Paul De Grauwe and Yuemei Ji

Inflation Targets and the Zero Lower Bound in a Behavioral Macroeconomic Model

Hotel "Grand Villa Argentina"

Dubrovnik

June 4 – 6, 2017

Draft version

Please do not quote

THE TWENTY-THIRD DUBROVNIK ECONOMIC CONFERENCE

Organized by the Croatian National Bank

INFLATIONTARGETSANDTHEZEROLOWERBOUNDINABEHAVIORALMACROECONOMICMODEL

PaulDeGrauwe(LondonSchoolofEconomics)

YuemeiJi

(UniversityCollegeLondon)AbstractWe analyse the relation between the level of the inflation target and the zerolowerbound(ZLB)imposedonthenominalinterestrateintheframeworkofabehavioralNew-Keynesianmacroeconomicmodelinwhichagents,experiencingcognitive limitations, use adaptive learning forecasting rules. The modelproduces endogenous waves of optimism and pessimism (animal spirits) thatleadtonon-normaldistributionsoftheoutputgap.

Wefindthatwhentheinflationtargetistooclosetozero,theeconomycangetgripped by “chronic pessimism” that leads to a dominance of negative outputgaps and recessions, and in turn feeds back on expectations producing longwaves of pessimism. Low inflation targets create the risk of persistence ofrecessions and low growth. We conclude that the 2% inflation target, nowpursuedbymanycentralbanks,istoolow.

JEL: E03, E31, E32

Keywords: animal spirits, monetary policy, inflation target, behavioraleconomics,zerolowerbound

*We are grateful to Francesco Caselli, DomenicoDelli Gatti,Wouter denHaan,Doyne Farmer, Eddie Gerba, Daniel Gros, Cars Hommes, Ricardo Reis, LarrySummers and participants in seminars at the London School of Economics,CESifo(UniversityofMunich),JaumeUniversity,UniversityCollegeLondonandthe University of Graz for helpful comments and suggestions. We alsoacknowledge the insightful comments of two anonymous referees that havehelpedtoimprovethequalityofourpaper.

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1. IntroductionAn inflation target tooclose tozeroriskspushing theeconomy intoanegative

inflation territoryevenwhenmildshocksoccur. Suchanoutcome isgenerally

consideredtobedangerous.Economistshaveidentifiedseveralrisksassociated

with negative inflation. Two of these have received much attention in the

economicliterature.First,duringperiodsofdeflationthenominalinterestrateis

likely to hit the lower zero bound. When this happens the real interest rate

cannot decline further. In factwhendeflation intensifies, the real interest rate

increases,furtheraggravatingthedeflationarydynamics.Insuchascenariothe

centralbanklosesitscapacitytostimulatetheeconomyinarecession,thereby

risking prolonged recessions (Eggertson and Woodford(2003), Aruoba, &

Schorfheide,F.(2013),Blanchard,etal.(2010),Ball(2014)).

Second, deflation raises the real valueofdebt leading to attempts of agents to

reducetheirdebtbysavingmore.Thisaddstothedeflationarydynamics. This

debt deflation dynamicswas first described by Fisher(1933) and has received

renewed attention since the financial crisis of 2007-08 (see Koo(2011),

EggertssonandKrugman(2012)).

Inthispaperwefocusonthefirstrisk.StandardlinearDSGEmodelshavetended

tounderestimatetheprobabilityofhittingtheZLBaswasshownbyChung,etal.,

(2012).Most of thesemodels have led to theprediction thatwhen the central

bankkeeps an inflation targetof2%, it is veryunlikely for the economy tobe

pushed into the ZLB (Reifschneider and Williams (1999), Coenen(2003),

Schmitt-GroheandUribe(2007)).

Building on a New Keynesian framework, we develop a behavioral

macroeconomicmodeltoshednewlightonthenatureofthisrisk.Thismodelis

characterizedbythefactthatagentsexperiencecognitivelimitationspreventing

them from having rational expectations. Instead they use simple forecasting

rules (heuristics) and evaluate the forecasting performances of these rules ex-

post.Thisevaluationleadsthemtoswitchtotherulesthatperformbest.Thus,it

can be said that agents use a trial-and-error learningmechanism. This is also

called“adaptivelearning”.

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This adaptive learning model produces endogenous waves of optimism and

pessimism(animal spirits) thatdrive thebusiness cycle ina self-fulfillingway,

i.e. optimism (pessimism) leads to an increase (decline) in output, and the

increase (decline) in output in term intensifies optimism (pessimism), see De

Grauwe(2012),andDeGrauweandJi(2016).

Animportantfeatureofthisdynamicsofanimalspiritsisthatthemovementsof

the output gap are characterized by periods of tranquility alternating in an

unpredictablewaywithperiodsofintensemovementsofboomsandbusts.More

technically,thedynamicsofanimalspiritsleadstoanon-normaldistributionof

theoutputgapwithexcesskurtosisand fat tails.This isamodel thatdoesnot

needlargeoutsideshockstogeneratelargemovementsinoutput.

We use this behavioral model to analyze how the level of the inflation target

chosenbythecentralbankaffectsthedynamicscreatedbyanimalspiritswhena

zero lower bound (ZLB) is imposed on the nominal interest rate. Our main

results can be summarized as follows. First, we find that when the inflation

targetistooclosetozero,theeconomycangetgrippedby“chronicpessimism”

that leads to a dominance of negative output gaps and recessions, and in turn

feedsbackonexpectationsproducinglongwavesofpessimism.Usingparameter

calibrations thataregenerally found in the literature,our results suggests that

an inflation target of 2%,which has become the standard followed by central

banks, is too low, i.e. it produces negative skewness in the distribution of the

output gap.We find that an inflation target in the range of 3% to 4% comes

closertoproducingasymmetricdistributionoftheoutputgap.

Second,when comparing the results obtained fromourbehavioralmodelwith

thoseobtainedfromtheNewKeynesianmodelunderrationalexpectations(RE)

wefindthatthebehavioralmodelproducessignificantlymoreZLB-hitsthanthe

RE-model. In additionwhen the interest rate is in the ZLB it remains stuck at

zeromuchlongerinthebehavioralthanintheRE-model.

Third,wefindthatwhentheeconomyispushedintoarecessionasaresultofa

negative demand shock, the high inflation target regime has better stabilizing

properties.Wefindthatinthehighinflationtargetregimethepersistenceofthe

recession is shorter than in the low inflation target regime. That is, when the

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centralbanksetsarelativelyhighinflationtarget,thecapacityofthesystemto

liftitselfoutoftherecessionisstrongerthanwhenitsetsalowinflationtarget.

This ismadepossibleby thestabilizingpropertiesofmonetarypoliciesandby

theensuingeliminationofself-fulfillingpessimism.

Fourth,aninflationtargetof3%to4%ismorecrediblethananinflationtarget

of2%.Thereasonisthatinthelattercasetheeconomyfindsitselfmoreoftenin

theZLB than in the former.This reduces thecentralbank’s capacity tocontrol

inflationandoutputgap.

The paper is organized as follows. Section 2 presents themodel and itsmain

characteristics. Sections 3 to 5 present the results of this model. Section 3

focusesonthefeaturesoftheoutputgapandanimalspiritsandthefrequencyof

hitting ZLB under different inflation target regimes. To understand the role of

the animal spirits, we also compare the behavioral model with the rational

expectationsmodel. Section4discusses the impulse responses of thedifferent

endogenous variables to demand and supply shocks. Section 5 discusses

credibility issues resulting from increasing the inflation target. Section 6

providessomeempiricalvalidationofthemainpredictionsofthemodel.Section

7containstheconclusion.

2.Thebehavioralmodel2.1Modelchoice

Mainstream macroeconomics has been based on two fundamental ideas. The

firstoneisthatmacroeconomicmodelsshouldbemicro-founded,i.e.theyshould

startfromindividualoptimizationandthenaggregatetheseindividuals’optimal

planstoobtainageneralequilibriummodel.Thisprocedureleadstoaggregation

problems that cannot easily be solved (Sonnenschein(1972), Kirman(1992)).

TheDSGEmodelbuildersdealwiththeaggregationproblemsbyintroducingthe

representativeagent, i.e.byassuming thatdemandandsupplydecisions in the

aggregatecanbereducedtodecisionsmadeattheindividuallevel.

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The second idea is that expectations are rational, i.e. take all available

information into account, including the information about the structure of the

economicmodelandthedistributionoftheshockshittingtheeconomy.

We make a different choice of model. First, we will bring at center stage the

heterogeneityofagentsinthattheyhavedifferentbeliefsaboutthestateofthe

economy. As will be shown, it is the aggregation of these diverse beliefs that

createsadynamicsofboomsandbustsinanendogenousway.Thepricewepay

isthatwedonotmicro-foundthemodelandassumetheexistenceofaggregate

demand and supply equations. Second, we assume that agents have cognitive

limitationspreventingthemfromhavingrationalexpectations.Insteadtheywill

be assumed to follow simple rules of thumb (heuristics). Rationality will be

introduced by assuming a willingness to learn frommistakes and therefore a

willingness to switchbetweendifferentheuristics. Inmaking these choiceswe

followtheroadtakenbyanincreasingnumberofmacroeconomists,whichhave

developed “agent-based models” and “behavioral macroeconomic models”

(Tesfatsion,L.(2001),Colander,etal.(2008),FarmerandFoley(2009),Gatti,et

al.(2011), Westerhoff(2012), De Grauwe(2012), Hommes and

Lustenhouwer(2016)).

2.2Basicmodel

The model consists of an aggregate demand equation, an aggregate supply

equationandaTaylorrule.

Weassumetheexistenceofanaggregatedemandequationinthefollowingway:

𝑦! = 𝑎!E!𝑦!!! + 1− 𝑎! 𝑦!!! + 𝑎! 𝑟! − E!𝜋!!! + 𝜀!(1)

whereytistheoutputgapinperiodt,rtisthenominalinterestrate,πtistherate

ofinflation,andεtisawhitenoisedisturbanceterm.ThetildeaboveErefersto

the fact that expectations are not formed rationally. How exactly these

expectationsareformedwillbespecifiedsubsequently.

WefollowtheprocedureintroducedinNewKeynesianDSGE-modelsofaddinga

laggedoutputinthedemandequation.Thiscanbejustifiedbyinvokinginertiain

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decision-making.Ittakestimeforagentstoadjusttonewsignalsbecausethere

ishabitformationorbecauseofinstitutionalconstraints.Forexample,contracts

cannotberenegotiatedinstantaneously.

We assume an aggregate supply equation of the New Keynesian Philips curve

typewithaforwardlookingcomponent,E!𝜋!!!,andalaggedinflationvariable1:

𝜋! = 𝑏!E!𝜋!!! + 1− 𝑏! 𝜋!!! + 𝑏!𝑦! + 𝜂!(2)

FinallytheTaylorruledescribesthebehaviorofthecentralbank

𝑟! = 𝑘! 𝜋∗ + 𝑘! 𝜋! − 𝜋∗ + 𝑘!𝑦! + (1− 𝑘!)𝑟!!! + 𝑢!(3)

where is the inflation target; The central bank is assumed to smooth the

interestrate.Thissmoothingbehaviorisrepresentedbyaddingafractionofthe

laggedinterestrate𝑟!!!inequation(3).Forsimplicityweassumethatthelong-

termequilibriuminterestrateiszeroandthusitdoesnotappearinequation(3).

2.3Normalizingthemodel

In order to solve the behavioral model following De Grauwe(2012), we first

normalizethemodel.Theinflationratesandinterestratescanbeexpressedas

deviationsfromtheinflationtarget𝜋∗.Tobespecific,wedefinethesedeviations

as:𝜋!! = 𝜋! − 𝜋∗,𝑟!! = 𝑟! − 𝜋∗and𝑟!!!! = 𝑟!!! − 𝜋∗,and𝐸!𝜋!!!! = 𝐸!𝜋!!! − 𝜋∗.

Equations(1)-(3)canbenormalizedasfollows:

𝑦! = 𝑎!E!𝑦!!! + 1− 𝑎! 𝑦!!! + 𝑎! 𝑟!! − E!𝜋!!!! + 𝜀! (1a)

𝜋!! = 𝑏!E!𝜋!!!! + 1− 𝑏! 𝜋!!!! + 𝑏!𝑦! + 𝜂!(2a)

𝑟!! = 𝑐!𝜋!! + 𝑐!𝑦! + 𝑐!𝑟!!!! + 𝑢!(3a)

where𝑐! = 𝑘!𝑘!,𝑐! = 𝑘!𝑘!,𝑐! = 1− 𝑘!.

*π

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2.4Introducingheuristicsinforecastingoutput

Agentsareassumedtousesimplerules(heuristics)toforecastthefutureoutput

E!𝑦!!!.Thewayweproceed is as follows.Weassume two typesof forecasting

rules. A first rule is called a “fundamentalist” one. Agents estimate the steady

statevalueoftheoutputgap(whichisnormalizedat0)andusethistoforecast

thefutureoutputgap2.Asecondforecastingruleisan“extrapolative”one.Thisis

a rule thatdoesnotpresuppose that agentsknow the steady stateoutputgap.

They are agnostic about it. Instead, they extrapolate the previous observed

outputgapintothefuture.Thetworulesarespecifiedasfollows:

ThefundamentalistruleisdefinedbyE!!y!!! = 0(4)

TheextrapolativeruleisdefinedbyE!!y!!! = 𝑦!!!(5)

This kind of simple heuristic has often been used in the behavioral finance

literature where agents are assumed to use fundamentalist and chartist rules

(see Brock and Hommes(1997), Branch and Evans(2006), De Grauwe and

Grimaldi(2006)). It isprobablythesimplestpossibleassumptiononecanmake

abouthowagentswhoexperience cognitive limitations,use rules that embody

limited knowledge to guide their behavior3. They only require agents to use

informationtheyunderstand,anddonotrequirethemtounderstandthewhole

picture.

Thusthespecificationoftheheuristicsin(4)and(5)shouldnotbeinterpreted

asarealisticrepresentationofhowagentsforecast.Ratheris itaparsimonious

representation of a world where agents do not know the “Truth” (i.e. the

underlyingmodel). Theuse of simple rules does notmean that the agents are

irrationalandthattheydonotwanttolearnfromtheirerrors.Wewillspecifya

learningmechanismlaterinthissectioninwhichtheseagentscontinuouslytry

tocorrectfortheirerrorsbyswitchingfromoneruletotheother.

Weassume that themarket forecast canbeobtainedas aweightedaverageof

thesetwoforecasts,i.e.

E!𝑦!!! = 𝛼!,!E!!y!!! + 𝛼!,!E!!y!!!(6)

E!𝑦!!! = 𝛼!,!0+ 𝛼!,!y!!!(7)

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and𝛼!,! + 𝛼!,! = 1(8)

where and are the probabilities that agents use a fundamentalist,

respectively,anextrapolativerule.

In order to obtain some intuition about themechanics arising from the use of

thesetworulesit isusefultosubstitute(7) intoequation(1a). Using(8)this

yields

𝑦! = 1− 𝑎!𝛼!,! 𝑦!!! + 𝑎! 𝑟!! − E!𝜋!!!! + 𝜀!

It can be seen that when𝛼!,! = 1, i.e. the probability of all agents using the

fundamentalistruleisequalto1,thecoefficientinfrontof𝑦!!!is1− 𝑎!,whileif

𝛼!,! = 0, theprobability of all agents using the extrapolative rule is equal to 1,

that coefficient is 1. Thismakes clear that the source of thepersistence in the

outputgapwillbecomingfromtheuseoftheextrapolativerule.

The forecastingrules (heuristics) introducedherearenotderivedat themicro

levelandthenaggregated.Instead,theyareimposedexpost,onthedemandand

supply equations. This has also been the approach in the learning literature

pioneeredbyEvansandHonkapohja(2001).Ideallyonewouldliketoderivethe

heuristics from themicro-level in an environment inwhich agents experience

cognitive problems. Our knowledge about how to model this behavior at the

micro levelandhowtoaggregate it is toosketchy,however.Psychologistsand

brain scientists struggle to understand how our brain processes information.

There isasyetnogenerallyacceptedmodelwecoulduse tomodel themicro-

foundations of information processing in a world in which agents experience

cognitivelimitations.Wehavenottriedtodoso4.

2.5Selectingtheforecastingrulesinforecastingoutput

As indicated earlier, agents in our model are willing to learn, i.e. they

continuouslyevaluatetheir forecastperformance.Thiswillingnessto learnand

to changeone’sbehavior is avery fundamentaldefinitionof rationalbehavior.

Thus our agents in themodel are rational, not in the sense of having rational

expectations. Insteadour agents are rational in the sense that they learn from

tf ,α te,α

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theirmistakes.Theconceptof“boundedrationality”isoftenusedtocharacterize

thisbehavior.

Thefirststepintheanalysisthenconsistsindefiningacriterionofsuccess.This

will be the forecast performance (utility) of a particular rule. We define the

utilityofusingthefundamentalistandextrapolativerulesasfollows:

𝑈!,! = − ω! y!!!!! − E!,!!!!!y!!!!!!!

!!! (9)

𝑈!,! = − ω! y!!!!! − E!,!!!!!y!!!!!!!

!!! (10)

whereUf,tandUe,taretheutilitiesofthefundamentalistandextrapolatingrules,

respectively.Thesearedefinedasthenegativeofthemeansquaredforecasting

errors(MSFEs)of theforecastingrules;ωkaregeometricallydecliningweights.

Wemaketheseweightsdecliningbecauseweassumethatagentstendtoforget.

Put differently, they give a lower weight to errors made far in the past as

comparedtoerrorsmaderecently.Thedegreeof forgettingturnsouttoplaya

majorroleinourmodel.ThiswasanalyzedinDeGrauwe(2012).

The next step consists in evaluating these utilities. We apply discrete choice

theory (see Anderson, de Palma, and Thisse, (1992) and Brock &

Hommes(1997)) in specifying the procedure agents follow in this evaluation

process.IfagentswerepurelyrationaltheywouldjustcompareUf,tandUe,tin(9)

and(10)andchoosetherulethatproducesthehighestvalue.Thusunderpure

rationality, agents would choose the fundamentalist rule ifUf,t >Ue,t, and vice

versa.However,psychologistshavestressedthatwhenwehavetochooseamong

alternativeswearealsoinfluencedbyourstateofmind(seeKahneman(2002)).

Thelatteristoalargeextentunpredictable.Itcanbeinfluencedbymanythings,

theweather,recentemotionalexperiences,etc.Onewaytoformalizethisisthat

theutilitiesofthetwoalternativeshaveadeterministiccomponent(theseareUf,t

andUe,t in(9)and(10))andarandomcomponentεf,tandεe,tTheprobabilityof

choosingthefundamentalistruleisthengivenby

𝛼!,! = 𝑃 (𝑈!,! + 𝜀!,!) > (𝑈!,! + 𝜀!,!) (11)

Inwords,thismeansthattheprobabilityofselectingthefundamentalistruleis

equal to the probability that the stochastic utility associated with using the

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fundamentalistruleexceedsthestochasticutilityofusinganextrapolativerule.

Inordertoderiveamorepreciseexpressiononehastospecifythedistribution

of the random variables εf,t and εe,t. It is customary in the discrete choice

literaturetoassumethattheserandomvariablesarelogisticallydistributed(see

Anderson, Palma, and Thisse(1992), p.35). One then obtains the following

expressionsfortheprobabilityofchoosingthefundamentalistrule:

𝛼!,! =𝑒𝑥𝑝 𝛾𝑈𝑓,𝑡

𝑒𝑥𝑝 𝛾𝑈𝑓,𝑡 +𝑒𝑥𝑝 𝛾𝑈𝑒,𝑡 (12)

Similarlytheprobabilitythatanagentwillusetheextrapolativeforecastingrule

isgivenby:

𝛼!,! =𝑒𝑥𝑝 𝛾𝑈𝑒,𝑡

𝑒𝑥𝑝 𝛾𝑈𝑓,𝑡 +𝑒𝑥𝑝 𝛾𝑈𝑒,𝑡= 1− 𝛼!,! (13)

Equation (12) says that as the past forecast performance (utility) of the

fundamentalistruleimprovesrelativetothatoftheextrapolativerule,agentsare

morelikelytoselectthefundamentalistrulefortheirforecastsoftheoutputgap.

Equation (13) has a similar interpretation. The parameter γ measures the

“intensity of choice”. It is related to the variance of the random components.

Definingεt=εf,t-εe,t.wecanwrite(seeAnderson,PalmaandThisse(1992)):

𝛾 = !!"#(!!)

.

Whenvar(εt) goes to infinity,γapproaches0. In that case agentsdecide tobe

fundamentalist or extrapolator by tossing a coin and the probability to be

fundamentalist (orextrapolator) isexactly0.5.Whenγ=∞ thevarianceof the

random components is zero (utility is then fully deterministic) and the

probabilityofusingafundamentalistrule iseither1or0.Theparameterγcan

alsobeinterpretedasexpressingawillingnesstolearnfrompastperformance.

Whenγ=0thiswillingnessiszero;itincreaseswiththesizeofγ.

As argued earlier, the selection mechanism used should be interpreted as a

learningmechanism based on “trial and error”.When observing that the rule

theyuseperformslesswellthanthealternativerule,agentsarewillingtoswitch

to the more performing rule. Put differently, agents avoid making systematic

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mistakesbyconstantlybeingwillingtolearnfrompastmistakesandtochange

theirbehavior.Thisalsoensuresthatthemarketforecastsareunbiased.

2.6Heuristicsandselectionmechanisminforecastinginflation

Agentsalsohavetoforecastinflation𝐸!𝜋!!!.Asimilarsimpleheuristicsisused

as in the case of output gap forecasting, with one rule that could be called a

fundamentalistruleandtheotheranextrapolativerule.(SeeBrazieretal.(2008)

forasimilarsetup).Weassumeaninstitutionalset-upinwhichthecentralbank

announcesanexplicit inflationtarget.Thefundamentalistrulethenisbasedon

thisannouncedinflationtarget,i.e.agentsusingthisrulehaveconfidenceinthe

credibilityof thisruleanduse it to forecast inflation. Agentswhodonot trust

the announced inflation target use the extrapolative rule, which consists in

extrapolatinginflationfromthepastintothefuture.

Thefundamentalistrulewillbecalledan“inflationtargeting”rule.Itconsistsin

usingthecentralbank’sinflationtargettoforecastfutureinflation,i.e.

E!!"#𝜋!!! = 𝜋∗(14)

orafternormalizationE!!!"𝜋!!!! = 0

wheretheinflationtargetis The“extrapolators”aredefinedby E!!"#𝜋!!! = 𝜋!!!(15)orafternormalizationE!!"#𝜋!!!! = 𝜋!!! − 𝜋∗Themarketforecastisaweightedaverageofthesetwoforecasts,i.e.E!𝜋!!!! = 𝛽!"#,!E!!"#𝜋!!!! + 𝛽!"#,!E!!"#𝜋!!!! (16)

orE!𝜋!!!! = 𝛽!"#,!0+ 𝛽!"#,! 𝜋!!! − 𝜋∗ = 𝛽!"#,!𝜋!!!! (17)

and 𝛽!"#,! + 𝛽!"#,! = 1(18)

The same selectionmechanism is used as in the case of output forecasting to

determinetheprobabilitiesofagentstrustingtheinflationtargetandthosewho

donottrustitandreverttoextrapolationofpastinflation,i.e.

*π

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(19)

(20)

whereUtar,tandUext,taretheforecastperformances(utilities)associatedwiththe

use of the fundamentalist and extrapolative rules in equation (21) and (22).

Thesearedefinedinthesamewayasin(9)and(10),i.e.theyarethenegatives

oftheweightedaveragesofpastsquaredforecasterrorsofusingfundamentalist

(inflationtargeting)andextrapolativerules,respectively.

𝑈!"#,! = − ω! π!!!!! − E!,!!!!!π!!!!!!!

!!! (21)

𝑈!"#,! = − ω! π!!!!! − E!,!!!!!π!!!!!!!

!!! (22)

Thisinflationforecastingheuristicscanbeinterpretedasaprocedureofagents

to findouthowcredible the central bank’s inflation targeting is. If this is very

credible,usingtheannouncedinflationtargetwillproducegoodforecastsandas

aresult,theprobabilitythatagentswillrelyontheinflationtargetwillbehigh.If

on the other hand the inflation target does not produce good forecasts

(comparedtoasimpleextrapolationrule)theprobabilitythatagentswilluseit

willbesmall.

Finally itshouldbementionedthat thetwopredictionrules for theoutputgap

and inflation are made independently. This is a strong assumption. What we

model is the use of different forecasting rules. The selection criterion is

exclusivelybasedontheforecastingperformancesoftheserules.Agents inour

modeldonothaveapsychologicalpredispositiontobecomefundamentalistsor

extrapolators. However, it is possible that despite the assumption of

independence, the realized choices generated from our model are actually

correlatedduetotheinteractionsofthedifferentvariablesinthemodel.Wewill

come back to this when we implement the model and we will compute the

realizedcorrelationbetweentheprobabilitiesofbeingafundamentalist forthe

outputgapandafundamentalistforinflation.

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2.7Defininganimalspirits

Theforecastsmadebyextrapolatorsandfundamentalistsplayanimportantrole

in the model. In order to highlight this role we define an index of market

sentiments,whichwecall“animalspirits”,andwhichreflectshowoptimisticor

pessimistictheseforecastsare.

Thedefinitionofanimalspiritsisasfollows:

𝑆! = 𝛼!,! − 𝛼!,! 𝑖𝑓 𝑦!!! > 0 −𝛼!,! + 𝛼!,! 𝑖𝑓 𝑦!!! < 0 (23)

where𝑆! istheindexofanimalspirits.Thiscanchangebetween-1and+1.There

aretwopossibilities:

• When𝑦!!! > 0,extrapolators forecastapositiveoutputgap.The fractionofagentswhomakesuchapositiveforecastsis𝛼!,! .Fundamentalists,however,thenmakeapessimisticforecastsincetheyexpectthepositiveoutputgapto

declinetowardstheequilibriumvalueof0.Thefractionofagentswhomake

suchaforecastis𝛼!,! .Wesubtractthisfractionofpessimisticforecastsfromthefraction𝛼!,!whomakeapositiveforecast.Whenthesetwofractionsare

equal to each other (both are then 0.5)market sentiments (animal spirits)

are neutral, i.e. optimists and pessimists cancel out and St = 0. When the

fractionofoptimists𝛼!,!exceeds the fractionofpessimists𝛼!,! , St becomespositive.Aswewillsee,themodelallowsforthepossibilitythat𝛼!,! movesto

1.InthatcasethereareonlyoptimistsandS! = 1.

• When𝑦!!! < 0,extrapolatorsforecastanegativeoutputgap.Thefractionofagents whomake such a negative forecasts is𝛼!,!. We give this fraction a

negative sign. Fundamentalists, however, thenmake an optimistic forecastsince they expect the negative output gap to increase towards the

equilibrium value of 0. The fraction of agentswhomake such a forecast is

𝛼!,! .Wegivethisfractionofoptimisticforecastsapositivesign.Whenthesetwofractionsareequaltoeachother(botharethen0.5)marketsentiments

(animalspirits)areneutral,i.e.optimistsandpessimistscanceloutandSt=0.

Whenthe fractionofpessimists 𝛼!,!exceeds the fractionofoptimists𝛼!,!St

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becomesnegative.Thefractionofpessimists, 𝛼!,! ,canmoveto1.Inthatcase

thereareonlypessimistsandSt=-1.

Wecanrewrite(23)asfollows:

𝑆! = 𝛼!,! − (1− 𝛼!,! ) = 2 𝛼!,! − 1 𝑖𝑓 𝑦!!! > 0 −𝛼!,! + (1− 𝛼!,!) = −2 𝛼!,! + 1 𝑖𝑓 𝑦!!! < 0 (24)

2.8Solvingthemodel

The solution of the model is found by first substituting (3a) into (1a) and

rewritinginmatrixnotation.Thisyields:

1 −𝑏!−𝑎!𝑐! 1− 𝑎!𝑐!

𝜋!!𝑦!

= 𝑏! 0−𝑎! 𝑎!

E!𝜋!!!!

E!𝑦!!!+ 1− 𝑏! 0

0 1− 𝑎!𝜋!!!!

𝑦!!!+ 0

𝑎!𝑐!𝑟!!!! +

𝜂!𝑎!𝑢! + 𝜀!

i.e.

𝑨𝒁𝒕 = 𝑩𝑬𝒕 𝒁𝒕!𝟏 + 𝑪𝒁𝒕!𝟏 + 𝒃𝑟!!!! + 𝒗𝒕(25)

whereboldcharactersrefertomatricesandvectors.ThesolutionforZtisgivenby

𝒁𝒕 = 𝑨!𝟏 𝑩𝑬𝒕 𝒁𝒕!𝟏 + 𝑪𝒁𝒕!𝟏 + 𝒃𝑟!!!! + 𝒗𝒕 (26)

The solution exists if thematrixA is non-singular, i.e. (1-a2c2)-a2b2c1 ≠ 0.The

system(26)describesthesolutionsforytand𝜋!! giventheforecastsofytand𝜋!! .The latter have been specified in equations (7) and (17) and therefore can be

substitutedinto(26).Finally,thesolutionfor𝑟!!!! isfoundbysubstitutingytand

πtobtainedfrom(26)into(3a).

The model has non-linear features making it difficult to arrive at analytical

solutions.Thatiswhywewillusenumericalmethodstoanalyzeitsdynamics.In

ordertodoso,wehavetocalibratethemodel,i.e.toselectnumericalvaluesfor

theparametersof themodel. InTable1 theparametersused in thecalibration

exercise are presented. The values of the parameters are based on what we

foundintheliterature(seeGali(2008)forthedemandandsupplyequationsand

BlattnerandMargaritov(2010)fortheTaylorrule).Themodelwascalibratedin

15

suchawaythatthetimeunitscanbeconsideredtobequarters.Thethreeshocks

(demandshocks,supplyshocksandinterestrateshocks)areindependentlyand

identically distributed (i.i.d.) with standard deviations of 0.5%. These shocks

produce standard deviations of the output gap and inflation that mimic the

standarddeviationsfoundintheempiricaldatausingquarterlyobservationsfor

the US and the Eurozone. We describe the procedure that led us to select

standarddeviationsof0.5%inAppendixC.

Table1:Parametervaluesofthecalibratedmodel

a1=0.5 coefficientofexpectedoutputinoutputequationa2=-0.2 interestelasticityofoutputdemandb1=0.5 coefficientofexpectedinflationininflationequationb2=0.05 coefficientofoutputininflationequationc1=1.5 coefficientofinflationinTaylorequationc2=0.5 coefficientofoutputinTaylorequationc3=0.8 interestsmoothingparameterinTaylorequation𝛾=2 intensityofchoiceparameter𝜎! =0.5 standarddeviationshocksoutput𝜎! =0.5 standarddeviationshocksinflation𝜎!=0.5 standarddeviationshocksTaylor𝜌 =0.5measuresthespeedofdecliningweightsinmeansquareserrors

(memoryparameter)3.InflationtargetingandthezerolowerboundInthissectionwepresenttheresultsofsimulatingthemodelfordifferentvalues

oftheinflationtarget(goingfrom0%to4%),andimposingazerolowerbound

(ZLB)onthenominalinterestrate:

r! ≥ 0(27)

WithouttheZLBcondition,thecentralbankisabletoadjustitsnominalinterest

ratetoachievearealinterestratethatstabilizestheoutputgap.However,with

the ZLB condition, the ability of the central bank to stabilize the output gap

(especiallyanegativeone)isverymuchhindered.

Westartbypresentingtheresultsusinganinflationtargetof2%.Thiswillallow

ustounderstandthemainfeaturesofthemodel.Wethenpresenttheresultsfor

alternativelevelsoftheinflationtarget.

16

3.1Thebasicresults

Figure1showsthemovementsoftheoutputgapandanimalspiritsinthetime

domain (left hand side panels) and in the frequency domain (right hand side

panels)assimulatedinourmodel.Weobservethatthemodelproduceswavesof

optimism and pessimism (animal spirits) that can lead to a situation where

everybody becomes optimist (St = 1) or pessimist (St = -1). These waves of

optimism and pessimism are generated endogenously and arise because

optimistic (pessimistic) forecasts are self-fulfilling and therefore attract more

agentsintobeingoptimists(pessimists).

As can be seen from the left hand side panels, the correlation of these animal

spirits and theoutput gap is high. In the simulations reported in Figure1 this

correlationreaches0.94.Underlying this correlation is theself-fulfillingnature

of expectations. When a wave of optimism is set in motion, this leads to an

increase in aggregate demand (see equation (1)). This increase in aggregate

demandleadstoasituationinwhichthosewhohavemadeoptimisticforecasts

arevindicated.Thisattractsmoreagentsusingoptimisticforecasts.Thisleadsto

aself-fulfillingdynamicsinwhichmostagentsbecomeoptimists.Itisadynamics

thatleadstoacorrelationofthesamebeliefs.Thereverseisalsotrue.Awaveof

pessimistic forecasts can set in motion a self-fulfilling dynamics leading to a

downturn ineconomicactivity(outputgap). Atsomepointmostof theagents

havebecomepessimists.

The right hand side panels show the frequencydistribution of output gap and

animal spirits. We find that the output gap is not normally distributed, with

excesskurtosisandfattails.Wefindakurtosis=4whichistoohighforanormal

distribution.AJarque-Beratestrejectsnormalityofthedistributionoftheoutput

gap.Theoriginofthenon-normalityofthedistributionoftheoutputgap(excess

kurtosisandfattails)canbefoundinthedistributionoftheanimalspirits.We

findthatthereisaconcentrationofobservationsofanimalspiritsaround0.This

means thatmuchof the time there isno clear-cut optimismorpessimism.We

can call these “normal periods”. There is also, however, a concentration of

extreme values at either -1 (extreme pessimism) and +1 (extreme optimism).

These extreme values of animal spirits explain the fat tails observed in the

17

distribution of the output gap. The interpretation of this result is as follows.

When the market is gripped by a self-fulfilling movement of optimism (or

pessimism) this can lead to a situation where everybody becomes optimist

(pessimist).Thisthenalsoleadstoanintenseboom(bust)ineconomicactivity.

InDeGrauwe(2012)andDeGrauweandJi(2016)empiricalevidenceisprovided

indicating that observed output gaps in industrial countries exhibit excess

kurtosisandfattails(seealsoFagiolo,etal.(2008)andAscari,etal(2015)),and

that the output gaps are highly correlated with empirical measures of animal

spirits.Ourmodelmimicstheseempiricalobservationsandisparticularlysuited

tounderstandthenatureofbusinesscycles,whichischaracterizedbyperiodsof

“tranquility”, alternated by periods of booms and busts. We return to the

empiricalvalidationofthemodelinsection6.

In order to improve understanding of themodel it is also useful to relate the

fractions of extrapolators and fundamentalists to the performance (utility) of

these rules (asmeasuredby thenegativeof the forecasterrors).Wedo this in

Figures A2 to A4 (see Appendix B).We observe the strong correlation of the

relative performance of the extrapolating rule versus the fundamentalist rule

(measuredbythedifferenceinutilitiesofusingtheserules)andthefractionof

theagentsusingtheextrapolatingrule.

It is also important to understand the nature of forecasting in this model by

focusingonthesystematicpatternsintheforecastingerrors.Wefindthatthere

is autocorrelation in the forecasting errors. For the output gap this

autocorrelation is 0.37, while for inflation this is 0.15. We also find that the

standarddeviationsoftheseforecastingerrorsamountsto0.60inthecaseofthe

output gap and 0.5 in the case of inflation. We also find a small negative

correlation of -0.2 between the forecasting errors of the output gap and of

inflation.Notethatthereissomevariationinthesenumbersastheydependon

therealizationofthestochasticshocks.Thisimpliesthatsomeofthesenumbers

arenotsignificantlydifferentfromzero.Thisisthecaseoftheautocorrelationin

theforecasterrorsofinflationandinthecaseofthecorrelationsoftheforecast

errorsoftheoutputgapandinflation.

18

Finally, we computed the correlation between the probabilities of being a

fundamentalist for the output gap and a fundamentalist for inflation (the

fractions𝛼!,!and𝛽!"#,!definedin(12)and(19),respectively).Wefindthatthis

correlationisapproximately0.3(dependingontherealizationofthestochastic

shocks). Thus, despite the assumptionof independence in the selectionof the

forecasting rules, the realized choices generated from our model are actually

correlated.Thisisduetotheinteractionsofthedifferentvariablesinthemodel.

Figure1:Outputgapandanimalspiritsintimeandfrequencydomains

(Inflationtarget=2%)

FrequencyZLB-hits=26%;meanZLB-spell=4.8quarters

WenowaskthequestionofhowtheresultsshowninFigure1areaffectedbythe

levelof the inflationtargetchosenbythecentralbank.Westartbynotingthat

theoutputgapinFigure1 isslightlyskewedtothe left. In facttheskewness is

19

foundtobe-0.66.Thisskewnessfindsitsorigininthefactthatthedistribution

of animal spirits is also skewed to the left, i.e. there are more periods of

pessimism thanoptimism.We find that on average animal spirits arenegative

(-0.03).WealsofindthattheinterestrateishittingtheZLB26%ofthetimeand

thatthemeanZLB-spellis4.8quarters.

In order to evaluate the importance of the inflation target we simulated the

modelundertwoalternativeandextremeassumptionsoftheinflationtargets.In

thefirstonewesettheinflationtargetequalto0%;inthesecondoneto4%.We

showtheresultsinFigures2and3.

Ourmajor findingsare the following.Weobserve fromFigure2 thatwhen the

inflation target iszeroweobtainaveryskeweddistributionofoutputgapand

animalspirits(skewness is -0.96andmeananimalspirits is -0.22).Mostof the

timeanimalspiritsarenegativewithmanyperiodsofextremepessimism.There

areveryfewperiodsofoptimism.Thiscanalsobeseenfromthesimulationsin

thetimedomain:theoutputgapisnegativemostofthetimeandanimalspirits

arealsonegativemostofthetime.TheprobabilityofhittingtheZLBis64%and

oncetheZLBishitthemeanlengthofstayingintheZLBismorethan9quarters.

Thusitcanbeconcludedwhenthecentralbanksetsaninflationtargetequalto

zeropessimismprevailsmostofthetimeandrecessionisachronicfeatureofthe

businesscyclewithveryfewperiodsofoptimism.

Weobtainverydifferent resultswithan inflation targetof4%.Theresultsare

presentedinFigure3.Wenowfindthatthedistributionsoftheoutputgapand

animalspiritsaresymmetric.Skewnessofoutputgapisnotstatisticallydifferent

from0andanimalspiritsare0onaverage.Periodsofoptimismandpessimism

occurequallyfrequently.ThenumberoftimestheZLBishitislessthan10%and

themeanlengthofaZLB-spellhasdroppedto3.2quarters.

20

Figure2:Outputgapandanimalspiritsintimeandfrequencydomains(Inflationtarget=0%)

FrequencyZLB-hits=64%;meanZLB-spell=9.1quarters

Figure3:Outputgapandanimalspiritsintimeandfrequencydomains(Inflationtarget=4%)

FrequencyZLB-hits=9%;meanZLB-spell=3.2quarters

21

Inordertoobtainamorepreciseideaabouttherelationbetweeninflationtarget

and the asymmetry in the distribution of output gap and animal spirits we

computed theskewnessof thedistributionofoutputgapand themeananimal

spiritsfordifferentvaluesoftheleveloftheinflationtarget.Weshowtheresults

in Figures 4 and 5. From Figure 4 we conclude that as the inflation target

increasestheskewnessofthedistributionoftheoutputgapdeclines.Itreaches

valuescloseto0whentheinflationtargetis3%.Wenotethenon-linearrelation

between inflation target and skewness. With an inflation target equal to 2%

skewness is reduced substantially but there is still a significant amount of

skewness, suggesting that an inflation target of 2% may not be optimal. We

returntothequestionofoptimalityinthenextsection.

Figure5showstherelationbetweeninflationtargetandthemeananimalspirits.

Wefindthatwhentheinflationtargetincreasesthemeanvalueofanimalspirits

increases in a non-linear way. Put differently with increasing inflation target

(starting from 0%) endemic pessimism is reduced significantly. When the

inflation target reaches 3% animal spirits are zero on average, i.e. periods of

optimismandpessimismareequallyprobable.

Figure4: Figure5

How can these results be interpreted? When the inflation target is 0% the

cyclicalmovementsinoutputgapandanimalspiritsinevitablyleadtorecessions

that also drive inflation into negative territory. When that happens the zero

22

boundconstraintthatappliestothenominal interestratesmakesit impossible

forthecentralbankto lowerthereal interestrate. If therecessionisdeepand

deflationintensetherealinterestrateislikelytoincreasesignificantly.Thusthe

recession becomes protracted. Pessimism sets in and amplifies the recession,

and validates pessimism. As the central bank loses its stabilizing capacity the

economygetsstuckinpessimism,recessionanddeflation.Weconcludethatan

inflationtargetof0%becomesabreedinggroundforpessimismandrecession.

The way out is to increase the inflation target. Our results suggest that an

inflation target of 3%-4% is probably better than 2% inmaking sure that the

economydoesnotgetstuckinthechronicpessimismtrap.

Finallywealsolookedatthenumberoftimesthezerolowerboundwashitfor

different levels of the inflation target.We show the results inFigure6.On the

horizontalaxisweshowtheinflationtarget;ontheverticalaxiswepresentthe

number of ZLB-hits (out of 10000 simulations). We observe a non-linear

relationship:whentheinflationtargetis0%thenumberoftimestheZLBishit

exceeds 60%. The number of hits then declines when the inflation target is

raised.Notethatwhentheinflationtargetis2%wehittheZLBabout26%ofthe

time.InthatcasewefindthatthetypicallengthofaZLB-episodeis5quarters.

Figure6

inflation target0 0.5 1 1.5 2 2.5 3 3.5 4

num

ber

of Z

LB

s

0

1000

2000

3000

4000

5000

6000

7000Number of ZLBs and inflation target

23

3.2Theroleofanimalspirits

Whatistheimportanceofanimalspiritsincreatingskewnessintheoutputgap?

The way we want to answer this question is to compare the results of the

behavioral model with the three-equations New Keynesian model (aggregate

demand, aggregate supply andTaylor rule) solvedunder rational expectations

(RE).Weknowthat inaRE-modelanimalspiritsplaynorole.Acomparisonof

our behavioral model where animal spirits play a central role with the same

structuralmodelunderrationalexpectationswillallowusto isolatetheroleof

animalspirits.

Weusedthemodelconsistingofequations(1a)to(3a)andsolvedtheseunder

rationalexpectationsimposingazerolowerboundonthenominalinterestrate.

Weassumedthesameparametersandthesamedistributionofshocksasinthe

behavioralmodel.We simulated themodel for different levels of the inflation

target (from 0% to 4%). We show the results in Figure 7. This presents the

simulated output gaps in the frequency domain. We observe that with an

inflation target of 0% the distribution has a negative skewness (-0.37). This

skewness tends to decline as the inflation target is raised.When the inflation

target is 4%, the skewness in the distribution of the output gap is not

significantly different from zero. When comparing these results with those

obtained in the behavioral model (Figures 1 to 3) we find that the degree of

skewnessismuchlesspronouncedintherationalexpectationsmodelthaninthe

behavioralmodel.

WealsocomputedthenumberofZLB-hitsandthemeanlengthoftimeofZLB-

episodesintheRE-model.TheseareshownbelowthechartsofFigure7.Withan

inflation target of 0% the number of ZLB-hits is 47% (versus 64% in the

behavioralmodel). It declines sharplywhen the inflation target increases (8%

ZLB-hitswhen inflation target=2% (versus 26% in the behavioralmodel), and

0.5%ZLB-hitswhen inflation target=4% (versus9% inbehavioralmodel).We

find that in each of these cases the number of ZLB-hits is lower than in the

behavioralmodel(seeFigure6).

This difference between the RE-model and the behavioral model is equally

pronouncedforthemeanlengthofZLB-episodes.IntheRE-model,theseare3.1,

24

1.6 and 1.3 quarters compared to 9.1, 4.8, and 3.2 quarters in the behavioral

model,forinflationtargetsincreasingfrom0%to4%.Theseresultsleadtothe

conclusion thatby amplifying themovementsof theoutput gap, animal spirits

push the output gapmore frequently and keep it longer in negative territory

than in a RE-model. As a result, animal spiritis produce more protracted

economic downturns when the inflation target is set too low than what is

obtainedinaRE-model.

Figure7:Frequencydistributionofoutputgapinrationalexpectationsmodel

Inflationtarget=0% Inflationtarget=2%

Skewness=-0.37 Skewness=-0.20ZLB=47%;meanZLB-spell=3.1 ZLB=8%;meanZLB-spell=1.6 Inflationtarget=4%

Skewness=0.01ZLB=0.5%;meanZLB-spell=1.3

-6 -4 -2 0 2 4 60

20

40

60

80

100

120

140histogram output gap

-6 -4 -2 0 2 4 60

20

40

60

80

100

120histogram output gap

-6 -4 -2 0 2 4 60

20

40

60

80

100

120

140histogram output gap

25

4.Impulseresponsestodemandandsupplyshocks

In this section we compute impulse responses to demand and supply shocks

assuming different levels of inflation target. We will assume two different

inflation targets 0% and 4% (the 2% inflation target regime is intermediate

between these two extremes). These impulse responses are expressed as

“multipliers”, i.e. the output responses to the shock are divided by the shock

itself (which is two standarddeviations of the error terms in thedemand and

supplyequations).

Incontrasttolinearrationalexpectationsmodelstheimpulseresponsesdepend

onthetimingoftheshock.Putdifferently,animpulseresponsecomputedwith

one realization of the stochastic shocks in the equations of themodel will be

different from an impulse response to exactly the same shock but performed

usinganotherrealizationofthesestochasticshocks.Thisisthecaseevenwhen

allparametersofthemodelareidentical.

In order to illustrate this we simulated 1000 impulse responses to the same

shock(demandrespectivelysupply),assumingeachtimeadifferentrealization

ofstochasticshocks.Wethencomputedthemeanresponseobtainedfromthese

1000 impulse responses together with the impulse responses two standard

deviationsaboveandbelowthemeanresponse.TheresultsareshowninFigures

8and9forthedemandshockandFigure10forthesupplyshock.Weshowthe

resultsfortwoinflationtargetregimes,0%and4%.

4.1Impulseresponsestoanegativedemandshock

Wefirstdiscuss the impulseresponses to thenegativedemandshock5. (Figure

8).Tworesultsstandout.First,thereisgreatuncertaintyaboutthetransmission

ofashockeveniftheparametersofthemodelareknownwithcertainty,asthis

transmission depends on “initial conditions”, i.e. on the configuration of

stochasticdisturbancesthatprevailatthemomentthedemandshockoccurs.For

example, the effect of the demand shock may be very different depending on

whethertheshockoccursduringarecession,aboomorinmorenormalbusiness

cycle conditions. Thus, in ourmodel the timing of the shockmatters.We also

note that this uncertainty is more pronounced in the 0% inflation targeting

regimethaninthe4%regime.

26

Figure8:Impulseresponsestonegativedemandshock

Second, after the initial negativedemand shock, the output gap recoversmore

quicklyandanimalspiritsbecomeneutralfasterwhentheinflationtargetis4%

thanwhenitis0%.Weobservethatinthe4%inflationtargetregime,theoutput

gapreacheszeroafter8quarters,whileittakes12quartersfortheoutputgapto

reach zero in the0% inflation target regime.For theanimal spirits it takes15

quarters tobecomeneutral in the latter regimeas compared to10quarters in

the4%inflation targetregime.Thus,whenthe inflation target issetat4%the

economyrecoversfasterfromanegativedemandshockthanwhentheinflation

targetissetatthelowlevelof0%.Thisdifferenceisofcourseassociatedwith

thefactthat ina4%inflationtargetregimethereismorescopeforthecentral

Time90 100 110 120 130 140 150 160

Level

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

impulse responses output to negative demand shock(inflation target = 0%)

Time90 100 110 120 130 140 150 160

Level

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

impulse responses output to negative demand shock(inflation target = 4%)

Time90 100 110 120 130 140 150 160

Level

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

mean impulse response animal spirits to negative demand shock(inflation target = 0%)

Time90 100 110 120 130 140 150 160

Level

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

mean impulse response animal spirits to negative demand shock(inflation target = 4%)

Time90 100 110 120 130 140 150 160

Level

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

impulse responses interest rate to negative demand shock(inflation target = 0%)

Time90 100 110 120 130 140 150 160

Level

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

impulse responses interest rate to negative demand shock(inflation target = 4%)

27

bank to stabilize thebusiness cycleby lowering the interest rate (see last two

chartsinFigure8).

It will be useful to focus on the short-term impulse responses of the demand

shock. Thewaywe do this is by presenting the frequency distributions of the

short-term effects of the demand shock. These frequency distributions are

obtainedbycollectingtheimpulseresponsesobtainedinthe4thquarterafterthe

shockoccurred.Insodoingweobtained1000short-termoutputresponsesand

1000 short-term responses of animal spirits. We plot these in the frequency

domain.TheresultsareshowninFigure9.

Themoststrikingresult inFigure9 is the fact thatwhenthe inflation target is

high(4%)thenegative impactonoutput followinganegativedemandshock is

significantlylower,onaverage,thanwhentheinflationtargetislow(0%).Atthe

same time, the short-term responses in animal spirits are on average more

negative with a low inflation target than with a high one. But as mentioned

earlier, there is awide variation in the short-term effect of the same demand

shockonoutputandanimalspirits.Thisvariationtendstobehigherwhenthe

inflationtargetislow.

Focusing on the short-term interest rate responses, we find that when the

inflationtargetiszero,theinterestrateresponsetothenegativedemandshock

iszero inmorethanhalfof thecases(1000replicationsof thesameshock). In

otherwordswhenthe inflationtarget iszerothenegativedemandshock leads

theinterestratetohittheZLBinmorethanhalfofthecases.Whentheinflation

targetis4%thenumberofZLB-hitsalmostcompletelydisappears.

Themechanismthatdrivestheseresults is thesameastheoneweunveiled in

theprevioussections.Withazeroinflationtargetthereislittlescopeforoutput

stabilizationfollowinganegativedemandshockbecausetheinterestrateisvery

likelytohittheZLB.Asaresult,thisshockwillhaveastrongernegativeoutput

effect in a low inflation target environment as comparedwith a high inflation

targetenvironment.Animalspiritsamplifythesedifferences.Whentheinflation

target=0%thenegativedemandshocktogetherwiththeinabilityofthecentral

bankstostimulateoutputbyreducingtheinterestratecreatesmorepessimism

thanwhentheinflationtarget=4%(seefigure8).Inthelattercasethecentral

28

bank is capable of stabilizing output. This in turn reduces pessimism and

reinforcesthestabilizationeffortofthecentralbank.

Figure 9: Frequency distribution of short-term responses to negative

demandshock

-1 -0.8 -0.6 -0.4 -0.2 00

10

20

30

40

50

60

70

80

histogram short term output response to negative demand shock(inflation target = 0%)

-1 -0.8 -0.6 -0.4 -0.2 00

10

20

30

40

50

60

70

80

histogram short term output response to negative demand shock(inflation targt = 4%)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.40

10

20

30

40

50

60

70

80

90

100

110

histogram short term animal spirits response to negative demand shock(inflation target = 0%)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.40

10

20

30

40

50

60

70

80

90

100

110

histogram short term animal spirits response to negative demand shock(inflation target = 4%)

-1.2 -1 -0.8 -0.6 -0.4 -0.2 00

100

200

300

400

500

600

histogram short term interest rate response to negative demand shock(inflation target = 0%)

-1.2 -1 -0.8 -0.6 -0.4 -0.2 00

10

20

30

40

50

60

70

80

histogram short term interest rate response to negative demand shock(inflation target = 4%)

29

4.2Impulseresponsestoapositivesupplyshock

Inthissectionweanalyzetheimpulseresponsestoapositivesupplyshock6.The

positive supply shock can be interpreted as a positive shock in productivity,

shifting the supply curve downwards and producing a decline in the rate of

inflation.Aninflationtargetingcentralbankwillreacttothisdeclineintherate

of inflation by lowering the interest rate. The capacity to do so, however, is

limitediftheinflationtargetislow(0%).Itincreaseswhentheinflationtargetis

raised.TheeffectsareshowninFigure10.

Weobserve that in a regimeofhigh inflation targeting (4%) the impactof the

positivesupplyshockontheoutputgapishigherthaninthecaseoflowinflation

target (0%). Similarly, animal spirits become more positive after the supply

shockinthehighinflationtargetthaninthelowinflationtargetregime.

Theunderlyingmechanismthatdrivesthesedifferencesisthedifferentinterest

rateresponsestothepositivesupplyshock.Inthehighinflationtargetingregime

thecentralbankreactsbyloweringtheinterestratetherebyfuelingthesupply

shock.Inthelowinflationtargetingregimethecentralbankisconstrainedbythe

ZLB and is prevented from fueling the supply shock by an expansionary

monetarypolicy.

Thetwoinflationtargetregimesalsodifferinthespeedwithwhichthesystem

returnstothesteadystate. Inthehighinflationtargetregimethereturntothe

steadystateisfasterthaninthelowinflationtargetregime.Thishastodowith

the fact that the central bank reverses its interest rate policy quicker in the

former than in the latter case. This then also has the effect of leading animal

spiritsoutoftheir“euphoric”statefasterinthehighinflationascomparedtothe

lowinflationtargetregime.Itfollowsthatinthehighinflationtargetregimethe

capacity of the central bank to steer the economy towards its steady state is

better than in the low inflation target regime when a positive supply shock

occurs.Thisisasimilarconclusionastheonederivedintheprevioussection.

30

Figure10:Impulseresponsestopositivesupplyshock

5.CredibilityofinflationtargetingandthezerolowerboundAnobjectiontotheideathatcentralbanksshouldadoptahigherinflationtarget

is that this would negatively affect their credibility (see e.g. Branch and

Evans(2014)).Weanalyzethisquestionofcredibilityinthissection.

Ourmodelallowsustogiveaprecisedefinitionofthecredibilityoftheinflation

target. This can be defined by the fraction of agents who use the announced

inflationtargetastheirforecastforfutureinflation.Wehavecalledtheseagents

Time90 100 110 120 130 140 150 160

Level

-1

-0.5

0

0.5

1

1.5

2

impulse responses output to positive supply shock(inflation target = 0%)

Time90 100 110 120 130 140 150 160

Level

-1

-0.5

0

0.5

1

1.5

2

impulse responses output to positive supply shock(inflation target = 4%)

Time90 100 110 120 130 140 150 160

Level

-1

-0.5

0

0.5

1

1.5

mean impulse response animal spirits to positive supply shock(inflation target = 0%)

Time90 100 110 120 130 140 150 160

Level

-1

-0.5

0

0.5

1

1.5

impulse response animal spirits to positive supply shock(inflation target = 4%)

Time90 100 110 120 130 140 150 160

Level

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

impulse responses interest rate to positive supply shock(inflation target = 0%)

Time90 100 110 120 130 140 150 160

Level

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

impulse responses interest rate to positive supply shock(inflation target = 4%)

31

the “targeters”. Since these agents use the announced inflation target as their

inflation forecast it can be said that they trust the central bank’s inflation

commitment.Incontrast,theextrapolatorsdonottrustthecentralbank.

Weusedthisinsighttocomputeanindexofinflationcredibility,whichwedefine

to be the fraction of “targeters”,𝛽!"#,! as shown in equation (19). We then

computedthisindexfordifferentvaluesoftheTayloroutputparameterandthe

inflationtarget.WeshowtheresultinFigure11.

Two results stand out. First, when the central bank increases its stabilization

effort(theTaylorparameterincreases)thishastheeffectoffirstincreasingthe

inflation credibility of the central bank. When the Taylor output parameter

reachesavalueofapproximately0.5furtherstabilizationeffortsleadtoadecline

in inflation credibility. This result can be given the following interpretation.

Whenthecentralbankstartsincreasingitsstabilizationefforts,thecentralbank

also reduces the amplitude of the waves of optimism and pessimism (animal

spirits) therebystabilizingnotonlyoutputbutalso inflation.This increases its

inflationcredibility.When thecentralbankuses toomuchoutput stabilization,

however, it undermines its inflation credibility and as a result the credibility

decreases.Note that theempirical literature reveals that centralbanks tend to

set the Taylor output parameter close to 0.5. We discuss this result in more

detailsinDeGrauwe(2012).

Second,anincreaseintheinflationtargethastheeffectofshiftingthecredibility

linesupwards,i.e.whenthecentralbankincreasesitsinflationtargetfrom0%to

4%itscredibilityinfightinginflationincreasesforallvaluesoftheTayloroutput

parameter. Put differently, by increasing the inflation target the central bank

improvesitsinflationcredibilityregardlessofwhetherthecentralbankapplies

little or much output stabilization. This result can be given the following

interpretation.When the inflation target is set too low, the rate of inflation is

morelikelytobepushedintonegativeterritory.Thisistheterritoryinwhichthe

centralbanklosesitscapacitytoinfluencebothinflationandoutputbyvarying

the interestrate.Asresult, itwill frequentlyfail toreachits inflationtarget.By

raising the inflation target it reduces the frequency of hitting the zero lower

bound. As a result, it maintains its capacity to affect inflation and output by

32

varying the interest rate. This is illustrated in Figure 12, which shows the

number of times (in simulations of 1000 periods) the ZLB is hit for different

valuesoftheTayloroutputparameterandlevelsofinflationtarget.Weobserve

thatraisingtheintensityofoutputstabilizationreducesthenumberoftimesthe

ZLB is hit for all levels of inflation target. Similarly, raising the inflation target

reduces the frequency of hitting the ZLB for all values of the Taylor output

parameter.

Figure11:Credibilityandinflationtargets

Figure12:NumberofZLBperiods,stabilizationandinflationtargets

33

6.EmpiricalverificationInthissectionweprovideempiricalverificationofsomeofthepredictionsmade

byourbehavioralmodel.Wefocusonthreepredictions.

6.1Excesskurtosisandfattailsinthedistributionoftheoutputgap

Ourmodelpredicts that thedistributionof theoutputgap isnon-normal, i.e. it

exhibitsexcesskurtosisandfattails.Thelatteraregeneratedwhentheeconomy

is gripped by extreme optimism or pessimism, leading to large positive or

negative movements in output. This feature is also what drives most of our

results.Theexistenceofnon-normalityinthedistributionoftheoutputgap(and

output growth) has been confirmed empirically formost OECD countries (see

Fagiolo,etal.,(2008),Fagiolo,etal.,(2009),DeGrauweandJi(2016)).Ascariet

al.(2015)findthatRBCandNKmodelscannotgeneratethefattailsobservedin

thedata(whichourpapercan).

Onecouldobjecttothisempiricalevidencethatthelargeshocksobservedinthe

output gaps can also be the result of large exogenous shocks (see Fernandez-

Villaverde andRubio-Ramirez(2007) and Justiniano andPrimiceri(2008)). The

claimthatismadehereisnotthattheeconomycannotsometimesbehitbylarge

shocks(itoftenis),butthatatheorythatcanexplainlargemovementsinoutput

gaps only as a result of large exogenous shocks is an incomplete one. This

createsanopeningforatheorylikeoursthatcanexplainlargemovementsinthe

outputgap(fattails)endogenously.

6.2Two-waycausalitybetweenanimalspiritsandoutputgap

Inourtheoreticalmodel,thereexistsatwo-waycausalitybetweenanimalspirits

and the output gap, i.e. positive (negative) animal spirits produce a positive

(negative) output gap; conversely, a positive (negative) output gap leads to

positive(negative)animalspirits.Thisisinfactakeyfeatureofourtheoretical

model, which produces a self-reinforcingmechanism that leads to booms and

busts,characterizedbyextremeoptimismandpessimism.

34

We illustrate this feature inTable3.WeappliedGrangercausality testsonour

simulated output gap (Y) and animal spirits (ANSPIRITS) obtained from

simulatingour theoreticalmodel.We find thatwecannotreject thehypothesis

thatinourmodeltheoutputgapGrangercausesanimalspiritsandviceversa.

Table3:PairwiseGrangerCausalityTests(inflationtarget4%)

Lag1(observation=1998) Lag2(observation=1997)

NullHypothesis: F-Statistic Prob. F-Statistic Prob.

YdoesnotGrangerCauseANSPIRITS

681.507 0.0000 162.508 0.0000

ANSPIRITSdoesnotGrangerCauseY

134.629 0.0000 50.0886 0.0000

In order to test this two-way causality between animal spirits and the output

gap,wehave to findanempirical counterpartof animal spirits.Wedecided to

usethebusinessconfidenceindex(BCI)asproducedbytheOECDasanindicator

fortheanimalspirits.TheBCIisbasedonenterprises'assessmentofproduction,

orders and stocks, as well as its current position and expectations for the

immediate future. This is very similar to what our index of animal spirits St

measures.Aswillberemembered,whenStispositiveitmeansthatthefraction

ofagentswithapositiveoutlookaboutthefutureoutputgapis largerthanthe

fraction of agentswith a negative outlook. This is alsowhat theBCImeasures

when it questions participants about their sentiments (forecasts) about future

businessconditions.

The BCI has been rescaled to yield a long-term average of 100. Themore the

index exceeds 100, themore optimistic (positive animal spirits) it shows. The

more the index is below100, themore pessimistic (negative animal spirits) it

shows.We performedGranger causality tests between theBCI and the output

gap for theEurozoneand for theUSduring theperiod1999-2015.The results

areshowninTable4.

35

Table4:GrangercausalitytestsBusinessConfidenceIndex(BCI)andoutputgap(Sampleperiod:1999Q1-2015Q4)

Lag=1(obs=71) Lag=2(obs=70) NullHypothesis Ftest Prob. Ftest Prob.U.S. Outputgapdoesnot

grangercauseBCI6.4923 0.0131 11.542 0.0011

BCIdoesnotgrangercauseoutputgap

23.113 0 23.683 0

Eurozone OutputgapdoesnotgrangercauseBCI

6.9791 0.0102 6.4346 0.0135

BCIdoesnotgrangercauseoutputgap

25.51 0 12.148 0.0009

Note:theDickey–FullertestsrejectunitrootinBCIandoutputgapDataresources:BCIisfromOECD,outputgapisfromoxfordeconomics.Datafrequency:quarterly.TheBCIquarterlydataisaveragedfrommonthlydata.

Wefindthatwecannotrejectthehypothesisofatwo-waycausalitybetweenthe

outputgapandtheindicatorsofbusinessconfidence(BCI)intheEurozoneand

theUS.Thisconfirmsoneofthekeypredictionsofourmodel,i.e.thedynamicsof

boomsandbustsischaracterizedbyaprocessbywhichwavesofoptimismand

pessimismdrivethebusinesscycle,whilethelatteralsoinfluencesoptimismand

pessimism.

6.3ProbabilitiesofhittingtheZLB

WenotedintheintroductionthatstandardlinearDSGEmodelshavetendedto

underestimatetheprobabilityofhittingtheZLBaswasshownbyChung,etal.,

(2012).Most of thesemodels have led to theprediction thatwhen the central

bankkeeps an inflation targetof2%, it is veryunlikely for the economy tobe

pushed into the ZLB (Reifschneider and Williams (1999), Coenen(2003),

Schmitt-Grohe and Uribe(2007) ). Reifschneider andWilliams(1999) came to

the conclusion that “if monetary policy followed the prescriptions of the

standardTaylorrulewithaninflationtargetof2percent,thefederalfundsrate

would be near zero about 5 percent of the time (p.1). Our model came to a

prediction thatwhen centralbanks set an inflation targetof2%andusing the

sameTaylor rule, the probability of hitting the ZLB ismuch higher and in the

orderof20to30%.

Itisnoteasytoverifythesedifferentpredictionsempirically.Butwehavesome

suggestiveevidencefromtheperiodsincecentralbanksswitchedtoaninflation

36

target of 2%. In the Eurozone countries this happened in 1999 when the

European monetary union was started. The ECB then announced an inflation

targetof2%.IntheUSanexplicitinflationtargetof2%wasonlyintroducedin

2012by the thenChairman of the Federal Reserve, BernBernanke. It is clear,

however,thattheFederalReservepursuedanimplicitinflationtargetofcloseto

2%fromtheendofthe1990s.

InFigure13weshowthepolicyratesintheUSandtheEurozonesince1999.We

observethattheserateswereclosetozeroforaconsiderabletime.InfacttheUS

policyratewasclosetozero45%ofthetime,intheEurozonethiswas41%.This

isevenhigherthanthepredictionofourmodel.This isprobablyrelatedtothe

fact that the recession of 2008-09was deeper and longer-lasting than normal

recessionsasitfollowedafinancialcrisis(seeReinhartandRogoff(2009)).

Source:ECB,StatisticalWarehouseandBoardofGovernorsoftheUSFederalReserveSystemInFigure14,wealso show thepolicy ratesof othermajor centralbanks since

1999. These are the central banks of Switzerland, Sweden, the UK, Canada,

Norway,AustraliaandNewZealand.InTable5,weshowthattheprobabilityof

hitting the ZLB in countries such as Switzerland and Sweden is significantly

higher than in countries such as Canada, Norway, Australia and New Zealand.

NotethatinthecaseoftheUKtheinterestratewaskeptconstantatthelevelof

0.5%from2009to2016.TheBankofEnglandseemstohaveconsidered0.5%to

beaneffectiveZLB.

-101234567

1999-01-01

1999-09-01

2000-05-01

2001-01-01

2001-09-01

2002-05-01

2003-01-01

2003-09-01

2004-05-01

2005-01-01

2005-09-01

2006-05-01

2007-01-01

2007-09-01

2008-05-01

2009-01-01

2009-09-01

2010-05-01

2011-01-01

2011-09-01

2012-05-01

2013-01-01

2013-09-01

2014-05-01

2015-01-01

2015-09-01

2016-05-01

Figure13:AverageMonthlyUSandEurozoneofricialrates(%)

USfederalfundrate Eurozonereporate

37

A factor that leads to theobserveddifference in theprobabilitiesofhitting the

ZLBmayberelatedtotheleveloftheinflationtargetofacentralbank.Asshown

inTable5,Switzerland,SwedenandtheUKhavearelativelylowinflationtarget

(about2%)whileCanada,Norway,AustraliaandNewZealandhaveaninflation

targetwhichcan reach3%.Thisalsoconfirms thepredictionofourmodel, i.e.

thatcountriesusingahigherinflationtargetarelesslikelytohittheZLB.

Table5.ProbabilityofhittingZLBandInflationTarget

Inflationtarget ProbabilityofHittingZLB

US 1.7-2% 45%Eurozone belowbutcloseto2% 41%Switzerland <2% 30%Sweden 2% 11%UK 2% 2%

Canada 1-3% 6%Norway 2.50% 0%Australia 2-3% 0%

NewZealand 1-3% 0%Source:Inflationtargetratesareobtainedfromofficialwebsitesofthecentralbanks.TheinterestratesareobtainedfromIMFdataandtheprobabilitiesofhittingZLBarefromauthors’owncalculation.

-2

0

2

4

6

8

10

1999-01-01

1999-09-01

2000-05-01

2001-01-01

2001-09-01

2002-05-01

2003-01-01

2003-09-01

2004-05-01

2005-01-01

2005-09-01

2006-05-01

2007-01-01

2007-09-01

2008-05-01

2009-01-01

2009-09-01

2010-05-01

2011-01-01

2011-09-01

2012-05-01

2013-01-01

2013-09-01

2014-05-01

2015-01-01

2015-09-01

2016-05-01

Figure14:Policyrates(%)ofothermajorcentralbanks

Sweden Australia Switzerland Canada

Norway UK NewZealand

38

7.Conclusion

In this paperwe have analyzed the relation between the level of the inflation

target and the ZLB constraint imposed on the nominal interest rate. We

analyzedthisrelationintheframeworkofabehavioralmacroeconomicmodelin

which agents experience cognitive limitations, preventing them from forming

rationalexpectations.Thisforcesthemtousesimplerulesofthumbtoforecast

theoutputgapandtherateofinflation.Rationalityisintroducedintothemodel

by allowing agents to learn from their mistakes and to switch to the better

performing forecasting rules. The model produces endogenous waves of

optimism and pessimism (animal spirits) that, because of their self-fulfilling

nature,drivethebusinesscycleandinturnareinfluencedbythebusinesscycle.

Theuseofthisbehavioralmodelhasallowedustoshednewlightontheoptimal

level of the inflation target in aworldwhere a ZLB constraint on the nominal

interestrateexists.Wefoundthatwhentheinflationtargetistooclosetozero,

theeconomycangetgrippedby“chronicpessimism”thatleadstoadominance

ofnegativeoutputgapsandrecessions,and in turn feedsbackonexpectations

producinglongwavesofpessimism.Themechanismthatproducesthischronic

pessimismcanbedescribedasfollows.Endogenousmovementsinanimalspirits

regularly produce recessions and negative inflation rates.When that happens,

thecentralbankcannotuse its interestrate toboost theeconomyandtoraise

inflation as the nominal interest rate cannot become negative.When inflation

becomesnegativethisalsoimpliesthattherealinterestrateincreasesduringthe

recession,aggravatingthelatter,andincreasingpessimism.Theeconomycanget

stuckforaverylongtimeinthiscycleofpessimismandnegativeoutputgap.

Thus, when the inflation target is set too close to zero the distribution of the

outputgapisskewedtowardsthenegativeterritory.Thequestiontheniswhat

“too close to zero” means. The simulations of our model, using parameter

calibrations that are generally found in the literature, suggests that 2% is too

low, i.e. produces negative skewness in the distribution of the output gap.We

findthataninflationtargetintherangeof3%to4%comesclosertoproducinga

symmetricdistributionoftheoutputgap.

39

Wealsofoundthatwhentheeconomyispushedintoarecessionasaresultofa

negative demand shock, the high inflation target regime has better stabilizing

properties.We foundthat in thehigh inflation targetregimethepersistenceof

therecessionisshorterthaninthelowinflationtargetregime.Thatis,whenthe

centralbanksetsarelativelyhighinflationtarget,thecapacityofthesystemto

liftitselfoutoftherecessionisstrongerthanwhenitsetsalowinflationtarget.

This ismadepossibleby thestabilizingpropertiesofmonetarypoliciesandby

theensuingeliminationofself-fulfillingpessimism.

Another major finding of our analysis is that with an inflation target of 2%

(whichhasbecomethestandardininflationtargeting)theeconomyhitstheZLB

withaprobabilityof25%comparedtolessthan10%withaninflationtargetof

4%.Thesearesignificantlyhigherprobabilitiesthanthoseobtainedinstandard

DSGE models. The reason we find significantly higher probabilities has to do

withthefactthatanimalspiritsamplifyshocksandincombinationwiththeZLB

cankeeptheeconomyinapersistentwayintonegativeterritory.

Thepreviousresultsleadstotheconclusionthatcentralbanksshouldraisethe

inflationtargetfrom2%toarangebetween3%to4%(seealsoBlanchard,etal.

(2010)andBall(2014)onthis).Onemightobjectherethatthisconclusiondoes

not take into account the potential negative effect on inflation credibility of

raising the inflation target to 3% or 4%. We analyzed this question in the

framework of our behavioral model. Our model gives a precise definition of

credibility, as the fraction of agents that use the announced inflation target as

theirruleofthumbtoforecastinflation.Itturnsoutthataninflationtargetof3%

or4%hasmorecredibilitythanatargetof2%.Thereasonhastodowithwhat

wesaidearlier.Withan inflation targetof2%theoutputgapand inflationare

moreoftenpushedintonegativeterritorythanwhentheinflationtargetis3%or

4%.Onceinflationandoutputgapareinthenegativeterritorythepowerofthe

centralbanktoaffecttheoutputgapandinflationisweakened.Asaresult, the

observedinflationratewilldeviatemoreoftenfromthetarget,whenthetarget

islow,therebyunderminingthecredibilityofthecentralbank.

One issuethatwehavenotanalysed in thispaper ishowperiodsofprolonged

pessimismthatareproducedbyaninflationtargetthatissettoolowaffectslong

40

termgrowth. It isnotunreasonabletobelievethat“chronicpessimism” lowers

investmentinapersistentwaytherebyloweringlong-termgrowth.Aswehave

not incorporated these long-term growth effects in ourmodel, it is difficult to

cometopreciseconclusions.Weleavethisissueforfurtherresearch.

41

APPENDIXA:ForecastingrulesareAR(1)

InthisappendixweshowsomeresultswhenweimposeanAR(1)structureon

the forecasting rules. This does not affect our main results. We show typical

simulations in the time domain (using the same parameter values and

distributionoftheshocksasinthemaintext).Weobtainsomemorepersistence

inthebusinesscycles.

FigureA1:Outputgapinfrequencyandtimedomains(differentinflationtargets)

Inflationtarget=2%

Skewness=-0.32ZLB=41%

Inflationtarget=0%

skewness=-0.72ZLB=63%

-8 -6 -4 -2 0 2 4 6 80

20

40

60

80

100

120

140histogram output gap

Time1200 1250 1300 1350 1400

Level

-10

-8

-6

-4

-2

0

2

4output gap (inflation target=2%)

-8 -6 -4 -2 0 2 4 6 80

20

40

60

80

100

120

140

160

180

200histogram output gap (inflation target=0%)

Time1200 1250 1300 1350 1400

Level

-6

-5

-4

-3

-2

-1

0

1

2

3output gap (inflation target=0%)

42

APPENDIXB:Relativeperformanceofforecastingrulesandtheiruse

Inthisappendixwepresenttherelativeperformanceoftheextrapolativeversus

thefundamentalist forecastingrules(FigureA2).This isobtainedbytakingthe

difference in the utilities of the extrapolative and the fundamentalist rule. In

FiguresA3andA4weshowthefractionsoftheagentsusetheextrapolativeresp.

thefundamentalistrules.

FigureA2

FigureA3

FigureA4

Correlationrelativeperformanceandfractionextrapolators=0.78

Time1000 1050 1100 1150 1200

Level

-2

0

2

4

6

8

10

12Relative performance extrapolators - fundamentalists

Time1000 1050 1100 1150 1200

Level

0.4

0.5

0.6

0.7

0.8

0.9

1fraction extrapolators

Time1000 1050 1100 1150 1200

Level

0

0.1

0.2

0.3

0.4

0.5

0.6Fraction fundamentalists

43

APPENDIXC:ChoosingthestandarddeviationsofshocksIn this appendix we describe howwe selected the standard deviations in the

errorterms.Wefirstcollectedempiricaldataonthestandarddeviationsofthe

outputgapand inflation in theUSand theEurozone.Theseareshown in table

C1.

TableC1:Standarddeviationsofoutputgapandinflation(quarterlyobservations) Outputgap Inflation

U.S. Eurozone U.S. Eurozone

Sampleperiod:2000-2016 1,6 1,7 1,3 1,0Sampleperiod:1990-2016 1,7 -- 1,3 --

Source: Author’s own calculations using output gap from Oxford Economics and inflationfromtheUSBureauofLaborStatisticsandEurostat.

Wethensimulatedourmodelfordifferentstandarddeviationsoftheexogenous

shocks in inflation (while keeping the standard deviation of output shocks

constant)andcomputedthestandarddeviationsofthesimulatedoutputgapand

inflation. We did this consecutively for increasing standard deviations of the

outputshocks.Weassumedaninflationtargetof2%.Theresultsareshownin

FigureC1. This shows the standarddeviation of the simulated output gap and

inflation for different standard deviations in the inflation shocks (and for a

standard deviation of the output gap equal to 0.5). We observe that with a

standarddeviationof theshocks in inflationandofoutputgapof0.5wecome

veryclosetotheempiricallyobservedstandarddeviationsoftheoutputgapand

inflation. It is also interesting to observe that while we assume the same

standarddeviation in the shocksof output and inflation themodel produces a

significantlyhigherstandarddeviationoftheoutputgapthanofinflation.Thisis

alsoconfirmedempirically.Thus,wedonotneedtoassumethattheexogenous

shocksintheoutputgaparehigherthantheshocksininflationtoproducethis

result.

44

FigureC1:

std shock inflation0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

stan

dard

dev

iatio

n ou

tput

gap

0.5

1

1.5

2

2.5

3

3.5

4

4.5standard deviation output gap and shock inflation

std shock inflation0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

stan

dard

dev

iatio

n in

flatio

n

0

0.5

1

1.5

2

2.5

3standard deviation inflation and shock inflation

45

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1ItisnowstandardinDSGE-modelstouseapricingequationinwhichmarginalcosts enter on the right hand side. Such an equation is derived from profit2InDeGrauwe(2012)morecomplexrulesareused,e.g.itisassumedthatagentsdo not know the steady state output gapwith certainty and only have biasedestimates of it. This is also the approach taken by Hommes andLustenhouwer(2016).3Notethataccordingto(4)fundamentalistsexpectadeviationoftheoutputgapfromtheequilibriumtobecorrectedinoneperiod.WehaveexperimentedwithlaggedadjustmentsusinganAR(1)process.Thesedonotaffecttheresults inafundamentalsense.WeshowanddiscusstheresultsinAppendix.4Therearesomeattemptstoprovidemicro-foundationsofmodelswithagentsexperiencingcognitivelimitations,though.Seee.g.Kirman,(1992),DelliGatti,etal.(2005).ArecentattemptisprovidedbyGabaix(2014).SeealsoHommesandLustenhouwer(2015)whoderivemicrofoundationsofamodelsimilartotheoneusedhere,butassumingquitestrongcognitivecapacitiesofagents.Wehavenotpursuedthishere.5Wedonotshow impulseresponses topositivedemandshocks.Thereason isthat when positive shocks occur the central bank, that wishes to raise theinterest rate, isnot constrainedby theZLB.Asa result, impulse responsesareverysimilarfordifferentlevelsoftheinflationtargets.6Weshowonlytheimpulseresponsesofpositivesupplyshocks,becauseonlyunderthisshockdoestheZLBconstraintbecomeeffective.