International and Domestic Trading and Wealth Distribution

Abstract

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 ...

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

“…This hypothesis can be further relaxed by assuming that the saving rate is a random quantity, with a statistical mean which is different for different social groups. Here we describe the model proposed in [37] which is based on CPT conservative model. A related problem, based on increasing wealth, has been recently introduced and numerically studied in [27].…”

“…In the same spirit, a twodimensional model of Fokker-Planck type has been recently described in [37].…”

“…Numerical experiments by means of Monte Carlo methods have been recently done in [34,35] for both the CPT and the CCM models. Kinetic system (5.95) has been subsequently investigated via analogous methods to show the (eventual) formation of a bimodal steady distribution [37]. In view of their interest in application, we present here the main results, together with a brief presentation of the underlying numerical method.…”

“…As a further approach, related to conservative models, we discuss a kinetic model for wealth distribution in a market which comprises a whole number of countries, or alternatively different social groups within the same country [37]. The goal is to verify analytically the existence of a bimodal stationary distribution [42].…”

Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution

“…Systems of Fokker-Planck equations of type (1.1) have been considered in Ref. [34] to model wealth distribution in different countries which are coupled by mixed trading. Further, the operator J(f ) in equation (1.1) and its equilibrium kernel density have been considered in a non homogeneous setting to obtain Euler-type equations describing the joint evolution of wealth and propensity to invest [33], to study the evolution of wealth in a society with agents using personal knowledge to trade [59], and to observe the consequences on the distribution of wealth by exercising a control at the level of microscopic interactions [32].…”