美联储-通货膨胀加速器 The In ation Accelerator

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

TheIn丑ationAccelerator

AndresBlanco,CorinaBoar,CallumJones,VirgiliuMidrigan

2024-078

Pleasecitethispaperas:

Blanco,Andres,CorinaBoar,CallumJones,andVirgiliuMidrigan(2024).“TheIn丑ationAccelerator,”FinanceandEconomicsDiscussionSeries2024-078.Washington:BoardofGovernorsoftheFederalReserveSystem,

/10.17016/FEDS.2024.078

.

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

TheInflationAccelerator*

AndresBlancotCorinaBoar❹CallumJones§VirgiliuMidrigan¶

June2024

Abstract

Wedevelopatractablestickypricemodelinwhichthefractionofpricechangesevolvesendogenouslyovertimeand,consistentwiththeevidence,increaseswithinfla-tion.Becauseweassumethatfirmssellmultipleproductsandchoosehowmany,butnotwhich,pricestoadjustinanygivenperiod,ourmodeladmitsexactaggregationandreducestoaone-equationextensionoftheCalvomodel.Thisadditionalequationdeterminesthefractionofpricechanges.Themodelfeaturesapowerfulinflationac-celerator–afeedbackloopbetweeninflationandthefractionofpricechanges–whichsignificantlyincreasestheslopeofthePhillipscurveduringperiodsofhighinflation.AppliedtotheU.S.timeseries,ourmodelpredictsthattheslopeofthePhillipscurverangesfrom0.02inthe1990sto0.12inthe1970sand1980s.

Keywords:Phillipscurve,inflation,pricerigidities.

*WethankMarkGertlerforusefulfeedback,HughMontagandDanielVillarforsharingthedataonthefrequencyofpricechanges,andSukJoonKimforexcellentresearchassistance.TheviewsexpressedarethoseoftheauthorsandnotnecessarilythoseoftheFederalReserveBankofAtlantaortheFederalReserveBoard.

FederalReserveBankofAtlanta,

julioablanco84@.

NewYorkUniversityandNBER,

corina.boar@.

§FederalReserveBoard,

callum.j.jones@.

¶NewYorkUniversityandNBER,

virgiliu.midrigan@.

1

1Introduction

Therecentriseininflationinmanyeconomieshasspurredconsiderableinterestinfurtherunderstandingthedynamicsofprices.IdentifyingthecausesofhighinflationhingescriticallyontheshapeofthePhillipscurve.OurgoalinthispaperistomeasurehowtheslopeofthePhillipscurvefluctuatesintheU.S.macroeconomictimeseries.Sinceakeydeterminantofthisslopeisthefractionofpricechanges,weuseamodelthatreproducesthewidelydocumentedevidencethatthefractionofpricechangesincreasesintimesofhighinflation

.1

Thoughthemenucostmodelisanaturalframeworktoendogenizethefractionofpricechanges,

2

itisdifficulttouseforempiricalandpolicyanalysis.Thisdifficultyarisesfromthecomputationalchallengesassociatedwithaggregatingindividualdecisionrulesinasettinginwhichthestateoftheeconomyischaracterizedbythedistributionofpricesacrossfirms.Thesechallengesareespeciallypronouncedwhenthefractionofpricechangesrespondstoaggregateshocks,whichgivesrisetoimportantnon-linearities,orinthepresenceofstrategiccomplementarities,whichgenerateaninteractionbetweenthepricesofcompetitors.

Ourpaperproposesanalternativetothemenucostmodelthatalsoendogenizesthefractionofpricechangesandallowsittovaryovertime,but,unlikethemenucostmodel,ishighlytractable.Themainchallengeinallowingthefractionofpricechangestoevolveendogenouslyovertimeisthatafirm’spriceadjustmentdecisiondependsonhowfarfromtheoptimumitspriceis:firmswhosepricesarefurtherfromtheoptimumhavestrongerincentivestoadjust.Equilibriumoutcomesarethereforeafunctionoftheentiredistributionofpricechanges,aninfinite-dimensionalobject.Wecircumventthischallengebyassumingthatfirmssellacontinuumofproductsandchoosehowmany,butnotwhich,pricestoadjustinanygivenperiod,subjecttoanadjustmentcost.Becausefirmscannotchoosewhichpricestoadjust,thedistributionofpricesisnolongernecessarytodescribetheincentivestoadjust,sotheeconomyadmitsexactaggregation

.3

Weshowthatourmodelreducestoaone-equationextensionoftheCalvomodel,withtheadditionalequationpinningdownhowmanyprices

1See

Gagnon

(2009),

Nakamuraetal.

(2018),

Alvarezetal.

(2018),

KaradiandReiff

(2019),

Montagand

Villar

(2023)and

Blancoetal.

(2024)forevidencethatthefrequencyofpricechangesincreaseswithinflation,

aswellas

Hazelletal.

(2022)and

Fitzgeraldetal.

(2024)forevidenceontheslopeofthePhillipscurveusing

state-leveldata.

2See,forexample,

Barro

(1972),

SheshinskiandWeiss

(1977),

Dotseyetal.

(1999),

GolosovandLucas

(2007),

GertlerandLeahy

(2008),

Midrigan

(2011),

AlvarezandLippi

(2014),

Alvarezetal.

(2016),

Alvarez

etal.

(2018),

Auclertetal.

(2022)

.However,

Blancoetal.

(2024)showthatthecanonicalmenucostmodel

hasdifficultiesreproducingtheextenttowhichthefrequencyofpricechangescomoveswithinflation.

3Inadditiontothemenucostliterature,

Romer

(1990)alsoendogenizesthefrequencyofpricechangesin

aCalvomodel,butcircumventsthecurseofdimensionalitybyassumingthatfirmschoosethefrequencyofpricechangesonceandforall.Inthatmodelthefrequencyofpricechangesisconstantovertime.

2

changeinagivenperiod

.4

OurmodelneststheCalvomodelinthelimitingcasewhentheadjustmentcostgoestoinfinity.Moregenerally,uptoafirst-orderapproximation,ourmodel’sdynamicsareidenticaltothoseoftheCalvomodelabsenttrendinflation.

OurkeyfindingisthattheslopeofthePhillipscurvefluctuatesconsiderablyintheU.S.timeseriesandincreasesintimesofhighinflationduetoafeedbackloopbetweeninflationandthefractionofpricechanges.Ononehand,anincreaseinthefractionofpricechangesincreasesinflation,moresothehighertheinflationratetobeginwith.Ontheotherhand,anincreaseininflationincreasesthefirms’incentivestoadjustprices,furtherincreasingthefractionofpricechanges.WerefertothisfeedbackloopastheinflationacceleratorandshowthatitisresponsibleforthebulkofthesteepeningofthePhillipscurveinperiodsofhighinflation.Ourfindingsthereforesuggestthatreducinginflationislesscostlywheninflationishighthanwhenitislow.

WestudyarelativelystandardNewKeynesianeconomyinwhichmulti-productfirmssellacontinuumofgoodsandchoosewhatfractionoftheirpricestochangeeachperiod,subjecttoanadjustmentcostthatisincreasingandconvexinthenumberofpricesthatthefirmadjusts.WeassumedecreasingreturnstoscaleinproductionwhichintroducestrategiccomplementaritiesinpricesettinganddampentheslopeofthePhillipscurve.Forclarity,westartbyassumingthatmonetarypolicytargetsnominalspending,andshowinanextensionthatourresultsarerobusttoconsideringaconventionalTaylorrule.Shockstothegrowthrateofnominalspendingaretheonlysourceofaggregatefluctuations.RelativetothestandardCalvomodel,endogenizingthefrequencyofpricechangesaddsasingleadditionalequationthatbalancesthemarginalcostofchangingpricesagainstthemarginalbenefit.Themarginalbenefitincreaseswithinflation,implyingthatthefrequencyofpricechangesincreaseswithinflation.Becauseendogenizingthefrequencyofpricechangesintroducesnon-linearitiesinthedynamicsofoutputandinflation,wesolvethemodelusingglobalprojectionmethods,butshowthatathird-orderperturbationprovidesanaccurateapproximation,suggestingthatthemodelcanbesolvedusingreadily-availablesolutiontechniques.

Wefirstbuildintuitionfortheworkingsofthemodelbystudyingimpulseresponsestoexpansionarymonetaryshocksinenvironmentswithlowandhightrendinflation.Weshowthattherealeffectsofmonetaryshocksareconsiderablysmallerinenvironmentswithhighinflationfortworeasons.First,thesteady-statefrequencyofpricechangesishigher

inenvironmentswithhighinflation.Second,thefrequencyofpricechangesincreasesin

4Theassumptionwemakeisreminiscentofthatin

Greenwald

(2018)whousesalargefamilyconstruct

toendogenizerefinancingdecisions.

3

responsetoshocks.Thoughthisincreaseisrelativelysmall,ithasalargeimpactonthe

pricelevelbecauseadjustingfirmsrespondtotheunderlyingtrendinflationandincreasepricesbylargeamounts,aneffectreminiscentof

CaplinandSpulber

(1987)

.

WebuildadditionalintuitionforthedynamicsofinflationandoutputbyderivingthePhillipscurveimpliedbyoureconomy.WeshowthattheslopeofthePhillipscurveisequaltothesumoftwoterms,oneidenticaltotheslopeintheCalvomodel,whichincreasesmechanicallywiththefrequencyofpricechanges,andanotherwhichcapturestheinflationaccelerator.ThissecondtermincreasesmuchmorerapidlywithinflationandthusaccountsforthebulkoftheincreaseintheslopeofthePhillipscurveinhigh-inflationenvironments. WeuseourframeworktocharacterizehowtheslopeofthePhillipscurveevolvesinthepost-warU.S.timeseries.Wedosobyfirstidentifyingthesequenceofmonetaryshocksthatallowsthemodeltoreproducethepathofinflationinthedata.Wethenconsideralog-linearperturbationaroundtheequilibriumpointateachdateandderivetheslopeofthePhillipscurve.WefindthattheslopeofthePhillipscurvevariesconsiderably,rangingfrom0.02inlow-inflationperiodssuchasthe1990sto0.12inhigh-inflationperiodssuchasthe1970sandthe1980s.Theinflationacceleratoraccountsforthebulkofthisincrease:initsabsence

thehigherfrequencyofpricechangesinthe1970sand1980swouldonlyincreasetheslopeofthePhillipscurveto0.04.WeshowthatourfindingsarerobusttoeliminatingstrategiccomplementaritiesandtoassumingthatmonetarypolicyfollowsaconventionalTaylorrule.

ThattheslopeofthePhillipscurvevariesovertimehasimportantimplicationsforthetradeoffbetweeninflationandoutputstabilization.Wegaugehowthistradeoffvariesovertimebycalculatingameasureofthesacrificeratio:thefallinoutputrequiredtoachieveaonepercentagepointreductionininflation.Thesacrificeratiovariesconsiderably,from1.4%inthelow-inflationperiodinthe1990sto0.4%inthehigh-inflationperiodsinthe1970sand1980s.Wethereforeconcludethatourmodelimpliesthatifinflationishightobeginwith,bringingitdownrequiresasmallerdropinoutputthanifinflationislow.Ourmodelthusrationalizestheviewthatreducinginflationfrom10%to9%isalotlesscostlythanreducingitfrom3%to2%.

Therestofthepaperproceedsasfollows.Section2presentsthemodel.Section3describestheparameterization.Section4analyzesthesteadystateofthemodel.Section5appliestheframeworktothetime-seriesU.S.data.Section6discussesseveralrobustnessexercises.Section7concludes.

4

2Model

Westudyaneconomyinwhichfirmsadjustpricesinfrequently.IncontrasttothestandardNewKeynesianmodel,weallowthefrequencyofpricechangestofluctuateendogenouslyovertimebyassumingthatmulti-productfirmschoosewhatfractionoftheirpricestoadjustinanygivenperiod.Wecircumventtheneedtokeeptrackofthedistributionofpricesbyassumingthatfirmschoosehowmany,butnotwhich,pricestochange.Owingtothisassumption,ourmodelreducestoaone-equationextensionofthestandardCalvomodel,withtheadditionalequationdescribinghowthefractionofpricechanges,andthereforetheslopeofthePhillipscurve,ispinneddowneachperiod.

Forclarity,westartbyassumingthatmonetarypolicytargetsnominalspending,whichevolvesovertimeaccordingtoarandomwalkprocess.Shockstothegrowthrateofnominalspendingaretheonlysourceofaggregatefluctuations.WethenshowintherobustnesssectionbelowthatassuminginsteadthatmonetarypolicyfollowsaTaylorruledoesnotchangeourkeyfindings.

2.1Consumers

Arepresentativeconsumerhaspreferencesoverconsumptionctandhoursworkedhtand

maximizeslife-timeutility

subjecttothebudgetconstraint

wherePtisthenominalpricelevel,Btareholdingsofarisk-freebondwhichpaysnominalinterestit,Dtarethedividendsfromthefirmstherepresentativeconsumerowns,andWtisthenominalwagerate.

2.2MonetaryPolicy

Weassumethatmonetarypolicytargetsnominalspending,Mt≡Ptct,whichfollowsarandomwalkwithdrift

whereµistheaveragegrowthrateofnominalspendingandεt+1areGaussianinnovationswithstandarddeviationσ.As

AfrouziandYang

(2021)pointout,thisspecificationofthe

5

monetarypolicyruleisequivalenttoaninterestrateruleinwhichthecentralbankassignsthesameweighttoinflationandoutputgrowth.

2.3Technology

Thereisacontinuumofintermediategoodsfirmsindexedbyi.Eachfirmproducesacon-tinuumofproductskwithtechnology

yikt=(likt)η,

whereyiktistheoutputofproductkproducedbyfirmi,liktisthelaborusedinproduc-tionandη≤1isthespan-of-controlparameterwhich,asin

BursteinandHellwig

(2008),

introducesamicro-levelstrategiccomplementarityinpricesetting.

AperfectlycompetitivefinalgoodssectoraggregatestheintermediategoodsyiktintoacompositefinalgoodusingaCESaggregator

whereθistheelasticityofsubstitution,whichweassumetobethesamebothacrossproductsandacrossfirms.Thisimpliesthatthedemandforanindividualproductis

yikt=−θyt,(1)

wherePiktisthepriceofanindividualproductand

istheaggregatepriceindex.

2.4ProblemofIntermediateGoodsProducers

Wenextdescribetheprofitmaximizationproblemofintermediategoodsproducers.

PeriodProfits.ThenominalprofitsoffirmifromproducingproductkarePiktyikt−τWtlikt,

whereτ=1−1/θisasubsidythatremovesthemarkupdistortionthatwouldariseevenintheabsenceofpricerigidities.Usingthedemandfunction(

1

),wecanexpressrealprofitsas

6

LossesfromMisallocation.Differencesinthepriceofproductssoldbyagivenfirmgeneratelossesfrommisallocation,reducingfirmproductivity.Toseethis,let

denotethecompositeoutputproducedfirmiandletlit=∫liktdk

denotethetotalamountoflaborthefirmuses.Wecanthenderiveafirm-levelproductionfunction

where1

denotesthepriceindexoffirmiand

determinestheextentofmisallocation.Absentdispersioninprices,Xit/Pit=1andproduc-tivityismaximized.Withpricedispersion,Xit/Pit<1andproductivityisreduced.

PriceAdjustmentCost.Weassumethatthefirmhasaconvexcostofchangingpricesdenominatedinunitsoflabor.Thiscostisincreasinginthenumberofpricesnitthefirmresetsandisequalto

2,ifnit>

andzerootherwise.Here,ξdeterminesthesizeoftheadjustmentcostandisthefractionoffreepricechanges.Thekeyassumptionwemakeisthatalthoughthefirmcanchoosehowmanypricestochangeinagivenperiod,itcannotchoosewhichpricestochange.Byendogenizingthefrequencyofpricechanges,themodelcancapturetheevidencethatfirmsaremorelikelytoadjustpricesintimesofhighinflation,asinmenucostmodels,butinamuchmoretractableway.Whenξ→∞,themodelcollapsestotheCalvomodelwithaconstantfrequency.

Ourmodelsharessimilaritieswiththatin

Romer

(

1990

)whichendogenizesthefrequencyofpriceschangesintheCalvomodel

.5

In

Romer

(1990)firmschooseaonce-and-for-allprice

5Seealso

Kiley

(2000),

DevereuxandYetman

(2002)and

Bakhshietal.

(2007)

.

7

adjustmentprobability,balancingthegainsfrommorefrequentadjustmentagainstthecostsofrepricing.Extendingthatmodeltoallowforatime-varyingadjustmentprobabilitywouldrequirekeepingtrackofthedistributionofpricesbecausethegainsfromadjustingwouldbehigherforpricesfurtherawayfromtheoptimum,justlikeinmenucostmodels

.6

Incontrast,ourassumptionthatfirmssellacontinuumofproductsandchoosehowmany,butnotwhich,pricestochange,impliesthatfirmsareex-postidenticalandthatasmallnumberofstatevariablesaresufficienttocharacterizeafirm’sincentivestoadjustprices.Thisfeatureallowsexactaggregationandrendersourmodelverytractable.

PriceSetting.Wenextdescribethefirms’problemindetail.Thevalueofthefirmisthepresentdiscountedsumofitsflowprofits(

2

).Thelog-linearspecificationofpreferences

impliesthatct=tt=ytandallowstowritethevalueofthefirmas

EtΣsβs(∫1−θ—τ−s]dk—(nit+s—)2),

or,usingthedefinitionsofPitandXitinequations(

3

)and(

4

),

ThefirmchooseswhatfractionofpricesnittoreseteveryperiodandtheresetpricePi.Becauseallproductsareidentical,Pikt=Piforallproductswhosepriceisreset.

Tocharacterizetheseoptimalchoices,wefirstdescribehowthechoiceofPiandnitaffect

firmprofitsinfutureperiods.Considerfirsttheterm(Pit+s)1−θinequation(

5

).Usingthedefinitionofthefirm’spriceindexinequation(

3

)andtheassumptionthatthefirmchooseshowmany,butnotwhich,pricestochange,allowsustowritethistermasafunctionofthehistoryofpreviouslychosenresetpricesandrepricingprobabilitiesasfollows

Thefirsttermontherighthandsiderepresentsthecontributionofthenit+snewlyresetpricesinperiodt+s.Thesecondtermrepresentsthecontributionofthe(1—nit+s)nit+s−1

6Seealso

Alvarezetal.

(2021)and

Cavalloetal.

(2024)forvariantsofthemenucostmodelinwhichfirms

choosethepriceadjustmentprobabilitysubjecttoaconvexadjustmentcost.

8

pricesthatwereresetinperiodt+s—1andwerenotresetinperiodt+s.Thispatterncontinueswitheachsubsequenttermaccountingforthecontributionofpricesresetineachperiodleadinguptot+s,includingthoseresetinperiodt,capturedbythefirstterminthelastlineoftheexpression,aswellasthoseresetpriortoperiodt,capturedbythelasttermoftheexpression.Inwritingthislasttermweusedthedefinitionofthepriceindexinequation(

3

)toexpressthehistoryofallresetpricespriortoperiodtusingasinglestate

ss

WecannowcharacterizetheoptimalchoiceofPiandnit.ToderivetheoptimalityconditionwithrespecttoPiwenotethatequations(

6

)and(

7

)implythat

and

Therefore,theresetpricePithatmaximizesthevalueofthefirmsatisfiesthefirstorder

condition

where

and

Thetermsb1itandb2itcapturethepresentvalueofrevenueandmarginalcostsinfutureperiods,weightedbytheprobabilitythatapriceresettodayinstillineffectinthatfutureperiod.Anincreaseinfutureinflationreducestherealvalueoftheresetpriceandincreasesrevenueandmarginalcostsandthereforetheweightthatthefirmplacesonthatperiod.

9

Strategiccomplementarities,capturedbytheterm1+θ,dampentheextentto

whichtheresetpricerespondstoaggregateshocks,generatingadditionalpricestickiness.TheseexpressionsthatdeterminetheoptimalresetpriceareanalogoustothoseobtainedinaCalvomodel,withtheonlydifferencebeingthatintheCalvomodelntisconstant.

Tobuildadditionalintuition,wecanalternativelyexpressthefirm’soptimalresetpriceasafunctionoftheexpectedpresentvalueoffuturemarginalcosts,

∞

where

isthemarginalcostinperiodt+sofproducingagoodwhosepricewaslastresetinperiodtand

denotestherelativeweightofperiodt+sindeterminingthefirm’soptimalprice,withtheweightreflectingtheprobabilitythatthepriceisstillineffectatthatpoint,aswellastheamountthefirmexpectstosellgiventheaggregatepriceindexinthatperiod.

Toderivetheoptimalityconditionwithrespecttonit,wefirstnotethatequations(

6

)and(

7

)implythat

and

Thefractionofpricechangesnitthatmaximizesthevalueofthefirmthereforesatisfiesthefirstordercondition

ξ(nit—)=b1it1−θ1−θ)—τb2it−−.

Inchoosingwhatfractionofpricestoadjust,thefirmbalancesthepriceadjustmentcostsagainstthebenefitsresultingfromchangingitspriceindex,capturedbythefirsttermontherighthandside,andreducingmisallocationinsidethefirm,capturedbythesecondtermontherighthandside.Noticethatthetermsb1itandb2itthatdeterminetheoptimal

10

resetpricealsodeterminethefirm’sincentivetoadjustprices.Forexample,thehigheris

outputyt+sinfutureperiods,thelargerisb2itandthereforethestrongertheincentivetoreducemisallocationandthuseconomizeonlaborcosts.Similarly,themorelikelythefirmistoadjustitspricesinthefuture,thelowerareb1itandb2itandthereforethebenefitsfromadjustingpricestoday.Notethatthefirms’incentivestoadjustareshapedbytwostatevariables,thefirm’spriceindex,Pit−1,andtheamountofmisallocationinsidethefirm,Xit−1,asinthemulti-productmenucostmodelof

Blancoetal.

(

2024

).Nevertheless,becausefirmsareex-postidentical,wedonotneedtokeeptrackofthejointdistributionofthesetwostatevariables.

2.5Equilibrium

Sinceallfirmsareidentical,nit=ntandPi=Pt*.Consequentlyallfirmshavethesame

priceindicesandlossesfrommisallocation.Letp=Pt*/Mt,pt=Pt/Mt,xt=Xt/Ptand

πt=Pt/Pt−1andrecallthat

wherewt=ttistherealwage.Theequilibriumofthemodelischaracterizedbythe

followingsystemofequations:

1.thedefinitionofthepriceindex,whichdeterminesinflationasafunctionoftherelativeresetpriceandthefractionofpricechanges

2.theoptimalresetprice

(9)

whereb1tandb2taredeterminedby

b1t=1+βEt(1—nt+1)(πt+1)θ−1b1t+1(10)

−1θ

b2t=ptη+βEt(1—nt+1)(πt+1)ηb2t+1,(11)

3.theoptimalchoiceofthefractionofpricechanges

11

4.theendogenousproductivitytermthatcapturesthelossesfrommisallocation

Noticethatinflationisequalto

where

isthepreviousperiod’spricelevelscaledbycurrentnominalspending.Becausethegrowthrateofnominalspendingisiid,theonlytwostatevariablesinthiseconomyarestandxt−1,sothesolutionofthemodelisgivenbyfunctionspt=P(st,xt−1),xt=X(st,xt−1)etc.,thatdeterminehowoutputandinflationevolveovertimeinresponsetomonetarypolicyshocks.

RelativetotheCalvomodel,theonlynewequationisequation(

12

)whichcharacterizeshowthefractionofpricechangesevolvesovertime.TheCalvomodelisaspecialcaseofourmodelthatcanbeobtainedbysettingξ→∞,inwhichcasent=isconstant.IntheCalvomodelthepreviousperiod’slossesfrommisallocationxt−1donotaffectthefrequencyofpricechanges,sothepricelevelonlydependsonasinglestatevariable,st.

Wesolvethesystemoffunctionalequationsthatcharacterizethesolutionofthemodelus-ingglobalprojectionmethods,byapproximatingtheequilibriumfunctionsusingChebyshevpolynomials.Wefound,however,thatathird-orderperturbationprovidesaveryaccurateapproximation,suggestingthatthemodelcanbeeasilysolvedusingreadily-availablesolutiontechniques.SeetheAppendixfordetails.

3Parameterization

Wenextexplainhowweparameterizethemodel.Wefirstdiscusstheparametersweassignvaluestoandthentheparameterswecalibrateendogenously.

3.1AssignedParameters

Aperiodinthemodelisaquarter.Wesetthreeparameterstovaluesconventionalintheliterature:aquarterlydiscountfactorβof0.99,ademandelasticityθof6andareturnstoscaleparameterηof2/3.Intherobustnesssectionbelowweshowthatourresultsarerobusttoperturbingθandη.

12

3.2CalibratedParameters

Theparameterswecalibrateendogenouslyarethosedeterminingtheaveragelevelandvolatil-ityofinflation,aswellastheaveragefrequencyofpricechangesanditscomovementwithinflation.Specifically,wesettheaveragegrowthrateofnominalspendingμ,thestandarddeviationofnominalspendinggrowthσ,thefractionoffreepricechanges,andthepriceadjustmentcostparameterξtoreproducethemeanandstandarddeviationofinflation,themeanfractionofpricechanges,andtheslopecoefficientfromregressingthefractionofpricechangesontheabsolutevalueofinflation.Thislaststatisticcapturestheextenttowhichthefractionofpricechangesandinflationcomoveinthetimeseries.

3.2.1TheData

OurmeasureofinflationisthegrowthrateoftheU.S.CPI,availablefrom1962:Q1to2023:Q4.Wefollow

Nakamuraetal.

(

2018

)inusingtheseriesexcludingsheltertoensurethattheinflationdataiscompatiblewiththedataonthefractionofpricechanges.ThefractionofpricechangesiscomputedfromthepricequotescollectedbytheBLSthatunderlietheconstructionoftheCPI

.7

Specifically,weusethemonthlymedianfractionofpricechanges,excludingsales,av