美联储-菜单成本经济中的非线性动力学？来自美国数据的证据 Nonlinear Dynamics in Menu Cost Economies Evidence from U.S. Data 2024

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

NonlinearDynamicsinMenuCostEconomies?Evidencefrom

U.S.Data

AndresBlanco,CorinaBoar,CallumJones,VirgiliuMidrigan

2024-076

Pleasecitethispaperas:

Blanco,Andres,CorinaBoar,CallumJones,andVirgiliuMidrigan(2024).“NonlinearDynamicsinMenuCostEconomies?EvidencefromU.S.Data,”FinanceandEconomicsDiscussionSeries2024-076.Washington:BoardofGovernorsoftheFederalReserveSystem,

/10.17016/FEDS.2024.076

.

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

NonlinearDynamicsinMenuCostEconomies?

EvidencefromU.S.Data*

AndresBlancotCorinaBoar‡CallumJones§VirgiliuMidrigan¶

July2024

Abstract

Weshowthatstandardmenucostmodelscannotsimultaneouslyreproducethedispersioninthesizeofmicro-pricechangesandtheextenttowhichthefractionofpricechangesincreaseswithinflationintheU.S.time-series.Thoughthe

Golosovand

Lucas

(2007)modelgeneratesfluctuationsinthefractionofpricechanges,itpredicts

toolittledispersioninthesizeofpricechangesandthereforelittlemonetarynon-neutrality.Incontrast,versionsofthemodelthatreproducethedispersioninthesizeofpricechangesandgeneratestrongermonetarynon-neutralitypredictanearlyconstantfractionofpricechanges.

Keywords:menucosts,inflation,fractionofpricechanges.

*WethankHughMontagandDanielVillarforsharingthedataonthefractionofpricechanges.TheviewsexpressedherearethoseoftheauthorsandnotnecessarilythoseoftheFederalReserveBankofAtlantaortheFederalReserveBoard.

FederalReserveBankofAtlanta,

julioablanco84@.

NewYorkUniversityandNBER,

corina.boar@.

§FederalReserveBoard,

callum.j.jones@.

¶NewYorkUniversityandNBER,

virgiliu.midrigan@.

1

1Introduction

Arobustfeatureofthedataisthatthefractionofpricechangesincreasesinperiodsofhighinflation

.1

Itiswidelybelievedthatmenucostmodelscanreproducethispatternbecausefirmschooseendogenouslythetimingofpricechangesandaremorelikelytorespondtolargershocks.WhetherthisisindeedthecaseornothasimportantimplicationsfortheextenttowhichtheslopeofthePhillipscurvevariesinthetime-seriesandthereforethetradeoffbetweeninflationandoutputstabilizationfacedbymonetarypolicy.

2

BecauseinmenucostmodelsthedistributionofpricechangesalsocriticallyshapestheslopeofthePhillipscurveandthedegreeofmonetarynon-neutrality,

3

weevaluatetheabilityofstandardmenucostmodelstoreproducethecomovementbetweeninflationandthefractionofpricechangesintheU.S.data,whilesimultaneouslyaccountingforthelargedispersioninthesizeofpricechanges.Weconsiderthreecommonly-usedspecificationsforthetechnologyofchangingprices:(i)afixedmenucost,asin

GolosovandLucas

(

2007

),(ii)aCalvo-plusspecificationinwhichwithsomeprobabilityfirmscanchangepricesforfree,andotherwiseneedtopayafixedcost,asin

NakamuraandSteinsson

(

2010

)and(iii)aCalvo-plusspecificationaugmentedwithauniformdistributionofmenucosts,asin

Blanco

etal.

(2024a)

.

Weusethe

KrusellandSmith

(1998)approachtocharacterizethenon-linearsolutionof

eachmenucosteconomyinthepresenceofaggregatemonetarypolicyshocksandthenbackoutthesequenceofshocksthatexactlyreproducesthetimepathofinflationintheU.S.timeseries.Wefindthatthe

GolosovandLucas

(

2007

)modelpredictsthatthefractionofpricechangesincreasesconsiderablyinperiodsofhighinflation,asfirstpointedoutby

Nakamuraetal.

(2018),thoughlessthaninthedata

.4

Asiswellunderstood,however,thismodelpredictstoolittledispersioninthesizeofpricechanges.Incontrast,theothertwovariantsofthemodelthatbetterreproducethedispersioninthesizeofpricechangesimplyanearlyconstantfractionofpricechanges.

Weillustratetheimplicationsofthesepredictionsbystudyinghoweachoftheseeconomies

1See,forexample,

Gagnon

(2009),

Alvarezetal.

(2018),

KaradiandReiff

(2019),

Blancoetal.

(2024a)

and

Cavalloetal.

(2024)

.

2See

Blancoetal.

(2024b)whoshowthatamodelthatreproducestheincreaseinthefractionofprice

changesinperiodsofhighinflationimpliesasharpincreaseintheslopeofthePhillipscurveinthoseperiods,considerablyreducingtheoutputcostsofreducinginflation.

3See

CaballeroandEngel

(2007),

Midrigan

(2011)and

Alvarezetal.

(2016)

.

4

GolosovandLucas

(2007)and

Alvarezetal.

(2018)showthatthe

GolosovandLucas

(2007)model

reproducestherelationshipbetweeninflationandthefractionofpricechangesinacross-sectionofcountries.

2

respondstomonetarypolicyshocksofdifferentsizes.Becausethefractionofpricechanges

increasesconsiderablyinthe

GolosovandLucas

(

2007

)modelinresponsetolargemonetaryshocks,themodelpredictsimportantnon-linearities:therealeffectsfromlargershocksareproportionallysmallerthanthosefromsmallershocks.However,becausethemodelpredictslittledispersioninthesizeofpricechanges,therealeffectsofmonetarypolicyshocksaresmalleveninresponsetosmallshocks.Forexample,thecumulativeimpulseresponsetoa1%monetaryshockis0.14ofthatpredictedbytheCalvomodelwiththesameaveragefractionofpricechanges,andthattoa5%monetaryshockis0.08ofthatintheCalvomodel.Thus,inthe

GolosovandLucas

(2007)moneyisapproximatelyneutralregardlessofthesize

oftheshock.

Incontrast,theothertwovariantsofthemodelthatbetterreproducethedispersioninthesizeofpricechangespredictlargerrealeffects.However,becauseintheseeconomiesthefractionofpricechangesbarelyresponds,eventolargeshocks,theoutputeffectsarenearlylinearinthesizeoftheshock.Specifically,the

NakamuraandSteinsson

(

2010

)economypredictsacumulativeimpulseresponseofoutputtoa1%monetaryshockequalto0.36ofthatintheCalvomodelandtoa5%shockequalto0.30ofthatintheCalvomodel

.5

Similarly,theCalvo-pluseconomyaugmentedwithuniformmenucostspredictsacumulativeimpulseresponseofoutputtoa1%monetaryshockequalto0.58ofthatintheCalvomodelandtoa5%shockequalto0.56ofthatintheCalvomodel.

Ourpaperthereforecorroboratesthefindingsof

Blancoetal.

(

2024a

),wherewemadeasimilarpointusingU.K.micro-pricedata.Relativetothatpaper,herewefocusonasimplermenucosteconomywithoutstrategiccomplementaritiesinprice-settingandcalibratethemodeltomatchU.S.micro-pricestatistics.Inaddition,herewestudythecomovementbetweenaggregateinflationandthefractionofpricechanges,whereasin

Blancoetal.

(

2024a

)westudiedthecomovementbetweensectoralinflationandthefractionofpricechanges.

Weconcludethatanimportantchallengeforthemenucostliteratureistodevelopmodelsthatcansimultaneouslyreproducethemicro-pricedata,aswellastheextenttowhichthefractionofpricechangescomoveswithinflationinthetime-series.In

Blancoetal.

(

2024a

)weproposedonepotentialsolutiontothischallengeusingamenucosteconomyinwhichthelossesfrommisallocationfrompricedispersionwithinmulti-productfirmsarelow.Because

thatmodelreproducesthedispersioninthesizeofpricechangesinthedata,itpredicts

5

Auclertetal.

(2023)alsofindlittleevidenceofnon-linearityinthe

NakamuraandSteinsson

(2010)but,

unlikeus,theydonotconfrontthemodel’spredictionsonthecomovementbetweeninflationandthefractionofpricechangesinthetimeseries.

3

considerablerealeffectsfrommonetarypolicyshocks.Moreover,becauseitpredictsthat

thefractionofpricechangesincreasesconsiderablyintimesofhighinflation,italsopredictshighlynon-linearoutputresponsesandthereforeatime-varyingslopeofthePhillipscurve.

2MotivatingEvidence

Webrieflydescribetheevidencethatmotivatesourpaper.Figure

1

comparesthetimeseriesofU.S.inflationwiththatofthefractionofpricechanges.Wefollow

Nakamuraetal.

(

2018

)inmeasuringinflationusingthegrowthoftheConsumerPriceIndexexcludingshelter.Weusethedataonthefractionofpricechangescomputedby

Nakamuraetal.

(

2018

)usingpricequotescollectedbytheBureauofLaborStatistics.Thisserieswasrecentlyupdatedby

MontagandVillar

(2023)

.6

Wereporttheyear-on-yearchangeinthepriceindexand,forconsistency,theaveragemonthlyfractionofpricechangesintheprecedingyear.

Thefigureshowsthatthefractionofpricechangesincreasessystematicallywithinflation,aspointedoutby

Nakamuraetal.

(2018)and

MontagandVillar

(2023)

.7

Forexample,inthe1990s,wheninflationwaslow,approximately10%ofpriceschangedinagivenmonth.Incontrast,inthe1980s,wheninflationwashigh,approximately17%ofpriceschangedinagivenmonth.Morerecently,thepost-Covidincreaseininflationwasassociatedwithanevenhigherfractionofpricechanges,approximately23%permonth.

Figure1:InflationandtheFractionofPriceChanges

15

10

5

0

-5

0.25

0.15

0.05

inAation

1960198020002020

0.2

0.1

fractionofpricechanges

1960198020002020

Notes:TheshadedareasrepresentNBERrecessions.

6WethankHughMontagandDanielVillarforsharingthedatawithus.

7Thisisarobustfeatureofthedatathathasalsobeendocumentedforothercountries.See,forexample,

Gagnon

(2009),

Alvarezetal.

(2018),

KaradiandReiff

(2019),

Blancoetal.

(2024a)and

Cavalloetal.

(2024)

.

4

3Model

Weevaluatetheabilityofthemenucostmodeltoreproducethecomovementbetweenin-flationandthefractionofpricechangesinthedata.Weconsideraneconomyinwhichacontinuumofmonopolisticallycompetitivefirmsaresubjecttoidiosyncraticandaggregateshocksandfaceamenucostofchangingprices.Weconsiderthreespecificationsofthetech-nologyforchangingprices,allwidelyusedintheliterature:afixedmenucostasin

Golosov

andLucas

(2007),aCalvo-plusspecificationinwhichwithsomeprobabilityfirmscanchange

pricesforfreeasin

NakamuraandSteinsson

(

2010

),andauniformdistributionofmenucostsasin

Blancoetal.

(2024a)

.

3.1Consumers

Wefollow

GolosovandLucas

(2007)inassumingthatpreferencesarelogarithmicincon

-sumptionandlinearinhoursworked,soarepresentativeconsumermaximizes

subjectto

wherectisconsumption,htishoursworked,Ptistheaggregatenominalpriceindex,Btistheamountofgovernmentbondsthatpaythenominalinterestrateit,andDtdenotesprofits.Theoptimallaborsupplychoiceimpliesthatthenominalwageisequaltonominalspending,Wt=Ptct.

3.2MonetaryPolicy

WeassumethatmonetarypolicytargetsnominalspendingMt=Ptct,whichevolvesaccord-ingto

whereµt+1N(µ,σ).OurassumptionsonpreferencesimplythatWt=Mt.

3.3Technology

Thereisacontinuumofmonopolisticallycompetitiveintermediategoodsproducersi∈(0,1),eachproducingadifferentiatedvarietyusingalineartechnology

yit=zitlit,

5

wherezitisthequalityofthevarietyproducedbythefirmandlitistheamountoflabor

usedinproduction.

AperfectlycompetitivefinalgoodssectoraggregatesindividualvarietiesusingaCESaggregatorwithelasticityofsubstitutionσ

Thefinalgoodisusedforconsumptiononly,soyt=ct.Inadditiontoaffectingthefirm’sproductivity,thequalityzitalsoaffectsdemand.Ifpriceswereflexible,firmswouldrespondtoanincreaseinzitbyreducingpricesone-for-one,leavingquality-adjustedpricesandrevenuesunchanged.Theseshocksgenerateanidiosyncraticmotiveforfirmstochangetheirpricesandarewidelyusedinthemenucostliteratureduetotheirtractability.Weassumethatthezitevolvesaccordingto

logzit+1=logzit+εit+1,

whereεit+1~N(0,σ).

LettingPitdenoteanindividualfirm’sprice,thedemandfunctionforthefirm’soutputisgivenby

where

istheaggregatepriceindex.

3.4PriceAdjustmentTechnology

As

CaballeroandEngel

(2007),

Midrigan

(2011)and

Alvarezetal.

(2016)pointout,the

aggregateimplicationsofmenucostmodelsarecriticallyshapedbythedistributionofpricechanges.Wethereforeconsiderthreedifferentspecificationsforthepriceadjustmenttech-nologythatincreasinglyallowthemodeltoreproducethedistributionofpricechangesinthedata.First,wefollow

GolosovandLucas

(

2007

)andassumethatresettingpricesrequires

payingafixedmenucost.Second,wefollow

NakamuraandSteinsson

(2010)andassume

thatwithprobability1−λfirmscanchangepricesforfreeandwithprobabilityλtheyhave

topayafixedmenucost.Third,wefollow

Blancoetal.

(2024a)andassumethat,in

additiontothepossibilityoffreepricechanges,themenucostisidiosyncraticanddrawn

6

eachperiodforauniformdistributionU[0,].Inallthesecases,weassumethattheprice

adjustmentcostisdenominatedinunitsoflabor.Fromnowon,werefertotheseeconomiesasGL,NSandUniform,respectively.

3.5FirmObjective

Thefirmmaximizestheexpectedpresentvalueofprofits

whereτ=1/(σ−1)isasubsidythateliminatesthemarkupdistortionthatwouldariseevenwithflexibleprices,ξitisthepossiblyrandomcostofchangingpricesandIitisanindicatorequaltooneifthefirmadjustsitspriceandzerootherwise.

Tocharacterizethefirm’soptimalchoices,wefirstexpressitsobjectiveintermsofitspricegapwhichwedefineas

Similarly,wedefinetheaggregatepricegapastheCESweightedaverageoffirmpricegaps

Withthisnotationinplace,wecanwritethefirm’sobjectiveasafunctionofonlyitsownandaggregatepricegaps

Tocharacterizethepriceadjustmentdecision,let

denotethefirm’spricegapintheabsenceofapricechange,whichisalsothefirm’sidiosyn-craticstate.Eachperiodthefirmchooseswhethertoadjustitspriceornot.Ifthefirmdoesnotadjust,itspricegapisxit=sit.Ifthefirmadjustsitsprice,itresetsthepricegapto

xit=x,whichiscommontoallfirmsthatadjust.Thefirmrecognizesthatitsstateevolves

overtimeaccordingto

sit+1=xitexp(εit+1−µt+1).

Becausethisisamodelwithaggregateuncertaintyinwhichheterogeneousfirmsfollownon-lineardecisionrules,theentiredistributionofpricegaps,aninfinitedimensionalobject,

7

isnecessarytocharacterizetheequilibrium.Toseewhythisisthecase,letvdenotethe

valueofadjustingthepriceinperiodtandv(s)thevalueofnotadjustingthepricefora

firmwithidiosyncraticstates.Thefirmadjustsitspricewithprobability

inGLinNS

inUniform.

LettingFt(s)denotethedistributionoffirms,theequilibriumaggregatepricegapsatisfies

Thus,computingtheaggregatepricegapneededtosolvethefirm’sprobleminequation(

2

)requiresinformationabouttheentiredistributionoffirmpricegaps.

Wecircumventthecurseofdimensionalitybyusingthe

KrusellandSmith

(

1998

)approachtocharacterizehowXtevolvesovertimeasafunctionofasinglemomentofthedistributionofFt(s),namely

Specifically,wesimulatealonghistoryofmonetarypolicyshocks,calculatethetimeseriespathforStandXtimpliedbythemodelanduseprojectionmethodstoupdatetheperceivedaggregatepricegapfunction

Xt=X(St),

whereX(·)isalinearcombinationofChebyshevpolynomialsthatallowustocapturepo-tentialnon-linearities.

3.6Parameterization

Aperiodisamonthandwesetthediscountfactorβtoanannualizedvalueof0.96andtheelasticityofsubstitutionσequalto3.Following

NakamuraandSteinsson

(

2010

)and

Auclert

etal.

(2022),intheNSandUniformspecificationsoftheadjustmentcostweassumethat

thevalueofλissuchthat75%ofpricechangesarefree.

Table

1

reportstheresultoftheparameterization.Wecalibratethemeanandvolatilityofthemoneygrowthrateµandσm,thevolatilityofidiosyncraticqualityshocksσzandthe

menucostparametertotargettheaveragefractionofpricechanges,themediansizeofa

pricechange,andthemeanandstandarddeviationofinflation.Thesestatisticsarecomputed

8

fortheperiod1979-2014forwhich

Nakamuraetal.

(

2018

)reportamediansizeofregular

pricechangesof0.075.UsingthedatainFigure

1

,wecomputeanaveragefrequencyofpricechangesof0.105,anaverageannualizedinflationrateof3.4%andastandarddeviationofannualizedinflationof2.6%.AsPanelAofthetableshows,alleconomiesmatchthesetargetsperfectly.

PanelBofthetablereportstheparametervaluesrequiredtomatchthemoments.Since,aspointedoutby

BilsandKlenow

(2004)and

GolosovandLucas

(

2007

),theaveragesizeofapricechangeislargerelativetoaggregateinflation,themodelimpliesthatidiosyncraticshocksareapproximately3–4timesmorevolatilethanaggregateshocks.ThemenucostparameterisrelativelysmallintheGLeconomyandincreasesasweintroducerandomnessinthepriceadjustmentcosts.

Table1:Parameterization

A.Moments

DataGLNSUniform

I.Targeted

fraction∆p

0.105

0.105

0.105

0.105

median|∆p|

0.075

0.075

0.075

0.075

meaninflation

0.034

0.034

0.034

0.034

stddev.inflation

0.026

0.026

0.026

0.026

II.Not

targeted

kurtosis∆p

1.466

2.402

3.346

10thpctile|∆p

0.065

0.012

0.013

25thpctile|∆p

0.068

0.032

0.034

75thpctile|∆p

0.084

0.174

0.136

90thpctile|∆p

0.095

0.196

0.202

B.CalibratedParameterValues

GLNSUniform

μ

meanmoneygrowthrate

0.034

0.034

0.034

σm

s.d.monetaryshocks

0.008

0.009

0.010

σz

s.d.idios.shocks

0.024

0.037

0.037

menucost

0.015

0.246

2.818

Note:Themoneygrowthrateisannualized.Themenucostparameterisexpressedrelativetototalrevenue

inagivenperiod.

9

3.7TheDistributionofPriceChanges

Wenextinvestigatethemodels’implicationsforbroadermomentsofthedistributionofpricechanges.Figure

2

plotsthedistributionofpricechangesimpliedbyeachofthethreeeconomies.Asiswellunderstood,theGLmodelproducesabi-modaldistributionofpricechangeswithneitherverysmallnorverylargepricechanges.ThisisalsoreflectedintheuntargetedstatisticsreportedinPanelAofTable

1

whichshowsthatinthiseconomythedistributionofthesize(absolutevalue)ofpricechangesisverycompressed,rangingfroma10thpercentileof6.5%toa90thpercentileof9.5%.Analternativewayofsummarizingthedispersioninthesizeofpricechangesisthekurtosisofpricechanges,whichisequalto1.5.

BoththeNSandtheUniformeconomiesgeneratemoredispersioninthesizeofpricechangesandahigherkurtosis,butthedistributionintheNSeconomyhasjumps,reflectingthattheadjustmenthazardisastepfunction,asintheGLeconomy.Incontrast,thesmootheradjustmenthazardinducedbytheuniformmenucostdistributionintheUniformeconomyproducesasmoothdistributionofpricechanges.ThoughwedonothavedirectaccesstotheBLSmicro-pricedata,theevidenceonthedistributionofpricechangesinexistingstudiesismostconsistentwiththepredictionsoftheUniformeconomy.Forexample,usingthesameBLSdata

KlenowandKryvtsov

(2008)reportaunimodaldistributionof

regularpricechanges,with25%ofpricechangesbelow2.5%.Thispatternhasalsobeendocumentedforothercountries(see

Alvarezetal.,

2016

forFrance,

Blancoetal.,

2024a

fortheUKand

Gautieretal.,

forthcoming

for11Europeancountries),andforspecificsectors(see

Midrigan,

2011)

.8

4NonlinearDynamics?

Becausethefractionofpricechangesvariesovertimeinmenucosteconomies,thesemodelspredictpotentiallynonlinearresponsestoaggregateshocks.Wenextinvestigatetheextentofthesenonlinearities.

8Arecentpaperby

Alvarezetal.

(2021)usesdatafromagrocerystoreanddocumentsthat,after

controllingformeasurementerror,thedistributionofpricechangesisbimodal,featuringaconsiderabledipnearzero.Thisistrueforallthecountriesintheirsample,exceptfortheU.S.,forwhichthedistributionisunimodal.Suchabimodaldistributioncanberationalizedwithamodelwithrandommenucostsbutnofreepricechanges.WestudythismodelintheAppendixandshowthatithassimilaraggregateimplicationsastheNSandUniformeconomies.

10

Figure2:DistributionofPriceChanges

GL

40

30

20

10

0

-0.4-0.200.20.4

"p

6

4

2

0

NS

8

6

4

2

0

-0.4-0.200.20.4

"p

Uniform

-0.4-0.200.20.4

"p

4.1TheFractionofPriceChanges

WestartbygaugingtheabilityofeachofthethreeeconomiesabovetoreproducethefluctuationsinthefractionofpricechangesobservedintheU.S.timeseries.Tothatend,weusethenonlinearsolutionofeachmodeltobackoutthehistoryofmonetaryshocksμtthatallowseachmodeltoexactlyreproducethetimeseriesofinflationinFigure

1

.9

Figure

3

contraststheseriesforthefractionofpricechangesinthemodelandinthedata.Astheleftpanelshows,theGLmodelpredictssignificantmovementinthefractionofpricechanges,butnotasmuchasinthedata.Forexample,in1980thefractionofpricechangesinthedataincreasedto0.17andinthemodelitincreasedto0.14

.10

Similarly,duringthepost-COVIDinflationepisode,thefractionofpricechangesincreasedto0.23inthedataandtoonly0.13inthemodel.Overall,aregressionofthefractionofpricechangesontheabsolutevalueofinflationforthe1979-2014sampleperiodin

Nakamuraetal.

(

2018

)impliesaslopecoefficientof0.63%inthedataand0.32%inthemodel,sotheGLmodelgenerateshalfofthecomovementbetweeninflationandthefractionofpricechangesinthedata.

ThemiddleandrightpanelsofFigure

3

showthattheNSandUniformeconomiesimplymuchsmallerfluctuationsinthefractionofpricechanges:thefractionofpricechangesneverexceeds12%ineithereconomy.Theslopecoefficientfromregressingthefractionofpricechangesontheabsolutevalueofinflationis0.1%intheNSeconomyand0.07%inthe

9Forsimplicity,weassumedthatthemonetarypolicyshocksaretheonlysourceofaggregateuncertainty.Ourresultswouldbeidenticalifwealsoallowedforaggregateproductivityshocks,providedthesealsofollowarandomwalk,sincewhatmattersistheprocessfornominalmarginalcosts.

10Thesenumbersaresimilartothoseobtainedby

Nakamuraetal.

(2018)

.

0.25

0.15

0.05

Figure3:FractionofPriceChanges

0.2

0.1

0.25

0.15

0.05

GL

data

mod

el

1960198020002020

0.2

0.1

0.25

0.15

0.05

NS

1960198020002020

0.2

0.1

Uniform

1960198020002020

Notes:TheshadedareasrepresentNBERrecessions.

Uniformeconomy,muchsmallerthanthe0.63%inthedata.

In

Blancoetal.

(2024a)weshowthattheresponseofthefractionofpricechangesto

aggregateshocksdependsonthesizeoftheshockrelativetothedispersioninpricechanges.Modelsthataccountforthedispersionofpricechangespredictthatidiosyncratic,ratherthanaggregateshocksdeterminemostoftheadjustmentdecisions,renderingthefractionofpricechangesnearlytime-invariantwhenweconfrontthemodelwithaggregateshocksofsimilarmagnitudetothoseneededtoaccountfortime-seriesfluctuationsintheU.S.inflationrate.Forlargershocks,eventheseversionsofthemenucostmodelgeneratefluctuationsinthefractionofpricechanges.However,suchlargeshockswouldgeneratealotmorevariationininflationthanobservedinthedata.

Thus,weconcludethatnoneofthemenucostmodelsstudiedinthispapercansimul-taneouslyreproducethemicro-pricedataandthecomovementbetweeninflationandthefractionofpricechanges.Ontheonehand,versionsofthemodelthatgeneratesubstantialfluctuationsinthefractionofpricechanges,suchasGL,predicttoolittledispersioninthesizeofpricechanges.Ontheotherhand,versionsofthemodelthatreproducethelargedispersioninthesizeofpricechanges,suchasNSandUniform,predictanearlyconstantfractionofpricechanges.

Asweillustratebelow,thedispersioninthesizeofpricechangescriticallydeterminestherealeffectsofmonetaryshocks,andfluctuationsinthefractionofpricechangesarecrucialindetermininghownonlineartheserealeffectsare.Therefore,animportantchallengeformenucostmodelsistosimultaneouslyaccountforbothofthesefeaturesofthedata.In

Blancoet

11

12

al.

(2024a)weprovideonepotentialsolutiontothischallengeusingamulti-productmenu

costmodelwithalowdegreeofmisallocationinsidethefirm.

4.2RealEffectsofMonetaryShocks

Wenextinvestigatetheextentofnonlinearitiesthatthethreemodelsimplybystudyinghoweachoftheseeconomiesrespondst