The Monomodal, Polymodal, Equilibrium and Nonequilibrium Distribution of Money

Toscani

2008

SSRN Journal

Abstract. We introduce and discuss a kinetic model for wealth distribution in a simple market economy which is built of a number of countries or social groups. Our approach is based on the model with risky investments introduced by Cordier, Pareschi and one of the authors in [13] and borrows ideas from the kinetic theory of mixtures of rarefied gases. Wealth is exchanged by individuals inside these countries (domestic trade) as well as in between different countries (international trade). Under a suitable scaling we derive a system of Fokker-Planck type equations and discuss its extension to a two-dimensional model with distributed trading propensity. Theoretical and numerical results for two groups show that the wealth distribution develops a bimodal (and in general, a polymodal) shape. IntroductionIn recent years, a number of models have been proposed to account for the evolution of the distribution of wealth in a simple market economy. One class that might be considered to constitute a mesoscopic approach is based on generalized Lotka-Volterra models [24,31]. A second more popular approach relies on methods borrowed from statistical mechanics for particle systems [23,14,8,7,22,29,11,13]. The founding idea behind this last approach is that a trading market composed by a sufficiently large number of agents can be described using the laws of statistical mechanics as it happens in a physical system composed of many interacting particles. If one agrees with the claim that there are deep analogies between economics and physics, then various well established physical methods can be applied to analyze wealth distributions in economies. In particular, by identifying wealth in a closed economy with energy, the application of statistical physics methods leads to a better understanding of the development of tails in wealth distributions of real economies. In kinetic models of simple market economies, in fact, the knowledge of the large-wealth behavior of the steady state density is of primary importance, since it determines a posteriori if the model fits data of real economies. By identifying wealth with energy it becomes clear that the problem of describing the large-time behavior of the wealth distribution in a kinetic model of the type considered in [23,14,8,7,22,29,11,13] is the analogue of the problem of describing the large-time behavior of the density in the spatially homogeneous Boltzmann equation. In particular, for nonconservative kinetic models, this analogy has been recently enlightened in [29,27], while convergence to steady wealth distributions in conservative models has been dealt with in [25,26,16].The features typically incorporated in kinetic trade models are saving effects and randomness. Saving means that agents never exchange their entire wealth in a trade, but are guaranteed to retain at least a certain minimal fraction of their wealth at the end of each trade. This concept has been introduced in [8], where a fixed saving rate for all agents has been proposed, and generalized in [9] by introducing ...

Section: Numerical Resultssupporting

confidence: 71%

Section: Introductionmentioning

confidence: 69%

International and Domestic Trading and Wealth Distribution

Toscani

2008

SSRN Journal

Section: Numerical Resultssupporting

confidence: 71%

International and domestic trading and wealth distribution

Communications in Mathematical Sciences

Toscani²

2008

Abstract. We introduce and discuss a kinetic model for wealth distribution in a simple market economy which is built of a number of countries or social groups. Our approach is based on the model with risky investments introduced by Cordier, Pareschi, and Toscani in [S. Cordier, L. Pareschi and G. Toscani, J. Stat. Phys., 120, 253-277, 2005], and borrows ideas from the kinetic theory of mixtures of rarefied gases. Wealth is exchanged by individuals inside these countries (domestic trade) as well as in between different countries (international trade). Under a suitable scaling we derive a system of Fokker-Planck type equations and discuss its extension to a two-dimensional model with distributed trading propensity. Theoretical and numerical results for two groups show that the wealth distribution develops a bimodal (and in general, a polymodal) shape.

“…The distance of the two peaks in the distribution decreases with decreasing difference between γ 1 and γ 2 . Such bimodal distributions (and a polymodal distribution, in general) are also reported with real data for the income distributions in Argentina [38,42]. This distribution features transport of wealth from one group to the other, which makes it different from the probability distribution for the union of two groups with the same parameters which do not interact, see Figure 7.8 (right plot).…”

Section: Bimodal Distributionssupporting

confidence: 71%

“…The goal is to verify analytically the existence of a bimodal stationary distribution [42]. Bimodal distributions (and a polymodal distribution, in general) are, in fact, reported with real data for the income distributions in Argentina [38]. In the proposed model, a bimodal steady state can indeed be obtained, e.g.…”

Section: Introductionmentioning

confidence: 85%

A Boltzmann-Type Approach to the Formation of Wealth Distribution Curves

Matthes²,

Toscani

2008

SSRN Journal

Kinetic market models have been proposed recently to account for the redistribution of wealth in simple market economies. These models allow to develop a qualitative theory, which is based on methods borrowed from the kinetic theory of rarefied gases. The aim of these notes is to present a unifying approach to the study of the evolution of wealth in the largetime regime. The considered models are divided into two classes: the first class is such that the society's mean wealth is conserved, while for models of the second class, the mean wealth grows or decreases exponentially in time. In both cases, it is possible to classify the most important feature of the steady (or self-similar, respectively) wealth distributions, namely the fatness of the Pareto tail. We shall also discuss the tails' dynamical stability in terms of the model parameters. Our results are derived by means of a qualitative analysis of the associated homogeneous Boltzmann equations. The key tools are suitable metrics for probability measures, and a concise description of the evolution of moments. A recent extension to economies, in which different groups of agents interact, is presented in detail. We conclude with numerical experiments that confirm the theoretical predictions.