# A Mathematical Model of Universal Basic Income and Its Numerical Simulations
## 3 Results from Numerical Simulations
More specifically, the three histograms in each panel of Figure 1 display the fraction of individuals in each of the 15, 14, 13 non-empty income classes relative to the model quoted in the caption of the panel, in the three cases \((i),(ii),(iii)\). A similar representation of the long-run distributions, in which account is taken also of the width of the income intervals characterising the income classes, is given in Figure 2.  
One may easily notice that when basic income is introduced in a specific model, the fraction of individuals belonging to the poorest classes diminishes. Even if in some panels in Figure 1 the first column in the second and third histogram is higher than that one in the first histogram, one should remember that the fraction of individuals in the poorest class of \((iii)\) (i.e., the class with average income \(r_{3}\)) must be compared with--and it is always smaller than--the fraction of individuals belonging to the two poorest classes of \((ii)\) (the classes with average income \(r_{2}\) an \(r_{3}\)) and, in turn, this latter one must be compared with--and it is always smaller than--the fraction of individuals belonging to the three poorest classes of \((i)\) (the classes with average income \(r_{1}\), \(r_{2}\) an \(r_{3}\)). This leads one to think that for each of the examined cases economic inequality is smaller provided basic income allocation is present. Additionally, a comparison between stationary distributions emerging in the same taxation model (namely for fixed values of \(\tau_{min}\), \(\tau_{max}\) and \(\beta\)), but in correspondence to different total income values is interesting. It can be achieved by looking at the histograms contained in two panels on the same row, one in the right column and the other in the left column, e.g., in Figure 1a,b or in Figure 1c,d, and so on. By looking at the left and right panels in Figure 1 one may observe that, if \(\mu=24\) (larger total income), the fraction of individuals belonging to a class at the bottom of the income ladder is systematically smaller than that one of the same income class, of the same model, with (smaller total income) \(\mu=16\). Here and in the Figure 2 also notice that the histograms on the left and on the right panel column are scaled differently. To obtain an explicit measure of the income inequality characterising the various cases, we calculate the Gini coefficient. Its definition, next recalled, is based on the Lorenz curve. This curve plots on the \(y\) axis the fraction of the cumulative total income of the lowest-income fraction of the population, represented on the \(x\) axis. In other words, if the population and the total income are normalised to 1, the poorest cumulative fraction \(x\) of the population possesses the cumulative fraction \(y(x)\) of the total money. The line at 45 degrees represents perfect income equality. The Lorenz curve satisfies \(y(0)=0,y(1)=1\) and \(y(x)\leq x\) for all \(x\in[0,1]\). The Gini coefficient is defined as the ratio \(A/B\) of the area \(A\) that lies between the line of equality and the Lorenz curve over the total area \(B\) of the triangle which lies under the line of equality. Consequently, it takes values in \([0,1]\). The larger the Gini coefficient is, the greater the income inequality is. Each panel of Figure 3 shows the Lorenz curves relative to the income distribution of the model quoted in its caption in the three cases \((i),(ii),(iii)\). All graphs in the eight panels confirm that the Gini coefficient for a society, and equivalently also the economic inequality, diminishes provided a basic income is introduced. This can be seen also quantitatively:the approximate values of the Gini coefficients in the cases here considered are reported in Table 1.  
Figure 1: The three histograms in each of the panels **(a–h)** refer to the three cases \((i),(ii),(iii)\) of the model quoted in the relative caption and represent the fraction of individuals in each of the non-empty income classes.