From microscopic taxation and redistribution models to macroscopic income distributions

Abstract

We present here a general framework, expressed by a system of nonlinear differential equations, suitable for the modelling of taxation and redistribution in a closed society. This framework allows to describe the evolution of the income distribution over the population and to explain the emergence of collective features based on the knowledge of the individual interactions. By making different choices of the framework parameters, we construct different models, whose long-time behavior is then investigated. Asymptotic stationary distributions are found, which enjoy similar properties as those observed in empirical distributions. In particular, they exhibit power law tails of Pareto type and their Lorenz curves and Gini indices are consistent with some real world ones.

Stochastic effects in a discretized kinetic model of economic exchange

“…for the case of 10 classes, a distribution with x 3 = 1 and all the other x i = 0 (µ = 30 if ∆r is taken to be equal to 10). A reassuring feature of this mathematical structure is the Γ = 0 deterministic limit, where the results all converge to the asymptotic equilibria obtained using the algorithm of [13,14,24] in absence of taxation, as expected.…”

“…This is valid for incomes r j which increase linearly, namely r j = j · ∆r (for more general expressions compare Ref. [13,14]). The δ hk appearing here denotes the Kronecker's delta and must be defined for indices which go from 0 to n + 1.…”

“…The fundamental variables are, in this case, the population fractions x i (i = 1, 2, ..., n) of the classes. The interclass interactions are non-linear in these variables and the evolution equations fit into a discretized kinetic framework [13,14].…”

Mathematical models describing the effects of different tax evasion behaviors

“…More details on the primary mechanism underlying it can be found in our papers [4,5,6,7,8]. We point out however that in [4,5,6] the phenomenon of tax evasion was not taken into account, whereas in [7,8] all individuals were assumed to engage to the same degree in a kind of evasion different from that dealt with here. 1 The main novelty here is given by the assumption of the existence of different degrees of evasion.…”

“…The fraction of individuals with a specific evasion behavior is assumed here to be the same in each income class 5. The choice of this particular subdivision is motivated by the desire to derive a balanced comparison of the evolution of the different groups.…”