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Physica A 335 (2004) 155–163www.elsevier.com/locate/physa

Pareto law in a kinetic model of market withrandom saving propensity

Arnab Chatterjeea ;∗, Bikas K. Chakrabartia, S.S. MannabaSaha Institute of Nuclear Physics, Block-AF, Sector-I, Bidhannagar, Kolkata 700064, IndiabSatyendra Nath Bose National Centre for Basic Sciences Block-JD, Sector-III, Salt Lake,

Kolkata 700098, India

Received 15 October 2003

Abstract

We have numerically simulated the ideal-gas models of trading markets, where each agentis identi0ed with a gas molecule and each trading as an elastic or money-conserving two-bodycollision. Unlike in the ideal gas, we introduce (quenched) saving propensity of the agents,distributed widely between the agents (06 �¡ 1). The system remarkably self-organizes to acritical Pareto distribution of money P(m) ∼ m−(�+1) with � � 1. We analyze the robustness(universality) of the distribution in the model. We also argue that although the fractional savingingredient is a bit unnatural one in the context of gas models, our model is the simplest so far,showing self-organized criticality, and combines two century-old distributions: Gibbs (1901) andPareto (1897) distributions.c© 2003 Elsevier B.V. All rights reserved.

PACS: 87.23.Ge; 89.90.+n; 02.50.−r

Keywords: Econophysics; Income distribution; Gibbs and Pareto laws

Considerable investigations have already been made to study the nature of incomeor wealth distributions in various economic communities, in particular, in di@erentcountries. For more than a hundred years, it is known that the probability distributionP(m) for income or wealth of the individuals in the market decreases with the wealthm following a power law, known as Pareto law [1]:

P(m)˙ m−(1+�) ; (1)

∗ Corresponding author.E-mail address: arnab@cmp.saha.ernet.in (A. Chatterjee).

0378-4371/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2003.11.014

156 A. Chatterjee et al. / Physica A 335 (2004) 155–163

where the value of the exponent � is found to lie between 1 and 2 [2–4]. It is alsoknown that typically less than 10% of the population in any country possesses about40% of the wealth and follow the above power law. The rest of the low-income grouppopulation, in fact the majority, clearly follows a di@erent law, identi0ed very recentlyto be the Gibbs distribution [5–7]. Studies on real data show that the high-incomegroup indeed follow Pareto law, with � varying from 1.6 for USA [6], to 1.8–2.2 inJapan [3]. The value of � thus seem to vary a little from economy to economy.We have studied here numerically a gas model of a trading market. We have consid-

ered the e@ect of saving propensity of the traders. The saving propensity is assumed tohave a randomness. Our observations indicate that Gibbs and Pareto distributions fallin the same category and can appear naturally in the century-old and well-establishedkinetic theory of gas (see e.g. Ref. [8]): Gibbs distribution for no saving and Paretodistribution for agents with quenched random saving propensity. Our model study alsoindicates the appearance of self-organized criticality [9] in the simplest model so far,namely in the kinetic theory of gas models, when the stability e@ect of savings [10] isincorporated.We consider an ideal-gas model of a closed economic system where total money M

and total number of agents N is 0xed. No production or migration occurs and the onlyeconomic activity is con0ned to trading. Each agent i, individual or corporate, possessmoney mi(t) at time t. In any trading, a pair of traders i and j randomly exchangetheir money [5,7,11], such that their total money is (locally) conserved and none endup with negative money (mi(t)¿ 0, i.e., debt not allowed):

mi(t) + mj(t) = mi(t + 1) + mj(t + 1) ; (2)

time (t) changes by one unit after each trading. The steady-state (t→∞) distributionof money is Gibbs one:

P(m) = (1=T ) exp(−m=T ); T =M=N : (3)

Hence, no matter how uniform or justi0ed the initial distribution is, the eventualsteady state corresponds to Gibbs distribution where most of the people have got verylittle money. This follows from the conservation of money and additivity of entropy:

P(m1)P(m2) = P(m1 + m2) : (4)

This steady-state result is quite robust and realistic too! In fact, several variations of thetrading, and of the ‘lattice’ (on which the agents can be put and each agent trade withits ‘lattice neighbors’ only), whether compact, fractal or small-world like [2], leavesthe distribution unchanged. Some other variations like random sharing of an amount2m2 only (not of m1 + m2) when m1¿m2 (trading at the level of lower economicclass in the trade), lead to even drastic situation: all the money in the market drifts toone agent and the rest become truely pauper [12,13].In any trading, savings come naturally [10]. A saving propensity factor � is therefore

introduced in the same model [7] (see [11] for model without savings), where eachtrader at time t saves a fraction � of its money mi(t) and trades randomly with therest:

mi(t + 1) = mi(t) + Nm; mj(t + 1) = mj(t)−Nm ; (5)

A. Chatterjee et al. / Physica A 335 (2004) 155–163 157

0 21 3m

0.0

0.5

1.0

1.5

2.0

2.5

P f(m)

λ = 0λ = 0.5λ = 0.9

0

0.5

1

1.5

2

2.5

P f(m)

λ = 0.6λ = 0.8λ = 0.9

0 0.2 0.4 0.6m (1-λ)

0

2

4

6

P f /(1-

λ)

(a) (b)

~

~

0 21 3m

Fig. 1. Steady-state money distribution (a) Pf(m) for the 0xed (same for all agents) � model, and (b) P̃f (m)for some typical values of � in the distributed � model. The data are collected from the ensembles withN = 200 agents. The inset in (b) shows the scaling behavior of P̃f(m). For all cases, agents start withaverage money per agent M=N = 1.

where

Nm= (1− �)[�{mi(t) + mj(t)} − mi(t)] ; (6)

� being a random fraction, coming from the stochastic nature of the trading.The market (non-interacting at � = 0 and 1) becomes ‘interacting’ for any non-

vanishing �(¡ 1): For 0xed � (same for all agents), the steady-state distribution Pf(m)of money is exponentially decaying on both sides with the most-probable money peragent shifting away from m = 0 (for � = 0) to M=N as �→ 1 (Fig. 1(a)). Thisself-organizing feature of the market, induced by sheer self-interest of saving by eachagent without any global perspective, is quite signi0cant as the fraction of paupersdecrease with saving fraction � and most people end up with some fraction of theaverage money in the market (for �→ 1, the socialists’ dream is achieved with justpeople’s self-interest of saving!). Interestingly, self-organization also occurs in suchmarket models when there is restriction in the commodity market [14]. Although this0xed saving propensity does not give yet the Pareto-like power-law distribution, theMarkovian nature of the scattering or trading processes (Eq. (4)) is lost and the systembecomes co-operative. Indirectly through �, the agents get to know (start interactingwith) each other and the system co-operatively self-organises towards a most-probabledistribution (mp �= 0).In a real society or economy, � is a very inhomogeneous parameter: the interest of

saving varies from person to person. We move a step closer to the real situation wheresaving factor � is widely distributed within the population. One again follows the sametrading rules as before, except that

Nm= �(1− �j)mj(t)− (1− �i)(1− �)mi(t) (7)

here; �i and �j being the saving propensities of agents i and j. The agents have0xed (over time) saving propensities, distributed independently, randomly and uni-formly (white) within an interval 0 to 1 (see [15] for preliminary results): agent isaves a random fraction �i (06 �i ¡ 1) and this �i value is quenched for each agent

158 A. Chatterjee et al. / Physica A 335 (2004) 155–163

10-3

10-2

10-1

100

101

102

103

m

10-8

10-6

10-4

10-2

100

P(m

)

0.001 0.01N

-10.05

0.07

0.1

mp

M / N = 1mp N = 1000

1 + = 2.0ν

Fig. 2. Steady-state money distribution P(m) in the model for distributed � (06 �¡ 1) for N=1000 agents.Inset shows that the most-probable peak mp shifts towards 0 (indicating the same power law for the entirerange of m) as N→∞: results for four typical system sizes N = 100, 200, 500, 1000 are shown. For allcases, agents play with average money per agent M=N = 1.

(�i are independent of trading or t). Starting with an arbitrary initial (uniform or ran-dom) distribution of money among the agents, the market evolves with the tradings. Ateach time, two agents are randomly selected and the money exchange among them oc-curs, following the above mentioned scheme. We check for the steady state, by lookingat the stability of the money distribution in successive Monte Carlo steps t. Eventu-ally, after a typical relaxation time (∼ 105 for N = 200 and uniformly distributed �)dependent on N and the distribution of �, the money distribution becomes stationary.After this, we average the money distribution over ∼ 103 time steps. Finally we takecon0gurational average over ∼ 105 realizations of the � distribution to get the moneydistribution P(m). It is found to follow a strict power-law decay. This decay 0ts toPareto law (1) with �=1:02± 0:02 (Fig. 2). Note, for 0nite size N of the market, thedistribution has a narrow initial growth upto a most-probable value mp after which itfalls o@ with a power-law tail for several decades. As can be seen from the inset ofFig. 2, this Pareto law (with � 1) covers the entire range in m of the distributionP(m) in the limit N→∞. We checked that this power law is extremely robust: apartfrom the uniform � distribution used in the simulations in Fig. 2, we also checked theresults for a distribution

�(�) ∼ |�0 − �|�; �0 �= 1; 0¡�¡ 1 ; (8)

of quenched � values among the agents. The Pareto law with �=1 is universal for all�. The data in Fig. 2 corresponds to �0 =0, �=0. For negative � values, however, weget an initial (small m) Gibbs-like decay in P(m) (see Fig. 3).In case of uniform distribution of saving propensity � (06 �¡ 1), the individ-

ual money distribution P̃f(m) for agents with any particular � value, although di@ersconsiderably, remains non-monotonic: similar to that for 0xed � market with mp(�)shifting with � (see Fig. 1). Few subtle points may be noted though: while for 0xed� the mp(�) were all less than of the order of unity (Fig. 1(a)), for distributed �

A. Chatterjee et al. / Physica A 335 (2004) 155–163 159

10-2

10-1

100

101

102

m

10-6

10-4

10-2

100

P(m

)

α = − 0.7

α = + 0.7 1 + ν = 2

α = − 0.3α = + 0.3

10110

-5

10-4

10-3

10-2

Fig. 3. Steady-state money distribution P(m) in the model for N=100 agents with � distributed as �(�) ∼ ��

with di@erent values of �. The inset shows the region of validity of Pareto law with 1+ �=2. For all cases,agents play with average money per agent M=N = 1.

case mp(�) can be considerably larger and can approach to the order of N for large �(see Fig. 1(b)). The other important di@erence is in the scaling behavior of P̃f(m), asshown in the inset of Fig. 1(b). In the distributed � ensemble, P̃f(m) appears to havea very simple scaling:

P̃f(m) ∼ (1− �)F(m(1− �)) (9)

for �→ 1, where the scaling function F(x) has non-monotonic variation in x. The0xed (same for all agents) � income distribution Pf(m) do not have any such com-parative scaling property. It may be noted that a small di@erence exists between theensembles considered in Fig. 1(a) and (b): while

∫mPf(m) dm =M (independent of

�),∫mP̃f(m) dm is not a constant and in fact approaches to order of M as �→ 1.

There is also a marked qualitative di@erence in Puctuations (see Fig. 4): while for0xed �, the Puctuations in time (around the most-probable value) in the individuals’money mi(t) gradually decreases with increasing �, for quenched distribution of �, thetrend gets reversed (see Fig. 4).We now investigate on the range of distribution of the saving propensities in a

certain interval a¡�i ¡b, where, 0¡a¡b¡ 1. For uniform distribution within therange, we observe the appearance of the same power law in the distribution but fora narrower region. As may be seen from Fig. 5, as a→ b, the power-law behavioris seen for values a or b approaching more and more towards unity: For the samewidth of the interval |b − a|, one gets power law (with same �) when b→ 1. Thisindicates, for 0xed �, � = 0 corresponds to Gibbs distribution, and one gets Paretolaw when � has got non-zero width of its distribution extending upto � = 1. This ofcourse indicates a crucial role of these high saving propensity agents: the power-lawbehavior is truely valid upto the asymptotic limit if �= 1 is included. Indeed, had weassumed �0 = 1 in (8), the Pareto exponent � immediately switches over to �= 1+ �.Of course, �0 �= 1 in (8) leads to the universality of the Pareto distribution with �= 1(independent of �0 and �). Indeed, this can be easily rationalized from the scaling

160 A. Chatterjee et al. / Physica A 335 (2004) 155–163

0

4

8m

i(t)

0

1

2

3

4

0

4

8

mi(t

)

0

1

2

3

4

0 5000 10000

t

0

4

8

mi(t

)

mi(t

)m

i(t)

mi(t

)

0 5000 10000

t

0

1

2

3

4

λ = 0

λ = 0.5

λ = 0.9

λ = 0.1

λ = 0.5

λ = 0.9

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4. Money of the ith trader with time. For 0xed � for all agents in the market: (a) with �= 0, (b) �=0:5, (c) � = 0:9. For distributed � (06 �¡ 1): for agents with (d) � = 0:1, (e) � = 0:5 and (f) � = 0:9 inthe market. All data are for N = 200 (M=N = 1).

behavior (9): P(m) ∼ ∫ 10 P̃f(m)�(�) d� ∼ m−2 for �(�) given by (8) and m−(2+�) if

�0 = 1 in (8) (for large m values).These model income distributions P(m) compare very well with the wealth distri-

butions of various countries: Data suggests Gibbs like distribution in the low-incomerange [6] (more than 90% of the population) and Pareto-like in the high-income range[3] (less than 10% of the population) of various countries (Fig. 6). In fact, we havecompared one model simulation of the market with saving propensity of the agentsdistributed following (8), with �0 = 0 and �=−0:7. This model result is shown in theinset of Fig. 6. The qualitative resemblance of the model income distribution with thereal data for Japan and USA in recent years is quite intriguing. In fact, for negative� values in (8), the density of traders with low saving propensity is higher and since�= 0 ensemble yields Gibbs-like income distribution (3), we see an initial Gibbs-likedistribution which crosses over to Pareto distribution (1) with �= 1:0 for large m val-ues. The position of the crossover point depends on the magnitude of �. The importantpoint to note is that any distribution of � near �= 1, of 0nite width, eventually givesPareto law for large m limit. The same kind of crossover behavior (from Gibbs toPareto) can also be reproduced in a model market of mixed agents where � = 0 fora 0nite fraction of population and � is distributed uniformly over a 0nite range near�= 1 for the rest of the population.

A. Chatterjee et al. / Physica A 335 (2004) 155–163 161

10-2

10-1

100

101

102

m

10-8

10-6

10-4

10-2

100

P(m

)

0.2 < λ< 0.40.75 < λ< 0.950.8 < λ < 1.0

10-6

10-4

10-2

100

P(m

) 0 < λ < 0.50.5 < λ< 1.0

(a)

(b)

Fig. 5. Money distribution in cases where saving propensity � is distributed uniformly within a range ofvalues: (a) When width of � distribution is 0:5, money distribution shows power law for 0:5¡�¡ 1:0;(b) When width of � distribution is 0:2, money distribution starts showing power law when 0:7¡�¡ 0:9.Note, the exponent � � 1 in all these cases when the power law is seen. All data are for N =100 (M=N =1).

We also considered annealed randomness in the saving propensity �: here �i forany agent i changes from one value to another within the range 06 �i ¡ 1, aftereach trading. Numerical studies for this annealed model did not show any power-lawbehavior for P(m); rather it again becomes exponentially decaying on both sides of amost-probable value.We have numerically simulated here ideal-gas like models of trading markets, where

each agent is identi0ed with a gas molecule and each trading as an elastic or money-conserving two-body collision. Unlike in the ideal gas, we introduce (quenched) sav-ing propensity of the agents, distributed widely between the agents (06 �¡ 1). Forquenched random variation of � among the agents the system remarkably self-organizesto a critical Pareto distribution (1) of money with � 1:0 (Fig. 2). The exponent isquite robust: for savings distribution �(�) ∼ |�0−�|�, �0 �= 1, one gets the same Paretolaw with �=1 (independent of �0 or �). It may be noted that the trading market modelwe have talked about here has got some apparent limitations. The stochastic nature oftrading assumed here in the trading market, through the random fraction � in (6), isof course not very straightforward as agents apparently go for trading with some def-inite purpose (utility maximization of both money and commodity). We are however,looking only at the money transactions between the traders. In this sense, the incomedistribution we study here essentially corresponds to ‘paper money’, and not the ‘real

162 A. Chatterjee et al. / Physica A 335 (2004) 155–163

Fig. 6. Cumulative distribution Q(m)=∫∞m P(m) dm of wealth m in USA [6] in 1997 and Japan [3] in 2000.

Low-income group follow Gibbs law (shaded region) and the rest (about 5%) of the rich population followPareto law. The inset shows the cumulative distribution for a model market where the saving propensityof the agents is distributed following (8) with �0 = 0 and � = −0:7. The dotted line (for large m values)corresponds to � = 1:0.

wealth’. However, even taking money and commodity together, one can argue (seeRef. [13]) for the same stochastic nature of the tradings, due to the absence of ‘justpricing’ and the e@ects of bargains in the market.Apart from the intriguing observation that Gibbs (1901) and Pareto (1897) dis-

tributions fall in the same category and can appear naturally in the century-old andwell-established kinetic theory of gas, that this model study indicates the appearanceof self-organized criticality in the simplest (gas) model so far, when the stability e@ectof savings incorporated, is remarkable.

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