Mathematics and Statistics

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Browsing Mathematics and Statistics by Subject "Middle class"

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    Mathematical Models, Analysis and Simulations of the Handy Model with Middle Class
    (2021-12-06) Al-Khawaja, Thanaa Ali Kadhim A; Shillor, Meir; Spagnuolo, Anna Maria; Ogunyemi, Theophilus; Andrews, Kevin
    This study presents three different mathematical versions of the HANDY (Human And Nature DYnamics) model for the socioeconomic dynamics of a large stratified society. The basic model was introduced in the ground breaking publications of Motesharrei (dissertation 2014) and Motesharrei et. al. (2016). The original model consists of a nonlinear system of four ordinary differential equations (ODEs) which describe the development, in time, of a ’very simple’ society consisting of two populations: the Elite (rich) and Commoners (workers). Included also are the use of natural (renewable and nonrenewable) resources and the accumulation of human wealth. The model’s solutions depict the dynamics of these variables. Motesharrei’s main impetus and interest was to use the model as a tool for evaluating the conditions that contribute to the flourishing, sustainability, or collapse of complex societies. This dissertation expands the basic HANDY model and studies its mathematical properties and those of its three extensions. It establishes the existence of solutions to the models, as well as their uniqueness, boundedness and positivity. Furthermore, it investigates the stability of the systems’ steady states, which describe the long-time behavior of the societies. It also presents a number of qualitatively different computer simulations, providing insights into potential behaviors of the societies described by these models. The main contributions of this work are the mathematical analysis of the basic HANDY model, its three expansions and their analysis, and computer simulations. The first extension, the HANDY-SM model, includes social mobility. Rich individuals may go bankrupt and become workers, and some workers may become rich. It also allows for two different aspects of inequality, through variations in salaries and the wealth structure. The second extension, the HANDY-MC-I model, includes the Middle Class population, making the model more practical when applied to modern societies. It expands the system into five ODEs, and allows for social mobility among the three populations. Finally, in the third extension, HANDY-MC-II, two variables describe the natural resources: the renewable resources (wood, solar and wind energies), and nonrenewable resources (coal, oil, gas). This particular extension makes the model more realistic, but it also adds considerable complexity since it consists of six nonlinear coupled ODEs. The model simulations depict the consequences of having three different populations with different income status, two natural resources, and unequal contributions to wealth structure. Analysis of the models’ steady states shows that the state is stable when the populations and wealth die out but nature (the resources) is at its equilibrium. The model has also asymptotically stable, nonzero steady states to which the populations, the resources and the wealth converge in the long-time limit. The simulations also show the existence of periodic solutions in which the populations, the natural resources and wealth undergo large oscillations, indicating cycles of ‘boom and bust.’ Finally, the simulations demonstrate that the models may have chaotic solutions, pointing to a high level of unpredictability. This dissertation describes three increasingly more complex HANDY models. It paves the way and raises mathematically interesting topics for their further study. In particular, the uniqueness of the solutions, and the questions of the existence of periodic solutions, limit cycles and chaos, remain unresolved, yet. Furthermore, it suggests the possibility of tailoring such models to existing societies, and then using them as tools for evaluation of the potential outcomes of various policy decision