**1. Introduction**

Income disparities and economic inequality are all but recent problems. They are, however, becoming worryingly and increasingly significant (for a few references in this regard see e.g. [1–7]). In fact, reducing inequality within and between countries is one of the sustainable development goals included in the 2030 Agenda of by the United Nations General Assembly. Certainly, such an objective is, as least in the first instance, a main responsibility and competence of economists and policy-makers. Nonetheless, mathematics can provide some hints and contributions towards it.

The goal of this chapter is to provide a brief illustration of the role that mathematics can play in this regard. By way of example, we will recall an elementary (first approximation) model first proposed by Bertotti in [8], together with some variants and extensions of it, developed and investigated in a series of other studies by Bertotti et al. [9–13].

The approach of this model can be seen as a contribution attempt in the spirit of complexity economics. This paradigm, which began to take shape in the late 1980s, looks at the economy as an evolving system, not in equilibrium, and puts emphasis on the process through which structures and patterns emerge from the microinteractions (see e.g. [14–16]). Similarly, the perspective of the model here discussed is to put the interactions among heterogeneous individuals at the very heart of the question. These interactions lead to self-organised collective features and macro-observables, which emerge from the system as a whole.

It should be noted here that during the last two/three decades a research line has been developing, not only but mostly among the physicist's community, which addresses socio-economic questions and phenomena using ideas, methods and tools, which have their roots in statistical mechanics and the kinetic theory of gases. An explanation for that comes from the existing analogy, for example, between complex systems composed by a number of individuals who interact exchanging money with each other (may be participating in a financial market) and physical systems consisting of a huge number of particles (atoms or molecules), which interact with each other undergoing collisions.

A variety of tools and techniques, including for example Boltzmann type equations, Fokker-Planck type equations, Ising type models and agent-based simulations, have been adapted and employed in this connection. See for example references [17–21] and the survey papers [22, 23] that offer an interesting historical perspective and also contain extensive bibliographies.

The model developed in the paper [8] and then further generalised and explored in subsequent work [9–13] differs to a great extent from those we know belonging to the mentioned literature strand. The motivation behind [8] was precisely to understand how the taxation process and diverse fiscal systems could affect the income distribution of a population. Aiming at modelling a fiscal system with taxes on personal income levied at a finite number of progressive rates, with high-income earners expected to pay more than low-income earners, as in the case of the Italian IRPEF (Imposta sul Reddito delle Persone Fisiche), the most natural approach seemed to be one dividing a population into a finite number of income classes. This is the reason behind the construction of a discrete framework, suitable for the formulation of the model, which will be briefly recalled in the next section. Albeit expressed using a system of ordinary differential equations (ODE)—as many as the income classes in the society—the model incorporates stochastic and probabilistic components. In a nutshell, the ODE system governs the evolution in time of the income distribution, generated by a whole of money exchanges expressing binary individual interactions, and a whole of withdrawals and earnings of the individuals, due to taxation and redistribution. Specifically, each differential equation in the system describes the variation in time of the fraction of individuals belonging to a certain class. As we will see in the next section, the framework allows possible tuning of various parameters (the frequencies with which the interactions are supposed to occur, the probability that in an encounter between two individuals the one who pays is one or the other, the tax rates or other) that give it an exploratory character.

The study of the dynamics of the model, supported by some analytical results and, inevitably, largely pursued through numerical simulations (performed using Mathematica software [24]), focuses on the asymptotic behaviour of the system. What all simulations suggest is that after a sufficiently long time the solutions of the equations reach a stationary state corresponding to an income distribution, which depends on the total wealth and the interaction parameters, but not on the specific initial distribution. This stationary state represents in fact a macro-observable feature. At the micro-individual level, the economic exchanges continue to take place and the situation is a non-equilibrium one.

The interest is to find and compare one with the other different shapes of the asymptotic income distributions corresponding to different fiscal policies. In this *Taxation and Redistribution against Inequality: A Mathematical Model DOI: http://dx.doi.org/10.5772/intechopen.100939*

connection, the model shows that over time redistributive policy leads to a reduction of economic inequality.

The rest of the chapter is organised as follows:

In Section 2, we recall the framework of the original model.

In Section 3, we revisit some insights and extensions of the model discussed in previous work. The issues include the following:


In Section 4, a novel application of the model is developed, which shows the impact of different fiscal policies.

Section 5 contains concluding considerations.
