applications of physics to economics and finance

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APPLICATIONS OF PHYSICS TO ECONOMICS AND FINANCE:MONEY, INCOME, WEALTH, AND THE STOCK MARKET ∗

Adrian A. Dr agulescu † Department of Physics, University of Maryland, College Park

(Dated: May 15, 2002, cond-mat/0307341 )

Abstract: Several problems arising in Economics and Finance are analyzed using concepts and quantitativemethods from Physics. The disertation is organized as follows:

In the rst chapter, it is argued that in a closed economic system, money is conserved. Thus, by analogywith energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbslaw characterized by an effective temperature equal to the average amount of money per economic agent. Theemergence of Boltzmann-Gibbs distribution is demonstrated through computer simulations of economic mod-els. A thermal machine which extracts a monetary prot can be constructed between two economic systemswith different temperatures. The role of debt and models with broken time-reversal symmetry for which theBoltzmann-Gibbs law does not hold, are discussed.

In the second chapter, using data from several sources, it is found that the distribution of income is describedfor the great majority of population by an exponential distribution, whereas the high-end tail follows a powerlaw. From the individual income distribution, the probability distribution of income for families with two earnersis derived and it is shown that it also agrees well with the data. Data on wealth is presentend and it is foundthat the distribution of wealth has a structure similar to the distribution of income. The Lorenz curve and Ginicoefcient were calculated and are shown to be in good agreement with both income and wealth data sets.

In the third chapter, the stock-market uctuations at different time scales are investigated. A model where

stock-price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic varianceis proposed. The corresponding Fokker-Planck equation can be solved exactly. Integrating out the variance, ananalytic formula for the time-dependent probability distribution of stock price changes (returns) is found. Theformula is in excellent agreement with the Dow-Jones index for the time lags from 1 to 250 trading days. Fortime lags longer than the relaxation time of variance, the probability distribution can be expressed in a scalingform using a Bessel function. The Dow-Jones data follow the scaling function for seven orders of magnitude.

Contents

Acknowledgements 1

I. Statistical mechanics of money 2A. Introduction 2

B. Boltzmann-Gibbs distribution 2C. Computer simulations 3D. Thermal machine 4E. Models with debt 4F. Boltzmann equation 5G. Non-Boltzmann-Gibbs distributions 6H. Nonlinear Boltzmann equation vs. linear master equation 7

I. Conclusions 7

II. Distribution of income and wealth 8A. Introduction 8B. Distribution of income for individuals 8

C. Exponential distribution of income 8D. Power-law tail and “Bose” condensation 10E. Geographical variations in income distribution 11F. Distribution of income for families 13

∗This document is a reformatted version of my PhD thesis. Advisor: Pro-fessor Victor M. Yakovenko. Committee: Professor J. Robert Dorfman, Pro-fessor Theodore L. Einstein, Professor Bei-Lok Hu, and Professor John D.Weeks.

G. Distribution of wealth 15H. Other distributions for income and wealth 16I. Conclusions 17

III. Distribution of stock-price uctuations 18A. Introduction 18

B. The model 18C. Solution of the Fokker-Planck equation 19D. Path-Integral Solution 20E. Averaging over variance 20F. Asymptotic behavior for long time t 21

G. Asymptotic behavior for large log-return x 23H. Comparison with Dow-Jones time series 24I. Conclusions 25

IV. Path-integral solution of the Cox-Ingersoll-Ross/Feller model

References 29

Acknowledgements

My six year apprenticeship in physics at the University of Maryland was a unique and remarkable leg of my life. And toa large degree this is due to my advisor, Professor Victor M.Yakovenko, who has been at the center of my PhD education.I will deeply miss his clear perspective, his sure hand, and theincommensurate thrill of doing physics together.

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I am grateful to Professors J. Robert Dorfman, TheodoreL. Einstein, Bei-Lok Hu, and John D. Weeks for agreeing toserve on my advisory committee.

Thanks are also due to Krishnendu Sengupta, NicholasDupuis, Hyok-Jon Kwon, Anatoley Zheleznyak, and Hsi-Sheng Goan all former students or post-docs of professor Vic-tor Yakovenko, for stimulating discussions in condensed mat-ter physics. My interests in physics were greatly molded by

professor Bei-Lok Hu, through his remarkable courses in non-equilibrium physics, and also through personal interaction formore than three years. I had proted a lot from the associationwith Charis Anastopoulos and Sanjiv Shresta while workingon quantum optics problems suggested by professor Bei-Lok Hu.

It is not without melancholy that I thank my rst truephysics teachers: Ion Cot˘aescu, Ovidiu Lipan, Dan Luca,Adrian Neagu, Mircea Rasa, Costel Rasinariu, Vlad Socol-iuc, and Dumitru Vulcanov. On an even deeper layer, I thank my family for all the support they provided me over the years,and for allowing me to pursue my interests.

All the people mentioned above have generously shared

with me some of their time and knowledge. I will do all Ican to represent them well.

I. STATISTICAL MECHANICS OF MONEY

A. Introduction

The application of statistical physics methods to economicspromises fresh insights into problems traditionally not asso-ciated with physics (see, for example, the recent review andbook 1). Both statistical mechanics and economics study bigensembles: collections of atoms or economic agents, respec-

tively. The fundamental law of equilibrium statistical mechan-ics is the Boltzmann-Gibbs law, which states that the proba-bility distribution of energy ε is P (ε) = Ce−ε/T , where T isthe temperature, and C is a normalizing constant 2. The mainingredient that is essential for the textbook derivation of theBoltzmann-Gibbs law 2 is the conservation of energy 3. Thus,one may generalize that any conserved quantity in a big sta-tistical system should have an exponential probability distri-bution in equilibrium.

An example of such an unconventional Boltzmann-Gibbslaw is the probability distribution of forces experienced bythe beads in a cylinder pressed with an external force 4. Be-cause the system is at rest, the total force along the cylinderaxis experienced by each layer of granules is constant and israndomly distributed among the individual beads. Thus theconditions are satised for the applicability of the Boltzmann-Gibbs law to the force, rather than energy, and it was indeedfound experimentally 4.

We claim that, in a closed economic system, the to-tal amount of money is conserved. Thus the equilibriumprobability distribution of money P (m) should follow theBoltzmann-Gibbs law P (m) = C e−m/T . Here m is money,and T is an effective temperature equal to the average amountof money per economic agent. The conservation law of

money 5 reects their fundamental property that, unlike mate-rial wealth, money (more precisely the at, “paper” money) isnot allowed to be manufactured by regular economic agents,but can only be transferred between agents. Our approachhere is very similar to that of Ispolatov et al. 6. However, theyconsidered only models with broken time-reversal symmetry,for which the Boltzmann-Gibbs law typically does not hold.The role of time-reversal symmetry and deviations from the

Boltzmann-Gibbs law are discussed in detail in Sec. I G.It is tempting to identify the money distribution P (m) withthe distribution of wealth 6. However, money is only one partof wealth, the other part being material wealth. Material prod-ucts have no conservation law because they can be manu-factured, destroyed, consumed, etc. Moreover, the monetaryvalue of a material product (the price) is not constant. Thesame applies to stocks, which economics textbooks explicitlyexclude from the denition of money 7. So, we do not ex-pect the Boltzmann-Gibbs law for the distribution of wealth.Some authors believe that wealth is distributed according to apower law (Pareto-Zipf), which originates from a multiplica-tive random process 8. Such a process may reect, amongother things, the uctuations of prices needed to evaluate themonetary value of material wealth.

B. Boltzmann-Gibbs distribution

Let us consider a system of many economic agents N ≫1,which may be individuals or corporations. In this thesis, weonly consider the case where their number is constant. Eachagent i has some money m i and may exchange it with otheragents. It is implied that money is used for some economicactivity, such as buying or selling material products; however,we are not interested in that aspect. As in Ref. 6, for us theonly result of interaction between agents i and j is that somemoney ∆ m changes hands: [m i , m j ] → [m ′i , m ′j ] = [m i −∆ m, m j + ∆ m]. Notice that the total amount of money isconservedin each transaction: m i + m j = m ′i + m ′j . This localconservation law of money 5 is analogous to the conservationof energy in collisions between atoms. We assume that theeconomic system is closed, i. e. there is no external ux of money, thus the total amount of money M in the system isconserved. Also, in the rst part of this chapter, we do notpermit any debt, so each agent’s money must be non-negative:m i ≥ 0. A similar condition applies to the kinetic energy of atoms: εi ≥0.

Let us introduce the probability distribution function of money P (m), which is dened so that the number of agentswith money between m and m + dm is equal to NP (m) dm .We are interested in the stationary distribution P (m) corre-sponding to the state of thermodynamic equilibrium. In thisstate, an individual agent’s money m i strongly uctuates, butthe overall probability distribution P (m) does not change.

The equilibrium distribution function P (m) can be derivedin the same manner as the equilibrium distribution functionof energy P (ε) in physics 2. Let us divide the system intotwo subsystems 1 and 2. Taking into account that moneyis conserved and additive: m = m1 + m2 , whereas the



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Money, m

P r o

b a b

i l i t y ,

P ( m )

N=500, M=5*10 5, time=4*10 5.

m , T

0 1000 2000 30000

1

2

3

Money, m

l o g

P ( m )

FIG. 1: Histogram and points: stationary probability distributionof money P (m). Solid curves: ts to the Boltzmann-Gibbs lawP (m) ∝ exp( − m/T ). Vertical lines: the initial distribution of money.

probability is multiplicative: P = P 1P 2 , we conclude thatP (m1 + m2) = P (m1)P (m2 ). The solution of this equa-tion is P (m) = Ce−m/T ; thus the equilibrium probabil-ity distribution of money has the Boltzmann-Gibbs form.From the normalization conditions ∞

0 P (m) dm = 1 and

∞0 m P (m) dm = M/N , we nd that C = 1 /T and

T = M/N . Thus, the effective temperature T is the aver-age amount of money per agent.

The Boltzmann-Gibbs distribution can be also derived bymaximizing the entropy of money distributionS = − ∞

0 dm P (m) ln P (m) under the constraint of mon-ey conservation 2. Following original Boltzmann’s argument,let us divide the money axis 0 ≤ m ≤ ∞into small bins of size dm and number the bins consecutively with the index b =1, 2, . . . Let us denote the number of agents in a bin b as N b,the total number being N = ∞b=1 N b. The agents in the binb have money mb, and the total money is M = ∞b=1 mbN b.The probability of realization of a certain set of occupationnumbers N bis proportional to the numberof ways N agentscan be distributed among the bins preserving the set

N b

.

This number is N !/N 1!N 2! . . . The logarithm of probabilityis entropy ln N ! − ∞b=1 ln N b!. When the numbers N b arebig and Stirling’s formula ln N ! ≈ N ln N −N applies, theentropy per agent is S = ( N ln N − ∞b=1 N b ln N b)/N =

− ∞b=1 P b ln P b, where P b = N b/N is the probability that anagent has money mb. Using the method of Lagrange multipli-ers to maximize the entropy S with respect to the occupationnumbers N bwith the constraints on the total money M andthe total number of agents N generates the Boltzmann-Gibbsdistribution for P (m)2.

0 500 1000 1500 20000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time, t

E n

t r o p y ,

S

FIG. 2: Time evolution of entropy. Top curve: for the exchange of a random fraction ν of the average money in the system: ∆ m =ν M/N . Bottom curve: for the exchange of a small constant amount∆ m = 1 . The time scale for the bottom curve is 500 times greaterthan indicated, so it actually ends at the time 106 .

C. Computer simulations

To check that these general argumentsindeed work, we per-formed several computer simulations. Initially, all agents aregiven the same amount of money: P (m) = δ (m −M/N ),which is shown in Fig. 1 as the double vertical line. One pairof agents at a time is chosen randomly, then one of the agentsis randomly picked to be the “winner” (the other agent be-comes the “loser”), and the amount ∆ m ≥ 0 is transferredfrom the loser to the winner. If the loser does not have enoughmoney to pay ( m

i < ∆ m), then the transaction does not take

place, and we proceed to another pair of agents. Thus, theagents are not permitted to have negative money. This bound-ary condition is crucial in establishing the stationary distribu-tion. As the agents exchange money, the initial delta-functiondistribution rst spreads symmetrically. Then, the probabilitydensity starts to pile up at the impenetrable boundary m = 0 .The distributionbecomes asymmetric (skewed) and ultimatelyreaches the stationary exponential shape shown in Fig. 1. Weused several trading rules in the simulations: the exchange of a small constant amount ∆ m = 1 , the exchange of a ran-dom fraction 0 ≤ ν ≤ 1 of the average money of the pair:∆ m = ν (m i + m j )/ 2, and the exchange of a random fractionν of the average money in the system: ∆ m = ν M/N . Fig-ures in this chapter mostly show simulations for the third rule;however, the nal stationary distribution was found to be thesame for all rules.

In the process of evolution, the entropy S of the systemincreases in time and saturates at the maximal value for theBoltzmannGibbs distribution. This is illustrated by the topcurve in Fig. 2 computed for the third rule of exchange. Thebottom curve in Fig. 2 shows the time evolution of entropyfor the rst rule of exchange. The time scale for this curve is500 times greater than for the top curve, so the bottom curve



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actually ends at the time 106. The plot shows that, for the rstrule of exchange, mixing is much slower than for the third one.Nevertheless, even for the rst rule, the system also eventuallyreaches the Boltzmann-Gibbs state of maximal entropy, albeitover a time much longer than shown in Fig. 2.

One might argue that the pairwise exchange of money maycorrespond to a medieval market, but not to a modern econ-omy. In order to make the model somewhat more realistic,

we introduce rms. One agent at a time becomes a “rm”.The rm borrows capital K from another agent and returns itwith an interest rK , hires L agents and pays them wages W ,manufactures Q items of a product and sells it to Q agentsat a price R . All of these agents are randomly selected. Therm receives the prot F = RQ −LW −rK . The net re-sult is a many-body exchange of money that still satises theconservation law.

Parameters of the model are selected following the pro-cedure described in economics textbooks 7. The aggregatedemand-supply curve for the product is taken to be R(Q) =V/Q η , where Q is the quantity people would buy at a priceR , and η = 0 .5 and V = 100 are constants. The produc-

tion function of the rm has the conventional Cobb-Douglasform: Q(L, K ) = Lβ K 1−β , where β = 0 .8 is a constant. Inour simulation, we set W = 10 . By maximizing rm’s protF with respect to K and L , we nd the values of the otherparameters: L = 20 , Q = 10 , R = 32 , and F = 68 .

However, the actual values of the parameters do not matter.Our computer simulations show that the stationary probabilitydistribution of money in this model always has the universalBoltzmann-Gibbs form independent of the model parameters.

D. Thermal machine

As explained in the Introduction, the money distributionP (m) should not be confused with the distribution of wealth.We believe that P (m) should be interpreted as the instanta-neous distribution of purchasing power in the system. Indeed,to make a purchase, one needs money. Material wealth nor-mally is not used directly for a purchase. It needs to be soldrst to be converted into money.

Let us consider an outside monopolistic vendor selling aproduct (say, cars) to the system of agents at a price p. Sup-pose that a certain small fraction f of the agents needs to buythe product at a given time, and each agent who has enoughmoney to afford the price will buy one item. The fraction f is assumed to be sufciently small, so that the purchase doesnot perturb the whole system signicantly. At the same time,the absolute number of agents in this group is assumed to bebig enough to make the group statistically representative andcharacterized by the Boltzmann-Gibbs distribution of money.The agents in this group continue to exchange money with therest of the system, which acts as a thermal bath. The demandfor the product is constantly renewed, because products pur-chased in the past expire after a certain time. In this situation,the vendor can sell the product persistently, thus creating asmall steady leakage of money from the system to the vendor.

What price p would maximize the vendor’s income?

To answer this question, it is convenient to introducethe cumulative distribution of purchasing power N (m) =N ∞

m P (m ′) dm ′ = Ne−m/T , which gives the number of agents whose money is greater than m . The vendor’s incomeis f p N ( p). It is maximal when p = T , i. e. the optimal priceis equal to the temperature of the system. This conclusionalso follows from the simple dimensional argument that tem-perature is the only money scale in the problem. At the price

p = T that maximizes the vendor’s income, only the fraction N (T )/N = e−1 = 0 .37 of the agents can afford to buy theproduct.

Now let us consider two disconnected economic systems,one with the temperature T 1 and another with T 2 : T 1 > T 2 .A vendor can buy a product in the latter system at its equi-librium price T 2 and sell it in the former system at the priceT 1 , thus extracting the speculative prot T 1 −T 2 , as in a ther-mal machine. This example suggests that speculative protis possible only when the system as a whole is out of equi-librium. As money is transferred from the high- to the low-temperature system, their temperatures become closer andeventually equal. After that, no speculative prot is possi-ble, which would correspond to the “thermal death” of theeconomy. This example brings to mind economic relationsbetween developed and developing countries, with manufac-turing in the poor (low-temperature) countries for export tothe rich (high-temperature) ones.

We will demonstrate in Ch. II that for the large majorityof the population the distribution of income is exponential.Hence, similar to the distribution of money, the distributionof income has a corresponding temperature. If in the previ-ous discussion about economic trade, the two countries have adifferent temperature for the distribution of income, the pos-sibility of constructing a thermal machine will still hold true.This is because purchasing power (total money) per unit timehas two components: a positive one, income and a negativeone, spending. Instead of making a one time purchase, thebuyer will buy one product per unit time, effectively creatingthe thermal engine. We will revisit this idea in Sec. II E whenwe give the values for income temperature between the statesof the USA and for the United Kingdom.

E. Models with debt

Now let us discuss what happens if the agents are permit-ted to go into debt. Debt can be viewed as negative money.Now when a loser does not have enough money to pay, hecan borrow the required amount from a reservoir, and hisbalance becomes negative. The conservation law is not vi-olated: The sum of the winner’s positive money and loser’snegative money remains constant. When an agent with anegative balance receives money as a winner, she uses thismoney to repay the debt until her balance becomes positive.We assume for simplicity that the reservoir charges no inter-est for the lent money. However, because it is not sensibleto permit unlimited debt, we put a limit md on the maximaldebt of an agent: mi > −md . This new boundary condi-tion P (m < −md ) = 0 replaces the old boundary condition



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Money, m

P r o

b a b i

l i t y , P ( m )

N=500, M=5*10 5, time=4*10 5.

Model without debt, T=1000

Model with debt, T=1800

FIG. 3: Histograms: stationary distributions of money with and with-out debt. Solid curves: ts to the Boltzmann-Gibbs laws with tem-peratures T = 1800 and T = 1000 .

P (m < 0) = 0 . The result of a computer simulation withmd = 800 is shown in Fig. 3 together with the curve for md =0. P (m) is again given by the Boltzmann-Gibbs law, but nowwith the higher temperature T = M/N + md , because thenormalization conditions need to be maintained including thepopulation with negative money: ∞

−m dP (m) dm = 1 and

∞−m d

m P (m) dm = M/N . The higher temperature makesthe money distribution broader, which means that debt in-creases inequality between agents.

In general, temperature is completely determined by the av-erage money per agent, m = M/N , and the boundary con-

ditions. Suppose the agentsare required to have no less moneythan m1 and no more than m2 : m 1 ≤m ≤ m2 . In this case,the two normalization conditions: m 2

m 1P (m) dm = 1 and

m 2

m 1m P (m) dm = m with P (m) = C e−m/T give the

following equation for T

coth∆ mT −

T ∆ m

= m − m

∆ m , (1)

where m = ( m1 + m2)/ 2 and ∆ m = ( m2−m1)/ 2. It followsfrom Eq. (1) that the temperature is positive when m > m ,negative when m < m , and innite ( P (m) = const ) whenm = m . In particular, if agents’ money are bounded fromabove, but not from below:

−∞ ≤ m

≤ m2 , the temper-

ature is negative. That means an inverted Boltzmann-Gibbsdistribution with more rich agents than poor.

Imposing a sharp cutoff at m d may be not quite realistic.In practice, the cutoff may be extended over some range de-pending on the exact bankruptcy rules. Over this range, theBoltzmann-Gibbs distribution would be smeared out. So weexpect to see the Boltzmann-Gibbs law only sufciently farfrom the cutoff region. Similarly, in experiment 4, some devi-ations from the exponential law were observed near the lowerboundary of the distribution. Also, at the high end of the dis-

tributions, the number of events becomes small and statisticspoor, so the Boltzmann-Gibbs law loses applicability. Thus,we expect the Boltzmann-Gibbs law to hold only for the in-termediate range of money not too close either to the lowerboundary or to the very high end. However, this range is themost relevant, because it covers the great majority of popula-tion.

Lending creates equal amounts of positive (asset) and neg-

ative (liability) money5,7

. When economics textbooks de-scribe how “banks create money” or “debt creates money” 7,they do not count the negative liabilities as money, and thustheir money is not conserved. In our operational denition of money, we include all nancial instruments with xed denom-ination, such as currency, IOUs, and bonds, but not materialwealthor stocks, and we count both assets and liabilities. Withthis denition, money is conserved, and we expect to see theBoltzmann-Gibbs distribution in equilibrium. Unfortunately,because this denition differs from economists’ denitions of money (M1, M2, M3, etc. 7), it is not easy to nd the appro-priate statistics. Of course, money can be also emitted by acentral bank or government. This is analogous to an externalinux of energy into a physical system. However, if this pro-cess is sufciently slow, the economic system may be able tomaintain quasi-equilibrium, characterized by a slowly chang-ing temperature.

We performed a simulation of a model with one bank andmany agents. The agents keep their money in accounts onwhich the bank pays interest. The agents may borrow moneyfrom the bank, for which they must pay interest in monthlyinstallments. If they cannot make the required payments, theymay be declared bankrupt, which relieves them from the debt,but the liability is transferred to the bank. In this way, the con-servation of money is maintained. The model is too elaborateto describe it in full detail here. We found that, depending onthe parameters of the model, either the agents constantly losemoney to the bank, which steadily reduces the agents’ temper-ature, or the bank constantly loses money, which drives downits own negative balance and steadily increases the agents’temperature.

F. Boltzmann equation

The Boltzmann-Gibbs distribution can be also derived fromthe Boltzmann equation 9, which describes the time evolutionof the distribution function P (m) due to pairwise interactions:

dP (m)dt = −w[m,m

′

]→[m −∆ ,m′

+∆] P (m)P (m ′) (2)+ w[m −∆ ,m ′ +∆] →[m,m ′ ]P (m −∆) P (m ′ + ∆) dm ′ d∆ .

Here w[m,m ′ ]→[m −∆ ,m ′ +∆] is the rate of transferring money∆ from an agent with money m to an agent with money m ′.If a model has time-reversal symmetry, then the transition rateof a direct process is the same as the transition rate of the re-versed process, thus the w-factors in the rst and second linesof Eq. (2) are equal. In this case, it can be easily checked thatthe Boltzmann-Gibbs distribution P (m) = C exp(−m/T )



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Money, m

P r o

b a b

i l i t y ,

P ( m )

N=500, M=5*10 5, α=1/3.

FIG. 4: Histogram: stationary probability distribution of money inthe multiplicative random exchange model studied in Ref. 6 . Solidcurve: the Boltzmann-Gibbs law.

nullies the right-hand side of Eq. ( 2); thus this distributionis stationary: dP (m)/dt = 0 9.

G. Non-Boltzmann-Gibbs distributions

However, if time-reversal symmetry is broken, the two tran-sition rates w in Eq. (2) may be different, and the system mayhave a non-Boltzmann-Gibbs stationary distribution or no sta-tionary distribution at all. Examples of this kind were stud-ied in Ref. 6. One model was called multiplicative random

exchange. In this model, a randomly selected loser i losesa xed fraction α of his money m i to a randomly selectedwinner j : [m i , m j ] → [(1 −α)m i , mj + αm i ]. If we tryto reverse this process and immediately appoint the winner j to become a loser, the system does not return to the orig-inal conguration [m i , m j ]: [(1 − α )m i , mj + αm i ] →[(1 − α )m i + α (m j + αm i ) , (1 − α )(m j + αm i )]. Ex-cept for α = 1 / 2, the exponential distribution function is nota stationary solution of the Boltzmann equation derived forthis model in Ref. 6 . Instead, the stationary distribution has theshape shown in Fig. 4 for α = 1 / 3, which we reproduced inour numerical simulations. It still has an exponential tail endat the high end, but drops to zero at the low end for α < 1/ 2.Another example of similar kind was studied in Ref. 10, whichappeared after the rst version of our paper was posted ascond-mat/0001432 on January 30, 2000. In that model, theagents save a fraction λ of their money and exchange a ran-dom fraction ǫ of their total remaining money: [m i , m j ] →[λm i + ǫ(1−λ)(m i + m j ) , λm j + (1 −ǫ)(1 −λ)(m i + m j )].This exchange also does not return to the original congura-tion after being reversed. The stationary probability distribu-tion was found in Ref. 10 to be nonexponential for λ = 0 witha shape qualitatively similar to the one shown in Fig. 4.

Another interesting example which has a non-Boltzmann-

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16N=500, M=5*10 5, tax=40%.

Money, m

P r o

b a b

i l i t y ,

P ( m )

FIG. 5: Histogram: stationary probability distribution of money inthe model with taxes and subsidies. Solid curve: the Boltzmann-Gibbs law.

Gibbs distribution occurs in a model with taxes and subsi-dies. Suppose a special agent (“government”) collects a frac-tion (“tax”) of every transaction in the system. The collectedmoney is then equally divided among all agents of the sys-tem, so that each agent receives the subsidy δm with the fre-quency 1/τ s . Assuming that δm is small and approximatingthe collision integral with a relaxation time τ r 9, we obtain thefollowing Boltzmann equation

∂P (m)∂t

+ δm

τ s∂P (m)

∂m = −

P (m) − P (m)τ r

, (3)

where P (m) is the equilibrium Boltzmann-Gibbs function.The second term in the left-hand side of Eq. (3) is analo-gous to the force applied to electrons in a metal by an ex-ternal electric eld 9. The approximate stationary solutionof Eq. (3) is the displaced Boltzmann-Gibbs one: P (m) =P (m−(τ r /τ s ) δm). The displacement of the equilibrium dis-tribution P (m) by (τ r /τ s ) δm would leave an empty gap nearm = 0 . This gap is lled by interpolating between zero pop-ulation at m = 0 and the displaced distribution. The curveobtained in a computer simulation of this model (Fig. 5) qual-itatively agrees with this expectation. The low-money popula-tion is suppressed, because the government, acting as an exter-nal force, “pumps out” that population and pushes the systemout of thermodynamic equilibrium. We found that the entropyof the stationary state in the model with taxes and subsidies isfew a percent lower than without.

These examples show that the Boltzmann-Gibbs distribu-tion is not fully universal, meaning that it does not hold for just any model of exchange that conserves money. Neverthe-less, it is universal in a limited sense: For a broad class of models that have time-reversal symmetry, the stationary dis-tribution is exponential and does not depend on the details of a model. Conversely, when time-reversal symmetry is bro-ken, the distribution may depend on model details. The dif-





7

ference between these two classes of models may be rathersubtle. For example, let us change the multiplicative randomexchange from a xed fraction of loser’s money to a xedfraction of the total money of winner and loser. This mod-ication retains the multiplicative idea that the amount ex-changed is proportional to the amount involved, but restorestime-reversal symmetry and the Boltzmann-Gibbs distribu-tion. In the model with ∆ m = 1 discussed in the next Section,

the difference between time-reversible and time-irreversibleformulations amounts to the difference between impenetra-ble and absorbing boundary conditions at m = 0 . Unlike inphysics, in economy there is no fundamental requirement thatinteractions have time-reversal symmetry. However, in the ab-sence of detailed knowledge of real microscopic dynamics of economic exchange, the semiuniversal Boltzmann-Gibbs dis-tribution appears to be a natural starting point.

Moreover, deviations from the Boltzmann-Gibbs law mayoccur only if the transition rates w in Eq. (2) explicitly de-pend on the agents’ money m or m ′ in an asymmetric man-ner. In another simulation, we randomly preselected winnersand losers for every pair of agents (i, j ). In this case, moneyows along directed links between the agents: i

→ j

→ k ,

and time-reversal symmetry is strongly broken. This modelis closer to the real economy, in which, for example, onetypically receives money from an employer and pays it to agrocer, but rarely the reverse. Nevertheless, we still foundthe Boltzmann-Gibbs distribution of money in this model, be-cause the transition rates w do not explicitly depend on m andm ′.

H. Nonlinear Boltzmann equation vs. linear master equation

For the model where agents randomly exchange the con-stant amount ∆ m = 1 , the Boltzmann equation is:

dP mdt

= P m +1∞

n =0P n + P m −1

∞

n =1P n

−P m∞

n =0P n −P m

∞

n =1P n (4)

= ( P m +1 + P m −1 −2P m ) + P 0(P m −P m −1), (5)

where P m ≡ P (m) and we have used ∞m =0 P m = 1 . Therst, diffusion term in Eq. ( 5) is responsible for broadening of the initial delta-function distribution. The second term, pro-portional to P 0 , is essential for the Boltzmann-Gibbs distri-bution P m = e−m/T (1

−e−1/T ) to be a stationary solution

of Eq. (5). In a similar model studied in Ref. 6, the secondterm was omitted on the assumption that agents who lost allmoney are eliminated: P 0 = 0 . In that case, the total numberof agents is not conserved, and the system never reaches anystationary distribution. Time-reversal symmetry is violated,since transitions into the state m = 0 are permitted, but notout of this state.

If we treat P 0 as a constant, Eq. ( 5) looks like a linearFokker-Planckequation 9 for P m , with therst term describingdiffusion and the second term an external force proportional

to P 0 . Similar equations were studied in Ref. 8. Eq. (5) can bealso rewritten as

dP mdt

= P m +1 −(2 −P 0 )P m + (1 −P 0 )P m −1 . (6)

The coefcient (1−P 0) in front of P m −1 represents the rate of increasing moneyby ∆ m = 1 , and the coefcient 1 in front of P m +1 represents the rate of decreasing money by ∆ m = −1.Since P 0 > 0, the former is smaller than the latter, whichresults in the stationary Boltzmann-Gibbs distributions P m =(1 −P 0)m . An equation similar to Eq. ( 6) describes a Markovchain studied for strategic market games in Ref. 11 . Naturally,the stationary probability distribution of wealth in that modelwas found to be exponential 11 .

Even though Eqs. (5) and (6) look like linear equations,nevertheless the Boltzmann equation ( 2) and (4) is a pro-foundly nonlinear equation. It contains the product of twoprobability distribution functions P in the right-hand side, be-cause two agents are involved in money exchange. Most stud-ies of wealth distribution 8 have the fundamentalaw that theyuse a single-particle approach. They assume that the wealthof an agent may change just by itself and write a linear mas-

ter equation for the probability distribution. Because onlyone particle is considered, this approach cannot adequatelyincorporate conservation of money. In reality, an agent canchange money only by interacting with another agent, thus theproblem requires a two-particle probability distribution func-tion. Using Boltzmann’s molecular chaos hypothesis, the two-particle function is factorized into a product of two single-particle distributions functions, which results in the nonlinearBoltzmann equation. Conservationof moneyis adequately in-corporated in this two-particle approach, and the universalityof the exponential Boltzmann-Gibbs distribution is transpar-ent.

I. Conclusions

Everywhere in this chapter we assumed some randomnessin the exchange of money. Our results would apply the best tothe probability distribution of money in a closed communityof gamblers. In more traditional economic studies, the agentsexchange money not randomly, but following deterministicstrategies, such as maximization of utility functions 5,12 . Theconcept of equilibrium in these studies is similar to mechan-ical equilibrium in physics, which is achieved by minimizingenergy or maximizing utility. However, for big ensembles,statistical equilibrium is a more relevant concept. When manyheterogeneous agentsdeterministically interact and spend var-ious amounts of money from very little to very big, the moneyexchange is effectively random. In the future, we would liketo uncover the Boltzmann-Gibbs distribution of money in asimulation of a big ensemble of economic agents following re-alistic deterministic strategies with money conservation takeninto account. That would be the economics analog of molecu-lar dynamics simulations in physics. While atoms collide fol-lowing fully deterministic equations of motion, their energyexchange is effectively random due to the complexity of thesystem and results in the Boltzmann-Gibbs law.



8

We do not claim that the real economy is in equilibrium.(Most of the physical world around us is not in true equilib-rium either.) Nevertheless, the concept of statistical equilib-rium is a very useful reference point for studying nonequilib-rium phenomena.

II. DISTRIBUTION OF INCOME AND WEALTH

A. Introduction

In Ch. I we predicted that the distribution of money shouldfollow an exponential Boltzmann-Gibbs law. Unfortunately,we were not able to nd data on the distribution of money. Onthe other hand, we found many sources with data on incomedistribution for the United States (USA) and United Kingdom(UK), as well as data on the wealth distribution in the UK,which are presented in this chapter. In all of these data, wend that the great majority of the population is described byan exponential distribution, whereas the high-end tail followsa power law.

The study of income distribution has a long history.Pareto 14 proposed in 1897 that income distribution obeys auniversal power law valid for all times and countries. Sub-sequent studies have often disputed this conjecture. In 1935,Shirras 15 concluded: “There is indeed no Pareto Law. It istime it should be entirely discarded in studies on distribu-tion”. Mandelbrot 16 proposed a “weak Pareto law” applica-ble only asymptotically to the high incomes. In such a form,Pareto’s proposal is useless for describing the great majorityof the population.

Many other distributions of income were proposed: Levy,log-normal, Champernowne, Gamma, and two other forms byPareto himself (see a systematic survey in the World Bank re-

search publication17

). Theoretical justications for these pro-posals form two schools: socio-economic and statistical. Theformer appeals to economic, political, and demographic fac-tors to explain the distribution of income (e. g. 18), whereas thelatter invokes stochastic processes. Gibrat 19 proposed in 1931that income is governed by a multiplicative random process,which results in a log-normal distribution (see also 20). How-ever, Kalecki 21 pointed out that the width of this distributionis not stationary, but increases in time. Levy and Solomon 22

proposed a cut-off at lower incomes, which stabilizes the dis-tribution to a power law.

Many researchers tried to deduce the Pareto law from a the-ory of stochastic processes. Gibrat 19 proposed in 1931 thatincome and wealth are governed by multiplicative randomprocesses, which result in a log-normal distribution. Theseideas were later followed, among many others, by Montrolland Shlesinger 20. However, already in 1945 Kalecki 21 pointedout that the log-normal distribution is not stationary, becauseits width increases in time. Modern econophysicists 22,33,34

also use various versions of multiplicative random processesin theoretical modeling of wealth and income distributions.

Unfortunately, numerous recent papers on this subject dovery little or no comparison at all with real statistical data,much of which is widely available these days on the Inter-

net. In order to ll this gap, we analyzed the data on in-come distribution in the United States (US) from the Bureauof Census and the Internal Revenue Service (IRS) in Ref. 35 .We found that the individual income of about 95% of popu-lation is described by the exponential law. The exponentiallaw, also known in physics as the Boltzmann-Gibbs distribu-tion, is characteristic for a conserved variable, such as energy.In Ref.36 , we argued that, because money (cash) is conserved,

the probability distribution of money should be exponential.Wealth can increase or decrease by itself, but money can onlybe transferred from one agent to another. So, wealth is notconserved, whereas money is. The difference is the same asthe difference between unrealized and realized capital gains instock market.

In Sec. IIB, we propose that the distribution of individualincome is given by an exponential function. This conjectureis inspired by the results of Ch. I,35 , where we argued that theprobability distribution of money in a closed system of agentsis given by the exponential Boltzmann-Gibbs function. Wecompare our proposal with the census and tax data for individ-ual income in USA. In Sec. II D we show that the exponentialdistribution has to be amended for the top 5% of incomes bya power-law function. In Sec. II F, we derive the distributionfunction of income for families with two earners and compareit with census data. In Sec. IIG, we discuss the distribution of wealth. In Sec. II H we critically examine several other alter-natives proposed in the literature for the distribution of incomeand wealth. Speculations on the possible origins of the expo-nential and power-law distribution of income and conclusionsfor this chapter are given in Sec. II I.

B. Distribution of income for individuals

C. Exponential distribution of income

We denote income by the letter r (for “revenue”). The prob-ability distribution function of income, P (r ), (called the prob-ability density in book 17) is dened so that the fraction of in-dividuals with income between r and r + dr is P (r ) dr . Thisfunction is normalized to unity (100%): ∞

0 P (r ) dr = 1 . Wepropose that the probability distribution of individual incomeis exponential:

P 1(r ) = exp( −r/R )/R, (7)

where the subscript 1 indicates individuals. Function (7)contains one parameter R, equal to the average income:

∞0 r P 1(r ) dr = R, and analogous to temperature in theBoltzmann-Gibbs distribution 35.

From the Survey of Income and Program Participation(SIPP) 23, we downloaded the variable TPTOINC (total in-come of a person for a month) for the rst “wave” (a four-month period) in 1996. Then we eliminated the entries withzero income, grouped the remaining entries into bins of thesize 10/3 k$, counted the numbers of entries inside each bin,and normalized to the total number of entries. The results areshown as the histogram in Fig. 6, where the horizontal scale



9

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

2%

4%

6%

8%

10%

12%

14%

16%

18%

Individual annual income, k$

P

r o b a

b i l i t y

C u m u

l a t i v e p r o

b a

b i l i t y

0 20 40 60 80 100 1200.01%

0.1%

1%

10%

100%

B

A

Individual annual income, k$

P r o

b a

b i l i t y

FIG. 6: Histogram: Probability distribution of individual incomefrom the U.S. Census data for 1996 23 . Solid line: Fit to the expo-nential law. Inset plot A: The same with the logarithmic verticalscale. Inset plot B: Cumulative probability distribution of individualincome from PSID for 1992 24 .

has been multiplied by 12 to convert monthly income to anannual gure. The solid line represents a t to the exponentialfunction ( 7). In the inset, plot A shows the same data withthe logarithmic vertical scale. The data fall onto a straightline, whose slope gives the parameter R in Eq. (7). The ex-ponential law is also often written with the bases 2 and 10:P 1 (r ) ∝ 2−r/R 2

∝ 10−r/R 10 . The parameters R , R2 andR10 are given in line (c) of Table I.

Source Year R ($) R2 ($) R10 ($) Set sizea PSID24 1992 18,844 13,062 43,390 1.39 × 103

b IRS26

1993 19,686 13,645 45,329 1.15 × 108

c SIPP p23 1996 20,286 14,061 46,710 2.57 × 105

d SIPP f 23 1996 23,242 16,110 53,517 1.64 × 105

e IRS25 1997 35,200 24,399 81,051 1.22 × 108

TABLE I: Parameters R , R2 , and R10 obtained by tting data fromdifferent sources to the exponential law (7) with the bases e, 2, and10, and the sizes of the statistical data sets.

Plot B in the inset of Fig. 6 shows the data from the PanelStudy of Income Dynamics (PSID) conducted by the Insti-tute for Social Research of the University of Michigan 24. Wedownloaded the variable V30821 “Total 1992 labor income”for individuals from the Final Release 1993 and processed thedata in a similar manner. Shown is the cumulative probabil-ity distribution of income N (r ) (called the probability dis-tribution in book 17). It is dened as N (r ) = ∞

r P (r ′) dr ′and gives the fraction of individuals with income greater thanr . For the exponential distribution ( 7), the cumulative dis-tribution is also exponential: N 1(r ) = ∞

r P 1(r ′) dr ′ =exp(−r/R ). Thus, R2 is the median income; 10% of popula-tion have income greater than R10 and only 1% greater than2R10 . The points in the inset fall onto a straight line in the

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100%

Adjusted gross income, k$

C u m u

l a t i v e p e r c e n

t o

f t a x r e

t u r n s

C u m u

l a t i v e p r o

b a

b i l i t y

0 20 40 60 80 1000.1%

1%

10%

100%

B

A

P r o

b a

b i l i t y


FIG. 7: Points: Cumulative fraction of tax returns vs income fromthe IRS data for 1997 25 . Solid line: Fit to the exponential law. Insetplot A: The same with the logarithmic vertical scale. Inset plot B:Probability distribution of individual income from the IRS data for199326 .

logarithmic scale. The slope is given in line (a) of Table I.The points in Fig. 7 show the cumulative distribution of

tax returns vs income in 1997 from column 1 of Table 1.1of Ref. 25 . (We merged 1 k$ bins into 5 k$ bins in the interval1–20 k$.) The solid line is a t to the exponential law. Plot Ain the inset of Fig. 7 shows the same data with the logarithmicvertical scale. The slope is given in line (e) of Table I. PlotB in the inset of Fig. 7 shows the distribution of individualincome from tax returns in 1993 26. The logarithmic slope isgiven in line (b) of Table I.

While Figs. 6 and 7 clearly demostrate the t of incomedistribution to the exponential form, they have the followingdrawback. Their horizontal axes extend to + ∞, so the high-income data are left outside of the plots. The standard way torepresent the full range of data is the so-called Lorenz curve(for an introduction to the Lorenz curve and Gini coefcient,see book 17). The horizontal axis of the Lorenz curve, x(r ),represents the cumulative fraction of population with incomebelow r , and the vertical axis y(r ) represents the fraction of income this population accounts for:

x(r ) = r

0P (r ′) dr ′, y(r ) = r

0 r ′P (r ′) dr ′

∞0 r ′P (r ′) dr ′

. (8)

As r changes from 0 to

∞, x and y change from 0 to 1, and

Eq. (8) parametrically denes a curve in the (x, y )-space.Substituting Eq. ( 7) into Eq. (8), we nd

x(r ) = 1 −exp(−r ), y(r ) = x(r ) −r exp(−r ), (9)

where r = r/R . Excluding r , we nd the explicit form of theLorenz curve for the exponential distribution:

y = x + (1 −x) ln(1 −x). (10)

R drops out, so Eq. ( 10) has no tting parameters.



10

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

10

20

30

40

50

60

70

80

90

100%

C u m u

l a t i v e p e r c e

n t o

f i n c o m e

Cumulative percent of tax returns

1980 1985 1990 19950

0.25

0.5

0.75

1

Year

Gini coefficient ≈ 12

FIG. 8: Solid curve: Lorenz plot for the exponential distribution.

Points: IRS data for 1979–199727

. Inset points: Gini coefcient datafrom IRS 27 . Inset line: The calculated value 1/2 of the Gini coef-cient for the exponential distribution.

The function (10) is shown as the solid curve in Fig. 8. Thestraight diagonal line represents the Lorenz curve in the casewhere all population has equal income. Inequality of incomedistribution is measured by the Gini coefcient G, the ratio of the area between the diagonal and the Lorenz curve to the areaof the triangle beneath the diagonal

G = 2

1

0(x −y) dx (11)

The Gini coefcient is conned between 0 (no inequality) and1 (extreme inequality). By substituting Eq. (10) into the inte-gral, we nd the Gini coefcient for the exponential distribu-tion: G1 = 1 / 2.

The points in Fig. 8 represent the tax data during 1979–1997from Ref. 27. With theprogress of time, the Lorenzpointsshifted downward and the Gini coefcient increased from 0.47to 0.56, which indicates increasing inequality during this pe-riod. However, overall the Gini coefcient is close to the value0.5 calculated for the exponential distribution, as shown in theinset of Fig. 8.

D. Power-law tail and “Bose” condensation

As Fig. 8 shows, the Lorenz curve deviates from the theo-retical Lorenz curve implied by the exponential distribution,mostly for the top 20% of tax returns. Moreover, as explainedin the previous section, the Lorenz curve for a pure exponen-tial distribution is independent of temperature (the scale of thedistribution). Therefore the variations in the Lorenz curvesover the period 1979-1997 suggest that the shape of the distri-bution changes.

1 10 100 10000.1%

1%

10%

100%

Adjusted Gross Income, k$

C u m u


t o

f r e

t u r n s

United States, IRS data for 1997

Pareto

Boltzmann−Gibbs

0 20 40 60 80 100

10%

100%

AGI, k$

FIG. 9: Cumulative probability distribution of US individual incomefor 1997 in log-log scale, with points (raw data) and solid lines (ex-ponential and power-law t). The inset shows the exponential regimeand the t with a Boltzmann-Gibbs distribution in the log-linearscale.

In Sec. II A we mentioned the early proposal of Pareto, whoclaimed that the income distribution obeys a universal powerlaw valid for all times and countries 14. The Pareto distributionP (r ) = A/r α has several undesirable properties. It divergesfor low incomes, and if α < 2 the distribution has a diverg-ing rst moment. These two properties are never found in realdata. Moreover, our study has shown that at least for the over-whelming majority (95%) of the population the distribution of income is exponential.

The data for the remaining top 5% percent of the populationis hard to get and often it is unreliable. Because the numberof people with income in the top few percents is small, censusdata is poor. The Internal Revenue Service (IRS) has reliabledata but it seldom reports it. We managed to nd a data setfrom IRS which contains high income data 25.

Fig. 9 shows United States IRS incomedata for 1997 25. Thedata points represent the cumulative probability distribution inlog-log scale. Although for large income values r > 120k$we have only three data points, the points align on a straightline in accordance with a Pareto power-law distribution.

From the plot it is evident that there is a discontinuityaround 120k$ where the exponential and power-law regimesintersect. We conclude that it is not possible to describethe entire income distribution with one single differentiablefunction. The two regimes of the probability distributionare clearly separated and this may be due to different in-come dynamics in the two regimes. Among others, theAdjusted Gross Income contains capital gains that is stock-market gains/losses. It is conceivable that for the top 5% of the population, capital gains rather than the labor income ac-count for the majority share of the total income. The capitalgains contribution to the Adjusted Growth Income may be re-sponsible for the observed power-law in the tail of the incomedistribution.

We plot the Lorenz curve for income in Fig. 10. As in Sec.



11

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

10

20

30

40

50

60

70

80

90

100%


C u m u


t o

f i n c o m e

US, IRS data for 1997

16%

FIG. 10: Lorenz plot with points (raw data) and solid line (function(12) with fraction b = 16% ).

II B the horizontal and vertical coordinates are the cumula-tive population x(r ) and the cumulative income y(r ) from (8).An imaginary line through the data shows an abrupt changein derivative for the last 2% of the tax returns. This suddenchange in derivative around the 98% mark on the horizontalaxis of Fig. 10 gives an almost innite slope for the Lorenzcurve. Physically, an innite slope in the Lorenz function forarguments x just slightly less than one, is equivalent to hav-ing a nite amount of total society’s income in the hand of avery few individuals. We have coined for this effect the name“Bose condensation of income” because of similarities withthe celebrated phase-transition from statistical mechanics.

To understand quantitatively the “Bose condensation” of

income, we calculate f the ratio of the total income of thesociety if all the population is described by the exponentiallaw, to the actual total income of the population. For theUSA tax income data for 1997 shown in Fig. 10 this frac-tion was f = 0 .84, which means that the “condensate” hasb = 1 −f = 16% of the total income of the population. Theanalytical formula for the Lorenz curve in this case is

y = (1 −b)[x + (1 −x) ln(1 −x)] + bδ (1 −x), (12)

where the rst term represents the contribution of theBoltzmann-Gibbs exponential regime and the second termrepresents the contribution of the Pareto power-law tail. It isremarkablethatas in the case of ( 10), Eq. 12 does not have anytting parameters. The condensate fraction b is completely de-termined by temperature, which in turn is determined from theprobability distribution.

The function ( 12) is plotted as a solid line in Fig. 9. One cansee that the data systematically deviate from the exponentiallaw because of the income concentrated in the power-law tail;however, the deviation is not very big. The inequality of theUS income distribution was also increasing during that timeperiod 35, which implies that the size of the “Bose condensate”has increased in time, too.

10 100 1000

0.1%

1%

10%

100%


C u m u


t o

f t a x r e

t u r n s

All states of the USA, IRS data for 1998

0 40 80 120

10%

100%


0 200 500 1000

0.1%

1%

10%

100%

CT

WV

AGI, k$

FIG. 11: Cumulative probability distributions of yearly individualincome for different states of the USA shown as raw data (top inset)and scaled data in log-log, log-linear (lower inset).

The Gini coefcient dened in (11) can be calculated for theLorenz curve given by (12). The effect of the power-law tailschanges the value of the Gini coefcient from G1 = 1 / 2 inthe case of a pure exponential distribution to Gb = (1 + b)/ 2.

E. Geographical variations in income distribution

In the previous sections we found that the distribution of income is an exponential for the large majority of the popu-lation followed by a power-law tail for the top few percent of the population. It would be interesting to establish the univer-sality of this shape for various other countries.

We obtained the data on distribution of the yearly individ-ual income in 1998 for each of the 50 states and the Districtof Columbia that constitute the USA from the Web site of theIRS42 . We plot the original raw data for the cumulative distri-bution of income in log-linear scale in the upper inset of Fig.11. The points spread signicantly, particularly at higher in-comes. For example, the fraction of individuals with incomegreater than 1 M$ varies by an order of magnitude betweendifferent states. However, after we rescale the data in the man-ner described in the preceding section, the points collapse ona single curve shown in log-log scale in the main panel andlog-linear scale in the lower inset. The open circles representthe US average, obtained by treating the combined data forall states as a single set. We observe that the distribution of higher incomes approximately follows a power law with theexponent α = 1 .7 ±0.1, where the ±0.1 variation includes70% of all states. On the other hand, for about 95% of indi-viduals with lower incomes, the distribution follows an expo-nential law with the average US temperature RUS = 36 .4 k$(R (2)

US = 25 .3 k$ and R (10)US = 83 .9 k$). The temperatures of

the individualstates differ from RUS by the amounts shown inTable II. For example, the temperature of Connecticut (CT) is1.25 times higher and the temperature of West Virginia (WV)



12

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

10

20

30

40

50

60

70

80

90

100%

WY

WA


C u m u


t o

f i n c o m e

All states of the USA, IRS data for 1998

FIG. 12: Lorenz plot with points (raw data) and solid line (function(10) calculated for the exponential law).

TABLE II: Deviations of the state temperatures from the average UStemperature.CT NJ MA MD VA CA NY IL CO NH AK

25% 24% 14% 14% 9% 9% 7% 6% 6% 5% 5%

DC DE MI WA MN GA TX RI AZ PA FL5% 4% 4% 2% 1% 0% -1% -3% -3% -3% -4%

KS OR HI NV NC WI IN UT MO VT TN-5% -6% -7% -7% -7% -8% -8% -9% -9% -9% -11%

NE OH LA AL SC IA WY NM KY ID OK-12% -12% -13% -13% -13% -14% -14% -14% -14% -15% -16%

ME MT AR SD ND MS WV-16% -19% -19% -20% -20% -21% -22%

is 0.78 times lower than the average US temperature.The Lorenz plot for all states is shown in Fig. 12 together

with the solid curve representing Eq. ( 10). The majority of points are well clustered and are not too far from the solidcurve. The exceptions are Wyoming (WY) with much higherinequality and the Washington state (WA) with noticeablylower inequality of income distribution. The average USdata, shown by open circles, is consistent with our previousresults 35 . Unlike in the UK case, we did not make any ad- justment in the US case for individuals with income below thethreshold, which appears to be sufciently low.

We obtained the data on the yearly income distribution inthe UK for 1997/8 and 1998/9 from the Web site of the IR 39 .The data for 1994/5, 1995/6, and 1996/7 were taken from theAnnual Abstract of Statistics derived from the IR 40 . The datafor these 5 years are presented graphically in Fig. 13. In theupper inset, the original raw data for the cumulative distribu-tion are plotted in log-linear scale. For not too high incomes,the points form straight lines, which implies the exponentialdistribution N (r ) ∝ exp(−r/R ), where r stands for income(revenue), and R is the income “temperature”. However, theslopes of these lines are different for different years. The tem-peratures for the years 1994/5, 1995/6, 1996/7, and 1997/8differ from the temperature for 1998/9, R (98 / 9)

UK = 11 .7 k £

10 1000.1%

1%

10%

100%

Individual income, kpounds

C u m u


t o

f t a x r e

t u r n s

United Kingdom, IR data for 1994−1998

0 10 20 30 40

10%

100%

Income, kpounds

0 50 100 2000.1%

1%

10%

100%

Income, kpounds

FIG. 13: Cumulative probability distributions of yearly individualincome in the UK shown as raw data (top inset) and scaled data inlog-log (left panel), log-linear (lower inset), and Lorenz (right panel)coordinates. Solid curve: t to function ( 10) calculated for the expo-nential law.

(R (2)UK = 8 .1 k £ and R (10)

UK = 26 .9 k £ ), by the factors 0.903,0.935, 0.954, and 0.943. To compensate for this effect, werescale the data. We divide the horizontal coordinates (in-comes) of the data sets for different years by the quoted abovefactors and plot the results in log-log scale in the main paneland log-linearscale in the lower inset. We observe scaling: thecollapse of points onto a single curve. Thus, the distributionsN i (r ) for different years i are described by a single functionf (r/R i ). The main panel shows that this scaling function f follows a power law with the exponent α = 2 .0-2.3 at high in-comes. The lower inset shows that f has an exponential formfor about 95% of individuals with lower incomes. These re-sults qualitatively agree with a similar study by Cranshaw 41.He proposed that the P (r ) data for lower incomes are bettertted by the Gamma distribution Γ(r )∝r β exp(−r/R ). Forsimplicity, we chose not to introduce the additional tting pa-rameter β .

We must mention that the individuals with income below acertain threshold are not required to report to the IR. That iswhy the data in the lower inset do not extend to zero income.We extrapolate the straight line to zero income and take theintercept with the vertical axis as 100% of individuals. Thus,we imply that the IR data does not account for about 25-27%of individuals with income below the threshold.

The Lorenz curve for the distribution of the UK incomeis shown in Fig. 14. We treat the number of individuals be-low the income threshold and their total income as adjustableparameters, which are the horizontal and vertical offsets of the coordinates origin relative to the lowest known data point.These parameters are chosen to t the Lorenz curve for theexponential law ( 10) shown as the solid line. The t is verygood, and is illustrated in Fig. 13. The horizontal offsets are28-34%, which is roughly consistent with the numbers quotedfor the lower inset of the left panel.



13

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

10

20

30

40

50

60

70

80

90

100%


C u m u


t o

f i n c o m e

United Kingdom, IR data for 1994−1998

FIG. 14: Cumulative probability distributions of yearly individualincome in the UK shown as raw data (top inset) and scaled data inlog-log (left panel), log-linear (lower inset), and Lorenz (right panel)coordinates. Solid curve: t to function ( 10) calculated for the expo-

nential law.

The income temperature for the UK in 1998/9 was RUK =11.7 k £ and for the US in 1998 was R US = 36 .4 k$. Us-ing the exchange rate as of December 31, 1998 to convertpounds into dollars 45, we nd that the UK temperature wasRUK = 19 .5 k$, which is 1.87 times lower than the US tem-perature. The difference in temperatures indicates nonequilib-rium, which can be exploited to create a thermal machine 36.The gain (prot) produced by such a thermal machine is pro-portional to the difference in temperatures. In agreement withthe second law of thermodynamics, money would ow from ahigh-temperature system to a low-temperature one. This mayexplain the huge trade decit of the USA in global interna-tional trade with other, lower-temperature countries. The vari-ation of temperatures between different states of the USA isshown in Table II.

F. Distribution of income forfamilies

Now let us discuss the distribution of income for familieswith two earners. The family income r is the sum of two indi-vidual incomes: r = r1 + r 2 . Thus, the probability distributionof the family income is given by the convolution of the indi-

vidual probability distributions28

. If the latter are given by theexponential function (7), the two-earners probability distribu-tion function P 2(r ) is

P 2 (r ) = r

0P 1(r ′)P 1(r −r ′) dr ′ = r

R2 e−r/R . (13)

The function P 2(r ) (13) differs from the function P 1(r ) (7) bythe prefactor r/R , which reects the phase space available tocompose a given total income out of two individual ones. It isshown as the solid curve in Fig. 15. Unlike P 1(r ), which has a

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

1%

2%

3%

4%

5%

6%

7%

8%

Family annual income, k$

P r o

b a

b i l i t y

0 20 40 60 80 100 1200%

2%

4%

6%

8%

Family annual income, k$

P r o

b a

b i l i t y

FIG. 15: Histogram: Probability distribution of income for familieswith two adults in 1996 23 . Solid line: Fit to Eq. (13). Inset histogram:Probability distribution of income for all families in 1996 23 . Insetsolid line: 0.45P 1 (r ) + 0 .55P 2 (r ).

maximum at zero income, P 2(r ) has a maximum at r = R andlooks qualitatively similar to the family income distributioncurves in literature 18.

From the same 1996 SIPP that we used in Sec. II B23 , wedownloaded the variable TFTOTINC (the total family incomefor a month), which we then multiplied by 12 to get annualincome. Using the number of family members (the variableEFNP) and the number of children under 18 (the variableRFNKIDS), we selected the families with two adults. Theirdistribution of family income is shown by the histogram inFig. 15. The t to the function ( 13), shown by the solid line,gives the parameter R listed in line (d) of Table I. The familieswith two adults and more than two adults constitute 44% and11% of all families in the studied set of data. The remaining45% are the families with one adult. Assuming that these twoclasses of families have two and one earners, we expect theincome distribution for all families to be given by the super-position of Eqs. ( 7) and (13): 0.45P 1(r ) + 0 .55P 2(r ). It isshown by the solid line in the inset of Fig. 15 (with R fromline (d) of Table I) with the all families data histogram.

By substituting Eq. ( 13) into Eq. (8), we calculate theLorenz curve for two-earners families:

x(r ) = 1 −(1 + r )e−r , (14)y(r ) = x(r )

−r 2 e−r / 2. (15)

It is shown by the solid curve in Fig. 17. Given that x −y = r 2 exp(−r )/ 2 and dx = r exp(−r ) dr , the Gini coef-cient for two-earners families is: G2 = 2 1

0 (x −y) dx =

∞0 r 3 exp(−2r ) dr = 3 / 8 = 0 .375. The points in Fig. 17

show the Lorenz data and Gini coefcient for family incomeduring 1947–1994 from Table 1 of Ref. 29. The Gini coef-cient is very close to the calculated value 0.375.

The fundamentalassumption which underlies the derivationof (13) is the independence of the two incomes r 1 and r 2 of



14

0 20 40 60 80 1000

20

40

60

80

100

Labor income of one earner, k$

L a

b o r

i n c o m e o f a n o

t h e r e a r n e r ,

k $

PSID data for families, 1999

FIG. 16: Points: The income for families with two earners in 1999 24 .One family is represented by two points (r 1 , r 2 ) and (r 2 , r 1 ). Theexistence of correlations between r 1 and r2 is equivalent to a cluster-

ing of points along the diagonal. The data shows little evidence forsuch correlation.

the two-earner family. While the good t of the function (13)is an implicit validation of this assumption, it would be goodto show that the assumption holds true directly fromdata. TheCensus database does not provide the individual values of r1and r 2 , but only their sum r = r1 + r2 . We found that thePSID survey 24 does have give for a family with two earners,each earners contribution to the total income of the family. Wedownloaded the variable ER16463 (total labor income, head)and ER16465 (total labor income, wife). We represent in Fig.16 each family by two points (r 1 , r 2) and (r 2 , r 1). The ex-istence of correlations between r 1 and r2 is equivalent to aclustering of points along the diagonal. Or stated otherwise,the independence of r1 and r2 should result in a uniform dis-tribution of points on innitesimal strips perpendicular to thediagonal. The plotted data shows little, if any evidence forsuch correlation.

The distributions of the individual and family income dif-fer qualitatively. The former monotonically increases towardthe low end and has a maximum at zero income (Fig. 6).The latter, typically being a sum of two individual incomes,has a maximum at a nite income and vanishes at zero (Fig.15). Thus, the inequality of the family income distributionis smaller. The Lorenz data for families follow the differentEq. (15), again without tting parameters, and the Gini coef-cient is close to the smaller calculated value 0.375 (Fig. 17).Despite different denitions of income by different agencies,the parameters extracted from the ts (Table I) are consistent,except for line (e).

The qualitative difference between the individual and fam-ily income distributions was emphasized in Ref. 26 , which splitup joint tax returns of families into individual incomes andcombined separately led tax returns of married couples intofamily incomes. However, Refs. 25 and27 counted only “indi-

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

10

20

30

40

50

60

70

80

90

100%

C u m u


t o

f f a m

i l y i n c o m e

Cumulative percent of families

1950 1960 1970 1980 19900

0.20.375

0.6

0.8

1

Year

Gini coefficient ≈ 38

FIG. 17: Solid curve: Lorenz plot ( 15) for distribution ( 13). Points:Census data for families, 1947–1994 29 . Inset points: Gini coefcientdata for families from Census 29 . Inset line: The calculated value 3/8of the Gini coefcient for distribution ( 13).

vidual tax returns”, which also include joint tax returns. Sinceonly a fraction of families le jointly, we assume that the lattercontribution is small enough not to distort the tax returns dis-tribution from the individual income distribution signicantly.Similarly, the denition of a family for the data shown in theinset of Fig. 15 includes single adults and one-adult familieswith children, which constitute 35% and 10% of all families.The former category is excluded from the denition of a fam-ily for the data 29 shown in Fig. 17, but the latter is included.Because the latter contribution is relatively small, we expectthe family data in Fig. 17 to approximately represent the two-earners distribution ( 13). Technically, even for the familieswith two (or more) adults shown in Fig. 15, we do not knowthe exact number of earners.

With all these complications, one should not expect perfectaccuracy for our ts. There are deviations aroundzero incomein Figs. 6, 7, and 15. The ts could be improved there by mul-tiplying the exponential function by a polynomial. However,the data may not be accurate at the low end because of under-reporting. For example, ling a tax return is not required forincomes below a certain threshold, which ranged in 1999from$2,750 to $14,400 30. As the Lorenz curves in Figs. 8 and 17show, there are also deviations at the high end, possibly wherePareto’s power law is supposed to work. Nevertheless, theexponential law gives an overall good description of incomedistribution for the great majority of the population.

Figs. 6 and 7 demonstrate that the exponential law ( 7) tsthe individual income distribution very well. The Lorenz datafor the individual income follow Eq. ( 10) without tting pa-rameters, and the Gini coefcient is close to the calculatedvalue 0.5 (Fig. 8).

It is interesting to study the distribution of the Gini indexacross the globe. We present in Fig. 18 data from a WorldBank publication 48. This is an question of utmost importancefor the World Bank since the distribution of the Gini index



15

0

0.1

0.2

0.3

0.4

0.5

0.6

W.Europe Asia S.America Africa E.EuropeN.America FSU

0.375

World distribution of Gini index, 1988 and 1993

1 9 8 8

1 9 9 3

G i n i i n d e x

f o r

h o u s e

h o

l d s

FIG. 18: Distribution of Gini index for households across the globefor two different years, 1988 and 1993 48 . For West Europe and NorthAmerica, the Gini index is close to 0.375, the value predicted in Sec.IIF. A sharp increase in inequality is observed in the Eastern Europeand the Former Soviet Union (FSU) republics after the fall of the“iron curtain” when socialist economies were replaced by market-like economies.

reects the degree of inequality (poverty) in various regionsof the globe. For West Europe and North America, the Giniindex is close to 0.375, the value we predicted in Sec. II F. Asharp increase in inequality is observed in the Eastern Europeand the Former Soviet Union (FSU) republics after the fall of the “iron curtain” when socialist economies where replacedby market-like economies. We conjecture that the high valueof the Gini index in Asia may be due to the fact that a typicalfamily in Asia has only one earner, so the Gini index is close

to a value of 0.5 as observed for individuals.As theevidence from Fig. 17 and Fig. 18 shows the value of the Gini index varies in time and across the globe, but for theWestern world it is close to the value of 0.375, as predicted inSec. IIF.

G. Distribution of wealth

In this section, we discuss the cumulative probability dis-tribution of wealth N (w)=(the number of people whose in-dividual wealth is greater than w)/(the total number of peo-ple). A plot of N vs. w is equivalent to a plot of a person’srank in wealth vs. wealth, which is often used for top rich-est people 37. We will use the power law, N (w) ∝ 1/w α ,and the exponential law N (w) ∝ exp(−w/W ), to t thedata. These distributions are characterized by the exponentα and the “temperature” W . The corresponding probabilitydensities, P (w) = −dN (w)/dw , also follow a power law oran exponential law. For the exponential law, it is also use-ful to dene the temperatures W (2) (also known as the me-dian) and W (10) using the bases of 1/2 and 1/10: N (w) ∝(1/ 2)w/W (2)

∝(1/ 10)w/W (10).

10 100 10000.01%

0.1%

1%

10%

100%

Total net capital (wealth), kpounds

C u m u


t o

f p e o p

l e

United Kingdom, IR data for 1996

Pareto

Boltzmann−Gibbs

0 20 40 60 80 10010%

100%

Total net capital, kpounds

FIG. 19: Left panel: Cumulative probability distribution of to-tal net capital (wealth) shown in log-log, log-linear (inset) coordi-nates. Points: the actual data. Solid lines: ts to the exponential(Boltzmann-Gibbs) and power (Pareto) laws.

The distribution of wealth is not easy to measure, becausepeople do not report their total wealth routinely. However,when a person dies, all assets must be reported for the pur-pose of inheritance tax. Using these data and an adjustmentprocedure, the British tax agency, the Inland Revenue (IR), re-constructed wealth distribution of the whole UK population.In Figs. 19 and 20, we present the 1996 data obtained fromtheir Web site 38 . Fig. 19 shows the cumulative probability asa function of the personal total net capital (wealth), which iscomposed of assets (cash, stocks, property, household goods,etc.) and liabilities (mortgages and other debts). The mainpanel illustrates in the log-log scale that above 100 k £ thedata follow a power law with the exponent α = 1 .9. Theinset shows in the log-linear scale that below 100 k £ thedata is very well tted by an exponential distribution withthe temperature W UK = 59 .6 k £ (W (2)

UK = 41 .3 k £ andW (10)

UK = 137 .2 k £ ).Since we have estabilished that the distribution of wealth

has an exponential regime followed by a power-law tail, weplot the Lorenz curve for wealth in the right panel of Fig. 20.As in Sec. II B the horizontal and vertical coordinates are thecumulative population x(w) and the cumulative wealth y(w):

x(w) =

w

0P (w′) dw′, (16)

y(w) = w

0w′P (w′) dw′ / ∞

0w′P (w′) dw′. (17)

As in the case of US income data Fig. 10, there is an abruptchange in the Lorenz curve for the top few percent of peo-ple. We interpret this fact in the same way as in Sec. IID,as a “Bose condensation of wealth”. Again, we calculate f the fraction of the total wealth of the society if all people aredescribed by the exponential law, to the actual total wealthof the society. For the United Kingdom in 1996, this frac-



16

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 people

C u m u


t o

f w e a

l t h


16%

FIG. 20: Points: Lorenz plot for wealth, United Kingdom 1996 38 .Solid curve: The Lorenz curve given by (18) with a condensate frac-tion of b = 16% .

tion was f = 0 .84, which means that the “condensate” hasb = 1 −f = 16% of the total wealth of the system. Thefunctional form for the Lorenz curve for wealth is

y = (1 −b)[x + (1 −x)ln(1 −x)] + bδ (1 −x). (18)

This function is plotted as a solid line in Fig. 20. One cansee that the data systematically deviate from the exponentiallaw because of the wealth concentrated in the power-law tail;however, the deviation is not very big. The so-called Ginicoefcient 17,35 , which measures the inequality of wealth dis-tribution, has increased from 64% in 1984 to 68% in 1996 38.This value is bigger than the Gini coefcient 50% for a purelyexponential distribution 35.

While there seems to be no controversy about the fact thata few people hold a nite fraction of the entire wealth, thespecic value of this fraction varies widely in the literature.Fractions as high as 80-90% were reported in the literature 43

without any support from data. We provide a clear procedureof how to calculate this factor, and we nd that b = 16% .

H. Other distributions for income and wealth

As we discussed in the introduction, Sec. II A there havebeen many proposed functional forms for the distribution of income and wealth. In this section we will investigate severalof them.

In Secs. IIB, IID, II E and II G we have shown ample evi-dence to support our ndings that the distribution on incomeand wealth have a similar structure with an exponential partfollowed by a power-law tail. These results immediately in-validate the proposed Pareto or Gamma distributions as thecorrect distribution of income.

Another popular distribution which has been proposed formany years as the distribution of income is the lognormal

1 10 100 10000.1%

1%

10%

100%

Adjusted Gross Income, r (k$)

P r o

b a

b i l i t y d e n s

i t y f u n c

t i o n ,

r P ( r )

United States, IRS data for 1997

FIG. 21: Points: Calculated probability density P (r ) for UnitedStates IRS income data for 1997 multiplied by income r , i.e. rP (r ).Curve: Fit of data with function P (r ) = rP LN (r ) (19) with aquadratic function. Clearly, the lognormal distribution fails to de-scribe the data points accurately.

distribution 20. The lognormal distribution has the expression

P LN (r ) = 1

r√ 2πσ 2exp −

(ln r −µ)2

2σ2 , (19)

where µ and σ are two parameters. As given by ( 19) thefunctional form of the lognormal distribution makes it hardto compare it with data. But the function P (r ) = rP LN (r )is a quadratic function in a log-log plot. In Fig. 21 we plot-ted ln P (r ) versus ln r for the 1997 income data from IRS 25 .As it can be seen from Fig. 21 a t of P (r ) with a quadratic

function cannot be made. The data points show the power-law regime for high incomes. We conclude that the lognormaldistribution does not describe the distribution of income.

The distribution of wealth has essentially the same struc-ture as the distribution of income. An exponential regimefor low wealth values and a power-law tail for high wealthvalues. Contrary to the distribution of income, the distribu-tion of wealth appears to have a continuous derivative in thecross-over between the two regimes (see the discussion at thebeginning of Sec. IID). The shape of the wealth distributionsuggests a modication of the regular exponential Boltzmann-Gibbs distribution for values of wealth w ≫ W . In the pastyears, in the context of non-extensive statistical mechanics,Tsallis has proposed a viable alternative to the Boltzmann-Gibbs distribution 46.

The Tsallis distribution can be obtained by maximizinga generalized entropy and is the subject of a lot of currentresearch. The appeal of the Tsallis distribution is that ithas power-law tails for large arguments, and that in certainlimit it becomes the exponential distribution. The Tsallisdistribution 46 has the expression

P q(w) = 1W T

1 + ( q −1) wW T

q1 − q

, (20)



17

10 100 1000

0.1%

1%

10%

100%

Total net capital (wealth), kpounds

C u m u


t o

f p e o p

l e


(a)(b)(c)

0 20 40 60 80 100

20%

40%

60%80%

100%

Total net capital, kpounds

FIG. 22: Points: Internal Revenue wealth data for individuals,199638 . Curve (a): Fit with the Kaniadakis distribution ( 21). Curve(b): Fit with the Tsallis distribution ( 20). Curve (c): Fit with theBouchaud-Mezard/Solomon-Richmond distribution ( 22). The inset

shows the same three curves in a log-linear scale. The deviations of model (22) from data are evident for small values of income.

where W T is the dimensional parameterof the Tsallis distribu-tion, analogous to the temperature of the exponential distribu-tion. In the limit q →1, the distribution goes to the exponen-tial form P q=1 (w)∝e−w/W T . For q > 1, the distribution hasa power-law decay P q(w)∼(1/w )q/q −1 for values w≫W .

In Fig. 22 we present the t of UK wealth data with theTsallis distribution. Overall, the quality of the t is good. Theparameters of the Tsallis distribution implied by the t are:q = 1 .42, and for the Tsallis temperature W T = 41 k£ .

Another distribution proposed by Kaniadakis 47 in the con-text of non-extensive statistical mechanics has the form of adeformed exponential

P κ (x) = 1 + κ2x2 −κx1/κ

, (21)

where x = w/W κ . For κ → 0, the distribution has the ex-ponential form P κ =0 (w) = e−w/W κ . For large argumentsw ≫ W , the distribution ( 21) has the power-law behaviorP κ (w) ≈ (1/w )1/κ . The t with the Kaniadakis distributionis even better than that with the Tsallis distribution, as shownin Fig. 22. The parameters of the t for the Kaniadakis dis-tribution are: κ = 0 .33 and for the Kaniadakis temperature

W κ = 45 k £

.A distribution for wealth put forward by Bouchaud andMezard 33, and independently by Solomon and Richmond 32,is

P α (x) = (α −1)α

Γ(α)exp(−α −1

x )xα +1 , (22)

where x = w/W c . This distribution has an exponentiallysharp cut-off for wealths w ≪ W c , and a power-law tailP α (w) ∝ 1/w α +1 for w ≫ W c . The t of the UK wealth

data with the distribution ( 22) is presented in Fig. 22. Thevalues we obtain for the parameters are: α = 1 .93 andW c = 74 k £ . The distribution ( 22) has its maximum atxmax = ( α −1)/ (α + 1) = 0 .32, which essentially statesthat for w > 0.32W c = 23 k £ the distribution is a power law.As seen in Fig. 22 this claim is not supported by data. More-over, the percent of people in the power-law tail predicted bythe distribution ( 22),

∞

x maxdxP α (x) = 81% , is clearly much

larger than what is observed from the Lorenz curve for wealth,Fig. 20.All three proposed distributions ( 20,21,22) seem to capture

well the power-law tail. In the inset of Fig. 22 a log-linearscale is used, and this makes the discrepancy between (22)and data much more evident. The distribution ( 22) is clearlyinappropriate for low incomes.

I. Conclusions

Our analysis of the data shows that there are two clearregimes in the distribution of individual income. For low and

moderate incomes up to approximately 95% of the total pop-ulation, the distribution is well described by an exponential,while the income of the top 5% individuals is described by apower-law (Pareto) regime.

The exponential Boltzmann-Gibbs distribution naturallyapplies to quantities that obey a conservation law, such as en-ergy or money 35. However, there is no fundamental reasonwhy the sum of incomes (unlike the sum of money) must beconserved. Indeed, income is a term in the time derivativeof one’s money balance (the other term is spending). Maybeincomes obey an approximate conservation law, or somehowthe distribution of income is simply proportional to the distri-bution of money, which is exponential 35.

Another explanation involves hierarchy. Groups of peo-ple have leaders, which have leaders of a higher order, andso on. The number of people decreases geometrically (ex-ponentially) with the hierarchical level. If individual incomeincreases linearly with the hierarchical level, then the incomedistribution is exponential. However, if income increases mul-tiplicatively, then the distribution follows a power law 31 . Formoderate incomes below $100,000, the linear increase maybe more realistic. A similar scenario is the Bernoulli trials 28 ,where individuals have a constant probability of increasingtheir income by a xed amount.

We found scaling in the cumulative probabilitydistributionsN (r ) of individual income r derived from the tax statistics fordifferent years in the UK and for different states in the US. Thedistributions N i (r ) have the scaling form f (r/R i ), where thescale R i (the temperature) varies from one data set i to an-other, but the scaling function f does not. The function f hasan exponential (Boltzmann-Gibbs) form at the low end, whichcovers about 95% of individuals. At the high end, it followsa power (Pareto) law with the exponents about 2.1 for the UKand 1.7 for the US. Wealth distribution in the UK also has aqualitatively similar shape with the exponent about 1.9 andthe temperature W UK = 60 k £ . Some of the other values of the exponents found in literature are 1.5 proposed by Pareto



18

himself ( α = 1 .5), 1.36 found by Levy and Solomon 37 for thedistribution of wealth in the Forbes 400 list, and 2.05found bySouma 44 for the high end of income distribution in Japan. Thelatter study is similar to our work in the sense that it also usestax statistics and explores the whole range of incomes, not justthe high end. Souma 44 nds that the probability density P (r )at lower incomes follows a log-normal law with a maximumat a nonzero income. This is in contrast to our results, which

suggest that the maximum of P (r ) is at zero income. The dis-repancy may be due to the high threshold for tax reporting inJapan, which distorts the data at the low end. On the otherhand, if the data is indeed valid, it may reect the actual dif-ference between the social stuctures of the US/UK and Japan.

III. DISTRIBUTION OF STOCK-PRICE FLUCTUATIONS

A. Introduction

Stochastic dynamics of stock prices is commonly describedby a geometric (multiplicative) Brownian motion, which gives

a log-normal probability distribution for stock price changes(returns) 50. However, numerous observations show that thetails of the distribution decay slower than the log-normaldistribution predicts (the so-called “fat-tails” effect) 51,52,53 .Particularly, much attention was devoted to the power-lawtails54,55 . The geometric Brownian motion model has two pa-rameters: the drift µ, which characterizes the average growthrate, and the volatility σ, which characterizes the noisiness of the process. There is empirical evidence and a set of stylizedfacts indicating that volatility, instead of being a constant pa-rameter, is driven by a mean-reverting stochastic process 56,57 .Various mathematical models with stochastic volatility havebeen discussed in literature 58,59,60,61 .

In this chapter, we study a particular stochastic volatil-ity model, where the square root of the stock-price volatil-ity, called the variance, follows a random process known innancial literature as the Cox-Ingersoll-Ross process and inmathematical statistics as the Feller process 57,60 . We solve theFokker-Planck equation for this model exactly and nd the joint probability distribution of returns and variance as a func-tion of time, conditional on the initial value of variance. Thesolution is obtained in two differentways: using the method of characteristics 62 and the method of path integrals 63,64,65 . Thelatter is more familiar to physicists working in nance 61,66 .

While returns are readily known from a nancial time se-ries data, variance is not given directly, so it acts as a hiddenstochastic variable. Thus, we integrate the joint probabilitydistribution over variance and obtain the marginal probabilitydistribution of returns unconditional on variance. The latterdistribution can be directly compared with nancial data. Wend excellent agreement between our results and the Dow-Jones data for the period of 1982–2001. Using only four t-ting parameters, our equations very well reproduce the proba-bility distribution of returns for time lags between 1 and 250trading days. This is in contrast to popular ARCH, GARCH,EGARCH, TARCH, and similar models, where the number of parameters can easily go to a few dozen 67.

Our result for the probability distribution of returns has theform of a one-dimensional Fourier integral, which is easilycalculated numerically or, in certain asymptotical limits, an-alytically. For large returns, we nd that the probability dis-tribution is exponential in log-returns, which implies a power-law distribution for returns, and we calculate the time depen-dence of the corresponding exponents. In the limit of longtimes, the probability distribution exhibits scaling. It becomes

a function of a single combination of return and time, withthe scaling function expressed in terms of a Bessel function.The Dow-Jones data follow the predicted scaling function forseven orders of magnitude.

The original theory of option pricing was developedby Black and Scholes for the geometric Brownian motionmodel 50. Numerous attempts to improve it using stochasticvolatility models have been made 57,58,59,60,61 . Particularly, op-tion pricing for the same model as in our paper 49 was inves-tigated by Heston 60. Empirical studies 68 show that Heston’stheory fares better than the Black-Scholes model, but stilldoes not fully capture the real-market option prices. Sinceour paper 49 gives a closed-form time-dependent expressionfor the probability distribution of returns that agrees with -nancial data, it can serve as a starting point for a better theoryof option pricing.

B. The model

We consider a stock, whose price S t , as a function of time t ,obeys the stochastic differential equation of a geometric (mul-tiplicative) Brownian motion 50:

dS t = µS t dt + σt S t dW (1)t . (23)

Here the subscript t indicates time dependence, µ is the driftparameter, W (1)t is the standard randomWiener process 81 , andσt is a time-dependent parameter, called the stock volatility,which characterizes the noisiness of the Wiener process.

Since any solution of (23) depends only on σ2t , it is conve-

nient to introduce the new variable vt = σ2t , which is called

the variance. We assume that vt obeys the following mean-reverting stochastic differential equation:

dvt = −γ (vt −θ) dt + κ√ vt dW (2)t . (24)

Here θ is the long-time mean of v, γ is the rate of relaxationto this mean, W (2)

t is the standard Wiener process, and κ is aparameter that we call the variance noise. Eq. ( 24) is knownin nancial literature as the Cox-Ingersoll-Ross (CIR) processand in mathematical statistics as the Feller process 57 (p. 42).Alternative equations for vt , with the last term in ( 24) replacedby κ dW (2)

t orby κvt dW (2)t , are also discussed in literature 58.

However, in this chapter, we study only the case given by Eq.(24).

We take the Wiener process appearing in (24) to be corre-lated with the Wiener process in ( 23):

dW (2)t = ρ dW (1)

t + 1 −ρ2 dZ t , (25)



19

where Z t is a Wiener process independent of W (1)t , and ρ ∈[−1, 1] is the correlation coefcient. A negative correlation

(ρ < 0) between W (1)t and W (2)

t is known as the leverageeffect 57 (p. 41).

It is convenient to change the variable in ( 23) from priceS t to log-return r t = ln( S t /S 0). Using Ito’s formula 69, weobtain the equation satised by r t :

dr t = µ − vt

2 dt + √ vt dW (1)t . (26)

The parameter µ can be eliminated from ( 26) by changing thevariable to xt = r t −µt , which measures log-returns relativeto the growth rate µ:

dx t = −vt

2 dt + √ vt dW (1)

t . (27)

Where it does not cause confusion with r t , we use the term“log-return” also for the variable xt .

Equations (27) and (24) dene a two-dimensional stochas-tic process for the variables xt and vt . This process is charac-terized by the transition probability P t (x, v |vi ) to have log-return x and variance v at time t given the initial log-returnx = 0 and variance vi at t = 0 . Time evolution of P t (x, v |vi )is governed by the Fokker-Planck (or forward Kolmogorov)equation 69

∂ ∂t

P = γ ∂ ∂v

[(v −θ)P ] + 12

∂ ∂x

(vP ) (28)

+ ρκ ∂ 2

∂x∂v(vP ) +

12

∂ 2

∂x 2 (vP ) + κ2

2∂ 2

∂v 2 (vP ).

The initial condition for ( 28) is a product of two delta func-tions

P t =0 (x, v |vi ) = δ (x) δ (v −vi ). (29)

The variance v is a positive quantity, so Eq. ( 28) is denedonly for v > 0. However, when solving (28), it is convenientto extend the domain of v so that v ∈(−∞, ∞). If 2γθ > κ 2 ,such an extension does not change the solution of the equa-tion, because, given that P = 0 for v < 0 at the initial timet = 0 , the condition P t (x,v < 0) = 0 is preserved for alllater times t > 0. In order to demonstrate this, let us consider(28) in the limit v →0:

∂ ∂t

P = − γθ − κ2

2∂P ∂v

+ γP + ρκ∂P ∂x

. (30)

Eq. (30) is a rst-order partial differential equation (PDE),which describes propagation of P from negative v to positive

v with the positive velocity γθ − κ2/ 2. Thus, the nonzerofunction P (x ,v > 0) does not propagate to v < 0, and

P t (x ,v < 0) remains zero at all times t. Alternatively, itis possible to show 50 (p. 67) that, if 2γθ > κ 2 , the randomprocess (24) starting in the domain v > 0 can never reach thedomain v < 0.

The probability distribution of the variance itself,Πt (v) = dx P t (x, v ), satises the equation

∂ ∂t

Πt (v) = ∂ ∂v

[γ (v −θ)Π t (v)] + κ2

2∂ 2

∂ 2v [vΠt (v)] , (31)

which is obtained from ( 28) by integration over x . Eq. (31)has the stationary solution

Π∗(v) =

αβ +1

Γ(β + 1)vβ e−αv , α =

2γ κ2 , β = αθ −1,

(32)which is the Gamma distribution. The maximum of Π

∗(v) is

reached at vmax = β/α = θ −κ2/ 2γ . The width w of Π∗(v)

can be estimated using the curvature at the maximum w ≈(κ2/ 2γ ) 2γθ/κ 2 −1. The shape of Π∗(v) is characterized

by the dimensionless ratio

χ = vmax

w = 2γθ

κ2 −1. (33)

When 2γθ/κ 2 → ∞, χ → ∞and Π∗(v) →δ (v −θ).

C. Solution of the Fokker-Planck equation

Since x appears in (28) only in the derivative operator∂/∂x , it is convenient to make the Fourier transform

P t (x, v |vi ) = 12π + ∞

−∞dpx eip x x P t,p x (v |vi ). (34)

Inserting ( 34) into (28), we nd

∂ ∂t

P = γ ∂ ∂v

(v −θ)P (35)

− p2

x −ipx

2 v −iρκp x

∂ ∂v

v − κ2

2∂ 2

∂v2 v P .

Eq. (35) is simpler than ( 28), because the number of variableshas been reduced to two, v and t, whereas px only plays therole of a parameter.

Since Eq. ( 35) is linear in v and quadratic in ∂/∂v , it canbe simplied by taking the Fourier transform over v

P t,p x (v |vi ) = 12π + ∞

−∞dpv eip v v P t,p x ( pv |vi ). (36)

The PDE satised by P t,p x ( pv |vi ) is of the rst order

∂ ∂t

+ Γ pv + iκ 2

2 p2

v + ip2

x + px

2 ∂ ∂pv P = −iγθp vP ,

(37)where we introduced the notation

Γ = γ + iρκp x . (38)

Eq. (37) has to be solved with the initial condition

P t =0 ,p x ( pv |vi ) = exp( −ipv vi ). (39)

The solution of the PDE ( 37) is given by the method of characteristics 62:

P t,p x ( pv |vi ) = exp −i ˜ pv (0)vi −iγθ t

0dτ ˜ pv (τ ) ,

(40)



20

where the function ˜ pv (τ ) is the solution of the characteristic(ordinary) differential equation

d˜ pv (τ )dτ

= Γ pv (τ ) + iκ 2

2 ˜ p2

v (τ ) + i2

( p2x −ipx ) (41)

with the boundary condition ˜ pv (t) = pv specied at τ = t.The solution ( 40) can be also obtained using the method of path integrals described in Sec. III D. The differential equa-tion (41) is of the Riccati type with constant coefcients 70,and its solution is

˜ pv (τ ) = −i2Ωk2

1ζe Ω( t−τ ) −1

+ iΓ −Ω

k2 , (42)

where we introduced the frequency

Ω = Γ2 + κ2( p2x −ipx ). (43)

and the coefcient

ζ = 1 −i 2Ω

κ2 pv −i(Γ −Ω). (44)

Substituting (42) into (40) and taking the Fourier trans-forms (34) and (36), we get the solution

P t (x, v |vi ) = 1(2π)2 + ∞

−∞dpx dpv eip x x + ip v v (45)

×exp −i ˜ pv (0)vi + γ θ(Γ −Ω)t

κ2 − 2γθ

κ2 ln ζ −e−Ωt

ζ −1

of the original Fokker-Planck equation (28) with the initialcondition ( 29), where ˜ pv (τ = 0) is given by (42).

D. Path-Integral Solution

The Fokker-Planck equation ( 35) can be though of as aSchrodinger equation in imaginary (Euclidean) time:

∂ ∂t

P t,p x (v |vi ) = −H px (ˆ pv , v)P t,p x (v |vi ) (46)

with the Hamiltonian

H = κ2

2 ˆ p2

v v −iγ ˆ pv (v −θ) + p2

x −ipx

2 v + ρκpx ˆ pv v. (47)

In (47) we treat ˆ pv and v as canonically conjugated op-erators with the commutation relation [v, ˆ pv ] = i. The

transition probability P px (v, t |vi ) is the matrix element of the evolution operator exp(−Ht ) and has a path-integralrepresentation 63,64,65

P t,p x (v |vi ) = v|e−Ht |vi = Dv D pv eS p x [ pv (τ ) ,v (τ )] .(48)

Here the action functional S px [ pv (τ ), v(τ )] is

S px = t

0dτ ipv (τ )v(τ ) −H px [ pv (τ ), v(τ )], (49)

and the dot denotes the time derivative. The phase-space pathintegral ( 48) is the sum over all possible paths pv (τ ) and v(τ )with the boundary conditions v(τ = 0) = vi and v(τ = t) =v imposed on v.

It is convenient to integrate the rst term on the r.h.s. of (49)by parts:

S px = i[ pv (t)v

− ˜ pv (0)vi ]

−iγθ

t

0

dτ pv (τ )

− t

0dτ i ˙ pv (τ ) +

δH δv(τ )

v(τ ), (50)

where we also separated the terms linear in v(τ ) from theHamiltonian ( 47). Because v(τ ) enters linearly in the action(50), taking the path integral over Dv generates the delta-functional δ [ pv (τ ) −˜ pv (τ )], where ˜ pv (τ ) is the solution of the ordinary differential equation ( 41) with a boundary con-dition specied at τ = t. Taking the path integral over D pvresolves the delta-functional, and we nd

P t,p x (v

|vi ) =

+ ∞

−∞

dpv

2π ei [ pv v−˜ pv (0) vi ]−iγθ

t

0dτ ˜ pv (τ ) , (51)

where ˜ pv (τ = t) = pv . Eq. (51) coincides with ( 40) after theFourier transform ( 36).

E. Averaging over variance

Normally we are interested only in log-returns x and do notcare about variance v. Moreover, whereas log-returns are di-rectly knownfrom nancial data, variance is a hiddenstochas-tic variable that has to be estimated. Inevitably, such an esti-mation is done with some degree of uncertainty, which pre-

cludes a clear-cut direct comparison between P t (x, v |vi ) andnancial data. Thus, we introduce the probability distribution

P t (x |vi ) = + ∞

−∞dv P t (x, v |vi ), (52)

where the hidden variable v is integrated out. The integrationof (45) over v generates the delta-function δ ( pv ), which ef-fectively sets pv = 0 . Substituting the coefcient ζ from (44)with pv = 0 into (45), we nd

P t (x |vi ) = 12π + ∞

−∞dpx eip x x−vi

p 2x − ip x

Γ+Ωcoth(Ω t/ 2)

×e−

2 γθκ 2 ln(cosh Ωt

2 + γΩ sinh Ωt

2 )+ γ Γ θtκ 2

. (53)To check the validity of ( 53), let us consider the case κ = 0 .

In this case, the stochastic term in (24) is absent, so the timeevolution of variance is deterministic:

v(t) = θ + ( vi −θ)e−γt . (54)

Then process (27) gives a Gaussian distribution for x,

P (κ =0)t (x |vi ) =

1√ 2πtv t

exp −(x + vt t/ 2)2

2vt t, (55)



21

with the time-averaged variance vt = 1t t

0 dτ v(τ ). Eq. (55)demonstrates that the probability distribution of stock price S is log-normal in the case κ = 0 . On the other hand, by takingthe limit κ →0 and integrating over px in (53), we reproducethe same expression (55).

Eq. (53) cannot be directly compared with nancial timeseries data, because it depends on the unknown initial variancevi . In order to resolve this problem, we assume that vi has the

stationary probability distribution Π∗(vi ), which is given by(32).82 Thus we introduce the probability distribution P t (x)by averaging ( 53) over vi with the weight Π

∗(vi ):

P t (x) = ∞0

dvi Π∗(vi ) P t (x |vi ). (56)

The integral over vi is similar to the one of the Gamma func-tion and can be taken explicitly. The nal result is the Fourierintegral

P t (x) = 12π + ∞

−∞dpx eip x x + F t ( px ) (57)

with

F t ( px ) = γ Γθt

κ2

− 2γθ

κ2 ln cosh Ωt

2 +

Ω2 −Γ2 + 2 γ Γ2γ Ω

sinh Ωt

2. (58)

The variable px enters (58) via the variables Γ from (38) andΩ from (43). It is easy to check that P t (x) is real, becauseReF is an even function of px and ImF is an odd one. Onecan also check that F t ( px = 0) = 0 , which implies that P t (x)is correctly normalized at all times: dx P t (x) = 1 . Thesecond term in the r.h.s. of ( 58) vanishes when ρ = 0 , i.e.

when there are no correlations between stock price and vari-ance. The simplied result for the case ρ = 0 is given in III Fby Eqs. (72), (73), and (74).

Eqs. (57) and (58) for the probability distribution P t (x) of log-return x at time t are the central analytical result of thechapter. The integral in (57) can be calculated numericallyor, in certain regimes discussed in Secs. IIIF and IIIG, an-alytically. In Fig. 23, the calculated function P t (x), shownby solid lines, is compared with the Dow-Jones data, shownby dots. (Technical details of the data analysis are discussedin Sec. IIIH.) Fig. 23 demonstrates that, with a xed set of the parameters γ , θ, κ, µ, and ρ, Eqs. (57) and (58) very wellreproduce the distribution of log-returns x of the Dow-Jonesindex for all times t . In the log-linear scale of Fig. 23, the tailsof ln P t (x) vs. x are straight lines, which means that that tailsof P t (x) are exponential in x. For short times, the distributionis narrow, and the slopes of the tails are nearly vertical. Astime progresses, the distribution broadens and attens.

F. Asymptotic behavior for long time t

Eq. (24) implies that variance reverts to the equilibriumvalue θ within the characteristic relaxation time 1/γ . In this

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

10 0

10 1

10 2

10 3

10 4

Log−return, x

P r o

b a

b i l i t y

d e n s

i t y , P

t ( x )

Dow Jones data, 1982−2001

1 day

5 days

20 days

40 days

250 days

−0.4 −0.2 0 0.2

1

10

100

FIG. 23: Probability distribution P t (x) of log-return x for differenttime lags t. Points: The Dow-Jones data for t = 1 , 5, 20, 40, and 250trading days. Solid lines: Fit of the data with Eqs. (57) and (58). Forclarity, the data points and the curves for successive t are shifted upby the factor of 10 each. The inset shows the curves without verticalshift.

section, we consider the asymptotic limit where time t is muchlonger than the relaxation time: γt ≫ 2. According to (38)and (43), this condition also implies that Ωt ≫ 2. Then Eq.(58) reduces to

F t ( px ) ≈ γθt

κ2 (Γ −Ω). (59)

Let us change of the variable of integration in ( 57) to

px = ω0

κ 1 −ρ2

˜ px + ip0 , (60)

where

p0 = κ −2ργ 2κ(1 −ρ2)

, ω0 = γ 2 + κ2(1 −ρ2 ) p20 . (61)

Substituting ( 60) into (38), (43), and (59), we transform (57)to the following form

P t (x) = ω0e− p0 x +Λ t

πκ 1 −ρ2 ∞0

d˜ px cos(A˜ px )e−B √ 1+ p2x , (62)

where

A = ω0

κ 1 −ρ2x + ρ γθt

κ, B = γθω0 t

κ2 , (63)

and

Λ = γθ2κ2

2γ −ρκ1 −ρ2 . (64)

According to 72 , the integral in (62) is equal toBK 1(√ A2 + B 2 )/ √ A2 + B 2 , where K 1 is the rst-ordermodied Bessel function.



23

explained in Sec. III H, the case relevant for comparison withthe data is ρ = 0 . In this case, by shifting the contour of integration in ( 57) as follows px → px + i/ 2, we nd

P t (x) = e−x/ 2 + ∞

−∞

dpx

2π eip x x + F t ( px ) , (72)

where

F t ( px ) = γ 2θtκ2 − 2γθ

κ2 ln cosh Ωt2

+ Ω2 + γ 22γ Ω

sinh Ωt2

(73)and

Ω = γ 2 + κ2( p2x + 1 / 4). (74)

Now the function F t ( px ) is real and symmetric: F t ( px ) =F t (− px ). Thus, the integral in ( 72) is a symmetric function of x. So it is clear that the only source of asymmetry of P t (x) inx is the exponential prefactor in (72), as discussed at the endof Sec. IIIH. Eqs. (72), (73), and (74) are much simpler thanthose for ρ = 0 .

Let us expand the integral in ( 72) for small x:

P t (x) ≈e−x/ 2 µ0 − 12

µ2x2 , (75)

where the coefcients are the rst and the second moments of exp[F t ( px )]

µ0(t) =+ ∞

−∞

dpx

2π eF t ( px ) , µ2(t) =

+ ∞ −∞

dpx

2π p2

x eF t ( px ) .

(76)On the other hand, we know that P t (x) is Gaussian for smallx. So we can write

P t (x) ≈µ0 e−x/ 2e−µ2 x 2 / 2µ0 , (77)

with the same coefcients as in ( 75). If we ignore the exis-tence of fat tails and extrapolate ( 77) to x ∈ (−∞, ∞), thetotal probability contained in such a Gaussian extrapolationwill be

G t =+ ∞

−∞dx µ0 e−x/ 2−µ2 x 2 / 2µ0 = 2πµ 3

0µ2

eµ 0 / 8µ2 . (78)

Obviously, G t < 1, because the integral ( 78) does not takeinto account the probability contained in the fat tails. Thus,the difference 1

−G t can be taken as a measure of how much

the actual distribution P t (x) deviates from a Gaussian func-tion.

We calculate the moments (76) numerically for the functionF given by (73), then determine the Gaussian weight G t from(78) and plot it in Fig. 25 as a function of time. For t → ∞,G t →1, i.e. P t (x) becomes Gaussian for very long time lags,which is known in literature 55. In the opposite limit t → 0,F t ( px ) becomes a very broad function of px , so we cannotcalculate the moments µ0 and µ2 numerically. The singularlimit t →0 requires an analytical study.

ps

p1+

p2+

p3+

p1−

p2−

Re(px)

I m

( p x

)

0

−iq−

iq+

FIG. 26: Complex plane of px . Dots: The singularities of F t ( px ).Circled crosses: The accumulation points ± iq ±∗ of the singularitiesin the limit γ t ≫ 2. Symbol × : Saddle point ps , which is locatedin the upper half-plane for x > 0. Dashed line: The contour of integration displaced from the real axis in order to pass through thesaddle point ps .

Fig. 25 shows that, at sufciently long times, the total prob-ability contained in the non-Gaussian tails becomes negligi-ble, which is known in literature 55. The inset in Fig. 25 illus-trates that the time dependence of the probability density atmaximum, P t (xm ), is close to t−1/ 2 , which is characteristicof a Gaussian evolution.

G. Asymptotic behavior for large log-return x

In the complex plane of px , function F ( px ) becomes sin-gular at the points px where the argument of any logarithmin (58) vanishes. These points are located on the imaginaryaxis of px and are shown by dots in Fig. 26. The singu-larity closest to the real axis is located on the positive (neg-ative) imaginary axis at the point p+

1 ( p−1 ). At these twopoints, the argument of the last logarithmin (58) vanishes, andwe can approximate F ( px ) by the dominant, singular term:F ( px ) ≈ −(2γθ/κ 2)ln( px − p±1 ).

For large |x|, the integrand of (57) oscillates very fast as afunction of px . Thus, we can evaluate the integral using themethod of stationary phase 70 by shifting the contour of inte-gration so that is passes through a saddlepoint of the argumentipx x + F ( px ) of the exponent in ( 57). The saddle point posi-tion ps , shown in Fig. 26 by the symbol ×, is determined bythe equation

ix = − dF ( px )

dpx px = ps≈

2γθκ2 ×

1 ps − p+

1, x > 0,

1 ps − p−

1, x < 0.

(79)

For a large |x| such that |xp±1 |≫2γθ/κ 2 , the saddle point psis very close to the singularity point: ps ≈ p+

1 for x > 0 and



24

0 10 20 30 40 50 60 70 800

20

40

60

80

100

120

140

160

180

T h e s

l o p e , q t +

= −

d l n P

t ( x ) / d x

Time lag, t (days)

0 1 2 3γ

FIG. 27: small Solid line: The slope q +t = − d ln P/dx of the expo-nential tail for x > 0 as a function of time. Points: The asymptoticapproximation (82) for the slope in the limit γt ≪ 2. Dashed line:The saturation value q +∗ for γt ≫ 2, Eq. (81).

ps ≈ p−1 for x < 0. Then the asymptotic expression for theprobability distribution is

P t (x)∼ e−xq +

t , x > 0,exq −

t , x < 0,(80)

where q ±t = ∓ip±1 (t) are real and positive. Eq. (80) showsthat, for all times t, the tails of the probability distributionP t (x) for large |x|are exponential. The slopes of the exponen-tial tails, q ± = ∓d ln P/dx , are determined by the positions p±

1 of the singularities closest to the real axis.These positions p±1 (t) and, thus, the slopes q ±t depend

on time t. For times much shorter than the relaxation time(γt ≪ 2), the singularities lie far away from the real axis.As time increases, the singularities move along the imaginaryaxis toward the real axis. Finally, for times much longer thanthe relaxation time ( γt ≫ 2), the singularities approach lim-iting points: p±1 → ±iq ±

∗ , which are shown in Fig. 26 by

circled ×’s. Thus, as illustrated in Fig. 27, the slopes q ±tmonotonously decrease in time and saturate at long times:

q ±t →q ±∗

= ± p0 + ω0

κ

1 −ρ2

for γt≫2. (81)

The slopes ( 81) are in agreement with Eq. ( 69) valid for γt ≫2. The time dependence q ±t at short times can be also foundanalytically:

q ±t ≈ ± p0 + p20 +

4γ κ2(1 −ρ2)t

for γt ≪2. (82)

The dotted curve in Fig. 27 shows that Eq. ( 82) works verywell for short times t, where the slope diverges at t →0.

82 84 86 88 90 92 94 96 98 00 02

103

104

Year

D o w −

J o n e s

I n d e x

Dow−Jones data, 1982−2001

FIG. 28: Log-linear plot of the Dow-Jones versus time, for the pe-riod 1982-2001. The straight solid line through the data points repre-sents a t of the Dow-Jones index with an exponential function withgrowth rate µ = 13 .3%/ year.

H. Comparison with Dow-Jones time series

To test the model against nancial data, we downloadeddaily closing values of the Dow-Jones industrial index for theperiod of 20 years from 1 January 1982 to 31 December 2001from the Web site of Yahoo 73. The data set contains 5049points, which form the time series S τ , where the integertime variable τ is the trading day number. We do not lter thedata for short days, such as those before holidays.

Given S τ , we use the following procedure to extract the

probability density P (DJ )t (r ) of log-return r for a given timelag t. For the xed t, we calculate the set of log-returns

r τ = ln S τ + t /S τ for all possible times τ . Then we par-tition the r -axis into equally spaced bins of the width ∆ r andcount the number of log-returns rτ belonging to each bin. Inthis process, we omit the bins with occupation numbers lessthan ve, because we consider such a small statistics unreli-able. Only less than 1% of the entire data set is omitted in thisprocedure. Dividing the occupation number of each bin by∆ r and by the total occupation number of all bins, we obtainthe probability density P (DJ )

t (r ) for a given time lag t . Tond P (DJ )

t (x), we replace r → x + µt . In Fig. 28 we plotthe value of the Dow-Jones index, and the t with an expo-

nential function corresponding to an ination rate µ. In Fig.29 we illustrate the histogram corresponding to the probabil-ity density P (DJ )

t (r ) for a time lag t = 5 days. The rst andlast bin in Fig. 29 show all the events outside the two verticaldashed lines. These events are very rare, contribute less than1% to the entire number of data points, and therefore they arediscarded in the tting process.

Assuming that the system is ergodic, so that ensemble aver-aging is equivalent to time averaging, we compare P (DJ )

t (x)extracted from the time series data and P t (x) calculated in



25

−0.09 −0.06 −0.03 0 0.03 0.06 0.090

50

100

150

200

250

300

350

400

450

500

Five day log−return, rt=5

B i n

c o u n

t

Dow−Jones data, 1982−2001

FIG. 29: Histogram of ve day log-returns r t =5 . The rst and thelast bin, which are separated from the rest by the two vertical dashedlines, are discarded in the tting process. The population of therst (last) bin contains all the extreme events from −∞ (+ ∞ ) to the

left(right) dashed line.

previous sections, which describes ensemble distribution. Inthe language of mathematical statistics, we compare our the-oretically derived population distribution with the sample dis-tribution extracted from the time series data. We determineparametersof the model by minimizing the mean-square devi-ation x,t | ln P (DJ )

t (x) −ln P t (x)|2 , where the sum is takenover all available x and t = 1 , 5, 20, 40, and 250 days.These values of t are selected because they represent differ-ent regimes: γ t≪ 1 for t = 1 and 5 days, γt ≈1 for t = 20days, and γ t

≫ 1 for t = 40 and 250 days. As Figs. 23 and

24 illustrate, our expression ( 57) and (58) for the probabilitydensity P t (x) agrees with the data very well, not only for theselected ve values of time t, but for the whole time intervalfrom 1 to 250 trading days. However, we cannot extend thiscomparison to t longer than 250 days, which is approximately1/20 of the entire range of the data set, because we cannotreliably extract P (DJ )

t (x) from the data when t is too long.The values obtained for the four tting parameters ( γ , θ, κ,

µ) are given in Table III. Within the scattering of the data, wedo not nd any discernible difference between the ts with thecorrelation coefcient ρ being zero or slightly different fromzero. Thus, we conclude that the correlation ρ between thenoise terms for stock price and variance in Eq. ( 25) is practi-cally zero, if it exists at all. Our conclusion is in contrast withthe value ρ = −0.58 found in74 by tting the leverage corre-lation function introduced in 75. Further study is necessary inorder to resolve this discrepancy. Nevertheless, all theoreticalcurves shown in this chapter are calculated for ρ = 0 , and theyt the data very well.

All four parameters ( γ , θ, κ, µ) shown in Table III have thedimensionality of 1/time. The rst line of the Table gives theirvalues in the units of 1/day, as originally determined in our t.The second line shows the annualized values of the parameters

Units γ θ κ µ1/day 4.50 × 10− 2 8.62 × 10− 5 2.45 × 10− 3 5.67 × 10− 4

1/year 11.35 0.022 0.618 0.143

TABLE III: Parameters of the model obtained from the t of theDow-Jones data. We also nd ρ = 0 for the correlation coefcientand 1/γ = 22 .2 trading days for the relaxation time of variance.

in the units of 1/year, where we utilize the average number of 252.5 trading days per calendar year to make the conversion.The relaxation time of variance is equal to 1/γ = 22 .2 tradingdays = 4.4 weeks ≈1 month, where we took into account that1 week = 5 trading days. Thus, we nd that variance has arather long relaxation time, of the order of one month, whichis in agreement with the conclusion of Ref. 74 .

Using the numbers given in Table III, we nd the valueof the parameter 2γθ/κ 2 = 1 .296. Since this parameter isgreater than one, the stochastic process (24) never reaches thedomain v < 0, as discussed in Sec. IIIB.

The effective growth rate of stock prices is determined by

the coordinate rm (t) where the probability density P t (r m ) ismaximal. Using the relation rm = xm + µt and Eq. (71), wend that the actual growth rate is µ = µ−γθ/ 2ω0 = 13 % peryear. This number coincides with the average growth rate of the Dow-Jones index obtained by a simple t of the time series

S τ with an exponential function of τ . The effective stock growth rate µ is comparable with the average stock volatilityafter one year σ = √ θ = 14 .7%. Moreover, the parameter(33), which characterizes the width of the stationary distribu-tion of variance, is equal to χ = 0 .54. This means that thedistribution of variance is broad, and variance can easily uc-tuate to a value twice greater than the average value θ. Thus,even though the average growth rate of stock index is posi-tive, there is a substantial probability

0

−∞dr P t (r ) = 17 .7%

to have negative growth for t = 1 year.According to (81), the asymmetry between the slopes of

exponential tails for positive and negative x is given by theparameter p0 , which is equal to 1/2 when ρ = 0 (see also thediscussion of Eq. (72) at the end of Sec. IIIF). The originof this asymmetry can be traced back to the transformationfrom (23) to (26) using Ito’s formula. It produces the term0.5vt dt in the r.h.s. of ( 26), which then generates the secondterm in the r.h.s. of ( 28). The latter term is the only source of asymmetry in x of P t (x) when ρ = 0 . However, in practice,the asymmetry of the slopes p0 = 1 / 2 is quite small (about2.7%) compared with the average slope q ±

∗ ≈ω0/κ = 18 .4.

I. Conclusions

We derived an analytical solution for the probability dis-tribution P t (x) of log-returns x as a function of time t forthe model of a geometrical Brownian motion with stochasticvariance. The nal result has the form of a one-dimensionalFourier integral ( 57) and (58). [In the case ρ = 0 , theequations have the simpler form (72), (73), and (74).] Nu-merical evaluation of the integral ( 57) is simple compared



27

×M

j =1

xj |e−H ( t j −t j − 1 ) |xj −1 . (88)

The contribution coming from interval (t j , t j +1 ) is evaluatedas

xj |e−H ( t j −t j − 1 ) |xj −1 =

dpj x j | pj

× pj |e−H ( t j −t j − 1 ) |x j −1 . (89)

The matrix element inside the integral in Eq.( 89) can be nowevaluated to

pj |e−H ( t j −t j − 1 ) |xj −1 ≈e−ǫH ( pj ,x j − 1 ) pj |x j −1 . (90)

Substitute the scalar product

x j | pj = eip j x j

√ 2π pj |xj −1 =

e−ip j x j − 1

√ 2π, (91)

and Eq. (90) back into ( 89), to obtain

P (xf , t f |xi , t i ) = M −1

j =1dx j

M

j =1

dpj

2π

×eM

j =1 ip j (x j −x j − 1 )−ǫH ( pj ,x j − 1 ) . (92)

where

H ( pj , x j −1 ) = κ2

2 p2

j xj −1 −iγp j (x j −1 −θ) (93)

By taking the limit M → ∞in Eq.(92), one arrives at the pathintegral expression

P (xf , t f |x i , t i ) = Dx D p eS [x,p ] (94)

where the action functional S [x t , pt ] is given by

S [x t , pt ] = t f

t i

dt ip t xt − κ2

2 p2

t xt + iγp t (x t −θ) .

(95)The phase-space path integral from Eq.( 94) has to be calcu-lated over all continuous paths p(t) and x(t) satisfying theboundary conditions x(t i ) = xi and x(t f ) = xf .

For most path-integrals in physics, the path integral isquadratic in momenta and because of that it is customary tointegrate rst the momenta p(t) to obtain the Lagrangean of the problem, and be left with the path integral over the coor-dinate x(t). This route can be taken here too, but the resultingLagrangean will have a position dependent mass, which un-necessary complicates the remaining integration over the co-ordinate. It is important to notice that the problem is linearin the coordinate x(t) and quadratic in p(t), so the originalSchrodinger formulation will be simpler in the momentumrepresentation. In the path-integral formulation, it turns outthat the integration over x(t) is trivial and gives a delta func-tional.

Both the continuous expression ( 94) and the discrete one(92) have their own computational advantages. In the follow-ing, I will use the discrete form (92), because for this problem,it allows for a quick and transparent solution. The exponentin (92) can be rearranged in the form

S = i( pM xf − p0x i ) −iǫγθM

j =1

pj (96)

+ iM −1

j =1

x j ( pj −1 − pj ) + ǫγpj + iǫκ2

2 p2

j

where I made the notation p0 = p1 −ǫ(γp 1 + iκ 2 p21 / 2).

The path integral over the coordinate can be taken one byone, starting with the one over dxM −1 . The result is a deltafunction 2πδ ( pM −1 − pM + ǫγpM + iǫκ2 p2

M / 2). The inte-gration over pM −1 resolves the delta function for the valueof pM −1 equal to pM −1 ≡ (1 −ǫT ) pM , where the operatorT = γ (·) + k2(·)2 / 2. The integral over dxM −2 is done next,with the result 2πδ ( pM −2−(1−T )2 pM ), followedby the inte-gral over dpM −2 . The procedure is repeated until the last pair

dx1 dp1 . In this way we get p0 = (1 −ǫT ) p1 = (1 −ǫT )M pM .The value of the transition probability is

P (xf , t f |x i , t i ) = ∞

−∞

dpM

2π ei ( pM x f − p0 x i )−iǫγθ M

j =1 pj

(97)where in (97), pj ≡(1−ǫT )(M −j ) pM for j = 0 , M . Becauseof the approximation used in (90), Eq.(97) is valid only whenthe number of time splittings M → ∞. In this limit we have

pj − pj −1

ǫ = T pj −→

dpt

dt = T pt with p(t f ) = pM

(98)The differential equation dpt /dt = T pt = γp t + i κ 2

2 p2t is a

Bernoulli equation and has the solution

pt = 1

( 1 pM

+ iκ 2

2γ )e−γ ( t−t f ) − iκ 2

2γ . (99)

The expression for the transition probability in theM → ∞limit is

P (xf , t f |x i , t i ) = ∞

−∞

dpM

2π e

i ( pM x f − p0 x i )−iγθ t f t i

dt p t

(100)where pt is given by Eq.(99). The time integral in ( 100) canbe evaluated as

t f

t i

dt p t =

−2i

κ2

γT

0

dy

ζey −1 =

−2i

κ2 ln

ζ −e−γT

ζ −1(101)

where

ζ = 1 − 2iγ κ2 pM

, and T = tf −t i . (102)

After minor rearrangements in the argument of the logarithm,the nal result is given in the form of a Fourier integral

P (xf , t f |x i , t i ) = ∞

−∞

dpM

2π ei ( pM x f − p0 x i )



28

0

C −

C +

iλ

Re(p)

I m ( p )

FIG. 30: The complex plane of p. The branch cut due to the denomi-nator of Eq. (106) is shown as a broken line. The contour C has beendeformed so that it encircles the branch cut.

×exp −2γθκ2 ln 1 + iκ

2

2γ (1 −e−γT ) pM (103)

where

p0 ≡ p(t = t i ) = pM e−γT

1 + iκ 2

2γ (1 −e−γT ) pM (104)

Although, the integral from Eq.( 103) seems complicated, itcan be evaluated exactly by integration in the complex plane.Introduce the following notations

ν = 2 γθ/κ 2 , and λ = 2 γ/κ 2(1 −exp(−γT )) . (105)

The integral in ( 103) becomes

P (xf , t f |xi , t i ) = +

∞−∞

dp2π

exp ipx f −i pe − γT

1+ ip/λ x i(1 + ip/λ )ν

(106)There is a branch point singularity in the complex plane of p which is situated on the imaginary axis, for p = iλ . Wetake the branch cut to extend from iλ to i∞. For positivexf , the contour of integration is deformed so that it encirclesthe branch cut. The complex plane of p, and the contour of integration are presented in FIG .30. The contribution fromthe contour C can be split into two parts, one from the partto the left of the cut C− and the other part, to the right of thecut C+ . The Bessel function J ν (z) has the following integralrepresentation due to Schl¨ai

J ν (z) = 12πi

z2

ν

(0+)

−∞dt t −ν −1 et−z 2 / 4t , (107)

where the integral is taken along a contour around the branchcut on the negative real axis, encircling it in the counterclock-wise direction. By making the change of variable p → −ip in(106), the transition probabilityP (xf , t f |x i , t i ) can be expressed in terms of the Bessel func-tion as

P (xf , t f |x i , t i ) = λν e−λ (x f + x i e − γT ) (108)

× −xi

xf λ2 e−γT

1 − ν2

J ν −1(2i xi xf λ2 exp(−γT )).

The modied Bessel function I ν (z) is dened in terms of theusual Bessel function J ν (z) of imaginary argument by the re-lation

I ν (z) = e− iπν2 J ν (eiπ/ 2z) for −π < arg (z) ≤π/ 2.

(109)This relation allows to write the nal result as

P (xf , t f |xi , t i ) = λν

e−λ (x f + x i e − γT )

(110)

×xi

xf λ2 e−γT

1 − ν2

I ν −1(2 x i xf λ2 exp(−γT )) ,

were λ and ν are given by (105), and T = tf −t i .The normalization of the probability density,

∞0

dxf P (xf , t f |x i , t i ) = 1 ,

can be proved easily by making use of the result

∞0 dx x ν +1 e−αx

2

I ν (βx ) = β ν

(2α)ν +1 eβ2

/ 4α . (111)

We can check the validity of expression ( 110) by takingseveral limiting cases:

• In the limit T → ∞, the transition probabilityP (xf , t f |xi , t i ) should become the stationaryprobability dis-tribution (given in our paper). The argument of the Besselfunction from ( 110) goes to zero in the limit T → ∞. Themodied Bessel function I ν (z) has the following small argu-ment behavior

I µ (z) ≈ 1

Γ(µ + 1)z2

µfor z →0+ . (112)

Substituting the expression ( 112) into (110), we obtain

P (xf , t f |x i , −∞) ≡P ∗(xf ) =

λν

Γ(ν )xν −1

f e−λx f . (113)

which is the stationary distribution.

• In the short time limit, T → 0 (λ → ∞), theprobability distribution should approach a delta function,P (xf , t i |x i , t i ) = δ (xf −x i ). The argument of the Besselfunction from (110) goes to innity in the limit T → 0. Themodied Bessel function I ν (z) has the following large argu-ment behavior

I µ (z) ≈ ez

√ 2πzfor z → ∞ (114)

Substituting ( 114) back into ( 110) , we obtain

P (xf , t f |x i , t i ) = f (xi /x f ) limλ →∞ λ

4πx ie−λ (√ x f −√ x i ) 2

= δ (xf −xi ) (115)



29

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mailto:[email protected]


http://ferret.bls.census.gov/

http://www.isr.umich.edu/src/psid/

http://www.irs.ustreas.gov/prod/tax_stats/soi/

http://ftp.fedworld.gov/pub/irs-soi/petasa98.pdf

http://ftp.fedworld.gov/pub/irs-soi/disindit.exe

http://www.census.gov/hhes/www/p60191.html



http://www.inlandrevenue.gov.uk/stats/distri-bu-tion-2000.pdf

http://www.inlandrevenue.gov.uk/stats/

http://www.inlandrevenue.gov.uk/stats/table3_3a

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http://www.xe.net/ict/

http://www.worldbank.org/poverty/ine-qual/ab-stracts/



http://www.worldbank.org/poverty/ine-qual/ab-stracts/

http://www.xe.net/ict/


http://www.irs.gov/tax_stats/soi/ind_st.html

http://www.statistics.gov.uk/

http://www.inlandrevenue.gov.uk/stats/table3_3a

http://www.inlandrevenue.gov.uk/stats/

http://www.inlandrevenue.gov.uk/stats/distri-bu-tion-2000.pdf



http://www.census.gov/hhes/www/p60191.html

http://ftp.fedworld.gov/pub/irs-soi/disindit.exe

http://ftp.fedworld.gov/pub/irs-soi/petasa98.pdf

http://www.irs.ustreas.gov/prod/tax_stats/soi/

http://www.isr.umich.edu/src/psid/

http://ferret.bls.census.gov/


mailto:[email protected]



30

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