# A Mathematical Model of Universal Basic Income and Its Numerical Simulations
## 1 Introduction  
A lively debate on the appropriateness of introducing some form of universal basic income (UBI) has been ongoing in several countries for a long time (for concepts, methodologies, practice and much more see for example the recent papers [1, 2, 3] and books [4, 5] which in turn also provide several further references). Additionally, economic difficulties which grow stronger in times of crisis spark the interest for the subject. The problem is in fact a complex one which involves not only political and economic, but also sociological and psychological aspects. Engaging in the discussion to endorse one or the other position towards is certainly not the goal of this paper. Rather, the goal is to illustrate, through a technical approach, how numerical simulations of mathematical models can help predict long-term effects of different policies, in particular related to UBI. This can then help to explore different possible scenarios emerging in correspondence of different political programs. Of course, a serious and attentive contribution (not to be found here) cannot ignore and should definitely take into account several related socio-economic facets and elements. The investigation and the simulations here described have roots in a mathematical \"micro-to-macro\" model proposed by this author in [6] and later generalised and studied in other works (see e.g., [7, 8]). The model describes the evolution in time, in the case of constant global income, of the income distribution over a population, resulting out of a whole of economic interactions which take place between individuals of the population. Individuals are grouped in a finite number of classes distinguished by their income and exchange money according to frequency and rules first suggested in [7]. A fiscal system with progressive taxation together with a redistribution process, integrated in paper [8] with a welfare policy, was considered in the mentioned papers. The mathematical framework developed for the purpose is provided by a set of nonlinear ordinary differential equations, each of which gives account of how the number of individuals in a certain income class changes in time. The analysis of the dynamics, based to a great extent on a large number of simulations performed in correspondence to different choices of initial conditions and different values of the model parameters allows the establishment of the existence, for fixed parameter values and for fixed global income, of a unique asymptotic (long-run) stationary distribution. Such distribution has been intended as a dynamic equilibrium, because, albeit the number of individuals in each class remains constant from a certain time on, single individuals can move from one class to another. It has also been shown in [7] that suitable choices of the parameters allow the recovery of distributions which are well in agreement with empirical data. Among other features, economic inequality, typically measured by means of the Gini coefficient, has been the object of particular attention. The mentioned framework turns out to be suitable also to investigate the effect of the introduction of a universal basic income. Indeed, if a basic income provides a guaranteed financial grant to everybody, then everybody has a minimal income which is different from zero. This can be expressed in the context of the framework by assuming that the lowest income class is empty, or the lowest income classes are empty. Specifically, distributions of the population can be considered for which the lowest income class/classes are empty.