Breakdown of the mean-field approximation in a wealth distribution model

arXiv:0809.4139v2 [q-fin.ST] 26 Nov 2008 Breakdown of the mean-field approximation in a wealth distribution model M Medo1,2 1 Physics Department, University of Fribourg, P´erolles, 1700 Fribourg, Switzerland 2 Department of Mathematics, Physics and Informatics, Mlynsk´a dolina, 842 48 Bratislava, Slovak republic E-mail: matus.medo@unifr.ch Abstract. One of the key socioeconomic phenomena to explain is the distribution of wealth. Bouchaud and M´ezard have proposed an interesting model of economy [Bouchaud and M´ezard (2000)] based on trade and investments of agents. In the mean-field approximation, the model produces a stationary wealth distribution with a power-law tail. In this paper we examine characteristic time scales of the model and show that for any finite number of agents, the validity of the mean-field result is time-limited and the model in fact has no stationary wealth distribution. Further analysis suggests that for heterogeneous agents, the limitations are even stronger. We conclude with general implications of the presented results. PACS numbers: 05.40.-a, 89.65.-s, 89.75.-k Keywords: stochastic processes, interacting agent models, fluctuations Submitted to: JSTAT Breakdown of the mean-field approximation in a wealth distribution model 2 1. Introduction Many empirical studies report broad distributions of income and wealth of individuals and these distributions are often claimed to have power-law tails with exponents around two for most countries [1, 2, 3, 4, 5]. The first models attempting to explain the observed properties appeared over fifty years ago [6, 7, 8]. Much more recently, physics-motivated kinetic models based on random pairwise exchanges of wealth by agents have attracted considerable interest [9, 10, 11, 12, 13]. An alternative point of view is adopted in the wealth redistribution model (WRM) where agents continuously exchange wealth in the presence of noise [14, 15, 16]. There are also several specific effects which can lead to broad wealth distributions [17, 18, 19]. (For reviews of power laws in wealth and income distributions see [20, 21, 22], while for general reviews of power laws in science see [23, 24].) In this paper we analyze the WRM with two complementary goals in mind. Firstly we investigate the simplest case when exchanges of all agents are identical, focusing on the validity of the mean-field approximation which is the standard tool to solve the model and derive the stationary wealth distribution. In particular, we show that for any finite number of agents there is no such stationary distribution (other finite-size effects are discussed for a similar model in [18]). Secondly we investigate the model’s behaviour when the network of agent exchanges is heterogeneous. Previous attempts to investigate the influence of network topology on the model [14, 25, 26, 27] were all based on the mean-field approximation. We show that this is questionable because heterogeneity of the exchange network strongly limits the validity of results obtained using the mean-field approximation. 2. Model and its mean field solution Adopting the notation used in [14], we study a simple model of an economy which is composed of N agents with wealth vi (i = 1, . . . , N). The agents are allowed to mutually exchange their wealth (representing trade) and they are also subject to multiplicative noise (representing speculative investments). The time evolution of agents’ wealth is given by the system of stochastic differential equations (SDEs) dvi(t) =  X j̸=i Jijvj(t) − X j̸=i Jjivi(t)  dt + √ 2σvi(t) dWi(t), (1) where σ ≥0 controls the noise strength. The coefficient Jij quantifies the proportion of the current wealth vj(t) that agent j spends on the production of agent i per unit time. We assume the Itˆo convention for SDEs and dWi(t) is standard white noise [29, 30]. Hence, denoting averages over realisations by ⟨·⟩, we have ⟨dWi(t)⟩= 0, ⟨dWi(t) dWj(t)⟩= δij dt, and ⟨vi(t) dWi(t)⟩= 0. By summing dvi(t) over all agents one can see that the average wealth vA(t) := 1 N PN i=1 vi(t) is not influenced by wealth exchanges and obeys the SDE dvA(t) = √ 2σ N PN i=1 vi(t) dWi(t). Therefore ⟨dvA(t)⟩= 0 and ⟨vA(t)⟩is constant. For simplicity we assume vi(0) = 1 (i = 1, . . . , N) and thus Breakdown of the mean-field approximation in a wealth distribution model 3 ⟨vi(t)⟩= 1 and ⟨vA(t)⟩= 1. (The influence of the initial conditions is discussed in Section 4.1.) The system behaviour is strongly influenced by the exchange coefficients Jij. The simplest choice is Jij = J/(N −1) where all exchanges are equally intensive—we say that the exchange network is homogeneous. By rescaling the time we can set J = 1 which means that during unit time agents exchange all their wealth. Consequently, (1) simplifies to dvi(t) = (˜vi(t) −vi(t)) dt + √ 2σvi(t) dWi(t) (2) where ˜vi(t) := 1 N−1 P j̸=i vj is the average wealth of all agents but agent i. In the limit N →∞, fluctuations of ˜vi(t) are negligible and one can replace ˜vi(t) →⟨˜vi(t)⟩= 1 as in [14].

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