Pareto law in a kinetic model of market with random saving propensity

Abstract

We have numerically simulated the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving two-body collision. Unlike in the ideal gas, we introduce (quenched) saving propensity of the agents, distributed widely between the agents (0 ≤ λ < 1). The system remarkably self-organizes to a critical Pareto distribution of money P (m) ∼ m −(ν+1) with ν ≃ 1. We analyse the robustness (universality) of the distribution in the model. We also argue that although the fractional saving ingredient is a bit unnatural one in the context of gas models, our model is the simplest so far, showing selforganized criticality, and combines two century-old distributions: Gibbs (1901) Considerable investigations have already been made to study the nature of income or wealth distributions in various economic communities, in particular, in different countries. For more than a hundred years, it is known that the probability distribution P (m) for income or wealth of the individuals in the market decreases with the wealth m following a power law, known as Pareto law [1]:where the value of the exponent ν is found to lie between 1 and 2 [2,3,4]. It is also known that typically less than 10% of the population in any country possesses about 40% of the wealth and follow the above power law. The rest of the low-income group population, in fact the majority, clearly follows a different law, identified very recently to be the Gibbs distribution [5,6,7]. Studies on real data show that the high income group indeed follow Pareto law, with ν varying from 1.6 for USA [6], to 1.8 − 2.2 in Japan [3]. The value of ν thus seem to vary a little from economy to economy. We have studied here numerically a gas model of a trading market. We have considered the effect of saving propensity of the traders. The saving propensity is assumed to have a randomness. Our observations indicate that Gibbs and Pareto distributions fall in the same category and can appear naturally in the century-old and well-established kinetic theory of gas [8]: Gibbs distribution for no saving and Pareto distribution for agents with quenched random saving propensity. Our model study also indicates the appearance of self-organized criticality [9] in the simplest model so far, namely in the kinetic theory of gas models, when the stability effect of savings [10] is incorporated.We consider an ideal-gas model of a closed economic system where total money M and total number of agents * Electronic address: arnab@cmp.saha.ernet.in N is fixed. No production or migration occurs and the only economic activity is confined to trading. Each agent i, individual or corporate, possess money m i (t) at time t. In any trading, a pair of traders i and j randomly exchange their money [5,7,11], such that their total money is (locally) conserved and none end up with negative money (m i (t) ≥ 0, i.e, debt not allowed):time (t) changes by one unit after each trading. The steady-state (t → ∞) distribution of money is Gibbs one:Hence...

Bipartite producer–consumer networks and the size distribution of firms

“…When a > 1, the firm size obeys Gamma distribution. These results are similar with money distribution in ref [9,10] gained by transferring model. The above results indicate that we can not get power law distribution by preferential attachment in a constant market.…”

“…Various kinds of power-law behaviors have been observed in a wide range of systems, including the wealth distribution of individuals [9,10,11] and the price-returns in stock markets [12,13]. Pareto-Zipf law in firm size provides another interesting example which exhibits some universal characteristics similar to those observed in physical systems with a large number of interacting units.…”

Inequality measures in kinetic exchange models of wealth distributions

“…In this model [19,24] (CCM model hereafter), agents are assumed to be heterogeneous, the saving fraction λ for each agent is different, drawn from a given distribution…”

“…It is important to note that, in reality, the richest follow a different dynamic where heterogeneity plays the key role. To obtain the power law distribution of wealth for the richest, one needs simply to consider each agent as different in terms of the fraction of wealth he/she saves in each trading [24], which is very natural to assume, because it is quite likely that agents in a market think differently from one another. With this very little modification, one can explain the whole range of wealth distribution [19].…”

“…With this very little modification, one can explain the whole range of wealth distribution [19]. When λ is distributed uniformly in [0, 1) and quenched, (CCM model hereafter), i.e., for heterogeneous agents, one obtains a Pareto law for the probability density of wealth P (m) ∼ m −ν with exponent ν = 2 [19,24]. Several variants of these models, find possible applications in a variety of trading processes [7,25].…”