# The Investigation of a Wealth Distribution Model on Isolated Discrete Time Domains

Authors: Chunhua Hu; Wenyi Huang; Tianhao Xie
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A wealth distribution model on isolated discrete time domains, which allows the wealth to exchange at irregular time intervals, is used to describe the effect of agent’s trading behavior on wealth distribution. 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 decreasing 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 transversality 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 the rational expectation model on isolated discrete time domains. A numerical example is given to highlight the advantages of the wealth distribution model.


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## Body

## 1. Introduction

In recent years, unequal distribution of wealth has caused a series of negative phenomena in the society. Empirical and theoretical research on wealth distribution has become an active branch of modern economics. The problem of wealth distribution is closely related to the optimal conditions of economic growth. Studying the distribution of wealth is important for analyzing the wealth inequality and formulating tax and redistributive policies.The dynamic distribution of wealth is a complex phenomenon, which is determined by many factors, including tax, welfare, innovation, education, consumption, and agents’ preferences. Guala [1] studies the redistribution of wealth among agents in the presence of taxation in the trading process. Pareschi and Toscani [2] consider the influence of agent’s knowledge level on wealth distribution and analyze the formation process of knowledge. Toscani et al. [3] discuss the influence of agents’ preferences on wealth distribution and get Pareto curve of wealth distribution. Brugna and Toscani [4] generalize the works of Toscani et al. [3] and further investigate the changes in wealth distribution of two types of agents with different trading strategies. A. S. Chakrabarti and B. K. Chakrabarti [5] propose a model based on consumers’ optimization which can give rise to those particular forms of asset exchange equations and discuss a problem of income distribution. In [6], the authors further investigate income and wealth inequality in the context of a generalized kinetic exchange model and conclude that the economy indeed shows inequality reversal under some certain conditions. Chakrabarti [7] discusses the laws of wealth distribution, firm size distribution, and the city size distribution in a common framework and shows that the equilibrium configurations of some general economic mechanisms are consistent with a power law in general and Zipf’s law in particular, in size distribution. A survey of wealth distribution is referred to [8].In fact, wealth distribution is not only related to the knowledge level and preference of agents but also to the transaction time. Therefore, it is necessary to divide the agent’s transaction into several periods and set up a discrete wealth distribution model. Chatterjee et al. [9] assume that the agent’s saving tendency satisfies a certain distribution and give a discrete wealth distribution model. Benhabib et al. [10] investigate the wealth distribution in Bewley economies with capital income risk and solve an infinite horizon consumption-saving problem with incomplete markets. In [10], the equilibrium distribution for consumption, savings, and wealth is obtained under the assumption that agents face saving restrictions. Mariacristina and Giulio [11] study the influence of agent’s savings on wealth distribution and find out the factors causing wealth inequality. Other discrete models on wealth distribution can be found in [12–16].In actual transactions, agents usually anticipate the price of assets or commodities in the market based on past trading information. This anticipated behavior affects the distribution of wealth among agents. Acedńnski [17] discusses the influence of agent’s expectation on wealth distribution. The results show that the effect of agent’s expectation on wealth distribution is obvious. In [17], a standard constant relative risk aversion (CRRA) utility function is used to describe the behaviors of agents, and a discrete wealth distribution model affected by agent’s expectation is given. In [10, 17], although the agent’s transaction is divided into several periods, the time interval of the transaction is uniform, without considering the wealth distribution when the transaction interval is uneven.Generally speaking, trading is not a uniformly discrete nor continuous behavior. Consider the stock market as an example. In a stock market, an agent buys shares of a particular company in varying time intervals. Suppose that on the first week of the year, the agent buys stocks every day, and in the next several weeks, the company generates a good profit, which makes the agent buy stocks several times a week. Unfortunately, due to the company’s poor performance and lower earnings for the rest of the year, the agent buys only one stock on the last week of the year. Since there is no rule to restrict the agent’s trading behavior, we do not know whether the time interval of trading is uniform. As a result, the agent’s trading behavior needs to be described in isolated discrete time domains where events may occur at unevenly spaced time points. Atici et al. [18] present a dynamic optimal problem from economics and construct a time scale model. Atici et al. [19] investigate a perfect-foresight utility maximisation problem by using a method for solving the dynamic optimisation problem in which the real-valued function and constraints depend on different times. Atici and Turhan [20] discuss the deterministic dynamic sequence problem on isolated discrete time domains. Atici et al. [21] introduce a nonlinear stochastic dynamic problem on isolated discrete time domains and give the Euler equation and transversality condition for the problem. The work in [21] provides a method for solving the nonlinear growth model on isolated discrete time domains. Other studies about the isolated discrete models can be found in [22–25].In [10, 17, 21], a CRRA utility function is utilized to describe the agent’s behavior. In actual transactions, the agent’s aversion for risk is complex. To better study the distribution of wealth among agents, we employ the hyperbolic absolute risk aversion (HARA) utility function to characterize the degrees of risk aversion of agents. The HARA utility function includes several agents’ aversion for risk, such as decreasing relative risk aversion (DRRA), increasing relative risk aversion (IRRA), and constant relative risk aversion (CRRA). In recent years, the HARA utility function has been applied in various economic models. Achury et al. [26] study a two-asset portfolio choice model with the HARA utility function which includes the subsistence consumption parameter. Levaggi and Menoncin [27] consider optimal dynamic tax evasion with the HARA utility function and a penalty function. Menoncin and Nembrini [28] find a closed form solution for the stochastic continuous time growth model with the HARA utility function.Some researchers show that the problem of wealth distribution is not only related to the optimal conditions of economic growth but also closely related to the wealth inequality and the formulation of tax policies. To reduce the inequality of wealth distribution, it is very important to analyze the factors that affect wealth distribution, such as the preferences of agents, trading strategies, and trading time. Motivated by the works in [10, 17, 21, 28], we assume that agents estimate the next saving rate and profit of assets based on past trading information. We allow agents to adopt certain trading strategies to maximize their utility, such as using part of the assets for trading and the rest for savings. Under these assumptions, we investigate the influence of agent’s trading behavior and average consumption level on the wealth distribution.The objective of this work is to employ the HARA utility function to describe the agents’ behavior and discuss the influence of average consumption level and trading time of agents on wealth distribution in the multiagent market. Compared with the works in [17, 21], we utilize the HARA utility function to characterize the agent’s behavior and divide the agent’s transaction into multiple periods. A wealth distribution model on isolated discrete time domains, which allows the wealth to exchange on unevenly spaced time points, is given. HARA utility function and the wealth distribution model are different from those in [17, 21], in which the CRRA utility function is used and the time domain of the wealth distribution model is a uniformly discrete set of points which allows the wealth to exchange at uniform time interval. HARA utility function contains several kinds of utility functions, e.g., DRRA utility function, IRRA utility function, and CRRA utility function. 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 nonuniform time interval. Consequently, we extend parts of results in [17, 21].The study is organized as follows. We introduce the calculus and dynamic equation on isolated discrete time domains in Section2. The Euler equation and transversality condition of a nonlinear stochastic dynamic model with a bivariate utility function are derived in Section 3. The main results about the wealth distribution model are given in Section 4. In Section 5, a numerical example is given to illustrate the advantages of the wealth distribution model on isolated discrete time domains.


## 2. Calculus and Dynamic Equation on Isolated Discrete Time Domains

In Sections3 and 4, we will set up the nonlinear stochastic dynamic model and the wealth distribution model on the isolated discrete time domains. The isolated discrete time domains are different from the classic discrete time domain. The classical discrete time domain contains only uniform time points, while the isolated discrete time domains contain not only uniform time points but also nonuniform time points. Therefore, we will introduce some definitions, theorems, properties, and dynamic equations on isolated discrete time domains.In this section, the basic concepts about delta-derivative, delta-integral, exponential function, and dynamic equation on isolated discrete time domains are introduced.LetT be any nonempty closed subset of the real numbers ℝ. The operators σt, pt, and μt are defined by σt=infs∈T:s>t, pt=sups∈T:s<t, and μt=σt−t, respectively, where σti is the forward jump operator, μti is the forward graininess operator, and pti is the backward jump operator. From the time scale T, the set Tk is obtained. For a point t∈T if σt>t, we say that t is right-scattered, and if pt<t, we say that t is left-scattered. When T has a left-scattered maximum m, we have Tk=T−m. Otherwise, Tk=T.It is worth noting that, in the caseT=ℝ, we obtain (see Merrell et al. [29])(1)σt=pt=t,μt≡0.WhenT=ℤ, where ℤ is the set of positive integers, for any t∈ℤ, we have(2)σt=t+1,pt=t−1,μt≡1.WhenPℕ=Pnn∈ℕ, where P>1 and ℕ is the natural number, we get(3)σt=Pt,pt=tP,μt=P−1t.P
ℕ
=
1
,
P
,
P
2
,
… is the called quantum time scale, which is very important in quantum theory (see Merrell et al. [29]).LetF:T⟶ℝ be a real-valued function on T. The delta-derivative [21, 30] of F is denoted by(4)FΔt=Fσt−Ftμt,where t∈T. The delta-integral [21, 30] of F is defined by(5)∫0TFτΔτ=∑s∈0,T∩TμsFs,and the exponential function [21, 25, 30] is given as(6)eqt,t0=∏s∈t0,t∩T1+μsqs,where 1+qtμt≠0 for all t∈T.We mention that whenT=ℤ, exponential function eqt,t0 becomes(7)eqt,t0=1+qt−t0,where 1+q≠0.WhenT=Pℤ, we get(8)eqt,t0=∏s∈t0,t∩T1+P−1qs,t>t0.For exponential functioneq−1/μt,t0, we have(9)eq−1/μt,t0=∏s∈t0,t∩Tqs,t≥t0,∏s∈t0,t∩T1qs,t<t0.Let0<q<1 be a constant and nt be a function of t, where nt indicates that the number of discrete points on the interval t0,t. Exponential function eq−1/μt,t0 is rewritten by(10)eq−1/μt,t0=∏s∈t0,t∩Tqs=qnt,where(11)ntt,s≔∫stΔτμτ.To calculate the wealth distribution model in Section4, we introduce an algorithm on time scales.Lemma 1 (see [30]).
LetQ be a n×n-matrix. If λ1, λ2, λ3,… , λn are the eigenvalues of Q, then(12)Qt=∑i=0n−1ri+1tUi,where rit i=1,2,…,n are chosen to satisfy the system(13)r1t+1r2t+1⋮rnt+1=λ100⋯01λ20⋯001λ3⋯0⋮⋯⋯⋯⋮0⋯01λnr1tr2t⋮rnt,r10r20r30⋮rn0=100⋮0,and Ui are defined by(14)U0=I,Ui=Q−λiIUi−1,1≤i≤n.The algorithm in Lemma 1 is called Putzer algorithm. The application of Lemma 1 is displayed in the numerical example.Definition 1 (see [30]).
Ifh∈ℛ and f:T⟶ℝ is the right-dense continuous function, then the dynamic equation(15)yΔt=htyt+ft,is called regressive.Definition 2 (see [30]).
Leth:T⟶ℝ be the right-dense continuous function. Then, the circle minus of ht for all t∈Tk is denoted by(16)Θht≔−ht1+μtht,where 1+μtht≠0, h∈ℛ is the set of all regressive.Lemma 2 (see [21]).
LetYt be an n×1 vector function on isolated discrete time domains and satisfy(17)Yt=AEtYtσ+Ft,Zt,where A is the n×n nonsingular matrix and F is the n×1 vector function. Then, the solution of equation (17) is(18)Yt=eI−AA−11/μt,0Mt−eI−AA−11/μt,0∫0TeΘI−AA−11/μs,01μsFs,ZsΔs,where t∈T and Mt is a n×n arbitrary matrix martingale on T and meets the martingale property EtMσt=Mt.For more information of dynamic equations on isolated discrete time domains, see [30].


## 3. Nonlinear Stochastic Dynamic Model with a Bivariate Utility Function

We assume that the isolated discrete time domainsT satisfy sup T=∞. Let T∩0,∞=0,∞. Then, the nonlinear stochastic dynamic model (NSDM) with a bivariate utility function on T is given by(19)supci,t,ai,tt=0∞E0∑t∈0,∞Teβi−1/μt,0Uci,t,c¯tμts.t.ci,t=fai,t,ai,σt,yi,σt,ri,σt,c¯t=fc¯t,c¯σt,y¯σt,r¯σt,ai,σ0,a¯σ0,ci,0,c¯0∈X,are given,where 0<βi<1, ai,t i=1,2,…,n, and a¯σt are the state variables, σ:T⟶T is the forward jump operator, yi,t i=1,2,…,n, y¯σt, ri,t i=1,2,…,n, and r¯σt are the random state variables, E0 represents the mathematical expectation, U:ℝ2⟶ℝ+ is a strictly increasing, concave, continuous, and differentiable utility function, X=κ×κ, where κ=xtt=0∞:xt≥0,t∈0,∞, and fx,y,z,u:ℝ4⟶ℝ+ is a concave in x,y, continuous, and differentiable real-valued function with fy≤0. The discount factor relating to NSDM (19) is eβi−1/μt,0. When T=ℤ, eβi−1/μt,0=βit, where βit is a standard discount factor in the case of uniform time domain.We mention that the concavity inx,y means concavity in two variables x and y jointly, not in each variable separately. The concavity of fx,y,z,u is used to find the optimal solution of NSDM (19), namely, function fx,y,z,u is concave in x,y if and only if for any pair of distinct points x,y,z,u and x¯,y¯,z¯,u¯ in the domain, we have(20)fx,y,z,u−fx∗,y∗,z,u≤fxx∗,y∗,z,ux−x∗+fxx∗,y∗,z,uy−y∗,(21)fx¯,y¯,z¯,u¯−fx∗¯,y∗¯,z¯,u¯≤fxx∗¯,y∗¯,z¯,u¯x¯−x∗¯+fyx∗¯,y∗¯,z¯,u¯y¯−y∗¯.To find optimal sequenceci,t,ai,tt=0∞ to maximize the expected utility in NSDM (19), the Euler equation and transversality condition for NSDM (19) need to be given.An Euler equation is the first-order condition for describing optimal choice of dynamic stochastic problems. There are several methods for the analysis and definition of Euler equation, namely, calculus of variations, optimal control theory, the Lagrangian method, and dynamic programming [31, 32]. We assume that NSDM (19) reaches the upper bound at ci,t,ai,tt=0∞. The Euler equation for NSDM (19) is defined as(22)μtUci,t∗′fyai,t∗,ai,σt∗,yi,σt,ri,σt+EtβiμσtUci,σt∗′fxai,σt∗,ai,σt+1∗,yi,σt+1,ri,σt+1=0,μtUc¯t∗′fya¯t∗,a¯σt∗,y¯σt,r¯σt+EtβiμσtUc¯σt∗′fxa¯σt∗,a¯σt+1∗,y¯σt+1,r¯σt+1=0,where Et represents the conditional expectation on the available information at time t, and the information is included in It which contains the observations on ci,t, yi,t, ri,t, and their past values; fx and fy are partial derivatives of function f which is given in NSDM (19) with respect to the first and second variables, respectively.The transversality condition and Euler equation are often used to describe the optimal paths of dynamic economic models. There are two main explanations for the transversality condition: (1) when the marginal utility is greater than 0, agents have the possibility of improving welfare near the end of life, and wealth is completely used for consumption. (2) When the marginal utility is equal to 0, the welfare of the agents cannot be improved. Usually, when the end point is fixed, there are several paths satisfying the Euler equation. The purpose of the transversality condition is to select the optimal path from the paths satisfying the Euler equation.The transversality condition for NSDM (19) is given by(23)limT⟶∞E0eβi−1/μT,0μTUci,T∗′fyai,T∗,ai,σT∗,yi,σT,ri,σTai,σT∗=0,limT⟶∞E0eβi−1/μT,0μTUa¯T∗′fya¯T∗,a¯σT∗,y¯σT,r¯σTa¯σT∗=0.From Euler equation (22) and transversality condition (23), the optimal solution of NSDM (19) is found. In other words, Euler equation (22) and transversality condition (23) are sufficient for the existence of the optimal solution of NSDM (19).Theorem 1.
If sequenceci,t∗,ai,t∗t=0∞ meets systems (22) and (23), then it is the optimal sequence for NSDM (19).Proof.
We assume that sequenceci,t∗,ai,t∗t=0∞ meets systems (22) and (23). To prove Theorem 1, it is sufficient to prove that the difference between the values of the objective function of NSDM (19) at ci,t∗,ai,t∗t=0∞ and ci,t,ai,tt=0∞ is nonnegative, namely,(24)NSDM∗−NSDM=E0limT⟶∞∑t∈0,σTTeβi−1/μt,0μtUci,t∗,c¯t∗−Uci,t,c¯t≥0,where NSDM∗ and NSDM are the values of the objective function of NSDM (19) at ci,t∗,ai,t∗t=0∞ and ci,t,ai,tt=0∞, respectively.
Using the Dominated Convergence Theorem [33], we get(25)NSDM∗−NSDM=limT⟶∞E0∑t∈0,σTTeβi−1/μt,0μtUci,t∗,c¯t∗−Uci,t,c¯t≥limT⟶∞E0∑t∈0,σTTeβi−1/μt,0μtUci,t∗′ci,t∗−ci,t+Uc¯t∗′c¯t∗−c¯t.
From equations (20) and (21), we obtain(26)ci,t∗−ci,t≥fxai,t∗,ai,σt∗,yi,σt,ri,σtai,t∗−ai,t+fyai,t∗,ai,σt∗,yi,σt,ri,σtai,σt∗−ai,σt,(27)c¯t∗−c¯t≥fxa¯t∗,a¯σt∗,y¯σt,r¯σta¯t∗−a¯t+fya¯t∗,a¯σt∗,y¯σt,r¯σta¯σt∗−a¯σt.
Substituting equations (26) and (27) into inequality (25), we have(28)NSDM∗−NSDM≥limT⟶∞E0∑t∈0,σTTeβi−1/μt,0μtUci,t∗′fxai,t∗,ai,σt∗,yi,σt,ri,σtai,t∗−ai,t+fyai,t∗,ai,σt∗,yi,σt,ri,σtai,σt∗−ai,σt+limT⟶∞E0∑t∈0,σTTeβi−1/μt,0μtUc¯t∗′fxa¯t∗,a¯σt∗,y¯σt,r¯σt×a¯t∗−a¯t+fya¯t∗,a¯σt∗,y¯σt,r¯σta¯σt∗−a¯σt=I.
Utilizing the concavity assumption on the functionfx,y,z,u, we obtain(29)I=limT⟶∞E0eβi−1/μ0,0μ0Uci,0∗′fxai,0∗,ai,σ0∗,yi,σ0,ri,σ0×ai,0∗−ai,0+eβi−1/μ0,0μ0Uc¯0∗′fxa¯0∗,a¯σ0∗,y¯σ0,r¯σ0×a¯0∗−a¯0+eβi−1/μσ0,0μσ0Uci,σ0∗′fxai,σ0∗,ai,σ1∗,yi,σ1,ri,σ1×ai,σ0∗−ai,σ0+eβi−1/μσ0,0μσ0Uc¯σ0∗′×fxa¯σ0∗,a¯σ1∗,y¯σ1,r¯σ1a¯σ0∗−a¯σ0+⋯+eβi−1/μT,0μTUci,T∗′fxai,T∗,ai,σT∗,yi,σT,ri,σTai,T∗−ai,T+eβi−1/μT,0μTUc¯t∗′fxa¯T∗,a¯σT∗,y¯σT,r¯σTa¯T∗−a¯T+limT⟶∞E0∑t∈0,σTTeβi−1/μt,0μtUci,t∗′fyai,t∗,ai,σt∗,yi,σt,ri,σt×ai,σt∗−ai,σt+Uc¯t∗′fya¯t∗,a¯σt∗,y¯σt,r¯σta¯σt∗−a¯σt.
Sinceai,0=ai,0∗ and a¯0=a¯∗0 at the initial state, equation (29) becomes(30)I=limT⟶∞eβi−1/μσ0,0μσ0Uci,σ0∗′fxai,σ0∗,ai,σ1∗,yi,σ1,ri,σ1×ai,σ0∗−ai,σ0+eβi−1/μσ0,0μσ0Uc¯σ0∗′×fxa¯σ0∗,a¯σ1∗,y¯i,σ1,r¯i,σ1a¯σ0∗−a¯σ0+⋯+eβi−1/μT,0μTUci,T∗′fxai,T∗,ai,σT∗,yi,σT,ri,σTai,T∗−ai,T+eβi−1/μT,0μTUc¯T∗′fxa¯T∗,a¯σT∗,y¯i,σT,r¯i,σTa¯T∗−a¯T+limT⟶∞E0∑t∈0,σTTeβi−1/μt,0μtUci,t∗′fyai,t∗,ai,σt∗,yi,σt,ri,σt×ai,σt∗−ai,σt+Uc¯T∗′fya¯t∗,a¯σt∗,y¯i,σt,r¯i,σta¯σt∗−a¯σt.
From the recurrence relation of the exponential function, we obtaineβi−1/μσt,0=βieβi−1/μt,0. Consequently, equation (30) is turned into(31)I=limT⟶∞E0∑t∈0,TTβieβi−1/μt,0μσtUci,σt∗′×fxai,σt∗,ai,σt+1∗,yi,σt+1,ri,σt+1ai,σt∗−ai,σt+Uc¯σt∗′fxa¯σt∗,a¯σt+1∗,y¯σt+1,r¯σt+1a¯σt∗−a¯σt+limT⟶∞E0∑t∈0,σTTeβi−1/μt,0μtUci,t∗′fya¯t∗,a¯σt∗,y¯σt,r¯σt×a¯σt∗−a¯σt+Uc¯t∗′fya¯t∗,a¯σt∗,y¯σt,r¯σta¯σt∗−a¯σtorI=limT⟶∞E0∑t∈0,TTeβi−1/μt,0βiμσtUci,σt∗′×fxai,σt∗,ai,σt+1∗,yi,σt+1,ri,σt+1ai,σt∗−ai,σt+μtUci,t∗′fyai,t∗,ai,σt∗,yi,σt,ri,σtai,σt∗−ai,σt+limT⟶∞E0∑t∈0,TTeβi−1/μt,0βiμσtUc¯σt∗′×fxa¯σt∗,a¯σt+1∗y¯σt+1,r¯σt+1a¯σt∗−a¯σt+μtUc¯t∗′fya¯σt∗,a¯σt+1∗,y¯σt+1,r¯σt+1a¯σt∗−a¯σt+limT⟶∞E0eβi−1/μT,0μTUci,T∗′fyai,T∗,ai,σT∗,yi,σT,ri,σT×ai,σT∗−ai,σT+eβi−1/μT,0μTUc¯T∗′×fya¯T∗,a¯σT∗,y¯σT,r¯σTa¯σT∗−a¯σT.
From system (22), we get(32)I=0+limT⟶∞E0eβi−1/μT,0μT×Uci,T∗′fyai,T∗,ai,σT∗,yi,σT,ri,σTai,σT∗−ai,σT+Uc¯T∗′fya¯T∗,a¯σT∗,y¯σT,r¯σTa¯σT∗−a¯σT.
Employing system (23) and fy≤0, we obtain(33)I≥limT⟶∞E0eβi−1/μT,0μT×Uci,T∗′fyai,T∗,ai,σT∗,yi,σT,ri,σTai,σT∗−ai,σT+Uc¯T∗′fya¯T∗,a¯σT∗,y¯σT,r¯σTa¯σT∗−a¯σT=0.
Thus,NSDM∗−NSDM≥0. This proof is completed.


## 4. The Wealth Distribution Model with the HARA Utility Function

Inspired by the works in [17, 21], the wealth distribution model with the HARA utility function is given by(34)maxci,t,ai,tt=0∞E0∑t=0∞βitci,t−γic¯t1−δi−11−δis.t.ai,t=1+rtλai,t−1−ci,t−1−1−λai,t−1−ci,t−1+yi,t,a¯t=1+rtλa¯t−1−c¯t−1−1−λa¯t−1−c¯t−1+y¯t,logyi,t=1−ϕilogyi,0+ϕilogyi,t−1+εi,t,logy¯t=1−ϕ¯logy¯0+ϕ¯logy¯t−1+ε¯t,logrt=φlogr0+1−φlogrt−1+ηt,εi,t,ε¯t,ηt,i.i.d.N0;σ2,ai,0,a¯0,ci,0,c¯0,yi,0,y¯0,r0,are given,where E0 is the conditional expectation, ci,t i=1,2,…,n denotes the consumption of agent i at time t, c¯t represents the average consumption of agents in the economy at time t, 0<φ<1, 0<ϕ¯<1, 0<ϕi<1, 0<γi<1 i=1,2,…,n, all agents’ consumption in the economy satisfies ci,t>γic¯t by limiting the value of γi, 0<λ<1 is the saving rate of agent i, ai,t i=1,2,…,n denotes the wealth of agent i at time t, a¯t is the average wealth of agents at time t, yi,t i=1,2,…,n represents the earnings of agent i at time t, y¯t is the average income of agents at time t, rt denotes the rate of return of agents on wealth process at time t, all agents are assumed to have the same rate of return at the same time, and the discount factor 0<βi<1 i=1,2,…,n and the relative risk aversion parameter 0<δi<1 i=1,2,…,n are different to each agent i and evolve stochastically.In model (34), the wealth of agent i in time t is divided into two parts. Part of the wealth comes from the income that the agent uses part of the remaining wealth in time t−1 for savings, that is, 1+rtλai,t−1−ci,t−1, where ai,t−1−ci,t−1 is the remaining wealth of the previous. The second part of the wealth comes from market transaction, namely, yi,t−1−λai,t−1−ci,t−1, where yi,t is the wealth obtained by agent i in the transaction, and 1−λai,t−1−ci,t−1 indicates that agent i uses part of the remaining wealth in time t−1 for trading.In model (34), agents are allowed to use part of the remaining wealth at time t−1 for trading and part for savings. Agents adopt specific trading strategies that determine whether the agent gains or loses in wealth exchange to maximize their utility. In model (34), yi,t−1−λai,t−1−ci,t−1>0 i=1,2,…,n denotes the net profits of agent i from wealth exchange, and yi,t−1−λai,t−1−ci,t−1<0 i=1,2,…,n represents the amount of wealth lost by agent i in the process of wealth exchange. When yi,t−1−λai,t−1−ci,t−1=0 i=1,2,…,n in model (34), agent i does not make profits in the process of wealth exchange.The utility function of model (34) is(35)Uci,t,c¯t=ci,t−γic¯t1−δi−11−δi,which describes the preference of agent i for risk. Equation (35) is so-called the HARA utility function which captures the idea of “keeping up with the Joneses,” namely, the consumption standards of agents are affected by average consumption level. When time t is fixed, c¯t becomes a constant parameter c¯ which plays an important role in determining agent’s preferences.(i)
Whenc¯>0, utility function (35) describes an agent with decreasing relative risk aversion (DRRA) preference, and c¯ is interpreted as the minimum standard of living for consumption. In utility function (35), when ci,t⟶c¯, the marginal utility tends to infinity, and the optimal consumption level will never reach c¯t=c¯ and always satisfies ci,t>c¯ (see Achury et al. [26]).(ii)
Whenc¯<0, utility function (35) characterizes an agent with increasing relative risk aversion (IRRA) preference. c¯ in utility function (35) indicates that agents become more conservative as their consumption increases. However, c¯ is no longer interpreted as the agent’s minimum consumption level. Levaggi and Menoncin [27] introduce the increasing relative risk aversion preference.(iii)
Whenc¯=0, utility function (35) describes an agent with constant relative risk aversion preference which is the CRRA utility function.We set the isolated discrete time domainsT to satisfy supT=∞. Let T∩0,∞=0,∞. From model (34), the wealth distribution model on isolated discrete time domains T has the following form:(36)supci,t,ai,tt=0∞E0∑t∈0,∞Teβi−1/μt,0ci,t−γic¯t1−δi−11−δiμt,s.t.ci,t=ai,t−ai,σt2λ+λrσt−1+yi,σt2λ+λrσt−1,c¯t=a¯t−a¯σt2λ+λrσt−1+y¯σt2λ+λrσt−1,logyi,t=1−ϕilogyi,0+ϕilogyi,pt+εi,t,logy¯t=1−ϕ¯logy¯0+ϕ¯logy¯pt+ε¯t,logrt=φlogr0+1−φlogrpt+ηt,εi,t,ε¯t,ηt∼i.i.d.N0,σ2,ai,0,ai,σ0,ci,0,c¯0,yi,0,y¯0,r0are given,where δi≠1, 2λ+λrσt−1≠0 i=1,2,…,n, σ:T⟶T is the forward jump operator, and ai,t and ai,σt are values of ai on two consecutive points on the isolated discrete time domains. p:T⟶T is the backward jump operator.Remark 1.
WhenT=ℤ, wealth distribution model (36) becomes model (34). Consequently, model (36) is a generalization of model (34).Remark 2.
WhenT=ℤ, λ=1, and yi,t=0, i.e., wealth exchange is not considered, model (36) is rewritten as(37)maxci,t,ai,tt=0∞E0∑t=0∞βitci,t1−σi−11−σis.t.ai,t=1+rtλai,t−1−ci,t−1ci,t−1<ai,t−1,where 1+rt is equivalent to the agent’s idiosyncratic rate of return on wealth process. This model is consistent with that in [10], and the stationary distribution for ai,t which satisfies the wealth accumulation equation in model (34) has a fat tail. This conclusion is proved by Benhabib et al. in [10].
To analyze the distribution of wealth among agents, the method in [21] is employed to find the optimal solution of model (36). The steps to find the optimal solution of model (36) are as follows: (I) constructing the Euler equation and transversality condition for model (36); (II) finding the initial state of the constraint conditions in model (36); (III) log-linearizing the constraints and the Euler equation in Step I; (IV) expressing the corresponding recursive equations by matrix operators; and (V) finding the optimal solution of model (36) by using the method of solving the rational expectation model on isolated discrete time domains.Step I.
Construction of the Euler equation and transversality condition.
From system (37), the Euler equation of model (36) is given by(38)Etμtci,t−γic¯t−δi−12λ+λrσt−1+βiμσtci,σt−γic¯σt−δi=0,Etμtci,t−γ¯c¯t−δγ¯2λ+λrσt−1−βμσtci,σt−γ¯c¯σt−δγ¯=0,where δ is a relative risk aversion parameter in the mean sense, β is the average discount factor, and γ¯ is the average of γi i=1,2,…,n.
Consequently, the first-order condition of model (36) is(39)Etβiμσtμtci,t−γic¯tci,σt−γic¯σtδikσt=1,Etβμσtμtci,t−γ¯c¯tci,σt−γ¯c¯σtδkσt=1,where kt=2λ+λrt−1, ci,σt−γic¯σt≠0, ci,σt−γ¯c¯σt≠0 i=1,2,…,n, and μt≠0.
Employing system (38), the transversality condition of model (36) becomes(40)limT⟶∞E0eβi−1/μT,0μTci,T−γic¯T−δi−1kσTai,σT=0,limT⟶∞E0eβi−1/μT,0μTci,T−γ¯c¯T−δγ¯kσta¯σT=0.Step II.
Expression of the initial state.
To seek out the initial state, the constraints of model (36) are rewritten as follows:(41a)ci,t=ai,t−12λ+λrσt−1ai,σt+12λ+λrσt−1yi,σt,(41b)c¯t=a¯t−12λ+λrσt−1a¯σt+12λ+λrσt−1y¯σt,(41c)kt=2λ+λrt−1,(41d)logyi,t=1−ϕilogyi,0+ϕilogyi,pt+εi,t,(41e)logy¯t=1−ϕ¯logy¯0+ϕ¯logy¯pt+ε¯t,(41f)logrt=φlogr0+1−φlogrpt+ηt.
From system (39) and equations (41a)–(41f), the initial states of model (36) are(42a)ci,0=ai,0−12λ+λrσ0−1ai,σ0+12λ+λrσ0−1yi,σ0,(42b)c¯0=a¯0−12λ+λrσ0−1a¯σ0+12λ+λrσ0−1y¯σ0,(42c)k0=2λ+λr0−1,(42d)E0βiμσ0μ0ci,0−γic¯0ci,σ0−γic¯σ0δikσ0=1,(42e)E0βμσ0μ0ci,0−γ¯c¯0ci,σ0−γ¯c¯σ0δkσ0=1,(42f)logyi,0=1−ϕilogyi,0+ϕilogyi,p0+εi,0,(42g)logy¯0=1−ϕ¯logy¯0+ϕ¯logy¯p0+ε¯0,(42h)logrt=φlogr0+1−φlogrp0+η0,where 2λ+λrσ0−1≠0 i=1,2,…,n, ci,σ0−γic¯σ0≠0 i=1,2,…,n, ci,σ0−γ¯c¯σ0≠0, and μ0≠0.Step III.
Log-linearization of the constraints and the Euler equation.
Letx∧t=xt−x0/x0. To log-linearize equation (41a), our idea is to linearize each part of the equation and then sum it up. For ci,t, we have(43)ci,t≈ci,01+c^i,t.
Through a series of algebraic calculations, we get (see AppendixA)(44)c^i,t≈ai,0ci,0a^i,t−ai,σ0ci,0kσ0a^i,σt+yi,σ0ci,0kσ0y^i,σt+λrσ0ai,σ0kσ02ci,0r^σt−λrσ0yi,σ0kσ02ci,0r^σt,where ci,0≠0 i=1,2,…,n and kσ0≠0.
Similarly, we log-linearize equation (41b) and obtain(45)c¯^t≈a¯0c¯0a¯^t−a¯σ0kσ0c¯0a¯^σt+y¯σ0kσ0c¯0y¯^σt+λ¯σ0rσ0a¯σ0kσ02c¯0r^σt−λ¯σ0rσ0y¯σ0kσ02c¯0r^σt,where c¯0≠0 and kσ0≠0.
We log-linearize two sides of equation (41c) and have(46)kt≈k01+k^t≈2λ+λr0−1+λrt−r0≈k0+λr0r^t.
Dividing two sides of equation (46) by k0, we obtain(47)k^t=λr0k0r^t,where k0≠0.
Log-linearizing system (39), we get (see Appendix B)(48)Etμσt−μt+ci,0δici,0−γic¯0c^i,t−δic¯0γici,0−γic¯0c¯^t−ci,σ0δici,σ0−γic¯σ0c^i,σt−δiγic¯σ0ci,σ0−γic¯σ0c¯^σt+k^σt=0,(49)Etμσt−μt+δci,0ci,0−γ¯c¯0c^i,t−δγ¯c¯0ci,0−γ¯c¯0c¯^t−δci,σ0ci,σ0−γ¯c¯σ0c¯i,σt+δγ¯c¯σ0ci,σ0−γ¯c¯σ0c¯^σt+k^σt=0.
Log-linearizing equation (41d), we obtain the following expression (see Appendix C):(50)y^i,σt≈ϕiy^i,t+εi,σt−εi,σ0,where εi,σ0=0.
Similarly, we log-linearize equation (41e) and get(51)y¯^σt=ϕ¯y¯^t+ε¯σt−ε¯σ0,where ε¯σ0=0.
Taking conditional expectation on both sides of equations (50) and (51) on information set It yields(52)Ety^i,σt≈ϕiy^i,t,(53)Ety¯^σt≈ϕ¯y¯^t.
Log-linearizing equation (41f), we obtain (see Appendix D)(54)r^σt≈1−φr^t+ησt−ησ0,where ησ0=0.
Adding conditional expectation on both sides of equation (54) on information set It, we have(55)Etr^σt≈1−φr^t.Step IV.
Expressing the corresponding recursive equations by matrix operators.
Taking conditional expectation on both sides of equation (44) on information set It, we get(56)ai,σ0ci,0kσ0Eta^i,σt≈ai,0ci,0a^i,t−c^i,t+yi,σ0ci,0kσ0Ety^i,σt+λi,σ0rσ0kσ02ci,0ai,σ0−yi,σ0Etr^σt.
From equations (52) and (55), equation (56) becomes(57)ai,σ0ci,0kσ0Eta^i,σt≈ai,0ci,0a^i,t−c^i,t+yi,σ0ci,0kσ0ϕiy^i,t+λi,σ0rσ0kσ02ci,0ai,σ0−yi,σ01−φr^t.
Likewise, equation (45) is rewritten by(58)a¯σ0c¯0kσ0Eta¯^σt≈a¯0c¯0a¯^t−c¯^t+y¯σ0c¯0kσ0ϕ¯y¯^t+λ¯σ0rσ0kσ02c¯0a¯σ0−y¯σ01−φr^t.
From equations (47), (48), and (55), we have(59)ci,σ0δici,σ0−γic¯σ0Etc^i,σt−δiγic¯σ0ci,σ0−γic¯σ0Etc¯^σt=μσt−μt+ci,0δici,0−γic¯0c^i,t−δic¯0γici,0−γic¯0c¯^t+λi,σ0rσ0kσ01−φr^t.
Calculating equation (49) gives rise to(60)ci,σ0δci,σ0−γ¯c¯σ0Etc^i,σt−δγ¯c¯σ0ci,σ0−γ¯c¯σ0Etc¯^σt=μσt−μt+ci,0δci,0−γ¯c¯0c^i,t−δc¯0γ¯ci,0−γ¯c¯0c¯^t+λrσ0kσ01−φr^t.
Equations (52), (53), (55), and (57)–(60) are rewritten by the matrix form, namely,(61)BEta^i,σta¯^σtc^i,σtc¯^σty^i,σty¯^σtr^σt=Ca^i,ta¯^tc^i,tc¯^ty^i,ty¯^tr^t+00μσt−μtμσt−μt000,where(62)B=ai,σ0ci,0kσ00000000a¯σ0c¯0kσ00000000ci,σ0δici,σ0−γic¯σ0−ci,σ0δiγici,σ0−γic¯σ000000ci,σ0δci,σ0−γ¯c¯σ0−ci,σ0δγ¯ci,σ0−γ¯c¯σ0000000010000000100000001,C=ai,0ci,00−10yi,σ0ci,0kσ00λrσ0kσ02ci,0W1W0a¯0c¯00−10y¯σ0c¯0kσ0λrσ0kσ02c¯0W2W00ci,0δici,0−γic¯0−δic¯0γici,0−γic¯000λrσ0kσ0W00ci,0δci,0−γ¯c¯0−δc¯0γ¯ci,0−γ¯c¯000λrσ0kσ0W0000ϕi0000000ϕ¯0000000W,and ci,0≠0, kσ0≠0, ci,σ0−γic¯σ0≠0, ci,σ0−γ¯c¯σ0≠0, ci,0−γic¯0≠0, ci,0−γ¯c¯0≠0 i=1,2,…,n, c¯0, W=1−φ, W1=ai,σ0−yi,σ0, W2=a¯σ0−y¯σ0, and k¯σ0 are nonzero constants.Step V.
Finding the optimal solution of equation (61).Theorem 2.
LetT be the isolated discrete time domains. If matrix C is nonsingular, then the optimal solution of equation (61) is(63)Jt=eI−DD−11/μt,0Mt−eI−DD−11/μt,0∫0TeΘI−DD−11/μs,01μsFsΔs,where(64)Jt=a^i,ta¯^tc^i,tc¯^ty^i,ty¯^tr^t,D=C−1B,Ft=C−100μσt−μtμσt−μt000,and Mt is a 7×1 arbitrary matrix martingale on T and satisfies the martingale property EtMσt=Mt.Here, we state that the proof of Theorem2 is similar to that of Lemma 2.3 in [21]. We omit its proof.Remark 3.
IfT=ℤ, then solution (63) becomes(65)Jt=D−tMt−D−t∑s=0t−1DsFs.
We mention that equation (65) is the optimal solution of model (34).Remark 4.
Whenc¯t=c¯, c¯t indicates the average consumption level of agents during the whole trading period, and equation (61) becomes(66)B1Eta^i,σtc^i,σty^i,σtr^σt=C1a^i,tc^i,ty^i,tr^t+0μσt−μt00,where(67)C1=ai,0ci,0−1yi,σ0ci,0kσ0λrσ0kσ02ci,0ai,σ0−yi,σ01−φ0δici,σ0ci,σ0−γic¯0λrσ0kσ01−φ00ϕi00001−φ,B1=ai,σ0ci,0kσ00000δici,σ0ci,σ0−γic¯0000100001,and ci,σ0−γic¯≠0. The optimal solution of equation (66) is(68)J¯t=eI−D1D1−11/μt,0M˜t−eI−D1D1−11/μt,0∫0TeΘI−D1D1−11/μs,01μsF1sΔs,where J¯t=a^i,tc^i,ty^i,tr^t, D1=C1−1B1, F1t=C1−10μσt−μt00, and M˜t is a 4×1 arbitrary matrix martingale and meets the martingale property EtM˜σt=M˜t.Remark 5.
Whenc¯=0, utility function (35) becomes the CRRA utility function which describes the agents with constant relative risk aversion preferences. Equation (66) becomes a wealth distribution model with the CRRA utility function.
From solution (63) of model (36), when the initial values ai,σ0, ci,σ0, yi,σ0, λi,σ0, rσ0, y¯σ0, a¯σ0, c¯σ0, and k¯σ0 are fixed, the wealth distribution is only related to the parameters σi, γi, σ, ϕi, φ, γ¯, and ϕ¯, namely, model (36) depicts the influence of agent’s trading behavior (the behavior of agent’s risk aversion and the behavior of forecasting market risk and saving tendency) on wealth distribution.


## 5. A Numerical Example

To explain solution (63), we choose parameter values from [21, 28] to calculate it. We set γi=0.9, λ¯=0.5, δi=0.2, δ=0.5, ci,0=ci,σ0, c¯0=c¯σ0=16, ai,0=ai,σ0=50, a¯0=a¯σ0=30, ki,σ0=0.98, a¯σ0=0.32, λi,σ0=0.9, rσ0=0.2, λ¯=0.6, φ=0.05, ϕi=0.85, ϕ¯=0.7, yi,σ0=20, and y¯σ0=15. Then, the matrices B, C, and D are obtained:(69)B=1.2700000005.8600000000.31−0.11000000.625−0.125000000010000000100000001,C=1.250−100.5100.13301.8750−101.171.04000.3125−0.1125000.174000.625−0.125000.11600000.8500000000.0700000000.95,D=1.016000.8080−0.0080−0.480000.122403.12530.02670.50670−8.91430.7186001.0100−0.0100000.2931000.05000.9500002.442100001.1765000000014.285700000001.0526.The Putzer algorithm on time scales [34] is employed to calculate the matrix eI−DD−11/μt,0. Consequently, solution (63) is rewritten as(70)a^i,t=d11M1t+d12M2t+d13M3t+d14M4t+d15M5t+d16M6t+d17M7t−d13∑t∈0,TTg32μσt−μt−d14∑t∈0,TTg42μσt−μt,a¯^t=d21M1t+d22M2t+d23M3t+d24M4t+d25M5t+d26M6t+d27M7t−d23∑t∈0,TTg32μσt−μt−d24∑t∈0,TTg42μσt−μt,c^i,t=d31M1t+d32M2t+d33M3t+d34M4t+d35M5t+d36M6t+d37M7t−d33∑t∈0,TTg32μσt−μt−d34∑t∈0,TTg42μσt−μt,c¯^t=d41M1t+d42M2t+d43M3t+d44M4t+d45M5t+d46M6t+d47M7t−d13∑t∈0,TTg42μσt−μt−d44∑t∈0,TTg42μσt−μt,y^i,t=d51M1t+d52M2t+d53M3t+d54M4t+d55M5t+d56M6t+d57M7t−d53∑t∈0,TTg42μσt−μt−d54∑t∈0,TTg42μσt−μt,y¯^t=d61M1t+d62M2t+d63M3t+d64M4t+d65M5t+d66M6t+d67M7t−d63∑t∈0,TTg42μσt−μt−d64∑t∈0,TTg42μσt−μt,r^t=d71M1t+d72M2t+d73M3t+d74M4t+d75M5t+d76M6t+d77M7t−d73∑t∈0,TTg42μσt−μt−d74∑t∈0,TTg42μσt−μt,where M1t, M2t, M3t, M4t, M5t, M6t, and M7t are arbitrary martingales and(71)d11=λ1nt,d12=0,d16=0,d13=−66.49λ1nt−3.69×10−3λ2nt−3.83λ4nt−2.77×10−15λ5nt−8.39×10−24λ6nt+6.74×10−18λ7nt+70.33,d14=10.23λ1nt+3.64×10−3λ2nt+3.84λ4nt+9.57×10−17λ5nt−2.50×10−23λ6nt−2.081×10−18λ7nt−14.07,d15=−10.82λ1nt+0.05λ2nt+0.001λ4n4+4.06λ5nt+1.37×10−23λ6nt+1.7×10−17λ7nt+6.701,d17=−253.22λ1nt−0.12λ2nt−220.75λ4nt−13.13λ5nt+3.301×10−7λ6nt+0.031λ7nt+1365.98,d21=0,d22=λ2nt−1.913×10−5λ1nt,d25=0,d23=−0.04λ1nt+0.017λ2nt−0.33λ4nt−1.22×10−13λ5nt+2.11×10−19λ6nt+3.15×10−16λ7nt+0.35,d24=−0.49λ1nt+0.23λ2nt+0.33λ4nt−2.87×10−14λ5nt+4.95×10−20λ6nt+7.43×10−17λ7nt−0.07,d26=13081293.87λ1nt+26451.40λ2nt+4751069.56λ4nt+336.031λ5nt−28057.56λ6nt−9.65×10−10λ7nt−17831521.75,d27=27.45λ1nt+0.63λ2nt+27.55λ4nt+3.51λ5nt−8.83×10−8λ6nt−0.008λ7nt−294.10,d31=0,d32=0,d35=0,d36=0,d33=2.13λ1nt+0.000265λ2nt+0.28λ4nt+2.44×10−17λ5nt−1.58×10−24λ6nt−578λ7nt−1.41,d34=−0.0013λ1nt−0.00026λ2nt−0.28λ4nt+5.02×10−17λ5nt+2.21×10−24λ6nt−1.24×10−19λ7nt+0.28,d37=−5.26λ1nt+0.0044λ2nt+0.022λ4nt−0.64λ5nt+1.62×10−8λ6nt+0.0015λ7nt+49.02,d41=0,d42=0,d45=0,d46=0,d43=0.00694λ1nt+0.00132λ2nt+1.4λ4nt−2.49×10−16λ5nt−1.1×10−23λ6nt+6.15×10−19λ7nt−1.41,d44=2.12λ1nt−0.00132λ2nt−1.4λ4nt+3.24×10−16λ5nt+1.16×10−23λ6nt−7.96×10−19λ7nt+0.281,d47=−111.0λ1nt−0.00503λ2nt−103.0λ4nt−13.5λ5nt+3.4×10−7λ6nt0.0322λ7nt+1133.0,d51=0,d52=0,d53=0,d54=0,d56=0,d57=0,d55=4.62λ1nt−0.0177λ2nt−3.84×10−4λ4nt−1.36λ5nt−2.24,d61=0,d62=0,d63=0,d64=0,d65=0,d67=0,d66=−1.64×107λ1nt−3.31×104λ2nt−5.95×106λ4nt−421.0λ5nt+3.51×104λ6nt+2.23×107,d71=0,d72=0,d73=0,d74=0,d75=0,d76=0,d77=−2.43λ1nt−0.00147λ2nt−5.57λ4nt−0.556λ5nt+1.4×10−8λ6nt+0.00132λ7nt+46.8,g32=1.35×106λant−3.44λbnt+1.56×105λdnt−1177.0λent+2.76×10−4λfnt−3.34×105λgnt−1.21×106,g42=1.35×106λant−3.44λbnt+1.56×105λdnt−1177.0λent+2.76×10−4λfnt−3.34×105λgnt−1.21×106,with the eigenvalues of matrix D−1 and D as λ1=1.016, λ2=3.1253, λ3=1, λ4=0.96, λ5=1.1765, λ6=14.2857, and λ7=1.0526 and λa=0.9843, λb=0.32, λc=1, λd=1.0417, λe=0.85, λf=0.07, and λa=0.95, respectively.From the calculation process of solution (63), the effect of isolated discrete time domains on model (34) is obvious. When T=ℤ, we get(72)ai,t=ai,01+∑i=17d1iMit,ci,t=ci,01+∑i=17d3iMit.It is worth noting that system (72) is a closed form optimal solution for model (34). System (72) explains the distribution of wealth among agents as the trading time interval is uniform. The risk aversion parameter δi depicts the influence of agent’s risk aversion on wealth distribution. ϕ and φ describe the distribution of wealth among agents as the agents predict market risks.


## 6. Conclusion

In this paper, a wealth distribution model on isolated discrete time domains is given to describe the process of wealth distribution. We assume that agents have different degrees of risk aversion. The HARA utility function is employed to characterize the degrees of risk aversion of agents, including decreasing relative risk aversion, increasing relative risk aversion, and constant relative risk aversion. We consider the influence of agent’s expectation and average consumption level on wealth distribution. In addition, we allow agents to adopt certain trading strategies to maximize their utility and improve their wealth status. The Euler equation and transversality condition of the nonlinear stochastic dynamic 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 the rational expectation model on isolated discrete time domains. A numerical example is given to illustrate the advantages of the wealth distribution model on isolated discrete time domains. The main advantage of the wealth distribution model is that, unlike the wealth distribution models in which wealth exchange is assumed to take place at evenly spaced intervals, the intervals between wealth exchange are of any arbitrary length.



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*Source: 4353025-2020-02-11.xml*