Cupid's Invisible Hand

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Cupid's Invisible Hand: Social Surplus and Identification in Matching Models Alfred Galichon1 Bernard Salanie2 April 6, 20113 1Economics Department, Ecole polytechnique; e-mail: 2Department of Economics, Columbia University; e-mail: . 3The authors are grateful to Pierre-Andre Chiappori for useful comments and discussions. Galichon gratefully acknowledges support from Chaire EDF-Calyon “Finance and Developpement Durable,” Chaire Axa “Assurance et Risques Majeurs” and FiME, Laboratoire de Finance des Marches de l'Energie. Part of the research underlying this paper was done when Galichon was visit- ing the University of Chicago Booth School of Business and Columbia University. Galichon thanks the Alliance program for its support.

  • optimal assignment algorithms

  • match partners

  • surplus function

  • siow

  • partners share

  • positive assortative

  • group can


Published : Tuesday, June 19, 2012
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Cupid’sInvisibleHand:SocialSurplusandIdentificationinMatchingModelsAlfredGalichon1BernardSalanie´2April6,201131EconomicsDepartment,E´colepolytechnique;e-mail:alfred.galichon@polytechnique.edu2DepartmentofEconomics,ColumbiaUniversity;e-mail:bsalanie@columbia.edu.3TheauthorsaregratefultoPierre-Andre´Chiapporiforusefulcommentsanddiscussions.GalichongratefullyacknowledgessupportfromChaireEDF-Calyon“FinanceandDe´veloppementDurable,”ChaireAxa“AssuranceetRisquesMajeurs”andFiME,LaboratoiredeFinancedesMarche´sdel’Energie.PartoftheresearchunderlyingthispaperwasdonewhenGalichonwasvisit-ingtheUniversityofChicagoBoothSchoolofBusinessandColumbiaUniversity.GalichonthankstheAllianceprogramforitssupport.
AbstractWeinvestigateamatchinggamewithtransferableutilitywhensomeofthecharacteristicsoftheplayersareunobservabletotheanalyst.Weallowforawideclassofdistributionsofunobservedheterogeneity,subjectonlytoaseparabilityassumptionthatgeneralizesChooandSiow(2006).Wefirstshowthatthestablematchingmaximizesasocialgainfunctionthattradesofftwoterms.Thefirsttermissimplytheaveragesurplusduetotheobservablecharacteristics;andthesecondonecanbeinterpretedasageneralizedentropyfunctionthatreflectstheimpactoftheunobservedcharacteristics.Weusethisresulttoderivesimpleclosed-formformulæthatidentifythejointsurplusineverypossiblematchandtheequilibriumutilitiesofallparticipants,givenanyknowndistributionofunobservedheterogeneity.Moreover,weshowthatiftransfersareobserved,thenthepre-transferutilitiesofbothpartnersarealsoidentified.Weconcludebydiscussingsomeempiricalapproachessuggestedbytheseresultsforthestudyofmarriagemarkets,hedonicprices,andthemarketforCEOs.Keywords:matching,marriage,assignment,hedonicprices.JELcodes:C78,D61,C13.
IntroductionSincetheseminalcontributionofBecker(1973),economistshavemodeledmarriagemarketsasamatchingprobleminwhicheachpotentialmatchgeneratesamaritalsurplus.Giventransferableutilities,thedistributionsoftastesandofdesirablecharacteristicsdetermineequilibriumshadowprices,whichinturnexplainhowpartnerssharethemaritalsurplusinanyrealizedmatch.Thisinsightisnotspecificofthemarriagemarket:itcharacterizesthe“assignmentgame”(ShapleyandShubik(1972)),i.e.modelsofmatchingwithtransferableutilities.Thesemodelshavealsobeenappliedtocompetitiveequilibriumwithhedonicpricing(Chiappori,McCann,andNesheim(2008))andthemarketforCEOs(GabaixandLandier(2008)).Wewillshowhowourresultscanbeusedinthesethreecontexts;butforconcreteness,weoftenrefertopartnersasmenandwomenintheexpositionofthemainresults.WhileBeckerpresentedthegeneraltheory,hefocusedonthespecialcaseinwhichthetypesofthepartnersareone-dimensionalandarecomplementaryinproducingsurplus.Asiswell-known,thesociallyoptimalmatchesthenexhibitpositiveassortativematching.Moreover,theresultingconfigurationisstable,itisinthecoreofthecorrespondingmatchinggame,anditcanbeefficientlyimplementedbyclassicaloptimalassignmentalgorithms.Thisresultisbothsimpleandpowerful;butitsimplicationsarealsoquiteunrealisticandatvariancewiththedata,inwhichmatchesareobservedbetweenpartnerswithquitedifferentcharacteristics.Toaccountforthiswidervarietyofmatchingpatterns,onecouldintroducesearchfrictions,asinShimerandSmith(2000).Buttheresultingmodelishardtohandle,andundersomeadditionalconditionsitstillimpliesassortativematching.Asimplersolutionconsistsinallowingthejointsurplusofamatchtoincorporatelatentcharacteristics—heterogeneitythatisunobservedbytheanalyst.ChooandSiow(2006)showedthatitcanbedoneinawaythatyieldsahighlytractablemodelinlargepopulations,providedthattheunobservedheterogeneitiesenterthemaritalsurplusquasi-additivelyandthattheyaredistributedasstandardtype-Iextremevalueterms.Thentheusualapparatusofmultinomiallogitdiscretechoicemodelsapplies,linkingmarriagepatternstomarital1
surplusinaverysimplemanner1.ChooandSiow(2006)usedthismodeltolinkthechangesingainstomarriageandabortionlaws;SiowandChoo(2006)appliedittoCanadiandatatomeasuretheimpactofdemographicchanges.Ithasalsobeenusedtostudyincreasingreturnsinmarriagemarkets(BotticiniandSiow(2008))andtotestforcomplementaritiesacrosspartnereducations(Siow(2009)).WerevisitherethetheoryofmatchingwithtransferableutilitiesinthelightofChooandSiow’sinsights.Ourcontributionisthreefold.First,weextendthisframeworktomoregeneraldistributionsofutilityshocks.Chiappori,Salanie´,andWeiss(2010)showedthatquasi-additivitybyitselfreducesthecomplexityofthematchingmodeltoaseriesofdis-cretechoiceproblems.Weprovethatwithquasi-additivesurplus,themarketequilibriummaximizesasocialsurplusfunctionthatconsistsoftwoterms:atermthatdescribesassor-tativenessontheobservedcharacteristics;andageneralizedentropictermthatdescribestherandomcharacterofmatchingconditionalonobservedcharacteristics.Whilethefirsttermtendstomatchpartnerswithcomplementaryobservedcharacteristics,thesecondonepullstowardsrandomlyassigningpartnerstoeachother.Thesocialgainfromanymatch-ingpatternstradesoffthesetwoterms.Inparticular,whenunobservedheterogeneityisdistributedasinChooandSiow(2006),thegeneralizedentropyissimplytheusualentropymeasure.Oursecondcontributionistoshowthatthemaximizationofthesocialsurplusfunctiondescribedabovehasverystraightforwardconsequencesintermsofidentification,bothwhenequilibriumtransfersareobservedandwhentheyandarenot.Infact,mostquantitiesofinterestcanbeobtainedfromderivativesofthetermsthatconstitutegeneralizedentropy.Weshowinparticularthatthejointsurplusfrommatchingis(minus)aderivativeofthegeneralizedentropy,computedattheobservedmatching.Theexpectedandrealizedutilitiesofalltypesofmenandwomenfollowjustasdirectly.Ifmoreoverequilibriumtransfersareobserved,thenwealsoidentifythepre-transferutilitiesonbothsidesofthemarket.1Fox(2010)reliesinsteadonarank-orderpropertytoidentifythesurplusfunctionfromthematchingpatterns.ThehandbookchapterbyGraham(2011)discussestheseandotherapproaches.2
Theseresultssuggestvariousempiricalstrategiesthatcanbeusedtoestimatetheparametersofmodelsofmatchingwithtransferableutilities.Weshowhowtheyfitwithintheframeworkofminimumdistanceestimation,andwediscusstheirapplicabilitytothethreeclassesofmarkets:marriagemarkets,wheretransfersbetweenspousesnotobserved;themarketforCEOsandcompetitivemarketwithhedonicprices,wheretransfers(CEOcompensation,theequilibriumpricesofdifferentvarietiesofproducts)maybeobserved.Section1setsupthemodelandthenotation.Weproveourmainresultsinsection2,andwespecializethemtoleadingexamplesinsection3.OurresultsverysignificantlyextendtheChooandSiowframework:theyallowforgeneralerrordistributionswithheteroskedasticityandcorrelationacrossalternatives,asingeneralizedextremevaluesmodelsormixedlogitmodelsforinstance.Theyopenthewaytonewandricherspecifications;section4explainshowvariousrestrictionscanbeimposedtoidentifyandestimatetheunderlyingparameters2.1TheAssignmentProblemwithUnobservedHeterogeneityThroughoutthepaper,wemaintainthebasicassumptionsofthetransferableutilitymodelofChooandSiow:utilitytransfersbetweenpartnersareunconstrained,matchingisfric-tionless,andthereisnoasymmetricinformation.WealsotrytostayascloseaspossibletothenotationChooandSiowused.MencanbelongtoIgroups,indexedbyi;andwomencanbelongtoJgroups,indexedbyj.Groupscanforinstancebedefinedbyeducation,race,andothercharacteristicswhichareobservedbyallmenandwomenandalsobytheanalyst.Ontheotherhand,menandwomenofagivengroupdifferalongsomedimensionsthattheyallobserve,butwhichdonotfigureintheanalyst’sdataset.ChooandSiowassumedthattheutilityofamanmofgroupiwhomarriesawoman2Thispaperbuildsonandsignificantlyextendsourearlierdiscussionpaper(GalichonandSalanie´(2010)),whichisnowobsolete.3
ofgroupjcanbewrittenasα˜ijτij+εijm,whereτijrepresentstheutilitythatthemanhastotransfertohispartnerinequilibrium,andεijmisastandardtype-Iextremevaluedisturbance.Ifsuchamanremainssingle,hegetsutilityα˜i0+εi0m.Similarly,theutilityofawomanwofgroupjwhomarriesamanofgroupicanbewrittensaandshegetsutilityissheissingle.Onlyutilitydifferencesmatterinthismodel;wedenoteγ˜ij+τij+ηijw,γ˜0j+η0jw.αij=α˜ijα˜i0andγij=γ˜ijγ˜0j.Thekeyassumptionhereisthattheutilityofamanmofgroupiwhomarriesawomanwofgroupjdoesnotdependonwhothiswomanis—withasimilarassumptionforwomen.Wewillreturntotheinterpretationofthisassumption,whichwewillcall“separability”.Whenthereareverylargenumbersofmenandwomenwithineachgroup,ChooandSiowshowedthatthereisasimpleequilibriumrelationshipbetweengrouppreferences,asdefinedbyαandγ,andequilibriummarriagepatterns.Denoteµijthenumberofmarriagesbetweenmenofgroupiandwomenofgroupj;µi0thenumberofsinglemenofgroupi;andµ0jthenumberofsinglewomenofgroupj.Denoteπij=αij+γij.2thetotalsystematicnetgainstomarriage;andnotethatbyconstruction,πi0andπ0jarezero.ChooandSiowprovedthefollowingresult:4
Theorem1(ChooandSiow)Inequilibrium,foralli,j1µexp(πij)=ij.µi0µ0jThereforemarriagepatternsµdirectlyidentifythegainstomarriageπinsuchamodel.Itturnsoutthattheassumptiononthedistributionoftheutilityshocksεandηisnotcrucial.AsshowninChiappori,Salanie´andWeiss(2010),someofthestructureoftheproblemispreservedifthisassumptionisrelaxed.Thecrucialassumptioniswhattheycall“separability”.Tostateit,letΦmwdenotethejointsurpluscreatedbyamatchbetweenamanmandawomenw.Assumption1(Separability)Ifmenmandm0belongtothesamegroupiandwomenwandw0belongtothesamegroupj,thenΦmwm0w0mw0m0w.ItiseasytoseethatunderAssumption1,thesurplusfromamatchbetweenamanmofgroupiandawomanwofgroupjmustdecomposeintoΦmw=2πij+εijm+ηijw,wheretheεandηcanbenormalizedtohavezeromean.Again,πi0=π0j=0:withoutlossofgenerality,singlesgetzeromeanutility.Thisassumptionrulesoutinteractionsbetweenunobservedcharacteristicsinthemaritaloutputfromamatch,giventheobservedcharacteristicsofbothpartners.Ontheotherhand,itdoesnotrestrictgrouppreferencesinanyway;anditalsoallowsforvariationinmaritaloutputwithingroups,aslongastheydonotinteractacrosspartners.Forinstance,menofagivengroupmaydifferinthemaritaloutputstheycanform,butonlyasrelatestothegroupoftheirpartner.5
Totakeananalogywithdiscretechoicemodelsofconsumerpurchases,takethefollowingstandardspecificationfortheutilityabuyerbderivesfromavarietyv:Ubv=π(Xb,Xv)+Xvεb+Xbεv+εbv.Inthiscontext,separabilitywouldallowforvariationintastesoverobservedcharacteristicsofproducts(throughεv),andforgroup-dependenttastesforunobservedproductcharac-teristicsεb.Ontheotherhand,itwouldruleouttheinteractiontermεbv.Wedenotepithenumberofmenofgroupi,andqjthenumberofwomenofgroupj;thenJIXXi1,µij=pi;j1,µij=qj.(1.1)j=0i=0Forfuturereference,wedenoteMthesetof(IJ+I+J)non-negativenumbers(µij)thatsatisfythese(I+J)equalities.EachelementofMiscalleda“matching”asitdefinesafeasiblesetofmatches(andsingles).LikeChooandSiow,weassumethatthepi’sandqj’sare“large”:therearealargenumberofmeninanygroupi,andofwomeninanygroupj.Moreprecisely,ourstatementsinthefollowingareexactlytruewhenthenumberofindividualsgoestoinfinityandtheproportionsofgendersandtypesconverge.Tosimplifytheexposition,weconsiderthelimitofasequenceoflargeeconomieswheretheproportionofeachtyperemainsconstant:PAssumption2(LargeMarket)ThenumberofindividualsonthemarketN=iI=1pi+JPj=1qjgoestoinfinity;andtheratios(pi/N)and(qj/N)areconstant.WithfiniteNwewouldneedtointroducecorrectiveterms;weleavethisforfurtherresearch.2SocialSurplus,Utilities,andIdenticationAsChooandSiow(2006)remindus(p.177):“Awell-knownpropertyoftransferableutilitymodelsofthemarriagemarketisthattheymaximizethesumofmaritaloutputinthe6
society”.Thisistruewhenmaritaloutputisdefinedasitisevaluatedbytheparticipants:themarketequilibriuminfactmaximizesXδmwΦmwwmoverthesetoffeasiblematchings(δmw).Ontheotherhand,thisisnotveryusefultotheanalyst:shedoesnotobservesomeofthecharacteristicsoftheplayers,andshecanonlycomputequantitiesthatdependontheobservedgroupsofthepartnersinamatch.Averynaiveevaluationofthesumofmaritaloutput,computedfromthegroupsofpartnersonly,wouldbeX2µijπij;(2.1)jibutthisisclearlymisleading.Realizedmatchesbynaturehaveavalueoftheunobservedmaritalsurplus(εijm+ηijw)thatismorefavorablethananunconditionaldraw;andasaconsequence,theequilibriummarriagepatterns(µ)donotmaximizethevaluein(2.1)over.MInordertofindtheexpressionofthevaluefunctionthat(µ)maximizes,weneedtoaccountfortermsthatreflecttheconditionalexpectationoftheunobservedpartsofthesurplus,givenamatchonobservabletypes.Tomakethismoreprecise,weneedtointroducesomenotation.Wecontinuetoassumeseparability(Assumption1)andalargemarket(Assumption2);butweallowforquitegeneraldistributionsofunobservedheterogeneity:Assumption3(DistributionofUnobservedVariationinSurplus)a)Foranymanmi,theεijmaredrawnfroma(J+1)-dimensionaldistributionPi;b)Foranywomanwj,theηijwaredrawnfroman(I+1)-dimensionaldistribution;Qjc)Thesedrawsareindependentacrossmenandwomen.Assumption3clearlyisasubstantialgeneralizationwithrespecttoChooandSiow7
(2006),whoassumethatPiandQjareindependentproductsofstandardtype-Iextremevaluesdistributions:Assumption4(Type-Iextremevaluesdistribution)a)Foranymanmi,the(εijm)j=0,...,Jaredrawnindependentlyfromastandardtype-Iextremevaluedistribution;b)Foranywomanwj,the(ηijm)i=0,...,Iaredrawnindependentlyfromastandardtype-Iextremevaluedistribution;c)Thesedrawsareindependentacrossmenandwomen.Assumption3generalizesassumption4inthreeimportantways:itallowsfordiffer-entfamiliesofdistributions,withanyformofheteroskedasticity,andwithanypatternofcorrelationacrosspartnergroups.2.1AHeuristicDerivationNowsupposethatthemenofgroupiexpecttogetmeanutilitieswjfrommarryingpartnertypej,forj=(0,...,J).Agivenmanofthisgroup,characterizedbyadraw(εj)fromPi,wouldthenchoosethepartnertypejthatmakes(wj+εj)largest.ThereforethesumoftheexpectedutilitiesofthesemenwouldbeGi(w)=piEPij=m0,a...x,J(wj+εj),wheretheexpectationistakenoverarandomvector(ε0,...,εJ)Pi.Similarly,thesumoftheexpectedutilitiesofthewomenofgroupjisHj(z)=qjEQjmax(zi+ηi).i=0,...,IThesocialsurplusissimplythesumoftheexpectedutilitiesofalltypesofmenandwomen.ThusifwedenoteUijandVijthemeanutilitiesofamanoftypeiandofawoman8
oftypejwhentheyarematched,thesocialsurplusisJIXXGi(Ui.)+Hj(Vj.),i=1j=1denotingUi.=(Ui0=0,Ui1,...,UiJ)andV.j=(V0j=0,V1j,...,VIj).Ofcoursethesemeanutilitiesareunobserved,andwemustfindawaytowritethemintermsofthematchingpatternsµ.Wewillgivehereaheuristicexplanationofhowweobtainsuchaformula3.LetusfocusonthefunctionGi.Byconstruction,JXGi(w)=piPr(j|i;w)(wj+e(j|i;w)),(2.2)0=jwherewedenotePr(j|i;w)theprobabilitythatthemaximumisachievedforachoiceofpartneringroupjwhenmeanutilitiesarew,ande(j|i;w)theconditionalexpectationofεjinthiscase.Inparticular,ifw=Ui.thenPr(j|i;w)=µij/pi;andweobtainJJXXGi(Ui.)=µijUij+piPr(j|i;Ui.)e(j|i;Ui.).(2.3)j=0j=0NowUij+Vij=2πijsincethemeanutilitiesofthepartnersmustadduptothemaritalsurplus;andwhenweaddthefirsttermin(2.3)tothecorrespondingtermforwomen,wewillfindIJJIXXXXXµijUij+µijVij=2µijπiji=1j=0j=1i=0i,j1whichisjustthe“naive”formulain(2.1).Westillhavetoevaluatetheconditionalterms.Todothis,notethatGiisconvexsinceitisalinearcombinationofthemaximaoflinearfunctions;assuchitisalmosteverywheredifferentiable,withderivatives∂Gi(w)=piPr(j|i;w).(2.4)wjButthismeansthatwecanrewritethesecondtermin(2.2)asXJXJ∂GpiPr(j|i;w)e(j|i;w)=Gi(w)wji(w)≡G(w).j=0j=0∂wj3AppendixAgivesrigorousproofsofallofourresults.9)5.2(
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