Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue5-7.Jul2000:

Journal of Economic Dynamics & Control
24 (2000) 761}798

Heterogeneous beliefs and the non-linear
cobweb model
Jacob K. Goeree!,*, Cars H. Hommes"
!Department of Economics, 114 Rouss Hall, University of Virginia, Virginia, Charlottesville, VA 22903,
USA
"Center for Nonlinear Dynamics in Economics and Finance (CeNDEF) and Tinbergen Institute,
University of Amsterdam, Roetersstraat 11, NL-1018 WB, Amsterdam, Netherlands
Accepted 30 April 1999

Abstract
This paper generalizes the evolutionary cobweb model with heterogeneous beliefs of
Brock and Hommes (1997. Econometrica 65, 1059}1095), to the case of non-linear
demand and supply. Agents choose between a simple, freely available prediction strategy
such as naive expectations and a sophisticated, costly prediction strategy such as rational
expectations, and update their beliefs according to an evolutionary &"tness' measure such
as past realized net pro"ts. It is shown that, for generic non-linear, monotonic demand
and supply curves, the evolutionary dynamics exhibits &rational routes to randomness',
that is, bifurcation routes to strange attractors occur when the traders' sensitivity to

di!erences in evolutionary "tness increases. ( 2000 Elsevier Science B.V. All rights
reserved.
JEL classixcation: E32; C60
Keywords: Endogenous #uctuations; Heterogeneous expectations; Evolutionary dynamics; Chaos; Bifurcations

* Corresponding author. We are much endebted to Buz Brock for his stimulating ideas and
support during the writing of this paper. The "rst author acknowledges "nancial support from the
NSF (SBR 9818683).

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 2 5 - 1

762 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

1. Introduction
Many models of economic behaviour are representative agent models, in
which all agents are assumed to be identical: they have the same endowments,
preferences, tastes and expectations. Homogeneity in expectations formation is
often justi"ed by the hypothesis that all agents are rational and that rationality is
common knowledge. Another frequently heard motivation for homogeneous

beliefs is that heterogeneity in expectations among agents would lead to analytically untractable models. In this paper we investigate heterogeneity in beliefs or
expectations in the well-known dynamic cobweb model. In particular, we show
that heterogeneity in expectation formation can lead to market instability and
to periodic, or even chaotic, price #uctuations.
Brock and Hommes (1997a) studied heterogeneity in expectation formation
by introducing the concept of adaptive rational equilibrium dynamics (ARED),
a coupling between market equilibrium dynamics and adaptive predictor selection. The ARED is an evolutionary dynamics between competing prediction
strategies. Agents can choose between di!erent prediction strategies and update
their beliefs over time according to a publically available &"tness' or &performance' measure such as (a weighted sum of) past realized pro"ts. Prediction
strategies with higher "tness in the recent past are selected more often than those
with lower "tness. Brock and Hommes (1997a), henceforth BH, present a detailed analysis of the cobweb model where agents can either buy a rational
expectations (perfect foresight) forecast at positive information costs, or freely
obtain the naive expectations forecast. BH show that a rational route to randomness, that is a bifurcation route to chaos and strange attractors, occurs when the
intensity of choice to switch prediction strategies increases. Stated di!erently,
when agents become more sensitive to di!erences in evolutionary "tness, equilibrium price #uctuations become more erratic. A high intensity of choice leads
to an irregular switching between cheap &free riding' and costly sophisticated
prediction, with prices moving on a strange attractor.
BH restrict their analysis to the special case of linear demand and supply. The
only non-linearity in their model comes from the heterogeneity in expectations
formation and updating of beliefs. The main purpose of the present paper is

to generalize the results of BH to non-linear demand and supply curves. In
particular, we address the question whether a rational route to randomness
arises in a general, non-linear demand-supply framework, and if so, what are the
typical bifurcation routes to chaos? We show that for a large class of non-linear,
monotonic demand and supply curves, derived from utility and pro"t maximization, in a market that is unstable under naive expectations, complicated endogenous price #uctuations on a strange attractor arise when the intensity of
choice to switch prediction strategies becomes high. BH's rational route to
randomness thus appears to be a general feature of the law of demand and
supply.

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 763

In the case of linear demand and supply, the bifurcation route to chaos
exhibits a non-generic secondary bifurcation, a so-called 1:2 strong resonance
Hopf bifurcation. We show that in the case of non-linear demand and supply,
this non-generic (co-dimension four) bifurcation &breaks' into three or four
di!erent co-dimension one bifurcations. For the case of non-linear demand and
supply, we "nd essentially two di!erent generic rational routes to randomness:
the period doubling route to chaos and the &breaking of an invariant circle'
bifurcation route to strange attractors.
Expectation formation and learning have been important themes in the recent

literature on bounded rationality. Some have focussed on stability and convergence of learning rules to rational expectations equilibria, e.g. Bray (1982), Bray
and Savin (1986) and Marcet and Sargent (1989); others have focussed on
conditions for instability and the possibility of endogenous #uctuations under
learning, e.g. Bullard (1994), Grandmont (1985, 1998), Grandmont and Laroque
(1986), Marimon et al. (1993), and Hommes and Sorger (1998). In particular,
a number of studies follow an evolutionary approach for selecting prediction
strategies, e.g. Arifovich (1994, 1996), Blume and Easley (1992), Bullard and
Du!y (1998), Arthur et al. (1997) and LeBaron et al. (1998). Nice recent surveys
of the bounded rationality literature are Sargent (1993), Marimon (1997), and
Evans and Honkapohja (1998). Our approach "ts into this bounded rationality
literature, emphasizing evolutionary selection of prediction strategies by
boundedly rational agents.
Recently, there have been a number of related studies investigating the dynamical behavior in heterogeneous belief models. In these studies, two typical classes
of agents are fundamentalists, expecting prices to return to their &fundamental
value', and chartists or technical analysts extrapolating patterns, such as trends, in
past prices. For example, De Grauwe et al. (1993) show that periodic and chaotic
exchange rate #uctuations arise due to an interaction between fundamentalism
and chartism. Chiarella (1992), Day and Huang (1990), Lux (1995), and Lux and
Marchesi (1998), Cabrales and Hoshi (1996), and Sethi (1996) study stock market
#uctuations due to the presence of chartists and fundamentalists. de Fontnouvelle

(1998) analyzes a "nancial market model with informed and uninformed traders,
which in fact "ts our non-linear cobweb framework, and presents numerical
evidence of a period doubling route to chaos. Arthur et al. (1997) and LeBaron et
al. (1998) run computer simulations of an evolutionary dynamics in an &Arti"cial
Stock Market', with an ocean of traders using di!erent trading strategies. Building
on Brock (1993), Brock and Hommes (1997b,1998) and also Gaunersdorfer (1998)
investigate the present discounted value asset pricing model with heterogeneous
beliefs, and detect several bifurcation routes to strange attractors. Brock and de
Fontnouvelle (1998) investigate heterogeneous beliefs in the overlapping generations model, and detect bifurcation routes to complicated dynamics as well.
From a methodological viewpoint, this paper applies recent mathematical
theory of homoclinic bifurcations and strange attractors. See Palis and Takens

764 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

(1993) for an excellent and extensive mathematical treatment; see also Vilder
(1995, 1996) and Yokoo (1998) for recent applications of homoclinic bifurcations
in two-dimensional versions of the overlapping generations model. The present
paper also applies the theory of local bifurcations. See e.g. Kuznetsov (1995) for
a recent and extensive mathematical treatment. In particular, local bifurcations
will be investigated by numerical analysis, using the sophisticated LOCBIF

bifurcation package (Khibik et al., 1992, 1993).
The paper is organized as follows. In Section 2 we discuss the cobweb model
with rational versus naive expectations. The main results of the paper are
summarized in Section 3. Section 4 focusses on the case of a quadratic cost
function and a non-linear decreasing demand curve, and sketches the proof that
homoclinic bifurcations and strange attractors arise, when the intensity of
choice to switch predictors becomes high. In Section 5, we use the LOCBIFprogram to detect the primary and secondary local bifurcations in generic
rational routes to randomness. In the "nal section we end with some conclusions. An appendix discusses the case of a general convex (non-quadratic)
cost function.

2. The cobweb model with rational versus naive expectations
In order to be self-contained, we brie#y recall the cobweb model with rational
versus naive expectations, as introduced in BH. The cobweb model describes
#uctations of equilibrium prices in an independent market for a non-storable
good, that takes one time period to produce, so that producers must form price
expectations one period ahead. Applications of the cobweb model mainly
concern agricultural markets, such as the classical examples of cycles in hog or
corn prices. Supply S(p%t ) is a function of the price expected by the producers, p%t ,
derived from expected pro"t maximization:
S(p%t )"argmax Mp%t qt!c(qt)N"(c@)~1(p%t ).

qt

(1)

The cost function c( ) ) is assumed to be strictly convex so that the marginal cost
function can be inverted, and supply is then strictly increasing in expected price.
The expected price may be some function of (publically known) past prices:
p%t "H(Pt~1), where Pt~1"(pt~1, pt~2,2, pt~L) denotes a vector of past prices
of lag-length ¸, and H( ) ) is called a predictor. In the case, when producers have
rational expectations, or perfect foresight, H(Pt~1) equals the actual price, pt, for
all times.
Consumer demand D depends upon the current market price pt. Demand will
be assumed to be strictly decreasing in price to ensure that its inverse is
well-de"ned. The demand curve D may be derived from consumer utility

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 765

maximization, but for our purposes it is not necessary to specify these preferences explicitly and we will simply work with general decreasing demand curves.
If beliefs are homogeneous, i.e., all producers use the same predictor, market
equilibrium price dynamics in the cobweb model is given by

D(pt)"S(H(Pt~1)), or pt"D~1(S(H(Pt~1))).
(2)
The actual equilibrium price dynamics thus depends upon the demand curve D,
the supply curve S as well as the predictor H used by the producers. For
example, if all producers were to use the perfect foresight, or rational expectations predictor HR(Pt~1)"pt, price dynamics would become extremely simple:
pt"pH in all periods, where pH is the unique price corresponding to the
intersection of demand and supply. If, on the other hand, all producers use the
naive, or myopic predictor HN(Pt~1)"pt~1, price dynamics is given by
pt"D~1(S(pt~1)), which is the familiar textbook cobweb system. If demand D is
decreasing and supply S is increasing, price dynamics in the cobweb model with
naive expectations is simple. When !1(S@(pH)/D@(pH)(0 prices converge to
the stable steady state pH; otherwise, they diverge away from the steady state and
either converge to a stable 2-cycle or exhibit unbounded up and down oscillations.1
In this paper we investigate the dynamics of the cobweb model with heterogeneous beliefs. Instead of all producers using the same predictor, we assume
that each producer can choose between the two predictors HR and HN. As in BH,
producers can either obtain the sophisticated, rational expectations predictor
HR at information cost C, or freely obtain the &simple rule of thumb', naive
predictor HN. Market equilibrium in the cobweb model with rational versus
naive expectations is determined by
D(pt)"ft~1

R S(pt)#ft~1
N S(pt~1),
(3)
where ft~1
R and ft~1
N denote the fractions of agents using the rational respectively
the naive predictor, at the beginning of period t. Notice that producers using the
rational expectations predictors have perfect foresight due to perfect knowledge
about the market equilibrium equations, past prices as well as the fractions of
both groups determining the market equilibrium price, i.e. perfect knowledge
about beliefs of all other agents. The di!erence C between the information costs
for rational and naive expectations represents an extra e!ort cost producers
incur over time when acquiring this perfect knowledge.
The cobweb model with rational versus naive expectations, may be seen
as an analytically tractable, stylized two predictor model in which rational
1 Note that for other predictors such as adaptive expectations or linear predictors with two or
three lags, price #uctuations in the cobweb model can become much more complicated. In
particular, chaotic price oscillations may arise even when both demand and supply are monotonic
(Hommes, 1994, 1998).

766 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

expectations represents a costly sophisticated (and stabilizing) predictor, and
naive expectations represent a cheap &habitual rule of thumb' (but destabilizing)
predictor. Other two predictor cases, such as fundamentalists (expecting prices
to return to the rational expectations fundamental steady-state price pH) versus
adaptive expectations, yield essentially the same results.
To complete the model, we have to specify how the fractions of traders using
rational c.q. naive expectations are determined. These fractions are updated over
time, according to a publically available &performance' or &"tness' measure
associated to each predictor. Here, we take the most recent realized net pro"t as
the performance measure for predictor selection.2 For the rational expectations
predictor, realized pro"t is given by
nRt "pt S(pt)!c(S(pt)).
(4)
The net realized pro"t for rational expectations is thus given by nRt !C, where
C is the information cost that has to be paid for obtaining the perfect forecast.
For the naive predictor the realized net pro"t is given by
nN
(5)

t "ptS(pt~1)!c(S(pt~1)).
The fractions of the two groups are determined by the Logit discrete choice
model probabilities. Anderson et al. (1993) contains an extensive discussion and
motivation of discrete choice modelling in various economic contexts; see also
Goeree (1996). BH provide motivation of discrete choice models for selecting
prediction strategies. The fraction of agents using the rational expectations
predictor in period t equals
exp(b(nR!C))
t
ftR"
,
(6)
exp(b(nR!C))#exp(bnN)
t
t
and the fraction of agents choosing the naive predictor in period t is then
(7)
f N"1!f R.
t
t
A crucial feature of this evolutionary predictor selection is that agents are
boundedly rational, in the sense that most but not all agent use the predictor
that has the highest "tness. Indeed, from (6) we have for instance that f R'f N
t
t
whenever nR!C'nN, although the optimal predictor is not chosen with
t
t
probability one. We expect that other models than discrete choice models that
satisfy this general feature lead to similar results as those presented below. The
2 The case where the performance measure is realized net pro"t in the most recent past period,
leads to a two-dimensional dynamic system. The more general case, with a weighted sum of past net
realized pro"ts as the "tness measure, leads to higher-dimensional systems, which are not as
analytically tractable as the two-dimensional case. In this more general case however, numerical
simulations suggest similar dynamic behaviour.

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 767

parameter b is called the intensity of choice; it measures how fast producers
switch between the two prediction strategies. Let us brie#y discuss the two
extreme cases b"0 and b"R. For b"0, both fractions are "xed over time
and equal to 1/2. The other extreme b"R, corresponds to the neoclassical
limit in which agents are unboundedly rational, and all producers choose the
optimal predictor in each period. Hence, the higher the intensity of choice the
more rational, in the sense of evolutionary "tness, agents are in choosing their
prediction strategies. The neoclassical limit b"R will play an important role
in what follows.
It will be convenient to de"ne the di!erence m of the two fractions:
t
(8)
m ,f R!f N,
t
t
t
so m "!1 corresponds to all producers being naive, whereas m "1 means
t
t
that all producers prefer the rational expectations predictor. The evolution of
the equilibrium price, p , and the di!erence of fractions, m , is then summarized
t
t
by the following two-dimensional, non-linear dynamical system
D(p )"1(1#m )S(p )#1(1!m ) S(p ),
(9)
2
t
2
t~1
t
t~1
t~1
m "tanh(b(nR!nN!C)/2).
(10)
t
t
t
The "rst equation de"nes p implicitly, in terms of (p , m ); the monotonicity
t
t~1 t~1
of demand and supply ensures that p is uniquely de"ned. The timing of
t
predictor selection in (9), (10) is important. In (9) the old (di!erence in) fractions
are used to determine the new equilibrium price p . Thereafter, this new equilibt
rium price p is used in the evaluation of predictors according to their evolutiont
ary "tness, through (4)}(7), and the new fractions are updated according to (10).
These new fractions are then used in determining the next equilibrium price
p , etc.
t`1
BH termed the coupling (9), (10) between the equilibrium price dynamics and
adaptive predictor selection an adaptive rational equilibrium (ARE) model. They
restricted their advanced analysis of the ARE-dynamics to the special case of
linear demand and supply curves. Our aim is to investigate local bifurcations as
well as global dynamics in the ARE model with general monotonic, non-linear
demand and supply functions. As will be seen, the analysis of the global
complicated dynamics in the model is considerably simpli"ed in the case of
a linear supply curve
(11)
S(p%)"bp%,
t
t
or equivalently a quadratic cost function c(q)"q2/(2b). In particular, for a linear
supply curve the di!erence in realized pro"ts is given by
b
nR!nN" (p !p )2,
t
t
t~1
2 t

(12)

768 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

that is, the di!erence in realized pro"ts is proportional to the squared prediction
error of naive expectations. In the case of a linear supply curve and a general
non-linear, decreasing demand curve, the ARE thus becomes
D(p )"1(1#m )bp #1(1!m )bp ,
2
t
t~1 t 2
t~1 t~1

AC

b
m "tanh b (p !p )2!C
t~1
t
2 t

DN B

2 .

(13)
(14)

In the sequel, we use the shorthand notation (p , m )"F (p , m ) for the
t t
b t~1 t~1
ARE-model (9), (10) or (13), (14). We are especially interested in the dynamics
when the °ree of rationality', that is, the intensity of choice, b, becomes high.

3. Main results
This section summarizes the main results concerning the price dynamics of
the cobweb model with rational versus naive expectations. First, we describe the
local (in)stability of the steady state. Second, we state the main result concerning
existence of strange attractors for high values of the intensity of choice. Finally,
we discuss possible generic bifurcation scenarios when the intensity of choice
increases.
3.1. The steady state and its stability
To "nd the steady state (p6 , m6 ) of the general ARE-model (9), (10), observe that
the "rst equation (9) dictates D(p6 )"S(p6 ). Since the left-hand side is strictly
decreasing in p6 , and the right-hand side is strictly increasing in p6 , the solution,
pH, to this equation is unique. The di!erence of the realized pro"ts for the two
predictors, evaluated at the steady price p6 , is zero, from which we infer that
m6 "!tanh(bC/2). The unique steady state is thus given by (p6 , m6 )"
(pH,!tanh(bC/2)).
The stability properties of the steady state are determined by the derivatives of
supply and demand at the steady state price pH. A straightforward computation
shows that the eigenvalues of the Jacobian evaluated at the steady state are
j "0, and
1
(1!m6 )S@(pH)
j "
(0.
2 2D@(pH)!(1#m6 )S@(pH)

(15)

Since m6 is less than or equal to one in absolute value, the value of the second
eigenvalue lies between S@(pH)/D@(pH) and zero. If DS@(pH)/D@(pH)D(1, then the
steady state is locally stable for all b. Prices close to the steady state pH converge

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 769

to pH, and the di!erence of fractions converges to m6 . To allow for the possibility
of an unstable steady-state and endogenous price #uctuations in the evolutionary ARE-model, from now on we assume the following.
Assumption ;. The market is locally unstable when all producers are naive, that is,
S@(pH)/D@(pH)(!1.
The stability properties of the steady state in the evolutionary ARE-model are
summarized as follows.
Proposition 1. Under Assumption U, the evolutionary ARE-model satisxes:
(i) When information costs are zero (C"0), the steady state is locally stable for all b.
(ii) When information costs are strictly positive (C'0), there exists a critical value
b such that the steady state is stable for 04b(b and unstable for b'b . At
1
1
1
b"b the second eigenvalue satisxes j "!1, and F in (9), (10) exhibits
1
2
b
a period doubling bifurcation.
Proof. When C"0 the steady state is given by (p6 , m6 )"(pH, 0) and the second
eigenvalue satis"es Dj D"S@(pH)/(S@(pH)#2DD@(pH)D)(1, so the steady state is
2
locally stable. For positive information cost the steady state is given by
(p6 , m6 )"(pH,!tanh(bC/2)) and the second eigenvalue by (15). When b"0 the
steady state reduces to (pH, 0) and j is then smaller than one in absolute value.
2
However, when b"R the steady state becomes (pH,!1) and the second
eigenvalue satis"es j "S@(pH)/D@(pH)(!1, by Assumption U. Both m6 and
2
j depend continuously on b. Since m6 is strictly decreasing in b and j is strictly
2
2
increasing in m6 , there exists a b such that !1(j (0 for b(b , j "!1
1
2
1 2
for b"b and j (!1 for b'b . h
1
2
1
3.2. Strange attractors
According to Proposition 1, the steady state (p6 , m6 )"(pH,!tanh(bC/2)) is
unstable for large values of the intensity of choice to switch predictors, b. In the
case of linear demand and supply, Brock and Hommes (1997a) have shown that
for a large value of the intensity of choice (corresponding to a high degree of
rationality) the ARE system does not settle down to simple (periodic) behavior,
but chaotic price #uctuations on a strange attractor arise.3 The next theorem
generalizes this result concerning the global dynamics to the case of a linear

3 For a de"nition of strange attractor and technical details, see Section 4.1.

770 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

supply curve (or equivalently a quadratic cost function) and any non-linear,
decreasing demand curve; the case of a general, non-linear increasing supply
curve is more subtle, and will be discussed in the appendix.
¹heorem. For any linear supply curve and generic non-linear decreasing demand
curves, such that Assumption U is satisxed, and for a suzciently low but positive
information cost C, the ARE-model (13), (14) has strange attractors for a set of
b-values of positive Lebesgue measure.
The proof of the theorem is given in Section 4. The result states that, in the
case of a quadratic cost function (or equivalently linear supply), for generic
non-linear, decreasing demand curves (and therefore for generic underlying
utility functions), when the cobweb dynamics under naive expectations is unstable, and when costs for rational expectations are low but positive, the evolutionary system exhibits chaotic price #uctuations for large values of the intensity of
choice. Notice that for the theorem to hold, the information costs for rational
expectations has to be positive, but should also not be too high, because
otherwise the evolutionary system might lock into a state far away from the
equilibrium steady state, e.g. into a 2-cycle, with almost all agents remaining
naive since it is still optimal not to buy the expensive rational expectations
forecast.
Fig. 1 shows an example of a strange attractor, with corresponding time
series of prices p and di!erence in fractions m . Numerical simulations
t
t
suggest that for (almost) all initial states (p , m ) the orbit (p , m ) converges
0 0
t t
to this strange attractor. Its intricate geometric shape explains why it is
called a strange attractor. The time series exhibit sensitive dependence upon
initial conditions. Price #uctuations are characterized by an irregular switching between a stable phase, with prices close to the steady state, and an unstable phase with #uctuating prices. During the stable phase most agents use
the cheap, naive predictor. As a result prices diverge from the steady state,
start #uctuating, and net realized pro"ts from the naive predictor decrease.
When the intensity of choice to switch predictors is high, most agents will
switch to costly rational expectations during the unstable phase, because
with highly #uctuating prices the rational expectations predictor yields higher
net realized pro"ts. As a result, prices are driven back close to the steady
state, and the story repeats. Irregular, chaotic price #uctuations thus result
from a (boundedly) rational choice between cheap &free riding' and costly
sophisticated prediction. In fact, the above economic mechanism already
suggests that for a large intensity of choice, the ARE-cobweb model will be close
to a homoclinic orbit associated to the unstable, saddle point steady state.
Mathematical details behind this economic mechanism will be discussed in
Section 4.

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 771

Fig. 1. Strange attractor (a) for b"6 and information cost C"1, when demand is
D(p)"!p/2!p2!p3 and supply S(p)"2p (both in deviations from the steady state), and
corresponding chaotic time series of price deviations from the steady state (b) and di!erence in
fractions (c).

772 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

3.3. Local bifurcations
In this subsection, we focus on generic bifurcation routes to complicated dynamics, as the intensity of choice increases. From Proposition 1 it follows that for any
non-linear demand and supply curves satisfying the unstable cobweb assumption,
the primary bifurcation towards instability in the evolutionary ARE-model is
a period doubling or #ip bifurcation. At the bifurcation value b"b , a 2-cycle
1
bifurcates from the steady state. The situation for the secondary bifurcation is much
more complicated however. For the case of linear supply and demand, BH have
shown that the secondary bifurcation is a so-called 1 : 2 strong resonance Hopf
bifurcation, in which the 2-cycle becomes unstable and four 4-cycles, two stable
4-cycles and two saddle 4-cycles, are created simultaneously. This secondary bifurcation is a highly degenerate bifurcation, occurring only in the special case of linear
demand and supply. In fact, it is a co-dimension four bifurcation.4 In this subsection
we present possible co-dimension one secondary and consecutive local bifurcations
in the general case of non-linear, monotonic supply and demand curves.
In order to discuss possible co-dimension one secondary bifurcations for the
general ARE-model (9), (10), it will be su$cient to consider the following simple,
but general enough example:
(16)
D(p )"a!d p !d p2!d p3
3 t
t
1 t
2 t
and supply is linear: S(p )"bp , as in (11). The parameters d are chosen such
t
t
i
that the demand in (16) is strictly decreasing. Notice that for d "d "0 (16)
2
3
reduces to the linear demand curve, so that the linear case investigated by BH is
nested as a special case.
In Section 5, using the LOCBIF bifurcation package, we will present a detailed numerical analysis of generic, co-dimension one bifurcation routes to
complicated dynamics in the case of linear supply (11) and non-linear demand
(16). The LOCBIF analysis may be summarized by the two-dimensional bifurcation diagram in the (d , b) parameter plane shown in Fig. 2. The other para2
meters have been "xed at d "0.5, d "0.1, b"2, C"1 (unit information cost)
1
3
and a"0.5 The striped curve is the Hopf bifurcation curve of the 2-cycle, the
4 Intuitively, the co-dimension of a bifurcation is the minimum number k of parameters such that
the bifurcation occurs in generic k parameter families. The fold, Hopf, #ip and pitchfork bifurcations
are well-known co-dimension one bifurcations; see, e.g., Guckenheimer and Holmes (1983) or
Kuznetsov (1995) for extensive mathematical treatments of local bifurcation theory. In the case with
linear demand and supply, at the secondary bifurcation of the 2-cycle, the Jacobian matrix JF2 of the
b
second iterate, at the points of the 2-cycle, equals minus the identity matrix, implying that the
co-dimension must be at least four (see Arrowsmith and Place, 1990, Exercise 5.1.5, p. 292).
5 In the case of a linear supply curve, the di!erence in fractions in (14) is given by
m "tanh((b/2)[(b/2)(p !p )2!C]). Therefore, without loss of generality, we can choose the
t
t
t~1
intersection of supply and demand as the origin and work in deviations from the steady state, or
equivalently we may set a"0 in the non-linear demand curve (16).

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 773

Fig. 2. Two-dimensional bifurcation diagram w.r.t. the intensity of choice b and d . The dotted line
2
represents a period-doubling bifurcation of the 2-cycle, the striped line a Hopf bifurcation of the
2-cycle, and the solid lines a saddle-node, or fold bifurcation of a 4-cycle.

dotted curves are two period doubling or #ip bifurcation curves of the 2-cycle
and the solid curves are saddle-node or fold bifurcation curves of the 4-cycles.
For b"1.8 the model has a stable 2-cycle, which becomes unstable as b increases, either through a Hopf or a #ip bifurcation. The vertical line segments in
Fig. 2 represent four di!erent generic bifurcation routes to complexity, as the
intensity of choice b increases:6
1.
2.
3.
4.

d "0.01: hopf-fold-fold;
2
d "0.04: #ip-#ip-hopf-fold;
2
d "0.05: #ip-#ip-fold;
2
d "0.08: #ip-#ip.
2
In each of the "rst three scenario's, as b increases, two stable coexisting 4-cycles
and two 4-saddles are created in a sequence of two or three consecutive local
bifurcations. In the fourth scenario, only one stable 4-cycle is created, as
b increases. Hence, far from the linear case (i.e., for large enough d -values)
2
co-existing stable 4-cycles do not necessarily arise as b increases. On the

6 As noted above, the primary bifuraction is always a #ip bifurcation of the steady state, so we
focus on the secondary bifurcation of the 2-cycle, and consecutive bifurcations to complexity.

774 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

other hand, close to the linear case (i.e., for small d ), at least three possible
2
co-dimension one bifurcation routes can occur, from a stable steady state to four
co-existing 4-cycles, two stable 4-cycles and two (unstable) 4-saddles.7 In the
special case of linear demand and supply considered by BH, the four 4-cycles are
created simultaneously in one single, co-dimension four bifurcation. Our numerical LOCBIF-analysis shows that, for generic non-linear demand and supply,
this co-dimension four bifurcation &breaks' into two or three consecutive codimension one bifurcations.
4. Global dynamics
This section presents the proof of existence of strange attractors for high
values of the intensity of choice, for a linear supply (or equivalently a quadratic
cost function) and generic non-linear, decreasing demand curves; the case of
a non-linear, increasing supply curve is discussed in the appendix. Since a large
part of the proof closely follows BH, we will only sketch the main (geometric)
ideas underlying the proof, emphasizing the di!erences with the case of linear
demand and supply. In order to be self-contained, we brie#y discuss homoclinic
bifurcations and recent mathematical results concerning strange attractors in
Section 4.1. Next, we consider the neoclassical limit (b"R) of the ARE-model
in Section 4.2. In Section 4.3, we investigate the geometric shape of the unstable
manifold of the steady state for high, but "nite, b-values. Finally, in Section 4.4
we prove existence of strange attractors for high, but "nite, values of b.
4.1. Homoclinic bifurcations and strange attractors
A key feature of chaotic dynamical behavior in two- and higher-dimensional
systems is the existence of so-called homoclinic points. This concept was introduced already by PoincareH (1890), in his prize winning essay on the stability of
the three-body system. Let us brie#y discuss this important notion.
Recall that after the primary bifurcation in our ARE-model, the steady state
S loses its stability and becomes a saddle point. The stable manifold and the
unstable manifold of the steady state are de"ned as

G K
G K

H

W4(S)" (p, m) lim Fn (p, m)"S ,
b
n?=

(17)

W6(S)" (p, m) lim Fn (p, m)"S .
b
n?~=

(18)

H

7 Brock and Hommes (1997a) contains an extensive discussion of the dynamic complexity due to
coexisting stable 4-cycles. In particular, the basin boundaries between the two stable 4-cycles may
have a complicated fractal structure.

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 775

For a periodic saddle point (p, m), with period k, the stable and unstable
manifold are de"ned similarly, by replacing F by Fk . If F is a di!eomorb
b
b
phism (a smooth invertible function) the stable and unstable manifolds
are smooth curves without self-intersections; if it is non-invertible the unstable manifold may have self-intersections and/or the stable manifold may
have more than one component. A transversal homoclinic point HOS, associated to the saddle S, is an intersection point of the stable and unstable
manifold of S. If the manifolds are tangent at H, it is called a point of homoclinic
tangency.
It was already pointed out by PoincareH that the existence of a homoclinic
intersection implies that the geometric structure of both the stable and unstable
manifold is quite complicated. Because they are both invariant under F ,
b
the existence of one homoclinic point H implies the existence of in"nitely
many such points, since Fn (H) is also an element of both the stable and unb
stable manifold for all n3Z. As a result the stable and unstable manifolds
have to intertwine an in"nite number of times, accumulating at the steady state,
and so-called homoclinic tangles arise. More recently, Smale (1965) has
shown that a homoclinic point implies that F has (in"nitely many) horseshoes,
b
that is, there exist rectangular regions R such that for some positive integer n,
the image Fn (R) is folded over R in the form of a horseshoe. Smale showed
b
that the occurence of a horseshoe implies that the map has in"nitely many
periodic points, an uncountable set of chaotic orbits, and exhibits sensitive
dependence with respect to initial states. A horseshoe is not an attractor
however, and chaos may occur only on a set of initial states of Lebesgue measure
zero. This situation is commonly referred to as topological chaos. (See for
example Guckenheimer and Holmes (1983) for more details on homoclinic
orbits and horseshoes.)
Recently, it has been shown that homoclinic bifurcations, that is, the creation
of homoclinic orbits as a parameter varies, is closely related to existence of
strange attractors. We say that the dynamical system, represented by the map
F , undergoes a homoclinic bifurcation associated to the (periodic) saddle S at
b
b"b , if W4(S)WW6(S)"S for b(b and W4(S)WW6(S) contains a (homoch
h
linic) point HOS for b5b . The importance of a homoclinic bifurcation is
h
spelled out by the next theorem due to Benedicks and Carleson (1991), and
Mora and Viana (1993); see also Palis and Takens (1993) for an extensive
mathematical treatment. Let G be any smooth two-dimensional non-linear
b
map, which undergoes a homoclinic bifurcation associated to a saddle (periodic)
point.
0Strange Attractor ¹heorem1. If G exhibits a homoclinic bifurcation associated
b
to a locally dissipative periodic point at b"b , then generically there exists a set
h
of b-values of positive Lebesgue measure for which the map G has a strange
b
attractor.

776 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

A strange attractor is an attractor that is the closure of an unstable manifold
of some periodic saddle point and contains a dense orbit with positive
Lyapunov exponent (see e.g. Palis and Takens, 1993, pp. 138}143). Recall that
locally dissipative means that the determinant of the Jacobian of Gk is less than
b
one at the period k points. The &strange attractor theorem' implies that, under
generic conditions, for a large set of b-values the dynamical behavior generated
by the map G is chaotic.8 A recent economic application of the &strange
b
attractor theorem' is due to de Vilder (1995, 1996), who showed the existence of
strange attractors in a two-dimensional overlapping generations model with
production. In this section, we will apply the theorem to show existence of
strange attractors in the cobweb ARE-model for linear supply and generic
non-linear, decreasing demand curves.
4.2. The neoclassical limit: b"R
In order to understand the dynamical behavior for a high, but "nite intensity
of choice, it is important to understand the neoclassical limit, that is, the case
b"R. In particular, for the neoclassical limit the stable and unstable manifolds of the steady state can be characterized analytically.
For b"R and C'0, the steady state S"(pH,!1), where pH is the price at
which demand and supply intersect. One component of the stable manifold of
the steady state is easily found: all points (pH, m) are mapped to the steady state
S"(pH,!1), so the vertical line p"pH is part of the stable manifold. The
unstable manifold requires more work to derive. First, note that when b"R
the fraction m , t51, is either !1 or 1, and the ARE system reduces to
t
1 if nR!nN'C,
t
t
m"
t
!1 if nR!nN4C.
t
t

G

(19)

Hence in the neoclassical limit, in each period all producers choose the optimal
predictor. Furthermore, p "D~1(S(p )), when all producers choose the naive
t
t~1
predictor, and p "pH when all use the rational expectations predictor.
t
For general demand and supply the di!erence in realized pro"ts is given by
¸(p ; p )"nR!nN"p S(p )!c(S(p ))!p S(p )#c(S(p )).
t
t t
t
t t~1
t~1
t
t t~1

(20)

This di!erence represents the loss producers face when their expected price is
p , whereas the actual price becomes p . Notice that under naive expectations
t~1
t
p%"p , the actual price becomes p "D~1(S(p )). We will refer to the
t
t~1
t
t~1
8 See Palis and Takens (1993, pp. 35}36) for technical details. See also Takens (1992, pp. 192}93)
for a considerable weakening of the generic conditions in the case of real analytic families.

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 777

di!erence (20) as the loss function ¸(p ;p ) under naive expectations, since it
t t~1
represents the pro"t loss due to forecasting error when all agents are naive. In
the case of a quadratic cost function c( ) ), or equivalently a linear supply curve
S"(c@)~1, the loss function has the particularly simple quadratic form
b
nR!nN" (p !p )2.
t
t
t~1
2 t

(21)

The next lemma states that the loss function ¸ has a unique minimum at the
steady state p "pH:
t~1
¸emma 1. The loss function ¸(p ;p ), with actual price p "D~1(S(p )), has
t t~1
t
t~1
a unique (global) minimum 0 at the steady state p "pH.
t~1
Proof. Di!erentiating ¸ w.r.t. p

, and using the fact that c@"S"id, yields
t~1

L¸
Lp
t (S(p )!S(p ))#S@(p ) (p !p ).
"
t
t~1
t~1 t~1
t
Lp
Lp
t~1
t~1
First of all, S@ is always positive and Lp /Lp (0, since p "D~1(S(p )) is
t t~1
t
t~1
decreasing. Obviously, p "pH implies p "pH and L¸/Lp "0. In the case
t~1
t
t~1
p (pH, we get p "D~1(S(p ))'pH and since S is increasing it follows that
t~1
t
t~1
L¸/Lp (0. Similarly, when p 'pH, we get p "D~1(S(p ))(pH and
t~1
t~1
t
t~1
L¸/Lp '0. We conclude that the loss function ¸ has a unique (and global)
t~1
minimum 0 at p "pH. h
t~1
Now suppose we start from a situation where all producers choose the naive
predictor, and the price p is larger than, but close to, the steady state pH. The
0
next Lemma shows that when the information cost for rational expectations is
low enough, the equilibrium price drifts away from its steady state value until
the di!erence in realized pro"ts for rational and naive expectations exceeds the
information cost C.
¸emma 2. For an inxnite intensity of choice, and a suzciently low information cost
C, the rational expectations predictor becomes the optimal predictor after a xnite
amount of time.
Proof. Suppose (in contradiction) that in every period all producers remain
naive. The steady state is unstable and D~1 " S is decreasing, so without loss
of generality we may assume that pH(p (p (p (2(p and pH'
0
2
4
`
p 'p 'p '2'p . The limits p and p may be unbounded, but if they
1
3
5
~
`
~
are both "nite, Mp ,p N forms the &smallest' two-cycle of D~1(S( ) )), i.e.,
` ~
p "D~1(S( p )) and p "D~1(S(p )). According to Lemma 1, the loss
~
`
`
~
function ¸(p ; p ) has a unique minimum at pH, or, in other words, the
t t~1
di!erence in pro"ts is larger the more the prices di!er from pH. In even periods

778 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

the pro"t di!erence thus increases over time, with (possibly "nite) limit n .
`
Likewise, it increases in odd periods with limit n . If the information cost C is
~
su$ciently low these limits are greater than C, contradicting the assumption
that the naive predictor is optimal at all times. h
In the sequel we assume that the information cost C is less than the minimum of
n and n (the long run di!erences between realized pro"ts for rational and naive
~
`
expectations in odd and even periods respectively), when all producers are naive.9
According to Lemma 2, this assumption implies that at some point in time all
producers will switch to rational expectations in the neoclassical limit. When all
producers use the rational expectations predictor (m"1), the equilibrium price is
forced to pH. In subsequent periods the system remains at the steady state: p "pH
t
and m "!1. This observation is the content of the next lemma:
t
¸emma 3. For an inxnite intensity of choice and a suzciently low (but positive)
information cost C, all time paths in the ARE system (9), (10) converge to the steady
state S"( pH,!1), even though the latter is a locally unstable saddle point.
When a small amount of noise is added to the neoclassical limit case, the
system will be driven close to the steady state, but it does not collapse exactly
onto the steady state. Instead the noisy neoclassical limit is characterized by an
irregular switching between an unstable phase in which all agents are naive and
prices diverge from the steady state, and a stable phase in which all agents
become rational and prices return close to the steady state. As we show below
the same behavior arises in the deterministic, noise free case for a high, but "nite,
intensity of choice.
4.3. The unstable manifold of the steady state
The unstable manifold of the steady state plays a crucial role for understanding the global characteristics of the evolutionary dynamics. In this subsection,
we investigate the (geometric shape of the) unstable manifold of the steady state
in the neoclassical limiting case (b"R), and for large but "nite b-values. In the
case of linear demand and supply considered in BH there were only two
(symmetric) possibilities for the unstable manifold, depending on the ratio of
marginal supply and demand. In this subsection, we focus on the case of a linear
supply curve and a non-linear, decreasing demand curve. As we will see, there

9 For a general non-linear demand curve the map D~1(S( ) )) may have a 2-cycle Mp , p N, for
` ~
which the di!erences in realized pro"ts, or the pro"t losses, are ¸(p ; p ) and ¸(p ; p ). The
` ~
~ `
assumption thus implies that the information cost is less than the minimum of the loss function at
the two points of the 2-cycle, i.e., the assumption excludes a &cheap 2-cycle'.

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 779

are now four possible cases for the unstable manifold for large values of the
intensity of choice, the two previous symmetric cases and two additional
asymmetric cases. Fig. 3 summarizes the four possible cases and will be helpful
in understanding the details of the construction below. The general case of
non-linear supply and demand is more complicated, allowing for additional
asymmetric cases, and is discussed in the appendix.
Recall from Section 2 that for the linear supply curve (11), the loss function
¸ in (20) reduces to the quadratic function
b
¸(p ;p )" (p !p )2.
t t~1
t~1
2 t

(22)

By Lemma 1 and the assumption that the information cost is su$ciently low, we
know that there exists a point A on the line m"!1, with price component
0
p (A )'pH, such that the di!erence in realized pro"ts equals the information
0
cost: ¸(A ; A )"C, where we de"ned A "F (A ).10 Likewise, there exists
1 0
1
= 0
a point AH, with p(AH)(pH, such that ¸(AH; AH)"C. The point A (AH)
0
0
0
1 0
0
corresponds to the unique price above (below) the steady state price pH, where
all agents will switch from naive to rational expectations. Let A be the second
2
iterate of A , and AH be the second iterate of AH, that is, A "F2 (A ) and
0
= 0
2
2
0
AH"F2 (AH). Since ¸(A ; A )"(b/2)(p(A )!p(A ))2'(b/2) ( p(A )!p(A ))2
= 0
2 1
2
1
1
0
2
"C, we must have p (A )(p (AH)(pH. Similarly we have p(AH)'p(A )'pH,
1
0
0
1
as illustrated in Fig. 3. Next, consider what happens to the points A and AH.
2
2
For these cases, all agents are rational, i.e. m(A )"m(AH)"#1, so that
2
2
p(A )"p(AH)"pH. Consider the pro"t di!erence or loss function
3
3
M(p )"¸(pH; p )"pH S(pH)!c(S(pH))!pH S(p )#c(S(p )).
t~1
t~1
t~1
t~1
(23)
M( ) ) may be interpreted as the hypothetical loss of a naive agent, when all
agents are rational so that the actual price p becomes pH. Therefore, we call
t
M the loss function (associated to the naive predictor) under rational expectations. For linear supply S this loss function M simpli"es to the quadratic
function
b
(24)
M( p )"¸( pH; p )" (pH!p )2.
t~1
t~1
t~1
2
There are four cases to be distinguished, depending on whether the pro"t loss
M(A )"¸( pH; A ) and M(AH)"¸( pH; AH) is less than or greater than the
2
2
2
2
10 Here ¸(A ; A ) is shorthand notation for ¸(p(A ); p (A )). We will also adopt the notation
1 0
1
0
X "F (X ) or XH "F (XH), for any point X or XH. Points X and XH will always be on
i
i
= i
i`1
= i
i
i
i`1
H
opposite sides of the steady state p .

780 J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798

Fig. 3. The four cases for the unstable manifold of the steady state for a high b, in the case of
a quadratic cost function or linear supply curve.

information cost. Consider for instance the case that the "rst pro"t loss
M(A )"¸(pH; A )(C, whereas the second loss M(AH)"¸(pH; AH)'C (see
2
2
2
2
Fig. 3a). The other cases are treated similarly (see Figs. 3b}d).
The unstable direction of the steady state S"(pH,!1) is the horizontal axis,
and the line segment SA is part of the unstable manifold. It is now straightfor0
ward to calculate the "rst few iterates of this unstable segment:
F (SA ) " SA .
=
0
1
The part of the line segment SA that lies to the left of AH will be mapped onto
1
0
the line m"1. Let B denote the point on the line m"1, with the same p-value
0
as AH. Including the vertical segment AHB , where a discontinuous jump occurs,
1
1 0
the second iterate of the unstable segment SA is given by (see Fig. 3a)
0
F2 (SA ) " SAH B A .
=
0
1 0 2
For the third iterate we de"ne BH and C , that lie on the line m"1, and have the
1
0
same p-value as A , and AH respectively. The image of B A is the steady state
1
0 2
0
since the prices are mapped to zero when m"1, and M(A )"¸(pH; A )(C by
2
2
assumption for this case. We claim that there exists a point C on the segment
0

J.K. Goeree, C.H. Hommes / Journal of Economic Dynamics & Control 24 (2000) 761}798 781

AHB , such that F (AHC )"AHC and F (C B )"AHS. This follows from the
= 1 0
= 0 0
1 0
2 1
0
fact that F (AH)"AH, F (B )"S and for a quadratic loss function
= 1
= 0
2
¸(C ;C )"¸(AH; AH)"C. Hence,
1 0
1 0
F3 (SA ) " SA BHAH C AHS.
=
0
1 0 2 1 0
The fourth iterate is determined in a similar way. De"ne CH on the line m"1
1
with the same p-value as A
and recall that by assumption
0
M(AH)"¸(pH; AH