Micro to macro models for income distribution in the absence and in the presence of tax evasion

Abstract

We investigate the effect of tax evasion on the income distribution and the inequality index of a society through a kinetic model described by a set of nonlinear ordinary differential equations. The model allows to compute the global outcome of binary and multiple microscopic interactions between individuals. When evasion occurs, both individuals involved in a binary interaction take advantage of it, while the rest of the society is deprived of a part of the planned redistribution. In general, the effect of evasion on the income distribution is to decrease the population of the middle classes and increase that of the poor and rich classes. We study the dependence of the Gini index on several parameters (mainly taxation rates and evasion rates), also in the case when the evasion rate increases proportionally to a taxation rate which is perceived by citizens as unfair. Finally, we evaluate the relative probability of class advancement of individuals due to direct interactions and welfare provisions, and some typical temporal rates of convergence of the income distribution to its equilibrium state.

Exchange Models for the Emergence of Income Distribution and Economic Inequality

“…In contrast, the model we are proposing, first developed in [12] and then variously extended and exploited e.g. in [13][14][15][16] to deal with different specific questions, is formulated by means of systems of several differential equations. These equations are nonlinear, of the kinetic discretized Boltzmann kind and involve in particular transition probabilities.…”

“…Issues like redistribution through means-tested welfare policies, the impact of tax evasion on the income distribution and the effects of taxpayer audits have been treated in [12][13][14][15][16]. In all these cases simulations show that after a sufficiently long time the solution (x 1 (t), ..., x n (t)) of the equations reaches an equilibrium state which depends on the total income and on the interaction parameters, but not on the initial distribution.…”

“…To do that, we must go into some more detail. Referring also to [12][13][14][15][16] for extensive motivations and comments, we fix next the following terms.…”

“…In particular, these expressions contain parameters which can be variously tuned and represent encounter frequencies, saving propensities, average incomes, tax rates, and also tax evasion rates. In any case, the C hk j and T hk j [ ] are constructed in such a way that the vector (x 1 (t), ..., x n (t)) (suitably normalized) is a discretized probability density function at time t. We point out that in the cases studied in [12][13][14][15][16], as also here, the total wealth is assumed to remain constant in time. Growth or variation of the wealth, however, can be considered as well and are in fact the object of ongoing investigation.…”

“…In the basic version, it takes the form [12][13][14][15][16], also depending on the different questions therein tackled. In particular, these expressions contain parameters which can be variously tuned and represent encounter frequencies, saving propensities, average incomes, tax rates, and also tax evasion rates.…”