multidimensionaltypes.
IvarEkeland
CanadaResearchChairinMathematicalEconomics,UBC
Firstversion,April2005;thisversion,August2008
Abstract
Westudyequilibriuminhedonicmarkets,whenconsumersandsuppli-ershavereservationutilities,andtheutilityfunctionsareseparablewithrespecttoprice.Thereisoneindivisiblegood,whichcomesindifferentqualities;eachconsumerbuys0or1unit,andeachsuppliersells0or1unit.Consumertypes,suppliertypesandqualitiescanbeeitherdiscreteofcontinuous,inwhichcasetheyareallowedtobemultidimensional.Pricesplayadoublerole:theykeepsomeagentsoutofthemarket,andtheymatchtheremainingonespairwise.Wedefineequilibriumpricesandequilibriumdistributions,andweprovethatequilibriaexist,weinvesti-gatetowhatextendequilibriumpricesanddistributionsareunique,andweprovethatequilibriaareefficient.Intheparticularcasewhenthereisacontinuumoftypes,andageneralizedSpence-Mirrleesconditionissatisfied,weprovetheexistenceofapureequilibrium,wheredemanddis-tributionsareinfactdemandfunctions,andweshowtowhatextentitisunique.Theproofsrelyonconvexanalysis,andcarehasbeengiventoillustratethetheorywithexamples.
1
Introduction.
1.1
Mainresults.
Inthispaper,weshowtheexistenceanduniquenessofequilibriuminahedonicmarket,andwegiveuniquenessresults.Themainfeaturesofourmodelareasfollows:
•Thereisasingle,indivisible,goodinthemarket,anditcomesindifferentqualitiesz
1
•Consumersandproducersareprice-takersandutility-maximizers.Theyarecharacterizedbythevaluesofsomevariables;eachsetofvaluesiscalleda(multidimensional)type.
•Consumersbuyatmostoneunitofthegood,andtheybuynoneiftheirreservationutilityisnotmet;producerssupplyatmostoneunitofthegood,andtheysupplynoneiftheirreservationutilityisnotmet.Inotherwords,agentsalwayshavetheoptionofstayingoutofthemarket.•Theutilitiesofconsumersandofproducersarequasi-linearwithrespecttoprice:theutilityconsumerswithtypexderivefrombuyingoneunitofqualityzatpricep(z)isu(x,z)−p(z),andtheutilityproducerswithtypeyderivefromsellingoneunitofqualityzatpricep(z)isp(z)−v(y,z)Ourresultsarevalidinthediscretecaseandinthecontinuouscase.Weshowthatthereisa(nonlinear)pricesystemp(z)suchthat,foreveryqualityz,thenumber(ortheaggregatemass)ofconsumerswhodemandzisequaltothenumber(ortheaggregatemass)ofsupplierswhoproducez.Inaddition,agentswhoarestayingoutofthemarketaredoingsobecausebyenteringtheywouldlowertheirutility.Inotherwordsthispricesystemexactlymatchesasubsetofconsumerswithasubsetofproducers,andtheremainingconsumersorproducersarepricedoutofthemarket.Thisiscalledanequilibriumprice,andtheresultingallocationofqualitiesiscalledanequilibriumallocation.Anexampleisgiveninsection4.4,andthereadermayproceedtheredirectly.Weshouldstress,however,thatweproveexistenceinfullgenerality,beyondtheone-dimensionalsituationdescribedinthatexample.
Everypricesystemp(z)createsamatchingbetweenconsumersandproduc-ers:foreveryunittraded,thereisapairconsistingofaconsumerwhobuysitandaproducerwhosellsit.Whensummingtheirutilities,thepriceofthetradeditemcancelsout,sothattheresultingutilityofthepairisindependentofthepricesystem.Unmatchedconsumersandproducers(singles)gettheirreservationutility.Itisthenmeaningfultotakethesocialplanner’spointofview,andtoaskforamatchingbetweenconsumersandproducerswhichwillmaximizeaggregateutility,wheretheutilityofmatchedpairsisthemaximumutilitytheycangetbytrading,andtheutilityofunmatchedagentsistheirreservationutility.Wewillshowthatthesolutionofthisproblemcoincideswiththeequilibriummatching.Thisimpliesthateveryequilibriumisefficient.Aninterestingfeatureofequilibriumpricingisthat,eventoughalltechno-logicallyfeasiblequalitiesarepriced,notallofthemwillbetradedinequilib-rium.Foreachnon-tradedquality,thereisanon-emptybid-askrange:allpriceswhichfallwithinthatrangeareequilibriumprices,thatis,theywillnotlurecustomersorsuppliersawayfromtradedqualities.Thismeansthatequilibriumpricescannotbeuniquelydefinedonnon-tradedqualities.Ontheotherhand,theyareuniquelydefinedontradedqualities.Thereisacorrespondingdegreeofuniquenessfortheequilibriumallocation.
Themaindrawbackofourmodelistheassumptionthatutilitiesarequasi-linear.Itisquitearestriction,fromtheeconomicpointofview,sinceitmeans
2
thatthemarginalutilityofmoneyisconstant,butourproofseemstorequireitinanessentialway.Ontheotherhand,italsoenablesustoprovesomeuniquenessresults,whichareprobablynottobeexpectedinthemoregeneralcase.
1.2Thelitterature.
Thispaperinheritsfromtwotraditionsineconomics.Ontheonehand,itcanbeseenasacontributiontotheresearchprogramonhedonicpricingthatwasoutlinedbyShervinRoseninhisseminalpaper[16].Theideaofdefiningagoodasabundleofattributes(originatingperhapswithHouthakker[9],anddevelopedbyLancaster[11],Becker[1]andMuth[12]),providesasystematicframeworkfortheeconomicanalysisofthesupplyanddemandforquality.Themaindirectionofinvestigation,however,hasbeentowardseconometricissues,suchastheconstructionofpriceindicesnetofchangesinquality;seeforinstancetheseminalworkofCourt[3]andthebook[8]).TheidentificationofhedonicmodelsraisesspecificquestionswhichhavebeenfirstdiscussedbyRosen[16],andmostrecentlybyEkeland,HeckmanandNesheim[4].Theoreticalquestion,suchastheexistenceandcharacterizationofequilibria,haveattractedlessat-tention.ThepapersbyRosen[16]andlaterMussaandRosen[14]studytheone-dimensionalsituation,thatis,thecasewhenagentsarefullycharacterizedbythevalueofasingleparameter.ThemultidimensionalsituationhasbeeninvestigatedbyRochetandChon´e[15],butitdealswithmonopolypricing.Theissueofequilibriumpricinginthemultidimensionalsituation,hadtomyknowledgenotbeenadresseduptonow(nor,forthatmatter,hastheissueofoligopolypricing).
OneofRosen’smainachievementhasbeentorecognizehedonicpricingasnonlinear,againsttheprevailingtraditionineconometricusage.Asnotedin[16],abuyercanforcepricestobelinearwithrespecttoqualityifcertaintypesofarbitrageareallowed.Inthepresentpaper,buyersandsellersarerestrictedtotradingoneunitofasinglequality,andthereisnosecond-handmarket,sothiskindofarbitrageisunavailable,andpriceswillbeinherentlynonlinear.Thiswouldnotbethecaseifconsumersandproducerswereallowedtobuyandsellseveralqualitiessimultaneously.
Ontheotherhand,thispaperalsobelongstothetraditionofassignmentproblems.Thistraditionhasseveralstrands,oneofwhichoriginateswithKoop-mansandBeckmann[10],andtheotherwithShapleyandShubik[17].WerefertothepapersbyGretzki,OstroyandZame[6]and[7],andto[13]formorerecentwork.Inthisliterature,producersarenotfreetochoosethequalitytheysell:eachqualityisassociatedwithasingleproducer,whocanproducethatoneandnotanyotherone.TheShapley-Shubikmodel,forinstance,describesamarketforhouses.Thereareacertainnumberofsellers,eachoneisendowedwithahouse,andacertainnumberofbuyers.Nosellercansellahouseotherthanhisown,butabuyercanbuyanyhouse.Thisisincontrastwiththesituationinthepresentpaper,wherebothbuyersandsellersarefreetochoosethequalitytheybuyorsell.
3
1.3Structureofthepaper
Section2describesthemathematicalmodelandthebasicassumptions.Aswementionedearlier,wedonotrequirethatthedistributionoftypesbecontinuous,northatthenumberofconsumersequalsthenumberofproducers.Mathemat-icallyspeaking,thereisapositivemeasureµonthesetofconsumertypesX,andameasureνonthesetofproducertypesY,bothµandνcanhaveatoms,andtypicallyµ(X)=ν(Y).Thesefeatures,althoughveryappealingfromthepointofviewofeconomicmodelling,introducegreatcomplicationsinthemathematicaltreatment.Inearlierwork[5],theauthorhasgivenastreamlinedproofintheparticularcasewhenµandνarenon-atomic,µ(X)=ν(Y)andanadditionalsortingassumptiononutilitiesissatisfied(extendingtomultidimen-sionaltypestheclassicalSpence-Mirrleessingle-crossingassumption),sothatallagentswiththesametypedothesamething.BesidethefactthatitdoesnotapplywhenXorYarefinite,suchamodeldoesnotcaptureoneoftheessentialroleofprices,whichservenotonlytomatchconsumersandproducerswhichenterthemarket(theremustnecessarilybeanequalnumberofboth)butalsotokeepoutofthemarketenoughagentssothatmatchingbecomespossible.Thelatterfunctionisanessentialfocusofthepresentpaper.
Inourmodel,thereisasingleindivisiblegood,consumersarerestrictedtobuyingoneorzerounit,andproducersarerestrictedtosupplyoneorzerounit.Thepriceisanonlinearfunctionp(z)ofthequalityz.Itisanequilibriumpriceifthemarketforeveryqualityclears.Thisimpliesthatthenumberofconsumerswhotradeisequaltothenumberofsupplierswhotrade.Theremaining,non-trading,agents,arekeptoutofthemarketbythepricesystem,whichiseithertoohigh(forconsumers)ortoolow(forproducer)toallowthemtomakemorethantheirreservationutility.
Itisimportanttonotethatinequilibriumconsumers(orproducers)whichhavethesametypemaynotbedoingthesamething.Thiswilltypicallyoccurwhenutilitymaximisationdoesnotresultinasinglequalitybeingselected.Tobeprecise,givenanequilibriumpricep(z),consumersoftypexmaximizeu(x,z)−p(z)withrespecttoz.Butthereisnoreasonwhythereshouldbeauniqueoptimalquality:evenifweassumedu(x,z)tobestrictlyconcavewithrespecttoz,thepricep(z)typicallyisnonlinearwithrespecttoz,andnoconclusioncanbederivedaboutuniqueness.
Ifp(z)isanequilibriumprice,andifthereisanon-trivialsubsetD(x)⊂Zsuchthatanyz∈D(x)isautilitymazimizerforx,therewillbeacertain
α
equilibriumprobabilityPxonD(x).Thismeansthat,givenA⊂D(x),the
α
numberPx[A]∈[0,1]istheproportionofagentsoftypexwhosedemandslie
β
foreveryproducerinA.Similarly,therewillbeanequilibriumprobabilityPy
y,andtheresultingdemandandsupplyforeveryqualityzwillbalanceout.Aformaldefinitionisgiveninsection3.Inotherwords,inequilibrium,wecannottellwhichagentofagiventypedoeswhat,butwecantellhowmanyofthemdothisorthat.
Themainresultsofthepaper,togetherwiththedefinitionofequilibrium,arestatedinsection3:equilibriaexist,equilibriumpricesarenotunique,there
4
isauniqueequilibriumallocation,anditisefficient(Paretooptimal).ProofsaredeferredtoAppendicesCandD.Theseproofscombinetwomathematicalingredients,theHahn-Banachseparationtheoremontheonehand,anddualitytechniqueswhichextendtheclassicalFencheldualityforconvexfunctions,andwhichhavebeendevelopedinthecontextofoptimaltransportation(see[18]forarecentsurvey).Everythingreliesinstudyingacertainoptimizationproblem(33),whichisnovel.
Section4givesadditionalassumptionswhichensurethatallagentsofthesametypedothesamethinginequilibrium:µandνshouldbenon-atomic,andconditions(9)and(10)shouldbesatisfied.Theseconditionsextendtomultidimensionaltypestheclassicalsingle-crossingassumptionofSpenceandMirrlees.Theresultingequilibriaarecalledpure,inreferencetopureandmixedequilibriaingametheory.Notehoweverthat,eveninthiscase,onecannotfullydeterminethebehaviourofagentsinequilibrium:ifconsumersoftypexareindifferentbetweenenteringthemarketornot(eitherdecisiongivingthemtheirreservationutility),then,evenwiththeseadditionalassumptions,wecannotsaywhichoneswillstayoutandwhichoneswillcomein.Theequilibriumrelationswillonlydeterminetheproportionofeach.
Subsection4.4describesanexplicitexample.Itisstrictlyone-dimensional(typesandqualitiesarerealnumbers),whichmakescalculationspossible,andacompletedescriptionoftheequilibriumisprovided.Unfortunately,themethodusesdoesnotextendtomultidimensionaltypes.
AppendixAgivesthemathematicalresultsonu-convexandv-concaveanal-ysiswhichwillbeinconstantuseinthetext.AppendixBgivesgeneralmathe-maticalnotations,andreferencesaboutRadonmeasures.AppendicesCandDcontainproofs.
2
2.1
Themodel.
Standingassumptions.
LetX⊂Rd1,Y⊂Rd2,andZ0⊂Rd3becompactsubsets.Wearegivennon-negativefinitemeasuresµonXandνonY.Theyareallowedtohavepointmasses.
Typically,wewillhaveµ(X)=ν(Y).
LetΩ1beaneighbourhoodofX×Z0inRd1+d3,andΩ2beaneighbourhood
d2+d3
.Wearegivencontinuousfunctionsu:Ω1→RandofY×Z0inR
v:Ω2→R.Itisassumedthatuisdifferentiablewithrespecttox,andthatthederivative:
∂u
Dxu=
∂xd1iscontinuouswithrespectto(x,z).Similarlyitisassumedthatvisdifferen-tiablewithrespecttoy,andthatthederivativeDyviscontinuouswithrespectto(y,z).
5
NotethatX,Yand/orZ0areallowedtobefinite.IfXisfinite,theas-sumptiononuissatisfied.IfYisfinite,theassumptiononvissatisfied.
2.2Bidandaskprices
Wearedescribingthemarketforaqualitygood:itisindivisible,andunitsdifferbytheircharacteristics(z1,...,zd3)∈Z0.Thebundlez=(z1,...,zd3)willbereferredtoasa(multidimensional)quality.SoZ0isthesetofalltechnologi-callyfeasiblequalities;itistobeexpectedthattheywillnotallbetradedinequilibrium.
PointsinXrepresentconsumertypes,pointsinYrepresentproducertypes.IfXisfinite,thenµ(x)isthenumberofconsumersoftypex.IfYisfinite,thenν(y)isthenumberofproducersoftypey.IfXisinfinite,thenµisthedistributionoftypesintheconsumerpopulation,andthesameinterpretationholdsfor(Y,ν).
Eachconsumerbuyszerooroneunit,andeachsuppliersellszerooroneunit.Thereisnosecond-handtrade.
Forthetimebeing,wedefineapricesystemtobeacontinuousmapp:Z0→R.Thisdefinitionwillbemodifiedinamoment,asthesetZ0willbeextendedtoalargersetZ.Typically,pricingisnonlinearwithrespecttothecharacteristics.Oncethepricesystemisannounced,agentsmaketheirdecisionsaccordingtothefollowingrules:
•Consumersoftypexmaximizeu(x,z)−p(z)overZ0.Ifthevalueofthatmaximumisstrictlypositive,theconsumerentersthemarketandbuysoneunitofthemaximizingqualityz.Ifthereareseveralmaximizingqualities,heisindifferentbetweenthem,andthewayhechooseswhichonetobuyisnotspecifiedatthisstage.Ifthevalueofthemaximumis0,heisindifferentbetweenstayingoutofthemarket,andenteringittobuyoneunitofthemaximizingquality.Again,thewayhechoosesisnotspecifiedatthisstage.
•Producersoftypeymaximizep(z)−v(y,z)overZ0.Ifthevalueofthatmaximumisstrictlypositive,theproducerentersthemarketandsellsoneunitofthemaximizingqualityz.Ifthereareseveralmaximizingqualities,heisindifferentbetweenthem.Ifthevalueofthemaximumis0,heisindifferentbetweenstayingoutofthemarket,andenteringittoselloneunitofthemaximizingquality.
Tomodelthisprocedurebyastraigthforwardmaximization,weintroducetwoextrapoints∅d∈/Z0and∅s∈/Z0,with∅d=∅s,andweextendutilitiesandpricesasfollows:
p(∅d)=u(x,∅d)=0∀x∈Xp(∅s)=v(y,∅s)=0∀y∈Yu(x,∅s)=−1,v(y,∅d)=1
6
(1)(2)(3)
Thesetofpossibledecisionsforagentsisnow
Z=Z0∪{∅d}∪{∅s}
sothat:
max{u(x,z)−p(z)|z∈Z}≥u(x,∅d)−p(∅d)=0max{p(z)−v(y,z)|z∈Z}≥p(∅s)−v(y,∅s)=0
andtheprocedurewejustdescribedamountstomaximizingoverZinsteadofZ0.Therelations(1)to(3)implythatconsumerswillneverchoose∅s(itisalwaysbettertochoose∅d),andproducerswillneverchoose∅d(itisalwaysbettertochoose∅s).Soourmodeldoescapturetheintendedbehaviour.
Notethatwehavenormalizedreservationutilitiesto0.Thisdoesnotcauseanylossofgenerality.Thebehaviourofconsumers,forinstance,isfullyspecifiedbyu(x,z)andu¯(x),thelatterbeingthereservationutility,andwegetthesamebehaviourbyreplacingu(x,z)byu(x,z)−u¯(x)andu¯(x)by0,theonly
1
restrictionbeingthatwewouldrequireu¯tobeC,topreservetheregularitypropertiesofu.
Normalizingreservationutilitiesto0,wefindthatu(x,z)isthebidpriceforqualityzbyconsumersoftypex,thatis,thehighestpricethattheyarewillingtopayforthatquality.Similarly,v(y,z)istheaskingpriceforqualityzbyproducersoftypey,thatis,thelowestpricetheyarewillingtoacceptforsupplyingthatquality.Foragivenqualityz∈Z,itisnaturaltoconsiderthehighestbidpricefromconsumersandthelowestaskpricefromproducers:Definition1Thehighestbidpriceb:Z→Risgivenby:
b(z)=maxu(x,z)
x
andthelowestaskpricea:Z→Risgivenby:
a(z)=minv(y,z)
y
Notethatb(∅d)=a(∅s)=0andthata(∅d)=−b(∅s)=1.
Itfollowsfromtheirdefinitionsthatbisu-convexandaisv-concave.More
♭
precisely,wehaveb(z)=0♯xanda(z)=0ywhere0xand0ydenotethemapsx→0andy→0onXandY.Conversely,wehave0=maxz{u(x,z)−b(z)}and0=minz{v(y,z)−a(z)},sothatb♯(x)=0anda♭(y)=0.
Notethatifthepricesystemissuchthatp(z)>b(z)forsomequalityz,thentherewillbenobuyersforthisquality,andsoitcannotbetradedatthatprice.Similarly,ifp(z) 7 2.3Demandandsupply Fromnowon,apricesystemwillbeacontinuousmapp:Z→Rsuchthatp(∅d)=p(∅s)=0. Givenapricesystemp,themapp:Z→RiscontinuousandthesetZiscompact,sothatthefunctionsu(x,z)−p(z)andp(z)−v(y,z)attaintheirmaximumonZ. Definition3Givenapricesystemp,wedefine: D(x)=argmax{u(x,z)−p(z)|z∈Z}S(y)=argmin{v(y,z)−p(z)|z∈Z} Botharecompactandnon-emptysubsetsofZ.WeshallrefertoD(x)asthedemandoftypexconsumers,andtoS(y)asthesupplyoftypeyproducers.Itfollowsfromthedefinitionsthatifaconsumeroftypexisoutofthemarket,thenwemusthave∅d∈D(x).IfthereisnootherpointinD(x),thenallconsumersofthesametypestayoutofthemarket.If,ontheotherhand,D(x)containssomepointz∈Z0,thenallconsumersoftypexareindifferentbetweenstayingoutorbuyingqualityz,andwemayexpectthatsomeofthemactuallybuyqualityzinsteadofstayingout.Thisremarkwillbeatthecoreofourequilibriumanalysis.Ofcourse,thesameobservationisvalidforproducers.ThefollowingresultclarifiestherelationbetweenD(x)andS(y)ontheonehand,andthesub-andsupergradients∂p♯(x)and∂p♭(y)ontheother.Recallthat: p♯(x)=max{u(x,z)−p(z)|z∈Z}p♭(y)=min{v(y,z)−p(z)|z∈Z} Proposition4WehaveD(x)⊂∂p♯(x)andS(y)⊂∂p♭(y).Moreprecisely: D(x)=z∈∂p♯(x)|p(z)=p♯♯S(y)=(z) z∈∂p♭(y)|p(z)=p♭♭ (z) Proof.Thepointx∈Xbeingfixed,considerthefunctionsϕ:Z→Randψ:Z→Rdefinedbyϕ(z)=u(x,z)−p(z)andψ(z)=u(x,z)−p♯♯(z).Thesubgradient∂p♯(x)isthesetofpointszwhereψattainsitsmaximum(seeappendixA),whileD(x)isthesetofpointszwhereϕattainsitsmaximum.Butψ≥ϕandmaxψ=maxϕ.Theresultfollows. 2.4Admissiblepricesystems Wehaveseenthat,ifa(z)>b(z)everywhere,thereisano-tradeequilibrium.Weareconcernedwiththemoreinterestingcasewhena(z)≤b(z)forsomez.Definition6Qualityz∈Zismarketableifa(z)≤b(z).Thesetofmar-ketablequalitieswillbedenotedbyZ1: Z1={z∈Z|a(z)≤b(z)} ={z∈Z|∃x,∃y:v(y,z)≤u(x,z)} Notethatstayingoutisnotamarketableoption:a(∅d)>b(∅d)anda(∅s)>b(∅s).Asmentionedearlier,thismeansthatconsumerswillneverchoose∅sandthatsupplierswillneverchoose∅d.Wehavethereforetheinclusions: Z1⊂Z0ZIfaqualityzisnotmarketable,onewillneverbeabletofindabuyer/sellerpairthattradez.Ifaqualityzismarketable,thereisnosenseinsettingitspricetobehigherthanb(z)(therewouldbenobuyers),orlowerthana(z)(therewouldbenosellers).Hence: Definition7Apricesystemp:Z→Rwillbecalledadmissibleif: ∀z∈Z1,a(z)≤p(z)≤b(z) Letpbeanadmissiblepricesystem,sothata(z)≤p(z)≤b(z).Recallthatp♯(x)istheindirectutilityoftypexconsumers,andthat−p♭(y)istheindirectutilityoftypeyproducers.Takingconjugates,weget: ∀x∈X,0≤p♯(x)∀y∈Y, 0≥p♭(y) whichmeansthatallconsumersandproducersachieveatleasttheirreservationutility. 3 3.1 Equilibrium Demanddistributionandsupplydistribution Assumeapricesystemp:Z→Risgiven.LetD(x)andS(y)betheassociateddemandandsupply.Recall.thattheirgraphsarecompactsets. WerefertoAppendixBfornotationsanddefinitionsconcerningRadonmeasuresandprobabilities. Definition8AdemanddistributionassociatedwithpisapositivemeasureαX×ZonX×Zsuchthat: 9 •αX×ZiscarriedbythegraphofD•itsmarginalαXisequaltoµ Similarly,asupplydistributionassociatedwithpisapositivemeasureβY×ZonY×Zsuchthat: •βY×ZiscarriedbythegraphofS•itsmarginalβYisequaltoν αβ TheconditionalprobabilitiesPxandPythenarecarriedbyD(x)and αβ S(y)respectively.GivenA⊂Z,thenumbersPx[A]andPy[A]arereadilyinterpretedastheprobabilitythatconsumersoftypexdemandsomez∈Aandtheprobabilitythatproducersoftypeysupplysomez∈A. IfS(y)isasingleton,sothatthesupplyoftypeyproducersisuniquely β defined,thenPyreducestoaDiracmass: β S(y)={s(y)}=⇒Py=δs(y) andsimilarlyforconsumers. 3.2Definitionofequilibrium Definition9Anequilibriumisatriplet(p,αX×Z,βY×Z),wherepisanad-missiblepricesystemandαX×ZandβY×Zaredemandandsupplydistributions associatedwithp,suchthat: αZ0=βZ0ByαZ0andβZ0wedenotethemarginalsofαX×ZandβY×ZonZ0.Letuswritedownexplicitlyalltheconditionson(p,α,β)impliedbythisdefinition:1.p:Z→Riscontinuous,andp(z)∈[a(z),b(z)]whenevera(z)≤b(z)2.themarginalαXisequaltoµ α 3.theconditionalprobabilityPxiscarriedbyD(x) 4.themarginalβYisequaltoν β5.theconditionalprobabilityPyiscarriedbyS(y) 6.themarginalsαZandβZcoincideonZ0: αZ[A]=βZ[A] ∀A⊂Z0 10 Theinterpretationisasfollows.Givenp,consumersoftypexmaximizetheirutility,therebydefiningtheirindividualdemandsetD(x).Ifthatsetisa α singleton,D(x)={d(x)},theprobabilityPxmustbetheDiracmasscarriedbyd(x),andallconsumersoftypexdothesamething:theystayoutofthemarketifd(x)=∅d,andtheybuyz∈Z0ifd(x)=z.IfD(x)containsseveralpoints,thenconsumersoftypexareindifferentamongthesealternatives,andtheyalldodifferentthings.ForanyBorelsubsetA⊂D(x),theprobabilityαPx[A]givesustheproportionofconsumersoftypexwhochoosesomez∈Ainequilibrium. Similarconsiderationsholdforsuppliers.Condition6juststatesthatmar-ketsclearinequilibrium:foreveryqualityz∈Z0,thenumber(ortheaggregatemass)ofbuyersequalsthenumber(ortheaggregatemass)ofsuppliers.Notethatthisnumber(orthismass)mightbezero,meaningthatthisparticularqualityisnottraded.Thiswillhappen,forinstance,ifa(z)>b(z),sothatqualityzisnotmarketable.Itfollowsthat,inequilibrium,demandandsupplyarecarriedbyZ1,thesetofmarketablequalities: αZ[Z1]=αZ[Z0]=βZ[Z0]=βZ[Z1] Thenumber(ortheaggregatemass)ofconsumerswhostayoutofthemarketisαZ({∅d}),andthenumber(ortheaggregatemass)ofproducerswhostayoutofthemarketisβZ({∅s}).Aswementionedseveraltimesbefore,wemusthaveαZ({∅s})=0andβZ({∅d})=0. 3.3Mainresults Webeginbyanexistenceresult: Theorem10(Existence)Underthestandingassumptions,thereisanequi-librium. Asnotedabove,ifthesetZ1ofmarketablequalitiesisempty,thereisanequilibrium,namelytheno-tradeequilibrium,anditisunique.FromnowonweassumeZ1=∅.TheExistenceTheoremwillbeprovedinsectionC. Thereisnouniquenessofequilibriumprices.Forinstance,ifaqualityz∈Z0isnon-marketable,itspricep(z)canbespecifiedarbitrarily.Moregenerally,insectionCwewillprovethefollowing(seeProposition37): Theorem11(Non-uniquenessofequilibriumprices)Thesetofallequi-libriumpricespisconvexandnon-empty.Ifp:Z→Risanequilibriumprice,thensoiseveryq:Z→Rwhichisadmissible,continuous,andsatisfies: p♯♯(z)≤q(z)≤p♭♭(z) ∀z∈Z (4) Forα-andβ-almosteveryqualityzwhichistradedinequilibrium,wehavep♯♯(z)=p(z)=p♭♭(z). 11 Notethatqisalsorequiredtobeadmissible,sothatinadditionto(4)ithastosatisfytheinequality: a≤q≤b(5)Theeconomicinterpretationisasfollows.If(p,αX×Z,βY×Z)isanequilib-rium,therewillbequalitieszwhicharemarketable,butwhicharenottraded inequilibrium,becauseeverysuppliertypeyandeveryconsumertypexpreferssomeotherquality,whichmeansthatthepricep(z)istoolowtointerestsup-pliers,andtoohightointerestconsumers.Formulas(4)and(5)givetherangeofpricesforwhichthissituationwillpersist.Aslongasthepricep(z)staysintheopeninterval ]maxa(z),p♯♯(z),minb(z),p♭♭ (z)[ thequalityzwillnotbetraded.Inotherwords,thepriceofnon-tradedqualitiescanbechanged,withinacertainrange,withoutaffectingαX×ZorβY×Z,thatis,theequilibriumdistributionofconsumersandsuppliers.Thisisthemajorsourceofnon-uniquenessinequilibriumprices.Ontheotherhand,ifaqualityzistradedinequilibrium,onecannotchangethepricep(z)withoutaffectingαX×ZandβY×Z,thatis,withoutdestroyingthegivenequilibrium. Theequilibriumpricepisnotunique,butthefollowingresultshowsthatthedemandandsupplymapsD(x)andS(y)almostare: Theorem12(Quasi-uniquenessofequilibriumallocations)Letandp2,α2 X×Z,β2Y×Zp1,α1X×Z,β1 Y×Zbetwoequilibria.DenotebyD1(x),D2(x)andS1(y),S2(y) thecorrespondingdemandandsupplymaps.DenotebyPsupply.x1,Py1andPThen: x2,P2 conditionalprobabilitiesofdemandandythecorrespondingPx2[D1(x)]=Px1[D1(x)]=1forµ-a.e.xPy2[S1(y)]=Py1[S1(y)]=1forν-a.e.y Inotherwords,anyzwhichtypesxdemandsinthesecondequilibrium, whenpricesarep2,mustbelongtothedemandsetofxwhenpricesarep1(eventhoughxmightnotdemanditinthesecondequilibrium) Corollary13Ifthedemandofconsumersoftypexissingle-valuedinthefirstequilibrium,D1(x)={d1(x)},thend1(x)∈D2(x).Iftheirdemandissingle-valuedinthesecondequilibriumaswell,thend1(x)=d2(x). Proof.WehavePx2[d1(x)]=1=Px2 [D2(x)].Sod1(x)mustbelongtoD2(x),andtheremaindermusthavezeroprobability: Px2[D2(x){d1(x)}]=0 Proof.SinceD1(x)={∅d},wemusthave∅d∈D2(x).Assumeconsumersoftypexareactiveinthesecondequilibrium.Wemusthaveu(x,z)−p(z)>0forallz∈D2(x),includingz=∅d.Sinceu(x,∅d)=p(∅d)=0,thisisacontradiction. Thehighestbidpriceforzisb=maxxu(x),andthelowestaskpriceisa=minyv(y). Ifb I1(p)={x∈X|u(x) anddefineJ1(p),J2(p),J3(p)inasimilarwayforproducers.Anequilibriumisaset(p,α,β)suchthat •α=(αx),x∈X,whereeachαxisaprobabilityon{z}∪{∅d}•β=(βy),y∈Y,whereeachβyisaprobabilityon{z}∪{∅s} α(z)=•yβy(z)xx Letustranslatethis.Ifx∈I1(p),thenconsumersoftypexstayoutofthe market,sothatαx(z)=0.Ifx∈I3(p),thenconsumersoftypexbuyz,sothatαx(z)=1.Ifi∈I2(p),thenαx(z)istheproportionofconsumersoftypexwhobuyzinequilibrium.Denoteby#[A]thenumberofelementsinafinitesetA.Theequilibriumconditionimpliesthat: #[I3(p)]≤#[J2(p)∪J3(p)]#[J3(p)]≤#[I2(p)∪I3(p)] (7)(8) Conversely,ifthesetwoinequalitiesaresatisfied,wewillalwaysbeabletofindnumbersαxandβysuchthat0≤αx≤1,αx=0ifx∈I1(p)andαx=1ifx∈I3(p),withcorrespondingconstraintsfortheβy.So,inthatparticularcase,theequilibriumconditionsboildowntotheinequalities(7)and(8). Notethatthereisnouniquenessoftheequilibriumpricep.Ifforinstanceux¯andvy>ux¯,thenanyprice¯>vy¯,withux 0≤αx(z)≤1,0≤βy(z)≤1,αx(z)=βy(z) x y (p,α,β)isanequilibriumallocation. 3.5Example2:moreonuniqueness WegiveanexampletoclarifytheuniquenessstatementinTheorem12.There arethreegoods,z1,z2,z3,twoconsumersx1,x2,threeproducersy1,y2,y3.The 14 utilityfunctionsare: u(x1,z1)=2,u(x1,z2)=1,u(x1,z3)=0.1u(x2,z1)=3,u(x2,z2)=2,u(x2,z3)=0.1 andthecostfunctionsare: v(y1,z1)=0,v(y1,z2)=5,v(y1,z3)=5v(y2,z1)=5,v(y2,z2)=0,v(y2,z3)=5v(y3,z1)=5,v(y1,z2)=5,v(y1,z3)=0 Itiseasytocheckthattherearetwoequilibria: 1.y1producesz1,y2producesz2,y3producesnothing;x1choosesz1,x2 choosesz2;pricesarep(z1)=1,p(z2)=0,p(z3)=02.y1producesz1,y2producesz2,y3producesnothing;x1choosesz2,x2 choosesz1;pricesarep(z1)=1.9,p(z2)=0.9,p(z3)=0Thedemandsetofx1is{z1,z2}:=D1(x1)inthefirstequilibriumand{z1,z2,z3}:=D2(x1)inthesecond.Thedemanddistribution,ontheother 1 (z)=δz1(Diracmassatz1)inthefirstequilibrium(simplyexpress-hand,isPx1 2 (z)=ingthefactthatx1choosesz1andnothingelseinherdemandset)andPx1 δz2inthesecond.Theorem12thenstatesthatδz2[D1(x1)]=δz1[D1(x1)]=1,whichsimplyexpressesthefactthatbothz1andz2belongtoD1(x1). Noteforinstancethatthesocialutilityisthesameforbothequilibria,namely4: 1.Inthefirstone: u(x1,z1)−v(y1,z1)+u(x2,z2)−v(y2,z2)=2−0+2−0=4 2.Inthesecondone: u(x1,z2)−v(y2,z2)+u(x2,z1)−v(y1,z1)=1−0+3−0=4 Thisisageneralfact:thesocialutilityisthesameatallequilibria.Indeed,equilibriumpricesarefoundbymaximizingtheright-handsideof(6):itmaybeachievedatdifferentp1andp2,butthevalueofthemaximumisthesame. 4 4.1 Pureequilibrium. Definition α Inequilibrium,consumersoftypexdemandqualityzwithprobabilityPx(z), β andsuppliersoftypeysupplyqualityzwithprobabilityPy(z).Theequilibriumispureifallagentsofthesametypewhoareinthemarketatthesametimearedoingthesamething(buyingorsellingthesamequality),sothattheseprobabilitiesareDiracmasses.Formally: 15 Definition16Anequilibrium(p,αX×Z,βY×Z)ispureif: •forµ-almosteveryx,thesetD(x)∩Z0containsatmostonepoint•forν-almosteveryy,thesetS(y)∩Z0containsatmostonepointDenotebyXpthesetofactiveorindifferentconsumers.If(p,αX×Z,βY×Z)isapureequilibrium,thereisaBorelmapd:Xp→Z0withd(x)∈D(x)suchthat,forµ-almosteveryx,oneandonlyoneofthefollowingholds:•eitherconsumersoftypexareinactive,sothatD(x)=∅d•orconsumersoftypexareindifferent;thenD(x)=∅d∪{d(x)}•orconsumersoftypexareactive;thenD(x)={d(x)} Wecanthenrewritethedefinitionofequilibriumdirectlyintermsofsandd. Definition17Apureequilibriumisatriplet(p,d,s)where: 1.disaBorelmapfromthesetXp=x|p♯(x)≥0intoZ0 2.sisaBorelmapfromthesetYp=y|p♭(y)≤0intoZ0 3.Forµ-almosteveryxwithp♯(x)>0,thefunctionz→u(x,z)−p(z)attainsitsmaximumatasinglepointz=d(x)∈Z04.Forν-almosteveryywithp♭(y)<0,thefunctionz→p(z)−v(y,z)attainsitsmaximumatasinglepointz=s(y)∈Z0 5.Forµ-almosteveryxwithp♯(x)=0,thefunctionz→u(x,z)−p(z)attainsitsmaximumattwopoints,∅dandz=d(x)∈Z06.Forν-almosteveryywithp♭(y)=0,thefunctionz→p(z)−v(y,z)attainsitsmaximumattwopoints,∅sandz=s(y)∈Z07.ThedemandandsupplydistributionsαandβassociatedwithdandshavethesamemarginalsonZ0: ∀A⊂Z0, µ[x|d(x)∈A]=ν[y|s(y)∈A] Forthesakeofsimplicity,weshallnowassumethata(z) Theorem18Let(p1,d1,s1)and(p2,d2,s2)betwopureequilibria.Everycon-sumerxwhoisactiveinoneequilibriumisactiveorindifferentintheother,andwehaved1(x)=d2(x).Similarly,everyproducerywhoisactiveinoneequilibriumisactiveorindifferentintheother,ands1(y)=s2(y). Proof.Itisanimmediateconsequenceoftheuniquenesstheoremforequi-libriumallocations. 4.3Existence Theorem19Assumethatthestandardassumptionshold.AssumemoreoverthatµandνareabsolutelycontinuouswithrespecttotheLebesguemeasure,andthatthepartialderivativesDxuandDyvwithrespecttozareinjective: ∀x∈X,Dxu(x,z1)=Dxu(x,z2)=⇒z1=z2(9)∀y∈Y,Dyv(y,z1)=Dyv(y,z2)=⇒z1=z2 (10) Thenanyequilibriumispure. Corollary20Intheabovesituation,thereisapureequilibrium. Proof.Weknowthatthereisanequilibrium,bytheExistenceTheorem,andweknowthatithastobepure. ∂x∂z =0 sothatcondition(9),or(10)forthatmatter,isamulti-dimensionalgeneraliza-tionoftheclassicalSpence-Mirrleesconditionintheeconomicsofassymmetric information(see[2]).Itissatisfied,forinstance,byu(x,z)=x−zα ,pro-videdα=0andα=1;ifα<1,oneshouldaddtherequirementthatX∩Z=∅,sothatuisdifferentiableonX×Z. 4.4 Example 4.4.1 AcasewhenZa=∅=Zb SetX=[1,2]andY=[2,3].BothareendowedwiththeLebesguemeasure.SetZ0=[0,1]and u(x,z)=− 1 2yz2,v¯(y)=0 sothatsuppliersareorderedonthelineaccordingtoefficiency,themostefficient ones(thosewiththelowestcost,neary=2)beingontheleft,andconsumersareorderedaccordingtotaste,themostavidones(thosewiththehighestutility,nearx=2)beingontheright(notetheorderreversal). Wecomputethelowestaska(z)andthehighestbidb(z): b(z)=u¯♯(z)=1max≤x≤2 − 1 2 z2+2za(z)=v¯♭(z)=2min ≤y≤3 1 Notethatb(z)isthebidpriceforconsumerx=2(themostavidone),anda(z)istheaskpriceforsuppliery=2(theleastefficientone).Wehavea≤basexpected. NotethatthegeneralizedSpence-Mirrleesassumptions(9)and(10)aresat-isfied: Dxu(x,z)=z D1 yv(y,z)= 1+yw(x,y)= 1 1+y wherew(x,y)istheresultingutilityforthepair.Wethenseekthemeasure-preservingmapσ:[1,2]→[2,3]whichmaximizestheintegral: 2dx= 1 2 x,σ(x))x2 w(1 ∂x∂y =− x 5−x(11) s(y)= 4−y andthesetoftradedqualitiesisZt=uniquelydefined,andisfound13by ,whichisastrictsubsetofZ0:again,notalltechnologicallyfeasiblequailitiesaretradedinequilibrium.OnZt,thepriceiswritingthefirst-orderconditionforoptimality,p′(z)=∂u 1 2 z2+5z−5ln(z+1)+cfor 3 (13) Wecannowtrytovalidateourassumptionthateveryagentisactive.Com-putetheindirectutilities: p♯(x)=u(x,d(x))−p(d(x))=x+5(ln5−ln(5−x))−c−p♭(y)=p(s(y))−v(y,s(y))= (4−y)(6+y) 8+5ln 5 4 =2.1157(14) Foranycinthatthefunctionp(z)givenbyformula(13)isrestrictiontoZt=given1interval,by3(12)theofanequilibriumprice,theequilibriumsupplyanddemandbeingand(11).Wehavetoz∈now0,1extendpttoZ0=[0,1]z= 1insuchawaythatthequalities3,1arenottraded.Forbesuchthateachofthemprefersstayingat 1 4must4 −v 3,1 4 −p 1z2+z+1−5ln5 z2+ 9 +cfor0≤z≤ 1 22 4 3, wegettheinequalities:− 1 8 3 +c≤p(z)≤z2+ 3 +cfor 2 4.4.2AcasewhenZaisnon-empty Letusnowincreasethenumberofconsumers:sayY=[2,3]isunchanged,whileX=[h,2]with0 z2+5z−5ln 4(z+1) 4≤z≤ 2 4 =s(3).For0 2 z2+z Consumersoftypex<1willhavealowerbidprice.Chooseacontinuousfunctionpsuchthat: − 1 4−v 3,1172z2+4(18)Theleftinequalityensuresthatconsumersortypex<1arenotbiddersfor qualityz,sotheyjustbuyquality0atprice0,thatis,theyreverttotheirreservationutility.Therightinequalityensuresthattheleastefficientproducerwillnotbecomeinterestedinproducingqualityz,sothatthemoreefficientoneswillnoteither. Anyfunctionp(z)satisfying(17),(18)and(16)(with1+5ln5 wheretisthequantityofthesecondgood,andπits(linear)price.Ourmethodsdonotreadilyapplytothissituation,andweplantoinvestigateitfurther.Finally,wewishtostressthatalthoughwehavewhatappearsasacompleteequilibriumtheoryformultidimensionalhedonicmodels,thenumericalaspectsarefarfrombeingaswellunderstood.Themethodweusedintheexampleisstrictlyone-dimensional,andthereisnoeasywaytoextendittothemul-tidimensionalcase.Theobviouswaytoproceedistofollowthetheoreticalargument,andtrytominimizetheintegralI(p)in(33),butwehavemadenoprogressinthatdirection.Itcertainlyisagoodtopicforfutureresearch.Sowillalltheeconometricaspects(characterizationandidentification).Thisinvestigationhasbeenstartedin[4],butisfarfrombeingcomplete. 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Afunctionf:X→R∪{+∞}willbecalledu-convexiffthereexistsanon-emptysubsetA⊂Z×Rsuchthat: ∀x∈X,f(x)=sup{u(x,z)+a} (z,α)∈A (19) Afunctionp:Z→R∪{+∞}willbecalledu-convexiffthereexistsanon-emptysubsetB⊂X×Rsuchthat: p(z)=sup{u(x,z)+b} (x,b)∈B (20) A.2Subconjugates Letf:X→R∪{+∞},notidentically{+∞},begiven.Wedefineitssubcon-jugatef♯:Z→R∪{+∞}by: f♯(z)=sup{u(x,z)−f(x)} x (21) Itfollowsfromthedefinitionsthatf♯isau-convexfunctiononZ(itmightbeidentically{+∞}). 22 Letp:Z→R∪{+∞},notidentically{+∞},begiven.Wedefineitssubconjugatep♯:X→R∪{+∞}by: p♯(x)=sup{u(x,z)−p(z)} z (22) Itfollowsfromthedefinitionsthatp♯isau-convexfunctiononX(itmightbeidentically{+∞}). Example21Setf(x)=u(x,z¯)+a.Then f♯(¯z)=sup{u(x,z¯)−u(x,z¯)−a}=−a x ♯♯ Conjugationreversesordering:iff1≤f2,thenf1≥f2,andifp1≤p2,♯♯ thenp1≥p2.Asaconsequence,iffisu-convex,notidentically{+∞},thenf♯isu-convex,notidentically{+∞},.Indeed,sincefisu-convex,wehavef(x)≥u(x,z)+aforsome(z,a),andthenf♯(z)≤−a<∞. Proposition22(theFenchelinequality)Foranyfunctionsf:X→R∪{+∞}andp:Z→R∪{+∞},notidentically{+∞},wehave: ∀(x,z),f(x)+f♯(z)≥u(x,z)∀(x,z)p(z)+p♯(x)≥u(x,z) A.3Subgradients Letf:X→R∪{+∞}begiven,notidentically{+∞}.Takesomepointx∈X.Weshallsaythatapointz∈ZisasubgradientoffatxifthepointsxandzachieveequalityintheFenchelinequality: f(x)+f♯(z)=u(x,z) (23) Thesetofsubgradientsoffatxwillbecalledthesubdifferentialoffatxanddenotedby∂f(x).Specifically: Definition23∂f(x)=argmaxzu(x,z)−f♯(z) Similarly,letp:Z→R∪{+∞}begiven,notidentically{+∞}.Takesomepointz∈Z.Weshallsaythatapointx∈Xisasubgradientofpatzif: p♯(x)+p(z)=u(x,z) (24) Proposition25Thefollowingareequivalent: 23 Thesetofsubgradientsofpatzwillbecalledthesubdifferentialofpatzanddenotedby∂p(z). Definition24∂p(z)=argmaxxu(x,z)−p♯(x) 1.z∈∂f(x) 2.∀x′,f(x′)≥f(x)+u(x′,z)−u(x,z) Ifequalityholdsforsomex′,thenz∈∂f(x′)aswell. Proof.Webeginwithprovingthatthefirstconditionimpliesthesecondone.Assumez∈∂f(x).Then,by(23)andtheFenchelinequality,wehave: f(x′)≥u(x′,z)−f♯(z)=u(x′,z)−[u(x,z)−f(x)] Wethenprovethatthesecondconditionimpliesthefirstone.Usingtheinequality,wehave: f♯(z)=sup{u(x′,z)−f(x′)} x′ ≤sup{u(x′,z)−f(x)−u(x′,z)+u(x,z)} x′ =u(x,z)−f(x) sof(x)+f♯(z)≤u(x,z).WehavetheconversebytheFenchelinequality,soequalityholds. Finally,ifequalityholdsforsomex′incondition(2),thenf(x′)−u(x′,z)=f(x)−u(x,z),sothat: ∀x′′,f(x′′)≥f(x)−u(x,z)+u(x′′,z) =f(x′)−u(x′,z)+u(x′′,z) whichimpliesthatz∈∂f(x′). Thisexamplegeneralizestoallu-convexfunctions.DenotebyCu(Z)thesetofallu-convexfunctionsonZ. Proposition27Foreveryfunctionp:Z→R∪{+∞},notidentically{+∞},wehave p♯♯(z)=sup{ϕ(z)|ϕ≤p,ϕ∈Cu(Z)} ϕ Proof.Denotebyp¯(z)theright-handsideoftheaboveformula.Wewanttoshowthatp♯♯(z)=p¯(z) ♯♯♯♯ Sincep≤pandpisu-convex,wemusthavep♯♯≤p¯.Ontheotherhand,p¯isu-convexbecauseitisasupremumofu-convexfunctions.SotheremustbesomeB⊂X×Rsuchthat: p¯(z)=sup{u(x,z)+b} (x,b)∈B Let(x,b)∈B.Sincep¯≤p,wehaveu(x,z)+b≤p¯(z)≤p(z).Taking biconjugates,asintheprecedingexample,wegetu(x,z)+b≤p♯♯(z).Takingthesupremumsover(x,b)∈B,wegetthedesiredresult. Definition29Weshallsaythatafunctionp:Z→R∪{+∞}isu-adaptedifitisnotidentically{+∞}andthereissome(x,b)∈X×Rsuchthat: ∀z∈Z,p(z)≥u(x,z)+b Itfollowsfromtheabovethatifpisu-adapted,thensoarep♯,p♯♯andallfurthersubconjugates.Notethatau-convexfunctionwhichisnotidentically{+∞}isu-adapted. 25 Corollary30Letp:Z→R∪{+∞}beu-adapted.Then: p♯♯♯=p♯ Proof.Ifpisu-adapted,thenp♯isu-convexandnotidentically{+∞}.Theresultthenfollowsfromcorollary28. toz′,weget: p♯(x)≤sup{u(x,z′)−ϕz(z′)}−ε z′ =supu(x,z′)−u(x,z′)+u(x,z)−p♯♯(z)−ε z′ =u(x,z)−p♯♯(z)−ε=p♯(x)−ε whichisacontradiction.Theresultfollows A.6v-concavefunctions. LetusnowconsiderthedualitybetweenYandZ.Givenv:Y×Z→R,wesaythatamapg:Y→R∪{−∞}isv-concaveiffthereexistsanon-emptysubsetA⊂Z×Rsuchthat: ∀y∈Y,g(y)= (z,a)∈A inf{v(y,z)+a}(29) andafunctionp:Z→R∪{−∞}willbecalledv-concaveiffthereexistsanon-emptysubsetB⊂X×Rsuchthat: p(z)= (x,b)∈B inf{v(y,z)+b}(30) Alltheresultsonu-convexfunctionscarryovertov-concavefunctions,withobviousmodifications.Thesuperconjugateofafunctiong:Y→R∪{−∞},notidentically{−∞},isdefinedby: g♭(z)=inf{v(y,z)−g(y)} y (31) andthesuperconjugateofafunctionp:Z→R∪{−∞},notidentically{−∞},isgivenby: p♭(y)=inf{v(y,z)−p(z)}(32) z Thesuperdifferential∂p♭isdefinedby: ∂p♭(y)=argmin{v(y,z)−p(z)} z andwehavetheFenchelinequality: p(z)+p♭(y)≤v(y,z)∀(y,z) withequalityiffz∈∂p♭(y).Notefinallythatp♭♭≥p,withequalityifpis v-concave 27 B B.1 Somenotationsanddefinitions. Radonmeasuresandprobabilities. WithalocallycompactsetΩ(suchasanopensubsetofthecompactsetZ)wewillassociatethefollowingsetsoffunctionsandmeasuresonΩ:•K(Ω),thespaceofcontinousfunctionsonΩwithcompactsupport•Cb(Ω),thespaceofboundedcontinousfunctionsonΩ•C+(Ω),theconeofnon-negativefunctions•M(Ω),thespaceofmeasuresonΩ •M+(Ω)⊂M(Ω),theconeofpositivemeasures•Mb(Ω)⊂M(Ω),theconeoffinitemeasures •Mb+(Ω)=Mb(Ω)∩M+(Ω),theconeofpositivefinitemeasures•P(Ω)⊂Mb+(Ω)thesetofprobabilitiesonΩ ThespaceK(Ω)willbeendowedwiththetopologyofuniformconvergenceoncompactsubsetsofΩ,andthespaceCb(Ω)withtheuniformnorm.ThenCb(Ω)isaBanachspace,butK(Ω)isnot,unlessΩiscompact,inwhichcaseallcontinuousfunctionsonΩarebounded,andwehaveC(Ω)=K(Ω)=Cb(Ω).WhenΩisfiniteandhasdelements,allthesespacescoincidewithRd. WetakemeasuresinthesenseofRadon,thatis,M(Ω)isdefinedtobethedualofK(Ω)andMb(Ω)isdefinedtobethedualofCb(Ω).SoMb(Ω)isaBanachspace,butM(Ω)isnot,unlessΩiscompact,inwhichcaseM(Ω)=Mb(Ω),thatis,allRadonmeasuresonΩarefinite.Forγ∈M(Ω)andϕ∈K(Ω),wewriteindifferently<γ,ϕ>orZϕdγ. Aprobabilityγ∈P(Z)isdefinedasanon-negativeboundedmeasuresuchthat<γ,1>=1.ThesetP(Z)isconvex,andiscompactintheweak*topology:γn→γif<γn,ϕ>→<γ,ϕ>foreveryϕ∈Cb(Ω). WesaythatameasureγiscarriedbyKif<γ,ϕ>=0forallϕ∈K(Ω)whichvanishonK.IfγiscarriedbyasubsetK,itisalsocarriedbyitsclosure.Thesupportofameasureγ,denotedbySupp(γ),isthesmallestclosedsetKsuchthatγiscarriedbyK. B.2Conditionalprobabilitiesandmarginals. GivenapositivemeasureαX×Z∈M+(X×Z)(whichhastobefinite,sinceX×Ziscompact)wedefineitsmarginalsαX∈M+(X)andαZ∈M+(Z)asfollows: 28 ϕ(x)dαX=ψ(z)dαZ= X Z X×Z X×Z ϕ(x)dαX×Zψ(z)dαX×Z ∀ϕ∈K(X)∀ψ∈K(Z) andwedenotetheprobabilityofthesecondcoordinatebeingzconditionalon α thefirstcoordinatebeingxbyPx(z).Themathematicalexpectationwith α respecttothisprobabilitywillbedenotedbyEx: ααEx[ψ]=ψ(z)dPx(z) Z Thisconditionalprobabilityisrelatedtothefirstmarginalbytheformula: αEx[f(x,z)]dαX∀f∈K(X×Z)f(x,z)dαX×Y= X×Z X SimilarconsiderationsholdforpositivemeasuresβY×Z∈M+(Y×Z),We have: ϕ(y)dβY×Z∀ϕ∈K(Y)ϕ(y)dβY= YY×Z ψ(z)dβZ=ψ(z)dβY×Z∀ψ∈K(Z)ZX×Z β g(y,z)dβY×Z=Ey[g(y,z)]dβY∀g∈K(Y×Z) Y×Z Y C C.1 ProofoftheExistenceTheorem Thedualproblem:existence RecallthatZ={∅d}∪Z0∪{∅s},withZ1={z|a(z)≤b(z)}acompact non-emptysubsetofZ0.DenotebyAthesetofalladmissiblepricesystemsonZ,thatis,thesetofallcontinuousmapsp:Z→Rwhichsatisfy: ∀z∈Z1, a(z)≤p(z)≤b(z) Aisanon-empty,convexandclosedsubsetofK(Z),thespaceofallcon-tinuousfunctionsonZ.NowdefineamapI:K(Z)→Rby: p♭(y)dν(33)p♯(x)dµ−I(p)= X Y Proposition33ThemapIisconvex 29 Proof.Takep1andp2inA.Takesandtin[0,1]withs+t=1.Then:(sp1+tp2)(x)=sup{u(x,z)−sp1(z)−tp2(z)} z ♯ =sup{s[u(x,z)−p1(z)]+t[u(x,z)−p2(z)]} z ≤ssup{u(x,z)−p1(z)}+tsup{u(x,z)−p2(z)}= z♯ sp1(x) z + tp♯2 (x) Similarly,wefindthat: ♭ (sp1+tp2)(y)≥sp♭1(x)+tp2(x) ♭ Integrating,wefindthatIisconvex,asannounced. Proposition36Problem(P)hasasolution. ♭♭♭♭♭♭ Similarly,wefindthatp♭♭=p,withp=minp,bbb ♭Proof.Takeaminimizingsequencepn.Sincethefunctionsp♯n(resp.pn),n∈N,areu-convex(resp.v-concave),theyareuniformlyLipschitzian(seesectionA),andhenceequicontinuous.ByAscoli’stheoremwecanextract ♭ uniformlyconvergentsubsequences(stilldenotedbyp♯nandpn): p♯n→fp♭n→g 30 sothat: f(x)dµ− (y)dν=inf X gY a≤p≤b p♯(x)dµ− ♭(y)dν X pY (35) Itiseasytoseethatfisu-convexandgisv-concave.Inaddition,p♯♯fandpgeverywhere(anduniformlyaswell,sincethefunctionsn→♯♭♭n→♭ are equicontinuous).Sincep♯♯n≤p♭♭n,wegetf♯≤g♭ inthelimit.Sincepn≤b, wehavep♯♯n≤b ♯♯ =b,andlettingn→∞,wefindthatf♯≤b.Sincef♯isu-convex,itiscontinuous(andevenLipschitzian,seesectionA).Similarly,g♭isv-concave,hencecontinuous,andsatisfiesg♭≥a.Nowtakeanycontinuouspriceschedulep¯suchthat f♯♯♯=maxf♯,a≤p¯≤ming♭ a,b=g♭♭♭b (36) forinstancep¯= 1 Theproofindicatesthatuniquenessisnottobeexpected.Thefollowing resultistheNon-UniquenessTheoremforprices: Proposition37Letpbeasolutionofproblem(P).Thenp♯♯♭♭ isanadmissiblepricescheduleaandpsuchthat: barealsosolutions.Moregenerally,ifqp♯♯a(z)≤q(z)≤p♭♭ b(z) ∀z∈Z1 thenqisasolutionofproblem(P). Proof.Fromp♯♯a≤q≤p♭♭b,we♭ q♯≤p♯♯ a deducethatp=p♭♭♯ b=p♯.Substitutingintotheintegral,weget:♭ ≤q♭andthatq♯ (x)dµ− q♭(y)dν≤X Y p♯(x)dµ−X p♭(y)dν=inf(P)Y andsinceqisadmissible,itmustbeaminimizer. ♭♭ Proof.♭♭BytheprecedingProposition,pbisasolutionofproblem(P),sothatIpb=I(p).Substitutingintheintegrals,weget: ♭♯ ♯♭♭♭♭ p♭(y)dνp(x)dµ−pbdν=pbdµ− X Y X Y ♭♭♯ Corollary38Letpbeasolutionofproblem(P).Thenp=pb,µ-almost ♭ ,ν-almosteverywhere.everywhere,andp♭=p♯♯a ♯ ♭ andsincep♭♭=p♭,thisreducesto:b ♯ ♭♭ p♯(x)dµpbdµ= X X ♭♭♯ Sincep♭♭≤p,wehavepb≥p♯,andsincetheintegralsareequal,itb ♯♯♯♭ ♭ followsthatp♯=p♭♭,µ-a.e.Thesameargumentshowsthatp=pa,bν-a.e. Notethatwealreadyhavep(z)=p♯♯(z)foreveryz∈D(x),andp(z)=p♭♭(z)foreveryz∈S(y) C.2Thedualproblem:optimalityconditions RecallthatwehavedefinedamapI:K(Z)→Rby: p♭(y)dνp♯(x)dµ−I(p)= X Y WehavecheckedthatthefunctionIisconvex.Itiseasilyseentobecon-♯♭♭ tinuous:ifpn→puniformlyonZ,thenp♯n→puniformlyonXandpn→puniformlyonY.Ontheotherhand,thesetAisnon-empty,convexandclosedinK(Z).Thismeansthattheconstraintqualificationconditionsholdinproblem(P):anecessaryandsufficientconditionforp¯tobeoptimalisthat: 0∈∂I(¯p)+NA(¯p) (37) where∂I(¯p)isthesubgradientofIatp¯inthesenseofconvexanalysis,andNA(¯p)isthenormalconetoAatp¯.Allwehavetodonowistocomputebothofthem. 32 C.2.1Computing∂I(p) Lemma40Letp∈K(Z)andϕ∈K(Z).Then,foreveryx∈Xandeveryy∈Y,wehave: 1hlim h>→0 0 (p+hϕ)♭ (y)−p♭ h (y) =−max{ϕ(z)|z∈S(y)} Proof.Letusprovethesecondequality;thefirstoneisderivedinasimilarway.Takez∈Sp(y)andzh∈Sp+hϕ(y).FromthedefinitionofSp(y)andSp+hϕ(y),wehave: v(y,zh)−p(zh)≥p♭(y)=v(y,z)−p(z) v(y,z)−p(z)−hϕ(z)≥(p+hϕ)♭ (y)=v(y,zh)−p(zh)−hϕ(zh)Substracting,wefindthat: −hϕ(z)≥(p+hϕ)♭(y)−p♭(y)≥−hϕ(zh) (38) SincezisanarbitrarypointinSp(y),wecantakeittobetheminimizerontheleft-handside,andthisinequalitybecomes: −hmax{ϕ(z)|z∈Sp(y)}≥(p+hϕ)♭(y)−p♭(y)≥−hϕ(zh) Nowleth→0.Thefamilyzh∈Sp+hϕ(y)musthaveclusterpoints,becauseZiscompact,andanyclusterpointz¯mustbelongtoSp(y).Takinglimitsininequality(38),wefindthat,forsomez¯∈Sp(y):−max{ϕ(z)|z∈Sp(y)}≥hlim 1 h>→0 0 Becauseofinequality(39),wecanapplytheLebesgueconvergencetheorem,andweget: hlim 1 h>→0 0Lemma41Foreveryϕ∈C(Z),wehave: min{ϕ(z)|z∈D(x)}dµ=min (41) X ϕ(d(x))dµ|d∈B(X,D)Xmax{ϕ(z)|z∈S(y)}dν=maxϕ(s(y))dµ|s∈B(Y,S) (42) Y Y Proof.Givenϕ∈C(Z),themultivaluedmapsΓ1andΓ2definedby: Γ1(x)=argmin{ϕ(z)|z∈D(x)}Γ2(y)=argmax{ϕ(z)|z∈S(y)} havecompactgraph.Formulas(41)and(42)thenfollowfromastandardmea-surableselectiontheorem. andbyintegratingwithrespecttoµ,wegetthedesiredresult: ϕdαZ≥min{ϕ(z)|z∈D(x)}dµZX =minϕ(z)dµ|z∈D(x) X=minϕ(d(x))dµ|d∈B(X,D) X h [I(p+hϕ)−I(p)] =maxϕdβZ|βY×Z∈M+(Y,S)−minϕdαZ|αX×Z∈M+(X,D) ZZ =maxϕdαZ|βY×Z∈M+(Y,S),αX×Z∈M+(X,D)ϕdβZ− Z Z Proposition43ThesubdifferentialofIatpisgivenby: ∂I(p)={βZ−αZ|βY×Z∈M+(Y,S),αX×Z∈M+(X,D)} Proof.Takeλ∈M(Z)=Mb(Z).Bydefinitionofthesubgradient,λ∈∂I(p)ifandonlyif,foreveryϕ∈K(Z)andh>0,wehave: ϕdλI(p+hϕ)≥I(p)+h Z SinceIisconvex,thisisequivalentto: h→0 h>0 lim 1 35 C.2.2ComputingNA(p) Takeλ∈M(Z)=Mb(Z).Bydefinition,λ∈NA(p)ifandonlyif,foreveryq∈A,wehave: (q−p)dλ≤0 Z Sinceq(∅d)=p(∅d)=0andq(∅s)=p(∅s)=0foreveryq∈A,thisconditionisequivalentto: (q−p)dλ≤0(45) Z0 Tointerpretthiscondition,weneedsomenotation.Set: Zb={z∈Z0|a(z) Za={z∈Z0|a(z) b sothatwehaveapartitionofZ0intosubsetsZ0=Za∪Za∪Zb∪M∪N,where b Za∪Za∪Zb∪M=Z1,thesetofmarketablequalities. bb Denotebyλb,λba,λa,λM,λNtherestrictionsofλtoZ,Za,Za,ZM,ZNrespectively.Notethatsinceλwasaboundedmeasure,soareλb,λba,λa,λMandλN.Con-dition(45)isequivalenttothefollowing: λb≥0,λba=0,λa≤0,λN=0 C.2.3 Concludingtheproof. (46) Letp¯beasolutionofproblem(P).Bycondition(37),wehave0∈∂I(¯p)+NA(¯p).ByProposition43,thismeansthatthereexistsβY×Z∈M+(Y,S),αX×Z∈M+(X,D)andλ∈M(Z)satisfying(46)suchthatαZ−βZ=λ. b ,ZarespectivelyareInotherwords,therestrictionofαZ−βZtoZb,Za positive,zeroandnegative: αZ≥βZonZb b αZ=βZonZa (47)(48)(49)(50) αZ≤βZonZaαZ=βZonN ThereisnoconditionontherestrictionofαZorβZto{∅d},{∅s}orM. αα SincePxiscarriedbyD(x),wemusthavePx(z)=0wheneverz∈/D(x), β whichcertainlyisthecasewhenp(z)>b(z).Similarly,Py(z)=0when 36 α p(z) orPy(z)=0.TheconditionαZ=βZonNthenimpliesthat: αZ=βZ=0onN ′ Wewillnowshowthatthereexistsα′X×Z∈M+(X,D)andβY×Z∈M+(Y,S) ′ suchthatα′Z0=βZ0.ThiswillbedonebysuitablymodifyingαX×ZandβY×ZonthesubsetsZbandZa(notethattheyarebothsubsetsofZ0).Inthesequel,wewilldenotebyαX×A(resp.βY×B)therestrictionofαX×Z(resp.βY×Z)toX×A(resp.Y×B),forA⊂X(resp.B⊂Y),andbyαA(resp.βB)themarginalonA(resp.B).OnX×Zb,wehave,by: αβ αX×Zb=PzdαZbandβX×Zb=PzdβZb Z Z withαZb≥βZbby(47).Defineα′X×Zby: αPzdβZbα′X×Zb= Z ′ α(X×{∅d})=α(X×{∅d})+αZb−βZbα′X×(Z−Zb∪{∅d})=αX×(Z−Zb∪{∅d}) Clearlyα′X×Zisapositivemeasure.Itfollowsfromthefirstequationthat ′ αZb=βZb,andfromthesecondthatα′x=αX=µ.Itremainstocheckthat′ αX×Z∈M+(X,D).WealreadyknowthatαX×Z∈M+(X,D),meaningthat α forPx[D(x)]=1forµ-a.e.x,anditdiffersfromα′X×Zonlyintheregion α wherez∈Zborz=∅d.IfD(x)∩Zb=∅thenPx[D(x)]=1aswell.IfD(x) bb intersectsZ,sothatz∈Z∩D(x),thenconsumerxispayingthehighestbidpriceforz,andsohemustbeindifferentbetweenzand∅d;thisshowsthat∅dalsobelongstoD(x).Inthenewallocationα′X×Z,someofthedemandmaybetransferedfromZb∩D(x)to∅dwithpositiveprobability,butthisredistributionoccurswithinD(x)anddoesnotaffectthetotalprobability,so α′ thatPx[D(x)]=1. Inwords,foreveryqualityzwherethehighestbidpriceispaid,weclearthemarketbylettingsomeofthedemandgoallproducersyhaveunsatisfied:bb sold,butthereistotalquantityαZ−βZofpotentialbuyerswhicharethrownoutofthemarket.However,theydon’tcare,becausethepriceaskedisthehighestbidprice,andtheyareindifferentbetweenbuyingornor. Wethenshiftsomeofthesupplyto∅s,aswedidforthedemand.Weendup ′ withα′X×Z∈M+(X,D)andβY×Z∈M+(Y,S)whichsatisfytheconclusionsoftheExistenceTheorem. 37 D D.1 Remainingproofs Paretooptimalityofequilibriumallocations Witheverypairofdemandandsupplydistributions,α′X×Z∈M+(X,D)and′ βY×Z∈M+(Y,S),weassociatethenumber: ′ ′ JαX×Z,βY×Z= u(x,z)dα′X×Z− ′ v(y,z)dβY×Z Y×ZX×Z β′α′ Ey[v(y,z)]dν(y)Ex[u(x,z)]dµ(x)−= X Y ′ Assumethatα′Z0=βZ0.Weclaimthat: β′α′ [p(z)]dν(y)=0EyEx[p(z)]dµ(x)− X Y (51) Indeed,theleft-handsidecanbewrittenas: ′′′′ +p(∅d)(α′p(z)dβZp(z)dα′Z[∅d]−βZ[∅d])+p(∅s)(αZ[∅s]−βZ[∅s])Z− Z0 Z0 ′ Thefirsttermvanishesbecauseα′Z0=βZ0,andthetwonexttermsvanishbecausep(∅d)=p(∅s)=0. Substracting(51)fromJ,weget: ′ β′α′′ [v(y,z)−p(z)]dν(y)Ex[u(x,z)−p(z)]dµ(x)−EyJαX×Z,βY×Z= (52) ByFenchel’sinequality,(u(x,z)−p(z))≤p(x)forallz∈Z.Taking α′ expectationswithrespecttotheprobabilityPx,weget: ♯ α Ex[u(x,z)−p(z)]≤p♯(x) ′ XY (53) withequalityifandonlyifu(x,z)−p(z)=p♯(x)(inotherwords,z∈D(x)) α′ forPx-almosteveryz∈Z.Similarly,wehave: β Ey[v(y,z)−p(z)]≥p♭(y) ′ (54) withequalityifandonlyifv(y,z)−p(z)=p♭(y)(inotherwords,z∈S(y)) β′ forPy-almosteveryz∈Z.Writingthisin(52),andtreatingthesecondterminthesameway,weget: ′ ♯′ p(x)dµ−p♭(y)dν(55)JαX×Z,βY×Z≤ X Y Theright-handsideisequaltoJ(αX×Z,βY×Z),foranyequilibriumalloca-tion(α,β).Thisprovesthatequilibriumallocationssolvetheplanner’sproblem, andassuchtheyareParetooptimal. 38 D.2Uniquenessofequilibriumallocations Observethatequalityholdsin(55)ifandonlyifequalityholdsin(53)forµ-almosteveryx,andequalityholdsin(54)forν-almosteveryy.Thismeans α′β′ thatPx[D(x)]=1forµ-almosteveryxandPy[S(y)]=1forν-almosteveryy. D.3ProofofTheorem19 Let(p,αX×Z,βY×Z)beanequilibrium.ByRademacher’stheorem,sincep♯:X→RisLipschitz,andµisabsolutelycontinuouswithrespecttotheLebesguemeasure,p♯isdifferentiableeverywhere.µ-almost♯ ConsiderthesetA=x|p(x)≥0.Letx∈Abeapointwherep♯isdifferentiable,withderivativeDxp♯(x).Sincexisactiveorindifferent,thesetD(x)∩Z0isnon-empty,andwemaytakesomez∈D(x)∩Z0.Considerthefunctionϕ(x′)=u(x′,z)−p(z).Byproposition25,sinceD(x)⊂∂p♯(x),wehaveϕ≤fandϕ(x)=f(x),sothatϕandfmusthavethesamederivativeatx: Dxf(x)=Dxu(x,z)(56)Bycondition(9),thisequationdefineszuniquely.Inotherwords,forµ-almosteverypointx∈A,thesetDZ0consists(x)∩ofonepointonly.Similarly,♭ forν-almosteverypointy∈B=y|p(y)≤0,thesetS(y)∩Z0consistsofonepointonly.Thisisthedesiredresult. 39 因篇幅问题不能全部显示，请点此查看更多更全内容