**2. General framework**

Referring the reader to the paper [8] for further explanations and details on the mechanism behind the formulation of the proposed framework, we recall here that, if *n* denotes the number of income classes of a population, characterised by their average incomes *r*<sup>1</sup> <*r*<sup>2</sup> < … < *rn*, and *xi*ð Þ*t* , with *xi* : **R** ! ½ Þ 0, þ∞ for *i* ¼ 1, 2, … , *n* denotes the fraction at the time *t* of individuals belonging to the *i*-th class (with the normalisation P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*xi* <sup>¼</sup> 1), the variation in time of the quantities *xi*ð Þ*<sup>t</sup>* may be thought to obey a system of differential equations of the form

$$d\boldsymbol{x}\_{i}d\boldsymbol{t} = \sum\_{h=1}^{n} \sum\_{k=1}^{n} \left(\mathbf{C}\_{hk}^{i} + T\_{[hk]}^{i}(\boldsymbol{\omega})\right) \mathbf{x}\_{h}\boldsymbol{\omega}\_{k} - \boldsymbol{\omega}\_{i} \sum\_{k=1}^{n} \boldsymbol{\omega}\_{k}, \qquad i = 1, 2, \ldots, n. \tag{1}$$

The coefficients *C<sup>i</sup> hk*'s and the continuous functions *T<sup>i</sup>* ½ � *hk* 's in (Eq. (1)) incorporate the instructions for the variation of the fraction of individuals (i.e. the movement of individuals) from one class to another. They keep into account "impoverishment and enrichment" due both to direct money exchanges taking place between pairs of individuals and to the (small) withdrawals and earnings of each individual, due to taxation and redistribution, processes that are here considered as occurring in correspondence to each transaction. More precisely, *Ci hk* ∈½ Þ 0, þ∞ expresses the probability density that an individual of the *h*-th class will belong to the *i*-th class after a direct interaction with an individual of the *k*-th class. Accordingly, the identity P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*C<sup>i</sup> hk* ¼ 1 has to be satisfied for any fixed *h* and *k*;

• *T<sup>i</sup>* ½ � *hk* : **<sup>R</sup>***<sup>n</sup>* ! **<sup>R</sup>** expresses the variation density in the *<sup>i</sup>*-th class due to an interaction between an individual of the *h*-th class with an individual of the *k*-th class. The functions *T<sup>i</sup>* ½ � *hk* are required to satisfy <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼1*T<sup>i</sup>* ½ � *hk* ð Þ¼ *<sup>x</sup>* 0 for any fixed *<sup>h</sup>*, *k* and *x*∈ **R***<sup>n</sup>*.

Specific expressions for these quantities have to be carefully calibrated if we want, as is the case in the model at hand, to treat a case in which the total amount of money is constant. Towards this, let


The ratio for the definition of *C<sup>i</sup> hk* and *T<sup>i</sup>* ½ � *hk* is that when an individual of the *<sup>h</sup>*-th class pays a quantity *S* to an individual of the *k*-th class, this one in turn has to pay a tax *Sτk*. The government, for its part, redistributes to the entire population the revenue collected by all taxes and this one in particular (this redistribution may be interpreted as public expenditure in health, education, security and defence, transports and so on). From a practical standpoint, the effect of a payment of *S* from an *h*-individual to a *k*-individual can be thought, bypassing the government, as the same of a payment of *S*ð Þ 1 � *τ<sup>k</sup>* from the *h*-individual to the *k*-individual and payment of *Sτ<sup>k</sup>* from the *h*-individual to the entire population.

Skipping here some technical details, we recall that the expressions proposed in paper [8] for *Ci hk* and *T<sup>i</sup>* ½ � *hk* are as follows: each *<sup>C</sup><sup>i</sup> hk* can be written as *C<sup>i</sup> hk* <sup>¼</sup> *ai hk* <sup>þ</sup> *bi hk*, where the only nonzero elements *ai hk* are *a<sup>i</sup> ij* ¼ 1 for *i*, *j* ¼ 1, 2, … , *n* and the only possibly nonzero elements *bi hk* are those of the form

$$\begin{aligned} b\_{i+1,k}^{i} &= p\_{i+1,k} \, \mathcal{S} \, \frac{\mathbf{1} - \tau\_k}{r\_{i+1} - r\_i}, \\ b\_{i,k}^{i} &= -p\_{k,i} \, \mathcal{S} \, \frac{\mathbf{1} - \tau\_i}{r\_{i+1} - r\_i} - p\_{i,k} \, \mathcal{S} \, \frac{\mathbf{1} - \tau\_k}{r\_i - r\_{i-1}}, \\ b\_{i-1,k}^{i} &= p\_{k,i-1} \, \mathcal{S} \, \frac{\mathbf{1} - \tau\_{i-1}}{r\_i - r\_{i-1}}, \end{aligned} \tag{2}$$

whereas *T<sup>i</sup>* ½ � *hk* ð Þ¼ *<sup>x</sup> <sup>U</sup><sup>i</sup>* ½ � *hk* ð Þþ *<sup>x</sup> <sup>V</sup><sup>i</sup>* ½ � *hk* ð Þ *<sup>x</sup>* , with

$$U\_{[hk]}^{\dot{t}}(\mathbf{x}) = \frac{p\_{h,k} \mathbf{S} \,\tau\_k}{\sum\_{j=1}^{n} \mathbf{x}\_j} \left( \frac{\mathbf{x}\_{i-1}}{r\_i - r\_{i-1}} - \frac{\mathbf{x}\_i}{r\_{i+1} - r\_i} \right), \tag{3}$$

and

$$\mathcal{V}\_{[hk]}^{i}(\mathbf{x}) = p\_{h,k} \mathcal{S} \tau\_k \left( \frac{\delta\_{h,i+1}}{r\_h - r\_i} - \frac{\delta\_{h,i}}{r\_h - r\_{i-1}} \right) \frac{\sum\_{j=1}^{n-1} \mathbf{x}\_j}{\sum\_{j=1}^{n} \mathbf{x}\_j}. \tag{4}$$

*Taxation and Redistribution against Inequality: A Mathematical Model DOI: http://dx.doi.org/10.5772/intechopen.100939*

In particular, *U<sup>i</sup>* ½ � *hk* ð Þ *<sup>x</sup>* keeps track of the advancement from a class to the subsequent one, due to the benefit of tax revenue redistribution and *V<sup>i</sup>* ½ � *hk* ð Þ *<sup>x</sup>* of the retrocession from a class to the preceding one, due to the payment of some tax. The symbol *δh*,*<sup>k</sup>* denotes the *Kronecker delta* and all expressions are to be thought as present only for meaningful values of the indices.

Well-posedness of the equation system (Eq. (1)) is proved in [8]: in correspondence to any initial condition *x*<sup>0</sup> ¼ ð Þ *x*01, … , *x*0*<sup>n</sup>* with *x*0*<sup>i</sup>* ≥0 for all *i* ¼ 1, 2, … , *n* and P*<sup>n</sup> <sup>i</sup>*¼1*x*0*<sup>i</sup>* <sup>¼</sup> 1, a unique solution *x t*ðÞ¼ ð Þ *<sup>x</sup>*1ð Þ*<sup>t</sup>* , … , *xn*ð Þ*<sup>t</sup>* of (Eq. (1)), satisfying *x*ð Þ¼ 0 *x*0, exists, defined for all *t*∈½ Þ 0, þ∞ , and such that for all *t*≥0, both *xi*ð Þ*t* ≥ 0 for *<sup>i</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>* and <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*xi*ðÞ¼ *<sup>t</sup>* 1 hold true. Hence, the solutions of (Eq. (1)) are distribution functions. Also, the expressions of *U<sup>i</sup>* ½ � *hk* ð Þ *<sup>x</sup>* and *<sup>V</sup><sup>i</sup>* ½ � *hk* ð Þ *<sup>x</sup>* above simplify becoming linear in the variables *x <sup>j</sup>* and the right-hand sides of (Eq. (1)) turn out to be polynomials of degree three.

A second result proved in [8] is that the scalar function *<sup>μ</sup>*ð Þ¼ *<sup>x</sup>* <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*rixi*, expressing the global income (total amount of money) and, due to the population normalisation, also the mean income, is a first integral for the system (Eq. (1)).

Also, the following empirical fact (not analytically proved) is recognised to be true according to a large number of numerical simulations. If the parameters in the model are fixed, for any fixed value of the global income *μ*, a unique asymptotic stationary solution of (Eq. (1)) exists to which all solutions *x t*ðÞ¼ ð Þ *x*1ð Þ*t* , … , *xn*ð Þ*t* satisfying *x*ð Þ¼ 0 *x*<sup>0</sup> with *μ*ð Þ¼ *x*<sup>0</sup> *μ* (i.e. all solutions evolving from initial conditions which share the same value *μ* of the global income) tend as *t* ! ∞.

As already emphasised, a great freedom remains for the choice of various parameters, namely the average incomes *r*1,*r*2, … ,*rn*, the tax rates *τ*1, *τ*2, … , *τn*, and the *ph*,*<sup>k</sup>* for *h*, *k* ¼ 1, 2, … , *n*. Different cases were already considered in [8].

### **3. Properties of the model and its variants**

The first result of interest from a socio-economic point of view, which is discussed in [8] is that for fixed parameters *r*1,*r*2, … ,*rn* and *ph*,*<sup>k</sup>* (*h*, *k* ¼ 1, 2, … , *n*) and fixed growth laws of the tax rates 0 ≤*τ*<sup>1</sup> ≤*τ*<sup>2</sup> ≤ … ≤*τ<sup>n</sup>* ≤1, the effect of an increase of the difference *τ<sup>n</sup>* � *τ*<sup>1</sup> between the maximum and the minimum tax rate in correspondence to the stationary income distribution is an increase of the fraction of individuals belonging to the middle classes, accompanied by a decrease of the fraction of individuals belonging to the poorest and the richest classes. We remark that only five income classes were considered in [8], the motivation being that the number of different tax rates generally foreseen in real world is similarly small (in Italy the number of the IRPEF tax rates relative to different income ranges is exactly five).

To try and see whether the model allows to obtain long-time stationary income distributions with shapes exhibiting fat tails as it occurs in real world, a larger number of classes in the model were considered in the work by Bertotti et al. [9]. Various choices of the parameters were evaluated. The purpose was to deal with cases as realistic as possible, and initial distributions of the population were chosen with a majority of individuals in lower-income classes and only a minority in higher income classes. In this way, stationary income distributions with Pareto-like behaviour were found.

Among other aspects to be explored, the curiosity remained to see whether one can find an analytic expression of a distribution, to which the stationary solutions of the model suit. A focus of the paper [10] by Bertotti et al. is on the search for such

an analytic expression. Several parameter choices as well as various distributions proposed in the literature are considered in that paper. What is found is that an excellent fitting can be obtained between distributions arising from numerical simulations of the model and the *κ*-generalised distribution proposed by Kaniadakis in [25]. And it is worth pointing out that, in turn, the *κ*-generalised distribution has proved to greatly perform when considered in connection with empirical data: for example, its agreement with data on personal income of Germany, Italy and the United Kingdom is discussed by Kaniadakis et al. in [26] and that one with data on personal income of Australia and the United States by Kaniadakis et al. in [27].

In real life, welfare policies provide benefits, in particular to the lowest income classes, in connection with health care, education, home, to help improve living conditions. To simulate a policy of this kind, a modified version of the model is treated in the paper by Bertotti et al. [11], where also the contribution of what can be considered as a welfare form is incorporated. This is achieved through some weights that differently measure the amount of tax revenue redistributed among classes. In the same paper, also a comparison is established between different ways to fight economic inequality. A specific result therein obtained is that, at least under certain hypotheses, inequality reduction is more efficiently reached by a policy of reduction of the welfare and subsidies for the rich classes than by an enlargement of the tax rate difference *τ<sup>n</sup>* � *τ*<sup>1</sup> aimed at taxing rich people much more than poor ones.

A further issue on which the model was tested relates to social mobility. Empirical data relative to several countries show the general existence of a negative correlation between economic inequality and mobility (a reference for that being e.g. the article by Corak [1]). This relevant topic is dealt with in the paper by Bertotti et al. [12]. Certainly, in the model at hand, one cannot distinguish different generations. Nonetheless, some indicators are introduced, useful to quantify mobility, which is meant here as a probability for individuals of a given class to climb up [respectively, down] the income ladder and pass to an upper [respectively, lower] class. Without entering technical details, we emphasise that a negative correlation between economic inequality and upward mobility turns out to be in fact a feature of the model.

Finally, the question of tax evasion, occurring as a matter of fact in a stronger or weaker form in several countries, can be and was investigated in the context of the model under consideration. In the work by Bertotti et al. [13], for instance, also the co-existence of different evasion levels among individuals was postulated and its consequences were explored. In particular, it was shown there that, besides leading to a reduction in tax revenue, the evasion misbehaviour too contributes to an increase of economic inequality.

#### **4. The impact of different fiscal policies towards economic inequality**

To give a further illustration of the impact of different fiscal policies on the shape of income distribution and economic inequality as suggested by the model, we develop in this section a novel application.

To solve numerically the differential equations, we have to fix the parameters that are so far free. We choose for example


*Taxation and Redistribution against Inequality: A Mathematical Model DOI: http://dx.doi.org/10.5772/intechopen.100939*

• the average incomes of the classes to be linearly growing according to

$$r\_j = \mathfrak{B}j,\tag{5}$$

• the tax rates relative to the different income classes to be of the form

$$
\tau\_j = \tau\_{\min} + \frac{j-1}{n-1} \left( \tau\_{\max} - \tau\_{\min} \right), \tag{6}
$$

for *j* ¼ 1, … , *n*, with *τmin* rand *τmax* respectively denoting the minimum and maximum tax rate.

Finally,

• with the purpose to define reasonable heterogeneous transaction and payment probabilities, we assume the coefficients *ph*,*<sup>k</sup>* to be given by

$$p\_{h,k} = \min\left\{r\_h, r\_k\right\}/4r\_n,\tag{7}$$

except for the terms

$$\begin{aligned} p\_{j,j} &= r\_j / 2r\_n \quad \text{for} \quad j = 2, \ldots, n-1, \\ p\_{h,1} &= r\_1 / 2r\_n \quad \text{for} \quad h = 2, \ldots, n, \\ p\_{n,k} &= r\_k / 2r\_n \quad \text{for} \quad k = 1, \ldots, n-1, \\ p\_{1,k} &= 0 \quad \text{for} \quad k = 1, \ldots, n, \\ p\_{h,n} &= 0 \quad \text{for} \quad h = 1, \ldots, n. \end{aligned} \tag{8}$$

Such a choice stands for the belief that poorer individuals usually spend and earn less than richer ones. The requirements for the coefficients with *h*, *k* ¼ 1 or *n* are of a technical nature, due to constraints on the extreme classes.

According to the empirical result recalled at the end of Section 2, for a specific given model (i.e. once parameters are fixed), the solutions of (Eq. (1)) evolving from all initial conditions *x*<sup>0</sup> with the same global income tend to a same asymptotic equilibrium.

The application we are going to discuss here includes four steps and is constructed as follows:


#### **Figure 1.**

*The four panels display the stationary income distributions in correspondence to the same given global income for the four different fiscal policies described in steps (i), (ii), (iii), and (iv). Even a simple look provides evidence of the fact that in passing from each panel to the next one the fraction of individuals in the poorest as well as in the richest class decreases while increasing in the intermediate classes. Correspondingly, economic inequality decreases.*

• Step (iii): To simulate the implementation of a more targeted fiscal policy, we now introduce another change amounting to the choice of a progressive taxation. Equivalently, we fix different tax rates, lower for low-income earners and higher for high-income earners. Specifically, we choose here *τmin* ¼ 20% and *τmax* ¼ 50%. The "asymptotic" stationary solution obtained in

correspondence to an initial condition coinciding with the asymptotic stationary solution of step (ii) is displayed in Panel (iii) in **Figure 1**.

• Step (iv): As a further focused fiscal policy, we also incorporate in the taxation algorithm what can be thought of as an addition of welfare provision. From a technical point of view, this can be achieved through the introduction of suitable weights in the terms *U<sup>i</sup>* ½ � *hk* ð Þ *<sup>x</sup>* and *<sup>V</sup><sup>i</sup>* ½ � *hk* ð Þ *<sup>x</sup>* in system (Eq. (1)). Such weights allow to differently measure the portion of redistributed tax revenue to individuals of different income classes. A formula able to realise this is given in [11], and we refer to that paper for further details. What is of interest here is the final "asymptotic" income distribution relative to the equations, which include this modification and to an initial condition coinciding with the asymptotic stationary solution of step (iii). This income distribution is shown in Panel (iv) in **Figure 1**.

Already a simple look at the panels in **Figure 1** provides evidence of the fact that the effect in the long run of each of the different fiscal policies adopted throughout the steps (i), (ii), (iii), and (iv) is to modify the income distribution over the population so as to lower the number of individuals in the poorest as well as in the richest classes, simultaneously increasing this number in the intermediate classes. Also, an alternative, unified representation of the four stationary income distributions corresponding to the four different taxation system fiscal policies (i), (ii), (iii), and (iv) is given in **Figure 2**. Lastly, in **Figure 3** the evolution in time of the fraction of individuals in the 15 income classes is displayed. Once again, together with others, one may notice that the fractions of individuals that are initially the largest and the smallest (fractions to which the poorest and the richest individuals belong) are both non-increasing in time.

We emphasise that economic inequality decreases in passing from the income distribution displayed in Panel (i) of **Figure 1** to the income distributions in Panel (ii). The same holds true in passing from the distribution in Panel (ii) to that one in Panel (iii), and from the distribution in Panel (ii) to that one in Panel (iv).

#### **Figure 2.**

*An alternative representation of the stationary income distributions in correspondence to a given global income for the four different taxation systems fiscal policies described in steps (i), (ii), (iii), and (iv). One clearly notices that the fraction of the poorest and the fraction of the richest individuals decrease when passing from the distribution for step (i) (corresponding to the strip [0,1]) to the distribution for step (iv) (corresponding to the strip [3, 4]).*

#### **Figure 3.**

*The evolution in time of the fraction of individuals in the 15 income classes for the model with fiscal policies as in steps (ii), (iii), and (iv). One may notice, in particular, that the fractions of individuals which are initially (in the stationary distribution reached in absence of taxes) the largest and the smallest—fractions to which the poorest and the richest individuals belong—are both non-increasing in time.*

#### **Figure 4.**

*The graph of the Gini coefficient as a function of time in correspondence to the application in the sequence of the three different fiscal policies adopted throughout the steps (ii), (iii), and (iv).*

A quantitative measure of economic inequality is given by the Gini coefficient *G* (named after the Italian statistician and economist C. Gini who introduced it in the early twentieth century, see [28]), whose definition we recall next: if the Lorenz curve expresses on the *y*-axis, the cumulative percentage of the total income of a population earned by the bottom percentage of individuals (represented, in turn, on the *x*-axis), denote *A*<sup>1</sup> the area between the Lorenz curve of the distribution at hand and the line of perfect equality *y* ¼ *x*, characterising a uniform distribution; also, denote *A*<sup>2</sup> the total area under the line of perfect equality. The Gini coefficient is defined as the ratio *A*1*=A*<sup>2</sup> and takes values in the interval 0, 1 ½ �. The extreme values 0 and 1 of *G* respectively represent complete equality and complete inequality.

The Gini coefficients relative to the income distributions in the Panels (i), (ii), (iii), and (iv) in **Figure 1** are

$$G = 0.453551, \quad G = 0.426308, \quad G = 0.386833, \quad G = 0.365182\tag{9}$$

respectively.

#### *Taxation and Redistribution against Inequality: A Mathematical Model DOI: http://dx.doi.org/10.5772/intechopen.100939*


*a refers the solution at time \$t\$ of the equation system with coefficients as in step (ii),*

*b refers to the solution at time \$t\$ of the equation system with coefficients as in step (iii),*

*c refers to the solution at time \$t\$ of the equation system with coefficients as in step (iv).*

#### **Table 1.**

*In this table, the Gini coefficients G relative to the income distributions are evaluated in correspondence of a finite number of times for the model systems described in steps (ii), (iii), and (iv). One sees here that each of the solutions G decreases. Accordingly, economic inequality is decreasing for each of the three models (ii), (iii), and (iv), models characterised respectively by the existence of a taxation system with a unique tax rate, the existence of a progressive taxation system with different tax rates, the existence of a taxation system integrated by welfare.*

#### **Figure 5.**

*The Lorenz curves corresponding to the stationary income distributions reached at the end of steps (i), (ii), (ii), and (iv). The variable on the x-axis denotes the bottom percentage of individuals and the variable on the y-axis is the cumulative percentage of the total income earned by the corresponding percentage of individuals. The Lorenz curves referring to the final distribution relative to steps (i), (ii), (ii), and (iv) are ordered from the lowest to the highest. Accordingly, in passing from step (i) to step (ii) to step (iii) to step (iv) the Gini coefficient decreases.*

It is also worth noting that the Gini coefficient decreases along with the solutions of the equation systems relative to the three models defined in steps (ii), (iii), and (iv). In particular, in **Table 1** some values of *G* are reported, relative to the income distributions in a finite number of instants during the evolution of the three dynamical systems. An overall picture of this behaviour is contained in **Figure 4**: there, the graph of the Gini coefficient as a function of time is shown, in correspondence to the application in the sequence of the three different fiscal policies adopted throughout the steps (ii), (iii), and (iv). Lastly, **Figure 5** displays the Lorenz curves corresponding to the stationary income distributions reached at the end of steps (i), (ii), (ii), and (iv).
