Optimal income crossover for a two-class model using particle swarm optimization

Personal income distribution may exhibit a two-class structure, such that the lower income class of the population (85-98%) is described by exponential Boltzmann-Gibbs distribution, whereas the upper income class (2-15%) has a Pareto power-law distribution. We propose a method, based on a theoretical and numerical optimization scheme, which allows us to determine the crossover income between the distributions, the temperature of the Boltzmann-Gibbs distribution, and the Pareto index. Using this method, the Brazilian income distribution data provided by the National Household Sample Survey was studied. The data was stratified into two dichotomies (sex/gender and color/race), so the model was tested using different subsets along with accessing the economic differences between these groups. Last, we analyze the temporal evolution of the parameters of our model and the Gini coefficient discussing the implication on the Brazilian income inequality. In this paper, we propose an optimization method to find a continuous two-class income distribution, which is able to delimit the boundaries of the two distributions. It also gives a measure of inequality which is a function that depends only on the Pareto index and the percentage of people in the high-income region. We found a temporal dynamics relation, that may be general, between the Pareto and the percentage of people described by the Pareto tail.