# A Mathematical Model of Universal Basic Income and Its Numerical Simulations
## 2 The Model  
We introduce here a modified version of a model first proposed in [6] and then further developed in [7; 8]. For the convenience of the reader we explicitly point out since the beginning that the novelties in the present version consist of a different subdivision of income ranges compared to the previous model and in the formalisation of the allocation of a basic income through the assumption of emptiness of the poorest class/classes (see below). We start by considering \(n+1\) numbers \(0=\rho_{0}<\rho_{1}<\ldots<\rho_{n}\), with \(\rho_{n}\) intended to be large enough to represent an upper limit (that cannot be exceeded) for the maximal conceivable income of an individual of a given society. In particular, differently from what was done in the mentioned previous works, we will take \(\rho_{k+1}-\rho_{k}=\alpha\left(\rho_{k}-\rho_{k-1}\right)\) with \(\alpha>1\), for \(k=1,\ldots,n-1\). In this way, \(n\) intervals can be considered, whose length increases with \(k\). These intervals identify \(n\) income classes. We emphasise that this provides a more realistic description. Indeed, in this way, one can, by suitably choosing \(\alpha\), describe societies in which the richest individuals (those belonging to the \(n\)-th income class) have an average income which is order of magnitude larger that of the poorest individuals (those in the first class). We suppose for simplicity that the fraction of individuals whose income falls between \(\rho_{k-1}\) and \(\rho_{k}\) (more precisely, in \([\rho_{k-1},\rho_{k})\)) is the same in all points of the interval. We denote by \(x_{k}(t)\), with \(x_{k}:\mathbf{R}\rightarrow[0,+\infty)\), this fraction at time \(t\), and denote \(r_{k}=\left(\rho_{k+1}-\rho_{k}\right)/2\) the average income of the \(k\)-th class. The overall dynamics, which determines at each instant the income distribution, results from a whole of economic exchanges between individuals. These exchanges represent both earnings (due for example to the provision of some good or service) and losses (due to the payment in exchange for some good or service). They induce \"small\" movements of the individuals undergoing the exchanges from a class to a closer one. Beside such earnings and losses, which occur as direct interactions of pairs of individuals, others can be thought to take place, representing the benefit from the tax revenue redistribution (provided through education, health care and similar supports) and the payment of taxes accompanying any gain respectively. Indeed, we postulate the existence of a fiscal system based on a progressive taxation. Specifically, denoting by \(\tau_{k}\in[0,1)\) the tax rate for the \(k\)-th class, we assume  
\[\tau_{k}=\frac{\left(r_{k}-r_{1}\right)\tau_{max}+\left(r_{n}-r_{k}\right)\tau _{min}}{r_{n}-r_{1}}\qquad\quad\text{for}\quad k=1,\ldots,n\,,\]  
where \(\tau_{min}=\tau_{1}\) and \(\tau_{max}=\tau_{n}\) respectively denote the minimum and maximum tax rates. The way how taxation (together with redistribution) is taken into account by the algorithm described below is as follows. Every time  
* an individual of the \(i\)-th class pays an amount \(S\left(S\ll\left(r_{k+1}-r_{k}\right)\right)\) for all \(k=1,\ldots,n\)) to one of the \(j\)-th class and
* the latter one pays to the government a tax on her earnings according to her income class (her tax rate being \(\tau_{j}\)) and
* the government redistributes the tax revenue to the population. * This process is equivalently described by the following actions:
* the \(i\)-individual pays an amount \(S\left(1-\tau_{j}\right)\) to the \(j\)-individual and
* the \(i\)-individual pays an amount \(S\tau_{j}\) which is divided among all individuals. Further details can be found in [6], where this mechanism was first introduced. We notice here that due to constraints inherent to the model, individuals of the \(n\)-th class are excluded by the redistribution and cannot in general receive money which would let them exit from the highest income class.