❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s
❊s❝♦❧❛ ❞❡
Pós✲●r❛❞✉❛çã♦
❡♠ ❊❝♦♥♦♠✐❛
❞❛ ❋✉♥❞❛çã♦
●❡t✉❧✐♦ ❱❛r❣❛s
◆
◦
✹✷✸
■❙❙◆ ✵✶✵✹✲✽✾✶✵
❆ ◆♦t❡ ♦♥ ▲❡❛r♥✐♥❣ ❈❤❛♦t✐❝ ❙✉♥s♣♦t ❊q✉✐✲
❧✐❜r✐✉♠
❆❧♦ís✐♦ P❡ss♦❛ ❞❡ ❆r❛ú❥♦✱ ❲✐❧❢r❡❞♦ ▲✳ ▼❛❧❞♦♥❛❞♦
▼❛✐♦ ❞❡ ✷✵✵✶
❖s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s sã♦ ❞❡ ✐♥t❡✐r❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ❛✉t♦r❡s✳ ❆s
♦♣✐♥✐õ❡s ♥❡❧❡s ❡♠✐t✐❞❛s ♥ã♦ ❡①♣r✐♠❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛
❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s✳
❊❙❈❖▲❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❈❖◆❖▼■❆
❉✐r❡t♦r ●❡r❛❧✿ ❘❡♥❛t♦ ❋r❛❣❡❧❧✐ ❈❛r❞♦s♦
❉✐r❡t♦r ❞❡ ❊♥s✐♥♦✿ ▲✉✐s ❍❡♥r✐q✉❡ ❇❡rt♦❧✐♥♦ ❇r❛✐❞♦
❉✐r❡t♦r ❞❡ P❡sq✉✐s❛✿ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r
❉✐r❡t♦r ❞❡ P✉❜❧✐❝❛çõ❡s ❈✐❡♥tí✜❝❛s✿ ❘✐❝❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❈❛✈❛❧❝❛♥t✐
P❡ss♦❛ ❞❡ ❆r❛ú❥♦✱ ❆❧♦ís✐♦
❆ ◆♦t❡ ♦♥ ▲❡❛r♥✐♥❣ ❈❤❛♦t✐❝ ❙✉♥s♣♦t ❊q✉✐❧✐❜r✐✉♠✴
❆❧♦ís✐♦ P❡ss♦❛ ❞❡ ❆r❛ú❥♦✱ ❲✐❧❢r❡❞♦ ▲✳ ▼❛❧❞♦♥❛❞♦ ✕ ❘✐♦ ❞❡ ❏❛♥❡✐r♦
✿ ❋●❱✱❊P●❊✱ ✷✵✶✵
✭❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s❀ ✹✷✸✮
■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳
sunspot equilibrium
AloisioP. Araujo 1
and Wilfredo L. Maldonado 2
1
EPGE/FGVand IMPA,Caixa Postal 34021 Jardim Bot^anio,
CEP 22470-050 , RJ, BRASIL
2
DepartamentodeEonomia,UniversidadeFederalFluminense,RuaTiradentes17,Niteroi,
CEP 24210-510, RJ, BRASIL
Summary. In this paper we prove onvergene to haoti sunspot equilibrium through
two learning rules used in the bounded rationality literature. The rst one shows the
onvergene ofthe atualdynamis generatedbysimple adaptive learningrulesto a
prob-abilitydistributionthatislosetothestationarymeasureofthesunspotequilibrium;sine
this stationary measure is absolutely ontinuous it results in a robust onvergene to the
stohasti equilibrium. The seond one is based on the E-stability riterion for testing
stability of rational expetations equilibrium, we show that the onditional probability
distribution dened by the sunspot equilibrium is expetational stable under a
reason-able updating rule of this parameter. We also report some numerial simulations of the
proesses proposed.
JEL Classiation Numbers: C61, E32
1. Introdution
Convergene of the dynamis generated by learning proesses is the subjet of many
papers in the intertemporal eonomyliterature (Chatterji (1995), Evans and Honkapohja
(1994,1995), Grandmont (1998), Guesnerie and Woodford (1991) and Woodford (1990)).
They developed methods (adaptive, reursive, Bayesian) for learning dierent types of
equilibria (steady states, yles and sunspots equilibria) whih are based on the feedbak
that the learning rule generates in the strutural equations. In this note we show two
types of onvergene to a sunspot equilibrium having an absolutely ontinuous invariant
stationary sunspot equilibrium, namely the reurrent behaviour of the atual dynamis
denes a probability distribution whih is lose to the stationary measure of the sunspot
equilibrium; in this sense we an say that the learning rule \onverges" to the sunspot
equilibrium. The seond type of onvergene results from the expetational stability of
the onditional probabilities of the sunspot equilibrium. The expetational stability of a
rationalexpetationsequilibrium(EvansandHonkapohja(1995),Guesnerie(1993))isused
as a seletion riterion of that type of equilibria. Speially, this onept haraterizes
the stability of some parameters whih dene the rational expetation equilibrium under
instantaneous orretions by some spei rule and the strutural equations.
The types of sunspot equilibria we will onsider in this work were proved to exist in
AraujoandMaldonado(2000)andtheytypiallyexistwhenthebakwardperfetprevision
map exhibits ergodi haos (see Majumdar and Mitra (1994)) it means that there exists
an absolutely ontinuous invariant and ergodi measure for it. Sometimes this measure is
alled \empirial", \observable" or \physial" (de Melo and van Strien (1993)). Here we
willdesribe themainresultsontheexistene ofthesetypesofsunspotequilibria(that we
willallhaotisunspotequilibria)andshowhowtheinvariantmeasuresanbeestimated.
Woodford(1990)proposedanotherlearningrulebasedonthealgorithmgivenbyLjung
and Soderstrom (1983); with this learning rule he obtained onvergene to the (nite)
support of the onsidered SE from any initial state lose to this support, so agents an
learn the (support of the) sunspot. In this paper we will prove that for almost all initial
state in a large interval: i) the agents an learn the stationaryprobability measure of the
sunspotwhentheyusesimpleadaptivelearningrulesandii)theyanlearntheonditional
probabilitydistributionofthatequilibriumifupdatingisinnotionalorvirtualtime. Inthis
sensethispaperwillprovidea theoretialfundationfor onsideringahaotideterministi
equilibriumasbeingastohastione(Hommes,vanStrienanddeVilder (1994),deVilder
(1996)).
This paper is divided in seven setions. In setion 2 we present the general framework
andshowtheonditionsfor existeneof haotisunspotequilibrium. In setion3wemake
a review of one-dimensional dynamial systems for unimodal maps whih will be used in
the next setion. In setion 4 we show OLG models with the type of sunspot equilibrium
onstruted insetion 2. In setion 5 we show a learning proess that generatesan atual
omplex dynamis whih mimis the SSE and in setion 6 we prove the expetational
stability of the onditional probabilities of the haoti sunspot equilibrium. Conlusions
are given in setion 7and the proofs are ontained in the appendix.
2. Chaoti sunspot equilibrium
Let X < n
be the state variable set and B(X) denote the Borel sets of X. The
equilibrium ondition of our model is represented by the zeros of the funtion:
~
Z : X P(X) ! < n
;
where P(X) = f : B(X) ! [0;1℄= is a probabilitymeasureg. We will all this map
a stohasti exess demand funtion beause in some models ~ Z(x
0
0
state variableis . Most of theoverlapping generations(OLG) models have thisstruture
and we are allowingfor no perfet prevision in general.
Thedeterministiexess demand funtion is: Z : XX ! < n
dened byZ(x 0 ;x 1 )= ~ Z(x 0 ;Æ x1 ).
In models that admit a representative agent there exists some type of linearity in the
rationalizing measures in the sense that they form a onvex set. For these ases we have
the next
(CVR)Property: Thestohastiexessdemandfuntion
~
Z hastheonvexvaluednessof
rationalizing measures (CVR) property if 8x 2X; 8 1
; 2
2P(X) suh that ~ Z(x; 1 )= ~ Z(x; 2
)=0 and 82[0;1℄we have ~
Z(x;
1
+(1 )
2 ) =0.
Sinewe will look for Markovian equilibria, let us remember that a transition funtion
dened on X is a funtion Q : X B(X) ! [0;1℄ suh that: i) 8x 2 X Q(x;:) 2 P(X)
and ii) 8A2B(X) Q(:;A)is a measurable funtion.
Denition 2.1.- Asunspot equilibrium (SE) is a pair (X 0
;Q) where X 0
X and Q is a
transition funtion on X 0
suh that:
i)9x 0
2X
0
suh that Q(x 0
;:) is not a Dira measure(it is truly stohasti).
ii)8x 2X 0
~
Z(x;Q(x;:))=0.
We are following the Chiappori and Guesnerie (1991) struture. They ompare this
denitionwith thestandard versionof the sunspot equilibrium onept. Woodford (1986)
presents another form for the sunspot equilibrium sine his exess demand depends on
\theories" fromthe total historyof the extrinsis.
Denition 2.2.- Asunspot equilibrium (X 0
;Q) is alled stationary (SSE) if there exists
2P(X) with support equal to X 0 suh that (A)= Z X 0
Q(x;A)(dx) 8A2B(X
0 ):
So, if a SE is stationary then the stohasti proess generated by the measure and
the transition funtion Q is a stationaryMarkov proess.
In orderto makethe onetionbeetween theexistene of SSEwith apositiveLebesgue
measuresupportandthehaotiityofthedeterministieonomy,wewillgivethefollowing
denition.
Denition 2.3.- A bakward perfet foresight (bpf) map is a funtion : X ! X suh
that: Z((x);x)=0 ; 8x2X.
This denition was also used by Grandmont (1986). It is easy to see that if (x t
) t0
is
a sequene suh that x t
= (x
t+1
) 8t 0 then it is a perfet foresight equilibrium. The
followingtheorem shows thatif the bpf map hasergodihaos then thereexistsa SSE for
the eonomy.
Theorem2.4. Supposethat ~
Z has theCVR property and thebpf map :X !X (where
X =[0;a℄)isaunimodalmapwith(0)=0. LetX 1
(X 2
)be theinterval whereisstritly
inreasing (dereasing) and suppose that the maps f : X ! X 1
and g : X ! X 2
C loal inverses of . If 9 2 P[0;1℄; << suh that is -invariant, then we have
that the following proess is a SSE:
Q(x;A)=
dÆf
d (x)Æ
f(x) (A)+
dÆg
d (x)Æ
g(x)
(A) ;x2supp():
The proof of theorem 2.4 in a more general setting is given in Araujo and Maldonado
(2000)butwewillprovethisversionintheappendixbeausewewilluseitinthefollowing
setions.
Remarks: If the measure is ergodi, the Birkho theoremimplies:
N = 1 N N 1 X n=0 Æ n (x) ! N!1
; a:e:x;
(the onvergene is in the weak topology). Therefore with the bakward perfet foresight
trajetory from -almost initial state we an obtain the histograms orresponding to the
measures
N
(the empirial measures) and they will approximate the density
(Radon-Nikodym derivative) of with respet to the Lebesgue measure (when << ). These
histogramsareonstrutedbytakingpartitions(I i
) 0in
,so anapproximationof(I i )is: N (I i )=
#fj; x j
2I i
; 0j N g
N +1
:
3. Bowen-Ruelle-Sinai measures for unimodal maps
This setion is devoted to desribe in whih ases a unimodal map has an absolutely
ontinuous invariant measure. Let X =[0;a℄ be a non-trivial interval.
A map : X ! X is alled unimodal if has a unique interior loal (maximum)
extremum.
If is the loal extremumof a unimodal map, we will all itnon-at if there existsa
C 2
loal dieomorphism h suh thath()=0 and (x)=()jh(x)j
, for some 2.
For example, if is C 1
with some derivative non-zero at then is a non-at ritial
point.
TheShwarzian derivative of 2C 3
is dened by:
S(x)= 000 (x) 0 (x) 3 2 ( 00 (x) 0 (x) ) 2 ; if 0
(x) 6=0:
Let us onsider thefollowing set of funtions:
F =f2C
3
(X); is a unimodalmap with non-at extremum;S<0; (0)=0; 0
(0)>1g
Givena funtion :X !X, we dene the !-limit setof theorbit from x2X as:
!(x)=fy 2X;there existsa subsequene n i
!1with
n i
(x)!yg:
In other words it is the set of aumulation points of the sequene f n
(x)g n0
. The
next theoremharaterizesthe !-limitset of Lebesgue almostall pointsof the interval. It
a:e:x. Moreover A is either a nite union of intervals or has Lebesgue measure zero.
Furthermore, if has an attrating periodi orbit then Ais this periodi orbit.
Asnoted in setion 2, if the measure is ergodi we an use the bakward trajetory to
estimate the stationary distribution. We will in fat workwith the onept of a \visible"
measure (also alled a Bowen-Ruelle-Sinai measure) where the bakward trajetory an
be used to estimate the stationary distribution from initial values in some set of positive
Lebesgue measure.
Denition 3.2.- Let : X !X and a -invariant probabilitymeasure. We say that
is a Bowen-Ruelle-Sinai (B-R-S) measure if there exists B X with (B) >0 suh that
for all x 2B:
1
N
N 1
X
n=0 Æ
n
(x) ! N!1
in the weak topology
i.e. for all F 2C 0
(X): 1 N
P
N 1
n=0
F(
n
(x)) !
N!1 R
X
F(x)(dx).
Thedierene between ergodi and B-R-Smeasures is that for the former theBirkho
theoremholdsjustfor theelementsinthesupportofthemeasureanditanhaveLebesgue
measure zero (for example, if the support of the measure is a yle) on the other hand
B-R-S measures satises the ergodi (Birkho) theorem for a set with positive Lebesgue
measure (a visibleset).
The next theorem gives onditions for the existene of B-R-S measures of funtions in
F. Itis provedin de Melo-van Strien (1993).
Theorem 3.3. if 2F, then:
1)There is at most one B-R-S measure.
2)If << and is -invariantthen:
i) is a B-R-S measure for and (B)=(X) (B as in denition 3.5).
ii)The unique set A given in theorem 3.4 is a nite union of transitive intervals (J is
a transitive interval if there exists N >0 suh that N
(J)\J 6=fg).
iii) The support of is equal to Aand is equivalent to j A
.
Theseond part of thetheorem above saysthat it is suÆient to nd an invariant and
absolutely ontinuous (with respet to the Lebesgue measure)measure for obtaininga
B-R-S measurewhihis supported ina unionof transitive intervals. The followingtheorems
state onditions for existene of absolutely ontinuous invariant measures.
Keller (1990) proved that there exists an absolutely ontinuous invariant probability
measure if and only if has a positive Liapounovexponent in almost every point and in
this wayrelated the existene of suh measures with the \haotiity"of .
Theorem 3.4. If 2F then there exists
2< suh that for almost every x:
=limsup 1
n
logjD n
(x)j
lim 1 n logjD n (x)j. 2)
<0 , has an hyperboli periodi attrator.
Inthefollowingsetionwewillshowthatalassof OLGmodelshasunimodalbpfmaps
assoiated with its stohasti exess demand and we analyze in whih ases it has B-R-S
measures for obtaining the SSE.
4. An OLG eonomy with haoti SE
In this setion we will show a spei eonomy that exhibit haoti SE for some set
of parameters. We will onsider the overlapping generations model with money transfer,
subsidies and publi expenditures treated in Grandmont (1986). The agents live two
periods andhave separable utilityfuntion V 1
( 1
)+V 2
( 2
) where t
is theonsumption of
theunique goodatage t=1;2and V i ()= 1 i =(1 i ); i
>0; i=1;2. Supposethat
oneunitofthegoodisproduedwithoneunitoftheuniqueprodutivefator(labor). The
agent's endowments at eah age t =1;2 are l 1
> 0 and l 2
> 0 and let =V 0 1 (l 1 )=V 0 2 (l 2 ). Dene: s t = M t 1 z t +S t M t 1 z t ; d t = M t 1 z t +S t +G t M t 1 z t : whereM t 1
>0represents the moneystokattheend of theperiodt 1,z t
isthe money
transfer fator (z t
1is thenominalinterestrate), S t
is thesubsidy andG t
isthe amount
of money issuedwhenthegovernment purhases(or sells)somequantityof thegood. The
dynami of themoney supply is given by the equation:
M t =M t 1 z t +S t +G t or M t =M t 1 z t d t ; M 0 given: s t and d t
are exogenous variables and we suppose s t
=s;d t
=d for all t.
Now let us analyze the agent's deision problem. Let p t
be the prie of the good in
period 1 and p t+1
the (random) prie of the good in period 2. Then the agent must
hose onsumptionplans
t
(deterministi), t+1
(random)and money demandm soas to
maximize:
V 1
( t
)+E[V 2
( t+1
)℄
with the budget onstraints:
p t
t
+m=p
t l 1 p t+1 t+1 =p t+1 l 2 +mz t +S t+1 :
The rst order onditionfor this problem (when themoney demandis positive) is:
1 p t V 0 1 (l 1 m p t
t x t = M t p t ; v 1
(x)=xV 0 1
(l 1
x) ; v 2
(x) =xV 0 2 (l 2 +x);
then we will havethe equation (1)equivalent to:
v 1
(x t
) =E[s 1 v 2 (sd 1 x t+1 )℄
(we have used M t z t+1 =d 1 M t+1
andtherefore M t z t+1 +S t+1 =sd 1 M t+1 ).
Remember that the expetedvalue is taken with respet to the probability measure of
x t+1
(orp t+1
)whih istheagent'sexpetationsofthefuturepries. Inthis asetheexess
demand funtion is:
~
Z(x;)=v 1 (x) E [s 1 v 2 (sd 1 x 0 )℄ =x(l 1 x) 1 E [d 1 x 0 (l 2 +sd 1 x 0 ) 2 ℄;
andithasthe(CVR)propertygivenindenition2.2. Thebakwardperfetforesightmap
is (x) = v 1 1 (s 1 v 2 (sd 1
x)), sine v 1
is stritly inreasing. In the following lemma we
give the parameter setthat guarantees 2F (see setion 3).
Lemma 4.1. If d
< 1, 1
2(0;1℄ and 2
2[2;+1) then 2F.
Finally, in orderto determineif thereexists aninvariant B-R-S measure (seedenition
3.2)wehaveto testthe onditiongivenin theorem3.4. Thefollowingtheorem shows how
it an be done.
Theorem 4.2. If the hypothesis of lemma 4.1 holds and
= limsup 1 n
jD n
(x)j > 0 ,
a:e then there exists SSE whose invariant measure is an absolutely ontinuous B-R-S
measure.
Theproof of this theoremfollows from theorem2.4 and theorem 3.4. Thetable bellow
gives estimates of
for some values of 1
and
2
and the orresponding histograms of
thefuntion . Theintervalin thex-axiswherethedynamis takesplaeis[0;x
℄. In this
interval we take a partition and estimate the frequeny of any orbit in eah interval. The
vertial axis shows these frequenies. So the histograms show an approximation for the
density of the invariant measure . Also we an note that for
> 0 the support of the
invariant measure is all the interval whereas for
<0 the support is an attrative yle.
In the seond ase there existsa nite SSE lose to the yle as proved byAzariadis and
Guesnerie (1986).
5. The onvergene of a learning proess to the haoti SSE
In Araujo and Maldonado (2000) it was proved that the haoti sunspot equilibrium
anbelearnedbyagentsif: i)They knowthe deterministiharateristisof theeonomy
(funtions f and g) and the state variable is following the perfet foresight path given
by the dynamial system x t
= (x
t+1
t+1
=R (z)Æ f(z)
+(1 R (z))Æ
g(z)
whereR (z)istherelativefrequenyinanysubinterval
of the points in the path (x 0
;:::;x t 1
;x t
= z) wih remains in I 1
(interval where is
inreasing); it results that R t
(z) ! dÆ f(z)=d almost surely when t ! 1. ii) The
eonomy is followingthe stationary equilibrium given by the haoti SSE, i.e. if the state
variableisarandomproess(~x t
) t0
givenby(Q;)(Prob[~x t+1
2Aj~x t
=x℄=Q(x;A)and
Prob[~x t
2 A℄ = (A)); in this ase the onvergene of R t
(z) results using the stohasti
ergodi theorem.
Inthissetionwewillprovideanadaptivelearningproesses thatallowsonvergeneto
the haoti SSE inthe sense that theatual dynamis mimis the haoti SSE. Woodford
(1990)proposedan adaptivelearningproessbased onthe stohastiapproximation
algo-rithmof RobinsandMonro(1951)forobtainingonvergenetothesupportoftheniteSE
heonsidered. Itisworthnotingthatsuhamethodusesthedeterminististrutureofthe
modeland the \projetion faility"desribed in Maret and Sargent (1989) for obtaining
thisonvergene,soWoodford'smethodonvergestoaniteSEforanyinitialonditionin
a(probablysmall)neighborhoodof thesteadystate,whileourlearningproesses onverge
to a SE whose support has positive Lebesgue measure for any initial ondition in a total
Lebesgue measureset.
Alsoweprovide aproof of theexpetationalstability(ES)of the onditional
probabili-tiesof thehaotiSSE.Expetationalstabilityof someparameterwhih denesarational
expetation equilibrium means that this equilibrium is stable for the dynamis generated
by the instantaneous orretions (in notional or virtual time) of this parameter. Suh a
onept was used for example by Evans and Honkapohja (1995) and Guesnerie (1993) in
dierentontextsbuttheideais thesame: Ifwesuppose someinitialvalueof the
parame-ter(whihanbeonstrutedinanyspeiway)theeetinthestruturalequationswill
give us a orretion of it. If the dynamis dened in this way onverges to some rational
expetationequilibriumthenwe willsaythatsuhanequilibrium isexpetationalystable.
First,letusdenethedynamisgeneratedbylearningproesses(seeforexample
Grand-mont (1998)). Let X be the state variable set. A learning proess is a sequene ( t ) t1 suh that t :X t
!P(X) and it anbe interpreted in the followingway: Ifthe eonomy
followed the path (x 0
;:::;x t 1
) then in period t agents will formulate expetations about
the state variable in period t+1 with t
(x 0
;:::;x t 1
).
The strutural equation is ~ Z(x
t ;
t+1
) = 0, so the atual dynamis under the learning
proess ( t
) t1
is given by:
~ Z(x t ; t (x 0 ;:::;x
t 1 ))=0
Let :[0;a℄