A model of coupled maps with Pareto behavior

1.2 MB·English

by  J. R. Sanchez

EPJ manuscript No.
(will be inserted by the editor)
A model of coupled maps with Pareto behavior
J. R. Sa´nchez1, J. Gonz´alez-Est´evez2,4 a, R. Lo´pez-Ruiz3, and M. G. Cosenza4
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0 1 Facultad deIngenier´ıa, Universidad Nacional deMar delPlata, Av.J. B. Justo 4302, 7600-Mar del Plata, Argentina.
0 2 Laboratorio deF´ısica AplicadayComputacional, UniversidadNacionalExperimentaldelT´achira, SanCrist´obal, Venezuela.
2 3 Facultad deCiencias, DIISand BIFI, Universidad deZaragoza, 50009-Zaragoza, Spain.
n 4 Centro deF´ısica Fundamental,Universidad deLos Andes,M´erida, Apartado Postal 26, M´erida 5251, Venezuela.
a
J
Received: date/ Revised version: date

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Abstract. A deterministic system of coupled maps is proposed as a model for economic activity among
]
D interactingagents.Thevaluesofthemapsrepresentthewealthoftheagents.Thedynamicsofthesystem
is controlled by two parameters. One parameter expresses the growth capacity of the agents and the
C
other describes the local environmental pressure. For some values of the parameters, the system exhibits
.
n nontrivial collective behavior, characterized by macroscopic periodic oscillations of the average wealth of
i the system, emerging out of local chaos. The probability distribution of wealth in the asymptotic regime
l
n shows a power law behaviorfor some ranges of parameters.
[
PACS. 05.45.-a – 05.45.Ra – 89.65.-s – 89.65.Gh
1
v
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The study of wealth distribution in western societies local field
1
hasbeenafocusofmuchattentionintheemergingareaof
0 1
1 research of Econophysics. A power law behavior is found Ψt(i)= [xt(i−1)+xt(i+1)], (1)
2
0 in many economic activities, for instance, in income dis-
7 tributions.Typically,highincomeearners,amountingtoa through a negative exponential with parameter a(i),
0 few percent of the population, are distributed following a
n/ Pareto-like distribution. [1,2,3,4,5]. Hence, inequality in xt+1(i)=r(i)xt(i) exp(−|xt(i)−a(i)Ψt(i)|). (2)
i the wealth distribution is a fact in most economic activi-
nl ties.Theoriginofsuchbehaviorseemstobecausedbythe Theparameterr(i)representsthecapacityofagentitoget
: interactionof the macrowith the microeconomy.Here we richerandthe parametera(i)describes the localselection
v propose a simple deterministic spatiotemporal model for pressure[7].Thismeansthatthelargestrateofgrowthfor
i
X economic dynamics where inequality emerges as a result agent i is obtained when x(i) ≃ a(i)Ψt(i), i.e., when the
ofthedynamicalprocessestakingplaceonlyatthemicro- agent has reached some kind of adaptation to the local
r
a scopic scale. That is, the microeconomy fully determines environment. In this paper we consider a homogeneous
the macroeconomic characteristics of the system. system with a uniform capacity r and a fixed selection
pressure a for all the agents.
The model [6] consists of N interacting agents repre-
Thesimplestcollectivebehavioroftheone-dimensional
senting companies, countries or other economic entities,
coupled map system described by Eqs. (1) and (2) cor-
placed as nodes on a network. For simplicity, we shall as-
responds to the synchronized or spatially uniform state.
sumethattheagentsaredistributedonaone-dimensional
This state satisfies x (i) = x and Ψ (i) = x , ∀i, and its
latticewithperiodicboundaryconditions.Thestateofan t t t t
evolution is determined by the single map
agent i, i = 1···N, is characterized by a real variable
x (i) ∈ [0,∞] denoting its wealth or richness at the dis-
t xt+1 =rxtexp(−|(1−a)xt |). (3)
crete time t. The system evolves in time synchronously.
Eachagentupdatesitsstatext(i)accordingtoitspresent For r < 1 the synchronized dynamics Eq. (3) relaxes to
state and the states of its nearest neighbors. Thus, in a the stable fixed point x= 0, while for r >1 the synchro-
firstapproach,weproposethatthevalueofxt+1(i)isgiven nized system displays a sequence of bifurcations through
by the product of two terms; the natural growth of agent different periodic and chaotic attractors, except for the
i given by r(i)xt(i) with positive local ratio r(i), and a singular case a = 1. For r > 1 the stable fixed point
control term that limits this growth with respect to the is x = logr/|1−a|. This point becomes unstable by a
flip bifurcation at r = e2. For increasing r, the entire
period-doubling cascade characteristic of unimodal maps
a E-mail address: [email protected] and other complex dynamical behaviors are generated by
2 S´anchez, Gonz´alez-Est´evez, L´opez-Ruiz, Cosenza: A Model of Coupled-Maps for Economic Dynamics
Eq.(3). However,it canbe shownthat suchsynchronized with α = −2.86, similarly to scaling behaviors of Pareto
states areunstable.When a perturbationis introducedin type directly obtained from actual economy data [8].
the initial uniform state, the asymptotic dynamical state
of the system is found to be more complex.
In general, the collective behavior of the system can 105
be characterized through the instantaneous mean field of
104
the network, defined as
103
N
1 P(x)
Ht = N Xxt(j). (4) 102
j=1
101
Figures1(a)and1(b)showabifurcationdiagramofH as
t
a function of the parameter r, for two different values of 100
theselectionpressurea.InFig.1(a)itcanbeseenthatH
t
reaches stationary values with some intrinsic fluctuations 10-1100 101 102 103
due to the local chaotic dynamics. However, for some pa- Agent wealth (x)
rameter values,nontrivialcollective behavior [9] canarise
Fig. 2. Log-log plot of the asymptotic probability distribution
inthissystemasshowninFig.1(b).Inthiscase,amacro-
of wealth P(x) vs. agent wealth x. The distribution P(x) is
scopic variable such as the mean field or average wealth calculatedaveragingtheoutcomesatt=104 of100realizations
in the system follows a periodic behavior coexisting with
of random initial conditions. Parameter values are r = 12,
chaos at the microscopic level. a = 0.67, N = 104. The slope obtained is α = −2.86 with a
correlation coefficient β =0.98.
(a)
2 Insummary,wehaveshownthatapowerlawbehavior
intheprobabilitydistributionofwealthcanariseforsome
valuesofparametersinadeterministicsystemofinteract-
1,5
ing economic agents, such as in the coupled map model
H
considered here. In addition, nontrivial collective behav-
t
1 ior, where macroscopic order coexists with local disorder,
can emerge in this system.
0,5 ThisworkwassupportedinpartbyDecanatodeInves-
tigaci´onofUniversidadNacionalExperimentaldelT´achira
(UNET) and by FONACIT, Venezuela, under grants 04-
0
001-2006 and F-2002000426, respectively. J.G.E. thanks
1 3 6 r 9 12 15
Decanatode Investigaci´onandVicerrectoradoAcad´emico
4
ofUNETfortravelsupporttotheUniversidaddeZaragoza,
(b)
Spain.R.L.-R.acknowledgessomefinancialsupportfrom
3 the spanish researchproject FIS2004-05073-C04-01.
H
t
2 References
1. V. Pareto, Le Cours d’Economie Politique (MacMillan,
1 London 1897).
2. M.O.Lorenz,PublicationsoftheAmericanStatisticalAs-
sociation 9, (1905) 209.
0 3. P. K. Rawlings, D. Reguera, H. Reiss, Physica A 343
1 3 6 r 9 12 15 (2004) 643.
4. O. S. Klass, O. Biham, M. Levy, O. Malcai, S. Solomon,
Fig. 1. Ht as a function of r, for two values of a. For each
Econ. Lett.90, (2006) 290.
valueofr,Htisplottedfor100iterations,afterdiscarding9900
transients. System size N =104. (a) a=0.67. (b) a=0.1. 5. W. J. Reed, arXiv:cond-mat/0412004 v3(2006).
6. J. R. S´anchez, R. L´opez-Ruiz, arXiv:nlin.AO/0507054
(2005).
7. M. Ausloos, P. Clippe and A. Pekalski, Physica A 324,
The statistical properties of the system can be ex-
(2003) 330.
pressed in terms of the probability distribution of wealth
8. L. A. Nunes-Amaral, S. V. Buldyrev, S. Havlin, H.
among the agents. For some values of the parameters r
Leschorn, P. Maass, M. A. Salinger, H. E. Stanley, M. H.
anda, the distribution ofwealthdisplaysa powerlaw be-
R. Stanley,J. PhysI (France) 7, (1997) 621.
havior, as shown in Fig. 2. For the values of parameters 9. H.Chat´e,P.Manneville,Prog.Theor.Phys.87,(1992) 1.
chosen, the probability distribution scales as P(x) ∼ sα,