What Can Measured Beliefs Tell Us About Monetary Non-Neutrality?

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) What Can Measured Beliefs Tell Us About Monetary Non-Neutrality? Hassan Afrouzi, Joel P. Flynn, Choongryul Yang 2024-053 Please cite this paper as: Afrouzi, Hassan, Joel P. Flynn, and Choongryul Yang (2024). “What Can Measured Beliefs Tell Us About Monetary Non-Neutrality?,” Finance and Economics Discussion Series 2024-053. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.053. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

What Can Measured Beliefs Tell Us About Monetary Non-Neutrality?* HassanAfrouzi† JoelP.Flynn‡ ChoongryulYang§ ColumbiaUniversity&NBER YaleUniversity FederalReserveBoard June5,2024 Abstract Thispaperstudieshowmeasuredbeliefscanbeusedtoidentifymonetarynon-neutrality.In a general equilibrium model with both nominal rigidities and endogenous information acquisition,weanalyticallycharacterizefirms’optimaldynamicinformationpoliciesandhow theirbeliefsaffectmonetarynon-neutrality. Wethenshowthatdataonthecross-sectional distributionsofuncertaintyandpricingdurationsarebothnecessaryandsufficienttoidentify monetarynon-neutrality.Finally,implementingourapproachinNewZealandsurveydata,we findthatinformationalfrictionsapproximatelydoublemonetarynon-neutralityandendogeneityofinformationisimportant:modelswithexogenousinformationwouldoverstatemonetary non-neutralitybyapproximately50%. JELCodes:E31;E32;E71 KeyWords:measuredbeliefs,nominalrigidities,rationalinattention,monetarynon-neutrality. *Thispaperwasformerlycirculatedunderthetitle: “SelectioninInformationAcquisitionandMonetaryNon- Neutrality.”WearegratefultoOlivierCoibionandYuriyGorodnichenkofortheirinvaluablefeedbackandforsharing data.WealsothankSarojBhattarai,MarkDean,ChenLian,GiuseppeMoscarini,KarthikSastry,LuminitaStevens,and MichaelWoodfordaswellasseminarparticipantsatColumbiaUniversity,UniversityofNotreDame,LMUMunich, PrincetonUniversity,UniversityofTexasatAustin,theFederalReserveBanksofCleveland,Richmond,andSt.Louis, theFederalReserveBoard,andtheNBERMonetaryEconomicsconferenceforthoughtfulcommentsandsuggestions. TheviewsexpressedhereinarethoseoftheauthorsanddonotreflecttheviewsoftheFederalReserveBoardorany personassociatedwiththeFederalReserveSystem. †DepartmentofEconomics,420W118thSt,NewYork,NY,10027.Email:hassan.afrouzi@columbia.edu. ‡DepartmentofEconomics,30HillhouseAvenue,NewHaven,CT,06511.Email:joel.flynn@yale.edu. §20thSt.andConstitutionAve.,NWWashington,DC20551.Email:choongryul.yang@frb.gov.

1 Introduction Recentsurveyevidencedemonstratesthattheaveragefirmholdshighlyinaccurateanddiffuse beliefsaboutitseconomicenvironmentandthatthereissubstantialheterogeneityinthesebeliefs across firms (see Candia, Coibion, and Gorodnichenko, 2023, for a review). While these facts areinterestingontheirown,howthesesurveysconnecttoandinformourcoremacroeconomic theories remains largely ambiguous both qualitatively and quantitatively. To make progress in mappingsurveyedbeliefstomacroeconomicoutcomes,weinvestigatewhatmeasuredbeliefstell usaboutoneofthemostcentralquestionsinmacroeconomics: theeffectsofmonetarypolicyon realaggregateoutput,orinshort,monetarynon-neutrality. Concretely,weask: whatdomeasured beliefstellusaboutmonetarynon-neutrality? Inansweringthisquestion,weprovideananalytical bridgebetweensurveymeasuresofbeliefsandourtheoreticalunderstandingofhowbeliefsmediate therealeffectsofmonetaryshocks. Theoretically,itiswellunderstoodthatmonetarynon-neutralitydependsontheresponsiveness offirms’beliefstomonetaryshocks(seee.g.,Woodford,2003,Nimark,2008,AngeletosandLa’O, 2009, Angeletos and Lian, 2018, Baley and Blanco, 2019). Moreover, beliefs are not something outsideofafirm’scontrol—firmsshouldacquireinformationtomakebetterdecisionswhenitis valuabletodoso(Sims,2003,Mac´kowiakandWiederholt,2009). Thisisparticularlyimportant because firms change their prices infrequently (see e.g., Bils and Klenow, 2004, Nakamura and Steinsson, 2008), tying the value of information to the arrival of price-setting opportunities. It followsthatforatheorytorelatesurveyedbeliefstomonetarynon-neutrality,nominalrigidities andendogenousinformationacquisitionareminimalnecessaryingredients. Accordingly,westudyageneralequilibriummonetaryeconomywithtime-dependenthazard rates for price adjustment that, for instance, nests both the Calvo (1983) and the Taylor (1979) models. In this otherwise standard model, we allow firms to acquire any dynamic information structure about their marginal costs of production, subject to a cost that is proportional to the flow of the information acquired, which is a standard way of modelling information costs (see Mac´kowiak,Mateˇjka,andWiederholt,2023,forareview). Usingthisframework, weprovidethreekeyresults. First, weanalyticallycharacterizefirms’ optimaldynamicinformationpoliciesandthegeneralequilibriumoutputresponsetoamonetary shock. Most importantly, we find that the interaction of nominal rigidities with endogenous informationacquisitionhasanovelandquantitativelyimportanteffectonmonetarynon-neutrality. Second, we derive theoretical results on how to identify the model using cross-sectional data 1

on the duration of pricing spells and firms’ subjective uncertainty about their desired prices. We further show that data on subjective uncertainty are not only sufficient but also necessary, makingsurveymeasuresofbeliefsessentialforidentification. Third,usingsurveydataonfirms’ expectationsfromNewZealand(Coibion,Gorodnichenko,andKumar,2018),wefindquantitatively that information rigidities with endogenous information acquisition approximately double the real effects of monetary shocks relative to the perfect information benchmark. Moreover, the endogeneity of information is important: if we estimated our model to match the data while imposingexogenousinformation,thenwewouldoverstatetherealeffectsofmonetarypolicyby approximately50%. TheoreticalResults:OptimalUncertaintyandMonetaryNon-Neutrality.Webeginouranalysisby theoreticallycharacterizingfirms’optimalinformationacquisition.Weshowthatthistakesasimple form: acquireinformationonlywhenchangingpricesandacquireexactlyenoughinformationto ∗ resetposterioruncertaintyabouttheoptimalpricetosomestate-independentlevel,U . Intuitively, whilebeingbetterinformedreducesthecostsofachievinganygivenlevelofuncertaintyinthefuture asyouneedtoacquirelessinformation,itdoesnotaffectthemarginalcost ofreduceduncertainty. Moreover,weshowthat:theoptimallevelofuncertaintyisdecreasinginthefirm’sdemandelasticity (asthisincreasesthelossesfromsettingthewrongprice);increasinginthevolatilityofmarginal costs(asthisreducesthevalueofinformationacquiredtodayforfuturedecisions);andambiguously affectedbypricestickiness(asthisbothincreasesthevalueofinformationforthispricingspelland decreasesthevalueofinformationforallfuturepricingspells). Akeyimplicationofthisresultisthatafirm’suncertaintyisincreasinginthedurationofits pricingspell. Thisimpliesthatprice-settingfirmsaretheleastuncertainfirmsintheeconomy. Wecallthisphenomenonselectionininformationacquisitionasprice-settingfirmsarethemost informed in the cross-section at any given point in time. This differentiates our model relative toalternativeswithexogenousinformationalfrictionsandnominalrigidities(asinNimark,2008, Angeletos and La’O, 2009) or models of endogenous information acquisition without nominal rigidities(e.g.,Sims,2003,Mac´kowiakandWiederholt,2009): inbothsuchcases,firms’uncertainty hasnorelationshipwiththedurationoftheirpricingspell. Next,westudytherealeffectsofmonetaryshocksbycharacterizingthecumulativeimpulse response(CIR)ofaggregateoutputtoaone-timeunexpectedincreaseinnominalmarginalcostsof firms. Normalizingtheshocksizesothattheimpactresponseofoutputis1percent,wedenotethe 2

CIRbyMb andderiveaclosed-formrepresentationforitas: ∗ U Mb=D¯ + (1) σ2 where D¯ is the average duration of (ongoing) pricing spells as measured in a cross-section of firms, σ2 is the variance of shocks to firms’ idiosyncratic productivity, andU ∗ is the subjective uncertaintyofprice-settingfirmsabouttheirmarginalcosts. Inthisformula,thefirsttermisthe usualonederivedinmodelsoftime-dependentpricestickiness(asinCarvalho,2006,Carvalho andSchwartzman,2015),andcapturesthenotionthat(allelseequal)monetarynon-neutrality increasesasfirms’nominalpricesarestickierforlonger. Thesecondtermisnewtoouranalysis andcapturesthelifetimelackofresponsivenessofallfirmsintheeconomyinresettingtheirprices inlightoftheiruncertainty. Intuitively,whenpriceresettingfirmsaremoreuncertain,theyrespond totheircurrentinformationtoalesserdegreeandsoadjusttheirpricesbylessinresponsetoa monetary shock. Moreover, when microeconomic volatility is higher, firms know that their old informationislesslikelytobeusefulasthingswillhavesincechangedbyalargeramount; this makesfirmsmoreresponsivetotheirinformationandlowerstheextentofmonetarynon-neutrality. ∗ Interestingly,asU movesambiguouslywhenpricestickinesschanges,increasesinpricestickiness haveanambiguouseffectonmonetarynon-neutralityinthepresenceofendogenousinformation. Thisresultestablishesthatuncertaintyamplifiestherealeffectsofmonetaryshocksrelative toafull-informationbenchmark. However,duetoselectionininformationacquisition,lookingat thedatathroughthelensofanexogenousinformationmodelwouldsystemicallyoverstatethereal effectsofmonetaryshocksbyrelatingthemtotheuncertaintyoftheaveragefirm. OurfinaltheoreticalresultsestablishthatthesufficientstatisticsthatdeterminetheCIR(asper Equation1)canbeestimatedgivencross-sectionaldataonfirms’uncertaintyandthetimesince theylastresettheirprice. Thus,surveydataonthesequantitiesaresufficienttoidentifythemodel. Moreover,wefindthatsuchdataarenecessary,incontrasttobenchmarkmodelsinwhichfirms havefullinformationabouttheirenvironment—itiswellknownthatcross-sectionaldataonthe distributionoffirms’pricechangesaresufficientforidentifyingtherealeffectsofmonetaryshocks insuchmodels(see,e.g.,CarvalhoandSchwartzman,2015,Alvarez,LeBihan,andLippi,2016). ThiscanindeedbeseeninEquation1,asU ∗=0underfullinformationandthesufficientstatistic collapsestoD¯,whichcanbemeasuredusingdataonpricechanges. However,weshowthat,in thepresenceofinformationcoststhatimplyapositivedegreeofuncertaintyforprice-setters,i.e., U ∗ >0, dataonthe distributionofpricechangescannot identifyU ∗ . Thisisbecausethe firm’s ∗ choiceofinformationrendersthedistributionofpricechangesinvariant toU . 3

UsingSurveyDatatoQuantifytheModel.Finally,weadopta“micro-to-macro”approachofcombiningmeasuredbeliefswiththestructureofthemodeltoquantifytheextenttowhichimperfect informationandendogenousinformationacquisitionmatterformonetarynon-neutrality. First,byintegratinganewquestionintoasurveyofNewZealandfirmsbetweenQ42017and Q22018(implmenetedbyCoibion,Gorodnichenko,andKumar,2018,Coibion,Gorodnichenko, Kumar,andRyngaert,2021),weobtaininformationonfirms’uncertaintyabouttheiroptimalreset pricesandthelengthoftheircurrentpricingspell. Second,applyingtheestimatorsfortheCIRfromthetheorytothesurveydata,wefindthat accountingforuncertaintyapproximatelydoublestheCIRthatonewouldobtainunderfullinformation. Moreover,ignoringtheendogeneityofinformationacquisitionwouldleadustooverstate thesizeoftheCIRbyapproximately50%. Thus, wearguethatbothimperfectinformationand endogeneityofinformationacquisitionarequantitativelyimportant. Finally,byusingthefirm’sfirst-orderconditionforitsoptimaluncertainty,wecanderiveand implementestimatorsoftheeffectofcounterfactuallyincreasingmicroeconomicvolatilityand pricestickinessontheCIR.Wefindthatgreatermicroeconomicvolatilitysignificantlydampens the real effects of monetary policy. This is because the direct effect of reducing firms’ reliance onpastinformationquantitativelydominatestheindirecteffectthatfirmsoptimallychooseto belessinformedinthefaceofthisincrease. Wealsofindthatgreaterpricestickinessincreases monetarynon-neutralitybutbyapproximately20%lessthanwithfullinformation. Thisisbecause wefindthatfirmswouldbecomebetterinformedinthefaceofincreasedstickiness. Thishappens asincreasingthedurationoverwhichinformationgatheredtodayisusedquantitativelydominates thereductioninthevalueofinformationforfuturepricingspells. Related Literature. At a broad level, our research connects the literature on the real effects of monetaryshocksinpricingmodelswithtime-dependentnominalrigidities(e.g.,Carvalhoand Schwartzman, 2015) and the field of rational inattention models that incorporate endogenous informationacquisition(e.g.,Sims,2003). Furthermore,ourfindingsregardingthenecessityand sufficiencyofmeasuredbeliefsforquantifyingtheaggregateeffectsofmonetaryshocksalignwith a broader body of literature advocating for the development and utilization of new datasets to measureattention(seeCaplin,2016,forareview). Ourkeycontributionistoshowhowmeasured beliefscanbeusedtoquantifytheimportanceofinformationalfrictionsandtheendogeneityof informationacquisitionfortheeconomy’sresponsetomonetaryshocks. More specifically, our focus on using firms’ measured beliefs connects our analysis to the recentliteraturestudyinghowfirmsformtheirexpectationsandhowtheirexpectationsaffecttheir 4

decisions. UsingthesurveyofNewZealandfirms’macroeconomicbeliefs(thatwealsouseinthis paper), Coibion, Gorodnichenko, and Kumar (2018) and Coibion, Gorodnichenko, Kumar, and Ryngaert(2021)studythedeterminantsoffirms’inattentivenesstoaggregateeconomicconditions, howfirmsupdatetheirbeliefsinresponsetonewinformation,andhowchangesintheirbeliefs affecttheirdecisions. Astheseanalysesdonotbridgetheoryanddata,theydonotspeaktothe quantitativerelevanceoffirms’beliefsformacroeconomicoutcomes. Inthissense,anotableand complementarycontributionisRoth,Wiederholt,andWohlfart(2023),whichusessurveydatato quantifytherealeffectsofmonetaryshocksinamodelwithheterogeneoushouseholdsbutno nominalrigidities.1 Within the domain of monetary shocks and their real effects, this paper builds on and contributestoseveralstrandsofliteraturethatstudytherealeffectsofmonetarypolicyshocksunder time-dependentpricestickinessorinformationalfrictions. First,onthenominalrigiditiesside, our work builds onthe work ofCarvalho andSchwartzman (2015) and the general equilibrium model of Alvarez, Le Bihan, and Lippi (2016), who study monetary non-neutrality in time- and state-dependentmodels,respectively. Wecontributetothisliteraturebyintroducingendogenous information acquisition into time-dependent models and showing that it has a quantitatively importanteffectontherealeffectsofmonetaryshocks. Second,ourworkisalsorelatedtotheliteratureonpricingmodelswithnonominalrigidities butwitheitherexogenousinformationalfrictions(MankiwandReis,2002,Woodford,2003,Nimark, 2008,AngeletosandLa’O,2009)orendogenousinformationacquisitionwithrationalinattention (e.g.,Sims,2003,Mac´kowiakandWiederholt,2009).2 Webuildonthisliteraturebyinvestigatingthe realeffectsofmonetarypolicyshocksinaunifiedframeworkfeaturingbothinformationacquisition andnominalrigiditiesandfindthattheirinteractionisbothqualitativelyandquantitativelyimportant.Inthiscontext,themostrelatedworksareWoodford(2009),Stevens(2020)andthemorerecent workofMorales-JiménezandStevens(2024),whomicro-foundstate-andtime-dependentnominal rigiditiesthroughrationalinattentiontothetimingofpricingdecisions. Tofocusonthedegreeof endogenousstate-dependence,thesepapersassumethatthe“referencedistribution”forthecostof informationistheunconditionaltime-invariantonethatemergesinthesteadystate,whichisa 1SeealsoAfrouzi(2024),whichusessurveydatatoquantifyamodelwherefirmsfacingmorecompetitorsarebetter informedaboutaggregateinflationaswellasYang(2022)whousessurveydatatostudyamodelwherefirmswitha greaterproductscopehavebetterinformationaboutaggregateeconomicconditions. 2SeealsoMoscarini(2004),Sims(2010),PacielloandWiederholt(2014),Mac´kowiak,Mateˇjka,andWiederholt(2018), AfrouziandYang(2021).WereferthereadertoMac´kowiak,Mateˇjka,andWiederholt(2023)foracomprehensivereview. AnothernotablecontributionbeyondtherationalinattentionframeworkisReis(2006),whichstudiestheoptimal informationacquisitionoffirmsunderfixedobservationcostsbutnonominalrigidities. 5

necessaryformulationtokeepthefirm’sproblemtractable(Morales-JiménezandStevens,2024). In thispaper,itisnecessaryforustokeeptrackofthesereferencedistributionsasstatevariablesfor firms(i.e.,thepriorsoffirmsabouttheirdesiredprices),whichdovaryendogenouslyovertime becausefirmsoptimallyacquiremoreinformationwhentheirpriorsaremorediffuse—aresultthat isintegraltoourfindingselectionininformationacquisitionandwhichhassignificantquantitative implications. Thus,tofocusonthesetime-varyingpriors,weinsteadtaketime-dependentnominal rigidities as given and abstract away from their state-dependence to keep the firms’ problems tractable. Futureworkthatallowsforbothtime-varyingpriorsandstate-dependencewouldbea naturalstepforwardforunderstandingtheimplicationsofthesemodels. In a broader sense than rational inattention with Shannon entropy costs, our paper is also relatedtotheworkthatstudiesnominalrigiditiesandinformationcostsjointly. Alvarez,Lippi,and Paciello(2011,2016),andBonomo,Carvalho,Garcia,Malta,andRigato(2023)studymodelswith bothmenucostsandobservationalcosts,wherefirmsdecidewhentheyobserveeitheridiosyncratic shocksoraggregateshocksbypayingafixedcost. Inthesemodels,firmscanperfectlyobservethe underlyingshockswhenevertheypaythefixedcost. Asaresult,theirpriorinformationsetatthe timeofinformationacquisitionbecomesirrelevantastheybecomefullyawareoftheirmarginal cost. In our framework, firms decide how much information they want to acquire. As a result, firmsdonotbecomefullyawareoftheirmarginalcostuponacquiringinformation. Thisrelates their current prices to the full history of their past information sets, which is not ex ante trivial tocharacterize. Moreover,beyondthetechnicalcontribution,thisisthenovelcomponentofour theorythatgivesrisetothequantitativelyimportantsubjectiveuncertaintyterminEquation1.3 2 Model: StickyPriceswithInformationAcquisition Westudyageneralequilibriummonetaryeconomywithendogenousinformationacquisitionby firmsthataresubjecttogeneral,time-dependentpricingfrictions. Tomaketheroleofinformation acquisitionasclearaspossible,themacroeconomicsideofthemodelfollowsGolosovandLucas (2007), Alvarez and Lippi (2014), and Alvarez, Le Bihan, and Lippi (2016). Conditional on this canonicalstructure,ourtheoreticalgoalistoanswertwoquestions: howdofirmsoptimallyacquire information? Howdoesthechoiceofinformationaffectmonetarynon-neutrality? 3Infact,iffirmsweretobecomefullyawareoftheirmarginalcostsuponacquiringinformation,thistermwouldbe zeroandthesufficientstatistic,inourframework,wouldcollapsebacktotheaveragedurationofongoingspells,D¯.In thissense,ourframeworkisalsorelatedtoGorodnichenko(2008)andYang(2022),whichstudymenucostmodels withpartialinformationacquisition.Ouranalyticalapproachcontributestothisliteraturebysheddinglightonhow measuredbeliefscanbeusedtoidentifytherealeffectsofmonetaryshocks. 6

2.1. Households Primitives.Timeiscontinuousandindexedbyt ∈[0,∞]. ArepresentativehouseholdhaspreferencesoverconsumptionC ,realmoneybalancesM /P (whereM ismoneyandP isthepriceof t t t t t consumption),andlaborL givenby: t (cid:90) ∞ (cid:34) C 1−γ (cid:181) M (cid:182) (cid:35) e −rt t +log t −αL dt (2) 1−γ P t 0 t wherer >0isthediscountrate,γ−1>0istheelasticityofintertemporalsubstitution,andα>0 indexestheextentoflabordisutility. Consumptionisaconstantelasticityofsubstitutionaggregate ofacontinuumofvarieties,indexedbyi ∈[0,1]: η (cid:181)(cid:90) 1 1 η−1 (cid:182)η−1 C = A η C η di (3) t i,t i,t 0 whereη>1istheelasticityofsubstitutionbetweenvarietiesand A isavariety-specifictasteshock. i,t The household can also trade a risk-free nominal bond in zero net supply that pays a nominal interestrateofR . Thus,thehousehold’slifetimebudgetconstraintis: t (cid:90) ∞ (cid:181) (cid:90) t (cid:182)(cid:183) (cid:90) 1 (cid:90) 1 (cid:184) M + exp − R ds w L + Π di− P C di−R M dt =0 (4) 0 s t t i,t i,t i,t t t 0 0 0 0 wherew isthewage,P isthepriceofvarietyi attimet,andΠ isthenetnominalprofitoffirmi t i,t i,t attimet. ThemoneysupplyisconstantandequaltoM¯. Later,whenwedomonetaryexperiments, wewillshockM¯ toM¯ +δforsomesmallvalueofδ∈R. OptimalityConditions.Asiswell-known,thissetupimpliesthefollowingoptimalityconditions, whichreduceunderstandingaggregatedynamicstounderstandingtheprice-settingdecisionsof eachfirmintheeconomy. First,thehousehold’sdemandforconsumptionvarietyi attime t is givenby: (cid:181) (cid:182)−η P C =A C i,t (5) i,t i,t t P t wheretheaggregatepriceindexisgivenby: P = (cid:181)(cid:90) 1 A P 1−η (cid:182) 1− 1 η (6) t i,t i,t 0 andaggregateconsumptiondemandisgivenby: C −γ =α P t (7) t w t Moreover,nominalwagesandtheinterestratearegivenby: w =αrM and R =r (8) t t t 7

2.2. Firms’Production,Pricing,andProfits ProductionTechnology.Eachvarietyi ∈[0,1]isproducedbyafirmwiththesameindex. Firms produceoutputY accordingtothelinearproductiontechnology: i,t 1 Y = L (9) i,t i,t Z i,t whereL isthelaborinputand Z isamarginalcostshocktothefirm. AsinAlvarezandLippi i,t i,t (2014),wemakethesimplifyingassumptionthat Z 1−η A =1,whichimpliesthat(log)marginal i,t i,t costisperfectlycorrelatedwith(log)demand. Moreover,weassumethat: Z =exp{σW } (10) i,t i,t where{W i,t } t≥0 isastandardBrownianmotionthatisindependentacrossi ∈[0,1]. Time-DependentPricing.Firmsarepricesettersandsubjecttotime-dependentpricingfrictions. Formally, price change opportunities for firm i are governed by the Poisson process N which i,t is independent across i ∈[0,1]. We assume that the distribution of price reset opportunities is exogenouslygivenbyG. WemoreoverassumethatG admitsadensityg anddefineitshazardrate asθ(h)≡g(h)/(1−G(h)). Thisgeneralmodeloftime-dependentpricingnestsseveralimportantbenchmarks,including Calvo(1983)pricinginwhichpriceresetopportunitiesariseataconstantrate: Example1(CalvoPricing). Priceresetopportunitiesariseataconstantrateθ(h)=θ. △ Amoregeneralformulation,inwhichG doesnotadmitadensity,alsoallowsforTaylor(1979) pricing,underwhichfirmsresettheirpricesperiodically. Allofourresultsholdunderthisspecification: Example2(TaylorPricing). Priceresetopportunitiesariseeveryk∈R +periodsandsog =δ k ,where δ isaDiracdeltafunctiononk. △ k ApproximatingFirms’Profits.Giventheirpriceatagiventime,firmscommittohiringenough labor to meet demand at their given price. Define the (log) optimal price of the firm as q ≡ i,t (cid:179) η (cid:180) log w Z and the (log) price of the firm as p ≡ logP . Approximating the firm’s profit η−1 t i,t i,t i,t functiontosecond-orderaroundp =q ,asiswell-known,thefirm’slossfrommispricingrelative i,t i,t totheoptimumisgivenby: L(p ,q )=− B (cid:161) p −q (cid:162)2 (11) i,t i,t i,t i,t 2 8

whereB =η(η−1). Intuitively,whenthefirmfacesmoreelasticdemand,thelossesfrommispricing arelarger. 2.3. Firms’CostlyInformationAcquisition Sofarwehavefollowedthetextbookmodeloffirmpricingingeneralequilibrium.Wenowintroduce thenovelfeatureofouranalysis: endogenousinformationacquisition. Weassumefirmsareaware oftheirpricechangeopportunities,i.e.,theyobservetheprocessN ,butcannotdirectlyobserve i,t theshocktotheirmarginalcostsandacquireinformationaboutthisprocesssubjecttoacost. Formally, given the joint measure for the process {(W ,N ): t ≥0}, firm i chooses a joint i,t i,t measure for {(W ,N ,s ):t ≥0}, observes realizations of the process s along with N and i,t i,t i,t i,t i,t makesdecisionsattimet giventheinformationsetSt ≡{(s ,N ):h≤t}∈S t. i i,h i,h We assume that the cost of acquiring information is given by mutual information à la Sims (2003). Formally,givenaninformationstructure{St :t ≥0},wemeasuretheamountofinformation i acquiredbyfirmi uptotimet asthemutualinformationbetweenthehistoryofthemarginalcost shock, W t ≡{W :h ≤t}, andthe informationsetSt. Thus, lettingµWS bethe measureforthe i i,h i i,t process{(W ,s ,N ):h ≤t},andµW ⊗µS betheproductmeasureinducedbyµWS,mutual i,h i,h i,h i,t i,t i,t informationisdefinedby: (cid:90) (cid:195) dµWS (cid:33) I(µWS)≡ log i,t dµWS (12) i,t d(µW ⊗µS ) i,t i,t i,t wheretheterminsidethelogarithmistheRadon-Nikodymderivativebetweenthejointmeasure µWS andtheproductmeasureµW ⊗µS . Wealsodefinetheamountofinformationprocessedinthe i,t i,t i,t timeinterval(h,t]asI(µWS)−I(µWS)andletdI(µWS)denotethedifferentialformofthisobject—i.e., i,t i,h i,t theamountofinformationprocessedatthe“instant”t. Asisstandardintherationalinattentionliterature(seee.g.,Mac´kowiak,Mateˇjka,andWiederholt, 2023),inourbaselinemodelweassumethatthecostoftheflowofinformationtothefirmislinear in the information that the firm acquires, with scaling parameter ω>0. That is, the cost of the informationflowisωdI. 2.4. TheFirm’sProblem Putting together the firm’s profit function and its information costs, we obtain that the firm’s problemistochooseapricingandinformationpolicytomaximizetheexpecteddiscountedvalue ofitsprofitsnetofitsinformationcosts. Formally,apricingpolicyforthefirmisamapthatreturns the price that the firm charges after each history at each time pˆ :St →R. A pricing policy is i,t i feasibleifitisconstantwheneverthefirmdoesnotreceiveapricechangeopportunity. Thefirm 9

choosesitsinformationpolicyµWS alongwithafeasiblepricingpolicytomaximizeitsexpected i,t discountedprofitsnetofinformationcosts: sup E (cid:183)(cid:90) ∞ e −rt (cid:181) − B (cid:161) p −q (cid:162)2 dt−ωdI (cid:182)(cid:175) (cid:175)S0 (cid:184) (13) {µW i,t S,pˆ i,t}t≥0 0 2 i,t i,t t (cid:175) i 2.5. Equilibrium An equilibrium is a path for all endogenous variables such that the household maximizes its expectedutility,thefirmmaximizesitsprofits,andallmarketsclear. Definition1(Equilibrium).Anequilibriumisasequenceofrandomvariables: (cid:110) (cid:111) C ,P ,L ,R ,w , (cid:169)Π ,P ,C ,L ,Y (cid:170) (14) t t t t t i,t i,t i,t i,t i,t i∈[0,1] t∈R + andacollectionofpolicyfunctions(µW i,t S ,pˆ i,t ) i∈[0,1],t∈R + suchthat: 1. ThepolicyfunctionssolveEquation13 2. ProductionoccursaccordingtoEquation9 3. Thehouseholdoptimizesitsexpecteddiscountedutility(Equation2)subjecttoitsintertemporal budgetconstraint(Equation4)andsoEquations5,6,7,8hold. 4. Themarketsforlabor,goods,bonds,andmoneyclear. Inthefollowingsections,wewillstudyequilibriumfirmpoliciesandcharacterizetheresulting implicationsformonetarynon-neutrality. 3 Firms’InformationAcquisition Wenowsolveforfirms’optimalpricingandinformationstrategies. Optimalinformationpolicies takeastrikingform: onlyacquireinformationwhenresettingpricesandalwaysacquireexactly enoughinformationtoresetuncertaintyaboutoptimalpricestosomefixedlevel,regardlessofthe currentstateofyouruncertainty. 3.1. OptimalInformationAcquisition Webeginbyfullycharacterizingfirms’optimalinformationandpricingpolicies. Oncetheinformationpolicyispinneddown,optimalpricingissimple,asfirmssimplysetpricesequaltotheir conditionalexpectationoftheiroptimalprice: p =E[q |St] (15) i,t i,t i Towardcharacterizingtheoptimalinformationpolicy,definefirmi’sposterioruncertaintyabout itsoptimalresetpriceattime t asU i,t =V[q i,t |S i t]. WeletU i,t− denotethecorrespondingprior 10

uncertaintyaboutq attimet. Thefollowingresultcharacterizesoptimalinformationacquisition. i,t Theorem1(Optimal Dynamic Information Policy).The firm only acquires information when it ∗ changesitsprice. Whenthefirmchangesitsprice,thereexistsathresholdlevelofuncertaintyU suchthat: 1. IfU i,t− ≤U ∗ ,thenthefirmacquiresnoinformationandU i,t =U i,t−. 2. IfU i,t− >U ∗ ,thenthefirmacquiresaGaussiansignalofitsoptimalpricesuchthatitsposterior uncertaintyisU =U ∗ . i,t ∗ Moreover,U istheuniquesolutionto: ω (cid:104) ω (cid:105) (cid:181) 1−Eh[e −rh] (cid:182) −Eh e −rh = B (16) U ∗ U ∗+σ2h r (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) marginalcostofinformation marginalbenefitofinformation Proof. SeeAppendixA.1. We prove this result in three steps. First, we show that firms should only wish to acquire informationwhentheychangetheirprices. Theintuitionforwhythisisoptimalcomesfromtwo observations: (i)becauseofdiscounting,acquiringinformationfurtherinthefutureispreferable, and (ii) as the firm’s marginal cost moves over time, information becomes stale over time. By acquiringinformationonlywhenitisused,thefirmpushesinformationacquisitionfurtherinto thefutureandneveracquiresinformationthatbecomesstale. Second,weshowthatthefirmshouldalwaysacquireGaussiansignalswhentheyresettheir prices. Intuitively, as the firm sets p =E [q |St], the firm’s expected per period loss until it i,t i,t i,t i resetsitspriceisproportionaltoV[q |St]. Thus,thefirm’spayoffsdependonlyonasequence i,t i ofconditionalvariancesofaGaussianrandomvariable. Undermutualinformation,thecheapest waytoachievesuchasequenceiswithasequenceofsignalsthatmaximizesentropy. Thehighest entropydistributionforanyexpectedvariance-covariancematrixistheGaussianone. Thus,the firmshouldalwaysacquireaGaussiansignalofitsoptimalresetprice: s =q +σˆ ε (17) i,t i,t i,t i,t whereε isanindependentandidenticallydistributedstandardnormalrandomvariableandσˆ i,t i,t isanadaptedsequenceofsignalstandarddeviations. Third, we characterize the optimal noise in signals. To do this, we observe that the firm’s posteriorvarianceaboutoptimalresetpricesisasufficientstatisticforthefirm’sdynamicproblem. 11

Thus,lettingU i,t−bethefirmi’sprioruncertaintyinperiodt,wehavethatfirmssolve: B (cid:183)(cid:90) h (cid:184) (cid:104) (cid:105) ω (cid:181) U (cid:182) V(U i,t−)= max −U i,t Eh e −rτ dτ +Eh e −rhV(U i,t +σ2h) + ln i,t (18) Ui,t ≤Ui,t− 2 0 2 U i,t− Thefirsttermistheexpectedlossfrommispricing,whichisU ×B perperiod,fortheexpected i,t 2 discounteddurationofthepricingspell.Thesecondtermisthecontinuationvalue.Ifyouresetyour priceinh periods,uncertaintyatthatpointisyourposterioruncertaintytodayplusthevolatilityof theidealpricemultipliedbyh. Thesetwotermsgiverisetoatrade-off: informationtodayismore valuablethemorelikelyitisthatyouresetyourpricesoonbecauseyouwillhavebetterinformation thenexttimeyousetyourprice,butlossesfrommispricingarelowerifyouresetyourpricessooner. Thefinaltermissimplythecostofachievingagivenlevelofposterioruncertaintygiventhemutual informationformofcosts. Thesetrade-offsyieldtheclaimedfirst-ordercondition. Importantly,theoptimallevelofposterior uncertaintydoesnotdependonprior uncertainty whenfirmscometoresetprices. Intuitively,havingbetterpriorinformationreducesthecostof obtainingbetterposteriorinformation. However,undermutualinformation,itdoesnotchangethe marginalcost ofbetterinformationandsotheoptimalpolicyisinvarianttoU i,t−. 3.2. TheEconomicForcesThatShapeOptimalUncertainty Wenowstudyhowchangesinpricestickiness,thevolatilityofmarginalcosts,andthecostsand benefitsofmorepreciseinformationaffecttheoptimallevelofuncertainty. ComparativeStatics.Thefollowingresultcharacterizeshowtheoptimalresetlevelofuncertainty dependsonvariousfeaturesoftheunderlyingeconomicenvironment: Corollary1(ComparativeStaticsforOptimalUncertainty).Theoptimallevelofuncertaintyupon ∗ resettingtheprice,U ,is: 1. Decreasinginthepriceelasticityofdemand,η 2. Increasinginthecostofinformation,ω 3. Increasinginthevolatilityofmarginalcosts,σ2 4. Increasinginthediscountrate,r. Changes in the distribution of price reset opportunities, G, in the sense of first-order stochastic ∗ dominance(FOSD),haveanambiguouseffectonU . Proof. SeeAppendixA.2 WeillustratethesecomparativestaticsinFigure1.Intuitively,agreaterpriceelasticityofdemand increasestheprofitlossesfrommispricingandleadsfirmstoacquiremorepreciseinformation. Moreover,whenmarginalcostsbecomemorevolatile,itbecomesmoreexpensivetotargetagiven 12

Figure1: ComparativeStaticsofOptimalResetUncertaintyinModelParameters MC MB MB(η↑) MC MC(ωorσ2↑) MB η↑ ωorσ2↑ U U U ∗ (η↑) U ∗ U ∗ U ∗ (ωorσ2↑) (a)AnIncreaseinDemandElasticityη (b)AnIncreaseinInfo.CostωorVolatilityσ2 MC MC(r↑) MC MC(g↑) MB MB(r↑) MB MB(g↑) r↑ U U U ∗ U ∗ (r↑) ∆U ∗ isambiguous (c)AnIncreaseinDiscountRater (d)AFOSDIncreaseinG levelofuncertaintyandthebenefitsdonotchange. Thus,whenmarginalcostvolatilityincreases, sotoodoesoptimaluncertainty. Whenthediscountrateincreases,futurelossesfrommispricing becomesmallerandthevalueofinformationforfuturedecisionsissmaller. Thus,higherdiscount rates lead to greater uncertainty. Changes in the flexibility of prices have ambiguous impacts becauseoftwocountervailingeffects. First,aspriceresetopportunitiesbecomelessfrequent,the valueofinformationuntilyounextresetpricesisgreaterbecauseyoukeepyourpricefixedbased onthisinformationforalongerperiodoftime. Second,whenpriceadjustmentislessfrequent, informationacquiredtodayislessvaluableforfuturepriceresettingopportunitiesbecausemarginal costsarelikelytohavechangedbymorewhenyounextcometoresetyourprices. Whichofthese effectsdominatesdependsontheotherparametersoftheproblemandthetotaleffectofprice flexibilityonoptimaluncertaintyisambiguous. SpecialCasesandBoundsonUncertainty.Toillustratetheseresults,itisinformativetoconsider thespecialcaseofTaylorpricing,inwhichoptimaluncertaintycanbesolvedinclosedform. 13

Corollary2(OptimalUncertaintyUnderTaylorPricing).UnderTaylorpricing,i.e.,firmsresetprices everyk∈R +periods,optimalresetuncertaintyisgivenby: (cid:114) − (cid:179) B1−e −rkσ2k−ω(cid:161) 1−e −rk(cid:162) (cid:180) + (cid:179) B1−e−rkσ2k−ω(cid:161) 1−e −rk (cid:162) (cid:180)2 +4B1−e−rkωσ2k U ∗= r r r (19) 2B1−e−rk r Moreover,inthespecialcaseinwhichdiscountingiszero,wehavethat: (cid:115) σ2k (cid:181)σ2k (cid:182)2 σ2 limU ∗=− + +ω (20) r→0 2 2 B Proof. SeeAppendixA.3. In the special case of no discounting, the general comparative statics from Corollary 1 are particularlysimpletoobserve: increasesinωandσ2anddecreasesinB increaseU ∗ . Moreover,in ∗ theTaylorspecialcasewithnodiscounting,weobservethatU isdecreasingink. Thisisbecause theeffectofusinginformationforlongerdominatestheeffectthatinformationacquiredtodayis lessusefulthenexttimethatthefirmresetsitsprice. Finally,evenwhenoptimaluncertaintydoesnotadmitanexplicitsolution,tightboundsonits valuecanbeattainedbyconsideringthespeciallimitcasesinwhichmarginalcostsareinfinitely volatileandmarginalcostsareconstantovertime. Inthesecases,wecansolvefortheoptimallevel ofuncertaintyinclosedform. Asperourearliercomparativestatics,thesecasesalsoprovideupper andlowerboundsonfirms’optimaluncertainty. Corollary3(SpecialCasesandBoundsforOptimalUncertainty).Inthelimitofinfinitevolatility, optimalresetuncertaintyis: ωr 1 lim U ∗= ≡UMax (21) σ2→∞ B 1−Eh[e −rh] Inthelimitofzerovolatility,optimalresetuncertaintyis: ωr lim U ∗= ≡UMin (22) σ2→0 B Moreover,anyoptimalresetuncertaintyissuchthatUMin≤U ∗≤UMax. Proof. ImmediatefromTheorem1andCorollary1. Intuitively,whenmarginalcostsareinfinitelyvolatile,informationacquiredtodayhasnovalue inmakingfutureprice-settingdecisionsbecausethecurrentstateofmarginalcostsiscompletely uninformativeaboutthefuturestateofmarginalcosts. Inthiscase,aspriceadjustmentbecomes morefrequent,firms’optimaluncertaintyincreases. Intuitively,becauseinformationtodayhasno continuationvalue,theonlyeffectofmorefrequentpriceadjustmentsisthatlossesfrommispricing 14

basedoninformationtodayoccurforfewerperiods. Thismakesinformationtodaylessvaluable andincreasestheoptimallevelofuncertainty. Asthiscaseminimizesthecontinuationvalueof information, this case also places an upper bound on the optimal uncertainty that a firm will choose. Conversely,whenmarginalcostsareclosetoconstant,informationtodayisequallyusefultoday asitwillbewhenthefirmresetsprices. Thus,thefrequencyofpriceadjustmentisirrelevantto optimaluncertainty. Asthiscasemaximizesthecontinuationvalueofinformation,thiscaseplaces alowerboundonfirms’optimaluncertainty. 3.3. SelectionandUncertainty Ourmodelofendogenousinformationacquisitionimpliesanimportantproperty: firmsthatare settingpricesaretheleastuncertain. Corollary4(UncertaintyandTimeSinceChangingPrice).Considerafirmi attimet thatchanged itspriceh periodsago. Thefirm’suncertaintyaboutitsoptimalpricefollows: U =U ∗+σ2h (23) i,t Proof. SeeAppendixA.4. Animportantimplicationofthisresultisthatitisnotaverageuncertaintythatisrelevantfor theprice-settingdecisionsoffirms,butrathertheoptimalresetlevelofuncertainty. Wecallthis phenomenonselectionininformationacquisition: itistheprice-settingfirmwhoseuncertainty matters and, as these are the firms that most recently acquired information, they are the least uncertainfirms. Thispredictedrelationshipbetweenafirm’suncertaintyandthedurationofitspricingspell distinguishesourtheoryfrommodelswithexogenousinformationprocessingcapacityorGaussian signals with constant precision. This is because, in models with Gaussian signals or constant capacity,thefirm’sbeliefsfollowaKalman-BucyfilterinwhichU isconstant. Thus,undereither i,t model,thefirm’slevelofuncertaintyisconstantanddoesnotdependonthetimesincethefirm resetitsprice. Aswewillshortlysee,thefactthattheextentoffirms’uncertaintydependsonthe durationoftheirpricingspellhasimportantqualitativeimplicationsformonetarynon-neutrality. 4 ImplicationsforMonetaryNon-Neutrality Havingcharacterizedfirms’optimaldynamicinformationpolicies,wenowexploretheimplications of endogenous information acquisition for the propagation of monetary shocks. We find that 15

uncertaintyaffectsthecumulativeimpulseresponseofoutputtoamonetaryshockinasurprisingly simpleway: itisequaltothebenchmarkwithperfectinformationplustheratiooftheuncertainty ofprice-settingfirmstotheinstantaneousvarianceoftheirmarginalcosts. Thishighlightstheimportanceoftheselectionmechanism: itisnotaverageuncertaintythatmatters,itistheuncertainty ofpricesetters. Thus,theeffectsofamonetaryshockwithendogenousinformationacquisition alwaysliebetweenthosewithperfectinformationandthebenchmarkunderexogenouslygiven imperfectinformation. 4.1. FromFirm-LevelPriceGapstoTheAggregateOutputGap Webeginbydecomposingtheaggregateresponsetoshocksintofirm-levelresponsestoshocks. Fromthehousehold’soptimalityconditions(Equations7and8),wehavethataggregateoutput follows: 1 y = (m −p ) (24) t γ t t where y ≡logY −logY , m ≡logM −logM , and p ≡logP −logP . Following the literature t t 0 t t 0 t t 0 on the propagation of monetary shocks (see e.g., Alvarez and Lippi, 2014), we will primarily be interestedinstudyingthecumulativeimpulseresponse(CIR)ofoutputtoamonetaryshockfrom thesteadystateattimet =0: (cid:90) ∞ M = y dt (25) t 0 TocomputethisCIR,wecanre-expresstheaggregateoutputgapasanintegraloffirm-level outputgapsandthenintegratethisovertime. Formally,bylog-linearizingtheidealpriceindex (Equation6),wehavethat: (cid:90) 1 p = p di (26) t i,t 0 Thus,wedecomposetheaggregateoutputgapastheintegraloffirm-leveloutputgaps,y =(cid:82)1 y di, t 0 i,t wherefirm-leveloutputgapsfollow: 1 y =− (p −q ) (27) i,t γ i,t i,t Hence,tocharacterizetheresponsetomonetaryshocks,weneedonlyconsiderhowfirms’prices respondtotheshock.Todothis,wedecomposefirms’outputgapsintotwocomponents.Thefirstis thebeliefgap, yb = 1(cid:161) q −E [q ] (cid:162) ,whichmeasurestheoutputeffectsoffirms’errorsinpricing i,t γ i,t i,t i,t from having incorrect information. The second is the perceived gap, yx =−1(cid:161) p −E [q ] (cid:162) , i,t γ i,t i,t i,t whicharisesfromafirm’spricenothavingadjustedsinceitreceivesinformation. Forafirmthat 16

lastchangeditspricehperiodsagoandthathasaninitialbeliefgapyb,perceivedgapyx,wedefine thefirm-levelcumulativeoutputgapas (cid:183)(cid:90) ∞ (cid:184) Y(yb,yx,h)=E y dt |yb =yb,yx =yx,D =h (28) i,t i,0 i,0 i,0 0 Followingthemonetaryshock,wedefinetheinitialjointdistributionofchangesinbeliefgaps andperceivedgapsandthelengthsofpricingspellsasF ∈∆(R3).Moreover,wedefinetherespective marginaldistributionsasFb,Fx,andFh. Aspricingistime-dependent,thedistributionofpricing durationsisexogenoustoanymonetaryshock. Thus,Fh=F,whichisthedistributionofpricing spelllengthsinthecross-sectionoffirms,and yb and yx areindependentofh. Wethereforehave thattheCIRisgivenby: (cid:90) M(F)= Y(yb,yx,h)dF(yb,yx,h) (29) R3 Thisreducesthequestionofhowmonetaryshocksaffectoutputtoansweringtwoquestions. First, howdofirms’lifetimeoutputgapsdependontheirinitialbeliefgap,initialperceivedgap,andthe timesincetheylastchangedtheirpriceviaY? Second,howdoweaggregatefirms’lifetimeoutput gapstocomputetheCIR? 4.2. CharacterizationofLifetimeOutputGaps Wefirstcharacterizeafirm’sexpectedlifetimeoutputgap. Todothis,wemakeuseofthefollowing definitions. WedefinetheaverageconditionaldurationasD¯ =Eh ′ [h ′|h],whichissimplyhowlong h g afirmthatresetitspriceh periodsagoexpectstowaitbeforeresettingitsprice. ByTheorem1, wehavethattheKalmangainforafirmthatresetsitspriceτperiodsafterlastresettingitsprice isκ τ = U∗ σ + 2 σ τ 2τ . WedefinetheaverageconditionalKalmangainasκ¯ h =Eh g ′ [κ h′+h |h],whichisthe expectedKalmangainatthenextpriceresetopportunityforafirmthatlastresetitspriceh periods ago. With these objects in hand, the following Proposition characterizes the expected lifetime outputgapofafirm Proposition1(LifetimeOutputGapCharacterization).Theexpectedlifetimeoutputgapofafirm withinitialpricingdurationh,initialbeliefgap yb,andinitialperceivedgap yx isgivenby: (cid:181) 1−κ¯ (cid:182) Y(yb,yx,h)=D¯ yx+ D¯ +D¯ h yb (30) h h 0 κ¯ 0 Proof. SeeAppendixA.5 Tounderstandthisresult,considerfirstthelifetimeoutputeffectofaperceivedgap.Importantly, ′ asthefirmknowsitsperceivedgap,itpersistsonlyuntilthefirmcanresetitspriceinh periods,at whichpointanyperceivedgapisresettozero. Weillustratethisinpanel(a)ofFigure2. Thus,as 17

Figure2: ContributionofaSingleFirmtoMonetaryNon-Neutrality MoneySupply/Price m=w =δ p =w =δ i,t Y =δ×h ′ =⇒ (cid:82) Y di =δ×(cid:82) h ′ di i i i i Time(t) −h t =0 h ′ i i (a)MonetaryNon-NeutralitywithFullInformation MoneySupply/Price m=w =δ Y i =δ×h i ′+δ×(h i ′′−h i ′ )×(1−κ hi +h i ′)+... ∆p i,t =κ h ′′−h ′ ×δ i i pp ii,,tt ==κκ hhii ++hh ii ′′ ××δδ Time(t) −h t =0 h ′ h ′′ i i i (b)MonetaryNon-NeutralitywithCostlyInformation thefirmonaveragewilltakeD¯ periodstoresetitsprice,thelifetimeeffectofaperceivedgap yx is h simplyD¯ yx. h Second,incontrasttoperceivedgaps,beliefgapspersistforever. Weillustratethedynamics forasamplepathofpriceadjustmentfollowingamonetaryshockthataffectsbeliefgapsinpanel (b)ofFigure2. Initially,abeliefgapoperatesinmuchthesamewayasaperceivedgap. Untilthe firmnextresetsitsprice,inexpectationitsbeliefgapremains yb andsountilthefirstpricereseta beliefgapalsocontributesD¯ yb totheexpectedlifetimeoutputgapofthefirm. Afterthispoint, h itsbehaviorbecomesmorecomplicated. Inparticular,whenafirmthatresetitspriceh periods ′ agocomestoresetitspriceinh periods,Theorem1impliesthatitacquiresaGaussiansignalof itsmarginalcostswithaKalmangainofκ h+h′. Hence,ifthisfirmhadabeliefgapof yb attimet, itwouldhaveanexpectedbeliefgapofEh g ′ [1−κ h+h′ |h]yb=(1−κ¯ h )yb attimet+h ′ . Moreover,on 18

average,thisbeliefgappersistsforD¯ periodsbeforethefirm’snextpriceresetopportunity. Thus, 0 betweenthefirstpriceresetandthesecond,theexpectedtotaloutputgapofafirmisD¯ (1−κ¯ )yb. 0 h ′′ Afterthispoint,ifafurtherh periodselapsebeforethefirmresetsitspricenext,itsKalmangainat thatpointwouldbeκ h′′ andsothefirm’sexpectedoutputgapatthesecondpriceresetopportunity ′′ ′ wouldbeEh g [1−κ h′′]Eh g [1−κ h+h′ |h]yb=(1−κ¯ 0 )(1−κ¯ h )yb. Thus,onceagainintegratingoverthe expected duration of the third pricing spell, this period contributes D¯ (1−κ¯ )(1−κ¯ )yb to the 0 0 h expectedlifetimeoutputgap. Thesameprocessnowhappensadinfinitumforallfuturespells: the initialbeliefgapgetsdown-weightedby1−κ¯ becauseoftheacquisitionofnewinformationand 0 eachspelllastsD¯ periodsonaverage. Hence,thetotaleffectofthebeliefgaponthelifetimeoutput 0 gapisgivenbythefollowinggeometricseries: D¯ yb+ (cid:88) ∞ D¯ (1−κ¯ )k(1−κ¯ )yb=D¯ yb+D¯ yb 1−κ¯ h (31) h 0 0 h h 0 κ¯ k=0 0 whichcollapsestotheclaimedexpressioninProposition1. 4.3. ThePropagationofMonetaryShocks Wenowcharacterizethepropagationofmonetaryshocksconditionalonthedistributionofoutput gapsthattheyinduceonimpact. Thisissimplytheintegraloftheexpectedlifetimeoutputgapsof firmsoverthejointdistributionofpricegapsandpricingspells. Aspricegapsandspellduration areindependent,Proposition1immediatelyimpliesthat: (cid:181) 1−κ¯ (cid:182) M(F)=E F[yx]D¯ +E F[yb] D¯ +D¯ 0 κ¯ (32) 0 whereD¯ =Eh[D¯ ]istheaveragepricingdurationinthepopulationandκ¯ =Eh[κ¯ ]istheaverage f h f h acrossallfirmsoftheexpectedKalmangainwhentheynextresettheirprices. Theseobjectsare in principle quite complicated: they are double integrals of Kalman gains and durations with respect to two different distributions—the conditional distribution of price reset opportunities G and the cross-sectional distribution of pricing spell durations F. However, Theorem 2 shows ∗ thattheycollapsetoasimpleformulaintermsofonlytheuncertaintyofprice-settersU andthe instantaneousvarianceofmarginalcostsσ2: Theorem2(CIRCharacterization).GivenaninitialdistributionF ∈∆(R3),theCIRisgivenby: (cid:181) ∗(cid:182) U M(F)=E F[yx]D¯ +E F[yb] D¯ + (33) σ2 Proof. SeeAppendixA.6. ThisresultfollowsfromshowingthatthenetpresentvalueoftheaverageKalmangaininthe cross-section is given by the ratio of price-setters’ uncertainty to the instantaneous variance of 19

marginalcosts.Moreover,ithastwoimportantimplications:imperfectinformationaboutmonetary shocks amplifies their real effects and selection effects in information acquisition dampen the importanceofimperfectinformation. ImperfectInformationAmplifiesMonetaryNon-Neutrality.Theorem2highlightsthattheeffects ofamonetarypolicyshockhingeonwhethermonetarypolicyshocksareobserved(thusaffecting perceived gaps) or unobserved (thus affecting belief gaps). Concretely, if there is a permanent monetaryexpansionofamountm=logM −logM anditisunobserved,thenallfirms’initialbelief t 0 gapschangeby y m b = m γ . WeletthenormalizedCIRinthiscasebegivenbyMb=M(δ 0 ,δ m γ ,F) (cid:177)m γ . Bycontrast,ifthemonetaryshockm isobserved,then yx = m andnofirm’sbeliefgapchanges. γ WeletthenormalizedCIRinthiscasebegivenbyMx =M(δ m,δ 0 ,F) (cid:177)m γ . Thefollowingcorollary γ characterizestherelativeexpansionoftheeconomyunderthesetwoscenarios: Corollary5(ImperfectInformationAmplifiesMonetaryNon-Neutrality).Thedifferencebetweenthe normalizedCIRstoapermanentandunobservedmonetaryshockandapermanentandobserved monetaryshockofthesamesizeis: ∗ U ∆Info≡Mb−Mx = >0 (34) σ2 Proof. ImmediatefromTheorem2. Theintuitionforthisresultissimple: iffirmsaremoresluggishintheiradjustmentofprices, thenmonetarypolicyhaslargereffects. Moreover,whenfirmshaveimperfectinformation,theyare slowertoadjustbecausetheyonlylearnabouttheshockovertime. SelectionDampensMonetaryNon-Neutrality. Importantly, Theorem 2 shows that it is the uncertaintyofprice-settersalonethatdeterminesthenon-neutralityofshocksandnottheaverage uncertaintyinthepopulation. WeletMexo betheCIRofanunobservedmonetaryshockwhen firms’uncertaintyisexogenouslyfixedatsomelevelU¯. Thefollowingresultcharacterizestheimportanceofselectionorthefactthatprice-setters’uncertaintyiswhatmattersandnottheaverage levelofuncertaintyinthepopulation: Corollary6(SelectionDampensMonetaryNon-Neutrality).Thedifferencebetweenthenormalized CIRstopermanentandunobservedmonetaryshocksunderexogenousuncertaintyandendogenous uncertaintyisgivenby: U¯ −U ∗ ∆Select≡Mexo−Mb= >0 (35) σ2 Proof. ImmediatefromTheorem2. 20

Intuitively, as uncertainty is lowest for price-setters by Theorem 1, and greater uncertainty amplifiesmonetarynon-neutrality,itisimmediatethatselectioneffectsininformationacquisition dampenmonetarynon-neutralityrelativetoabenchmarkmodelinwhichallfirmshaveexogenous uncertaintyequaltosomelevelU¯.Moreover,ourcharacterizationfromTheorem2givesusasimple formulabywhichselectioneffectscanbequantifiedinthedata. 4.4. ComparativeStaticsforMonetaryNon-Neutrality Finally,westudyhowchangesinuncertainty,microeconomicvolatility,andpricestickinessaffect theCIR.Toaidintuition,wealsoprovideanexplicitformulafortheCIRinthespecialcaseofTaylor pricing. UncertaintyShocksDampenMonetaryNon-Neutrality. First, we gauge how changes in firms’ uncertaintyaffectthepropagationofmonetarypolicyshocks. Concretely,supposethatattime t =0,eachfirmissubjecttoashockthatincreasestheirprioruncertaintyabouttheiroptimalreset pricebyU˜ >0. BycomputingthechangesintheprofileofKalmangainsacrossfirms,wefindthe followingformulafortheeffectofanuncertaintyshockontheCIR: Proposition2(UncertaintyShocksDampenMonetaryNon-Neutrality).Theeffectofanuncertainty shockU˜ >0ontheCIRisgivenby: ∂+Mb(cid:175) (cid:175) (cid:175) =− 1 U ∗ Eh (cid:34) κ2 h (cid:35) <0 (36) ∂+ U˜ (cid:175) U˜=0 κ¯ 0 σ2 g σ2h where∂+ denotestherightpartialderivativeofafunction. Proof. SeeAppendixA.7. Intuitively,iffirmsaremoreuncertain,theyrelylessontheirpriorinformationandsoupdate theirpricesmoreaggressivelyinresponsetotheinformationtheyacquire. Asaresult,pricesadjust morerapidlyandtherealeffectsofmonetarypolicyaredampenedfollowinganuncertaintyshock. GreaterMicroeconomicVolatilityDampensMonetaryNon-Neutrality.Sofar,wehaveseenhow exogenousuncertaintyshocksaffectmonetarynon-neutrality. Wenowstudythemorecomplicated questionofhowchangesinmicroeconomicvolatilityaffectmonetarynon-neutrality. Thisismore subtlebecausewhileσ2 decreasestheCIRallelseequal,weknowfromCorollary1thatU ∗ will increaseinresponsetoanincreaseinσ2. Thispotentialambiguitynotwithstanding,bycombining Theorems1and2,wefindthatincreasesinmicroeconomicvolatilityalwaysdampenmonetary non-neutrality: 21

Proposition3(MicroeconomicVolatilityDampensMonetaryNon-Neutrality).Theeffectofgreater microeconomicvolatilityσ2ontheCIRisgivenby: ∂Mb U ∗ 1−Eh(cid:163) e −rh(1−κ ) (cid:164) =− h <0 (37) ∂σ2 σ4 1−Eh (cid:163) e −rh(1−κ )2 (cid:164) h Proof. SeeAppendixA.8. Intuitively,thedirecteffectofgreatermicroeconomicvolatilitymakingfirmsrelylessonprior informationalwaysdominatesthefactthatfirmsacquirelessinformationwhenmicroeconomic volatilityrises. Thisisqualitativelydifferentfromtheroleofmicroeconomicvolatilityinmodels thatdonotfeaturenominalrigidities,suchasMoscarini(2004),inwhichmarginalcostvolatility canhavenon-monotoneeffectsonprice-responsiveness. PriceStickinessHasanAmbiguousEffectonMonetaryNon-Neutrality.Moreover,weobservein thefollowingresultthatchangesinthestickinessofpriceshaveanambiguouseffectonthereal effectsofamonetaryshock: Proposition4(AmbiguousEffectsofPriceStickinessonMonetaryNon-Neutrality).Forε>0,let Gε(h) ≡G(h−ε),∀h ≥ ε denote a distribution for the arrival of price change opportunities that increasesthedurationofallpricespellsbyε. Then,theeffectofagreaterpricestickinessontheCIRas ε↓0isgivenby: ∂+ ∂ M +ε b(cid:175) (cid:175) (cid:175) (cid:175) ε=0 = 1−Eh (cid:163) e −r 1 h(1−κ h )2 (cid:164) (cid:183) 1−r U σ2 ∗(cid:181) U U M ∗ in −1 (cid:182)(cid:184) ⋛ 0 (38) Proof. SeeAppendixA.9. Thisambiguityarisesbecausetherearetwo(potentially)opposingforcesatplay. First,more stickypricesincreasetheaveragedurationofpricingspellsD¯,whichincreasestherealeffectsof ∗ monetaryshocks. Second,morestickypricesaffectfirms’optimalchoiceofuncertaintyU . As wesawinCorollary2forthespecialcaseofTaylorpricing,morestickypricescandecreasefirms’ optimaluncertainty. Thisisquiteintuitive: ifthepriceisstuckforlonger,it’smoreimportantto makethatpriceagoodonesoit’sbettertoacquiremoreinformation. Thus,thesignandmagnitude ofhowchangesinthestickinessofpricesaffecttheCIRisaquantitativequestiontowhichwewill returninSection7. TheCIRUnderTaylorPricing.Finally,toaidintuitionfortheeconomicforcesthatshapetheCIR, wesolveinclosedformfortheCIRinthespecialcaseofTaylorpricingwithnodiscounting: 22

Corollary7.UnderTaylorpricingandzerodiscounting,i.e.,firmsresetpriceseveryk∈R +periods andr =0,theCIRisgivenby: (cid:115) (cid:181) k (cid:182)2 ω Mb= + (39) 2 Bσ2 Proof. ImmediatefromcombiningTheorem2andCorollary2. Itisimmediatefromthisformulathat: increasesinmicroeconomicvolatilitylowertheCIR; increases in the price elasticity of demand lower the CIR; increases in the cost of information increasetheCIR;andincreasesinpricestickinessincreasetheCIRbutbylessthanone-for-one becauseoftheendogenousresponseoffirmsacquiringmoreinformationwhenstickinessincreases. 5 IdentificationoftheRealEffectsofMonetaryPolicy Inourfinaltheoreticalresults,weturntowhichdataarenecessaryandsufficienttoidentifythereal effectsofmonetarypolicyinthepresenceofendogenousinformationacquisition. Weshowthat thecross-sectionaldistributionsofuncertaintyandpricingdurationsacrossfirmsaresufficientto identifytheCIR.Moreover,weshowthataccesstostandarddataonpricechangesisinsufficient toidentifythecomponentofCIRthatstemsfromfirms’subjectiveuncertainty. Thus,inaformal sense,accesstoinformationaboutfirms’uncertaintyisnecessaryfortheidentificationoftheCIR. 5.1. TheDistributionsofUncertaintyandPricingDurationsAreSufficientforIdentification We first show how data on firms’ uncertainty about their optimal reset prices and the duration of their pricing spells are sufficient to identify the CIR. Formally, let l be the density of firms’ uncertainty. An implication of Theorem 1 is that the distribution of firms’ uncertainty and the distributionoffirms’spelllengths f arecloselyrelated: Proposition5(CharacterizationoftheDistributionofUncertainty).Thecross-sectionaldensityof uncertaintyaboutoptimalresetpricesl ∈∆(R +)isgivenby:   0, z<U ∗ , l(z)= (40)   1 f (cid:179) z−U ∗(cid:180) , z≥U ∗ . σ2 σ2 where f(·)= 1 (1−G(·))isthedensityofongoingspelllengthsinthecross-section. D¯ 0 Proof. SeeAppendixA.10 Thisresulttellsusthatknowledgeofthedistributionofuncertaintyl andthelengthofongoing ∗ pricing spells f is sufficient to identify the uncertainty of price-setters U , the instantaneous 23

varianceofmarginalcostsσ2,andtheaverageexpecteddurationofpricingspellD¯,whichinturn identifytheCIRM(F)foranyF. FromIdentificationtoEstimation.Moreover,thisresultsuggestsasimplemethodologybywhich U ∗ andσ2canbeestimatedfromdata. First,observethattheuncertaintyofprice-settersisgiven bythemodeoftheuncertaintydistributionU ∗=mode [U]. Thus,givenanempiricalestimateof l theuncertaintydistributionlˆ,weobtainthefollowingestimatorforU ∗ : Uˆ∗=mode [U] (41) lˆ Second, given an empirical estimate fˆ of the distribution of ongoing spell lengths and our estimate of the uncertainty of price-settersUˆ∗ , by Proposition 5 we can determine the model implieduncertaintydistributionas: 1 (cid:181) z−Uˆ∗(cid:182) lM(z;σ2)=I [z≥Uˆ∗]σ2 fˆ σ2 (42) whichdependsonasingleparameter,thevolatilityofmarginalcostsσ2. Wecanthentherefore estimateσ2byminimizingthedistancebetweenlM(σ2)andlˆ: (cid:90) ∞ σˆ2∈argmin (cid:161) lˆ(z)−lM(z;σ2) (cid:162)2 dz (43) Uˆ∗ Thus, wenowhaveapracticalmethodthatwouldallowustoleveragedataonuncertaintyand durationstoestimatetheCIR. 5.2. DataonPriceChangesAreInsufficientforIdentification Wefinallyshowthatdataonuncertaintyisnecessaryinthesensethatdataonpricechangesand pricingdurationsareinsufficienttoidentifytheCIRintheabsenceofinformationaboutuncertainty. Asiswellknown(seee.g.,Alvarez,LeBihan,andLippi,2016),dataonpricechangesaresufficientto identifytheCIRinmanymodelswithbothstate-dependentpricingandtime-dependentpricing frictions. Thus,itisnaturaltoaskifdataonpricechanges(potentiallyalongsidedataonpricing durations)aresufficienttoidentifytheCIRinthepresenceofendogenousinformationacquisition. Thefollowingresultanswersthisquestioninthenegative: Theorem3(InvariancetoUncertaintyoftheDistributionofPriceChanges).Thedistributionof pricechangesconditionalonafirmchangingitspriceH ∈∆(R)isinvarianttoU ∗ andfollows: (cid:90) ∞ (cid:181) ∆p (cid:182) H(∆p)= Φ (cid:112) dG(h) (44) 0 σ h whereΦisthestandardnormalCDF. Proof. SeeAppendixA.11. 24

Weprovethisresultbyfirstderivingtheconditionaldistributionofpricechangesconditional on a firm’s last pricing spell lasting h periods and conditional on a firm’s information set at the beginningofitslastpricingspellSt−h. Weshowthattheconditionalvarianceofsuchpricechanges i isinvarianttotheinformationsetofthefirm.Intuitively,thenatureofthefirm’soptimalinformation acquisitionmakesitspricechangeindependentofthepricesthatitpreviouslycharged. Moreover, from the form of the firm’s optimal information policy derived in Theorem 1, the conditional varianceofpricechangesdependsonlyonthevolatilityofmarginalcostsσandthelengthofthe pricing spell h and is given by σ2h. By mixing this distribution over the distribution of pricing durations,weobtainthedistributionofpricechanges. Theimportantupshotofthisresultisthatdataonpricechanges,eveninconjunctionwithdata ∗ on pricing durations, are insufficient to identifyU and, therefore, the real effects of monetary policywhenthereisendogenousinformationacquisition. Thus,dataonuncertaintyarenotonly ∗ sufficientforidentifyingU ,buttheyarealsonecessary. 6 UsingSurveyDatatoQuantifyandTesttheModel Wehaveshownhowtoidentifytheeffectsofuncertaintyontherealeffectsofmonetaryshocks given information about firms’ uncertainty and the volatility of their marginal costs. We now showhowtousesurveymicrodataonfirms’uncertaintyandthedurationoftheirpricingspellsto identifyandestimatethesesufficientstatistics. UsingasurveyofNewZealandfirmsfromCoibion, Gorodnichenko,Kumar,andRyngaert(2021),weperformthisestimation. TurningtotheCIRtoa monetaryshock,wefindthattheeffectofuncertaintyisofcomparablemagnitudetotheeffectof pricestickinessitselfandthattheeffectofselectionisofacomparablemagnitudeagain. Fromthis, weconcludethatuncertaintyiscriticalforunderstandingtherealeffectsofmonetarypolicyand thattheendogeneityofinformationacquisitionisequallyimportant. 6.1. SurveyDataonFirms’UncertaintyandPricingDuration Motivatedby our identificationresults, we need dataonfirms’uncertainty abouttheiroptimal resetpricesandhowlongagotheylastresettheirprice. Toobtainthesedata,weusethesurvey of firm managers in New Zealand described in Coibion, Gorodnichenko, Kumar, and Ryngaert (2021), implemented between 2017Q4 and 2018Q2. The survey included 515 firms with six or moreemployees. ThesefirmswerearandomsampleoffirmsinNewZealandwithbroadsectoral coverage.4 4Previousworkshaveusedthesurveydatatocharacterizehowfirmsformtheirexpectations.Forexample,Afrouzi (2024)showsthatstrategiccomplementaritydecreaseswithcompetitionandreportsthatfirmswithmorecompetitors 25

Thesedatacontaintwoquestionsthatallowustomeasurethekeyobjectsofinterest. First, firmsareaskedabouttheirsubjectiveuncertaintyabouttheiridealprices: Q1:Ifyourfirmwasfreetochangeitsprice(i.e.supposetherewasnocosttorenegotiating contractswithclients,nocostsofreprintingcatalogues,etc...) today,whatprobability wouldyouassigntoeachofthefollowingcategoriesofpossiblepricechangesthefirm wouldmake? Pleaseprovideapercentageanswer.5 As the survey was conducted by phone, firms’ answers are consistent in that they feature no probabilities below zero and all probabilities sum to one. To compute an estimate of the firm’s uncertainty,wefirstcomputeanestimateofthefirm’sexpectationofitsoptimalpricebytakingthe midpointofeachbinandcomputingitsexpectedvalueundertheprobabilitiesthefirmmanager provides. Then,weconstructanestimateofthefirm’suncertaintybycomputingthevarianceunder theelicitedprobabilitydistribution.6 ThisgivesusameasureU offirmi’suncertaintyaboutits i optimalresetpriceforeachofthefirmsinoursample. Second,firmsareaskedthetimethathaselapsedsincetheylastchangedtheirprice: Q2:Whendidyourfirmlastchangeitsprice(inmonths)andbyhowmuch(in%change)? ThisstraightforwardlygivesusameasureD ofthedurationoffirmi’spricingspell. i 6.2. TheQuantitativeImpactofUncertaintyandSelection Wenowusethesedatatoquantifytheimportanceofbothuncertaintyandselectionformonetary non-neutrality. Wefirstestimatethedensityofpricingdurationsanduncertaintyusingstandard kerneldensitymethodstoobtain fˆandlˆ.7 WethenobtainUˆ∗ andσˆ2usingourestimatorsfrom Equations41and43. Forallestimatedobjects,weconstructstandarderrorsusingthebootstrap.8 havemorecertainposteriorsaboutaggregateinflation.Also,Coibion,Gorodnichenko,Kumar,andRyngaert(2021) evaluatetherelationbetweenfirst-orderandhigher-orderexpectationsoffirms,includinghowtheyadjusttheirbeliefs inresponsetoavarietyofinformationtreatments.Yang(2022)showsthatfirmsproducingmoregoodshavebothbetter informationaboutinflationandmorefrequentbutsmallerpricechanges.SeeCoibion,Gorodnichenko,Kumar,and Ryngaert(2021)foracomprehensivedescriptionofthesurvey. 5Firmsassignedprobabilitiestothefollowing16bins:lessthan-25%,from-25%to-15%,from-15%to-10%,from -10%to-8%,from-8%to-6%,from-6%to-4%,from-4%to-2%,from-2%to0%,from0%to2%,from2%to4%,from 4%to6%,from6%to8%,from8%to10%,from10%to15%,from15%to25%,morethan25%. 6Whenwecalculatethevariance,weassumeauniformdistributionwithineachbin.Forexample,ifafirmassigns 100%onthebin“2-4percent”,thentheimpliedvarianceis 1 (4−2)2=1/3. 12 7Weestimatelˆusingastandardkerneldensityfunctionwithabandwidthof0.34on[0,50].WethenobtainUˆ∗ as themodeoflˆandreestimatethekerneldensityon[Uˆ∗ ,50].Weestimate fˆwithabandwidthof2.4on[0,80]. 8Formally,ford =1,...,10,000,weuniformlyresampleN =515datapoints(thenumberofobservationsinthe surveydata).Were-estimate fˆ andlˆ usingthesedata.Wethenre-estimateanymodel-impliedquantityunderthese d d distributionsandcomputethedistributionoftheresultingestimatesoverthe10,000bootstrapsamples. Wethen computethestandarderrorasthestandarddeviationoftheresultingdistribution. 26

Figure3: DistributionsofFirms’SubjectiveUncertaintyintheDataandtheModel Notes:Thisfigureshowsthedistributionoffirms’subjectiveuncertaintyabouttheiridealprices.Theblackvertical solidlineshowsthemodeoftheempiricaldistributionofsubjectiveuncertainty(Uˆ∗ )andtheblackverticaldashed lineshowsthemeanofthesubjectiveuncertaintyobservedinthesurveydata. Thebluesolidlineistheempirical distributionofuncertaintylˆ(z). Thereddashedlineshowstheestimateddistributionofuncertainty(lM(z))from Equation42usingtheempiricaldistributionoftimesincethelastpricechanges(fˆ)andtheestimateduncertaintyof shocks(σˆ2). Fromthisexercise,weobtainthatUˆ∗=1.17(S.E.: 0.02)andσˆ2=0.21(S.E.: 0.03). InFigure3,we plottheestimateduncertaintydistribution(inred)alongsidetheempiricaluncertaintydistribution (inblue). Thefit,whilenotperfect,issurprisinglygoodgivenweonlyhaveonedegreeoffreedom (thevolatilityofmarginalcostsσ2)tomatchtheentiredistribution. InAppendixFigureB.1,weplot theestimatedconditionaldurationsofpricingspellsD¯ aswellastheestimatedconditionalKalman h gainsκ¯ thattheseestimatesimply. InAppendixFigureB.2,weplottheestimateddistributionof h priceresetopportunitiesG andthecorrespondinghazardfunctionθ,whichisincreasinginthe durationofthepricingspell. Using Theorem 2, we now estimate the extent to which uncertainty affects monetary nonneutralityaswellastheextenttowhichselectioneffectsininformationacquisitionmatter. Figure4 showsthemonthlyCIRofa1percentagepoint(pp)shocktooutputgapsunderdifferentscenarios (i.e., to obtain the annual CIRs simply divide the following numbers by 12). First, we recall as a baseline that the output effect of a 1pp perceived gap is simply the average duration of firms’ pricingspells∆Sticky =D¯, whichweestimatetobe5.95pp(S.E.: 0.17). The effectofa1ppbelief ∗ gap is the effect of a perceived gap plus ∆Info = U , which we estimate to be 5.59pp (S.E.: 0.59). σ2 Thus,accountingforuncertaintyisapproximatelyasimportantformonetarynon-neutralityas 27

Figure4: EstimatedMonthlyCumulativeImpulseResponsestoanInitial1PercentagePointOutput GapunderDifferentScenarios Notes: Thisfigureshowstheoutputeffectsofa1percentagepointshocktoperceivedgaps(leftbar),tobeliefgaps (middlebar),andbeliefgapsignoringtheselectioneffect(rightbar).Theoutputeffectofa1ppperceivedgapisthe averagedurationoffirms’pricingspells∆Sticky=D¯,theeffectofa1ppbeliefgapistheeffectofaperceivedgapplus ∆Info=U∗ ,andtheeffectof1ppbeliefgapwithoutselectioneffectis∆Sticky+∆Infoplus∆Select=U¯−U∗ .Wepresent95% σ2 σ2 confidenceintervalsasblackverticallines. accounting for the mechanical effects of price stickiness. We also estimate the importance of selection∆Select=U¯−U ∗ ,whichistheerrorinwhatwewouldhaveestimated∆Infotobeifwenaively σ2 usedfirms’averageuncertaintyratherthantheuncertaintyofprice-setters,whichwefindtobe 6.71pp(S.E.: 0.80). Thus,explicitlyaccountingforuncertaintyisaboutasimportantasaccounting for price stickiness itself. Moreover, accounting for selection is slightly more important than accountingforpricestickinessitself. Indeed,computingtheeffectsofshocksignoringselection wouldmassivelyoverstatethenon-neutralityofmonetaryshocks. FurtherModelPredictionsintheData.Giventhatendogenousandexogenousinformationmodels havesignificantlydifferentimplicationsformonetarypolicy,herewediscusswhatfeaturesofthe dataareconsistentwiththeendogenousinformationmodelgivenotherpredictionsofthesetwo theories. First,onesignificantpredictionoftheendogenousinformationmodelisthatthedistributionof uncertaintyshouldinherittheshapeofthedistributionofpricingdurationsuptoascalingfactor. AswesawinFigure3,thesetwoestimateddistributionsfromthesurveydataarequiteclose,despite thefactthatourmodelonlyallowsforonefreeparametertorelatethem. Wesuggestthatsucha closefitwouldbeunlikelytobeobtainedinamodelwithexogenousinformation,asthereshould 28

benorelationshipbetweenthetwodistributions. Second,astarkquantitativelytestableimplicationofthetheoryisthatthemagnitudeofselection effectsshouldalwaysbeequaltotheaveragedurationoffirms’pricingspells. Toseethis,wecan combineCorollary6withCorollary4toobservethat: ∆Select= U¯ −U ∗ = E f [U ∗+σ2h]−U ∗ =E [h]=D¯ (45) σ2 σ2 f Asnothinginourestimationapproachimposessucharelationship,thepredictionthat∆Select=D¯ representsastrongoveridentifyingtestofthetheory. Weestimatethat∆S(cid:225)elect−D¯ =0.759witha 95%confidenceinterval(computedviathebootstrap)of(−1.44,1.54). Weplottheseestimatesin AppendixFigureB.6. Thet−statisticagainstthenullthat∆Select=D¯ is0.689. Thus,wecannotreject thisoveridentifyingrestrictionatanyconventionallevelofstatisticalsignificance. Thisprovides additionalevidenceinfavorofthetheory. Finally,theendogenousinformationmodelalsoimpliesanupward-slopingrelationshipbetween uncertainty and time since the last price change at the firm level. We show in Appendix FigureB.5andAppendixTableB.1thatthesurveyevidenceisconsistentwiththisprediction. Wedo notwishtoover-emphasizethisresultasmanyfactorsthatvaryatthefirmlevelcouldpotentially drivesucharesult. Thatsaid,itprovidesfurthersuggestiveevidenceinfavorofthemodel.9 6.3. Robustness:Heterogeneity,MeasurementError,andGeneralTime-Dependence In three further analyses, we first probe the quantitative robustness of our findings when firms areheterogeneousintheirnominalrigiditiesandmarginalcostvolatilities. Second,weperform a deconvolution analysis to explicitly account for the potential impact that measurement error in the survey might have on our findings. Finally, we examine the importance of allowing for time-dependentpricingfrictionsthataremoregeneralthanthoseofCalvo(1983). ExAnteHeterogeneity.Wehaveassumedinouranalysisthatallfirmsareexanteidenticalanddiffer onlybecausetheyexperiencedifferentproductivityshocksandpricingspells. Ofcourse,firmsmay beheterogeneousinseveralrespectsandthiscouldmatterforthepropagationofmonetaryshocks. However,Theorem2tellsushowheterogeneitycanmatterinverypreciseways. Inparticular,ifwe augmentthemodeltoallowforarbitrarycross-firmheterogeneityinallrelevantprimitives(pricing durationsG ,thecostsofmispricingB ,thecostsofinformationacquisitionω ,andthevolatility i i i 9Givenourdatasetislimitedbythenumberofobservationsthatwehaveinthesurvey,whilewedofindthatfirms thatresettheirpricesmorethanoneyearagoaremoreuncertain,alternativespecificationsyieldunsurprisinglynoisy estimates. 29

Table1: EstimatesofSectoralHeterogeneityinUncertaintyandMarginalCostVolatility GDP Obs. Uˆ∗ σˆ2 Share ManufacturingandConstruction 0.284 195 1.209 0.161 Trade,Transportation,Accommodation,andFoodServices 0.290 150 1.107 0.302 FIREandProfessionalServices 0.426 170 1.090 0.241 GDP-WeightedAverageofThreeSectors 1 515 1.129 0.236 Allsector(Baseline) 1 515 1.173 0.210 Notes:ThistableshowstheestimationresultsforUˆ∗ andσˆ2forthreegroupsofsectors.WealsopresenttheGDPweightedaverageoftheseestimatesaswellasthebaselineestimateswithallsectors.TheGDPshareiscomputed usingthe2018NewZealandGDPbysectors. FIREstandsforFinancialActivities,Information,andRealEstate servicessectors. ofmarginalcostsσ ),wehavethattheCIRtoabeliefshockisgivenby: i (cid:34) ∗(cid:35) U Mb=E[D¯ ]+E i (46) i σ2 i whereD¯ istheaverageexpecteddurationimpliedbyG andU ∗ istheposterioruncertaintyofprice i i i setteri. Moreover,asE[D¯ ]=D¯,heterogeneitydoesnotmatterforthemechanicaltermcoming i frompricestickiness. Heterogeneitythereforematterspreciselyinsofarasthereisheterogeneity ∗ U in i . Moreover,byallowingforunrestrictedheterogeneityinpricinghazardsacrossfirms,this σ2 i formulaholdsundermanyrecentlydevelopedextensionsofthesimpleCalvomodel,suchasthe mixedproportionalhazardmodelproposedbyAlvarez,Borovicˇková,andShimer(2021). To gauge the potential importance of such heterogeneity, we re-estimateU ∗ and σ2 across i i different sectors, which are potentially quite likely to differ along each of the possible margins ∗ highlighted above. We present the results of this analysis in Table 1. We find estimates ofU thatareverysimilaracrosssectors,rangingbetween1.1and1.2,whilefindingmoresubstantial heterogeneity in the instantaneous variance of marginal costs, ranging between 0.16 and 0.30. (cid:183) (cid:184) WeightingeachsectorbyitsGDPcontribution,wefindthat∆Info=Eˆ Uˆ i ∗ =4.98(S.E.: 0.29),which σˆ2 i isclosetoourbaselineestimateof5.59withoutsectoralheterogeneity. Moreadvancedmodelingof heterogeneouspricinghazardsacrossfirms,suchasthatperformedbyAlvarez,Borovicˇková,and Shimer(2021),wouldrequirepaneldatatowhichwedonothaveaccessfromthissurvey. Extending theanalysistoaccountforheterogeneityofthissortisaninterestingavenueforfuturework. Measurement Error. As uncertainty is a complex variable to elicit, it is of course possible that firms’measureduncertaintyiscontaminatedwithmeasurementerror. Toexaminetherobustness of our results to the possibility of measurement error in firms’ uncertainty, we use a standard 30

deconvolutionapproach. Formally,weassumethatmeasurementerrorisadditiveinlogarithms: logU =logU ′+ζ (47) i i i whereU istheuncertaintythatwemeasure,U ′ istrueuncertaintyforfirmi,andζ ∼N(0,σ2)is i i i ζ measurementerrorwithmeanzeroandvarianceσ2.Wethenestimatethedistributionoffirms’true ζ ′ uncertaintyl byusingthedeconvolutionkerneldensityapproachofStefanskiandCarroll(1990) andselectingthetheoreticallyoptimalbandwidthforaGaussiandistributionfromtheobserved data. From the estimated distribution of true uncertainty lˆ′ (σ2), we compute its mode as our ζ estimateoftheoptimalresetuncertainty,Uˆ∗ (σ2)=Mode [U ′ ]. FollowingProposition5,wethen ζ lˆ′ havethatthemodel-implieduncertaintydistributionisgivenby: (cid:195) z−Uˆ∗ (σ2) (cid:33) lM(z;σ2,σ2)=I[z≥Uˆ∗ (σ2)] 1 fˆ ζ (48) ζ ζ σ2 σ2 Wecanthenestimatethevarianceofmarginalcostsσ2andtheextentofmeasurementerrorσ2by ζ minimizingthedistancebetweenthemodel-implieduncertaintydistributionandtheestimated distributionoftrueuncertainty: (cid:179) σˆ2,σˆ2 (cid:180) ∈argmin (cid:90) ∞ (cid:179) l ′ (z;σ2)−lM(z;σ2,σ2) (cid:180)2 dz (49) ζ ζ ζ Uˆ∗(σ2) ζ Theestimatedvarianceofmeasurementerrorisσˆ2=0.369andtheestimatedvarianceofmarginal ζ cost is σˆ2 =0.25, which is larger than the baseline estimate of 0.21. Moreover,Uˆ∗ (σˆ2)=0.73 is ζ smaller than the baseline estimate of 1.17. In Appendix Figure B.3, we compare the estimated uncertaintydistributionswithandwithoutmeasurementerror. InAppendixFigureB.7,weshow thequantitativeeffectsofaccountingformeasurementerrorontheCIR.Wefindasmaller,but quantitativelysimilar,valueof∆Infoandalarger,butquantitativelysimilar,valueof∆Select. PerformanceoftheModelwithCalvo(1983)Pricing.Wehaveallowedforgeneraltime-dependence ofpricing. ItisinterestingtoinspecttheextenttowhichrestrictingtoCalvo(1983)pricingfromthe outsetwouldhaveaffectedourquantitativeconclusions. Concretely,weestimatethefrequencyof price-settingasθˆ= 1. Wethentake fˆasanexponentialdistributionwithhazardθˆandre-estimate D¯ theCIRasbefore. InAppendixFigureB.4,comparetheestimateduncertaintydistributiontothe empiricalone. InAppendixFigureB.7,weshowtheeffectsofrestrictingtoCalvopricingonthe CIRandfindthathasminimalquantitativeeffects. Asaresult,wearguethatrestrictingtoCalvo (1983)pricingiswithlittlequantitativelossforquantifyingtherealeffectsofmonetarypolicyin thisempiricalsetting. 31

7 Counterfactuals: HowMicroeconomicVolatilityandPriceStickinessAffectMonetaryNon-Neutrality Inafinalquantitativeanalysis,westudyhowchangesinmicroeconomicvolatilityandpricestickinessaffectthedegreeofmonetarynon-neutrality. Weleverageourempiricalestimatestobothsign andquantifytheextenttowhichgreatermicroeconomicvolatilityandpricestickinesswouldaffect theefficacyofmonetarypolicy. 7.1. MicroeconomicVolatility We first use the model and data to ask how changes in microeconomic volatility matter for the propagationofmonetaryshocks. AsevidencefromBloom,Floetotto,Jaimovich,Saporta-Eksten, andTerry(2018)showsthatmicroeconomicvolatilityissignificantlyhigherinrecessions,thisallows ustogaugetheimplicationsofthisfact,throughthelensofourmodel,fortherelativeefficacyof monetarypolicyinboomsversusrecessions. Theeffectofσ2 ontheoutputCIRisregulatedbytwoopposingforces. First,thereisadirect effectofincreasingthevolatilityoffirms’marginalcosts. Thismakesthempayattentionlessto theirpriorsastheyknowthattheirpastinformationislessaccurate. Thismeansthatfirmspay moreattentiontotheirinformation,whichdampenstherealeffectsofmonetaryshocks. Second, thereisanindirecteffectonfirms’optimalinformationchoice. ByProposition3,weknowthatthe firsteffecttheoreticallydominatesandtheCIRisalwaysdecreasingwithσ2;however,sincethis effectismitigatedbytheoptimalchoiceofU ∗ ,ourobjectivehereistoquantifytheneteffectofσ2. ∂Mb Givenouridentificationresults,wecanestimateboththesignandmagnitudeof byusing ∂σ2 thestructureofourmodelandourestimatesoffirms’pricingdurationsgˆ,optimaluncertaintyUˆ∗ , andmicroeconomicvolatilityσˆ2.Together,thisinformationpinsdowntheeffectsofmicroeconomic volatilityontheCIRuptoasingleparameter,thediscountrateoffirmsr. The left panel of Figure 5 shows the value of the CIR for different values of microeconomic volatilityσ2,wherewehavecalibratedthevalueofr to0.0034tomatchanannualinterestrateof4 percent. WeobservethatdoublingmicroeconomicvolatilitydecreasestheCIRfromitsbenchmark valueof11.54pptoaround9pp;i.e.,amonetaryshockthatincreasesthevalueofoutputbyonepercentonimpacthasaround2.5percentagepointslessimpactonCIRwhenmicroeconomicvolatility is doubled. This effect is not symmetric as the relationship is convex: cutting microeconomic uncertaintytohalfitsestimatedvalueof0.21increasestheCIRtoaround15.4pp,increasingthe realeffectsofmonetaryshocksbyaround4percentagepoints. Finally,sincer istheonlyexternally calibratedparameterinthissetting,thetoppanelsofAppendixFigureB.8showthattheseeffects 32

Figure5: MicroeconomicVolatility,PriceStickiness,andMonetaryNon-Neutrality Notes:Thisfigureshowstwocounterfactualanalysesofhowmicrouncertaintyandpricestickinessaffectmonetary non-neutrality. Theleftpanelshowstheeffectofmicroeconomicuncertaintyonmonetarynon-neutrality. The rightpanelshowstheeffectofpricestickinessonmonetarynon-neutrality.Redstarsshowourbaselineestimates σˆ2=0.21andε=0.Wepresent95%confidenceintervalsasbluedashedlines. arerobustandlargelyinsensitivetoalternativecalibrationsofthediscountrater,whenitranges from0(equivalenttoanannualdiscountfactorof1)to0.02(equivalenttoanannualdiscountfactor ofapproximately0.8). Thus, we find thathigher microeconomic volatility significantly dampens the real effects of monetary policy. This parallels a similar point that has been made in the context of models of lumpyadjustment(seee.g.,Vavra,2014),inwhichhighervolatilityaffectsthefrequencyofprice adjustments. However,themechanismthatunderliesthisresultinourmodelisentirelydifferent andindependentofitseffectonthefrequencyofpriceadjustments. Inoursetting,frequencyis governedbythetime-dependentarrivalhazardthatisnotaffectedbyvolatility. Instead,thiseffect followsbecausefirmspaylessattentiontopriorinformationwhenmarginalcostsaremorevolatile andarethereforemoreresponsivetocurrentinformationandmonetaryshocks. Moreover, the currentliteratureonmonetarynon-neutralitywithinformationalfrictions(seee.g.,Afrouziand Yang,2021)largelyemphasizestheroleofmacroeconomicvolatilityformonetarynon-neutrality, while this result emphasizes the importance of microeconomic volatility (see e.g., Lucas, 1972, Flynn,Nikolakoudis,andSastry,2023). 33

7.2. PriceStickiness Wenowusethemodeltoanalyzehowchangesinpricestickinessaffectmonetarynon-neutrality. Aswehavemodeledgeneraltime-dependentpricing,therearemanywaystoperturbthestickiness ofprices. Forthisexercise,tomaximizetransparency,wesimplyincreasethedurationofallpricing spellsbyaconstantamountε>0,i.e.,afirmthatwouldhaveresetitspriceattimeh nowresetsits priceattimeh+ε. Moreformally,thedistributionofpriceresettimeschangesfromG toG˜,where G˜(x)=G(x−ε)forallx≥ε. Theorem2thenimpliesthattheeffectsonmonetarynon-neutralityof suchanincreaseinpricestickinessaregivenby: U ∗ (ε) Mb(ε)=D¯ +ε+ (50) σ2 wherethefirsttermisthedirecteffectofanincreaseinstickiness, whichincreasestheaverage expecteddurationone-for-one. Thesecondtermistheindirecteffect,whichcomesfromhowprice stickinessaffectstheoptimallevelofuncertainty. Theorem1impliesthatthisindirecteffecthasa theoreticallyambiguoussignbecauseoftwocountervailingeffectsofεonU ∗ (asperProposition 4). First, longerpricingdurationsmakeinformationmorevaluableforthecurrentpricingspell by increasing its duration. Thus, a marginally better pricing decision now yields higher profits for a longer time. This encourages a lower level of optimal uncertainty. Second, longer pricing durationsmakeinformationlessvaluableforallfuturepricingspellsbecausetoday’sinformationis lessvaluablefurtherintothefuture. Thisservestoincreasethemarginalcostofinformationinthe futureandencouragesahigherlevelofoptimaluncertainty. Despitethistheoreticalambiguity,wecanestimateboththesignandmagnitudeoftheseeffects usingourdata,uptoacalibrationofthediscountrater. Thisrequiresustoestimatetheratioofthe lossesfrommispricingparameterB totheinformationcostparameterω, (cid:161) (cid:100)B(cid:162) ,whichwecandoby ω findingthevalueof B thatrationalizestheU ∗ weseeinthedata. Thatis,wefindtheexactvalueof ω B thatsolvesthefirm’sfirst-orderconditionfortheoptimalchoiceofU ∗ (fromTheorem1)given ω theU ∗ ,σ2andg thatweseeinthedataandanyfixedvalueforr: (cid:181) (cid:182) (cid:181) (cid:183) (cid:184)(cid:182) (cid:100)B (r)= r 1 −Eh e −rh 1 (51) ω 1−Eh[e −rh] U ˆ∗ gˆ U ˆ∗+σˆ2h gˆ Withthisinhand,asεmoves,wecanuseTheorem1tosolveforU ∗ (ε)andthenuseTheorem2to computehowtheCIRdependsonε. TherightpanelofFigure5plotsCIRasafunctionofεunderthecalibratedvalueofr. Wefind thatU ∗ decreases with ε but that the direct effect of price duration mostly dominates the sign ofthesechangesonCIR;e.g.,increasingthedurationofpricingspellsby4percentincreasesthe 34

∗ CIR from its calibrated value of 11.54pp to around 14.74pp. Noting that ifU were insensitive toεthisincreaseshouldhavebeenone-for-one,weseethatthetotaleffectofthedeclineinU ∗ within this range is that it mitigates the direct effect of ε on pricing duration by about 50 basis ∗ points. Therefore,quantitatively,thedeclineinU offsetsapproximately20%oftheincreasein monetarynon-neutralitythatstickinesswouldinduceinamodelwithoutendogenousinformation acquisition. Finally,inthebottompanelsofAppendixFigureB.8,toproberobustnesstothesole externallycalibratedparameter,weplottheresultsofthisexerciseaswevaryr from0to0.02. 8 Conclusion In this paper, we study how to use firms’ measured beliefs to quantify the degree of monetary non-neutralityinageneralequilibriummodelwithnominalrigiditiesandendogenousinformationacquisition. Weshowedthatthecombinationofthesetwoingredientsleadstoselectionin informationacquisition: theprice-settingfirmsarethemostinformedinthecross-sectionatany giventimeanditistheirbeliefsthatultimatelydeterminethedegreeofmonetarynon-neutrality. Implementing our approach in a survey of firms’ beliefs in New Zealand, we estimate that endogenousinformationacquisitiondoublesthedegreeofmonetarynon-neutralityrelativetothe benchmarkmodelwithnoinformationcosts,whileamodelwithexogenousinformationwould overstatemonetarynon-neutralitybyapproximately50%. Finally,weshowedthatdataonbeliefs arenotonlysufficienttoidentifytherealeffectofmonetarypolicybutalsonecessary: commonly useddataonthedistributionofpricechangesareinsufficientforidentificationinthepresenceof endogenousinformationacquisition. Morebroadly,ourframeworkhasimplicationsforhowmeasuredbeliefs(e.g.,fromsurveys)can beusedtouncoverthemacroeconomicimpactsofimperfectandendogenousinformation. Thisis usefulbecauseitisexanteunclearwhosebeliefs,andwhichaspectsofthosebeliefs,matterforany givenoutcome. Forinstance,withinastandardgeneralmodelofprice-settingwithendogenous informationacquisition,weshowedthattherelevantmomentofbeliefsformonetarynon-neutrality isprice-setters’uncertaintyabouttheiroptimalprices. Thishighlightshow,foragivenoutcomeof interest,onecanusetheorytonarrowdownwhosebeliefstomeasure,whataspectsofthesebeliefs tomeasure,andhowtousethesemeasuredbeliefstounderstandmacroeconomicphenomena at both quantitative and qualitative levels. Interestingly, in our case, our results imply that the idealsurveywoulduseaselectedsampleofprice-setters—asopposedtoarepresentativesample ofall firms, whichisusuallythetargetedpoolforfirmsurveys—andmeasuretheiruncertainty abouttheirdesiredprices. Webelievethisimplicationshouldalsoholdinsomeformforsettings 35

whereeconomicagentsmakeinfrequentdecisions,suchashouseholdsbuyinghousesorother durable goods or firms making lumpy investment decisions. In all such settings, agents might prefertoacquireinformationwhenthedecisionisrelevantandsoaveragesofuncertaintyfrom representativesamplesmightexaggeratethedegreeofinformationrigiditiesthatarerelevantfor macroeconomicoutcomes. Our analysis also highlights several questions for future research. Our model shows how a givenandexogenousprocessforthearrivalofpriceadjustmentsaffectsthedynamicinformation acquisitionpolicyoffirms. Nonetheless,theprocessforadjustmentofpricescanitselfbeaffected bytheinformationacquisitionpoliciesoffirms. Whileweabstractedawayfromthisfeedbackin thispapertofocusonhowthearrivalprocessaffectsincentivesforacquiringinformationovertime, studyingthisfeedbackeffectisanopenquestionforfutureresearch, whichcanbeachievedby extendingourformulationofnominalrigiditiesbyincludingmenucostsforchangingprices. In thisregard,previousworkbyAlvarez,Lippi,andPaciello(2011,2016)showshowsuchinteractions work in models where agents can pay a fixed cost and update their information set to that of a fullyinformedagent. However,ouranalysisshowsthatwhenupdatingtosuchinformationsets is not cost-effective and information acquisition is flexible, these interactions could take more complicatedformsasthehistoriesofpreviousbeliefsnowmatterbyformingtheagents’priors. Whilethesemodelsareanalyticallycomplextosolve,wethinkthatourmainmodelmechanism wouldstilloperateinamodelwithstate-dependentpricingfrictions. Indeed,previousworkon menucostmodelswithflexibleinformationacquisitioncostsdemonstratesthatfirmsdoacquire additionalinformationwhentheychangetheirprices(Gorodnichenko,2008,Yang,2022).Thus,how state-dependentpricingfrictionsaffecttheimplicationsofuncertaintyformonetarynon-neutrality isaquantitativequestionthatweleavetofutureresearch. References AFROUZI,H.(2024):“Strategicinattention,inflationdynamics,andthenon-neutralityofmoney,”JournalofPolitical Economy,Forthcoming. AFROUZI,H.,ANDC.YANG(2021):“DynamicrationalinattentionandthePhillipscurve,”CESifoWorkingPaper. ALVAREZ,F.,H.LEBIHAN,ANDF.LIPPI(2016):“Therealeffectsofmonetaryshocksinstickypricemodels:Asufficient statisticapproach,”AmericanEconomicReview,106(10),2817–2851. ALVAREZ,F.,ANDF.LIPPI(2014):“Pricesettingwithmenucostformultiproductfirms,”Econometrica,82(1),89–135. ALVAREZ, F. E., K. BOROVICˇKOVÁ, AND R. SHIMER (2021): “Consistentevidenceondurationdependenceofprice changes,”NBERWorkingPaperNo.29112. ALVAREZ, F. E., F. LIPPI, AND L. PACIELLO (2011): “Optimalpricesettingwithobservationandmenucosts,” The QuarterlyJournalofEconomics,126(4),1909–1960. (2016):“Monetaryshocksinmodelswithinattentiveproducers,”TheReviewofEconomicStudies,83(2),421–459. 36

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Appendices A Proofs A.1. ProofofTheorem1 Proof. Wefirstcharacterizeoptimalpricingconditionalonanarbitraryinformationpolicyµi,t . WS Letν bethefirm’sbeliefregardingW attime t. Supposethatthefirmhasreceivedapricing i,t it opportunityatsomedatet. Thefirm’spricepolicyproblemisgivenby: (cid:183)(cid:90) h (cid:183) B (cid:184) (cid:184) J(ν i,t )=supEh e −rτ − (p−q i,τ)2 dτ+e −rhJ(ν i,t+h )|ν it (A.1) p 0 2 Thus,anyoptimalpricesolves: (cid:183)(cid:90) h (cid:184) (cid:183)(cid:90) h (cid:184) p i,t (ν i,t )Eh e −rτ dτ =Eh e −rτE[q i,τ |ν i,t ]dτ (A.2) 0 0 UsingthefactthatE[q i,τ |ν i,t ]=q¯+σE[W i,t |ν i,t ],weobtain: p (ν )=q¯+σE[W |ν ] (A.3) it it i,t i,t Wecanthereforecomputethevaluefunction J(ν )as: it (cid:183)(cid:90) h (cid:183) B (cid:184) (cid:184) (cid:104) (cid:105) J(ν i,t )=Eh e −rτ − σ2V[W i,τ |ν i,t ] dτ +E e −rhJ(ν i,t+h )|ν i,t (A.4) 2 0 Wenowshowthatthefirmonlyacquiresinformationwhenitchangesitsprice. Fixatimet atwhich thefirmcannotchangeitsprice. Thevalueofagiveninformationpolicyisgivenby: V˜(ν i,t )=Eh (cid:183) −ω (cid:90) h e −rτ d d I i τ ,τ dτ+e −rhJ(ν i,t+h )|ν i,t (cid:184) (A.5) 0 Fix the horizon at which the firm next adjusts its price h. For each such h, suppose that the informationpolicyyieldsν i,t+h andlettheinformationbeI i,τunderthispolicy. Considerinstead aninformationpolicythatacquiresnoinformationuntiltimet+h andachievesthesameν i,t+h andlettheinformationbe˜I i,τ underthepolicy. Asbothpoliciesattainthesameposterioratthe nextprice-settingopportunity,thedifferenceinthevaluesofthesepoliciesisjustthedifferencein theinformationcosts. Moreover,wehavethatthisdifferenceininformationcostssatisfies: ω (cid:181)(cid:90) h e −rτ d d I i τ ,τ dτ−e −rh(˜I i,t+h −˜I i,t ) (cid:182) ≥ωe −rh (cid:181)(cid:90) h d d I i τ ,τ dτ−(˜I i,t+h −˜I i,t ) (cid:182) 0 0 =ωe −rh(cid:161) (I i,t+h −I i,t )−(˜I i,t+h −˜I i,t ) (cid:162) (A.6) =ωe −rh(cid:161)I i,t+h −˜I i,t+h (cid:162) wheretheinequalityfollowsase −rh≤e −rτ forτ≤h,thefirstequalityfollowsbythefundamental theoremofcalculus,andthefinalequalityfollowsastheinitialinformationunderbothpoliciesis thesame. Thus,acquiringinformationonlywhenthereisapriceresetopportunityyieldsahigher valueifthispolicyleadstoacquiringlessinformationintotal. Considerthefollowinggarblingof 39

thesignalsobtainedunderthebaselineinformationpolicy: receiveaperfectsignalaboutν i,t+h , i.e.,garble{s i,τ}τ∈[t,t+h] intotheinducedposteriorattimet+h. Asthisisagarbling,andmutual informationismonotoneintheBlackwellorder,wehavethatI i,t ≥˜I i,t+h . Itremainstocharacterizeoptimalinformationacquisitionwhenfirmsresettheirprice. First, weshowthatanyoptimalinformationstructureisGaussian. FixapathofpriceresettimesR,let sucharesettimebet,andletν i,t−bethebeliefatthestartoftimet. Wehavethatν i,0 =N(0,σ2 0 ). Let{p t } t∈R bethesequenceofrandomvariablescorrespondingtothefirm’sresetpricesateach resetdateandletSt betheinformationsetimpliedbythispricesequence. Nowdefineasequence of Gaussian random variables {pˆ t } t∈R such that for all t ∈ R: V[W i,t |pˆ t ] = E(cid:163)V[W i,t |St] (cid:164) . The expected nominal profits of the firm are the same under both policies. Thus, {pˆ t } t∈R yields a payoffimprovementifandonlyifitstotalmutualinformationislesser. Thisisimmediateas,for anygivenexpectedvariance-covariancematrix,theGaussianrandomvariablemaximizesentropy (seeChapter12inCoverandThomas,1991). Thus,asR wasarbitrary,thefirmshouldacquirea Gaussiansignalateachpriceresetopportunityregardlessofthesequenceofpriceresettimes. Second,wewritetheirdynamicoptimizationproblemusingthisstructure. Weobservethat, W i,t+h = W i,t+h −W i,t +W i,t −W i,0 , W i,t+h −W i,t ⊥ W i,t −W i,0 , and W i,t+h −W i,t |ν i,t ∼ N(0,h). Thus,ν i,t+h−istheconvolutionmeasureofν i,t withN(0,h),whichwewilldenotebyν i,t ∗N(0,h). Moreover,weknowthatV[W i,τ |ν i,t ]=τ+V[W i,t |ν i,t ]. AsthefirmacquiresaGaussiansignal,we havethattheirproblemreducesto: B (cid:183)(cid:90) h (cid:184) (cid:104) (cid:105) ω (cid:181) U (cid:182) V(U i,t−)= max −U i,t Eh e −rτ dτ +Eh e −rhV(U i,t +σ2h) + ln i,t (A.7) Ui,t ≤Ui,t− 2 0 2 U i,t− Takingthefirst-orderconditionwehavethat(iftheconstraintthatU i,t ≤U i,t−isslack): B (cid:183)(cid:90) h (cid:184) (cid:104) (cid:105) ω 1 0=− Eh e −rτ dτ +Eh e −rhV ′ (U +σ2h) + (A.8) i,t 2 2U 0 i,t Bytheenvelopetheorem,wealsohavethat: ω 1 V ′ (U +h)=− (A.9) i,t 2U +σ2h i,t Thus,weobtainthefollowingconditionfortheoptimalityofU : i,t 1 Bσ2 (cid:183)(cid:90) h (cid:184) (cid:183) 1 (cid:184) = Eh e −rτ dτ +Eh e −rh (A.10) U ω U +σ2h i,t 0 i,t Toseethatthisequationhasauniquesolution,werewriteitas: B (cid:183)(cid:90) h (cid:184) (cid:183) U (cid:184) 1−U Eh e −rτ dτ =Eh e −rh i,t (A.11) i,tω U +σ2h 0 i,t Theright-handsideisastrictlypositiveandstrictlyincreasingfunctionofU andtheleft-hand i,t sideisastrictlydecreasingfunctionthatattainsavalueof1atU =0andattainsavalueof0at i,t z¯= 1 . Thus,thisequationhasauniquesolutionU ∗ ,whichmoreoversatisfiesU ∗≤z¯. BEh (cid:104)(cid:82)he−rτdτ (cid:105) ω 0 40

Moreover,computingthesecondderivativeoftheobjectivefunction,weobtain: (cid:195) (cid:33) (cid:34) (cid:195) (cid:33)(cid:35) ω 1 (cid:183) 1 (cid:184) ω 1 1 − −Eh e −rh <− Eh e −rh − ≤0 (A.12) 2 U2 (U +σ2h)2 2 U2 (U +σ2h)2 i,t i,t i,t i,t Thus, astheproblemisstrictlyconcave, wehavethissolutionissimplytheminimumbetween U i,t−andU ∗ . Asaresult,ifU i,t− ≤U ∗ thefirmacquiresnoinformation,andifU i,t− >U ∗ ,thefirm ∗ acquiresaGaussiansignalofW thatresetsitsposterioruncertaintyaboutZ toU . i,t it A.2. ProofofCorollary1 Proof. ByTheorem1,theoptimallevelofuncertaintysolves: B (cid:183)(cid:90) h (cid:184) LHS(U ∗ ;B,ω,r,G)≡1−U ∗ Eh e −rτ dτ ω 0 (A.13) (cid:183) ∗ (cid:184) U =Eh e −rh ≡RHS(U ∗ ;r,σ2,G) U ∗+σ2h ∗ Given the existence of a unique solutionU (from Theorem 1), the results are immediate from theobservationsthat: LHSisdecreasinginU ∗ ,B (whichisincreasinginη),andG (inthesenseof first-orderstochasticdominance)andincreasinginωandr,andRHSisdecreasinginσ2,r,andG ∗ ∗ andincreasinginU . SeeProposition4foranexplicitexampleofhowU movesambiguouslywith respecttoFOSDchangesinG. A.3. ProofofCorollary2 Proof. CombiningEquation16withtheassumptionofTaylorpricing,wehavethat: ω ω (cid:181) 1−e −rk(cid:182) −e −rk =B (A.14) U ∗ U ∗+σ2k r Rewritingthisequation,weobtainthat: (cid:183) (cid:181) 1−e −rk(cid:182)(cid:184) (cid:183) (cid:181) 1−e −rk(cid:182) (cid:179) (cid:180) (cid:184) U ∗2 B +U ∗ B σ2k−ω 1−e −rk −ωσ2k=0 r r Applicationofthequadraticformulaandnotingthatonlythegreatersolutionisvalidcompletes theproof. A.4. ProofofCorollary4 ∗ Proof. ByTheorem1,thefirm’suncertaintyataprice-settingopportunityisresettoU andthey acquirenoinformationbetweenprice-settingopportunities. Thus,inh periods,theiruncertainty isgivenby: (cid:104) (cid:105) U i,t =V[q i,t |S i t]=V[q i,t |S i t−h]=V σ(cid:161) W i,t −W i,t−h (cid:162)+σW i,t−h |S i t−h (A.15) (cid:104) (cid:105) =σ2h+V σW i,t−h |S i t−h =σ2h+U ∗ asclaimed. 41

A.5. ProofofProposition1 Proof. FromTheorem1,weknowthatfirmsdonotacquireinformationbetweenpriceresetting opportunities. Thus,E [q ]=E [q ]untilthefirmnextresetsitsprice,whichwewillsuppose i,t i,t i,0 i,0 ′ happens in h periods. As firms’ marginal costs follow a martingale, this implies that the firm’s expectedbeliefgapuntilperiodh ′ issimplythefirm’sinitialbeliefgap, yb. FromTheorem1,we havethatwhenfirmsresettheirprices,theyacquireaGaussiansignaloftheirmarginalcostswitha signalnoiseσ˜ h+h′ thatresetstheirposterioruncertaintytoU ∗ : s i,t+h′ =W i,t+h′ +σ˜ h+h′ ε i,t+h′ (A.16) whereε i,t+h′ ∼N(0,1). Becauseofthis,aresettingfirmhasaconditionalexpectationoftherandom componentoftheirmarginalcoststhatisgivenby: E i,t+h′[W i,t+h′]=κ h+h′s i,t+h′ +(1−κ h+h′)E i,t [W i,t ] (A.17) =W i,t+h′ +(1−κ h+h′)(E i,t [W i,t ]−W i,t+h′)+κ h+h′ σ˜ h+h′ ε i,t+h′ Thisimpliesthatthebeliefgapisgivenby: 1 σ y i b ,t+h′ =(1−κ h+h′)y i b ,t +(1−κ h+h′)(W i,t+h′ −W i,t ) γ − γ κ h+h′ σ˜ h+h′ ε i,t+h′ (A.18) =(1−κ h+h′)y i b ,t +Z i,t+h′ whereZ i,t+h′ ∼N(0,σˆ2 h+h′ ). Wecanthenproceedrecursivelytocharacterizeexpectedlifetimeoutputgapsbyobserving that: ′ (cid:34) (cid:90) h ′ (cid:90) h ′ (cid:179) (cid:180) (cid:35) Y(yb,yx,h)=Eh,Z ybdτ+ yxdτ+Y (1−κ h+h′)yb+Z h′,0,0 (A.19) 0 0 WenowguessandverifythatY(yb,yx,U,h)=β(h)yx+m(h)yb. PluggingthisguessintoEquation A.19andmatchingcoefficients,weobtainthatβ(h)andm(h)mustsatisfy: β(h)=E [h ′|h]=D¯ (A.20) g h m(h)=E g [h ′|h]+m(0)Eh g ′ [1−κ h+h′ |h]=D¯ h +m(0)(1−κ¯ h ) (A.21) m(0)= E g [h ′ ] =D¯ 1 (A.22) 1−E g [1−κ h′] 0κ¯ 0 completingtheproof. A.6. ProofofTheorem2 Proof. First, by Proposition 1, we have that the CIR is given by Equation 32. We now show that 42

D¯ 1−κ¯ =U ∗ . Bydefinition,wehavethat: 0 κ¯0 σ2 (cid:183) ′ (cid:183) σ2(h+h ′ ) (cid:184)(cid:184) (cid:183) ′ (cid:183) U ∗ (cid:184)(cid:184) 1−κ¯ =E [1−κ¯ ]=E 1−Eh |h =E Eh |h f h f g U ∗+σ2(h+h ′ ) f g U ∗+σ2(h+h ′ ) =E (cid:34) Eh ′ (cid:34) U σ2 ∗ |h (cid:35)(cid:35) = (cid:90) ∞ (cid:34) (cid:90) ∞ U σ2 ∗ g(τ) dτ (cid:35) f(h)dh (A.23) f g U σ2 ∗ +(h+h ′ ) 0 h τ+U σ2 ∗ 1−G(h) Wenowstateandproveanancillaryresultthatcharacterizesthecross-sectionaldistributionof durationsintermsoftheexpecteddurationofapricesettingfirmandthedistributionofpricesettingopportunities.10 LemmaA.1.Thedistributionofpricingdurationsinthecross-sectionisgivenby: 1 f(h)= (1−G(h)) (A.26) D¯ 0 Proof. Toderive f,definep h =P[h˜ ∈[h−δ,h]]andobservethatp h =p h−δ ×(1−P[Resetbetweenh− δandh|Notresetbyh−δ]). Thus,wehavethat: G(h)−G(h−δ) p h −p h−δ =−p h−δ 1−G(h−δ) (A.27) divingbyδandtakingthelimitδ→0,weobtain: f ′ (h)=−f(h)θ(h) (A.28) Integratingthisexpressionyields: (cid:189) (cid:90) h (cid:190) (cid:189) (cid:90) h g(s) (cid:190) f(h)∝exp − θ(s)ds =exp − ds =1−G(h) (A.29) 1−G(s) 0 0 UsingthefactthatG(0)=0,wethenhavethat f(h)= f(0)(1−G(h)). Integratingbothsidesofthis expression,wethenhavethat: (cid:90) ∞ (cid:90) ∞ 1= f(h)dh= f(0) (1−G(h))dh= f(0)E [h]= f(0)D¯ (A.30) g 0 0 0 whichimpliesthat f(h)= 1 (1−G(h)),asclaimed. D¯ 0 10AsthisresultusesthefactthatG admitsadensity,itdoesnotnestTaylorpricing. However,ourresultstillgoes through.Concretely,weobservethath ′=k−hand f isuniformover[0,k].Thus,wehavethat: Eh (cid:104) Eh′ [h ′|h] (cid:105) =Eh[k−h]= k (A.24) f g f 2 Moreover,wehavethat: Eh (cid:104) Eh′(cid:104) U∗ |h (cid:105)(cid:105) U∗ ∗ E [h ′|h=0] f g U∗+σ2(h+h′) =k U∗+σ2k = U (A.25) g 1−Eh g ′(cid:104) U∗ U +σ ∗ 2h′ |h=0 (cid:105) 1− U∗ U + ∗ σ2k σ2 AndtheconclusionofTheorem2stillholds. 43

CombiningEquationsA.23andA.26,weobtainthat: 1−κ¯ = (cid:90) ∞ (cid:34) (cid:90) ∞ U σ2 ∗ g(τ) dτ (cid:35) 1 (1−G(h))dh 0 h τ+U σ2 ∗ 1−G(h) D¯ 0 (cid:90) ∞(cid:90) ∞ U ∗ (cid:90) ∞ (cid:34) (cid:90) τ U ∗ (cid:35) = 1 σ2 g(τ)dτdh= 1 σ2 g(τ)dh dτ D¯ 0 0 h τ+U σ2 ∗ D¯ 0 0 0 τ+U σ2 ∗ (A.31) ∗ = 1 (cid:90) ∞ U σ2 τ g(τ)dτ= 1 U ∗(cid:90) ∞ τ g(τ)dτ D¯ 0 0 τ+U σ2 ∗ D¯ 0 σ2 0 τ+U σ2 ∗ ∗ 1 U = κ¯ D¯ σ2 0 0 whichimpliesthatD¯ 1−κ¯ =U ∗ . SubstitutingthisintoEquation32yieldstheresult. 0 κ¯0 σ2 A.7. ProofofProposition2 Proof. AfteranuncertaintyshockofU˜ >0,wehavethatafirmwithpricingdurationofh nowhasa prioruncertaintyofU ∗+U˜ +σ2h atthetimethemonetaryshockhits. Moreover,byTheorem1,we ∗ havethatatthefirm’snextpriceresetopportunity,itwillresetitsposterioruncertaintytoU . Thus, it’sKalmangainmustsolveU ∗=(1−κ (U˜))(U ∗+U˜ +σ2h)andso: h U˜ +σ2h κ (U˜)= (A.32) h U ∗+U˜ +σ2h ByAdaptingtheargumentsofProposition1,wethenobtainthatTheCIRisgivenby: 1−κ¯(U˜) Mb=D¯ +D¯ (A.33) 0 κ¯ 0 Thus,theimpactofanuncertaintyshockisgivenby: ∂ ∂ + + M U˜ b(cid:175) (cid:175) (cid:175) (cid:175) U˜=0 =− D κ¯ ¯ 0 0κ¯ ′ (U˜)| U˜=0 (A.34) wherewehavethat: (cid:34) (cid:34) ∗ (cid:35)(cid:35) κ¯ ′ (U˜)| =Eh (cid:104) Eh ′(cid:163)κ′ (U˜)| |h (cid:164) (cid:105) =Eh Eh ′ U |h U˜=0 f g h+h′ U˜=0 f g (cid:161) U ∗+σ2(h+h ′ ) (cid:162)2 (cid:90) ∞ (cid:34) (cid:90) ∞ U ∗ g(τ) (cid:35) 1 (cid:90) ∞ (cid:34) (cid:90) ∞ U ∗ (cid:35) = dτ f(h)dh= g(τ)dτ dh 0 h (cid:161) U ∗+σ2τ(cid:162)21−G(h) D¯ 0 0 h (cid:161) U ∗+σ2τ(cid:162)2 1 (cid:90) ∞ (cid:34) (cid:90) τ U ∗ (cid:35) 1 (cid:90) ∞ U ∗τ = g(τ)dh dτ= g(τ)dτ D¯ 0 0 0 (cid:161) U ∗+σ2τ(cid:162)2 D¯ 0 0 (cid:161) U ∗+σ2τ(cid:162)2 ∗ (cid:34) κ2 (cid:35) 1 U = Eh h D¯ σ2 g σ2h 0 (A.35) Completingtheproof. 44

A.8. ProofofProposition3 Proof. ByTheorem2,wehavethat: ∂Mb ∂U ∗ −U ∗ = ∂σ2 σ2 (A.36) ∂σ2 σ2 Moreover,implicitlydifferentiatingEquation16fromTheorem1,weobtainthat: (cid:195) 1 ∂U ∗ (cid:34) (cid:181)∂U ∗ (cid:182) 1 (cid:35)(cid:33) 0=ω − +Eh e −rh +h U ∗2 ∂σ2 ∂σ2 (cid:161) U ∗+σ2h (cid:162)2 (A.37) ∂U ∗(cid:195) (cid:34) 1 (cid:35) 1 (cid:33) (cid:34) h (cid:35) = Eh e −rh − +Eh e −rh ∂σ2 (cid:161) U ∗+σ2h (cid:162)2 U ∗2 (cid:161) U ∗+σ2h (cid:162)2 Thus,wecanwrite: (cid:183) (cid:184) ∂U ∗ = Eh e −rh (U∗+ h σ2h)2 = U ∗ Eh (cid:104) e −rh U∗ U + ∗ σ2U∗ σ + 2 σ h 2h (cid:105) = U ∗ Eh(cid:163) e −rhκ h (1−κ h ) (cid:164) (A.38) ∂σ2 1 −Eh (cid:183) e −rh 1 (cid:184) σ2 1−Eh (cid:183) e −rh (cid:179) U∗ (cid:180)2 (cid:184) σ2 1−Eh (cid:163) e −rh(1−κ h )2 (cid:164) U∗2 (U∗+σ2h)2 U∗+σ2h CombiningthiswithEquationA.36,weobtainthat: ∂Mb U ∗(cid:195) Eh(cid:163) e −rhκ (1−κ ) (cid:164) (cid:33) U ∗ 1−Eh[e −rh(1−κ )] =− h h −1 =− h (A.39) ∂σ2 σ4 1−Eh (cid:163) e −rh(1−κ )2 (cid:164) σ4 1−E[e −rh(1−κ )2] h h Observing that e −rh ≤ 1 and κ h ∈ [0,1) for all h ∈ R +, we obtain that Eh[e −rh(1−κ h )] < 1 and Eh[e −rh(1−κ )2]<1andso ∂Mb <0. h ∂σ2 A.9. ProofofProposition4 ∗ Proof. IffollowsfromM =D¯ +U that b σ2 ∂M b (cid:175) (cid:175) = ∂D¯ (cid:175) (cid:175) + 1 ∂U ∗(cid:175) (cid:175) (cid:175) (cid:175) (cid:175) ∂ε (cid:175) ∂ε(cid:175) σ2 ∂ε (cid:175) ε=0 ε=0 ε=0 Tocalculate ∂ ∂ D ε ¯(cid:175) (cid:175) ε=0 . LetusdefineD¯(ε)astheaveragedurationunderGε(h),whichisgivenby: (cid:90) ∞ 1−Gε(h) (cid:90) ∞ 1−G(h) D¯(ε)= h dh= (h+ε) dh 0 (cid:82) 0 ∞ (1−Gε(h ′ ))dh ′ 0 (cid:82) 0 ∞ (1−G(h ′ ))dh ′ Thus,wehavethat: ∂D¯ (cid:175) (cid:175) (cid:90) ∞ 1−G(h) ∂ε (cid:175) (cid:175) ε=0 = 0 (cid:82) 0 ∞ (1−G(h ′ ))dh ′ dh=1 (A.40) As for the second term, let U ∗ (ε) be the reset uncertatiny defined under Gε(h). Then, by the definitionofGε(h),Theorem1impliesthatU ∗ (ε)solves: ω (cid:183) ω (cid:184) (cid:181) 1−Eh[e −r(h+ε)] (cid:182) −Eh e −r(h+ε) =B U ∗ (ε) U ∗ (ε)+σ2(h+ε) r (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) ≡MC(ε) ≡MB(ε) 45

Differentiatingeachsidewithrespecttoεandevaluatingatε=0wehave: ∂MC(ε) (cid:175) (cid:175) (cid:175) =− ω ∂U ∗(cid:175) (cid:175) (cid:175) +Eh (cid:34) e −rh ω (cid:35) (cid:181)∂U ∗(cid:175) (cid:175) (cid:175) +σ2 (cid:182) +rEh (cid:104) e −rh ω (cid:105) ∂ε (cid:175) ε=0 U ∗2 ∂ε (cid:175) ε=0 (cid:161) U ∗+σ2h (cid:162)2 ∂ε (cid:175) ε=0 U ∗+σ2h ∂MB(ε) (cid:175) (cid:175) (cid:175) =BEh[e −rh] ∂ε (cid:175) ε=0 Equatingthesetwoequations,wearriveat (cid:183) (cid:184) σ2Eh e −rh (cid:179) U ∗ (cid:180)2 +rU ∗2 (cid:179) Eh (cid:104) e −rh ω (cid:105) −BE[e −rh] (cid:180) ∂U ∗(cid:175) (cid:175) U∗+σ2h ω U∗+σ2h r = ∂ε (cid:175) (cid:175) ε=0 1−Eh (cid:183) e −rh (cid:179) U∗ (cid:180)2 (cid:184) U∗+σ2h σ2Eh(cid:163) e −rh(1−κ )2(cid:164)−rU ∗ (cid:179) BU ∗ −1 (cid:180) h rω = 1−Eh (cid:163) e −rh(1−κ )2 (cid:164) h Thus,wehave ∂M b (cid:175) (cid:175) = ∂D¯ (cid:175) (cid:175) + 1 ∂U ∗(cid:175) (cid:175) (cid:175) (cid:175) (cid:175) ∂ε (cid:175) ∂ε(cid:175) σ2 ∂ε (cid:175) ε=0 ε=0 ε=0 ∗(cid:179) ∗ (cid:180) 1−rU U −1 = σ2 UMin 1−Eh (cid:163) e −rh(1−κ )2 (cid:164) h whereUMin= ωr istheminimumuncertaintyasdefinedinthemaintext,sothat U ∗ −1>0. B UMin A.10. ProofofProposition5 Proof. ByCorollary4,afirm’suncertaintyh periodsafterchangingitspriceisU =U ∗+σ2h≥U ∗ . Thus,L(z)=P[U ≤z]=P (cid:104) h≤ z−U ∗(cid:105) =F (cid:179) z−U ∗(cid:180) . Differentiatingthisexpressionyieldstheclaimed σ2 σ2 formulaforl(z). A.11. ProofofTheorem3 Proof. Toderivethedistributionofpricechanges,westartbyfindingtheconditionaldistribution ofpricechangesforfirmswhohadagivendurationofh periodswhohadafixedinformationsetat theirlastpricechangeopportunity. Wethenmarginalizeoverthedistributionofpricedurations andinformationsetstoobtainthepricechangedistribution. Tothisend,considerafirmi thatis changingitspriceattimet thatchangeditspriceh periodsagoanddefine: ∆hp i,t ≡p i,t −p i,t−h =σ(cid:161)E i,t [W i,t ]−E i,t−h [W i,t−h ] (cid:162) (A.41) Moreover,wehavethat: E[W i,t ]=κ h s i,t +(1−κ h )E i,t−h [W i,t−h ] (A.42) where: s =W +σ˜ ε (A.43) i,t i,t h i,t 46

Combiningtheseequations,wecanwrite: ∆hp i,t =σκ h (cid:161) W i,t +σ˜ h ε i,t −E i,t−h [W i,t−h ] (cid:162) (A.44) Therefore,wehavethat: (cid:179) (cid:180) ∆hp |St−h∼N 0,σˇ2(St−h) (A.45) i,t i i where: (cid:104) (cid:105) σˇ2(St−h)=κ2V σW +σσ˜ ε |St−h (A.46) i h i,t h i,t i whereweknowthat: (cid:104) (cid:105) (cid:104) (cid:105) V σW i,t |S i t−h =V σ(W i,t −W i,t−h )+σW i,t−h |S i t−h =σ2h+U ∗ (A.47) as, byTheorem1, wehavethatatatimeofpricereset(which t−h isbyassumption)thefirm’s posterioruncertaintyisalwaysequaltoV[σW i,t−h |S i t−h]=U ∗ . Thus,wehavethat: σˇ2(St−h)=κ2(cid:161)σ2h+U ∗+σ2σ˜2(cid:162) (A.48) i h h Moreover,thesignalnoiseσ˜2 thatachievestheKalmangainκ solves: h h 1−κ σ2σ˜2 =(U ∗+σ2h) h (A.49) h κ h andsowehavethat: (cid:181) 1−κ (cid:182) σˇ2(St−h)=κ2(U ∗+σ2h) 1+ h =κ (U ∗+σ2h)=σ2h (A.50) i h κ h h Thus, we have that conditioning on the firm’s information set is irrelevant and the conditional distributionofpricechangesisthemarginaldistributionofpricechanges: ∆hp |St−h∼N(0,σ2h) =⇒ ∆hp ∼N(0,σ2h) (A.51) i,t i i,t Finally,integratingoverthedistributionofpricedurations,G,weobtainthatthedistributionof pricechangesis: (cid:104) (cid:105) H(∆p)=P[∆p ≤∆p|∆p ̸=0]=Eh P[∆hp ≤∆p|∆p ̸=0] i,t i,t g i,t i,t (cid:183) (cid:181) ∆p (cid:182)(cid:184) (cid:90) ∞ (cid:181) ∆p (cid:182) (A.52) =Eh Φ (cid:112) = Φ (cid:112) dG(h) g σ h 0 σ h whichdependsonσandG butdoesnotdependonU ∗ . 47

B AdditionalFiguresandTables 10 0.85 8 0.8 6 0.75 4 0.7 2 0.65 0 0.6 0 5 10 15 20 25 30 0 5 10 15 20 25 30 FigureB.1: ExpectedDurationofNextPriceChangesandKalmanGains Notes:Theleftpanelshowstheaverageconditionalduration,D¯ =Eh′ [h ′|h],whichishowlongafirmthatresetitsprice h g hperiodsagoexpectstowaitbeforeresettingitsprice(bluesolidline),aswellastheaverageduration,D¯ =Eh[D¯ ], f h whichishowlongthefirmsexpecttowaitonaveragebeforeresettingtheirprices(bluedashedline).Therightpanel showstheaverageconditionalKalmangain,κ¯ h =Eh g ′ [κ h′+h |h],whichistheexpectedKalmangainatthenextprice resetopportunityforafirmthatlastresetitspricehperiodsago(bluesolidline),aswellastheaverageKalmangain, κ¯=Eh[κ¯ ],whichistheaverageacrossallfirmsoftheexpectedKalmangainwhentheynextresettheirprices(blue f h dashedline) 0.12 1 2 0.1 0.8 1.5 0.08 0.6 0.06 1 0.4 0.04 0.5 0.2 0.02 0 0 0 0 10 20 30 0 10 20 30 FigureB.2: DistributionofPriceResetOpportunitiesandtheHazardRate Notes: TheleftpanelshowstheempiricallyestimateddistributionofpriceresetopportunitiesG,givenbyG(h)= 1−fˆ(h)/fˆ(0)where fˆistheempiricaldistributionoftimesincefirms’lastpricechanges.g isthedensityfunction.The rightpanelshowsthehazardrate,θ(h)=g(h)/(1−G(h)). 48

FigureB.3: UncertaintyDistributionwithMeasurementErrors Notes:Thisfigureshowsthedistributionoffirms’subjectiveuncertaintyabouttheiridealpricesunderourbaseline approachandtheapproachtoaccountformeasurementerrorthatwedescribeinSection6.3.Thelabelingfollows Figure3. FigureB.4: UncertaintyDistributionunderCalvo Notes:Thisfigureshowsthedistributionoffirms’subjectiveuncertaintyabouttheiridealpricesunderourbaseline approachandwhenweimposethatthepricinghazardisconstantasinCalvo(1983)thatwedescribeinSection6.3. ThelabelingfollowsFigure3. 49

FigureB.5: FirmsThatRecentlyChangedTheirPricesAreLessUncertain 5 4 3 2 1 ytniatrecnU evitcejbuS segnahC ecirP laedI tuoba Linear Fit 66% Confidence Interval 0 2 4 6 8 10 12 14 16 18 20 Time Since Last Price Changes (Months) Notes:Thisfigureplotsthetimeelapsedsincefirms’lastpricechangesversusfirms’subjectiveuncertaintyabouttheir idealpricechanges.Theblacklineisalinearfittedlineandtheshadedareais66%confidenceinterval.Wedropthe outlierswithimpliedsubjectiveuncertaintygreaterthan20.Thesizeofthebinsrepresentstheaverageemploymentof firmsineachpercentile. TableB.1: TheRelationshipBetweenUncertaintyandTimeSinceChangingPrice (1) (2) (3) Dependentvariable:Subjectiveuncertaintyaboutfirms’idealpricechanges Dummyforpricechangesinthelast3months 0.0495 (0.0862) Dummyforpricechangesinthelast6months 0.0306 (0.0850) Dummyforpricechangesinthelast12months -0.643*** (0.151) Observations 467 467 467 R-squared 0.114 0.114 0.153 IndustryControls Yes Yes Yes Firm-levelControls Yes Yes Yes ManagerControls Yes Yes Yes Notes:ThistablereportsresultsfortheHuberrobustregression.Thedependentvariableisthesubjectiveuncertainty aboutfirms’idealpricechangesinthe2018Q1survey,whichismeasuredbythevarianceimpliedbyeachfirm’sreported probabilitydistributionoverdifferentoutcomesoftheiridealpricechangesiffirmsarefreetochangetheirprices.Each Columnusesdifferentthresholdsforthedummyforthelastpricechanges.Industryfixedeffectsincludedummies for13sub-industries.Firm-levelcontrolsincludealogoffirms’age,alogoffirms’employment,foreigntradeshare, numberofcompetitors,theslopeoftheprofitfunction,firms’expectedsizeofpricechangesin3months,andfirms’ subjectiveuncertaintyabouttheiridealpricesinnextthreemonthsreportedinthe2017Q4survey.Managercontrols includetheage,education,andtenureatthefirmoftherespondent(eachfirm’smanager).Sampleweightsareapplied toallspecifications.Robuststandarderrors(clusteredatthe3-digitAustralianandNewZealandStandardIndustrial Classificationlevel)arereportedinparentheses.***denotesstatisticalsignificanceatthe1%level. 50

6.71 5.95 0.76 0.00 FigureB.6: OveridentificationTestfor∆Select=D¯ Notes:Thisfigureshowsthebaselineestimatesoftheinformationselectioneffect(∆Select)andtheaveragepricing duration(D¯).Wealsopresentthedifference∆Select−D¯ toimplementtheoveridentificationtestofthetheoryderived inEquation45.Wepresent95%confidenceintervalsasblackverticallines. FigureB.7: CIRDecompositionwithMeasurementErrorsandCalvoPricing Notes:Thisfigureshowstheoutputeffectsofa1percentagepointshocktoperceivedgaps(leftbar),tobeliefgaps (middlebar),andtobeliefgapsignoringtheselectioneffect(rightbar).Wecompareourbaselineestimates(blue bars)totheestimatesthatweobtainwhenweaccountformeasurementerror(graybars)andimposethatfirmshave aconstantpricinghazardasinCalvo(1983)(redbars).WithCalvopricing,theoutputeffectofashocktobeliefgaps is11.28pp(middleredbar)andtheoutputeffectofashocktobeliefgapsignoringtheselectioneffectis17.66pp (rightredbar).Afteraccountingformeasurementerror,theoutputeffectofashocktobeliefgapsis8.85pp(middle graybar)andtheoutputeffectofashocktobeliefgapsignoringtheselectioneffectis16.24pp(rightgraybar).The estimationapproachesforthesetwocomparisonsaredescribedinSection6.3.ThelabellingfollowsFigure4. 51

FigureB.8: MicroeconomicVolatility,PriceStickiness,DiscountRate,andMonetaryNon-Neutrality Notes:Thisfigureshowscounterfactualanalysesonhowmicrouncertainty,pricestickiness,andthediscountrate affectmonetarynon-neutrality.Thetwoleftpanelsshowtheeffectofpricestickinessandthediscountrateonfirms’ ∗ optimalresetuncertainty(U )asafunctionofthevolatilityofmarginalcost.Thetworightpanelsshowtheeffect ofpricestickinessanddiscountrateonmonetarynon-neutrality(Mb)asafunctionofthevolatilityofmarginal cost.Redstarsshowtheestimateswithε=0,σˆ2=0.21,andthebaselinediscountrater =0.0034,whichimpliesan annualdiscountfactorofβ= 1 =0.96(1/12). 1+r 52

-9.66 0.83 0.79 -20.76 -21.36 0.77 0 0.0034 0.01 0.02 0 0.0034 0.01 0.02 FigureB.9: MicroeconomicVolatility,PriceStickiness,andMonetaryNon-Neutrality Notes:Thisfigureshowstwocounterfactualanalysesonhowmicrouncertaintyandpricestickinessaffectmonetary non-neutrality.Theleftpanelshowstheeffectofmicroeconomicuncertaintyonmonetarynon-neutralityinduced byinformationfriction,∂ (cid:131) Mb/∂σ2asafunctionofthediscountrate(r).Therightpanelshowstheeffectofprice stickinessonmonetarynon-neutrality, ∂ (cid:131) Mb/∂ε| ε=0 asafunctionofthediscountrate(r). Redstarsshowthe estimateswiththebaselinediscountrater =0.0034,whichimpliesβ= 1 =0.96(1/12).Wepresent95%confidence 1+r intervalsasbluedashedlines. 53