美联储-测度通货膨胀与新凯恩斯模型 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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MeasuredInflationandtheNew-KeynesianModel*

LawrenceJ.ChristianoMartinEichenbaumBenjaminK.JohannsenNovember2024t

Abstract

ResearcherstypicallycompareinflationinthenewKeynesian(NK)modeltopublishedinfla-tionmeasuresconstructedfromindicesliketheCPI.InflationinthestandardNKmodelwithoutpriceindexationisboundedabove.Themodelanalogueoffixed-weightinflationmeasures,liketheCPI,isnot.Wheninflationisintherangeofvaluesobservedafter2021,thereisasubstan-tialdifferencebetweenmodel-basedandfixed-weightmeasuresofinflation.ThisfindingposesachallengetousinglinearapproximationstotheNKmodelinenvironmentswithmoderatelyhighinflationandimpliesthatanalystsshouldconstructdata-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-stylefrictionsaccordingtowhichfirmsupdatepriceswithsomeexogenousprobability.TheCalvo(1983)modelimpliesthatinflationcalculatedusingthemodel’sconsumptiongoodpriceindex(whichisanidealpriceindex)isbounded(seeAndreasenandKronborg(2022)).Thebounddependsontwokeymodelparameters:averagemarkupsandtheprobabilityofupdatingprices.Thefirstparametercharacterizestheconstant-elasticity-of-substitution(CES)aggregatorinhouseholds’utilityfunc-tions.Thesecondparametercharacterizesthefractionofmonopolistconsumptiongoodproducersthatcannotadjusttheirpriceinanygivenperiod.

Weshowthattoafirst-orderapproximation,thepriceoftheconsumptiongoodintheCalvo(1983)modelisthesameasafixed-weightpriceindexwithequalweightsonallprices.Inafirst-orderapproximation,thereisnoupperboundonthelevelofinflationintheidealpriceindex.AspointedoutbyAndreasenandKronborg(2022),theboundfromthemodel’snonlinearequationsimpliesthatafirst-orderapproximationmustbreakdownforhighinflationlevels.Toexaminehowhighinflationhastobebeforeafirst-orderapproximationbreaksdown,weuserecentU.S.ConsumerPriceIndex(CPI)datatoshowthatthenonlineardynamicsofafixed-weightindexcandiffermeaningfullyfromthepercentchangeinthepriceoftheconsumptiongoodinthemodel.Ourresultsofferananalystonewaytoascertainwhenthefirst-orderapproximationisnolongerreasonablyaccurate.Ourresultsalsoimplyanalystsusingnonlinearmodelsshouldconstructdata-consistentmodelanaloguestoassesstheNKmodelwheninflationisevenmoderatelyhigh.

OurresultsaredistinctfromaseparatecriticismoftheCalvomodel,whichisthatthepriceadjustmentfrequencychangesathighinflationlevels.Itisanopenquestionhowhighthatlevelmustbebeforethemodelbreaksdown.Ouranalysisindicatesthat,dependingonparametervalues,theCalvomodelmaynotbeabletogeneratethatlevelofinflationintheidealpriceindex.

OurresultsarerelatedtoKocherlakota(2024),whoanalyzesshort-rundifferencesbetweenLaspeyresandidealpriceindicesinthestandardNKmodel.Byshort-run,Kocherlakota(2024)meansthatpeople’sexpectationsaboutfutureinflationandrealmarginalcostsareheldfixedatsteadystatevalues.Incontrast,ouranalysisdoesnotdependonproductiontechnology,governmentpolicy,orevenrationalexpectations.Ourresultsarealsorelatedtothetrendinflationliterature

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(see,forexample,Ascari(2004)).Theboundthatwestudydiffersfromandcanbelowerthantheboundonsteady-stateinflationanalyzedinthatliterature,thoughtheboundwestudyalsoappliestosteady-stateinflation.

Theremainderofthispaperisorganizedasfollows.Section2presentsthestandardNKmodel.Section3presentstheinflationboundsandcalculatesthemforparametersusedintheNKliterature.Section4relatesouranalysistothesteady-stateinflationboundofAscari(2004).Section5discussesthepropertiesofafixed-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-sumprofits.

ThehouseholdpurchasesCi,tdirectlyfromamonopolistproducerofthatgood.Anecessaryconditionforhouseholdoptimizationis

Ci,t=-εCt,(3)

where

Here,PtdenotestheidealpriceindexandCtdenotestheutility-relevantconsumptionbundle.

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

Ouranalysisdoesnotrequirethatwespecifyafunctionalformofu,thetechnologyforproducingCi,t,thenatureofgovernmentmonetarypolicy,themonopolists’maximizationproblem,orevenhowEtin(1)isformed.Wedorequirethatusatisfytheusualpropertiesthatitisincreasingandconcaveinitsfirstargument,decreasingandconcaveinthesecond,andcontinuouslydifferentiable.Otherthanthat,weonlyusethedefinitionoftheidealindex,Pt,theCalvofriction,theassumptionthatallmonopoliststhatupdatetheirpricechoosethesameresetprice,andthefactthathouseholdsmaximize(1).

3Anupperboundoninflationintheidealpriceindex

Letπt=Pt/Pt-1denoteinflationintheidealpriceindex.Inthissection,wederiveanupperboundonπt,asinAndreasenandKronborg(2022).ItfollowsfromthedefinitionofPtandtheCalvofrictionthat

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)representsanupperboundonthelevelofperiodinflationinanyequilibriumofthemodel.

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.Differentauthorsassumedifferentvaluesfortheseparameters.So,wecalculatetheupperboundforπtinthestandardNKmodelforvariousvaluesofθandε,seeFigure1.Thehorizontalaxisofeachpanelcorrespondstothesteadystatemarkupimpliedbyε(whichisgivenbyε/(ε-1)-1)expressedinpercent.Theverticalaxiscorrespondstotheupperboundonπt,expressedinannualizedpercent.

FromFigure1weseethattheboundonπtisincreasinginthesizeofthesteady-statemarkup(lowerε).Highvaluesofthesteadystatemarkupmeanthatgoodsarelesssubstitutableforeachother.Astrises,householdsarelesswillingtosubstitutetowardlower-pricedgoodsandπtishigher.Notsurprisingly,thesamefigureshowsthattheboundonπtisincreasingintheflexibilityofprices(lowerθ).

4Relationshiptoboundsonsteady-stateinflation

KingandWolman(1996),Ascari(2004),andAscariandSbordone(2014)arguethatthestandardNKmodel,inwhichfirmshaverationalexpectations,hasanupperboundonsteady-statepriceinflation.AscariandSbordone(2014)showthatforthefirmproblemtobewelldefinedinsteadystate,itmustbethatβθπε<1andβθπε-1<1.2Theseconditionsimpliesthat

Dependingonparametervalues,theboundonπin(8)maybehigherorlowerthantheboundon

2KingandWolman(1996)arguethatiftheboundontrendinflationisnotsatisfiedthentheprobabilityofnotadjustingpricesishighenoughthatfirmswouldrathernotoperatethanriskhavingtheirpricebelowforaverylongtime.Ouranalysis,(6),doesnotdependonoptimalpricingbyfirms.

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

Inthedata,thelevelandvolatilityofinflationareoftenpositivelycorrelated(seeBalletal.(1988)).So,eveniftheaveragelevelofinflationsatisfies(6),theremaybelongperiodsinwhichmeasuredinflationexceedsthebound.

5Measuredinflation

MostresearchersusingNKmodelscomparethepercentchangeintheidealpriceindextomeasuredinflation.However,measuredpriceindicesarenotperfectanalogstotheidealpriceindexinthemodel.Indeed,overshorthorizons,measuredpriceindices–liketheCPI–arewellapproximatedbyafixed-weightindex.Inthissection,weanalyzeafixed-weightpriceindexinthestandardNKmodelusingequalweightsontheintermediategoodstoconstructthepriceindex.Ourassumptionofequalweightscorrespondstousingthetime-seriesaverageofweightsplacedontheintermediategoodsintheidealpriceindex.WeusedataoncoreCPIinflationasthedataanalogueforthisfixed-weightindexandshowthatitcandiffermeaningfullyfrominflationintheidealpriceindexwhencoreCPIinflationismoderatelyhigh.3

Wedefinetheequal-weightpriceindex,Zt,by

IncontrasttoPt,Zt→∞ast→∞becausetheweightsonhighpricesdonotdeclineasconsumerssubstitutetowardlower-pricedgoods.Letztdenotetheratioofthefixed-weightpriceindexandtheidealizedpriceindex(Zt/Pt).Inaddition,letζtdenoteinflationinthefixed-weightpriceindex

3Bycontrast,AndreasenandKronborg(2022)usecoreCPIinflationasthedataanalogueoftheNKmodel’sidealpriceindex.

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(Zt/Zt-1).Then

(11)

(12)

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

Itisstraightforwardtoshowthatinafirst-orderapproximationtothemodelwithzerosteadystateinflation,t=0andt=t.5However,inthenonlinearversionofthemodelorinlinearizedversionsofthemodelwithpositivesteady-stateinflationtherecanbesubstantialdifferencesbetweenζtandπt.

WetakecoreCPIinflationasthedatacounterparttothefixed-weightindexdiscussedabove(Zt).6Weusethenonlinearequations(5),(11),and(12)toconstructatimeseriesforπt.7Theblack,solidlineinFigure2correspondstothecoreCPIinflationratefor2021:Q1through2024:Q3.Figure2alsodisplaystheinflationrateintheidealpriceindexfordifferentvaluesofθandε.ThefigureshowsthattheseriesforπtdiffersmorefromtheCPIinflationratewhentheboundonπtislower.Recallthattheboundonπtincreasesinthemarkupanddecreasesinθ.Moreover,thedifferenceoverthepastfewyearscanbesubstantialforvaluesofθandεusedintheliterature.

6Conclusion

Wearguethat,especiallyovershorthorizons,inflationinafixed-weightpriceindexisaclosermodelanaloguetomeasuredinflationthaninflationintheNKmodel’sidealpriceindex.Weshowthatduringaperiodofmoderatelyhighinflation,inflationinmodel-basedfixed-weightpricesindicesmaydiffermarkedlyfromthemodel-basedmeasureofinflation(usingtheidealpriceindex)

5Here,tdenotesthelogdeviationofxtfromitssteady-statevalue.Tounderstandthisresult,notethatt=

4Though,thereisalowerboundequaltoθ.~

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

beawell-definedsteadystate,itmustbethattheboundonπissatisfiedsothatpiswelldefined.Whenπ>1,it

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

inflationrateinbothindicesisthesameforthesteadystaterealresetprice,p,eventhoughtheweightsonthereset

pricearedifferentinthetwoindices.

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

/series/CPILFESL

,November7,2024.

7Wesetzt=1in2020:Q4.Ourresultsarelittlechangedifwesetztin2020:Q4equaltoitssteadystatevalueundertheassumptionof2percentsteady-stateinflation.

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typicallyusedtoempiricallyassessNKmodels.OuranalysisindicatesthatresearcherswhousethestandardNKmodelwheninflationismoderatelyhigh(asitwasinthepost-Covidperiod)shouldusemodel-basedmeasuresofinflationthatareconsistentwiththewayinflationismeasuredinthedata.

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GuidoAscariandArgiaM.Sbordone.TheMacroeconomicsofTrendInflation.JournalofEconomicLiterature,52(3):679–739,September2014.

LaurenceBall,NGregoryMankiw,DavidRomer,GeorgeAAkerlof,AndrewRose,JanetYellen,an