Discussion Paper No. 988 - PDF document

Discussion Paper No. 988 January 2004 Email: iza@iza.org This Discussion Paper is issued within the framework of IZA’s research area General Labor IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available on the IZA website ( www.iza.org ) or directly from the author. Williamson(1975)andGroult(1984)showthathold-upproblemsariseduetoalackofcompletecontingentcontracts;withcompletecontractsallthosewhobenetfroman thesameprobabilitytoattractasingleapplicant.Sincethesurplusofajobincreaseswiththecapitalstock,thisleadstoapositiverelationbetweentheexpectedwagecostsandtheinvestmentlevelofarm.Firmswillthereforenotbeabletoappropriatetheentiremarginalreturnsoncapitalandthisleadstounder-investment.Ournextresultshowsthatthehold-upproblemdisappearswhenworkerscandirecttheirsearchtowardsdierenttypesofrms.Asucientconditionfortheequilibriumwithdirectedsearchisthatworkers(costlessly)observethecapitalstockofallrmsbeforetheymaketheirapplications.Thisas-sumptioncreatesacompetitiveenvironmentinwhicheachjobopeningneedstooerunemployedworkersthesameexpectedincome.Armthatincreasesherinvestmentlevelmaythereforestillneedtopayahigherwage,butthisisnowexactlycompensatedbyalargernumberofexpectedapplicantsandhenceasmallerprobabilitythatthermwillfaceasingleapplicant.Fromtheviewpointofanindividualrmtheexpectedwagecostsarethereforeindependentofherinvestmentlevel,resultinginecientinvestments.Finally,inalaststepweallowrmstoannouncedierentreservationts.Inthisso-calledcompetingauctionssetup,applicationdecisionsre-spondbothtocapitalandthereservationstrategyofrms.Aslongasthesevariablesarefreelyobservable,wendthattheequilibriumisecientandthatrmsannouncetheirtrueoutsideoptionvalue.Oureciencyresultisthereforenotcontingentonourchoiceoftheauctionrule.Whenrmshaveachoice,theyselectanauctionruleintheclassofecientrules.Ourresultsillustratetheimportanceofjobcompetition.Nonetheless,theintroductionofjobcompetitionisnotsucient.Inordertoattaina(con-strained)ecientresourceallocation,thelabourmarketneedstogeneratestrongercompetitionformoreattractivejobs.Itisthereforecrucialthatunemployedworkersobserveallthepayo-relevantinformationaboutjobs.Theconclusionthatdirectedsearchmaypreventhold-upproblemsisnot 2TheModel2.1MainassumptionsThereisacontinuumofworkerswithmeasurenormalizedtooneandalargercontinuumofrms.Allagentsarerisk-neutral,liveforeverindiscretetimeandhaveacommondiscountfactor.Theutilityofanagentinperiodisequaltoherconsumptionoftheuniquenalgood.Allagentsthereforemaximizethevalueoftheirexpectedlifetimeincome.Workersarehomogeneousandmaybeinoneoftwostates,employedandproducingorunemployedandsearching.Firms,onthecontrary,areinactiveuntiltheybuysomecapitalatconstantmarginalcost,whichallowsthemtoattempttohireaworkerbypostingavacancy.Ifarmemploysaworkerandunitsofcapital,itproducesunitsofoutputperperiod.ThepriceofthisgoodisnormalizedtooneandisassumedtobestrictlyincreasingandconcaveandsatisestheusualInadaconditionswith(0)=0Finally,attheendofeachperiodthecapitalstockofanactivebreaksdownwithprobabilityAfterthisshockthermreturnstothepoolofinactiverms.Likewise,aworkerwhosejobisdestroyedbecomesunemployedwithaowincomethatisnormalizedtozero.2.2AmodelofjobauctionsOurmodelofthelabourmarketisbasedonShimer(1999).Inthissequentialsearchmodelrmsauctiontheirjobstoapplicantsandworkersmayhavetocompeteforjobs.Letusstartwiththeapplicationdecisions.Atthestartofaperiodeachunemploymentworkercanapplyforonejob.Anapplicationstrategynestheprobabilitythataworkerappliesforajobateachmeasurablesetofrms.Moreover,tocapturethenotionofalargemarketeconomy,we Accordingto(1),avacantrmwillattractatleastoneapplicantwithprob-ability.Thisexpressionisknownasthe“urnballmatchingfunction”.rmsrandomizeoverapplicants,asimilarexpressionmaybederivedforworkers.Theprobabilitythataworkerishiredisthengivenby .Itiseasytoverifythattheurnballmatchingtech-nologysatisesalltheusualproperties:themassofvacancieswithatleastoneapplicant,islinearlyhomogeneousinthemassofunemployedworkersandvacantjobs,)=lim)=0andlim)=limThematchingprobabilitieswithdirectedsearcharederivedinSection5.Inthiscaseworkerscanperfectlydiscriminateamongjobs.Asaresult,thelabourmarketmaydivideinseveralsubmarkets,eachwithaparticulartypeofajobandtheassociated(optimal)queuelength. TheurnballmatchingtechnologywasrstusedbyButters(1977).Forrecentappli-cationsseeMoen(1999)orShimer(1999). concavityofFinally,allpayosarediscountedtotheinitialperiod,whiletheevolutionoftheunemploymentratesatiseseqn.(4).Belowwerestrictattentiontostationaryallocations.Inasteadystatetheplanner’sproblemreducestothemaximizationofthe(constant)shadowvalueofunemployedworkers.ThisresultissummarizedinthePropositionbelow,whichreformulatesProposition1inAcemogluandShimer(1999):Proposition1Anecientsteadystatesolutionexists.Itischaracterizedbyapairsolving:maxk,q 1s)s Proof.SeeAppendix. Maximizationproblem(5)isnotjointlyconcaveinand.Hence,theorderconditionsarenotsucientforamaximization.Nonetheless,becausetheecientsolutionisaninteriorsolution,therst-orderconditionsarenecessary,andsotheyareusefulinrecognizinginecientallocations.Usingtheresultthat)=1,thisyields:Corollary2Anyecientallocationes: 1sseqS)µf0(kS) 1s)¶=ps)³1eqSqSeqS´ 1ss)¡1qSeqS¢µf(kS) 1s)¶= AppendixAprovidesaformalproofofthisassertion. TheecientallocationisillustratedinFigure1.Thecurvesassociatedwitheqs.(6)and(7)aredenotedbyand,respectively.Bothcurvesstartintheoriginandarestrictlyincreasinginq,kspace.Moreover,forlargeenoughvaluesofliesaboveandthetwocurvesintersectatleastonceontheinterior.Hence,asstatedinProposition,anecientallocationalwaysexists.Finally,ifthereismorethanoneecientcombina-ifworkerscandiscriminateamongjobs,theplannermaydecidetoopenmorethanone(sub)market.Inthatcasetheplannerwillassignalongerqueuetormswithalargercapitalstock.Inanecientallocation,capital-intensivejobsarethereforelledatafasterratethanlesscapital-intensivejobs.3.1ecientvsdecentralizedallocationsInthenextsectionswecomparetheecientallocationtothedecentralizedoutcome.Theinvestmentsincapitalarenownancedbyrmsandwagesaredeterminedbyauctions.Toconcentrateontheinvestmentdecisionsofrms,weinitiallyimposetwoconditionsonthesetofauctionrules.First,weassumethattheauctionrulesarefair.Ajobisthereforeawardedtotheworkerwhodemandsthelowestwage.Second,weassumethatrmscannotcrediblyannounceareservebidabovethevalueoftheirendogeneousoutsideoption.Togetherthesetwoconditionsdeneasetofrevenueequivalentauctionrules.Moreover,eachoftheseruleswouldinduceanecientresourceal-locationifthevalueofproductivitywereexogeneous(e.g.Shimer,1999).Forconvenience,weshallconsidertheexampleofasealedbidsecondprize Thisfollowsimmediatelyfromthefeaturethatandbothhaveastrictlypositiveslope.Thus,ifthereexisttwoecientallocations,say,andthenitmustbetruethat Themodelissolvedbackwardsstartingwithwages,andforthemomentIassumethatcondition(iv)isalwayssatis4.1WagesConsideravacantrmwithunitsofcapitalthatattractsapplicants.Inthefollowingwedenotethecontractualpayosofarmandaworkerk,nandk,nrespectively.Thesepayosarediscountedtothestartoftherstperiodofproduction.Moreover,wedenotethejointpayoofarm-workerpairbysothatk,nk,nwhiledenotestheassetvalueofanunemployedworkerItiseasytodemonstratethatisalsotheuniqueweaklydominantwagebidofanapplicant(e.g.Shimer1996).Hence,sinceworkersareiden-tical,therearetwopossibleoutcomes.Intherstcase,thermhasasingleapplicantwhosubmitsawagebidequalto.Thermhirestheapplicantandagreestoawagestreamwithvalue1)=Inthesecondcase,thermhasseveraljobcandidates.Eachoftheseapplicantssubmitsawagebidequaltoandthermselectsonecandidateatran-domandoersthisworkeracontractwithvaluek,nSincek,nk,n,theoutcomeoftheauctioncanthusbesummarizedasfollows: WhenIcharacterisetheinvestmentdecisions,Iwillshowthatthisisindeedanequi-libriumfeature.Inequilibriumitisneveroptimalforarmtocreateajobthatissubsequentlyrefusedbyallworkers.Giventhatagentsareriskneutral,theyareindierentbetweencontractswiththesameexpectedvaluesk,nk,nMoreover,thetimingofthepaymentsisirrelevantbecausecontractsareperfectlyenforceableandcannotberenegotiated.ThislastassumptionisrelaxedinSection6.2. Theprobabilitythatamatchsurvivesisequalto,inwhichcasebothpartiescontinuetoobtain.Withcomplementaryprobability,however,thecapitalequipmentisdestroyed.Thematchwillbedissolvedandthebecomesinactivewithpayosequaltozero,whiletheworkerreturnstothepoolofunemployedwithpayosequalto.Rearrangingterms,thisyieldsthefollowingexpression: Next,considertheexpressionfor.Withrandomsearchthenumberofapplicantsofarmisdistributedaccordingto(1).Moreover,theexpectedpayosforeachrealizationofaredenedby(9).Theassetvalueofavacantrmcanthereforebeexpressedastheweightedsumofthreecontingentpayo)+(1Firstofall,withprobabilitythermreceivesnoapplicationsandremainsvacant.Second,withprobabilitythermreceivesexactlyoneapplica-tionandsettlesforapayoequalto1)=,leavingthesurplustotheworker.Finally,withcomplementaryprobabilitythehasmorethanoneapplicant.Inthiscasethermretainsthesurplusandreceivesk,n.Finally,inallthreeeventsthepayosneedtobediscountedtoaccountfortimepreferenceandtheriskofbreakdown.Thevaluefunctionofanunemployedworkerisslightlymorecomplicated,asvacantrmsmayhavedierentcapitalstocks::(−s)(S(k)−JV(k))+ TheaboveequationdenesastheexpectedprotfromhiringaworkeratherreservationwageTheprobabilityofthiseventisequalto.Moreover,fromourdiscussionof(13)weknowthatthereservationwageofworkersisnotaectedbytheinvestmentdecisionofasingleThederivationoftheequilibriuminvestmentsisnowstraightforward.Eachrmthatentersthemarketmaximizesexpectedprotakingandtheinvestmentdecisionsofallotherrmsasgiven.Thisleadstothefollowingrstordercondition:)(1 )(1 Equation(15)isourrstequilibriumcondition.Itdenestheinvestmentlevelofrmsasastrictlyincreasingfunctionofthelikelihoodthataattractsatleasttwoapplicants.Inanyequilibriumrmsthereforemakethesameinvestmentandisdegenerate.Furthermore,whatismoreimportant,acomparisonwith(6)showsthatrmsunderinvestincapital.Lemma3Foranygivenvalueofrmsunder-investincapitalProof.SeeAppendix. Theintuitionforthisresultisstraightforward.Withhomogeneousagentsonbothsidesofthemarket,alljobsthatattractatleastoneapplicantwillbelled.Thesocialmarginalreturnsfromaninvestmentincapitalarethereforeproportionalto)=1.However,rmsonlyconsidertheirownprivatemarginalreturnswhichareproportionaltoFirmstherefore isstrictlyconcave,thesecondorderconditionsarealwayssatis Next,weneedtosubstitute(16)intotheright-handsideof(14).Theresultingexpressioncanbesolvedforandinvokingthefreeentryconditionweobtain:)(1 1ssq)µf(k) Thisfreeentryconditioncoincideswith(7).Hence,conditionalonanecientvalueforthecapitalstock,thelabourmarketwillgeneratetheecientnumberofjobs.However,fromtheprecedingdiscussionofinvestmentsweknowthatthereversedoesnothold.Givenanecientvalueforthemarketqueuelengthrmsarenotwillingtoadopttheecientcapitalstock.Theequilibriumwithrandomsearchisthereforeneverecient. qRkRKR qkKSQS qRkRKR Figure2:TheequilibriumwithrandomsearchThisfeatureisillustratedinFigure.Thecurveassociatedwithfreeentrycondition(17)isdenotedbywhiletheinvestmentlocusassoci-atedwith(15)isdenotedby.Duetothehold-upproblemthislocus TheexplanationfortheeciententrymarginisprovidedinShimer(1996).Namely,withasecondprizeauctionapplicantsarepaidtheirac-tualmarginalproduct.Supposethatarmreceivesasingleapplicant.Bycontactingtherm,theapplicantraisesthejointwealthoftherm-workerpairfrom.InordertohiretheworkerthermwouldthusbewillingtopayatmostIncontrast,ifthermreceivesmorethanoneapplicants,onlyonenewjobiscreated.Themarginalproductivityofeachindividualworkeristhusequaltozero,andthermwouldnotbewillingtopayanyoftheseworkersmorethanherreservationvalue.Therealsoexistsaclearrelationwiththeso-calledHosioscondition.Accordingtothiscondition,wagescorrectlyreectproductivityifworkers’shareofthematchsurplusisequaltotheelasticityofthematchingfunctionwithrespectto.Withtheurnballmatchingtechnologythiselasticityisgivenby (q)=q .Butthisisnothingelsethantheconditionalprobabilitythatthesurplusofajobaccruestotheworker(seefootnote11).Hosios’conditionisthereforesatised.Nonetheless,withrandomsearch ThisgeneralisesanimportantresultinMortensen(1982).Forthecaseofapairwiselinearmatchingtechnology,Mortensenshowedthattheecientallocationcanbedecen-tralisedifthepropertyrightstoamatchareassignedtotheagentwhoiniatesthecontact.Theusefullnessofthisresultwaslaterquestionedontwogrounds(.Pissarides,2000).First,withpairwisematchingitisoftennotpossibletodeterminewhoinitiatedthematch.Second,Mortensen’smatchingtechnologydoesnotexhibitcongestionexternalities.Theurnballmatchingtechnologydoesexhibittheseexternalitiesandeciencyisobtainedun-derasimilarrule.Namely,whenthepropertyrightsareassignedtotheagentontheshortside.ThereasonthatMortensen’srulestillgenerateseciencyisalsoeasilyexplained.Whencreatingajob,rmsdisregardthereturnsthataccruetoasingleapplicant.Asaresult,theprivatemarginalreturnsareequaltotimesthesurplusofthematch.However,thisisexactlythesocialmarginalreturnfromjobcreationsince=(1.Inotherwords,thefactthatrmsdisregardthere-turnsofasingleapplicantcompensatesforthereductioninthematchingrateofexistingvacancies.Thecongestionexternalityisthereforeperfectlyinternalised. compensatedbythelongerqueueandthelowerprobabilitytomeetasin-gleapplicant–anddeviantrmsareabletoappropriatethefullmarginalreturnsontheirinvestmentBelowweshowthatthismechanismcreatesacompetitiveenvironmentinwhichrmsmakeconstrained-ecientinvestments.Theanalysisexploitsthereduced-formcharacterizationofcompetitivesearchequilibriadevelopedbyMoen(1997).Wedenotetheexpectednumberofapplicantsofawithunitsofcapitalby.Moreover,comparedtotheprevioussection,weimposetwostrongerconditionsontheequilibriumallocation:ThecommonapplicationstrategyofunemployedworkersmaximizestheiractualexpectedincomeforanyarbitrarydistributionFirms’beliefsaboutareconsistentwithrationalexpectationsbeginningatanydecisionnodeandforallfeasiblevaluesof.Thislastrestrictionisneededtoruleoutsituationsinwhichrmswouldfailtodeviatetoaprotableinvestmentlevelbecausetheyincorrectlyconjecturetoofewworkerswouldapply.Conditions(i-a)andreplaceconditioninSection.Therestoftheconditionsisthesameasbefore.5.1AnalysisWestarttheanalysisbywritingtheBellmanequations.Withtargetedap-plications,thevalueofavacantjobwithunitsofcapitalsatissatis(e3q(k)+q(k)e3q(k))JV(k)+(1−e3q(k)+q(k)e3q(k))(S(k)−JU)],(18)whereisthejob-specicqueuelength.Similarly,thevalueofanapplicationatarmwithcapital-intensitysatises: cestosubstitute(21)intothezeroprotcondition.Thisyields 1ssq(k)eq(k))·f(k) whichcoincideswith(7).Thedierencewiththeprevioussectionconcernsthechoiceofcapital.Namely,withdirectedsearchrmscorrectlyanticipatethebestreplyofunemployedworkerstochangesintheirinvestmentstrategy.Thisresponseisgovernedby(22).Foragivenvalueof,thisequationdenesastrictlypositiverelationshipbetweenthecapitalstockandthequeuelengthineachsub-market.Moreover,sinceisnotaectedbythedecisionsofasinglerm,thesamerelationshipalsodenesthequeuelengththatresultsifsomermdeviatesbyoeringajobwithunitsofcapital.Denotethisrelationshipby.From(22)itfollowsimmediatelythatcontinuous,strictlyincreasinginandstrictlydecreasingin where isdenedby Furthermore,forcapitallevelsbelowthisthreshold,forallInequilibriumnoworkerwillthereforeapplyforthesejobs,andso)=0kk WecannowformalizeConditionbyrequiringthatrms’beliefsaboutareconsistentwithforallvaluesof,includingthevaluesofcapitalthatarenotactuallyobservedinequilibrium.Individualrmstakethisrelationshipasgivenandchoosetomaximizeprots.Letk,qdenotetheassetvalueofavacancygiventheequilibriumrelation Sinceworkershaveperfectinformation,thequeuelengthassociatedwitheachcapitallevelbecomesinstantlyobservableassoonasarmdeviatestooerthiscapitalstock.Itseemsreasonablethereforetoassumethattheout-of-equilibriumbeliefsofrmsareconsistentwith indierencecurveofworkers,denotedby)=(Thisfeatureofthemodelallowsustocharacterizeequilibriumallocationastheoutcomeofasimpleconstrainedoptimizationproblem:Lemma7Thepairwithelementsandisasteady-stateequilibriumwithdirectedsearchisolves:=maxk,q 1ssq)µf(k) subjectto:)(1 1ssq)µf(k) (P1)TheproofofLemma7isstandard.Supposethatrmsoerjobswithalowercapital-intensityThiscaseisillustratedbypointinFigure4.Itiseasytoshowthatthisoutcomecannotbeanequilibrium.Considerarmthatdeviatesfromtheallegedequilibriumbyinvestinganamount.Fromourpreviousdiscussion,weknowthatthermwillattractaqueueofapplicantsoflengthDEV.Moreover,thepointDEVliesbelowthefreeentrylocuswhichshowsthattheexpectedprotsofthedeviantrmarepositive.Hence,thecombinationcannotbeanequilibriumandrmswillcontinuetocreatejobswithunitsofcapitaluntilthe Ourrestrictionsontheproductiontechnologyarenotsucienttoguaranteethatthezero-protcurveisconcave.Asaresult,theremaybemultipleequilibriumallocations.However,thisisnotimportantfortheanalysissinceeachoftheseallocationsmustcor-respondtoapointoftangencybetweenthefreeentrylocusandthehighestattainableerencecurve.Theequilibriumwelfarelevelofworkersisthusuniquelydetermined. Proposition8If()isanequilibriumwithdirectedsearchwithelementsand,then()isanecientallocationasdenedinPropositionConversely,if()isanecientallocationascharacterizedinProposition,thenthereexistsanequilibriumwithdirectedsearchsuchthatandProof.SeeAppendix. ThedierencewithPropositionisthatrmsmakeecientinvestments.Theabilityofworkerstodirecttheirsearchtormswithdierentcapi-talstocksthereforeresolvestheholdupproblem.Namely,itallowsrmstoincreasetheirinvestmentsandattractmoreapplicantswithoutincurringhigherexpectedwagecosts.Inaneighbourhoodaroundtheequilibriumrmsthereforeacquiretheentiremarginalreturnsontheirinvestments.Togetherwiththeeciencyoftheentrymarginthisensuresthatrmsmakeecientinvestments.6PostingandrenegotiationTheprevioussectionsidentiedthetwonecessaryconditionsforeciency.Inequilibriumworkersneedtobepaidtheir(actualorexpected)shadowvalueandunemployedworkersneedtobeabletodirecttheirsearchtormswithdierentcapitalstocks.Sofar,therstconditionissatisedbecausermscouldnotcommittoareservationbidabovethevalueoftheiroutsideoption.Furthermore,weassumedthatcontractsareperfectlyenforceableandruledoutrenegotia-tion.Inthissectionweshowthatbothassumptionscanberelaxedwithoutchangingourmaineciencyresult. theirreturnsoncapitalbypostinghighreservationprots.Areservationbid�pkwoulddecreasethewageofasingleapplicantandpreventexpostrent-sharing.Buttheoveralleectonprotswouldbenegativeasrmswouldattractfewerapplicants.6.2RenegotiationbymutualconsentThelastissueconcernstheenforceabilityofthewageagreements.Undertheequilibriumwageruleoneofthetwopartiesiskeptatherreservationvaluethroughouttheentirerelationship.Thesecond-prizeauctionthereforecre-atesastrongincentiveforrenegotiationoncethepartieshaveconsummatedtheirmatch.Forinstance,anemployeewhowaschosenamongatotalofapplicants,maytrytonegotiateahigherwageoncethermhasdismissedthealternativeapplicants.Sofar,renegotiationwasruledoutbyassumption.However,thisassump-tionisunnecessarilystrong.Allthatisneededisthatrenegotiationcannotbeimposedunilaterallybyoneoftheparties.Withapropertreatmentoftheoutsideoptionsthiseliminatesrenegotiationinourmodel(e.g.Malcomsonetal.(1993)orMalcomson(1998)).Theonlyrequirementisthatthecon-tractualpaymentsneedtosatisfytheparticipationconstraintofbothagentsateachmomentduringtheirrelationship.Whenthisconditionissatised,renegotiationisapureredistributionthatwillberefusedbythepartythatisentitledtothesurplus.Hence,whenweallowforrenegotiationbymutualconsent,thetimingofpaymentsmatters,butthisdoesnotchangeanyofourciencyresults. hirearandomlyselectedcandidate.Thisequivalencebreaksdownifwork-ersareriskaverse–becauseauctionsaremoreriskythanposting–,orifthenumberofagentsoneithersideofthemarketisnite(Julien,KennesandKing(2001)).Sofar,however,theroleoftheinformationstructurehasnotbeenstudiedindetail.Itiseasytoseethattheinformationstructurematters.Considerforexamplethecasethatunemployedworkersobservewages,butnotinvestments.Inthatcasermswilloptforwageposting.Thismechanismallowsrmstoappropriatetheentiremarginalreturnsfrominvestmentsincapital,whileauctionsleadtohold-ups.Reversely,ifworkersmakeexanteinvestmentsandiftheresultingskilllevelisprivateinforma-rmsmaypreferauctions.Thereasonisthatauctionselicittheprivateinformationfromworkers.Withwageposting,onthecontrary,thisisonlyfeasibleifrmscanpostamenuofcontractsthatinducesworkerstoself-selectindierentsubmarkets.Finally,onemayconsidertheroleofeducationasapurescreeningdevice.Inthisenvironmenteducationiswastefulinthesensethatitdoesnotaectproductivity.Nonetheless,itmayimprovetheallocationofexanteheteroge-neousworkersoverjobsbecauseitprovidesasignalabouttheinnateabilityofworkers.Thisandothertopicsareonouragendaforfutureresearch. [10]Hosios,Arthur(1990),“OntheEciencyofMatchingandRelatedMod-elsofSearchandUnemployment”,ReviewofEconomicStudies,57,279-[11]Julien,Benoit,JohnKennesandIanKing(2000),“BiddingforLabor”,ReviewofEconomicDynamics,3(4),619-649.[12]–––(2001),“AuctionsandPostedPricesinDirectedSearchEqui-librium”,TopicsinMacroeconomics(BEJournalinMacroeconomics),1(1),Article1.[13]Kultti,Klaus(1998),”EcientAllocationofAgentsandtheTradingMechanisms”,HelsinkiSchoolofEconomicsandBusinessAdministra-tion,WorkingPaperNo.W-206.[14]–––(1999),“EquivalenceofAuctionsandPostedPrices”,GamesandEconomicBehavior,27(1),106-113.[15]Laing,Derek,TheodorePalivosandPingWang(1995),“Learning,MatchingandGrowth”,ReviewofEconomicStudies,62,115-129.[16]MacLeod,W.BentleyandJamesM.Malcomson(1993),“Investments,HoldupandtheFormofMarketContracts”,AmericanEconomicRe-,83,811-837.[17]Malcomson,JamesM.(1998),“NewDevelopmentsintheStudyofCon-tractsinLabourMarkets”,Chapter3inOrlyAshenfelterandDavidCard(eds.):HandbookofLaborEconomics,Vol.3b,North-HollandPublishers,Amsterdam.[18]Masters,AdrianM.(1998),“EciencyofInvestmentinHumanandPhysicalCapitalinaModelofBilateralSearchandBargaining”,Inter-nationalEconomicReview,39,477-493. 8AppendixProofofProposition1(i)CharacterizationofecientstationaryallocationsConsidertheLagrangianfunctionassociatedwitheqs.(2)-(4): u(t+1)))+(1whereisthe(undiscounted)shadowvalueofanunemployedworkerinperiodandsatises(4).Sinceweareinterestedinsteadystates,wehenceforthsuppressalltimeindices.Therst-orderconditionsforandcanthenbewrittensuccinctlyas: 1s)p¸1 (25) (26) (27)Ataninternalsolution,theaboveconditionswillholdwithequality,whiletheemploymentratesatis)=(1Theexistenceofaninternalmaximumisdemonstratedbelow.Inthissectionwewanttoshowthatthesolutionofconditions(25)-(27)coincideswiththe yieldseqn.(5)inthemaintext.Thesolutiontothisstaticoptimizationproblemthuscharacterizestheecient(steadystate)capitalinvestmentandqueuelength,whiletheassociatedunemploymentratecanbefoundbysubstitutinginto(28).(ii)Existenceofaninteriorsolutionk,qdenotethevalueofmaximand(5)whichiscontinuousonk,q.Thefollowingresultsareimmediate:k,qk,qk,q)=0limk,qExtremalvaluesofand/orcanthereforeneverbeasolutionaslongaswecanshowthatthereexistpositiveandnitevaluesofandforwhichk,qisstrictlypositive.FollowingAcemogluandShimer(1999)thiscanbeestablishedusingtheFundamentalTheoremofCalculus.Fixanypositivevalueofsuchthatanddenethevalueofby: )=(1Suchastrictlypositivevalueforalwaysexists,sinceweassumedthat.Thenbythefundamentaltheoremofcalculus,itfollowsthatq,kisequalto:))(1)+(1 ))(1)+(1 (32)whichisstrictlypositive.ThemaximumisthereforeaninteriorextremumandtheKuhn-Tuckerconditionsarenecessaryconditions.Inaddition,the withrespecttoandonthewholedomainin,1]2.Theconditionsoftheimplicitfunctiontheoremarethereforesatisedandeqn.(15)implicitlynesafunctionthatismonotonicallyincreasing,continuousanderentiablewith(0)=0andwhereimplicitlydenedby: Next,dividing(17)by(15)wearriveatthefollowingexpression: )(1 Theleft-handsideof(34)isastrictlydecreasingfunctionofthatmapss,1]onto Hence,wheneverthereexistsavalueofthatsolves(34).LetusdenotethissolutionbyUnderCondition,therearethreepossiblecases.CaseI:Theproductionfunctionisiso-elastic.InthiscasenesaverticallineatsomestrictlypositiveandnitevalueofCaseII:Theelasticityoftheproductionfunctionismonotonicallydecreasing.Inthissecondcase,nesadownward-slopingcurve.Thiscurvestartsataverticalinterceptq,kwhereisimplicitlydenedby: 1s)=f(ek) (A13)anditcutsthe-axis()atsomestrictlypositivevalueofCaseIIITheelasticityoftheproductionfunctionislocallyincreasing.Asaresult,thegraphofhasapositiveslope.However,giventhatisconcave, Equation(37)denesthevalueof s+seq)1s) Substitutingthissolutioninto(36)yields:)(1 1sseq)µf0(k) Thisexpressioncoincideswitheq.(6).Hence,inanyequilibriumwithdirectedsearchtheinvestmentmarginisecient.Theeciencyoffreeentrycondition(38)wasshownalreadyinSection.TheequilibriumallocationthereforecoincideswithanecientallocationandPropositionshowsthatsuchanecientallocationexists.ProofofProposition9denotethevalueofavacantrmwithcapital-intensityreservationbidForanarbitraryvalueofq,Jsatis+(1Similarly,aworkerwhoappliesatthisrmobtainsapayothatsatises:Thedierencewithbeforeisthatasingleapplicantnowreceivesratherthanrmcanthereforereducetheextentof(expost) thatischaracterizedinLemmaandfromPropositionweknowthatthisproblemhasaninternalsolutionthatcoincideswithanecientallocation.Inequilibriumthereservebidofrmsisthereforegivenbyandtheresultingallocationisconstrainedecient. 3 12/03 975 A. 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