沒有承諾的貨幣政策 Monetary Policy without Commitment

NBERWORKINGPAPERSERIESONETARYPOLICYWITHOUTCOMMITMENTsanAfrouziinaHalacKennethS.RogoffrreYaredWorkingPaper31207http//papers/w31207NATIONALBUREAUOFECONOMICRESEARCHCambridgeMA8May23TheviewsexpressedhereinarethoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheNationalBureauofEconomicResearch.NBERworkingpapersarecirculatedfordiscussionandcommentpurposes.Theyhavenotbeenpeer-reviewedorbeensubjecttothereviewbytheNBERBoardofDirectorsthataccompaniesofficialNBERpublications.?2023byHassanAfrouzi,MarinaHalac,KennethS.Rogoff,andPierreYared.Allrightsreserved.Shortsectionsoftext,nottoexceedtwoparagraphs,maybequotedwithoutexplicitpermissionprovidedthatfullcredit,including?notice,isgiventothesource.MonetaryPolicywithoutCommitmentHassanAfrouzi,MarinaHalac,KennethS.Rogoff,andPierreYaredNBERWorkingPaperNo.31207May23JELNoD,E02,E52,E58,E61STRACTThispaperstudiestheimplicationsofcentralbankcredibilityforlong-runinflationandinflationdynamics.Weintroducecentralbanklackofcommitmentintoastandardnon-linearNewKeynesianeconomywithsticky-pricemonopolisticallycompetitivefirms.Inflationisdrivenbytheinteractionoflackofcommitmentandtheeconomicenvironment.Weshowthatlong-runinflationincreasesfollowinganunanticipatedpermanentincreaseinthelaborwedgeordecreaseintheelasticityofsubstitutionacrossvarieties.Inthetransition,inflationovershootsandthengraduallydeclines.Quantitatively,theinflationresponseislarge,asisthewelfarelossfromlackofcommitmentrelativetoinflationtargeting.HassanAfrouziKennethS.RogoffDepartmentofEconomicsThomasDCabotProfessorofPublicPolicyColumbiaUniversityEconomicsDepartment420W118thStreetHarvardUniversityNewYork,NY10027LittauerCenter216andNBERCambridge,MA02138-3001ha2475@andNBERkrogoff@inaHalacYaleUniversityPierreYaredDepartmentofEconomicsColumbiaUniversity30HillhouseAvenue,Room22BGraduateSchoolofBusinessNewHaven,CT065113022Broadway,UrisHall823andCEPRNewYork,NY10027marina.halac@andNBERpyared@11IntroductionIn?ationacrossadvancedeconomiesintheaftermathoftheCOVID-19pandemicrosetolevelsnotseensincetheearlysThishasbroughtaboutaresurgenceofinterestinthesubjectofcentralbankcredibilitytothemaintenanceoflowandstablein?ation.Thesereactionfunctionsandin?ationtargets—forunderstandingthepost-pandemicenvironment.Inthispaper,weapproachtheanalysisofcentralbankcredibilitybydevelopingaframeworkwherepolicyisnotexogenous,butisinsteaddynamicallychosenbyacentrallwelfareineverydateWefocusontheimplicationsofthecentralbank’sinabilitytomakeex-antecommitments.OuranalysisbuildsontheseminalworkofBarroandGordon(1983)andRogo?(1985),whostudytheimplicationsofthecentralbank’sin?ation-outputtradeo?fortheeconomy.ThisworkandmostofthevastliteratureBytheirconstruction,theseanalysesdonotinformhowcentralbankcredibilityimpactslong-runin?ationortransitiondynamics.2WebuildonthispriorworkbystudyingcentralbankcredibilityinastandardNewKeynesianmodel.3Inordertoallowforananalysisoflong-runin?ationandtransitiondynamics,wedonotperformalog-linearizationaroundthezero-in?ationsteadystatebutinsteadexaminethefullynon-linearmodel.Fortractability,wetakeadeterministiciscomposedofmonopolisticallycompetitive?rmswithstickyprices:ineveryperiod,arandomfractionof?rmshavetheabilityto?exiblychoosetheirprice,whiletheremaining?rmsareconstrainedtochoosingtheirpreviousperiod’sprice.Wagesarefully?exibleandtakingintoaccountcurrenteconomicconditionsandpoliciesandtheirexpectationsoffutureeconomicconditionsandpolicies.Asisstandardintheliterature,weallowfortheexistenceofanexogenouslaborwedgethattakestheformofaproportionalpositiveornegativetaxonlabor.Thiswedgecapturesstatutorytaxesonlaborandotherlabormarketdistortions,sofregulationandunionization1See,forexample,BackusandDri?ll(1985),Canzoneri(1985),CukiermanandMeltzer(1986),Athey,Atkeson,andKehoe(2005),andHalacandYared(2020,2022).2Linearizedenvironmentsareusefulforconsideringtransitiondynamicsaroundanassumedsteadystate;however,suchastatemaynotcoincidewiththeactualsteadystateoftheeconomy,whichcanonlybedeterminedbyanalyzingthenon-linearenvironment.3SeeClarida,Galí,andGertler(1999),Woodford(2003),andGalí(2015).Aswedescribenext,weconsiderpricestickinesswithCalvopricing(Calvo,1983),butourresultsareidenticalunderwagestickiness.2Oureconomyadmitstwotypesofdistortions.First,theexistenceofmonopolypowermeansthatabsentasu?cientlynegativelaborwedge,?rmsunderproduceandunderhire.Toexaminetheroleofthisdistortion,weassumethatthelaborwedgeislargeenoughthatwithtoomuchlabordrawntotheproductionoflow-pricevarietiesandtoolittletotheproductionofhigh-pricevarieties.Ouranalysishighlightshowmonopolydistortionsandlabormisallocationimpactthein?ation-outputtradeo?andguidetheconductofmonetarypolicy.Beforedescribingourmain?ndings,itisinstructivetonotethatmonetarypolicyinourenvironmentisnotneutralinthelongrun.Di?erentpolicypathscanleadtoacontinuumofpotentialsteadystates.Comparingacrosshypotheticalsteadystates,thosewithhigherthedivergenceinpricesbetween?exible-price?rms(whichraisepricesbymoreunderhigherin?ation)andsticky-price?rmsisrelativelylarger.Moreover,steadystateswithhigherin?ationalsoadmitrelativelylowermonopolydistortions,sinceoverhiringbysticky-price?rmsisrelativelyhigher.WestudycentralbankcredibilitybyanalyzingtheMarkovPerfectCompetitiveEquilibriaofourmodel,wherecentralbankstrategiesandprivatesectorbeliefsareafunctionofesthecentralbanksetstheinterestrate,andmarketsopenandclear.Thecentralbanklackscommitmentandsgiventhedistributionofprices.Firmssettingprices?exiblyanticipatethatthecentralbanklackscommitmenttodayandinthefuturewhenformingexpectationsaboutfuturepolicy.Weshowthatanequilibriumischaracterizedbytwodi?erenceequations.Thedynamicpathofin?ationischaracterizedbyanequationthatisforwardlooking;i.e.,anon-linearPhillipscurvewithcurrentin?ationbeingafunctionofexpectationsoffuturein?ation.Thedynamicpathofpricedispersionischaracterizedbyanequationthatisbackwardlooking;i.e.,withcurrentdispersionbeingafunctionofpastdispersion.Theseequationsyieldthatichallowsustoanalyzetransitiondynamicsaroundsuchasteadystate.Thetractableformoftheequilibriumowespartlytothetiminginourmodel,wherepricesarechosenpriortothechoiceofinterestrates.Acentralbanksettinginterestratesattimettakesaspredeterminedthedistributionofpricesattimet,andthereforealsothecontinuationequilibriumatt+1(sincetheequilibriumisMarkov).Becausethecentralbankcannotchangefuturewelfareo?theequilibriumpath,itoptimallychoosesaninterest3ratethatmaximizesstaticwelfareconditionalonthelevelofpricedispersion.4Theresultisapolicythateliminatesthemonopolydistortionandsetsthelaborsharetoone,5andanequilibriumthatcanbesimpli?edtoasystemoftwoequations.Ourmainresultsareasfollows.First,weshowthatlong-runin?ationisdeterminedbytheinteractionofthecentralbank’slackofcommitmentandtheeconomicenvironment.Speci?cally,long-runin?ationishigherthehigheristhelaborwedgeandtheloweristheelasticityofsubstitutionacrossvarieties.Tounderstandthesecomparativestatics,consider?rsttheincentivesofthecentralbanktocutinterestrateso?theequilibriumpath.Aratecutincreasesconsumptionatthecostofincreasedlabore?ort,soitsmarginalbene?tisesslaboranddecreasinginpricedispersionandlabormisallocation(whichreduceaggregatelaborproductivity).wedgeorapermanentdecreaseintheelasticityofsubstitutionacrossvarieties(withthesechangesbeingunanticipated).Acentralbankwithcommitmentwouldbeabletorespondinawaythatpreservesin?ationstability.However,acentralbankwithoutcommitmenthasanincentivetoundotheresultingincreaseinmonopolydistortionsbycuttinginterestratesandstimulatingoutput.Flexible-price?rmsanticipatethisandrationallyforecasthigherfuturelabordemandandhigherfuturerealwages(relativetothecommitmentcase),whichertimeexiblepricermsthusincreasepricesleadingtorisingpricedispersion.Theeconomyconvergestoanewsteadystateoncetheriseindispersionreducesaggregateproductivitysu?cientlythatthecentralbank’sbene?tsultofthehigherlaborwedgeorlowerelasticityofsubstitutionacrossvarieties,pricedispersionandin?ationarebothhigherinthenewsteadystate.aninitialsteadystatetoonewithhigherin?ation.Weshowthatthetransitionfeaturesin?ationovershooting.Startingfromagivensteadystate,considerapermanentincreaseinthelaborwedgeorapermanentdecreaseintheelasticityofsubstitutionacrossvarieties.In?ationovershootingemergesbecauseoftheevolutionofcentralbankincentivesaspricearelativelylargerbene?ttostimulatingoutputearlyinthetransitionwhenpricedispersiontly4Ourresultsareunchangedif?rmsandthecentralbankmovesimultaneously,asinBarroandGordon(1983).However,ifthecentralbanksetstheinterestratebefore?rmssetprices,theno?-equilibriumexpectationsby?rmscanplayanimportantroleinequilibriumselection,complicatingtheanalysis.5Inotherwords,average?rmpro?ts(netoflabortaxes)acrosstheeconomyequalzero,withsomesticky-price?rmsmakingnegativepro?tsandall?exible-price?rmsmakingpositivepro?ts.4itisnolongerasworthwhiletostimulatelabortogenerateadditionalconsumption.Flexible-price?rmsinturnrealizethatmonetarystimuluswillbelargerearlierinthetransition,sotheyo?settheensuinghigherwagecostswithpriceincreasesthatarealsorelativelylargerearlierinthetransition.Theimplicationofthesetransitiondynamicsisin?ationOur?nalmainresultshowsthatthequantitativemagnitudesimpliedbyourmodelarelarge.UsingstandardparameterizationsoftheNewKeynesianmodel,weevaluatetheintheelasticityofsubstitutionacrossvarieties.Inbothcases,in?ationjumpsupfollowingtheshockandthengraduallydeclinestowardsanewhighersteady-statelevel.Nominalinterestratesjumpupandgraduallyincreasetoahigherlevel,whilerealinterestratesjumpdownandgraduallyincreasebacktotheiroriginallevel.6Outputfallsgraduallyaspricedispersionandlabormisallocationincreaseinthetransition.Nominalwagein?ationrisestowardsthenewsteadystate,convergingtothelevelofpricein?ationfrombelow.Thesedynamicsunderpinaneventualpermanentreductionintherealwage.We?ndthatsmallchangesinthelaborwedgeortheelasticityofsubstitutionacrossvarietieshavelargeimpactsonthelong-runlevelofin?ationandthedegreeofin?ationponsetoapercentincreaseinthelaborwedgestartingfroma2-percent-in?ationsteadystate,in?ationovershootsto10.11percentandeventuallyconvergesto8.76percent.Thisovershootingispersistent:ittakes12monthsforin?ationtodeclinewithin25basispointsofitsnewsteady-statelevel.Weobtainsimilarlylargemagnitudesforchangesintheelasticityofsubstitution.Furthermore,wecomparetheresponseoftheeconomyundercentralbanklackofcommitmenttotheresponseundercommitmenttoin?ationtargeting,andwe?ndthatthewelfarelossfromlackofcommitmentisquantitativelylarge.inadvancedeconomies.First,ourresultssuggestthatchangesintheglobaleconomycana?ectlong-runin?ationthroughtheirinteractionwithcentralbankincentives.Considerforinstanceadecreaseinimmigrationwhichchangesthecompositionoflabortowardsmorethemarketpowerofdomestic?rms(lowerelasticityofsubstitutionacrossvarieties).Ourmodelsaysthatifthesechangesintheglobaleconomyarepermanent,thentheycanresultinpermanentlyhigherin?ation.Thesechangesreduceoutputandconsumptionandmakeitmorechallengingforacentralbankwithoutcommitmenttomaintainlowin?ation.Moreover,6Thehighersteady-statelevelfornominalinterestratesre?ectstheFisheriane?ect,whichispresentinthenon-linearNewKeynesianmodel.5underthisview,asecondimplicationofouranalysisisthatthelargespikeinin?ationfollowingthepandemiccanbepartiallyunderstoodasaconsequenceoftheprivatesectoranticipatingthecentralbank’sresponse.Firmsrationallyanticipatethatthecentralbankwouldbemoreaccommodativeearlierinthetransition(whenthecostofpricedispersionislow)relativetolater(whenthecostofpricedispersionishigh),resultinginin?ationovershootingastheeconomymovestoahigher-in?ationsteadystate.RelatedLiterature.Ourpaper?tsintotheliteratureoncentralbankcredibilityandreputationpioneeredbyBarroandGordon(1983)andRogo?(1985).7Asdiscussed,ourdeparturefromthisliteratureisinanalyzingtheequilibriumofthefullynon-linearNewKeynesianmodel.Thisdepartureallowsustoexaminetheendogenousdynamicevolutionofthecentralbank’sin?ation-outputtradeo?aswellasthequantitativeimplicationsofcentralbankcredibility.AnapproachthatconsidersaMarkovianequilibriumunderlackofcommitmentaroundalinearized(distorted)zero-in?ation,zero-dispersionsteadystate(asinHalacandYared,2022,forexample)notonlyfeaturesnotransitiondynamics,butalsoationrelativetothenon-linearmodel.8Previousworkhasconsideredlackofcommitmenttomonetarypolicyinnon-linearenvironments.Forexample,alargenumberofmodelsof?scalpolicyareconcernedwiththecentralbank’scommitmenttonotin?atingawaypublicdebt(e.g.,Alvarez,Kehoe,andNeumeyer,2004;Aguiar,Amador,Farhi,andGopinath,2015).DávilaandSchaab(2023)showthatlackofcommitmenttomonetarypolicyhasdistributionalimplicationsinteconomiesOurdeparturefromtheseliteraturesisinconsideringthedispersioncostthatresultsfrompricestickinessinstandardNewKeynesianmodels,andinexamininghowthisdispersioncostdynamicallya?ectsthein?ation-outputtradeo?forthecentralbank.Inthisregard,wecontributetotheliteraturethatstudiesthecentralbank’sin?ation-outputtradeo?underlackofcommitmentinnon-linearsettings.Thefocusofthisliteraturehasbeenondeterminingconditionsforequilibriummultiplicityundermonetarydiscretion(see,e.g.,Albanesi,Chari,andChristiano,2003;KingandWolman,2004;ZandwegheandWolman,2019).Theseconsiderationsdonotariseinoursettingwhereweobtainauniqueequilibrium.Wedepartfromthisworkbyprovidingananalyticalcharacterizationofthe7SeeadditionallytheworkcitedinFootnote1.8Thereasonisthatthelinearizedmodelunderestimatesthewelfarecostsofrisingpricedispersionduetoin?ation.9Theirmodelhasstickypriceswithexogenouscostsofpriceadjustment,asopposedtoCalvopricingasinourframework.Asaconsequence,thereisnopricedispersionintheirequilibrium.6uniquesteadystateandthetransitiondynamicsofthenon-linearmodel,andbyanalyticallyowthesedependontheeconomicenvironmentFinally,ourpaperalsorelatestotheliteratureonoptimalcommitmentpolicyinthenon-weprovideanovelrecursiverepresentationofthenon-linearPhillipscurvebyde?ninganewauxiliaryvariablewhichcanitselfberepresentedrecursively.Thisvariablecapturesthepassthroughofrealwagestocurrentin?ation,holdingfuturein?ationexpectations?xed.ethattheycouldbeusefulinfutureanalysesofthenon-linearNewKeynesianmodel.ronmentWeconsiderastandardnon-linearNewKeynesianmodel(Clarida,Galí,andGertler,1999;Woodford,2003;Galí,2015).Thereisaunit-massofmonopolisticallycompetitive?rmsthatsetpricesunderCalvo-stylerigidity(Calvo,1983):arandomfractionof?rmsineveryperiodhavetheabilityto?exiblychoosetheirprice,whiletheremainingfractionof?rmsareconstrainedtochoosingthesamepriceasinthepreviousperiod.Wagesarefully?exibleandhouseholdsmakeconsumption,labor,andsavingsdecisions.11Firmsandhouseholdsoptimizetakingintoaccountcurrenteconomicconditionsandpoliciesandtheirexpectationsoffutureeconomicconditionsandpolicies.Ineveryperiod,?exible-price?rmssetprices,thecentralbanksetstheinterestrate,andmarketsopenandclear.Akeyfeatureofourenvironmentisthatthecentralbanklackstakingasgiventhedistributionofprices.Firmssettingprices?exiblyanticipatethatthecentralbanklackscommitmenttodayandinthefuturewhenformingexpectationsaboutfuturepolicy.Ateverydatete{0,1,2,...},therepresentativehouseholdchoosesitsconsumptionCj,tofeach?rmvarietyje[0,1],itslaborsupplyLt,itsholdingsBtofarisk-freenominaltofsharesofeachrmjeFor10ThemodelofAlbanesi,Chari,andChristiano(2003)hasstickypricesonlywithintheperiod,whilethatofKingandWolman(2004)haspricesthatarestickyacrosstwoperiods.TheframeworkinZandwegheandWolman(2019)istheclosesttoourswithCalvopricingacrossperiods,buttheirtimingisdi?erentandtheirresultsunderlackofcommitmentarenumericalratherthananalytical.11Ourresultsareunchangedifweinsteadconsiderastickywagemodel.7eachj,denotethe?rm’svarietypricebyPj,1,itsnominalsharepricebyP,anditsnominalolvesthefollowingproblem:Ct,Lt,Bt,a,tj,t)je[0,1]81!log(C1)_{subjectto/01Pj,1Cj,1dj+B1<w1L1+(1+i1-1)B1-1+/01sj,1Ⅹj,1dj+/01(sj,1-1_sj,1)Pdj_T1,1C1=╱/01C口-1dj、1-σ-1.C1denotestheaggregateconsumptionbundleandT1isalumpsumtaxatdatet.Wehavetaken8e(0,1)tobethediscountfactor,口>1istheelasticityofsubstitutionacrossvarieties,andw>1istheinverseelasticityoflaborsupply.Thehousehold’soptimizationyieldsthatthedemandforvarietyjsatis?esCj,1=C1╱、-口,1wherePPdjThuswecanwritePjCjdj=P1C1inthebudgetconstraintje[0,1]asafunctionof(Pj,1)j∈[0,1].Theintratemporalconditionis=C1L.TheintertemporalconditionisP1+1C1+1.P1+1C1+1.Thetransversalityconditionsrequirethatforeachdatetand?rmj,12h→&|=0(1+i1+e)h→&|=0(1+i1+e)lim1Eh→&|=0(1+i1+e)h→&|=0(1+i1+e)12Thistransversalityconditioncombinesahouseholdoptimalityconditionandano-PonziconditioninacompletemarketenvironmentthatallowsforArrow-Debreusecurities,includingsecuritiesthatpayo?atafuturedateanamountproportionaltothepro?tsofany?rmconditionalonanygivenhistory.8TheexpectationoperatorE[.]operatesover?rmj’sfutureidiosyncraticshocks,whichwewilldiscussinthenextsection.NoarbitrageforstocksrequiresP=Ⅹj,t+E[P+1]/(1+it),whichcombinedwith(5)yields&P=Ⅹj,t+E[Ⅹj,t+1+h].&Combining(6)withtheintertemporalconditionforarbitraryhorizont+h,weobtainthatthenominalsharepriceof?rmjsatis?es13P=8hE[Ⅹj,t+h].2.2FirmsA?rmsellingvarietyjproduceswithtechnologyxj,t=Lj,t.GiventhelaborLj,tdemandedbyeach?rmje[0,1],ineveryperiodtwehaveLt=|01Lj,tdj.FirmssetpricesasinCalvo(1983).Ineveryperiod,arandomfraction1_9e(0,1)ofusperiod’sprice.Theexogenousinitialdistributionof?rmpricesisgivenby{Pj,-1}j∈[0,1].FirmscommittoproduceenoughtomeetdemandgiventheirpricePj,t,evenifthatmeansmakingnegativepro?ts.Firmsaresubjecttoaproportionalpayrolltaxre(0,1),whichwerefertoasthelaborasthepervasivenessofregulationandunionization.14Wemakethefollowingassumption:Assumption1.Thelaborwedgesatis?esr>_1/口.Thisassumptionimpliesthatmonopolydistortionsariseinaneconomywith?exiblescussinthenextsectionAssumptionguaranteesthatthelaborwedgeissu?cientlylargethatmonopolydistortionsdonotdisappearinthesteadystateofourFirmpro?tsatanydatetsatisfyⅩj,t=Pj,txj,t_(1+r)wtLj,t,13Notethatinequilibrium,allhouseholdsholdequalportionsofall?rms,so|01(sj,t_1_sj,t)Ptdj=0.14Underthislatterinterpretation,_Ttcanbethoughtofasmeasuringunionpro?ts.9whichcombinedwith(7)impliesP=8hE[Pj,t+hxj,t+h_(1+r)wt+hLj,t+h].A?rmthatresetsitspricePj,tattimetmaximizesitssharepricePgivenby(8),takingintoaccountthatthepricePj,twillprevailatdatet+hwithprobability9h.The?exible-pricethusberepresentedasfollowsmPa*tx(89)h[Pt*_(1+r)wt+h]Ct+h╱、-口,wherewehavetakenintoaccountthatthetransversalityconditionimpliesthat,forallt,(89)h[Pt*_(1+r)wt+h]Ct+h╱、-口=0.2.3GovernmentAteverydatet,thecentralbanksetstheinterestrateittomaximizesocialwelfarewhichisgivenby(1).Wedescribethecentralbank’sprobleminSection4.1.The?scalauthoritysetstaxesTtanddebtBttosatisfyitsbudgetconstraint:(1+it-1)Bt-1=Bt+Tt+rwtLt.TheexogenousinitiallevelofgovernmentdebtisgivenbyB-1.2.4OrderofEventsTheorderofeventsatadatet,givenadistributionofpricesPj,t-1,isasfollows:1.Flexible-price?rmschoosepricePj,t=Pt*.Sticky-price?rmschoosepricePj,t=Pj,t-1.2.Thecentralbankchoosesmonetarypolicy,i.e.theinterestrateit.4.The?scalauthoritychooses?scalpolicy,i.e.taxesTtanddebtBt.Observethatmonetarypolicyischosenafter?rmshavechosentheirprices.Hence,ifthecentralbankdeviatesfromequilibriumpolicy,?rmswillnolongerbeoptimizingduringtheperiodofthedeviationo?theequilibriumpath.Thefactthat?scalpolicyischosenattheendoftheperiodisforexpositionalsimplicityandwithoutlossgivenourequilibriumde?nitionwhichwedescribeinthenextsection.2.5EquilibriumDe?nitionOursolutionconceptisMarkovPerfectCompetitiveEquilibrium(MPCE),implyingthathepresenceoflumpsumtaxesoureconomyfeaturesRicardianEquivalence,whichmeansthatthelevelofdebtBtcanbetreatedaspayo?irrelevantandthatattheendofeachperiodthe?scalauthoritychoosesdebtBt=0andtaxesTttobalanceitsbudget,bothonando?theequilibriumpath.15Formally,let?t-1correspondtothedistributionofpricesacross?rmsatdatet_1.Conditionalon?t-1,?exible-price?rmschoosepricePt*whichdetermines?t.LetΓdenotecorrespondingmappingwithttGiventthecentralbankchoosesmonetarypolicyit=Ψ(?t),whereΨisthecentralbank’sreactionfunction.Finally,given?tandit,oseconsumptionlaborandsavingsCtLtBtsjtCjtjutitwhereuisthehouseholds’reactionfunction.Inequilibrium,Bt=0andsj,t=1,sincehouseholdscanbeidenticallytreatedwithoutlossofgenerality.AnMPCEisacollectionofmappings{Γ,Ψ,u}suchthat,ateverydatetandgiven{Γ,Ψ,u},themappingΓ(?t-1)satis?es?exible-price?rmoptimality,Ψ(?t)maximizessocialwelfare,andu(?t,it)satis?eshouseholdoptimality.