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JobSearch: TheWorkersproblemSearchTheoryoftheLabourMarketJobSearchTheoryCarlosCarrillo-TudelaUniversityofEssexFebruary,2013CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemMotivatingevidence
Modernlabourmarketsarecharacterisedbylargeowsofworkersbetweenemployment,unemploymentandnon-participationstates.
Furtheremployedworkerscommonlychangejobswithoutandinterveningspellofnon-employment.
Typicallyworkersthatchangejobsobtainawageincrease.
However,animportantproportionofthemobtainawagedecrease.
Whenemployedthelikelihoodofmakingajob-to-jobtransitiondecreaseswithjobduration.
DurationofunemploymentdependsonthethelevelofUIbenets.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemMotivatingevidence
Modernlabourmarketsarecharacterisedbylargeowsofworkersbetweenemployment,unemploymentandnon-participationstates.
Furtheremployedworkerscommonlychangejobswithoutandinterveningspellofnon-employment.
Typicallyworkersthatchangejobsobtainawageincrease.
However,animportantproportionofthemobtainawagedecrease.
Whenemployedthelikelihoodofmakingajob-to-jobtransitiondecreaseswithjobduration.
DurationofunemploymentdependsonthethelevelofUIbenets.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemMotivatingevidence
Modernlabourmarketsarecharacterisedbylargeowsofworkersbetweenemployment,unemploymentandnon-participationstates.
Furtheremployedworkerscommonlychangejobswithoutandinterveningspellofnon-employment.
Typicallyworkersthatchangejobsobtainawageincrease.
However,animportantproportionofthemobtainawagedecrease.
Whenemployedthelikelihoodofmakingajob-to-jobtransitiondecreaseswithjobduration.
DurationofunemploymentdependsonthethelevelofUIbenets.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEvidencefromJolivetetal(EER,2006)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEvidencefromJolivetetal(EER,2006)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEvidencefromTatsiramos(JEEA,2009)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEvidencefromTatsiramos(JEEA,2009)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEvidencefromRubinsteinandWeiss(HBE,2011)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEvidencefromRubinsteinandWeiss(HBE,2011)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemWhatisSearchTheoryabout?
SearchTheorystartswiththepostulatethatthereareinformationalfrictionsinthelabourmarket.
Workersandrmsdonotknowthelocationofeachotherandmustinvestresourcesinordertoacquirethisinformation.
Onceameetingtakesplace:
Therearematch-specicrentstobeshared.
Thewaytheserentsareshareddependsontheinformationstructureandthewagedeterminationmechanism.
Ifbothpartiescometoanagreementonhowtosharetherents,productiontakesplace.
Overtime,workerscanchangeemployerorexperiencedisplacementduetojobdestruction.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemWhatisSearchTheoryabout?
SearchTheorystartswiththepostulatethatthereareinformationalfrictionsinthelabourmarket.
Workersandrmsdonotknowthelocationofeachotherandmustinvestresourcesinordertoacquirethisinformation.
Onceameetingtakesplace:
Therearematch-specicrentstobeshared.
Thewaytheserentsareshareddependsontheinformationstructureandthewagedeterminationmechanism.
Ifbothpartiescometoanagreementonhowtosharetherents,productiontakesplace.
Overtime,workerscanchangeemployerorexperiencedisplacementduetojobdestruction.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemWhySearchTheoryisrelevant?
Searchtheoryprovidesacoherent,dynamictheorythatcanbeusedtoanalyse:
Laboursupplyanddemandinamoresatisfactorywaythantheperfectcompetitiveapproach.
Thenatureofunemploymentandworkersandvacancyows.
Thedynamicofwagesoverworkerslabourmarkethistories.
Thedistributionofwagesacrossworkers.
Providesastrongtheoreticalbackgroundforempirical/quantitativeimplementation.
Usefultoolforpolicy.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemWhatarewegoingtostudy?
JobSearchtheory(McCallmodelofjobsearch).
Thebasicstools: Bellmanequations,Poissonprocesses,Optimalstoppingrules.
Unemploymentandemploymentduration.
Averagewage-experienceprole.
Equilibriumwagepostingmodels(BurdettandMortensenframework).
Wagedierentialsamongsimilarworkers.
Thewagedistributionandwagedynamics.
Humancapitalandon-the-jobsearch
Optimalwagecontracts.
Returnstoexperienceandtenure.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemWhatarewegoingtostudy?
SearchandMatchingmodels(Diamond-Mortensen-Pissaridesframework).
Thematchingfunction.
EquilibriumunemploymentandtheBeverigecurve.
Unemployment,vacanciesandtheBusinesscycle.
Othermodelsofsearchfrictions:
Stock-Flowmatchingmodels.
Islandsmodels
DirectedandCompetitiveSearch.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemWhatarewegoingtostudy?
SearchandMatchingmodels(Diamond-Mortensen-Pissaridesframework).
Thematchingfunction.
EquilibriumunemploymentandtheBeverigecurve.
Unemployment,vacanciesandtheBusinesscycle.
Othermodelsofsearchfrictions:
Stock-Flowmatchingmodels.
Islandsmodels
DirectedandCompetitiveSearch.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemWhatarewegoingtostudy?
SearchandMatchingmodels(Diamond-Mortensen-Pissaridesframework).
Thematchingfunction.
EquilibriumunemploymentandtheBeverigecurve.
Unemployment,vacanciesandtheBusinesscycle.
Othermodelsofsearchfrictions:
Stock-Flowmatchingmodels.
Islandsmodels
DirectedandCompetitiveSearch.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemWhatarewegoingtostudy?
SearchandMatchingmodels(Diamond-Mortensen-Pissaridesframework).
Thematchingfunction.
EquilibriumunemploymentandtheBeverigecurve.
Unemployment,vacanciesandtheBusinesscycle.
Othermodelsofsearchfrictions:
Stock-Flowmatchingmodels.
Islandsmodels
DirectedandCompetitiveSearch.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJobSearch: BasicFramework
Consideraninnitehorizoneconomyinwhichtimeisdiscrete;t= 0, 1, 2, ...
Thereisaunitmassofidenticalworkers.
TheobjectiveofanyworkeristomaximiseexpectedlifetimeutilityE0
t=0tu(yt),where (0, 1)isthediscountfactor,ytistheinstantaneousincomeatdatet,u(yt)istheinstantaneousutilityfunctionandE0istheexpectationconditionalontheinformationavailableatdatet= 0.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJobSearch: BasicFramework
Consideraninnitehorizoneconomyinwhichtimeisdiscrete;t= 0, 1, 2, ...
Thereisaunitmassofidenticalworkers.
TheobjectiveofanyworkeristomaximiseexpectedlifetimeutilityE0
t=0tu(yt),where (0, 1)isthediscountfactor,ytistheinstantaneousincomeatdatet,u(yt)istheinstantaneousutilityfunctionandE0istheexpectationconditionalontheinformationavailableatdatet= 0.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJobSearch: BasicFramework
Forsimplicityassumethatworkersareriskneutralsuchthatu(yt) = yt.
Instantaneousincomeisgivenby
Ifemployedatwagew,yt= w.
Ifunemployed,yt= z.
Fixthehoursofworktounitysothatwrepresentsthehourlywageorthetotal(perperiod)associatedwithajob.
Theopportunitycostofemploymentz,canbeinterpretedtoinclude
thevalueofleisureandhomeproductionactivities
plusunemploymentbenets
minusout-of-thepocketsearchcosts.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJobSearch: BasicFramework
Forsimplicityassumethatworkersareriskneutralsuchthatu(yt) = yt.
Instantaneousincomeisgivenby
Ifemployedatwagew,yt= w.
Ifunemployed,yt= z.
Fixthehoursofworktounitysothatwrepresentsthehourlywageorthetotal(perperiod)associatedwithajob.
Theopportunitycostofemploymentz,canbeinterpretedtoinclude
thevalueofleisureandhomeproductionactivities
plusunemploymentbenets
minusout-of-thepocketsearchcosts.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJobSearch: BasicFramework
Forsimplicityassumethatworkersareriskneutralsuchthatu(yt) = yt.
Instantaneousincomeisgivenby
Ifemployedatwagew,yt= w.
Ifunemployed,yt= z.
Fixthehoursofworktounitysothatwrepresentsthehourlywageorthetotal(perperiod)associatedwithajob.
Theopportunitycostofemploymentz,canbeinterpretedtoinclude
thevalueofleisureandhomeproductionactivities
plusunemploymentbenets
minusout-of-thepocketsearchcosts.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJobSearch: BasicFramework
Forsimplicityassumethatworkersareriskneutralsuchthatu(yt) = yt.
Instantaneousincomeisgivenby
Ifemployedatwagew,yt= w.
Ifunemployed,yt= z.
Fixthehoursofworktounitysothatwrepresentsthehourlywageorthetotal(perperiod)associatedwithajob.
Theopportunitycostofemploymentz,canbeinterpretedtoinclude
thevalueofleisureandhomeproductionactivities
plusunemploymentbenets
minusout-of-thepocketsearchcosts.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemBasicFramework
Anunemployedworkerisnotworkingbutactivelysearchingforajob.
Consistentwiththedenitionusedtomeasureunemployment.
Theworkersproblemistochoosewhentoacceptemployment(leaveunemployment)suchthathemaximisesexpectedlifetimeutility.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemBasicFramework
Anunemployedworkerisnotworkingbutactivelysearchingforajob.
Consistentwiththedenitionusedtomeasureunemployment.
Theworkersproblemistochoosewhentoacceptemployment(leaveunemployment)suchthathemaximisesexpectedlifetimeutility.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemAsequential(random)searchprocess
AnunemployedworkersamplesoneiidwageoereachperiodfromaknownwageoerdistributionF.
Letwandwdenotethe(nite)inmumandsupremumofthesupportofF.
Ifthewageoerisaccepted,theworkerisemployedatthatwageforever.
Ifthewageoerisrejected,theworkerremainsunemployedandcontinuesthesearch.
Thereisnorecallofpastoers.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemAsequential(random)searchprocess
AnunemployedworkersamplesoneiidwageoereachperiodfromaknownwageoerdistributionF.
Letwandwdenotethe(nite)inmumandsupremumofthesupportofF.
Ifthewageoerisaccepted,theworkerisemployedatthatwageforever.
Ifthewageoerisrejected,theworkerremainsunemployedandcontinuesthesearch.
Thereisnorecallofpastoers.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemAsequential(random)searchprocess
AnunemployedworkersamplesoneiidwageoereachperiodfromaknownwageoerdistributionF.
Letwandwdenotethe(nite)inmumandsupremumofthesupportofF.
Ifthewageoerisaccepted,theworkerisemployedatthatwageforever.
Ifthewageoerisrejected,theworkerremainsunemployedandcontinuesthesearch.
Thereisnorecallofpastoers.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemBellmanequations
LetV(w)denotethelifetimeutilityofemploymentatwagew.Sincejobsareretainedforever,V(w) =
t=0tw=w1 ,V(w) = w+V(w).
LetUdenotetheexpectedlifetimeutilityofunemployment.Sincethewageoersareiid,U= z +
wwmax[V(w), U]dF(w).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemBellmanequations
LetV(w)denotethelifetimeutilityofemploymentatwagew.Sincejobsareretainedforever,V(w) =
t=0tw=w1 ,V(w) = w+V(w).
LetUdenotetheexpectedlifetimeutilityofunemployment.Sincethewageoersareiid,U= z +
wwmax[V(w), U]dF(w).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOptimalstoppingrule
AworkerwillacceptemploymentatwagewifandonlyifV(w) Uorequivalentlyw1 z +
wwmax[V(w), U]dF(w).
TheminimumacceptablewageRsuchthatforallw Riscalledthereservationwage.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOptimalstoppingrule
AworkerwillacceptemploymentatwagewifandonlyifV(w) Uorequivalentlyw1 z +
wwmax[V(w), U]dF(w).
TheminimumacceptablewageRsuchthatforallw Riscalledthereservationwage.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOptimalstoppingrule
RthensolvesV(R) = UandhenceR= z(1 ) +
wwmax[w, R]dF(w).
Theoptimalstoppingpolicythenimpliesareservationstrategy.
WeusethecontractionmappingtheoremtoprovethatthereexistsauniqueR.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOptimalstoppingrule
RthensolvesV(R) = UandhenceR= z(1 ) +
wwmax[w, R]dF(w).
Theoptimalstoppingpolicythenimpliesareservationstrategy.
WeusethecontractionmappingtheoremtoprovethatthereexistsauniqueR.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContractionMappingTheorem
LetC[w,w]denotethesetofcontinuousandboundedfunctionson[w, w] [w, w].
DenethemappingT: C[w,w] C[w,w]asT max
w1 , z +
ww(w
)dF(w
)
.
ThusTtakesafunctionandmapsitintoanotherfunctionT.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContractionMappingTheorem
Usingthesup-normmetricd(, ) = supwww| (w) (w) |,guaranteesthemetricspaceis(C[w,w], d)iscomplete;
TisdeneonaBanachspace.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContractionMappingTheorem
TheBlackwellsucientconditionsthenguaranteesthatTisacontractionwithmodulus:
Tismonotone: Given(w) (w)forallw [w, w],itfollowsthatT T.
Tsatisesthediscountingproperty: Foranypositiveconstantm,T( + m) = T +m.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContractionMappingTheorem
TheContractionMappingTheorem(BanachFixedPointTheorem)guaranteesthatthereexistsauniquesolutionsuchthat= T.
Let = max[V(w), U]
NotethatVisincreasinginwandUisindependentofit
ThereexistsauniqueRsuchthatV(w) Uifandonlyifw R.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContractionMappingTheorem
TheContractionMappingTheorem(BanachFixedPointTheorem)guaranteesthatthereexistsauniquesolutionsuchthat= T.
Let = max[V(w), U]
NotethatVisincreasinginwandUisindependentofit
ThereexistsauniqueRsuchthatV(w) Uifandonlyifw R.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemUsefulrepresentationofR
NotethatthevaluefunctionsUandVcanbeexpressedasV(w) = w+V(w)U= z +U +
wR[V(w) U]dF(w).
SinceU= V(R),thereservationwageisgivenbyR= z +1
wR[w R]dF(w).
IntegrationbypartsthenimpliesR= z +1
wR[1 F(w)]dw.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemUsefulrepresentationofR
NotethatthevaluefunctionsUandVcanbeexpressedasV(w) = w+V(w)U= z +U +
wR[V(w) U]dF(w).
SinceU= V(R),thereservationwageisgivenbyR= z +1
wR[w R]dF(w).
IntegrationbypartsthenimpliesR= z +1
wR[1 F(w)]dw.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemUsefulrepresentationofR
NotethatthevaluefunctionsUandVcanbeexpressedasV(w) = w+V(w)U= z +U +
wR[V(w) U]dF(w).
SinceU= V(R),thereservationwageisgivenbyR= z +1
wR[w R]dF(w).
IntegrationbypartsthenimpliesR= z +1
wR[1 F(w)]dw.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJoboersandlayos
Supposethatattheendofeachperiodthereisapositiveprobabilitythatajoboerarrives.
TheexpectedvalueofunemploymentisgivenbyU= z +[(1 )U +
wwmax[V(w), U]dF(w)],CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJoboersandlayos
Supposethatattheendofeachperiodthereisapositiveprobabilitythatajoboerarrives.
TheexpectedvalueofunemploymentisgivenbyU= z +[(1 )U +
wwmax[V(w), U]dF(w)],CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJoboersandlayos
Nextsupposethatonceemployed,theworkerfacesaprobabilityofbeingdisplacedattheendofeachperiod.
TheexpectedvalueofemploymentisV(w) = w+[(1 )V(w) +U],
Ex. 1: ShowthatRexists,isuniqueandsolvesR= z +1 +
wR[1 F(w)]dw.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJoboersandlayos
Nextsupposethatonceemployed,theworkerfacesaprobabilityofbeingdisplacedattheendofeachperiod.
TheexpectedvalueofemploymentisV(w) = w+[(1 )V(w) +U],
Ex. 1: ShowthatRexists,isuniqueandsolvesR= z +1 +
wR[1 F(w)]dw.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemJoboersandlayos
Nextsupposethatonceemployed,theworkerfacesaprobabilityofbeingdisplacedattheendofeachperiod.
TheexpectedvalueofemploymentisV(w) = w+[(1 )V(w) +U],
Ex. 1: ShowthatRexists,isuniqueandsolvesR= z +1 +
wR[1 F(w)]dw.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemSurplusfunctionandcomparativestatics
DenethesurplusfunctionS(R) =
wR[1 F(w)]dw.
Dierentiation(LeibnizsRule)impliesS
(R) = [1 F(R)] < 0forR< wS
(R) = F
(R) > 0whenFisdierentiableandhasapositivedensity.
Ex. 2: Usingthesurplusfunctionre-doEx. 1.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemSurplusfunctionandcomparativestatics
DenethesurplusfunctionS(R) =
wR[1 F(w)]dw.
Dierentiation(LeibnizsRule)impliesS
(R) = [1 F(R)] < 0forR< wS
(R) = F
(R) > 0whenFisdierentiableandhasapositivedensity.
Ex. 2: Usingthesurplusfunctionre-doEx. 1.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemSurplusfunctionandcomparativestatics
DenethesurplusfunctionS(R) =
wR[1 F(w)]dw.
Dierentiation(LeibnizsRule)impliesS
(R) = [1 F(R)] < 0forR< wS
(R) = F
(R) > 0whenFisdierentiableandhasapositivedensity.
Ex. 2: Usingthesurplusfunctionre-doEx. 1.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemSurplusfunctionandcomparativestatics
DenetheimplicitfunctionG(R, z, , ) R z 1 +S(R) = 0.
Ex. 3: VerifythatthefollowinginequalitiesaretrueRz> 0,R> 0,R< 0.Discusstheeconomicintuitionoftheseresults.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemSurplusfunctionandcomparativestatics
DenetheimplicitfunctionG(R, z, , ) R z 1 +S(R) = 0.
Ex. 3: VerifythatthefollowinginequalitiesaretrueRz> 0,R> 0,R< 0.Discusstheeconomicintuitionoftheseresults.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessPoissonprocess
Consideraninnitehorizoneconomywithtimeperiodsoflengthdt.
Supposethatineachperiodtheworkerreceivesn = 0, 1, 2, ...iidwageoerswithprobabilitya(n, dt).
Property1: LetN()denotethenumberofwageoersreceivedbytime 0,suchN(0) = 0.
Property2: Assumethata(n, dt)followsaPoissondistributionsuchthata(n, dt) =edt(dt)nn!.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessPoissonprocess
Consideraninnitehorizoneconomywithtimeperiodsoflengthdt.
Supposethatineachperiodtheworkerreceivesn = 0, 1, 2, ...iidwageoerswithprobabilitya(n, dt).
Property1: LetN()denotethenumberofwageoersreceivedbytime 0,suchN(0) = 0.
Property2: Assumethata(n, dt)followsaPoissondistributionsuchthata(n, dt) =edt(dt)nn!.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessPoissonprocess
Consideraninnitehorizoneconomywithtimeperiodsoflengthdt.
Supposethatineachperiodtheworkerreceivesn = 0, 1, 2, ...iidwageoerswithprobabilitya(n, dt).
Property1: LetN()denotethenumberofwageoersreceivedbytime 0,suchN(0) = 0.
Property2: Assumethata(n, dt)followsaPoissondistributionsuchthata(n, dt) =edt(dt)nn!.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessPoissonprocess
Property3: Assumeindependentincrements: Thenumberofoersreceivedisindependentacrossperiods.
Thesepropertiesimplythatthecountingprocess{N(), 0}isaPoissonprocesswithrate > 0.
TheusefulnessofthePoissonprocessisthatprob[N(dt) = 1] = dt + O(dt)prob[N(dt) 2] = O(dt),whereafunctionf issaidtobeO(dt)iflimdt0f (dt)dt= 0.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessPoissonprocess
Property3: Assumeindependentincrements: Thenumberofoersreceivedisindependentacrossperiods.
Thesepropertiesimplythatthecountingprocess{N(), 0}isaPoissonprocesswithrate > 0.
TheusefulnessofthePoissonprocessisthatprob[N(dt) = 1] = dt + O(dt)prob[N(dt) 2] = O(dt),whereafunctionf issaidtobeO(dt)iflimdt0f (dt)dt= 0.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessPoissonprocess
Property3: Assumeindependentincrements: Thenumberofoersreceivedisindependentacrossperiods.
Thesepropertiesimplythatthecountingprocess{N(), 0}isaPoissonprocesswithrate > 0.
TheusefulnessofthePoissonprocessisthatprob[N(dt) = 1] = dt + O(dt)prob[N(dt) 2] = O(dt),whereafunctionf issaidtobeO(dt)iflimdt0f (dt)dt= 0.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessBellmanequations
AssumethatlayosalsofollowaPoissonprocesswithrate> 0.
Letr denotethediscountrate.
TheexpectedvalueofunemploymentisthengivenbyU= zdt+11 + rdt
(1 dt)U +dt
wwmax[V(w), U]dF(w)
+O(dt).
TheexpectedvalueofemploymentatawagewisgivenbyV(w) = wdt +11 + rdt[(1 dt)V(w) +dtU] + O(dt)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessBellmanequations
AssumethatlayosalsofollowaPoissonprocesswithrate> 0.
Letr denotethediscountrate.
TheexpectedvalueofunemploymentisthengivenbyU= zdt+11 + rdt
(1 dt)U +dt
wwmax[V(w), U]dF(w)
+O(dt).
TheexpectedvalueofemploymentatawagewisgivenbyV(w) = wdt +11 + rdt[(1 dt)V(w) +dtU] + O(dt)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessBellmanequations
AssumethatlayosalsofollowaPoissonprocesswithrate> 0.
Letr denotethediscountrate.
TheexpectedvalueofunemploymentisthengivenbyU= zdt+11 + rdt
(1 dt)U +dt
wwmax[V(w), U]dF(w)
+O(dt).
TheexpectedvalueofemploymentatawagewisgivenbyV(w) = wdt +11 + rdt[(1 dt)V(w) +dtU] + O(dt)CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessBellmanequations
Re-arranginganlettingdt 0,wehavethatrU= z +
wwmax[V(w) U, 0]dF(w)rV(w) = w+[U V(w)].
TheseBellmanequationscanbeinterpretedasassetpricingequationsinperfectcapitalmarkets.
Duetoarbitragetheexpectedreturnofanassetelsewhere(atrater )equalstheowpayooftheassetduringthenextperiod(e.g. dividends)plusanycapitalgainsduetopricechanges.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessBellmanequations
Re-arranginganlettingdt 0,wehavethatrU= z +
wwmax[V(w) U, 0]dF(w)rV(w) = w+[U V(w)].
TheseBellmanequationscanbeinterpretedasassetpricingequationsinperfectcapitalmarkets.
Duetoarbitragetheexpectedreturnofanassetelsewhere(atrater )equalstheowpayooftheassetduringthenextperiod(e.g. dividends)plusanycapitalgainsduetopricechanges.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessBellmanequations
Re-arranginganlettingdt 0,wehavethatrU= z +
wwmax[V(w) U, 0]dF(w)rV(w) = w+[U V(w)].
TheseBellmanequationscanbeinterpretedasassetpricingequationsinperfectcapitalmarkets.
Duetoarbitragetheexpectedreturnofanassetelsewhere(atrater )equalstheowpayooftheassetduringthenextperiod(e.g. dividends)plusanycapitalgainsduetopricechanges.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessReservationwage
InthisnewenvironmentthereservationwageisgivenbyR= z +
wR[V(w) U]dF(w).
IntegrationbypartsthenimpliesR = z +r+
wR[1 F(w)]dw= z +r+S(R).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemContinuoustimeapproximationandthePoissonprocessReservationwage
InthisnewenvironmentthereservationwageisgivenbyR= z +
wR[V(w) U]dF(w).
IntegrationbypartsthenimpliesR = z +r+
wR[1 F(w)]dw= z +r+S(R).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheort
Uptonowwehaveassumethatworkerscannotcontroltherateatwhichjoboersarrive.
Supposethatthewageoerarrivalrateis = (s),wheres 0denotesthesearchintensityortheamountofeortdevotedintosearch. Assumethat(0) = u> 0and
> 0,
< 0.
Supposethereisapercostk(s)ofsearchsuchk(0) = 0andthatk
> 0,k
> 0.
Theopportunitycostofemploymentisthengivenbyz= b k(s).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheort
Uptonowwehaveassumethatworkerscannotcontroltherateatwhichjoboersarrive.
Supposethatthewageoerarrivalrateis = (s),wheres 0denotesthesearchintensityortheamountofeortdevotedintosearch. Assumethat(0) = u> 0and
> 0,
< 0.
Supposethereisapercostk(s)ofsearchsuchk(0) = 0andthatk
> 0,k
> 0.
Theopportunitycostofemploymentisthengivenbyz= b k(s).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheort
Uptonowwehaveassumethatworkerscannotcontroltherateatwhichjoboersarrive.
Supposethatthewageoerarrivalrateis = (s),wheres 0denotesthesearchintensityortheamountofeortdevotedintosearch. Assumethat(0) = u> 0and
> 0,
< 0.
Supposethereisapercostk(s)ofsearchsuchk(0) = 0andthatk
> 0,k
> 0.
Theopportunitycostofemploymentisthengivenbyz= b k(s).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheort
TheexpectedvalueofunemploymentisnowgivenbyrU= maxs0
b k(s) +(s)
wwmax[V(w) U, 0]dF(w)
.
Letsdenotetheoptimalsearchintensity. Sincethereservationwagecanbederivedgivenanysearchintensity,wehavethatR= b k(s) +(s)r+S(R).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheort
MaximisingthevalueofunemploymentthenimpliesthatssatisestheKhun-Tuckercondition(r+)k
(s)
(s)S(R) =ifs> 0.
Giventhepropertiesofandk,thereexistsauniques> 0thatmaximisesU.
Theworkeroptimalsearchstrategyisthengivenbythepair(R, s).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheort
MaximisingthevalueofunemploymentthenimpliesthatssatisestheKhun-Tuckercondition(r+)k
(s)
(s)S(R) =ifs> 0.
Giventhepropertiesofandk,thereexistsauniques> 0thatmaximisesU.
Theworkeroptimalsearchstrategyisthengivenbythepair(R, s).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheort
MaximisingthevalueofunemploymentthenimpliesthatssatisestheKhun-Tuckercondition(r+)k
(s)
(s)S(R) =ifs> 0.
Giventhepropertiesofandk,thereexistsauniques> 0thatmaximisesU.
Theworkeroptimalsearchstrategyisthengivenbythepair(R, s).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOn-the-jobsearch
ThisextensionisbasedonBurdett,(AER1978).
Supposethattheoerarrivalratetounemployedworkersisu.
Supposethatemployedworkerscancontinuesearchingfromthesameoerdistribution.
Assumethatoersarrivetoemployedworkersatratee.
TheexpectedvalueofunemploymentisrU= z +u
wwmax[V(w) U, 0]dF(w).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOn-the-jobsearch
ThisextensionisbasedonBurdett,(AER1978).
Supposethattheoerarrivalratetounemployedworkersisu.
Supposethatemployedworkerscancontinuesearchingfromthesameoerdistribution.
Assumethatoersarrivetoemployedworkersatratee.
TheexpectedvalueofunemploymentisrU= z +u
wwmax[V(w) U, 0]dF(w).CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOn-the-jobsearch
TheexpectedvalueofemploymentatwagewisrV(w) = w+e
wwmax[V(w
)V(w), 0]dF(w
)+[UV(w)]
TheoptimalsearchstrategyforanunemployedworkerisstilldescribedbyareservationwageRsuchthattheworkeracceptsemploymentifandonlyifw R.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOn-the-jobsearch
TheexpectedvalueofemploymentatwagewisrV(w) = w+e
wwmax[V(w
)V(w), 0]dF(w
)+[UV(w)]
TheoptimalsearchstrategyforanunemployedworkerisstilldescribedbyareservationwageRsuchthattheworkeracceptsemploymentifandonlyifw R.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemOn-the-jobsearch
Theoptimalsearchstrategyforanemployedworkeristomovetoadierentjobifandonlyifw
> w.
ThereservationwageinthiscaseisR= z + (ue)
wR[1 F(w)]r+ +e(1 F(w))dw.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheortwithOJS
Letsuandsedenotethesearcheortofanunemployedandemployedworker,respectively.
Theoerarrivalratesaregivenbyi= (si)with(0) = and
> 0,
< 0fori = u, e.
Thesearchcostki= k(si)suchthatki(0) = 0andk
i> 0,k
i> 0fori = u, e.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheortwithOJS
Letsuandsedenotethesearcheortofanunemployedandemployedworker,respectively.
Theoerarrivalratesaregivenbyi= (si)with(0) = and
> 0,
< 0fori = u, e.
Thesearchcostki= k(si)suchthatki(0) = 0andk
i> 0,k
i> 0fori = u, e.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheortwithOJS
Letsuandsedenotethesearcheortofanunemployedandemployedworker,respectively.
Theoerarrivalratesaregivenbyi= (si)with(0) = and
> 0,
< 0fori = u, e.
Thesearchcostki= k(si)suchthatki(0) = 0andk
i> 0,k
i> 0fori = u, e.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheortwithOJS
ThevaluefunctionsinthiscasearegivenbyrU= maxsu0
b ku(su) +(su)
wR[V(w) U]dF(w)
rV(w) = maxse0
w ke(se) +(se)
ww[V(w
) V(w)]dF(w
)
+[U V(w)]CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheortwithOJS
SolvingthemazimisationproblemyieldsthefollowingKuhn-Tuckerconditions
(su)
wR[V(w) U]dF(w) k
u(su) with =ifsu> 0,
(se)
ww[V(w
)V(w)]dF(w
) = k
e(se) with =ifse> 0.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheortwithOJS
Thepropertiesofandkimplythatsu> 0andse> 0.
Itiseasytoverifythatifk
e> k
uforallsthensu> se.
Further,workerswithhigherwageswillspendlesseortinsearch.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheortwithOJS
Thepropertiesofandkimplythatsu> 0andse> 0.
Itiseasytoverifythatifk
e> k
uforallsthensu> se.
Further,workerswithhigherwageswillspendlesseortinsearch.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemEndogenoussearcheortwithOJS
Thepropertiesofandkimplythatsu> 0andse> 0.
Itiseasytoverifythatifk
e> k
uforallsthensu> se.
Further,workerswithhigherwageswillspendlesseortinsearch.CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemDurationofUnemploymentandEmployment
ThePoissonprocessassumptionimpliesthatthedurationofunemploymentisexponentiallydistributedwithparameteru(su)[1 F(R)].
Theaveragedurationofunemploymentis1(su)[1 F(R)]
Similarly,thedurationofemploymentatwagewisexponentiallydistributedwithparameter +(se)[1 F(w)].
Theaveragedurationofemploymentatwagewis1 +(se)[1 F(w)].CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemDurationofUnemploymentandEmployment
ThePoissonprocessassumptionimpliesthatthedurationofunemploymentisexponentiallydistributedwithparameteru(su)[1 F(R)].
Theaveragedurationofunemploymentis1(su)[1 F(R)]
Similarly,thedurationofemploymentatwagewisexponentiallydistributedwithparameter +(se)[1 F(w)].
Theaveragedurationofemploymentatwagewis1 +(se)[1 F(w)].CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemDurationofUnemploymentandEmployment
Animportantimplicationofthistheoryisthatthereis:
Apositiverelationshipbetweenwagesandlabourmarketexperience
Tenureonthejobandwagesarepositivelyrelated.
Quitratesandwagesarenegativelyrelated.
Ex. 4:Whataretheeectsof,iandzonthedurationofemploymentandunemployment?CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemDurationofUnemploymentandEmployment
Animportantimplicationofthistheoryisthatthereis:
Apositiverelationshipbetweenwagesandlabourmarketexperience
Tenureonthejobandwagesarepositivelyrelated.
Quitratesandwagesarenegativelyrelated.
Ex. 4:Whataretheeectsof,iandzonthedurationofemploymentandunemployment?CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemDurationofUnemploymentandEmployment
Animportantimplicationofthistheoryisthatthereis:
Apositiverelationshipbetweenwagesandlabourmarketexperience
Tenureonthejobandwagesarepositivelyrelated.
Quitratesandwagesarenegativelyrelated.
Ex. 4:Whataretheeectsof,iandzonthedurationofemploymentandunemployment?CarlosCarrillo-Tudela SearchTheoryoftheLabourMarketJobSearch: TheWorkersproblemDurationofUnemploymentandEmployment
Animportantimplicationofthistheoryisthatthereis:
Apositiverelationshipbetweenwagesandlabourmarketexperience
Tenureonthejobandwagesarepositivelyrelated.
Quitratesandwagesarenegativelyrelated.
Ex. 4:Whataretheeectsof,iandzonthedurationofemploymentandunemployment?CarlosCarrillo-Tudela SearchTheoryoftheLabourMarket