A simple and efficient kinetic model for wealth distribution with saving propensity effect: based on lattice gas automaton
aa r X i v : . [ phy s i c s . s o c - ph ] S e p A simple and efficient kinetic model for wealthdistribution with saving propensity effect: based onlattice gas automaton
Lijie Cui a , Chuandong Lin b,c, ∗ a School of Labor Economics, Capital University of Economics and Business, Beijing100070, China b Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University,Zhuhai 519082, China c Key Laboratory for Thermal Science and Power Engineering of Ministry of Education,Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China
Abstract
The dynamics of wealth distribution plays a critical role in the economic market,hence an understanding of its nonequilibrium statistical mechanics is of great im-portance to human society. For this aim, a simple and efficient one-dimensional(1D) lattice gas automaton (LGA) is presented for wealth distribution of agentswith or without saving propensity. The LGA comprises two stages, i.e., randompropagation and economic transaction. During the former phase, an agent eitherremains motionless or travels to one of its neighboring empty sites with a certainprobability. In the subsequent procedure, an economic transaction takes placebetween a pair of neighboring agents randomly. It requires at least 4 neigh-bors to present correct simulation results. The LGA reduces to the simplestmodel with only random economic transaction if all agents are neighbors andno empty sites exist. The 1D-LGA has a higher computational efficiency thanthe 2D-LGA and the famous Chakraborti-Chakrabarti economic model. Finally,the LGA is validated with two benchmarks, i.e., the wealth distributions of in-dividual agents and dual-earner families. With the increasing saving fraction,both the Gini coefficient and Kolkata index (for individual agents or two-earner ∗ Corresponding author
Email address: [email protected] (Chuandong Lin)
Preprint submitted to Elsevier September 14, 2020 amilies) reduce, while the deviation degree (defined to measure the differencebetween the probability distributions with and without saving propensities) in-creases. It is demonstrated that the wealth distribution is changed significantlyby the saving propensity which alleviates wealth inequality.
Keywords:
Lattice gas automaton, Agent-based model, Wealth distribution,Wealth inequality, Saving propensity
1. Introduction
In econophysics, various economic and financial issues can be analyzed andsolved with probabilistic methods of statistical physics [1, 2, 3]. As an openproblem in economics and econophysics, the fundamental dynamics of wealthdistribution has been widely studied due to its key role in human as well asnonhuman society [4, 5, 6]. For the description of wealth distribution, a fa-mous analytical tool is the Boltzmann-Gibbs exponential function [7] (see Eq.(2)), another well-known empirical approach is the Pareto power-law function P ( m ) ∝ m − α in terms of the Pareto index α and individual wealth m [8]. His-torical data indicate that the Boltzmann-Gibbs function is usually reasonablefor the low and middle ranges of wealth distribution [7, 9], while the Paretofunction provides a good fit to the high range [10, 11]. The Gaussian-like distri-bution which is observed for the lower wealth of the population (around 90%)is due to the additive process of lower wealth accumulation [8, 12]. While thepower-law tail for the top (10% or so) is because of the multiplicative way of thewealth in a higher group [8, 12]. Recently, to further explain the datasets for 67countries, Tao et al. presented a theoretical model within the standard frame-work of modern economics and demonstrated that free competition and Rawls’fairness are the underlying mechanisms producing the exponential pattern [9].Besides the aforementioned analytical and empirical studies, numerical re-search provides convenient insight into the wealth distribution with the emer-gence of various versatile computational methods [13, 14, 15, 16, 17, 18]. In 2000,2r˘agulescu and Yakovenko presented both analytical arguments and computa-tional simulations for the exponential distribution that emerges in computersimulations of economic models, and discussed the role of debt and models withbroken time-reversal symmetry for which the Boltzmann-Gibbs law does nothold [13]. In the same year, the Chakraborti-Chakrabarti (CC) economic modelwas proposed for a closed economic system with a fixed number of agents andtotal money, and the saving propensity influence upon the statistical mechan-ics of wealth distribution was studied [14]. In 2006, Bourguignon and Spadaroreviewed microsimulation techniques and their theoretical background as a toolfor the investigation of public policies [15]. In 2014, Pareschi and Toscani ex-tended a nonlinear kinetic equation of Boltzmann type that describes the effectof knowledge on the wealth of agents who interact through binary trades [16].In 2018, an agent-based model was considered to investigate the wealth distri-bution where the interchange was determined with a symmetric zero-sum game[17]. In 2019, Alves and Monteiro modified a spatial evolutionary version of theultimatum game as a toy model suitable for wealth distribution [18].As an effective stochastic methodology, the lattice gas automaton (LGA) isa simple kinetic model that is applicable to the hydrodynamics [19], chemistry[20], electromagnetics [21], thermoacoustics [22], and economics [23, 24], etc.The LGA was pioneered by the Hardy-Pomeau-de Pazzis model [25] and thelater Frisch-Hasslacher-Pomeau model [19]. In 2013, Cerd´a et al. presented atwo-dimensional (2D) LGA for income distribution in a market with charity reg-ulations [23]. Very recently, a modified LGA economic model was developed forthe income distribution under the conditions of the Matthew effect, income taxand charity [24]. In fact, the LGA is based on the mechanism that there is a one-to-infinite mapping between a macroscopic performance and various microscopicdetails, thus the realistic phenomenon can be manifested by a collective group ofartificial particles evolving on lattices in an appropriate way [26, 27]. This ideaalso enlightened the development of other versatile methodologies, such as thelattice Boltzmann method (LBM) [27, 28, 29, 30, 31] and discrete Boltzmannmethod (DBM) [32, 33, 34, 35, 36, 37, 38, 39, 40]. Actually, these mesoscopic3inetic models (including the LGA, LBM, DBM) have attracted great attentiondue to their simple schemes, flexible applications, easy programing, and highparallel computing efficiency, etc.Motivated by previous investigations [14, 23, 24], an effective 1D-LGA isproposed for wealth distribution in an economic society where people have savingpropensities or not. Compared with the 2D-LGA [23, 24], the current model issimpler and faster, and the factor of individual saving propensity is taken intoaccount as well. Moreover, the LGA has a higher computational efficiency thanthe famous CC-model [14]. The rest of the paper is organized as follows. InSec. 2, the LGA economic model is introduced in detail. In Sec. 3, the model isvalidated and then used to study the wealth distribution with saving propensityeffect. Finally, conclusions are drawn in Sec. 4.
2. Lattice gas automaton
In practice, the wealth distribution of agents in a closed free market takesthe form of an exponential Boltzmann-Gibbs function, which is analogous to theenergy distribution in statistical physics [3]. Here, the LGA is constructed todescribe an artificial society where a monetary exchange may occur if two agentsencounter after random movements. Similarly to statistical physics, the agents,wealth and human society are equivalent to the ideal gas molecules, internalenergy and particle system, respectively.Let us consider a simple economic system where the number of agents N a is fixed and the total amount of money M is conserved. An agent A i thatrepresents an individual or a corporation owns money m i , with the subscript i = 1, 2, · · · , N a . Initially, the total money M is divided amongst N a agents,hence each agent possesses the same amount of money m = M/N a . All agentsare randomly located in a circle with sites N c (under the condition N c ≥ N a ),see Fig. 1. The spatial and temporal steps are ∆ x = 1 and ∆ t = 1, respectively.In the evolution of the LGA, there are two key stages, i.e., the randompropagation and economic transaction.4 N c N - c N - D1N4 D1N2 D1N6
Figure 1: Computational domain with N c sites and discrete models (D1N2, D1N4 and D1N6). Stage 1: Propagation
An agent can move to its neighboring empty sites (with probability P m )or keep resting (with probability 1 − P m ) in the stage of random propagation.For simplicity, the left and right neighboring sites are symmetrical, namely,the number of neighbors is even. For example, there are 1 left and 1 rightneighboring sites in the model 1-dimensional-2-neighbor (D1N2), while thereare 2 neighboring positions on each side in D1N4, see Fig. 1. Hence, thereare 2, 4, and 6 neighboring sites for D1N2, D1N4, and D1N6, respectively.Numerical tests show that it needs at least 4 neighbors for the LGA to obtainright simulation results, see Fig. 2. That is to say, the D1N2 model presentsincorrect results while the D1N4 and D1N6 are satisfactory. Stage 2: Transaction
In the phase of economic transaction, two neighboring agents A i and A j trade with probability P t . (For example, an agent may deal with one of 4neighbors in D1N4.) Each agent’s money is always non-negative, namely, nodebt is permitted. Conservation of the total money is obeyed in each exchange,as earlier. An arbitrary pair of agents A i and A j get engaged in an exchange5ith trading volume ∆ m , i.e., m ′ i = m i − ∆ m,m ′ j = m j + ∆ m, (1)where m i and m ′ i are the money amounts of A i before and after the trans-action, and similar to m j and m ′ j for A j . In this work, two types of trademodels are adopted for agents with or without saving propensity [14, 23, 24](see Appendix A for details).Remark: For the case N c > N a , the sequence could be “Propagation +Transaction” or “Transaction + Propagation” in the main loop of the program;For the special case N c = N a , there is no empty site, so all agents remainmotionless and no propagation takes place. In the latter case, the LGA becomesa reduced model with only economic transaction. The LGA is extremely robustand independent of a specific initial condition, see Appendix B.
3. Numerical simulations
In theory [7], for arbitrary and random trades with local money conservationin a market, the wealth distribution approaches the equilibrium Boltzmann-Gibb distribution of statistical mechanics. It is proved that the stationarywealth distribution functions of individual agents and two-earner families inan ideal free market take the Dr˘agulescu-Yakovenko (DY) forms [7], P ( m ) = 1 m exp (cid:18) − mm (cid:19) , (2)and P ( m ) = mm exp (cid:18) − mm (cid:19) , (3)respectively. Despite its simplicity, the theoretical model misses a very naturalingredient for realistic transactions: Almost no economic agent exchanges withthe entire wealth without saving some parts; The saving propensity is a naturaltendency for any normal economic agent. This defect also exists in the trademodel-I without saving propensity, which is equivalent to the trade model-IIwith saving propensity in the case λ = 0.6 a) (b) Figure 2: Wealth distributions of individual agents (on the left axis) and dual-earner fam-ilies (on the right axis), respectively. The symbols denote the simulation results of D1N2-I(squares), D1N2-II (circles), D1N4-I (upper triangles), D1N4-II (lower triangles), D1N6-I (lefttriangles), D1N6-II (right triangles), Reduced-I (diamonds), Reduced-II (pentagons), D2N4-I(hexagons), and D2N4-II (stars). The lines represent the corresponding exact solutions.
In the following subsections, we first consider the above simple case, i.e.,transaction without saving propensity, and compare simulation results to exactsolutions in Eqs. (2) and (3). Next, the transaction with saving propensity istaken into account.
The site number is chosen as N c = 1500 for D1N4 and D1N6, and N c = 600for the reduced model. The other parameters are P m = 0 . P t = 0 . N A = 600, m = 1. Figure 2 illustrates the wealth distributions of individual agents anddual-earner families, respectively. In the legend, for convenience, “I” refers tothe first trade model without saving propensity, and “II” refers to the secondtrade model with saving propensity in the case λ = 0. In Fig. 2 (a), the squaresand circles indicate the simulation results of D1N2 using the first trade model(D1N2-I) and the second one (D1N4-II), respectively. In Fig. 2 (b), the upper,lower, left and right triangles stand for D1N4-I, D1N4-II, D1N6-I, and D1N6-II,respectively. The diamonds and pentagons are for the reduced model (withoutpropagation) using the first and second trade models, respectively. Besides, thetwo-dimensional models, D2N4-I (diamonds) and D2N4-II (stars), are used as7ell [23, 24]. Meanwhile, the solid lines are for the corresponding exact solutionsin Eqs. (2) and (3).It is apparent in Fig. 2 that the market is non-interacting and the result-ing individual wealth distribution takes the equilibrium Boltzmann-Gibb form.Most agents own little wealth, the maximum probable money is zero, and thepopulation becomes lower for a larger fortune. Meanwhile, the density of dual-earner families firstly increases then reduces with increasing wealth, and themaximum is located at m = m . Figure 2 (a) depicts that the simulation re-sults of D1N2-I and D1N2-II have a relatively large departure from the exactsolutions (2) and (3). While all numerical results in Fig. 2 (b) agree well withthe exact solutions (2) and (3). It is confirmed that, except D1N2, both 1D-and 2D-LGA could present the correct simulation results of wealth distributionin an ideal free market. The trade model-I is consistent with the trade model-IIfor λ = 0. In addition, the average money exactly remains constant m duringall simulations, which demonstrates that the money conservation is guaranteedin the LGA.Model Temporal step Interval Relaxation time Computing timeD1N4-I 5 × × × × ×
10 218 sReduced-II 500 1 1 36 sD2N4-I 5 × × Table 1: Computational time (steps) taken by various models
It should be mentioned that the simulated smooth stationary distributionsin Fig. 2 are determined by an average over a sequence of dynamic probabilitiesat set intervals. Table 1 shows the temporal step, interval, relaxation time and8omputing time taken by those models in Fig. 2 (b). From table 1 the followingpoints can be obtained.(i) The number of probability distributions is n d = t s /t b in terms of thetemporal step t s and interval t b . Clearly, there are n d = 500 sets of probabilitydistributions in each simulation. Note that the distributions under considerationare in near equilibrium states after an early relaxation process, during whichthe economic system starts to approach the (near) equilibrium state from aninitial configuration [14, 24].(ii) Numerical tests show that the LGA has a very high computational effi-ciency. For example, to conduct the above simulations, it only takes 208, 332,and 218 seconds (s) for models D1N4-I, D1N6-I, and Reduced-I, respectively.(The computing time has a narrow variation for different runs of the same pro-gram due to its random nature.) Here the computational facility is a personalcomputer with Intel(R) Core(TM) i7-8750H CPU @ 2.20 GHz and RAM 16.0GB.(iii) Model-I takes more relaxation time and computing time than model-II.For example, the relaxation time is 2 × and 12 temporal steps for D1N4-Iand D1N4-II, respectively. The running time needs 208 and 9 s for D1N4-I andD1N4-II, respectively. The reason is that the trade volume in the first trademodel (without saving propensity) is smaller than the mean exchange volumein the second trade model (with saving propensity). Consequently, the lattermodel is more efficient than the former one.(iv) The 1D-models have less relaxation time and computing time than the2D-models. For instance, the relaxation time is 2 × and 3 × temporalsteps for D1N4-I and D2N4-I, respectively. The running time needs 208 and438 s for D1N4-I and D2N4-I, respectively. In other words, the 1D-model has ahigher calculation efficiency than the 2D-model. Additionally, it is obvious thatthe former is simpler than the latter in the program as well.(v) It takes longer computing time per temporal step and shorter relaxationtime for a model with more neighbors. The computing time per temporal stepis t p = t c /t s with the computing time t c and temporal step t s . For example,9he results are t p = 4 . × − s and t p = 6 . × − s for D1N4-I and D1N6-I,respectively. The relaxation time is t r = 2 × and 1 . × temporal stepsfor D1N4-I and D2N6-I, respectively.(vi) It requires less temporal steps and intervals for a model with moreneighbors. The computing time is almost the same for two models with differentneighbors to achieve the (near) equilibrium state. For example, the computingtime within the relaxation process is t r × t p = 8 . . Now, let us consider the wealth distribution under the condition of individualsaving propensity. Figure 3 displays the wealth probability distributions andcumulative wealth shares with various saving fractions from λ = 0 . .
8. Thesymbols indicate the simulation results of the LGA, whose parameters are thesame as those of D1N4-II in Fig. 2 (b). The solid lines denote the correspondingresults of the CC-model [14]. The two sets of numerical results coincide exactlywith each other. It is numerically verified that the saving propensity of agentsis incorporated appropriately within the LGA.In addition, the saving propensity destroys the multiplicative property of thedistributions in Eqs. (2) and (3). As shown in Fig. 3 (a), the wealth distributionchanges from the Boltzmann-Gibb form to the asymmetric Gaussian-like formwith a finite λ introduced. The agents with zero wealth gradually decrease andeven disappear with the increasing λ . A peak of P emerges as λ is large enough.Figure 3 (b) shows that there is a peak of P for any λ . For either P or P ,the peak becomes thinner and higher, and moves rightward for a larger savingfraction. It can be found in Figs. 3 (c) and (d) that the cumulative sharesof wealth rise from 0 to 1 as the cumulative shares of either individual agentsand dual-earner families increase from 0 to 1. With increasing saving fractions,10 ! " ! " $ %&%’ ( ) * + , )*+,-./0121 ! " ! " $ %&%’ ( ) * + , )*+,)- ).) & /)*+,)- ).) &"/)*+,)- ).) &$/)*+,)- ).) &(/)*+,)- ).) &’/))00)- ).) & /))00)- ).) &"/))00)- ).) &$/))00)- ).) &(/))00)- ).) &’/ / (a) (b) ! !" ! ! " ! $ %& ’ ( ) * + $ , ( ) - . ) / ($ + ’()(*+,-./012+3/04506(+*7/+38/305+)-*-/1 ! !" ! ! " ! $ %& ’ ( ) * + $ , ( ) - . ) / ($ + ’()(*+,-./012+3/0450-67-.-7(+*0+8/6,1 (c) (d) Figure 3: Wealth distributions of individual agents (a) and dual-earner families (b), and cumu-lative wealth shares of individual agents (c) and dual-earner families (d) with various savingfractions. The symbols denote the LGA results with λ = 0 . . . . . i = 1 to N a in the LGA, while the pairundertaking the transaction are chosen in an arbitrary and random way fromall N a agents in the CC-model [14]. Therefore, the LGA requires less (mean)times for that all agents have traded.To measure the wealth inequality under saving propensity, we introduce theGini coefficient expressed by G = 12 N a w X N w i =1 X N w j =1 | w i − w j | , (4)which theoretically ranges from 0 (complete equality) to 1 (complete inequality).Specifically, the Gini coefficient G depends on the parameters w = m , N w = N a , w i = m i for individual agents; And G is a function of w = 2 m , N w = N a / w i = m i + m i + N a / for two-earner families.Apart from the Gini coefficient, another important parameter is the Kolkata( k ) index that gives an intuitive measure of wealth inequality [41, 42]. It isdefined as follows: 1 − k fraction of population possess top k fraction of wealthin the society, namely, the cumulative wealth of 1 − k fraction of people exceedthose earned by the rest k fraction of the people [41, 42]. The k index is from0 . K = 2 k − k .Besides, to describe the departure of probability distribution with savingpropensity from that without saving propensity (i.e., the DY form (2) or (3)),12e define the deviation degree as∆ = 12 Z ∞ | f − f eq | dm, (5)which is between 0 (complete overlap) and 1 (no overlap), see Appendix C formore details. In particular, the symbols f and f eq denote the wealth distribu-tions for an arbitrary value of λ and λ = 0, respectively. Namely, f = P ( λ )and f eq = P ( λ = 0) for individual agents; f = P ( λ ) and f eq = P ( λ = 0) fortwo-earner families.Figure 4 (a) plots the Gini coefficients versus saving fractions. The lines withsquares and circles represent the Gini coefficients G for individual agents and G for two-earner families, respectively. It is apparent that the Gini coefficient G for individual agents is not lower than G for dual-earner families. The Ginicoefficients G , G , and their differences are smaller for a larger saving fraction.Compared to the theoretical solutions G = 1 / G = 3 / λ = 0 [7], thecorresponding calculation results G = 0 .
499 and G = 0 .
371 are satisfactory.And the LGA results G = G = 0 coincide well with the exact solutions at thepoint λ = 0 [7]. Moreover, it is interesting to obtain the relationship 3 G ≈ G for all saving fractions.Figure 4 (b) illustrates the Kolkata indices versus saving fractions. The lineswith squares and circles denote the Kolkata indices k and k for individualagents and dual-earner families, respectively. Obviously, the Kolkata index k for individual agents is greater than or equal to k for dual-earner families. TheKolkata indices k , k , and their differences decrease with the increasing savingfraction. In comparison with the analytic solutions k = 0 . k =0 . λ = 0 [7, 41, 42], the corresponding simulation results k = 0 . k = 0 .
633 are satisfying. Meanwhile, the simulation results k = k = 0 . λ = 1 [7, 41, 42]. Additionally,comparison between Figs. 4 (a) and (b) indicates a linear relationship betweenthe Gini coefficients and Kolkata indices, i.e., k = 0 . γ G , k = 0 . γ G ,and γ ≈ γ ≈ .
36, which are close to the results in Ref. [42]. Similar to theGini coefficients, the indices K = 2 k − K = 2 k − !" !"" ! ! " $ % $ & ’ () * + %&’%&( ! !" ! ! " $ % &’ ( ) ! * + !! ’()*+,-./(01*2+ - & - " ! !" ! ! " ! $ % & ’’ ! $ ! & " ( ’( " ’( (a)(b)(c) Figure 4: The Gini coefficients (a), Kolkata indices (b), and deviation degrees (c) versus savingfractions. The lines with squares and circles are for individual agents and two-earner families,respectively. K ≈ K within 0 ≤ λ ≤ that describes the depar-ture of the wealth distribution of individual agents from the DY expression (2);The line with circles is for the deviation degree ∆ which measures the differencebetween the wealth distribution of two-earner families and the DY formula (3).The calculation results ∆ = 0 .
004 and ∆ = 0 .
009 for λ = 0 are satisfactory bycomparison with the corresponding theoretical solutions ∆ = ∆ = 0. Mean-while, the numerical results ∆ = 0 . = 0 . = ∆ = 1 at the point λ = 1. Remarkably, with the increasingsaving fraction λ , the deviation degrees ∆ and ∆ increase. That is to say, thewealth inequality is alleviated by the saving propensity, and the wealth distri-bution is affected by the human factor. Additionally, it is interesting to find therelation ∆ ≈ ∆ (with only slight differences) between them within the wholerange of λ .
4. Conclusion
We proposed a quite simple, robust and effective kinetic method, 1D-LGA,for the wealth distribution in a closed economic market where the amount ofmoney and the number of agents are fixed. Analogously to statistical physics,the agents, wealth and human society are equivalent to the ideal gas molecules,internal energy and particle system, respectively. The LGA includes two keystages, i.e., random propagation and economic transaction. During the propa-gation stage, an agent either remains motionless or travels to one of its neigh-boring empty sites with a certain probability. In the subsequent procedure,an economic transaction takes place randomly when two agents are located inthe neighboring sites. Two types of transaction models are introduced. Oneis model-I for agents without saving propensity [23, 24], the other is model-IIwith saving propensity [14]. The former is equivalent to the latter if the savingfraction is zero. 15umerical tests indicate that to obtain right simulation results requires atleast four neighbors. The LGA reduces to the simplest coarse-grained modelwith only random economic transaction if all agents are neighbors and no emptysites exist. For a model with more neighbors, it takes longer computing timeper temporal step, shorter relaxation time, less temporal steps and intervals.However, the total computing time is almost the same for two models withdifferent neighbors to achieve the (near) equilibrium state. Because there aremore economic transactions and longer running time for a model with moreneighbors during one main loop of the program. And the artificial economicsystem requires approximately the same transaction times to obtain an equilib-rium state. Furthermore, model-I takes more relaxation time and computingtime than model-II, because the trade volume in the former is smaller than themean exchange volume in the latter. Consequently, the latter model is moreefficient than the former one. The 1D-LGA is more efficient and simpler thanthe 2D-LGA [23, 24], and also takes less computing time than the CC-model[14], although all these models have a quite high computational efficiency.Next, the LGA is validated with two benchmarks, i.e., the wealth distribu-tions of individual agents and two-earner families. The LGA is extremely robustand independent of a specific initial condition. It presents the numerical resultsof wealth distributions with various saving propensity factors exactly the sameas the CC-model [14]. To be specific, the wealth distribution changes from theBoltzmann-Gibb form to the asymmetric Gaussian-like form with a finite λ in-troduced. The agents with zero wealth gradually decrease and even disappearwith the increasing λ . A peak of individual wealth distribution emerges as λ is large enough, while there is a peak of wealth distribution of two-earner fam-ilies for any λ . For either of them, the peak becomes thinner and higher, andmoves rightward for a larger saving fraction. It is noteworthy that the LGA hasthe potential to describe the main feature of the wealth distribution in humansociety.Finally, the Gini coefficient and Kolkata index are used to measure wealthinequality under saving propensity. Meanwhile, the deviation degree is defined16o describe the departure of probability distribution with saving propensity fromthat without saving propensity. With the increasing λ , the Gini coefficients G for individual agents and G for two-earner families decrease, the Kolkataindices k for individual agents and k for two-earner families reduce, whilethe deviation degrees ∆ for individual agents and ∆ for two-earner familiesincrease. The transformation K = 2 k − K = 2 k − G ≈ G , 3 K ≈ K , and ∆ ≈ ∆ . TheGini coefficients and Kolkata indices satisfy the linear relations k = 0 . γ G and k = 0 . γ G with γ ≈ γ ≈ .
36, which are similar to the results in Ref.[42]. It is demonstrated that the wealth inequality is alleviated by the savingpropensity, and the wealth distribution is influenced by the human factor.
Acknowledgments
This work is supported by the National Natural Science Foundation of China(NSFC) under Grant No. 51806116.
Appendix A.
Here two types of transaction models are introduced for agents with or with-out saving propensity, respectively. (i) Trade model-I without saving propensity
The trading volume between two agents is fixed as ∆ m = m /N m . Mostof our simulations are for m = 1 and N m = 100. There are three cases ofexchange under consideration [23, 24].Case A: m i = m j = 0. No exchange takes place between two agents withoutpersonal possessions.Case B: m i = 0 and m j = 0. An agent without any wealth can only stayunchanged or win money during an economic exchange.Case C: m i = 0 and m j = 0. An agent either earns or loses money withprobability P t / (ii) Trade model-II with saving propensity ! " ) " ) ! ! " $ %&%’ ( ) * + , *+,-*. /’(($$*01/2 *03/" *456765**** **88 ? & &" &$ &’ ! " $ %&%’ ( ) * - , Figure B.5: Wealth distributions of individual agents (on the left axis) and dual-earner families(on the right axis) for λ = 0 .
4. The symbols denote the simulation results of the LGA fora different m , N a , N c , and an uneven configuration. The lines represent the correspondingCC-model results [14]. Assume that each economic agent saves a fraction λ of its wealth m i beforetrading, where λ is a fixed value between zero and unity. The parameter λ , alsocalled the “marginal propensity to save”, remains fairly constant, independentof economic agents [14]. After the transaction, the wealth of agents A i and A j becomes m ′ i = λm i + ∆ m i ,m ′ j = λm j + ∆ m j , (A.1)in terms of ∆ m i = ǫ (1 − λ ) ( m i + m j ) and ∆ m j = (1 − ǫ ) (1 − λ ) ( m i + m j ),where ǫ represents a random number between zero and unity [14]. Via straight-forward substitution, it can be derived that Eq. (A.1) is equivalent to Eq. (1)for ∆ m = (1 − λ ) [ m i − ε ( m i + m j )]. The random exchange amount is lessthan the total money because of the saving by each agent. Appendix B.
It is worth mentioning that the LGA is extremely robust and independentof a specific initial configuration. For this purpose, Fig. B.5 delineates thewealth distributions of individual agents and dual-earner families. The squaresand lines correspond to the LGA and CC-model results for λ = 0 . ! " ! " $ %&%’ ( ) * + , )*+,-./0121 ! " ! " $ %&%’ ( ) * + , )*+,-./0121 S L S G S O S L S G S O (a) (b) Figure C.6: Sketch of the wealth distributions of individual agents (a) and dual-earner families(b). The solid lines stand for λ = 0, and the dotted lines for λ = 0. respectively. The circles, pentagons, and triangles stand for the LGA resultsfor m = 68844, N a = 900 and N c = 2000, respectively. The stars are for anuneven initial configuration, m i = m + A sin (2 πi/N a ), with i = 1, 2, · · · , N a .Here the perturbation amplitude A = m / Appendix C.
Here we introduce a useful parameter to describe the departure of the wealthdistribution with saving propensity from the one without saving propensity.To give an intuitive description, Fig. C.6 delineates the sketch of the wealthdistributions of individual agents (a) and dual-earner families (b). The solidlines stand for the case without saving propensity, and the dotted lines forthe other case. The area below each line equals one because of the followingformulas, S f = Z ∞ f dm = 1 , (C.1)19 f eq = Z ∞ f eq dm = 1 . (C.2)The overlap between the two regions S f and S f eq is S O , and the areas are S L and S G for f < f eq and f > f eq , respectively. To be specific, S L = Z ∞ ( f eq − f ) dm (cid:12)(cid:12)(cid:12)(cid:12) f
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