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Please do not quote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.
( ))exp()exp(
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,,
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ttarttar UU
Uγγ
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texttext UU
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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.