Microscopic models for the study of economic inequality, tax evasion and welfare measures
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We formulate a family of models for the description of the micro-processes of money exchange, taxation and re-distribution in a closed market society. These models are expressed by systems of nonlinear differential equations of the kinetic discretized-Boltzmann kind. While traditional treatments of these subjects in mainstream economics rely on the assumption of a representative rational agent, our approach fits in with a complex system perspective. Society is described as an ensemble of individuals divided into a finite number of income classes; the individuals exchange money through binary and ternary interactions, the total wealth being a conserved quantity. The ternary interactions represent taxation and redistribution processes: they express the subtraction, in correspondence to each binary transfer, of an amount whose percentage (tax rate) depends on the income classes of the individuals involved in the interaction; and they define the redistribution (possibly weighted according to a means-tested welfare system) of this amount to all individuals. The frequencies with which the interactions occur as well as other parameters can be tuned to provide a probabilistic representation as realistic as possible. E.g., we can fix the probability that in an encounter between two individuals the one who pays is the rich or the poor, we can postulate that the exchanged amount depends on saving propensity, etc. We show the emergence from the interplay of the individual interactions of observable collective patterns: all computational simulations suggests that after a sufficiently long time the solutions of the equations reach an equilibrium state corresponding to an income distribution, which depends on the total wealth and on the interaction parameters, but not on the specific initial distribution. Distributions arising from initial conditions for which the majority of individuals belong to lower income classes exhibit fat tails as do the real world ones. An excellent fit for these distributions obtained through computational simulations is provided by the κ-generalized distribution introduced by Kaniadakis. We also consider the occurrence of tax evasion and explore its effect on the income distribution and the inequality index of the society. We also investigate the behavior of the Gini index in dependence on taxation rates gap and welfare means-testing. Concerning tax evasion, our findings support the idea that a fair fiscal policy and individual correctness contribute to the overcoming of economic inequalities. The models under study provide a flexible tool which could lead to the identification of parameters and policies producing desirable trends.