美聯(lián)儲-測度通貨膨脹與新凱恩斯模型 Measured In

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

MeasuredIn丑ationandtheNew-KeynesianModel

LawrenceJ.Christiano,MartinEichenbaum,andBenjaminK.Johannsen

2024-095

Pleasecitethispaperas:

Christiano,LawrenceJ.,MartinEichenbaum,andBenjaminK.Johannsen(2024).“MeasuredIn丑ationandtheNew-KeynesianModel,”FinanceandEconomicsDiscus-sionSeries2024-095.Washington:BoardofGovernorsoftheFederalReserveSystem,

/10.17016/FEDS.2024.095

.

NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.

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MeasuredIn?ationandtheNew-KeynesianModel*

LawrenceJ.ChristianoMartinEichenbaumBenjaminK.JohannsenNovember2024t

Abstract

Researcherstypicallycomparein?ationinthenewKeynesian(NK)modeltopublishedin?a-tionmeasuresconstructedfromindicesliketheCPI.In?ationinthestandardNKmodelwithoutpriceindexationisboundedabove.Themodelanalogueof?xed-weightin?ationmeasures,liketheCPI,isnot.Whenin?ationisintherangeofvaluesobservedafter2021,thereisasubstan-tialdi?erencebetweenmodel-basedand?xed-weightmeasuresofin?ation.This?ndingposesachallengetousinglinearapproximationstotheNKmodelinenvironmentswithmoderatelyhighin?ationandimpliesthatanalystsshouldconstructdata-consistentmodelanalogueswhenassessingtheNKmodel.

*TheviewsexpressedherearethoseoftheauthorsandnottheBoardofGovernorsoftheFederalReserveSystem,theFOMC,oranyoneelseassociatedwiththeFederalReserveSystem.

tChristiano:NorthwesternUniversityandNBER,l-christiano@,Eichenbaum:NorthwesternUni-versityandNBER,eich@,Johannsen:FederalReserveBoard:benjamin.k.johannsen@.

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1Introduction

Thestandardnew-Keynesian(NK)modelincorporatesnominalrigiditiesviaCalvo-stylefrictionsaccordingtowhich?rmsupdatepriceswithsomeexogenousprobability.TheCalvo(1983)modelimpliesthatin?ationcalculatedusingthemodel’sconsumptiongoodpriceindex(whichisanidealpriceindex)isbounded(seeAndreasenandKronborg(2022)).Thebounddependsontwokeymodelparameters:averagemarkupsandtheprobabilityofupdatingprices.The?rstparametercharacterizestheconstant-elasticity-of-substitution(CES)aggregatorinhouseholds’utilityfunc-tions.Thesecondparametercharacterizesthefractionofmonopolistconsumptiongoodproducersthatcannotadjusttheirpriceinanygivenperiod.

Weshowthattoa?rst-orderapproximation,thepriceoftheconsumptiongoodintheCalvo(1983)modelisthesameasa?xed-weightpriceindexwithequalweightsonallprices.Ina?rst-orderapproximation,thereisnoupperboundonthelevelofin?ationintheidealpriceindex.AspointedoutbyAndreasenandKronborg(2022),theboundfromthemodel’snonlinearequationsimpliesthata?rst-orderapproximationmustbreakdownforhighin?ationlevels.Toexaminehowhighin?ationhastobebeforea?rst-orderapproximationbreaksdown,weuserecentU.S.ConsumerPriceIndex(CPI)datatoshowthatthenonlineardynamicsofa?xed-weightindexcandi?ermeaningfullyfromthepercentchangeinthepriceoftheconsumptiongoodinthemodel.Ourresultso?erananalystonewaytoascertainwhenthe?rst-orderapproximationisnolongerreasonablyaccurate.Ourresultsalsoimplyanalystsusingnonlinearmodelsshouldconstructdata-consistentmodelanaloguestoassesstheNKmodelwhenin?ationisevenmoderatelyhigh.

OurresultsaredistinctfromaseparatecriticismoftheCalvomodel,whichisthatthepriceadjustmentfrequencychangesathighin?ationlevels.Itisanopenquestionhowhighthatlevelmustbebeforethemodelbreaksdown.Ouranalysisindicatesthat,dependingonparametervalues,theCalvomodelmaynotbeabletogeneratethatlevelofin?ationintheidealpriceindex.

OurresultsarerelatedtoKocherlakota(2024),whoanalyzesshort-rundi?erencesbetweenLaspeyresandidealpriceindicesinthestandardNKmodel.Byshort-run,Kocherlakota(2024)meansthatpeople’sexpectationsaboutfuturein?ationandrealmarginalcostsareheld?xedatsteadystatevalues.Incontrast,ouranalysisdoesnotdependonproductiontechnology,governmentpolicy,orevenrationalexpectations.Ourresultsarealsorelatedtothetrendin?ationliterature

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(see,forexample,Ascari(2004)).Theboundthatwestudydi?ersfromandcanbelowerthantheboundonsteady-statein?ationanalyzedinthatliterature,thoughtheboundwestudyalsoappliestosteady-statein?ation.

Theremainderofthispaperisorganizedasfollows.Section2presentsthestandardNKmodel.Section3presentsthein?ationboundsandcalculatesthemforparametersusedintheNKliterature.Section4relatesouranalysistothesteady-statein?ationboundofAscari(2004).Section5discussesthepropertiesofa?xed-weightpriceindexandcomparesittothemodel’sidealpriceindex.Wecomparethebehaviorofthesepriceindicessince2021forparametersusedintheNKliterature.Finally,section6containsconcludingremarks.

2ThestandardNKmodel

InthissectionwelayoutthestandardNKmodel(seeClaridaetal.(1999)).Arepresentativehouseholdmaximizesexpectedutility

,0<β<1,1<ε

whereCi,tdenotestheithconsumptiongoodandNtdenotesemployment.Thehousehold’stimetbudgetconstraintis

Pi,tCi,tdi+Bt≤Rt-1Bt-1+WtNt+Tt.(2)

Here,Pi,tdenotesthepriceofCi,t,Btdenotesholdingsofaone-periodrisk-freenominalbondwithgrossnominalreturn,Rt.Also,Wtdenotesthewagerate,andTtdenoteslump-sumpro?ts.

ThehouseholdpurchasesCi,tdirectlyfromamonopolistproducerofthatgood.Anecessaryconditionforhouseholdoptimizationis

Ci,t=-εCt,(3)

where

Here,PtdenotestheidealpriceindexandCtdenotestheutility-relevantconsumptionbundle.

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Monopolistsproducedi?erentiatedconsumptiongoods.Theithmonopolistsetsitsprice,Pi,t,subjectto(3),andtheCalvofriction.Inparticular,θrandomlyselectedmonopolistscannotchangetheirpriceandsetPi,t=Pi,t-1whiletheothermonopolistscanchangetheirprice.Wedenotetheresetpricebyt.1

Ouranalysisdoesnotrequirethatwespecifyafunctionalformofu,thetechnologyforproducingCi,t,thenatureofgovernmentmonetarypolicy,themonopolists’maximizationproblem,orevenhowEtin(1)isformed.Wedorequirethatusatisfytheusualpropertiesthatitisincreasingandconcaveinits?rstargument,decreasingandconcaveinthesecond,andcontinuouslydi?erentiable.Otherthanthat,weonlyusethede?nitionoftheidealindex,Pt,theCalvofriction,theassumptionthatallmonopoliststhatupdatetheirpricechoosethesameresetprice,andthefactthathouseholdsmaximize(1).

3Anupperboundonin?ationintheidealpriceindex

Letπt=Pt/Pt-1denotein?ationintheidealpriceindex.Inthissection,wederiveanupperboundonπt,asinAndreasenandKronborg(2022).Itfollowsfromthede?nitionofPtandtheCalvofrictionthat

Pt1-ε=(1-θ)t1-ε+θPt1-ε.(5)

Becauseε>1,Ptandπtareincreasingintheresetprice,t.ConsiderthelimitofPtast→∞.

Becauseε>1itfollowsthat(1-θ)t1-ε→0,soPt1-ε→θPt1-ε.Afterrearranging,weconclude

thatπt→

(6)

forallt.

AspointedoutbyAndreasenandKronborg(2022),theright-handsideoftheinequalityin(6)representsanupperboundonthelevelofperiodin?ationinanyequilibriumofthemodel.

Theupperboundarisesbecauset→∞meansthattheresetpriceisveryhigh,andhouseholds

shiftdemandawayfromthoseproductstowardlower-pricedgoods.Toseethis,notethattheideal

1Wemaintainthestandardassumptionthatallmonopoliststhatupdatetheirpricechoosethesameresetprice.

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priceindexcanbeexpressedasaweightedaverageovertheindividualconsumptiongoodprices.Combiningequations(3)and(4),weobtain

Pi;tdi.(7)

ThepreviousequationimpliesthattheweightonPi;tintheidealpriceindexistheratioofCi;ttoCt.ThatratioisadecreasingfunctionofPi;t.

Theupperboundin(6)dependsontheparametersθandε.Di?erentauthorsassumedi?erentvaluesfortheseparameters.So,wecalculatetheupperboundforπtinthestandardNKmodelforvariousvaluesofθandε,seeFigure1.Thehorizontalaxisofeachpanelcorrespondstothesteadystatemarkupimpliedbyε(whichisgivenbyε/(ε-1)-1)expressedinpercent.Theverticalaxiscorrespondstotheupperboundonπt,expressedinannualizedpercent.

FromFigure1weseethattheboundonπtisincreasinginthesizeofthesteady-statemarkup(lowerε).Highvaluesofthesteadystatemarkupmeanthatgoodsarelesssubstitutableforeachother.Astrises,householdsarelesswillingtosubstitutetowardlower-pricedgoodsandπtishigher.Notsurprisingly,thesame?gureshowsthattheboundonπtisincreasinginthe?exibilityofprices(lowerθ).

4Relationshiptoboundsonsteady-statein?ation

KingandWolman(1996),Ascari(2004),andAscariandSbordone(2014)arguethatthestandardNKmodel,inwhich?rmshaverationalexpectations,hasanupperboundonsteady-statepricein?ation.AscariandSbordone(2014)showthatforthe?rmproblemtobewellde?nedinsteadystate,itmustbethatβθπε<1andβθπε-1<1.2Theseconditionsimpliesthat

Dependingonparametervalues,theboundonπin(8)maybehigherorlowerthantheboundon

2KingandWolman(1996)arguethatiftheboundontrendin?ationisnotsatis?edthentheprobabilityofnotadjustingpricesishighenoughthat?rmswouldrathernotoperatethanriskhavingtheirpricebelowforaverylongtime.Ouranalysis,(6),doesnotdependonoptimalpricingby?rms.

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πtgivenin(6).Because(6)musthold,theactualboundonπisgivenby

Inthedata,thelevelandvolatilityofin?ationareoftenpositivelycorrelated(seeBalletal.(1988)).So,eveniftheaveragelevelofin?ationsatis?es(6),theremaybelongperiodsinwhichmeasuredin?ationexceedsthebound.

5Measuredin?ation

MostresearchersusingNKmodelscomparethepercentchangeintheidealpriceindextomeasuredin?ation.However,measuredpriceindicesarenotperfectanalogstotheidealpriceindexinthemodel.Indeed,overshorthorizons,measuredpriceindices–liketheCPI–arewellapproximatedbya?xed-weightindex.Inthissection,weanalyzea?xed-weightpriceindexinthestandardNKmodelusingequalweightsontheintermediategoodstoconstructthepriceindex.Ourassumptionofequalweightscorrespondstousingthetime-seriesaverageofweightsplacedontheintermediategoodsintheidealpriceindex.WeusedataoncoreCPIin?ationasthedataanalogueforthis?xed-weightindexandshowthatitcandi?ermeaningfullyfromin?ationintheidealpriceindexwhencoreCPIin?ationismoderatelyhigh.3

Wede?netheequal-weightpriceindex,Zt,by

IncontrasttoPt,Zt→∞ast→∞becausetheweightsonhighpricesdonotdeclineasconsumerssubstitutetowardlower-pricedgoods.Letztdenotetheratioofthe?xed-weightpriceindexandtheidealizedpriceindex(Zt/Pt).Inaddition,letζtdenotein?ationinthe?xed-weightpriceindex

3Bycontrast,AndreasenandKronborg(2022)usecoreCPIin?ationasthedataanalogueoftheNKmodel’sidealpriceindex.

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(Zt/Zt-1).Then

(11)

(12)

t=t/Pt.Equations(11)and(12)implythatthereisnoupperboundonζt.4

Itisstraightforwardtoshowthatina?rst-orderapproximationtothemodelwithzerosteadystatein?ation,t=0andt=t.5However,inthenonlinearversionofthemodelorinlinearizedversionsofthemodelwithpositivesteady-statein?ationtherecanbesubstantialdi?erencesbetweenζtandπt.

WetakecoreCPIin?ationasthedatacounterparttothe?xed-weightindexdiscussedabove(Zt).6Weusethenonlinearequations(5),(11),and(12)toconstructatimeseriesforπt.7Theblack,solidlineinFigure2correspondstothecoreCPIin?ationratefor2021:Q1through2024:Q3.Figure2alsodisplaysthein?ationrateintheidealpriceindexfordi?erentvaluesofθandε.The?gureshowsthattheseriesforπtdi?ersmorefromtheCPIin?ationratewhentheboundonπtislower.Recallthattheboundonπtincreasesinthemarkupanddecreasesinθ.Moreover,thedi?erenceoverthepastfewyearscanbesubstantialforvaluesofθandεusedintheliterature.

6Conclusion

Wearguethat,especiallyovershorthorizons,in?ationina?xed-weightpriceindexisaclosermodelanaloguetomeasuredin?ationthanin?ationintheNKmodel’sidealpriceindex.Weshowthatduringaperiodofmoderatelyhighin?ation,in?ationinmodel-based?xed-weightpricesindicesmaydi?ermarkedlyfromthemodel-basedmeasureofin?ation(usingtheidealpriceindex)

5Here,tdenotesthelogdeviationofxtfromitssteady-statevalue.Tounderstandthisresult,notethatt=

4Though,thereisalowerboundequaltoθ.~

-t.Thenequation(11)impliest=0(solongas0=0)andequation(12)im~pliesthatt=t.Forthereto

beawell-de?nedsteadystate,itmustbethattheboundonπissatis?edsothatpiswellde?ned.Whenπ>1,it

isstraightforwardtoshowthatz>1andζ=π.Intuitively,becauseZtislar~gerthanPtinsuchasteadystate,the

in?ationrateinbothindicesisthesameforthesteadystaterealresetprice,p,eventhoughtheweightsonthereset

pricearedi?erentinthetwoindices.

6U.S.BureauofLaborStatistics,ConsumerPriceIndexforAllUrbanConsumers:AllItemsLessFoodandEnergyinU.S.CityAverage[CPILFESL],retrievedfromFRED,FederalReserveBankofSt.Louis;

/series/CPILFESL

,November7,2024.

7Wesetzt=1in2020:Q4.Ourresultsarelittlechangedifwesetztin2020:Q4equaltoitssteadystatevalueundertheassumptionof2percentsteady-statein?ation.

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typicallyusedtoempiricallyassessNKmodels.OuranalysisindicatesthatresearcherswhousethestandardNKmodelwhenin?ationismoderatelyhigh(asitwasinthepost-Covidperiod)shouldusemodel-basedmeasuresofin?ationthatareconsistentwiththewayin?ationismeasuredinthedata.

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