Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue5-7.Jul2000:
Journal of Economic Dynamics & Control
24 (2000) 1027}1046
Exponentially fading memory learning
in forward-looking economic modelsq
Emilio Barucci*
DIMADEFAS, Facoltà di Economia, Universita% di Firenze, Via C. Lombroso 6/17, 50134 Firenze, Italy
Abstract
In this paper we analyze forward-looking economic models under bounded rationality.
We consider a learning mechanism characterized by exponentially fading memory with
a learning step not vanishing in the limit. The dynamics of the model under bounded
rationality can be parametrized with respect to the memory of the learning mechanism.
We show that memory plays a stabilizing role in a local sense: it induces local convergence towards a stationary rational expectations equilibrium and in some cases it does
not allow for nonperfect foresight attractors obtained through local bifurcations. We
discuss this learning mechanism in pure exchange overlapping generations models. We
also analyze models with predetermined state variables, in this setting the e!ect of
memory on the learning dynamics is controversial. ( 2000 Elsevier Science B.V. All
rights reserved.
JEL classixcation: D83; D84; E21; E32
Keywords: Forward looking models; Overlapping generations; Rational expectations
equilibria; Learning; Cycles
q
I thank A. Naimzada, M. Posch for useful discussions during the preparation of the paper
and G.I. Bischi for useful conversations at the beginning of the project. I thank two anonymous
referees and in particular Cars Hommes for useful suggestions and comments. The usual disclaimers
apply.
* Tel.: 0039-55-4223936; fax: 0039-55-4223944.
E-mail address: [email protected]".it (E. Barucci)
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 3 5 - 4
1028
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1. Introduction
In this paper we investigate forward-looking economic models assuming that
the agents are not fully rational, i.e., they do not know the complete economic
model and they employ a recursive learning mechanism to update their beliefs.
Speci"cally we consider a learning mechanism characterized by fading memory,
i.e., at time t the agents evaluate the time t#1 state as a weighted average of the
observed values of the state up to time t!1. The weights are described by
a geometric progression with ratio smaller than one and therefore the weights
for older observations are smaller than the weights for recent observations.
Di!erently from other learning mechanisms proposed in the literature, the
learning step does not go to zero as time goes. The model with learning is
characterized by the stationary solutions obtained under perfect foresight,
whereas the other attractors (cycles, complex attractors) do not coincide with
those of perfect foresight. We may have forward-looking models characterized
by cycles and chaotic dynamics under perfect foresight and not under learning
and vice-versa. The existence of cycles under bounded rationality depends on
the shape of the backward perfect foresight map and on the memory parameter.
In what follows, we will consider the following cases: a forward-looking map
increasing or decreasing and a single-peaked map with a maximum or a
minimum.
The learning mechanism proposed in this paper is similar to the one analyzed
in Bray (1982), Evans and Honkapohja (1995) and Balasko and Royer (1996),
where the agents' expectation is computed as the arithmetic average of past
observations. In that framework there is full memory (the weight for each
observation is a constant) and the learning step goes to zero as time goes
(vanishing learning step). In our framework there is not full memory, remote
observations are less relevant than recent observations and the learning step
does not go to zero as time goes. These features of our learning mechanism are
appealing because the vanishing of the learning step and the assumption of
a constant weight for past observations are implausible from a behavioral point
of view.
The main result of the paper is that memory in the learning process plays
a stabilizing role in a local sense. The learning dynamics can be parametrized
with respect to the agents' memory. Increasing the memory with a backward
perfect foresight map decreasing in a neighborhood of an unstable stationary
rational expectations equilibrium (REE) we have two e!ects: the REE becomes
locally stable and the conditions on the parameters of the model to observe
cycles and chaotic dynamics through local bifurcations become stronger. In
some cases stability is reached only for a very long memory. In a pure exchange
overlapping generations model we obtain that to have a cycle and a chaotic
dynamics under bounded rationality learning we need a coe$cient of risk
aversion for old agents higher than under perfect foresight. The lower-bound on
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1029
the coe$cient of risk aversion for old agents to observe a cycle of period two
under bounded rationality is an increasing function of the memory. As far as the
global analysis is concerned we cannot establish such a general result. Among
the models considered below, only for a strictly decreasing backward perfect
foresight map we can have complex dynamics under bounded rationality and
not under perfect foresight. Simulations and bifurcation diagrams with respect
to the memory parameter show that memory dampens the oscillations but not
necessarily yields a simpler dynamics.
Models with predetermined variables are considered. Considering the stability of a REE, it is shown that a large enough memory allows for a strong
negative dependence of the state on the expected state. Memory plays a stabilizing role eliminating in some cases Flip bifurcations (the emergence of a two
period non perfect foresight attracting cycle) but not Neimark}Hopf bifurcations (the emergence of an attracting invariant set). Therefore, under bounded
rationality, we have that the agents may learn a nonperfect foresight invariant
set if they have a long memory. This type of models has been analyzed in
Grandmont and Laroque (1991) and Gauthier (1997), where it is assumed that
the agents estimate the rate of growth of the state variable in order to learn
autoregressive perfect foresight equilibria; in our analysis we concentrate our
attention on a learning mechanism which aims to predict the state variable itself
(e.g., the price level instead of the in#ation rate), so the results cannot be
compared directly.
The analysis of the role played by memory in a learning mechanism calls for
a discussion of the results with respect to those obtained in Bischi and Gardini
(1995), Balasko and Royer (1996), Hommes (1991, 1994). In Hommes
(1991, 1994) the cobweb model under adaptive expectations is analyzed. The
&destabilizing' e!ect of the parameter regulating the adaptation of expectations is
pointed out: as the parameter is increased, the stability region of a "xed point
becomes smaller and the oscillations become larger. Our analysis extends this
type of results to a di!erent class of economic models (forward-looking economic models with and without predetermined variables) and to a di!erent
learning mechanism which identi"es in a more precise way the e!ect of memory
in the learning process. In Balasko and Royer (1996) memory is identi"ed by the
number of past observations (h) considered by the agents and then each observation is weighted in the same way (least squares with "nite/in"nite memory). It is
shown that an equilibrium which is stable under learning with a "nite memory
h is also stable for a "nite memory h@ with h@'h. Our results go in the same
direction: given a memory parameter for which a stationary REE is expectationally stable then the equilibrium is still stable augmenting the memory.
The paper is organized as follows. In Section 2 we present fading memory
learning. In Section 3 we analyze the dynamics of the economic model assuming
that the agents update their beliefs with the fading memory learning mechanism.
In Section 4 we analyze the classical pure exchange overlapping generations
1030
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
model. In Section 5 we analyze forward-looking models with predetermined
state variables.
2. Fading memory learning
Let us consider the following class of forward-looking economic models:
x "F(x% ),
(1)
t
t`1
where x is a scalar variable and F( ) ) is a twice continuous di!erentiable map. As
usual we restrict our attention to economic models for which x*0. Eq. (1)
describes the backward law of motion associated with many economic models
characterized by forward-looking agents, e.g. overlapping generations models,
see Grandmont (1985) and Guesnerie and Woodford (1992).
At time t, the agents forecast the state at time t#1 as a weighted average of
the values of the state observed in the past:
t~1
x% " + a x ,
t`1
tk k
k/0
where
(2)
ot~1~k
t~1
a "
with = " + ok o3[0,1].
(3)
tk
t
=
t
k/0
Note that +t~1 a "1. The weights decrease exponentially as the terms of
k/0 tk
a geometric progression with ratio o3[0, 1], implying that old observations are
less relevant than the latest observations. The parameter o regulates the memory
of the learning mechanism, i.e., augmenting o we have more memory in the
learning mechanism (there is little di!erence between the weight for the state
observed at time t!1 and that observed at time 0). Two limiting cases are given,
for o"0 we have myopic expectations, x% "x , for o"1 we have the
t`1
t~1
uniform distribution of the weights, a "1/t for each 04k4t!1, so that we
tk
have the learning rule proposed in Bray (1982) and Evans and Honkapohja
(1995).
Assuming that the agents employ a learning rule like that described above,
then the dynamical system in (1) becomes a dynamical system of the form
A
B
t~1
+ a x
(4)
tk k
k/0
which is known as a Mann iteration, see Mann (1953), Aicardi and Invernizzi
(1992), Borwein and Borwein (1991) and Franks and Marzec (1971). The properties of this type of map with the weights as in (3) have been analyzed in Bischi
x "F
t
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1031
and Gardini (1995). It is shown that the dynamics can be reduced to that of
a two-dimensional autonomous map, whose attractors are those of the following one dimensional limiting map:
g (x)"ox#(1!o)F(x), 04o(1.
(5)
o
To show this fact we observe that the agents' expectation can be written as
a "rst-order non autonomous map
o= t~2 ot~2~k
1
1
o=
t~1 +
t~1 x%# F(x%),
x% "
(6)
x# x "
t`1
t =
t
k = t~1
=
=
=
t k/0
t
t~1
t
t
where = can be written recursively as
t
= "1#o= , = "1.
(7)
t
t~1
1
From (6) and (7), the following two-dimensional autonomous map is obtained
o=
1
t~1 x%#
x% "
F(x%)
t`1 1#o=
t 1#o=
t
¹:
t~1
t~1
= "1#o=
t
t~1
G
(8)
with an initial condition (x% ,= )"(x ,1). This map is equivalent to (4) with the
2 1
0
weights in (3): the sequence of the "rst coordinates of a phase trajectory of (8)
generates a sequence for the state variable starting from x , obtained as the
0
images of the function in (1). We observe that the second di!erence equation in
(8) is independent of x% and gives a monotonically increasing sequence. If 04o
t
(1, then the sequence = converges to
t
1
=H"
,
(9)
1!o
which de"nes an invariant and globally attracting line for the map ¹. Along
=H the dynamics is given by the "rst map of ¹, i.e., the map g . In Bischi and
o
Gardini (1995), the following Proposition is stated.
Proposition 2.1. (i) Every k-cycle A"MxH, xH,2, xHN of the limiting map g (x) is
1 2
k
o
in a one to one correspondence with a k-cycle A"A]M=HN"
M(xH, =H),2, (xH, =H)N of the map T;
1
k
(ii) if A is attracting, or attracting from one side, for g (x) then the set A is an
o
attracting cycle of the map T, and hence F(A) is an attracting set of the model with
learning;
(iii) the basin of attraction D of the attractor F(A) of the model with learning is
given by the intersection of the two-dimensional basin D of the cycle A of the map
T with the line of initial conditions ="1.
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E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
This learning mechanism can be interpreted as the "rst-order autoregressive
learning mechanism with a time varying adaptive coe$cient. Set a "1/(= ),
t
t`1
then the di!erence equation (6) becomes
x% "x%#a (x !x%),
t
t
t~1 t~1
t`1
where a converges towards 1!o as tPR. Our learning mechanism is similar
t
to the one analyzed in a deterministic setting in Evans and Honkapohja (1995),
a "0, a condition which is
where it is required that += a "#R and lim
t/1 t
t?= t
not veri"ed in our case. Note that only in the limit our learning mechanism
coincides with the "rst-order autoregressive learning scheme with a constant
learning step (a modi"ed version of adaptive expectations). Therefore, considering the autoregressive learning scheme with a constant learning step as the "nal
outcome of an exponentially fading memory learning process, we have that the
learning step is inversely related to the memory of the learning process.
3. The learning dynamics
In this section we study the dynamics of the forward-looking economic model
(1) under the learning mechanism described in the above section. We classify the
dynamics with respect to the shape of F(x). The following cases are considered:
F(x) strictly increasing, strictly decreasing, single-peaked with a maximum/minimum. To avoid degenerate cases we restrict our attention to the class
of functions F(x) de"ned for every x3[0,R).
The map g (x) is a convex combination of F(x) and of the identity function,
o
the weight of the combination is given by o, see Hommes (1994) for a similar
interpretation of the map obtained for the cobweb model with adaptive expectations. This fact implies that the graph of g (x) always belongs to the region
o
de"ned by the bisectrix and the graph of F(x): if o is next to zero then we have
a graph similar to F(x), if o is next to one then we have a graph similar to the
identity function. Note that a o large enough does not assure the monotonicity
of g (x). The two maps F(x) and g (x) share the same "xed points, so a stationary
o
o
REE is a stationary solution also for the dynamics with learning and vice versa.
As far as the other attractors is concerned we have that they are not those
obtained under perfect foresight.
In the two limiting cases, o"0 and o"1, the dynamics can be easily
analyzed. For o"0 (myopic expectations), the forward dynamics with learning
is exactly the backward perfect foresight dynamics (the time evolution is reversed), i.e., x "F(x ); therefore, a stationary/periodic solution which is stable
t
t~1
for the backward perfect foresight dynamics is stable for the forward dynamics
with learning and vice versa. For o"1, convergence can only occur at a "xed
point xH of the map F( ) ) such that F@(xH)(1, see Franks and Marzec (1971) and
Borwein and Borwein (1991).
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1033
The following Proposition can be easily stated about the local stability of
a stationary REE.
Proposition 3.1. Let xH be a stationary REE then we have the following local
stability results under the learning mechanism (2)}(3):
f if xH is stable for the backward perfect foresight dynamics, i.e., DF@(xH)D(1, then
it is also stable under learning for every o3[0, 1];
f if xH is unstable for the backward perfect foresight dynamics with F@(xH)'1 then
xH is unstable under learning for every o3[0, 1];
f if xH is unstable for the backward perfect foresight dynamics with F@(xH)(!1
then xH is stable under learning for o3(oH,1] and unstable for o3[0, oH), where
oH"(F@(xH)#1)/(F@(xH)!1).
A su$cient condition for the local stability of a REE under learning is its
stability for the backward perfect foresight dynamics, i.e., the stationary REE is
determinate. If F is increasing in a neighborhood of a stationary REE then it is
stable/unstable for the backward perfect foresight dynamics if and only if it is
stable/unstable for the dynamics under learning. If F@(xH)(!1 then a o large
enough (enough memory in the learning process) plays a stabilizing role inducing local convergence towards the REE, memory allows for a strong negative
dependence of the state on the expected state. Our results contrast in part with
Proposition 2 of Evans and Honkapohja (1995), where it is shown that if
lim
a "0 (o"1) then learning with a decreasing map is always charact?= t
terized by local stability of the stationary REE. The di!erence is due to
the fact that for our learning mechanism lim
a "1!o50. Note that the
t?= t
equilibrium can become stable for a value of o next to one yielding the
implausible assumption that the agents weight past observations almost in the
same way.
If F is strictly increasing then also g is strictly increasing and therefore in
o
both cases we do not observe the emergence of cycles. Monotonicity of the
solution is assured in both cases. Increasing the memory parameter o we observe
a smaller local rate of convergence towards the stationary REE. If the basin of
attraction of a stationary REE under the backward perfect foresight dynamics is
de"ned by two unstable "xed points then it coincides with the basin of attraction
under learning.
Exploiting the theorem about the existence of a Flip or period doubling
bifurcation we have that parametrizing the perfect foresight map through
a parameter k, i.e., F , then a condition for the emergence of a cycle of period
k
two under perfect foresight is that a bifurcating value kH exists such that at the
stationary REE xH we have F@ H(xH)"!1, see Theorem 12.7 in Devaney (1987).
k
If this condition is satis"ed and the bifurcation is supercritical then for k in
a neighborhood of kH we have F@ (xH)(!1 and a repelling "xed point with an
k
1034
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
attracting cycle of period two, see p. 1018 in Grandmont (1985). The following
result can be easily stated about our learning mechanism.
Proposition 3.2. Let kH be a bifurcating value for the map F and the stationary REE
xH, i.e., F@ H(xH)"!1. Then kH is not a bifurcating value for the map g (x)
k
o
(g@ H(xH)"2o!1'!1). If F@ (xH)(!1 and a perfect foresight cycle of period
ok
k
two exists, then for o high enough a cycle of period two for the map g does not
ok
exist, i.e., g@ (xH)'!1 for o'!(1#F@ (xH))/(1!F@ (xH)).
k
k
ok
The Proposition says that a long enough memory rules out the emergence of
a stable cycle of period two through a Flip bifurcation; moreover if a perfect
foresight cycle of period two exists then it can be eliminated under learning with
a o large enough. Note that this result precludes the emergence of a Feigenbaum
cascade through a sequence of Flip-period doubling bifurcations, a quite common scenario to generate complex dynamics.
Let us consider now a single-peaked map with a maximum and a unique
strictly positive stationary REE. The following Proposition can be stated.
Proposition 3.3. Assume that F is C1, single-peaked with a critical point x0
(maximum point) and a unique xxed point at xH, then we have the following:
o
f if F@(x)'!
∀x'0 then g (x) is increasing ∀x'0;
o
1!o
f if 0(xH(x0 then ∀o3(0,1) we have xH(x0(x0;
o
f if 0(x0(xH then xH(x0 if o'!F@(xH)/(1!F@(xH));
o
where x0 is the critical point of g (x), if it exists.
o
o
A necessary condition for the existence of a cycle and of a chaotic dynamics
for a single-peaked map with a maximum is that the critical point is smaller than
the "xed point; if it is larger than the "xed point then the REE is a global
attractor. The above Proposition states that for o large enough the map under
learning is increasing if F@(x) is bounded from below; if the stationary REE is
smaller than the critical point of the map F(x) then it is also smaller than the
critical point of the map g (x) (when it exists). If it is larger than the critical point
o
of F(x) then there is a o large enough such that the REE is smaller than the
critical point of g (x), and therefore it is attractive eliminating all kinds of cycles
o
and chaotic dynamics under learning. These results can be established for
a single-peaked map with multiple stationary REE considering the highest
among them in the above Proposition. Increasing memory in the learning
mechanism, we observe a smaller local rate of convergence towards the stationary REE xH if g@ (xH)'0 and a higher rate of convergence if g@ (xH)(0 with
o
o
damped oscillations.
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1035
A Proposition similar to Proposition 3.3 can be stated for a map F(x)
single-peaked with a minimum point and a unique stationary REE. If F@(x) is
bounded from below then there exists a o high enough such that g is increasing.
o
If o is not so high or F@(x) is unbounded from below, then there is a critical point
for g which is on the left of the stationary REE for o high enough ruling out all
o
kinds of cycles-complex dynamics.
Let us consider now a strictly decreasing map F. Many di!erent scenarios are
possible in this case. In what follows, looking also at the economic examples
presented below, we restrict our attention to a map F with FA(x)'0,∀x3(0,R),
and lim
F@(x)"0, see Barucci and Bischi (1996) and Evans and Honkapohja
x?=
(1995) for some economic examples. The following Proposition can be stated.
Proposition 3.4. Assume that F is C1, F@(x)(0, FA(x)'0, ∀x3(0,R). Let inf
F@(x)"!k'!R then we have the following cases:
f if o'k/(1#k) then g is monotonically increasing;
o
f if o(k/(1#k) then g is single-peaked with a minimum point x0.
o
o
Let inf F@(x)"!R, then g is single-peaked with a minimum point x0.
o
o
About the critical point x0 we have the following:
o
f if o'!F@(xH)/(1!F@(xH)) then x0(xH;
o
f if o(!F@(xH)/(1!F@(xH)) then x0'xH.
o
In every case lim
(g (x))/x"o and x0 (if it exists) is decreasing in o.
o
x?= o
A necessary condition for the existence of a chaotic dynamics for a singlepeaked map with a minimum and a unique "xed point is that the critical point is
larger than the "xed point (the Li-Yorke scenario can be observed), otherwise
the stationary REE is stable and monotonicity of the expectations path is
obtained after a few steps. From the above Proposition we have that
if inf F@(x)"!k'!R and F@(x)(0, ∀x'0, then g is increasing if o is
o
high enough. If o is not so high or inf F@(x)"!R, then the perfect foresight
map may be characterized by a cycle of period two but not by a cycle of period
three, whereas g is a single-peaked map with a critical point which is on the
o
right of the stationary REE for o small and on the left for o large enough.
Therefore, the dynamics under learning may be characterized by a cycle of every
period and by complex dynamics if o is small, but all kinds of these phenomena
disappear for o large enough. About the local rate of convergence towards the
stationary REE, the discussion developed above for the single-peaked map with
a maximum holds.
The basins of attraction of the stationary REE can be fully analyzed in this
setting by looking at the intersection between the two-dimensional basin of
attraction of the map ¹ and the line of the initial conditions ="1. The
boundary of the basin of attraction is an invariant set for the map ¹.
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E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
4. Pure exchange overlapping generations models
Let us consider the classical pure exchange overlapping generations model in
the Samuelson case, see Grandmont (1985). There is one non storable consumption good and money which is employed to transfer wealth from one period to
the next; agents are identical and live two periods, their endowment of the good
in the two periods is lH'0, i"1, 2. Let c , c denote consumption in the "rst
i
1 2
and in the second period, the agents are characterized by separable utility
functions ;(c , c )"< (c )#< (c ), where < (c ), i"1, 2 , satis"es the classical
1 2
1 1
2 2
i i
conditions. The money stock is constant over time, i.e., M'0. Let h "p /(pe ),
t
t t`1
then the optimal demand of the consumption good at time t is z (h )"
i t
c !lH, i"1, 2. Assuming market equilibrium, then the demand of money for
i
i
young/old agents is m(h )"!p z (h )"pe z (h ). Let hM "(
24 (2000) 1027}1046
Exponentially fading memory learning
in forward-looking economic modelsq
Emilio Barucci*
DIMADEFAS, Facoltà di Economia, Universita% di Firenze, Via C. Lombroso 6/17, 50134 Firenze, Italy
Abstract
In this paper we analyze forward-looking economic models under bounded rationality.
We consider a learning mechanism characterized by exponentially fading memory with
a learning step not vanishing in the limit. The dynamics of the model under bounded
rationality can be parametrized with respect to the memory of the learning mechanism.
We show that memory plays a stabilizing role in a local sense: it induces local convergence towards a stationary rational expectations equilibrium and in some cases it does
not allow for nonperfect foresight attractors obtained through local bifurcations. We
discuss this learning mechanism in pure exchange overlapping generations models. We
also analyze models with predetermined state variables, in this setting the e!ect of
memory on the learning dynamics is controversial. ( 2000 Elsevier Science B.V. All
rights reserved.
JEL classixcation: D83; D84; E21; E32
Keywords: Forward looking models; Overlapping generations; Rational expectations
equilibria; Learning; Cycles
q
I thank A. Naimzada, M. Posch for useful discussions during the preparation of the paper
and G.I. Bischi for useful conversations at the beginning of the project. I thank two anonymous
referees and in particular Cars Hommes for useful suggestions and comments. The usual disclaimers
apply.
* Tel.: 0039-55-4223936; fax: 0039-55-4223944.
E-mail address: [email protected]".it (E. Barucci)
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 3 5 - 4
1028
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1. Introduction
In this paper we investigate forward-looking economic models assuming that
the agents are not fully rational, i.e., they do not know the complete economic
model and they employ a recursive learning mechanism to update their beliefs.
Speci"cally we consider a learning mechanism characterized by fading memory,
i.e., at time t the agents evaluate the time t#1 state as a weighted average of the
observed values of the state up to time t!1. The weights are described by
a geometric progression with ratio smaller than one and therefore the weights
for older observations are smaller than the weights for recent observations.
Di!erently from other learning mechanisms proposed in the literature, the
learning step does not go to zero as time goes. The model with learning is
characterized by the stationary solutions obtained under perfect foresight,
whereas the other attractors (cycles, complex attractors) do not coincide with
those of perfect foresight. We may have forward-looking models characterized
by cycles and chaotic dynamics under perfect foresight and not under learning
and vice-versa. The existence of cycles under bounded rationality depends on
the shape of the backward perfect foresight map and on the memory parameter.
In what follows, we will consider the following cases: a forward-looking map
increasing or decreasing and a single-peaked map with a maximum or a
minimum.
The learning mechanism proposed in this paper is similar to the one analyzed
in Bray (1982), Evans and Honkapohja (1995) and Balasko and Royer (1996),
where the agents' expectation is computed as the arithmetic average of past
observations. In that framework there is full memory (the weight for each
observation is a constant) and the learning step goes to zero as time goes
(vanishing learning step). In our framework there is not full memory, remote
observations are less relevant than recent observations and the learning step
does not go to zero as time goes. These features of our learning mechanism are
appealing because the vanishing of the learning step and the assumption of
a constant weight for past observations are implausible from a behavioral point
of view.
The main result of the paper is that memory in the learning process plays
a stabilizing role in a local sense. The learning dynamics can be parametrized
with respect to the agents' memory. Increasing the memory with a backward
perfect foresight map decreasing in a neighborhood of an unstable stationary
rational expectations equilibrium (REE) we have two e!ects: the REE becomes
locally stable and the conditions on the parameters of the model to observe
cycles and chaotic dynamics through local bifurcations become stronger. In
some cases stability is reached only for a very long memory. In a pure exchange
overlapping generations model we obtain that to have a cycle and a chaotic
dynamics under bounded rationality learning we need a coe$cient of risk
aversion for old agents higher than under perfect foresight. The lower-bound on
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1029
the coe$cient of risk aversion for old agents to observe a cycle of period two
under bounded rationality is an increasing function of the memory. As far as the
global analysis is concerned we cannot establish such a general result. Among
the models considered below, only for a strictly decreasing backward perfect
foresight map we can have complex dynamics under bounded rationality and
not under perfect foresight. Simulations and bifurcation diagrams with respect
to the memory parameter show that memory dampens the oscillations but not
necessarily yields a simpler dynamics.
Models with predetermined variables are considered. Considering the stability of a REE, it is shown that a large enough memory allows for a strong
negative dependence of the state on the expected state. Memory plays a stabilizing role eliminating in some cases Flip bifurcations (the emergence of a two
period non perfect foresight attracting cycle) but not Neimark}Hopf bifurcations (the emergence of an attracting invariant set). Therefore, under bounded
rationality, we have that the agents may learn a nonperfect foresight invariant
set if they have a long memory. This type of models has been analyzed in
Grandmont and Laroque (1991) and Gauthier (1997), where it is assumed that
the agents estimate the rate of growth of the state variable in order to learn
autoregressive perfect foresight equilibria; in our analysis we concentrate our
attention on a learning mechanism which aims to predict the state variable itself
(e.g., the price level instead of the in#ation rate), so the results cannot be
compared directly.
The analysis of the role played by memory in a learning mechanism calls for
a discussion of the results with respect to those obtained in Bischi and Gardini
(1995), Balasko and Royer (1996), Hommes (1991, 1994). In Hommes
(1991, 1994) the cobweb model under adaptive expectations is analyzed. The
&destabilizing' e!ect of the parameter regulating the adaptation of expectations is
pointed out: as the parameter is increased, the stability region of a "xed point
becomes smaller and the oscillations become larger. Our analysis extends this
type of results to a di!erent class of economic models (forward-looking economic models with and without predetermined variables) and to a di!erent
learning mechanism which identi"es in a more precise way the e!ect of memory
in the learning process. In Balasko and Royer (1996) memory is identi"ed by the
number of past observations (h) considered by the agents and then each observation is weighted in the same way (least squares with "nite/in"nite memory). It is
shown that an equilibrium which is stable under learning with a "nite memory
h is also stable for a "nite memory h@ with h@'h. Our results go in the same
direction: given a memory parameter for which a stationary REE is expectationally stable then the equilibrium is still stable augmenting the memory.
The paper is organized as follows. In Section 2 we present fading memory
learning. In Section 3 we analyze the dynamics of the economic model assuming
that the agents update their beliefs with the fading memory learning mechanism.
In Section 4 we analyze the classical pure exchange overlapping generations
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E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
model. In Section 5 we analyze forward-looking models with predetermined
state variables.
2. Fading memory learning
Let us consider the following class of forward-looking economic models:
x "F(x% ),
(1)
t
t`1
where x is a scalar variable and F( ) ) is a twice continuous di!erentiable map. As
usual we restrict our attention to economic models for which x*0. Eq. (1)
describes the backward law of motion associated with many economic models
characterized by forward-looking agents, e.g. overlapping generations models,
see Grandmont (1985) and Guesnerie and Woodford (1992).
At time t, the agents forecast the state at time t#1 as a weighted average of
the values of the state observed in the past:
t~1
x% " + a x ,
t`1
tk k
k/0
where
(2)
ot~1~k
t~1
a "
with = " + ok o3[0,1].
(3)
tk
t
=
t
k/0
Note that +t~1 a "1. The weights decrease exponentially as the terms of
k/0 tk
a geometric progression with ratio o3[0, 1], implying that old observations are
less relevant than the latest observations. The parameter o regulates the memory
of the learning mechanism, i.e., augmenting o we have more memory in the
learning mechanism (there is little di!erence between the weight for the state
observed at time t!1 and that observed at time 0). Two limiting cases are given,
for o"0 we have myopic expectations, x% "x , for o"1 we have the
t`1
t~1
uniform distribution of the weights, a "1/t for each 04k4t!1, so that we
tk
have the learning rule proposed in Bray (1982) and Evans and Honkapohja
(1995).
Assuming that the agents employ a learning rule like that described above,
then the dynamical system in (1) becomes a dynamical system of the form
A
B
t~1
+ a x
(4)
tk k
k/0
which is known as a Mann iteration, see Mann (1953), Aicardi and Invernizzi
(1992), Borwein and Borwein (1991) and Franks and Marzec (1971). The properties of this type of map with the weights as in (3) have been analyzed in Bischi
x "F
t
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1031
and Gardini (1995). It is shown that the dynamics can be reduced to that of
a two-dimensional autonomous map, whose attractors are those of the following one dimensional limiting map:
g (x)"ox#(1!o)F(x), 04o(1.
(5)
o
To show this fact we observe that the agents' expectation can be written as
a "rst-order non autonomous map
o= t~2 ot~2~k
1
1
o=
t~1 +
t~1 x%# F(x%),
x% "
(6)
x# x "
t`1
t =
t
k = t~1
=
=
=
t k/0
t
t~1
t
t
where = can be written recursively as
t
= "1#o= , = "1.
(7)
t
t~1
1
From (6) and (7), the following two-dimensional autonomous map is obtained
o=
1
t~1 x%#
x% "
F(x%)
t`1 1#o=
t 1#o=
t
¹:
t~1
t~1
= "1#o=
t
t~1
G
(8)
with an initial condition (x% ,= )"(x ,1). This map is equivalent to (4) with the
2 1
0
weights in (3): the sequence of the "rst coordinates of a phase trajectory of (8)
generates a sequence for the state variable starting from x , obtained as the
0
images of the function in (1). We observe that the second di!erence equation in
(8) is independent of x% and gives a monotonically increasing sequence. If 04o
t
(1, then the sequence = converges to
t
1
=H"
,
(9)
1!o
which de"nes an invariant and globally attracting line for the map ¹. Along
=H the dynamics is given by the "rst map of ¹, i.e., the map g . In Bischi and
o
Gardini (1995), the following Proposition is stated.
Proposition 2.1. (i) Every k-cycle A"MxH, xH,2, xHN of the limiting map g (x) is
1 2
k
o
in a one to one correspondence with a k-cycle A"A]M=HN"
M(xH, =H),2, (xH, =H)N of the map T;
1
k
(ii) if A is attracting, or attracting from one side, for g (x) then the set A is an
o
attracting cycle of the map T, and hence F(A) is an attracting set of the model with
learning;
(iii) the basin of attraction D of the attractor F(A) of the model with learning is
given by the intersection of the two-dimensional basin D of the cycle A of the map
T with the line of initial conditions ="1.
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E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
This learning mechanism can be interpreted as the "rst-order autoregressive
learning mechanism with a time varying adaptive coe$cient. Set a "1/(= ),
t
t`1
then the di!erence equation (6) becomes
x% "x%#a (x !x%),
t
t
t~1 t~1
t`1
where a converges towards 1!o as tPR. Our learning mechanism is similar
t
to the one analyzed in a deterministic setting in Evans and Honkapohja (1995),
a "0, a condition which is
where it is required that += a "#R and lim
t/1 t
t?= t
not veri"ed in our case. Note that only in the limit our learning mechanism
coincides with the "rst-order autoregressive learning scheme with a constant
learning step (a modi"ed version of adaptive expectations). Therefore, considering the autoregressive learning scheme with a constant learning step as the "nal
outcome of an exponentially fading memory learning process, we have that the
learning step is inversely related to the memory of the learning process.
3. The learning dynamics
In this section we study the dynamics of the forward-looking economic model
(1) under the learning mechanism described in the above section. We classify the
dynamics with respect to the shape of F(x). The following cases are considered:
F(x) strictly increasing, strictly decreasing, single-peaked with a maximum/minimum. To avoid degenerate cases we restrict our attention to the class
of functions F(x) de"ned for every x3[0,R).
The map g (x) is a convex combination of F(x) and of the identity function,
o
the weight of the combination is given by o, see Hommes (1994) for a similar
interpretation of the map obtained for the cobweb model with adaptive expectations. This fact implies that the graph of g (x) always belongs to the region
o
de"ned by the bisectrix and the graph of F(x): if o is next to zero then we have
a graph similar to F(x), if o is next to one then we have a graph similar to the
identity function. Note that a o large enough does not assure the monotonicity
of g (x). The two maps F(x) and g (x) share the same "xed points, so a stationary
o
o
REE is a stationary solution also for the dynamics with learning and vice versa.
As far as the other attractors is concerned we have that they are not those
obtained under perfect foresight.
In the two limiting cases, o"0 and o"1, the dynamics can be easily
analyzed. For o"0 (myopic expectations), the forward dynamics with learning
is exactly the backward perfect foresight dynamics (the time evolution is reversed), i.e., x "F(x ); therefore, a stationary/periodic solution which is stable
t
t~1
for the backward perfect foresight dynamics is stable for the forward dynamics
with learning and vice versa. For o"1, convergence can only occur at a "xed
point xH of the map F( ) ) such that F@(xH)(1, see Franks and Marzec (1971) and
Borwein and Borwein (1991).
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1033
The following Proposition can be easily stated about the local stability of
a stationary REE.
Proposition 3.1. Let xH be a stationary REE then we have the following local
stability results under the learning mechanism (2)}(3):
f if xH is stable for the backward perfect foresight dynamics, i.e., DF@(xH)D(1, then
it is also stable under learning for every o3[0, 1];
f if xH is unstable for the backward perfect foresight dynamics with F@(xH)'1 then
xH is unstable under learning for every o3[0, 1];
f if xH is unstable for the backward perfect foresight dynamics with F@(xH)(!1
then xH is stable under learning for o3(oH,1] and unstable for o3[0, oH), where
oH"(F@(xH)#1)/(F@(xH)!1).
A su$cient condition for the local stability of a REE under learning is its
stability for the backward perfect foresight dynamics, i.e., the stationary REE is
determinate. If F is increasing in a neighborhood of a stationary REE then it is
stable/unstable for the backward perfect foresight dynamics if and only if it is
stable/unstable for the dynamics under learning. If F@(xH)(!1 then a o large
enough (enough memory in the learning process) plays a stabilizing role inducing local convergence towards the REE, memory allows for a strong negative
dependence of the state on the expected state. Our results contrast in part with
Proposition 2 of Evans and Honkapohja (1995), where it is shown that if
lim
a "0 (o"1) then learning with a decreasing map is always charact?= t
terized by local stability of the stationary REE. The di!erence is due to
the fact that for our learning mechanism lim
a "1!o50. Note that the
t?= t
equilibrium can become stable for a value of o next to one yielding the
implausible assumption that the agents weight past observations almost in the
same way.
If F is strictly increasing then also g is strictly increasing and therefore in
o
both cases we do not observe the emergence of cycles. Monotonicity of the
solution is assured in both cases. Increasing the memory parameter o we observe
a smaller local rate of convergence towards the stationary REE. If the basin of
attraction of a stationary REE under the backward perfect foresight dynamics is
de"ned by two unstable "xed points then it coincides with the basin of attraction
under learning.
Exploiting the theorem about the existence of a Flip or period doubling
bifurcation we have that parametrizing the perfect foresight map through
a parameter k, i.e., F , then a condition for the emergence of a cycle of period
k
two under perfect foresight is that a bifurcating value kH exists such that at the
stationary REE xH we have F@ H(xH)"!1, see Theorem 12.7 in Devaney (1987).
k
If this condition is satis"ed and the bifurcation is supercritical then for k in
a neighborhood of kH we have F@ (xH)(!1 and a repelling "xed point with an
k
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E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
attracting cycle of period two, see p. 1018 in Grandmont (1985). The following
result can be easily stated about our learning mechanism.
Proposition 3.2. Let kH be a bifurcating value for the map F and the stationary REE
xH, i.e., F@ H(xH)"!1. Then kH is not a bifurcating value for the map g (x)
k
o
(g@ H(xH)"2o!1'!1). If F@ (xH)(!1 and a perfect foresight cycle of period
ok
k
two exists, then for o high enough a cycle of period two for the map g does not
ok
exist, i.e., g@ (xH)'!1 for o'!(1#F@ (xH))/(1!F@ (xH)).
k
k
ok
The Proposition says that a long enough memory rules out the emergence of
a stable cycle of period two through a Flip bifurcation; moreover if a perfect
foresight cycle of period two exists then it can be eliminated under learning with
a o large enough. Note that this result precludes the emergence of a Feigenbaum
cascade through a sequence of Flip-period doubling bifurcations, a quite common scenario to generate complex dynamics.
Let us consider now a single-peaked map with a maximum and a unique
strictly positive stationary REE. The following Proposition can be stated.
Proposition 3.3. Assume that F is C1, single-peaked with a critical point x0
(maximum point) and a unique xxed point at xH, then we have the following:
o
f if F@(x)'!
∀x'0 then g (x) is increasing ∀x'0;
o
1!o
f if 0(xH(x0 then ∀o3(0,1) we have xH(x0(x0;
o
f if 0(x0(xH then xH(x0 if o'!F@(xH)/(1!F@(xH));
o
where x0 is the critical point of g (x), if it exists.
o
o
A necessary condition for the existence of a cycle and of a chaotic dynamics
for a single-peaked map with a maximum is that the critical point is smaller than
the "xed point; if it is larger than the "xed point then the REE is a global
attractor. The above Proposition states that for o large enough the map under
learning is increasing if F@(x) is bounded from below; if the stationary REE is
smaller than the critical point of the map F(x) then it is also smaller than the
critical point of the map g (x) (when it exists). If it is larger than the critical point
o
of F(x) then there is a o large enough such that the REE is smaller than the
critical point of g (x), and therefore it is attractive eliminating all kinds of cycles
o
and chaotic dynamics under learning. These results can be established for
a single-peaked map with multiple stationary REE considering the highest
among them in the above Proposition. Increasing memory in the learning
mechanism, we observe a smaller local rate of convergence towards the stationary REE xH if g@ (xH)'0 and a higher rate of convergence if g@ (xH)(0 with
o
o
damped oscillations.
E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
1035
A Proposition similar to Proposition 3.3 can be stated for a map F(x)
single-peaked with a minimum point and a unique stationary REE. If F@(x) is
bounded from below then there exists a o high enough such that g is increasing.
o
If o is not so high or F@(x) is unbounded from below, then there is a critical point
for g which is on the left of the stationary REE for o high enough ruling out all
o
kinds of cycles-complex dynamics.
Let us consider now a strictly decreasing map F. Many di!erent scenarios are
possible in this case. In what follows, looking also at the economic examples
presented below, we restrict our attention to a map F with FA(x)'0,∀x3(0,R),
and lim
F@(x)"0, see Barucci and Bischi (1996) and Evans and Honkapohja
x?=
(1995) for some economic examples. The following Proposition can be stated.
Proposition 3.4. Assume that F is C1, F@(x)(0, FA(x)'0, ∀x3(0,R). Let inf
F@(x)"!k'!R then we have the following cases:
f if o'k/(1#k) then g is monotonically increasing;
o
f if o(k/(1#k) then g is single-peaked with a minimum point x0.
o
o
Let inf F@(x)"!R, then g is single-peaked with a minimum point x0.
o
o
About the critical point x0 we have the following:
o
f if o'!F@(xH)/(1!F@(xH)) then x0(xH;
o
f if o(!F@(xH)/(1!F@(xH)) then x0'xH.
o
In every case lim
(g (x))/x"o and x0 (if it exists) is decreasing in o.
o
x?= o
A necessary condition for the existence of a chaotic dynamics for a singlepeaked map with a minimum and a unique "xed point is that the critical point is
larger than the "xed point (the Li-Yorke scenario can be observed), otherwise
the stationary REE is stable and monotonicity of the expectations path is
obtained after a few steps. From the above Proposition we have that
if inf F@(x)"!k'!R and F@(x)(0, ∀x'0, then g is increasing if o is
o
high enough. If o is not so high or inf F@(x)"!R, then the perfect foresight
map may be characterized by a cycle of period two but not by a cycle of period
three, whereas g is a single-peaked map with a critical point which is on the
o
right of the stationary REE for o small and on the left for o large enough.
Therefore, the dynamics under learning may be characterized by a cycle of every
period and by complex dynamics if o is small, but all kinds of these phenomena
disappear for o large enough. About the local rate of convergence towards the
stationary REE, the discussion developed above for the single-peaked map with
a maximum holds.
The basins of attraction of the stationary REE can be fully analyzed in this
setting by looking at the intersection between the two-dimensional basin of
attraction of the map ¹ and the line of the initial conditions ="1. The
boundary of the basin of attraction is an invariant set for the map ¹.
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E. Barucci / Journal of Economic Dynamics & Control 24 (2000) 1027}1046
4. Pure exchange overlapping generations models
Let us consider the classical pure exchange overlapping generations model in
the Samuelson case, see Grandmont (1985). There is one non storable consumption good and money which is employed to transfer wealth from one period to
the next; agents are identical and live two periods, their endowment of the good
in the two periods is lH'0, i"1, 2. Let c , c denote consumption in the "rst
i
1 2
and in the second period, the agents are characterized by separable utility
functions ;(c , c )"< (c )#< (c ), where < (c ), i"1, 2 , satis"es the classical
1 2
1 1
2 2
i i
conditions. The money stock is constant over time, i.e., M'0. Let h "p /(pe ),
t
t t`1
then the optimal demand of the consumption good at time t is z (h )"
i t
c !lH, i"1, 2. Assuming market equilibrium, then the demand of money for
i
i
young/old agents is m(h )"!p z (h )"pe z (h ). Let hM "(