MyopiaandAnchoring*

George-MariosAngeletos† ZhenHuo‡

September26, 2018

Abstract

Weconsiderastationarysettingfeaturingforward-lookingbehavior, strategiccomplementarity,

andincompleteinformation. Weobtainanobservationalequivalenceresultthatrecaststheag-

gregatedynamicsofthissettingasthatofarepresentative-agentmodelfeaturingtwodistortions:

myopia, asinmodelswithimperfectforesight; andanchoringofthecurrentoutcometothepast

outcome, asinmodelswithhabitpersistenceandadjustmentcosts. Wefurthershowthattheas-if

distortionsarelargerwhenthegeneral-equilibriumfeedback, orthestrategiccomplementarity, is

stronger. Theseresultsofferafreshperspectiveontheobservableimplicationsofinformational

frictions; buildausefulbridgetotheDSGE literature; andhelpreduceanuncomfortablegapbe-

tweentheprevailingstructuralinterpretationsofthemacroeconomictimeseriesandtherelated

microeconomicevidence. Finally, anempiricalevaluationisofferedinthecontextofinflation,

whereinitisshownhowourresultscanrationalizeexistingestimatesoftheHybridNKPC while

alsomatchingsurveyevidenceonexpectations.

*WethankAlexanderKohlhasandLuigi Iovino fordiscussingourpaper in, respectively, the2018Cambridge/INET

conferenceandthe2018Hydra/CEPR workshop. WealsoacknowledgeusefulcommentsfromJaroslavBorovicka, Simon

Gilchrist, JenniferLa’O,StephenMorris, MikkelPlagborg-Mollerandseminarparticipantsattheaforementionedconfer-

ences, Columbia, NYU,CUHK,the2018DukeMacroJamboree, the2018BarcelonaGSE SummerForumEGF Group, the

SED 2018, the2018NBER SummerInstituteImpulseandPropagationMechanismsGroupandMacroPerspectivesGroup.†MIT andNBER;angelet@mit.edu.‡YaleUniversity; zhen.huo@yale.edu.

1 Introduction

Forward-lookingbehaviorandequilibriumfeedbacksarecentral toourunderstandingofbusiness

cycles, asset-pricefluctuations, industrydynamics, andmore. Inthispaper, weopenanewwindow

intotheinterplayofthesefeatureswitharealisticfrictionintheamountofknowledgeagentshave

aboutthefundamentalsandaboutoneanother’sbeliefsandactions.

Wefirstestablishanobservationalequivalencebetweenarational-expectationssettingfeaturing

suchaninformationalfrictionandabehavioralvariantfeaturingtwodistortions:

• myopia, orextradiscountingoffutureoutcomes; and

• anchoringofcurrentoutcomestopastoutcomes, orbackward-lookingbehavior.

Wefurthershowthattheas-ifdistortionsarelargerwhenthegeneral-equilibrium(GE) feedback, or

thestrategiccomplementarity, isstronger, andelaborateontheunderlyingtheoreticalprinciples. We

finallybuildusefulconnectionstomultiplestrandsoftheliterature, addressadisturbingdisconnect

betweentheprevailingstructuralinterpretationsofthemacroeconomictimeseriesandtherelated

microeconomicevidence, andofferanempiricalevaluationinthecontextofinflation.

Framework. Westudyadynamicsettinginwhichtheoptimalaction(orbestresponse)ineach

perioddependspositivelyontheexpecteddiscountedpresentvaluesofanexogenousfundamental,

denotedby ξt, andtheaverageaction, denotedby at. Intheabsenceoftheinformationalfriction,

thissettingreducestoarepresentative-agentmodel, inwhich at obeysthefollowinglawofmotion:

at = φξt + δEt [at+1] , (1)

where φ > 0, δ ∈ (0, 1], and Et[·] denotestherationalexpectationoftherepresentativeagent.Condition(1)neststheEulerconditionoftherepresentativeconsumer, whichcorrespondstoag-

gregatedemand(ortheDynamicIS curve)intheNewKeynesianmodel, aswellastheNewKeynesian

PhilipsCurve(NKPC),whichdescribesaggregatesupply. Alternatively, thisconditioncanbereadas

anasset-pricingequation, with ξt standingfortheasset’sdividendand at foritsprice. Theseexamples

indicatethebroaderapplicabilityoftheresultswedevelopinthispaper.1

Wedepartfromtherepresentative-agentbenchmarkbyallowinginformationtobeincomplete

(i.e., noisyandheterogeneous). Thiscanbetheproductofeitherdispersedprivateinformation(Lucas,

1972; MorrisandShin, 2002)or rational inattention (Sims, 2003). Eitherway, thekey is thatwe

accommodate, notonly first-order uncertainty(imperfectknowledgeoftheunderlyingfundamental),

butalso higher-order uncertainty(uncertaintyaboutthebeliefsandactionsofothers). Thatis, welet

agentsfacerealisticdoubtsabouttheawareness, attentiveness, orresponsivenessof others.

1ApplicationsnotexploredheremayincludedynamicBertrandcompetitionandmarketswithnetworkexternalities.

1

Mainresult. Underappropriateassumptions, theincomplete-informationeconomyisshownto

beobservationallyequivalenttoavariant, complete-information, representative-agenteconomyin

whichcondition(1)ismodifiedasfollows:

at = φξt + δωfEt [at+1] + ωbat−1 (2)

forsome ωf < 1 and ωb > 0. Thefirstmodification(ωf < 1)representsmyopiatowardsthefuture, the

second(ωb > 0)anchorsthecurrentoutcometothepastoutcome. Theonedullstheforward-looking

behavior, theotheraddsabackward-lookingelement. Furthermore, bothdistortionsareshownto

increasewiththestrengthoftheGE feedback, orthedegreeofthestrategiccomplementarity.2

Theexactformofthisresultreliesonstrongassumptionsaboutthestochasticprocessfor ξt and

theinformationstructure. Itneverthelessstylizesafewmoregeneralinsights, whoserobustnesswe

documentinSection 7 underaflexiblespecificationofthestochasticityandtheinformation.

Onekeyinsight, whichbuildson AngeletosandLian (2018), explainthemyopia. Recallthat, in

oursetting, behaviordependscriticallyonexpectationsofthe future actionsofothers—e.g., current

aggregatespendingdependsonexpectationsoffutureaggregatespending, currentassetpricesdepend

onexpectationsoffutureassetprices, etc. Inequilibrium, suchexpectationsarepinneddownbythe

currentbeliefsof the futurebeliefsofothers. Because suchhigher-orderbeliefs tend tovary less

thanthecorrespondingfirst-orderbeliefs inresponsetoanyinnovationin ξt, it is as if theagents

discountthefutureoutcomesmoreheavilythaninthefrictionless, representative-agentbenchmark.

Furthermore, becausethedependenceofequilibriumbehavioronhigher-orderbeliefsincreaseswith

theGE feedback, orthestrategiccomplementarity, thiskindofmyopiaalsoincreaseswithit.3

Another key insight, which builds on Woodford (2003), Morris and Shin (2006) and Nimark

(2008), regardstheroleplayedbylearning. Learninginducesextrapersistenceinthebeliefsofthe

fundamentalandofthefutureoutcomes. Furthermore, thispersistenceisstrongerinthelatterbeliefs

thanintheformer, reflectingthesmallerdependenceofhigher-orderbeliefsonrecentinformation.

Thisexplainswhythecurrentoutcomeappearstobeanchoredtothepastoutcomeataratethat, like

ourformofmyopia, increaseswiththestrengthoftheGE feedback, orthestrategiccomplementarity.

Weaddtotheliterature, notonlybyblendingtheseinsightsandofferingafewmore, butalsoby

operationalizingthemintermsofourobservational-equivalenceresult, whichisnew. Alsonewisthe

analysisinSection 7. Thisallowsforaflexiblespecificationofthestochasticityandthehigher-order

2SuchGE feedbacksareoften“hidden”behindthekindofrepresentative-agentequilibriumconditionshereinstylizedbycondition(1). TheyincludetheKeynesianincome-spendingmultiplierinthecontextoftheDynamicIS curve, thedynamicstrategiccomplementarityinthefirms’price-settingdecisionsinthecontextoftheNKPC,andthepositivefeedbackfromexpectationsoffuturetradestocurrenttradesinthecontextofassetpricing.

3Tobeprecise, ourobservational-equivalence resultconfounds theaforementionedeffectwithanadditionaleffect,whichisthatthefirst-orderbeliefsofthefundamentalalsomovelessthaninthefrictionlessbenchmark. AsexplainedinSection 7, thiseffectisnotstrictlyneeded: higher-orderuncertaintyisalonesufficientforthedocumentedformofmyopia.However, thetwoeffectsmaynaturallycometogetherinapplicationsandonlycomplementeachother.

2

beliefdynamics, whilealsocuttingtheGordianknotofcomplexsignal-extractionproblems, soasto

developasharpunderstandingoftheprinciplesthatunderlyourobservational-equivalenceresult.

Together, these resultsoffera freshperspectiveon theaggregate implicationsof informational

frictions. Theyallowustodrawanumberofusefulconnectionstotheliterature. Andtheyfacilitate

our empirical exercise in the context of inflation. Weelaborateon these applied aspects of our

contributionbelow.

DSGE. Baselinemacroeconomicmodelsemphasizeforward-lookingbehaviorbuthavehardtime

capturingasalientfeatureoftheaggregatetimeseries: inflation, consumptionandinvestmentalike

appeartorespondsluggishlytotheunderlyinginnovations, asifthereisastrongbackward-looking

component in their lawsofmotion. Toaddress thischallengeandprovideasuccessfulstructural

interpretationofthedata, theDSGE literaturehassacrificedonthemicro-foundations.

Forinstance, tomatchtheinflationdynamics, theliteraturehasfollowed GaliandGertler (1999)

and Christiano, Eichenbaum, andEvans (2005)inreplacingthestandardNKPC withtheso-called

HybridNKPC,abackward-lookingvariantthatfindsnosupportintherelatedmenu-costliterature

(AlvarezandLippi, 2014; GolosovandLucas Jr, 2007; NakamuraandSteinsson, 2013). Similarly, to

matchtheaggregateinvestmentdynamics, theliteraturehasemployedaformofadjustmentcostthat

isincompatiblewithbothstandardQ theory(Hayashi, 1982)andtheliteraturethatstudiesinvestment

atthemicrolevel(Bachmann, Caballero, andEngel, 2013; CaballeroandEngel, 1999).

Ourobservational-equivalenceresultoffersthesharpestpossibleillustrationofhowinformational

frictionscanprovideaplausiblemicro-foundationfortheDSGE add-ons: accommodatingincom-

pleteinformationisakintoaddinghabitpersistenceinconsumption, adjustmentcostsininvestment,

andabackward-lookingelementintheNKPC.Atthesametime, ourresultindicatesthattheas-if

distortionsmaybeendogenoustoGE feedbackmechanismsandtherebyalsotopoliciesthatregulate

thesuchmechanisms.4

MicrovsMacro. TheGE feedbackisactive, andhigher-orderbeliefsarerelevant, onlywhen

agentsrespondtoaggregateshocks. Itfollowsthatthedocumentedformsofmyopiaandanchoring

mayloomlargeatthemacroleveleveniftheyappeartobesmallinmicrodata. Thisprovidesasimple,

unifiedexplanationtowhythemacroeconomicestimatesofboththehabitpersistenceinconsumption

andtheadjustmentcostsininvestmentaremuchhighertheirmicroeconomiccounterparts(Havranek,

Rusnak, andSokolova, 2017; GrothandKhan, 2010; Zorn, 2018); whythepersistenceofinflation

ishigherintheaggregatetimeseriesthanindisaggregateddata(Altissimoet al., 2010); andperhaps

evenwhythemomentuminassetpricesismorepronouncedatthestock-marketlevelthanatthe

individual-stocklevel(JungandShiller, 2005).

4Eearlierworkssuchas Sims (2003), MankiwandReis (2002, 2007), MackowiakandWiederholt (2009, 2015), andNimark (2008)share the idea that informational frictionscansubstitute for theDSGE add-ons, butdonotcontainourobservational-equivalenceresultandourinsightsaboutGE effectsandthemicro-to-macrogap, whichwediscussnext.

3

BoundedRationality. Theformofmyopia, orimperfectforesight, documentedhereisrationalized

byaninformationalfriction, asin AngeletosandLian (2018). Similarformsofimperfectforesight

areobtainedin Garcıa-SchmidtandWoodford (2018)and FarhiandWerning (2017)byreplacing

rationalexpectationswithLevel-kThinking, andin Gabaix (2017)byintroducingabeliefbiascalled

“cognitivediscounting.” Theseworks, however, donotallowtheimperfectiontodiminishwiththe

accumulationofinformationovertimeand, asaresult, donotproducethesecondkeyfeatureof

ourapproach, theanchoringof thecurrentoutcometotheprevious-periodoutcome. Intermsof

condition(2), theyeffectivelylet ωf < 1 butrestrict ωb = 0. Bycontrast, thedataappeartodemand

both ωf < 1 and ωb > 0.5 Thisfavorsofourapproachovertheaforementionedalternatives, butalso

invitestheconsiderationofvariantsthat combine incompleteinformationwithboundedrationality.

AsexplainedinSection 7, suchvariantsmayintensifythedocumenteddistortions.

ApplicationtoInflation. Ourcontributioniscompletedwithanempricalexerciseinthecontext

of inflation. We revisit themicro-foundationsof theNKPC,adding incomplete information. We

assumeamodestdegreeofprice stickiness, one in linewith textbookparameterizationsand the

relatedmicroeconomicevidence. Wenextshowthatourresultshelprationalizeexistingestimatesof

theHybridNKPC,suchasthosefoundin GaliandGertler (1999)and Gali, Gertler, andLopez-Salido

(2005). Wefinallyshowthat this isachievedwithaninformational frictionthat also matchesthe

evidenceoninflationexpectationsprovidedby CoibionandGorodnichenko (2015).

Here, it isusefultoclarifythefollowingpoint. Bymeasuringthepredictabilityof theaverage

forecasterrorinsurveys, CoibionandGorodnichenko (2015)providedakeyempiricalmomentthat

helpsgaugetheleveloftheinformationalfriction. Yet, bytreatinginflationasanexogenousstochas-

ticprocess, thisworkcouldnotpossiblyquantifytheequilibriumimpactofthisfrictionontheactual

inflationdynamics. Ourpaperfillsthisgapbysolvingthefixedpointbetweeninflationexpectations

andactualinflation, byconnectingtheaforementionedempiricalmomenttoitstheoreticalcounter-

part, andbyevaluatingtheimpliedbiteontheequilibriumoutcomes.

Layout. Therestofthepaperisorganizedasfollows. Section 2 expandsontherelatedliterature.

Section 3 introducesourframework. Section 4 developstheobservational-equivalenceresult. Sec-

tion 5 illustratestheapplicabilityofthisresultanddiscusseditsvalue-added. Section 6 containsour

quantitativeexerciseinthecontextofinflation. Section 7 illustratestherobustnessoftheinsightsun-

derlyingourobservational-equivalenceresult. Section 8 concludes. TheAppendicescontainproofs

andafewadditionalresults.5Whiletheliteratureontheforward-guidancepuzzle(Del Negro, Giannoni, andPatterson, 2015; McKay, Nakamura,

andSteinsson, 2016; AngeletosandLian, 2018; FarhiandWerning, 2017)hasfocusedongettingextradiscounting, theDSGE andSVARsliteratureshavelongpointedouttheneedofastrongbackward-lookingcomponent. Furthermore, theavailableevidenceonexpectations(e.g., CoibionandGorodnichenko, 2012, 2015; Coibion, Gorodnichenko, andKumar,2015)suggests, notonlythattheaverageforecastsoffutureoutcomesrespondlittleonimpacttoaggregateshocks, butalsothattheyadjustmoreandmorewiththepassageoftime. Thefirstpropertymapsto ωf < 1, thesecondto ωb > 0.

4

2 RelatedLiterature

Asalreadynoted, ourpaperbuildsheavilyontheexistingliteratureoninformationalfrictions, most

notablyon MorrisandShin (2002, 2006), Woodford (2003), Nimark (2008), and AngeletosandLian

(2018); see AngeletosandLian (2016)forareviewandadditionalreferences. Ourmaincontributions

vis-a-visthisliteratureare: (i)theobservational-equivalenceresultanditsapplications; (ii)thebridges

ithelpsbuildtothreeotherstrandsoftheliterature, namelyonDSGE,onmicro-to-macro, andon

bounded-rationality; (iii) theempiricalexercise in thecontextof inflation; and(iv) theanalysis in

Section 7, whichusesanunconventionalyetflexiblespecificationoftheinformationstructuresoas

toovercomethefamiliardifficultiesinthecharacterizationofthehigher-orderbeliefdynamicsand

shedamplelightonthebroaderunderlyingprinciples.

Onthemethodologicalfront, ourpaperbuildson HuoandTakayama (2015). Inparticular, we

utilizethemethodsofthatpaperinordertosolveexplicitlyfortherational-expectationsequilibrium

under our baseline specification. We then translate theobtained equilibrium in the formof our

observational-equivalenceresultandstudyitscomparativestaticswithrespecttothestrengthofGE

feedbackandotherparameters. Whatisnewhereisthesecondstep, whichisintegraltoourapplied

contribution, andtheanalysisinSection 7, whichfollowsadifferentmethodologicalroute.

Ontheappliedfront, ourpaperismostcloselyrelatedto Nimark (2008). Thispaperisthefirst

tostudyanincomplete-informationversionoftheNKPC.Itsharesouremphasisonforward-looking

beliefsandtheinsightthatincompleteinformationcanrationalizeabackward-lookingcomponent

in the inflationdynamics. It doesnot, however, contain eitherour analytical results or the tight

connectionwedrawbetweentheestimatesoftheHybridNKPC (GaliandGertler, 1999)andthe

evidenceoninflationexpectations(CoibionandGorodnichenko, 2012, 2015).6

MackowiakandWiederholt (2009)arguethatrationalinattentioncanreconcilepricerigidityat

themacrolevelwithpriceflexibilityatthemicrolevel.7 Althoughthismessagesoundssimilarto

oursregardingthegapbetweenmicroandmacro, themechanismisdifferent. Inthatwork, thegap

isexplainedbytheagentspayinglessattentionto, orotherwisefacinggreaterfirst-orderuncertainty

about, aggregateshocksrelativetoidiosyncraticshocks. Inourpaper, instead, thegapisattributedto

GE effectsandhigher-orderuncertainty; itisthereforepresent evenif theleveloffirst-orderuncertainty

about, ortheattentionto, thetwokindsofshocksisthesame. Thetwomechanismsarenevertheless

complementaryandnaturallycometogetherinapplications.6Thesepointsdistinguishmorebroadlyourcontributionfromalargerliterature, including MankiwandReis (2002), Reis

(2006), Kiley (2007), Melosi (2016), and Matejka (2016), thatstudiesprice-settinginthepresenceofinformationalfrictions.A fewotherpapersdirectlyreplacetherepresentativeagent’sinflationexpectationswiththeaverageforecastofinflationinsurveydatawhenestimatingthestandardNKPC;asnotedin Mavroeidis, Plagborg-Møller, andStock (2014), thisapproachlackssatisfactorymicro-foundationsandisindeedinconsistentwithouranalysis.

7Relatedly, Zorn (2018)arguesthatrationalinattentioncanhelpexplainwhysectoralinvestmentrespondssluggishlytoaggregateshocksbutnottosectorspecificshocks, and Carrollet al. (2018)arguethatinformationalfrictionscanreconcilethemicroandmacroestimatesofconsumptionhabit.

5

Bordaloet al. (2018)and KohlhasandWalther (2018a,b)arguethattheexpectationsdatapainta

morevariedpicturethantheonecontainedinourpaperandintheaforementionedliterature: while

averageforecaststendtounder-reacttoaggregateshocks, individualforecaststendtoover-reactto

idiosyncraticnews, suggestingeitheraparticulardeparturefromrationalexpectations(Bordaloet al.,

2018)oramoreelaborateinformationalfriction(KohlhasandWalther, 2018a,b). Theinterplayof

theseideaswiththemechanismswestudyisanimportantquestion, beyondthescopeofthispaper.

Nevertheless, themost relevant fact forourpurposes is theunder-reactionofaverage forecasts to

aggregateshocks, whichisconsistentwithourtheory.

TherelationofourpapertotheliteratureonLevel-kThinkinghasalreadybeencommentedon

andisfurtherdiscussedinthelastpartofSection 5. Relatedisalsotheliteratureonadaptivelearning

(Sargent, 1993; EvansandHonkapohja, 2012; MarcetandNicolini, 2003). Thisliteratureallowsfor

theanchoringofcurrentoutcomestopastoutcomes; see, inparticular, Carvalhoet al. (2017)foran

applicationinthecontextofinflation. Theanchoringfoundinourpaperhasthreedistinctqualities:

itisconsistentwithrationalexpectations; itistiedtothestrengthoftheGE feedback; anditisdirectly

comparabletothatfoundintheDSGE literature.

3 TheAbstractFramework

Inthissectionwesetupourframework, reviewthefrictionless, complete-informationbenchmarkwe

departfrom, andillustratetheinteractionofforward-lookingbehaviorandhigher-orderbeliefs.

Setup. Timeisdiscrete, indexedby t ∈ 0, 1, ..., andthereisacontinuumofplayers, indexed

by i ∈ [0, 1]. Ineachperiod t, eachagent i choosesanaction ait ∈ R. Wedenotethecorresponding

averageactionby at. Wenextspecifythebestresponseofplayer i inperiod t asfollows:

ait = Eit [φξt + βait+1 + γat+1] (3)

where ξt istheexogenousfundamental, Eit[·] istherational-expectationoperatorconditionalontheperiod-t informationofplayer i, and (φ, β, γ) areparameters, with φ > 0, β, γ ∈ [0, 1), and β+γ < 1.

Condition(3)specifiesbestresponsesinrecursiveform. Toseemoreclearlyhowcurrentbehavior

dependsonexpectationsoftheentirefuturepathsofthefundamentalandoftheaverageaction, we

iteratethisconditionforwardandreachthefollowingextensive-formrepresentation:

ai,t =

∞∑k=0

βkEi,t [φξt+k] + γ

∞∑k=0

βkEi,t [at+k+1] , (4)

Aggregatingtheaboveconditionacrossagents, wethenalsoobtainthefollowingequilibriumrestric-

6

tionbetweenoutcomesandexpectations:

at = φ

∞∑k=0

βkEt [ξt+k] + γ

∞∑k=0

βkEt [at+k+1] , (5)

where Et[.] denotestheaverageexpectationinthecross-sectionofthepopulation.8

Thelastconditionisuseful, notonlybecauseitfacilitatesthecharacterizationoftheaggregate

outcomeasafunctionoffirst-andhigher-orderbeliefs(seebelow), butalsobecauseithelpsnest

applicationsinwhichadirectanaloguetotheindividual-levelbest-responsecondition(3)isunavail-

able, asincaseswhere at correspondstothepricedeterminedinaWalrasianmarket. Finally, either

oftheseconditionsmakesclearthatthescalars β and γ parameterizetwodistinctaspectsofforward-

lookingbehavior. Ontheonehand, β determinestheextenttowhichanindividualdiscountsthe

futurevaluesofeithertheexogenousfundamentalortheendogenousoutcome. Ontheotherhand, γ

regulatestheextenttowhichanindividualconditionshiscurrentbehavioronherexpectationsofthe

futureactionsof others. Inapplications, thiskindofdynamicstrategiccomplementaritycorresponds

toGE effectssuchasthepositivefeedbackfromexpectationsoffutureinflationtocurrentinflation,

orthatfromexpectationsoffutureaggregatespendingtocurrentaggregatespending.

Frictionless, Representative-AgentBenchmark. Suppose, momentarily, thatinformationiscom-

plete, bywhichwemeanthatallagentssharethesameinformationandthereforefacenouncertainty

aboutoneanother’sbeliefs. Inthiscase, wecanreplace Et[·] incondition(5)withtheexpectationofarepresentativeagent, thatis, theexpectationconditionalonthecommoninformation. Regardless

ofhownoisythatcommoninformationmightbe, wecanthenusetheLawofIteratedExpectations

toreducecondition(5)tothefollowing, representative-agent, Euler-likecondition:

at = Et[φξt + δat+1], (6)

where Et[·] denotestheexpectationoftherepresentativeagentand δ ≡ β + γ ∈ (0, 1).

Itisthenimmediatetoseethatthecomplete-informationversionofourframeworkneststhetwo

buildingblocksoftheNewKeynesianmodel: theNKPC isnestedwith at standingforinflationand

ξt fortherealmarginalcostortheoutputgap; andtheDynamicIS Cure(thatis, theEulercondition

8Thebest-responsebehaviorassumedabove is similar to thatassumed in AngeletosandLian (2018). Butwhereasthatpaperconsidersanon-stationarysettingwhere ξt isfixedatzero inall t = T , for somefixed T ≥ 1, ourpaperaccommodatesastationarysettinginwhich ξt variesinall t and, inaddition, thereisgraduallearningovertime. Thesefeaturesareessentialforourobservational-equivalenceresult, aswellasfortheappliedlessonswedrawinthispaper. Ourframeworkalsoresemblesthebeauty-contestgamesconsideredby, interalia, MorrisandShin (2002), Woodford (2003),AngeletosandPavan (2007), BergemannandMorris (2013), and HuoandPedroni (2017). Becausebehavioris not forward-lookinginthesesettings, therelevanthigher-orderbeliefsarethoseregardingthe concurrent beliefsofothers. Bycontrast,therelevanthigher-orderbeliefsinoursettingarethoseregardingthe future beliefsofothers. Theimplicationsofthissubtledifferencearediscussedasweproceed; theyincludethedependenceofthedocumentedformofmyopiaonthepersistenceofthefundamentalandontheanticipationofthefuturelearningofothers.

7

oftherepresentativeconsumer)isnestedwith at standingforconsumptionand ξt fortherealinterest

rate. Alternatively, condition(6)canrepresentanasset-pricingequationwith at standingfortheasset

priceand ξt forthenext-perioddividend.

Byiteratingcondition(6), wecanobtainthecomplete-informationoutcomeasfollows:

at = φ∞∑k=0

δkEt[ξt+k]. (7)

Thisstylizeshow, intheaforementionedapplicationsandbeyond, outcomesarepinneddownbyfirst-

orderbeliefsoffundamentals. Asforextensionsoftheseapplicationsthataddincompleteinformation,

wewilllatershowhowsuchextensionscanindeedbenestedincondition(5).

IncompleteInformationandHigher-OrderBeliefs. Onceinformationisincomplete, condition

(7)ceasestoholdand, instead, theaggregateoutcomehingesonacertainkindofforward-looking,

higher-orderbeliefs. Lateron, appropriateassumptionsabouttheinformationstructurewillpermit

anexplicitcharacterizationofthesebeliefs. Fornow, weexplainhowtheaggregateoutcomecanbe

expressedasafunctionofthesebeliefs regardless oftheinformationstructure.

Toillustrate, let β = 0 < γ,9 whichmeansthattheequilibriumoutcomesatisfies

at = φEt [ξt] + γEt [at+1] . (8)

Iteratingthisconditiononcegives

at = φEt [ξt] + γφEt

[Et+1 [ξt+1]

]+ γ2Et

[Et+1 [at+2]

],

fromwhichitisevidentthattheequilibriumoutcomedependsonaparticularkindofforward-looking,

second-orderbeliefs, namelythecurrentbeliefsofthenext-periodbeliefsofthenext-periodfunda-

mentalandthenext-periodoutcome(seesecondandthirdtermintheabovecondition, respectively).

Byiteratingcondition(8)againandagain, wecanultimatelyexpresstheequilibriumoutcomeasa

functionofaninfinitehierarchyofbeliefsaboutthecurrentandfuturevaluesofthefundamental:

at = φ

∞∑h=0

γhFh+1t [ξt+h] (9)

where, foranyrandomvariable X, Fht [X] isdefinedrecursivelyby

F1t [X] ≡ Et [X] and Fh

t [X] ≡ Et

[Fh−1t+1 [X]

]∀h ≥ 2.

9Thiscaseistoonarrowfortheapplicationsofinterest, butitisusefulbecauseofitsrelativesimplicity.

8

Considernextthemoregeneralcaseinwhichboth γ > 0 and β > 0. Inthiscase, whichisrelevant

fortheapplicationsofinterest, theclassofhigher-orderbeliefsthatdrivetheequilibriumoutcomeis

muchricherthantheonedescribedabove. Toseethis, let φ1−ρβ = 1 (thisiscompletelyinnocuous)

andrewritecondition(8)asfollows:

at = Et [ξt] + γ

∞∑k=1

βk−1Et [at+k]

Applyingthisconditiontoperiod t+ k, forany k ≥ 1, andtakingtheexpectationsasofperiod t, we

obtainthefollowingrepresentationoftheperiod-t beliefsofthefutureoutcomes:

Et[at+k] = Et

[Et+k [ξt+k]

]+ γ

∞∑j=1

βj−1Et

[Et+k [at+k+j ]

]Combiningandrearranging, wereachthefollowingcharacterizationoftheperiod-t outcome:

at = Et [ξt] + γ

∞∑k=1

βkEt

[Et+k [ξt+k]

]+ γ2

∞∑k=1

βk−1∞∑j=1

βj−1Et

[Et+k [at+k+j ]

]Therelevantsecond-orderbeliefsarethereforethoseregardingthebeliefsofothers, notonlyinthe

nextperiod, butalsoin all futureperiods, namely Et[Et+k[ξt+k]] forevery k ≥ 1.

Asweiteratethisargumentagainandagain, thesetofhigher-orderbeliefsthatemergegetsricher

andricher. Inparticular, fixa t andpickany k ≥ 2, any h ∈ 2, ..., k, andany t1, t2, ...th suchthatt = t1 < t2 < ... < th = t + k. Then, theperiod-t outcomedependsonallofthefollowingtypesof

forward-lookinghigher-orderbeliefs:

Et1 [Et2 [· · · [Eth [ξt+k] · · · ]].

Forany t andany k ≥ 2, thereare k − 1 typesofsecond-orderbeliefs, plus (k − 1)× (k − 2)/2 types

ofthird-orderbeliefs, plus (k − 1)× (k − 2)× (k − 3) /6 typesoffourth-orderbeliefs, andsoon.

What’snext. Theabovederivationsindicatethepotentialcomplexityofdynamic, incomplete-

informationmodels. Anintegralpartofourcontributionisthebypassingofthiscomplexityandthe

developmentofsharpanalyticalresults. Thisisachievedbyfollowingtwodifferent, butcomplemen-

tary, paths. InSections 4-6, weemployasomewhatrigidspecificationofthefundamentalprocess

andtheinformationstructuresoastofacilitateourobservation-equivalenceresultanditsvariousap-

plications. InSection 7, wethenuseamoreflexiblespecificationbutalsocuttheGordianknotof

complexsignal-extractionproblemssoastoshedfurtherlightontheunderlyingprinciplesandtheir

robustness; thissacrificestheeleganceofourobservational-equivalenceresultbutnotitsessence.

9

4 TheEquivalenceResult

Inthissectionwedevelopourobservation-equivalenceresult. Theprinciplesunderlyingthisresult

arebestunderstoodbystudying thepropertiesofhigher-orderbeliefs, whichwedo inSection 7.

Here, webypassesthisstepand, instead, characterizedirectlytherational-expectationsfixedpoint.

Thisservesalsothegoalofclarifyingthefollowingbasicpoint. Eventhoughtheanalystmayfind

itinsightfultounderstandtherational-expectationsequilibriumintermsofhigher-orderbeliefs, the

agentsintheeconomyneednotthemselvesengageinhigher-orderreasoning. Instead, inthetradition

ofMuthandLucas, itsufficesthattheyhaveinmindthecorrectstatisticalmodelofthefundamental,

ξt, andtheaggregateoutcome, at. Whatismore, thisstaticalmodelcanberelativelysimpledespite

thecomplexityoftheunderlyinghigher-orderbeliefdynamics.

4.1 Specification

Wehenceforthmaketwoassumptions. First, weletthefundamental ξt followanAR(1)process:

ξt = ρξt−1 + ηt =1

1− ρLηt, (10)

where ηt ∼ N (0, 1) istheperiod-t innovation, L isthelagoperator, and ρ ∈ (0, 1) parameterizes

thepersistenceofthefundamental. Second, weassumethatplayer i receivesanewprivatesignalin

eachperiod t, givenby

xit = ξt + uit, uit ∼ N (0, σ2) (11)

where σ ≥ 0 parameterizestheinformationalfriction(thelevelofnoise). Theplayer’sinformationin

period t isthehistoryofsignalsuptothatperiod.

Theseassumptionsarerestrictive. Theyruleout, interalia, endogenoussignalssuchasprices. The

mainjustificationforthemisthattheyguaranteetheexactvalidityofourobservational-equivalence

result. Thisresultmayneverthelessserveasagoodproxyoftheequilibriumeveninextensionsthat

allowforendogenoussignals; weillustratethisinAppendixC.Furthermore, thepresenceofidiosyn-

craticnoiseintheavailableinformationcanbemotivatedasthebyproductofrationalinattention,

subjecttothecaveatthatwedonotstudytheproblemoffindingtheoptimalsignal.

Allinall, weviewourbaselinespecificationasausefulandempiricallyplausibleperturbation

ofthecomplete-informationbenchmark. Thisbenchmark, whichishereinnestedbysetting σ = 0,

imposes, notonlythateveryagentknowsthecurrentvalueof ξt, butalsothatsheisconfidentthat

everyotheragentsharesthesamebeliefswithhimabouttheentirefuturepathofboth ξt and at. By

contrast, setting σ > 0 letsagentsface, notonlyuncertaintyabouttheunderlyingaggregateshocks,

butalsodoubtsabouttheattentiveness, awareness, andresponsivenessofothers.

10

4.2 SolvingtheRational-ExpectationsFixedPoint

Considerfirstthefrictionlessbenchmark(σ = 0), inwhichcasetheoutcomeispinneddownbyfirst-

orderbeliefs, asincondition(7). ThankstotheAR(1)specificationassumedabove, Et[ξt+k] = ρkξt, for

all t, k ≥ 0. Wethusreachthefollowingresult, whichstatesthatthecomplete-informationoutcome

followsthesameAR(1)processasthefundamental, rescaledbythefactor φ1−ρδ .

Proposition 1. Inthefrictionlessbenchmark (σ = 0), theequilibriumoutcomeisgivenby

at = a∗t ≡φ

1− ρδξt =

φ

1− ρδ

1

1− ρLηt. (12)

Considernextthecaseinwhichinformationisincomplete(σ > 0). Asalreadyexplained, the

outcome is thena functionofan infinitenumberofhigher-orderbeliefs. Despite thesimplifying

assumptionsmadehere, thedynamicstructureofthesebeliefsisquitecomplex. Indeed, usingthe

Kalmanfilter, wecanreadilyshowthattherelevantfirst-orderbelief, Et[ξt], followsanAR(2)process:

Et[ξt] =

(1− λ

ρ

)(1

1− λL

)ξt =

(1− λ

ρ

)(1

1− λL

)(1

1− ρL

)ηt, (13)

where λ = ρ(1−G) and G istheKalmangain. Itthenfollowsthattherelevantsecond-orderbelief,

Et[Et+1[ξt+1]], followsanARMA(3,1). Andbyinduction, wecanalsoshowthat, forany h ≥ 1, the

relevant h-thorderbelief, Et[Et+1[...Et+h[ξt+h]], followsanARMA(h+ 1, h− 1).

Inshort, beliefsofhigherorderexhibitincreasinglycomplexdynamicsandthestatespaceneeded

to track theentirebeliefhierarchy is infinite. Yet, asanticipated in thebegginingof this section,

thiscomplexityis not inheritedbytherational-expectationsfixedpoint. Themethodsof Huoand

Takayama (2015)guaranteethat, insofarasthefundamentalandthesignalsfollowfiniteARMA pro-

cesses, thefixedpointweareinterestedinisalsoafiniteARMA process. Undertheassumptions

madehere, thisismerelyanAR(2)process, whoseexactformischaracterizedbelow.

Proposition 2. Theequilibriumexists, isuniqueandissuchthattheaggregateoutcomeobeysthe

followinglawofmotion:

at =

(1− ϑ

ρ

)(1

1− ϑL

)a∗t , (14)

where a∗t isthefrictionlesscounterpart, obtainedinProposition 1, andwhere ϑ isascalarthatsatisfies

ϑ ∈ (0, ρ) andthatisgivenbythereciprocalofthelargestrootofthefollowingcubic:

C(z) ≡ −z3 +(ρ+

1

ρ+

1

ρσ2+ β

)z2 −

(1 + β

(ρ+

1

ρ

)+β + γ

ρσ2

)z + β,

Condition(14)expressestheincomplete-informationdynamicsasasimpletransformationofthe

complete-informationcounterpart. Thistransformationisindexedbythescalar ϑ, whichplaysadual

11

role: relativetothefrictionlessbenchmark(whichishereinnestedby ϑ = 0), ahigher ϑ meansboth

asmallerimpacteffect, capturedbythefactor 1− ϑρ incondition(14), andamoresluggishbuildup

overtime, capturedbythelagterm ϑL.

Todevelopsomeintuitionfortheresult, considermomentarilythespecialcaseinwhich γ = 0.

Byshuttingdownthestrategiccomplementarity, thiscaseisolatestheroleoffirst-orderuncertainty.

Usingcondition(5)alongwith γ = 0 (andhence δ ≡ β + γ = β)andthefactthat Et[ξt+k] = ρkEt[ξt]

forall k ≥ 0, weinferthattheaggregateoutcomeisgivenby

at = φ∞∑k=0

βkEt[ξt+k] =φ

1− δρEt[ξt]. (15)

This is thesameas thecomplete-informationoutcome, modulo the replacementof ξt, theactual

fundamental, with Et[ξt], theaveragefirst-orderforecastofit. AndsincethelatterfollowstheAR(2)

processgivenincondition(13), weinferthatProposition 2 holdswith ϑ = λ when γ = 0.

Whathappenswhen γ > 0? Inthiscase, higher-orderbeliefsbecomerelevant. Asalreadynoted,

suchbeliefs followARMA processesofever increasingorder. Andyet, theequilibriumoutcome

continuestofollowanAR(2)process, asinthecasewith γ = 0. Thisisthe“magic”oftherational-

expectationsfixedpoint. Butwhereas ϑ equaled λ inthatcase, itisnowstrictlyhigherthan λ.10 That

is, theequilibriumdynamicsexhibitslessamplitudeandmorepersistence, notonlyrelativetothe

complete-informationcounterpart, butalsorelativetothefirst-orderbeliefofthefundamental.

Thispropertyreflectsthefactthathigher-orderbeliefsthemselvesdisplaylessamplitudeandmore

persistencethanfirst-orderbeliefs. InSection 7, wemakethislogicclear, andelaborateonitsrobust-

ness, byworkingwithamoreflexiblespecificationofthefundamentalprocessandtheinformation

structure. Thebottomlineisthatthemorestringentspecificationassumedhereguaranteesthatthe

equilibriuminheritsthekeyqualitativepropertiesofhigherbeliefswithout, however, inheritingtheir

complexity. Thisinturnfacilitatesourobservational-equivalenceresult, whichwepresentnext.

4.3 TheEquivalenceResult

Letusmomentarilyput aside the economyunder considerationand, instead, consider a variant,

representative-agenteconomyinwhichtheaggregateEulercondition(6)ismodifiedasfollows:

at = φξt + δωfEt [at+1] + ωbat−1 (16)

forsome ωf < 1 and ωb > 0. Theoriginalrepresentative-agenteconomyisnestedwith ωf = 1 and

ωb = 0. Relativetothisbenchmark, alower ωf representsahigherdiscountingofthefuture, orless

10Thefactthat ϑ > λmaynotbeobviousfromlookingatProposition 2, butfollowsfromthepropertythat ϑ isincreasingin γ, whichisestablishedasapartoftheproofofProposition 4.

12

forward-lookingbehavior; ahigher ωb representsagreateranchoringofthecurrentoutcometothe

pastoutcome, ormorebackward-lookingbehavior.

Condition(16)neststheEulerconditionofarepresentativeconsumerwhoexhibitshabit; avariant

oftheQ theorythathastherepresentativefirmfaceacostforadjustingitsrateofinvestmentrather

thanacostforadjustingitscapitalstock; andtheso-calledHybridNKPC.Withthelatterexamplein

mind, wehenceforthrefertotheeconomydescribedaboveasthe“hybrideconomy.”

ItiseasytoverifythattheequilibriumoutcomeofthiseconomyisgivenbyanAR(2)process,

whosecoefficients (ζ0, ζ1) arefunctionsof (ωf , ωb) and (φ, δ, ρ). Incomparison, theequilibriumout-

comeinourincomplete-informationeconomyisanAR(2)processwithcoefficientsdeterminedasin

Proposition 2. MatchingthecoefficientsofthetwoAR(2)processes, andcharacterizingthemapping

fromthelattertotheformer, wereachthefollowingresult.

Proposition 3 (ObservationalEquivalence). Fix (φ, β, γ, ρ). Forany σ > 0 intheincomplete-information

economy, thereexistsauniquepair (ωf , ωb) inthehybrideconomy, with ωf < 1 and ωb > 0, such

that the twoeconomiesgenerate thesame jointdynamics for the fundamentaland theaggregate

outcome. Furthermore, ahigher σ mapstoalower ωf andahigher ωb.

Thisproposition, whichis themainresultofourpaper, allowsonetorecast theinformational

frictionasthecombinationoftwobehavioraldistortions: extradiscountingofthefuture, ormyopia,

intheformof ωf < 1; andbackward-lookingbehavior, oranchoringofthecurrentoutcometopast

outcome, intheformof ωb > 0.Wecomplimentthisresultwiththefollowing, whichstudiesthecom-

parativestaticsoftheas-ifdistortionswithrespecttotheGE effect, orthestrategiccomplementarity.

Proposition 4 (GE). A strongerGE feedback (higher γ) mapstobothgreatermyopia(lower ωf )and

greateranchoring(higher ωb)inthehybridmodel.

WeexpandontheapplicabilityandtheusefulnessoftheseresultsinSections 5 and 6. Before

that, wenextexplainthebroaderprinciples thatunderly them. Wetherebyalsoexplainwhywe

prefertheperspectivedevelopedaboveoverthecharacterizationprovidedinProposition 2: whereas

thelatterdependscriticallyonthedetailsoftheassumedspecificationforfundamentalprocessand

theinformationstructure, theinsightsencapsulatedbyPropositions 3 and 4 aremoregeneral.

4.4 UnderlyingPrinciples

Tounderstandwhatdrives ωf < 1, itsufficestoabstract fromlearningand, instead, focusonthe

effectsoffirst-andhigher-orderuncertainty. Asevidentfromconditions(4)and(5), theoptimalbe-

haviorinoursettingispinneddownbytwoforward-lookingobjects: theexpectedpresentdiscounted

valueoftheexogenousfundamental, andtheexpectedpresentdiscountedvalueoftheendogenous

outcome. Whenaninnovationoccursinthefundamental, bothoftheseobjectsmove, butlessso

13

underincompleteinformationthanundercompleteinformation. First-orderuncertaintyarreststhe

movementoftheformer. Higher-orderuncertaintyarreststhemovementofthelatter, indeedatan

evengreaterdegreethanthatcharacterizingtheformer. Botheffectscausetheeconomytorespond

less tonewsabout the future that in the frictionlessbenchmark, explaining thedocumentedform

ofmyopia. Andbecausetherelevanceofthesecondeffect(i.e., thatregardinghigher-orderbeliefs)

increaseswiththestrengthoftheGE feedback, thismyopiaalsoincreaseswithit.

Tounderstandwhatdrives ωb > 0, ontheotherhand, itisnecessarytoallowforlearning. The

arrivalofnewinformationastimepassesinducesextrapersistenceinthebeliefsofthefundamental

andofthefutureoutcomesrelativetothepersistenceintheunderlyingfundamental. Furthermore,

thispersistenceinhigher-orderbeliefsisstrongerthanthatinfirst-orderbeliefs, reflectingthesmaller

dependenceofhigher-orderbeliefsonrecentinformation. Thisexplainswhythecurrentoutcome

appearstobeanchoredtothepastoutcomeataratethat, likeourformofmyopia, increaseswiththe

strengthoftheGE feedback.

TheseinsightsaremadecrystalclearinSection 7 withthehelpofaflexiblespecificationthat, not

onlyillustratestherobustnessoftheseinsights, butalsodisentanglestheleveloffirst-andhigher-order

uncertaintyfromthespeedoflearning. Bycontrast, suchadisentanglingisnotpossibleunderthe

morerigidspecificationconsideredhere, because σ, asingleparameter, regulates both thespeedof

learningandtheleveloffirst-andhigher-orderuncertainty.

Theanalysis inSection 7 also reveals that learning shapes theequilibriumbehavior, notonly

inthemannerdescribedabove, butalsoinanother, moresubtle, manner: throughtheanticipation

thatotheragentswill learnin thefuture. Thisanticipatoryeffectmattersonlybecauseagentsare

forward-lookingandisthereforeabsentinstaticbeautycontests.

Theforward-lookingnatureoftheproblemunderconsiderationalsoexplainswhy ϑ, theequi-

libriumpersistenceoftheaggregateoutcome, isincreasingin ρ, theexogenouspersistenceofthe

fundamental. Holding σ constant, anincreasein ρ raisesthehorizonofthe“newscomponent”of

anygiveninnovationinthefundamental: thehigher ρ is, themoreinformationanysuchinnovation

containsabout, notonlyaboutthefundamental, butalsoabouttheequilibriumoutcomefurtherinto

thefuture. Butrecallthatforecastingtheequilibriumoutcomefurtherandfurtherintothefuturein-

volvesbeliefsofhigherandhigherorder. Itfollowsthatanincreasein ρ raisestherelativeimportance

ofhigher-orderuncertainty, inthesamewayasanincreaseinthedegreeofstrategiccomplementarity.

TheseinsightscannotbeunderstoodbymerelyinspectingtheAR(2)solutionobtainedinPropo-

sition 2. Thisunderscoreshowourcontributionhingeson thecombinationof theobservational-

equivalenceresultpresentedaboveandthemoreelaborateanalysisoffered inSection 7. For the

appliedpurposesofthenexttwosections, however, onecansidestepthatanalysis, trustthesummary

ofinsightsprovidedabove, andrelyontheobservational-equivalenceresultalone.

14

4.5 TestableRestrictions

AlthoughProposition 3 guaranteesthatanincomplete-informationeconomycanalwaysbemapped

toahybrideconomy, theconverseisnottrue: ahybrideconomycanbereplicatedbyanincomplete-

informationeconomyonlywhen ωf and ωb satisfyacertainrestriction.

Proposition 5. Theequilibriumdynamicsofahybrideconomycanbereplicatedbythatofanincomplete-

informationeconomyforsome σ > 0 ifandonlyif ωb > 0 and

ωf = 1− 1

δρ2ωb. (17)

Furthermore, foranypair (ωb, ωf ) thatsatisfiestheaboverestriction, thereexistsaunique σ > 0 such

thatthetwoeconomiesareobservationallyequivalent.

Thisresultoffersasimpletestforourtheory. Supposethatoneusesatimesseriesof ξt and at to

estimate ρ, thepersistenceofthefundamental, andthepair (ωf , ωb), whichgovernsthelawofmotion

(16)oftheoutcome. Supposefurtherthatoneknows β and γ, andhencealso δ, fromindependent

sources. Onecanthentestwhethercondition(17)issatisfied. Ifitdoes, thenandonlythenthedata

iscompatiblewithourtheory.

Additionaltestablepredictions, oroveridentifyingrestrictions, canbeobtainedbylookingatthe

forecastsoffutureoutcomes. Let ϵkt ≡ at+k−Et [at+k] betherealizedaverage k-periodaheadforecast

error. Aslongasinformationisincomplete, ϵkt isseriallycorrelated. Andbecausethemagnitudeof

theserialcorrelationdependson σ, thisprovidesuswithanadditionalrestrictionthatcanbeusedto

identify σ and/ortotestthemodel. WeputtheseideasatworkinSection 6.

5 ImplicationsandDiscussion

TheDSGE literaturethatfollows Christiano, Eichenbaum, andEvans (2005)and SmetsandWouters

(2007)hasaddedthreedistinctbackward-lookingelementstothekeyforward-lookingequationsof

baselinemacroeconomicmodels: habitpersistenceinconsumption, adjustmentcoststoinvestment,

andautomaticpast-priceindexation. Thesemodificationsarecrucialforthisliterature’scapacityto

offerasuccessfulstructuralinterpretationofthemacroeconomictimesseries,11 butlackindependent

empiricalsupport. Theyarealsoinconsistentwiththemodelsemployedinstrandsoftheliterature

thataimatunderstanding themicroeconomicdata.12 All inall, thesekindsofbackward-looking

elementsareconsideredascrudeproxiesforother, unspecifiedmechanisms.11Notonlydotheyallowthetheorytomatchthesluggishnessinthedynamicresponsesofconsumption, investment,

andinflationtoavarietyofidentifiedshocks, butalsohelpfixthecomovementpropertiesoftheNewKeynesianmodel.12Forinstance, althoughtheinflationdynamicsimpliedbythestandardNKPC arebroadlyconsistent—qualitativelyifnot

quantitatively—withmenu-costmodels(AlvarezandLippi, 2014; GolosovandLucas Jr, 2007; NakamuraandSteinsson,2013), suchmodelsdonotproducethekindofbackward-lookingbehaviorimpliedbytheHybridNKPC.Similarly, the

15

Priorworkhasalreadypushed the idea that informational frictionscanbesuchamechanism

(Sims, 2003; Woodford, 2003; MankiwandReis, 2002, 2007; MackowiakandWiederholt, 2009,

2015; Nimark, 2008). Ouranalysisaddstothislineofworkinfourways:

1. Itoffersthesharpest, uptodate, illustrationoftheaforementionedidea.

2. Ithighlights that thebackward-lookingelements featuredin theDSGE literaturemaybeen-

dogenous, notonlytotheleveloftheinformationalfriction, butalsotoGE mechanisms—and

therebyalsotomarketstructuresandpoliciesthatregulatethestrengthofsuchGE mechanisms.

Forinstance, afiscal-policyreformthatalleviatesliquidityconstraintsandreducestheincome-

spendingmultipliermayalsoreducetheas-ifhabitandmyopiaintheconsumptiondynamics.

3. It helps reduce adiscomforting gapbetween themacroeconomic and themicroeconomics

estimatesoftheseelements, apointwediscussnext.

4. Itblendsthesebackward-lookingDSGE elementswithaformofimperfectforesight.

5. Itclarifieshowanemergingliteratureonboundedrationalityrelatestoourapproachandthe

relatedliteratureoninformationalfrictions.

6. ItfacilitatesthequantitativeevaluationweconductinSection 6.

Inthesequel, weelaborateoneachoneoftheseaspectsofourcontribution, startingwithasketchof

howourresultscanbeappliedtoaggregatedemandintheNewKeynesianmodel.

5.1 Applications

InthetextbookNewKeynesianmodel, whichabstractsfrominvestment, aggregatedemandisgiven

bytheEulerconditionoftherepresentativeconsumer:13

ct = −rt + Et [ct+1] , (18)

where ct isaggregateconsumptioninperiod t, rt istherealinterestratebetweenperiod t and t+ 1,

and Et istherationalexpectationconditionalonperiod-t information.

Asshownin AngeletosandLian (2018), anincomplete-informationextensionofcondition(18)is

givenbythefollowing:

ct = −∞∑k=0

θkEt[rt+k] + (1− θ)

∞∑k=1

θk−1Et[ct+k]. (19)

formofinvestmentadjustmentcostassumedintheDSGE literatureisatoddswiththeliteraturethatstudiesinvestmentattheplantorfirmlevel(Bachmann, Caballero, andEngel, 2013; Bloomet al., 2018; CaballeroandEngel, 1999).

13Throughout, weworkwiththelog-linearizedmodel: allvariablesareinlog-deviationsfromsteadystate.

16

where Et stand for theaverageexpectationof theconsumers, and θ ∈ (0, 1) for their subjective

discountfactor. Tounderstandwherethisconditioncomesfrom, considerthetextbookversionof

PermanentIncomeHypothesis. Thisgivesconsumptionasafunctionoftheexpectedpresentdis-

countedvalueofincome. Extendingthissoastoaccommodatevariationintherealinterestrateand

heterogeneityininformation, andusingthefactthataggregateincomeequalsaggregateconsumption

inequilibrium, resultstocondition(19).

Thisconditionrecaststheaggregate-demandblockoftheNewKeynesianmodelasadynamic

gameamongtheconsumers. Thisgameisnestedinourabstractframeworkbymapping rt and ct to

ξt and at, respectively, andbyletting

φ = −1, β = θ, and γ = 1− θ.

ThefollowingresultisthenanimmediatecorollaryofPropositions 3 and 4, providedofcoursethat

wemaintaintheassumptionsintroducedinthebeginningofSection 4.14

Corollary 1. Wheninformationisincomplete, thereexistscalars ωf < 1 and ωb > 0 suchthatthe

equilibriumprocessforaggregateconsumptionsolvesthefollowingequation:

ct = −rt + ωfEt[ct+1] + ωbct−1 (20)

Furthermore, alower θ, whichmeansastrongerincome-spendingmultiplier, mapstobothalower

ωf andahigher ωb.

Itisthereforeasiftheeconomyispopulatedbyarepresentativeagentwhoseconsumptionex-

hibitshabitpersistence, ofthekindassumedin Christiano, Eichenbaum, andEvans (2005)and Smets

andWouters (2007). Thereare, though, twosubtledifferences. First, whereasthetruehabitmodel

imposes ωf + ωb = 1, ourmodelimplies ωf + ωb < ρ < 1. Thismeansthattheoverallmovement

inconsumptionissmaller, reflectingthemyopiaproducedbyincompleteinformation. Second, and

perhapsmostimportantly, thecoefficients ωf and ωb dependscriticallyon θ, becausethisparameter

governsthestrengthoftherelevantGE feedback, namelytheKeynesianincome-spendingmultiplier.

Itisuseful not tointerpret θ literally, asthesubjectivediscountfactor. Forinstance, wecanreadily

extendtheanalysistoaperpetual-youth, overlapping-generationsmodelalongthelinesof Del Negro,

Giannoni, andPatterson (2015)and, underappropriateassumptions, replicatetheresultsreported

abovewith θ replacedby χθ, where χ isthesurvivalprobability. Thelattercaninturnbethought14OnlineAppendixC of AngeletosandLian (2018)developsa“discounted”Eulerequationthatresemblescondition

(19), butalsodiffersfromitintwocrucialrespects. First, itimposes ωb = 0, rulingoutthehabit-likeelement. Second, itonlydescribestheparticularpathofconsumptiontriggeredbyaonce-and-for-allshiftintheexpectationsoftherealinterestratethatwillprevailatasingle, andfixed, futuredate. Bycontrast, ourresultdescribestheentirestochasticprocessofconsumptioninastationarysettingwithrecurrentshocks. ThesamepointsdistinguishthediscountedNKPC foundinthatpaperfromtheversionoftheHybridNKPC wedeveloplateron, inSection 6.

17

ofasameasureofthelengthofplanninghorizons, eitherinthesenseofexpectedlifespansorinthe

sensedescribedin Woodford (2018). Alternatively, asin FarhiandWerning (2017), 1 − χ canserve

asaproxyfortheprobabilityofbindingliquidityconstraints. Forourpurposes, thekeyobservation

isthat, evenifsuchfeatureshappentobeirrelevantundercompleteinformationduetooffsetting

PE andGE effects,15 theycanbecrucialunderincompleteinformationbecausetheydeterminethe

strengthoftherelevantGE effectandtheconsequentimportanceofhigher-orderuncertainty.

Thisinturnbuildsabridgetoagrowingtheoreticalandempiricalliteraturethatstudiesthede-

terminantsofaggregatedemandandoftheKeynesianmultiplierinsettingsthatallowforincomplete

marketsandrichheterogeneity.16 Inthelightofourresults, theinteractionofthesefeatureswithin-

formationalfrictionscanperhapsrationalizebothsignificantmyopiavis-a-visthefutureandastrong,

habit-like, backward-lookingforceintheaggregateconsumptiondynamics. Thedependenceofthese

distortionsonpoliciesthatregulatethestrengthoftheKeynesianmultiplier, suchasfiscalreformsthat

alleviateliquidityconstraints, isanotherimplicationofouranalysisthatwarrantsfurtherinvestigation.

Intheabove, wefocusedonconsumption. InAppendixB,weturntoinvestment. Wetakea

modelthatfeaturesaconventionalformofadjustmentcoststocapital, asin Hayashi (1982)and Abel

andBlanchard (1983), andshowhowtheintroductionofincompleteinformationtothismodelcan

makeinvestmentbehave asif theadjustmentcosttakesthemoreexoticformassumedintheDSGE

literature. AndinSection 6, weshowhowtheHybridNKPC canbeobtainedbyaugmentingthe

standardNKPC withincompleteinformation. Together, theseapplicationsexplainhowouranalysis

connectstoeachofthethreebuildingblocksofthemodernmacroeconomicframework.17 Finally,

intheConclusionandAppendixC,wediscussanapplicationtoassetpricing.

5.2 Micro-vsMacro-levelDistortions

AsmentionedintheIntroduction, themacroeconomicestimatesofthehabitinconsumptionandof

theadjustmentcostsininvestmentaremuchlargerthanthecorrespondingmicroeconomicestimates;

see Havranek, Rusnak, andSokolova (2017)forameta-analysisofmultiplestudiesinthecontextof

consumptionhabit, and GrothandKhan (2010)and Zorn (2018)forinvestment. Ourresultsalsooffer

asimpleresolutiontothisdisconnect.

Toillustrate, considertheapplicationtoconsumptionstudiedaboveandallowforidiosyncratic

incomeorpreference shocks. Suppose further thateachconsumerhasperfectknowledgeofher

idiosyncraticshocks, whilemaintainingtheinformationalfrictionregardingtherealinterestrate(the

aggregatefundamental)andaggregatespending(theaggregateoutcome). Inthiscontext, Corollary 1

15Inthepresentcontext, thisoffsettingisevidentthepropertythatthesumofthecoefficients γ and β (whichparameterize,respectively, thePE andtheGE effects)isinvariantto θ and χ.

16E.g., Auclert (2017), KaplanandViolante (2014), Kaplan, Moll, andViolante (2016), and Werning (2015).17Theseapplicationstreateachblockinisolationofeachother. Accommodatingtheirinteractionmaybreaktheexact

observationalequivalence, but, asinthecaseofricherinformationstructures, neednotupsetthekeyinsights.

18

continuestohold: thedynamicsofaggregateconsumptionexhibithabit-likebehavior. Atthesame

time, theresponseof individualconsumptiontoidiosyncraticshocksexhibitnosuchbehavior. It

followsthataneconometricianmayestimateapositivehabitatthemacrolevel(i.e., intheresponse

ofaggregateoutcomestoaggregateshocks)alongwithazerohabitatthemicrolevel(i.e., inthe

responseofindividualoutcomestoidiosyncraticshocks).

In thecasejustdescribed, theabsenceofhabit-likebehaviorat themicrolevelhingesonthe

assumptionthatagentsobserveperfectlytheiridiosyncraticshocks. Relaxingthisassumption—for

example, lettingagentsberationallyinattentivetobothaggregateandidiosyncraticshocks—allows

themicroresponsestodisplayasimilarformofanchoringasthemacroresponses. Yet, thedistortion

islikelytoremainmorepronouncedatthemacrolevelthanatthemicroonefortworeasons, the

onehighlightedin MackowiakandWiederholt (2009)andtheonehighlightedhere.

Insofarasthefrictionistheproductofcostlyinformationacquisitionorrationalinattention, itis

naturaltoexpectthatthetypicalagentwillcollectrelativemoreinformationabout, orallocaterela-

tivelymorecognitivecapacityto, idiosyncraticshocks, simplybecausesuchshocksaremorevolatile

andthereishigherreturninreducinguncertaintyaboutthem. Thisisthemechanismarticulatedin

MackowiakandWiederholt (2009)andboilsdowntohavinglessfirst-orderuncertaintyaboutid-

iosyncraticthanaggregateshocks. Butevenifthefirst-orderuncertaintyaboutthetwokindofshocks

werethesame, thedistortionatthemacrolevelwouldremainlargerinsofarastherearepositiveGE

feedbackeffects, suchastheKeynesianincome-spendingmultiplierorthedynamicstrategiccom-

plementarityinprice-settingdecisionsofthefirms. Inshort, themechanismidentifiedinourpaper

andtheoneidentifiedintheaforementionedworkcomplementeachothertowardsgeneratingmore

pronounceddistortionsatthemacrolevelthanatthemicrolevel.18

5.3 ImperfectForesightandBoundedRationality

Asalreadynoted, theideathatincompleteinformationcanrationalizeacertainkindofmyopiawas

firstputforwardin AngeletosandLian (2018). Butwhereasthatpaperfocusedonanon-stationary

environmentfeaturingasingle, once-and-for-allanticipatedchangeinthevalueofthefundamentalat

somepredeterminedfuturedate(aparticulartypeof“MIT shock”), ouranalysisconsidersastationary

settingwithrecurringshocks. Thisstepiscrucialforthedevelopmentandtheapplicabilityofour

observational-equivalenceresult. Furthermore, byaccommodatinglearningovertime, ouranalysis

blendsthemyopiawiththebackward-lookingelementsoughtafterbytheDSGE literature.

Thelastpointalsohelpsdistinguishourcontributionfromthoseof Gabaix (2017)and Farhiand

Werning (2017). Theseworksdepart fromrationalexpectations inamanner thathelpscapturea

similarformofimperfectforesightasours. Theformerachievesthisbyassumingthattheperceived18WeverifyalltheseintuitionsinAppendixD withavariantthatletsbothaggregateandidiosyncraticshocksbeobserved

withnoise. Thequantitativepotentialofthisparticularidea, however, isleftopenforfutureresearch.

19

lawofmotionofalltherelevanteconomicvariablesexhibitlessamplitudeandlesspersistencethan

thetrueone(anassumptioncalled“cognitivediscounting”), thelatterbylettingagentshavelimited

depthofreasoninginthesenseofLevel-kThinking.19 Theseworksdonot, however, provideatheory

ofwhythecurrentoutcomemaybeanchoredtothepastoutcome, orwhytheequilibriumdynamics

mayexhibitmomentum. Intermsofourobservational-equivalenceresult, theyaccommodate ωf < 1

butrestrict ωb = 0. Itfollowsthatanelementarytestabledifferencebetweenincompleteinformation

andthesealternativesiswhether ωb ispositiveorzero.20

Fromthisperspective, theverdict seems tobeclear. First, themacroeconomicdatademands

ωb > 0, whichispreciselythereasonwhytheDSGE literaturedepartedfrombaseline, forward-looking

macroeconomicmodelsbyaddinghabitpersistenceinconsumption, adjustmentcoststoinvestment,

etc. Second, andperhapsmoretellingly, theavailableevidenceonexpectationsalsodemands ωb > 0:

asshownin, interalia, CoibionandGorodnichenko (2012, 2015), Coibion, Gorodnichenko, and

Kumar (2015)and VellekoopandWiederholt (2017), theaverageforecasterrorsexhibitpositiveserial

autocorrelation, bothunconditionallyandconditionalonidentifiedshocks. Finally, theexercisewe

conductinthenextsectionsuggeststhat, atleastinthecontextofinflation, thetwokindsofempirical

regularities—thebackward-lookingforceinoutcomesandthemomentuminbeliefs—aremutually

consistent, notonlyqualitatively, butalsoquantitatively.

Notwithstandingthesepoints, afruitfuldirectionforfutureresearchisperhapsonethatcombines

incomplete informationwithbounded rationality. Weexplore twosuchextensions in theendof

Section 7. Thefirstrelaxestheassumptionthatagentscananticipate, as it isrational, thatothers

willlearninthefuture.21 ThesecondmergesourapproachwithLevel-kThinking.22 Bothofthese

extensionspreservetheessenceofourresults, butalsointensifythedocumentedmyopia. Another

interestingdirectionforfutureresearchmaybeonethataugmentsourworkwiththekindofbelief

extrapolationstudiedin Bordaloet al. (2018)and KohlhasandWalther (2018a,b).

6 ApplicationtotheNKPC

In this section, we study theapplicationofour theory to theaggregate-supplyblockof theNew

Keynesianmodel, thatis, theNKPC.Thisapplicationisshowntomatch jointly existingestimatesof

19Thisfollowstheleadof Garcıa-SchmidtandWoodford (2018), whosesolutionconcept(“reflectiveequilibrium”)isessentiallythesameasLevel-kThinking. Seealso IovinoandSergeyev (2017)foranother, topicalapplication.

20Tobemoreprecise, itmaybepossibletoreconcilethesealternativeswith ωb > 0 byassumingthattheotherwisearbitrary“defaultpoints”intheseworksareincreasingfunctionsofthepastoutcome. Butthisbegsthequestionofwhythedefaultpointsoughttodisplaysuchapositivedependence. Ourapproachoffersasimpleanswer: iftheeffectivedefaultpointistheaverageBayesianbeliefconditionalonhistoricalinformation, itnaturallyexhibitssuchapositivedependence.

21Wefindthis“behavioral”varianttobeplausiblebecauseitimposeslessontheknowledgeandthecognitivecapabilitiesofagents. Inaddition, thisvariantservesausefulpedagogicalpurposebyclarifyingtheroleplayedbytheinteractionoflearningandforward-lookingbehavior.

22WethankAlexanderKohlhasforsuggestingustoexploresuchavariant.

20

theHybridNKPC (GaliandGertler, 1999; Gali, Gertler, andLopez-Salido, 2005)andindependent

evidenceoninflationexpectations(CoibionandGorodnichenko, 2015). Furthermore, theimplied

quantitativebiteoftheinformationalfrictionontheinflationdynamicsisnon-trivial.

SetupandMappingtoAbstractFramework. Apartfortheintroductionofincompleteinformation,

themicro-foundationsarethesameasinfamiliartextbooktreatmentsoftheNKPC (e.g., Galí, 2008).

Thereisacontinuumoffirms, eachproducingadifferentiatedcommodity. Firmssetpricesoptimally,

but canadjust themonly infrequently. Eachperiod, afirmhas theoption to reset itspricewith

probability 1 − θ, where θ ∈ (0, 1); otherwise, itisstuckattheprevious-periodprice. Technologyis

linear, sothattherealmarginalcostofafirmisinvarianttoitsproductionlevel.

Regardlessofhowinformationthefirmhas, theoptimalresetpricesolvesthefollowingproblem:

P ∗it = argmax

Pit

∞∑k=0

(δθ)kEit

Qt|t+k

(PitYit+k|t − Pt+kΨt+kYi,t+k|t

)

subjecttothedemandequation, Yit+k =(

PitPt+k

)−ϵYt+k,whereQt|t+k isthestochasticdiscountfactor

between t and t+ k, Yt+k and Pt+k are, respectively, aggregateincomeandtheaggregatepricelevel

inperiod t+ k, Pit isthefirm’sprice, assetinperiod t, Yi,t+k|t isthefirm’squantityinperiod t+ k,

conditionalonnothavingchangedthepricesince t, andΨt+k istherealmarginalcostinperiod t+k.

Takingthefirst-orderconditionandlog-linearizingaroundasteadystatewithnoshocksandzero

inflation, wegetthefollowingcharacterizationoftheoptimalrestprice:

p∗it = (1− δθ)∞∑k=0

(δθ)kEit[ψt+k + pt+k]. (21)

Thisconditionmeansthattheoptimalresetpriceofafirmisaweightedaverageofherbeliefofthe

currentandfuturenominalmarginalcosts. Wenextmakethesimplifyingassumptionthatthefirms

observethatcurrentpricelevelbutdonotextractinformationfromit.23 Thispermitsustorestate

condition(21)as

p∗it − pt−1 = (1− δθ)

∞∑k=0

(δθ)kEit[ψt+k + πt+k], (22)

Sinceonlyafraction 1 − θ ofthefirmsadjusttheirpriceseachperiod, thepricelevelinperiod t is

23Asin VivesandYang (2017), thisassumptioncanbeinterpretedasaformofinattentionorboundedrationality. Itcanalsobemotivatedonempiricalgrounds: inthedata, inflationcontainslittlestatisticalinformationaboutcurrentandfuturerealmarginalcosts. Inanyevent, thisassumptionsharpenstheexpositionbutisnotessential. AsshowninAppendixC,ourobservational-equivalenceresultcanbeagoodapproximationofthetrueequilibriuminsettingsthatallowagentstoextractinformationfromcurrentorpastaggregateoutcomes.

21

givenby pt = (1− θ)∫p∗itdi+ θpt−1. Bythesametoken, inflationisgivenby

πt ≡ pt − pt−1 = (1− θ)

∫(p∗it − pt−1) .

Combiningthiswithcondition(22)andrearranging, wearriveatthefollowingexpressionforinflation:

πt =(1− δθ)(1− θ)

θ

∞∑k=0

(δθ)kEt [ψt+k] + δ(1− θ)

∞∑k=0

(δθ)kEt [πt+k+1] . (23)

Wheninformationiscomplete, wecanreplace Et[·] with Et[·], theexpectationoperatorcondi-tional thecommon informationset. Wecan thenuse theLawof IteratedExpectations to reduce

condition(23)tothefollowing:

πt = κψt + δEt[πt+1], (24)

where κ ≡ (1−δθ)(1−θ)θ . ThisthestandardNKPC.

When, instead, informationisincomplete, theaboveconditionnomoreholds. Instead, inflation

mustbeunderstoodasasolutiontoadynamicgameofthetypestudiedinSections 3 and 4. This

gameisnestedinourframeworkbymapping ψt and πt to ξt and at, respectively, andbyletting

φ = κ β = δθ and γ = δ(1− θ).

ThefollowingisthenanimmediateapplicationofPropositions 3 and 4, providedofcoursethatwe

maintaintheassumptionsintroducedinSection 4.

Proposition 6. (i)Thereexist ωf < 1 and ωb > 0 suchthat, wheninformationis incomplete, the

equilibriumprocessforinflationsolvesthefollowingequation:

πt = κψt + ωfδEt[πt+1] + ωbπt−1 (25)

(ii)Foranygivenlevelofnoise, increasingthedegreeofpriceflexibility(i.e., reducing θ)results

toalower ωf andahigher ωb.

Part(i)establishesthat, wheninformationisincomplete, itis asif inflationisgovernedbyavariant

oftheNKPC thatintroducesmyopia, intheformof ωf < 1, alongwithabackward-lookingcompo-

nent, intheformof ωb > 0. ThisissimilartotheHybridNKPC consideredin, interalia, Galiand

Gertler (1999), Christiano, Eichenbaum, andEvans (2005)and SmetsandWouters (2007).

Part(ii)addsthefollowinginterestinglesson. Wheninformationiscomplete, higherpriceflexibil-

itycontributesmerelytoasteeperNKPC,thatis, toahigher κ incondition(24). Thisisgenerallybad

fortheempiricallyfitoftheNewKeynesianmodel, whichinturnexplainswhytheliteraturehastried

22

hardtojustifyadegreeofpricestickinessattheaggregatelevelthatishigherthantheonethatappears

totopresentatthemicro-economiclevelunderthelensofmenu-costmodels. Butonceinformation

isincomplete, amoderatedegreeofpriceflexibilitycanbegoodinthesensethatitcontributesto

moresluggishnessintheinflationdynamicsbyreinforcingtheroleofhigher-orderuncertainty. This

pointhelpsexplainwhyourquantitativeimplementationreconcilessalientfeaturesoftheinflation

dynamicswitharelativelymodestdegreeofpricestickiness.

Testing theTheory. TheHybridNKPC estimatedin GaliandGertler (1999)and Gali, Gertler,

andLopez-Salido (2005), issimilar to theoneseen in (25). Thereare, however, twodifferences.

First, ourtheoryrestrictsthepair (ωf , ωb) inthewaydescribedinProposition 5, whereasunrestricted

estimationsoftheHybridNKPC allowtheseparameterstobefree. Andsecond, ourtheorytiesthe

pair (ωf , ωb) tothedynamicsofinflationforecasts. Wenowusetheserestrictionstotestourtheory.

MatchingEstimatesoftheHybridNKPC. Gali, Gertler, andLopez-Salido (2005)synthesizethe

literatureontheestimationoftheHybridNKPC andprovideafewestimatesofthepair(ωf , ωb). A

quicktestofourtheoryiswhethertheseestimatessatisfytherestrictioninProposition 5.

Thispropositiongivesthelocusofthepairs (ωf , ωb) thatarecompatiblewithourtheoryforsome

levelofnoise. Toconstructthislocus, andtobeabletoidentify σ byinvertingtheprovidedestimates

of (ωf , ωb), weneedtospecify δ, θ, and ρ. Weset δ = 0.99, θ = 0.6, and ρ = 0.95. Thevalueof θ

correspondstoamodestdegreeofpricestickiness, broadlyinlinewithtextbookcalibrationsofthe

NewKeynesianmodelandwiththemicrodata. Thevalueof ρ isobtainedbyestimatinganAR(1)

processonthelaborshare, astandardempiricalproxyfortherealmarginalcost. Thelocusimplied

underthisparameterizationofourmodelisthenrepresentedbythesolidredlineinFigure 1.

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Figure 1: TestingtheTheory

Gali, Gertler, andLopez-Salido (2005)providethreebaselineestimatesof (ωf , ωb). Theseesti-

matesandtheirconfidenceregionsarerepresentedbythebluecrossesandthesurroundingdisks

23

inFigure 1. A priori, thereisnoreasontoexpectthattheestimatesobtainedin Gali, Gertler, and

Lopez-Salido (2005)shouldfallon, orcloseto, thelocusimpliedbyourtheory. Andyet, asevident

inthefigure, that’spreciselythecase. Inotherwords, ourmodelmatchestheexistingestimateson

theHybridNKPC andallowsonetorationalizethemastheproductofinformationalfrictions.

MatchingSurveyEvidenceonInformationalFrictions. Althoughthetheorypassestheaforemen-

tioned test, it isnotclearat thispointwhether this successhingesonanempirically implausible

magnitudefortheinformationalfriction. Wenowaddressthisquestion, andimposethetheorytoan

additionaltest, byexaminingwhethertheinformationalfrictionrequiredinordertorationalizethe

existingestimatesof ωf and ωb isconsistentwithsurveyevidenceonexpectations.

Tothisgoal, weutilizethefindingsof CoibionandGorodnichenko (2015). Thatpaperusesdataon

inflationforecastsfromtheSurveyofProfessionalForecasterstomeasureakeymomentthatcanhelp

gaugethemagnitudeoftheinformationalfriction. Thebasicideaisthatthefrictionshouldmanifest

itselfinthepredictabilityoftheaverageforecasterrors. Inparticular, CoibionandGorodnichenko

(2015)runthefollowingregression:

πt+k − Et[πt+k] = K(Et[πt+k]− Et−1[πt+k]

)+ vt+k,t (26)

Withcompleteinformation, K iszero, becausethecurrentforecastcorrectionisindependentofpast

information. Bycontrast, wheninformationisincomplete, averageforecastsadjustsluggishlytowards

thetruth, implyingthatpastinnovationsinforecastspredictfutureforecastcorrections, thatis,K > 0.

Furthermore, K islargerthelargerthenoiseandtheslowerthespeedoflearning.

CoibionandGorodnichenko (2015) illustrate this logicwithanexample inwhich inflation is

followsanexogenousAR(1)processand K isadirecttransformationofthelevelofnoise. Clearly,

thisnotthecasehere. Becauseactualinflationandforecastsofinflationarejointlydeterminedin

equilibrium, theregressioncoefficientK impliedbyourtheoryismorecomplicatedthanthatintheir

exampleandis indeedendogenoustotheGE interactionamongthefirms. Nevertheless, wecan

usethetheorytocharacterize K asafunctionof σ andof (δ, θ, ρ). Withthelatterfixedintheway

describedearlier, thisgivesusamappingfromthe90%confidenceintervalofK providedin Coibion

andGorodnichenko (2015)toanintervalfor σ inourmodel. Thatis, wehaveaconfidenceinterval

fortheinformationalfrictionitself. Forany σ inthisinterval, wecanthencomputethepair (ωf , ωb)

predictedbyourtheory.

Wecanthusmaptheevidencereportedin CoibionandGorodnichenko (2015)toasegmentof

the (ωf , ωb) locusweobtainedearlieron. ThissegmentisidentifiedbytheredcrossesinFigure 1 and

givesthepairsof (ωf , ωb) thatareconsistentwiththeconfidenceintervalfor K providedin Coibion

andGorodnichenko (2015). Itisthenevidentfromthefigurethatourmodelcanpassjointlythetest

24

Figure 2: ImpulseResponseFunctionofInflation

ofmatchingthatevidenceandthetestofmatchingtheexistingestimatesoftheHybridNKPC.24

QuantitativeBite. Thequantitativeimplicationsoftheexerciseconductedabovearefurtheril-

lustratedinFigure 2. Thisfigurecomparestheimpulseresponsefunctionof inflationunderthree

scenarios. Thesolidblacklinecorrespondstofrictionlessbenchmark, withperfectinformation. The

dashedbluelinecorrespondstothefrictionalcase, withaninformationalfrictionthatmatchesthe

baselineestimationof CoibionandGorodnichenko (2015). Thedottedredlineisexplainedlater.

Asevidentbycomparing thedashedblue line to the solidblackone, thequantitativebiteof

theinformational frictionissignificant: theimpacteffectoninflationisabout60%lowerthanits

complete-informationcounterpart, andthepeakoftheinflationresponseisattained5quartersafter

impactratherthanonimpact. Thisissuggestiveofhowinformationalfrictionsmayhelpreconcile

quantitativemacroeconomicmodels, whichcanaccount for thebusinesscycleonlybyassuming

significantsluggishnessintheinflationdynamics, withrealisticmenu-costmodels, whichappearto

beunabletoproducesuchsluggishness.25

Let us now explain the dotted red line in the figure. Using condition (23), the incomplete-

informationinflationdynamicscanbedecomposedintotwocomponents: thebeliefofthepresent

discountedvalueof realmarginal costs, φ∑∞

k=0 βkEt[ψt+k]; and thebelief of of thepresentdis-

24Tobeprecise, theabovestatementistruefortwoofthethreebaselineestimatesprovidedin Gali, Gertler, andLopez-Salido (2005). Butthesehappentobetheestimatesassociatedwiththesmallest ωf andthelargest ωb. Thisisgoodnewsforthequantitativesignificanceofourtheory: onceourtheoryisdisciplinedbythesurveyevidence, itrationalizessignificantlevelsofbothmyopiaandanchoring.

25See, forexample, GolosovandLucas Jr (2007), Midrigan (2011), AlvarezandLippi (2014), and NakamuraandSteinsson(2013). AlthoughdifferentspecificationscanrationalizeadegreeofpricerigidityeithermuchsmallerthanoralmostaslargeastheonepredictedbythestandardNKPC,thisliteraturehasnotproducedthekindofhump-shapedinflationdynamicsthattheDSGE literaturehascapturedwiththeHybridNKPC.

25

countedvalueofinflation, γ∑∞

k=0 βkEt[πt+k+1]. Thesamedecompositioncanalsobeappliedwhen

agentshaveperfectinformation:

π∗t = φ∞∑k=0

βkEt [ψt+k|ψt] + γ∞∑k=0

βkEt

[π∗t+k+1|ψt

], (27)

A naturalquestioniswhichcomponentcontributesmoretotheanchoringofinflationaswemove

fromthecompletetoincompleteinformation.

Toanswerthisquestion, wedefinethefollowingauxiliaryvariable:

πt = φ

∞∑k=0

βkEt [ψt+k] + γ

∞∑k=0

βkEt

[π∗t+k+1|ψt

]. (28)

Thedifferencebetween π∗t and πt measurestheimportanceofbeliefsaboutrealmarginalcosts, and

thedifferencebetween πt and πt measurestheimportanceofbeliefsaboutinflation. Thedottedred

lineinFigure 2 correspondsto πt. Clearly, mostofthedifferencebetweencompleteandincomplete

informationisduetheanchoringofbeliefsaboutfutureinflation. Thesebeliefsaretiedwithhigher-

orderbeliefs, whichdisplaylessresponsivenessandmoreinertiathanthefirst-orderbeliefs.

7 Robustness: IncompleteInformationasMyopiaandAnchoring

Earlier onewe claimed that, although our observational-equivalence result depends on stringent

assumptionsabouttheprocessofthefundamentalandtheavailablesignals, itencapsulatesafew

broaderinsights, whichinturnjustifytheperspectiveputforwardinourpaper. Inthissection, we

substantiatethisclaimbyclarifyingtheseinsightsandbyelaboratingontheirrobustness.

Setup. Wehenceforthletthefundamental ξt followaflexible, possiblyinfinite-order, MA process:

ξt =

∞∑k=0

ρkηt−k, (29)

wherethesequence ρk∞k=0 isnon-negativeandsquaresummable. Clearly, theAR(1)processas-

sumedearlieronisnestedasaspecialcasewhere ρk = ρk forall k ≥ 0. Thepresentspecification

allowsforricher, possiblyhump-shaped, dynamicsinthefundamental, aswellasfor“newsshocks,”

thatis, forinnovationsthatshiftthefundamentalonlyafteradelay.

Next, forevery i and t, welettheincrementalinformationreceivedbyagent i inperiod t begiven

bytheseries xi,t,t−k∞k=0, where

xi,t,t−k = ηt−k + ϵi,t,t−k ∀k,

26

where ϵi,t,t−k ∼ N (0, (τk)−2) isi.i.d. across i and t, uncorrelatedacross k, andorthogonaltothepast,

current, andfutureinnovationsinthefundamental, andwherethesequence τk∞k=0 isnon-negative

andnon-decreasing. Inplainwords, whereasourbaselinespecificationhas theagentsobservea

signalabout ξt ineachperiod, thenewspecificationletsthemobserveaseriesofsignalsaboutthe

entirehistoryoftheunderlyinginnovations.

Thisspecificationissimilartoourbaselineinthatitallowsformoreinformationtobeaccumulated

astimepasses. Itdiffers, however, intworespects. First, it“orthogonalizes”theinformationstructure

inthesensethat, forevery t, every k, andevery k′ = k, thesignalsreceivedatorpriortodate t about

theshock ηt−k areindependentofthesignalsreceivedabouttheshock ηt−k′ . Second, itallowsfor

moreflexiblelearningdynamicsinthesensethattheprecision τk doesnothavetobeflatin k: the

qualityoftheincrementalinformationreceivedinanygivenperiodaboutapastshockmayeither

increaseordecreasewiththelagsincetheshockhasoccurred.

Thefirstpropertyisessentialfortractability. Thepertinentliteraturehasstruggledtosolvefor, or

accuratelyapproximate, thecomplexfixedpointbetweentheequilibriumdynamicsandtheKalman

filteringthatobtainsindynamicmodelswithincompleteinformation, especiallyinthepresenceof

endogenoussignals; see, forexample, Nimark (2017). Byadoptingtheaforementionedorthogonal-

ization, wecuttheGordianknotandfacilitateaclosed-formsolutionoftheentiredynamicstructure

ofthehigher-orderbeliefsandoftheequilibriumoutcome. Thesecondpropertythenpermitsus, not

onlytoaccommodateamoreflexiblelearningdynamics, butalsotodisentanglethespeedoflearning

fromlevelofnoise—adisentanglingthatwasnotpossibleinSection 4 becauseasingleparameter,

σ, controledbothobjectsatonce.

DynamicsofHigher-OrderBeliefs. Theinformationregarding ηt−k thatanagenthasaccumulated

upto, andincluding, period t canberepresentedbyasufficientstatistic, givenby

xki,t =

k∑j=0

τjπkxi,t−j,t−k

where πk ≡∑k

j=0 τj . Thatis, thesufficientstatisticisconstructedbytakingaweightedaverageof

alltheavailablesignals, withtheweightofeachsignalbeingproportionaltoitsprecision; andthe

precisionofthestatisticisthesumoftheprecisionsofthesignals. Letting λk ≡ πk

σ−2η +πk

, wehavethat

Eit[ηt−k] = λkxki,t, whichinturnimplies Et[ηt−k] = λkηt−k andtherefore

Et [ξt] = Et

[ ∞∑k=0

ρkηt−k

]=

∞∑k=0

f1,kηt−k, with f1,k = λkρk. (30)

Thesequence F1 ≡ f1,k∞k=0 = λkρk∞k=0 identifiestheIRF oftheaveragefirst-orderforecastto

aninnovation. Bycomparison, theIRF ofthefundamentalitselfisgivenbythesequence ρk∞k=0 . It

27

followsthattherelationofthetwoIRFsispinneddownbythesequence λk∞k=0, whichdescribes

thedynamicsoflearning. Inparticular, thesmaller λ0 is(i.e., thelessprecisetheinitialinformation

is), thelargertheinitialinitialgapbetweenthetwoIRFs(i.e., alargertheinitialforecasterror). And

theslower λk increaseswith k (i.e., theslowerthelearningovertime), thelongerittakesforthatgap

(andtheaverageforecast)todisappear.

Thesepropertiesareintuitiveandaresharedbythespecificationstudiedintherestofthepaper.

IntheinformationstructurespecifiedinSection 4, theinitialprecisionistiedwiththesubsequent

speedoflearning. Bycontrast, thepresentspecificationdisentanglesthetwo. Asshownnext, italso

allowsforasimplecharacterizationoftheIRFsofthehigher-orderbeliefs, whichiswhatweareafter.

Considerfirsttheforward-lookinghigher-orderbeliefs. Applyingcondition(30)toperiod t + 1

andtakingtheperiod-t averageexpectation, weget

F2t [ξt+1] ≡ Et

[Et+1 [ξt+1]

]= Et

[ ∞∑k=0

λkρkηt+1−k

]=

∞∑k=0

λkλk+1ρk+1ηt−k

Noticehere, agentsinperiod t understandthatinperiod t+1 theaverageforecastwillbeimproved,

andthisiswhy λk+1 showsupintheexpression. Byinduction, forall h ≥ 2, the h-thorder, forward-

lookingbeliefisgivenby

Fht [ξt+h−1] =

∞∑k=0

fh,kηt−k, with fh,k = λkλk+1...λk+h−1ρk+h−1. (31)

Theincreasingcomponentsintheproduct λkλk+1...λk+h−1 seenabovecapturetheanticipationof

learning. Werevisitthispointattheendofthissection.

Thesetofsequences Fh = fh,k∞k=0, for h ≥ 2, providesacompletecharacterizationoftheIRFs

oftherelevant, forward-looking, higher-orderbeliefs. Notethat ∂E[ ξt+h|ηt−k]∂ηt−k

= ρk+h−1. Itfollowsthat

theratio fh,kρk+h−1

measurestheeffectofaninnovationontherelevant h-thorderbeliefrelativetoits

effectonthefundamental. Wheninformationiscomplete, thisratioisidentically 1 forall k and h.

When, instead, informationisincomplete, thisratioisgivenby

fh,kρk+h−1

= λkλk+1...λk+h−1.

Thefollowingresultisthusimmediate.

Proposition 7. Considertheratio fh,kρk+h−1

, whichmeasurestheeffectatlag k ofaninnovationonthe

h-thorderforward-lookingbeliefrelativetoitseffectonthefundamental.

(i)Forall k andall h, thisratioisstrictlybetween0and1.

(ii)Forany k, thisisdecreasingin h.

28

(iii)Forany h, thisratioisincreasingin k.

(iv)As k → ∞, thisratioconvergesto 1 forany h ≥ 2 ifandonlyifitconvergesfor h = 1, and

thisinturnistrueifandonlyif λk → 1.

Thesepropertiesshedlightonthedynamicstructureofhigher-orderbeliefs. Part(i)statesthat, for

anybelieforder h andanylag k, theimpactofashockonthe h-thorderbeliefislowerthanthaton

thefundamentalitself. Part(ii)statesthathigher-orderbeliefsmovelessthanlower-orderbeliefsboth

onimpactandatanylag. Part(iii)statesthatthatthegapbetweenthebeliefofanyorderandthe

fundamentaldecreasesasthelagincreases; thiscapturestheeffectoflearning. Part(iv)statesthat,

regardlessof h, thegapvanishesinthelimitas k → ∞ ifandonlyif λk → 1, thatis, ifandonlyifthe

learningisboundedawayfromzero.

MyopiaandAnchoring. Toseehowthesepropertiesdrivetheequilibriumbehavior, wehence-

forthrestrict β = 0 andnormalize φ = 1. Asnotedearlier, thelawofmotionfortheequilibriumout-

comeisthengivenby at = Et[ξt] + γEt[at+1], whichinturnimpliesthat at =∑∞

h=1 γh−1Fh

t [ξt+h−1] .

Fromtheprecedingcharacterizationofthehigher-orderbeliefs Fht [ξt+h−1], itfollowsthat

at =

∞∑k=0

gkηt−k, with gk =

∞∑h=1

γh−1fh,k =

∞∑h=1

γh−1λkλk+1...λk+h−1ρk+h−1

. (32)

ThismakesclearhowtheIRF oftheequilibriumoutcomeisconnectedtotheIRFsofthefirst-and

higher-orderbeliefs. Importantly, thehigher γ is, themorethedynamicsoftheequilibriumoutcome

tracksthedynamicshigher-orderbeliefsrelativetothedynamicsoflower-orderbeliefs. Ontheother

hand, whenthegrowthrateoftheIRF ofthefundamental ρk+1

ρkishigher, italsoincreasestherelative

importanceofhigher-orderbeliefs. Thisargumentisparticularlyclearwhenconsideringtheinitial

response g0 andsetting ρk = ρk (ξt followsanAR(1)process)

g0 =

∞∑h=1

(γρ)h−1λ0λ1 . . . λh−1.

Here, γ and ρ playasimilarrole. ThisanalysisfurtherillustratesthepointmadeinSection 4.3.

Wearenowreadytoexplainourresultregardingmyopia. Forthispurpose, itisbesttoabstract

fromlearningandfocusonhowthemerepresenceofhigher-orderuncertaintyaffectsthebeliefsabout

thefuture. Intheabsenceoflearning, λk = λ forall k andforsome λ ∈ (0, 1). Theaforementioned

formulafortheIRF coefficientsthenreducestothefollowing:

gk =

∞∑h=1

(γλ)h−1ρk+h−1

λ.

Clearly, this thesame IRF as thatofacomplete-information, representative-economyeconomyin

29

whichtheequilibriumdynamicssatisfy

at = ξ′t + γ′Et[at+1], (33)

where ξ′t ≡ λξt and γ′ ≡ γλ. Itisthereforeasifthefundamentalislessvolatileand, inaddition, the

agentsarelessforward-looking. Thefirsteffectstemsfromfirst-orderuncertainty: itispresentsimply

becausetheforecastofthefundamentalmovelessthanone-to-onewiththetruefundamental. The

secondeffectoriginatesinhigher-orderuncertainty: itispresentbecausetheforecastsoftheactions

ofothersmove even lessthantheforecastofthefundamental.

Thisisthecruxoftheforward-lookingcomponentofourobservational-equivalenceresult(thatis,

theoneregardingmyopia). Noteinparticularthattheextradiscountingofthefutureremainspresent

even ifwhen if control for the impactof the informational frictiononfirst-orderbeliefs. Indeed,

replacing ξ′t with ξt intheaboveshutsdowntheeffectoffirst-orderuncertainty. Andyet, theextra

discountingsurvives, reflectingtheroleofhigher-orderuncertainty. Thiscomplementstherelated

pointswemakeinSection 4.4 and 5.2.

Sofar, weshedlightonthesourceofmyopia, whileshuttingdowntheroleoflearning. Wenext

elaborateontherobustnessoftheaboveinsightstothepresenceoflearningand, mostimportantly,

onhowthepresenceoflearninganditsinteractionwithhigher-orderuncertaintydrivethebackward-

lookingcomponentofourobservational-equivalenceresult.

Tothisgoal, andasabenchmarkforcomparison, weconsideravarianteconomyinwhichall

agentssharethesamesubjectivebeliefabout ξt, thisbeliefhappenstocoincidewiththeaverage

first-orderbeliefintheoriginaleconomy, andthesefactsarecommonknowledge. Theequilibrium

outcomeinthiseconomyisproportionaltothesubjectivebeliefof ξt andisgivenby

at =∞∑k=0

gkηt−k, with gk =∞∑h=1

γh−1λkρk+h−1.

Thisresemblesthecomplete-informationbenchmarkinthattheoutcomeispineddownbythefirst-

orderbeliefof ξt, butallowsthisbelieftoadjustsluggishlytotheunderlyinginnovationsin ξt.

Byconstruction, thevarianteconomypreservestheeffectsoflearningonfirst-orderbeliefsbut

shutsdowntheinteractionoflearningwithhigher-orderuncertainty. Itfollowsthatthecomparison

ofthiseconomywiththeoriginaleconomyrevealstheroleofthisinteraction.

Proposition 8. Let gk and gk denotetheImpulseResponseFunctionoftheequilibriumoutcome

inthetwoeconomiesdescribedabove.

(i) 0 < gk < gk forall k ≥ 0

(ii)If ρkρk−1

≥ ρk+1

ρkand ρk > 0 forall k > 0, then gk+1

gk>

gk+1

gkforall k ≥ 0

30

Considerproperty(i), inparticularthepropertythat gk < gk. Thispropertymeansthatourecon-

omyexhibitsauniformlysmallerdynamicresponsefortheequilibriumoutcomethantheaforemen-

tionedeconomy, inwhichhigher-orderuncertaintyisshutdown. Butnotethatthetwoeconomies

sharethefollowinglawofmotion:

at = φEt[ξt] + γEt[at+1]. (34)

Furthermore, the twoeconomies share the samedynamic response for Et[ξt]. It follows that the

responsefor at inoureconomyissmallerthanthatofthevarianteconomybecause, andonlybecause,

theresponseof Et[at+1] isalsosmallerinoureconomy. Thisverifiesthatthepreciseroleofhigher-

orderuncertaintyistoarresttheresponseoftheexpectationsofthefutureoutcome(thefutureactions

ofothers)beyondandabovehowmuchthefirst-orderuncertainty(theunobservabilityof ξt)arrests

theresponseoftheexpectationsofthefuturefundamental.

A complementarywayofseeingthispointistonotethat gk satisfiesthefollowingrecursion:

gk = f1,k + λkγgk+1. (35)

Thefirsttermintheright-handsideofthisrecursioncorrespondstotheaverageexpectationofthe

futurefundamental. Thesecondtermcorrespondstheaverageexpectationofthefutureoutcome(the

actionsofothers). Theroleoffirst-orderuncertaintyiscapturedbythefactthat f1,k islowerthan

ρk. Theroleofhigher-orderuncertaintyiscapturedbythepresenceof λk inthesecondterm: itis

asifthediscountfactor γ hasbeenreplacedbyadiscountfactorequalto λkγ, whichisstrictlyless

than γ. Thisrepresentsageneralizationoftheformofmyopiaseenincondition(33). There, learning

wasshutdown, sothatthat λk andtheextradiscountingofthefuturewereinvariantinthehorizon k.

Here, theadditionaldiscountingvarieswiththehorizonbecauseoftheanticipationoffuturelearning

(namely, theknowledgethat λk willincreasewith k).

Considernextproperty(ii), namelythepropertythat

gk+1

gk>gk+1

gk

Thispropertyhelpsexplainthebackward-lookingcomponentofourobservational-equivalenceresult

(thatis, theoneregardinganchoring).

Tostartwith, considerthevarianteconomy, inwhichhigher-orderuncertaintyisshutdown. The

impactofashock k + 1 periodsfromnowrelativetoitsimpact k periodsfromnowisgivenby

gk+1

gk=λk+1

λk

∑∞h=0 γ

hρk+h+1∑∞h=0 γ

hρk+h>

∑∞h=0 γ

hρk+h+1∑∞h=0 γ

hρk+h.

31

Theinequalitycapturestheeffectoflearningonfirst-orderbeliefs. Hadinformationbeingperfect,

wewouldhavehad gk+1

gk=

∑∞h=0 γ

hρk+h+1∑∞h=0 γ

hρk+h; now, weinsteadhave gk+1

gk>

∑∞h=0 γ

hρk+h+1∑∞h=0 γ

hρk+h. Thismeans

that, inthevarianteconomy, theimpactoftheshockontheequilibriumoutcomecanbuildforce

overtimebecause, andonlybecause, learningallowsforagradualbuildupinfirst-orderbeliefs.26

Considernowoureconomy, inwhichhigher-orderuncertaintyispresent. Wenowhave

gk+1

gk>gk+1

gk

Thismeansthathigher-orderuncertaintyamplifiesthebuild-upeffectoflearning: astimepasses, the

impactoftheshockontheequilibriumoutcomebuildsforcemorerapidlyinoureconomythanin

thevarianteconomy. Butsincetheimpactisalwayslowerinoureconomy,27 thismeansthattheIRF

oftheequilibriumoutcomeislikelytodisplayamorepronouncedhumpshapeinoureconomythan

inthevarianteconomy. Indeed, thefollowingisadirectlycorollaryoftheaboveproperty.

Corollary 2. Supposethatthevarianteconomydisplaysahump-shapedresponse, namely gk is

singlepeakedat k = kb forsome kb ≥ 1. Then, theequilibriumoutcomealsodisplaysahump-shaped

response, namely gk isalsosinglepeakedat k = kg. Furthermore, thepeakof theequilibrium

responseisafterthepeakofthevarianteconomy: kg ≥ kb necessarily, and kg > kb foranopenset

of λk sequences.

Tointerpretthisresult, thinkmomentarilyof k asacontinuousvariableand, similarly, thinkof λk,

gk, and gk asdifferentiablefunctionsof k. If gk ishump-shapedwithapeakat k = kb > 0, itmustbe

that gk isweaklyincreasingpriorto kb andlocallyflatat kb. Butsincewehaveprovedthatthegrowth

rateof gk isstrictlyhigherthanthatof gk, thismeansthat gk attainsitsmaximumatapoint kg thatis

strictlyabove kb. Intheresultstatedabove, thelogicisthesame. Theonlytwististhat, because k is

discrete, wemusteitherrelax kg > kb to kg ≥ kb orputrestrictionson λk soastoguaranteethatkg ≥ kb + 1.

Summingup, learningbyitselfcontributestowardsagradualbuildupoftheimpactofanygiven

shockontheequilibriumoutcome; butitsinteractionwithhigher-orderuncertaintymakesthisbuild

upevenmorepronounced. Itispreciselythesepropertiesthatareencapsulatedinthebackward-

lookingcomponentofourobservationalequivalenceresult: thecoefficient ωb, whichcapturesthe

endogenousbuildupintheequilibriumdynamics, ispositivebecauseoflearninganditishigherthe

highertheimportanceofhigher-orderuncertainty.

TwoFormsofBoundedRationality. Wenowshedlightontwoadditionalpoints, whichwere

26Thisiseasiesttoseewhen ρk = 1 (i.e., thefundamentalfollowsarandomwalk), forthen gk+1 isnecessarilyhigherthan gk forall k. IntheAR(1)casewhere ρk = ρk with ρ < 1, gk+1 canbeeitherhigherorlowerthan gk, dependingonthebalancebetweentwoopposingforces: thebuild-upeffectoflearningandthemean-reversioninthefundamental.

27Recall, thisisbyproperty(i)ofProposition 8.

32

anticipatedearlieron: theroleplayedbytheanticipationthatotherswilllearninthefuture; andthe

possibleinteractionofincompleteinformationwithLevel-kThinking.

Toillustratethefirstpoint, weconsiderabehavioralvariantwhereagentsfailtoanticipatethat

otherswilllearninthefuture. Tosimplify, wealsoset β = 0. Recallfromequation(31), whenagents

arerational, theforwardhigher-orderbeliefsare

Fht [ξt+h−1] =

∞∑k=0

λkλk+1...λk+h−1ρk+h−1ηt−k.

In thevarianteconomy, byshuttingdown theanticipationof learning, thenatureofhigher-order

beliefschanges, as Eit

[Et+k [ξt+q]

]= Eit

[Et [ξt+q]

]for k, q ≥ 0, andthecounterpartof Fh

t [ξt+h−1]

becomes

Eht [ξt+h−1] ≡ Et

[Et[. . .Et [[ξt+h−1] . . .]

]=

∞∑k=0

λhkρt+h−1ηt−k.

Learningimplies λk+1 > λk, andtheanticipationoflearningimplies λkλk+1...λk+h−1 > λhk . Asa

result, higher-orderbeliefsinthebehavioralvariantunderconsiderationvary less thanthoseunder

rationalexpectations. Bythesametoken, theaggregateoutcomeinthiseconomy, whichisgiven

at =

∞∑h=1

γh−1Eht [ξt+h−1] ,

behavesasifthemyopiaandanchoringarestrongerthanintherational-expectationscounterpart. In

linewiththeseobservations, itcanbeshownthat, ifwegobacktoourbaselinespecificationand

imposethatagentsfailtoanticipatethatotherswilllearninthefuture, Proposition 3 continuesto

holdwiththefollowingmodification: ωf issmallerand ωb ishigher.

Toillustratethesecondpoint, weconsideravariantthatletsagentshavelimiteddepthofreasoning

inthesenseofLevel-kThinking. Withlevel-0thinking, agentsbelievethattheaggregateoutcomeis

fixedatzeroforall t, butstillformrationalbeliefsaboutthefundamental. Therefore, a0it = Eit[ξt],

andtheimpliedaggregateoutcomeforlevel-0thinkingis a0t = Et[ξt].

Withlevel-1thinking, agent i’sactionchangesto

a1it = Eit[ξt] + γEit[a0t+1] = Eit[ξt] + γEit

[Et+1[ξt+1]

],

wherethesecond-orderhigher-orderbeliefshowsup. Byinduction, thelevel-k outcomeisgivenby

akt =k+1∑h=1

γh−1Fht [ξt+h−1] .

33

Inanutshell, Level-kThinkingsimplytruncatesthehierarchyofbeliefsatanexogenouslyspecified,

finiteorder.

Comparedwiththerational-expectationseconomythathasbeenthefocusofouranalysis, the

GE feedbackeffectsinbothoftheaforementionedtwovariantsareattenuated, andtheresultingas-

ifmyopiaisstrengthened. Furthermore, byselectingthedepthofthinking, wecanmakesurethat

thesecondvariantproducesasimilardegreeofmyopiaasthefirstone.28 Thatsaid, thesourceof

theadditionalmyopiaisdifferent. Inthefirst, therelevantforward-lookinghigher-orderbeliefshave

beenreplacedbymyopiccounterparts, whichmoveless. Inthesecond, theright, forward-looking

higher-orderbeliefsarestillatwork, buttheyhavebeentruncatedatafinitepoint.

8 Conclusion

Weshowedhowtheaccommodationofincompleteinformation, higher-orderuncertaintyandlearn-

inginforward-looking, general-equilibriummodelsisakintotheintroductionoftwobehavioraldis-

tortions: myopia, orextradiscountingofthefuture; andbackward-lookingbehavior, oranchoringof

currentoutcomestopastoutcomes. Weformalizedthisperspectivewiththehelpofanobservational-

equivalenceresult, whichrestedonstrongassumptions, butalsoelaboratedontherobustnessofthe

underlyinginsightsandtheofferedperspective.

Ourobservational-equivalenceresultwasinstrumental, notonlyfortheformalizationoftheabove

perspective, butalsoforthefollowingadditional, appliedpurposes:

1. Itillustratedhowincompleteinformationcan, notonlyofferasubstituteforthemoreadhoc

backward-lookingfeaturesoftheDSGE literature, butalsohelpresolvethegapbetweenthe

macroeconomicandmicroeconomicestimatesofsuchfeatures.

2. Itblendthesebackward-lookingfeatureswithaformofimperfectforesight.

3. IthighlightedhowbothoftheseelementsmaybeendogenoustoGE mechanismsandthereby

alsotomarketstructuresandpoliciesthatregulatethem.

4. Itfacilitatedtheempiricalevaluationofourtheoryinthecontextofinflationdynamics.

5. Itletusrelateourapproachandtheexistingliteratureoninformationalfrictionstoanemerging

literatureonboundedrationality.

Althoughourpaperwas focusedonmacroeconomicapplications, ourresultsand insightsare

relevantmorebroadly. Weconcludethepaperbyillustratingthisinthecontextofassetpricing.

28Thisfollowsdirectlyfromthefactthatimpactofeffectofaninnovationinthefirstvariantisboundedbetweenthoseofthelevel-0 andthelevel-∞ outcomeinthesecondvariant.

34

InAppendix C,we take a settingwith dispersed information and overlapping generations of

traders, along the linesof Singleton (1987). We then show, under appropriateassumptions, that

thissettingcanbenestedinourframeworkandcanthusgiverisetothefollowingequilibriumasset-

pricingcondition:

pt = Et[dt+1] + ωfδEt[pt+1] + ωbpt−1,

where pt istheasset’sprice, dt isitsdividend, Et[·] isthefull-information, rational-expectationsop-

erator, δ isthestandarddiscountfactor, and (ωf , ωb) areourfamiliarcoefficients.

Theaboveoffersasharpillustrationofhowincompleteinformationcanrationalizemomentum

andpredictabilityinassetprices (ωb > 0), inlinewith Kasa, Walker, andWhiteman (2014). Butitalso

highlightsthepossibilityofexcessivediscountingofnewsaboutfuturefundamentals (ωf < 1). This

inturnhintstoapossiblefragilityofthepredictionsoftheliteraturethatemphasizeslong-termrisks

totheaccommodationofhigher-orderuncertainty. Finally, ourinsightaboutthedependenceofthe

as-ifdistortionsonstrategiccomplementarityandGE feedbacksaddsanewangletotheSamuelson

dictum(JungandShiller, 2005): totheextentthatthepositivefeedbackinassettradingisstrongerat

thestock-marketlevelthanattheindividual-stocklevelbecauseoffire-saleexternalitiesandliquidity

blackholes, our resultsmayhelpexplainwhyasset-priceanomaliesaremorepronouncedat the

formerlevelthanatthelatter.

35

References

Abel, Andrew B.andOlivier J.Blanchard.1983. “AnIntertemporalModelofSavingandInvestment.”

Econometrica 51 (3):675–692. 5.1, 8

Allen, Franklin, StephenMorris, andHyun SongShin.2006. “BeautyContestsandIteratedExpecta-

tionsinAssetMarkets.” ReviewoffinancialStudies 19 (3):719–752. 8

Altissimo, Filippo, LaurentBilke, AndrewLevin, ThomasMatha, andBenoitMojon.2010. “Sectoral

andAggregateInflationDynamicsintheEuroArea.” JournaloftheEuropeanEconomicAssociation

4 (2-3):585–593. 1

Alvarez, FernandoandFrancescoLippi.2014. “PriceSettingwithMenuCostforMultiproductFirms.”

Econometrica 82 (1):89–135. 1, 12, 25

Angeletos, George-MariosandChenLian.2016. “IncompleteInformationinMacroeconomics: Ac-

commodatingFrictionsinCoordination.” HandbookofMacroeconomics 2:1065–1240. 2

———.2018. “ForwardGuidancewithoutCommonKnowledge.” American EconomicReview

108 (9):2477–2512. 1, 5, 2, 8, 5.1, 14, 5.3

Angeletos, George-MariosandAlessandroPavan.2007. “EfficientUseof InformationandSocial

ValueofInformation.” Econometrica 75 (4):1103–1142. 8

Auclert, Adrien.2017. “MonetaryPolicyandtheRedistributionChannel.” NBER WorkingPaperNo.

23451 . 16

Bachmann, Rüdiger, Ricardo J.Caballero, andEduardoM.R. A.Engel.2013. “Aggregate Impli-

cationsofLumpyInvestment: NewEvidenceandaDSGE Model.” AmericanEconomicJournal:

Macroeconomics 5 (4):29–67. 1, 12, 8

Bergemann, DirkandStephenMorris.2013. “RobustPredictionsinGameswithIncompleteInfor-

mation.” Econometrica 81 (4):1251–1308. 8

Bloom, Nicholas, MaxFloetotto, NirJaimovich, ItaySaporta-Eksten, andStephen J.Terry.2018. “Re-

allyUncertainBusinessCycles.” Econometrica forthcoming. 12

Bordalo, Pedro, NicolaGennaioli, YueranMa, andAndreiShleifer.2018. “Over-reactioninMacroe-

conomicExpectations.” NBER WorkingPaperNo.24932 . 2, 5.3

Caballero, Ricardo J.andEduardoM.R. A.Engel.1999. “ExplainingInvestmentDynamicsinU.S.

Manufacturing: A Generalized(S,s)Approach.” Econometrica 67 (4):783–826. 1, 12, 8

36

Carroll, Christopher D., EdmundCrawley, JiriSlacalek, KiichiTokuoka, andMatthew N.White.2018.

“StickyExpectations andConsumptionDynamics.” WorkingPaper24377, NationalBureauof

EconomicResearch. 7

Carvalho, Carlos, StefanoEusepi, EmanuelMoench, andBrucePreston.2017. “AnchoredInflation

Expectations.”. 2

Christiano, Lawrence J,MartinEichenbaum, andCharles L Evans.2005. “NominalRigiditiesandthe

DynamicEffectsofaShocktoMonetaryPolicy.” JournalofPoliticalEconomy 113 (1):1–45. 1, 5,

5.1, 6, 8

Coibion, OlivierandYuriyGorodnichenko.2012. “WhatCanSurveyForecastsTellUsaboutInfor-

mationRigidities?” JournalofPoliticalEconomy 120 (1):116–159. 5, 2, 5.3

———.2015. “InformationRigidityandtheExpectationsFormationProcess: A SimpleFramework

andNewFacts.” AmericanEconomicReview 105 (8):2644–78. 1, 5, 2, 5.3, 6, 6, 6, 6

Coibion, Olivier, YuriyGorodnichenko, andSatenKumar.2015. “HowDoFirmsFormTheirExpec-

tations? NewSurveyEvidence.” NBER WorkingPaperSeries . 5, 5.3

Del Negro, Marco, Marc P Giannoni, andChristinaPatterson.2015. “TheForwardGuidancePuzzle.”

FRB ofNewYorkmimeo . 5, 5.1

Evans, George W andSeppoHonkapohja.2012. Learningandexpectations inmacroeconomics.

PrincetonUniversityPress. 2

Farhi, EmmanuelandIvánWerning.2017. “MonetaryPolicy, BoundedRationality, andIncomplete

Markets.” NBER WorkingPaperNo.23281 . 1, 5, 5.1, 5.3

Gabaix, Xavier.2017. “A BehavioralNewKeynesianModel.” Harvardmimeo . 1, 5.3

Galí, Jordi.2008. MonetaryPolicy, Inflation, andtheBusinessCycle: AnIntroductiontotheNew

KeynesianFrameworkandItsApplicationsSecondedition. PrincetonUniversityPress. 6

Gali, JordiandMarkGertler.1999. “Inflationdynamics: A structuraleconometricanalysis.” Journal

ofMonetaryEconomics 44 (2):195–222. 1, 2, 6, 6

Gali, Jordi, MarkGertler, andJ DavidLopez-Salido.2005. “Robustnessoftheestimatesofthehybrid

NewKeynesianPhillipscurve.” JournalofMonetaryEconomics 52 (6):1107–1118. 1, 6, 6, 6, 24

Garcıa-Schmidt, MarianaandMichaelWoodford.2018. “AreLowInterestRatesDeflationary? A

ParadoxofPerfect-ForesightAnalysis.” AmericanEconomicReview forthcoming. 1, 19

37

Golosov, MikhailandRobert E Lucas Jr.2007. “MenuCostsandPhillipsCurves.” JournalofPolitical

Economy 115 (2):171–199. 1, 12, 25

Groth, CharlotaandHashmatKhan.2010. “InvestmentAdjustmentCosts: AnEmpiricalAssessment.”

JournalofMoney, CreditandBanking 42 (8):1469–1494. 1, 5.2

Havranek, Tomas, MarekRusnak, andAnnaSokolova.2017. “HabitFormationinConsumption: A

Meta-Analysis.” EuropeanEconomicReview 95:142–167. 1, 5.2

Hayashi, Fumio.1982. “Tobin’sMarginalqandAverageq: A NeoclassicalInterpretation.” Econo-

metrica 50 (1):213–224. 1, 5.1, 8

Huo, ZhenandMarcelo ZouainPedroni.2017. “InfiniteHigherOrderBeliefsandFirstOrderBeliefs:

AnEquivalnceResultInfiniteHigherOrderBeliefsandFirstOrderBeliefs: AnEquivalnceResult.”

miméo, YaleUniversity/UniversityofAmsterdam. 8

Huo, ZhenandNaokiTakayama.2015. “RationalExpectationsModelswithHigherOrderBeliefs.”

Yalemimeo . 2, 4.2, 31

Iovino, LuigiandDmitriySergeyev.2017. “QuantitativeEasingwithoutRationalExpectations.” Work

inprogress . 19

Jung, JeemanandRobert J.Shiller.2005. “Samuelson’sDictumandtheStockMarket.” Economic

Inquiry 43 (2):221–228. 1, 8, 8

Kaplan, Greg, BenjaminMoll, andGiovanni L Violante. 2016. “Monetary PolicyAccording to

HANK.” NBER WorkingPaperNo.21897 . 16

Kaplan, GregandGiovanni L.Violante. 2014. “A Modelof theConsumptionResponse toFiscal

StimulusPayments.” Econometrica 82 (4):1199–1239. 16

Kasa, Kenneth, Todd B.Walker, andCharles H.Whiteman.2014. “HeterogeneousBeliefsandTests

ofPresentValueModels.” TheReviewofEconomicStudies 81 (3):1137–1163. 8, 8

Khan, AubhikandJulia K.Thomas.2008. “IdiosyncraticShocksandtheRoleofNonconvexitiesin

PlantandAggregateInvestmentDynamics.” Econometrica 76 (2):395–436. 8

Kiley, Michael T.2007. “A QuantitativeComparisonofSticky-PriceandSticky-InformationModels

ofPriceSetting.” JournalofMoney, CreditandBanking 39 (s1):101–125. 6

Kohlhas, AlexandreandAnsgarWalther.2018a. “AsymmetricAttention.” IIES miméo . 2, 5.3

———.2018b. “SparseExpectations: A UnifiedExplanationofForecastData.” IIES miméo . 2, 5.3

38

Lucas, RobertE. Jr.1972. “ExpectationsandtheNeutralityofMoney.” JournalofEconomicTheory

4 (2):103–124. 1

Mackowiak, BartoszandMirkoWiederholt.2009. “OptimalStickyPricesunderRationalInattention.”

AmericanEconomicReview 99 (3):769–803. 4, 2, 5, 5.2, 8

———.2015. “BusinessCycleDynamicsunderRationalInattention.” ReviewofEconomicStudies

82 (4):1502–1532. 4, 5

Mankiw, N. GregoryandRicardoReis.2002. “StickyInformationversusStickyPrices: A Proposalto

ReplacetheNewKeynesianPhillipsCurve.” QuarterlyJournalofEconomics 117 (4):1295–1328.

4, 6, 5

Mankiw, N GregoryandRicardoReis.2007. “Stickyinformationingeneralequilibrium.” Journalof

theEuropeanEconomicAssociation 5 (2-3):603–613. 4, 5

Marcet, AlbertandJuan P Nicolini.2003. “Recurrenthyperinflationsandlearning.” AmericanEco-

nomicReview 93 (5):1476–1498. 2

Matejka, Filip.2016. “RationallyInattentiveSeller: SalesandDiscretePricing.” ReviewofEconomic

Studies 83 (3):1125–1155. 6

Mavroeidis, Sophocles, Mikkel Plagborg-Møller, and James H Stock. 2014. “Empirical evidence

oninflationexpectations in theNewKeynesianPhillipsCurve.” JournalofEconomicLiterature

52 (1):124–88. 6

McKay, Alisdair, EmiNakamura, andJónSteinsson.2016. “ThePowerofForwardGuidanceRevis-

ited.” AmericanEconomicReview 106 (10):3133–3158. 5

Melosi, Leonardo.2016. “Signallingeffectsofmonetarypolicy.” TheReviewofEconomicStudies

84 (2):853–884. 6

Midrigan, Virgiliu.2011. “MenuCosts, MultiproductFirms, andAggregateFluctuations.” Economet-

rica 79 (4):1139–1180. 25

Morris, StephenandHyun SongShin.2002. “SocialValueofPublicInformation.” AmericanEconomic

Review 92 (5):1521–1534. 1, 2, 8

———.2006. “InertiaofForward-lookingExpectations.” TheAmericanEconomicReview :152–157.

1, 2

Nakamura, EmiandJónSteinsson.2013. “PriceRigidity: MicroeconomicEvidenceandMacroeco-

nomicImplications.” AnnualReviewofEconomics 5 (1):133–163. 1, 12, 25

39

Nimark, Kristoffer.2008. “DynamicPricingandImperfectCommonKnowledge.” JournalofMonetary

Economics 55 (2):365–382. 1, 4, 2, 5

———.2017. “DynamicHigherOrderExpectations.” CornellUniveristymimeo . 7, 8, 31

Reis, Ricardo.2006. “Inattentiveproducers.” TheReviewofEconomicStudies 73 (3):793–821. 6

Sargent, Thomas J.1993. Boundedrationalityinmacroeconomics. OxfordUniversityPress. 2

Sims, Christopher A.2003. “ImplicationsofRationalInattention.” JournalofMonetaryEconomics

50 (3):665–690. 1, 4, 5

Singleton, Kenneth J.1987. “AssetPricesinaTime-seriesModelwithDisparatelyInformed, Com-

petitiveTraders.” In NewApproachestoMonetaryEconomics, editedbyWilliam A.Barnettand

Kenneth J.Singleton.CambridgeUniversityPress, 249–272. 8, 8

Smets, FrankandRafaelWouters.2007. “ShocksandFrictionsinUS BusinessCycles: A Bayesian

DSGE Approach.” AmericanEconomicReview 97 (3):586–606. 5, 5.1, 6, 8

Vellekoop, NathanaelandMirkoWiederholt.2017. “InflationExpectationsandChoicesofHouse-

holds.” GoetheUniversityFrankfurtmimeo . 5.3

Vives, XavierandLiyanYang.2017. “CostlyInterpretationofAssetPrices.” mimeo, IESE/University

ofToronto. 23

Werning, Iván.2015. “IncompleteMarketsandAggregateDemand.” NBERWorkingPaperNo.21448

. 16

Woodford, Michael.2003. “ImperfectCommonKnowledgeand theEffectsofMonetaryPolicy.”

Knowledge, Information, andExpectationsinModernMacroeconomics: InHonorofEdmundS.

Phelps . 1, 2, 8, 5

———.2018. “MonetaryPolicyAnalysisWhenPlanningHorizonsAreFinite.” In NBER Macroeco-

nomicsAnnual2018, Volume33.UniversityofChicagoPress. 5.1

Zorn, Peter.2018. “InvestmentunderRationalInattention: EvidencefromUS SectoralData.” miméo,

UniversityofMunich. 1, 7, 5.2

40

AppendixA:Proofs

ProofofProposition 1. Followsdirectlyfromtheanalysisinthemaintext.

ProofofProposition 2. Supposethattheagent’sequilibriumpolicyfunctionisgivenby

ait = h(L)xit

forsomelagpolynomial h(L). Theaggregateoutcomecanthenbeexpressedasfollows:

at = h(L)ξt =h(L)

1− ρLηt.

Inthesequel, weverifythattheaboveguessiscorrectandcharacterize h(L).

First, welookforthefundamentalrepresentationofthesignals. Define τη = σ−2η and τu = σ−2

asthereciprocalsofthevariancesof, respectively, theinnovationinthefundamentalandthenoise

inthesignal. (Inthemaintext, wehavenormalized ση = 1.) Thesignalprocesscanberewrittenas

xit = M(L)

[ηt

uit

], with M(L) =

[τ− 1

2η

11−ρL τ

− 12

u

].

Let B(L) denotethefundamentalrepresentationofthesignalprocess. Bydefinition, B(L) needsto

beaninvertibleprocessanditneedstosatisfythefollowingrequirement

B(L)B(L−1) = M(L)M′(L−1) =τ−1η + τ−1

u (1− ρL)(L− ρ)

(1− ρL)(L− ρ), (36)

whichleadsto

B(L) = τ− 1

2u

√ρ

λ

1− λL

1− ρL,

where λ istheinsiderootofthenumeratorinequation(36)

λ =1

2

ρ+ 1

ρ

(1 +

τuτη

)−

√(ρ+

1

ρ

(1 +

τuτη

))2

− 4

.Next, wecharacterizethebeliefsof ξt, ai,t+1, and at+1, thatis, thebeliefsthatshowupinthe

best-responseconditionoftheagent. Theforecastofarandomvariable

ft = A(L)

[ηt

uit

]

41

canbeobtainedbyusingtheWiener-Hopfpredictionformula:

Eit[ft] =[A(L)M′(L−1)B(L−1)−1

]+B(L)−1xit.

Considertheforecastofthefundamental. Notethat

ξt =[τ− 1

2η

11−ρL 0

] [ ηtuit

],

fromwhichitfollowsthat

Eit[ξt] = G1(L)xit, G1(L) ≡λ

ρ

τuτη

1

1− ρλ

1

1− λL.

Considertheforecastofthefutureownandaverageactions. Usingtheguessthat ait+1 = h(L)xi,t+1

and at+1 = h(L)ξt+1, wehave

at+1 =[τ− 1

2η

h(L)L(1−ρL) 0

] [ ηtuit

], ait+1 − at+1 =

[0 τ

− 12

u h(L)

] [ ηtuit

],

andtheforecastsare

Eit [at+1] = G2(L)xit, G2(L) ≡λ

ρ

τuτη

(h(L)

(1− λL)(L− λ)− h(λ)(1− ρL)

(1− ρλ)(L− λ)(1− λL)

),

Eit [ait+1 − at+1] = G3(L)xit, G3(L) ≡λ

ρ

(h(L)(L− ρ)

L(L− λ)− h(λ)(λ− ρ)

λ(L− λ)− ρ

λ

h(0)

L

)1− ρL

1− λL

Now, turntothefixedpointproblemthatcharacterizestheequilibrium:

ait = Eit[φξt + βait+1 + γat+1]

Usingourguess, wecanreplacetheleft-handsidewith h(L)xit. Usingtheresultsderivedabove,

ontheotherhand, wecanreplacetheright-handsidewith [G1(L) + (β + γ)G2(L) + βG3(L)]xit. It

followsthatourguessiscorrectifandonlyif

h(L) = G1(L) + (β + γ)G2(L) + βG3(L)

42

Equivalently, weneedtofindananalyticfunction h(z) thatsolves

h(z) = φλ

ρ

τuτη

1

1− ρλ

1

1− λz+

+ (β + γ)λ

ρ

τuτη

(h(z)

(1− λz)(z − λ)− h(λ)(1− ρz)

(1− ρλ)(z − λ)(1− λz)

)+ β

λ

ρ

(h(z)(z − ρ)

z(z − λ)− h(λ)(λ− ρ)

λ(z − λ)− ρ

λ

h(0)

z

)1− ρz

1− λz,

whichcanbetransformedas

C(z)h(z) = d(z;h(λ), h(0))

where

C(z) ≡ z(1− λz)(z − λ)− λ

ρ

β(z − ρ)(1− ρz) + (β + γ)

τuτηz

d(z;h(λ), h(0)) ≡ φ

λ

ρ

τuτη

1

1− ρλz(z − λ)− 1

ρ

(τuτη

λ(β + γ)

1− ρλ+ β(λ− ρ)

)z(1− ρz)h(λ)

− β(z − λ)(1− ρz)h(0)

Notethat C(z) isacubicequationandthereforecontainswiththreeroots. Wewillverifylaterthat

therearetwoinsiderootsandoneoutsideroot. Tomakesurethat h(z) isananalyticfunction, we

choose h(0) and h(λ) sothatthetworootsof d(z;h(λ), h(0)) arethesameasthetwoinsiderootsof

C(z). Thispinsdowntheconstants h(0), h(λ), andthereforethepolicyfunction h(L)

h(L) =

(1− ϑ

ρ

)φ

1− ρδ

1

1− ϑL,

where ϑ−1 istherootof C(z) outsidetheunitcircle.

NowweverifythatC(z) hastwoinsiderootsandoneoutsideroot. NotethatC(z) canberewritten

as

C(z) = λ

− z3 +

(ρ+

1

ρ+

1

ρ

τuτη

+ β

)z2 −

(1 + β

(ρ+

1

ρ

)+β + γ

ρ

τuτη

)z + β

.

Withtheassumptionthat β > 0, γ > 0, and β+γ < 1, itisstraightforwardtoverifythatthefollowing

43

propertieshold:

C(0) = β > 0

C(λ) = −λγ 1ρ

τuτη

< 0

C(1) =τu(1− β − γ)

τηρ+ (1− β)

(1

ρ+ ρ− 2

)> 0

Therefore, thethreerootsareallreal, twoofthemarebetween0and1, andthethirdone ϑ−1 islarger

than1.

Finally, toshowthat ϑ islessthan ρ, itissufficienttoshowthat

C

(1

ρ

)=τu(1− ρβ − ργ)

τηρ3> 0.

Since C(ϑ−1) = 0, ithastobethat ϑ−1 islargerthan ρ−1, or ϑ < ρ.

ProofofProposition 3. Theequilibriumoutcomeinthehybrideconomyisgivenbythefollowing

AR(2)process:

at =ζ0

1− ζ1Lξt

where

ζ1 =1

2ωfδ

(1−

√1− 4δωfωb

)and ζ0 =

φζ1ωb − ρωfδζ1

(37)

and δ ≡ β + γ. Thesolutiontotheincomplete-informationeconomyis

at =

(1− ϑ

ρ

)(φ

1− ρδ

)(1

1− ϑLξt

),

Tomatchthehybridmodel, weneed

ζ1 = ϑ and ζ0 =

(1− ϑ

ρ

)φ

1− ρδ. (38)

Combining(37)and(38), andsolvingforthecoefficientsof ωf and ωb, weinferthatthetwoeconomies

generatethesamedynamicsifandonlyifthefollowingtwoconditionshold:

ωf =δρ2 − ϑ

δ(ρ2 − ϑ2)(39)

ωb =ϑ(1− δϑ)ρ2

ρ2 − ϑ2(40)

44

Since δ ≡ β + γ andsince ϑ isa functionof theprimitiveparameters (σ, ρ, β, γ), theabove two

conditionsgivethecoefficients ωf and ωb asasfunctionsoftheprimitiveparameters, too.

Itisimmediatetocheckthat ωf < 1 and ωb > 0 if ϑ ∈ (0, ρ), whichinturnisnecessarilytruefor

any σ > 0; andthat ωf = 1 and ωb = 0 if ϑ = ρ, whichinturnisthecaseifandonlyif σ = 0. The

proofofthecomparativestaticsintermsof σ iscontainedintheproofofProposition 4.

ProofofProposition 4. Wefirstshowthat ωf isdecreasingin ϑ and ωb isincreasingin ϑ. Thiscan

beverifiedasfollows

∂ωf

∂ϑ=

−δ(ρ2 + ϑ2) + 2δ2ρ2ϑ

(δ(ρ2 − ϑ2))2<

−δ(ρ2 + ϑ) + 2δρϑ

(δ(ρ2 − ϑ2))2=

−δ(ρ− ϑ)2

(δ(ρ2 − ϑ2))2< 0

∂ωb

∂ϑ=ρ2(ρ2 + ϑ2 − 2δϑρ2)

(ρ2 − ϑ2)2>ρ2(ρ2 + ϑ2 − 2ϑρ)

(ρ2 − ϑ2)2=

(ρ

ρ+ ϑ

)2

> 0

Nowitissufficienttoshowthat ϑ isincreasingin γ. Notethat

C

(1

ρ

)=τu(1− ρβ − ργ)

τηρ3> 0 and C

(1

λ

)= −τu

τη

γβ

ρλ2< 0

Bythecontinuityof C(z), itmustbethecasethat C(z) admitsarootbetween 1ρ and 1

λ . Recallfrom

theproofofProposition 2, ϑ−1 istheonlyoutsideroot, anditfollowsthat λ < ϑ < ρ. Italsoimplies

that C(z) isdecreasingin z intheneighborhoodof z = ϑ−1, apropertythatweuseinthesequelto

characterizecomparativestaticsof ϑ.

Next, usingthedefinitionof C(z), namely

C(z) ≡ −z3 +(ρ+

1

ρ+

1

ρ

τuτη

+ β

)z2 −

(1 + β

(ρ+

1

ρ

)+β + γ

ρ

τuτη

)z + β,

takingitsderivativewithrespectto γ, andevaluatingthatderivativeat z = ϑ−1, weget

∂C(ϑ−1)

∂γ= − τu

ρτη< 0

Combiningthiswiththeearlierobservationthat ∂C(ϑ−1)∂z < 0, andusingtheImplicitFunctionTheo-

rem, weinferthat ϑ isanincreasingfunctionof γ.

Similarly, takingderivativewithrespectto τu, wehave

∂C(ϑ−1)

∂τu=

1

ρτηϑ−1(ϑ−1 − β − γ) >

1

ρτηϑ−1(1− β − γ) > 0.

Since σ =√τu, ϑ isalsoincreasingin σ.

45

ProofofProposition 5. Thehybridandtheincomplete-informationeconomiesgeneratethesame

dynamicsifandonlyifconditions(39)and(40)hold. Using(40), wecanrewrite(39)asfollows:

ωf = Ω(ωb; δ, ρ) ≡ 1− 1

δρ2ωb. (41)

Furthermore, any (β, γ, ρ), theequilibriumoftheincomplete-informationeconomygivesaninvertible

mappingfrom σ ∈ (0,∞) to ϑ ∈ (0, ρ), whereascondition(40)givesaninvertiblemappingfrom

ϑ ∈ (0, ρ) to ωb ∈ (0,∞). Itfollowsthatthereexistsa σ ∈ (0,∞) suchthattheequilibriumdynamics

oftheincomplete-informationeconomyreplicatesthatofthehybrideconomyifandonlyifthepair

(ωb, ωf ) satisfiescondition(41)alongwith ωb ∈ (0,∞). Finally, theleveloftheinformationalfriction

thatachievesthisreplicationisobtainedbyinvertingcondition(40)toobtain ϑ, andtherebyalso σ,

asanimplicitfunctionof ωb.

ProofofProposition 8. First, letusprove gk < gk. Recallthat gk isgivenby

gk =

∞∑h=0

γhλkλk+1...λk+hρk+h

Clearly,

0 < gk <

∞∑h=0

γhλkρk+h = gk,

whichprovesthefirstproperty. If limk→∞ λk = 1 and∑∞

h=0 γhρk+h existsforall k, thenitfollows

that

limk→∞

gkgk

=limk→∞

∑∞h=0 γ

hρk+h

limk→∞∑∞

h=0 γhρk+h

= 1.

Next, letusprovethat gk+1

gk>

gk+1

gk. Bydefinition,

gk+1

gk=λk+1

λk

∑∞h=0 γ

hρk+h+1∑∞h=0 γ

hρk+h

gk+1

gk=λk+1

λk

∑∞h=0 γ

hλk+2...λk+h+1ρk+h+1∑∞h=0 γ

hλk+1...λk+hρk+h

Since λk isstrictlyincreasingand ρk > 0, wehave

gk+1

gk

/gk+1

gk>

∑∞h=0 γ

hλk+1...λk+hρk+h+1∑∞h=0 γ

hλk+1...λk+hρk+h

/∑∞h=0 γ

hρk+h+1∑∞h=0 γ

hρk+h

Itremainstoshowthatthetermontheright-handsideisgreaterthan1. Toproceed, westartwith

46

thefollowingobservation. If θ1 ≥ θ2 > 0, and y2y1+y2

≥ x2x1+x2

, then

x1θ1 + x2θ2x1 + x2

≥ y1θ1 + y2θ2y1 + y2

Notethat ∑∞h=0 γ

hλk+1...λk+hρk+h+1∑∞h=0 γ

hλk+1...λk+hρk+h=ρk+1

ρk

1 + γλk+1ρk+2

ρk+1+ γ2λk+1λk+2

ρk+3

ρk+1+ . . .

1 + γλk+1ρk+1

ρk+ γ2λk+1λk+2

ρk+2

ρk+ . . .

and ∑∞h=0 γ

hρk+h+1∑∞h=0 γ

hρk+h=ρk+1

ρk

1 + γρk+2

ρk+1+ γ2

ρk+3

ρk+1+ . . .

1 + γρk+1

ρk+ γ2

ρk+2

ρk+ . . .

Basedontheobservation, wewillshowthat

1 + γλk+1ρk+2

ρk+1+ γ2λk+1λk+2

ρk+3

ρk+1+ . . .

1 + γλk+1ρk+1

ρk+ γ2λk+1λk+2

ρk+2

ρk+ . . .

≥1 + γ

ρk+2

ρk+1+ γ2

ρk+3

ρk+1+ . . .

1 + γρk+1

ρk+ γ2

ρk+2

ρk+ . . .

byinduction. Wefirstestablishthefollowing

1 + γλk+1ρk+2

ρk+1

1 + γλk+1ρk+1

ρk

≥1 + γ

ρk+2

ρk+1

1 + γρk+1

ρk

This inequality isobtainedby labeling θ1 = 1, θ2 =ρkρk+2

ρ2k+1, x1 = y1 = 1, x2 = γλk+1

ρk+1

ρk, and

y2 = γρk+1

ρk. Byassumption, ρkρk+2

ρ2k+1≤ 1. Meanwhile,

x2x1 + x2

=γλk+1

ρk+1

ρk

1 + γλk+1ρk+1

ρk

≤γλk+1

ρk+1

ρk

λk+1 + γλk+1ρk+1

ρk

=y2

y1 + y2

Nowsupposethat

1 + γλk+1ρk+2

ρk+1+ . . .+ γn−1λk+1 . . . λk+n−1

ρk+n

ρk+1

1 + γλk+1ρk+1

ρk+ . . .+ γn−1λk+1 . . . λk+n−1

ρk+n−1

ρk

≥1 + γ

ρk+2

ρk+1+ . . .+ γn−1 ρk+n

ρk+1

1 + γρk+1

ρk+ . . .+ γn−1 ρk+n−1

ρk

Wewanttoshow

1 + γλk+1ρk+2

ρk+1+ . . .+ γn−1λk+1 . . . λk+n−1

ρk+n

ρk+1+ γnλk+1 . . . λk+n

ρk+n+1

ρk+1

1 + γλk+1ρk+1

ρk+ . . .+ γn−1λk+1 . . . λk+n−1

ρk+n−1

ρk+ γnλk+1 . . . λk+n

ρk+n

ρk

≥1 + γ

ρk+2

ρk+1+ . . .+ γn−1 ρk+n

ρk+1+ γn

ρk+n+1

ρk+1

1 + γρk+1

ρk+ . . .+ γn−1 ρk+n−1

ρk+ γn

ρk+n

ρk

47

Let θ1 =1+γ

ρk+2ρk+1

+...+γn−1 ρk+nρk+1

1+γρk+1ρk

+...+γn−1ρk+n−1

ρk

, θ2 =ρkρk+n+1

ρk+1ρk+n, x1 = 1+γλk+1

ρk+1

ρk+. . .+γn−1λk+1 . . . λk+n−1

ρk+n−1

ρk,

x2 = γnλk+1 . . . λk+nρk+n

ρk, y1 = 1 + γ

ρk+1

ρk+ . . .+ γn−1 ρk+n−1

ρk, y2 = γn

ρk+n

ρk. Wehave

1 + γλk+1ρk+2

ρk+1+ . . .+ γn−1λk+1 . . . λk+n−1

ρk+n

ρk+1+ γnλk+1 . . . λk+n

ρk+n+1

ρk+1

1 + γλk+1ρk+1

ρk+ . . .+ γn−1λk+1 . . . λk+n−1

ρk+n−1

ρk+ γnλk+1 . . . λk+n

ρk+n

ρk

=

x11+γλk+1

ρk+2ρk+1

+...+γn−1λk+1...λk+n−1ρk+nρk+1

1+γλk+1ρk+1ρk

+...+γn−1λk+1...λk+n−1ρk+n−1

ρk

+ x2θ2

x1 + x2

≥x1θ1 + x2θ2x1 + x2

and

1 + γρk+2

ρk+1+ . . .+ γn−1 ρk+n

ρk+1+ γn

ρk+n+1

ρk+1

1 + γρk+1

ρk+ . . .+ γn−1 ρk+n−1

ρk+ γn

ρk+n

ρk

=y1θ1 + y2θ2y1 + y2

Itremainstoshowthat θ1 ≥ θ2 and x2x1+x2

≤ y2y1+y2

. Notethat

θ1θ2

=1 + γ

ρk+1

ρk

ρk+2ρkρ2k+1

+ . . .+ γn−1 ρk+n−1

ρk

ρk+nρkρk+1ρk+n−1

θ2 + γρk+1

ρkθ2 + . . .+ γn−1 ρk+n−1

ρkθ2

Byassumption, θ2 < 1 and θ2 ≤ ρkρk+i+1

ρk+1ρk+iwhen i ≤ n, whichleadsto θ1 ≥ θ2. Alsonotethat

x2x1 + x2

=γnλk+1 . . . λk+n

ρk+n

ρk

1 + γλk+1ρk+1

ρk+ . . .+ γn−1λk+1 . . . λk+n−1

ρk+n−1

ρk+ γnλk+1 . . . λk+n

ρk+n

ρk

≤γnλk+1 . . . λk+n

ρk+n

ρk

λk+1 . . . λk+n + γλk+1 . . . λk+nρk+1

ρk+ . . .+ γn−1λk+1 . . . λk+n

ρk+n−1

ρk+ γnλk+1 . . . λk+n

ρk+n

ρk

=y2

y1 + y2

Thiscompletestheproofthat gk+1

gk>

gk+1

gk.

AppendixB:Investment

A longtraditioninmacroeconomicsthatgoesbackto Hayashi (1982)and AbelandBlanchard (1983)

hasstudiedrepresentative-agentmodelsinwhichthefirmsfaceacostinadjustingtheircapitalstock.

48

Inthisliterature, theadjustmentcostisspecifiedasfollows:

Costt = Φ

(It

Kt−1

)(42)

where It denotestherateofinvestment, Kt−1 denotesthecapitalstockinheritedfromtheprevious

period, and Φ isaconvexfunction. Thisspecificationgivesthelevelof investmentasadecreas-

ingfunctionofTobin’sQ.Italsogeneratesaggregateinvestmentresponsesthatarebroadlyinline

withthosepredictedbymorerealistic, heterogeneous-agentmodelsthataccountforthedynamicsof

investmentatthefirmorplantlevel(CaballeroandEngel, 1999; Bachmann, Caballero, andEngel,

2013; KhanandThomas, 2008).29

Bycontrast, theDSGE literaturethatfollows Christiano, Eichenbaum, andEvans (2005)and Smets

andWouters (2007)assumesthatthefirmsfaceacostinadjusting, nottheircapitalstock, butrather

theirrateofinvestment. Thatis, thisliteraturespecifiestheadjustmentcostasfollows:

Costt = Ψ

(ItIt−1

)(43)

AswiththeHybridNKPC,thisspecificationwasadoptedbecauseitallowsthetheorytogenerate

sluggishaggregateinvestmentresponsestomonetaryandothershocks. Butithasnoobviousanalogue

intheliteraturethataccountsforthedynamicsofinvestmentatthefirmorplantlevel.

Inthesequel, wesetupamodelofaggregateinvestmentwithtwokeyfeatures: first, theadjust-

mentcosttakestheformseenincondition(42); andsecond, theinvestmentsofdifferentfirmsare

strategiccomplementsbecauseofanaggregatedemandexternality. Wethenaugmentthismodelwith

incompleteinformationandshowthatitbecomesobservationallyequivalenttoamodelinwhichthe

adjustmentcosttakestheformseenincondition(43). Thisillustrateshowincompleteinformation

canmergethegapbetweenthedifferentstrandsoftheliteratureandhelpreconcilethedominant

DSGE practicewiththerelevantmicroeconomicevidenceoninvestment.

Letusfillinthedetails. WeconsideranAK modelwithcoststoadjustingthecapitalstock. There

is a continuumofmonopolistic competitivefirms, indexedby i andproducingdifferent varieties

of intermediate investmentgoods. Thefinal investmentgood isaCES aggregatorof intermediate

investmentgoods. Letting Xit denote the investmentgoodproducedbyfirm i, wehave that the

aggregateinvestmentisgivenby

It =

[∫X

σ−1σ

it

] σσ−1

.

29Theseworksdifferontheimportancetheyattributetoheterogeneity, lumpiness, andnon-linearities, butappeartosharethepredictionthattheimpulseresponseofaggregateinvestmentispeakedonimpact. Theythereforedonotprovideamicro-foundationofthekindofsluggishinvestmentdynamicsfeaturedintheDSGE literature.

49

Andletting Qit denotethepricefacedbyfirm i, wehavethattheinvestmentpriceindexisgivenby

Qt =

[∫Q1−σ

it

] 11−σ

.

A representativefinalgoodsproducerhasperfect informationandpurchases investmentgoods to

maximizeitsdiscountedprofit

maxKt,It

∞∑t=0

χtE0

[exp(ξt)AKt −QtIt − Φ

(ItKt

)Kt

],

subjectto

Kt+1 = Kt + It.

Here, thefundamentalshock, ξt, isanexogenousproductivityshocktothefinalgoodsproduction,

and Φ(

ItKt

)Kt represents thequadraticcapital-adjustmentcost. Thefollowingfunctional formis

assumed:

Φ

(ItKt

)=

1

2ψ

(ItKt

)2

.

Let Zt ≡ ItKt

denotetheinvestment-to-capitalratio. Onabalancedgrowthpath, thisratioandthe

pricefortheinvestmentgoodsremainconstant, i.e., Zt = Z and Qt = Q. Thelog-linearizedversion

ofthefinalgoodsproducer’soptimalconditionaroundthebalancedgrowthpathcanbewrittenas

Qqt + ψZzt = χEt

[Aξt+1 +Qqt+1 + ψZ(1 + Z)zt+1

]. (44)

Whentheproducersoftheintermediateinvestmentgoodschoosetheirproductionscale, theymay

notobservetheunderlyingfundamental ξt perfectly. Asaresult, theyhavetomaketheirdecision

basedontheirexpectationsaboutfundamentalsandothers’decisions. Letting

maxXit

Eit [QitXit − cXit] ,

subjectto

Qit =

(Xit

It

)− 1σ

Qt.

Define Zit ≡ XitKt

asthefirm-specificinvestment-to-capitalratio, andthelog-linearizedversionofthe

optimalchoiceof Xit is

zit = Eit [zt + σqt] .

50

Insteadystate, theprice Q simplyequalsthemarkupovermarginalcost c,

Q =σ

σ − 1c,

andtheinvestment-to-capitalratio Z solvesthequadraticequation

Q+ ψZ =χ

(A+Q+ ψZ + ψZ2 − 1

2ψZ2

).

FrictionlessBenchmark. Ifallintermediatefirmsobserve ξt perfectly, thenwehave

zit = zt + σqt

Aggregationimpliesthat zit = zt and qt = 0. Itfollowsthat zt obeysthefollowingEulercondition:

zt = φξt + δEt [zt+1]

where

φ =ρχA

ψZand δ = χ(1 + Z).

IncompleteInformation. Supposenowthatfirmsreceiveanoisysignalaboutthefundamental ξtasinSection 3. Here, wemakethesamesimplifyingassumptionasintheNKPC application. We

assumethatfirmsobservecurrent zt, butprecludethemfromextractinginformationfromit. Together

withthepricingequation(44), theaggregateinvestmentdynamicsfollow

zt =ρχA

ψZ

∞∑k=0

χkEt[ξt+k] + χZ

∞∑k=0

χkEt[zt+k+1]

Theinvestmentdynamicscanbeunderstoodasthesolutiontothedynamicbeautyconteststudiedin

Section 3 byletting

φ =ρχA

ψZ, β = χ, and γ = χZ.

Thefollowingisthenimmediate.

Proposition 9. Wheninformationisincomplete, thereexist ωf < 1 and ωb > 0 suchthattheequilib-

riumprocessforinvestmentsolvesthefollowingequation:

zt = φξt + ωfδEt[zt+1] + ωbzt−1

Finally, it straightforward toshowthat theaboveequation isof thesame typeas theone that

governsinvestmentinacomplete-informationmodelwheretheadjustmentcostis intermsofthe

51

investmentrate, namelyamodelinwhichthefinalgoodproducer’sproblemismodifiedasfollows:

maxKt,It

∞∑t=0

χtE0

[exp(ξt)AKt −QtIt −Ψ

(It

It−1

)It

]

where It istheaggregateinvestment.

AppendixC:AssetPrices

Consideralog-linearizedversionofthestandardasset-pricingconditioninaninfinitehorizon, representative-

agentmodel:

pt = Et[dt+1] + δEt[pt+1],

where pt isthepriceoftheassetinperiod t, dt+1 isitsdividendinthenextperiod, Et istheexpec-

tationoftherepresentativeagent, and δ ishisdiscountfactor. Iteratingtheaboveconditiongivesthe

equilibriumpriceastheexpectedpresentdiscountedvalueofthefuturedividends.

Byassumingarepresentativeagent, theaboveconditionconcealstheimportanceofhigher-order

beliefs. A numberofworkshavesoughttounearththatrolebyconsideringvariantswithheteroge-

neouslyinformed, short-termtraders, inthetraditionof Singleton (1987); see, forexample, Allen,

Morris, andShin (2006), Kasa, Walker, andWhiteman (2014), and Nimark (2017). Wecancapture

theseworksinoursettingbymodifyingtheequilibriumpricingconditionasfollows:

pt = Et[dt+1] + δEt[pt+1] + ϵt,

where Et isthe average expectationofthetradersinperiod t and ϵt isani.i.dshockinterpretedas

thepriceeffectofnoisytraders. Thekeyideaembeddedintheaboveconditionisthat, aslongasthe

tradershavedifferentinformationandtherearelimitstoarbitrage, assetmarketsarelikelytobehave

like(dynamic)beautycontests.

Letusnowassumethat thedividendisgivenby dt+1 = ξt + ut+1, where ξt followsanAR(1)

processand ut+1 isi.i.d. overtime, andthattheinformationofthetypicaltradercanberepresented

bya seriesofprivate signalsas incondition (11).30 Applyingour results, andusing the fact that

ξt = Et[dt+1], wethenhavethatthecomponentoftheequilibriumassetpricethatisdrivenby ξt30Here, weareabstractingfromthecomplicationsoftheendogenousrevelationofinformationandwethinkofthesignals

in(11)asconvenientproxiesforalltheinformationofthetypicaltrader. Onecanalsointerpretthisasasettinginwhichthedividendisobservable(andhencesoistheprice, whichismeasurableinthedividend)andtheassumedsignalsaretherepresentationofaformofrationalinattention. Lastbutnotleast, wehaveverifiedthatthesolutionwithendogenousinformationcanbeapproximatedverywellbythesolutionobtainedwithexogenousinformation.

52

obeysthefollowinglawofmotion, forsome ωf < 1 and ωb > 0:

pt = Et[dt+1] + ωfδEt[pt+1] + ωbpt−1,

where Et[·] isthefully-information, rationalexpectations. Wethushavethatassetpricescandisplay

bothmyopia, intheformof ωf < 1, andmomentum, orpredictability, intheformof ωb > 0.

Kasa, Walker, andWhiteman (2014)havealreadyemphasizedhowincompleteinformationand

higher-orderuncertaintycanhelpexplainmomentumandpredictabilityinassetprices. Ourresult

offersasharpillustrationofthisinsightandblendsitwiththeinsightregardingmyopia. Inthepresent

context, thelatterinsightseemstochallengetheasset-priceliteraturethatemphasizeslong-runrisks:

newsaboutthelong-runfundamentalsmaybeheavilydiscountedwhenthereishigher-orderuncer-

tainty. Finally, ourresultsuggeststhatbothkindsofdistortionsarelikelytobegreateratthelevelof

theentirestockmarketthanatthelevelofthestockofaparticularfirminsofarasfinancialfrictions

andGE effectscausethetradestobestrategiccomplementsatthemacroleveleveniftheyarestrate-

gicsubstitutesatthemicrolevel, whichinturnmayhelprationalizeSamuelson’sdictum(Jungand

Shiller, 2005).

Weleavetheexplorationofthese—admittedlyspeculative—ideasopenforfutureresearch. We

concludethisappendixbyillustratinghowourobservational-equivalenceresult, whichreliesonas-

sumingawaytheendogenousrevelationofinformationthroughtheequilibriumprice, canbeseen

asanapproximationofthedynamicsthatobtainwhenthisassumptionisrelaxed.

Allowinglearningfrompricesaddsmorerealism, buttypicallyrulesoutananalyticcharacteriza-

tionoftheequilibrium.31 Suppose, inparticular, thatthetradersinoursettingcanperfectlyobserve

thecurrentpriceaswellasthelast-perioddividend. Inthiscase, theequilibriumpricingdynamics

doesnotadmitafinitestate-spacerepresentation. Toillustrate, set δ = 0.98, ρ = 0.95, σu = 2, and

σϵ = σν = 5, andapproximatetheequilibriumdynamicswithanMA(100)process. Thesolidbluein

Figure 3 givestheresultingIRF oftheequilibriumpricetoaninnovationin ξt. Thedashedredline

isobtainedbytakingourhybrideconomy, whichassumesawaythelearningfromeitherthepriceor

thepastdividend, andrecalibratingtheleveloftheidiosyncraticnoisesothattheimpliedIRF isclose

aspossibletotheoneobtainedintheeconomyinwhichsuchlearningisallowed. Asevidentinthe

figure, thehybrideconomydoesaverygoodjobinreplicatingthedynamicsofthelattereconomy.

Wehaveverifiedthatthissimilarityextendstoawiderangeofvaluesfortheparametersofthe

assumedsetting. Thissimilaritymay, ofcourse, bebrokenbyassumingamorecomplexstochastic

processfor thefundamentalandamoreconvolutedlearningdynamics. However, theanalysisof

Section 7 togetherwiththeexamplepresentedhereillustratewhyouranalysiscanbethoughtofasa

31See Nimark (2017)and HuoandTakayama (2015)foramoredetaileddiscussion.

53

0 2 4 6 8 10 12 14 164

6

8

10

12

14

16

Figure 3: ImpulseResponseFunctionofAssetPrice

convenientproxyofsettingswithendogenousinformationaggregation.

AppendixD:VariantwithIdiosyncraticShocks

Inthisappendix, weextendtheanalysistoasettingthatfeaturesbothaggregateandidiosyncratic

shocks. Thisservestoillustratehowourtheoryoffersanaturalexplanationofwhysignificantlevelsof

as-ifmyopiaandanchoringcanbepresentatthemacroeconomiclevel(i.e., intheresponsetoaggre-

gateshocks)eveniftheyareabsentatthemicroeconomiclevel(i.e., intheresponsetoidiosyncratic

shocks), whichcomplementsthediscussioninSection 5 .

Toaccommodateidiosyncraticshocks, weextendthemodelsothattheoptimalbehaviorofagent

i obeysthefollowingequation:

ait = Eit[φξit + βait+1 + γat+1]

where

ξit = ξt + ζit

andwhere ζit isapurelyidiosyncraticshock. WeletthelatterfollowasimilarAR(1)processasthe

aggregateshock: ζit = ρζit−1 + ϵit, where ϵit isi.i.d. acrossboth i and t.32

Wethenspecifytheinformationstructureasfollows. First, weleteachagentobservethesame

signal xit abouttheaggregateshock ξt asinourbaselinemodel. Second, weleteachagentobserve

32Therestrictionthatthetwokindsofshockshavethesamepersistenceisonlyforexpositionalsimplicity.

54

thefollowingsignalabouttheidiosyncraticshock ζit :

zit = ζit + vit,

where vit isindependentof ζit, of ξt, andof xit.

Becausethesignalsareindependent, theupdatingofthebeliefsabouttheidiosyncraticandthe

aggregateshocksarealsoindependent. Let 1− λρ betheKalmangainintheforecastsoftheaggregate

fundamental, thatis,

Eit[ξt] = λEit−1[ξt] +

(1− λ

ρ

)xit

Next, let 1− λρ betheKalmangainintheforecastsoftheidiosyncraticfundamental, thatis,

Eit[ζit] = λEit−1[ζit] +

(1− λ

ρ

)zit

ItisstraightforwardtoextendtheresultsofSection 4.2 tothecurrentspecification. Itcanthusbe

shownthattheequilibriumactionisgivenbythefollowing:

ait =

(1− λ

ρ

)φ

1− ρβ

1

1− λLζit +

(1− ϑ

ρ

)φ

1− ρδ

1

1− ϑLξt + uit

where ϑ isdeterminedinthesamemannerasinourbaselinemodelandwhere uit isaresidualthat

isorthogonaltoboth ζit and ξt andthatcapturesthecombinedeffectofalltheidiosyncraticnoisesin

theinformationofagent i. Finally, itisstraightforwardtocheckthat ϑ = λ when γ = 0; ϑ > λ when

γ > 0; andthegapbetween ϑ and λ increaseswiththestrengthoftheGE effect, asmeasuredwith γ.

Incomparison, thefull-informationequilibriumactionisgivenby

a∗it =φ

1− ρβζit +

κ

1− ρδξt.

Itfollowsthat, relativetothefull-informationbenchmark, thedistortionsofthemicro-andthemacro-

levelIRFsaregivenby, respectively,(1− λ

ρ

)1

1− λLand

(1− ϑ

ρ

)1

1− ϑL.

Themacro-leveldistortionsisthereforehigherthanitsmicro-levelcounterpartifandonlyif ϑ > λ.

Following MackowiakandWiederholt (2009), itisnaturaltoassumethat λ islowerthan λ, be-

causethetypicalagentislikelytoallocatemoreattentiontoidiosyncraticshocksthantoaggregate

shocks. Thisguaranteesalowerdistortionatthemicrolevelthanatthemacrolevelevenifweabstract

55

fromGE interactions(whichamountstosetting γ = 0, orabstractingfromrolehigher-orderuncer-

tainty). Butoncesuchinteractionsaretakenintoaccount, wehavethat ϑ remainshigherthan λ even

if λ = λ. Inshort, themacro-levelresponsecandisplayabiggerdistortion, notonlybecauseofthe

mechanismidentifiedintheaforecitedpaper, butalsobecauseoftheroleofhigher-orderuncertainty

identifiedhere.

56