J Bioecon (2009) 11:43–63 DOI 10.1007/s10818-009-9059-8
Diversity, persistence and chaos in consumption patterns Francisco Fatás-Villafranca · Dulce Saura · Francisco J. Vazquez
Published online: 20 February 2009 © Springer Science+Business Media, LLC. 2009
Abstract In this paper we present a model from which discretionary consumption dynamics can be analyzed as global properties emerging from the endogenous transformation of a society inhabited by boundedly rational interactive consumers. By considering local and global interactions among consumers, we show that behavioral diversity plays a central role in the evolution of consumption patterns. The analysis of the model reveals the existence of a regime characterized by the persistence of different social standards, and a time evolution of the social distribution of behavioral patterns towards a heteroclinic cycle. In some cases the evolution seems to be chaotic, generating unpredictable, erratic dynamics of the aggregate social indices (average or social propensity for discretionary consumption). Keywords Chaos
Discretionary consumption · Externalities · Evolutionary dynamics ·
JEL Classification
E21 · C61 · B52
F. Fatás-Villafranca (B) · D. Saura Universidad de Zaragoza, Gran Via 2, 50005 Zaragoza, Spain e-mail:
[email protected] D. Saura e-mail:
[email protected] F. J. Vazquez Universidad Autonoma Madrid, F. Economicas y empresariales, Cantoblanco, 28049 Madrid, Spain e-mail:
[email protected]
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1 Introduction It is a well known fact in economics and sociology, that certain consumption activities which are not strictly necessary for life (let us call them ‘discretionary consumption activities’), combine the potential to provide consumers with material welfare, with the characteristic of helping them in conforming their personal and social identities (Veblen 1899; Stigler and Becker 1977; Baudrillard 1981; Becker 1996; Witt 2001). The significance of discretionary consumption activities in explaining consumption dynamics is remarkable, since they include a wide range of human activities such as buying ‘positional goods’ (Leiss 1983), deciding on key aspects of housing expenditure (Schor 1999), recreation, hobbies, entertainment, restaurants, shopping, or tourism, for example (Cowan et al. 1997; Corneo and Jeanne 1999). An important feature of discretionary consumption activities is the existence among them of socially created and reinforced complementarities, so that they jointly play the role of strong lifestyle-shapers (Earl 1986; Aversi et al. 1999; Gualerzi 2001). It is for this reason that we will herein distinguish between the consumption of a composite good Q δ , which we shall call ‘discretionary consumption good’ (the principal feature of which being that it offers consumers both material welfare and social reputation or status), and the consumption of ‘less discretionary goods or services’ (which we shall designate as Q −δ ). The model that we propose allows discretionary consumption decisions to be analysed under the assumption that, at least partially, they are the result of evolving processes of social interaction. This implies the use of a theoretical framework that is significantly different from the classical one. Our model is not based on fully rational agents with constant preferences, that face well defined intertemporal budget constraints—assumptions that have been recently questioned; see, for example, Aversi et al. (1999), Loasby (2001), Metcalfe (2001), but we consider that consumers modify their habits through social learning, thereby transforming the framework in which they carry out these decisions. In this case, the dynamics of consumption will be the result of a self-organizing process within a social system that is comprised of heterogeneous and boundedly rational consumers learning from each other, adapting themselves and their consumption patterns in accordance with the behavior of others, and thereby transforming the social framework itself (Hodgson 2000). From this point of view, we will assume that there exist global and local interdependencies between consumers who belong to different social groups that differ in their members’ propensity for discretionary consumption. Drawing upon the contributions by Scitovsky (1976), Bourdieu (1979), Sobel (1982) or Granovetter and Soong (1986), among others, we assume that the social reputation from discretionary consumption can be strengthened or weakened depending on the people that consumers may meet while engaging in these consumption activities. On the other hand, and as resources are scarce, although consumers enjoy discretionary consumption, they also perceive (through their own experience and by looking at the behavior of others) the fact that intensifying discretionary consumption means reducing the resources available for less discretionary consumption activities (such as health, education, financial services etc.). In this way, in our model we can see that consumers have doubts about the most
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suitable consumption pattern at any given moment (in terms of propensity to engage in discretionary versus less discretionary consumption activities). Unlike more standard approaches to human rationality, we characterize consumers in the model as being boundedly-rational, in the sense that they do not have a definite answer to the question of which is the most suitable consumption pattern at any moment. Uncertainty, the ongoing social change in consumption patterns, and the effect of externalities produced by the interaction with other consumers from certain ‘reference groups’ maintain our consumers’ awareness of the possibility of improving their satisfaction through social learning. Thus, consumers in the model might be conceived as ‘novelty- seekers’ (instead of utility maximizers) looking for a ‘good move’ at any time in terms of satisfaction obtained from discretionary consumption. They update their consumption patterns according to a changing environment and, as they learn and adapt their behavior, they influence the behavior of others. In this way, a social process of cognitive self-organization is generated, which is the essence of our vision of consumption dynamics. Starting out from this point, our model highlights to what extent this process of social transformation can be complex. The paper is structured as follows: in Sect. 2 we describe the assumptions of the model, analyzing the importance of externalities in discretionary consumption activities, and how they may affect consumption patterns in a framework of social interaction that is endogenously updated by individual decisions. So we propose an evolutionary model, along the lines of Cowan et al. (1997) or Turab et al. (1998) in which the endogenous transformation of the social distribution of behavioral standards gives rise to certain emergent properties as the determinants of the trajectories of the social system. The dynamic properties of the model are analyzed in Sect. 3 for the reduced twodimensional case. Section 4 tackles the n-dimensional case. Finally Sect. 5 presents the conclusions. 2 An evolutionary model of consumption patterns 2.1 Discretionary consumption Let us consider a society in which income is uniformly distributed,1 and individuals and households spend their entire income on two types of consumption activities: (1) Discretionary consumption, which includes activities that are not strictly necessary for life, but provide consumers both with material welfare and social reputation. This reputation comes from distinction, popularity, style, originality, prestige or a certain image of himself that the consumer gives the more he spends on discretionary consumption. Social reputation can be strengthened or 1 We leave aside for future research the case of unequal income distribution in society. If the supposition of uniformly distributed income is restrictive, we can interpret the model as an analysis of the change in consumption patterns in a population of individuals (or consumers) which belong to the same income group (middle class or upper-middle class or low income people, etc.). In this sense we will assume that all the consumption patterns that we may consider in the model are compatible with the budget constraints typical of the class or group under consideration.
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weakened depending on the people that consumer may meet while engaging in these consumption activities (Cowan et al. 1997), so that positive or negative externalities may be derived from social interaction. Discretionary consumption includes important aspects of housing decisions (like choosing a neighborhood); entertainment, joining clubs, restaurants; shopping; hobbies; tourism and choosing holiday destinations. In these activities, defining goods as bearers of different characteristics that provide alternative solutions to perceived individual problems (like defining self-identity while satisfying material needs (Lancaster 1966)) is highly significant. (2) Less-discretionary consumption activities (such as education, health, basic nutrition or technical/financial services); these activities are necessary to maintain a suitable quality of life and do not share the aforementioned features. In our model, we assume the existence of a composite good/service Q δ ‘discretionary consumption good’ with the characteristics presented in 1, and another one Q −δ characterized by those features mentioned in 2. For the sake of simplicity, lessdiscretionary consumption only appears in our approach indirectly, through a negative externality induced by what we will call the ‘prevention effect’. That is, we suppose that consumers can question the convenience of maintaining high levels of discretionary consumption as they socially interact with individuals with a greater propensity to spend on less discretionary activities (financial or technical services, health care, education). Furthermore, in this paper we do not deal with savings choices, since they depend on such variables as interest rates, expectations, time preference parameters or consumers’ wealth (Deaton 1992), which do not appear explicitly in our model. Given that our main concern here is to analyze discretionary consumption dynamics, we leave aside these aspects for future research.
2.2 The co-existence of different lifestyles Let us assume that consumers can be classified as belonging to one of n different social groups (i = 1, . . . , n) exclusively differentiated with respect to their lifestyles, tastes and consumption habits (Sobel 1982), and not with respect to their absolute level of income. We consider that the representative consumer lifestyle within group i can be identified by the proportion of income (ci , with 0 < c1 < · · · < cn < 1) that nthey spend xi = 1) on consumption of Q δ . In addition, let us denote by xi (0 ≤ xi ≤ 1 and i=1 the proportion of individuals in the entire population that have a propensity ci for Q δ -consumption. This assumption requires several comments. Although in actual societies it is debatable that there are two individuals with an identical ci , we must recognize that there are propensities that are sufficiently similar, as well as propensities sufficiently different, to recognize them as being typical of different social groups. Therefore, the consideration of a finite number n of social groups implies the existence of a threshold that defines similarity and difference concerning consumption patterns.
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We propose that this threshold should be defined in relative terms, so we consider different levels of ci such as ci =a ci with a being constant and such that 0 < a 0 β β
(2)
where the parameter ρ represents the relative influence of the “intrinsic utility” derived from material characteristics of Q δ , and the social influence: if ρ > 1 the first effect will dominate; the social influence and externalities will be predominant when ρ < 1. Note that v(ci ) does not determine a change of lifestyle; it only reflects the satisfaction that consumers within group i derive from their consumption pattern. In fact, we will assume below that although individual valuations are influenced just by local externalities derived from the habitual interactions with consumers from the reference groups, behavioral changes will take place in the model as a consequence of interactions that may occur throughout the whole society. We do so because it seems realistic to consider that although frequent interactions with close individuals may specially condition individual perceptions and valuations of currently held consumption patterns, individuals may meet and learn from each other throughout society, even pertaining to different social groups. Thus, in order to state how changes in behavior take place in the model, let us start by mentioning that cognitive limitations and imperfect knowledge on the part of consumers, together with the fact that society evolves, lead us to propose that consumers are boundedly rational (Simon 1983; Langlois 2001). This means that although purposive, they are not driven by conventionally-specified rational choice on (c1 , . . . , cn ). Therefore, let us assume that consumers can modify their lifestyles through the emulation of people from distinct social groups, with the unique condition that behavioral change only takes place towards better valued behavioral patterns. According to Scitovsky (1976), Earl (1998), Witt (2001) and others, the consumers are restless novelty seekers that keep ongoing learning processes, in search of new consumption patterns and better conditions for life. In their personal evolution within a global social environment, personal interaction and the emulation of others is an efficient way to continuous improvement. Thus, let us denote by f i j the rate at which consumers with lifestyle c j switch to lifestyle ci , in their pursuing of more satisfactory behavioral patterns. Let us consider that this switching rate is given by f i j = γ vi − v j + = γ max vi − v j ; 0 , γ > 0
(3)
whereγ is a constant that captures the ease with which consumers can transform their life-styles passing from one social group to another. This expression implies that consumers switch only to more highly valued lifestyles. Note that we are not requiring the existence of interpersonal comparisons of utility. What we are assuming is that, because of the existence of a homogenous valuation criterion in society—given by Eq. 1—when a consumer from group i meets another from j, he gets to know the possibility of adopting a behavioral pattern c j . Then, by comparing his present satisfaction vi with the satisfaction v j enjoyable in case of
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consuming c j and joining the j-social space, the consumer may decide changing his lifestyle through emulation. Note also that, in the switching process proposed in (3), emulation is not an automatic process. Actually, we are not stating that every consumer from i who meets another from j will always emulate c j if it is a better valuated pattern. The process in (3) just determines a certain flow of consumers in the ‘better-valuation’ direction, but not a total instantaneous emulation. In fact, this assumption is plausible given the empirics of consumption activities, which often are strongly dependent on habits, routines and personal fears to change (Hodgson 2000; Langlois 2001). Then, assuming that the product δxi x j (with δ being a proportionality constant such that 0 < δ < 1) gives the probability for a random and independent interaction between one consumer with lifestyle i and other one with pattern j, in a small time interval t, the flow of consumers from j to i would be given by (Hofbauer and Sigmund 1998)6 : δxi x j f i j t and the change in the proportion of consumers with lifestyle ci (i = 1, . . . , n) would be: xi =
n
δxi x j ( f i j − f ji )t
j=1
where f i j − f ji = γ vi − v j + − γ v j − vi + = γ vi − v j Therefore, the continuous evolution of the proportion of consumers with lifestyle i is described by the differential equation x˙i =
n j=1
δxi x j f i j − f ji = δxi
n j=1
⎛ ⎞ n x j γ vi − v j = γ δxi ⎝vi − xjvj⎠ j=1
or equivalently (taking xi (t) = xi (δγ τ ), which only represents a change in velocity) we can represent the evolving social distribution of lifestyles, driven by boundedly rational purposive consumers, interacting in pursuit of higher satisfaction levels, by the replicator dynamics system:
6 We would like to suggest two possible extensions of our present contribution concerning parameters γ and δ. These parameters affect, respectively, the ease with which consumers can transform their life-styles passing from one social group to another and the probability of interactions on a global level between consumers from different groups. The possible appearance of learning costs when radically transforming a lifestyle, as well as the cognitive distance between consumers with very different life-styles, lead to the possibility of supposing that γ and δ depend inversely on the distance between any two groups i, j or on the difference between their respective propensities to consume. The formal complexity associated to this idea forces us to leave this question for future research.
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⎛ x˙i = xi ⎝vi −
n
51
⎞ x j v j ⎠ = xi (vi − v) ¯
(4)
j=1
In this way, we have modeled a changing bundle of behaviors such that, better valued consumption patterns attract new individuals—increasing their relative importance within the population—and those patterns providing unsatisfactory valuation levels, progressively decline, eventually disappearing from the range of lifestyles. As long as n evolves, and this endogenous individuals are adapting their behavioral patterns, {vi }i=1 change reshapes what is perceived as satisfactory or unsuitable within society. 3 Two-dimensional dynamics We begin by analyzing the dynamic behavior of the evolutionary model proposed above for n = 3 social groups. In this case, the evolution of the relative sizes of the groups xi with respective propensities ci is determined by the following system of nonlinear differential equations: ⎧ ¯ = x1 [ρc1 + x2 c1 − v] ¯ ⎨ x˙1 = x1 (v1 − v) ¯ = x2 [ρc2 + (x3 − x1 )c2 − v] ¯ x˙2 = x2 (v2 − v) (5) ⎩ ¯ = x3 [ρc3 − x2 c3 − v] ¯ x˙3 = x3 (v3 − v) where v¯ = x1 v1 + x2 v2 + x3 v3 = x1 [ρc1 + x2 c1 ] + x2 [ρc2 + (x3 − x1 )c2 ] + x3 [ρc3 − x2 c3 ] . Given that the simplex S3 = {(x1 , x2 , x3 ) ∈ R3 : x1 + x2 + x3 = 1; x1 , x2 , x3 ≥ 0} is not altered by the flow implied by the system (5) (x˙1 + x˙2 + x˙3 = 0), the dynamics on S3 are described by the plane system (x3 = 1 − x1 − x2 )
ˆ x˙1 = x1 [ρc1 + x2 c1 − v] ˆ x˙2 = x2 [ρc2 + (1 − 2x1 − x2 )c2 − v]
(6)
with vˆ = x1 [ρc1 + x2c1 ] + x2 [ρc2 + (1 − 2x1 − x2 )c2 ] + (1 −x1 − x2 )[ρc3 − x2 c3 ] on the simplex Sˆ2 = (x1 , x2 ) ∈ R2 : x1 + x2 ≤ 1; x1 , x2 ≥ 0 . The analysis of the behavior of the orbits of the two-dimensional model (6) over the invariant simplex Sˆ2 is done in the following steps: (1) Stationary points in Sˆ2 . There are six possible solutions x˙1 = 0, of theaρsystem , aa(a+2)ρ x˙2 = 0 : P1 = (0, 0), P2 = (1, 0), P3 = (0, 1), P4 = 21 − a 2 +2a+2 2 +2a+2 ,
P5 = (0, ρ − a1 ) and P6 = (ρ − a1 , 1 − ρ + a1 ). In particular, we obtain the four regions corresponding to different possibilities of stationary points shown on Table 1, where the stability (deduced from the eigenvalues of the Jacobian matrix) of each stationary point is also indicated. (2) We use the Bendixson–Dulac method (see Hofbauer and Sigmund 1998) for proving that periodic orbits do not exist in the interior of Sˆ2 : if there exists a positive function B defined on a connected G ⊂ R2 such that the vector field BF has positive divergence at every point, then x˙ = F(x) admits no periodic
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Table 1 Stationary points of the 2-d reduced model Stationary points (I) (II) (III) (IV)
0 < ρ < a1
P1 , P2 and P3 saddle points; P4 source
1 a 2 +2a+2 a < ρ < 2a(a+1) a 2 +2a+2 < ρ < 1 + 1 a 2a(a+1) ρ > 1 + a1
P1 and P3 sinks; P2 , P5 and P6 saddle points; P4 source P1 and P3 sinks; P2 and P5 saddle points; P6 source P1 sink; P2 source; P3 saddle point
orbit in G (such a function B is said to be a Dulac function). Let us consider the function B(x1 , x2 ) =
1 x1 x2 (1 − x1 − x2 )
which is positive on the interior of Sˆ2 . We have div(BF) = traceD(BF) =
ac1 (1 − x2 ) ac1 [(1 + a)(1 − x2 ) − ax1 ] > >0 x1 (1 − x1 − x2 ) x1 (1 − x1 − x2 )
and thus we conclude that no periodic orbits exist in the interior of Sˆ2 . (3) Finally, the direct application of the Poincaré–Bendixson theorem assures that, for ρ < a1 , the ω-limit points of the non-stationary orbits of (6) are located on the simplex boundary, and then the heteroclinic cycle on the boundary (made up of P1 , P2 , P3 and the three invariant edges that join them and determine their stable and unstable manifolds) is the attractor of the system: all the interior orbits converge to this heteroclinic cycle. On the other hand, the same arguments guarantee that, for ρ > a1 , (almost) every interior orbit tends to a frontier stationary point (P1 or P3 ). The four possible phase portraits (corresponding to the four regions for ρ described in Table 1) of the plane non linear system (6) are shown in Fig. 1. Region (I) corresponds to what we call the ‘Dynamic Diversity’ regime (D-regime): all of the non-stationary orbits of (4) converge to the heteroclinic cycle of the border of the simplex. In this regime (corresponding to ρ < a1 ), not only do we get three different types of behavior (c1 , c2 , c3 ) persisting at all times, but also the individual spending patterns are continually being adapted. This process emerges as a continuous flow of consumers who modify their lifestyles by passing from one group to another. In Fig. 1 we can observe the persistence (which is formally defined as lim sup xi (t) > δ for every i = 1, . . . , n; see Hofbauer and Sigmund (1998)) of each of the consumption patterns through time, with a cyclic evolution (non-periodic) of the sizes of each group. The condition ρ < a1 tells us that if the relative influence of the social interactions on the individual valuations goes beyond a given minimum critical level, the local and global effects produced by the interactions between individuals of different social
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Fig. 1 Different phase portraits of the 2-d reduced model
groups will maintain the system in an indefinite process of endogenous transformation of the structure of behavior. Thus the system will be in a D-regime when the social influence on the individual valuations is sufficiently intense in comparison with the intrinsic utility. Note that it is possible to be in this regime even though the intrinsic utility effect is greater than the social influence effect (ρ > 1) since must a be sufficiently small so that c3 = (1 + a)2 c1 is less than 1. In this D-regime, the average (or social) propensity for discretionary consumption c¯ =
n
xi ci
i=1
shows a cyclic (non-periodic) evolution, going from c1 to c3 (and viceversa) and taking more time when it passes near c2 (see Fig. 2 where it has been drawn with a continuous line). Note that the evolution takes more time in each new cycle. On the other hand, regions (II), (III) and (IV) correspond to the ‘Conformity’ regime (C-regime). Now, the dynamics of the system change notably (see Fig. 1): two new stationary points may appear on the border of Sˆ2 , the interior stationary point may disappear, and the orbits end up having a stationary point on the border (except for the orbits in the regions (II) and (III) with initial conditions over the heteroclinics which join either the aforementioned stationary points). Hence, in this case (ρ > a1 ) we get a long term concentration on one consumption pattern (‘conformity’ of lifestyles), which can depend on the initial conditions as occurs in regions (II) and (III), and the average propensity for discretionary consumption (drawn in Fig. 2 with a discontinuous line) tends to it.
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Fig. 2 Time evolutions of the average propensity for discretionary consumption in the 2-d reduced model
4 N-dimensional dynamics The evolution of the model with n social groups is determined by the differential equations: ¯ = xi [ρci + (Ax)i − x(ρc + Ax)], i = 1, . . . , n x˙i = gi (x) = xi f i (x) = xi [vi − v] (7) where x = (x1 , . . . , xn ), c = (c1 , . . . , cn ), A is the n × n matrix ⎛
0 ⎜ −c2 ⎜ ⎜0 ⎜ A = ⎜. ⎜ .. ⎜ ⎝0 0
c1 0 −c3 .. . 0 0
0 c2 0 .. .
0 0 c3 .. .
··· ··· ··· .. .
· · · −cn−1 0 ··· 0 −cn
0 0 0 .. .
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ cn−1 ⎠ 0
(8)
and (Ax)i denotes the i-th element of Ax. Equation 7 can be easily reduced to a linear replicator equation x˙i = xi [h i (x) − h(x)] where h i (x) = (Mx)i and M any square matrix. These equations arise in a variety of different fields (sociobiology, theoretical immunology, evolutionary games, etc.) and disguised as Lotka-Volterra equations they play a major role in mathematical ecology. The simplex Sn = (x1 , . . . , xn ) ∈ R : n
n
xi = 1; xi ≥ 0, i = 1, . . . , n
i=1
is invariant under (7), and so the phase space of the evolutionary model with n social groups is given by an (n −1)-dimensional simplex. The surface of the simplex consists of n hyperplanes xi = 0, each of which gives an invariant set. Therefore, any intersection of these hyperplanes is also an invariant set. As a direct consequence we conclude
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k
that every vertex ek = (0, . . . , 0, 1, 0, . . . , 0) of the simplex Sn is an equilibrium point of system (7). If (7) has interior equilibrium points they are given by the solutions of
ρc1 + (Ax)1 = ρc2 + (Ax)2 = · · · = ρcn + (Ax)n x1 + · · · + xn = 1
(9)
satisfying xi > 0. Although the exact number of equilibrium points of (7) (and, as we will see, their stability) depends on the values of the parameters a and ρ, its orbits always converge to bdSn . This can be proved by transforming (7) into the (n − 1)-dimensional system x˙i = gˆi (ˆx) = gi (ˆx, xn ), i = 1, . . . , n − 1 with xˆ = (x1 , . . . , xn−1 ) and xn = 1 − Sˆn−1 = (x1 , . . . , xn−1 ) ∈ R
n−1
:
n−1 i=1 n−1
xi , on the simplex
xi ≤ 1; xi ≥ 0, i = 1, . . . , n − 1
i=1
and determining the sign of the divergence of the vector field gˆ = (gˆ 1 , . . . , gˆ n−1 ) in int Sˆn−1 , which is the same as that corresponding to Bˆ gˆ for any real-valued positive ˆ x) defined in int Sˆn−1 . Taking function B(ˆ B(x) =
1 n
ˆ x) = ⇒ B(ˆ xi
i=1
n−1 i=1
1
xi 1 −
n−1 i=1
xi
we obtain div Bˆ gˆ =
n−1 ∂( Bˆ gˆi ) i=1
∂ xi
=B·
n−1
xi
i=1 n−1
= −B(x) · x Ax = B(x)
∂ fi ∂ fi − ∂ xi ∂ xn
+
n−1 i=1
fi xi xn
(ci+1 − ci )xi xi+1 > 0
i=1
So the divergence of f is positive at any point of Sn , that is, Eq. 7 is volume-expanding in intSn . Therefore, every orbit of (7) starting in intSn converges to bdSn . 4.1 Convergence to a frontier stationary point We begin by analyzing the case ρ > 1 + a1 . It is not difficult to prove that ek , k = 1, . . . , n, are the only equilibrium points of (7). Indeed, let p be an equilibrium point of (7), i the first integer such that pi = 0, and j the last integer such that p j = 0 (1 ≤ i ≤ j ≤ n). Let us suppose i < j, then
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ρci + (Ap)i = ρc j + (Ap) j ⇒ (ρ + pi+1 )ci = (ρ − p j−1 )c j ⇒ ρ(c j − ci ) = pi+1 ci + p j−1 c j ≤ c j ⇒ ρ[1 − (1 + a)i− j ] ≤ 1
and, if ρ > 1 + a1 , this implies 1 − (1 + a)i− j
1, which is not possible since i < j. Hence every p equilibrium point of (7) must satisfy i = j and therefore p = ek for some k = 1, . . . , n. In order to analyze the asymptotic behavior of the orbits of (7) we study the (local) stability of ek as follows. The Jacobian of (7) at ek is of the form ∂ fi k ∂gi k (e ) = δi j f i (ek ) + eik (e ) ∂x j ∂x j (where δi j is the Kronecker delta: δi j = 1 if i = j; δi j = 0 otherwise). If eik = 0 (i = k), this reduces to f i (ek ) for i = j and to 0 for i = j, so f i (ek ) is an eigenvalue for the (left) eigenvector ei . λk,i = f i (ek ) is called a transversal eigenvalue at the corner ek belonging to the eigenvector pointing towards the corner ei . For our system, we get λk,i = ρci + (Aek )i − ρek c − ek Aek = ρci + (Aek )i − ρck (note that ek Aek = 0 since the main diagonal of A consists of zeros) which leads to λk,i = (ρ + 1)ck−1 − ρck = (1 − aρ)ck−1 , i = k − 1 λk,i = (ρ − 1)ck+1 − ρck = (aρ − 1 − a)ck , i = k + 1
(10)
λk,i = ρ(ci − ck ), i = k − 1, k, k + 1 When ρ > 1 + a1 , we obtain that λk,i < 0 for i < k and λk,i > 0 for i > k. Then e1 is a source, en is a sink, every ek with k = 2, . . . , n − 1 is a saddle point, and the edges of the simplex Sn are isoclines of the system that determine the stable and unstable manifolds of the saddle points. In consequence, every orbit of (7) starting in intSn tends to en and the evolution of consumption patterns shows a long term concentration towards the biggest one (see Fig. 3a. This corresponds to the ‘Conformity’ regime described above for n = 3 social groups: the average propensity for discretionary consumption tends to cn , and the diversity of consumption patterns gradually disappears. On the other hand, for a1 < ρ < 1 + a1 , system (7) has equilibrium points on the simplex boundary that are different from ek , which can be deduced from the fact that p is an equilibrium point of (7) if p1 = (0, p) and p2 = (p, 0) are equilibrium points
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Fig. 3 Some phase portraits of the 3-d reduced model
of the (n + 1)-dimensional system, and reciprocally (note that, more precisely, the flow on the faces x1 = 0 and xn = 0 is described by the same equations but with n − 1 social groups). So we have a new equilibrium point on each edge ek ek+1 , and it is easy to prove that no others exist on the other edges. The study of the transversal eigenvalues at ek reveals that now e1 ,…, en−2 are saddle points and en−1 , en are sinks. Then, the orbits of (7) starting at (almost) any point of intSn converge to either en−1 or en . We have represented a typical phase portrait in Fig. 3b, corresponding to the region (III) of Table 1; the case of region (II) generates a very similar evolution of the orbits. In consequence, as before, the model exhibits a long term concentration on one consumption pattern, which can depend on the initial conditions, and the average propensity for discretionary consumption tends to it. Hence, the ‘Conformity’ regime, where the diversity of consumption patterns is gradually disappearing, arises when ρ > a1 . 4.2 Heteroclinic networks and chaotic evolution Now let us consider the replicator Eq. 7 for ρ < a1 . Above we have proved that every orbit starting in intSn tends to bdSn , but now, as we will see, showing a qualitatively very different evolution from the one generated in the ‘Conformity’ regime. First of all let us note that (as can be easily proved using methods similar to the previous section) the unique equilibrium points on the edges of Sn are their own vertices ek , k = 1, . . . , n, and so each edge (without the corners) is exactly one orbit of system (7). The transversal eigenvalues at every ek —given by formulae (10)—can be represented in an n × n matrix, where the entry in row k and column i is λk,i = f i (ek ) (and it is 0 if xi > 0 at ek ); so if element λk,i of this matrix is positive then the orbit ek ei evolves from ek to ei , and from ei to ek when λk,i < 0. For ρ < a1 the sign structure of the (characteristic) matrix takes the form
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⎛
··· ⎜ ⎜ + 0 − ... ⎜ ⎜ ⎜ − + 0 ... ⎜ ⎜. . .. ... ... ⎝ .. − ··· − + 0
−
+
⎞ + .. ⎟ . ⎟ ⎟ ⎟ +⎟ ⎟ ⎟ −⎠ 0
and so we deduce that a heteroclinic cycle appears on bdSn . Indeed, the edges connecting ek and ek−1 (k = 2, . . . , n) are orbits of (7) which evolve from ek towards ek−1 since the transversal eigenvalues λk,k−1 = (1 − aρ)ck−1 are all positive; and the edge e1 en is an orbit evolving from e1 to en since λ1,n = ρ(cn − c1 ) is also positive. So n,1 = {en → en−1 → · · · → e2 → e1 → en } is a heteroclinic cycle of system (7), but it is not the only one: any other sequence i, j = {ei → ei−1 → · · · → e j → ei }, with i ≥ j + 2 is also a heteroclinic cycle (note that however there is no ‘planar’ heteroclinic cycle, connecting only two corners). Thus, system (7) has a heteroclinic network on bdSn consisting of one n-cycle, two (n − 1)-cycles, three (n − 2)-cycles, …, and (n − 2) 3-cycles; so the number of heteroclinic cycles in the network is 21 (n − 1)(n − 2). As an illustration of these networks we have drawn in Fig. 4 all the existing heteroclinic cycles for n = 4 and n = 5. In spite of the fact that heteroclinic cycles and networks frequently appear in many other areas (ecological models, evolutionary biology, chemical kinetics, etc), they are very unusual in economic models. The analysis of the stability of these cycles is not an easy task, especially if they belong to a heteroclinic network. In this case, there exists at least one positive ‘transverse’ eigenvalue (corresponding to a direction which does
Fig. 4 Heteroclinic networks in system (7) with n = 4 and n = 5 social groups
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not belong to a connecting orbit) and therefore the cycle can not be an asymptotically stable attractor. However, the heteroclinic cycle, although unstable, can be attracting for quite a large set of initial points or even for a whole neighbourhood.7 Currently, stability conditions for cycles in heteroclinic networks are only known in very specific situations (see Brannath 1994; Hofbauer and Sigmund 1998). So henceforth we investigate the asymptotic behavior of system (7) for ρ < a1 by means of numerical simulations, which has to be done carefully because a small noise in the calculations may drastically change the behavior of the orbits, especially in the neighbourhood of the saddle points (where the evolution takes so much time). The simulations were done with the transformed variables yi = log xi (that is, over the equivalent system y˙ = G(y)) to avoid numerical underflow: the error for values of xi near 0 is drastically reduced by keeping track of its logarithm. For n = 4 social groups, the results of the simulations suggest that the interior orbits always approach one cycle of the heteroclinic network, although the concrete attracting cycle depends on the values of the parameters: so, for example, e4 e3 e2 is an attracting cycle if ρ = 3, a = 0.1, c1 = 0.3; the attracting cycle is e3 e2 e1 when ρ = 1, a = 0.1, c1 = 0.3; and e4 e3 e2 e1 is an attracting cycle for ρ = 2.1, a = 0.1, c1 = 0.3. We have drawn a typical orbit of each case on the left-hand side of Fig. 5; the right-hand side shows the order in which the saddle points of the cycle are visited: the abscissas represent time (we have dropped 2000 initial transients) and the ordinates display the nearest saddle (although the transition between saddles is omitted, the process is so fast that it would be practically zero on the time scale used anyway). Although in some cases the system is not persistent (according to the definition above introduced: one of the consumption patterns can gradually disappear), it always exhibits a continuous flow of consumers who modify their lifestyles by passing from one group to another. Then socio-diversity is permanently present over time, which characterizes the ‘Dynamic Diversity’ regime, and the average propensity for discretionary consumption shows a cyclic (non-periodic) evolution, analogous to that corresponding for n = 3 social groups (see Fig. 2). For n ≥ 5 social groups, the simulations suggest that the behavior can be far more complex. All the interior orbits approach the heteroclinic network, but their evolution presents some new interesting features. The behavior looks like the attraction to an heteroclinic cycle: the orbits visit several quasi-stable states (corners of the simplex), and the duration of the visits there shows a geometrical expansion. However, it seems that there is irregularity in the order and duration of visits, which is the major difference from typical cases of the attraction to an heteroclinic cycle. This irregularity can be observed in Fig. 6 where we have drawn on the left-hand side the order in which an orbit of system (7), with n = 5, ρ = 1, a = 0.1 and c1 = 0.3, visits the saddle points of the network, and on the right-hand side the durations (in the log scale) of its visits to a neighborhood of the saddle e5 . As is shown in Fig. 6, the evolution of the orbit seems to switch ‘randomly’ between the cycles of the heteroclinic network in each new return, which would constitute the 7 This peculiar phenomena concerning the existence of attracting unstable heteroclinic cycles was firstly reported by Melbourne (1991).
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Fig. 5 Attracting heteroclinic cycles in system (7) with n = 4 social groups
source of a special kind of chaotic behavior. Chaotic behavior in heteroclinic networks was firstly found by Chawanya (1995) in a game dynamics system, where the interior orbits evolve towards the simplex frontier approaching an heteroclinic cycle
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Fig. 6 Chaotic evolution in system (7) with n = 5 social groups
Fig. 7 Erratic time evolution of the average propensity to discretionary consumption in system (7) with n = 5 social groups
and exhibiting an erratic sequence of the equilibrium points visited (belonging to the heteroclinic cycle) and an unpredictable duration of the visits in their neighborhoods. An example of switching appears in Guckenheimer and Worfolk (1992). Recently, Aguiar et al. (2004) have proved the existence of switching on a heteroclinic network and horseshoe dynamics in a neighbourhood of the heteroclinic cycles. The study of switching dynamics on networks requires a rather complex mathematical framework (see, for example, (Aguiar et al. 2004), which exceeds the scope of this paper, and so a mathematical proof of the chaotic behavior of our model is left for future research. This chaotic behavior would imply that the ‘Dynamic Diversity’ regime (characterized by the persistence of socio-diversity in consumption patterns, which happens when ρ < a1 ) can generate an erratic, unpredictable evolution of the average propensity for discretionary consumption, as seems to be observed in Fig. 7 for n = 5, ρ = 1, a = 0.1, c1 = 0.3.
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5 Conclusions We have presented an evolutionary model which allows discretionary consumption decisions to be analysed under the assumption that, at least partially, they are the result of evolving processes of social interaction. The consideration of local and global effects underlying social interactions between boundedly rational agents, together with the relative importance with which consumers perceive the intrinsic utility of discretionary consumption and social connotations, seem to be crucial aspects in understanding certain consumption dynamics. Although our model precludes the analysis of significant realities (such as the influence on consumption of uneven income distributions), it uncovers interesting features of the evolution of socio-diversity in consumption and lifestyles. The analysis of the model (carried out for the cases n = 3, n = 4 and n ≥ 5) reveals the existence of two regimes (depending on the values of the parameters). The first, which we have called the ‘Conformity’ regime, is characterized by the long term concentration on one consumption pattern and the disappearance of socio-diversity. The second regime (that we have called the ‘Dynamic Diversity’ regime) is characterized by the persistence of socio-diversity in consumption patterns: the orbits of consumption patterns evolve towards an heteroclinic cycle and exhibit a continuous flow of consumers who modify their lifestyles by passing from one group to other. Moreover, for n ≥ 5 and certain values of the parameters, it seems that there is an erratic, unpredictable evolution of the consumption patterns, generating a chaotic evolution of the average propensity for discretionary consumption. To sum up, in our model the qualitative behavior of the time evolution of consumption patterns depends on some critical values of the implied parameters: the intrinsic utility of discretionary consumption and the intensity of social externalities generated by distinction, the spread of information, prestige or prevention effects. In some cases the persistence of diversity in consumption patterns is possible; in others, a gradual homogenization of standards is produced. Given the formal complexity of the model, interpreting intuitively the dynamics obtained is an adventurous task. Nevertheless, an important socio-economic lesson that might be learnt from our results is that, in a diverse society where consumers search for social reputation through discretionary consumption, but also have to cover certain less-discretionary needs, if the influence of social interaction on individual utility is strong enough (ρ < a1 ), and the number of coexisting lifestyles is sufficiently high (n ≥ 3), then the spread of any particular lifestyle leads to its future erosion. This is so, because of the contradictory effects of positive and negative local externalities on individual satisfaction levels. As we have seen, under certain specified conditions these contradictory effects keep the system in an ever-changing persistence of sociodiversity, and the irregularity of the evolutionary path increases with the range of coexisting lifestyles. Finally, considering this model as a first step within a future research strategy on consumption dynamics, we can conclude by saying that what it clearly shows is that certain patterns of consumption may be the result of self-organizing processes involving, in an essential way, local and global interactions between boundedly rational heterogeneous consumers.
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Acknowledgements We would like to thank Stan Metcalfe, P. P. Saviotti, Gerald Silverberg, Ulrich Witt, the editors and two referees of the Journal of Bioeconomics for their comments and suggestions on different versions of this paper.
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