# A Mathematical Model of Universal Basic Income and Its Numerical Simulations
## Appendix A  
We give here some details on the numerical method employed to find the solutions of the equations (numbered as (1) in the text)  
\[\frac{dx_{k}}{dt}=\sum_{i=1}^{n}\sum_{j=1}^{n}\Big{(}C_{ij}^{k}+T_{[ij]}^{k}( x)\Big{)}x_{i}x_{j}-x_{k}\sum_{j=1}^{n}x_{j}\,,\qquad k=1\,,\dots,n\,. \tag{1}\]  
We can write more concisely this differential equation system as  
\[\frac{d\mathbf{x}}{dt}=\mathbf{F}(\mathbf{x})\,, \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\tau_{min}\)-\(\tau_{max}\) & \(\boldsymbol{\beta}\) & \((\boldsymbol{i})\) & \((\boldsymbol{ii})\) & \((\boldsymbol{iii})\) \\ \hline
00-00 & 50 & 0.79 & 0.70 & 0.62 \\ \hline
20-20 & 50 & 0.69 & 0.65 & 0.58 \\ \hline
20-45 & 50 & 0.64 & 0.61 & 0.56 \\ \hline
20-45 & 67 & 0.63 & 0.60 & 0.55 \\ \hline \(\mu=16\) & & & & \\ \hline
00-00 & 50 & 0.74 & 0.68 & 0.63 \\ \hline
20-20 & 50 & 0.63 & 0.61 & 0.58 \\ \hline
20-45 & 50 & 0.58 & 0.57 & 0.55 \\ \hline
20-45 & 67 & 0.57 & 0.56 & 0.54 \\ \hline \(\mu=24\) & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: The two tables display the Gini coefficients for the models whose minimal and maximal tax rates \(\tau_{min}\) and \(\tau_{max}\) and welfare coefficient \(\beta\) are as in the first column, in the cases \((i)\), \((ii)\) or \((iii)\) as in the first row. The top table refers to total income \(\mu=16\) and the bottom table to total income \(\mu=24\).  
with obvious definition of the vector field \(\mathbf{F}=(F_{1},\ldots,F_{n})\) and with  
\[\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\,.\]  
If the two vectors  
\[\mathbf{v}_{1} =(1,1,\ldots,1)\in\mathbf{R}^{n}\,,\] \[\mathbf{v}_{2} =(r_{1},r_{2},\ldots,r_{n})\in\mathbf{R}^{n}\]  
are introduced, the constraints (3) and (4) can be expressed as  
\[\sum_{k=1}^{n}x_{k}(t)=\mathbf{x}\cdot\mathbf{v}_{1}=1\,, \tag{40}\]  
\[\sum_{k=1}^{n}r_{k}x_{k}(t)=\mathbf{x}\cdot\mathbf{v}_{2}=\mu\,. \tag{41}\]  
The Cauchy problem in correspondence to a fixed value of the total income (and for a population normalised to 1) amounts to search the solution \(\mathbf{x}(t)\) of (41) which satisfies  
\[\mathbf{x}(0)=\mathbf{x}_{0} \tag{42}\]  
for a given vector \(\mathbf{x}_{0}=(x_{0,1},x_{0,2},\ldots,x_{0,n})\), where \(\mathbf{x}_{0}\cdot\mathbf{v}_{1}=1\) and \(\mathbf{x}_{0}\cdot\mathbf{v}_{2}=\mu\).  
The Equation (40) (or, equivalently (41)) together with the constrains (40) and (41) form a semi-explicit DAE (differential algebraic equation). By directly applying a standard numerical solver to the \(n\) Equation (40) together with some initial condition \(\mathbf{x}(0)=\mathbf{x}_{0}\), one would not obtain the conservation of the population and of the total income; namely the constraints (40) and (41) would not be satisfied. In view of that, the numerical results described in the paper have been obtained by first performing a reduction of the number of system equations in the following way.  
The constraints (40) and (41) allow the explicit expression of two of the components of a vector \(\mathbf{x}\) satisfying them in terms of the others. For instance, given the vector  
\[\mathbf{x}^{*}=(x_{1},x_{2},\ldots,x_{n-2})\,,\]  
one can obtain from (40) and (41) (and this was done here)  
\[x_{n-1} =\xi_{n-1}(\mathbf{x}^{*})\,,\] \[x_{n} =\xi_{n}(\mathbf{x}^{*})\,. \tag{43}\]  
The Equation (41) can then be reformulated as  
\[\frac{dx_{k}^{*}}{dt}=F_{k}(\mathbf{x}^{*},\xi_{n-1}(\mathbf{x}^{*}),\xi_{n}( \mathbf{x}^{*}))\,,\qquad k=1,\ldots,n-2\,, \tag{44}\]  
and is studied in conjunction with the initial condition  
\[\mathbf{x}_{0}^{*}=(x_{0,1},x_{0,2},\ldots,x_{0,n-2})\,. \tag{45}\]  
Once the unique solution \(\mathbf{x}^{*}(t)\) of (44) and (45) has been found, by invoking the Formula (43), one finds the solution  
\[\mathbf{x}(t)=(\mathbf{x}^{*}(t),\xi_{n-1}(\mathbf{x}^{*}(t)),\xi_{n}( \mathbf{x}^{*}(t))) \tag{46}\]  
of (40), (40), (41) and (42).  
Calculations have been performed by employing a numerical solver from the software Wolfram Mathematica 12.3.0.0.