Econophysics: A brief introduction to modeling wealth distribution

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1 Econophysics: A brief introduction to modeling wealth distribution Anirban Chakraborti Department of Physics, Banaras Hindu University, Varanasi-22005, India (Dated: January 7, 2006) We discuss a few studies of wealth distributions in Econophysics, the developing interdisciplinary field, culminating ideas from Economics and Physics. The topic discussed here should illustrate how simple tools and methods in physics can be applied to study economic problems. PACS numbers: Gh,89.75.Da, y Keywords: Economics, econophysics, markets, power-laws, statistical mechanics I. INTRODUCTION Readers might wonder at the onset that how the theories which try to eplain the physical world of electrons, protons, atoms and molecules could be applied to understand the comple social structure and economic behavior of human beings. The electrons are identical (we do not have to tackle big or small electrons!) and indistinguishable. Each electron has a charge of C, mass of kg, and is a spin-/2 particle. These properties of electrons are universal (identical for all electrons), and this amount of information is sufficient to eplain many physical phenomena concerning electrons, once you know the interactions. But, are such precise universal properties known for human beings (who are certainly not identical in any respect), and is such little information sufficient to infer the comple behavior of human beings? Is it possible to quantify the nature of the interactions between human beings? The answers to these questions are on the negative. Nevertheless, during the past decade physicists have made attempts in studying problems in Economics, the Social science that analyzes and describes the consequences of choices made concerning scarce resources []. Here, we will not try to review all the attempts, but instead briefly describe a few undertaken by the author and his collaborators in modeling wealth distributions, which should motivate the readers to go and fend for themselves. wealth and income. A. Definition of money, wealth and income The most common definition of money suggests that money is the Commodity accepted by general consent as medium of economics echange []. In fact money circulates from one economic agent (which can be an individual, firm, country, etc.) to another, thus facilitating trade. It is something which all other goods or services are traded for (for details see Ref. [2]). Throughout history various commodities have been used as money, termed usually as commodity money which include rare seashells or beads, and cattle (such as cow in India). Since the 7th century the most common forms have been the metal coins, paper notes, and book-keeping entries. However, this is not the only important point about money. It is worth recalling the four functions of money according to standard economic theory: (i) to serve as a medium of echange universally accepted in trade for goods and services (ii) to act as a measure of value, making possible the determination of the prices and the calculation of costs, or profit and loss (iii) to serve as a standard of deferred payments, i.e., a tool for the payment of debt or the unit in which loans are made and future transactions are fied II. MODELING WEALTH DISTRIBUTION (iv) to serve as a means of storing wealth not immediately required for use. It is of much interest to the common people to find out where they fair in the society, in terms of monetary power. The distributions of money, wealth or income, i.e., how they are divided among the population of a given country and among different countries, have thus been long studied by economists. Since definitions in Economics are different in nature than in Physics, let us first try to understand the economic quantities: money, Electronic address: achakraborti@yahoo.com Another main feature which emerges from these properties and relevant for us, is that money is the medium in which prices or values of all commodities as well as costs, profits, and transactions can be determined or epressed. As for the wealth, it usually refers to things that have economic utility (monetary value or value of echange), or material goods or property. It also represents the abundance of objects of value (or riches) and the state of having accumulated these objects. For our purpose, it is important to bear in mind that wealth can be measured in terms of money. Finally, income is defined as The amount of money or its equivalent received

2 platform for understanding of the dependence of the distributions on the underlying mechanisms and parameters is yet to arise. 2 C. Application of Kinetic theory FIG. : Income distributions in the US (left) and Japan (right). Adapted from cond-mat/ during a period of time in echange for labor or services, from the sale of goods or property, or as profit from financial investments [3]. Therefore, it is also a quantity which can be measured in terms of money (per unit time). B. Pareto s law and Gibrat s law It was first observed by Pareto [4], more than a century ago, that in an economy the higher end of the distribution of income f() follows a power-law f() α, () and α is an eponent (now known as the Pareto eponent) which Pareto estimated to be 3/2. For the last hundred years the value of α 3/2 changed little in time and across the various capitalist economies. In 93, Gibrat [5] clarified that while Pareto s law is valid only for the high income range, the middle income range is given by the probability density f() { } 2πσ ep log2 (/ 0 ) 2 2σ 2, (2) where 0 is a mean value and σ 2 is the variance. The factor β = / 2σ 2 is also know an as the Gibrat inde, and a small Gibrat inde corresponds to a uneven income distribution. Detailed empirical studies on income distribution have been done by physicists like Dragulescu and Yakovenko for countries like the UK and the US [6], and by Fujiwara et al. for Japan [7, 8] recently. The distributions have been shown to be Gibb s (log-normal) and powerlaw types as shown in Fig.. Recently, several studies have been made investigating the characteristics of the real income distribution and providing theoretical models or eplanations. From these studies it is quite clear that it is possible to obtain power law distributions in the framework of some economy models, whereas other models predict eponential tails of the income distribution. However, a common It is our general aim to study a statistical model of closed economy, analogous to the kinetic theory model of ideal gases, which can be either solved eactly or simulated numerically. In such models, N agents echange a quantity, that has sometimes been defined as wealth and other times as money. To avoid confusion, in the following we will use only the term wealth. The states of agents are characterized by the wealths { n }, n =, 2,...,N, and the total wealth X = n n is conserved. The evolution of the system is then carried out according to a prescription, which defines the trading rule between agents. The evolution can be interpreted both as an actual time evolution or a Monte Carlo optimization procedure, aimed at finding the equilibrium distribution. At every time step two agents i and j are etracted randomly and an amount of wealth is echanged between them, i = i, j = j +. (3) It can be noticed that in this way, the quantity is conserved during the single transactions: i + j = i + j (see Fig. 2). Here i and j are the agent wealths after the transaction has taken place. Several rules have been studied for the model.. Basic model without saving: Boltzmann distribution In the first version of the model, the money difference is assumed to have a constant value [9 ], = 0. (4) This rule, together with the constraint that transactions can take place only if i > 0 and j > 0, provides a Boltzmann distribution, see the curve for λ = 0 in Fig. 3. Alternatively, can be a random fraction of the wealth of one of the two agents, = ǫ i or = ǫ j, (5) where ǫ is a random number uniformly distributed between 0 and. A trading rule based on the random redistribution of the sum of the wealths of the two agents had been introduced by Dragulescu and Yakovenko [2], i = ǫ( i + j ), j = ǫ( i + j ), (6) where ǫ is the complementary fraction of unity defined by the random number ǫ, i.e. ǫ + ǫ =. Equations (6)

3 3 N particles/agents i j Random collisions/trades i j j λ = 0 λ = 0.2 λ = 0.5 λ = 0.7 λ = 0.9 X energy/wealth f() i FIG. 2: Analogy of the minimal economic model with a classical isolated system of ideal gas, where the particles are randomly undergoing Elastic collisions, and echanging kinetic energy. In the closed economy, the economic agents randomly trade with each other according to some rule and echange wealth. are easily shown to correspond to the trading rule (3), with = ǫ i ǫ j. (7) In the following, we will concentrate on the latter version of the model and its generalizations, though all the versions of the basic model lead to an equilibrium Boltzmann distribution, given by f() = ( ep ), (8) where the effective temperature of the system is just the average wealth [9 2]. The result (8) is found to be robust; it is largely independent of various factors. Namely, it is obtained for the various forms of mentioned above, for a pair-wise as well as multi-agent interactions, for arbitrary initial conditions [3], and finally, for random or consecutive etraction of the interacting agents. The Boltzmann distribution thus obtained has been sometimes referred to as an unfair distribution, since it is characterized by a majority of poor agents and very few rich agents, as evident from the zero mode and the eponential tail. In the following, when we refer more specifically to the Mawell-Boltzmann distribution originally introduced for the velocities of molecules, we mean a distribution formally analogous to that of kinetic energy, which differs from Eq. (8) for a power of, and is mathematically equivalent to the so-called Gamma distribution. 2. Model with constant global saving propensity: Gamma distribution A step toward generalizing the basic model and making it more realistic, is the introduction of a saving criterion regulating the trading dynamics. This can be practically achieved by defining a saving propensity 0 < λ <, which represents the fraction of wealth which is saved and not reshuffled during a transaction. The dynamics f() e-05 e-06 λ = 0 λ = 0.2 λ = 0.5 λ = 0.7 λ = 0.9 e FIG. 3: Probability density for wealth. The curve for λ = 0 is the Boltzmann function f() = ep( / ) for the basic model of Sec. IIC. The other curves correspond to a global saving propensity λ > 0 (see Sec. IIC2). of the model is as follows [3, 4]: i = λ i + ǫ( λ)( i + j ), j = λ j + ǫ( λ)( i + j ), (9) with ǫ = ǫ, corresponding to a in Eq. (3) given by = ( λ)[ ǫ i ǫ j ]. (0) This model leads to a qualitatively different equilibrium distribution. In particular, it has a mode m > 0 and a zero limit for small, i.e. P( 0) 0, see Fig. 3. The functional form of such a distribution has been conjectured to be a Gamma distribution on the base of an analogy with the kinetic theory of gases, which is consistent with the ecellent fitting provided to numerical data [5, 6]. 3. Models with a continuous distribution of saving propensity The basic model and the model with a global saving propensity produce equilibrium distribution functions the Boltzmann and the Gamma distributions respectively

4 4 that match well with real data of income distributions at small and intermediate values [6, 7 22]. As a further generalization, the agents have been assigned different saving propensities λ i [23 30]. In particular, uniformly distributed λ i in the interval (0, ) have been studied numerically in Refs. [23, 24]. This model is described by the trading rule f () (a) i = λ i i + ǫ[( λ i ) i + ( λ j ) j ], j = λ j j + ǫ[( λ i ) i + ( λ j ) j ], () or, equivalently, by a as defined in Eq. (3) given by = ǫ( λ i ) i ǫ( λ j ) j. (2) One of the main features of this model, which is supported by theoretical considerations [25, 26, 28, 30], is that the wealth distribution ehibits a robust power-law at large values of, f() α, (3) with a Pareto eponent α = largely independent of the details of the λ-distribution. 4. Constructing a more realistic distribution In real wealth distributions, an eponential form at intermediate values of wealth is known to coeist with a power-law tail at larger values [9]. The power-law is due to a small percentage of population, of the order of a few per cent, while the majority of the population with smaller average wealth contribute to the eponential part. One can construct a more realistic eample of wealth distribution than the eponential or power-law form alone, starting from the following information: A global saving propensity λ 0 > 0 is associated to an equilibrium Gamma distribution with a mode m > 0 (but an eponential tail). Agents with high λ s [λ (0.9, )] produce a powerlaw tail, even when there is a finite cut-off in the λ-distribution. These considerations when taken together suggest that one can construct a more realistic wealth distribution by choosing a suitable hybrid λ-distribution, similar to what had been done in Ref. [23]: A small fraction of agents p 0 with saving propensities λ i uniformly distributed in the interval (0, ) according to Eq. (A4) and the remaining fraction p 0 with a constant value of the saving propensity λ 0. The distribution corresponding to p 0 = 0.0 and λ 0 = 0 is shown in Fig. 4. Both an eponential shape at small -scale and a power-law with eponent α = at large are observed. It is noteworthy that the condition for the coeistence of an eponential and power-law form sets p 0 to a few percents, in agreement f () F () e e-05 e-06 e-07 e e-06 e-08 e (b) FIG. 4: Equilibrium wealth distribution of a population of 0 5 agents, in which one per cent (0 3 agents) have uniformly distributed saving propensities in λ = (0, ), while the rest have λ = 0. (a) (semi-log scale): The numerical distribution f() (circles) is compared in the small -region with the eponential function ep( 3/5) (dotted line). (b) (log scale): The distribution f() (continuous line) is compared with the power-law 2 (dashed line) and the same eponential function (dotted line). The peaks visible at high are due to the discreteness effects discussed in the tet. (c): The cumulative distribution function F() = R dy f(y) (continuous line) is compared with the power-law (dashed line) and the same eponential function (dotted line). (c)

5 5 f () f () F () e e-05 e-06 e-07 e e-06 e-08 e-0 (a) (b) FIG. 5: Equilibrium wealth distribution of a population of 0 6 agents, in which % (0 4 agents) have uniformly distributed saving propensities in the interval (0, ), while the rest have λ = 0.2. With respect to the distribution in Fig. 4, this distribution has a mode m > 0. The meaning of symbols is the same as in Fig. 4. with real data on wealth distributions [3]. In fact, for larger values of p 0 the eponential part shrinks and the power-law dominates the distribution. It is also to be noticed that, due to the choice λ 0 = 0, the distribution in Fig. 4 has a mode m = 0. By choosing a more realistic value λ 0 = 0.2 for ( p 0 ) = 99% of the agents and a uniform λ-distribution for the remaining p 0 = %, one (c) still obtains a distribution in which an eponential and a power-law tail coeist, but also with a mode m > 0, see Fig. 5. The observed -cutoff is determined by the λ-cutoff of the saving propensity distribution. We want to emphasize by this eample that a realistic wealth distribution can be generated by a suitable tuning of the distribution parameters: (a) The large fraction of agents with a small saving propensity λ 0 > 0, producing a mode m > 0 and an eponential shape at intermediates values of ; (b) The small fraction of agents with λ distributed over the whole interval (0, ), leading to a power-law tail; (c) The cut-off λ M < of the saving propensity distribution inducing the corresponding finite cut-off M of the wealth distribution. III. ACKNOWLEDGEMENTS The author would like to thank all his collaborators. This article took its shape during the lectures given at IPST (Maryland, USA), Bose Institute (Kolkata) and MMV (BHU, Varanasi). APPENDIX A: DETERMINISTIC VERSUS RANDOM GENERATION OF A DISCRETE ENSEMBLE WITH A GIVEN CUMULATIVE DISTRIBUTION We consider how to generate a sequence of N numbers λ i (i =,...,N), assumed in the following eamples to be defined in the interval (0, ), that in the continuous limit (N ) becomes distributed according to a given distribution function g(λ) = dg(λ)/dλ, where G(λ) is the cumulative distribution function. Two methods are considered: () random and (2) deterministic, both based on the inversion of the cumulative distribution function G(λ), with G(0) = 0 and G() =. Despite being equivalent to each other in the continuous limit, for finite N they provide different distributions.. Random etraction A well known and simple method for the random etraction of a variable λ with probability distribution g(λ) = dg(λ)/dλ employs a generator of uniform random numbers. We recall it for completeness. If the variable λ was uniformly distributed in (0, ), the probability to etract a value in the subinterval (λ, λ + dλ) (0, ) would simply be dg = dλ, i.e. it would be given by the interval length itself. Instead, for a variable λ with a generic probability distribution g(λ) = dg(λ)/dλ, the corresponding probability is dg = g(λ)dλ, (A) i.e. the interval dλ is weighted by the probability density g(λ). Note that on the left hand side the probability

6 6 to obtain a value of the cumulative distribution between G and G + dg, corresponding to the probability to find the independent variable between λ and λ + dλ, is equal to the interval dg, just as in the case of a uniformly distributed variable. That is, if G is etracted uniformly in (0, ), then the variable λ obtained by inverting G = G(λ) will be automatically distributed according to the distribution function g(λ). 2. Deterministic etraction The same distribution can be obtained in the limit N through a suitable deterministic assignment of the values λ i which does not make use of random number generators. If the sequence {λ i } is labeled in order of increasing value, i.e. 0 λ < λ 2 < < λ N <, the function of i Λ(i) = λ i, (A2) increases monotonously with i. Then it is possible to uniquely invert Λ(i) and epress i as a function of λ i to define the function G(λ i ) = i N i = 0,...,N, (A3) which represents the fraction of agents with saving propensity less than or equal to λ i, and so by definition, it is the (lower) cumulative distribution function. In fact, in the continuous limit, 0 < G(λ) < for every λ, G(λ 0) 0, and G(λ ). In order to obtain the desired sequence λ i, one has to only invert Eq. (A3) for i =,...,N. Alternatively, this procedure can also be understood in this way: The variable G is still uniformly distributed in (0, ), but instead of being a random variable it takes only and all the discrete values /N, 2/N,..., (one could still shift all these values by λ/2 = /2N to make the distribution more homogeneous in (0, ) but this small correction is neglected here). In doing so, one automatically obtains the most homogeneous λ-distribution available, since in this problem the order of the assignment of the λ-values is not relevant. A few illustrations of the method: (i) Uniform distribution in (0, ): The cumulative distribution function of a variable λ defined and uniformly distributed in (0, ) is G(λ) = λ. Then Eq. (A3) directly provides the values of λ i as λ i = i, i =,...,N. (A4) N (ii) Uniform distribution with a an upper cutoff λ M : In this case the cumulative distribution is G(λ) = λ/λ M for 0 λ < λ M, and G(λ) = for λ M λ <. Equation (A3) gives λ i = i N λ M, i =,...,N. (A5) (iii) Uniform distribution with a lower and an upper cutoff, λ m and λ M : Here the lower cumulative distribution is G = 0 in λ (0, λ m ), G(λ) = (λ λ m )/(λ M λ m ) in λ (λ m, λ M ), and G = in λ (λ M, ). From Eq. (A3) λ i = λ m + i N (λ M λ M ), i =,...,N. (A6) [] Encyclopaedia Britannica, (????). [2] F. Shostak, Quarterly J. Australian Econ. 3, 69 (2000). [3] (????). [4] V. Pareto, Cours d economie politique (Rouge, Lausanne, 897). [5] R. Gibrat, Les Inegalites Economiques (Recueil Sirey, Paris, 93). [6] A. Dragulescu and V. M. Yakovenko, Physica A 299, 23 (200). [7] H. Aoyama, W. Souma, and Y. Fujiwara, Physica A 324, 352 (2003). [8] Y. Fujiwara, W. Souma, H. Aoyama, T. Kaizoji, and M. Aoki, Physics A 32, 598 (2003). [9] E. Bennati, La simulazione statistica nell analisi della distribuzione del reddito: modelli realistici e metodo di Montecarlo (ETS Editrice, Pisa, 988). [0] E. Bennati, Rivista Internazionale di Scienze Economiche e commerciali pp (988). [] E. Bennati, Rassegna di lavori dell ISCO 0, 3 (993). [2] A. Dragulescu and V. M. Yakovenko, Eur. Phys. J. B 7, 723 (2000). [3] A. Chakraborti and B. K. Chakrabarti, Eur. Phys. J. B 7, 67 (2000). [4] A. Chakraborti, Int. J. Mod. Phys. C 3, 35 (2002). [5] M. Patriarca, A. Chakraborti, and K. Kaski, Physica A 340, 334 (2004). [6] M. Patriarca, A. Chakraborti, and K. Kaski, Phys. Rev. E 70, 0604 (2004). [7] A. Dragulescu and V. M. Yakovenko, Eur. Phys. J. B 20, 585 (200). [8] J. C. Ferrero, Physica A 34, 575 (2004). [9] A. C. Silva and V. M. Yakovenko, Europhysics Letters 69, 304 (2005). [20] J. Angle, in Proceedings of the Social Statistics Section of the American Statistical Association (983), p [2] J. Angle, Social Forces 65, 293 (986). [22] J. Angle, J. Math. Soc. 8, 27 (993). [23] A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica Scripta T06, 367 (2003). [24] A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica A 335, 55 (2004). [25] A. Chatterjee, B. K. Chakrabarti, and R. B. Stinchcombe, Master equation for a kinetic model of trading

7 7 market and its analytic solution: cond-mat/ [26] A. Das and S. Yarlagadda, A distribution function analysis of wealth distribution. [27] J. Angle, J. Math. Soc. 26, 27 (2002). [28] P. Repetowicz, S. Hutzler, and P. Richmond, Physica A 356, 64 (2005). [29] M. Patriarca, A. Chakraborti, K. Kaski, and G. Germano, in Econophysics of Wealth Distributions, edited by A. Chatterjee, S.Yarlagadda, and B. K. Chakrabarti (Springer, 2005), p. 93. [30] A. Das and S. Yarlagadda, Physica A 353, 529 (2005). [3] A. Chatterjee, S.Yarlagadda, and B. K. Chakrabarti, eds., Econophysics of Wealth Distributions (Springer, 2005).