Myopia and Anchoring - London School of Economicsmyopia,as in models with imperfect foresight;and...
Transcript of Myopia and Anchoring - London School of Economicsmyopia,as in models with imperfect foresight;and...
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;[email protected].‡YaleUniversity; [email protected].
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.
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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.
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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.
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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
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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