EUROPEANCENTRALBANKEUR○SYSTEMPeterKaradi,AntonNakov,GaloNu?o,Strikewhiletheironishot–optimalErnestoPastén,DominikThalermonetarypolicyunderstate-dependentpricingchaMPchaMPDisclaimer:ThispapershouldnotbereportedasrepresentingtheviewsoftheEuropeanCentralBank(ECB).TheviewsexpressedarethoseoftheauthorsanddonotnecessarilyreflectthoseoftheECB.ChallengesforMonetaryPolicyTransThispapercontainsresearchcoChaMPseekstorevisitourknowledgeofmonetmultipleongoingstructursteepinflationwaveanditsreversal.Moreinformationisprovidedonitswebsite.AbstractWecharacterizeoptimalmonetarypolicyunderstate-dependentpricing.Theframeworkgivesrisetononlinearinflationdynamics:Theflexibilityofthepricelevelincreasesafterlargeshocksduetoanendogenousriseinthefrequencyofpricechanges.Inresponsetolargecost-pushshocks,optimalpolicyleveragesthelowersacrificeratiotocurbinflation.Whenfacedwithtotalfactorproductivityshocks,anefficientdisturbance,theoptimalpolicycommitstostrictpricestability.Theoptimallong-runinflationrateisjustabovezero.Keywords:State-dependentpricing,largeshocks,nonlinearPhillipscurve,optimalmonetarypolicyNon-TechnicalSummaryRecentglobaleconomicdevelopments,particularlythesurgeininflationfollowingtheCOVID-19pandemic,havepresentedsignificantchallengesforcentralbanks.Traditionalmodelsofinflation,whichassumealinearandstablerelationshipbetweeninflationandeconomicactivity,havestruggledtoexplainthesedevelopments.Thispaperaddressesakeyquestion:Howshouldcentralbanksadjusttheirmonetarypolicieswhentherelationshipbetweeninflationandeconomicoutputbecomesmorecomplex,particularlyduringperiodsoflargeeconomicshocks?Thepaperarguesthatinperiodsofsignificanteconomicshocks–suchassuddenincreasesincosts–firmstendtochangetheirpricesmorefrequently.Thisbehaviorleadstowhatisknownasa“nonlinearPhillipscurve,”whereinflationbecomesmoresensitivetochangesineconomicoutput.Thecentralfindingisthat,insuchscenarios,thecostsofreducinginflationthroughtightermonetarypolicyarelowerthanpre-viouslythoughtbecausetheeconomyadjustsmorequicklywhenpricesarealreadychangingfrequently.Thepaperalsodemonstratesthatinresponsetoproductivityshocks,thebestpolicyisforcentralbankstomaintainstableprices.Thisfindingalignswithtraditionalmod-elsandsupportstheideathatpricestabilityshouldremainacorefocusofmonetarypolicyduringthesetypesofshocks.Theresearchusesatheoreticalmodelthatincorporatesstate-dependentpric-ing—wherefirms’decisionstoadjustpricesdependontheeconomicenvironment.ThemodeliscalibratedusingdatafromtheUnitedStates.Bycomparingthismodeltoatraditionallinearmodel,theauthorsareabletohighlightthedifferencesinoptimalpolicyresponsesunderdifferenteconomicconditions.Thefindingshaveimportantimplicationsforcentralbanks.Duringperiodsofhighinflation,wherefirmsfrequentlyadjusttheirprices,centralbanksmightconsideramoreaggressivestanceagainstinflation.Thisisbecausetheeconomiccostofsuchpolicies–measuredintermsofreducedoutputorincreasedunemployment–maybelowerthaninperiodsofstableprices.Duringproductivityshocks,maintainingpricestabilityshouldalsoremainthepriority.Inconclusion,thispaperprovidesnewinsightsintohowcentralbankscanopti-mizetheirmonetarypolicyinaworldwheretherelationshipbetweeninflationandeconomicactivityisnotstraightforward.Byrecognizingwhenfirmsarelikelytoad-justpricesmorefrequently,centralbankscanbettertailortheirpoliciestominimizeeconomicdisruptionsandmaintainstableinflation.1.IntroductionWhatistheoptimaldesignofmonetarypolicy?ThetraditionalanswerofferedbytheNewKeynesianliteraturereliesonthepricesettingmodelbyCalvo(1983),whichdisregardsendogenousvariationinthefrequencyofpricechanges:firmsupdatepricesatrandomtimesirrespectiveofmacroeconomicconditions.1Incontrast,arapidlygrowingliteratureonstate-dependentpricesetting,sometimesreferredtoas“menucostmodels”,recognizesthatfirmsdecidewhentoadjustpricesendogenously,takingintoaccountmonetarypolicy.Highinflationperiods,suchastherecentinflationsurgeepisode,haveforcefullyillustratedthatthefrequencyofpricechangesisindeedresponsivetomacroeconomicconditions:intheU.S.,itmorethandoubledatthe2022inflationpeak.2Yet,thenormativeaspectsofthesestate-dependentmodelshavereceivedlimitedattention.Tobridgethisimportantgap,ourpapercharacterizesoptimalmonetarypolicyundercommitmentinthecanonicalmenucostmodelofGolosovandLucas(2007)andshowsrobustnessintheCalvoPlusmodelofNakamuraOuranalysisarrivesatanovelinsight:optimalpolicyleansagainstinflationdis-proportionatelystronglyinresponsetolargecost-pushshocks,whichexertupwardpressureontherepricingfrequency–a“strikewhiletheironishot”policy.Therea-sonsaretwofold.First,thecostoftheanti-inflationarypolicyintermsofoutputissmallerwhenthefrequencyofpricechangesincreasesinresponsetotheshocks–asthepricelevelbecomesmoreflexible,thesacrificeratiofalls.Second,asweexplainbelow,inourstate-dependentframeworktherelativeimportanceofinflationversusoutputinthecentralbank’sobjectivestaysclosetothatintheCalvomodel.Atthesametime,asweshowanalytically,optimalpolicyrequiresfullinflationstabilizationaftertotalfactorproductivityshocks–aversionofthe“divinecoincidence”resultafterefficiencyshocks,asinthecanonicalCalvomodel. Ourbaselinestate-dependentpricesettingmodelcloselyfollowstheseminalpaperofGolosovandLucas(2007).Inthemodel,arepresentativehouseholdconsumesacontinuumofdifferentiatedgoodsandsupplieslaborinacentralized,frictionlessmarket.Eachconsumptiongoodisproducedbyasinglefirmwithlaborastheonlyinput.Productiontechnologyissubjecttoaggregateproductivityandcost-pushshocks,andidiosyncraticqualityshocks.3Firmsmustincurasmall,fixed“menucost”toadjusttheirprices.Thus,firms’pricingdecisionsarecharacterizedbyan(S,s)rule:Whenpricesarewithinanendogenousbandaroundtheoptimalresetprice,firmskeepthemconstant;otherwise,theypaythemenucostandupdatetheirprice.Thecentralbanksetsthenominalinterestrate. 1Woodford(2003);Galí(2008)2SeeMontagandVillar(2023).TheempiricalrelationshipbetweeninflationandfrequencyhasbeenwellestablishedbothintheU.S.andonothercountries.Theliteraturereviewbelowlistsreferences.3WedepartfromtheGolosovandLucas(2007)modelinthisregard,which,insteadofidiosyncraticqualityshocks,assumesproductivityshocks.Thisfacilitatesthecomputation,whileitsimplicationsareinnocuous(seealsoMidrigan2011;Alvarezetal.2021).Westudytheoptimaldesignofmonetarypolicyinthismodel.Tothisend,weproposeanewalgorithmtosolvetheRamseyproblemnonlinearly,sothatitissuitableforassessingtheimpactoflargeaggregateshocks.Inparticular,weapproximatethevalueanddistributionfunctionsovertheendogenouslydeterminedrelevantrangeandsolvethesetofequilibriumconditionsunderperfectforesightoverthesequencespace.WecalibratethemodelparameterstomatchthemonthlyfrequencyofpricechangesintheU.S.beforetheinflationsurge4,aswellasa20%frequencylevelaccompanyinga10%inflationrateasexperiencedduringtheinflationsurgeof2022-2023(MontagandVillar2023).Wecontrasttheimplicationsofourstate-dependentmodelwiththoseofatime-dependentCalvomodel.5Themodeleconomyissubjecttothreewelfare-relevantdistortions.Thefirsttwoarecausedbyactualmarkupsdeviatingfromtheefficientmarkup:thefirstdistortioniscausedbytheaveragemarkup,andthesecondbythedispersionofmarkups.Thethirddistortionistheresourcecostsrelatedtopriceadjustment.ThedistortioncausedbyaveragemarkupisconventionalandispresentbothinourbaselinemodelandinthecanonicalCalvomodel.6Itincentivizesthecentralbanktominimizethevariation,causedbyaggregateshocks,intheaveragemarkup,whichitcanaffectduetopricerigidities.Thebehaviorofthesecondandthirddistortionsisdistinctinthetwoframe-works.Inourframework,aggregateshockscanreducemarkupdispersiononimpact,asnewadjustersareselectedfromthosewiththemostmisalignedmarkups;whileaggregateshocksincreasemarkupdispersionintheCalvoframework.Furthermore,resourcecostsofpricechangesbecomearelevantfactorinourframework,whiletheyarealwayszerointheCalvoframeworkbyconstruction.Thepolicymaker’staskistominimizetheeffectofthosedistortions.Wefindthatoptimalmonetarypolicyshouldleanmoreaggressivelyagainstinfla-tionafterlargecost-pushshockswhenthefrequencyofpricechangesisendogenouslyhighthaneitherafterasmallshockorinafixed-frequencyCalvosetting:itisoptimalto“strikewhiletheironishot”.7ThisnonlinearityestablishesakeydifferencebetweenourmodelandthestandardCalvomodel.Ourcalibrationimpliesthatthisnewpolicyprescriptionisrelevantforthe2022-2023inflationsurge:Alreadyforinflationandfre-quencyvaluesofthemagnitudeobservedduringthisperiod,optimalpolicyrequiresasignificantlymoreaggressiveanti-inflationarystancethanforasmallshockorunderCalvopricing.Whatexplainsthemodifiedpolicyprescription?Togaininsightintothisresult,weintroduceasimplifiedmodel.Inthesimplifiedmodel,weintroduceasub-period4See,forexample,NakamuraandSteinsson(2008).5TheCalvomodelisrecalibratedtogeneratethesameprice-flexibilityasourbaselinemodelforsmallshocks(Auclertetal.2024).Thisrecalibrationcompensatesfortheendogenous“selection”oflargepricechanges,whichsubstantiallyraisestheflexibilityoftheaggregatepricelevel.6Asisstandardinoptimalmonetarypolicyanalysis,weoffsetsteady-stateaveragemarkupdistortionduetothemarketpowerwithsuitablesubsidies.Wereintroducesteady-statemarkupdistortionsonlytoanalyzetheimpactoftime-inconsistency.7WeanalyzetimelessRamseypolicyálaWoodford(2003).ofnight,whenonlythefirmsareawake,andwhenthepricesarefullyflexible.Theassumptionimprovesthetractabilityofthemodelbyturningthedynamicproblemofthefirmsintoaseriesofstaticproblems,butitkeepsthekeyunderlyingchannelactive:therepricingraterespondsendogenouslytoaggregateshocks.Inthesimplifiedmodel,bothwelfareandtheplanner’schoicesetcanbeexpressedinthespaceof(i)theoutputgap,whichmeasuresthedistancebetweenoutputanditsefficientlevel,and(ii)inflation,asisconventionalinoptimalmonetarypolicyanalysisundertheCalvopricesetting.Welfaredependsonthesetwovariablesbecausetheoutputgapisrelatedtotheaveragemarkup,andinflationisrelatedtothemarkupdispersionandtheresourcecostsofpricechanges.Thechoicesetcanalsobeexpressedinthespaceofinflationandoutputgap–andtakestheformofanonlinearPhillipscurve.Itisnonlinearbecauseinflationbecomesmoresensitivetochangesintheoutputgapaslargeshocksraisethefrequencyofpricechangesand,thereby,increasepriceflexibility.8Optimalpolicyleansmoreaggressivelyagainstinflationafterlargecost-pushshocksthanaftersmallshocksinthesimplifiedmodel,justasinthefullmodel.Inthesimplifiedmodel,therelationshipbetweentheoutputgapandinflationunderoptimalpolicycanbeillustratedbyastructural“targetrule.”Theoptimal“strikewhiletheironishot”policytranslatesintoanonlineartargetrule:largeroutputgapsareassociatedwithrelativelylowerinflationratesthansmalleroutputgaps.ThisisinstarkcontrasttothecorrespondingtargetruleintheCalvoframework,whichisalmostlinear.Tounderstandthekeydrivingforcesbehindthispolicyprescription,itisinstructivetostartwiththequestionofwhythetargetruleisalmostlinearintheCalvomodel.There,welfarecanbewellapproximatedbyaquadraticfunctionoftheoutputgapandinflationwithafixedweight9,whilethePhillipscurveisalmostlinear.Optimalpolicythusmaximizesanear-quadraticobjectivesubjecttothenear-linearPhillipscurve.Theresultingpolicyisthusnearlinear.Whyisthesamerelationshipbetweeninflationandoutputgapnonlinearunderstate-dependentpricing?Wefindthatthisisalmostexclusivelydrivenbythenonlineartrade-offbetweeninflationandoutputgap–thenonlinearPhillipscurve.Intuitively,reducinginflationinthisframeworkischeaperafterlargeshocks,whenthefrequencyishigherandthepricelevelismoreflexible,thatis,whenthesacrificeratioislow.Toshowthatthisisthedominantdrivingforce,wecombinethenonlinearPhillipscurveofoursimplifiedframeworkwithacounterfactualquadraticwelfarefunctionapproximatingtheCalvomodelandderiveacounterfactualtargetrule.Weshowthattheensuingtargetruleisclosetothetruetargetruleandissimilarlycharacterizedbythe“strikewhiletheironishot”policy.Astherelativewelfareweightofinflation 8OtherpaperspointtocomplementaryreasonswhythePhillipscurvecanbenonlinear,suchasstate-dependentwagerigidity(BenignoandEggertsson2023)orKimballaggregators(Ercegetal.2024).9Woodford(2003)showsthatwelfarecanbeapproximatedbyaweightedsumofsquaresofoutputgapandinflation,theweightbeingdeterminedbystructuralparametersofthemodel.Thisapproximationappliesintheneighborhoodofanefficientsteadystateuptoasecondorder,i.e.forsmallshocks.Wefindthatthisapproximationworkswellnumericallyalsoforlargeshocks.andoutputgapisindependentoftheshockinthiscounterfactualbyconstruction,theresultshereareclearlydrivenbytheshapeofthePhillipscurve–thenonlinearsacrificeratio.10Thisresultgeneralizestothefullmodel.Therefore,weconcludethatthekeydrivingforcebehindtheaggressiveanti-inflationarystanceafterlargeshocksisthelowersacrificeratio.Weestablishaseriesofadditionalresultsinthefullmodel.First,weshowthatthemodelfeaturesaslightlypositiveRamseyoptimalsteady-stateinflationrate,ataround0.07%perannum.ThiscontrastswiththestandardCalvomodel,whereoptimalinflationisexactlyzero.Inourmenucostmodel,slightlypositivesteadystateinflationreducesthefrequencyandthushelpsfirmstoeconomizeoncostlypriceadjustments.Inparticular,itcounterbalancestheimpactoftoofrequentpriceincreasesrelativetopricedecreases,whichisaconsequenceoftheasymmetryoftheprofitfunction:firmsdislikemorenegativepricemisalignmentswhenthedemandfortheirproductishigh,relativetopositivemisalignmentswhenthedemandislow.Second,wealsofindthatforsmallcost-pushshocks,optimalpolicy“l(fā)eansagainstthewind”:thecentralbanktemporarilydrivesoutputbelowitsefficientleveltocontaintheinflationaryimpactofapositivecost-pushshock.Thisisverysimilar,thoughnotidentical,totheCalvomodel.However,thereasonisdifferent.IntheCalvomodel,thekeydistortioncausedbyinflationisthemarkupdispersion,whileinourbaselinemodelitistheresourcecostsduetomenucosts.Third,weshowanalyticallythattheoptimalresponsetoTFPshocksischaracterizedbythe“divinecoincidence”(BlanchardandGalí2007).Inotherwords,optimalpolicystabilizesbothinflationandtheoutputgap.Finally,weshowthatthewell-knowntimeinconsistencyproblemofoptimalmonetarypolicyisalsopresentinourmenucostmodel,althoughitisattenuatedrelativetoCalvo.Inbothmodels,whenthesteadystateisinefficient,monetarypolicyhastheincentivetostimulateoutputviaanunexpectedlyeasypolicy(Galí2008).However,inthemenucostmodel,suchapolicyislesseffectiveonoutputandmoreinflationarybecausetheensuingincreaseintherepricingrateraisestheflexibilityoftheaggregatepricelevel.Thetime-inconsistentmotivetoeaseisthusconsiderablyweaker. Ourresultsarerobusttoalternativeparameterizations,andalsoholdinthe“Calvo-Plus”model(NakamuraandSteinsson2010).Thelatterframeworkassumesthatthepriceadjustmentcostisstochastic:ittakesapositivevaluewithsomeexogenousprobability,anditiszerootherwise.Thismodelcanbettermatchthefractionofsmallpricechangesinthedata(seealsoMidrigan2011;Alvarezetal.2021),andcanachievearealisticdegreeofmonetarynon-neutralityforsmallshocks.However,asweshow,themodelalsoprescribesamoreaggressiveanti-inflationarystanceafterlargecost-pushshocksthanaftersmallshocks,withanevenhighernonlinearitythanourbaseline.Thisisprimarilybecausethesacrificeratioexhibitsevenmorenonlinearityinthisframeworkthaninourbaseline. 10Thedeviationofthetruewelfarefromthequadraticapproximationactuallysomewhatmitigatesthenonlinearityofthetruetargetrule,butitsimpactisquantitativelysmall.Relatedliterature.OurpaperbuildsontheseminalarticlebyGolosovandLucas(2007).Theyproposeamenucostmodel(Barro1972;SheshinskiandWeiss1977;CaballeroandEngel1993)thatprovidesamicro-foundedstate-dependentalternativetothecanonicaltime-dependentCalvo(1983)model.Theframeworkhasbecomethebackboneofapositiveliterature(GertlerandLeahy2008;Midrigan2011;CostainandNakov2011;Alvarezetal.2016;Auclertetal.2024)11andisshowntodescribefirms’price-settingbehaviorwellindiverseenvironmentswithbothlowandhighinflation(NakamuraandSteinsson2008;Gagnon2009;Alvarezetal.2019;Nakamuraetal.2018),andasresponsetolargeaggregateshocks(KaradiandReiff2019;Alexandrov2020;Aueretal.Tothebestofourknowledge,ourpaperisthefirsttosolveforoptimalmonetarypolicyinthiscanonicalmenucostmodel.Itsmaindistinctivefeatureistheendo-geneityoftheextentofpricestickiness:thefrequencyofpricechangescanvarywithmacroeconomicconditionsandthusitisendogenoustoshocksandpotentiallytomonetarypolicyitself.ThisisdifferentfromthecanonicaltextbookanalysisofoptimalmonetarypolicybasedonCalvo(1983),suchasinWoodford(2003)andGalí(2008).Changesinfrequencyhavebeendocum(KaradiandReiff2019;AlvarezandNeumeyer2020;Aueretal.2021;Gagliardoneetal.2025;Gautieretal.2025),andinhigh-inflationenvironments(Gagnon2009;Alvarezetal.2019;NakamuraandSteinsson2018),andhavereceivednewempiricalsupportfollowingtherecentU.S.inflationsurge(MontagandVillar2023;Cavalloetal.2024;Blancoetal.2024a).Variationinfrequencyimpliesastate-dependent,nonlinearrelationshipbetweeninflationandtheoutputgap(Vavra2014;Blancoetal.2024b).Ourconclusionprescribinganaggressiveanti-inflationarypolicyafterlargeshocksisadirectconsequenceofthisnonlinearity,whichimpliesafavorableinflation-outputtrade-offthatoptimalpolicyshouldexploit.Solvingdynamicoptimalpolicyinresponsetoaggregateshocksinthisframeworkcomplementspreviousresearchonoptimalmonetarypolicy,whichhasrestricteditsattentiontomenucostsettingswitharepresentativefirmandsmallaggregateshocks(NakovandThomas2014),sector-specificproductivityshocks(CaratelliandHalperin2023)ortooptimalsteady-stateinflation(AdamandWeber2019;Blanco2021;NakovandThomas2014).12ThispaperproposesanewalgorithmtosolveRamseyoptimalpolicyinheterogeneous-agentmodels,buildingonGonzálezetal.(2024).Thealgorithm(i)makestheinfinite-dimensionalplanner’sproblemfinite-dimensionalbyapproximat-11Akeyquestionoftheliteratureistherelationshipbetweenmonetarynon-neutralityandthedistri-butionofpricechangesatthemicrolevel.Thisisnotthefocusofourpaper.Weshowthatourresultsonoptimalmonetarypolicyafterlargeshocksarerobustacrossmodelswithverydifferentimplica-tionsaboutmonetarynon-neutrality,liketheGolosovandLucas(2007)modelontheonehandandtheCalvoPlusmodelofNakamuraandSteinsson(2010)ontheother.12NakovandThomas(2014)findnosignificantdifferencebetweenCalvoandarandommenucostmodel.CaratelliandHalperin(2023)showthat,inthefaceofsector-specificshocks,optimalpolicycanbecharacterizedasnominalwagetargeting.ingtheinfinite-dimensionalvalueanddistributionfunctionsbypiece-wiselinearfunctionsonagrid;(ii)accountsforthediscreteprice-adjustmentchoiceusinganendogenousgrid;(iii)derivestheFOCsoftheplanner’sproblembysymbolicdifferentiation;and(iv)solvestheresultingsetofequilibriumconditionsnonlinearlyunderperfectforesightoverthesequencespace.OurapproachcomplementsothermethodstosolveforRamseypolicyinheterogeneous-agentmodels(Bhandarietal.2021;LeGrandetal.2022;DávilaandSchaab2022;Nu?oandThomas2022;Smirnov2.ModelInthebaselineeconomyarepresentativehouseholdconsumesabasketofdifferenti-atedgoodsandsupplieslabor;monopolisticfirmsproduceusingatechnologythatisaffectedbybothaggregateandidiosyncraticshocksandmustpayafixedmenucosttochangeprices;andacentralbanksetsinterestrates.Timeisdiscreteandthereisnoaggregateuncertainty.WecompareourbaselineeconomytoaCalvoeconomy,whichisidenticaltoourbaselineexceptfirmsadjusttheirpriceswithanexogenousprobability.ArepresentativehouseholdconsumesCt,suppliesworkinghoursNtandsavesinone-period,nominalbondsBt,whichareinzeronetsupply.ThehouseholdmaximizessubjecttowhereTtarelump-sumtaxes,Wtisthenominalwage,Dtarelump-sumdividendsfromfirms,andQt三e-itisthepriceofthenominalbondanditisthenominalinterestrate.AggregateconsumptionCtiswhereCt(j)isthequantityofproductj∈[0,1]andAt(j)istheidiosyncraticqualityofproductj,whichfollowsarandomwalkinlogswithvolatilityσ:logAt(j)=logAt-1(j)+σεt(j),(4)εtisani.i.dGaussianinnovationandσisaparameter.ThedemandforproductjiswherePt(j)isthepriceofproductj,andtheaggregatepriceindexisWeassumeseparableutilityu(Ct,Nt)=logCt-NtasinMidrigan(2011).Thus,equilibriuminthelabormarketrequires:wherewt=Wt/Ptistherealwage.TheEulerequationis1=[Λt,t+1eit?πt+1],(8)whereitisthenominalinterestrate,andthethestochasticdiscountfactoris2.2.MonopolisticproducersGoodj∈[0,1]isproducedbyfirmjaccordingtoaconstant-returnstoscaletechnologywhereNt(j)islaborhours,AtisaggregateproductivityandAt(j)isidiosyncraticquality.Firmsmaximizethesumofthediscountedfutureprofits,usingthehousehold’sdiscountfactor.Theytakethedemandfunction(5)asgiven.Firmj’snominalprofitfunctiongivenitsnominalpricePt(j)isDt(j)=Pt(j)Yt(j)-(1-τt)WtNt(j)whereτtisanemploymentsubsidyfinancedbylump-sumtaxes,andwhereforthesecondlinewehaveusedthegoodsmarket-clearingconditionYt(j)=Ct(j)andconditions(5)and(10).Crucially,firmjmustincurafixed“menucost”ηinlaborunitstochangeitsprice.ThefirmchooseseachperiodwhethertoupdateitsnominalpricetoanewoneP(j),ortokeepthepricefromlastperiodPt?1(j).Thisisthesourceofendogeneityofpricestickinessinthemodel.Itisusefultoexpressthefirms’optimalpricingdecisionasafunctionofthepricegapxt(j)三pt(j)?p(j),whichisthelogdistancebetweenthecurrent,三,andtheoptimalquality-adjustedrelativepriceofgoodj, 三.Thankstotheconvenientassumptionsofrandom-walkqualityshocksasinMidrigan(2011),thepricegapistheonlyfirm-levelstatevariablethatthepricingdecisiondependsupon.Thefirms’optimalpricingpolicyfollowsaSsrulesuchthatafirmjkeepsitsnominalpricePt(j)constantifxt(j)∈[st,St],andresetsittotheoptimalpriceP(j)otherwise(equivalenttosetxt(j)=0).Subindextsubsumesallaggregatestates.IdiosyncraticqualityshocksgeneratesheterogeneityacrossnominalresetpricesP(j);however,quality-adjustedrelativeresetpricesp(j)areallidentical.Thus,wedropsubindexjtosimplifynotation.Whenafirmkeepsitsnominalpriceconstant,itspricegapevolvesaccordingtowhereistheinflationofthequality-adjustedrelativeoptimalresetprice.TheoptimalityconditionsforthepricingrulerequireV(0)=0,Vt(0)?ηwt=Vt(st),(15)Vt(0)?ηwt=Vt(St),(16)wherefirms’endofperiodvaluefunctionVt(.)isexpressedonlyintermsofpricegapsasallotherstatesareaggregateandaresubsumedbyatimesubindex.Equation(14)requiresthatthevaluefunctionismaximizedattheoptimalresetprice(x=0),andequations(15,16)requireindifferencebetweenresettingthepriceandpayingthemenucostversuskeepingpricesconstantattheendogenousSsthresholds.ThevaluefunctionequalswherethecurrentrealprofitsΠt(x)aregivenbyThevaluefunctionisthesumofcurrentprofitsandthediscountedcontinuationvalue.Thelatterdependsonthefirms’pricegapnextperiod,which,unlesschanged,evolvesaccordingto