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Power Law Distribution of Wealth in a Money-Based Model Yan-Bo Xie, Bo Hu, Tao Zhou and Bing-Hong Wang" Department of Modern Physics and The Nonlinear Science Center, University of Science and Technology of China, Hegel Anhui, 230026, PR China (Dated: February 2, 2008) 8 0 A money-based model for the power law distribution (PLD) of wealth in an economically inter- acting population is introduced. The basic feature of our model is concentrating on the capital movements and avoiding the complexity of micro behaviors of individuals. It is proposed as an extension of the Equfluz and Zimmennsum's (EZ) model for crowding and information transmission in financial markets. Still, we must emphasize that in EZ model the PLD without exponential correction is obtained only for a particular parameter, while our pattern will give it within a wide range. The Zipf exponent depends on the parameters in a nontrivial way and is exactly calculated cn in this paper. PACS numbers: 89.90.+n, 02.50.Le, 64.60.Cn, 87.10.-1-e 77 a.) I. INTRODUCTION we shall establish a money-based model which is essen- 794 tially an extension of the Eguiluz and Zimmermann's 0 (EZ) model for crowding and information transmission Many real life distributions, including wealth alloca- cd in financial markets126, 27]. The size of a cluster there is tion in individuals, sizes of human settlements, website now identified as the wealth of an agent here. However, E popularity, words ranked by frequency in a random cor- pus of tat, observe the Zipf law. Empirical evidence of analytical results will show that our model is quite dif- ferent from EZ's [27], which gives PLD with an exponen- the Zipf distribution of wealth [I-9] has recently attracted tial cut-off that vanishes only for a particular parameter. a lot of interest of economists and physicists. 'lb under- Here, a Zipf distribution of wealth is obtained within C.) stand the micro mechanism of this challenging problem, a wide range of parameters, and surprisingly, without various models have been proposed. One type of them is exponential correction. The Zipf exponent can be an- based on the so-called multiplicative random process110- alytically calculated and is found to have a nontrivial 21]. In this approach, individual wealth Ls multiplica- dependence on our model parameters. 00 tively updated by a random and independent factor. A This paper is organized as follows: In section 2, the 00 very nice power law is given, however, this approach es- model is described and the corresponding master equa- sentially does not contain interactions among individu- als. which are responsible for the economic structure and tion is provided directly. In section 3, we shall present 0 aggregate behavior. Another pattern takes into account our analytical calculation of the Zipf exponent. Next, 7t. the interaction between two individuals that results in a we give numerical studies for the master equation, which are in excellent agreement with analytic results. In sec- redistribution of their assetsl22-25]. Unfortunately, some tion 5, the relevance of our model to the real world are ct attempts only give Boltzmann-Gibbs distribution of as- sets1241,25], while some othersI23], though exhibiting Zipf discussed. distributions, fail to provide a stationary state. In this paper, we shall introduce a new perspective to understand this problem. Our model is based on the II. THE MODEL 0 following observations: (i) In order to minimize costs and maximize profits, two corporations/economic enti- The money-based model contains N units of money, ties may combine into one. This phenomenon occurs fre- where N is fixed. Though in real economic environment •-1 >< quently in real economic world. Simply fixing attention on capital movements, we can equally say that two cap- the total wealth is quite possible to fluctuate, our as- sumption is not oversimplified but reasonable, given that itals combine into one.(ii) The dignsneiation of an eco- the production and consumption processes are simulta- nomic entity into many small sections or individuals is neous and the resource is finite. The N units of money also commonplace. The bankruptcy of a corporation, for are then allocated to Al agents (or say, economic enti- instance, can be effectively classified into this category. ties), where Al is changeable with the passage of time. Allocating a fraction of assets for the employee's salary, a For simplicity, we may choose the initial state containing company also serves as a good example for the fragmen- just N agents, each with one unit of capital. The state tation of capitals. Under some appropriate assumptions, of system is mainly described by na, which denotes the number of agents with s units of money. The evolution of the system is under following rules: At each time step, a unit of money, instead of an agent, is selected at random. •Eleetronic address: bliwangttuste.edu.cn Notice that our model is much more concentrating on the EFTA01069603  2 capital movement among agents rather than the agents has been used. We must point out that Eq.(1) is almost themselves. With probability aryls, the agent who owns the same as the master equation derived in Ref.)27] for this unit of money is disassociated, here s is the amount the EZ model except for an additional factor -I/s in the of capitals owned by this agent and •-y is a constant which third term on the right hand side of Eq.(1). Notice that implies the relative magnitude of dissociative possibility this term is significant because otherwise the frequency at a macro level. After disassociation, this s units of of the disintegration for large a agents would be too high. money are redistributed to s new agents, each with just Now we introduce h, = sn,IN, which indicates the one unit. It must be illuminated that an real economic ratio of wealth occupied by agents in rank s to the total entity in most cases does not separate in such an equally wealth, and a = ay/2(1 — a), that represents the maxi- minimal way. However, with a point of statistical view mum ratio of the disintegration possibility to the merger and considering analytical facility, this simplified hypoth- probability in the whole economic environment. Then, esis is acceptable for original study. Now, continue with one can give the equations for the stationary state in a our evolution rules. With probability a(1 — Ws), noth- terse form: ing is done. And with probability 1 — a, another unit of money is selected randomly from the wealth pool. If 5-I these two units are occupied by different agents, then the two agents with all their money combine into one; ha - E kits, 2(s ÷ a) r=i. (4) otherwise, nothing occurs. Thus, 1 — a in our model is a factor reflecting the possibility for incorporation at a macro level. and One may find that as a is close to 1 and y is not too a small, a financial oligarch is almost forbidden to emerge hi - (5) in the evolution of the system; but, if the initial state 1 -I- a contains any figure such as Henry Ford or Bill Gates, he is preferentially protected. Note that the bankruptcy According to the definition of h,, it should satisfy the probability of moneybags is inverse proportional to their normalization condition Eq.(3) wealth ranks, and the possibility of being chosen is pro- portional to sus, thus, the Doomsday of a tycoon comes with possibility ans7/N, which is extremely small for large s. Meanwhile, the vast majority, if initially poor, =1 (6) is perpetually in poverty, with no chance to raise the eco- nomic status any way. In addition, if middle class exists When a is less than a critical value a, = 4 which will at first, it will not disappear or expand in the foreseeable be determined numerically in section 4, one can show future. Again, it may be interesting to argue that when that Eqs.(4-5) does not satisfy the normalization condi- a is slightly above zero, the merger process is prevailing tion Eq.(3). This inconsistency implies that when a < and overwhelming, and all the capitals are inclined to the state with one agent who has all the N units of money converge. In this case, though the rich are preferentially becomes important[28, 29]. In this ease, the finite-size ef- protected, the trend in the long run is to annihilate them fect and the fluctuation effect become nontrivial and the until the last. Of course, one-agent game is trivial. Like- master equations (1-3) is no longer applicable to describe wise, it is not appealing to observe the system when if the system[28, 29). In this paper, we shall restrict our goes to 0 and a to 1, since both merger and disintegration discussion to the case a > are nearly impassible-in other words, all the capitals are locked, thus the wealth pool is dead at any time. Following Refs.I27, 28, 29] in the case of N » 1, we give the master equation for n, III. ANALYTIC RESULTS On, 1—a 'Y = —Ern s(s — r)n,_r — 2(1 — a)sn, — ants— at r=1 When a > a c, one can show that h, —) A/s'7 for suffi- (1) ciently large s with for s > 1 and 00 8ni at = —2(1 — a)ni + a E s2ns2 (7) a=2 Er=17.6, = —2(1 — a)ni + ary(N — ) (2) where the identity Notice that this equation is only consistent when > 2 because otherwise the sum r r= 1 rh,. would be divergent, and thus h, -. Ale becomes an inconsistent formula. an, = N (3) a=1 The derivation of Eq.(7) is described as follows: When EFTA01069604  3 s is sufficiently large TABLE I: The results of H for various value of a. s-1 a H S h, 3.0 0.9940886 2(s + a) 44-" r=1 3.5 0.9997818 6 3.6 0.9999214 3.7 0.9999743 s+cr(E ha- rh, + hoO( 326, _,I )) 3.8 0.9999922 r=1 3.9 0.9999977 6 5 dh, 4.0 0.9999995 s er E(h a — tc p)h, 4.1 0.9999999 r=1 4.2 1.0000000 02 dh, 4.3 1.0000000 Ar, (1 — )111,E — —Erh r] + 4.4 1.0000000 1sr= ds r=1 h„O(OO1_11) 4.5 1.0000000 5.0 1.0000000 A-- (1— links dh, rc s E Hid (8) 6.0 1.0000000 r=1 where 6 < 1 but is close to 1, 8(r7-1) > 1 and 26ri-1-1/ > One immediately obtains that 1. Therefore CO dh, ds h, a s Er. 1 .rh, E rh, a— riot 2 (14) r=1 which gives that ass —> oo and the exponent A ha — (9) 9 (15) 1- Vri a The value of E tr l rh, can be further evaluated: Introducing the generating function which is a positive real number for a ≥ 4. Notice that co when a = 4, the exponent rl = 2. This implies that our G(x). ExThr (10) calculation is self-consistent, provided Eq.(6). In sum, we find from the master equation that h, obeys PLD when r=1 s is sufficiently large and a > 4. It may be important one can rewrite Eq.(4) as to point out that when s is small, h, also approximately obeys the PLD, and the restriction a > 4, introduced for x(G' — hi) + a(G — hi x) = + a(G — x) = xG'G the sake of discussing master equation, can be actually relaxed. This argument has been tested by the simulator or investigation, which supplies the gap of analytical tools G'x(G —1). a(G — x) (11) and verifies the analytical outcome. with the initial condition IV. NUMERICAL RESULTS G(0) = 0 (12) Since h„ A/sq as s co, G is only defined in the We have numerically calculated the number interval Ixl < 1. Front Eq.(6), we also have G(1) = 1. 00 What we need to calculate is just H= Eh, ao r=1 G'(1) = Erh, r=1 based on the recursion formula Eq.(4) with the initial condition Eq.(5). Table.1 lists the results of H for vari- Since the left and the right hand sides of Eq.(11) are both ous value of a. From Table.1, one immediately find that zero at x = 1, we differentiate both sides by x and obtain the normalization condition is satisfied for a > = 4, which, again, indicates consistency of related equations. G"x(1 — G)+ G'(1 - G) -x0' 2 = G') Fig.1-2 show h, as a function of s in a log-log scale for Let x 1 and one finds that G"(1— G) vanishes in this a = 10, a = 4.5, respectively. From Fig.1, one can see limit provided rl > 2, thus that h, conforms to PLD for s > 1 with the exponent ri given by Eq.(15). Fig.2 indicates that h., observes the G12(1) — a01(1) +a = 0 (13) Zipf law for nearly all s with,/ = 3.0. EFTA01069605  4 I ' ' ' ' ' ' ' 50 0 4.5_ -10_ a=10 4.0_ 35_ r -30 3.0_ -90 2.5_ -SO 20 i i i i i -1 0 1 2 3 4 5 6 7 4.0 4.6 60 5.5 6.0 In 8 a FIG. 1: The dependence of h, on s in a log-log scale for FIG. 3: The calculated exponent q for different values of a. a = 10. Black squares represent the numerical results of q obtained from by using the extrapolation method, see text. The solid line represents the analytic result Eq.(15). 0. -2- a=8 C 4- A • • 0 • 0 1 2 3 4 6 6 7 0 In S • 6 12 0.0 PP 20 26 FIG. 2: The dependence of ha on s in a log-log scale for a = 4.5. FIG. 4: h, for a = 8 from both numerical calculation and computer simulation. Black stars represent outcome of com- The fitted exponents for various values of a are plotted puter simulation for N = 2.5 x 105, y = 2 and a = 0.88889. in Fig.3. They are given by Total 2 x 106 time steps were run and the final 5 x 105 time steps were used to count nt, statistically. The circles represent the theoretical results derived from Eqs.(4-5). In(h9oo/hiaco) In(1000/900) V. DISCUSSIONS Fig.3 also exhibits the analytic results from Eq.(15). The analytic outcome fits the exponents calculated from re- In this paper, we have introduced a so-called money- cursion quite well for a > 4.2. However, when a -, 4.0, based model to mimic and study the wealth allocation discrepancy is obvious, since the convergence of hz, to the process. We find for a wide range of parameters, the correct power law is then very slow. wealth distribution n, A/0+1 with q given by Eq.(15) We have also performed computer simulation, which for sufficiently large s. The crucial difference between our gives excellent agreement with theoretical results derived model and the EZ model is that the dissociative proba- from Eqs.(4-5) for a = 8 and s ≤ 10, see Fig.4. For more bility I'd of an economic entity, after he/she is picked about our simulator investigation and further analysis for up, is proportional to 1/s in our model. However, the a < 4, see Ref. [30]. corresponding probability in the EZ model is simply pro- EFTA01069606  5 portional to 1. This difference gives rise to divergent capitals is not as restricted by space and time as between behaviors of Its . In the EZ model, n, •-•-• Rir82'5 exp(—as) agents. Therefore, when econophysics is much more in- for large s [27). When ne is interpreted as the number terested in the behaviors of capitals than that of agents, of individuals who own s units of assets, the choice of it is recommendable to adopt such a money-based model. rd o(1/s) is reasonable. Actually, since at the first The methodology to fix our attention on the capital step, we randomly picked up a unit of money, the indi- movements, instead of interactions among individuals, vidual who owns s units of assets is picked up with a will bring a lot of facility for analysis; moreover, using probability proportional to s. According to the obser- such random variables as sy and a to represent the macro vation in real economic life, large companies or rich men level of the micro mechanism also help us find a possi- are often much more robust than small or poor ones when ble bridge between the evolution of the system and the confronting economic impact and fierce competition. If protean behaviors of individuals. Whether the bridge is 0(1), the overall dissociation frequency would be steady or not can only be tested by further investigation. proportional to s which is totally unreasonable. In real economic environment, capitals and agents be- have similarly at some point. For instance, they both ceaselessly display integration and disintegration, driven Acknowledgments by the motivation to maximize profits and efficiency. This mechanism updates the system every time, and gives rise to clusters and herd behaviors. Furthermore, in This work has been partially supported by the State an agent-based model, it is usually indispensable to con- Key Development Programme of Basic Research (973 sider the individual diversity that is all too often hard to Project) of China, the National Natural Science Founda- deal with. When it conies to the money-based model, tion of China under Grant No.70271070 and the Special- this micro complexity may be considerably simplified. ized Research Rind for the Doctoral Program of Higher Finally, the conceptual movement and interaction among Education (SRFDP No.20020358009) [1] G.K.Zipf, Human Behavior and the Principle of Least [16] S. Solomon and M. Levy, Int. J. Mod. Phys. C 7, Effort (Addison-Wesley Press, Cambridge, MA, 1949). 745(1996). [2] V. Pareto, Cours d'Economique Politigue (Macmillan, [17] O. Malcai, O. Biham and S. Solomon, Phys. Rev. E 60, Paris, 1897), Vol 2. 1299(1999). [3] B. Mandelbrot, Economietrica 29,517(1961). [18] B.B. Mandelbrot, Int. Economic Rev. 1, 79(1960). [4] B.B. Mandelbrot, Comptes Rendus 232, 1638(1951). [19] E.W. Montroll and M.F. Shlesinger, Noneguilibrium Phe- [5] B.B. Mandelbrot, J. Business 36, 394(1963). nomena 11. Prom Stochastics to Hydrodynamics, edited [6] A.B. Atkinson and A.J. Harrison, Distribution of Total by J.L. Lebowitz, E.W. Montroll(North-Holland, Ams- Wealth in Britain (Cambridge University Press, Cam- terdtun, 1984). bridge, 1978). [20] D. Sornette and R. Cont, J. Phys. I Fiance 7, 431(1997). [7] H. Takayasu, A.-H. Sato and M. Takayasu, [21] H. Takayasu and K. Okuysuna, Fractals 6(1998). Phys.Rev.Lett. 79,966(1997). [22] Z.A. Melzak, Mathematical Ideas, Modeling and Applica- [8] P.W. Anderson, in The Economy as an Evolving Complex tions, Volume 11 of Companion to Concrete Mathematics System II, edited by W.B. Arthur, S.N. Durlauf and D.A. (Wiley, New York, 1976), p279. Lane (Addison-Wesley, Reading, MA,1997). [23] S. Ispolatov, P.L. Krapivsky and S. Redner, Euro. Phys. [9] The Theory of Income and Wealth Distribution, edited J. B 2, 267(1998). by Y.S. Brenner et aL,(New York, St. Martin's Press, [24] A. Dragulescu and V.M. Yakovenlco, Euro. Phys. J. B 1988). 17, 723(2000). [10] H.A. Simon and C.P. l3onini, Am. Econ. Rev. [25] C.B. Yang, to be published in Chinese Phys. Lett. 48,607(1958). [26] V.M. Egu0uz and M.G. Zimmermann, Phys. Rev. Lett. [11] U.G. Yule, Philos. nen. R. Soc. London, Ser. 85,5659(2000). Bc213,21(1924). [27] R. D'Hulst and G.J. Rodgers, Euro. Phys. J. B [12] D.G. Champernowne, Econometrica 63, 318(1953). 20,619(2001). [13] H. Kesten, Acta Math. 131,207(1973). [28] Y.B. Xie, B.H. Wang, H.J. Quan, W.S. Yang and P.M. [14] S. Solomon, in Annual Reviews of Computational Physics Hui, Phys. Rev. E 65, 046130(2002). II, edited by D. Stauffer (World Scientific, Singapore, [29] Y.B. Xie, B.H. Wang, H.J. Quan, W.S. Yang and W.N. 1995), p.243. Wang, Acta Physica Sinica (Chinese), 52,2399(2003). [15] M. Levy and S. Solomon, Int. J. Mod. Phys. C 7, [30] B. Hu, Y.B. Xie, T. Zhou and B.H. Wang (unpublished). 595(1996). EFTA01069607