Exploiting the flexibility of a family of models for taxation and redistribution

Abstract

We discuss a family of models expressed by nonlinear differential equation systems describing closed market societies in the presence of taxation and redistribution. We focus in particular on three example models obtained in correspondence to different parameter choices. We analyse the influence of the various choices on the long time shape of the income distribution. Several simulations suggest that behavioral heterogeneity among the individuals plays a definite role in the formation of fat tails of the asymptotic stationary distributions. This is in agreement with results found with different approaches and techniques. We also show that an excellent fit for the computational outputs of our models is provided by the κ-generalized distribution introduced by Kaniadakis in [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.…”

“…-in (6), C (j,α) (j−1,α);(k,β) is defined only for j ≥ 2 and k ≥ 2; -in (4) − (6), the indices α and β take any value in {1, ..., m}. We also emphasize that the coefficients p h,k enter in the formulae (4), (5), (6) in such a way that their effect can be also interpreted as weighting the amount of money exchanged. In other words, the situation is the same one would have assuming the frequency of payment of individuals independent on the income class, but with the amount of money paid in each transaction by individuals of the h-th income class to individuals of the k-th income class equal to p h,k S instead of S. The specific choice of p h,k adopted here is suggested by the phenomenological observation that typically poor people pay and earn less than rich people.…”

Stochastic effects in a discretized kinetic model of economic exchange

“…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].…”

“…In [13], it was shown numerically that in a discretized kinetic model including taxation and redistribution, the Gini index of the equilibrium income distribution is an increasing function of the total income defined by the initial condition for a closed system. In a remarkable connection to statistical physics, it was also found [14] that the equilibrium income distribution can be well fitted by the κ-generalized distribution of Kaniadakis [15,16] which displays the same behavior as a function of the temperature and average energy (whereas the Gini index of the Boltzmann-Gibbs distribution is independent of them).…”