heterogeneous preferences
A. K. Naimzada, and M. Pireddu
Citation: Chaos 28, 055907 (2018); doi: 10.1063/1.5024931 View online: https://doi.org/10.1063/1.5024931
View Table of Contents: http://aip.scitation.org/toc/cha/28/5
Fashion cycle dynamics in a model with endogenous discrete evolution
of heterogeneous preferences
A. K.Naimzada1,a)and M.Pireddu2,b)
1
Department of Economics, Management and Statistics, University of Milano-Bicocca, Piazza dell’Ateneo Nuovo 1, 20126 Milano, Italy
2
Department of Mathematics and Its Applications, University of Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy
(Received 5 February 2018; accepted 18 April 2018; published online 21 May 2018)
We propose a discrete-time exchange economy evolutionary model, in which two groups of agents are characterized by different preference structures. The reproduction level of a group is related to its attractiveness degree, which depends on the social visibility level, determined by the consumption choices of the agents in that group. The attractiveness of a group is initially increasing with its visibility level, but it becomes decreasing when its visibility exceeds a given threshold value, due to a congestion effect. Thanks to the combined action of the price mecha-nism and of the share updating rule, the model is able to reproduce the recurrent dynamic behav-ior typical of the fashion cycle, presenting booms and busts both in the agents’ consumption choices and in the population shares. More precisely, we investigate the existence of equilibria and their stability, and we perform a qualitative bifurcation analysis on varying the parameter describing the group’s heterogeneity degree. From a global viewpoint, we detect, among others, multistability phenomena in which the group coexistence is dynamic, either regular or irregular, and the fashion cycle occurs. The existence of complex dynamics is proven via the method of the turbulent maps, working with homoclinic orbits. Finally, we provide a social and economic inter-pretation of the main scenarios.Published by AIP Publishing.https://doi.org/10.1063/1.5024931
According to Simmel (1957), two contrasting tendencies operate in determining peoples’ behavior towards fashion: imitation and distinction. Those two opposite forces drive the fashion cycle, i.e., the oscillating behavior of the vari-able describing the consumed or purchased amount of a certain good, characterized by booms and busts. In the past decades, some dynamical models have been proposed to give a formal representation of the fashion cycle [see, for instance, Karni and Schmeidler (1990); Matsuyama (1991); Coelho and McClure (1993); Pesendorfer (1995); Frijters (1998); Corneo and Jeanne (1999); Bianchi (2002); Caulkins et al. (2007); Di Giovinazzo and Naimzada (2015); andZhang (2016; 2017)]. Unlike those works, we here present an evolutive general equilibrium model, in which we may investigate the joint effects of the price formation mechanism and of the population share updating mechanism, which describe the socio-economic interaction of heterogeneous agents exhibiting both band-wagon and snob behaviors. Indeed, our starting point is given by a simple general equilibrium model, which we make dynamical by introducing an evolutive rule, portray-ing the imitative and snob social behaviors. We investigate the existence and local stability of the equilibria for our model and, in order to illustrate the possible scenarios, we perform a qualitative bifurcation analysis in terms of the parameter describing the heterogeneity degree between groups. From a global viewpoint, in addition to multi-stability phenomena involving just equilibria, where the
coexistence between the groups is static in nature, we also find oscillatory behaviors in the population shares and in the consumption quantities. Indeed, thanks to the com-bined effect of the evolutionary and of the price formation mechanisms, the group coexistence may be dynamic in nature, either regular or irregular, due to the presence of chaotic attractors. The dynamic coexistence between groups then displays the recurrent dynamic behavior typi-cal of the fashion cycle, characterized by booms and busts. We complete our analysis by proving the existence of com-plex dynamics for the model via the method of the turbu-lent maps in Block and Coppel (1992), working with homoclinic orbits. Finally, we give a social and economic interpretation of the main scenarios we found.
I. INTRODUCTION
In human societies, individuals wish to distinguish themselves from others and fashion allows them to do so, as it signals their status. On the other hand, fashion also induces imitative behaviors. Indeed, according to Simmel (1957), two contrasting tendencies operate in determining people behavior towards fashion:
Just as soon as the lower classes begin to copy their style, thereby crossing the line of demarcation the upper classes have drawn and destroying the uniformity of their coherence, the upper classes turn away from this style and adopt a new one, which in its turn differentiates them from the masses; and thus the game goes merrily on (Simmel, 1957, p. 545).
a)Electronic mail: ahmad.naimzada@unimib.it.
b)Author to whom correspondence should be addressed: marina.pireddu@ unimib.it.
Namely, to follow fashion makes people feel accepted and socially integrated, answering to their innate tendency for conformity. This produces imitation of others, that is, the so called bandwagon effect. However, when too many indi-viduals choose the same commodities, e.g., the positional goods, then those become less desirable, because they confer minor distinction, and need to be replaced by new ones. Such phenomenon is known as snob effect. Those two oppo-site forces drive the fashion cycle, i.e., the oscillating behav-ior of the variable describing the consumed or purchased amount of a certain good, characterized by booms and busts.
In the past decades, some dynamical models have been proposed to give a formal representation of the fashion cycle [see, for instance,Karni and Schmeidler (1990);Matsuyama (1991); Coelho and McClure (1993); Pesendorfer (1995); Frijters (1998);Corneo and Jeanne (1999); Bianchi (2002); Caulkinset al. (2007);Di Giovinazzo and Naimzada (2015); andZhang (2016; 2017)]. Oscillatory dynamics in the con-sumption activities are found, in different contexts, also in Antoci et al. (2004); Bischi and Radi (2012); Naimzada et al. (2013); andOnozaki (2018).
Differently from such works, we here present an evolu-tive general equilibrium model, in which we may investigate the combined effects of the price formation mechanism and of the population share updating mechanism, which describe the socio-economic interaction of heterogeneous agents exhibiting both bandwagon and snob behaviors. Indeed, our starting point is given by a simple general equilibrium model, which we make dynamical by introducing an evolu-tive rule, portraying the imitaevolu-tive and snob social behaviors. Thanks to the joint action of the two mechanisms, our model is able to reproduce the recurrent dynamic behavior typical of the fashion cycle. More precisely, following the line of research started in Chang and Stauber (2009) and further developed inNaimzada and Pireddu (2016; 2018b; 2018c), we deal with an exchange economy evolutionary model with agents heterogeneous in the structure of preferences. Namely, the weights assigned to the two consumption goods in the Cobb-Douglas utility functions do not coincide across groups. Along the paper, we shall call the difference between such weights degree of heterogeneity. However, unlike Chang and Stauber (2009)andNaimzada and Pireddu (2016; 2018b;2018c), we here consider a setting in which the share updating rule is based on sociological, rather than biological, aspects. Furthermore, in Naimzada and Pireddu (2016; 2018b), it was assumed that time is continuous, while we now deal with a discrete-time framework. Also inNaimzada and Pireddu (2018a), a discrete-time exchange economy evo-lutionary model with agents heterogeneous in the structure of preferences is introduced. On the other hand, in that paper, the share updating rule is based on the agents’ relative satis-faction degree, measured in terms of the realized utility lev-els. Hence, the evolutionary mechanism in Naimzada and Pireddu (2018a) is merely economic and strongly differs from the one we consider here, which is instead grounded on the social interaction among agents.
In more detail, in this work, we deal with an evolution-ary mechanism in which the reproduction level of a group is based on a comparison between the attractiveness degree of
the two structures of preferences. The attractiveness of a group depends on its social visibility level, which is deter-mined as a linear combination of the amount of the two goods consumed by the agents in that group, multiplied by commodity-specific visibility factors. In particular, the attractiveness of a group is initially increasing with its visi-bility level, and this describes the bandwagon regime, char-acterized by an imitative behavior. However, when the visibility of a group exceeds a given threshold value, a con-gestion effect is produced. The attractiveness of that group then becomes a decreasing function of its visibility level, and this describes the snob regime, characterized by a pre-dominating wish for distinction. We find that the alternation between the bandwagon and snob regimes, together with the price formation mechanism, may induce periodic or chaotic behaviors in the population shares dynamics.
The choice of assuming that time is discrete, rather than continuous, comes from the consideration that the former framework is more suitable to represent the sequence of actions and decisions which lead to the formation of the pop-ulation shares. Namely, in view of embracing a new prefer-ence structure, agents need to perform the consumption activity, to evaluate the satisfaction degree resulting from their previous choice, to gather information on the other life-styles and to compare the various satisfaction levels, in order to make their next choice. The modeling representation of those actions and decisions requires a time structure which considers an interval between two consecutive time instants. In particular, following Taylor and Jonker (1978); Nachbar (1990); and Sandholm (2010), we do consider a discrete exponential replicator rule to describe the share updating mechanism.
With the aid of our model we are interested in under-standing what happens, from a dynamical viewpoint, when the heterogeneity level between groups increases. In order to answer such question, in addition to a stability analysis of the equilibria, we perform a bifurcation analysis on varying the parameter representing the heterogeneity degree in the preference structure of groups. The scenarios that we con-sider differ in the threshold value at which the social attrac-tiveness becomes a decreasing function of the visibility level.
multistability phenomena involving also a nontrivial equilib-rium, where, however, the coexistence between groups is just static in nature, in our model, we also find oscillatory behav-iors both for the population shares and the consumed quanti-ties. Indeed, thanks to the joint effect of the evolutionary mechanism and of the price formation mechanism, when the initial conditions are not too close to the extreme values 0 and 1, the group coexistence may be dynamic in nature, both regu-lar, just after the flip bifurcation, or irreguregu-lar, due to the pres-ence of chaotic attractors. The dynamic coexistpres-ence between groups and the oscillatory nature of the consumption activities over the two goods for the agents belonging to the two groups then display the recurrent dynamic behavior typical of the fashion cycle, with booms and busts.
We remark that multistability may be considered as a source of richness for the framework under analysis because, other parameters being equal, i.e., under the same institu-tional, cultural and social conditions, it allows to explain dif-ferent trajectories and evolutionary paths. The initial conditions, leading to the various attractors, represent indeed a summary of the past history, which in the presence of mul-tistability phenomena does matter in determining the evolu-tion of the system. Such property, in the literature on complex systems, is also called “path-dependence” (see Arthur (1994)). Moreover, in the specific context we deal with, the presence of multiple stable equilibria well repre-sents the variety of historical experiences across different countries in relation to the approach they adopt towards con-sumption choices and fashion.
We complete our analysis by proving the existence of complex dynamics for the model via the method of the tur-bulent maps in Block and Coppel (1992), working with homoclinic orbits. Finally, we give a social and economic interpretation of the main scenarios we found, in terms of visibility and attractiveness.
We stress that our results, starting with existence of non-trivial equilibria, hold true when visibility and attractiveness are produced by consumption of both goods. Nonetheless, the case generally considered in the literature in which the fashion cycle is generated by the consumption of a single good is well approximated in our model when dealing with polarized values for the parameters describing the degree of visibility that each agent derives from the consumption of a unit of a certain commodity, i.e., both values are positive but one is much larger than the other.
The remainder of the paper is organized as follows. In Sec.II, we present our model and we analyze the existence and local stability of its equilibria. In Sec.III, we perform a qualitative bifurcation analysis, showing the possible dynamic behaviors for our system, and we rigorously prove the presence of chaotic dynamics. Moreover, we give an eco-nomic interpretation of the main scenarios we find. In Sec. IV, we briefly discuss our results and describe possible extensions of the model.
II. THE MODEL
Let us consider an exchange economy with a continuum of agents, which may be of type a or of type b. There are
two consumption goods,x and y, and agent preferences are described by Cobb-Douglas utility functions, i.e., Uiðx; yÞ
¼ xiy1i; for i2 fa; bg; with 0 < b < a < 1: The quantity of
good x (y) consumed by an agent of type i2 fa; bg is denoted by xi (yi). Both kinds of agents have the same
endowments of the two goods, denoted bywxandwy,
respec-tively. The analysis is performed in terms of the relative pricepðtÞ ¼ pyðtÞ=pxðtÞ; where pxðtÞ and pyðtÞ are the prices
at timet for goods x and y, respectively. The size of the pop-ulation of kind a (b) at timet is denoted by A(t) (B(t)). The normalized variableaðtÞ ¼ AðtÞ=ðAðtÞ þ BðtÞÞ represents the population fraction composed by the agents of type a and bðtÞ ¼ 1 aðtÞ ¼ BðtÞ=ðAðtÞ þ BðtÞÞ represents the popula-tion fracpopula-tion composed by the agents of type b.
We now present the definition of market equilibrium, and we will refer to in the remainder of the paper.
Definition II.1 Given the economy and the population share a(t), a market equilibrium at time t is a vector ðpðtÞ; x
iðtÞ; yiðtÞÞ; with i 2 fa; bg; such that:
- every kind of agent chooses a utility-maximizing consump-tion bundle, given pðtÞ;
- the markets for the two goods clear.
Simple computations show that, solving the consumer maxi-mization problems for agents of type a and b and using a market clearing condition, the market equilibrium price is given by
pðtÞ ¼½1 ðaðtÞa þ ð1 aðtÞÞbÞwx ðaðtÞa þ ð1 aðtÞÞbÞwy
;
and the consumer equilibrium quantities of the two goods for an agent of typei2 fa; bg are
xiðtÞ ¼ iðwxþ pðtÞwyÞ ¼ iwx aðtÞa þ ð1 aðtÞÞb; yiðtÞ ¼ ð1 iÞ wx pðtÞþ wy ¼ ð1 iÞwy 1 ðaðtÞa þ ð1 aðtÞÞbÞ : (2.1) See Chang and Stauber (2009) and Naimzada and Pireddu (2016)for further mathematical details.
Once we specify a dynamical rule for the population share evolution, it is also possible to give the definition of market stationary equilibrium as follows:
Definition II.2: Given the economy, the vector ða; p; x
i; y
iÞ; i 2 fa; bg; is a market stationary equilibrium
if a* is constant and if, given a;ðp; x
i; yiÞ; i 2 fa; bg; is a
market equilibrium for every t.
For the sake of brevity, we shall identify market station-ary equilibria just with the population sharea, since it deter-mines all other equilibrium components.
The market stationary equilibria, at which for everyt the population shares, and thus also the market equilibrium price and the consumer equilibrium quantities, are constant, will be called trivial if they are not characterized by the coexistence between the two groups of agents, and nontrivial otherwise.
In a social interaction framework, consumption choices produce visibility. Since the various goods induce different visibility levels, we introduce the positive parameters vxand
vy describing the degree of visibility that each agent derives
from the consumption of a unit of commodityx and of com-modityy, respectively. Given such assumptions, we define the social visibility levelViðtÞ of an agent of type i 2 fa; bg at
timet as a linear combination of the units xiðtÞ and yiðtÞ of
goodsx and y he consumes, weighted respectively with vxand
vy; i.e., ViðtÞ ¼ vxxiðtÞ þ vyyiðtÞ: Recalling(2.1), it holds that
ViðtÞ ¼ i vxwx aðtÞa þ ð1 aðtÞÞbþ ð1 iÞvywy 1 aðtÞa ð1 aðtÞÞb: (2.2) Agents’ consumption choices, deriving by the underlying preference structures, give rise to different attractiveness degrees for the various preference structures. Indeed, in a social interaction setting, the attractiveness of a preference structure depends on its visibility in a non-monotone man-ner: the social attractiveness of a preference structure is increasing in its visibility as long as the latter is not exces-sive, and then such dependence becomes decreasing.
Introducing the attractivenessAiðtÞ of group i as a
func-tion ofViðtÞ; i 2 fa; bg; suitable hypotheses on the map are
that it is positive and bell-shaped, increasing with the visibil-ity level up to a certain threshold value ^V ; above which it becomes decreasing. A possible formalization is given by
AiðtÞ ¼ f ðViðtÞ; ^VÞ ¼
1 1þ rðViðtÞ ^VÞ2
; (2.3)
where r is a positive parameter describing the sensitivity of the attractiveness with respect to the distance between the visi-bility levelViðtÞ induced by the preference structure i 2 fa; bg
and the desirable visibility level ^V ; which allows maximizing the attractiveness degree. We stress that all the results we shall derive along the present section are robust with respect to the choice of the functional form employed to describe the bell-shaped attractiveness function. Indeed, modeling attractive-ness, e.g., via a Gaussian map, rather than via the expression in (2.3), it holds that the number of the equilibria, their expres-sion, as well as their stability conditions do not change. Just the threshold value for the flip bifurcation in Proposition II.2 varies according to the chosen formulation.
From (2.3), it follows that the attractiveness AiðtÞ is
inversely proportional to the distance betweenViðtÞ and ^V ;
and thatAiðtÞ 2 ð0; 1: In particular,AiðtÞ tends towards 0
when the visibility levelViðtÞ is too small [we recall that, by
definition, the visibility levelViðtÞ is positive] or too large,
or if r! þ1: Indeed, in the latter case even a small, but still positive, distance betweenViðtÞ and ^V leads to a very
low attractiveness level because of an excessive sensitivity to such distance. On the other hand, we reach the maximum attractiveness level, i.e., AiðtÞ ¼ 1; when ViðtÞ coincides
with ^V ; while AiðtÞ tends towards 1 when r ! 0; and thus
agents are insensitive even to a large distance betweenViðtÞ
and ^V ; when r raises, the social interaction degree increases andAiðtÞ starts decreasing.
The formulation in(2.3)is suitable to describe in rela-tion to the preference structure i2 fa; bg both the band-wagon behavior, which occurs as long asViðtÞ < ^V ; and the
snob behavior, which occurs when ViðtÞ > ^V : Indeed,
according to Simmel (1957), two contrasting tendencies operate in determining people behavior towards fashion. On the one hand, to follow fashion makes people feel accepted and socially integrated, answering to their innate tendency for conformity. This generates imitation of others, i.e., the so called bandwagon behavior. Such phenomenon is reproduced by (2.3)as long asViðtÞ is smaller than ^V ; when increasing
values for ViðtÞ imply higher and higher attractiveness
degrees AiðtÞ: On the other hand, when the visibility of a
group becomes excessive, a congestion effect arises and the people wish to distinguish themselves predominates. That phenomenon is known as snob behavior and it is reproduced by(2.3)when ViðtÞ is larger than ^V ; because in this regime
an increase in the visibility levelViðtÞ leads to a decrease in
AiðtÞ: Namely, according toVigneron and Johnson (1999),
on the basis of the empirical literature, too, in the context of luxury and prestige-seeking consumption agents may oscil-late between snob and bandwagon behaviors. Such two opposite forces, imitation and distinction, drive the fashion cycle, which for us emerges at the aggregate level as a con-tinuous oscillation of the shares of the two groups, corre-sponding to the different preference structures, while on the individual level it is characterized by oscillatory consump-tion choices, presenting booms and busts, over the two goods for the agents belonging to the two groups. We shall find evi-dence for it in Sec.III[see Figs.6(c)and14].
The share of agents which adopt a given preference structure in the next period depends on the present attractive-ness levels of the two preference structures. More precisely, following Taylor and Jonker (1978); Nachbar (1990); and Sandholm (2010), we do consider a discrete exponential rep-licator mechanism to formalize the population share updat-ing rule, so that the evolution of the fractiona(t) of agents of type a is described by the discrete choice model
aðt þ 1Þ ¼ aðtÞ exp lð AaðtÞÞ
aðtÞ exp lð AaðtÞÞ þ ð1 aðtÞÞ exp lAbðtÞ
;
¼ aðtÞ
aðtÞ þ ð1 aðtÞÞ exp l AbðtÞ AaðtÞ ;
(2.4) where l is a positive parameter measuring the sensitivity of the share formation mechanism to the preference structures attractiveness levels. By(2.3), we obtain
aðt þ 1Þ ¼ gðaðtÞÞ; (2.5)
where the one-dimensional mapg :½0; 1 ! R is defined as
We stress that if l! 0; independently of Va and Vb;
thenaðt þ 1Þ ¼ aðtÞ; for all t, and thus there is no evolution, the population shares remain unchanged with respect to the initial ones, as agents are insensitive to the attractiveness degrees of the preference structures; when instead l! þ1; agents are extremely sensitive to social attractiveness of the preference structures and they instantaneously move towards the “best” one, i.e., that which gave a visibility level closer to ^V :
Let us then start our analysis by deriving the expressions of the market stationary equilibria for(2.5).
Proposition II.1 Equation (2.5) admits as market sta-tionary equilibria, in addition to the trivial a¼ 0; a ¼ 1; also a¼ a; a¼ a 1, and a¼ a2; with a¼ ð1 bÞvxwx bvywy ða bÞðvxwxþ vywyÞ ; (2.7) a1;2¼2 ^Vð1 2bÞ þ vxwxða þ bÞ vywyð2 a bÞ6 ffiffiffi n p 4 ^Vða bÞ ; (2.8) where n¼ 4 ^V2þ ðvxwxða þ bÞ vywyð2 a bÞÞ 2 4 ^Vðvxwxða þ bÞ þ vywyð2 a bÞÞ;
as long as they are real and belong toð0; 1Þ:
Proof. The conclusion immediately follows by observ-ing thata¼ 0; a ¼ 1; a ¼ a in(2.7), anda¼ a
1;2 in(2.8)
are all the solutions to the fixed-point equation gðaÞ ¼ a;
withg as in(2.6). ⵧ
We notice that ata* in(2.7)it holds thatVa ¼ Vb; while
ata1;2 in(2.8) it holds thatðVaþ VbÞ=2 ¼ ^V ; i.e., ^V is the
midpoint betweenVa and Vb; with Va and Vb lying at the
same distance from ^V ; but on its opposite sides. Moreover, simple computations show thata* belongs to (0, 1) if and only if vxwx vywy 2 b 1 b; a 1 a : (2.9)
When introducing the heterogeneity degree between groups D¼ a b; measuring the difference in the preference struc-tures, it is possible to rewrite(2.9)as
vxwx vywy 2 b 1 b; bþ D 1 b D : (2.10)
Since the lower bound does not depend on D and the upper bound is increasing in D, we can infer that for large values of the heterogeneity degree it becomes easier to have a 2 ð0; 1Þ: Rewriting a* in(2.7)in terms of D as
a¼ð1 bÞvxwx bvywy Dðvxwxþ vywyÞ
;
we also deduce thata* is decreasing in D: A confirmation of this can be found, e.g., in Figs.1,3, and5.
Such decreasing behavior of a* with respect to D may be explained by looking at the expression of the visibility for the two groups of agents, i.e., ViðtÞ ¼ vxxiðtÞ þ vyyiðtÞ;
i2 fa; bg: Indeed, if D is raised, since b would not change and a¼ b þ D would increase, the preference of the agents of type a for commodity x would become stronger. Hence, the individual demand of the agents of type a for commod-ity x would raise and, if the population shares are fixed, also the aggregate demand for commodity x would raise. This would make the corresponding pricepxincrease, with
a resulting decrease in the relative price p¼ py=px: The
market closes and the optimal consumption quantities for the two goods are determined. Due to the increase in px;
agents of type a would consume a lower amount of com-modityx and a decrease in xa would induce a lower visibil-ity level for the agents of type a, violating the visibilvisibil-ity balance conditionVa ¼ Vbcharacterizinga¼ a: Hence, if
we want to remain in the steady state a¼ a; the share of
agents of type a has to decrease. In this way, the aggregate demand for commodity x falls and consequently the price of good x tends to decrease. Such opposite price effect makes the equilibrium consumption levels of good x for both groups increase and prevents the visibility level of agents of type b from exceeding that of agents of type a, so that the visibility balance condition characterizinga¼ ais
maintained.
Given that(2.10)is not satisfied for D¼ 0; by continu-ity, it follows that for small values of D we havea62 ð0; 1Þ: Indeed, as we will see in Sec.III,a* enters the interval (0, 1) through a transcritical bifurcation occurring ata¼ 1 when
vxwx vywy ¼ bþ D 1 b D; i.e., for D¼ ~D¼ 1 b vywy vxwxþ vywy (2.11) (cf. Proposition III.1).
FIG. 1. The bifurcation diagram of g for ^V¼ 0:2 and D 2 ð0; 0:9Þ: We denote in magentaa¼ 0, in green a ¼ 1, and in red a ¼ a: Solid (dashed)
We also notice that, due to(2.9), the parameters vxand
vy cannot vanish. Nonetheless, the case of polarized values
for vxand vy; in which both of them are positive but one is
much larger than the other, is allowed, and well approxi-mates those frameworks in which visibility and attractive-ness are produced by the consumption of a single good. In Sec.III, we will consider polarized values for vxand vy; even
if our results hold true also in the case of more balanced val-ues for such parameters.
Due to the heavy expressions ofa1;2in(2.8), in the next
proposition, we will analytically investigate just the local stability ofa¼ 0; a ¼ aanda¼ 1 for map g in(2.6).
Proposition II.2 The equilibrium a¼ 0 is locally asymp-totically stable for map g in (2.6) if ðvxwxþ vywy ^VÞ
2 <ðavxwx b þ ð1aÞvywy 1b ^VÞ 2 :
The equilibrium a¼ a in(2.7)is locally asymptotically
stable for map g if(2.9)is satisfied, if vxwxþ vywy< ^V and
l < l¼ vxwxvywyð1 þ rðvxwxþ vywy ^VÞ
2
Þ2
rðvxwxþ vywyÞðð1 bÞvxwx bvywyÞðavywy ð1 aÞvxwxÞð ^V vxwx vywyÞ
:
The equilibrium a¼ 1 is locally asymptotically stable for map g ifðvxwxþ vywy ^VÞ 2 <ðbvxwx a þ ð1bÞvywy 1a ^VÞ 2 : In particular, a flip bifurcation occurs at a¼ a if
l¼ l:
Proof. Since g0ð0Þ > 1 and g0ð1Þ > 1; with g as in (2.6), are always fulfilled, the stability conditions follow by imposingg0ð0Þ < 1; g0ðaÞ 2 ð1; 1Þ and g0ð1Þ < 1,
respec-tively. In regard to l; we stress that the factorsðð1 bÞvxwx
bvywyÞ and ðavywy ð1 aÞvxwxÞ are positive if and only
if(2.9)is satisfied, i.e., whena2 ð0; 1Þ: The condition for the flip bifurcation follows by settingg0ðaÞ ¼ 1: ⵧ
We remark that, when a2 ð0; 1Þ; simple computations show that, fora¼ 0; Vb¼ vxwxþ vywy<avxbwxþð1aÞvy
wy
1b ¼ Va
and, for a¼ 1; Va¼ vxwxþ vywy<bvxawxþ ð1bÞvywy
1a ¼ Vb:
Thus, according to Proposition II.2, in order to have stability both of a¼ 0 and a ¼ 1 for map g in (2.6), we need ^V to be closer, at the two endpoints, to the smaller betweenVa
andVb:
With reference to a1;2 in (2.8), we notice that both of
them can lie in (0, 1) (see, e.g., Figs.3and5). Even if they are always unstable in the frameworks, we will consider in Sec.III, with the only possibly stable nontrivial equilibrium
being a¼ a [cf. Fig.6(b), wherea¼ a is stable, and Fig. 6(c), wherea¼ a is unstable, too], this is not true in
gen-eral. Taking, for instance, as in Sec. III, b¼ 0:1; l ¼ 6:5; r ¼ 8; and vx¼ 0:8; wx¼ 0:1; vy¼ 0:5; wy¼ 2;
D¼ 0:8, and ^V ¼ 0:85; the only nontrivial equilibria are a¼ a
1 anda¼ a
2; of which the former is locally stable, as
well asa¼ 1. Considering instead b ¼ 0:1; l ¼ 6:5; r ¼ 8; and vx¼ 0:2; wx¼ 0:3; vy¼ 0:25; wy¼ 1:4; D ¼ 0:8 and
^
V ¼ 0:36; we find that also a ¼ abelongs toð0; 1Þ; and the
locally stable equilibria are given by a¼ 0; a ¼ a
2 and
a¼ 1. Hence, both with a ¼ abelonging to (0, 1) or not, we
may have multistability phenomena involving a¼ a
1 and
a¼ a 2; too.
We will illustrate some possible dynamic scenarios for our system in Figs. 1–6, where we take the heterogeneity degree D¼ a b as bifurcation parameter. With this respect, we stress that the stability conditions found in Proposition II.2 fora¼ amay be rewritten in terms of D as
stated in the following result, which highlights the destabiliz-ing role of the heterogeneity degree.
Corollary II.1 The equilibrium a2 ð0; 1Þ in (2.7) is locally asymptotically stable for map g in(2.6)if(2.9)is sat-isfied, if vxwxþ vywy< ^V and D < D¼ vxwxvywyð1 þ rðvxwxþ vywy ^VÞ 2 Þ2 lðð1 bÞvxwx bvywyÞrðvxwxþ vywyÞð ^V vxwx vywyÞ þ ð1 bÞvxwx bvywy vxwxþ vywy :
In particular, a flip bifurcation occurs at a¼ aif D¼ D: III. BIFURCATION ANALYSIS, CHAOTIC DYNAMICS, AND POSSIBLE SCENARIOS
A. Bifurcation analysis, static, and oscillatory group coexistence
In this subsection, we perform a qualitative bifurcation analysis, investigating the emergence/disappearance and stability
gain/loss of equilibria on varying both the parameter D measur-ing the heterogeneity degree in the structure of preferences between groups and the parameter ^V describing the threshold value at which the social attractiveness becomes a decreasing function of the visibility level. More precisely, in all figures below, we fix the remaining parameter values as follows We here consider polarized values for vxand vy; as both of them are
are produced by the consumption of a single good. However, our results hold true also in the case of more balanced values for vx and vy: vx ¼ 0:9; wx¼ 0:2; vy¼ 0:15; wy¼ 2; b
¼ 0:1; l ¼ 6:5; r ¼ 8:
In Fig.1, we report the bifurcation diagram ofg for ^V ¼ 0:2 and D 2 ð0; 0:9Þ; where we notice that the only attrac-tors are given by the trivial equilibria a¼ 0 and a ¼ 1. Indeed, as we shall see below, for small values of D just a¼ 0 is stable in the interval ½0; 1; while for larger values of D also a¼ 1 becomes locally stable via a transcritical bifur-cation. In order to better understand this transition, in Fig.2, we report the graph of mapg, with a2 ½0; 1:5; for ^V ¼ 0:2; and D¼ 0:200 in (a), D ¼ 0:275 in (b), D ¼ 0:400 in (c), and D¼ 0:600 in (d). In Fig.2(a), the only equilibria in the interval½0; 1 are a ¼ 0 and a ¼ 1, denoted by black squares, of which the former is stable and the latter is unstable, leading to the eventual extinction of the agents of group a: The non-trivial equilibriuma¼ a; denoted by a black dot, does not belong toð0; 1Þ: However, as noticed in Sec.II,a* is decreas-ing with D : hence, when raisdecreas-ing D, it approachesa¼ 1 and indeed for D¼ 0:275 [we remark that such value for D is coherent with the formulation for ~D obtained in (2.11)], a
transcritical bifurcation occurs [see Fig. 2(b)], at which a ¼ a loses stability in favor of a¼ 1. Namely, in Fig.2(c),
a¼ abelongs toð0; 1Þ; it is unstable and separates the basins
of attraction ofa¼ 0 and a ¼ 1, which are locally stable, the latter with a small basin of attraction. In (D) we just observe an increased basin of attraction of a¼ 1, with a consequent reduction of the basin of attraction ofa¼ 0.
We prove the occurrence of the just described transcriti-cal bifurcation in the next result.
Proposition III.1 For the map g¼ gða; DÞ in (2.6) a transcritical bifurcation occurs at ~a¼ 1 for ~D¼ 0:275:
Proof. According to Wiggins (2003), p. 507, for the occurrence of a transcritical bifurcation for the map g at a pointa¼ ~a for a certain D¼ ~D we have to check the follow-ing conditions: gð~a; ~DÞ ¼ ~a; @g @að~a; ~DÞ ¼ 1; @g @Dð~a; ~DÞ ¼ 0; @2 g @a @Dð~a; ~DÞ 6¼ 0; @2 g @a2ð~a; ~DÞ 6¼ 0:
Direct (software-assisted) computations show that the above conditions are satisfied. In particular, it holds that
@2g
@a @Dð~a; ~DÞ ¼ 6:194 and @2g
@a2ð~a; ~DÞ ¼ 3:407: This
com-pletes the proof. ⵧ
In Fig.3, we report the bifurcation diagram ofg for ^V ¼ 0:5 and D 2 ð0; 0:9Þ: We notice that for small values of D, onlya¼ 1 is stable while, for larger values of D, first a ¼ 0 and then also a¼ a become locally stable. As concerns
a¼ 1, it remains locally stable, except for an interval of intermediate values of D: To better understand such changes, in Fig.4, we report the graph of mapg, with a2 ½0; 1:25; for ^V ¼ 0:5; and D ¼ 0:250 in (a), D ¼ 0:275 in (b), D¼ 0:300 in (c), D ¼ 0:336 in (d), and D ¼ 0:370 in (e). In
(a), in addition to the locally stable trivial equilibria a¼ 0 anda¼ 1, inside (0, 1), there is an unstable nontrivial equi-librium separating their basins of attraction. However, instead of a¼ a; as it was in Figs.2(c)and2(d), this time the nontrivial equilibrium inside (0, 1) is given by a¼ a
1;
denoted by an empty square. Indeed, the nontrivial equilib-riuma¼ a does not belong to the interval (0, 1) yet,
enter-ing it for D¼ 0.275 via a transcritical bifurcation at which a¼ 1 loses stability in favor of a ¼ a [see Figs. 4(b) and 4(c)]. Increasing D further to 0.336, in Fig.4(d), we observe a transcritical bifurcation at which a¼ a
2; denoted by an
empty square, too, enters the interval (0, 1) and loses stabil-ity in favor of a¼ 1, that recovers its local stability. We notice that, since the stability condition for a ¼ 1 in Proposition II.2 is of the second degree in D, no other bifur-cations can occur ata¼ 1. Although the stability condition fora¼ 0 is of the second degree in D, too, in the framework considered in Fig. 3, we find just a bifurcation ata¼ 0 for D2 ð0; 0:9Þ: In Fig.4(e), for still larger values of D, in addi-tion to the locally stablea¼ 0 and a ¼ 1, we then have three internal equilibria, the locally stablea¼ a, and the unstable
a¼ a
1 anda¼ a2; separating the basins of attraction of the
locally stable equilibria. We remark that also when a¼ a 1
enters the interval (0, 1) a transcritical bifurcation of the map g occurs at a¼ 0 for D ¼ 0:027; at which a ¼ a1loses
stabil-ity in favor ofa¼ 0, that from such moment on is locally sta-ble (cf. Fig.3). Namely, from the bifurcation diagram in Fig. 3, it is immediate to see that the three fixed points undergo a transcritical bifurcation each. We stress that, since for
^
V ¼ 0:5, the map g is increasing, there can be only alternat-ing stable/unstable fixed points, and just fold and transcritical bifurcations can occur in such a scenario. In particular, we
FIG. 3. The bifurcation diagram ofg for ^V¼ 0:5 and D 2 ð0; 0:9Þ: We denote in magentaa¼ 0, in green a ¼ 1, in red a ¼ a; and in blue a¼ a 1
anda¼ a
2: Solid (dashed) curves refer to stable (unstable) equilibria.
here observe only the latter kind of bifurcation. The same remark applies to the framework in Figs.1and2, where for
^
V ¼ 0:2 the map g is increasing, too.
In Fig. 5, we report the bifurcation diagram of g for ^
V ¼ 0:8 and D 2 ð0; 0:9Þ; where for small values of D only a¼ 1 is stable, while for intermediate values of D just a ¼ a
is stable. When D increases, we observe first that also a¼ 0 becomes locally stable and then a destabilization ofa¼ a;
leading to a one-piece chaotic attractor, which coexists with the locally stable equilibria a¼ 0 and a ¼ 1 and that disap-pears for still larger values of D, when the unique attractors are given by the trivial equilibria.
To start understanding the various transitions, in Fig.6, we report the graph of map g for ^V ¼ 0:8; and D ¼ 0:2 in (a), D¼ 0:4 in (b), and D ¼ 0:7 in (c). In Fig.6(a), the only equilibria are a¼ 0 and a ¼ 1, of which the former is unsta-ble and the latter is staunsta-ble, leading to the eventual extinction of the agents of group b. In (b), the nontrivial equilibrium a¼ a belongs to (0, 1) and it is stable, while a¼ 0 and a¼ 1 are unstable. In (c), also a ¼ a
1 and a¼ a2 have
entered ð0; 1Þ; but they are unstable, as well as a ¼ a; and FIG. 5. The bifurcation diagram ofg for ^V¼ 0:8 and D 2 ð0; 0:9Þ: We
denote in magentaa¼ 0, in green a ¼ 1, in red a ¼ a; and in blue a¼ a 1
anda¼ a
2: Solid (dashed) curves refer to stable (unstable) equilibria.
separate the basins of attraction ofa¼ 0; a ¼ 1 and of the chaotic attractor surroundinga¼ a:
We stress that, whena¼ aenters the interval (0, 1) by
increasing the value of D from (a) to (b), a transcritical bifur-cation occurs for the mapg at a¼ 1 for D ¼ 0:275; at which a¼ 1 loses stability in favor of a ¼ a; as it happens in Figs. 3and 4. Raising further the value of D from (b) to (c), we observe the emergence ofa¼ a
1anda¼ a
2 inð0; 1Þ: More
precisely, whena¼ a
1enters the intervalð0; 1Þ; a
transcriti-cal bifurcation of the mapg occurs at a¼ 0 for D ¼ 0:436; at whicha¼ a
1 loses stability in favor of a¼ 0, which from
that moment on is locally stable; similarly, when a¼ a 2
enters the interval ð0; 1Þ; a transcritical bifurcation of the mapg occurs at a¼ 1 for D ¼ 0:654; at which a ¼ a
2 loses
stability in favor ofa¼ 1, which becomes locally stable. When increasing the value of D in Fig.6from (b) to (c), in addition to the two just described transcritical bifurcations ofg, it also happens that a¼ aloses stability via a subcritical
flip bifurcation occurring for D¼ 0:628; as shown in the next result. See Figs.5,7, and9(b), where we illustrate such phe-nomenon by analyzing the shape of the second iterate ofg.
Proposition III.2 For the map g¼ gða; DÞ in (2.6), a subcritical flip bifurcation occurs at ^a¼ a¼ 0:438 for
^
D¼ 0:628:
Proof. According to Wiggins (2003), p. 516, for the occurrence of a flip bifurcation for the map g at a point a¼ ^a for a certain D¼ ^D, we have to check the following conditions: fð^a; ^DÞ ¼ ^a; @f @að^a; ^DÞ ¼ 1; @f2 @Dð^a; ^DÞ ¼ 0; @2 f2 @a @Dð^a; ^DÞ 6¼ 0; @2 f2 @a2 ð^a; ^DÞ ¼ 0; @3 f2 @a3 ð^a; ^DÞ 6¼ 0:
Moreover, if@a @D@2f2 ð^a; ^DÞ > 0 and @@a3f32ð^a; ^DÞ > 0, we have a
subcritical flip bifurcation. Direct (software-assisted) compu-tations show that the needed conditions are satisfied at ^a ¼ 0:438 for ^D¼ 0:628: In particular, it holds that
@2f2
@a @Dð^a; ^DÞ ¼ 11:327 and @3f2
@a3ð^a; ^DÞ ¼ 44:903: This
com-pletes the proof. ⵧ
In order to better understand a few further crucial phe-nomena highlighted by Fig.5, we report in Fig.7a magnifi-cation of that bifurmagnifi-cation diagram, focusing now on D2 ð0:60; 0:76Þ:
At first, we notice that for D2 ð0:603; 0:628Þ a stable period-two cycle coexists with a¼ a: Indeed, as illustrated
in Fig.8, where we report the graph of the second iterate of g for ^V ¼ 0:8; and D ¼ 0:590 in (a), D ¼ 0:603 in (b), and D¼ 0:610 in (c), a stable and an unstable period-two cycles emerge for D¼ 0:603 via a fold bifurcation of g leading to a two-cycle with periodic points fa1; a2g ¼ f0:231; 0:761g:
We denote the stable period-two cycle by black diamonds and the unstable period-two cycle by empty diamonds.
We prove the occurrence of the fold bifurcation in the next result.
Proposition III.3 Given the map g¼ gða; DÞ in (2.6), for D^ ¼ 0:603 a fold bifurcation of a two-cycle of map g occurs, leading to a two-cycle with periodic points fa1; a2g
¼ f0:231; 0:761g:
Proof. According to Wiggins (2003), p. 503, for the occurrence of a fold bifurcation of g2 at a point a¼ a^
for a certain D¼ D^, we have to check the following conditions: g2ða^ ;D^Þ ¼ a^ ; @g 2 @aða ^ ;D^Þ ¼ 1; @g2 @Dða ^ ;D^Þ 6¼ 0; @ 2g2 @a2 ða ^ ;D^Þ 6¼ 0:
Direct (software-assisted) computations show that the above conditions are satisfied for D^ ¼ 0:603 at a1 ¼ 0:231: In
par-ticular, it holds that @g@D2ð a1;D ^
Þ ¼ 2:357 and @@a2g22ð a1;D ^
Þ
¼ 15:323: This completes the proof. ⵧ
We stress that the stable period-two cycle, emerging via the fold bifurcation, is the one we see in green in Fig. 7for D2 ð0:603; 0:628Þ; and that the companion unstable period-two cycle in red bounds the immediate basin of attraction of a¼ a: However, as illustrated in Fig. 9(a) for D¼ 0:620;
the basin of attraction of a¼ a is composed also by two
other intervals, and it is colored in dark green on thex-axis. Increasing the value of D, the basin of attraction of a¼ a
becomes smaller and smaller, until it reduces to a* for D¼ 0:628; when the two elements of the unstable period-two cycle collide with a; as shown in Fig.9(b), where we report the graph of the second iterate ofg. We recall that, as proved in Proposition III.2, just for D¼ 0:628, the map g undergoes a subcritical flip bifurcation at a¼ a; which
becomes unstable. After that bifurcation, when a* is repel-ling, the only attracting set is the period-two cycle previously coexisting with a; as illustrated in Fig. 7. Such period-two cycle loses stability for D¼ 0:656 via a supercritical flip bifurcation [we stress that for D¼ 0:645 there is an abrupt coordinate change for the stable period-two cycle, but there no bifurcations occur], followed by a sequence of period-doubling bifurcations leading to a two-piece chaotic attrac-tor, which for D¼ 0:687 is replaced by a period-two cycle. In view of understanding such sudden transition, we report in Fig. 10the graph of the second iterate ofg for ^V ¼ 0:8;
FIG. 7. The bifurcation diagram ofg for ^V¼ 0:8 and D 2 ð0:60; 0:76Þ: We denote in magentaa¼ 0, in red an unstable period-two cycle, and in blue a¼ aanda¼ a
1: Solid (dashed) curves refer to stable (unstable) equilibria
FIG. 8. The graph of the second iterate ofg for ^V¼ 0:8; and D ¼ 0:590 in (a), D ¼ 0:603 in (b), and D ¼ 0:610 in (c).
FIG. 9. In (a), the graph of mapg for ^V¼ 0:8 and D ¼ 0:620; with the basin of attraction of a ¼ acolored in dark green on thex-axis. In (b), the graph of the
and D¼ 0:660 in (a), D ¼ 0:687 in (b), and D ¼ 0:720 in (d). We notice that a stable and an unstable period-two cycles emerge via a fold bifurcation of g for D¼ 0:687 leading to a two-cycle with periodic points f0:152; 0:840g: We denote the stable period-two cycle by black stars and the unstable period-two cycle by empty stars. We show the graph of the second iterate ofg for ^V ¼ 0:8 and D¼ 0:687; together with the stable period-two cycle, in Fig.10(c).
Turning back to the bifurcation diagram in Fig. 7, we notice that the stable period-two cycle born for D¼ 0:687 undergoes a cascade of flip bifurcations leading to a two-piece chaotic attractor, followed by a one-piece chaotic attractor, which disappears for D¼ 0:746; so that for larger values of D the only attractors are given by the trivial equilibria a¼ 0 anda¼ 1 (see Fig.5). In order to explain the change occurring for D¼ 0:746; we report in Fig. 11the graph of map g for
^
V¼ 0:8; and D ¼ 0:740 in (a), D ¼ 0:746 in (b), and D¼ 0:750 in (c), together with the forward iterates of the minimum pointam: We find that, for D < 0:746; such iterates
do not overcome the unstable equilibrium a¼ a
1; whose
stable set (which includes also its rank-1 preimage) separates the basins of attraction of a¼ 0 and of the chaotic attractor surrounding a¼ a: For D¼ 0:746; it holds that gða
mÞ
¼ gð0:717Þ ¼ a
1¼ 0:051; and the first homoclinic
bifurca-tion of a1 occurs. For D > 0:746, the forward iterates of
am belong to ð0; a1Þ and converge to a ¼ 0. Hence, for
D > 0:746 the chaotic attractor surrounding a¼ a does not
exist anymore and, according to the initial condition, the generic orbit tends towarda¼ 0 or toward a ¼ 1, as shown in Fig.5. We remark however that, after the contact bifurcation with a1; the latter admits homoclinic orbits and a chaotic
repellor survives.
Summarizing, our bifurcation analysis has highlighted that the nontrivial equilibria emerge via transcritical bifurca-tions. In particular, for the considered parameter configura-tions, the only nontrivial equilibrium which may be locally stable is given by a¼ a; but, as remarked in Sec.II, this is
not true in general. We proved thata¼ a loses stability via
a subcritical flip bifurcation. The subcritical flip suggests the appearance of two-cycles and coexistence phenomena before the bifurcation, as we have shown.
Reconsidering, from a global viewpoint, the scenarios investigated above when increasing D, in Fig.1, we find just a coexistence phenomenon between the two trivial equilibria, while in Fig.3, we find multistability involving also the non-trivial equilibrium a¼ a: However, the most interesting
multistability phenomena occur in Fig.5and are better visi-ble in Fig.7, wherea¼ 0 first coexists just with a ¼ a; then
witha¼ aand with a stable period-two cycle, next with the
stable period-two cycle only, which undergoes a cascade of flip bifurcations, leading to a two-piece chaotic attractor, interrupted by a periodicity window, followed by another two-piece chaotic attractor, and finally by a one-piece cha-otic attractor. Together witha¼ 0 and the chaotic attractor, multistability involves alsoa¼ 1, which may be locally sta-ble. For larger values of D, the chaotic attractor disappears, and the only attractors are the trivial equilibria.
In more detail, when looking at the graph of mapg, we stress that in Fig.2(a) and in Figs. 6(a) and6(b) no multi-stability phenomena arise. Multimulti-stability in Figs. 2(c) and 2(d)and in Fig. 4(a) involves just the trivial equilibria and thus the most interesting scenarios are those in Figs.4(c)and 4(e), where the nontrivial equilibrium a¼ a is involved,
and in Fig.6(c), where there is a chaotic attractor surround-inga¼ a:
Thus, similarly to the framework considered in Naimzada and Pireddu (2016), at most one of the multiple locally stable equilibria is characterized by the coexistence between the two groups of agents, while in the other ones a group completely prevails and remains alone. As explained above, the unstable equilibria in Figs.4(c)and4(e)play the role of separating the basins of attraction of the locally stable equilibria: trajectories will be attracted by one or the other of the three locally stable equilibria according to the chosen ini-tial conditions. In particular, when the iniini-tial conditions, which represent a summary of the past history, are exces-sively close to the extreme valuesa¼ 0 and a ¼ 1, the trajec-tories tend towards one of the two trivial equilibria, and a preference structure totally prevails over the other. However, also when initial conditions are more balanced, in such frameworks, the coexistence between groups is just static in nature, confined to the nontrivial equilibrium.
On the other hand, differently from Naimzada and Pireddu (2016), when the initial conditions are not too close to a¼ 0 and a ¼ 1, for our model, we also find oscillatory
behaviors [see, for instance, Figs.5and6(c)], thanks to the combined effect of the evolutionary mechanism and of the price formation mechanism. Indeed, the group coexistence may here be dynamic in nature, both regular, just after the flip bifurcation, and irregular, due to the presence of the cha-otic attractor. The dynamic coexistence between groups then displays the recurrent dynamic behavior typical of the fash-ion cycle, characterized by booms and busts. See the corre-sponding time series for all the relevant variables in Fig.14.
We conclude this section by investigating the possible Pareto optimality of the various stationary equilibria. Indeed, it is well known that externalities in general equilibrium models typically lead to inefficiencies, as the first theorem of welfare does not hold and competitive equilibria are not Pareto optimal [see, e.g.,,Villanacciet al. (2002)]. Although in our exchange economy (evolutionary) model, the con-sumption externality are not present directly into the agent’s utility function, we find that all stationary equilibria are Pareto inefficient as well. Since the results we obtain are analogous in the three scenarios considered so far, character-ized by ^V¼ 0:2; ^V ¼ 0:5 and ^V ¼ 0:8; respectively, we focus just on the second one, in which the desirable visibility level ^V assumes an intermediate value. In order to illustrate the Pareto inefficiency of the stationary equilibria, we report in Fig.12the graph ofUaðxa; yaÞ in (a) and of Ubðxb; ybÞ in
(b), as a function of D2 ð0; 0:9Þ; computed in correspon-dence ofa¼ 0 (in magenta), of a ¼ a
1 (in cyan), of a¼ a
(in red), ofa¼ a
2 (in blue), and of a¼ 1 (in green), for ^V
¼ 0:5: We recall indeed that the optimal consumption quan-tities in(2.1) depend on a, which in turn, at the nontrivial equilibria, depends on the endowment values (see(2.7)and (2.8)). In Fig. 12(a), we find that, for any value of D;Ua is
decreasing witha. In Fig.12(b), we find that, for any value of D;Ubis increasing witha, or equivalently, since the
pop-ulation share of agents of type b is given byb¼ 1 a; that Ub is decreasing with b. This means that both groups of
agents are better off when their share is low. Such fact may be easily understood in terms of the reduced agents’ pressure on the price formation mechanism and of the resulting better consumption opportunities for that group (cf. the explanation of why a* is decreasing with D in Sec. II, where similar
arguments are employed). Due to the decreasing behavior of bothUaandUbwith respect to the corresponding population
share, and due to the inverse proportionality between the two groups’ shares, no stationary equilibrium may be Pareto opti-mal, as each change which makes a group better off, would make the other group worse off.
B. Analysis of the chaotic dynamics
Still considering the parameter values employed in Fig. 6(c), we conclude our analysis by proving the existence of complex dynamics for the system in(2.5)via the method of the turbulent maps in Block and Coppel (1992). Such tech-nique requires to find for a given continuous mapf : J! J; where Ø6¼ J R is a compact interval, two nonempty com-pact sub-intervalsJ0andJ1ofJ, with at most one common
point, such that
J0[ J1 f ðJ0Þ \ f ðJ1Þ: (3.1)
If the latter property is fulfilled, then the mapf is called tur-bulent inBlock and Coppel (1992)and it is therein shown to display some of the typical features associated with the con-cept of chaos as, e.g., existence of periodic points of each period, semi-conjugacy to the Bernoulli shift, and thus posi-tive topological entropy. In particular, the condition in (3.1) is fulfilled whenever ak-cycle, with k 2; is homoclinic.
We will use the just described methodology in the proof of Proposition III.4, working with homoclinic orbits. We recall that, given an unstable fixed point p of a one-dimensional map f at which the latter is locally invertible, p becomes homoclinic when we can find preimages of p arbi-trarily close to p itself. When this occurs the fixed point is called a snap-back repeller [seeMarotto (1978)]. More pre-cisely, a pointq is called homoclinic to p if there exist a posi-tive integer j such that fjðqÞ ¼ p; as well as a sequence of
preimages of q which tends to p. Under a nondegeneracy condition on the orbit of p, Theorem 1 in Marotto (1978) shows the presence of chaos for the mapf on suitable invari-ant sets.
In Proposition III.4 below, we limit ourselves to show that for the choice of the intervals J0(in green) and J1 (in
FIG. 12. The graph ofUaðxa; yaÞ in (a) and of Ubðxb; ybÞ in (b), as a function of D 2 ð0; 0:9Þ; computed in correspondence of a ¼ 0 (in magenta), of a ¼ a1(in
cyan), ofa¼ a(in red), ofa¼ a
red) in Fig. 13, condition (3.1) is fulfilled for f ¼ g10:
Indeed, ^a2 J1 is homoclinic to a2 J0 in Fig. 13, as
g6ð^aÞ ¼ a; the preimages of a* are internal to ½a m; aM;
whereamandaM are the minimum and maximum points of
g, respectively, and in any neighborhood U of a* it is possi-ble to find two intervalsJ0andJ1such that(3.1)is satisfied
for f ¼ gk; for some k 1: The intervals J
0andJ1 can be
constructed as explained in the proof of Theorem 2 in the Appendix ofGardiniet al. (2011), starting from a compact neighborhoodU of aand following the homoclinic orbit in a backward way. We address the interested reader toGardini (1994) and Gardini et al. (2011) for further mathematical details, as well as for an extension of (Marotto, 1978, Theorem 1).
Proposition III.4 Let us consider the map g in(2.6)for the following parameter configuration: vx¼ 0:9; wx¼ 0:2;
vy¼ 0:15; wy¼ 2; D ¼ 0:7; b ¼ 0:1; ^V¼ 0:8; l ¼ 6:5, and
r¼ 8: Setting J0¼ ½0:375; 0:405 and J1¼ ½0:41; 0:44;
it holds that(3.1)is satisfied for f ¼ g10
: Hence, the map g10 is turbulent and, in particular, it follows that htopðgÞ
logðp10ffiffiffi2Þ; where we denote by h
topðgÞ the topological
entropy of g.
Proof. The map g10is turbulent because it is continuous on R; and thus on J0 and J1: Moreover, direct
(software-assisted) computations show that g3ð0:375Þ ¼ 0:446 and g3ð0:405Þ ¼ 0:360; i.e., g3ðJ
0Þ J0[ J1; as well as that
g10ð0:41Þ ¼ 0:299 and g10ð0:44Þ ¼ 0:834; i.e., g10ðJ 1Þ J0
[ J1: Hence, g10ðJ0Þ \ g10ðJ1Þ J0[ J1; as required by (3.1). Then, from Block and Coppel (1992), Corollary 15, p. 200, it follows that htopðg10Þ logð2Þ and, from Adler et al. (1965), Theorem 2, we can conclude that htopðgÞ
logð10pffiffiffi2
Þ: ⵧ
We stress that the intervalsJ0andJ1are not unique, as
there are many possible choices for them which satisfy con-dition(3.1). In particular, the procedure leading to their con-struction can be repeated for any homoclinic point, in any neighborhood ofa* contained in the interval where the local inverse does not change.
Moreover, although in the statement of Proposition III.4 we have fixed some particular parameter values, the result is
robust, as the same conclusions hold for several different sets of parameter values, as well.
We also remark that, calling Dhthe value of the
parame-ter D for which it holds thatgðamÞ ¼ a; where amis the
min-imum point ofg, for D¼ Dh the first homoclinic orbit toa*
emerges and for D > Dh; even after the contact bifurcation
occurring for D¼ 0.746 (see Fig.11),a* still admits homo-clinic orbits and there is a chaotic repellor.
C. Economic and social scenarios
We now give an economic interpretation of the main scenarios we found, i.e., those in Fig. 2(a)and in Figs.6(a) and6(c), observing that the other frameworks locally repro-duce them, according to the chosen initial condition.
As we shall see, depending on the considered parameter configuration, and in particular, depending on the value of the social parameters and of the heterogeneity degree, we may be in a strong or in a weak visibility framework, in which the visibility levels for both groups of agents respec-tively exceed or lie below the desirable visibility level ^V ; and which lead to the extinction of the group with the lower attractiveness degree, caused by a social visibility level more distant from the desirable visibility level ^V ; for other param-eter configurations there is instead no persistence of a given ordering among the visibility levels for groups a; b and the threshold value ^V ; and we observe oscillatory, more or less regular, dynamics. Indeed, the first setting we focus on describes a strong visibility situation, which leads to the extinction of the group with the higher visibility level. On the contrary, the second setting describes a weak visibility situation, leading to the extinction of the group with the lower visibility level. The third setting describes amixed vis-ibility framework, in which the oscillatory dynamics are pro-duced by the interruption of the persistence of a given ordering amongVa; Vb, and ^V :
In regard to Fig.2(a), for instance, takinga(0)¼ 0.9, it holds that ^V < Vb< Vaand thus we expect a decrease in the
share of the agents of group a, i.e., for the next period, a smaller value fora as, according to(2.5)and(2.6), the rela-tive distances ofVaandVbfrom ^V determine the population
shares. We foresee this will produce a decrease in the aggre-gate demand [we stress that, although it is not possible to directly observe aggregate demand, we may infer its behav-ior from the population shares and we can find a confirma-tion of it by looking at the value of p] of commodity x, agents of group a have a stronger preference for, and that in turn this will lead to a lower value for px; producing an
increase in the relative pricep¼ py=px; which indeed we do
observe. The market closes and the optimal consumption quantities for the two goods are determined, which allow to computeVaandVband, consequently, the population shares.
In particular, due to the increase in p, both the values of xa andxbincrease, but the latter less than the former. Since vxis
much larger than vy; both Va and Vb raise and the ordering
^
V < Vb< Va remains unchanged, but the distance between
Vb andVa increases. Then, the same process repeats again,
leading to smaller and smaller values fora and to the even-tual extinction of the agents of group a:
As concerns Fig.6(a), for instance, takinga(0)¼ 0.5, we have thatVb< Va< ^V : Thus, since Va is closer to ^V than
Vb; we expect an increase in the share of the agents of group
a and, consequently, a raise in the aggregate demand of com-modityx. Indeed, we do observe a decrease in the relative price p. The optimal consumption quantities for the two goods are determined and, since the value of xa decreases more thanxband vxis still much larger than vy; the value of
Vadecreases, while the value ofVbincreases, so that the
dis-tance betweenVbandVais reduced, but the ordering among
Vb; Va and ^V is maintained. The repetition of such process
eventually leads to the extinction of the agents of group b. In regard to Fig.6(c), we slightly change the value of D from 0.7 to 0.651, in order to observe the simplest oscillatory behavior, i.e., a period-two cycle. Let us start from an initial condition close to the smallest between the two elements of the period-two cycle, taking, for instance, a(0)¼ 0.125. In such framework, the agents of type a, with their stronger preference for commodityx, are few and thus we expect that the aggregate demand for that good is low. Indeed, we do observe a high value for the relative price p. The optimal consumption quantities for the two goods are determined and, sincep is high, the value of xa is high, too. Due to the fact that vxis much larger than vy; we find that Vb< ^V < Va:
Since the distance betweenVaand ^V is smaller than the
dis-tance betweenVband ^V ; the share of agents of type a raises
and approaches a¼ 0:801: We then expect that the aggre-gate demand for commodityx is higher than before. Indeed, we observe a lower value for the relative pricep. The opti-mal consumption quantities for the two goods are determined and, sincep is lower now, the value of xadecreases. As vxis
still much larger than vy; we find that Va< Vb< ^V : Since
Vb is closer to ^V than Va; the share of agents of type a
decreases and is again near a¼ 0:125; giving rise to a period-two cycle.
We stress that, when dealing with the original parameter configuration considered in Fig. 6(c), we find the fashion cycle, which for us emerges at the aggregate level as a con-tinuous oscillation of the shares of the two groups, while on the individual level, it is characterized by oscillatory con-sumption choices over the two goods, presenting booms and busts, for the agents of the two groups. We report in Fig. 14 the corresponding time series of the main variables for the time periodst2 ½200; 300:
IV. CONCLUSIONS
We presented an exchange economy evolutionary discrete-time model with agents heterogeneous in the struc-ture of preferences. The reproduction level of a group is related to its attractiveness degree, which depends on the social visibility level, determined by the consumption choices of the agents in that group. In particular, the attrac-tiveness of a group is initially increasing with its visibility level, but it becomes decreasing, due to a congestion effect, when its visibility exceeds a given threshold value. Thanks to the combined action of the price formation mechanism and of the share updating rule, with the alternation between the bandwagon and snob regimes, the model is able to repro-duce the recurrent, periodic or chaotic, behavior typical of the fashion cycle, presenting booms and busts both in the agents’ consumption choices and in the population shares.
More precisely, we investigated the existence and local stability of the equilibria and, in order to illustrate the possi-ble scenarios, we performed a qualitative bifurcation analysis in terms of the heterogeneity degree parameter, which highlighted that the nontrivial equilibria emerge via transcrit-ical bifurcations and may lose stability via a flip bifurcation. Moreover, we showed that, far from the equilibrium, period two-cycles may arise via fold bifurcations of the map
FIG. 14. The time series corresponding to the periodst2 ½200; 300 for a(t) (in blue) and bðtÞ ¼ 1 aðtÞ (in green) in (a), for x
aðtÞ (in blue) and xbðtÞ (in
governing the dynamical system. From a global viewpoint, in addition to coexistence between the two trivial equilibria, and to multistability phenomena involving also a nontrivial equilibrium, where, however, the coexistence between groups is just static in nature, for our model we also found oscillatory behaviors in the population shares and in the con-sumption quantities. Indeed, thanks to the joint effect of the evolutionary mechanism and of the price formation mecha-nism, when the initial conditions are not too close to the extreme values 0 and 1, the group coexistence may be dynamic in nature, too, both regular, just after the flip bifur-cation, or irregular, due to the presence of chaotic attractors. The dynamic coexistence between groups then displays the recurrent dynamic behavior typical of the fashion cycle [see Simmel (1957)], characterized by booms and busts. We com-pleted our analysis by proving the existence of complex dynamics for the model via the method of the turbulent maps inBlock and Coppel (1992), working with homoclinic orbits. Finally, we gave a social and economic interpretation of the main scenarios we found.
As concerns future study directions, we could generalize the setting analyzed in this work by allowing the threshold value ^V ; at which the social attractiveness of a preference structure becomes a decreasing function of the visibility level, to possibly vary across the different preference struc-tures. Moreover, our framework could be employed to repre-sent the fashion cycle in settings with capital accumulation, such as the OLG model byDiamond (1965).
ACKNOWLEDGMENTS
The authors thank the anonymous reviewers for the helpful and valuable comments.
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