24 (2000) 1027}1046
Exponentially fading memory learning
in forward-looking economic models
qEmilio Barucci
*
DIMADEFAS, Facolta` 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 conver-gence 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)
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 byfading memory, i.e., at timetthe agents evaluate the timet#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 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 stabil-ity 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 stabiliz-ing role eliminatstabiliz-ing in some cases Flip bifurcations (the emergence of a two period non perfect foresight attracting cycle) but not Neimark}Hopf bifurca-tions (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 eco-nomic 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 observa-tion 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 his also stable for a"nite memoryh@with h@'h. Our results go in the same direction: given a memory parameter for which a stationary REE is expectation-ally stable then the equilibrium is still stable augmenting the memory.
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
t"F(x%t`1), (1)
wherexis a scalar variable andF()) 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 timet, the agents forecast the state at timet#1 as a weighted average of the values of the state observed in the past:
x%t`1"t~1+
tk"1. The weights decrease exponentially as the terms of a geometric progression with ratioo3[0, 1], implying that old observations are less relevant than the latest observations. The parameteroregulates 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 timet!1 and that observed at time 0). Two limiting cases are given, for o"0 we have myopic expectations, x%
t`1"xt~1, for o"1 we have the uniform distribution of the weights,a
tk"1/tfor each 04k4t!1, so that we 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
x
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 follow-ing one dimensional limitfollow-ing map:
go(x)"ox#(1!o)F(x), 04o(1. (5)
To show this fact we observe that the agents' expectation can be written as a"rst-order non autonomous map
x%
tcan be written recursively as =
t"1#o=t~1, =1"1. (7)
From (6) and (7), the following two-dimensional autonomous map is obtained
¹:
G
weights in (3): the sequence of the"rst coordinates of a phase trajectory of (8) generates a sequence for the state variable starting from x0, obtained as the images of the function in (1). We observe that the second di!erence equation in (8) is independent ofx%
t and gives a monotonically increasing sequence. If 04o (1, then the sequence=
tconverges to
=H" 1
1!o, (9)
which de"nes an invariant and globally attracting line for the map¹. Along =Hthe dynamics is given by the"rst map of¹, i.e., the mapgo. In Bischi and Gardini (1995), the following Proposition is stated.
Proposition2.1. (i)Every k-cycleA"MxH
attracting cycle of the map T,and henceF(A)is an attracting set of the model with learning;
(iii)the basin of attraction D of the attractorF(A)of the model with learning is given by the intersection of the two-dimensional basinDof the cycleAof the map
This learning mechanism can be interpreted as the"rst-order autoregressive learning mechanism with a time varying adaptive coe$cient. Seta
t"1/(=t`1), then the di!erence equation (6) becomes
x%
t`1"x%t#at~1(xt~1!x%t), wherea
tconverges towards 1!oastPR. Our learning mechanism is similar to the one analyzed in a deterministic setting in Evans and Honkapohja (1995), where it is required that+=
t/1at"#Rand limt?=at"0, a condition which is 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, consider-ing the autoregressive learnconsider-ing scheme with a constant learnconsider-ing 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 ofF(x). The following cases are considered: F(x) strictly increasing, strictly decreasing, single-peaked with a max-imum/minimum. To avoid degenerate cases we restrict our attention to the class of functionsF(x) de"ned for everyx3[0,R).
The mapg
o(x) is a convex combination ofF(x) and of the identity function, the weight of the combination is given byo, see Hommes (1994) for a similar interpretation of the map obtained for the cobweb model with adaptive expecta-tions. This fact implies that the graph of g
o(x) always belongs to the region de"ned by the bisectrix and the graph ofF(x): ifois next to zero then we have a graph similar toF(x), ifo is next to one then we have a graph similar to the identity function. Note that aolarge enough does not assure the monotonicity ofgo(x). The two mapsF(x) andgo(x) share the same"xed points, so a stationary 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. Foro"0 (myopic expectations), the forward dynamics with learning is exactly the backward perfect foresight dynamics (the time evolution is rever-sed), i.e.,x
t"F(xt~1); therefore, a stationary/periodic solution which is stable for the backward perfect foresight dynamics is stable for the forward dynamics with learning and vice versa. Foro"1, convergence can only occur at a"xed pointxHof the mapF()) such thatF@(xH)(1, see Franks and Marzec (1971) and
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 ifxHis stable for the backward perfect foresight dynamics,i.e.,DF@(xH)D(1,then
it is also stable under learning for everyo3[0, 1];
f ifxHis unstable for the backward perfect foresight dynamics withF@(xH)'1then
xHis unstable under learning for everyo3[0, 1];
f ifxHis unstable for the backward perfect foresight dynamics withF@(xH)(!1
thenxHis stable under learning foro3(oH,1]and unstable foro3[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. IfFis 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. IfF@(xH)(!1 then aolarge enough (enough memory in the learning process) plays a stabilizing role induc-ing 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
t?=at"0 (o"1) then learning with a decreasing map is always charac-terized by local stability of the stationary REE. The di!erence is due to the fact that for our learning mechanism lim
t?=at"1!o50. Note that the 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.
IfF is strictly increasing then alsog
ois strictly increasing and therefore in both cases we do not observe the emergence of cycles. Monotonicity of the solution is assured in both cases. Increasing the memory parameterowe 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 parameterk, i.e.,F
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. LetkHbe a bifurcatingvalue for the mapFand the stationary REE xH, i.e., F@kH(xH)"!1. Then kH is not a bifurcating value for the map go(x) (g@okH(xH)"2o!1'!1).IfF@k(xH)(!1and a perfect foresight cycle of period
two exists,then forohigh enough a cycle of period two for the map gokdoes not exist,i.e., g@ok(xH)'!1foro'!(1#F@
k(xH))/(1!F@k(xH)).
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 aolarge enough. Note that this result precludes the emergence of a Feigenbaum cascade through a sequence of Flip-period doubling bifurcations, a quite com-mon 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 uniquexxed point atxH,then we have the following:
f ifF@(x)'! o
1!o∀x'0thengo(x)is increasing∀x'0;
f if0(xH(x0then∀o3(0,1)we havexH(x0(x0
o;
f if0(x0(xHthenxH(x0
o ifo'!F@(xH)/(1!F@(xH)); wherex0ois the critical point ofg
o(x),if it exists.
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. IfF@(x) is bounded from below then there exists aohigh enough such thatg
ois increasing. Ifois not so high orF@(x) is unbounded from below, then there is a critical point forgowhich is on the left of the stationary REE forohigh enough ruling out all kinds of cycles-complex dynamics.
Let us consider now a strictly decreasing mapF. 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 mapFwithFA(x)'0,∀x3(0,R), and lim
x?=F@(x)"0, see Barucci and Bischi (1996) and Evans and Honkapohja (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'!Rthen we have the following cases:
f ifo'k/(1#k)theng
ois monotonically increasing;
f ifo(k/(1#k)theng
ois single-peaked with a minimum pointx0o. Let infF@(x)"!R,thengois single-peaked with a minimum pointx0o.
About the critical pointx0owe have the following:
f ifo'!F@(xH)/(1!F@(xH))thenx0
o(xH;
f ifo(!F@(xH)/(1!F@(xH))thenx0
o'xH. In every caselim
x?=(go(x))/x"oandx0o(if it exists)is decreasing ino.
A necessary condition for the existence of a chaotic dynamics for a single-peaked 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 go is increasing if o is
high enough. Ifois not so high orinf 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, whereasgois a single-peaked map with a critical point which is on the 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 ifois small, but all kinds of these phenomena disappear forolarge enough. About the local rate of convergence towards the stationary REE, the discussion developed above for the single-peaked map with a maximum holds.
4. Pure exchange overlapping generations models
Let us consider the classical pure exchange overlapping generations model in theSamuelson case, see Grandmont (1985). There is one non storable consump-tion 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 islH
i'0,i"1, 2. Letc1,c2denote consumption in the"rst and in the second period, the agents are characterized by separable utility functions;(c
1,c2)"<1(c1)#<2(c2), where<i(ci),i"1, 2 , satis"es the classical conditions. The money stock is constant over time, i.e.,M'0. Letht"p
t/(pet`1), then the optimal demand of the consumption good at time t is z
i(ht)" c
i!lHi, i"1, 2. Assuming market equilibrium, then the demand of money for young/old agents is m(h
t)"!ptz1(ht)"pet`1z2(ht). Let hM"(<@1(l1H)/<@2(lH2)), considering the case where the agents in the"rst period save and transfer money to the second period we have that z
1(h)"z2(h)"0 if h4hM and
!lH
1(z1(h)(0, z2(h)'0 if h'hM, in this case we have that the "rst order conditions for the agents'optimization problem hold along the price sequence, i.e.,<@
1(lH1#z1(h))"h<@2(lH2#z2(h)). As we restrict our attention to the Samuel-soncase (young agents have a positive demand for money in equilibrium), then we require thathM(1.
Set as state variable at timetthe price ratioh
tor the real money balanceM/pt, then the forward-looking model (1) becomes F(x)"z~1
2 (!z1(x)) and F(x)"x/z~12 (x), respectively. Assuming that the coe$cient of relative risk aver-sion for old agents is a non decreasing function of wealth then the mapF is strictly increasing or single-peaked with a maximum, it depends on the inter-temporal substitution e!ect and on the wealth e!ect. To obtain a single-peaked map, the coe$cient of relative risk aversion for old agents, R
2(c2)"
!c
2(<A2(c2)/<@2(c2)), should be greater than one for some c2'0. Let a2"sup
c2R2(c2), if a241 thenF is strictly increasing, ifa2'1 andR2(c2) is nondecreasing then the mapFis single-peaked.
The analysis developed in Grandmont (1985) about the emergence of a cycle of period two (F@(1)(!1) is con"rmed under the conditions put forward in Proposition 3.2. To have a cycle of period two we require that old agents are su$ciently risk averse as well that they are myopic with respect to the past (short memory). The degree of risk aversion to guarantee the existence of a cycle of period two under bounded rationality is larger than under perfect foresight; under perfect foresight the condition is z@1(1)'z@
2(1) which becomes
Note that foro"0 the condition required under perfect foresight is obtained, see Condition 4.10 in Grandmont (1985). The constraint on the coe$cient of risk aversion for old agents (the right-side of the above inequality) is increasing ino: augmenting the memory in the learning process we need a higher degree of risk aversion for old agents to "nd out a cycle of period two under bounded rationality. This result also implies that under bounded rationality we need a degree of risk aversion higher than under perfect foresight to obtain a complex dynamics as the outcome of a sequence of Flip bifurcations.
Conditions for the emergence of a period three cycle can be established considering as state variableM/p, see Grandmont (1985). To this end it is useful to consider the functions v
1(x)"x<@1(lH1!x), ∀x3[0,lH1) and v2(x)" x<@
2(lH2#x), ∀x50. The forward-looking map F becomes F(x)"v~11 (v2(x)), ∀x50. A Li-Yorke scenario for the critical pointx0of the backward perfect foresight map is obtained, i.e., F2(x0)(F3(x0)(x0(F(x0), if the following conditions are satis"ed: there exists an x6'x0(x0 is also the critical point of v
2(x)) such that v2(x0)'v1(x6) that satis"es v2(x6)4v1(hMx0) or the stronger conditionv
2(x6)4<@1(lH1)hMx0. Under bounded rationality, witho"xed, the condi-tions become the following: there exists anx6'x0
o'x0(x0ois the critical point of go(x)) such thatv
2(x0)'v1(x6) ando((x6!F(x0))/(x0!F(x0))(1 that satis"es the conditiongo(x6)(hM/(hMo#1!o)x0. Note that we have an upper-bound to the memory parameter to obtain this type of scenario for the critical point of F(x); the upper-bound depends on the shape ofF(x), it is a decreasing function of F(x0)!x0which describes the hump of the single-peaked map and therefore is negatively related to the coe$cient of risk aversion for old agents. Summing up to have a cycle of period three we need a coe$cient of risk aversion for old agents larger than a parameter which is an increasing function of the memory parametero.
second period then the income e!ect may prevail over the substitution e!ect for every h and therefore only the decreasing part of the single-peaked map is obtained. In Barucci and Bischi (1996) we have analyzed the model proposed in Marimon et al. (1994) where the agents' utility function is ;(c
1,c2)" k
1J(c1/a)!k2a2/c22and the endowment islH1'0,lH2"0. In what follows we summarize the main results, see Barucci and Bischi (1996) for a detailed analysis. The associated forward-looking map for the pricex
tis function, it has a"xed point and in case a cycle of period two, either the"xed point or the period two cycle are stable under the perfect foresight dynamics. Considering the parameters used in Marimon et al. (1993) we have a"xed point xH"5 and a period two cycle (x6,x
6)"(2.56, 14.75);xHis stable and the period two cycle is repelling for the backward perfect foresight dynamics. The map g
o has a unique attractor (xH or a periodic attractor), it is characterized by a unique minimum point,x0o.x0omoves to the left asoincreases, fromx0o'xHto x0o(xH, moreoverx0oP0 asoP1 andx0oP#RasoP0.
If o"0, then xH is unstable and the period two cycle (x6,x
6
) is globally asymptotically stable. Asois increased, the attracting period two cycle becomes smaller until a value o"o
f is reached at which the cycle disappears and xHbecomes stable. This occurs wheng@o
f(xH)"!1. For 0(o(ofthe
attract-ing cycle of period two is di!erent from that obtained under perfect foresight. Therefore, if agents have a short memory, then they can learn a period two cycle which is not a perfect foresight cycle. Asois increased aboveofa backward#ip bifurcation occurs at which the period two cycle disappears and the"xed point xH becomes attracting. For a su$ciently long memory, the dynamics of the model with learning converges to the stationary REE. Foro'o
f,xHis globally attracting for the model with learning, whereas, under perfect foresight it only attracts initial conditions inside the repelling period two cycle. Moreover, while under perfect foresight convergence towards xHoccurs through large oscilla-tions, under learning as o is increased the oscillations become smaller and smaller and then monotonic.
In the last example only a period two cycle under bounded rationality has been obtained. A simple example can be constructed yielding a period three cycle and therefore a chaotic dynamics under bounded rationality and not under perfect foresight.1 It is enough to take as utility function ;(c
t,ct`1)" log(c
t)#c1~at`1/(1!a) with a zero endowment in the second period andlH1'0 in
the "rst period. Then the forward-looking map becomes F(x% t`1)" M/lH
1#Ma/lH1(x%t`1)a~1. Fora'1 the mapFis strictly decreasing, for a mem-ory parameterosu$ciently low we have a minimum point for the mapgowhich is on the right of the stationary REE yielding the Li-Yorke scenario, under some conditions. SetM"1, l
1"2,a"27, o"0.1, we have a critical point for the mapgoatx0o"1.18 which is larger than the stationary REExH"1. For this set of parameters, starting from x"15 we obtain go(x)"1.95, g2o(x)"0.65, g3o(x)"16.68, i.e., the Li-Yorke scenario g3o(x)'x'go(x)'g2
o(x). Increasing othe critical point goes to the left of the stationary REE and a cycle of period three is not obtained.
As we pointed out in the Introduction, memory stabilizes the dynamics in a local sense by enlarging the stability region of stationary REE and by eliminating non perfect foresight attractors generated through local bifurca-tions. To evaluate the e!ect of memory in a global sense we can give a look at the bifurcation diagrams of the mapg
o. The bifurcation diagram for the"rst model described above shows that increasing memory the dynamics becomes simpler and the oscillations are damped. However, while the reduction of the oscilla-tions is obtained in almost every model, it is easy to build models showing that memory does not yield a simpler dynamics, for a similar phenomenon in the cobweb model with adaptive expectations see Hommes (1994). As an example we can consider the last model witha"6. The bifurcation diagram shows that increasingowe have"rst a region with a two period cycle, then a region with a more complex dynamics and then"nally a series of reverse period doubling bifurcations yielding the stationary REE as unique attractor, see Fig. 1. This type of scenario can also be obtained for a single-peaked map with a maximum.
5. Forward-looking models with predetermined state variables
The class of models considered in (1) simply relates the state at timetwith the agents'expectation of the state at timet#1. In some overlapping generations models, e.g. models with production, we end up with a dynamical system de"ned by an implicit function relating the state today with the state yesterday and the agents'expectation:
Z(x
t~1,xt,x%t`1)"0 , (10)
wherexis still a positive scalar variable andZ(),),)) is assumed to have enough
regularity. The analysis of the perfect foresight dynamics can be developed in a neighborhood of a stationary REE, an xH such that Z(xH,xH,xH)"0, by enlarging the dimension of the state variables and linearizing equation (10) at the stationary REE, see Guesnerie and Woodford (1992) and Kehoe and Levine (1985). Set a"Z
Fig. 1. Bifurcation diagram foro.
a,b,cO0, the partial derivatives ofZ(),),)) evaluated at (xH,xH,xH), then the
linearized system becomes
ax
t~1#bxt#cxt`1"0 ,
which can be rewritten in backward form as a two-dimensional system:
x
t~1"! b
axt!
c axt`1, x
t"xt
yielding the following conditions for local stability of the REE with respect to the backward perfect foresight dynamics, see Guesnerie and Woodford (1992) and Kehoe and Levine (1985):c/a'!1!b/a, c/a'!1#b/a,c/a(1.
The analysis under bounded rationality is more complex than the one ob-tained for the model (1). In what follows, we restrict our attention to the local analysis in a neighborhood of a stationary REE and we do not classify the bounded rationality dynamics with respect to the shape ofZ. Given the assump-tions done above about the mapZ(),),)) we can write it in an explicit form at
least locally in a neighborhood of the stationary REExHas follows:
x
We observe thatH@1(xH,xH)"!a/band H@2(xH,xH)"!c/b. Assuming that the agents employ the learning mechanism described in (2) and (3) we end up with a three-dimensional system:
In the limit, the dynamics of this system is governed by the map
¹H be the stationary REE. Linearizing the map¹H
1at the stationary REE (xH,xH)
The trace and the determinant of D¹H
1(xH,xH) are ¹r"o! (1!o)c/b!a/b, Det"!o(a/b). The following Proposition can be stated about the stability of the REE with respect to the bounded rationality dynamics.
Proposition 5.1. The stationary REExHis locally stable under bounded rationality if the following conditions are satisxed:
a
Proof. The statement of the proposition is easily obtained by looking at the su$cient conditions to have two eigenvalues inside the unit circle: 1!¹r#Det'0, 1#¹r#Det'0, 1!Det'0. h
The stability conditions of the stationary REE under learning give us the following restrictions on the economic model, see Fig. 2:
H@1(xH,xH)(1
o, H@1(xH,xH)'!1! 1!o
1#oH@2(xH,xH) ,
Fig. 2. Stability region of a Stationary REE: (a)o"0, (b) 0(o(1, (c)o"1.
Note that in the limit for H@1(xH,xH) small enough we obtain the stability conditions established in Proposition 3.1. The two limiting caseso"0 (myopic expectations) ando"1 (full memory) give us the following stability conditions:
f o"0:DH@
1(xH,xH)#H@2(xH,xH)D(1,
f o"1:DH@
1(xH,xH)D(1,H@1(xH,xH)#H@2(xH,xH)(1.
Foro"0 we havex%t`1"x
t~1and thereforext"H(xt~1,xt~1) ,xHis locally stable for the dynamics with learning if the derivative ofHwith respect tox
andH@1(xH,xH)#H@2(xH,xH)(1. Leto3(0,1), we observe thatoonly a!ects the
"rst and the second stability condition, the third condition does not depend on o. Increasingo we have two di!erent e!ects: the constraint obtained from the condition on the determinant becomes stronger and the slope for the second inequality becomes smaller in absolute value. This means that increasing mem-ory in the learning process we allow for a stronger negative dependence on the expected state and for a smaller positive dependence on the lagged-sate variable. This type of results generalizes those obtained for the models of class (1): memory stabilizes stationary REE characterized by a negative relationship between the state today and the expected state. As in the forward-looking model analyzed in Section 3, we have that enough memory allows for all kind of negative dependence of the state from the expected state provided a small positive dependence on the lagged variable. Instead, a positive dependence on the expected state is&stabilized'by a small enough memory parameter if the state negatively depends on the lagged state. From Fig. 2 we have that there is a region which is stable under learning for everyo3[0,1], the region is given by the following conditions DH@1(xH,xH)D(1, DH@1(xH,xH)#H@2(xH,xH)D(1. The area of the stability region is not monotonic inoas for the model (1), because of the trade-o! described above we have that the stability region reaches its maximum in o"1/2. The region for which we have stability both under the backward perfect foresight dynamics and under the forward learning dynamics is small and is increasing in o, for o"0 the intersection of the two stability region is the null set.
The third condition for the stability of a stationary REE is not a!ected byo. The "rst condition de"nes a Neimark}Hopf bifurcation locus. Leaving the stability region through this line we have two eigenvalues with non null im-maginary part on the unit circle, (xH,xH) changes from a stable focus to an unstable focus and a closed invariant set with periodic or quasi-periodic dynam-ics is observed around (xH,xH) in a neighborhood of the bifurcation locus. The second condition de"nes a Flip-period doubling bifurcation locus at which an eigenvalue exits the unit circle with the value!1 and (xH,xH) changes from an attracting node to a saddle-point and a period two cycle is observed in its neighborhood. This type of analysis only holds close to the bifurcating loci; increasing the distance from the bifurcating locus we are not able to handle easily the problem. Some theoretical facts are known, for example if a period two cycle is created increasing a parameter through a Flip-bifurcation then increas-ing further the parameter we may observe a sequence of such bifurcations yielding cycles of period 2k, k"1, 2,2, and in the limit even chaotic dynamics. This type of scenario depends on the global dynamics of the model and in particular on the emergence of a global bifurcation which is related to the coexistence of di!erent attractors.
rationality do not coincide with those obtained under perfect foresight. In what follows, we concentrate our attention on a neighborhood of the bifurcation locus, see Fig. 2. The role played by memory is not univocal. Just outside the Neimark}Hopf bifurcation line we can have an attracting invariant set. This type of non perfect foresight attractor is eliminated by decreasing the memory parameter, increasing the memory we have that the agents can learn a non perfect foresight invariant set. Memory plays a destabilizing e!ect. The e!ect of memory with respect to the Flip bifurcation is not univocal. Increasing othe slope regulating the third inequality decreases in absolute value. This fact implies that a two period attracting cycle obtained through a Flip bifurcation is eliminated by aolarge enough ifH@2(xH,xH)(0 and is created ifH@2(xH,xH)'0. So memory plays a stabilizing role eliminating a period two non perfect foresight cycle if the state negatively depends on the expectation. If the state positively depends on the expectation then memory plays a destabilizing e!ect. In some cases we may observe that increasingo the period two stable cycle disappears and an attracting invariant set surrounding (xH,xH) is obtained. Summing up, increasing memory we eliminate in some cases non perfect fore-sight stable period two cycles as in the forward-looking models analyzed in Section 3 (see Proposition 3.2), but not attracting closed invariant sets which are not obtained when the agents are myopic.
6. Conclusions
In this paper we have proposed a learning mechanism characterized by two main features: exponentially fading memory and a non vanishing learning step. Past observations are weighted with a decreasing weight and in the limit the learning process is characterized by a constant strictly positive learning step. This constant is inversely related to the memory of the learning process.
The bounded rationality assumption and the associated analysis of a for-ward-looking model can be interpreted in two di!erent perspectives: to provide an evolutive learning explanation of the perfect foresight dynamics, see Gues-nerie (1993), and per se as a plausible behavioral assumption. In the "rst perspective the bounded rationality analysis usually plays a subsidiary role with respect to the rational expectations paradigm, the main point addressed in the literature is the local stability of a REE solution (Expectational Stability, see Evans and Honkapohja, 1998). In the second perspective the bounded rational-ity assumption has a complete dignrational-ity in alternative to the full rationalrational-ity assumption. As far as the"rst approach is concerned, the analysis developed in this paper has shown that the Expectational Stability argument is only a part of the story. In fact the dynamics under bounded rationality can be characterized by the coexistence of perfect foresight and non perfect foresight attractors and therefore a closer look at the basins of attraction should be given to establish the relevance of REE. However, we have shown that memory reinforces the evolu-tive learning exaplanation of a REE. Considering the bounded rationality assumption as a plausible behavioral assumption, we have identi"ed attractors which are not of perfect foresight but are still a restpoint of the learning activity and so a solution of some interest, on this see also Hommes and Sorger (1998). This fact is particularly relevant in case of a non perfect foresight chaotic attractor because agents may not not recognize any regularity in the time series and therefore there is no reason to change the expectation formation mecha-nism.
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