# A Mathematical Model of Universal Basic Income and Its Numerical Simulations
## 3 Results from Numerical Simulations  
To find numerical solutions of the equation system (1), we obtain insight on its dynamics and make comparisons between the emerging scenarios in the presence of different fiscal systems, and in particular, in the presence of a basic income allocation, we need to fix the values of the parameters entering in the model. We want to be able to acquire a graphical representation of the income distribution, somehow resembling, albeit within a discrete approach, one of a continuous distribution. We hence choose here a rather high number of classes, \(n=15\). We then choose \(\rho_{1}=1\) and \(\alpha=1.3\), which yields (rounding off numbers)  
\[(\rho_{0}\,,\,\rho_{1}\,,\,\rho_{2}\,,\,\rho_{3}\,,\,\ldots\,,\rho_{n})=(0.0\,, \,1.0\,,\,2.3\,,\,3.99\,,\,\ldots\,,\,167.286)\,,\]  
and, correspondingly,  
\[(r_{1}\,,\,r_{2}\,,\,r_{3}\,,\,\ldots\,,r_{n})=(0.5\,,\,1.65\,,\,3.145\,,\, \ldots\,,147.599)\,.\]  
As for the minimum and maximum tax rates, we consider three cases:  
* \((I)\quad\tau_{min}=0.0\quad\) and \(\quad\tau_{max}=0.0\quad\) (case in which there is no taxation),
* \((II)\quad\tau_{min}=0.20\quad\) and \(\quad\tau_{max}=0.20\quad\) (case with a single taxation rate),
* \((III)\quad\tau_{min}=0.20\quad\) and \(\quad\tau_{max}=0.45\quad\) (case with a progressive taxation). Furthermore, whereas for the cases \((I)\) and \((II)\) we assume \(\beta=0.50\) (i.e., no additional welfare), for the case \((III)\) we consider both the possibility that \(\beta=0.50\) (no additional welfare) and \(\beta=0.67\) (additional welfare). We also fix \(S=1\). We consider, in correspondence of each of the cases \((I)\), \((II)\), \((III)\) above (and in the case \((III)\) in correspondence of the two versions, with \(\beta=0.50\) and \(\beta=0.67\)), relative to societies ruled by different fiscal systems, the time evolution of system (1) for two different values of the total income, here \(\mu=16\) and \(\mu=24\) (to exemplify a \"poor\" and a \"rich\" population), in three cases:  
* \((i)\) when no basic income allocation is present,
* \((ii)\) with basic income allocation and the class with average income \(r_{1}\) empty,
* \((iii)\) with basic income allocation and the classes with average income \(r_{1},r_{2}\) empty. Summarising, the cases here considered are those in the following list, where we also introduce a suitable label for each of them, to be then employed in the captions of the panels of some figures:  
- model [00-00, 50, 16]:  
\(\tau_{min}=0.00\), \(\tau_{max}=0.00\), \(\beta=0.50\), \(\mu=16\), in the cases \((i),(ii),(iii)\),  
- model [00-00, 50, 24]:  
\(\tau_{min}=0.00\), \(\tau_{max}=0.00\), \(\beta=0.50\), \(\mu=24\), in the cases \((i),(ii),(iii)\),  
- model [20-20, 50, 16]:  
\(\tau_{min}=0.20\), \(\tau_{max}=0.20\), \(\beta=0.50\), \(\mu=16\), in the cases \((i),(ii),(iii)\),  
- model [20-20, 50, 24]:\(\tau_{min}=0.20,\ \ \tau_{max}=0.20,\ \ \beta=0.50,\ \ \mu=24,\ \ \text{in the cases}\ (i),(ii),(iii),\)  
- model [20-45, 50, 16]:  
\(\tau_{min}=0.20,\ \ \tau_{max}=0.45,\ \ \beta=0.50,\ \ \mu=16,\ \ \text{in the cases}\ (i),(ii),(iii),\)  
- model [20-45, 50, 24]:  
\(\tau_{min}=0.20,\ \ \tau_{max}=0.45,\ \ \beta=0.50,\ \ \mu=24,\ \ \text{in the cases}\ (i),(ii),(iii),\)  
- model [20-45, 67, 16]:  
\(\tau_{min}=0.20,\ \ \tau_{max}=0.45,\ \ \beta=0.67,\ \ \mu=16,\ \ \text{in the cases}\ (i),(ii),(iii),\)  
- model [20-45, 67, 24]:  
\(\tau_{min}=0.20,\ \ \tau_{max}=0.45,\ \ \beta=0.67,\ \ \mu=24,\ \ \text{in the cases}\ (i),(ii),(iii).\)  
As recalled in Section 2.1, for any choice of model parameters, and any conceivable value of the total income \(\mu\), a unique stationary income distribution exists to which solutions tend in the long run. The \"asymptotic\" distributions relative to the mentioned cases are displayed in the Figures 1 and 2.