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Econophysics, Statistical Mechanics Approach to

Victor M. Yakovenko

Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA

(arXiv:0709.3662, v.1 September 23, 2007, v.3 November 2, 2007)

This is a review article for Encyclopedia of Complexity and System Science, to be published by

Springer http://refworks.springer.com/complexity/ . The terms highlighted in bold in Sec.

I refer to other articles in this Encyclopedia. This paper reviews statistical models for money,

wealth, and income distributions developed in the econophysics literature since late 1990s.

“Money, it’s a gas.” Pink Floyd

Contents

Glossary 1

I. Definition 1

II. Historical Introduction 2

III. Statistical Mechanics of Money Distribution 3

A. The Boltzmann-Gibbs distribution of energy 3

B. Conservation of money 4

C. The Boltzmann-Gibbs distribution of money 5

D. Models with debt 6

E. Proportional money transfers and saving propensity 7

F. Additive versus multiplicative models 8

IV. Statistical Mechanics of Wealth Distribution 9

A. Models with a conserved commodity 9

B. Models with stochastic growth of wealth 10

C. Empirical data on money and wealth distributions 11

V. Data and Models for Income Distribution 12

A. Empirical data on income distribution 12

B. Theoretical models of income distribution 15

VI. Other Applications of Statistical Physics 16

A. Economic temperatures in different countries 16

B. Society as a binary alloy 17

VII. Future Directions, Criticism, and Conclusions 17

A. Future directions 17

B. Criticism from economists 18

C. Conclusions 20

VIII. Bibliography 20

Books and Reviews 24

Glossary

Probability density P (x) is defined so that the prob-ability to find a random variable x in the interval from xto x + dx is equal to P (x) dx.

Cumulative probability C(x) is defined as the in-tegral C(x) =

∫ ∞

xP (x) dx. It gives the probability that

the random variable exceeds a given value x.The Boltzmann-Gibbs distribution gives the

probability of finding a physical system in a state withthe energy ε. Its probability density is given by the ex-ponential function (1).

The Gamma distribution has the probability den-sity given by a product of an exponential function and apower-law function, as in Eq. (9).

The Pareto distribution has the probability densityP (x) ∝ 1/x1+α and the cumulative probability C(x) ∝1/xα given by a power law. These expressions apply onlyfor high enough values of x and not for x → 0.

The Lorenz curve was introduced by Americaneconomist Max Lorenz to describe income and wealthinequality. It is defined in terms of two coordinates x(r)and y(r) given by Eq. (19). The horizontal coordinatex(r) is the fraction of population with income below r,and the vertical coordinate y(r) is the fraction of incomethis population accounts for. As r changes from 0 to ∞,x and y change from 0 to 1, parametrically defining acurve in the (x, y)-plane.

The Gini coefficient G was introduced by Italianstatistician Corrado Gini as a measure of inequality in asociety. It is defined as the area between the Lorenz curveand the straight diagonal line, divided by the area of thetriangle beneath the diagonal line. For perfect equality(everybody has the same income or wealth) G = 0, andfor total inequality (one person has all income or wealth,and the rest have nothing) G = 1.

The Fokker-Planck equation is the partial differ-ential equation (22) that describes evolution in time t ofthe probability density P (r, t) of a random variable r ex-periencing small random changes ∆r during short timeintervals ∆t. It is also known in mathematical literatureas the Kolmogorov forward equation. Diffusion equationis an example of the Fokker-Planck equation.

I. DEFINITION

Econophysics is an interdisciplinary research field ap-plying methods of statistical physics to problems in eco-nomics and finance. The term Econophysics was firstintroduced by the prominent theoretical physicist Eu-gene Stanley in 1995 at the conference Dynamics of Com-plex Systems, which was held in Calcutta (later knownas Kolkata) as a satellite meeting to the STATPHYS–19conference in China [1, 2]. The term appeared in printfor the first time in the paper by Stanley et al. [3] in

2

the proceedings of the Calcutta conference. The paperpresented a manifesto of the new field, arguing that “be-havior of large numbers of humans (as measured, e.g., byeconomic indices) might conform to analogs of the scalinglaws that have proved useful in describing systems com-posed of large numbers of inanimate objects” [3]. Soonthe first econophysics conferences were organized: Inter-national Workshop on Econophysics, Budapest, 1997 andInternational Workshop on Econophysics and StatisticalFinance, Palermo, 1998 [2], and the book An Introduc-

tion to Econophysics [4] was published.

The term Econophysics was introduced by analogywith similar terms, such as Astrophysics, Geophysics,and Biophysics, which describe applications of physicsto different fields. Particularly important is the paral-lel with Biophysics, which studies living creatures, whichstill obey the laws of physics. It should be emphasizedthat Econophysics does not literally apply the laws ofphysics, such as Newton’s laws or quantum mechanics,to humans, but rather uses mathematical methods devel-oped in statistical physics to study statistical propertiesof complex economic systems consisting of a large numberof humans. So, it may be considered as a branch of ap-plied theory of probabilities. However, statistical physicsis distinctly different from mathematical statistics in itsfocus, methods, and results.

Originating from physics as a quantitative science,Econophysics emphasizes quantitative analysis of largeamounts of economic and financial data, which becameincreasingly available with massive introduction of com-puters and Internet. Econophysics distances itself fromthe verbose, narrative, and ideological style of PoliticalEconomy and is closer to Econometrics in its focus.Studying mathematical models of a large number of inter-acting economic agents, Econophysics has much commonground with the Agent-Based Modeling and Sim-

ulation. Correspondingly, it distances itself from therepresentative-agent approach of traditional Economics,which, by definition, ignores statistical and heteroge-neous aspects of the economy.

Two major directions in Econophysics are applicationsto Finance and Economics. Applications to finance aredescribed in a separate article, Econophysics of Finan-

cial Markets, in this Encyclopedia. Observational as-pects are covered in the article Econophysics, Obser-

vational. The present article, Econophysics, Statis-

tical Mechanics Approach to, concentrates primarilyon statistical distributions of money, wealth, and incomeamong interacting economic agents.

Another direction related to Econophysics has been ad-vocated by theoretical physicist Serge Galam since early1980 under the name of Sociophysics [5], with the firstappearance of the term in print in Ref. [6]. It echoes theterm “physique sociale” proposed in the 19th century byAuguste Comte, the founder of Sociology. Unlike Econo-physics, the term Sociophysics did not catch on when firstintroduced, but it is coming back with the popularity ofEconophysics and active promotion by some physicists

[7, 8]. While the principles of both fields have a lot incommon, Econophysics focuses on the narrower subjectof economic behavior of humans, where more quantitativedata is available, whereas Sociophysics studies a broaderrange of social issues. The boundary between Econo-physics and Sociophysics is not sharp, and the two fieldsenjoy a good rapport [9]. A more detailed description ofhistorical development in presented in Sec. II.

II. HISTORICAL INTRODUCTION

Statistical mechanics was developed in the second halfof the 19th century by James Clerk Maxwell, LudwigBoltzmann, and Josiah Willard Gibbs. These physi-cists believed in existence of atoms and developed math-ematical methods for describing their statistical prop-erties, such as the probability distribution of velocitiesof molecules in a gas (the Maxwell-Boltzmann distribu-tion) and the general probability distribution of stateswith different energies (the Boltzmann–Gibbs distribu-tion). There are interesting connections between the de-velopment of statistical physics and statistics of socialphenomena, which were recently brought up by the sci-ence journalist Philip Ball [10, 11].

Collection and study of “social numbers”, such as therates of death, birth, and marriage, was growing pro-gressively since the 17th century [11, Ch. 3]. The term“statistics” was introduced in the 18th century to de-note these studies dealing with the civil “states”, andits practitioners were called “statists”. Popularization ofsocial statistics in the 19th century is particularly cred-ited to the Belgian astronomer Adolphe Quetelet. Be-fore 1850s, statistics was considered an empirical arm ofpolitical economy, but then started to transform into ageneral method of quantitative analysis suitable for alldisciplines. It stimulated physicists to develop statisticalmechanics in the second half of the 19th century.

Rudolf Clausius started development of kinetic theoryof gases, but it was James Clerk Maxwell who madea decisive step of deriving the probability distributionof velocities of molecules in a gas. Historical studiesshow [11, Ch. 3] that, in developing statistical mechan-ics, Maxwell was strongly influenced and encouraged bythe widespread popularity of social statistics at the time.This approach was further developed by Ludwig Boltz-mann, who was very explicit about its origins [11, p. 59]:

“The molecules are like individuals, . . . andthe properties of gases only remain unalteredbecause the number of these molecules whichon the average have a given state is constant.”

In his book “Populare Schrifen” from 1905 [12], Boltz-mann praises Josiah Willard Gibbs for systematic devel-opment of statistical mechanics. Then, Boltzmann says(cited from [13]):

“This opens a broad perspective if we do notonly think of mechanical objects. Let’s con-

3

sider to apply this method to the statistics ofliving beings, society, sociology and so forth.”

(The author is grateful to Prof. Michael E. Fisher forbringing this quote to his attention.)

It is worth noting that many now-famous economistswere originally educated in physics and engineering. Vil-fredo Pareto earned a degree in mathematical sciencesand a doctorate in engineering. Working as a civil engi-neer, he collected statistics demonstrating that distribu-tions of income and wealth in a society follow a powerlaw [14]. He later became a professor of economics atLausanne, where he replaced Leon Walras, also an engi-neer by education. The influential American economistIrving Fisher was a student of Gibbs. However, mostof the mathematical apparatus transferred to economicsfrom physics was that of Newtonian mechanics and clas-sical thermodynamics. It culminated in the neoclassicalconcept of mechanistic equilibrium where the “forces” ofsupply and demand balance each other. The more gen-eral concept of statistical equilibrium largely eluded themainstream economics.

With the time, both physics and economics becamemore formal and rigid in their specializations, and thesocial origin of statistical physics was forgotten. The sit-uation is well summarized by Philip Ball [11, p. 69]:

“Today physicists regard the application ofstatistical mechanics to social phenomena asa new and risky venture. Few, it seems, re-call how the process originated the other wayaround, in the days when physical scienceand social science were the twin siblings of amechanistic philosophy and when it was notin the least disreputable to invoke the habitsof people to explain the habits of inanimateparticles.”

Some physicists and economists attempted to connectthe two disciplines during the 20th century. Theoreticalphysicist Ettore Majorana argued in favor of applyingthe laws of statistical physics to social phenomena in apaper published after his mysterious disappearance [15].Statistical physicist Elliott Montroll co-authored a bookIntroduction to Quantitative Aspects of Social Phenom-ena [16]. Several economists applied statistical physicsto economic problems [17, 18, 19, 20]. An early attemptto bring together the leading theoretical physicists andeconomists at the Santa Fe Institute was not entirely suc-cessful [21]. However, by the late 1990s, the attempts toapply statistical physics to social phenomena finally co-alesced into the robust movements of Econophysics andSociophysics, as described in Sec. I.

The current standing of Econophysics within thePhysics and Economics communities is mixed. Althoughan entry on Econophysics has appeared in the New Pal-grave Dictionary of Economics [22], it is fair to say thatEconophysics is not accepted yet by the mainstreamEconomics. Nevertheless, a number of open-minded,

non-traditional economists have joined this movement,and the number is growing. Under these circum-stances, econophysicists publish most of their papers inphysics journals. The journal Physica A: Statistical Me-

chanics and its Applications emerged as the leader ineconophysics publications and even attracting submis-sions from some bona fide economists. The mainstreamPhysics community is generally sympathetic to Econo-physics, although it is not uncommon for econophysicspapers to be rejected from Physical Review Letters on thegrounds that “it is not physics”. There are regular confer-ence in Econophysics, such as Applications of Physics in

Financial Analysis (sponsored by the European PhysicalSociety), Nikkei Econophysics Symposium, and Econo-

physics Colloquium. Econophysics sessions are includedin the annual meetings of physical societies and statis-tical physics conferences. The overlap with economistsis the strongest in the field of Agent-Based Simulation.Not surprisingly, the conference series WEHIA/ESHIA,which deals with Heterogeneous Interacting Agents, reg-ularly includes sessions on Econophysics.

III. STATISTICAL MECHANICS OF MONEY

DISTRIBUTION

When the modern Econophysics started in the middleof 1990s, its attention was primarily focused on analy-sis of financial markets. However, three influential pa-pers [23, 24, 25], dealing with the subject of money andwealth distributions, were published in year 2000. Theystarted a new direction that is closer to Economics thanFinance and created an expanding wave of follow-up pub-lications. We start reviewing this subject with Ref. [23],whose results are the most closely related to the tradi-tional statistical mechanics in physics.

A. The Boltzmann-Gibbs distribution of energy

The fundamental law of the equilibrium statistical me-chanics is the Boltzmann-Gibbs distribution. It statesthat the probability P (ε) of finding a physical system orsub-system in a state with the energy ε is given by theexponential function

P (ε) = c e−ε/T , (1)

where T is the temperature, and c is a normalizing con-stant [26]. Here we set the Boltzmann constant kB tounity by choosing the energy units for measuring thephysical temperature T . Then, the expectation value ofany physical variable x can be obtained as

〈x〉 =

∑

k xke−εk/T

∑

k e−εk/T, (2)

where the sum is taken over all states of the system.Temperature is equal to the average energy per particle:T ∼ 〈ε〉, up to a numerical coefficient of the order of 1.

4

Eq. (1) can be derived in different ways [26]. All deriva-tions involve the two main ingredients: statistical char-acter of the system and conservation of energy ε. Oneof the shortest derivation can be summarized as follows.Let us divide the system into two (generally unequal)parts. Then, the total energy is the sum of the parts:ε = ε1 + ε2, whereas the probability is the product ofprobabilities: P (ε) = P (ε1)P (ε2). The only solution ofthese two equations is the exponential function (1).

A more sophisticated derivation, proposed by Boltz-mann himself, uses the concept of entropy. Let us con-sider N particles with the total energy E. Let us dividethe energy axis into small intervals (bins) of the width∆ε and count the number of particles Nk having the en-ergies from εk to εk + ∆ε. The ratio Nk/N = Pk givesthe probability for a particle to have the energy εk. Letus now calculate the multiplicity W , which is the numberof permutations of the particles between different energybins such that the occupation numbers of the bins donot change. This quantity is given by the combinatorialformula in terms of the factorials

W =N !

N1! N2! N3! . . .. (3)

The logarithm of multiplicity of called the entropy S =lnW . In the limit of large numbers, the entropy perparticle can be written in the following form using theStirling approximation for the factorials

S

N= −

∑

k

Nk

Nln

(

Nk

N

)

= −∑

k

Pk lnPk. (4)

Now we would like to find what distribution of particlesbetween different energy states has the highest entropy,i.e. the highest multiplicity, provided that the total en-ergy of the system, E =

∑

k Nkεk, has a fixed value.Solution of this problem can be easily obtained using themethod of Lagrange multipliers [26], and the answer givesthe exponential distribution (1).

The same result can be derived from the Ergodic

Theory, which says that the many-body system occu-pies all possible states of a given total energy with equalprobabilities. Then it is straightforward to show [27, 28]that the probability distribution of the energy of an in-dividual particle is given by Eq. (1).

B. Conservation of money

The derivations outlined in Sec. III.A are very generaland use only the statistical character of the system andthe conservation of energy. So, one may expect that theexponential Boltzmann-Gibbs distribution (1) may applyto other statistical systems with a conserved quantity.

The economy is a big statistical system with millionsof participating agents, so it is a promising target forapplications of statistical mechanics. Is there a con-served quantity in economy? The paper [23] argues that

such a conserved quantity is money m. Indeed, theordinary economic agents can only receive money fromand give money to other agents. They are not permit-ted to “manufacture” money, e.g. to print dollar bills.When one agent i pays money ∆m to another agent jfor some goods or services, the money balances of theagents change as follows

mi → m′i = mi − ∆m,

mj → m′j = mj + ∆m. (5)

The total amount of money of the two agents before andafter transaction remains the same

mi + mj = m′i + m′

j , (6)

i.e. there is a local conservation law for money. The rule(5) for the transfer of money is analogous to the transferof energy from one molecule to another in molecular col-lisions in a gas, and Eq. (6) is analogous to conservationof energy in such collisions.

Addressing some misunderstandings developed in eco-nomic literature [29, 30, 31, 32], it should be emphasizedthat, in the model of Ref. [23], the transfer of money fromone agent to another happens voluntarily, as a paymentfor goods and services in a market economy. However,the model only keeps track of money flow, but does notkeep track of what kind of goods and service are deliv-ered. One reason for this is that many goods, e.g. foodand other supplies, and most services, e.g. getting a hair-cut or going to a movie, are not tangible and disappearafter consumption. Because they are not conserved andalso are measured in different physical units, it is notvery practical to keep track of them. In contrast, moneyis measured in the same unit (within a given country witha single currency) and is conserved in transactions, so itis straightforward to keep track of money flow.

Unlike, ordinary economic agents, a central bank or acentral government can inject money into the economy.This process is analogous to an influx of energy into a sys-tem from external sources, e.g. the Earth receives energyfrom the Sun. Dealing with these situations, physicistsstart with an idealization of a closed system in thermalequilibrium and then generalize to an open system sub-ject to an energy flux. As long as the rate of money influxfrom central sources is slow compared with relaxationprocesses in economy and does not cause hyperinflation,the system is in quasi-stationary statistical equilibriumwith slowly changing parameters. This situation is anal-ogous to heating a kettle on a gas stove slowly, where thekettle has a well-defined, but slowly increasing tempera-ture at any moment of time.

Another potential problem with conservation of moneyis debt. This issue is discussed in more detail in Sec.III.D. As a starting point, Ref. [23] first considered sim-ple models, where debt is not permitted. This means thatmoney balances of agents cannot go below zero: mi ≥ 0for all i. Transaction (5) takes place only when an agenthas enough money to pay the price: mi ≥ ∆m, otherwise

5

the transaction does not take place. If an agent spendsall money, the balance drops to zero mi = 0, so the agentcannot buy any goods from other agents. However, thisagent can still produce goods or services, sell them toother agents, and receive money for that. In real life,cash balance dropping to zero is not at all unusual forpeople who live from paycheck to paycheck.

The conservation law is the key feature for the success-ful functioning of money. If the agents were permittedto “manufacture” money, they would be printing moneyand buying all goods for nothing, which would be a dis-aster. The physical medium of money is not essential, aslong as the conservation law is enforced. Money may bein the form of paper cash, but today it is more often rep-resented by digits in computerized bank accounts. Theconservation law is the fundamental principle of account-ing, whether in the single-entry or the double-entry form.More discussion of banks and debt is given in Sec. III.D.

C. The Boltzmann-Gibbs distribution of money

Having recognized the principle of money conserva-tion, Ref. [23] argued that the stationary distribution ofmoney should be given by the exponential Boltzmann-Gibbs function analogous to Eq. (1)

P (m) = c e−m/Tm . (7)

Here c is a normalizing constant, and Tm is the “moneytemperature”, which is equal to the average amount ofmoney per agent: T = 〈m〉 = M/N , where M is the totalmoney, and N is the number of agents.

To verify this conjecture, Ref. [23] performed agent-based computer simulations of money transfers betweenagents. Initially all agents were given the same amountof money, say $1000. Then, a pair of agents (i, j) wasrandomly selected, the amount ∆m was transferred fromone agent to another, and the process was repeated manytimes. Time evolution of the probability distribution ofmoney P (m) can be seen in computer animation videoson the Web pages [33, 34]. After a transitory period,money distribution converges to a stationary form shownin Fig. 1. As expected, the distribution is very well fittedby the exponential function (7).

Several different rules for ∆m were considered in Ref.[23]. In one model, the transferred amount was fixedto a constant ∆m = $1. Economically, it means thatall agents were selling their products for the same price∆m = $1. Computer animation [33] shows that the ini-tial distribution of money first broadens to a symmet-ric, Gaussian curve, characteristic for a diffusion process.Then, the distribution starts to pile up around the m = 0state, which acts as the impenetrable boundary, becauseof the imposed condition m ≥ 0. As a result, P (m) be-comes skewed (asymmetric) and eventually reaches thestationary exponential shape, as shown in Fig. 1. Theboundary at m = 0 is analogous to the ground state en-ergy in statistical physics. Without this boundary con-

dition, the probability distribution of money would notreach a stationary state. Computer animation [33, 34]also shows how the entropy of money distribution, de-fined as S/N = −

∑

k P (mk) lnP (mk), grows from theinitial value S = 0, when all agents have the same money,to the maximal value at the statistical equilibrium.

While the model with ∆m = 1 is very simple and in-structive, it is not very realistic, because all prices aretaken to be the same. In another model considered inRef. [23], ∆m in each transaction is taken to be a randomfraction of the average amount of money per agent, i.e.∆m = ν(M/N), where ν is a uniformly distributed ran-dom number between 0 and 1. The random distributionof ∆m is supposed to represent the wide variety of pricesfor different products in real economy. It reflects the factthat agents buy and consume many different types ofproducts, some of them simple and cheap, some sophis-ticated and expensive. Moreover, different agents like toconsume these products in different quantities, so thereis variation of paid amounts ∆m, even though unit priceof the same product is constant. Computer simulationof this model produces exactly the same stationary dis-tribution (7), as in the first model. Computer animationfor this model is also available on the Web page [33].

The final distribution is universal despite different rulesfor ∆m. To amplify this point further, Ref. [23] alsoconsidered a toy model, where ∆m was taken to be arandom fraction of average amount of money of the twoagents: ∆m = ν(mi + mj)/2. This model produced thesame stationary distribution (7), as the two other models.

The pairwise models of money transfer are attractivein their simplicity, but they represent a rather primitivemarket. Modern economy is dominated by big firms,which consist of many agents, so Ref. [23] also studieda model with firms. One agent at a time is appointed

0 1000 2000 3000 4000 5000 60000

2

4

6

8

10

12

14

16

18

Money, m

Pro

babi

lity,

P(m

)

N=500, M=5*105, time=4*105.

⟨m⟩, T

0 1000 2000 30000

1

2

3

Money, m

log

P(m

)

FIG. 1 Histogram and points: Stationary probability distri-bution of money P (m) obtained in agent-based computer sim-ulations. Solid curves: Fits to the Boltzmann-Gibbs law (7).Vertical lines: the initial distribution of money. Reproducedfrom Ref. [23].

6

to become a “firm”. The firm borrows capital K fromanother agent and returns it with an interest hK, hiresL agents and pays them wages W , manufactures Q itemsof a product, sells them to Q agents at a price R, andreceives the profit F = RQ − LW − hK. All of theseagents are randomly selected. Parameters of the modelare optimized following a procedure from economics text-books [35]. The aggregate demand-supply curve for theproduct is taken in the form R(Q) = v/Qη, where Qis the quantity consumers would buy at a price R, andη and v are some parameters. The production func-tion of the firm has the traditional Cobb-Douglas form:Q(L, K) = LχK1−χ, where χ is a parameter. Then theprofit of the firm F is maximized with respect to K andL. The net result of the firm activity is a many-bodytransfer of money, which still satisfies the conservationlaw. Computer simulation of this model generates thesame exponential distribution (7), independently of themodel parameters. The reasons for the universality ofthe Boltzmann-Gibbs distribution and its limitations arediscussed from a different perspective in Sec. III.F.

Well after the paper [23] appeared, Italian econophysi-cists [36] found that similar ideas were published ear-lier in obscure journals in Italian by Eleonora Bennati[37, 38]. They proposed to call these models the Bennati-Dragulescu-Yakovenko (BDY) game [39]. The Boltz-mann distribution was independently applied to socialsciences by Jurgen Mimkes using the Lagrange principleof maximization with constraints [40, 41]. The exponen-tial distribution of money was also found in Ref. [42] us-ing a Markov chain approach to strategic market games.A long time ago, Benoit Mandelbrot observed [43, p 83]:

“There is a great temptation to consider theexchanges of money which occur in economicinteraction as analogous to the exchanges ofenergy which occur in physical shocks be-tween gas molecules.”

He realized that this process should result in the expo-nential distribution, by analogy with the barometric dis-tribution of density in the atmosphere. However, he dis-carded this idea, because it does not produce the Paretopower law, and proceeded to study the stable Levy distri-butions. Ironically, the actual economic data, discussedin Secs. IV.C and V.A, do show the exponential distribu-tion for the majority of population. Moreover, the datahave finite variance, so the stable Levy distributions arenot applicable because of their infinite variance.

D. Models with debt

Now let us discuss how the results change when debt ispermitted. Debt may be considered as negative money.When an agent borrows money from a bank (consideredhere as a big reservoir of money), the cash balance ofthe agent (positive money) increases, but the agent alsoacquires a debt obligation (negative money), so the total

balance (net worth) of the agent remains the same, andthe conservation law of total money (positive and neg-ative) is satisfied. After spending some cash, the agentstill has the debt obligation, so the money balance ofthe agent becomes negative. Any stable economic sys-tem must have a mechanism preventing unlimited bor-rowing and unlimited debt. Otherwise, agents can buyany products without producing anything in exchangeby simply going into unlimited debt. The exact mecha-nisms of limiting debt in real economy are complicatedand obscured. Ref. [23] considered a simple model wherethe maximal debt of any agent is limited by a certainamount md. This means that the boundary conditionmi ≥ 0 is now replaced by the condition mi ≥ −md forall agents i. Setting interest rates on borrowed money tobe zero for simplicity, Ref. [23] performed computer sim-ulations of the models described in Sec. III.C with thenew boundary condition. The results are shown in Fig.2. Not surprisingly, the stationary money distributionagain has the exponential shape, but now with the newboundary condition at m = −md and the higher moneytemperature Td = md + M/N . By allowing agents to gointo debt up to md, we effectively increase the amountof money available to each agent by md. So, the moneytemperature, which is equal to the average amount of ef-fectively available money per agent, increases. A modelwith non-zero interest rates was also studied in Ref. [23].

We see that debt does not violate the conservation lawof money, but rather modifies boundary conditions forP (m). When economics textbooks describe how “bankscreate money” or “debt creates money” [35], they countonly positive money (cash) as money, but do not countliabilities (debt obligations) as negative money. Withsuch a definition, money is not conserved. However, ifwe include debt obligations in the definition of money,

0 2000 4000 6000 80000

2

4

6

8

10

12

14

16

18

Money, m

Pro

babi

lity,

P(m

)

N=500, M=5*105, time=4*105.

Model without debt, T=1000

Model with debt, T=1800

FIG. 2 Histograms: Stationary distributions of money withand without debt. The debt is limited to md = 800. Solidcurves: Fits to the Boltzmann-Gibbs laws with the “temper-atures” T = 1800 and T = 1000. Reproduced from Ref. [23].

7

then the conservation law is restored. This approach is inagreement with the principles of double-entry accounting,which records both assets and debts. Debt obligationsare as real as positive cash, as many borrowers painfullyrealized in their experience. A more detailed study ofpositive and negative money and book-keeping from thepoint of view of Econophysics is presented in a series ofpapers by physicist Dieter Braun [44, 45, 46].

One way of limiting the total debt in economy is theso-called required reserve ratio r [35]. Every bank is re-quired by law to set aside the fraction r of money de-posited into the bank, and this reserved money cannotbe further loaned. If the initial amount of money in thesystem (the money base) is M0, then with loans and bor-rowing the total amount of positive money available tothe agents increases to M = M0/r, where the factor 1/ris called the money multiplier [35]. This is how “bankscreate money”. Where do this extra money come from?They come from the increase of the total debt in the sys-tem. The maximal total debt is equal to D = M0/r−M0

and is limited by the factor r. When the debt is maxi-mal, the total amounts of positive, M0/r, and negative,M0(1 − r)/r, money circulate between the agents in thesystem, so there are effectively two conservation laws foreach of them [47]. Thus, we expect to see the exponentialdistributions of positive and negative money character-ized by two different temperatures: T+ = M0/rN andT− = M0(1 − r)/rN . This is exactly what was found incomputer simulations in Ref. [47] shown in Fig. 3. Similartwo-sided distributions were also found in Ref. [45].

-50 0 50 100 150

0

10

20

30

40

-50 0 50 100 150

0.1

1

10

log

(P(m

)) (

1x

10

-3)

Monetary Wealth,m

Pro

babi

lity,

P(m

) (1x

10-3)

Monetary Wealth,m

FIG. 3 The stationary distribution of money for the requiredreserve ratio r = 0.8. The distribution is exponential for pos-itive and negative money with different “temperatures” T+

and T−, as illustrated by the inset in log-linear scale. Repro-duced from Ref. [47].

E. Proportional money transfers and saving propensity

In the models of money transfer considered thus far,the transferred amount ∆m is typically independent ofthe money balances of agents. A different model was in-troduced in physics literature earlier [48] under the nameMultiplicative Asset Exchange model. This model alsosatisfies the conservation law, but the transferred amountof money is a fixed fraction γ of payer’s money in Eq. (5):

∆m = γmi. (8)

The stationary distribution of money in this model,shown in Fig. 4 with an exponential function, is close,but not exactly equal, to the Gamma distribution:

P (m) = c mβ e−m/T . (9)

Eq. (9) differs from Eq. (7) by the power-law prefactormβ. From the Boltzmann kinetic equation (discussed inmore detail in Sec. III.F), Ref. [48] derived a formularelating the parameters γ and β in Eqs. (8) and (9):

β = −1 − ln 2/ ln(1 − γ). (10)

When payers spend a relatively small fraction of theirmoney γ < 1/2, Eq. (10) gives β > 0, so the low-moneypopulation is reduced and P (m → 0) = 0, as in Fig. 4.

Later, Thomas Lux brought to the attention of physi-cists [30] that essentially the same model, called the In-equality Process, was introduced and studied much ear-lier by the sociologist John Angle [49, 50, 51, 52, 53], seealso review [54] for additional references. While Ref. [48]did not give much justification for the proportionality law(8), Angle [49] connected for this rule with the SurplusTheory of Social Stratification [55], which argues thatinequality in human society develops when people can

0 1000 2000 3000 4000 50000

2

4

6

8

10

12

14

16

Money, m

Pro

babi

lity,

P(m

)

N=500, M=5*105, α=1/3.

FIG. 4 Histogram: Stationary probability distribution ofmoney in the multiplicative random exchange model (8) forγ = 1/3. Solid curve: The exponential Boltzmann-Gibbs law.Reproduced from Ref. [23].

8

produce more than necessary for minimal subsistence.This additional wealth (surplus) can be transferred fromoriginal producers to other people, thus generating in-equality. In the first paper by Angle [49], the parameterγ was randomly distributed, and another parameter δgave a higher probability of winning to the agent with ahigher money balance in Eq. (5). However, in the follow-ing papers, he simplified the model to a fixed γ (denotedas ω by Angle) and equal probabilities of winning forhigher- and lower-balance agents, which makes it com-pletely equivalent to the model of Ref. [48]. Angle alsoconsidered a model [53, 54] where groups of agents havedifferent values of γ simulating the effect of education andother “human capital”. All of these models generate aGamma-like distribution, well approximated by Eq. (9).

Another model with an element of proportionality wasproposed in Ref. [24]. (This paper originally appeared asa follow-up preprint cond-mat/0004256 to the preprintcond-mat/0001432 of Ref. [23].) In this model, theagents set aside (save) some fraction of their money λmi,whereas the rest of their money balance (1 − λ)mi be-comes available for random exchanges. Thus, the rule ofexchange (5) becomes

m′i = λmi + ξ(1 − λ)(mi + mj),

m′j = λmj + (1 − ξ)(1 − λ)(mi + mj). (11)

Here the coefficient λ is called the saving propensity, andthe random variable ξ is uniformly distributed between 0and 1. It was pointed out in Ref. [54] that, by the changeof notation λ → (1 − γ), Eq. (11) can be transformed tothe same form as Eq. (8), if the random variable ξ takesonly discrete values 0 and 1. Computer simulations [24]of the model (11) found a stationary distribution closeto the Gamma distribution (9). It was shown that theparameter β is related to the saving propensity λ by theformula β = 3λ/(1 − λ) [36, 56, 57, 58]. For λ 6= 0,agents always keep some money, so their balances nevergo to zero and P (m → 0) = 0, whereas for λ = 0 thedistribution becomes exponential.

In the subsequent papers by the Kolkata school [1] andrelated papers, the case of random saving propensity wasstudied. In these models, the agents are assigned randomparameters λ drawn from a uniform distribution between0 and 1 [59]. It was found that this model produces apower-law tail P (m) ∝ 1/m2 at high m. The reasonsfor stability of this law were understood using the Boltz-mann kinetic equation [58, 60, 61], but most elegantlyin the mean-field theory [62]. The fat tail originatesfrom the agents whose saving propensity is close to 1,who hoard money and do not give it back [36, 63]. Aninteresting matrix formulation of the problem was pre-sented in Ref. [64]. Ref. [65] studied relaxation rate in themoney transfer models. Ref. [23] studied a model withtaxation, which also has an element of proportionality.The Gamma distribution was also studied for conserva-tive models within a simple Boltzmann approach in Ref.[66] and using much more complicated rules of exchangein Ref. [67, 68].

F. Additive versus multiplicative models

The stationary distribution of money (9) for the mod-els of Sec. III.E is different from the simple exponentialformula (7) found for the models of Sec. III.C. The originof this difference can be understood from the Boltzmannkinetic equation [26, 69]. This equation describes timeevolution of the distribution function P (m) due to pair-wise interactions:

dP (m)

dt=

∫∫

{−f[m,m′]→[m−∆,m′+∆]P (m)P (m′) (12)

+f[m−∆,m′+∆]→[m,m′]P (m − ∆)P (m′ + ∆)} dm′ d∆.

Here f[m,m′]→[m−∆,m′+∆] is the probability of transfer-ring money ∆ from an agent with money m to an agentwith money m′ per unit time. This probability, multi-plied by the occupation numbers P (m) and P (m′), givesthe rate of transitions from the state [m, m′] to the state[m−∆, m′ + ∆]. The first term in Eq. (12) gives the de-population rate of the state m. The second term in Eq.(12) describes the reversed process, where the occupationnumber P (m) increases. When the two terms are equal,the direct and reversed transitions cancel each other sta-tistically, and the probability distribution is stationary:dP (m)/dt = 0. This is the principle of detailed balance.

In physics, the fundamental microscopic equations ofmotion of particles obey the time-reversal symmetry.This means that the probabilities of the direct and re-versed processes are exactly equal:

f[m,m′]→[m−∆,m′+∆] = f[m−∆,m′+∆]→[m,m′]. (13)

When Eq. (13) is satisfied, the detailed balance condi-tion for Eq. (12) reduces to the equation P (m)P (m′) =P (m − ∆)P (m′ + ∆), because the factors f cancels out.The only solution of this equation is the exponential func-tion P (m) = c exp(−m/Tm), so the Boltzmann-Gibbsdistribution is the stationary solution of the Boltzmannkinetic equation (12). Notice that the transition prob-abilities (13) are determined by the dynamical rules ofthe model, but the equilibrium Boltzmann-Gibbs distri-bution does not depend on the dynamical rules at all.This is the origin of the universality of the Boltzmann-Gibbs distribution. It shows that it may be possible tofind out the stationary distribution without knowing de-tails of the dynamical rules (which are rarely known verywell), as long as the symmetry condition (13) is satisfied.

The models considered in Sec. III.C have the time-reversal symmetry. The model with the fixed moneytransfer ∆ has equal probabilities (13) of transferringmoney from an agent with the balance m to an agentwith the balance m′ and vice versa. This is also truewhen ∆ is random, as long as the probability distributionof ∆ is independent of m and m′. Thus, the stationarydistribution P (m) is always exponential in these models.

However, there is no fundamental reason to expect thetime-reversal symmetry in economics, so Eq. (13) may

9

be not valid. In this case, the system may have a non-exponential stationary distribution or no stationary dis-tribution at all. In the model (8), the time-reversal sym-metry is broken. Indeed, when an agent i gives a fixedfraction γ of his money mi to an agent with the balancemj , their balances become (1 − γ)mi and mj + γmi. Ifwe try to reverse this process and appoint the agent jto be the payer and to give the fraction γ of her money,γ(mj + γmi), to the agent i, the system does not returnto the original configuration [mi, mj]. As emphasized byAngle [54], the payer pays a deterministic fraction of hismoney, but the receiver receives a random amount froma random agent, so their roles are not interchangeable.Because the proportional rule typically violates the time-reversal symmetry, the stationary distribution P (m) inmultiplicative models is typically not exactly exponen-tial.1 Making transfer dependent on the money balanceof the payer effectively introduces a Maxwell’s demon inthe model. That is why the stationary distribution is notexponential, and, thus, does not maximize entropy (4).Another view on the time-reversal symmetry in economicdynamics is presented in Ref. [70].

These examples show that the Boltzmann-Gibbs dis-tribution does not hold for any conservative model. How-ever, it is universal in a limited sense. For a broad classof models that have the time-reversal symmetry, the sta-tionary distribution is exponential and does not dependon details of a model. Conversely, when the time-reversalsymmetry is broken, the distribution may depend onmodel details. The difference between these two classesof models may be rather subtle. Deviations from theBoltzmann-Gibbs law may occur only if the transitionrates f in Eq. (13) explicitly depend on the agents moneym or m′ in an asymmetric manner. Ref. [23] performed acomputer simulation where the direction of payment wasrandomly selected in advance for every pair of agents(i, j). In this case, money flows along directed links be-tween the agents: i→ j → k, and the time-reversal sym-metry is strongly violated. This model is closer to thereal economy, where one typically receives money froman employer and pays it to a grocery store. Neverthe-less, the Boltzmann-Gibbs distribution was found in thismodel, because the transition rates f do not explicitlydepend on m and m′ and do not violate Eq. (13).

In the absence of detailed knowledge of real micro-scopic dynamics of economic exchanges, the semiuniver-sal Boltzmann-Gibbs distribution (7) is a natural startingpoint. Moreover, the assumption of Ref. [23] that agentspay the same prices ∆m for the same products, indepen-dent of their money balances m, seems very appropriatefor the modern anonymous economy, especially for pur-chases on the Internet. There is no particular empirical

1 However, when ∆m is a fraction of the total money mi+mj of the

two agents, the model is time-reversible and has the exponential

distribution, as discussed in Sec. III.C.

evidence for the proportional rules (8) or (11). However,the difference between the additive (7) and multiplicative(9) distributions may be not so crucial after all. From themathematical point of view, the difference is in the im-plementation of the boundary condition at m = 0. Inthe additive models of Sec. III.C, there is a sharp cut-offof P (m) at m = 0. In the multiplicative models of Sec.III.E, the balance of an agent never reaches m = 0, soP (m) vanishes at m → 0 in a power-law manner. At thesame time, P (m) decreases exponentially for large m forboth models.

By further modifying the rules of money transfer andintroducing more parameters in the models it is possi-ble to obtain even more complicated distributions [71].However, one can argue that parsimony is the virtue ofa good mathematical model, not the abundance of ad-ditional assumptions and parameters, whose correspon-dence to reality is hard to verify.

IV. STATISTICAL MECHANICS OF WEALTH

DISTRIBUTION

In the econophysics literature on exchange models, theterms money and wealth are often used interchangeably.However, economists emphasize the difference betweenthese two concepts. In this Section, we review the modelsof wealth distribution, as opposed to money distribution.

A. Models with a conserved commodity

What is the difference between money and wealth? Oncan argue [23] that wealth wi is equal to money mi plusthe other property that an agent i has. The latter mayinclude durable material property, such as houses andcars, and financial instruments, such as stocks, bonds,and options. Money (paper cash, bank accounts) is gen-erally liquid and countable. However, the other propertyis not immediately liquid and has to be sold first (con-verted into money) to be used for other purchases. Inorder to estimate the monetary value of property, oneneeds to know the price p. In a simplest model, let usconsider just one type of property, say stocks s. Thenthe wealth of an agent i is given by the formula

wi = mi + p si. (14)

It is assumed that the price p is common for all agentsand is established by some kind of market process, suchas an auction, and may change in time.

It is reasonable to start with a model where both thetotal money M =

∑

i mi and the total stock S =∑

i si

are conserved [72, 73, 74]. The agents pay money to buystock and sell stock to get money, and so on. AlthoughM and S are conserved, the total wealth W =

∑

i wi

is generally not conserved, because of price fluctuation[73] in Eq. (14). This is an important difference from themoney transfers models of Sec. III. Here the wealth wi

10

of an agent i, not participating in any transactions, maychange when transactions between other agents establisha new price p. Moreover, the wealth wi of an agent i doesnot change after a transaction with an agent j. Indeed,in exchange for paying money ∆m, the agent i receivesthe stock ∆s = ∆m/p, so her total wealth (14) remainsthe same. In principle, the agent can instantaneously sellthe stock back at the same price and recover the moneypaid. If the price p never changes, then the wealth wi ofeach agent remains constant, despite transfers of moneyand stock between agents.

We see that redistribution of wealth in this model isdirectly related to price fluctuations. One mathematicalmodel of this process was studied in Ref. [75]. In thismodel, the agents randomly change preferences for thefraction of their wealth invested in stocks. As a result,some agents offer stock for sale and some want to buyit. The price p is determined from the market-clearingauction matching supply and demand. Ref. [75] demon-strated in computer simulations and proved analyticallyusing the theory of Markov processes that the stationarydistribution P (w) of wealth w in this model is given bythe Gamma distribution, as in Eq. (9). Various modifica-tions of this model [30], such as introducing monopolisticcoalitions, do not change this result significantly, whichshows robustness of the Gamma distribution. For modelswith a conserved commodity, Ref. [73] found the Gammadistribution for a fixed saving propensity and a powerlaw tail for a distributed saving propensity.

Another model with conserved money and stock wasstudied in Ref. [76] for an artificial stock market wheretraders follow different investment strategies: random,momentum, contrarian, and fundamentalist. Wealth dis-tribution in the model with random traders was foundhave a power-law tail P (w) ∼ 1/w2 for large w. How-ever, unlike in most other simulation, where all agentsinitially have equal balances, here the initial money andstock balances of the agents were randomly populatedaccording to a power law with the same exponent. Thisraises a question whether the observed power-law distri-bution of wealth is an artifact of the initial conditions,because equilibrization of the upper tail may take a verylong simulation time.

B. Models with stochastic growth of wealth

Although the total wealth W is not exactly conservedin the models considered in Sec. IV.A, nevertheless Wremains constant on average, because the total money Mand stock S are conserved. A different model for wealthdistribution was proposed in Ref. [25]. In this model,time evolution of the wealth wi of an agent i is given bythe stochastic differential equation

dwi

dt= ηi(t)wi +

∑

j( 6=i)

Jijwj −∑

j( 6=i)

Jjiwi, (15)

where ηi(t) is a Gaussian random variable with the mean〈η〉 and the variance 2σ2. This variable represents growthor loss of wealth of an agent due to investment in stockmarket. The last two terms describe transfer of wealthbetween different agents, which is taken to be propor-tional to the wealth of the payers with the coefficients Jij .So, the model (15) is multiplicative and invariant underthe scale transformation wi → Zwi. For simplicity, theexchange fractions are taken to be the same for all agents:Jij = J/N for all i 6= j, where N is the total number ofagents. In this case, the last two terms in Eq. (15) canbe written as J(〈w〉 − wi), where 〈w〉 =

∑

i wi/N is theaverage wealth per agent. This case represents a “mean-field” model, where all agents feel the same environment.It can be easily shown that the average wealth increases

in time as 〈w〉t = 〈w〉0e(〈η〉+σ2)t. Then, it makes more

sense to consider the relative wealth wi = wi/〈w〉t. Eq.(15) for this variable becomes

dwi

dt= (ηi(t) − 〈η〉 − σ2) wi + J(1 − wi). (16)

The probability distribution P (w, t) for the stochasticdifferential equation (16) is governed by the Fokker-Planck equation

∂P

∂t=

∂[J(w − 1) + σ2w]P

∂w+ σ2 ∂

∂w

(

w∂(wP )

∂w

)

. (17)

The stationary solution (∂P/∂t = 0) of this equation isgiven by the following formula

P (w) = ce−J/σ2w

w2+J/σ2. (18)

The distribution (18) is quite different from theBoltzmann-Gibbs (7) and Gamma (9) distributions. Eq.(18) has a power-law tail at large w and a sharp cutoff atsmall w. Eq. (15) is a version of the generalized Lotka-Volterra model, and the stationary distribution (18) wasalso obtained in Ref. [77, 78]. The model was generalizedto include negative wealth in Ref. [79].

Ref. [25] used the mean-field approach. A similar resultwas found for a model with pairwise interaction betweenagents in Ref. [80]. In this model, wealth is transferredbetween the agents using the proportional rule (8). Inaddition, wealth the agents increases by the factor 1 + ζin each transaction. This factor is supposed to reflectcreation of wealth in economic interactions. Because thetotal wealth in the system increases, it makes sense toconsider the distribution of relative wealth P (w). In thelimit of continuous trading, Ref. [80] found the same sta-tionary distribution (18). This result was reproducedusing a mathematically more involved treatment of thismodel in Ref. [81]. Numerical simulations of the modelswith stochastic noise η in Ref. [67, 68] also found a powerlaw tail for large w.

Let us contrast the models discussed in Secs. IV.A andIV.B. In the former case, where money and commodity

11

are conserved and wealth does not grow, the distribu-tion of wealth is given by the Gamma distribution withthe exponential tail for large w. In the latter models,wealth grows in time exponentially, and the distributionof relative wealth has a power law tail for large w. Theseresults suggest that the presence of a power-law tail isa nonequilibrium effect that requires constant growth orinflation of economy, but disappears for a closed systemwith conservation laws.

Refs. [82, 83] also give reviews of the discussed mod-els. For lack of space, we omit discussion of models withwealth condensation [25, 48, 84, 85, 86], where few agentsaccumulate a finite fraction of total wealth, and studies ofwealth distribution on networks [87, 88, 89, 90]. Here wediscussed the models with long-range interaction, whereany agent can exchange money and wealth with any otheragent. A local model, where agents trade only with thenearest neighbors, was studied in Ref. [91].

C. Empirical data on money and wealth distributions

It would be very interesting to compare theoretical re-sults for money and wealth distributions in various mod-els with empirical data. Unfortunately, such empiricaldata is difficult to find. Unlike income, which is discussedin Sec. V, wealth is not routinely reported by the major-ity of individuals to the government. However, in manycountries, when a person dies, all assets must be reportedfor the purpose of inheritance tax. So, in principle, thereexist good statistics of wealth distribution among deadpeople, which, of course, is different from the wealth dis-tribution among live people. Using an adjustment proce-dure based on the age, gender, and other characteristicsof the deceased, the British tax agency, the Inland Rev-enue (IR), reconstructed wealth distribution of the wholepopulation of the United Kingdom [92]. Fig. 5 shows theUK data for 1996 reproduced from Ref. [93]. The figureshows the cumulative probability C(w) =

∫ ∞

wP (w′) dw′

as a function of the personal net wealth w, which is com-posed of assets (cash, stocks, property, household goods,etc.) and liabilities (mortgages and other debts). Be-cause statistical data is usually reported at non-uniformintervals of w, it is more practical to plot the cumulativeprobability distribution C(w) rather than its derivative,the probability density P (w). Fortunately, when P (w)is an exponential or a power-law function, then C(w) isalso an exponential or a power-law function.

The main panel in Fig. 5 is plotted in the log-log scale,where a straight line represents a power-law dependence.The figure shows that the distribution follows a power lawC(w) ∝ 1/wα with the exponent α = 1.9 for the wealthgreater than about 100 k£. The inset in Fig. 5 shows thedata in the log-linear scale, where a straight line repre-sents an exponential dependence. We observe that below100 k£ the data is well fitted by the exponential distri-bution C(w) ∝ exp(−w/Tw) with the effective “wealthtemperature” Tw = 60 k£ (which corresponds to the me-

dian wealth of 41 k£). So, the distribution of wealth ischaracterized by the Pareto power law in the upper tailof the distribution and the exponential Boltzmann-Gibbslaw in the lower part of the distribution for the great ma-jority (about 90%) of the population. Similar results arefound for the distribution of income, as discussed in Sec.V. One may speculate that wealth distribution in thelower part is dominated by distribution of money, be-cause these people do not have other significant assets,so the results of Sec. III give the Boltzmann-Gibbs law.On the other hand, the upper tail of wealth distributionis dominated by investment assess, where the results ofSec. IV.B give the Pareto law. The power law was studiedby many researchers for the upper-tail data, such as theForbes list of 400 richest people [94, 95], but much less at-tention was paid to the lower part of wealth distribution.Curiously, Ref. [96] found that wealth distribution in theancient Egyptian society is consistent with Eq. (18).

For direct comparison with the results of Sec. III, itwould be very interesting to find data on the distributionof money, as opposed to the distribution of wealth. Mak-ing a reasonable assumption that most people keep mostof their money in banks, the distribution of money canbe approximated by the distribution of balances on bankaccounts. (Balances on all types of bank accounts, suchas checking, saving, and money manager, associated withthe same person should be added up.) Despite imperfec-tions (people may have accounts in different banks or notkeep all money in banks), the distribution of balances onbank accounts would give valuable information about thedistribution of money. The data for a big enough bankwould be representative of the distribution in the wholeeconomy. Unfortunately, it was not possible to obtainsuch data thus far, even though it would be completelyanonymous and not compromise privacy of bank clients.

10 100 10000.01%

0.1%

1%

10%

100%

Total net capital (wealth), kpounds

Cum

ulat

ive

perc

ent o

f peo

ple

United Kingdom, IR data for 1996

Pareto

Boltzmann−Gibbs

0 20 40 60 80 10010%

100%

Total net capital, kpounds

FIG. 5 Cumulative probability distribution of net wealth inthe UK shown in log-log (main panel) and log-linear (inset)scales. Points represent the data from IR, and solid lines arefits to the exponential (Boltzmann-Gibbs) and power (Pareto)laws. Reproduced from Ref. [93].

12

Measuring the probability distribution of money wouldbe very useful, because it determines how much fundspeople can, in principle, spend on purchases without go-ing into debt. This is different from the distribution ofwealth, where the property component, such as house,car, or retirement investment, is effectively locked and,in most cases, is not easily available for consumer spend-ing. So, although wealth distribution may reflect the dis-tribution of economic power, the distribution of moneyis more relevant for consumption. Money distributioncan be useful for determining prices that maximize rev-enue of a manufacturer [23]. If a price p is set too high,few people can afford it, and, if a price is too low, thesales revenue is small, so the optimal price must be in be-tween. The fraction of population who can afford to paythe price p is given by the cumulative probability C(p), sothe total sales revenue is proportional to pC(p). For theexponential distribution C(p) = exp(−p/Tm), the max-imal revenue is achieved at p = Tm, i.e. at the optimalprice is equal to the average amount of money per person[23]. Indeed, the prices of mass-market consumer prod-ucts, such as computers, electronics, and appliances, re-main stable for many years at a level determined by theiraffordability to the population, whereas technical param-eters of these products at the same price level improvedramatically due to technological progress.

V. DATA AND MODELS FOR INCOME DISTRIBUTION

In contrast to money and wealth distributions, a lotmore empirical data is available for the distribution ofincome r from tax agencies and population surveys. Inthis Section, we first present empirical data on incomedistribution and then discuss theoretical models.

A. Empirical data on income distribution

Empirical studies of income distribution have a longhistory in economic literature [97, 98, 99]. Following thework by Pareto [14], much attention was focused on thepower-law upper tail of income distribution and less onthe lower part. In contrast to more complicated functionsdiscussed in literature, Ref. [100] introduced a new ideaby demonstrating that the lower part of income distribu-tion can be well fitted with a simple exponential functionP (r) = c exp(−r/Tr) characterized by just one parame-ter, the “income temperature” Tr. Then it was recog-nized that the whole income distribution can be fitted byan exponential function in the lower part and a power-law function in the upper part [93, 101], as shown in Fig.6. The straight line in the log-linear scale in the insetof Fig. 6 demonstrates the exponential Boltzmann-Gibbslaw, and the straight line in the log-log scale in the mainpanel illustrates the Pareto power law. The fact thatincome distribution consists of two distinct parts revealsthe two-class structure of the American society [102, 103].

Coexistence of the exponential and power-law distribu-tions is known in plasma physics and astrophysics, wherethey are called the “thermal” and “superthermal” parts[104, 105, 106]. The boundary between the lower andupper classes can be defined as the intersection point ofthe exponential and power-law fits in Fig. 6. For 1997,the annual income separating the two classes was about120 k$. About 3% of population belonged to the upperclass, and 97% to the lower class.

Ref. [103] studied time evolution of income distributionin the USA during 1983–2001 based on the data fromthe Internal Revenue Service (IRS), the government taxagency. The structure of income distribution was foundto be qualitatively the same for all years, as shown in Fig.7. The average income in nominal dollars approximatelydoubled during this time interval. So, the horizonal axisin Fig. 7 shows the normalized income r/Tr, where the“income temperature” Tr was obtained by fitting of theexponential part of the distribution for each year. Thevalues of Tr are shown in Fig. 7. The plots for 1980s and1990s are shifted vertically for clarity. We observe that,the data points in the lower-income part of the distribu-tion collapse on the same exponential curve for all years.This demonstrates that the shape of income distributionfor the lower class is extremely stable and does not changein time, despite gradual increase of the average income innominal dollars. This observation suggests that the lowerclass distribution is in statistical, “thermal” equilibrium.

On the other hand, Fig. 7 shows that income distribu-tion in the upper class does not rescale and significantlychanges in time. Ref. [103] found that the exponent α ofthe power law C(r) ∝ 1/rα decreased from 1.8 in 1983to 1.4 in 2000. This means that the upper tail became“fatter”. Another useful parameter is the total incomeof the upper class as the fraction f of the total incomein the system. The fraction f increased from 4% in 1983

1 10 100 10000.1%

1%

10%

100%

Adjusted Gross Income, k$

Cum

ulat

ive

perc

ent o

f ret

urns

United States, IRS data for 1997

Pareto

Boltzmann−Gibbs

0 20 40 60 80 100

10%

100%

AGI, k$

FIG. 6 Cumulative probability distribution of tax returns forUSA in 1997 shown in log-log (main panel) and log-linear (in-set) scales. Points represent the IRS data, and solid lines arefits to the exponential and power-law functions. Reproducedfrom Ref. [101].

13

to 20% in 2000 [103]. However, in year 2001, α increasedand f decreases, indicating that the upper tail was re-duced after the stock market crash at that time. Theseresults indicate that the upper tail is highly dynamicaland not stationary. It tends to swell during the stockmarket boom and shrink during the bust. Similar resultswere found for Japan [107, 108, 109, 110].

Although relative income inequality within the lowerclass remains stable, the overall income inequality inthe USA has increased significantly due to tremendousgrowth of the upper class income. This is illustrated bythe Lorenz curve and the Gini coefficient shown in Fig. 8.The Lorenz curve [97] is a standard way of representingincome distribution in economic literature. It is definedin terms of two coordinates x(r) and y(r) depending ona parameter r:

x(r) =

∫ r

0

P (r′) dr′, y(r) =

∫ r

0 r′P (r′) dr′∫ ∞

0 r′P (r′) dr′. (19)

The horizontal coordinate x(r) is the fraction of popu-lation with income below r, and the vertical coordinatey(r) is the fraction of income this population accountsfor. As r changes from 0 to ∞, x and y change from 0 to1 and parametrically defines a curve in the (x, y)-plane.

Fig. 8 shows the data points for the Lorenz curves in1983 and 2000, as computed by the IRS [111]. Ref. [100]analytically derived the Lorenz curve formula y = x +(1 − x) ln(1 − x) for a purely exponential distributionP (r) = c exp(−r/Tr). This formula is shown by the redline in Fig. 8 and describes the 1983 data reasonably well.However, for year 2000, it is essential to take into accountthe fraction f of income in the upper tail, which modifies

0.1 1 10 1000.01%

0.1%

1%

10%

100%

1983, 19.35 k$1984, 20.27 k$1985, 21.15 k$1986, 22.28 k$1987, 24.13 k$1988, 25.35 k$1989, 26.38 k$

1990, 27.06 k$1991, 27.70 k$1992, 28.63 k$1993, 29.31 k$1994, 30.23 k$1995, 31.71 k$1996, 32.99 k$1997, 34.63 k$1998, 36.33 k$1999, 38.00 k$2000, 39.76 k$2001, 40.17 k$

Cum

ulat

ive

perc

ent o

f ret

urns

Rescaled adjusted gross income

4.017 40.17 401.70 4017

0.01%

0.1%

1%

10%

100%

Adjusted gross income in 2001 dollars, k$

100%

10% Boltzmann−Gibbs

Pareto

1980’s

1990’s

FIG. 7 Cumulative probability distribution of tax returnsplotted in log-log scale versus r/Tr (the annual income r nor-malized by the average income Tr in the exponential part ofthe distribution). The IRS data points are for 1983–2001, andthe columns of numbers give the values of Tr for the corre-sponding years. Reproduced from Ref. [103].

for the Lorenz formula as follows [101, 102, 103]

y = (1 − f)[x + (1 − x) ln(1 − x)] + f Θ(x − 1). (20)

The last term in Eq. (20) represent the vertical jump ofthe Lorenz curve at x = 1, where a very small percentageof population in the upper class accounts for a substantialfraction f of the total income. The blue curve represent-ing Eq. (20) fits the 2000 data in Fig. 8 very well.

The deviation of the Lorenz curve from the straightdiagonal line in Fig. 8 is a certain measure of incomeinequality. Indeed, if everybody had the same income,the Lorenz curve would be the diagonal line, because thefraction of income would be proportional to the fractionof population. The standard measure of income inequal-ity is the so-called Gini coefficient 0 ≤ G ≤ 1, which isdefined as the area between the Lorenz curve and the di-agonal line, divided by the area of the triangle beneaththe diagonal line [97]. Time evolution of the Gini coeffi-cient, as computed by the IRS [111], is shown in the insetof Fig. 8. Ref. [100] derived analytically that G = 1/2for a purely exponential distribution. In the first approx-imation, the values of G shown in the inset of Fig. 8 areindeed close to the theoretical value 1/2. If we take intoaccount the upper tail using Eq. (20), the formula for theGini coefficient becomes G = (1+f)/2 [103]. The inset inFig. 8 shows that this formula very well fits the IRS datafor the 1990s using the values of f deduced from Fig. 7.The values G < 1/2 in the 1980s cannot be captured bythis formula, because the Lorenz data points are slightlyabove the theoretical curve for 1983 in Fig. 8. Overall,we observe that income inequality was increasing for thelast 20 years, because of swelling of the Pareto tail, butdecreased in 2001 after the stock market crash.

Thus far we discussed the distribution of individual

0 10 20 30 40 50 60 70 80 90 100%0

10

20

30

40

50

60

70

80

90

100%

Cumulative percent of tax returns

Cum

ulat

ive

perc

ent o

f inc

ome

US, IRS data for 1983 and 2000

1983→

←2000

4%

19%

1980 1985 1990 1995 2000 0

0.5

1

Year

Gini from IRS dataGini=(1+f)/2

FIG. 8 Main panel: Lorenz plots for income distribution in1983 and 2000. The data points are from the IRS [111], andthe theoretical curves represent Eq. (20) with f from Fig.7. Inset: The closed circles are the IRS data [111] for theGini coefficient G, and the open circles show the theoreticalformula G = (1 + f)/2. Reproduced from Ref. [103].

14

0 10 20 30 40 50 60 70 80 90 100 110 1200%

1%

2%

3%

4%

5%

6%

Annual family income, k$

Pro

babi

lity

United States, Bureau of Census data for 1996

FIG. 9 Histogram: Probability distribution of family incomefor families with two adults (US Census Bureau data). Solidline: Fit to Eq. (21). Reproduced from Ref. [100].

income. An interesting related question is the distribu-tion P2(r) of family income r = r1 + r2, where r1 andr2 are the incomes of spouses. If individual incomes aredistributed exponentially P (r) ∝ exp(−r/Tr), then

P2(r) =

∫ r

0

dr′P (r′)P (r − r′) = c r exp(−r/Tr), (21)

where c is a normalization constant. Fig. 9 shows thatEq. (21) is in a good agreement with the family incomedistribution data from the US Census Bureau [100]. InEq. (21), we assumed that incomes of spouses are uncor-related. This simple approximation is indeed supportedby the scatter plot of incomes of spouses shown in Fig.10. Each family is represented in this plot by two points

0 20 40 60 80 1000

20

40

60

80

100

Labor income of one earner, k$

Labo

r in

com

e of

ano

ther

ear

ner,

k$

PSID data for families, 1999

FIG. 10 Scatter plot of the spouses incomes (r1, r2) and(r2, r1) based on the PSID data. Reproduced from Ref. [101].

0 10 20 30 40 50 60 70 80 90 100%0

10

20

30

40

50

60

70

80

90

100%

Cum

ulat

ive

perc

ent o

f fam

ily in

com

e

United States, Bureau of Census data for 1947−1994

Cumulative percent of families

1950 1960 1970 1980 19900

0.2

0.375

0.6

0.8

1

Year

Gini coefficient ≈ 38

FIG. 11 Main panel: Lorenz plot for family income calculatedfrom Eq. (21), compared with the US Census data points.Inset: The US Census data points for the Gini coefficientfor families, compared with the theoretically calculated value3/8=37.5%. Reproduced from Ref. [100].

(r1, r2) and (r2, r1) for symmetry. We observe that thedensity of points is approximately constant along thelines of constant family income r1 + r2 = const, whichindicates that incomes of spouses are approximately un-correlated. There is no significant clustering of pointsalong the diagonal r1 = r2, i.e. no strong positive corre-lation of spouses incomes.

The Gini coefficient for the family income distribution(21) was calculated in Ref. [100] as G = 3/8 = 37.5%.Fig. 11 shows the Lorenz quintiles and the Gini coeffi-cient for 1947–1994 plotted from the US Census Bureaudata. The solid line, representing the Lorenz curve cal-culated from Eq. (21), is in a good agreement with thedata. The systematic deviation for the top 5% of earnersresults from the upper tail, which has less pronounced ef-fect on family income than on individual income, becauseof income averaging in the family. The Gini coefficient,shown in the inset of Fig. 11, is close to the calculatedvalue 37.5%. Moreover, the average G for the developedcapitalist countries of North America and Western Eu-rope, as determined by the World Bank [101], is also closeto the calculated value 37.5%.

Econophysics papers examined income distribution indifferent countries: Japan [66, 107, 108, 109, 110, 112,113, 114], Germany [115, 116], the United Kingdom[66, 83, 114, 115, 116], Italy [116, 117, 118], the UnitedStates [115, 119], India [95], Australia [89, 118, 120], andNew Zealand [66, 114]. The distributions are qualita-tively similar to the results presented in this Section.The upper tail follows a power law and comprises a smallfraction of population. To fit the lower part of the distri-bution, different papers used exponential, Gamma, andlog-normal distributions. Unfortunately, income distri-bution is often reported by statistical agencies for house-

15

holds, so it is difficult to differentiate between one-earnerand two-earner income distributions. Some papers usedinterpolating functions with different asymptotic behav-ior for low and high incomes, such as the Tsallis function[114] and the Kaniadakis function [116]. However, tran-sition between the lower and upper classes is not smoothfor the US data shown in Figs. 6 and 7, so such func-tions would not be useful in this case. The special caseis income distribution in Argentina during the economiccrisis, which shows a time-dependent bimodal shape withtwo peaks [114].

B. Theoretical models of income distribution

Having examined the empirical data on income distri-bution, let us now discuss theoretical models. Incomeri is the influx of money per unit time to an agent i.If the money balance mi is analogous to energy, thenthe income ri would be analogous to power, which is theenergy flux per unit time. So, one should conceptuallydistinguish between the distributions of money and in-come. While money is regularly transferred from oneagent to another in pairwise transactions, it is not typi-cal for agents to trade portions of their income. Never-theless, indirect transfer of income may occur when oneemployee is promoted and another demoted while the to-tal annual budget is fixed, or when one company getsa contract whereas another one loses it, etc. A reason-able approach, which has a long tradition in economicliterature [121, 122, 123], is to treat individual incomer as a stochastic process and study its probability dis-tribution. In general, one can study a Markov processgenerated by a matrix of transitions from one income toanother. In the case where income r changes by a smallamount ∆r over a time period ∆t, the Markov processcan be treated as income diffusion. Then one can ap-ply the general Fokker-Planck equation [69] to describeevolution in time t of the income distribution functionP (r, t) [103]

∂P

∂t=

∂

∂r

[

AP +∂(BP )

∂r

]

, A = −〈∆r〉

∆t, B =

〈(∆r)2〉

2∆t.

(22)The coefficients A and B in Eq. (22) are determined bythe first and second moments of income changes per unittime. The stationary solution ∂tP = 0 of Eq. (22) obeysthe following equation with the general solution

∂(BP )

∂r= −AP, P (r) =

c

B(r)exp

(

−

∫ r A(r′)

B(r′)dr′

)

.

(23)For the lower part of the distribution, it is reasonable

to assume that ∆r is independent of r, i.e. the changes ofincome are independent of income itself. This process iscalled the additive diffusion [103]. In this case, the coeffi-cients in Eq. (22) are constants A0 and B0. Then Eq. (23)gives the exponential distribution P (r) ∝ exp(−r/Tr)

with the effective income temperature Tr = B0/A0. [No-tice that a meaningful stationary solution of Eq. (23) re-quires that A > 0, i.e. 〈∆r〉 < 0.] The coincidence of thisresult with the Boltzmann-Gibbs exponential law (1) and(7) is not accidental. Indeed, instead of considering pair-wise interaction between particles, Eq. (1) can be derivedby considering energy transfers between a particle and abig reservoir, as long as the transfer process is “additive”and does not involve a Maxwell-demon-like discrimina-tion. Stochastic income fluctuations are described bya similar process. So, although money and income aredifferent concepts, they may have similar distributions,because they are governed by similar mathematical prin-ciples. It was shown explicitly in Refs. [23, 80, 81] thatthe models of pairwise money transfer can be describedin a certain limit by the Fokker-Planck equation.

On the other hand, for the upper tail of income distri-bution, it is reasonable to expect that ∆r ∝ r, i.e. incomechanges are proportional to income itself. This is knownas the proportionality principle of Gibrat [121], and theprocess is called the multiplicative diffusion [103]. In thiscase, A = ar and B = br2, and Eq. (23) gives the power-law distribution P (r) ∝ 1/rα+1 with α = 1 + a/b.

Generally, the lower-class income comes from wagesand salaries, where the additive process is appropriate,whereas the upper-class income comes from bonuses, in-vestments, and capital gains, calculated in percentages,where the multiplicative process applies [124]. However,the additive and multiplicative processes may coexist.An employee may receive a cost-of-living raise calculatedin percentages (the multiplicative process) and a meritraise calculated in dollars (the additive process). In thiscase, we have A = A0+ar and B = B0+br2 = b(r2

0 +r2),where r2

0 = B0/b. Substituting these expressions into Eq.(23), we find

P (r) = ce−(r0/Tr) arctan(r/r0)

[1 + (r/r0)2]1+a/2b. (24)

The distribution (24) interpolates between the exponen-tial law for low r and the power law for high r, becauseeither the additive or the multiplicative process domi-nates in the corresponding limit. The crossover betweenthe two regimes takes place at r ∼ r0, where the addi-tive and multiplicative contributions to B are equal. Thedistribution (24) has three parameters: the “income tem-perature” Tr = A0/B0, the Pareto exponent α = 1+a/b,and the crossover income r0. It is a minimal model thatcaptures the salient features of the empirical income dis-tribution shown in Fig. 6. A mathematically similar, butmore economically oriented model was proposed in Refs.[112, 113], where labor income and assets accumulationare described by the additive and a multiplicative pro-cesses correspondingly. A general stochastic process withadditive and multiplicative noise was studied numericallyin Ref. [125], but the stationary distribution was not de-rived analytically. A similar process with discrete timeincrements was studied by Kesten [126]. Recently, a for-mula similar to Eq. (24) was obtained in Ref. [127].

16

To verify the multiplicative and additive hypothesesempirically, it is necessary to have the data on incomemobility, i.e. the income changes ∆r of the same peo-ple from one year to another. The distribution of incomechanges P (∆r|r) conditional on income r is generally notavailable publicly, although it can be reconstructed byresearchers at the tax agencies. Nevertheless, the mul-tiplicative hypothesis for the upper class was quantita-tively verified in Refs. [109, 110] for Japan, where taxidentification data is published for the top taxpayers.

The power-law distribution is meaningful only whenit is limited to high enough incomes r > r0. If allincomes r from 0 to ∞ follow a purely multiplicativeprocess, then one can change to a logarithmic variablex = ln(r/r∗) in Eq. (22) and show that it gives a Gaus-sian time-dependent distribution Pt(x) ∝ exp(−x2/2σ2t)for x, i.e. the log-normal distribution for r, also known asthe Gibrat distribution [121]. However, the width of thisdistribution increases linearly in time, so the distributionis not stationary. This was pointed out by Kalecki a longtime ago [122], but the log-normal distribution is stillwidely used for fitting income distribution, despite thisfundamental logical flaw in its justification. In the classicpaper [123], Champernowne showed that a multiplicativeprocess gives a stationary power-law distribution when aboundary condition is imposed at r0 6= 0. Later, thisresult was rediscovered by econophysicists [128, 129]. Inour Eq. (24), the exponential distribution of the lowerclass effectively provides such a boundary condition forthe power law of the upper class. Notice also that Eq.(24) reduces to Eq. (18) in the limit r0 → 0, which cor-responds to purely multiplicative noise B = br2.

There are alternative approaches to income distribu-tion in economic literature. One of them, proposed byLydall [130], involves social hierarchy. Groups of peoplehave leaders, which have leaders of a higher order, and soon. The number of people decreases geometrically (ex-ponentially) with the increase of the hierarchical level.If individual income increases by a certain factor (i.e.multiplicatively) when moving to the next hierarchicallevel, then income distribution follows a power law [130].However, the original Lydall’s argument can be easilymodified to produce the exponential distribution. If indi-vidual income increases by a certain amount, i.e. incomeincreases linearly with the hierarchical level, then incomedistribution is exponential. The latter process seems tobe more realistic for moderate incomes below $100,000.A similar scenario is the Bernoulli trials [131], where in-dividuals have a constant probability of increasing theirincome by a fixed amount. We see that the deterministichierarchical models and the stochastic models of additiveand multiplicative income mobility represent essentiallythe same ideas.

VI. OTHER APPLICATIONS OF STATISTICAL PHYSICS

Statistical physics was applied to a number of othersubjects in economics. Because of lack of space, only twosuch topics are briefly discussed in this Section.

A. Economic temperatures in different countries

As discussed in Secs. IV.C and V.A, the distributionsof money, wealth, and income are often described by ex-ponential functions for the majority of population. Theseexponential distributions are characterized by the param-eters Tm, Tw, Tr, which are mathematically analogousto temperature in the Boltzmann-Gibbs distribution (1).The values of these parameters, extracted from the fitsof the empirical data, are generally different for differentcountries, i.e. different countries have different economic“temperatures”. For example, Ref. [93] found that theUS income temperature was 1.9 times higher than theUK income temperature in 1998 (using the exchange rateof dollars to pounds at that time). Also, there is ±25%variation between income temperatures of different stateswithin the United States [93].

In physics, a difference of temperatures allows to setup a thermal machine. In was argued in Ref. [23] that thedifference of money or income temperatures between dif-ferent countries allows to extract profit in internationaltrade. Indeed, as discussed at the end of Sec. IV.C, theprices of goods should be commensurate with money orincome temperature, because otherwise people cannot af-ford to buy those goods. So, an international tradingcompany can buy goods at a low price T1 in a “low-temperature” country and sell them at a high price T2 toa “high-temperature” country. The difference of pricesT2 − T1 would be the profit of the trading company. Inthis process, money (the analog of energy) flows fromthe “high-temperature” to the “low-temperature” coun-try, in agreement with the second law of thermodynam-ics, whereas products flow in the opposite direction. Thisprocess very much resembles what is going on in globaleconomy now. In this framework, the perpetual tradedeficit of the United States is the consequence of the sec-ond law of thermodynamics and the difference of tem-peratures between the USA and the “low-temperature”countries, such as China. Similar ideas were developed inmore detail in Refs. [132, 133], including a formal Carnotcycle for international trade.

The statistical physics approach demonstrates thatprofit originates from statistical non-equilibrium (the dif-ference of temperatures), which exists in the global econ-omy. However, it does not answer the question what isthe origin of this difference. By analogy with physics, onewould expect that the money flow should reduce the tem-perature difference and, eventually, lead to equilibriza-tion of temperatures. In physics, this situation is knownas the “thermal death of the Universe”. In a completelyequilibrated global economy, it would be impossible to

17

make profit by exploiting differences of economic tem-peratures between different countries. Although global-ization of modern economy does show a tendency towardequilibrization of living standards in different countries,this process is far from straightforward, and there aremany examples contrary to equilibrization. This interest-ing and timely subject certainly requires further study.

B. Society as a binary alloy

In 1971, Thomas Schelling proposed the now-famousmathematical model of segregation [134]. He considereda lattice, where the sites can be occupied by agents of twotypes, e.g., blacks and whites in the problem of racial seg-regation. He showed that, if the agents have some prob-abilistic preference for the neighbors of the same type,the system spontaneously segregates into black and whiteneighborhoods. This mathematical model is similar tothe so-called Ising model, which is a popular model forstudying phase transitions in physics. In this model, eachlattice site is occupied by a magnetic atom, whose mag-netic moment has only two possible orientations, up ordown. The interaction energy between two neighboringatoms depends on whether their magnetic moments pointin the same or in the opposite directions. In physics lan-guage, the segregation found by Schelling represents aphase transition in this system.

Another similar model is the binary alloy, a mixtureof two elements which attract or repel each other. Itwas noticed in Ref. [135] that behavior of actual binaryalloys shows a picture strikingly similar to social segre-gation. In the following papers [40, 136], this mathemat-ical analogy was developed further and compared withsocial data. Interesting concepts, such as the coexistencecurve between two phases and the solubility limit, werediscussed in this work. The latter concept means thata small amount of one substance dissolves into anotherup to some limit, but phase separation (segregation) de-velops for higher concentrations. Recently, similar ideaswere rediscovered in Refs. [137, 138, 139]. The vast expe-rience of physicists in dealing with phase transitions andalloys may be helpful for practical applications of suchmodels [140].

VII. FUTURE DIRECTIONS, CRITICISM, AND

CONCLUSIONS

Statistical models described in this review are quitesimple. It is commonly accepted in physics that theoret-ical models are not intended to be photographic copies ofreality, but rather be caricatures, capturing the most es-sential features of a phenomenon with a minimal numberof details. With only few rules and parameters, the mod-els discussed in Secs. III, IV, and V reproduce sponta-neous development of stable inequality, which is presentin virtually all societies. It is amazing that the calculatedGini coefficients, G = 1/2 for individuals and G = 3/8 for

families, are actually very close to the US income data,as shown in Fig. 8 and 11. These simple models establisha baseline and a reference point for development of moresophisticated and more realistic models. Some of thesefuture directions are outlined below.

A. Future directions

a. Agents with a finite lifespan. The models discussed inthis review consider immortal agents who live forever, likeatoms. However, humans have a finite lifespan. They en-ter economy as young people and exit at an old age. Evo-lution of income and wealth as functions of age is studiedin economics using the so-called overlapping-generationsmodel. The absence of the age variable was one ofthe criticisms of econophysics by economist Paul Anglin[29]. However, the drawback of the standard overlapping-generations model is that there is no variation of incomeand wealth between agents of the same age, because itis a representative-agent model. It would be the best tocombine stochastic models with the age variable. Also,to take into account inflation of average income, Eq. (22)should be rewritten for the relative income, in the spiritof Eq. (17). These modifications would allow to study theeffects of demographic waves, such as the baby boomers,on the distributions of income and wealth.

b. Agent-based simulations of the two-class society. Theempirical data presented in Sec. V.A show quite convinc-ingly that the US population consists of two very distinctclasses characterized by different distribution functions.However, the theoretical models discussed in Secs. IIIand IV do not produce two classes, although they doproduce broad distributions. Generally, not much atten-tion is payed in the agent-based literature to simulationof two classes. One exception is Ref. [141], which simu-lated spontaneous development of employers and employ-ees classes from initially equal agents [34]. More work inthis direction would be certainly desirable.

c. Access to detailed empirical data. Great amount of sta-tistical information is publicly available on the Internet,but not for all types of data. As discussed in Sec. IV.C,it would be very interesting to obtain data on the distri-bution of balances on bank accounts, which would giveinformation about the distribution of money (as opposedto wealth). As discussed in Sec. V.B, it would be use-ful to obtain detailed data on income mobility, to verifythe additive and multiplicative hypotheses for income dy-namics. Income distribution is often reported as a mixof data on individual income and family income, whenthe counting unit is a tax return (joint or single) or ahousehold. To have a meaningful comparison with theo-retical models, it is desirable to obtain clean data where

18

the counting unit is an individual. Direct collaborationwith statistical agencies would be very useful.

d. Economies in transition. Inequality in developed cap-italist countries is generally quite stable. The situationis very different for the former socialist countries makingtransition to market economy. According to the WorldBank data [101], the average Gini coefficient for family in-come in the Eastern Europe and the former Soviet Unionjumped from 25% in 1988 to 47% in 1993. The Gini coef-ficient in the socialist countries before the transition waswell below the equilibrium value of 37.5% for the marketeconomies. However, the fast collapse of socialism leftthese countries out of market equilibrium and generateda much higher inequality. One may expect that, withthe time, their inequality would decrease to the equilib-rium value of 37.5%. It would be very interesting to tracehow fast this relaxation takes place. Such a study wouldalso verify whether the equilibrium level of inequality isuniversal for all market economies.

e. Relation to physical energy. The analogy between en-ergy and money discussed in Sec. III.B is a formal math-ematical analogy. However, the actual physical energywith low entropy (typically in the form of fossil fuel)also plays a very important role in modern economy, asthe basis of the current human technology. In the viewof looming energy and climate crisis, it is imperative tofind realistic ways for making a transition from the cur-rent “disposable” economy based in “cheap” and “unlim-ited” energy and natural resources to a sustainable one.Heterogeneity of human society is one of the importantfactors affecting such a transition. Econophysics, at theintersection of energy, entropy, economy, and statisticalphysics, may play a useful role in this quest [142].

B. Criticism from economists

As econophysics is gaining popularity, some criticismhas appeared from economists [29], including those whoare closely involved with the econophysics movement[30, 31, 32]. This reflects a long-standing tradition in eco-nomic and social sciences of writing critiques on differentschools of thought. Much of the criticism is useful andconstructive and is already being accommodated in theeconophysics work. However, some criticism results frommisunderstanding or miscommunication between the twofields and some from significant differences in scientificphilosophy. Several insightful responses to the criticismhave been published [143, 144, 145], see also [7, 146]. Inthis Section, we briefly address the issues that are directlyrelated to the material discussed in this review.

a. Awareness of previous economic literature. One com-plaint of Refs. [29, 30, 31, 32] is that physicists are not

well aware of the previous economic literature and eitherrediscover known results or ignore well-established ap-proaches. Addressing this issue, it is useful to keep inmind that science itself as a complex system, and sci-entific progress as an evolutionary process with naturalselection. The sea of scientific literature is enormous,and nobody knows it all. Recurrent rediscovery of reg-ularities in the natural and social world only confirmstheir validity. Independent rediscovery usually brings adifferent perspective, broader applicability range, higheraccuracy, and better mathematical treatment, so there isprogress even when some overlap with previous resultsexists. Physicists are grateful to economists for bringingrelevant and specific references to their attention. Sincethe beginning of modern econophysics, many old refer-ences have been uncovered and are now routinely cited.

However, not all old references are relevant to thenew development. For example, Ref. [31] complains thateconophysics literature on income distribution ignoresthe so-called “Kuznets hypothesis” [147]. The Kuznetshypothesis postulates that income inequality first risesduring an industrial revolution and then decreases, pro-ducing an inverted U-shaped curve. Ref. [147] admitsthat, to date, the large literature on the Kuznets hy-pothesis is inconclusive. Ref. [147] mentions that thishypothesis applies to the period from colonial times to1970s. However, the empirical data for this period aresparse and not very reliable. The econophysics literaturedeals with the reliable volumes of data for the secondhalf of the 20th century, collected with the introductionof computers. It is not clear what is the definition of in-dustrial revolution and when exactly it starts and ends.The chain of technological progress seems to be contin-uous (steam engine, internal combustion engine, cars,plastics, computers, Internet), so it is not clear wherethe purported U-curve is supposed to be placed in time.Thus, the Kuznets hypothesis appears to be, in princi-ple, unverifiable and unfalsifiable. The original paperby Kuznets [147] actually does not contain any curves,but it has one table filled with made-up, imaginary data!Kuznets admits that he has “neither the necessary datanor a reasonably complete theoretical model” [147, p 12].So, this paper is understandably ignored by econophysicsliterature. In fact, the data analysis for 1947–1984 showsamazing stability of income distribution [148], consistentwith Fig. 11. The increase of inequality in the 1990s re-sulted from growth of the upper tail relative to the lowerclass, but the relative inequality within the lower classremains very stable, as shown in Fig. 7.

b. Reliance on visual data analysis. Another complaint ofRef. [31] is that econophysicists favor graphic analysis ofdata over the formal and “rigorous” testing prescribed bymathematical statistics, as favored by economists. Thiscomplaint goes against the trend of all sciences to use in-creasingly sophisticated data visualization for uncoveringregularities in complex system. The thick IRS publica-

19

tion 1304 [149] is filled with data tables, but has virtu-ally no graphs. Despite the abundance of data, it gives areader no idea about income distribution, whereas plot-ting the data immediately gives an insight. However,intelligent plotting is the art with many tools, which notmany researchers have mastered. The author completelyagrees with Ref. [31] that too many papers mindlesslyplot any kind of data in the log-log scale, pick a finiteinterval, where any smooth curved line can be approxi-mated by a straight line, and claim that there is a powerlaw. In many cases, replotting the same data in the log-linear scale converts a curved line into a straight line,which means that the law is actually exponential.

Good visualization is extremely helpful in identifyingtrends in complex data, which can be then fitted to amathematical function. However, for complex system,such a fit should not be expected with an infinite preci-sion. The fundamental laws of physics, such as Newton’slaw of gravity or Maxwell’s equations are valid with enor-mous precision. However, the laws in condensed matterphysics, uncovered by experimentalists with a combina-tion of visual analysis and fitting, usually have muchlower precision, at best 10% or so. Most of these lawswould fail the formal criteria of mathematical statistics.Nevertheless these approximate laws are enormously use-ful in practice, and the everyday devices, engineered onthe basis of these laws, work very well for all of us.

Because of the finite accuracy, different functions mayproduce equally good fits. Discrimination between theexponential, Gamma, and log-normal functions may notbe always possible [120]. However, the exponential func-tion has fewer fitting parameters, so it is preferable onthe basis of simplicity. The other two functions cansimply mimic the exponential function with a particu-lar choice of the additional parameters [120]. Unfortu-nately, many papers in mathematical statistics introducetoo many fitting parameters into complicated functions,such as the generalized beta distribution mentioned inRef. [31]. Such over-parametrization is more misleadingthan insightful for data fitting.

c. Quest for universality. Ref. [31] criticizes physicists fortrying to find universality in economic data. It also seemsto equate the concepts of power law, scaling, and uni-versality. These are three different, albeit overlapping,concepts. Power laws usually apply only to a small frac-tion of data at the high ends of various distributions.Moreover, the exponents of these power laws are usuallynonuniversal and vary from case to case. Scaling meansthat the shape of a function remains the same when itsscale changes. However, the scaling function does nothave to be a power law function. A good example ofscaling is shown in Fig. 7, where income distributions forthe lower class collapse on the same exponential line forabout 20 years of data. We observe amazing universal-ity of income distribution, unrelated to a power law. Ina general sense, diffusion equation is universal, because

it describes a wide range of systems, from dissolution ofsugar in water to a random walk in stock market.

Universalities are not easy to uncover, but they formthe backbone of regularities in the world around us. Thisis why physicists are so much interested in them. Uni-versalities establish the first-order effect, and deviationsrepresent the second-order effect. Different countries mayhave somewhat different distributions, and economists of-ten tend to focus on these differences. However, this focuson details misses the big picture that, in the first approx-imation, the distributions are quite similar and universal.

d. Theoretical modeling of money, wealth, and income.

Refs. [29, 31, 32] point out that many econophysics pa-pers confuse or misuse the terms for money, wealth, andincome. It is true that terminology is sloppy in many pa-pers and should be refined. However, the terms in Refs.[23, 24] are quite precise, and this review clearly distin-guishes between these concepts in Secs. III, IV, and V.

One contentious issue is about conservation of money.Ref. [31] agrees that “transactions are a key eco-nomic process, and they are necessarily conservative”,i.e. money is indeed conserved in transactions betweenagents. However, Refs. [29, 31, 32] complain that themodels of conservative exchange do not consider produc-tion of goods, which is the core economic process andthe source of economic growth. Material production isindeed the ultimate goal of economy, but it does not vi-olate conservation of money by itself. One can grow cof-fee beans, but nobody can grow money on a money tree.Money is an artificial economic device that is designed tobe conserved. As explained in Sec. III, the money trans-fer models implicitly assume that money in transactionsis voluntarily payed for goods and services generated byproduction for the mutual benefit of the parties. In prin-ciple, one can introduce a billion of variables to keep trackof every coffee bean and other product of the economy.What difference would it make for the distribution ofmoney? Despite claims in Refs. [29, 31], there is no con-tradiction between models of conservative exchange andthe classic work of Adam Smith and David Ricardo. Thedifference is only in the focus: We keep track of money,whereas they keep track of coffee beans, from produc-tion to consumption. These approaches address differentquestions, but do not contradict each other. Becausemoney constantly circulates in the system as paymentfor production and consumption, the resulting statisticaldistribution of money may very well not depend on whatis exactly produced and in what quantities.

In principle, the models with random transfers ofmoney should be considered as a reference point for de-veloping more sophisticated models. Despite the totallyrandom rules and “zero intelligence” of the agents, thesemodels develop well-characterized, stable and stationarydistributions of money. One can modify the rules tomake the agents more intelligent and realistic and seehow much the resulting distribution changes relative to

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the reference one. Such an attempt was made in Ref.[30] by modifying the model of Ref. [75] with variousmore realistic economic ingredients. However, despitethe modifications, the resulting distributions were essen-tially the same as in the original model. This exampleillustrates the typical robustness and universality of sta-tistical models: Modifying details of microscopic rulesdoes not necessarily change the statistical outcome.

Another misconception, elaborated in Ref. [30, 32], isthat the money transfer models discussed in Sec. III im-ply that money is transferred by fraud, theft, and vio-lence, rather than voluntarily. One should keep in mindthat the catchy labels “theft-and-fraud”, “marriage-and-divorce”, and “yard-sale” were given to the money trans-fer models by the journalist Brian Hayes in a populararticle [150]. Econophysicists who originally introducedand studied these models do not subscribe to this termi-nology, although the early work of Angle [49] did mentionviolence as one source of redistribution. In the opinionof the author, it is indeed difficult to justify the propor-tionality rule (8), which implies that agents with highbalances pay proportionally greater amounts in transac-tions than agents with low balances. However, the addi-tive model of Ref. [23], where money transfers ∆m areindependent of money balances mi of the agents, doesnot have this problem. As explained in Sec. III.C, thismodel simply means that all agents pay the same pricesfor the same product, although prices may be different fordifferent products. So, this model is very well consistentwith voluntary transactions in a free market.

Ref. [143] argued that conservation of money is vio-lated by credit. As explained in Sec. III.D, credit doesnot violate conservation law, but creates positive andnegative money without changing net worth. Negativemoney (debt) is as real as positive money. Ref. [143]claims that money can be easily created with the tap of acomputer key via credit. Then why wouldn’t an employertap the key and double salaries, or a funding agency dou-ble research grants? Because budget constraints are real.Credit may provide a temporary relief, but sooner or laterit has to be paid back. Allowing debt may produce adouble-exponential distribution as shown in Fig. 3, butit does not change the distribution fundamentally.

As discussed in Sec. III.B, a central bank or a cen-tral government can inject new money into economy. Asdiscussed in Sec. IV, wealth is generally not conserved.As discussed in Sec. V, income is different from moneyand is described by a different model (22). However, theempirical distribution of income shown in Fig. 6 is qual-itatively similar to the distribution of wealth shown inFig. 5, and we do not have data on money distribution.

C. Conclusions

“Invasion” of physicists into economics and financeat the turn of the millennium is a fascinating phe-nomenon. Physicist Joseph McCauley proclaims that

“Econophysics will displace economics in both the uni-versities and boardrooms, simply because what is taughtin economics classes doesn’t work” [151]. Although thereis some truth in his arguments [143], one may consider aless radical scenario. Econophysics may become a branchof economics, in the same way as games theory, psycho-logical economics, and now agent-based modeling becamebranches of economics. These branches have their owninterests, methods, philosophy, and journals. The maincontribution from the infusion of new ideas from a differ-ent field is not in answering old questions, but in raisingnew questions. Much of the misunderstanding betweeneconomists and physicists happens not because they aregetting different answers, but because they are answeringdifferent questions.

The subject of income and wealth distributions andsocial inequality was very popular at the turn of an-other century and is associated with the names of Pareto,Lorenz, Gini, Gibrat, and Champernowne, among others.Following the work by Pareto, attention of researcherswas primarily focused on the power laws. However, whenphysicists took a fresh, unbiased look at the empiricaldata, they found a different, exponential law for the lowerpart of the distribution. The motivation for looking atthe exponential law, of course, came from the Boltzmann-Gibbs distribution in physics. Further studies provideda more detailed picture of the two-class distribution ina society. Although social classes are known in politi-cal economy since Karl Marx, realization that they aredescribed by simple mathematical distributions is quitenew. Demonstration of the ubiquitous nature of the ex-ponential distribution for money, wealth, and income isone of the new contributions produced by econophysics.

VIII. BIBLIOGRAPHY

[1] Chakrabarti BK (2005) Econophys-Kolkata: a shortstory. In: Chatterjee A, Yarlagadda S, Chakrabarti BK(ed) Econophysics of Wealth Distributions, Springer,Milan, pp 225–228

[2] Carbone A, Kaniadakis G, Scarfone AM (2007) Wheredo we stand on econophysics? Physica A 382:xi–xiv

[3] Stanley HE et al (1996) Anomalous fluctuations in thedynamics of complex systems: from DNA and physiol-ogy to econophysics. Physica A 224:302–321

[4] Mantegna RN, Stanley HE (1999) An Introduction toEconophysics: Correlations and Complexity in Finance.Cambridge University Press, Cambridge, UK

[5] Galam S (2004) Sociophysics: a personal testimony.Physica A 336:49–55

[6] Galan S, Gefen Y, Shapir Y (1982) Sociophysics: a newapproach of sociological collective behaviour. I. Mean-behaviour description of a strike. Journal of Mathemat-ical Sociology 9:1–13

[7] Stauffer D (2004) Introduction to statistical physics out-side physics. Physica A 336:1–5

[8] Schweitzer F (2003) Brownian Agents and Active Par-ticles: Collective Dynamics in the Natural and SocialSciences, Springer, Berlin

21

[9] Chakrabarti BK, Chakraborti A, Chatterjee A (ed)(2006) Econophysics and Sociophysics: Trends and Per-spectives, Wiley-VCH, Berlin

[10] Ball P (2002) The physical modelling of society: a his-torical perspective. Physica A 314:1–14

[11] Ball P (2004) Critical Mass: How One Thing Leads toAnother. Farrar, Straus and Giroux, New York

[12] Boltzmann L (1905) Populare Schriften. Barth, Leipzig,p S.360f

[13] Ludwig Boltzmann 1844–1906. Austrian Central Li-brary for Physics, Vienna (2006), ISBN 3-900490-11-2,p 113

[14] Pareto V (1897) Cours d’Economie Politique,L’Universite de Lausanne

[15] Majorana E (1942) Il valore delle leggi statistiche nellafisica e nelle scienze sociali. Scientia 36:58–66. Englishtranslation by Mantegna RN in: Bassani GF (ed) (2006)Ettore Majorana Scientific Papers, Springer, Berlin, pp250–260

[16] Montroll EW, Badger WW (1974) Introduction toQuantitative Aspects of Social Phenomena. Gordon andBreach, New York

[17] Follmer H (1974) Random economies with many inter-acting agents. Journal of Mathematical Economics 1:51–62

[18] Blume LE (1993) The statistical mechanics of strageticinteraction. Games and Economic Behavior 5:387–424

[19] Foley DK (1994) A statistical equilibrium theory of mar-kets. Journal of Economic Theory 62:321–345

[20] Durlauf SN (1997) Statistical mechanics approaches tosocioeconomic behavior. In: Arthur WB, Durlauf SN,Lane DA (ed) The Economy as a Complex EvolvingSystem II, Addison-Wesley, Redwood City, CA, pp 81–104

[21] Anderson PW, Arrow KJ, Pines D (ed) (1988) TheEconomy as an Evolving Complex System, Addison-Wesley, Reading, MA

[22] Rosser JB (2008) Econophysics. In: Blume LE, DurlaufSN (ed) New Palgrave Dictionary of Economics, 2ndedition, Macmillan, London, in press.

[23] Dragulescu AA, Yakovenko VM (2000) Statistical me-chanics of money. The European Physical Journal B17:723–729

[24] Chakraborti A, Chakrabarti BK (2000) Statistical me-chanics of money: how saving propensity affects its dis-tribution. The European Physical Journal B 17:167–170.

[25] Bouchaud JP, Mezard M (2000) Wealth condensationin a simple model of economy. Physica A 282:536–545

[26] Wannier GH (1987) Statistical Physics, Dover, NewYork

[27] Lopez-Ruiz R, Sanudo J, Calbet X (2007) Geomet-rical derivation of the Boltzmann factor, preprinthttp://arxiv.org/abs/0707.4081

[28] Lopez-Ruiz R, Sanudo J, Calbet X (2007) On theequivalence of the microcanonical and the canon-ical ensembles: a geometrical approach, preprinthttp://arxiv.org/abs/0708.1866

[29] Anglin P (2005) Econophysics of wealth distribu-tions: a comment. In: Chatterjee A, Yarlagadda S,Chakrabarti BK (ed) Econophysics of Wealth Distribu-tions, Springer, New York, pp 229–238

[30] Lux T (2005) Emergent statistical wealth distributionsin simple monetary exchange models: a critical review.

In: Chatterjee A, Yarlagadda S, Chakrabarti BK (ed)Econophysics of Wealth Distributions, Springer, Milan,pp 51–60

[31] Gallegati M, Keen S, Lux T, Ormerod P (2006) Worry-ing trends in econophysics. Physica A 370:1–6

[32] Lux T (2008) Applications of statistical physics in fi-nance and economics. In: Rosser JB (ed) Handbook ofComplexity Research, Edward Elgar, Cheltenham, UKand Northampton, MA, in print

[33] Computer animation videos of money-transfer modelshttp://www2.physics.umd.edu/∼yakovenk/econophysics/animation.h

[34] Wright I (2007) Computer simulations of sta-tistical mechanics of money in Mathematica,http://demonstrations.wolfram.com/StatisticalMechanicsOfMoney

[35] McConnell CR, Brue SL (1996) Economics: Principles,Problems, and Policies. McGraw-Hill, New York

[36] Patriarca M, Chakraborti A, Kaski K, Germano G,(2005) Kinetic theory models for the distribution ofwealth: Power law from overlap of exponentials. In:Chatterjee A, Yarlagadda S, Chakrabarti BK (ed)Econophysics of Wealth Distributions, Springer, Milan,pp 93–110

[37] Bennati E (1988) Un metodo di simulazione statis-tica per l’analisi della distribuzione del reddito. Riv-ista Internazionale di Scienze Economiche e Commer-ciali 35:735–756

[38] Bennati E (1993) Il metodo di Montecarlo nell’analisieconomica. Rassegna di Lavori dell’ISCO (IstitutoNazionale per lo Studio della Congiuntura), Anno X,No. 4, pp 31–79

[39] Scalas E, Garibaldi U, Donadio S (2006) Statistical equi-librium in simple exchange games I: Methods of solutionand application to the Bennati-Dragulescu-Yakovenko(BDY) game. The European Physical Journal B 53:267–272

[40] Mimkes J (2000) Society as a many-particle system.Journal of Thermal Analysis and Calorimetry 60:1055–1069

[41] Mimkes J (2005) Lagrange principle of wealth distribu-tion. In: Chatterjee A, Yarlagadda S, Chakrabarti BK(ed) Econophysics of Wealth Distributions, Springer,Milan, pp 61–69

[42] Shubik M (1999) The Theory of Money and FinancialInstitutions, vol 2, The MIT Press, Cambridge, MA, p192

[43] Mandelbrot B (1960) The Pareto-Levy law and thedistribution of income. International Economic Review1:79–106

[44] Braun D (2001) Assets and liabilities are the momen-tum of particles and antiparticles displayed in Feynman-graphs. Physica A 290:491–500.

[45] Fischer R, Braun D (2003) Transfer potentials shapeand equilibrate monetary systems. Physica A 321:605–618.

[46] Fischer R, Braun D (2003) Nontrivial bookkeeping: amechanical perspective. Physica A 324:266–271

[47] Xi N, Ding N, Wang Y (2005) How required reserveratio affects distribution and velocity of money. PhysicaA 357:543–555

[48] Ispolatov S, Krapivsky PL, Redner S (1998) Wealthdistributions in asset exchange models. The EuropeanPhysical Journal B 2:267–276

[49] Angle J (1986) The surplus theory of social stratifica-tion and the size distribution of personal wealth. Social

22

Forces 65:293–326.[50] Angle J (1992) The inequality process and the distribu-

tion of income to blacks and whites. Journal of Mathe-matical Sociology 17:77–98

[51] Angle J (1992) Deriving the size distribution of personalwealth from ’the rich get richer, the poor get poorer’.Journal of Mathematical Sociology 18:27–46

[52] Angle J (1996) How the Gamma Law of income distri-bution appears invariant under aggregation. Journal ofMathematical Sociology 21:325–358

[53] Angle J (2002) The statistical signature of pervasivecompetition on wage and salary incomes. Journal ofMathematical Sociology 26:217–270

[54] Angle J (2006) The Inequality Process as a wealth max-imizing process. Physica A 367:388–414

[55] Engels F (1972) The origin of the family, private prop-erty and the state, in the light of the researches of LewisH. Morgan. International Publishers, New York

[56] Patriarca M, Chakraborti A, Kaski K (2004) Gibbs ver-sus non-Gibbs distributions in money dynamics. Phys-ica A 340:334–339

[57] Patriarca M, Chakraborti A, Kaski K (2004) Statisticalmodel with a standard Gamma distribution. PhysicalReview E 70:016104

[58] Repetowicz P, Hutzler S, Richmond P (2005) Dynamicsof money and income distributions. Physica A 356:641–654

[59] Chatterjee A, Chakrabarti BK, Manna SS (2004) Paretolaw in a kinetic model of market with random savingpropensity. Physica A 335:155-163

[60] Das A, Yarlagadda S (2005) An analytic treatment ofthe Gibbs-Pareto behavior in wealth distribution. Phys-ica A 353:529–538

[61] Chatterjee S, Chakrabarti BK, Stinchcombe RB (2005)Master equation for a kinetic model of a trading marketand its analytic solution. Physical Review E 72:026126

[62] Mohanty PK (2006) Generic features of the wealth dis-tribution in ideal-gas-like markets. Physical Review E74:011117

[63] Patriarca M, Chakraborti A, Germano G (2006) Influ-ence of saving propensity on the power-law tail of thewealth distribution. Physica A 369:723–736

[64] Gupta AK (2006) Money exchange model and a generaloutlook. Physica A 359:634–640

[65] Patriarca M, Chakraborti A, Heinsalu E, Germano G(2007) Relaxation in statistical many-agent economymodels. The European Physical Journal B 57:219–224

[66] Ferrero JC (2004) The statistical distribution of moneyand the rate of money transference. Physica A 341:575–585

[67] Scafetta N, Picozzi S, West BJ (2004) An out-of-equilibrium model of the distributions of wealth. Quan-titative Finance 4:353–364

[68] Scafetta N, Picozzi S, West BJ (2004) A trade-investment model for distribution of wealth. Physica D193:338–352

[69] Lifshitz EM, Pitaevskii LP (1981) Physical Kinetics,Pergamon Press, Oxford

[70] Ao P (2007) Boltzmann-Gibbs distribution of fortuneand broken time reversible symmetry in econodynamics.Communications in Nonlinear Science and NumericalSimulation 12:619–626

[71] Scafetta N, West BJ (2007) Probability distributionsin conservative energy exchange models of multiple in-

teracting agents. Journal of Physics Condensed Matter19:065138

[72] Chakraborti A, Pradhan S, Chakrabarti BK (2001) Aself-organising model of market with single commodity.Physica A 297:253–259

[73] Chatterjee A, Chakrabarti BK (2006) Kinetic marketmodels with single commodity having price fluctuations.The European Physical Journal B 54:399–404

[74] Ausloos M, Pekalski A (2007) Model of wealth andgoods dynamics in a closed market. Physica A 373:560–568

[75] Silver J, Slud E, Takamoto K (2002) Statistical equi-librium wealth distributions in an exchange economywith stochastic preferences. Journal of Economic The-ory 106:417–435

[76] Raberto M, Cincotti S, Focardi SM, Marchesi M (2003)Traders’ long-run wealth in an artificial financial mar-ket. Computational Economics 22:255–272

[77] Solomon S, Richmond P (2001) Power laws of wealth,market order volumes and market returns. Physica A299:188–197

[78] Solomon S, Richmond P (2002) Stable power laws invariable economies; Lotka-Volterra implies Pareto-Zipf.The European Physical Journal B 27: 257-261

[79] Huang DW (2004) Wealth accumulation with randomredistribution. Physical Review E 69:057103

[80] Slanina F (2004) Inelastically scattering particles andwealth distribution in an open economy. Physical Re-view E 69:046102

[81] Cordier S, Pareschi L, Toscani G (2005) On a kineticmodel for a simple market economy. Journal of Statis-tical Physics 120:253–277

[82] Richmond P, Repetowicz P, Hutzler S, Coelho R (2006)Comments on recent studies of the dynamics and dis-tribution of money. Physica A: 370:43–48

[83] Richmond P, Hutzler S, Coelho R, Repetowicz P (2006)A review of empirical studies and models of income dis-tributions in society. In: Chakrabarti BK, ChakrabortiA Chatterjee A (ed) Econophysics and Sociophysics:Trends and Perspectives, Wiley-VCH, Berlin

[84] Burda Z, Johnston D, Jurkiewicz J, Kaminski M, NowakMA, Papp G, Zahed I (2002) Wealth condensation inPareto macroeconomies. Physical Review E 65:026102

[85] Pianegonda S, Iglesias JR, Abramson G, Vega JL(2003) Wealth redistribution with conservative ex-changes. Physica A 322:667–675

[86] Braun D (2006) Nonequilibrium thermodynamics ofwealth condensation. Physica A 369:714–722

[87] Coelho R, Neda Z, Ramasco JJ, Santos MA (2005) Afamily-network model for wealth distribution in soci-eties. Physica A 353:515–528

[88] Iglesias JR, Goncalves S, Pianegonda S, Vega JL,Abramson G (2003) Wealth redistribution in our smallworld. Physica A 327:12–17

[89] Di Matteo T, Aste T, Hyde ST (2004) Exchanges incomplex networks: income and wealth distributions. In:Mallamace F, Stanley HE (ed) The Physics of ComplexSystems (New Advances and Perspectives), IOS Press,Amsterdam, p 435

[90] Hu MB, Jiang R, Wu QS, Wu YH (2007) Simulating thewealth distribution with a Richest-Following strategy onscale-free network. Physica A 381:467–472

[91] Bak P, Nørrelykke SF, Shubik M (1999) Dynamics ofmoney. Physical Review E 60:2528–2532

23

[92] Her Majesty Revenue and Customs(2003) Distribution of Personal Wealth.http://www.hmrc.gov.uk/stats/personal wealth/wealth oct03.pdf.

[93] Dragulescu AA, Yakovenko VM (2001) Exponential andpower-law probability distributions of wealth and in-come in the United Kingdom and the United States.Physica A 299:213–221,

[94] Klass OS, Biham O, Levy M, Malcai O, SolomonS (2007) The Forbes 400, the Pareto power-law andefficient markets. The European Physical Journal B55:143–147

[95] Sinha S (2006) Evidence for power-law tail of the wealthdistribution in India. Physica A 359:555–562

[96] Abul-Magd AY (2002) Wealth distribution in an ancientEgyptian society. Physical Review E 66:057104

[97] Kakwani N (1980) Income Inequality and Poverty. Ox-ford University Press, Oxford

[98] Champernowne DG, Cowell FA (1998) Economic In-equality and Income Distribution. Cambridge Univer-sity Press, Cambridge

[99] Atkinson AB, Bourguignon F (ed) (2000) Handbook ofIncome Distribution. Elsevier, Amsterdam

[100] Dragulescu AA, Yakovenko VM (2001) Evidence for theexponential distribution of income in the USA. The Eu-ropean Physical Journal B 20:585–589

[101] Dragulescu AA, Yakovenko VM (2003) Statistical me-chanics of money, income, and wealth: a short survey.In: Garrido PL, Marro J (ed) Modeling of Complex Sys-tems: Seventh Granada Lectures, American Institute ofPhysics (AIP) Conference Proceedings 661, New York,pp 180–183

[102] Yakovenko VM, Silva AC (2005) Two-class structureof income distribution in the USA: Exponential bulkand power-law tail. In: Chatterjee A, Yarlagadda S,Chakrabarti BK (ed) Econophysics of Wealth Distribu-tions, Springer, Milan, pp 15–23

[103] Silva AC, Yakovenko VM (2005) Temporal evolution ofthe ‘thermal’ and ‘superthermal’ income classes in theUSA during 1983-2001. Europhysics Letters 69:304–310

[104] Hasegawa A, Mima K, Duong-van M (1985) Plasmadistribution function in a superthermal radiation field.Physical Review Letters 54:2608–2610

[105] Desai MI, Mason GM, Dwyer JR, Mazur JE, Gold RE,Krimigis SM, Smith CW, Skoug RM (2003) Evidencefor a suprathermal seed population of heavy ions accel-erated by interplanetary shocks near 1 AU. The Astro-physical Journal, 588:1149–1162

[106] Collier MR (2004) Are magnetospheric suprathermalparticle distributions (κ functions) inconsistent withmaximum entropy considerations? Advances in SpaceResearch 33:2108–2112

[107] Souma W (2001) Universal structure of the personalincome distribution. Fractals 9:463–470

[108] Souma W (2002) Physics of personal income. In:Takayasu H (ed) Empirical Science of Financial Fluc-tuations: the Advent of Econophysics, Springer, Tokyo,pp 343–352

[109] Fujiwara Y, Souma W, Aoyama H, Kaizoji T, AokiM (2003) Growth and fluctuations of personal income.Physica A 321:598–604

[110] Aoyama H, Souma W, Fujiwara Y (2003) Growth andfluctuations of personal and company’s income. PhysicaA 324:352–358

[111] Strudler M, Petska T, Petska R (2003) An analysis of

the distribution of individual income and taxes, 1979–2001. The Internal Revenue Service, Washington DC,http://www.irs.gov/pub/irs-soi/03strudl.pdf.

[112] Souma W, Nirei M (2005) Empirical study and modelof personal income. In: Chatterjee A, Yarlagadda S,Chakrabarti BK (ed) Econophysics of Wealth Distribu-tions, Springer, Milan, pp 34–42

[113] Nirei M, Souma W (2007) A two factor model of incomedistribution dynamics. Review of Income and Wealth53:440–459

[114] Ferrero JC (2005) The monomodal, polymodal, equi-librium and nonequilibrium distribution of money. In:Chatterjee A, Yarlagadda S, Chakrabarti BK (ed)Econophysics of Wealth Distributions, Springer, Milan,pp 159–167

[115] Clementi F, Gallegati M (2005) Pareto’s law of incomedistribution: evidence for Germany, the United King-dom, the United States. In: Chatterjee A, YarlagaddaS, Chakrabarti BK (ed) Econophysics of Wealth Distri-butions, Springer, Milan, pp 3–14

[116] Clementi F, Gallegati M, Kaniadakis G (2007) κ-generalized statistics in personal income distribution.The European Physical Journal B 57:187–193

[117] Clementi F, Gallegati M (2005) Power law tails inthe Italian personal income distribution. Physica A350:427–438

[118] Clementi F, Di Matteo T, Gallegati M (2006) Thepower-law tail exponent of income distributions. Phys-ica A 370:49–53

[119] Rawlings PK, Reguera D, Reiss H (2004) Entropic basisof the Pareto law. Physica A 343:643–652

[120] Banerjee A, Yakovenko VM, Di Matteo T (2006) Astudy of the personal income distribution in Australia.Physica A 370:54–59

[121] Gibrat R (1931) Les Inegalites Economiques, Sirely,Paris

[122] Kalecki M (1945) On the Gibrat distribution. Econo-metrica 13:161–170.

[123] Champernowne DG (1953) A model of income distribu-tion. The Economic Journal 63:318–351.

[124] Milakovic M (2005) Do we all face the same constraints?In: Chatterjee A, Yarlagadda S, Chakrabarti BK (ed)Econophysics of Wealth Distributions, Springer, Milan,pp 184–191

[125] Takayasu H, Sato AH, Takayasu M (1997) Stable infinitevariance fluctuations in randomly amplified Langevinsystems. Physical Review Letters 79:966–969

[126] Kesten H (1973) Random difference equations and re-newal theory for products of random matrices, ActaMathematica 131:207–248.

[127] Fiaschi D, Marsili M (2007) Distribu-tion of wealth: Theoretical microfounda-tions and empirical evidence. Working paperhttp://www.dse.ec.unipi.it/persone/docenti/fiaschi/Lavori/dis

[128] Levy M, Solomon S (1996) Power laws are logarith-mic Boltzmann laws. International Journal of ModernPhysics C 7:595–751

[129] Sornette D, Cont R (1997) Convergent multiplicativeprocesses repelled from zero: power laws and truncatedpower laws. Journal de Physique I (France) 7:431–444

[130] Lydall HF (1959) The distribution of employment in-comes. Econometrica 27:110–115

[131] Feller W (1966) An Introduction to Probability Theoryand Its Applications, vol 2, John Wiley, New York, p 10

24

[132] Mimkes J, Aruka Y (2005) Carnot process ofwealth distribution. In: Chatterjee A, Yarlagadda S,Chakrabarti BK (ed) Econophysics of Wealth Distribu-tions, Springer, Milan, pp 70–78

[133] Mimkes J (2006) A thermodynamic formulation of eco-nomics. In: Chakrabarti BK, Chakraborti A ChatterjeeA (ed) Econophysics and Sociophysics: Trends and Per-spectives, Wiley-VCH, Berlin, pp 1–33

[134] Schelling TC (1971) Dynamic models of segregation.Journal of Mathematical Sociology 1:143–186

[135] Mimkes J (1995) Binary alloys as a model for the mul-ticultural society. Journal of Thermal Analysis 43:521–537

[136] Mimkes J (2006) A thermodynamic formulation of socialscience. In: Chakrabarti BK, Chakraborti A ChatterjeeA (ed) Econophysics and Sociophysics: Trends and Per-spectives, Wiley-VCH, Berlin

[137] Jego C, Roehner BM (2007) A physicist’sview of the notion of “racism”, preprinthttp://arxiv.org/abs/0704.2883

[138] Stauffer D, Schulze C (2007) Urban and scien-tific segregation: The Schelling-Ising model, preprinthttp://arxiv.org/abs/0710.5237

[139] Dall’Asta L, Castellano C, Marsili M (2007) Statisticalphysics of the Schelling model of segregation, preprinthttp://arxiv.org/abs/0707.1681

[140] Lim M, Metzler R, Bar-Yam Y (2007) Global pat-tern formation and ethnic/cultural violence. Science317:1540–1544

[141] Wright I (2005) The social architecture of capitalism.Physica A 346:589–620

[142] Defilla S (2007) A natural value unit — Econophysicsas arbiter between finance and economics. Physica A382:42–51

[143] McCauley JL (2006) Response to ‘Worrying Trends inEconophysics’. Physica A 371:601–609

[144] Richmond P, Chakrabarti BK, Chatterjee A, AngleJ (2006) Comments on ‘Worrying Trends in Econo-physics’: income distribution models. In: Chatterjee A,

Chakrabarti BK (ed) Econophysics of Stock and OtherMarkets, Springer, Milan, pp 244–253

[145] Rosser JB (2006) Debating the Role of Econophysics.Working paper, http://cob.jmu.edu/rosserjb/

[146] Rosser JB (2006) The nature and future of econophysics.In: Chatterjee A, Chakrabarti BK (ed) Econophysics ofStock and Other Markets, Springer, Milan, pp 225–234

[147] Kuznets S (1955) Economic growth and income inequal-ity. The American Economic Review 45:1–28.

[148] Levy F (1987) Changes in the distribution of Americanfamily incomes, 1947 to 1984. Science 236:923–927

[149] Internal Revenue Service (1999) Statistics of Income–1997, Individual Income Tax Returns. Publication 1304,Revision 12-99, Washington DC

[150] Hayes B (2002) Follow the money. American Scientist90:400–405

[151] Ball P (2006) Econophysics: Culture Crash. Nature441:686–688

Books and Reviews

• McCauley J (2004) Dynamics of Markets: Econo-physics and Finance. Cambridge University Press,Cambridge

• Farmer JD, Shubik M, Smith E (2005) Is economicsthe next physical science? Physics Today, Septem-ber issue, p 37

• Samanidou E, Zschischang E, Stauffer D, Lux T(2007) Agent-based models of financial markets.Reports on Progress in Physics 70:409–450

• Web resource: Econophysics Forumhttp://www.unifr.ch/econophysics/