没有承诺的货币政策 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@11IntroductionInflationacrossadvancedeconomiesintheaftermathoftheCOVID-19pandemicrosetolevelsnotseensincetheearlysThishasbroughtaboutaresurgenceofinterestinthesubjectofcentralbankcredibilitytothemaintenanceoflowandstableinflation.Thesereactionfunctionsandinflationtargets—forunderstandingthepost-pandemicenvironment.Inthispaper,weapproachtheanalysisofcentralbankcredibilitybydevelopingaframeworkwherepolicyisnotexogenous,butisinsteaddynamicallychosenbyacentrallwelfareineverydateWefocusontheimplicationsofthecentralbank’sinabilitytomakeex-antecommitments.OuranalysisbuildsontheseminalworkofBarroandGordon(1983)andRogoff(1985),whostudytheimplicationsofthecentralbank’sinflation-outputtradeofffortheeconomy.ThisworkandmostofthevastliteratureBytheirconstruction,theseanalysesdonotinformhowcentralbankcredibilityimpactslong-runinflationortransitiondynamics.2WebuildonthispriorworkbystudyingcentralbankcredibilityinastandardNewKeynesianmodel.3Inordertoallowforananalysisoflong-runinflationandtransitiondynamics,wedonotperformalog-linearizationaroundthezero-inflationsteadystatebutinsteadexaminethefullynon-linearmodel.Fortractability,wetakeadeterministiciscomposedofmonopolisticallycompetitivefirmswithstickyprices:ineveryperiod,arandomfractionoffirmshavetheabilitytoflexiblychoosetheirprice,whiletheremainingfirmsareconstrainedtochoosingtheirpreviousperiod’sprice.Wagesarefullyflexibleandtakingintoaccountcurrenteconomicconditionsandpoliciesandtheirexpectationsoffutureeconomicconditionsandpolicies.Asisstandardintheliterature,weallowfortheexistenceofanexogenouslaborwedgethattakestheformofaproportionalpositiveornegativetaxonlabor.Thiswedgecapturesstatutorytaxesonlaborandotherlabormarketdistortions,sofregulationandunionization1See,forexample,BackusandDriffill(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,theexistenceofmonopolypowermeansthatabsentasufficientlynegativelaborwedge,firmsunderproduceandunderhire.Toexaminetheroleofthisdistortion,weassumethatthelaborwedgeislargeenoughthatwithtoomuchlabordrawntotheproductionoflow-pricevarietiesandtoolittletotheproductionofhigh-pricevarieties.Ouranalysishighlightshowmonopolydistortionsandlabormisallocationimpacttheinflation-outputtradeoffandguidetheconductofmonetarypolicy.Beforedescribingourmainfindings,itisinstructivetonotethatmonetarypolicyinourenvironmentisnotneutralinthelongrun.Differentpolicypathscanleadtoacontinuumofpotentialsteadystates.Comparingacrosshypotheticalsteadystates,thosewithhigherthedivergenceinpricesbetweenflexible-pricefirms(whichraisepricesbymoreunderhigherinflation)andsticky-pricefirmsisrelativelylarger.Moreover,steadystateswithhigherinflationalsoadmitrelativelylowermonopolydistortions,sinceoverhiringbysticky-pricefirmsisrelativelyhigher.WestudycentralbankcredibilitybyanalyzingtheMarkovPerfectCompetitiveEquilibriaofourmodel,wherecentralbankstrategiesandprivatesectorbeliefsareafunctionofesthecentralbanksetstheinterestrate,andmarketsopenandclear.Thecentralbanklackscommitmentandsgiventhedistributionofprices.Firmssettingpricesflexiblyanticipatethatthecentralbanklackscommitmenttodayandinthefuturewhenformingexpectationsaboutfuturepolicy.Weshowthatanequilibriumischaracterizedbytwodifferenceequations.Thedynamicpathofinflationischaracterizedbyanequationthatisforwardlooking;i.e.,anon-linearPhillipscurvewithcurrentinflationbeingafunctionofexpectationsoffutureinflation.Thedynamicpathofpricedispersionischaracterizedbyanequationthatisbackwardlooking;i.e.,withcurrentdispersionbeingafunctionofpastdispersion.Theseequationsyieldthatichallowsustoanalyzetransitiondynamicsaroundsuchasteadystate.Thetractableformoftheequilibriumowespartlytothetiminginourmodel,wherepricesarechosenpriortothechoiceofinterestrates.Acentralbanksettinginterestratesattimettakesaspredeterminedthedistributionofpricesattimet,andthereforealsothecontinuationequilibriumatt+1(sincetheequilibriumisMarkov).Becausethecentralbankcannotchangefuturewelfareofftheequilibriumpath,itoptimallychoosesaninterest3ratethatmaximizesstaticwelfareconditionalonthelevelofpricedispersion.4Theresultisapolicythateliminatesthemonopolydistortionandsetsthelaborsharetoone,5andanequilibriumthatcanbesimplifiedtoasystemoftwoequations.Ourmainresultsareasfollows.First,weshowthatlong-runinflationisdeterminedbytheinteractionofthecentralbank’slackofcommitmentandtheeconomicenvironment.Specifically,long-runinflationishigherthehigheristhelaborwedgeandtheloweristheelasticityofsubstitutionacrossvarieties.Tounderstandthesecomparativestatics,considerfirsttheincentivesofthecentralbanktocutinterestratesofftheequilibriumpath.Aratecutincreasesconsumptionatthecostofincreasedlaboreffort,soitsmarginalbenefitisesslaboranddecreasinginpricedispersionandlabormisallocation(whichreduceaggregatelaborproductivity).wedgeorapermanentdecreaseintheelasticityofsubstitutionacrossvarieties(withthesechangesbeingunanticipated).Acentralbankwithcommitmentwouldbeabletorespondinawaythatpreservesinflationstability.However,acentralbankwithoutcommitmenthasanincentivetoundotheresultingincreaseinmonopolydistortionsbycuttinginterestratesandstimulatingoutput.Flexible-pricefirmsanticipatethisandrationallyforecasthigherfuturelabordemandandhigherfuturerealwages(relativetothecommitmentcase),whichertimeexiblepricermsthusincreasepricesleadingtorisingpricedispersion.Theeconomyconvergestoanewsteadystateoncetheriseindispersionreducesaggregateproductivitysufficientlythatthecentralbank’sbenefitsultofthehigherlaborwedgeorlowerelasticityofsubstitutionacrossvarieties,pricedispersionandinflationarebothhigherinthenewsteadystate.aninitialsteadystatetoonewithhigherinflation.Weshowthatthetransitionfeaturesinflationovershooting.Startingfromagivensteadystate,considerapermanentincreaseinthelaborwedgeorapermanentdecreaseintheelasticityofsubstitutionacrossvarieties.Inflationovershootingemergesbecauseoftheevolutionofcentralbankincentivesaspricearelativelylargerbenefittostimulatingoutputearlyinthetransitionwhenpricedispersiontly4Ourresultsareunchangediffirmsandthecentralbankmovesimultaneously,asinBarroandGordon(1983).However,ifthecentralbanksetstheinterestratebeforefirmssetprices,thenoff-equilibriumexpectationsbyfirmscanplayanimportantroleinequilibriumselection,complicatingtheanalysis.5Inotherwords,averagefirmprofits(netoflabortaxes)acrosstheeconomyequalzero,withsomesticky-pricefirmsmakingnegativeprofitsandallflexible-pricefirmsmakingpositiveprofits.4itisnolongerasworthwhiletostimulatelabortogenerateadditionalconsumption.Flexible-pricefirmsinturnrealizethatmonetarystimuluswillbelargerearlierinthetransition,sotheyoffsettheensuinghigherwagecostswithpriceincreasesthatarealsorelativelylargerearlierinthetransition.TheimplicationofthesetransitiondynamicsisinflationOurfinalmainresultshowsthatthequantitativemagnitudesimpliedbyourmodelarelarge.UsingstandardparameterizationsoftheNewKeynesianmodel,weevaluatetheintheelasticityofsubstitutionacrossvarieties.Inbothcases,inflationjumpsupfollowingtheshockandthengraduallydeclinestowardsanewhighersteady-statelevel.Nominalinterestratesjumpupandgraduallyincreasetoahigherlevel,whilerealinterestratesjumpdownandgraduallyincreasebacktotheiroriginallevel.6Outputfallsgraduallyaspricedispersionandlabormisallocationincreaseinthetransition.Nominalwageinflationrisestowardsthenewsteadystate,convergingtothelevelofpriceinflationfrombelow.Thesedynamicsunderpinaneventualpermanentreductionintherealwage.Wefindthatsmallchangesinthelaborwedgeortheelasticityofsubstitutionacrossvarietieshavelargeimpactsonthelong-runlevelofinflationandthedegreeofinflationponsetoapercentincreaseinthelaborwedgestartingfroma2-percent-inflationsteadystate,inflationovershootsto10.11percentandeventuallyconvergesto8.76percent.Thisovershootingispersistent:ittakes12monthsforinflationtodeclinewithin25basispointsofitsnewsteady-statelevel.Weobtainsimilarlylargemagnitudesforchangesintheelasticityofsubstitution.Furthermore,wecomparetheresponseoftheeconomyundercentralbanklackofcommitmenttotheresponseundercommitmenttoinflationtargeting,andwefindthatthewelfarelossfromlackofcommitmentisquantitativelylarge.inadvancedeconomies.First,ourresultssuggestthatchangesintheglobaleconomycanaffectlong-runinflationthroughtheirinteractionwithcentralbankincentives.Considerforinstanceadecreaseinimmigrationwhichchangesthecompositionoflabortowardsmorethemarketpowerofdomesticfirms(lowerelasticityofsubstitutionacrossvarieties).Ourmodelsaysthatifthesechangesintheglobaleconomyarepermanent,thentheycanresultinpermanentlyhigherinflation.Thesechangesreduceoutputandconsumptionandmakeitmorechallengingforacentralbankwithoutcommitmenttomaintainlowinflation.Moreover,6Thehighersteady-statelevelfornominalinterestratesreflectstheFisherianeffect,whichispresentinthenon-linearNewKeynesianmodel.5underthisview,asecondimplicationofouranalysisisthatthelargespikeininflationfollowingthepandemiccanbepartiallyunderstoodasaconsequenceoftheprivatesectoranticipatingthecentralbank’sresponse.Firmsrationallyanticipatethatthecentralbankwouldbemoreaccommodativeearlierinthetransition(whenthecostofpricedispersionislow)relativetolater(whenthecostofpricedispersionishigh),resultingininflationovershootingastheeconomymovestoahigher-inflationsteadystate.RelatedLiterature.OurpaperfitsintotheliteratureoncentralbankcredibilityandreputationpioneeredbyBarroandGordon(1983)andRogoff(1985).7Asdiscussed,ourdeparturefromthisliteratureisinanalyzingtheequilibriumofthefullynon-linearNewKeynesianmodel.Thisdepartureallowsustoexaminetheendogenousdynamicevolutionofthecentralbank’sinflation-outputtradeoffaswellasthequantitativeimplicationsofcentralbankcredibility.AnapproachthatconsidersaMarkovianequilibriumunderlackofcommitmentaroundalinearized(distorted)zero-inflation,zero-dispersionsteadystate(asinHalacandYared,2022,forexample)notonlyfeaturesnotransitiondynamics,butalsoationrelativetothenon-linearmodel.8Previousworkhasconsideredlackofcommitmenttomonetarypolicyinnon-linearenvironments.Forexample,alargenumberofmodelsoffiscalpolicyareconcernedwiththecentralbank’scommitmenttonotinflatingawaypublicdebt(e.g.,Alvarez,Kehoe,andNeumeyer,2004;Aguiar,Amador,Farhi,andGopinath,2015).DávilaandSchaab(2023)showthatlackofcommitmenttomonetarypolicyhasdistributionalimplicationsinteconomiesOurdeparturefromtheseliteraturesisinconsideringthedispersioncostthatresultsfrompricestickinessinstandardNewKeynesianmodels,andinexamininghowthisdispersioncostdynamicallyaffectstheinflation-outputtradeoffforthecentralbank.Inthisregard,wecontributetotheliteraturethatstudiesthecentralbank’sinflation-outputtradeoffunderlackofcommitmentinnon-linearsettings.Thefocusofthisliteraturehasbeenondeterminingconditionsforequilibriummultiplicityundermonetarydiscretion(see,e.g.,Albanesi,Chari,andChristiano,2003;KingandWolman,2004;ZandwegheandWolman,2019).Theseconsiderationsdonotariseinoursettingwhereweobtainauniqueequilibrium.Wedepartfromthisworkbyprovidingananalyticalcharacterizationofthe7SeeadditionallytheworkcitedinFootnote1.8Thereasonisthatthelinearizedmodelunderestimatesthewelfarecostsofrisingpricedispersionduetoinflation.9Theirmodelhasstickypriceswithexogenouscostsofpriceadjustment,asopposedtoCalvopricingasinourframework.Asaconsequence,thereisnopricedispersionintheirequilibrium.6uniquesteadystateandthetransitiondynamicsofthenon-linearmodel,andbyanalyticallyowthesedependontheeconomicenvironmentFinally,ourpaperalsorelatestotheliteratureonoptimalcommitmentpolicyinthenon-weprovideanovelrecursiverepresentationofthenon-linearPhillipscurvebydefininganewauxiliaryvariablewhichcanitselfberepresentedrecursively.Thisvariablecapturesthepassthroughofrealwagestocurrentinflation,holdingfutureinflationexpectationsfixed.ethattheycouldbeusefulinfutureanalysesofthenon-linearNewKeynesianmodel.ronmentWeconsiderastandardnon-linearNewKeynesianmodel(Clarida,Galí,andGertler,1999;Woodford,2003;Galí,2015).Thereisaunit-massofmonopolisticallycompetitivefirmsthatsetpricesunderCalvo-stylerigidity(Calvo,1983):arandomfractionoffirmsineveryperiodhavetheabilitytoflexiblychoosetheirprice,whiletheremainingfractionoffirmsareconstrainedtochoosingthesamepriceasinthepreviousperiod.Wagesarefullyflexibleandhouseholdsmakeconsumption,labor,andsavingsdecisions.11Firmsandhouseholdsoptimizetakingintoaccountcurrenteconomicconditionsandpoliciesandtheirexpectationsoffutureeconomicconditionsandpolicies.Ineveryperiod,flexible-pricefirmssetprices,thecentralbanksetstheinterestrate,andmarketsopenandclear.Akeyfeatureofourenvironmentisthatthecentralbanklackstakingasgiventhedistributionofprices.Firmssettingpricesflexiblyanticipatethatthecentralbanklackscommitmenttodayandinthefuturewhenformingexpectationsaboutfuturepolicy.Ateverydatete{0,1,2,...},therepresentativehouseholdchoosesitsconsumptionCj,tofeachfirmvarietyje[0,1],itslaborsupplyLt,itsholdingsBtofarisk-freenominaltofsharesofeachrmjeFor10ThemodelofAlbanesi,Chari,andChristiano(2003)hasstickypricesonlywithintheperiod,whilethatofKingandWolman(2004)haspricesthatarestickyacrosstwoperiods.TheframeworkinZandwegheandWolman(2019)istheclosesttoourswithCalvopricingacrossperiods,buttheirtimingisdifferentandtheirresultsunderlackofcommitmentarenumericalratherthananalytical.11Ourresultsareunchangedifweinsteadconsiderastickywagemodel.7eachj,denotethefirm’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’soptimizationyieldsthatthedemandforvarietyjsatisfiesCj,1=C1╱、-口,1wherePPdjThuswecanwritePjCjdj=P1C1inthebudgetconstraintje[0,1]asafunctionof(Pj,1)j∈[0,1].Theintratemporalconditionis=C1L.TheintertemporalconditionisP1+1C1+1.P1+1C1+1.Thetransversalityconditionsrequirethatforeachdatetandfirmj,12h→&|=0(1+i1+e)h→&|=0(1+i1+e)lim1Eh→&|=0(1+i1+e)h→&|=0(1+i1+e)12Thistransversalityconditioncombinesahouseholdoptimalityconditionandano-PonziconditioninacompletemarketenvironmentthatallowsforArrow-Debreusecurities,includingsecuritiesthatpayoffatafuturedateanamountproportionaltotheprofitsofanyfirmconditionalonanygivenhistory.8TheexpectationoperatorE[.]operatesoverfirmj’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,weobtainthatthenominalsharepriceoffirmjsatisfies13P=8hE[Ⅹj,t+h].2.2FirmsAfirmsellingvarietyjproduceswithtechnologyxj,t=Lj,t.GiventhelaborLj,tdemandedbyeachfirmje[0,1],ineveryperiodtwehaveLt=|01Lj,tdj.FirmssetpricesasinCalvo(1983).Ineveryperiod,arandomfraction1_9e(0,1)ofusperiod’sprice.Theexogenousinitialdistributionoffirmpricesisgivenby{Pj,-1}j∈[0,1].FirmscommittoproduceenoughtomeetdemandgiventheirpricePj,t,evenifthatmeansmakingnegativeprofits.Firmsaresubjecttoaproportionalpayrolltaxre(0,1),whichwerefertoasthelaborasthepervasivenessofregulationandunionization.14Wemakethefollowingassumption:Assumption1.Thelaborwedgesatisfiesr>_1/口.ThisassumptionimpliesthatmonopolydistortionsariseinaneconomywithflexiblescussinthenextsectionAssumptionguaranteesthatthelaborwedgeissufficientlylargethatmonopolydistortionsdonotdisappearinthesteadystateofourFirmprofitsatanydatetsatisfyⅩj,t=Pj,txj,t_(1+r)wtLj,t,13Notethatinequilibrium,allhouseholdsholdequalportionsofallfirms,so|01(sj,t_1_sj,t)Ptdj=0.14Underthislatterinterpretation,_Ttcanbethoughtofasmeasuringunionprofits.9whichcombinedwith(7)impliesP=8hE[Pj,t+hxj,t+h_(1+r)wt+hLj,t+h].AfirmthatresetsitspricePj,tattimetmaximizesitssharepricePgivenby(8),takingintoaccountthatthepricePj,twillprevailatdatet+hwithprobability9h.Theflexible-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.ThefiscalauthoritysetstaxesTtanddebtBttosatisfyitsbudgetconstraint:(1+it-1)Bt-1=Bt+Tt+rwtLt.TheexogenousinitiallevelofgovernmentdebtisgivenbyB-1.2.4OrderofEventsTheorderofeventsatadatet,givenadistributionofpricesPj,t-1,isasfollows:1.Flexible-pricefirmschoosepricePj,t=Pt*.Sticky-pricefirmschoosepricePj,t=Pj,t-1.2.Thecentralbankchoosesmonetarypolicy,i.e.theinterestrateit.4.Thefiscalauthoritychoosesfiscalpolicy,i.e.taxesTtanddebtBt.Observethatmonetarypolicyischosenafterfirmshavechosentheirprices.Hence,ifthecentralbankdeviatesfromequilibriumpolicy,firmswillnolongerbeoptimizingduringtheperiodofthedeviationofftheequilibriumpath.Thefactthatfiscalpolicyischosenattheendoftheperiodisforexpositionalsimplicityandwithoutlossgivenourequilibriumdefinitionwhichwedescribeinthenextsection.2.5EquilibriumDefinitionOursolutionconceptisMarkovPerfectCompetitiveEquilibrium(MPCE),implyingthathepresenceoflumpsumtaxesoureconomyfeaturesRicardianEquivalence,whichmeansthatthelevelofdebtBtcanbetreatedaspayoffirrelevantandthatattheendofeachperiodthefiscalauthoritychoosesdebtBt=0andtaxesTttobalanceitsbudget,bothonandofftheequilibriumpath.15Formally,letΩt-1correspondtothedistributionofpricesacrossfirmsatdatet_1.ConditionalonΩt-1,flexible-pricefirmschoosepricePt*whichdeterminesΩt.LetΓdenotecorrespondingmappingwithttGiventthecentralbankchoosesmonetarypolicyit=Ψ(Ωt),whereΨisthecentralbank’sreactionfunction.Finally,givenΩtandit,oseconsumptionlaborandsavingsCtLtBtsjtCjtjutitwhereuisthehouseholds’reactionfunction.Inequilibrium,Bt=0andsj,t=1,sincehouseholdscanbeidenticallytreatedwithoutlossofgenerality.AnMPCEisacollectionofmappings{Γ,Ψ,u}suchthat,ateverydatetandgiven{Γ,Ψ,u},themappingΓ(Ωt-1)satisfiesflexible-pricefirmoptimality,Ψ(Ωt)maximizessocialwelfare,andu(Ωt,it)satisfieshouseholdoptimality.