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 dynam-ics; 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).
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 arerationaland that rationality is common knowledge. Another frequently heard motivation for homogeneous beliefs is that heterogeneity in expectations among agents would lead to analyti-cally 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 selec-tion. 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 & perfor-mance' 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 de-tailed 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 arational route to random-ness, 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, equi-librium 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.
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 conver-gence 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 dynam-ical behavior in heterogeneous belief models. In these studies, two typdynam-ical 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 genera-tions model, and detect bifurcation routes to complicated dynamics as well.
(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 LOCBIF-program to detect the primary and secondary local bifurcations in generic rational routes to randomness. In the "nal section we end with some con-clusions. 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. SupplyS(p%t) is a function of the price expected by the producers,p%t, derived from expected pro"t maximization:
S(p%t)"argmax
qt
Mp%tqt!c(qt)N"(c@)~1(p%t). (1) The cost functionc()) 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), wherePt~1"(pt~1,pt~2,2,pt~L) denotes a vector of past prices
of lag-length¸, andH()) is called apredictor. In the case, when producers have rational expectations, or perfect foresight,H(Pt~1) equals the actual price,pt, for all times.
Consumer demandDdepends upon the current market pricep
t. Demand will
1Note 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 aremonotonic (Hommes, 1994, 1998).
maximization, but for our purposes it is not necessary to specify these prefer-ences 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 curveD, 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 expecta-tions predictorHR(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)"p
t~1, price dynamics is given by
p
t"D~1(S(pt~1)), which is the familiar textbook cobweb system. If demandDis decreasing and supplySis 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 statepH; otherwise, they diverge away from the steady state and either converge to a stable 2-cycle or exhibit unbounded up and down oscilla-tions.1
In this paper we investigate the dynamics of the cobweb model with
hetero-geneous beliefs. Instead of all producers using the same predictor, we assume
that each producer can choose between the two predictorsHRandHN. 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(p
t)"ftR~1S(pt)#ftN~1S(pt~1), (3)
whereftR~1andftN~1denote the fractions of agents using the rational respectively the naive predictor, at the beginning of periodt. 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!erenceCbetween the information costs for rational and naive expectations represents an extra e!ort cost producers incur over time when acquiring this perfect knowledge.
2The 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.
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 pricepH) 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.2For the rational expectations predictor, realized pro"t is given by
nRt"p
tS(pt)!c(S(pt)). (4)
Thenetrealized pro"t for rational expectations is thus given bynRt!C, where
Cis 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
nNt"ptS(pt~1)!c(S(pt~1)). (5)
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 periodtequals
ftR" exp(b(nRt!C))
exp(b(nR
t!C))#exp(bnNt)
, (6)
and the fraction of agents choosing the naive predictor in periodtis then
ftN"1!ftR. (7)
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 thatfR
t 'ftN
whenever nR
t!C'nNt, although the optimal predictor is not chosen with
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 casesb"0 andb"R. Forb"0, both fractions are"xed over time and equal to 1/2. The other extreme b"R, corresponds to the neoclassical
limitin which agents are unboundedly rational, andallproducers 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 limitb"Rwill play an important role in what follows.
t"!1 corresponds to all producers being naive, whereasmt"1 means
that all producers prefer the rational expectations predictor. The evolution of the equilibrium price,p
t, and the di!erence of fractions,mt, is then summarized
by the following two-dimensional, non-linear dynamical system
D(p
t)"12(1#mt~1)S(pt)#12(1!mt~1)S(pt~1), (9)
m
t"tanh(b(nRt!nNt!C)/2). (10)
The"rst equation de"nesp
timplicitly, in terms of (pt~1,mt~1); the monotonicity of demand and supply ensures that p
t is uniquely de"ned. The timing of
predictor selection in (9), (10) is important. In (9) the old (di!erence in) fractions are used to determine the new equilibrium pricep
t. Thereafter, this new
equilib-rium pricep
tis used in the evaluation of predictors according to their
evolution-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
t`1, etc.
BH termed the coupling (9), (10) between the equilibrium price dynamics and adaptive predictor selection anadaptive rational equilibrium(ARE) model. They restricted their advanced analysis of the ARE-dynamics to the special case of
lineardemand 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
S(p%
t)"bp%t, (11)
or equivalently a quadratic cost functionc(q)"q2/(2b). In particular, for a linear supply curve the di!erence in realized pro"ts is given by
nR
t!nNt"
b
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
t)"12(1#mt~1)bpt#12(1!mt~1)bpt~1, (13)
m
t"tanh
A
bC
b
2(pt!pt~1)2!C
DN
2B
. (14)In the sequel, we use the shorthand notation (p
t,mt)"Fb(pt~1,mt~1) for the 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 inp6, and the right-hand side is strictly increasing inp6, 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 pricepH. A straightforward computation shows that the eigenvalues of the Jacobian evaluated at the steady state are
j1"0, and
j2" (1!m6)S@(pH)
2D@(pH)!(1#m6 )S@(pH)(0. (15)
3For a de"nition of strange attractor and technical details, see Section 4.1.
topH, and the di!erence of fractions converges tom6. To allow for the possibility of an unstable steady-state and endogenous price#uctuations in the evolution-ary 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.
Proposition1. Under Assumption U,the evolutionary ARE-model satisxes:
(i) When information costs are zero(C"0),the steady state is locally stable for allb.
(ii) When information costs are strictly positive(C'0),there exists a criticalvalue
b1such that the steady state is stable for04b(b1and unstable forb'b1.At
b"b1the second eigenvalue satisxes j2"!1,andF
bin (9), (10)exhibits a period doubling bifurcation.
Proof. WhenC"0 the steady state is given by (p6,m6)"(pH, 0) and the second eigenvalue satis"es Dj2D"S@(pH)/(S@(pH)#2DD@(pH)D)(1, so the steady state is locally stable. For positive information cost the steady state is given by (p6,m6)"(pH,!tanh(bC/2)) and the second eigenvalue by (15). Whenb"0 the steady state reduces to (pH, 0) andj2is then smaller than one in absolute value. However, when b"R the steady state becomes (pH,!1) and the second eigenvalue satis"es j2"S@(pH)/D@(pH)(!1, by Assumption U. Both m6 and
j2depend continuously onb. Sincem6 is strictly decreasing inbandj2is strictly increasing inm6, there exists ab1such that!1(j2(0 forb(b1,j2"!1 forb"b1andj2(!1 forb'b1. h
3.2. Strange attractors
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 unsta-ble, and when costs for rational expectations are low but positive, the evolution-ary 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
t and di!erence in fractions mt. Numerical simulations
suggest that for (almost) all initial states (p0,m0) the orbit (p
t,mt) converges
4Intuitively, the co-dimension of a bifurcation is the minimum numberkof parameters such that the bifurcation occurs in generickparameter 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 matrixJF2bof the 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).
5In the case of a linear supply curve, the di!erence in fractions in (14) is given by m
t"tanh((b/2)[(b/2)(pt!pt~1)2!C]). Therefore, without loss of generality, we can choose the intersection of supply and demand as the origin and work in deviations from the steady state, or equivalently we may seta"0 in the non-linear demand curve (16).
3.3. Local bifurcations
In this subsection, we focus on generic bifurcation routes to complicated dynam-ics, 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 valueb"b
1, a 2-cycle bifurcates from the steady state. The situation for thesecondarybifurcation 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 bifur-cation is a highly degenerate bifurbifur-cation, occurring only in the special case of linear demand and supply. In fact, it is a co-dimension four bifurcation.4In this subsection we present possible co-dimension one secondary and consecutive local bifurcations in the general case ofnon-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:
D(p
t)"a!d1pt!d2p2t!d3p3t (16)
and supply is linear:S(p
t)"bpt, as in (11). The parametersdiare chosen such
that the demand in (16) is strictly decreasing. Notice that ford
2"d3"0 (16) 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 de-tailed 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 bifurca-tion diagram in the (d
2,b) parameter plane shown in Fig. 2. The other para-meters have been"xed atd
Fig. 2. Two-dimensional bifurcation diagram w.r.t. the intensity of choicebandd
2. The dotted line 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.
6As 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.
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 in-creases, 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 choicebincreases:6
1. d
2"0.01: hopf-fold-fold; 2. d
2"0.04:#ip-#ip-hopf-fold; 3. d
2"0.05:#ip-#ip-fold; 4. d
2"0.08:#ip-#ip.
In each of the"rst three scenario's, asbincreases, 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
7Brock 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.
other hand, close to the linear case (i.e., for small d
2), at least three possible 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.7In 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 numer-ical LOCBIF-analysis shows that, for generic non-linear demand and supply, this co-dimension four bifurcation &breaks' into two or three consecutive co-dimension 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 ofb. 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-calledhomoclinic points. This concept was intro-duced 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 manifoldof the steady state are de"ned as
W4(S)"
G
(p,m)K
limn?=
Fnb(p,m)"S
H
, (17)W6(S)"
G
(p,m)K
limn?~=
For a periodic saddle point (p,m), with period k, the stable and unstable manifold are de"ned similarly, by replacing F
b by Fkb. If Fb is a di!
eomor-phism (a smooth invertible function) the stable and unstable manifolds are smooth curves without self-intersections; if it is non-invertible the un-stable manifold may have self-intersections and/or the un-stable manifold may have more than one component. A transversal homoclinic point HOS, asso-ciated to the saddle S, is an intersection point of the stable and unstable manifold ofS. If the manifolds are tangent atH, it is called apoint 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 Fnb(H) is also an element of both the stable and un-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 thatF
bhas (in"nitely many) horseshoes,
that is, there exist rectangular regionsRsuch that for some positive integern, the image Fnb(R) is folded over R in the form of a horseshoe. Smale showed 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
b, undergoes ahomoclinic bifurcation associated to the (periodic) saddleSat
b"b
h, if W4(S)WW6(S)"Sforb(bhandW4(S)WW6(S) contains a
(homoc-linic) point HOS for b5b
h. The importance of a homoclinic bifurcation is
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
b be any smooth two-dimensional non-linear
map, which undergoes a homoclinic bifurcation associated to a saddle (periodic) point.
0Strange Attractor ¹heorem1. If G
bexhibits a homoclinic bifurcation associated
to a locally dissipative periodic point atb"b
h,then generically there exists a set
of b-values of positive Lebesgue measure for which the map G
b has a strange
8See 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.
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 ofGkbis less than one at the periodkpoints. The&strange attractor theorem'implies that, under generic conditions, for a large set ofb-values the dynamical behavior generated by the map G
b is chaotic.8 A recent economic application of the &strange
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 mani-folds of the steady state can be characterized analytically.
Forb"RandC'0, the steady stateS"(pH,!1), wherepHis 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 whenb"R
the fractionm
Hence in the neoclassical limit, in each periodallproducers choose the optimal predictor. Furthermore,p
t"D~1(S(pt~1)), when all producers choose the naive predictor, andp
t"pHwhen all use the rational expectations predictor.
For general demand and supply the di!erence in realized pro"ts is given by
¸(p
t;pt~1)"nRt!nNt"ptS(pt)!c(S(pt))!ptS(pt~1)#c(S(pt~1)). (20) This di!erence represents the loss producers face when their expected price is
p
t~1, whereas the actual price becomespt. Notice that under naive expectations
p%t"p
di!erence (20) as theloss function¸(p
t;pt~1) under naive expectations, since it represents the pro"t loss due to forecasting error when all agents are naive. In the case of a quadratic cost functionc()), or equivalently a linear supply curve
S"(c@)~1, the loss function has the particularly simple quadratic form
nR
t!nNt"
b
2(pt!pt~1)2. (21)
The next lemma states that the loss function¸has a unique minimum at the steady statep
t~1"pH:
¸emma1. The loss function ¸(p
t;pt~1), with actual price pt"D~1(S(pt~1)), has
a unique(global)minimum 0 at the steady statep
t~1"pH.
t~1'0. We conclude that the loss function¸has a unique (and global) minimum 0 atp
t~1"pH. h
Now suppose we start from a situation where all producers choose the naive predictor, and the pricep
0is larger than, but close to, the steady statepH. The 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 costC.
¸emma2. For an inxnite intensity of choice,and a suzciently low information cost
C,the rational expectations predictor becomes the optimal predictor after axnite
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
9For a general non-linear demand curve the mapD~1(S())) may have a 2-cycleMp`,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'.
the pro"t di!erence thus increases over time, with (possibly "nite) limit n
`.
Likewise, it increases in odd periods with limitn~. If the information costCis 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 costCis less than the minimum of
n~andn
`(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 topH. In subsequent periods the system remains at the steady state:p
t"pH
andm
t"!1. This observation is the content of the next lemma:
¸emma3. For an inxnite intensity of choice and a suzciently low (but positive)
information costC,all time paths in the ARE system(9), (10)converge to the steady
stateS"(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
10Here ¸(A
1;A0) is shorthand notation for¸(p(A1);p(A0)). We will also adopt the notation X
i`1"F=(Xi) orXHi`1"F=(XHi), for any pointXiorXHi. PointsXiandXHi will always be on
opposite sides of the steady statepH.
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
¸(p
t;pt~1)"
b
2(pt!pt~1)2. (22)
By Lemma 1 and the assumption that the information cost is su$ciently low, we know that there exists a pointA
0on the linem"!1, with price component
p(A
0)'pH, such that the di!erence in realized pro"ts equals the information cost: ¸(A
1;A0)"C, where we de"ned A1"F=(A0).10 Likewise, there exists a point AH
0, with p(AH0)(pH, such that ¸(AH1;AH0)"C. The point A0 (AH0) corresponds to the unique price above (below) the steady state pricepH, where all agents will switch from naive to rational expectations. LetA
2be the second as illustrated in Fig. 3. Next, consider what happens to the pointsA
2andAH2. For these cases, all agents are rational, i.e. m(A
2)"m(AH2)"#1, so that
M()) may be interpreted as the hypothetical loss of a naive agent, when all agents are rational so that the actual price p
t becomes pH. Therefore, we call
Mthe loss function (associated to the naive predictor) under rational expecta-tions. For linear supply S this loss function M simpli"es to the quadratic function
M(p
t~1)"¸(pH;pt~1)"
b
2(pH!pt~1)2. (24)
There are four cases to be distinguished, depending on whether the pro"t loss
M(A
Fig. 3. The four cases for the unstable manifold of the steady state for a highb, 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
2)"¸(pH;A2)(C, whereas the second loss M(AH2)"¸(pH;AH2)'C (see Fig. 3a). The other cases are treated similarly (see Figs. 3b}d).
The unstable direction of the steady stateS"(pH,!1) is the horizontal axis, and the line segmentSA
0is part of the unstable manifold. It is now straightfor-ward to calculate the"rst few iterates of this unstable segment:
F
=(SA0)"SA1.
The part of the line segmentSA
1that lies to the left ofAH0will be mapped onto the linem"1. LetB
0denote the point on the linem"1, with the samep-value asAH
1. Including the vertical segmentAH1B0, where a discontinuous jump occurs, the second iterate of the unstable segmentSA
0is given by (see Fig. 3a)
F2=(SA
0)"SAH1B0A2. For the third iterate we de"neBH
0andC1, that lie on the linem"1, and have the samep-value asA
1, andAH0respectively. The image ofB0A2is the steady state since the prices are mapped to zero whenm"1, andM(A
2)"¸(pH;A2)(Cby assumption for this case. We claim that there exists a pointC
AH
The fourth iterate is determined in a similar way. De"neCH
1 on the linem"1 with the same p-value as A
0 and recall that by assumption
M(AH
2)"¸(pH;AH2)'C, so that the image ofAH2is the pointP,(pH,1). We have (vertical segments where discontinuous jumps occur included again)
F4=(SA0)"SAH
1B0A2CH1A0S P S AH1S. (25) The fourth iterate of SAH
0 may be found following a similar reasoning. In contrast to the symmetric case of linear supply and demand considered by BH,
F4
b(SAH0) is not simply the mirror image ofF4b(SA0). In fact, in the asymmetric case under consideration, the assumption thatM(A
2)"¸(pH;A2)(C, implies
The geometric structure of the unstable manifold in the neoclassical limit
b"Ris illustrated in Fig. 3a, where we also plotted the unstable manifold for
blarge but"nite. Note how the shape of the strange attractor in Fig. 1 resembles that of the unstable manifold of the steady state.
Other possibilities for the geometric structure of the unstable manifold follow immediately, by replacingPbyS, and/or vice versa, in (25) and (26). Figs. 3a}d illustrate all four possible cases for the unstable manifold of the steady state for highb. The two&symmetric'cases in Figs. 3c and d are the only two cases arising for linear supply and linear demand, as in BH. In the case of linear supply and arbitrary non-linear, decreasing demand, we thus"nd two additional asymmet-ric cases (Figs. 3a and b), as discussed above.
4.4. Strange attractors for high,butxniteb-values
In this section, we sketch the proof of the existence of strange attractors in the ARE-model (13), (14) with a linear supply and a non-linear, decreasing demand curve, for high but "nite values of the intensity of choice. Using a continuity argument as in BH, it follows that forbsu$ciently large, the unstable manifold of the steady state gets arbitrarily close to the piecewise linear segments
F4=(SAH
non-linear, decreasing demand curve, and a high intensity of choice b, the ARE-system (13), (14) is thus close to a homoclinic tangency between the stable and the unstable manifolds of the steady state. On the other hand, there can be nohomoclinic intersection point between the stable and unstable manifolds of the steady state. This follows from the observation that, for "nite b, the only points (p
0,m0) mapped onto the stable segmentp"pHhave to satifyp0"pHor
m
0"#1. It can be shown however, that the non-linear ARE model displays homoclinic bifurcations associated toperiodicsaddle points:
Homoclinic Bifurcation Lemma. For a linear supply curve and any non-linear,
decreasing demand curve, a suzciently low information costC,and N52large
enough,there exists abhfor which the ARE-model(13), (14)exhibits a homoclinic
bifurcation associated to a dissipative period2N saddle point.
This lemma can be proven by applying the same&horseshoe'construction, as in BH for the case of linear demand and supply. From this geometric construc-tion it follows that horseshoes and homoclinic bifurcaconstruc-tions of periodic saddle points arise when the intensity of choice becomes su$ciently large. In each of the two&symmetric'cases in Figs. 3c and d, the unstable manifolds are essentially the same as in the two possible perfectly symmetric cases for linear demand and supply, so that BH's horseshoe construction can be applied immediately. In each of the two asymmetric cases in Figs. 3a and b, we have indicated a rectangular regionRwhich is mapped byFNb, for someN52 su$ciently large, over itself in the form of a horseshoe. BH's&horseshoe'construction applies here, because in each of the asymmetric cases, either to the right or to the left of the linep"pH, the geometric shape of the unstable manifold of the steady state is essentially the same as in one of the two possible&symmetric'cases (Figs. 3c and d) that occur for linear demand and supply.
Applying the &strange attractor theorem' discussed in Section 4.1, we thus conclude that:
Corollary. In the case of a linear supply curve and generic non-linear decreasing
demand curves, such that assumption U is satisxed, and a suzciently low but
positive information cost C,the ARE-model (13), (14)has a strange attractor for
a set ofb-values of positive Lebesgue measure.
market instability and diverging prices. Price#uctuations on the strange attrac-tors are characterized by an irregular switching between cheap free riding and costly sophisticated prediction.
5. Local bifurcation analysis
In the previous section, we have shown that for high values of the intensity of choice b, the ARE dynamics becomes very complicated. In this section we investigate the following problem: what are the generic primary and secondary bifurcations in the bifurcation routes to complicated dynamics asbincreases, in the general case of non-linear, monotonic demand and supply curves?
We use the LOCBIF-computer program for numerical analysis of bifurcation routes to complexity. The LOCBIF-program is a powerful tool for local bifurca-tion analysis of dynamical systems depending upon parameters; see Khibnik et al. (1992, 1993) for a description. The program is able to detect steady states and periodic points, and more importantly is able to compute, follow and draw bifurcation curves (e.g., Hopf, saddle node or fold, period doubling or #ip bifurcation curves, etc.) of steady states and cycles in parameter space. The program is based upon continuation techniques for the relevant local bifurca-tion curves.
According to Proposition 1 in Section 3.1, the primary bifurcation is always a period doubling or #ip bifurcation of the steady state, so we focus on the secondary and consecutive bifurcations. BH (1997, Theorem 3.4, pp.1080}81) have shown that, in the case of linear demand and supply, asb increases, the secondary bifurcation is a 1:2 strong resonance Hopf bifurcation of the 2-cycle, in which four coexisting 4-cycles, (two stable 4-cycles and two 4-saddles) are created simultaneously, as illustrated in Fig. 4. Fig. 4b illustrates the behavior of the corresponding eigenvalues of the Jacobian matrix of the second iterateF2
bat
the period 2 points, as b crosses the bifurcation value b2. For b(b2 the eigenvalues are complex, inside the unit circle, forb"b2both eigenvalues are real and equal to!1 and forb'b2the eigenvalues are complex, outside the unit circle. At the bifurcation valueb2, the Jacobian matrixJF2
bof the second
Fig. 4. The bifurcation scenario for linear supply and demand. (a) The vertical axis represents the bifurcation parameter, the intensity of choiceb, and the horizontal axes represents the state variable, the pricep, of the attractor of the ARE system. Period doubling of the steady state followed by a 1:2 strong resonance Hopf bifurcation of the 2-cycle atb"b
2+1.84; (b) eigenvalues at the 2-cycle for
bclose to the secondary bifurcation valueb
2.
In order to investigate generic co-dimension one bifurcation routes it turns out to be su$cient to focus on the case of a linear supply S(p%t)"bp%t and a non-linear demand curve D(p
t)"a!d1pt!d2p2t!d3p3t. Notice that for
d
the parametersd
1"0.5,d3"0.1,b"2,C"1 (unit information cost) anda"0 and investigate the bifurcation scenario with respect to the intensity of choice
bfor di!erent values ofd
2(see footnote 5).
5.1. Symmetric non-linear demand
For linear supply and demand the mapF
bis symmetric with respect to the
m-axis. More precisely,f(!p,m)"!f(p,m), wherefdenotes the"rst compon-ent of the mapF
b. Before dealing with the case of a general non-linear demand
curve,"rst consider the case with non-linear, but symmetric demand (16), i.e., the case d2"0 and d3O0. For a symmetric non-linear demand curve, the sym-metry w.r.t. them-axis is preserved, i.e.f(!p,m)"!f(p,m). Fig. 5 illustrates the bifurcation scenario in this case and Fig. 5b the behavior of the correspond-ing eigenvalues at the 2-cycle. As before, the primary bifurcation is a #ip bifurcation of the steady state in which a stable 2-cycle is created. However, in the non-linear symmetric case, the secondary 1:2 strong resonance Hopf bifurca-tion of the 2-cycle does not occur, but instead two bifurcabifurca-tions occur at di!erent bifurcation values. First, forb+1.92 the usual (weak resonance) Hopf bifurca-tion of the 2-cycle occurs in which an attractor consisting of a symmetric pair of closed curves is created, with periodic or quasi-periodic dynamics. At the bifurcation value b+1.92, the corresponding eigenvalues are complex conju-gates on the unit circle. Thereafter, forb+1.96 a saddle node or fold bifurcation occurs in which two stable 4-cycles and two saddle 4-cycles are created simulta-neously.
In the non-linear, symmetric case, the 1:2 strong resonance Hopf bifurcation of the linear case thus&breaks'into two di!erent bifurcations, a Hopf and a fold bifurcation. The Hopf bifurcation is a generic co-dimension one bifurcation. However, the fold bifurcation in which four 4-cycles are created simultaneously is non-generic. Due to the symmetry w.r.t. the pH-axis, in the fold bifurcation a symmetric pair of 4-sinks and a symmetric pair of 4-saddles are created simultaneously.
5.2. Asymmetric non-linear demand
Next consider the general case of non-linear, non-symmetric demand, i.e., when bothd
2andd3in (16) are non-zero. Using LOCBIF, we investigate which bifurcation scenarios can arise as the intensity of choice increases, for di!erent values ofd2with the other parameters"xed as before. As stated above, in all cases the primary bifurcation is a#ip bifurcation of the steady state leading to the creation of a stable 2-cycle.
Fig. 5. The bifurcation scenario for symmetric non-linear demand. (a) Hopf bifurcation followed by single saddle-node bifurcation leading to four 4-cycles. (b) eigenvalues at the 2-cycle close to the Hopf bifurcation.
case, i.e., to an unstable, saddle point steady state, an unstable, repelling 2-cycle (a source), two stable 4-cycles (sinks) and two 4-saddles. The corresponding "gures will be helpful in understanding the di!erent bifurcation scenarios.
1. Hopf-fold-fold(d
Fig. 6. Bifurcation scenario 1: Hopf-fold-fold. (a) Hopf bifurcation followed by two di!erent saddle-node bifurcations. (b) eigenvalues at the 2-cycle close to the Hopf bifurcation.
followed by a second fold bifurcation in which the second 4-saddle-sink pair is created. The di!erence with the non-linear symmetric case is thus that the two 4-sinks and 4-saddles are not created in one single, but in two indepen-dent fold bifurcations.
2. Flip-yip-Hopf-fold(d
Fig. 7. Bifurcation scenario 2:#ip-#ip-Hopf-fold. (a) two consecutive#ip bifurcations, followed by a Hopf and a fold bifurcation. (b) eigenvalues at the 2-cycle close to the two#ip and the Hopf bifurcation.
of the 2-cycle, as b increases, is much more complicated, as illustrated in Fig. 7b. In particular, asbincreases the stability of the 2-cycle changes three times in this scenario.
3. Flip-yip-fold (d
2"0.05). This scenario is similar to scenario 2, except that a Hopf bifurcation of the 2-cycle does not occur and the stability of the 2-cycle changes only once. The secondary bifurcation is a #ip, in which the 2-cycle becomes a saddle and a stable 4-cycle is created. Thereafter, the saddle 2-cycle exhibits a second#ip bifurcation (the second eigenvalue also crosses
!1), becomes a repellor and at the same time a 4-saddle is created. Finally, the second 4-saddle-sink pair is created in a fold bifurcation.
These three di!erent bifurcation scenarios have been detected by the LOCBIF numerical analysis, for linear supply and non-linear demand. In this three parameter family of non-linear demand curves, these are the only co-dimension one bifurcation routes for b that we have observed. In addition, we present a fourth scenario which we did not observe in our family, but which would be another possible co-dimension one bifurcation route from a stable steady state to co-existing pairs of 4-saddles and 4-sinks.
1. Flip-yip-yip-yip. This scenario consists of four consecutive#ip bifurcations of
the 2-cycle (after the primary#ip bifurcation of the steady state). In the"rst, the 2-cycle becomes a saddle and a stable 4-cycle is created. In the second#ip bifurcation, the 2-cycle becomes stable again and a 4-saddle is created. These "rst two#ip bifurcations are the same as in the second scenario, but this time they are followed by two more#ip bifurcations. In the third#ip bifurcation, the 2-cycle becomes a saddle again and a second stable 4-cycle is created. Finally, in the fourth#ip bifurcation, the two-saddle becomes a repellor and a second 4-saddle is created.
5.3. A two-dimensional bifurcation diagram
We have found at least three di!erent co-dimension one bifurcation routes, with respect to the intensity of choice b, from a stable steady state to two co-existing 4-sinks and two 4-saddles. To get a more comprised overview of these di!erent bifurcation scenarios, the reader should have a look again at Fig. 2 in Section 3.3. The Figure shows a two-dimensional bifurcation diagram in the (d
2,b) parameter plane, with the other parameters"xed as before (a"0,
#ip-curves are 1:2 strong resonance Hopf bifurcation points. Furthermore, both fold bifurcation curves of the 4-cycles end on the#ip bifurcation curves of the 2-cycle in co-dimension two&generalized#ip'bifurcation points (see e.g. Kuznet-sov, 1995, pp. 354}357).
Fig. 2 gives a nice overview of the di!erent scenarios of the previous subsec-tion. For instance, the line d
2"0 represents the non-linear symmetric case. Notice that this vertical line crosses the two fold-curves exactly at their intersec-tion point, so that two 4-saddle-sink pairs are created simultaneously in one single bifurcation. The bifurcation scenarios 1, 2 and 3 discussed above follow by considering the vertical lines d
2"0.01, d2"0.04 and d2"0.05 respectively. For large values ofDd2Donly two#ip bifurcations of the 2-cycle occur and only one 4-sink and one 4-saddle are created. Hence, far from the linear case co-existing stable 4-cycles do not necessarily arise asbincreases. On the other hand, close to the linear case (i.e., for Dd
2D small), at least three possible co-dimension one bifurcation routes can occur, from a stable steady state to two co-existing 4-sinks and two 4-saddles. In the linear case, the four 4-cycles are created simultaneously in one single, co-dimension four bifurcation. Our numer-ical LOCBIF-analysis shows that, for non-linear demand and supply, this co-dimension four bifurcation&breaks'into three or four consecutive co-dimen-sion one bifurcations.
6. Concluding remarks
We have investigated the evolutionary dynamics in the cobweb model with costly rational versus cheap naive expectations, in the case of non-linear, monotonic demand and supply curves. Predictor choice is determined by the logit discrete choice model. As the intensity of choice to switch prediction strategies increases, bifurcation routes to complexity from a stable steady state, through co-existing stable 4-cycles, to strange attractors arise. For non-linear monotonic demand and supply, a high rationality (i.e., high intensity of choice) thus leads to an irregular switching between cheap, destabilizing free riding and costly, stabilizing sophisticated prediction. The rational route to randomness, detected by Brock and Hommes (1997a) for the linear case, thus appears to be a general feature of the law of demand and supply.
the rational route to randomness arises when the intensity of choice, i.e. the parameterbin (6), tends to in"nity. A large intensity of choice simply means that the traders'sensitivity to di!erences in"tness is high, and consequently the mass of traders clumps onto the&best'predictor. We expect that for any evolutionary updating rule with the same ranking as the discrete choice rule, when the sensitivity to di!erences in"tness tends to in"nity, a similar rational route to randomness will occur in the corresponding evolutionary cobweb model.
We also would like to discuss the application of the adaptive evolutionary framework to other equilibrium models. An ARE-model is a coupling between equilibrium dynamics and evolutionary selection of prediction strategies. Ap-plication to di!erent equilibrium models should thus in general yield di!erent dynamical behavior. It seems that the number of rational expectations equilibria (REE) in the equilibrium model is closely related to the characteristic features of the corresponding ARE evolutionary dynamics. In the cobweb model with monotonic demand and supply, the rational expectations (or perfect foresight) equilbrium is unique: it is the unique steady state equilibrium price pH corre-sponding to the intersection of demand and supply. In order for the evolution-ary system to generate endogenous price #uctuations at all, a cost for sophisticated prediction rules is crucial. If there were freely available prediction strategies leading to the unique REE, in the evolutionary system all traders will clump to that rule, and the system will simply converge to the unique steady state REE. With information costs for sophisticated, stabilizing prediction strategies, the evolutionary system will exhibit irregular switching between cheap free riding with diverging prices and sophisticated prediction strategies with prices pushed back close to the unique REE steady state.
In contrast, for example in the well-known present discounted value asset pricing model there are two di!erent types of REE, namely the (constant) REE-fundamental price, given by the discounted sum of expected future divi-dends, and the rational, perfect foresight&bubble solutions', growing exponenti-ally at the risk free rate of return. The corresponding ARE evolutionary system, considered in Brock and Hommes (1997b,1998), can generate endogenous #uctuations when the traders'sensitivity to di!erences in"tness becomes high, even when costs for fundamentalists are zero. Moreover, chaotic asset price #uctuations, characterized by an irregular switching between phases of close to the fundamental prices and phases of temporary bubbles, can arise. In the asset pricing model, the complicated ARE evolutionary dynamics is thus character-ized by an irregular switching between the two di!erent REE equilibria.
would be interesting to apply the ARE-framework to this model. It is tempting to conjecture that the ARE-dynamics in this exchange rate equilibrium model exhibits a rational route to randomness, as the sensitivity to di!erences in"tness increases, with chaotic exchange rate #uctuations characterized by irregular switching between the continuum of REE equilibria.
Adaptive rational equilibrium dynamics is a way of modeling evolutionary competition in a market with heterogeneous traders. Sensitivity to di!erences in "tness may lead to market instability and endogenous#uctuations. Evolution-ary pressure between competing trading or prediction strategies may thus explain (part of) the excess volatility observed so frequently, especially in "nancial markets.
Appendix
In the analysis of the global evolutionary dynamics we have focussed on the case of a linear supply curve, or equivalently a quadratic cost function. In this Appendix we discuss the more general case of anon-linear, increasing supply curve, or equivalently a non-quadratic, convex cost function. We start with two general remarks. First, as long as a non-linear, increasing supply curve is close to a linear one, or equivalently as long as the cost function is close to a quadratic function, all results concerning the global dynamics (e.g. homoclinic bifurcations and strange attractors) of Section 4 remain valid. Second, we note that through-out the paper we have used (most recent) realized pro"ts as the&"tness measure' for predictor selection. Another possibility would have been the (most recent)
squared prediction error, as in Brock and deFontnouvelle (1998) in the case of an
OLG-model. With squared prediction error as the "tness measure, for any non-linear, increasing supply curve and any non-linear, decreasing demand curve, all results concerning the global dynamics in Section 4 remain valid, with four possibilities for the geometric shape of the unstable manifold of the steady state, as illustrated in Fig. 3. As argued below, with realized pro"ts as the"tness measure and a non-linear, increasing supply curve, the extra complications in the analysis of the global dynamics arise because the "tness measure is not necessarily (close to) a quadratic function.
The geometric shape of the unstable manifold of the steady state, for high values of the intensity of choice, plays an important role in the analysis of the global dynamics. This geometric shape is in fact determined by the di!erence in pro"ts between rational and naive agents (see the construction of the unstable manifold in Section 4.3). The global shape of the graphs of the loss functions
¸andM, determine the global geometric shape of the unstable manifold of the steady state. Recall that the loss functions¸andMare given by
¸(p
and
M(p
t~1)"¸(pH;pt~1)"pHS(pH)!c(S(pH))!pHS(pt~1)#c(S(pt~1)). (A.2)
In the case of a quadratic cost functionc()), or equivalently a linear supply curve
S"(c@)~1, these loss functions have the particularly simple quadratic forms
¸(p
Due to this simple quadratic form of the loss functions, the unstable manifold of the steady state is as in one of the four cases in Fig. 3.
For a general, non-linear supply curveSthe loss functions¸andMare not quadratic. In particular, the loss functions may be strongly asymmetric, leading to additional asymmetric cases for the unstable manifold of the steady state. The question which geometric shapes of the unstable manifold of the steady state can arise, thus boils down to the question of all possible global shapes of these loss functions. According to Lemma 1 of Section 4.2, the loss function ¸ has a unique, global minimum atp
t~1"pH. The following Lemma states that, for general, non-linear monotonic supply and demand curves, the loss function
M also has a unique minimum 0 at p
t~1"pH, and the loss ¸ under naive expectations is always larger or equal than the lossMunder rational expecta-tions:
SinceS@ is always positive, it follows immediately that LM/Lp
S@is always positive andLp ence in loss functions¸!Mhas a unique, global minimum 0 atp
t~1"pH, and consequently¸(p
t;pt~1)4M(pt~1), for allpt~1. h
Fig. 8 illustrates three cases of the loss function¸andM, and the correspond-ing unstable manifolds of the steady state for high values of the intensity of choice. Figs. 8a and b represent the case of a (close to) linear supply curve, so that¸andMare (close to) quadratic functions. This case is characterized by the following properties (see the construction in Section 4.3):
(a)¸(A
These properties are satis"ed foranyquadratic loss functions¸andM. Recall from Section 4.3 thatA
0and AH0 are the points where all agents switch from naive to rational expectations, that is, ¸(A
1;A0)"¸(AH1;AH0)"C. For general loss functions¸andM, with their minimum 0 atpH, either (a) or (b) is satis"ed, but not necessarily both. Below we discuss an example where (a) is not satis"ed, and only (b) is satis"ed. In general, properties (c) and (d) need not be satis"ed; below we discuss another example for which neither (c) nor (d) is satis"ed.
Figs. 8c and d illustrate the case ¸(A the intensity of choice, the unstable manifold of the steady state has a strongly asymmetric geometric shape, as illustrated in Fig. 8d.
Figs. 8e and f illustrate the case where properties (c) and (d) are not satis"ed, i.e. whenM(A
1)'CandM(AH1)'C. The"rst part of the construction of the unstable manifold for high intensity of choice is as in Section 4.3. In particular, the second iterate of the unstable segmentSA
11It is remarkable that the same unstable manifold of the steady state can also arise (as one of the three possibilities) in the case of costly fundamentalists (expecting prices to return to the steady state pH) versus free naive expectations, with linear supply and demand, considered in the working paper version Brock and Hommes (1995).
12There now seem to be at least 15 cases for the unstable manifold of the steady state, obtained by combining the four possibilities of the unstable manifold to the left ofpHwith the four possibilities to the right ofpH, with the case of an unstable manifold close to the linem"!1 on both sides of pHexcluded when information costs are su$ciently low.
follows that, for high values of the intensity of choice, the unstable manifold of the steady state is as illustrated in Fig. 8f.11
For general non-linear monotonic supply curves, we thus get additional (asymmetric) cases for the unstable manifold of the steady state. Does in each of these additional cases a homoclinic bifurcation route to strange attractors arise, as the intensity of choice increases?
In order to address this problem,"rst observe that in all cases, for high values of the intensity of choice, the system must be close to a homoclinic orbit. In fact, this follows already from Lemma 3, stating that forb"Rall orbits converge to the saddle point steady state. This lemma implies that forblarge, the unstable manifold of the steady state"rst moves away from the steady state, along the unstable segmentsSA
0andSAH0, and thereafter returns close to the steady state; see Figs. 8b, d and f. Since the stable manifold contains the vertical segment
p"pH, for large b-values the system must be close to a homoclinic tangency. However, to prove that this close to homoclinic tangency implies homoclinic bifurcations (associated to periodic saddle points) is a delicate matter and requires a modi"cation of BH's horseshoe construction to each of the additional asymmetric cases for the unstable manifold of the steady state.12In most cases, such a modi"cation of the horseshoe construction is straightforward, in particu-lar, when the left and/or right part of the unstable manifold is as in Figs. 3a}d. Furthermore, for example in the strongly asymmetric case in Fig. 8d, a modi" ca-tion of the horseshoe construcca-tion is also fairly straightforward. In Fig. 8d, we indicated a rectangular region which, after a number of iterates will be folded over itself in the form of a horseshoe, and thus can be used for a horseshoe construction as in BH. Unfortunately, in the case of Figs. 8e}f it is not clear whether a horseshoe construction as in BH is possible. Horseshoes are charac-terized by a mixture of stretching in one direction, contraction in another direction and folding. Stretching and contraction seems to occur in Fig. 8f, but it is not clear whether in general there will be enough folding to produce chaotic dynamical behaviour in this particular case.
Fig. 8. Graphs of the loss functions¸andMand the corresponding unstable manifolds, for (a,b) (close to) linear supply, and (c,f) non-linear (asymmetric) supply curves.
evolutionary#uctuations in this particular case thus remains unclear. Neverthe-less, except for this one particular case, BH's rational route to randomness is a characteristic feature of the non-linear law of demand and supply.
References
