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ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016
Unemployment and Matching in the Labor Market
A Model of Search and Matching in the Labor Market
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 2
AFullyCompe99veLaborMarketCannotAccountforUnemployment• Inafullycompe99velabormarket,withoutuncertaintyandfric9ons,
employerswouldbeindifferentaboutwhetheranemployeeleavesherjob,sincetheycanreplaceherimmediatelyandatnocost,andatthesamecompe99vewage,withanotheremployee.
• Accordingly,anemployeewouldbeindifferentaboutlosingherjob,sinceshecanreadilyfindanotheroneatthesamecompe99verealwage.
• Moreover,insuchamarket“involuntary”unemploymentcannotexist,becausetheexcesssupplyofworkerswouldcauseanimmediatedeclineinrealwages,whichwouldleadtotheelimina9onofunemployment.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 3
UnemploymentandVacancies• Inalmostalleconomiesthereisaposi9veandnon-trivialunemploymentrateeven
inboomperiods.Therearemanyunemployedpeopleseekingjobssimilartothoseheldbyworkerswithsimilarcharacteris9cs,atwagesequivalenttothosegenerallyprevailinginthelabormarket.
• Atthesame9me,therearemanyfirmswithvacancies,seekingtofillthemwithemployees,possessingcharacteris9cssimilartothoseoftheunemployed,andatprevailingrealwages.Howcanthenoneexplaintheexistenceandthefluctua9onsofbothunemploymentandvacancies?
• Theexplana9onofunemploymentisoneofthecentraltasksofmacroeconomics.Therearetwotypesofques9onsthatarebeingasked.First,whatdeterminestheequilibriumrateofunemploymentinaneconomy,whatareitsimplica9ons,andtowhatextenttheequilibriumunemploymentratereflectslabormarketdistor9ons.Thesecondkeyques9onconcernsthecyclicalfluctua9onsofunemploymentduringtheeconomiccycle.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 4
TheUnemploymentRateintheUSA
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Recession UnemploymentRate
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 5
EuropeanUnemployment:GermanyandtheUK
0.0%
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6.0%
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195019521954195619581960196219641966196819701972197419761978198019821984198619881990199219941996199820002002200420062008201020122014
Germany UK
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 6
TheUnemploymentRateinGreece
0.0%
5.0%
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195619581960196219641966196819701972197419761978198019821984198619881990199219941996199820002002200420062008201020122014
Greece EuroArea
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 7
Alterna9veViewsoftheLaborMarketandEquilibriumUnemployment
Therearemanyalterna9veapproachestothemodelingofthelabormarketthatdifferfromthecompe99vemodelorthemodelofmonopolis9ccompe99on.Alltheseapproachesofferanalterna9veexplana9onastowhy,despiteunemploymentandvacancies,realwagesdonotadjustinordertoabsorbtheunemployedandeliminateunemployment.
• EfficiencyWagetheories.• TheoriesofLongTermLaborContracts.• SearchandMatchingtheories.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 8
EfficiencyWageTheories• Inthesetheoriesthereisasymmetricinforma3on.Firmscannot
observeeithertheproduc9vityortheeffortofworkersdirectly.Thus,firmsofferwagesabovetheaverageproduc9vityofjobseekers,orexis9ngemployees,inorder,eithertoabractworkerswithaboveaverageproduc9vity,ortoprovideincen9vestotheiremployeestoworkmoreintensively.
• Theyarethereforenotpreparedtoreducerealwages,ortoreplaceworkersalreadyinjobswiththeunemployed,eveniftheunemployedofferedtoworkatlowerwages.
• Weiss(1980)andShapiroandS9glitz(1984)arethetwomostimportantearlymodelsbasedonthisapproach.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 9
TheoriesofLongTermContracts
• Thesecontractspreventfirmsfromundertakingunilateralchangestowagesinresponsetoshocks,ifthisisnotprovidedforinthelong-termcontract.Thecontractscanbeexplicit,suchascollec9ve,industryandindividualemploymentcontractsorinformalandimplicit.
• FortheoriginalimplicitcontracttheoriesseeAzariadis(1975),Baily(1974)andGordon(1974).Thesetheoriesviewedemploymentcontractsasinsurancecontractsbetweenriskneutralemployersandriskaverseemployees,againstadverseshockstoemployment.Fortheoriesthatarebasedonexplicitnego9a9onsseeMcDonaldandSolow(1981).Fortheoriesthatdis9nguishbetweeninsidersandoutsidersinthelabormarketseeLindbeckandSnower(1986),BlanchardandSummers(1986),Gregory(1986)andGoiries(1992).
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 10
SearchandMatchingTheories• Thesetheorieshighlightthesearchcostsoflookingforan
appropriatejob,byunemployedjobseekers,andthesearchcostsoflookingforanappropriateemployee,byfirmswithvacancies.
• Inthesetheories,acostlysearchprocessisrequiredforthematchingofjobseekerswithappropriatevacanciesinordertocreateanewjob.Consequently,jobsentailrents,somethingthatdoesnotapplyinfullycompe99velabormarketmodels.
• Theearlysearchtheoriesdatebackto1970.SeePissarides(2000)foranextensiveanalysisofsuchmodels.Otherrela9velyrecentpapersinclude,Mortensen(1986),Pissarides(1985)andMortensenandPissarides(1994).
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 11
UnemploymentandMatchingintheLaborMarket:TheMortensen-PissaridesModel
• Inthismodel,employersareinves9nginordertocreatejobvacanciesandtheprocessoffillingavacancyinvolvesmatchingofafirm,whichhascreatedavacancywithanunemployedjobseeker.
• Ateachinstant,therearetwoflowsintoandoutofunemployment.Someworkerslosetheirjobsandmovefromjobsintounemployment,andsomeoftheunemployedfindjobs,throughthematchingprocess,withfirmswhichhavecreatedvacancies.
• Inthesimplerversionsofthemodeltheprobabilityofjobtermina3onsisexogenous.Thisparameterdescribesthestructuralorcyclicalshocksaffec9ngtheeconomy,andleadingtothedestruc9onofjobs.
• Theprobabilityoffillingavacancy,aswellastheprobabilityofanunemployedjobseekertofindajob,areendogenousvariablesinthismodel,anddependonthedegreeoflabormarket3ghtness,whichisdefinedbythera9oofvacanciestotheunemployed.Thehigherthe9ghtnessofthelabormarket,thegreatertheprobabilityofanunemployedjobseekertofindajob,andthelowertheprobabilityofafirmtofillavacancy.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 12
EquilibriumUnemploymentintheMortensen-PissaridesModel• Inthesteadystate,theflowstoandfrom
unemploymentareequalized,andtheequilibriumunemploymentratedependsposi9velyontheexogenousprobabilityofjobtermina9ons,andnega9velyontheendogenousprobabilityofanunemployedjobseekertofindajob.
• Theequilibriumunemploymentratethereforedependsnega9velyonlabormarket9ghtness,andis,ofcourse,determinedendogenously.
• Thenega9verela9onshipbetweentheunemploymentrateandthevacancyratewhichisimpliedbythisdependenceisknownastheBeveridgecurve.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 13
TheBehaviorofFirmsandtheCrea9onofVacanciesandJobs
• Firmsandtheunemployedmaketheirdecisionsra9onally,maximizingtheexpectedpresentvalueoftheirprofitsandincome.
• Firmscreatenewvacanciesaslongastheexpectedprofitsfromtheinvestmentrequiredtocreateavacancyareposi9ve.Thecondi9onforavacancytobefilled,andforanewjobtobecreatedisthattherealwageshouldbeequaltolaborproduc9vityminusthecostofcrea9ngandmaintainingavacancy.Byfillingavacancy,afirmmustinequilibriumcoverboththewagecostsandthecostsofitsinvestmentinthecrea9onofthevacancy.
• Thejobcrea9oncondi9onimpliesanega9verela9onshipbetweenthewagethatthefirmiswillingtopayandlabormarket9ghtness.Thehigherislabormarket9ghtness,theloweristheprobabilityoffillingavacancyandthegreaterthetotalcostofmaintainingavacancy,sincevacanciesremainunfilledforlonger.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 14
TheBehaviorofUnemployedJobSeekersandtheDetermina9onofRealWages
• Anunemployedjobseekerwillagreetogetajobiftheexpectedpresentvalueofincomeofanemployedworkerisgreaterthantheexpectedpresentvalueofincomeofanunemployedjobseeker.Thiscondi9onissa9sfiedinthismodel,aslongastherealwageishigherthanunemploymentbenefits.
• Realwagesaredeterminedinequilibriumbydecentralizedbargainingbetweenfirmsthathavevacanciesandunemployedjobseekers.Theequilibriumrealwageistheresultofthisnego9a9on,anddependsposi9velyontherela9vebargainingpoweroftheunemployed,thelevelofunemploymentbenefits,laborproduc9vity,thecostofmaintainingavacancyandlabormarket9ghtness.
• Theequilibriumrealwagedependsposi9velyonlabormarket9ghtness,asthisincreasestheaveragerecruitmentcostperunemployedperson,therebyincreasingthe“threat”pointofprospec9veemployeesversusprospec9veemployers,andweakeningtheeffec9vebargainingposi9onofprospec9veemployers.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 15
EquilibriumRealWagesandUnemployment
• Theposi9verela9onshipbetweenrealwagesandlabormarket9ghtness,resul9ngfromthenego9a9onbetweenfirmswithvacanciesandtheunemployed,andthenega9verela9onshipbetweenrealwagesandandlabormarket9ghtnessimpliedbythejobcrea9oncondi9onforfirms,jointlydeterminetheequilibriumrealwageandequilibriumlabormarket9ghtness.
• Forgivenequilibriumlabormarket9ghtness,theequilibriumunemploymentrateisthendeterminedthroughtheBeveridgecurve,whichimpliesanega9verela9onshipbetweentheunemploymentandvacancyrates.
• Intheequilibriumofthismodel,theunemployedareworseoffthantheemployed.Consequentlyunemploymentisanundesirableandinvoluntarycondi9on,andnottheresultofchoicebytheunemployed,asincompe99vemodelsofthelabormarketwithoutfric9ons.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 16
TheMatchingFunc9onAkeyassump9onofthisclassofmodelsisthatthenumberofjobscreatedateachmomentisaposi9vefunc9onofthenumberoffirmslookingforemployeesandthenumberofunemployedjobseekers.Theoutcomeofthisprocessisdescribedbytheso-calledmatchingfunc3on,determiningthenumberofjobscreatedateachmomentin9me.Thematchingfunc9onisassumedtobeincreasingineveryoneofitsarguments,concaveandlinearlyhomogeneous.Thus,itischaracterizedbyconstantreturnstoscale.
mL = m(uL,vL)
L sizeofthelaborforceu unemploymentrate(unemployedasapropor9onofthelaborforce)v vacancyrate(vacanciesasapropor9onofthelaborforce)m thejobcrea9onrate(newjobsasashareofthelaborforce)
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 17
TheProbabilityofFillingaVacancy
Assumingthatallvacancieshavethesameprobabilityofbeingfilled,andthatalltheunemployedhavethesameprobabilityofbeingemployed,theprobabilityoffillingavacancyshallbeequaltothera9oofthenumberofnewjobscreatedoverallexis9ngvacancies.
Theprobabilityoffillingavacancyisafunc9ononlyofthera9ooftheunemployedtojobvacancies.Thisisduetotheassump9onthatthematchingfunc9onislinearlyhomogeneous.Themoretheunemployedpervacancy,thegreaterwillbethelikelihoodoffillinganypar9cularvacancy.
q = m(uL,vL)vL
= m uv,1⎛
⎝⎜⎞⎠⎟
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 18
LaborMarketTightnessandtheProbabilityofFillingaVacancy
Wedefineasθ,thera9oofvacanciestotheunemployed.θmeasuresthedegreeof9ghtnessinthelabormarket.Thehigherthenumberofvacanciesrela9vetotheunemployed,thegreaterthe9ghtnessofthelabormarket.
θ = vu
Theprobabilityoffillingavacancydependsnega9velyonθ.
q = q(θ )
′q (θ ) ≤ 0 −1<η(θ ) = θ ′q (θ )q(θ )
< 0
η(θ)istheelas9cityofqwithrespecttoθ.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 19
TheExpectedDura9onofaVacancy
BecauseofthePoissonassump9onthatallvacancieshavethesameprobabilityofbeingfilled,theexpecteddura9onofavacancyisequalto1/qanddependsposi9velyonθ.Thus,thehigherislabormarket9ghtness,thehighertheexpecteddura9onofavacancy.
1q= 1q(θ )
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 20
TheProbabilityofFindingaJobforanUnemployedJobSeekerandtheExpected
Dura9onofUnemployment
Theprobabilityoffindingajob,foranunemployedjobseekerdependsposi9velyonlabormarket9ghtnessθ.Theelas9citywithrespecttoθisequalto1-η(θ)>0.
m(uL,vL)uL
= vum(uL,vL)
vL= θq(θ )
1θq(θ )Theexpecteddura9onofunemploymentisgivenby,
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 21
ThePriceMechanismandAdjustmentintheLaborMarket
• Itisworthno9ngthat,inmodelsofthistype,thepricemechanismcannotleadtheprobabilityoffillingavacancyortheprobabilityoffindingajobtounity,asthelabormarketdoesnotfunc9ononlyviathepricemechanism,butalsoviathedegreeof9ghtnessofthelabormarket,whichdeterminestheprobabili9esoffirmstofilltheirvacanciesoroftheunemployedtofindjobsinanypar9cularinstance.
• Thedependenceoftheprobabilityoffillingavacancyandtheprobabilityoffindingajobontherela9venumberofvacanciestotheunemployed(9ghtness)isanexampleofatradingexternality.Thesesearchexternali9esareimportantfortheproper9esofequilibriumunemploymentinthesemodels.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 22
FlowsIntoandOutofUnemployment
• Themodelassumesthatapropor9onofexis9ngjobsareterminatedineveryinstant.Thedestruc9onofjobsandtheflowfromemploymenttounemployment,isduetoeithercyclicalorstructuralrealdisturbancesthatmakethemunprofitable.Itisassumedthatatanyinstanttheprobabilityofdestruc9onofanyjobisequaltoλ,whereλisanexogenousparameter.
• Ontheotherhand,jobcrea9onoccurswhenafirmandanemployeeagreetosignacontractwithawagewhichistheresultofabilateralnego9a9on.Thisleadstoaflowoutofunemployment.
• Therefore,atanygivenmomenttherearetwoflowsinthelabormarket.Oneflowisfromexis9ngjobsintounemployment,becauseofjobdestruc9on,andtheotherfromunemploymenttonewlycreatedjobs.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 23
ChangesintheUnemploymentRate
Thechangeintheunemploymentrateateachpointin9medependsonthedifferencebetweenflowsintounemploymentandflowsoutofunemployment.Inotherwords,itdependsonthedifferenceinthepropor9onofjobsdestroyedfromthepropor9onofnewjobscreated,thepropor9onsbeingdefinedrela9vetothelaborforce.
u•= λ(1− u)−θq(θ )u
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 24
TheDetermina9onofEquilibriumUnemployment
Inthesteadystate,theunemploymentratewillbeconstant.Consequently,theequilibriumunemploymentrateisdeterminedbythecondi9on,
λ(1− u) = θq(θ )u
Thiscondi9onimplies,
u = λλ +θq(θ )
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 25
TheBeveridgeCurve• Theequilibriumunemploymentratedependsposi9velyonλ,
theexogenousrateofjobdestruc9onandnega9velyonlabormarket9ghtnessθ.Labormarket9ghtnessisanendogenousvariableinthismodel,andisdeterminedinthelabormarket.
• Thenega9verela9onshipbetweentheunemploymentrateandlabormarket9ghtnessθ,or,equivalently,betweentheunemploymentrateandthevacancyrate,isusuallycalledtheBeveridgecurve.
• TheBeveridgecurvedefinesjustanega9verela9onbetweenvacanciesandunemployment.Inorderforunemploymenttobedetermined,oneneedstoknowlabormarket9ghtness,whichisoneoftheendogenousvariablesinthismodel.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 27
TheBeveridgeCurveandEquilibriumUnemployment
• TheBeveridgecurvecandetermineequilibriumunemploymentifweknowhow9ghtisthelabormarket.
• Inordertoexaminehow9ghtnessinthelabormarketisdeterminedwehavetoexaminethebehaviororprospec9veemployers(firms)whocreatevacancies,andunemployedjobseekers.
• Themodelassumesthatbothfirmsandunemployedjobseekersbehavera9onally,maximizingthepresentvalueoftheirexpectedincome.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 28
JobCrea9onandVacancies• Wehavethecrea9onofanewjobwhenaprospec9veemployerand
aprospec9veemployeegettogetherandagreetoanemploymentcontract.
• Ofcourse,beforethishappens,thepoten9alemployerhastocreateavacancyandsearchforanemployeeandtheprospec9veemployeehastobeunemployedandlookingforanewjob.
• Allthisinvolves9meandcostsandisdescribedbythematchingfunc9on.
• Weassumethattheinstantaneousvalueoftheproductofajobisconstantandequaltop>0.Theinstantaneouscostofavacantposttoaprospec9veemployerisequaltopcwhere0<c<1.Duringtheperiodofsearch,theemployerfacesaprobabilityq(θ)offindingaasuitableemployee,whichisindependentofherac9ons.
• Thenumberofvacanciesisendogenousandisdeterminedbyprofitmaximiza9on.Anyfirmcancreateavacancyandsearchforemployees.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 29
ThePresentValueofNetExpectedProfitsfromanExis9ngJob
WedenotebyJthepresentvalueofafirm’snetexpectedprofitsfromanexis9ngjob,bywtherealwageandbyrtherealinterestrate.
Theinstantaneousexpectednetprofitfromanexis9ngjobisequalto,
p −w − λJ
Assumingaperfectcapitalmarketandaninfinitehorizon,thepresentvalueofthisexpectednetprofitisequalto,
J = e−rt p −w − λJ( )t=0
∞
∫ dt = p −w − λJr
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 30
Implica9onsofaPosi9veNetPresentValueofExpectedProfitsfromanExis9ngJob
ThepresentvalueJofajobfortheemployeristhusdefinedas,
J = p −wr + λ
Thiswillonlybeposi9ve,ifproduc9vityexceedstherealwage,p>w.
Rearrangingthedefini9onofJ,weget,
w = p − (r + λ)J
Thiswillonlybeposi9ve,ifproduc9vityexceedstherealwage,p>w.Thewedgebetweenproduc9vityandtherealwagemustreflectthecapitalandinsurancecostsofmaintainingthejob.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 31
Prospec9veEmployersandthePresentValueofNetExpectedProfitsfromaVacancy
WedenotebyVthepresentvalueofexpectednetprofitsfromajobvacancy.Theinstantaneousnetexpectedprofitfromavacancyisequaltotheprobabilityoffillingthevacancy,andearningthedifferencebetweenthepresentvalueofajobandthepresentvalueofthevacancy,minusthemaintenancecostofthevacancypc.Itisdefinedby,
q(θ )(J −V )− pc
Assumingaperfectcapitalmarketandaninfinite9mehorizon,thepresentvalueofexpectednetprofitsfromavacancyVisdefinedby,
V = e−rt q(θ )(J −V )− pc( )t=0
∞
∫ dt = q(θ )(J −V )− pcr
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 32
TheCondi9onfortheCrea9onofVacancies
Inequilibriumwithfreecrea9onofvacancies(freeentry),allprofitopportuni9esfromcrea9ngnewvacancieswillbeexploited,andtheexpectedprofitsfromthecrea9onofanaddi9onalvacancywillbeequaltozero.Soinequilibriumwithfreeentry,V=0.This,impliesthat,
J = pcq(θ )
Thisisanimportantpredic9onofthismodel.Inequilibriumwithfreeentry,theexpectednetpresentvalueofajobwillbeequaltotheexpectedcostofhiringanemployee.Thisisequaltotheinstantaneouscostpcofmaintainingavacancy,9mestheexpecteddura9onofthevacancy1/q(θ).Thus,compe99onforthecrea9onofvacanciesandfreeentryreducestheexpectednetpresentvalueofprofitsfromajobtotheleveloftheexpectedcostofhiringaworker.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 33
TheJobCrea9onCondi9on
w = p − (r + λ)pcq(θ )
= 1− (r + λ)cq(θ )
⎛⎝⎜
⎞⎠⎟p
Combiningtheimplica9onsoffreeentry,thattheexpectednetpresentvalueofajobisequaltoexpectedcostofhiring,andthedefini9onoftheexpectednetpresentvalueofajob,weget,
Thisisthesecondkeyequa9onofthismodel,andcanbecalledthejobcrea3oncondi3on.Thefirmwillonlyhireanewworkerandcreateajobiftherealwageissmallerthanorequaltotheproduc9vityoftheworker,minusthemarginalhiringcost,whichisdefinedontheRHS.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 34
RealWages,Produc9vity,theMarginalHiringCostandtheCrea9onofJobs
• Firmscreatevacanciesaslongastheexpectedpresentvalueofnetprofitsfromthecrea9onofavacancyareposi9ve.
• Thecondi9onforthecrea9onofajobisthattherealwagemustbeequaltoproduc9vityminustheopportunitycostofmaintainingajob.Becauseoffreeentrythisisequaltothemarginalhiringcost.
• Thejobcrea9oncondi9onimpliesanega9verela9onbetweenlabormarket9ghtnessandtherealwage.Thehigherislabormarket9ghtness,thehighertheexpecteddura9onofavacancy,andthehigherthemarginalhiringcost.Thus,therealwagethatthefirmwouldbepreparedtopay,rela9vetoproduc9vity,willbelower.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 36
TheBehaviorofUnemployedJobSeekers
• Anunemployedjobseekerwillacceptajobofferiftheexpectedpresentvalueofincomewhenemployedishigherthantheexpectedpresentvalueofincomewhenunemployed.
• Thetypicalworkerearnsarealwagewwhenemployed,andislookingforajobwhenunemployed.
• Forthedura9onofunemployment,shehasaninstantaneousrealincomez,whichdependsonunemploymentbenefitsandanyotherincomeorbenefitfromtheuseoffree9me.Forsimplicityweshallcallztheunemploymentbenefit.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 37
TheExpectedPresentValueofIncomeofanUnemployedJobSeekerAnunemployedjobseekerhasaninstantaneousrealincomeequaltozandaninstantaneousprobabilityoffindingajobequaltoθq(θ).Asaresult,U,theexpectedpresentvalueofherincome,isdefinedby,
U = e−rtt=0
∞
∫ z +θq(θ )(W −U )( )dt = z +θq(θ )(W −U )r
whereUandWarethecorrespondingnetpresentvaluesofexpectedincomeofanunemployedjobseekerandanemployedworket.ThepermanentincomeofanunemployedjobseekerisequaltorU.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 38
TheExpectedPresentValueofIncomeofanEmployedWorker
Anemployedworkerhasaninstantaneousrealincomew,therealwage,butateachinstantalsofacestheriskoflosingherjob,withprobabilityλ.Therefore,thepresentvalueofherexpectedincomeWisequalto,
W = e−rt w + λ(U −W )( )t=0
∞
∫ dt = w + λ(U −W )r
whereUandWarethecorrespondingnetpresentvaluesofexpectedincomeofanunemployedjobseekerandanemployedworket.ThepermanentincomeofanemployedworkerisequaltorU.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 39
ThePermanentIncomeoftheEmployedandtheUnemployed
Fromthetwoequa9onsdefiningthepresentvalueofexpectedincomeoftheunemployedandtheemployed,wecansolveforthepermanentincomeoftheunemployedandtheemployedasfunc9onsoftheparametersofthemodel.Thesearedefinedby,
rU = (r + λ)z +θq(θ )wr + λ +θq(θ )
rW = λz + [r +θq(θ )]wr + λ +θq(θ )
Ifw≥z,thenW≥U,andthepermanentincomeofanemployedworkercannotbelowerthanthepermanentincomeofanunemployedjobseeker.Inwhatfollows,weshallassumethatwexceedsz,whichindeedturnsouttobethecaseinequilibrium.Asaresult,noworkerwishestoleaveherjobandallunemployedjobseekerswishtofindajob.Unemploymentisinvoluntary.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 40
Ra9onalBehaviorofFirmswithVacanciesandUnemployedJobSeekersandJobCrea9on
• Anunfilledvacancyimpliesalowerexpectednetpresentvalueofprofitsthanajob,despitethefactthat,withsomeprobability,ajobmaydisappear.
• Unemploymentimpliesalowerpermanentincomeforthosewhoexperienceit,thanforemployedworkers,despitethefactthateventheworkerswhoareemployedmay,withsomeprobability,losetheirjobsandbecomeunemployedinthefuture.
• Afirmwithavacancywillacceptajobofferifthejobcrea9oncondi9onissa9sfied,i.eiftherealwageislowerthanorequaltoproduc9vityminusthemarginalhiringcost.
• Anunemployedjobseekerwillacceptajobofferiftherealwageishigherthantheunemploymentbenefit.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 41
RealWageDetermina9on• Therealwageforapar9cularjobisdeterminedbyanego9a9on
betweenaprospec9veemployer,afirmwithavacancy,andaprospec9veemployee,anunemployedjobseeker.
• Becausealljobsareequallyproduc9veandalltheunemployedreceivethesameunemploymentbenefit,therealwagethatisdeterminedbyanindividualnego9a9onwillbethesameastherealwagethatprevailsintherestoftheeconomy.
• Fromtheassump9onsthatwehavemadeaboutproduc9vityandaggregatedisturbances,anyoneemployerandanyoneworker,whentheygettogetherthroughthematchingprocess,willcertainlyagreetoanemploymentcontractandcreateajob.Otherwisetheymustcon9nuesearching,withaddi9onalcostsforbothsides.
• Anemploymentcontractbetweenanemployerandanemployeeisdefinedbyarealwageandtheprovisionthatemploymentwillbeterminatedifthereisadisturbancethatmakesthejobuntenable.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 42
TheImpactofRealWagesontheOpportunityCostofaJobandthePermanentIncomeofa
Worker
Forarealwagewi,theexpectedreturnforaprospec9veemployerandaprospec9veemployeearegivenby
rJi = p −wi − λ(Ji −V )
rWi = wi + λ(U −Wi )
Wehavenotmadeuseoftheassump9onthatcompe99onhasreducedtheexpectedpresentvalueofavacancyVtozero.Wehaveassumedthough,asitappropriate,thattheexpectedpresentvalueofavacancy,andoftheincomeoftheunemployedUdependsonrealwagesintherestoftheeconomy,andisthusindependentofi.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 43
BargainingOvertheRealWage
Therealwageisdeterminedbya(generalized)Nashbargain,thatmaximizestheweightedproductofthesurplusoftheprospec9veemployerandtheprospec9veemployeefromtheagreementtocreateajob.Anagreementmeansthatthesurplusoftheprospec9veemployerisequaltoJi-V,andthatsurplusoftheprospec9veemployeeisWi-U.Thus,therealwagewillsa9sfy,
wi = argmax(Wi −U )β (Ji −V )
1−β
βisameasureoftherela9vebargainingpoweroftheprospec9veemployee,overandabovewhatresultsfromthe“threat”pointsUandV.Insymmetricbargaining,liketheoneweanalyze,areasonablevalueofβis½.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 44
FirstOrderCondi9on
Thefirstordercondi9onforamaximumimpliesthattheworkergetsashareβofthetotalsurplus.
Wi −U = β(Ji +Wi −V −U )
Subs9tu9ngforJiandWi,itfollowsthattheemployeegetsarealwagewhichexceedsthepermanentincomeofanunemployedworkerbyamul9pleβofthedifferenceofproduc9vityfromthepermanentincomeofanunemployedworker.
wi = rU + β(p − rU ) = w
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 45
RealWageDetermina9on
Fromthefirstordercondi9onforthedetermina9onoftherealwage,andasduetofreeentryV=0,itfollowsthat.
W −U = β1− β
J = β1− β
pcq(θ )
Asaresult,
rU = z +θq(θ )(W −U ) = z + β1− β
pcθ
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 46
TheWageFunc9onSubs9tu9ngforrUinthewagedetermina9onequa9on,
w = (1− β )z + β p(1+ cθ ) = z + β(p − z)+ β pcθ
Therealwageexceedstheunemploymentbenefitz.Itexceedsitbyapropor9onβofthedifferencebetweenproduc9vityandtheunemploymentbenefit,plusapropor9onβoftheaveragerecruitmentcostperunemployedworker.
Higherlabormarket9ghtnessθresultsinahigherrealwage,asthisincreasestheaveragerecruitmentcostperunemployedperson,therebyincreasingthe“threat”pointofprospec9veemployeesversusprospec9veemployers,andweakeningtheeffec9vebargainingposi9onofprospec9veemployers.
Thewagefunc9onplaysaroleanalogoustoalaborsupplyfunc9oninacompe99velabormarketmodel.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 47
Determina9onofRealWages,LaborMarketTightnessandUnemployment• Inthismodel,thelevelofrealwageswandlabor
market9ghtnessθ,isdeterminedindependentlyoftheunemploymentrateu.Thejobcrea9oncondi9onandthewageequa9onsufficeforthedetermina9onofrealwagesandlabormarket9ghtness.
• Oncelabormarket9ghtnessθisdetermined,wecansubs9tuteforitintheBeveridgecurve,anddetermineequilibriumunemployment.
• Thus,themodelhasarecursivestructure.Thedetermina9onofrealwagesandlabormarket9ghtnessisindependentoftheunemploymentrate,andtheunemploymentrateisdeterminedthroughtheBeveridgecurve.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 48
Determina9onofRealWages,LaborMarketTightnessandEquilibriumUnemployment
Insteadystatethismodeldeterminesthethreeendogenousvariables,(u,θ,w)whichsa9sfytheBeveridgecurve,i.e.thecondi9onofequalityofflowsinandoutofunemployment,thejobcrea3oncondi3on,andthewageequa3on.
u = λλ +θq(θ )
w = 1− (r + λ)cq(θ )
⎛⎝⎜
⎞⎠⎟p w = z + β(p − z)+ β pcθ
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 49
TheModelundertheAssump9onz=ρw
Ifweassumethattheunemploymentbenefitzisaconstantshareoftherealwageρ<1,thenthewageequa9onismodifiedandthemodeltakestheform,
u = λλ +θq(θ )
w = β(1+ cθ )1− (1− β )ρ
pw = 1− (r + λ)cq(θ )
⎛⎝⎜
⎞⎠⎟p
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 50
TheDetermina9onofRealWagesandLaborMarketTightness
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 53
Implica9onsofanIncreaseinUnemploymentBenefitsortheReplacementRate
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 54
AnIncreaseinUnemploymentBenefitsortheReplacementRateandEquilibriumUnemployment
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 55
Implica9onsofanIncreaseintheRealInterestRate
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 56
AnIncreaseintheRealInterestRateandEquilibriumUnemployment
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 57
Implica9onsofanIncreaseintheProbabilityofJobDestruc9on
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 58
AnIncreaseintheProbabilityofJobDestruc9onandEquilibriumUnemployment
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 59
TheDeterminantsofEquilibriumUnemployment
• Ifunemploymentbenefitsarepropor9onaltotherealwage,laborproduc9vitydoesnotaffecttheequilibriumunemploymentrateinthismodel.
• Thehigherthepercentageofrealwagespaidoutasunemploymentbenefits,thehighertheequilibriumrealwage,thelowerislabormarket9ghtness,andthehighertheequilibriumunemploymentrate.
• Higherrealinterestratesalsohaveaposi9veimpactonunemploymentinthismodel,becausetheyincreasethecostofmaintainingavacancy,resul9nginthecrea9onoffewerjobvacancies.
• Ahigherexogenousprobabilityofjobtermina9onshasaposi9veimpactonunemploymentfortworeasons.Firstbecauseitdirectlyincreasestheflowsfromexis9ngjobstounemployment,and,second,becauseit,indirectly,reducestheflowsfromunemploymenttojobs.Thesecondeffecttakesplacebecausetheexpectedprofitfromthecrea9onandfillingofavacancyfalls,resul9nginfewervacanciesandreducedflowsfromunemploymenttojobs.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 60
DynamicAdjustmenttotheSteadyState
Ouranalysissofarwasonlyconcernedwithsteadystateunemployment,andalmostnothingwassaidaboutthedynamicadjustmentofthelabormarketintheshortrun.
Vacanciesandrealwagesarenonpredeterminedvariablesintheshortrun,sincetheydependonforwardlookingexpecta9onsoffirmsandjobseekers.Unemploymentisapredeterminedvariable,andintheshortrunevolvesaccordingto,
u•(t) = λ − (λ +θEq(θE ))u(t)
whereθEisthesteadystatelabormarket9ghtnessdeterminedthroughthewagenego9a9onsoffirmswithvacanciesandunemployedjobseekers
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 61
ConvergenceoftheUnemploymentRatetotheSteadyState
Sincevacanciesareajumpvariable,andθisdeterminedindependentlyoftheunemploymentrate,θandrealwageswjumpimmediatelytotheirsteadystatevalues.Thevacancyratejumpstomaintainθatitssteadystatevalue,andtheunemploymentrateconvergestoitssteadystatevalueaccordingto,
whereu0istheunemploymentrateat9me0,anduEisthesteadystateunemploymentrate,definedby,
u(t) = uE + (u0 − uE )e−(λ+θEq(θE ))t
uE =λ
λ +θEq(θE )
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 62
TheDynamicAdjustmentofUnemploymentandVacancies
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 63
ANumericalExample
Weassumethefollowingini3alparametervalues:
λ=2.5%,p=1,c=1/2,r=3%,ρ=1/2,β=1/2,M=1/2,μ=1/2
Undertheseassump9ons,itfollowsthat,
w=0.95,θ=0.85,u=5.1%,v=4.4%
Weassumethefollowingformforthematchingfunc9on:
mL = M (uL)µ (vL)1−µ
whichimplies,
q(θ ) = mLvL
= M uv
⎛⎝⎜
⎞⎠⎟µ
= Mθ −µ
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 64
AnIncreaseintheProbabilityofTermina9onofaJob
Ifλweretodoublefrom2.5%to5%,itfollowsthat,
w=0.93,θ=0.79,u=10.1%,v=7.9%
Rela9vetotheoriginalequilibrium,therealwagefallsby2.1%,to93%ofproduc9vity,andthesteadystateunemploymentratealmostdoubles,to10.1%.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 65
DynamicAdjustmentofUnemploymentandVacanciesFollowinganIncreaseintheProbabilityofTermina9on
ofaJobfrom2.5%to5%
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 66
AnIncreaseintheReplacementRateofUnemploymentBenefits
Ifρweretorisefrom50%oftherealwageto70%,itfollowsthat,
w=0.96,θ=0.50,u=6.6%,v=3.3%
Rela9vetotheoriginalequilibrium,therealwagerisesbyroughly1%,andtheunemploymentraterisesby1.5percentagepoints,to6.6%.Withaconstantlaborforce,thisisequivalenttoa29.4%riseinthenumberoftheunemployed.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 67
DynamicAdjustmentofUnemploymentandVacanciesFollowinganIncreaseintheReplacementRatefrom
50%to70%
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 68
AnIncreaseoftheRealInterestRate
Ifrweretodoublefrom3%to6%,itfollowsthat,
w=0.925,θ=0.78,u=5.4%,v=4.2%
Rela9vetotheini9alequilibrium,therealwagefallsbyabout2.5%,andtheunemploymentraterisesby0.3percentagepoints,to5.4%.Withaconstantlaborforce,thisisequivalenttoa5.9%riseinthenumberoftheunemployed.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 69
DynamicAdjustmentofUnemploymentandVacanciesFollowinganIncreaseintheRealInterestRatefrom
3%to6%
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 70
TheNatureoftheMatchingProcessandFlowsintoandoutofUnemployment
• Inthismodel,employersareinves9nginordertocreatejobvacanciesandtheprocessoffillingavacancyinvolvesmatchingofafirmwithavacancywithanunemployedjobseeker.
• Ateachinstant,therearetwoflowsintoandoutofunemployment.Someworkerslosetheirjobsandmovefromjobsintounemployment,andsomeoftheunemployedfindjobs,throughthematchingprocess,withfirmswhichhavecreatedvacancies.
• Theprobabilityoffillingavacancy,aswellastheprobabilityofanunemployedjobseekertofindajob,areendogenousvariablesinthismodel.Theydependonthedegreeoflabormarket9ghtness,whichisdefinedbythera9oofvacanciestotheunemployed.Thehigherthe9ghtnessofthelabormarket,thegreatertheprobabilityofanunemployedjobseekertofindajob,andthelowertheprobabilityofafirmtofillavacancy.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 71
EquilibriumUnemployment• Inthesteadystate,theflowstoandfromunemploymentare
equalized,andtheequilibriumunemploymentratedependsposi9velyontheexogenousprobabilityoftermina9onofajob,andnega9velyontheendogenousprobabilityofanunemployedjobseekertofindajob.
• Theequilibriumunemploymentratethereforedependsnega9velyonlabormarket9ghtness,andis,ofcourse,determinedendogenously.Thenega9verela9onshipbetweentheequilibriumunemploymentrateandthevacancyratewhichisimpliedbythisdependenceisknownastheBeveridgecurve.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 72
FirmsandUnemployedJobSeekers
• Firmsandtheunemployedmaketheirdecisionsra9onally,maximizingtheexpectedpresentvalueoftheirprofitsandincome.
• Firmscreatenewvacanciesaslongastheexpectedprofitsfromtheinvestmentrequiredtocreateavacancyareposi9ve.
• Thecondi9onforavacancytobefilled,andforanewjobtobecreatedisthattherealwageshouldbeequaltolaborproduc9vityminusthecostofcrea9ngandmaintainingavacancy.Thejobcrea9oncondi9onimpliesanega9verela9onshipbetweenthewagethatthefirmiswillingtopayandlabormarket9ghtness.Thehigherislabormarket9ghtness,theloweristheprobabilityoffillingavacancyandthegreaterthetotalcostofmaintainingavacancy,sincevacanciesremainunfilledforlonger.
• Ontheotherhand,anunemployedjobseekerwillagreetogetajobiftheexpectedpresentvalueofincomeofanemployedworkerisgreaterthantheexpectedpresentvalueofincomeofanunemployedjobseeker.Thiscondi9onissa9sfiedinthismodel,aslongastherealwageishigherthanunemploymentbenefits.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 73
TheDetermina9onofRealWagesandLaborMarketTightness
• Realwagesaredeterminedinequilibriumbydecentralizedbargainingbetweenfirmsthathavevacanciesandunemployedjobseekers.
• Theequilibriumrealwageistheresultofthisnego9a9on,anddependsposi9velyontherela9vebargainingpoweroftheunemployed,thelevelofunemploymentbenefits,laborproduc9vity,thecostofmaintainingavacancyandlabormarket9ghtness.
• Theequilibriumrealwagedependsposi9velyonlabormarket9ghtness,asthisincreasestheaveragerecruitmentcostperunemployedperson,therebyincreasingthe“threat”pointofprospec9veemployeesversusprospec9veemployers,andweakeningtheeffec9vebargainingposi9onofprospec9veemployers.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 74
SimultaneousDetermina9onofRealWagesandLaborMarketTightnessandtheDetermina9on
ofEquilibriumUnemployment
• Theposi9verela9onshipbetweenrealwagesandlabormarket9ghtness,resul9ngfromthenego9a9onbetweenfirmswithvacanciesandtheunemployed,andthenega9verela9onshipbetweenrealwagesandandlabormarket9ghtnessimpliedbythejobcrea9oncondi9onforfirms,jointlydeterminetheequilibriumrealwageandequilibriumlabormarket9ghtness.
• Onceequilibriumlabormarket9ghtnesshasbeendetermined,theequilibriumunemploymentrateisthendeterminedthroughtheBeveridgecurve,whichimpliesanega9verela9onshipbetweentheunemploymentandvacancyrates.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2016 75
TheDependenceofEquilibriumUnemploymentonProduc9vity,UnemploymentBenefits,RealInterest
RatesandtheFrequencyofJobTermina9ons• Ifunemploymentbenefitsarepropor9onaltotherealwage,laborproduc9vitydoesnot
affecttheequilibriumunemploymentrateinthismodel.
• However,thehigherthepercentageofrealwagespaidoutasunemploymentbenefits,thehighertheequilibriumrealwageandthehighertheequilibriumunemploymentrate.Thereasonisthathigherrealwagesreduceincen9vesforcrea9ngnewjobs,thusreducingthenumberofvacancies,reducinglabormarket9ghtnessandincreasingunemployment.
• Higherrealinterestratesalsohaveaposi9veimpactonunemploymentinthismodel,becausetheyincreasethecostofmaintainingavacancy,resul9nginthecrea9onoffewerjobvacancies,lowerlabormarket9ghtnessandhigherunemployment.
• Ahigherexogenousprobabilityoftermina9onofajobhasaposi9veimpactonunemploymentfortworeasons.Firstbecauseitdirectlyincreasestheflowsfromexis9ngjobstounemployment,and,second,becauseit,indirectly,reducestheflowsfromunemploymenttojobs.Thesecondeffecttakesplacebecausetheexpectedprofitfromthecrea9onandfillingofavacancyfalls,resul9nginfewervacanciesandreducedflowsfromunemploymenttojobs.