# A Mathematical Model of Universal Basic Income and Its Numerical Simulations
## 2 The Model
Analogously, individuals of the poorest class cannot pay money to end up with a negative income. What we want to emphasise here is that this elementary model treats society as a collection of individuals who interact directly (through money exchange) or indirectly (through taxation and redistribution). No central government is effectively present in the algorithm. A system of \(n\) nonlinear ordinary differential equations, one for each income class in which the population is divided, describes the variation in time of the fraction of individuals in these classes:  
\[\frac{dx_{k}}{dt}=\sum_{i=1}^{n}\sum_{j=1}^{n}\left(C_{ij}^{k}+T_{[ij]}^{k}(x) \right)x_{i}x_{j}-x_{k}\sum_{j=1}^{n}x_{j}\,,\qquad k=1\,,\ldots,n\,. \tag{1}\]  
The meaning and the form of the \(C_{ij}^{k}\)'s and \(T_{[ij]}^{k}\)'s appearing on the r.h.s. of (1) are detailed next. For the convenience of the reader, we repeat here a description necessarily similar to one already given in [6; 7; 8]. A warning is however in order that some terms below are here different than the corresponding ones in those papers. * Each \(C_{ij}^{k}\in[0,+\infty)\) expresses the probability density that an individual of the \(i\)-th class will belong to the \(k\)-th class after a direct interaction with an individual of the \(j\)-th class. The identity \(\sum_{k=1}^{n}C_{ij}^{k}=1\) has hence to hold true for any fixed \(i\) and \(j\);
* Each \(T_{[ij]}^{k}:\mathbf{R}^{n}\to\mathbf{R}\) expresses the variation density in the \(k\)-th class due to an interaction between an individual of the \(i\)-th class with an individual of the \(j\)-th class. The functions \(T_{[ij]}^{k}\) are continuous and need to satisfy \(\sum_{k=1}^{n}T_{[ij]}^{k}(x)=0\) for any fixed \(i\), \(j\) and \(x\in\mathbf{R}^{n}\). To specify the form of the \(C_{ij}^{k}\)'s and \(T_{[ij]}^{k}\)'s we still need some more definitions:
* Denote by \(p_{i,j}\) (for \(i,j=1,2,\ldots,n\)) the probability that in an interaction between an individual of the \(i\)-th class with an individual of the \(j\)-th class, the one who pays is the former one. Specifically, we assume \[p_{i,j}=\min\{r_{i},r_{j}\}/4r_{n}\,,\] with the exception of the terms \(p_{h,h}=r_{h}/2r_{n}\) for \(h=2,\ldots,n-1\), \(p_{i,1}=r_{1}/2r_{n}\) for \(i=2,\ldots,n\), \(p_{n,j}=r_{j}/2r_{n}\) for \(j=1,\ldots,n-1\), \(p_{1,j}=0\) for \(j=1,\ldots,n\) and \(p_{i,n}=0\) for \(i=1,\ldots,n\). * Denote by \(S\) the minimal amount of money that individuals may exchange;
* Denote by \(w_{h}\in\mathbf{R}\) (for \(h=1,2,\ldots,n\)) \(n\) coefficients playing the role of weights, suitable to incorporate welfare in the redistribution terms. Specifically, we assume here (this expression is different from that one adopted in [8]) \[w_{h}=\frac{1}{2}\left(1+\frac{(1-2\beta)\left(2r_{h}-r_{1}-r_{n}\right)}{r_{ n}-r_{1}}\right),\] (2) with \(1/2\leq\beta\leq 1\). From (2) one easily checks that if \(\beta=1/2\), then \(w_{h}\) takes the same value, \(1/2\), for each \(h=1,\ldots,n\), whereas if \(\beta>1/2\), then \(w_{1}=\beta\), \(w_{n}=1-\beta\) and \(w_{h}\) is decreasing as a function of \(h\).