IMF-全球價值鏈與通貨膨脹動態(tài)-2024-03-宏觀大勢

VuChau,MarinaConesaMartinez,TaehoonKim,JohnSprayWP/24/62IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicitcommentsandtoencouragedebate.TheviewsexpressedinIMFWorkingPapersarethoseoftheauthor(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement.2024?2024InternationalMonetaryFundWP/24/62IMFWorkingPaperResearchandWesternHemisphereDepartmentsGlobalValueChainsandInflationDynamicsPreparedbyVuChau,MarinaConesaMartinez,TaehoonKim,andJohnSpray*AuthorizedfordistributionbyNanLiandCeydaOnerMarch2024IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicitcommentsandtoencouragedebate.TheviewsexpressedinIMFWorkingPapersarethoseoftheauthor(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement.ABSTRACT:Westudytheinflationaryimpactsofpandemiclockdownshocksandfiscalandmonetarystimulusduring2020-2022usinganovelharmonizeddatasetofsectoralproducerpriceinflationandinput-outputlinkagesformorethan1000sectorsin53countries.TheinflationaryimpactofshocksisidentifiedviaaBartikshift-sharedesign,wheresharesreflecttheheterogeneoussectoralexposuretoshocksandarederivedfromamacroeconomicmodelofinternationalproductionnetwork.Wefindthatpandemiclockdowns,andsubsequentreopeningpolicies,werethemostdominantdriverofglobalinflationinthisperiod,especiallythroughtheirimpactonaggregatedemand.Weprovideadecompositionoflockdownshockbysources,andfindthatbetween20-30percentofthedemandeffectoflockdown/reopeningisduetospilloverfromabroad.Finally,whilefiscalandmonetarypoliciesplayedanimportantroleinpreventingdeflationin2020,theireffectsdiminishedintherecoveryyears.RECOMMENDEDCITATION:Chau,V.,ConesaMartinez,M.,Kim,T.,&Spray,J.(2024).GlobalValueChainsandInflationDynamics.IMFWorkingPapers,2024/62.JELClassificationNumbers:E31;F10;F41Keywords:Inflation;globalvaluechains;network;pandemic;spillovers.Author’sE-MailAddress:vchau@;mconesamartinez@;tkim@;jspray@.feedback*AllauthorsareaffiliatedwiththeInternationalMonetaryFund.TheviewsinthispaperarepersonalviewsoftheauthorsanddonotnecessarilyrepresentthoseoftheIMF,itsExecutiveBoard,orIMFManagement.WethankChrisPapageorgiou,CeydaOner,PrachiMishra,RodrigoValdés,FilizUnsal,AdrianPeralta-Alva,RuiMano,Niels-JakobHansen,YanCarriere-Swallow,AlasdairScott,AbdoulWane,CianRuane,MarcoArena,andparcipantsattheIMFAPD,WHD,RESDMSeminarSeriesforthoughtfulcommentsand.feedbackPreparedbyVuChau,MarinaConesaMartinez,TaehoonKim,JohnSpray21IntroductionTheriseofglobalinflationaftertheCOVID-19pandemiccaughttheworldbysurpriseandpushedinflationtoitshighestlevelsindecades.Whiletraditionaldriversofinflation,suchasloosefiscalandmonetarypolicies,havecontinuedtobecitedasthecausesofhighinfla-tion,therecentinflationaryperiodhasalsocalledintoquestionsthevulnerabilitiesoftheglobalvaluechains(GVCs)todisruptions(causedbylockdowns,disasters,andwars)anditsroleintransmittingshocksacrosscountries.1Toproperlydrawlessonsfromthisinflation-aryperiodandensurecostlymistakesarenotrepeatedinthefuture,itisimportantthentoknowtherelativeimportanceofdifferentinflationdrivers.Inthispaper,weprovideanovelharmonizeddatasetofsectoralproducerprice,mergedwithinput-outputlinkagesandshocks,formorethan1000sectorsin53countries,andaframeworktoidentifyinflationaryimpactofdifferentinflationdriversusingthesectoraldata.Identifyingtheinflationaryimpactofshocksisempiricallychallenging.First,aggre-gateconsumerpriceindicesarefunctionsofmanyconcurrentshocks–demandandsupply,domesticandforeignspillovers–andrelyingonlyonthisdatawouldnotprovidesufficientstatisticalpowerforidentification.ThisisevidentintheUSinflationdebate,wherediffer-entpolicymakersandscholarsattributethehighinflationratetodifferentdrivers.Second,thesameshockcanappeartobebotha“supplyshock”anda“demandshock”thataffectinflationinopposingways.Lockdownpoliciescanbeanegativesupplyshockthatincreasesthepricelevelaslimitedproductiondepletesinventoriesandmakesgoodsscarce.Atthesametime,lockdownscanbethoughtofanegativedemandshockthatdepressesprices,asworkersreceivelowerincomeanddemandlessgoodsandservices.2Furthermore,lock-downscanbepredominantlyademandshockforonecountryandasupplyshockforanother,manifestingthroughhigherimportedinputprices.Thus,studyingtheinflationaryimpactofpandemiclockdownrequirestakingaholistic,structuredviewofhowlockdowncanaffecttheunitsofobservationofinterest.Inthispaper,wetrytoovercometheaforementionedempiricalchallengesandstudythedriversofglobalinflationusingasetofBartik(1991)shift-shareinstrumentsderivedfromageneralequilibriummodelwithinternationalproductionnetwork.Ourempiricalworkhastwocomponents.First,wecollectandharmonizePPIinflationdataforacross-sectionof1134internationalproductionsectorsduringtheperiod2020-2022,ensuringconsistencywithinternationalinput-outputdataandotherrelevantdatasets.Second,weconstructBartikinstrumentsofinflationdrivers,leveragingthelargechangesinlockdownandfis-calpoliciesworldwideduringthisperiodasshiftsandthedifferentialdegreesofsectoralexposuretoshocksthroughthetradenetworkasshares.The“networkexposureshare”isderivedfromamacroeconomicmodelofinternationalproductioninanetworkedsupply1SeeElliottandGolub(2022);Carvalhoetal.(2021);AcemogluandTahbaz-Salehi(2020)2SeeGuerrierietal.(2022)forhowashockcanbebothsupplyanddemandinamulti-sectoreconomy.3chain,whichcloselyfollowsBaqaeeandFarhi(2019).Inparticular,identificationoftheimpactofsupply-shocksreliesonthedifferentialuseofinputsinproduction:whenasectorinaparticularcountryenterslockdown(e.g.mi-crochipsinThailand),adownstreamsectorthatusesthoseinputsmoreextensively,eitherdirectly(e.g.automobileproductioninMexico)orindirectly(e.g.,automobileretailintheUS),shouldseealargerincreaseininflation.Symmetrically,identificationoftheimpactofdemand-shocksreliesonthedifferentialdemandforproducts:alockdownintheUSwilldirectlyimpactdemandforautomobilesfromMexico,andindirectlyimpactdemandformi-crochipsinMalaysiabutwillhaveanegligibleimpactonsectorswithsmallbilateraltradeshares(e.g.tobaccoproductionfromCuba).Theseparateidentificationofthesupplyeffectversusthedemandeffectoflockdownscomesfromthefactthataproductionsectorinaparticularcountrymightimportintermediategoodsfromgeographicallocationsthatdifferfromwheretheyexport.Usingtheseinstruments,werunLocalProjection(LP)regressionsàlaJordà(2005)tostudythecontemporaneousanddynamiceffectsoflockdownsandfiscalstimulionproducerprices.Theaimofthebaselineexerciseistostudytheimpulseresponsesofvariousdriversofinflation.Asanextension,welookatheterogeneityoftheseeffectsviaallowingtheimpulseresponsestodifferdependingonthestageofthepandemic,aswellasown-versusnetwork-effects.Ourmainresultsarethefollowing.First,lockdown(andreopening)policiesarestrongandpersistentdriversofglobalinflation,boththroughsupplyanddemandchannels.Ourestimatesimplylargeeconomicimpacts.Onthesupplyside,thefulllockdownofamajorsupplierthataccountsfor50%oftotalexpenditureraisesasector’sPPIby14.5%.Onthedemandside,afulllockdownofaneconomythataccountsfor50%ofasector’stotaloutputdepressesitsPPIby15%.Theinflationaryeffectsoflockdownarelargeandpersistent,peakingafter3quartersanddissipatingonlyafterayear.Allofourspecificationsincludeafullsetofcountry,sectorandtimefixedeffects.Theresultsarequalitativelyinvarianttocontrollingforlaggedinflation,changesintheexchangeratevis-a-vistheUSdollar,andtransportationcosts.Second,fiscalstimuluspackagesduringthepandemicsalsohadsignificantinflationaryimpact.Ourestimatespointtoafiscalmultiplierof0.75,i.e.aonedollarincreaseinfiscalspendingwouldincreasedomesticdemandbyabout75centsonimpact.Theinflationaryimpactonasector’sproducerpricedependsonits"exposure"totheeconomyreceivingthestimulus.Toillustratethepotentialmagnitude,considerastimuluspackageworth10%ofGDPandasectorthatsells50%ofitsvalueaddeddomestically.Theinflationaryimpactofthefiscalstimulusonthissectorwouldbe0.75×10%×0.5=3.8%.However,fiscalshockstendtobelesspersistentcomparedtolockdownshocks,peakingonimpactandquicklydissipatingafter2quarters.Third,inrelativecontribution,wefindthatpandemiclockdowns,andsubsequentre-4openingpolicies,werethemostdominantdriverofglobalinflationinthisperiod,espe-ciallythroughtheirimpactonaggregatedemand.Whilefiscalandmonetarypoliciesplayedanimportantroleinpreventingdeflationin2020,theireffectsdiminishedintherecoveryyears.Fourth,theinflationaryimpactsofshocksaredifferentinthebeginningversusrecoveryphaseofthepandemic.Inparticular,wefindthatthesupplydisruptionsduetolockdownweremoresevereandlong-lastingatthebeginningofthepandemicthaninthethere-coveryphase.Bycontrast,thedemandeffectoflockdown/reopeningwasthesameinbothphases.Thissuggeststhepossibilitythatfirmswereabletoadapttorisingcostsduetosupplydisruptionsduringtherecoveryphase,e.g.byswitchingtoalternativesources.Forfiscalpolicies,wefindthatfiscalstimulushadamuchmorepersistentinflationaryeffectintherecoveryphase,whichsuggeststhatwhilefiscalpackageswerehelpfulinsupport-ingbusinessesandworkerswithoutalong-lastinginflationaryimpactatthetroughofthepandemic,continuingtooverheattheeconomywithfiscalstimuluswhentheeconomyhasrecoveredcouldleadtoverypersistentinflation.Finally,wefindthatglobalsupplychainswereimportantpropagationchannelsofinter-nationalshockstoproducerprices.Inthesubsequentanalysis,wedecomposetheimpulseresponseintotheonedrivenbythenetworkeffectanditsowneffect.Thenetworkeffect–theimpactofshockstoconnectedsectors–accountsfor50%ofthetotaleffectfor“supplyshocks”andalmosttheentireeffectof“demandshocks.”Thisresultsuggeststhattheproductionnetworkisintegraltotheunderstandingofinternationalspillovereffectsofashocksuchasthepandemic.Ouridentificationleveragesheterogeneousexposureofdifferentindustriestoshocks.Theidentificationassumptionisthattheexposureweights(pre-COVIDnetworksupplyanddemandshares)donotpredictchangesinsectoralinflationratesthroughchannelsotherthanthenetworkofintermediateinputtradesaslaidoutinourtheory.3Whileas-sumingthatthenetworktradeshares,whichareendogenousobjects,areuncorrelatedwiththelevelsofsectoralinflationratesmightnotbeplausible,assumingthatthesharesareuncorrelatedwiththechangesininflationratesisamuchweakerassumption(Goldsmith-Pinkhametal.,2020).TheconstructedBartikinstrumentsareappliedasinstrumentalvariablestotheLocalProjectionframework(seeJordà(2023)foraliteraturesummary).Aviolationoftheidentificationassumptionwouldoccur,forexample,iftwocountrieshavehighnetworksupplyanddemandsharesduetogeographicalproximity;however,beingclosealsomeansthatCOVIDinfections—andtheconsequentialsupplydisruptions—canspreadacrossborders.Arelatedconcernisthattradeshareswerecorrelatedwithtransportcostshocksinducedbythepandemic.ThiscouldoccurifCOVIDinfectionsarepassedalongtraderoutesandleadtodisruptionstotransportationinfrastructure.Toaddressbothof3Moreformally,weassumethattheinitialperiodsharesarenotcorrelatedwiththestructuralerrorterms,i.e.,changesintheoutcomevariable(PPIinflation).5thesethreatwerunrobustnesstestscontrollingforthelevelandchangeofCOVIDinfectionsineachcountry.Wealsocontrolforroute-sector-specificmaritimetransportcosts.Todothisweinteractmonthlyroute-specificmaritimefreightcostsfromFreightosBalticContainerPriceIndexwithameasureofsectorspecificexposuretochangesintransportationcostsusingdatafromtheOECDMaritimeTransportCostdatasetwhichcontainsbilateraladvaloremmaritimetransportcostsfor43importingcountriesfrom218countriesoforiginataproductlevel.Inallinstances,theresultsarerobusttothesecontrolsandboththesignandmagnitudeofcoefficientsonthemainexplanatoryvariablesremainlargelyunchanged.Theseresultshaveimportantimplicationsforourunderstandingofinternationalspilloversofpolicythroughglobalsupplynetworks.Firstly,wecontributetotheinflationdebatebyshowingthattheriseininflationacrosstheworldisindeedmostlydemand-based.How-ever,contrarytoconventionalwisdomwefindthisdemandisstimulatedviaageneralre-openingofeconomiesinsteadoffiscalstimuli.Secondly,inaworldwhichisincreasinglysubjecttosupply-shocksandwiththethreatofeconomicfragmentationourresultsindicatethatpolicymakerswilla)needtoconsidertheindirectimpactsofeconomicpoliciesinma-jortradingpartners,b)shouldinternalizetheglobalimpactoftheirowneconomicpolicythroughsupply-networks,andc)shouldincentivizethebuildingofresilientsupply-chains,tominimizetheimpactofshocks.OurpaperisinthespiritofAcemogluetal.(2016)andBaqaeeandFarhi(2019),whichstudythetransmissionofmacro-economicshocksthroughinput-outputproductionlinkages.Weextendthisframeworktotheinternationalsettingandtothestudyofinflationdynam-ics.Theuseofthestructuralmodelhereisnecessary:aswearestudyinghundredsofdifferentshocks(lockdown,fiscal,monetaryshocksfor60+countries),themodelprovidesaparsimoniousandintuitivewaytocombinethoseshocksintoafewBartikinstruments.Furthermore,themodelalsoallowsustodecomposeashock,likelockdown,intodifferentpropagationchannels.Anotherclosely-relatedpaperisDiGiovannietal.(2023),whichalsostudiesthedriversofpandemic-erainflationusingamulticountrymodelwithinput-outputnetwork.Themodelinthatpaperisslightlyricherbutiscalibratedonlytofourlargecountries/areas,foregoingtherichnessofcross-countrygranulardata.OurpaperinsteadreliesonthemodeltoconstructtheBartikinstruments,appliedtomanymorecountriesandsectors.OurpapershedslightonsupplychaindisruptionscausedbytheCOVID-19lockdowns.Recentpapersintheliterature,whichfocusonthesupply-chainissuesofthepandemic,includeBonadioetal.(2021),MeierandPinto(2022),BaqaeeandFarhi(2022),Lafrogne-Joussieretal.(2022),Carrière-Swallowetal.(2023)andAlessandriaetal.(2023).Lau-renceBallandMishra(2022)decomposesthepost-pandemicinflationtothesupply-sideanddemandsidefactors.Onthetheoryfront,BenignoandEggertsson(2023)andHardingetal.(2023)attempttoexplainthenon-linearityofthePhillipsCurvethatwasobservedduringthepandemicperiod.Ourpaperalsocontributesalargeempiricalliteratureoninflation.6Inparticular,ourpaperrelatestoagrowinginterestintheidentificationofinflationdriversviacross-sectionalvariations.Forexample,FitzgeraldandNicolini(2014)andMcLeayandTenreyro(2019)showthatusingregionaldatahelpsdisentangledemand-andsupply-sideshockstoinflation.OtherpaperssuchasBerajaetal.(2019)andHazelletal.(2022)makeuseofstate-levelorcity-levelpriceindexesintheUStoestimatetheslopeoftheregionalPhillipsCurve.Debetal.(2023)constructedanovelmonetarypolicyshocksdatabasetostudythedynamicsofthetransmissionofmonetarypolicyoninflationandoutput,andthecountrycharacteristicsthatdrivethem.Ourpaperbuildsonaliteratureusingshift-sharedesignstoidentifytheimpactofmacroeconomicshocks.Buildingfromseminalappliedworkusingdifferentialexposuretoshockstoidentifycausalrelationships(Bartik,1991;Card,2001;Autoretal.,2013),are-centbodyofliteraturehasmadeadvancesinunderstandingbias,inferenceandaggregationinshiftsharedesigns(Jaegeretal.,2018;Adaoetal.,2019;BroxtermanandLarson,2020;Goldsmith-Pinkhametal.,2020;Borusyaketal.,2022).Ourpapermostcloselyrelatestoaliteratureusingdifferentialexposuretoforeignmarket(Hummelsetal.,2014;Bermanetal.,2015;Aghionetal.,2018)andwheredifferentproductsareexposedtodifferenttypesofconsumers(Jaravel,2019).Ourpaperusesamodelofinternationalproductionnetworkstoshowasector’sexposuretodemandandsupplyshocksaredeterminedbytheimpactofup-anddown-streamsectorsdirectandindirectexposuretotheshock.Usingthisinsight,webuildashift-shareem-piricalframeworkwhichallowsustoseparatelyidentifytheimpactofdemandandsupplyshocksonPPIinflationandtoindependentlyidentifythedirecteffectfromindirectnetworkeffects.Unlikethepreviousworksthatintegrateinput-outputlinkagestoacalibratedNewKeynesianmodel(Rubbo,forthcoming;La’OandTahbaz-Salehi,2022),ourfocusisoncon-structinganempiricalspecificationthatenablesmeasuringcross-sectionalvariationsofashock.Theremainderofthepaperisorganizedasfollows.Section2presentsamodel.Section3developsanempiricalmethodologyandpresentsdatadescription.Section4discussesthebaselineresults.Section5shedslightonadditionalissuesincludingidentificationissues,robustnesschecksandgeographicvariations.Section6concludes.2TheoreticalFrameworkInthissection,wewritedownastandardmulti-countrymodelfeaturingaglobalinput-outputnetworkandderivetheinternationalinflationaryresponsetovarioussupplyanddemandshocksinclosed-form.Themodel-basedelasticitiesofcountryandsector-levelin-flationratestoshocks,whicharefunctionsoftheunderlyingproductionnetwork,willserveasthebasisfortheconstructionofourempiricalBartikinstruments.72.1EnvironmentOurworldeconomyhasNcountries,eachwitharepresentativeconsumerwithsomepref-erenceui(Ci,Li)overconsumingacountry-specificfinalgood,denotedbyCi,andprovidingLiunitsoflabor.ThereareNfinalconsumptiongoodswhichareaggregatorsofSinternationallypro-ducedintermediategoodsunderaconstantreturnstoscale(CRS)technology:Di=Di(Di1,...,DiS)(1)ThefinalgoodoutputDiisequaltototaldomesticdemand,whichincludesprivateandpublicconsumption:4Di=Ci+Gi(2)LetQidenotethepriceoffinalgoodi(consumerpriceindex).Asasimplewaytoan-alyzedemandeffectoffiscalandmonetarypoliciesoninflation,weassumethatnominalpublicconsumptioni=QiGiisgivenexogenouslyandfinancedviaalump-sumtaxonhouseholds,whileprivateconsumptionCi=QiCiisachoicevariablesetbythemonetaryauthority.5Eachintermediategoodsrepresentsaspecificcountry-sector–forexample,Koreanchem-icalmanufacturing–inthedata.Anintermediategoodonceproducedcanbeusedeitherfortheproductionoffurtherintermediategoodsorforfinalconsumption.GoodsE{1,...,S}isproducedusingdomesticvalueaddedLsandotherintermediateinputsunderaCRStech-nology:Ys=Fs(ZsLs,{Xsk}=1）(3)whereLsandXskaretheamountsoflaborandintermediategoodkusedinsectors,re-spectively.Zsisthevalueadded-augmentedproductivityinsectors.6Intermediategoodscanbeimportedfromanothercountry.Laborisperfectlymobilebetweendifferentsectorswithinacountrybutcannotmoveacrosscountries.WeassumetheLawofOnePrice,whichmeansthepriceofintermediategoodsisthesameinallcountries.4Ourframeworkdoesnotincludeinvestmentandthusmightnotadequatelycaptureintertemporalchan-nelsofdemandandsupplyeffectsoninflation(forexample,ifanincreaseininvestmentthisperiodraisesdemandandinflationtoday,butwillleadtomoregoodsbeingavailableandthuscheaperpricesinthelongerterms).5Onfiscalpolicy,wefollowAcemogluetal.(2016)inusingexogenousvariationindomesticdemand(ourtermGi)asatooltotraceoutthenetworkimpactofdemandshock.Onmonetarypolicy,theassumptionthatthemonetaryauthoritydirectlysetnominaldemand,whichwefollowLucas(1973),mightappearad-hoc,butonecangiveamodelwithmoremicro-foundation,e.g.withcash-in-advanceconstraints,andbackoutthepathofmoneysupplyorinterestratethatwouldimplementthedesiredconsumptionpath.6Theassumptionthatproductivityisvalueadded-augmentingiswithoutlossofgenerality.ThemodeleconomyisidenticaltoonewithonlyHicks-neutralTFP,withvalueaddedbeingfirstproducedinonesectorbeforebeingsuppliedtotheintermediate-goodproducingsectors.8Themarketclearingconditionsforintermediategoodsandlaboraregivenby:Dis+Xks=Ys,s=1,2,...,S(4)ΣLs=Li,i=1,2,...,N(5)sEyiwhereyidenotethesetofintermediategoodsproducedincountryi.2.2EquilibriumInflationLetWsbethewagethatsectorshastopayforitslabor,andPsthepriceofgoods.Through-outthepaper,weuselowercasenotationstodenotethelogofcorrespondinguppercasevariablesx=log(X),andhatsindicatedeviationfromthesteadystate:三x一x.Thecostminimizationprobleminsectorsis:MCs(Zs,Ws,{Pk}=1)=L,=1{WsL+PkXk}s.t.⑨s(ZsL,{Xk}=1）=1(6)ApplyingShephard’sLemma,weobtainawell-knownoptimalequilibriumconditionthattheexpenditureshareofaninputinmarginalcostisequaltotheoutputelasticityofthatinput:=?logMCs?logWs=WsLsMCsYs=?logMCs=?logPkPkXskMCsYsWedenotebyθsandωsktheshareofvalueaddedandintermediateinputkintotalcost,re-spectively.Thesesharesarekeyobjectsthatdescribeinput-outputrelationshipsandsatisfyθs+Σkωsk=1byconstruction.Usingthisequilibriumcondition,wecanapproximatethemarginalcostfunctionofeachsectortothefirstorder:dlogMCs=?s·(dlogWs一dlogZs)+?s·dlogPk=θs(dlogWs一dlogZs)+ωskdlogPk.Theinputsharesareevaluatedatthepointofapproximation,whichcanbetheinput-outputmatrixofaparticularyearoritslong-termaverage.Weassumethatmarkupistime-invariant,sothatchangesinmarginalcostsarelinkedone-to-onewithchangesin9prices:7dlogPs=θs(dlogWs?dlogZs)+：ωskdlogPkkEquation(7)canbewritteninmatrixform:dlogP=Θ(dlogW?dlogZ)+ΩdlogPwhereΘ=({θs}s)andΩ=({ωsk}s,k).SolvingforequilibriuminflationdlogP,weget:dlogP=[I?Ω]?1Θ(dlogW?dlogZ)Equation(9)hasnotyetdescribedinflationasafunctionofexogenousdrivers,butisnonethelessusefultoexplainthemechanismsatplay.TheequationsaysthatthePPIinfla-tionrateforintermediategoods,dlogP,isaweightedaverageofthechangesinvalueaddedunitcost,dlogW?dlogZ,acrossallsectors.8TheweightthatsectorsplacesonsectorkisgivenbytheLeontiefinverse[I?Ω]?1,scaledbythevalueaddedshare,anddescribestheimportanceofsectorkasanupstreaminputproducer.Ifsectorsdoesnotuseintermediateinput,ithasa100%weightonitsownvalueadded,anditsoutputpriceisonlyaffectedbythevalueaddedcostinitsownsector.However,asectorthatusesmoreintermediateinputswillalsobeaffectedbyinputpricechanges,whicharedependentoncostconditionsintheupstreamsectors.Inequation(9),wagesareendogenousobjectsanddeterminedbyunderlyingexogenousshocks.Inaclosedeconomy,wecanthinkofwagesasbeingthedirectchoicevariableformonetarypolicy(Baqaeeetal.(2021)).Inaninternationalsetting,wageswillbedeterminedbyinternationalshocksandpoliciesaswell,soitisimportanttocaptureinternationallinkages.Foreaseofexposition,weassumethatintheshortrun,laborisperfectlyinelasticwithineachsector:Ls=Ls.9IftheproductionfunctionisCobb-Douglaswithrespecttolabor,7Inthecaseofvariablemarkup,theequilibriumconditionsimplybecomes:dlogPs≈dlogμs+(1?γs)dlogWs+：γsωskdlogPkkOnecanformulateamodelofvariablemarkupandstudyitsimplicationforinflation,whichwedonotattemptinthispaper.8Wecanshowthatthepre-multipliedmatrix(I?Ω)?1Θinequation(9)isindeedaweightmatrix.Let1denoteavectorof1.Wehave:[I?Ω]?11=Ωi)1=1+Ωi1=1+Ωi(I?Θ)1=1+[I?Ω]?1(I?Θ)1wherewehaveusedtheidentityΩ1=I?Θ.Re-arranging,wehave[I?Ω]?1Θ1=1,i.e.,eachrowofthematrix[I?Ω]?1Θsumstoone.9Wealsohavegeneralresultsforthecaseofendogenouslaborsupply,wherethewageresponsewillbejointlydeterminedbythelabordemandandsupplycurves,whichrequiremorestructuralassumptionssuchthewagebillisalwaysaconstantfractionofnominaloutput:WsLs=(1?γs)s,wheres=PsYs.Inthisspecialcaseofperfectlyinelasticlaborsupply,fluctuationsinnominaloutputtranslateone-to-oneintonominalwagechanges:dlogWs=dlog(s).Finally,werelatenominaloutputtoexpenditures.Nominaloutputorsalesofasec-torisequaltothespendingonthatgoodfromallintermediategoodpurchasersandfinalconsumers:s=is+ks(10)whereweremindthatDnotatestotaldomesticdemand,whichincludesprivateandpub-licconsumption.Let工is≡is/ibetheshareofgoodsincountryifinalconsumptionexpenditure.Wecanrewritetheequationaboveinmatrixform: