 ¡¯) · p¡R r+± ¸ wherea=c0
f(v)is the constantcostofposting a vacancy and the aggregate number of vacancies posted by all employers is implied byµ =V =s0u from
equation(22).B ecausesearchintensitywhen unemployed solves the¯rstorder condition c0w(s0)=m(µ)[W (w(p))¡U]=m(µ) · ¯(p¡R) r+ ± ¸
from equation (9),thereservation wagecan beexpressed as R =b¡cw(s0)+ s0c0w(s0):
A non-trivial equilibrium is any triple (µ;s0;R)that satis¯es these three
equations and theassociated steadystateunemploymentrateis u= ±
±+ s0m(µ)
: (30 )
G iven thatm(µ)is increasing and concave, the ¯rstequation implies a nega-tivelysloped relation betweenR andµwhich lies in thepositivequadrant.A s s0c0w(s0)¡c(s0)is non-negative and increasingins0 underA ssumption 1 ,the
second and third equation implyan increasingrelationship between theR and µ such thatR =bwhenµ = 0:Consequently, a unique positive solution for (µ;s0;R)exists ifx¡b¸0 .
3.
5 W age P ostin
gE quil
ib rium
Instead of sharing the rent, B urdett and M ortensen (1 998)assume that em-ployers postthe wage they pay allemployees and thateach workeronly have 4How ever, thisequivalence result holdsonly inthe c ase ofhom ogenousem ployers.Inthe
heterogenousc ase, w orkersem ployed at lessthanthe highest w age o®ered w illsearc h w hile em ployed . B ec ause em ployed w orker'sd on't ac c ount for the lossinrentssu®ered b y their em ployer w hena quit takesplac e, the Nash b argaining solutionw illb e set to partially o®set thisexternale®ec t ofsearch onthe job .
the powertoacceptorreject.B ecause the employersets the wage as wellas vacancies to maximize pro¯tfrom recruiting activities as de¯ned in equation (6),thelinearrentsharingrule,equation (1 0 ),is replaced by thewage posting rule,equation(1 1 ),whichmaximizes theemployer's expectedfuturediscounted pro¯tattributabletoemployingaworker.T hatcondition can bewritten as
w(p)= argmax
w¸0 f(p¡w)`(w)g: (31 )
wheretheemployer's steadystate`laborsupply'is `(w) = ´0s0 + ´1 Rw ws(z)dG(z) r+ ±+ ¸s(w)[1 ¡F (w)] (32) = µ m(µ)=µ r+ ±+ ¸s(w)[1 ¡F (w)] ¶ 2 4±+ m(µ) Rw ws(z)dG(z) ±+ m(µ)Rwws(z)dG(z) 3 5 (33)
from equations (21 )and(??).5 T hefactthatahigherwageincreases
theprob-abilitythatan employed applicantwillacceptand decreases therateatwhich existingemployeesquittotakehigherpayingjobs,theemployerhasandexploits dynamicmonopsonypower.
A labormarketequilibrium with wagepostingis de¯ned as follows:
D e¯nition4 A steadystatewagepostingequilibrium isafunctiontuple(s(w);v(p);w(p))
and the associated labormarketsteady state thatsatisfy the optimality condi-tions,equations (2),(8),and(31 ).
A s demonstrated byB urdettandM ortensen(1 998)and byB ontemps etal. (1 998),w(p)is unique,strictlyincreasing,w=R ;andR < w(p)< pwhen the distributionofpisatomlessjustasinthelinearrentsharingcase.6 Furthermore,
theoptimalvacancychoice,which solvesc0
f(v(x))= maxw¸0 f(p¡w)`(w)g;is
also increasing inp:Consequently, the numberofvacancies posted by a ¯rm o®ering wagewis also increasing. B ecause the wage posting wage rule and vacancypolicysatis¯esallthesamequalitativepropertiesasunderrentsharing, theexistenceofanon-trivialwagepostingequilibrium follows byessentiallythe sameargumentusedabovegivenaslightlystrongerconditiononthedistribution ofproductivity.
5O fc ourse,`(w) isthe prod uc t ofthe ac c eptance prob ab lity and the present value ofa unit
future stream ofincom e d isc ounted at a rate equalto the sum ofthe interest and separation rates.How ever, inthe c ase ofr= 0; `(w) isequalto the expec ted longrunsiz e ofthe lab or forc e ofanestab lishm ent that paysw agew b y equation(15).T hisfac t tiesthe m od elto that stud ied inB urd ett and M ortensen(199 8).
6O ne d i®erence isthat the low est w age isequalto the reservationw age,R ;evenifthe
low er support ofthe d istrib utionofem ployer prod uc tivity islarger. T he reasonisthat the em ployer payingthe low est w age hiresno em ployed w orkersand losesallem ployeesto higher paying ¯rm s. Hence, that em ployershasno incentive to pay m ore thanthe w age thanall unem ployed w orkers¯nd ac c eptab le.
T heorem 5 Ifthe measure ofemployers whowith productivityporless,¹(p);
is di®erentiable, ifthe di®erence between the uppersupportofthe distribution,
p;and the unemploymentbene¯t,R =b;is positive and ¯nite, and ifthe o®er arrivalrate per unitof search e®ort,m(µ);is non-negative, increasing, con-tinuous and concave, then a non-trivialwage postingequilibrium exists under A ssumptions 1 and2.
P roof.SeetheA ppendix.
4
E stim ation
A n estimation strategy thatuses matched employer-employee data which in-clude the size ofan employer's laborforce and employerspeci¯c worker° ows is sketched in this section. A s an example, consider the D anish Integrated D atabase forL aborM arketR esearch (ID A )¯le. It includes annualobserva-tions ontheemploymentofallD anish establishments,employeeidenti¯ers,the earningofeach employee, and employee characteristics forallyear1 980 -1 995. (SeeB ingleyetal.(1 999).)T heseobservations overasingleyeararesu±cient toidentifythestructureofthemodelconstructed above.
Speci¯cally,non-parametricsearchandrecruitingpolicyfunctionss(w)and v(p)=v(p)togetherwith the transition parametervector(±;¸;´0;´1)can be
estimatedusingamaximum likelihoodprocedureforgiveno®erandwagedistri-butionfunctionsF andG.Forthis purpose,itis naturaltousenon-parametric estimates ofthe twodistribution functions based on the wages received by all workers hiredduringtheyearandwages earnedbyallworkers employedatthe beginning,bothofwhichareobservableintheID A data.O ncetheseinitiales-timates areobtained,theassociatedwageando®erdistributions impliedbythe aggregate labormarketand steady state conditions, equations (22)-(25), can be computed. O bviously, one can use these computed distribution functions and the samemaximum likelihood proceduretoobtain second stage estimates ofthepolicy functions and transition parameters.Ifthesequenceofestimates obtainedbycontinuediterationinthis fashionconverges,thenthelimitingesti-mates incorporateallthestructureimplied bythemodel.A comparison ofthe o®erand wagedistributions actuallyobserved with the¯nalestimates provide agoodness of¯tness test.Furthermore,acostofsearch function and acostof recruitingfunctioncanbeinferred.T heinteractiveprocedurerepresents acom-putationallysimplewaytocomputetheequilibrium wageando®erdistribution whileatthesametimeestimatingthestructurethatunderlies thedistributions.
4 .
1
M aximum Likel
ihood E stim ates
Foreachi2 f1;:::;N gin arepresentativesampleofemployers in agiven labor market,
whereniis thenumberofemployees atthebeginningoftheyear,nsi is acount ofthe\ stayers,"workers presentinthe¯rm's laborforcebothatthebeginning andattheendoftheyear,nd
i is thenumberofworkers presentatthebeginning ofthe yearwholeave duringthe year, andnh
i is the numberhired duringthe yearwhoreported in the employer's laborforce atthe year.L etwi represent the average wage paid by the employerand letpi denote outputperworker.. U nderthe assumption thatworkers are identicalbutare paid di®erentwages across employingestablishments,themodel`explains'cross employervariation in thetriple(n0;ns;n;)bycross employervariation inwandp.7L et
L i(hi;di)= P rfn0=n0ijns=nsi;n=nigP rfns=nsijn=nigP rfn=nig (35) representthecontribution oftheithobservation totheoveralllikelihood func-tion.
A s demonstrated above, an employer's establishment size is a stochastic birth-death process de¯ned by the birth frequencyhi and death ratedi:T he invariantdistributionofthis continuous time¯rstorderM arkovprocess on the non-negative integers is the P oison with mean equaltohi=d.H ence, the size distribution forestablishmentiis P rfn=nig= e¡hidi ni! µh i di ¶ni (36) G iven thefactthatajob spellwith aparticularemployeris exponentially dis-tributed with parameterequaltotheseparation rated,the conditionalproba-bility thatexactlynsofthe originalnworkers stayed the entire yearand the remainderdid notis distributed accordingtothebinomialwithparametere¡d
P rfns=ns ijn=nig= µ ni ns i ¶ e¡dinsi(1 ¡e¡di)ni¡nsi: (37) O fcourse,nh=n0¡nsrepresentonlymeasuredhires,alowerboundonthe numberactuallyhired duringayeartotheextentthatsomeofthosehired left beforetheyearended Still,iftheseparation ratedwere`small'
P rfn0=n0ijns=nsi;n=nig ¼ e¡hi (n0
i¡nsi)!
(hi)n0i¡nsi
sincenewhires arriveattheP oissonratehovertheunitinterval.H owever,one canusethemodeltoimproveon this approximation.
A ssign the integers 1 ton0¡ns to those hired and presentatthe end of theintervaland assign largernumbers tothosewhowerehired and left.L etj
7T o em peric allyim plem ent thisassum ption,one need sto d ivid e the sam ple into sub sam ples
ofw orkersthat are ashom ogenousaspossib le and w ho partic ipate inseparate lab or m arkets. Inthe c ase ofthe IDA, w orkers are id enti¯ed b y oc c upationc ategories, m anagers, o± c e w orkers, skilled and unskilled w orkers,that approxim ate thisrequirem ent.
denote this index.T hen, any workerwith indexj·n0¡nsmusthave been hired atsome unobserved instantt2[0;1 ]within the period and stayed fora periodoflength[t;1 ]whileanyonewithindexjgreaterdidnotstay. Sincethe instanthiredintheintervalis uniformlydistributed,theprobabilityofactually observingsomeonewhowas hiredduringtheintervalstillinthe¯rm attheend is Z1 0 e¡(1¡t)ddt=e¡d Z1 0 edtdt=e¡d µedt ¡e d ¶ =1 ¡e¡ d d : H ence, P rfn0=n0ijns=ns i;n=nig= 1 X j=n0 i¡nsi e¡hi i j!(hi) jµ1 ¡e¡di di ¶n0i¡ns iµ 1¡1 ¡e¡ di di ¶j¡(n0i¡ns i) becauseweknowthatatleastj¸n0¡nswerehired,thosethatwereobserved atthe ¯rm stayed, and those hired butnotobserved left.Finally, aftersome tedious algebra,oneobtains P rfn0 = n0ijns=nsi;n=nig=e¡hi à 1¡e¡di di 1 ¡1¡e¡di di !n0i¡nsi £¡ µ n0i¡nsi;hi µ 1¡1 ¡e¡ di di ¶¶ (38) £ µ e¡hi (n0 i¡nsi)! (hi)n0i¡nsi ¶0 @e¡ hi di ni! µh i di ¶ni 1 A
where¡(x;y)is theincomplete G ammafunction,theCD F ofthe gammaran-dom variate.8
Conditionalonnon-parametricestimatesofthewageando®erdistributions, F andG,thehiringandseparation ratearerelatedtoeachemployer's produc-tivityand wageby hi = [´0s0 + ´1 Zwi R s(z)dG(z)]v(pi;wi) (39) di = ±+ ¸s(wi)[1 ¡F (wi)]: Forpurposeofidenti¯cation,thefollowingnormalizations areimposed. s0 =s(R)= 1 =v(p;w) (40 )
Essentially, these assumptions determine the units in which search e®ortand vacancies aremeasured.Finally,the¯rststageestimates solve
³ b±;b´0;b´1;v(bp;w);b¸;bs(w) ´ = argmax ( N X i=1 ln(L i(hi;di)) ) s.t.(39)and (40 ). (41 ) Callthe parameterand policy function estimates obtained in this case by ap-plying(41 )the¯rststageestimates.
4 .
2
Distrib utionan
d M atc hin
gCost E stim ates
T he equilibrium wage o®ered and wage earned distributions implied by the ¯rststage estimates ofthe transition parameters and policy functions can be constructedusingequations (23)-(25):Speci¯cally,theequilibrium sampleo®er distribution implied bytheestimates oftherecruitingpolicyfunction is
b F (w)= P wi·wbv(pi;wi) P i2N bv(pi;wi) (42) wherewi is the actualwage paid by employeriandpi is thesame employer's laborproductivity.T heassociated steadystatewagedistributionis theunique solutionto
±Gb(w)+ ¸[1 ¡Fb(w)]
Zw
R b
s(z)dGb(w)=±Fb(w): (43) Secondstageestimates oftheparameters andpolicyfunctions canbecomputed by usingthese functions in theequations of(39)and then resolvingthe maxi-mum likelihood estimation problem de¯ned byequation (41 ).T helimitofthe sequence ofestimates obtained by iterating between these two equations and equations (42)and(43)satisfyallthestructureimposedbythemodel.Finally, these estimates can becompared with theactualdistributions observed in the data todetermine the extenttowhich the modelexplains the wage and o®er distributions.
T hatequations (42)and (43)are the o®erand wage distributions implied bythemodelforboth therentsharingand wagepostinghypotheses is worthy ofnote.T hereasonfortheindependenceis thatbothwagedeterminationmod-els havethe same qualitativeimplications aboutthevacancy function,namely thatthe numberofvacancies posted increases with the wage paid.G iven an estimateofthis functionwhichis monotone,boththeo®erdistributionandthe steadystatewagedistributionimpliedbythemodelis thesameunderanywage determination mechanism thatimplies thatvacancies increasewith thewage.
T he fundamentalassumption ofthe search equilibrium `°ow'approach to labormarketanalysis is thatfrictions intheform ofsearchandrecruitingcosts are important.A ccountingdata simply cannotprovide a testofthis assump-tionbecauseaccountingpracticedoes notseparateoutthesecosts from general
overhead.O ne advantage ofthe structuralframework outlined here is thatit permits inferenceaboutthemagnitudeofthesecosts.Speci¯cally,themarginal searchcostfunction implied bytheestimates,
c0w(s(bwi))=b¸
Zw wi
[1 ¡Fb(we)]dwe
r+b±+ b¸bs(we)[1 ¡Fb(we)] (44) byequation(26),whichis determinedup toaspeci¯cationofthediscountrate r. B ecause empirically the separation rate is much largerthan any estimate ofthe discountrate, even the upperbound on the marginalcostobtained by settingr= 0 is agoodapproximation.Similarly,equation(28)implies thatthe marginalcostofrecruitingfunction estimatein therentsharingcaseis
c0f(bv(pi;wi))=
[b´0 + b´1
Rwi
w bs(z)dGb(z)](pi¡wi)
r+b±+ b¸bs(w)[1 ¡Fb(w)] : (45) O fcourse,intherentsharingcase,this estimateofrecruitingcostis onlyiden-ti¯edup toachoices oftheshareparameter¯ and reservation wageR =b:
5 Future R esearc h
T hemodelexplored in this paperis aprototypeforamoregeneralframework thatmightbe usefulforthe structuralinterpretation and analysis ofmatched employer-employeedata.A morecomplexversionisneededtoconfronttherich-ness ofthedataactuallyavailableand thesetofquestions thatareofinterest.
Extendingtheempiricalapproachtothefullpanelofobservations available in mostmatched employer-employee data sets is an obvious nextstep. T ime variationinemployerproductivitywillbeafeatureofthesedatathatonecan-notignore.H owdoes one modelthis feature? Should the observed variations beregarded as noiseoras therealizations ofastochasticprocess with alawof evolution known to the decision makers in the model? O fcourse, births and deaths ofemploying establishments willalso be observed. H owdoes one ac-knowledgetheseevents in themodeland in theestimationstrategy? A lthough the formalmodelis much more complicated when the employerproductivity parameteris treated as aprocess,ithas thepotentialforprovidinginsightinto the interpretation ofjob and worker° owdata, particularly those components thatareassociated with thebirths and deaths ofestablishments.
A nother generalization would explicitly account for within establishment workerheterogeneity.In the case ofthe ID A , the usualdemographic and ed-ucation ofeach workeris known as wellas occupation.T he availability ofthe data permitajointtheoreticaland empiricalstudy oflaborforce composition e®ects. Forexample, the modelcan be modi¯ed to include an establishment levelproduction function that could be estimated using observed time series variation in agiven employers laborforce composition as wellas cross section di®erences.T heextenttowhichemployeecompositionexplains cross employer productivity di®erences is also a question thatone should be able to address from astructuralperspective.
6 App end ix: E xistence P roof
s
6.
1
R en
t SharingCase
P roof.U nderthehypothesis,thelowestwageo®eredandtheleastproductive employer's productivity both equal the reservation wage, i.e.,w= p = R : H ence, any ¯xed pointofthe mapT :z ! z de¯ned in the text is a rent sharing equilibrium whenw(p)is the wage rule (9)wheref= (s(w);v(p))is an element ofz, a subset of continuous bounded real valued function pairs de¯nedonthecompactinterval[R ;w(p)]£[R ;p]:H ence,Schauder's ¯xedpoint theorem,as statedbyL ucas andStokey(1 989,p.520 ),implies existenceifz is non-empty,closed,bounded and convex,theoperatorT is continuous and the familyoffunctionsT(z)is equicontinuous.
T hevacancyfunctionv(p)maps[R ;p]into[0;v]andthesearchfunctions(w) maps [R ;w(p)]into[s;s]wheresis speci¯ed in A ssumption 1 ,vis speci¯ed in A ssumption 2, andsis uniquely determined by equations (27)and (29). A s ¹(R)is increasing inR,sis monotone decreasinginR :H ence, forany ¯nite w > R,1 > s > sforalls > 0 chosen su±ciently smallfrom equation (26). Consequently,z is nonempty,closed,bounded and convex.
T o showthatT is continuous onz itis necessary and su±cientto prove thatbothofitscomponentoperators,T1 andT2;arecontinuous.First,consider
T1 :z! <+£¢ de¯nedbyequations(23)-(25).W ithoutbelaboringthedetails,
continuity of the transformation is implied by that fact that the integration operatoron any continuous bounded realvalued function is continuous in the spaceofcontinuous boundedfunctions with thesup norm.T hetransformation T2 :<+ £¢ ! z de¯ned by the integral equation (26)is continuous and
equation (28)de¯nes a pointwise continuous functional relationship between v(w)and (µ;s(w);F (w))forallw2[R ;w(p)]:
T hefamilyT(F )isequicontinuousifeveryf2T(z)isuniformlycontinuous on[R ;w(p)]£[R ;p]and ifthecontinuity is uniform forallfunctionsT(z);i.e., ifforeveryx2[R ;w(p)]£[R ;p]and"0 > 0 thereexists a±0 > 0 suchthat
jx¡yj< ±0 impliesjjf(x)¡f(y)jj< "0;allf2T(z) (46)
where
jjf(x)¡f(y)jj´maxhjv(x)¡v(y)j;js(x)¡s(y)ji:
T hefollowingdemonstrationthatthis conditionholds amounts toshowingthat thegradientoff(x)is uniformlybounded forallfunctionsT(z).
B yequation (26),
c00w(s(w))s0(w)= ¡
m(µ)[1 ¡F (w)] r+ ±+ m(µ)s(w)[1 ¡F (w)]: H ence,m(µ)increasingandµ·µimply
js0(w)j·c00 m(µ)
whereµis de¯ned in equation (29).A s aconsequence, jx¡yj < ±0 ) js(x)¡s(y)j· max w2[b;w]js 0(w)jgjy¡xj (47) < ±0 max w2[R ;w] ½ m(µ) c00 w(s(w))(r+ ±) ¾ ="0
whereµis the uniform upperbound on markettightness de¯ned in (29)given thatthesearch costfunction is strictlyconvex.
Eitherv(p)equals theupperboundvorv(w)is an interiorsolution.In the lattercase,
c0f(v(p))= (1 ¡¯)(p¡R)`(w(p)) byequations (28)wherew(p)=¯p+ (1 ¡¯)R and
`(w)= µ m(µ)=µ r+ ±+ ¸s(w)[1 ¡F (w)] ¶2 4±+ m(µ) Rw ws(z)dG(z) ±+ m(µ)Rwws(z)dG(z) 3 5: (48)
B ecauses(w)andF (w)areboth di®erentiablegiven anyf= (s(w);v(p))2z by equations (2)and (23), so is`(w):N ow in general,m(µ)=µ is monotone increasinginµandm(µ)=µ! 1 asµ! 0:Ifso,then a1 > µ(p)> 0 exists such thatv(p)=vforallµ·.µ(p):H ence,
jx¡yj< ±0 ) jv(x)¡v(y)j· max p2[R ;p]fv 0(p)gjx¡yj(49) < ±0 max p2[b;p] ( à m(µ(p)) c00 f(v(p))µ(p) ! (1¡¯)[`(w(p))+ (p¡w)¯`0(w(p))] ) ="0
given thatthecostofrecruitingis strictlyconvex.
6.
2
W age P ostin
gCase
P roof.Inthis casethewagerulew(p)is endogenous andsatis¯es (31 ).H ence, itis appropriatetothinkofzasasubsetofthespaceofvectorfunctionsf(x)= (w(p);v(p);s(w(p)))de¯nedontheproductivityinterval[R ;p]togetherwiththe sup norm where
w(p)= argmax
w¸Rf(p¡w)`(w)g (50 ) v(p)= argmax
and s(w)= max s¸0 ( m(µ)s Zw w [1 ¡F (we)]dwe r+ ±+ ¸s(we)[1¡F (we)]¡cw(s) ) : (52) T hesede¯nitionstogetherwiththemarketequationsandsteadystateconditions (23)-(25)de¯ne a mapT :z ! z wherez is the subspace of continuous boundedrealvectorfunctionwiththesup norm andany¯xedpointofthemap is a non-trivialequilibrium underA ssumptions 1 and 2. A sz is non-empty, compactandconvexandT is continuous onz forthereasons givenintherent sharing case, Schauder's theorem applies in this case as wellif(46)holds for this extended de¯nition ofz:
In the rentsharingcase,we used the factthatT(z)is in the setofdi®er-entiablefunctions giventhatF is asubsetofthecontinuous boundedfunctions toprove equicontinuity in the rentsharingcase.T oapply the same argument in the wage postingcase wherew(p)is acomponentofanyf2F weneed to showthatw(p)is di®erentiableas wellasv(p)ands(p):T headdedassumption thatthemeasureofemployers with productivityless than orequaltop;callit ¹(p);is di®erentiableis needed forthis purpose.U nderthis assumption,z can beregardedas asubsetofthedi®erentiablefunctions de¯nedon[R ;p].N amely, given any di®erentiablef(p)= (w(p);v(p);s(w(p));itfollows thatthe implied F (w)is twicedi®erentiablebyequation (23).A s aconsequence,s(w )byequa-tion (52)is alsotwice di®erentiable and sois`(w)by equation (48)given any di®erentiablef(p).H ence,w(p)is di®erentiablebyequation (50 )sothat
jx¡yj< ±0 ) jw(x)¡w(y)j· max z2[R ;:p]fw 0(z)gjx¡yj< ± 0 max z2[R :p]fw 0(z)g=" 0; (53) jx¡yj < ±0 ) js(w(x))¡s(w(y))j· max z2[R ;p]js 0(w(z))w0(z)jgjy¡xj(54) < ±0 max z2[R ;p] ½ m(µ)w0(z) c00 w(s(z))(r+ ±) ¾ ="0; and jx¡yj < ±0 ) jv(x)¡v(y)j·max z2[b;p]fv 0(z)gjy¡xj (55) < ±0 max z2[R ;p] ( m(µ)`(w(z)) c00 f(v(z))µ ) ="0:
6.
3 Non
-trivialE quil
ib rium
P roof.Insum,wehaveshownthatT fsatis¯ed(46)andconsequentlyT :z !
postingcases.T heclaim thatany¯xed pointis non-trivialfollows:B ecauseas constructed everyf2z is such thats(w)¸s > 0 forallw¸bequation (22) implies thatµ=V =S < 1:B utgiventhatfact,equation(28)requiresv(w)> 0 forallw > band consequentlyµ=V =S > 0 givenw> b:In short,the trivial equilibrium thatalways exists is nota¯xed pointoftheconstructed map.
R ef
eren
c es
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