| For a long time,the problem of wealth distribution has been one of the hot topics concerned by all sectors of society.The inequality of wealth distribution is not conducive to social harmony and stability.The investigation of wealth distribution requires knowledge involving mathematics,physics,economics,psychology and other disciplines.Factors such as the agent’s knowledge and education level,decision-making ability and health state affect their trading behavior,and then have an impact on wealth distribution.From the social perspective,international relations,the tax system,government redistribution regulation,the social market environment and other factors affect social wealth distribution.Using the knowledge of rarefied gas dynamical theory in statistical physics,the exchange of wealth between agents is analogous to the energy exchange of rarefied gas molecules during collisions,and the Boltzmann-type equation is used to describe the change of wealth distribution.By constructing wealth exchange models and using random variables(noise term)to characterize market risk,this thesis investigates the influence of people’s trading decisions,the government’s economic regulation and emergencies(such as infectious diseases)on wealth distribution.Firstly,in an agent group involving government economic regulation,we use a Boltzmann equation to describe the change in wealth distribution,and investigate the wealth distribution which is affected by agents,decision-making competence.In the wealth exchange model,the exchange propensities and the risk diffusion are modified as functions of wealth and decision-making competence.The government’s control of wealth inequality is aimed to reduce the wealth gap between agents.Adopting the wealth exchange model and an improved decision updating model,and using the quasi-invariant asymptotic limit method,we obtain the Fokker-Planck equation which depicts the joint distribution of wealth and decision-making ability.The main properties of the kinetic model are analyzed.Through several examples,we discuss the effects of exchange propensity and risk diffusion on the marginal distributions of wealth and decision-making competence.The results illustrate that the decision-making competence of agents affects the wealth distribution,the increase of risk may aggravate wealth distribution inequality,and the government’s economic regulation can alleviate the inequality of wealth.Secondly,in a market composed of two agent groups,we use two Boltzmann equations to describe the change in wealth distribution caused by the transfer of personnel and the wealth exchange behavior between agents.We construct a wealth exchange model with varying trading propensity and controls of wealth inequality,including individual donation and government feedback control on wealth inequality,to investigate the wealth distribution problem.We use three different trading propensity functions to describe the trading decision-making psychology of agents,and solve the Fokker-Planck equations describing the time evolution of wealth distribution.We find that trading propensity affects the large time characteristics of wealth distribution.Using the Monte Carlo method,it is verified that the equilibrium solution of the Fokker-Planck equations of wealth distribution asymptotically tends to the solution of the Boltzmann system with personnel transfer.Numerical results illustrate that the transfer of agents affects wealth distribution,encouraging agents to participate in transactions or decreasing economic risks are conducive to reducing wealth inequality.Additionally,individual donations and government redistribution control measures can make wealth distribution more equal.In particular,encouraging the rich to donate more would help to optimize the social wealth structure and increase the wealth of the middle and lower classes.Thirdly,in the situation of epidemics,agents are divided into three groups:susceptible group,infected group,and recovered group.We use three Boltzmann equations to describe the change in wealth distribution caused by the wealth exchange behavior of agents and the transfer of people.The transfer of agents follows the law of epidemic transmission.We use a value function that satisfies the property of prospect theory to represent the psychological characteristics of agents and embed this value function into the binary wealth exchange rule.Through calculation,we get the Fokker-Planck equations describing the evolution of wealth distribution.The steady-state solution of each Fokker-Planck equation is the product of the inverse-gamma function and the exponential function,indicating that the existence of agents,psychological factors affects wealth inequality.Under appropriate parameters,we show that the steady-state wealth distribution curve of the agent system possesses bimodal.We introduce a transport term in the Boltzmann equations to represent the conversion of susceptible persons to recovered persons through vaccination,and the numerical simulation results illustrate that vaccination is crucial to controlling the spread of epidemic and improving economies.Finally,on the background of epidemics,we further consider the government’s contact control measures to curb the spread of infectious diseases.We construct a wealth exchange model involving varying trading propensity and control of wealth inequality to investigate wealth distribution.We introduce two cost functions into the epidemic control model,and use the feedback control method to obtain two nonlinear incidence rates which satisfy the biologically feasible condition and the saturating effect.We find that the contact control measures may reduce the peak of the infected fraction and make more people remain uninfected,but prolong the duration of the epidemic and increase wealth inequality.When the agent’s trading propensity depends on wealth,by calculating the stationary solution of the Fokker-Planck equations of wealth distribution,we obtain that the large-time behavior of wealth distribution presents a Pareto tail.We find that the increase in the rich’s savings may aggravate wealth inequality.In addition,numerical simulation results suggest that the government’s tax and redistribution measures can alleviate the inequality of wealth distribution caused by contact control. |
