PHYSICAL REVIEW E 72, 026126 共2005兲

Master equation for a kinetic model of a trading market and its analytic solution 1

Arnab Chatterjee,1,* Bikas K. Chakrabarti,1,2,† and Robin B. Stinchcombe2,1,‡

Theoretical Condensed Matter Physics Division and Centre for Applied Mathematics and Computational Science, Saha Institute of Nuclear Physics, Block-AF, Sector-I Bidhannagar, Kolkata-700064, India 2 Rudolf Peierls Centre for Theoretical Physics, Oxford University, 1 Keble Road, Oxford, OX1 3NP, United Kingdom 共Received 18 January 2005; published 22 August 2005兲 We analyze an ideal-gas-like model of a trading market with quenched random saving factors for its agents and show that the steady state income 共m兲 distribution P共m兲 in the model has a power law tail with Pareto index ␯ exactly equal to unity, confirming the earlier numerical studies on this model. The analysis starts with the development of a master equation for the time development of P共m兲. Precise solutions are then obtained in some special cases. DOI: 10.1103/PhysRevE.72.026126

PACS number共s兲: 89.90.⫹n, 05.20.Dd, 02.50.⫺r, 87.23.Ge

I. INTRODUCTION

The distribution of wealth among individuals in an economy has been an important area of research in economics for more than a hundred years. Pareto 关1兴 first quantified the high-end of the income distribution in a society and found it to follow a power law P共m兲 ⬃ m−共1+␯兲, where P gives the normalized number of people with income m, and the exponent ␯, called the Pareto index, was found to have a value between 1 and 3. Considerable investigations with real data during the last ten years revealed that the tail of the income distribution indeed follows the above mentioned behavior and the value of the Pareto index ␯ is generally seen to vary between 1 and 2.5 关2–4兴. It is also known that typically less than 10% of the population in any country possesses about 40% of the total wealth of that country and they follow the above law. The rest of the low income population, in fact, the majority 共90% or more兲, follow a different distribution which is debated to be either Gibbs 关3,5兴 or log-normal 关4兴. Much work has been done recently on models of markets, where economic 共trading兲 activity is analogous to some scattering process 关5–11兴. We put our attention to models where introducing a saving factor for the agents, a wealth distribution similar to that in the real economy can be obtained 关6,7兴. Savings do play an important role in determining the nature of the wealth distribution in an economy and this has already been observed in some recent investigations 关12兴. Two variants of the model have been of recent interest; namely, where the agents have the same fixed saving factor 关6兴, and where the agents have a quenched random distribution of saving factors 关7兴. While the former has been understood to a certain extent 共see, e.g, Refs. 关13,14兴兲, and argued to resemble a gamma distribution 关14兴, attempts to analyze the latter model are still incomplete 共see, however, Ref. 关15兴兲. Further numerical studies 关16兴 of time correlations in the model seem to indicate even more intriguing features of the model. In this

paper, we intend to analyze the second market model with randomly distributed saving factor, using a master equation type approach similar to kinetic models of condensed matter. II. MODEL

The market consists of N 共fixed兲 agents, each having money mi共t兲 at time t 共i = 1 , 2 , … , N兲. The total money M关=兺Ni mi共t兲兴 in the market is also fixed. Each agent i has a saving factor ␭i 共0 艋 ␭i ⬍ 1兲 such that in any trading 共considered as a scattering兲 the agent saves a fraction ␭i of its money mi共t兲 at that time and offers the rest 共1 − ␭i兲mi共t兲 for random trading. We assume each trading to be a two-body 共scattering兲 process. The evolution of money in such a trading can be written as mi共t + 1兲 = ␭imi共t兲 + ⑀ij关共1 − ␭i兲mi共t兲 + 共1 − ␭ j兲m j共t兲兴, 共1兲 m j共t + 1兲 = ␭ jm j共t兲 + 共1 − ⑀ij兲关共1 − ␭i兲mi共t兲 + 共1 − ␭ j兲m j共t兲兴, 共2兲 where each mi 艌 0 and ⑀ij is a random fraction 共0 艋 ⑀ 艋 1兲. Typical numerical results for the steady state money distribution in such a model is shown in Fig. 1共a兲 for uniform distribution of ␭i 共0 艋 ␭i ⬍ 1兲 among the agents. III. DYNAMICS OF MONEY EXCHANGE

We will now investigate the steady state distribution of money resulting from the above two equations representing the trading and money dynamics. We will now solve the dynamics of money distribution in two limits. In one case, we study the evolution of the mutual money difference among the agents and look for a self-consistent equation for its steady state distribution. In the other case, we develop a master equation for the money distribution function. A. Distribution of money difference

*Electronic address: [email protected] †

Electronic address: [email protected] ‡ Electronic address: [email protected] 1539-3755/2005/72共2兲/026126共4兲/$23.00

Clearly in the process as considered above, the total money 共mi + m j兲 of the pair of agents i and j remains constant, while the difference ⌬mij evolves as 026126-1

©2005 The American Physical Society

PHYSICAL REVIEW E 72, 026126 共2005兲

CHATTERJEE, CHAKRABARTI, AND STINCHCOMBE

共⌬mij兲t+1 ⬅ 共mi − m j兲t+1 =

冉 冊

冉 冊

␭i + ␭ j ␭i − ␭ j 共⌬mij兲t + 共mi 2 2

+ m j兲t + 共2⑀ij − 1兲关共1 − ␭i兲mi共t兲 + 共1 − ␭ j兲m j共t兲兴.

follows a similar power law variation, P共m兲 ⬃ m−共1+␯兲 and ␯ = ␥. We will now show in a more rigorous way that indeed the only stable solution corresponds to ␯ = 1, as observed numerically 关7–9兴.

共3兲 Numerically, as shown in Fig. 1, we observe that the steady state money distribution in the market becomes a power law, following such tradings when the saving factor ␭i of the agents remain constant over time but varies from agent to agent widely. As shown in the numerical simulation results for P共m兲 in Fig. 1共b兲, the law, as well as the exponent, remains unchanged even when ⑀ij = 1 / 2 for every trading. This can be justified by the earlier numerical observation 关6,7兴 for fixed ␭ market 共␭i = ␭ for all i兲 that in the steady state, criticality occurs as ␭ → 1 where of course the dynamics becomes extremely slow. In other words, after the steady state is realized, the third term in Eq. 共3兲 becomes unimportant for the critical behavior. We therefore concentrate on this case, where the above evolution equation for ⌬mij can be written in a more simplified form as 共⌬mij兲t+1 = ␣ij共⌬mij兲t + ␤ij共mi + m j兲t , where ␣ij = 21 共␭i + ␭ j兲 and − 21 ⬍ ␤ ⬍ 21 .

and

␤ij = 21 共␭i − ␭ j兲.

共4兲

As such, 0 艋 ␣ ⬍ 1

The steady state probability distribution D for the modulus ⌬ = 兩⌬m兩 of the mutual money difference between any two agents in the market can be obtained from Eq. 共4兲 in the following way provided ⌬ is very much larger than the average money per agent =M / N. This is because, using Eq. 共4兲, large ⌬ can appear at t + 1, say, from “scattering” from any situation at t for which the right hand side of Eq. 共4兲 is large. The possibilities are 共at t兲 mi large 共rare兲 and m j not large, where the right hand side of Eq. 共4兲 become ⬃共␣ij + ␤ij兲共⌬ij兲t; or m j large 共rare兲 and mi not large 关making the right hand side of Eq. 共4兲 becomes ⬃共␣ij − ␤ij兲共⌬ij兲t兴; or when mi and m j are both large, which is a much rarer situation than the first two and hence is negligible. Then if, say, mi is large and m j is not, the right hand side of Eq. 共4兲 becomes ⬃共␣ij + ␤ij兲共⌬ij兲t and so on. Consequently for large ⌬ the distribution D satisfies D共⌬兲 =

冕

=2

d⌬⬘D共⌬⬘兲具␦共⌬ − „␣ + ␤兲⌬⬘… + ␦„⌬ − 共␣ − ␤兲⌬⬘…典

冓 冉 冊 冉 冊冔 1 ⌬ D ␭ ␭

,

共5兲

B. Master equation and its analysis

We now proceed to develop a Boltzmann-like master equation for the time development of P共m , t兲, the probability distribution of money in the market. We again consider the case ⑀ij = 21 in Eqs. 共1兲 and 共2兲 and rewrite them as

冉 冊 冉 冊 mi mj

where A=

冕

d␭ ␭␥ = 2共1 + ␥兲−1 ,

t+1

␮+i ␮−j ␮−i ␮+j

冊

mi , mj t

共7兲

␮± = 21 共1 ± ␭兲.

;

共8兲

Collecting the contributions from terms scattering in and subtracting those scattering out, we can write the master equation for P共m , t兲 as 共cf. Ref. 关11兴兲 P共m,t + ⌬t兲 − P共m,t兲 = 具兰 dmi 兰 dm j P共mi,t兲P共m j,t兲 ⫻ 兵关␦共兵A m其i − m兲 + ␦共兵A m其 j − m兲兴 − 关␦共mi − m兲 + ␦共m j − m兲兴其典 = 具兰 dmi 兰 dm j P共mi,t兲P共m j,t兲 ⫻ 关␦共␮+i mi + ␮−j m j − m兲 + ␦共␮−i mi + ␮+j m j − m兲 − ␦共mi − m兲 + ␦共m j − m兲兴典.

共9兲

The above equation can be rewritten as

⳵ P共m,t兲 + P共m,t兲 = 具兰 dmi 兰 dm j P共mi,t兲P共m j,t兲␦共␮+i mi ⳵t + ␮−j m j − m兲典,

共10兲

which in the steady state gives P共m兲 = 具兰 dmi 兰 dm j P共mi兲P共m j兲␦共␮+i mi + ␮−j m j − m兲典. 共11兲 mi␮+i = xm,

Writing we can decompose the range 关0,1兴 of x into three regions: 关0 , ␬兴 , 关␬ , 1 − ␬⬘兴, and 关1 − ␬⬘ , 1兴. Collecting the relevant terms in the three regions, we can rewrite the equation for P共m兲 above as

where we have used the symmetry of the ␤ distribution and the relation ␣ij + ␤ij = ␭i, and have suppressed labels i , j. Here 具¯典 denote average over ␭ distribution in the market. Taking now a uniform random distribution of the saving factor ␭ , ␳共␭兲 = 1 for 0 艋 ␭ ⬍ 1, and assuming D共⌬兲 ⬃ ⌬−共1+␥兲 for large ⌬, we get 1=2

冉

=A

共6兲

giving ␥ = 1. No other value fits the above equation. This also indicates that the money distribution P共m兲 in the market also 026126-2

P共m兲 =

=

冓 冕 冓 再冉 m ␮ ␮− +

1

dxP

0

冉 冊冉 冊 冕

m共1 − x兲 xm P ␮+ ␮−

m m ␮+ P ␮ +␮ − ␮− m

+

␬m/␮+

冉 冊 冕 冕 冉 冊冉 m ␮− ␮+ m

+P

1−␬⬘

␬

dxP

␬⬘m/␮−

冊冔

dyP共y兲

0

dyP共y兲

0

m共1 − x兲 xm P ␮+ ␮−

冊冎冔

,

共12兲

PHYSICAL REVIEW E 72, 026126 共2005兲

MASTER EQUATION FOR A KINECTIC MODEL OF A…

IV. SUMMARY AND DISCUSSIONS

FIG. 1. Steady state money distribution P共m兲 against m in a numerical simulation of a market with N = 200, following Eq. 共1兲 and 共2兲 with 共a兲 ⑀ij randomly distributed in the interval 0–1 and 共b兲 ⑀ij = 1 / 2. The dotted lines correspond to m−共1+␯兲 ; ␯ = 1.

where the result applies for ␬ and ␬⬘ sufficiently small. If we take m Ⰷ 1 / ␬ , m Ⰷ 1 / ␬⬘, and ␬ , ␬⬘ → 0共m → ⬁兲, then P共m兲 =

冓 再冉

冊

冉 冊

m m ␮+ m ␮− P + P ␮ +␮ − ␮− m ␮+ m

+

冕

1−␬⬘

␬

dxP

冉 冊冉

m共1 − x兲 xm + P ␮ ␮−

冊冎冔

.

共13兲

Assuming now as before, P共m兲 = A / m1+␯ for m → ⬁, we get 1 = 具共␮+兲␯ + 共␮−兲␯典 ⬅ 兰 兰 d␮+d␮− p共␮+兲q共␮−兲关共␮+兲␯ + 共␮−兲␯兴 , 共14兲 as the ratio of the third term in Eq. 共13兲 to the other terms vanishes like 共m␬兲−␯ , 共m␬⬘兲−␯ in this limit and p共␮+兲 and q共␮−兲 are the distributions of the variables ␮+ and ␮−, which vary uniformly in the ranges 关 21 , 1兴 and 关0 , 21 兴, respectively 关cf. Eq. 共8兲兴. The i , j indices, for ␮+ and ␮− are again suppressed here in Eq. 共14兲 and we utilize the fact that ␮+i and ␮−j are independent for i ⫽ j. An alternative way of deriving Eq. 共14兲 from Eq. 共11兲 is to consider the dominant terms 关⬀x−r for r ⬎ 0, or ⬀ ln共1 / x兲 for r = 0兴 in the x → 0 limit of the integral 兰⬁0 m共␯+r兲 P共m兲exp共−mx兲dm 共see the Appendix兲. We therefore get from Eq. 共14兲, after integrations, 1 = 2 / 共␯ + 1兲, giving ␯ = 1.

In our models 关6–9兴, we consider the ideal-gas-like trading markets where each agent is identified with a gas molecule and each trading as an elastic or money-conserving 共two-body兲 collision. Unlike in a gas, we introduce a saving factor ␭ for each agent. Our model, without savings 共␭ = 0兲, obviously yields a Gibbs law for the steady state money distribution. Our numerical results for various widely distributed 共quenched兲 saving factor ␭ showed 关7–9兴 that the steady state income distribution P共m兲 in the market has a power-law tail P共m兲 ⬃ m−共1+␯兲 for large income limit, where ␯ ⯝ 1.0. This observation has been confirmed in several later numeri⬁ P共m兲dm can be cal studies as well 关15,16兴. Since Q共m兲 = 兰m identified with the inverse rank, our observation in the model with ␯ = 1 suggests that the rank of any agent goes inversely with his or her income or wealth, fitting very well with Zipf’s original observation 关17兴. It has been noted from these numerical simulation studies that the large income group people usually have larger saving factors 关7兴. This, in fact, compares well with observations in real markets 关12,18兴. The time correlations induced by the random saving factor also has an interesting power-law behavior 关16兴. A master equation for P共m , t兲, as in Eq. 共9兲, for the original case 关Eqs. 共1兲 and 共2兲兴 was first formulated for fixed ␭ 共 ␭i same for all i兲, in Ref. 关13兴 and solved numerically. Later, a generalized master equation for the same, where ␭ is distributed, was formulated and solved in Ref. 关15兴. We have formulated here a Boltzmann-type master equation for the distributed saving factor case in Eq. 共1兲 and 共2兲. Based on the observation that even in the case with ⑀ = 1 / 2 共with ␭ distributed in the range 0 艋 ␭i ⬍ 1 , ␭i ⫽ ␭ j兲, in Eqs. 共1兲 and 共2兲, the steady state money distribution has the same power-law behavior as in the general case and shows the same Pareto index, we solve the master equation for this special case. We show that the analytic results clearly support the power law for P共m兲 with the exponent value ␯ = 1. Although our analysis of the solution of the master equation is for a special case and it cannot be readily extended to explore the wide universality of the Pareto exponent as observed in the numerical simulations of the various versions of our model 关7,15兴, let alone the quasiuniversality for other ␯ values as observed in the real markets 关2–4兴, the demonstration here that the master equation admits of a Pareto-like power-law solution 共for large m兲 with ␯ = 1, should be significant. Apart from the intriguing observation that Gibbs 共1901兲 and Pareto 共1897兲 distributions fall in the same category of models and can appear naturally in the century-old and wellestablished kinetic theory of gas, our study indicates the appearance of self-organized criticality in the simplest 共gaslike兲 models so far, when the stability effect of savings is incorporated. This remarkable effect can be analyzed in terms of master equations developed here and can also be studied analytically in the special limits considered. ACKNOWLEDGMENTS

B.K.C. is grateful to the INSA-Royal Society Exchange Programme for financial support to visit the Rudolf Peierls

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PHYSICAL REVIEW E 72, 026126 共2005兲

CHATTERJEE, CHAKRABARTI, AND STINCHCOMBE

Centre for Theoretical Physics, Oxford University, UK, and R.B.S. acknowledges EPSRC support under Grant Nos. GR/ R83712/01 and GR/M04426 for this work and wishes to thank the Saha Institute of Nuclear Physics for hospitality during a related visit to Kolkata, India. APPENDIX: ALTERNATIVE SOLUTION OF THE STEADY STATE MASTER EQUATION (11)

Let Sr共x兲 = 兰⬁0 dmP共m兲m␯+r exp共−mx兲 ; r 艌 0 , x ⬎ 0. P共m兲 = A / m1+␯, then Sr共x兲 = A

冕

⬁

0

⬃ A ln

dm mr−1 exp共− mx兲 ⬃ A

冉冊

Sr共x兲 =

冓冕

dmi

冕

− 共mi␮+i + m j␮−j 兲x兴 ⯝

冕

⬁

0

⫻

冋冕

0

d␮+i p共␮+i 兲

1/2

dmi P共mi兲

0

共A2兲

冉冕

1/2

d␮−j q共␮−j 兲

0

共A1兲

⬁

dmiAmr−1 exp共− mi␮+i x兲 i

0

+

冉冕

⬁

冊共

␮+i 兲␯+r

dm jAmr−1 exp共− m j␮−j x兲 j

0

⫻共␮−j 兲␯+r ,

冊 共A3兲

since for small x, the terms in the square brackets in Eq. 共A2兲 approach unity. We can therefore rewrite Eq. 共A3兲 as

冔

dm j P共m j兲具exp共− m j␮−j x兲典

冕 冕 1

Sr共x兲 = 2

冋冕

+

+ + ␯+r dmiAmr−1 典 i 具exp共− mi␮i x兲共 ␮i 兲 ⬁

册

冋冕

⬁

or Sr共x兲 =

dm j P共mi兲P共m j兲共mi␮+i + m j␮−j 兲␯+rexp关

0

0

− − ␯+r dm jAmr−1 典 j 具exp共− m j␮ j x兲共 ␮ j 兲

⫻具exp共− mi␮+i x兲典

x−r if r ⬎ 0 r

1 if r = 0. x

⬁

冕

⬁

0

If

From Eq. 共11兲, we can write ⬁

+

冕

1/2

1/2

0

册

1

d␮+共␮+兲␯+rSr共x␮+兲

册

d␮−共␮−兲␯+rSr共x␮−兲 .

共A4兲

Using now the forms of Sr共x兲 as in Eq. 共A1兲, and collecting terms of order x−r 共for r ⬎ 0兲 or of order ln共1 / x兲 共for r = 0兲 from both sides of Eq. 共A4兲, we get Eq. 共14兲.

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关8兴 A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Phys. Scr. T106, 36 共2003兲. 关9兴 B. K. Chakrabarti and A. Chatterjee, in Application of Econophysics, Proceedings of the 2nd Nikkei Econophysics Symposiam, Tokyo, 2002, edited by H. Takayasu 共Springer, Tokyo, 2004兲, pp. 280–285. 关10兴 B. Hayes, Am. Sci. 90, 400 共2002兲; S. Sinha, Phys. Scr. T106, 59 共2003兲;J. C. Ferrero, Physica A 341, 575 共2004兲;J. R. Iglesias, S. Gonçalves, G. Abramson, and J. L. Vega, ibid. 342, 186 共2004兲;N. Scafetta, S. Picozzi, and B. J. West, Physica D 193, 338 共2004兲. 关11兴 F. Slanina, Phys. Rev. E 69, 046102 共2004兲. 关12兴 G. Willis and J. Mimkes, cond-mat/0406694. 关13兴 A. Das and S. Yarlagadda, Phys. Scr. T106 39 共2003兲. 关14兴 M. Patriarca, A. Chakraborti, and K. Kaski, Phys. Rev. E 70, 016104 共2004兲. 关15兴 P. Repetowicz, S. Hutzler, and P. Richmond, cond-mat/ 0407770. 关16兴 N. Ding, N. Xi, and Y. Wang, Eur. Phys. J. B 36, 149 共2003兲. 关17兴 G. K. Zipf, Human Behavior and the Principle of Least Effort 共Addison-Wesley, Reading MA, 1949兲. 关18兴 K. E. Dynan, J. Skinner, and S. P. Zeldes, J. Polit. Econ. 112, 397 共2004兲.

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