Analysis of Solidarity Effect for Entropy, Pareto, and Gini Indices on Two-Class Society Using Kinetic Wealth Exchange Model

It is well known that two different underlying dynamics lead to different patterns of income/wealth distribution such as the Boltzmann–Gibbs form for the lower end and the Pareto-like power-law form for the higher-end. The Boltzmann–Gibbs distribution is naturally derived from maximizing the entropy of random interactions among agents, whereas the Pareto distribution requires a rational approach of economics dependent on the wealth level. More interestingly, the Pareto regime is very dynamic, whereas the Boltzmann–Gibbs regime is stable over time. Also, there are some cases in which the distributions of income/wealth are bimodal or polymodal. In order to incorporate the dynamic aspects of the Pareto regime and the polymodal forms of income/wealth distribution into one stochastic model, we present a modified agent-based model based on classical kinetic wealth exchange models. First, we adopt a simple two-class society consisting of the rich and the poor where the agents in the same class engage in random exchanges while the agents in the different classes perform a wealth-dependent winner-takes-all trading. This modification leads the system to an extreme polarized society with preserving the Pareto exponent. Second, we incorporate a solidarity formation among agents belonging to the lower class in our model, in order to confront a super-rich agent. This modification leads the system to a drastic bimodal distribution of wealth with a varying Pareto exponent over varying the solidarity parameter, that is, the Pareto-regime becomes narrower and the Pareto exponent gets larger as the solidarity parameter increases. We argue that the solidarity formation is the key ingredient in the varying Pareto exponent and the polymodal distribution. Lastly, we take two approaches to evaluate the level of inequality of wealth such as Gini coefficients and the entropy measure. According to the numerical results, the increasing solidarity parameter leads to a decreasing Gini coefficient not linearly but nonlinearly, whereas the entropy measure is robust over varying solidarity parameters, implying that there is a trade-off between the intermediate party and the high end.

Lotka’s law for the Brazilian scientific output published in journals

Lotka’s law is a power law for the frequency of scholarly publications. We show that Lotka’s law cannot be dismissed after considering a massive sample of the number of publications of Brazilian researchers in journals listed on the SCImago Journal Rank and the Journal Citation Reports. For the SCImago Journal Rank, we found a power law with the Pareto exponent of 0.4 beyond the threshold of 50 papers. This means computing the ‘average number of publications’ of either a researcher or a discipline is of no practical significance.

THE SPATIAL MEANING OF PARETO'S SCALING EXPONENT OF CITY-SIZE DISTRIBUTIONS

The scaling exponent of a hierarchy of cities used to be regarded as a fractional dimension. The Pareto exponent was treated as the fractal dimension of size distribution of cities, while the Zipf exponent was considered to be the reciprocal of the fractal dimension. However, this viewpoint is not exact. In this paper, I will present a new interpretation of the scaling exponent of rank-size distributions. The ideas from fractal measure relation and the principle of dimension consistency are employed to explore the essence of Pareto's and Zipf's scaling exponents. The Pareto exponent proved to be a ratio of the fractal dimension of a network of cities to the average dimension of city population. Accordingly, the Zipf exponent is the reciprocal of this dimension ratio. On a digital map, the Pareto exponent can be defined by the scaling relation between a map scale and the corresponding number of cities based on this scale. The cities of the United States of America in 1900, 1940, 1960, and 1980 and Indian cities in 1981, 1991, and 2001 are utilized to illustrate the geographical spatial meaning of Pareto's exponent. The results suggest that the Pareto exponent of city-size distributions is a dimension ratio rather than a fractal dimension itself. This conclusion is revealing for scientists to understand Zipf's law on the rank-size pattern and the fractal structure of hierarchies of cities.

On Pareto’s law and the determinants of Pareto exponents

A stochastic model for the generation of observed income distributions is used to provide an explanation for the Pareto law of incomes. The basic assumptions of the model are that the evolution of individual incomes follows Gibrat's law and that the population or workforce is growing at a fixed (probabilistic) rate. Analysis of the model suggests that Paretian behaviour can occur in either or both tails of an income distribution. It is shown that the magnitude of the upper-tail Pareto exponent depends on the interaction between the distribution of the growth in incomes and the growth in the size of the earning population. In particular a small Pareto exponent can be expected to occur for a population exhibiting fast or highly variable growth in incomes coupled with relatively slow population growth.

PARETO'S LAW FOR INCOME OF INDIVIDUALS AND DEBT OF BANKRUPT COMPANIES

We analyze the distribution of income and income tax of individuals in Japan for the fiscal year 1998. From the rank-size plots, we find that the accumulated probability distribution of both data obey a power law with a Pareto exponent very close to -2. We also present an analysis of the distribution of the debts owed by bankrupt companies from 1997 to March 2000, which is consistent with a power law behavior with a Pareto exponent equal to -1. This power law is the same as that of the income distribution of companies. Possible implications of these findings for model building are discussed.