Investigation On Wealth Distribution Influenced By Knowledge Level,Preference,Trading Time And Saving Propensity Of Agents

For a long time,unequal distribution of wealth has caused a series of negative phenomena in society.Thus,the research on wealth distribution has become one of the hot spots of current research.The dynamic distribution of wealth is a complex phenomenon,which is determined by many factors,includ-ing tax,welfare,innovation,education,consumption and agents' preferences etc.In the transaction,the distribution of wealth among agents is affected by many factors,such as agent's knowledge level,agent's preference for goods and trading time.In this thesis,we employ kinetic theory of rarefied gases to inves-tigate the influence of agent's knowledge level,preference,trading strategies and saving propensity on wealth distribution.In addition,we set up a wealth distribution model on isolated discrete time domains,and analyze the impact of the uncertainty of trading time on commodity price and wealth distribution.Firstly,to study the influence of agent's knowledge level on wealth distri-bution,we add parameters that affect individual traits to binary interaction rules which are given in[54,97].Using the improved binary interaction rules,we set up a nonlinear kinetic equation of Boltzmann type to describe the influ-ence of knowledge level on wealth distribution,and obtain the corresponding Fokker-Planck equation by using the asymptotic method.The numerical sim-ulation experiments of the kinetic equation and Fokker-Planck equations are displayed by using the Monte Carlo method.The results show that knowledge has potential to increase the wealth of agents and takes a key role in wealth inequality.Secondly,motivated by the works in[26,122],we assume that a closed market consists of two groups(A and B)of agents,who have the same prefer-ences for goods.The constant elasticity of substitution(CES)utility function which contains Cobb-Douglas utility function and Leontief utility function,is used to describe the agent's preferences.A system of linear Boltzmann type equation is given to describe the probability distributions of goods for agents of the two groups,in which the agents are allowed to adopt certain trading strate-gies to maximize their utility and improve their wealth conditions.We assume that agents in group A put all their goods into the market for maximum utility,while agents in group B aim to acquire maximal utility from goods exchange by suitably selecting the percentage of goods which are exchanged.The effects of different trading strategies on commodity price and wealth distribution are discussed.The results show that the trading strategies of agents in group B have effectively improved their wealth status under certain assumptions.Also,a general system of nonlinear Boltzmann equation for the probability distri-butions of goods is given to describe the pricing issues of goods exchange for agents.The numerical experiments of nonlinear Boltzmann equations demon-strate how the trading strategies and preferences of agents modify the price of goods.Thirdly,motivated by the works in Acednnski[1]and Atici et al.[13],we set up a discrete wealth distribution model on isolated discrete time domains.The time domain of the wealth distribution model in our work is a collection of points along the real number,which describes the distribution of wealth in non-uniform time interval.We assume that the agents have different degrees of risk aversion.The hyperbolic absolute risk aversion(HARA)utility function is employed to describe the degrees of risk aversion of agents,including decreas-ing relative risk aversion(DRRA),increasing relative risk aversion(IRRA)and constant relative risk aversion(CRRA).The effect of agent's expectation on wealth distribution is taken into account in our wealth distribution model,in which the agents are allowed to adopt certain trading strategies to maximize their utility and improve their wealth status.The Euler equation and transver-sality condition for the model on isolated discrete time domains are given to prove the existence of the optimal solution of the model.The optimal solution of the wealth distribution model is obtained by using the method of solving rational expectation model on isolated discrete time domains.A numerical example is given to highlight the advantages of the wealth distribution model on isolated discrete time domains.Finally,based on the works in Albi et al.[4]and During et al.[50],we utilize the kinetic theory to investigate the optimal control problem of wealth distribution in a multi-agent system,in which the control term is to minimize the variance of wealth density among agents.Assume that the market consists of N agents,each of whom uses part of their wealth for trading and the rest for saving.For this optimal control problem,a kinetic equation of Boltzmann type based on model predictive control is given to describe the process of wealth distribution.To analyze the wealth inequality among agents,the asymptotic method is utilized to derive Fokker-Planck equation corresponding to kinetic equation of Boltzmann type.The stationary solution of Fokker-Planck equa-tion reflects how the agent's saving propensity affects the formulation of con-trol strategies.The results show that the control term of measuring wealth inequality is closely related to the agent's saving propensity.The numerical experiments are given to prove that the Fokker-Planck equation obtained by the asymptotic method retains the properties of Boltzmann equation.We con-clude that it is effective to analyze the optimal control of wealth distribution by the stationary solution of Fokker-Planck equation.