Money Exchange Model and a general outlook

The Kinetic Gas theory like two-agent money exchange model, recently introduced in the Econophysics of wealth distributions, is revisited. The emergence of BoltzmannGibbs like distribution of money to Pareto’s law in the tail of the distribution is examined in terms of 2 × 2 transition matrix with a general and simplified outlook. Some additional interesting results are also reported. Introduction Econophysics of Wealth distributions [1] is an emerging area where some Statistical Physicists and Economists have been engaged in interpreting real economic data of money, wealth and income of all kinds of people pertaining to different societies and nations. Economic activities have been assumed to be analogous to elastic scattering processes [3, 4, 5]. Analogy is drawn between Money (m) and Energy (E) where temperature (T ) is average money (< m >) of any individual at equilibrium. Some early attempts [2] have been made to understand the income distributions which follow Pareto’s law (P (m) ∝ 1/m) at the tail of the distributions with the index (α) varying from 1 to 2.5. Kinetic Gas theory like models are introduced to exploit apparent similarities between a many particle system and a social system of many agents. The primary attempt is to understand the distributions of money/ income corresponding to different classes of people. We revisit such a Kinetic theory like two-agent money exchange model as recently proposed by Chakrabarti and group [4, 6]. In this model any two agents chosen randomly from a number of agents (N) are allowed to interact (trade) stochastically and thus money is exchanged. Stochasticity is introduced in terms of a parameter 0 < ǫ < 1 into the interaction. One arrives at a Boltzmann-Gibbs (exponential)-type distribution (P (m) ∝ exp(−βm)) of individual money. In the next stage, a saving propensity factor (λ) is incorporated to show that the distribution shifts away from the exponential distribution. A peak appears at a value other than at zero. In the later stage the saving propensity factor is made random (among the agents) but frozen in time. This brings a distribution qualitatively different from the earlier: one gets a power law at the tail of the distribution. This indicates the emergence of Pareto’s Law (P (m) ∝ 1/m) at the tail of the distribution which is well known among Economists/ Econophysicists. In the above class of models total money (M = ∑ i mi) of all the agents is invariant in time. Also the money (m) is conserved locally which means the sum of money of two agents before and after trade (interaction) remains constant: mi(t+ 1) +mj(t+ 1) = mi(t) +mj(t).