Applications of statistical mechanics to economics: Entropic origin of theprobability distributions of money, income, and energy consumption

Victor M. Yakovenko

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

(Dated: 29 April 2012 post on arXiv)

This Chapter is written for the Festschrift celebrating the 70th birthday of the distinguishedeconomist Duncan Foley from the New School for Social Research in New York. This Chapterreviews applications of statistical physics methods, such as the principle of entropy maximization,to the probability distributions of money, income, and global energy consumption per capita.The exponential probability distribution of wages, predicted by the statistical equilibriumtheory of a labor market developed by Foley in 1996, is supported by empirical data onincome distribution in the USA for the majority (about 97%) of population. In addition, theupper tail of income distribution (about 3% of population) follows a power law and expandsdramatically during financial bubbles, which results in a significant increase of the overallincome inequality. A mathematical analysis of the empirical data clearly demonstrates thetwo-class structure of a society, as pointed out Karl Marx and recently highlighted by the OccupyMovement. Empirical data for the energy consumption per capita around the world are closeto an exponential distribution, which can be also explained by the entropy maximization principle.

“Money, it’s a gas.” Pink Floyd, Dark Side of the Moon

I. HOW I MET DUNCAN FOLEY

Although I am a theoretical physicist, I have been al-ways interested in economics and, in particular, in appli-cations of statistical physics to economics. These ideasfirst occurred to me when I was an undergraduate studentat the Moscow Physical-Technical Institute in Russia andstudied statistical physics for the first time. However, itwas not until 2000 when I published my first paper onthis subject (Dragulescu and Yakovenko, 2000), joiningthe emerging movement of econophysics (Farmer, Shu-bik, and Smith, 2005; Hogan, 2005; Shea, 2005). At thattime, I started looking for economists who may be inter-ested in a statistical approach to economics. I attendeda seminar by Eric Slud, a professor of mathematics atthe University of Maryland, who independently exploredsimilar ideas and eventually published them in Silver,Slud, and Takamoto (2002). In this paper, I saw a ref-erence to the paper by Foley (1994). So, I contactedDuncan in January 2001 and invited him to give a talkat the University of Maryland, which he did in March.Since then, our paths have crossed many times. I visitedthe New School for Social Research in New York severaltimes, and we also met and discussed at the Santa Fe In-stitute, where I was spending a part of my sabbatical inJanuary–February 2009, hosted by Doyne Farmer. Dur-ing these visits, I also met other innovative economistsat the New School, professors Anwar Shaikh and WilliSemler, as well as Duncan’s Ph.D. student at that timeMishael Milakovic, who is now a professor of economicsat the University of Bamberg in Germany.

As it is explained on the first page of Foley (1994),Duncan learned statistical physics by taking a course onstatistical thermodynamics at Dartmouth College. Forme, the papers by Foley (1994, 1996, 1999) conjure an

image of people who meet on a bridge trying to reach forthe same ideas by approaching from the opposite banks,physics and economics. Over time, I learned about otherpapers where economists utilize statistical and entropicideas, e.g. Aoki and Yoshikawa (2006); Golan (1994);Molico (2006), but I also realized how exceedingly rarestatistical approach is among mainstream economists.Duncan is one of the very few innovative economists whohas deeply studied statistical physics and attempted tomake use of it in economics.

One of the puzzling social problems is persistent eco-nomic inequality among the population in any society.In statistical physics, it is well known that identical(“equal”) molecules in a gas spontaneously develop awidely unequal distribution of energies as a result of ran-dom energy transfers in molecular collisions. By analogy,very unequal probability distributions can spontaneouslydevelop in an economic system as a result of random in-teractions between economic agents. This is the mainidea of the material presented in this Chapter.

First, I will briefly review the basics of statisticalphysics and then discuss and compare the applicationsof these ideas to economics in Foley (1994, 1996, 1999)and in my papers, from the first paper (Dragulescuand Yakovenko, 2000) to the most recent review pa-pers (Banerjee and Yakovenko, 2010; Yakovenko andRosser, 2009) and books (Cockshott, Cottrell, Michael-son, Wright, and Yakovenko, 2009; Yakovenko, 2011). Aswe shall see, statistical and entropic ideas have a multi-tude of applications to the probability distributions ofmoney, income, and global energy consumption. Thelatter topic has relevance to the ongoing debate on theeconomics of global warming (Rezai, Foley, and Taylor,2012). Duncan Foley and Eric Smith of the Santa FeInstitute have also published a profound study of the

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FIG. 1 A quantum paramagnet, where each atom has thespin s = 1/2 and is in one of the two possible states (q = 2):k = 1 =↑ or k = 2 =↓. The atoms are labeled consecutivelyby the integer index j = 1, 2, . . . , N .

connection between phenomenological thermodynamicsin physics and the utility formalism in economics (Smithand Foley, 2008). This paper will not be reviewed herebecause of a limited volume of this Chapter. Unfortu-nately, there is still no complete understanding of a con-nection between statistical mechanics and phenomeno-logical thermodynamics as applied to economics. Theseissues remain open for a future study. I will only focuson statistical mechanics in this Chapter.

II. ENTROPY AND THE BOLTZMANN-GIBBSDISTRIBUTION OF ENERGY IN PHYSICS

In this section, I briefly review the basics of statisti-cal physics, starting from a discrete model of a quantumparamagnet. Let us consider a collection of N atomsand label the individual atoms with the integer indexj = 1, 2, . . . , N . It is convenient to visualize the atoms assitting on a lattice, as shown in Fig. 1. Each atom hasan internal quantum degree of freedom called the spin sand can be in one of the q discrete states.1 These statesare labeled by the index k = 1, 2, . . . , q. The system isplaced in a external magnetic field, so each discrete statehas a different energy εk. Then, the state of an atom jcan be characterized by the energy εj that this atom has.(Throughout the paper, I use the indices i and j to labelindividual atoms or agents, and the index k to label pos-sible states of the atoms or agents.) A simple case withq = 2 is illustrated in Fig. 1, where each atom can bein one of the two possible states k = 1 =↑ or k = 2 =↓,which are depicted as the spin-up and spin-down states.Alternatively, one can interpret each atom as a bit ofcomputer memory with two possible states 0 and 1. Forq = 8, each atoms would correspond to a byte with 8possible values. The atoms can be also interpreted aseconomic agents, where the index k indicates an internalstate of each agent.

For a given configuration of atoms, let us count howmany atoms are in each state k and denote these numbersas Nk, so that

∑qk=1Nk = N . Let us introduce the

multiplicity Ω, which is the number of different atomicconfigurations that have the same set of the numbers

1 In quantum mechanics, s takes integer or half-integer values, andq = 2s + 1, but this is not important here.

Nk. Calculating Ω, we treat the atoms as distinguishable,because they are localized on distinguishable lattice siteslabeled by the index j in Fig. 1. For example, let usconsider the case of N = 2 and q = 2, i.e. two atomswith two possible states. The set N1 = 1 and N2 =1 can be realized in two possible ways, so Ω = 2. Inone configuration, the first atom j = 1 is in the statek = 1 =↑ with the energy ε1 = ε↑, whereas the secondatom j = 2 is in the state k = 2 =↓ with the energyε2 = ε↓. In another configuration, the states of the atomsare reversed. On the other hand, the set N1 = 2 andN2 = 0 can be realized in only one way, where both atomsare in the state k = 1 =↑ with the energies ε1 = ε2 = ε↑,so Ω = 1. In general, the multiplicity Ω can be calculatedcombinatorially as the number of different placements ofN atoms into q boxes with the fixed occupations Nk ofeach box:

Ω =N !

N1!N2!N3! . . . Nq!. (1)

The logarithm of multiplicity is called the entropy S =ln Ω. Using the Stirling approximation for the factorialsin the limit of large numbers Nk, we obtain

S = N lnN −q∑

k=1

Nk lnNk = −q∑

k=1

Nk ln

(NkN

). (2)

In the absence of further information, we assume thatall microscopic configurations of atoms are equally prob-able. Then, the probability of observing a certain set ofthe numbers Nk is proportional to the number of possiblemicroscopic realizations of this set, i.e. to the multiplic-ity Ω. So, the most probable set of the numbers Nk isthe one that maximizes Ω or S subject to certain con-straints. The typical constraints for a closed system arethat the total number of atoms N and the total energyE are fixed:

N =

q∑k=1

Nk, E =

q∑k=1

εkNk =

N∑j=1

εj . (3)

To implement the constraints, we introduce the Lagrangemultipliers α and β and construct the modified entropy

S = S + α

q∑k=1

Nk − βq∑

k=1

εkNk. (4)

Maximization of S is achieved by setting the derivatives∂S/∂Nk to zero for each Nk. Substituting Eq. (2) intoEq. (4) and taking the derivatives,2 we find that the oc-cupation numbers P (εk) = Nk/N for the states k dependexponentially on the energies of these states εk

P (εk) =NkN

= eα−βεk = e−(εk−µ)/T . (5)

2 Notice that N =∑

kNk in Eq. (2) should be also differentiated

with respect to Nk.

3

Here the parameters T = 1/β and µ = αT are calledthe temperature and the chemical potential. The valuesof α and β are determined by substituting Eq. (5) intoEq. (3) and satisfying the constraints for given N and E.The temperature T is equal to the average energy perparticle T ∼ 〈ε〉 = E/N , up to a numerical coefficient ofthe order of one. The interpretation of Eq. (5) is thatthe probability for an atom to occupy a state with theenergy ε depends exponentially on the energy: P (ε) ∝exp(−ε/T ).

This consideration can be generalized to the case wherethe label k becomes a continuous variable. In this case,the sums in Eqs. (2), (3), and (4) are replaced by inte-grals with an appropriate measure of integration. Thetypical physical example is a gas of atoms moving in-side a large box. In this case, the atoms have closely-spaced quantized energy levels, and the number of quan-tum states within a momentum interval ∆p and a spaceinterval ∆x is ∆p∆x/h, where h is the Planck constant.Then, the measure of integration in Eqs. (2), (3), and(4) becomes the element of volume of the phase spacedp dx/h in one-dimensional case and d3p d3r/h3 in three-dimensional case. The energy in Eq. (5) becomes thekinetic energy of an atom ε = p2/2m, which is boundedfrom below ε ≥ 0.

Eq. (5) is the fundamental law of equilibrium statis-tical physics (Wannier, 1987), known as the Boltzmann-Gibbs distribution.3 However, one can see that the abovederivation is really an exercise in theory of probabilitiesand, as such, is not specific to physics. Thus, it can beapplied to statistical ensembles of different nature, sub-ject to constraints similar to Eq. (3). Interpreting thediscrete states of the atoms as bits of information, onearrives to the Shannon entropy in the theory of informa-tion. Entropy concepts have been applied to such dis-parate fields as ecology (Banavar, Maritan, and Volkov,2010; Kelly, Blundell, Bowler, Fox, Harvey, Lomas, andWoodward, 2011) and neuroscience (Varshney, Sjostrom,and Chklovskii, 2006; Wen, Stepanyants, Elston, Gros-berg, and Chklovskii, 2009).

The economy is also a big statistical system consist-ing of a large number of economic agents. Thus, it istempting to apply the formalism presented above to theeconomy as well. However, there are different ways ofimplementing such an analogy. In the next section, Ibriefly summarize an analogy developed in Foley (1994,1996, 1999).

3 In physics, the elementary particles are indistinguishable andobey the Fermi-Dirac or Bose-Einstein statistics, rather than theBoltzmann statistics. In contrast, the economic agents are distin-guishable because of their human identity. Thus, the Boltzmannstatistics for distinguishable objects is appropriate in this case.

III. STATISTICAL EQUILIBRIUM THEORY OFMARKETS BY DUNCAN FOLEY

In Foley (1994), Duncan proposed a statistical equi-librium theory of markets. This theory is an alternativeto the conventional competitive equilibrium theory orig-inated by Walras and by Marshall. In the conventionaltheory, an auctioneer collects orders from buyers and sell-ers and determines an equilibrium price that clears themarket, subject to budget constraints and utility prefer-ences of the agents. In this theory, all transactions takeplace at the same price, but it is not quite clear how thereal markets would actually converge to this price. Incontrast, Duncan proposed a theory where is a probabil-ity distribution of trades at different prices, and marketclearing is achieved only statistically.

Foley (1994) studied an ensemble of N economic agentswho trade n types of different commodities. The state ofeach agent j = 1, 2, 3, . . . , N is characterized by the n-

component vector xj = (x(1)j , x

(2)j , x

(3)j , . . . , x

(n)j ). Each

component of this vector represents a possible trade thatthe agent j is willing to perform with a given commodity.Positive, negative, and zero values of xj represent an in-crease, a decrease, and no change in the stock of a givencommodity for the agent j. All trades in the system aresubject to the global constraint∑

jxj = 0, (6)

which represents conservation of commodities in the pro-cess of trading, i.e. commodities are only transferred be-tween the agents. An increase xi > 0 of a commoditystock for an agent i must be compensated by a decreasexj < 0 for another agent j, so that the algebraic sum (6)of all trades is zero. However, Eq. (6) does not requirebilateral balance of transactions between pairs of agentsand allows for multilateral trades.

The vector xj can be considered as an n-dimensionalgeneralization of the variable εj introduced in Sec. II tocharacterize the state of each atom. Different kinds oftrades xk are labeled by the index k, and the number ofagents doing the trade xk is denoted as Nk. Then theconstraint (6) can be rewritten as∑

kxkNk = 0. (7)

For a given set of Nk, the multiplicity (1) gives the num-ber of different assignments of individual agents to thetrades xk, such that the numbers Nk are fixed. Maxi-mizing the entropy (2) subject to the constraints (7), wearrive to an analog of Eq. (5), which now has the form

P (xk) =NkN

= c e−π·xk . (8)

Here c is a normalization constant, and π is the n-component vector of Lagrange multipliers introduced tosatisfy the constraints (7). Foley (1994) interpreted π asthe vector of entropic prices. The probability for an agent

4

to perform the set of trades x depends exponentially onthe volume of the trades: P (x) ∝ exp(−π · x).

In Foley (1996), the general theory (Foley, 1994) wasapplied to a simple labor market. In this model, thereare two classes on agents: employers (firms) and employ-ees (workers). They trade in two commodities (n = 2):x(1) = w is wage, supplied by the firms and taken by theworkers, and x(2) = l is labor, supplied by the workersand taken by the firms. For each worker, the offer setincludes the line x = (w > w0,−1), where −1 is thefixed offer of labor in exchange for any wage w greaterthan a minimum wage w0. The offer set also includesthe point x = (0, 0), which offers no labor and no wage,i.e. the state of unemployment. For each firm, the offerset is the line x = (−K, l > l0), where −K is the fixedamount of capital spent on paying wages in exchange forthe amount of labor l greater than a minimum value l0.

A mathematically similar model was also developedin Foley (1999) as a result of conversations with PerryMehrling. This model also has two commodities (n = 2),one of which is interpreted as money and another as afinancial asset, such as a bond or a treasury bill.

Let us focus on the labor market model by Foley(1996). According to Eq. (8), the model predicts theexponential probability distributions for the wages w re-ceived by the workers and for the labor l employed bythe firms. Let us try to compare these predictions withempirical data for the real economy. The probability dis-tribution of wages can be compared with income distri-bution, for which a lot of data is available and which willbe discussed in Sec. VI. On the other hand, the distribu-tion of labor employed by the firms can be related to thedistribution of firm sizes, as measured by their numberof employees. However, Foley (1996) makes an artificialsimplifying assumption that each firm spends the sameamount of capital K on labor. This is a clearly unre-alistic assumption because of the great variation in theamount of capital among the firms, including the capitalspent on labor. Thus, let us focus only on the probabilitydistribution of wages, unconditional on the distributionof labor. To obtain this distribution, we take a sum inEq. (8) over the values of x(2), while keeping a fixed valuefor x(1). In physics jargon, we “integrate out” the degreeof freedom x(2) = l and thus obtain the unconditionalprobability distribution of the remaining degree of free-dom x(1) = w, which is still exponential

P (w) = c e−w/Tw . (9)

Here c is a normalization constant, and Tw = 1/π(1) isthe wage temperature.

Now let us discuss how the constraint (7) is satisfiedwith respect to wages. The model has Nf firms, whichsupply the total capital for wages W = KNf , which en-ters as a negative term into the sum (7). Since only thetotal amount matters in the constraint, we can take Was an input parameter of the model. Given that the un-employed workers have zero wage w = 0, the constraint

(7) can be rewritten as∑wk>w0

wkNk = W, (10)

where the sum is taken over the employed workers, whosetotal number we denote as Ne. The average wage per em-ployed worker is 〈w〉 = W/Ne. Using Eq. (9) and replac-ing summation over k by integration over w in Eq. (10),we relate 〈w〉 and Tw

〈w〉 =W

Ne=

∫∞w0wP (w) dw∫∞

w0P (w) dw

, Tw = 〈w〉 − w0. (11)

So, the wage temperature Tw is a difference of the averagewage per employed worker and the minimal wage.

The model by Foley (1996) also has unemployed work-ers, whose number Nu depends on the measure of statis-tical weight assigned to the state with w = 0. Becausethis measure is an input parameter of the model, onemight as well take Nu as an input parameter. I will notfurther discuss the number of unemployed and will focuson the distribution of wages among the employed work-ers. It will be shown in Sec. VI that the exponentialdistribution of wages (9) indeed agrees with the actualempirical data on income distribution for the majority ofpopulation.

IV. STATISTICAL MECHANICS OF MONEY

The theory presented in Sec. III focused on markettransactions and identified the states of the agents withthe possible vectors of transactions xj . However, an anal-ogy with the Boltzmann-Gibbs formalism can be also de-veloped in a different way, as proposed by Dragulescuand Yakovenko (2000). In this paper, the states of theagents were identified with the amounts of a commoditythey hold, i.e. with stocks of a commodity, rather thanwith fluxes of a commodity. This analogy is closer to theoriginal physical picture presented in Sec. II, where thestates of the atoms were identified with the amounts ofenergy εj held by the atoms (i.e. the stocks of energy),rather than with the amounts of energy transfer in colli-sions between the atoms (i.e. the fluxes of energy). In thissection, I summarize the theory developed by Dragulescuand Yakovenko (2000).

Similarly to Sec. III, let us consider an ensemble ofN economic agents and characterize their states by theamounts (stocks) of commodities they hold. Let us intro-duce one special commodity called money,4 whereas allother commodities are assumed to be physically consum-able, such as food, consumer goods, and services. Moneyis fundamentally different from consumable goods, be-cause it is an artificially created, non-consumable object.

4 To simplify consideration, I use only one monetary instrument,commonly known to everybody as “money”.

5

I will not dwell on historical origins and various physicalimplementations of money, but will proceed straight tothe modern fiat money. Fundamentally, money is bitsof information (digital balances) assigned to each agent,so money represents an informational layer of the econ-omy, as opposed to the physical layer consisting of variousmanufactured and consumable goods. Although moneyis not physically consumable, and well-being of the eco-nomic agents is ultimately determined by the physicallayer, nevertheless money plays an extremely importantrole in the modern economy. Many economic crises werecaused by monetary problems, not by physical problems.In the economy, monetary subsystem interacts with phys-ical subsystem, but the two layers cannot transform intoeach other because of their different nature. For this rea-son, an increase in material production does not result inan automatic increase in money supply.

Let us denote money balances of the agents j =1, 2, . . . , N by mj . Ordinary economic agents can onlyreceive money from and give money to other agents, andare not permitted to “manufacture” money, e.g. to printdollar bills. The agents can grow apples on trees, butcannot grow money on trees. Let us consider an eco-nomic transaction between the agents i and j. When theagent i pays money ∆m to the agent j for some goods orservices, the money balances of the agents change as

mi → m′i = mi −∆m,

mj → m′j = mj + ∆m. (12)

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

mi +mj = m′i +m′j , (13)

i.e. there is a local conservation law for money. The trans-fer of money (12) is analogous to the transfer of energyin molecular collisions, and Eq. (13) is analogous to con-servation of energy. Conservative models of this kind arealso studied in some economic literature (Kiyotaki andWright, 1993; Molico, 2006).

Enforcement of the local conservation law (13) is cru-cial for successful functioning of money. If the agentswere permitted to “manufacture” money, they would beprinting money and buying all goods for nothing, whichwould be a disaster. In a barter economy, one consumablegood is exchanged for another (e.g. apples for oranges), sophysical contributions of both agents are obvious. How-ever, this becomes less obvious in a monetary economy,where goods are exchanged for money and money forgoods. The purpose of the conservation law (12) is toensure that an agent can buy goods from the society onlyif he or she contributed something useful to the societyand received monetary payment for these contributions.Money is an accounting device, and, indeed, all account-ing rules are based on the conservation law (12).

Unlike ordinary economic agents, a central bank or acentral government, who issued the fiat money in the firstplace, can inject money into the economy, thus changing

the total amount of money in the system. This process isanalogous to an influx of energy into a system from exter-nal sources. As long as the rate of money influx is slowcompared with the relaxation rate in the economy, thesystem remains in a quasi-stationary statistical equilib-rium with slowly changing parameters. This situation isanalogous to slow heating of a kettle, where the kettle hasa well defined, but slowly increasing, temperature at anymoment of time. Following the long-standing tradition inthe economic literature, this section studies the economyin equilibrium, even though the real-world economy maybe totally out of whack. An economic equilibrium im-plies a monetary equilibrium too, so, for these idealizedpurposes, we consider a model where the central author-ities do not inject additional money, so the total amountof money M held by all ordinary agents in the system isfixed. In this situation, the local conservation law (13)becomes the global conservation law for money∑

jmj = M. (14)

It is important, however, that we study the system in astatistical equilibrium, as opposed to a mechanistic equi-librium envisioned by Walras and Marshall. For this rea-son, the global constraint (14) is actually not as crucialfor the final results, as it might seem. In physics, weusually start with the idealization of a closed system,where the total energy E in Eq. (3) is fixed. Then, afterderiving the Boltzmann-Gibbs distribution (5) and theconcept of temperature, we generalize the considerationto an open system in contact with a thermal reservoir ata given temperature. Even though the total energy of theopen system is not conserved any more, the atoms stillhave the same Boltzmann-Gibbs distribution (5). Sim-ilarly, routine daily operations of the central bank (asopposed to outstanding interventions, such as the recentquantitative easing by the Federal Reserve Bank) do notnecessarily spoil the equilibrium distribution of moneyobtained for a closed system.

Another potential problem with conservation of moneyis debt, which will be discussed in more detail in Sec. V.To simplify initial consideration, we do not allow agentsto have debt in this section. Thus, by construction,money balances of the agents cannot drop below zero.i.e. mi ≥ 0 for all i. Transaction (12) takes place onlywhen an agent has enough money to pay the price, i.e.mi ≥ ∆m. An agent with mi = 0 cannot buy goodsfrom other agents, but can receive money for deliveringgoods or services to them. Most econophysics models, seereviews by Chakraborti, Tokea, Patriarca, and Abergel(2011a,b); Yakovenko and Rosser (2009), and some eco-nomic models (Kiyotaki and Wright, 1993; Molico, 2006)do not consider debt, so it is not an uncommon sim-plifying assumption. In this approach, the agents areliquidity-constrained. As the recent economic crisis bru-tally reminded, liquidity constraint rules the economy,even though some players tend to forget about it duringdebt binges in the bubbles.

6

The transfer of money in Eq. (12) from one agent toanother represents payment for delivered goods and ser-vices, i.e. there is an implied counterflow of goods for eachmonetary transaction. However, keeping track explicitlyof the stocks and fluxes of consumable goods would be avery complicated task. One reason is that many goods,e.g. food and other supplies, and most services, e.g. get-ting a haircut or going to a movie, are not tangible anddisappear after consumption. Because they are not con-served, and also because they are measured in many dif-ferent physical units, it is not practical to keep track ofthem. In contrast, money is measured in the same unit(within a given country with a single currency) and isconserved in local transactions (13), so it is straightfor-ward to keep track of money.

Thus, I choose to keep track only of the money bal-ances of the agents mj , but not of the other commoditiesin the system. This situation can be visualized as follows.Suppose money accounts of all agents are kept on a cen-tral computer server. A computer operator observes themultitude of money transfers (12) between the accounts,but he or she does not have information about the reasonsfor these transfers and does not know what consumablegoods were transferred in exchange. Although purpose-ful and rational for individual agents, these transactionslook effectively random for the operator. (Similarly, instatistical physics, each atom follows deterministic equa-tions of motion, but, nevertheless, the whole system iseffectively random.) Out of curiosity, the computer op-erator records a snapshot of the money balances mj ofall agents at a given time. Then the operator countsthe numbers of agents Nk who have money balances inthe intervals between mk and mk + m∗, where m∗ is areasonably small money window, and the index k labelsthe money interval. The operator wonders what is thegeneral statistical principle that governs the relative oc-cupation numbers P (mk) = Nk/N . In the absence ofadditional information, it is reasonable to derive thesenumbers by maximizing the multiplicity Ω and the en-tropy S of the money distribution in Eqs. (1) and (2),subject to the constraint

∑kmkNk = M in Eq. (14).

The result is given by the Boltzmann-Gibbs distributionfor money

P (m) = c e−m/Tm , (15)

where c is a normalizing constant, and Tm is the moneytemperature. Similarly to Eq. (11), the temperature isobtained from the constraint (14) and is equal to theaverage amount of money per agent: Tm = 〈m〉 = M/N .

The statistical approach not only predicts the equilib-rium distribution of money (15), but also allows us tostudy how the probability distribution of money P (m, t)evolves in time t toward the equilibrium. Dragulescuand Yakovenko (2000) performed computer simulationsof money transfers between the agents in different mod-els. Initially all agents were given the same amount ofmoney, say, $1000. Then, a pair of agents (i, j) was ran-domly selected, the amount ∆m was transferred from

0 1000 2000 3000 4000 5000 60000

2

4

6

8

10

12

14

16

18

Money, m

Prob

abilit

y, P

(m)

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

!m", T

0 1000 2000 30000

1

2

3

Money, m

log

P(m

)

FIG. 2 Histogram and points: The stationary probability dis-tribution of money P (m) obtained in computer simulationsof the random transfer models (Dragulescu and Yakovenko,2000). Solid curves: Fits to the exponential distribution (15).Vertical line: The initial distribution of money.

one agent to another, and the process was repeated manytimes. Time evolution of the probability distribution ofmoney P (m, t) is shown in computer animation videosby Chen and Yakovenko (2007); Wright (2007). Aftera transitory period, money distribution converges to thestationary form shown in Fig. 2. As expected, the distri-bution is well fitted by the exponential function (15).

In the simplest model considered by Dragulescu andYakovenko (2000), the transferred amount ∆m = $1 wasconstant. Computer animation (Chen and Yakovenko,2007) shows that the initial distribution of money firstbroadens to a symmetric Gaussian curve, typical for adiffusion process. Then, the distribution starts to pile uparound the m = 0 state, which acts as the impenetrableboundary, because of the imposed condition m ≥ 0. As aresult, P (m) becomes skewed (asymmetric) and eventu-ally reaches the stationary exponential shape, as shownin Fig. 2. The boundary at m = 0 is analogous to theground-state energy in physics, i.e. the lowest possibleenergy of a physical system. Without this boundarycondition, the probability distribution of money wouldnot reach a stationary shape. Computer animations(Chen and Yakovenko, 2007; Wright, 2007) also showhow the entropy (2) of the money distribution, definedas S/N = −

∑k P (mk) lnP (mk), grows from the initial

value S = 0, where all agents have the same money, tothe maximal value at the statistical equilibrium.

Dragulescu and Yakovenko (2000) performed simula-tions of several models with different rules for moneytransfers ∆m. As long as the rules satisfy the time-reversal symmetry (Dragulescu and Yakovenko, 2000),the stationary distribution is always the exponential one(15), irrespective of initial conditions and details of trans-fer rules. However, this symmetry is violated by themultiplicative rules of transfer, such as the proportional

7

rule ∆m = γmi (Angle, 1986; Ispolatov, Krapivsky, andRedner, 1998), the saving propensity (Chakraborti andChakrabarti, 2000), and the negotiable prices (Molico,2006). These models produce Gamma-like distributions,as well as a power-law tail for a random distribution ofsaving propensities (Chatterjee and Chakrabarti, 2007).Despite some mathematical differences, all these modelsdemonstrate spontaneous development of a highly un-equal probability distribution of money as a result of ran-dom money transfers between the agents (Chakraborti,Tokea, Patriarca, and Abergel, 2011a,b). The moneytransfer models can be also formulated in the language ofcommodity trading between the agents optimizing theirutility functions (Chakrabarti and Chakrabarti, 2009),which is closer to the traditional language of the eco-nomic literature. More involved agent-based simulationswere developed by Wright (2005, 2009) and demonstratedemergence of a two-classes society from the initially equalagents. This work was further developed in the book byCockshott, Cottrell, Michaelson, Wright, and Yakovenko(2009), integrating economics, computer science, andphysics. Empirical data on income distribution discussedin Sec. VI show direct evidence for the two-class society.

Let us compare the results of Secs. III and IV. Inthe former, the Boltzmann-Gibbs formalism is appliedto the fluxes of commodities, and the distribution ofwages (9) is predicted to be exponential. In the latter,the Boltzmann-Gibbs formalism is applied to the stocksof commodities, and the distribution of money balances(15) is predicted to be exponential. There is no contra-diction between these two approaches, and both fluxesand stocks may have exponential distributions simulta-neously, although this is not necessarily required. Thesepapers were motivated by different questions. The pa-pers by Foley (1994, 1996, 1999) were motivated by thelong-standing question of how markets determine pricesbetween different commodities in a decentralized man-ner. The paper by Dragulescu and Yakovenko (2000)was motivated by another long-standing question of howinequality spontaneously develops among equal agentswith equal initial endowments of money. In the neoclassi-cal thinking, equal agents with equal initial endowmentsshould forever stay equal, which contradicts everyday ex-perience. Statistical approach argues that the state ofequality is fundamentally unstable, because it has a verylow entropy. The law of probabilities (Farjoun and Ma-chover, 1983) leads to the exponential distribution, whichis highly unequal, but stable, because it maximizes theentropy. Some papers in the economic literature mod-eled the distributions of both prices and money balanceswithin a common statistical framework (Molico, 2006).

It would be very interesting to compare the theoreticalprediction (15) with empirical data on money distribu-tion. Unfortunately, it is very difficult to obtain suchdata. The distribution of balances on deposit accountsin a big enough bank would be a reasonable approxima-tion for the distribution of money among the population.However, such data are not publicly available. In con-

trast, plenty of data on income distribution are availablefrom the tax agencies and surveys of population. Thesedata will be compared with the theoretical prediction (9)in Sec. VI. However, before that, I will discuss in the nextsection how the results of this section would change whenthe agents are permitted to have debt.

V. MODELS OF DEBT

From the standpoint of individual economic agents,debt may be considered as negative money. When anagent borrows money from a bank,5 the cash balanceof the agent (positive money) increases, but the agentalso acquires an equal debt obligation (negative money),so the total balance (net worth) of the agent remainsthe same. Thus, the act of money borrowing still sat-isfies a generalized conservation law of the total money(net worth), which is now defined as the algebraic sum ofpositive (cash M) and negative (debt D) contributions:M −D = Mb, where Mb is the original amount of moneyin the system, the monetary base (McConnell and Brue,1996). When an agent needs to buy a product at a price∆m exceeding his money balance mi, the agent is nowpermitted to borrow the difference from a bank. Afterthe transaction, the new balance of the agent becomesnegative: m′i = mi − ∆m < 0. The local conservationlaw (12) and (13) is still satisfied, but now it involves neg-ative values of m. Thus, the consequence of debt is nota violation of the conservation law, but a modification ofthe boundary condition by permitting negative balancesmi < 0, so m = 0 is not the ground state any more.

If the computer simulation with ∆m = $1 describedin Sec. IV is repeated without any restrictions on thedebt of the agents, the probability distribution of moneyP (m, t) never stabilizes, and the system never reaches astationary state. As time goes on, P (m, t) keeps spread-ing in a Gaussian manner unlimitedly toward m = +∞and m = −∞. Because of the generalized conservationlaw, the first moment of the algebraically defined moneym remains constant 〈m〉 = Mb/N . It means that someagents become richer with positive balances m > 0 atthe expense of other agents going further into debt withnegative balances m < 0.

Common sense, as well as the current financial crisis,indicate that an economic system cannot be stable if un-limited debt is permitted. In this case, the agents canbuy goods without producing anything in exchange bysimply going into unlimited debt. Eq. (15) is not appli-cable in this case for the following mathematical reason.The normalizing coefficient in Eq. (15) is c = 1/Z, where

5 Here we treat the bank as being outside of the system consistingof ordinary agents, because we are interested in money distribu-tion among these agents. The debt of the agents is an asset forthe bank, and deposits of cash into the bank are liabilities of thebank (McConnell and Brue, 1996).

8

0 2000 4000 6000 80000

2

4

6

8

10

12

14

16

18

Money, m

Prob

abilit

y, P

(m)

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

Model without debt, T=1000

Model with debt, T=1800

FIG. 3 Histograms: Stationary distributions of money withand without debt (Dragulescu and Yakovenko, 2000). Thedebt is limited to md = 800. Solid curves: Fits to the expo-nential distributions with the money temperatures Tm = 1800and Tm = 1000.

Z is the partition function

Z =∑

ke−mk/Tm , (16)

and the sum is taken over all permitted states k of theagents. When debt is not permitted, and money balancesare limited to non-negative values mk ≥ 0, the sum inEq. (16) converges, if the temperature is selected to bepositive Tm > 0. This is the case in physics, where ki-netic energy takes non-negative values ε ≥ 0, and there isa ground state with the lowest possible energy, i.e. energyis bounded from below.6 In contrast, when debt is per-mitted and is not limited by any constraints, the sum inthe partition function (16) over both positive and nega-tive values of mk diverges for any sign of the temperatureTm.

It is clear that some sort of a modified boundary con-dition has to be imposed in order to prevent unlimitedgrowth of debt and to ensure overall stability and equi-librium in the system. Dragulescu and Yakovenko (2000)considered a simple model where the maximal debt ofeach agent is limited to md. In this model, P (m) againhas the exponential shape, but with the new boundarycondition at m = −md and the higher money tempera-ture Td = md +Mb/N , as shown in Fig. 3. This result isanalogous to Eq. (11) with the substitution w0 → −md.

6 When the energy spectrum εk is bounded both from above andbelow, the physical temperature T may take either positive ornegative values. For a quantum paramagnet discussed in Sec II,the negative temperature T corresponds to the inverse popula-tion, where the higher energy levels εk have higher populationNk than the lower energy levels.

FIG. 4 The stationary distribution of money for the requiredreserve ratio R = 0.8 (Xi, Ding, and Wang, 2005). The dis-tribution is exponential for both positive and negative moneywith different temperatures T+ and T−, as shown in the inseton log-linear scale.

By allowing the agents to go into debt up to md, we ef-fectively increase the amount of money available to eachagent by md.

Xi, Ding, and Wang (2005) considered a more realis-tic boundary condition, where a constraint is imposed onthe total debt of all agents in the system, rather than onindividual debt of each agent. This is accomplished viathe required reserve ratio R (McConnell and Brue, 1996).Banks are required by law to set aside a fraction R ofthe money deposited into bank accounts, whereas the re-maining fraction 1 − R can be lent to the agents. If theinitial amount of money in the system (the money base)is Mb, then, with repeated lending and borrowing, thetotal amount of positive money available to the agents in-creases up to M = Mb/R, where the factor 1/R is calledthe money multiplier (McConnell and Brue, 1996). Thisis how “banks create money”. This extra money comesfrom the increase in the total debt D = Mb/R −Mb ofthe agents. Now we have two constraints: on the pos-itive money M and on the maximal debt D. Applyingthe principle of maximal entropy, we find two exponen-tial distributions, for positive and negative money, char-acterized by two different temperatures, T+ = Mb/RNand T− = Mb(1−R)/RN . This result was confirmed incomputer simulations by Xi, Ding, and Wang (2005), asshown in Fig. 4.

However, in the real economy, the reserve requirementis not effective in stabilizing debt in the system, becauseit applies only to retail banks insured by FDIC, but notto investment banks. Moreover, there are alternative in-struments of debt, including derivatives, credit defaultswaps, and various other unregulated “financial innova-tions”. As a result, the total debt is not limited in prac-tice and can reach catastrophic proportions. Dragulescu

9

and Yakovenko (2000) studied a model with different in-terest rates for deposits into and loans from a bank, butwithout an explicit limit on the total debt. Computersimulations by Dragulescu and Yakovenko (2000) showthat, depending on the choice of parameters, the totalamount of money in circulation either increases or de-creases in time, but never stabilizes. The interest am-plifies the destabilizing effect of debt, because positivebalances become even more positive and negative evenmore negative. Arguably, the current financial crisis wascaused by the enormous debt accumulation, triggered bysubprime mortgages and financial derivatives based onthem. The lack of restrictions was justified by a mis-guided notion that markets will always arrive to an equi-librium, as long as government does not interfere withthem. However, an equilibrium is possible only whenthe proper boundary conditions are imposed. Debt doesnot stabilize by itself without enforcement of boundaryconditions.

Bankruptcy is a mechanism for debt stabilization. In-terest rates are meaningless without a mechanism fortriggering bankruptcy. Bankruptcy erases the debt ofan agent (the negative money) and resets the balanceto zero. However, somebody else (a bank or a lender)counted this debt as a positive asset, which also be-comes erased. In the language of physics, creation ofdebt is analogous to particle-antiparticle generation (cre-ation of positive and negative money), whereas cancella-tion of debt corresponds to particle-antiparticle annihila-tion (annihilation of positive and negative money). Theformer and the latter dominate during booms and bustsand correspond to monetary expansion and contraction.

Besides fiat money created by central governments,numerous attempts have been made to create alterna-tive community money from scratch (Kichiji and Nishibe,2007). In such a system, when an agent provides goodsor services to another agent, their accounts are creditedwith positive and negative tokens, as in Eq. (12). Be-cause the initial money base Mb = 0 is zero in this case,the probability distribution of money P (m) is symmetricwith respect to positive and negative m, i.e. “all moneyis credit”. Unless boundary conditions are imposed, themoney distribution P (m, t) would never stabilize in thissystem. Some agents would try to accumulate unlimitednegative balances by consuming goods and services andnot contributing anything in return, thus underminingthe system.

VI. EMPIRICAL DATA ON INCOME DISTRIBUTION

In this section, the theoretical prediction (9) for theprobability distribution of wages will be compared withthe empirical data on income distribution in the USA.The data from the US Census Bureau and the InternalRevenue Service (IRS) were analyzed by Dragulescu andYakovenko (2001a). One plot from this paper is shown inFig. 5. The data agree very well with the exponential dis-

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$

Prob

abilit

y

Cum

ulat

ive

prob

abilit

y

0 20 40 60 80 100 120 0.01%

0.1%

1%

10%

100%

B

A

Individual annual income, k$

Prob

abilit

y

FIG. 5 Histogram: Probability distribution of individual in-come from the US Census Bureau data for 1996 (Dragulescuand Yakovenko, 2001a). Solid line: Fit to the exponentiallaw. Inset plot A: The same with the logarithmic verticalscale. Inset plot B: Cumulative probability distribution ofindividual income from the PSID data for 1992.

tribution, and the wage temperature Tw = 20.3 k$/yearfor 1996 is obtained from the fit. Eq. (9) also has theparameter w0, which represents the minimal wage. Al-though the data deviate from the exponential distribu-tion in Fig. 5 for incomes below 5 k$/year, the proba-bility density at w = 0 is still non-zero, P (0) 6= 0. Onewould expect difficulties in collecting reliable data forvery low incomes. Given limited accuracy of the datain this range and the fact that the deviation occurs wellbelow the average income Tw, we can set w0 = 0 for thepractical purposes of fitting the data. Although there isa legal requirement for minimal hourly wage, there is nosuch requirement for annual wage, which depends on thenumber of hours worked.

The upper limit of the income data analyzed byDragulescu and Yakovenko (2001a) was about 120k$/year. Subsequently, Dragulescu and Yakovenko(2001b, 2003) analyzed income data up to 1 M$/yearand found that the upper tail of the distribution fol-lows a power law. Thus, income distribution in theUSA has a two-class structure, as shown in Fig. 6.This figure shows the cumulative distribution functionC(r) =

∫∞rP (r′) dr′, where P (r) is the probability den-

sity, and the variable r denotes income. When P (r) isan exponential or a power-law function, C(r) is also anexponential or a power-law function. The straight lineon the log-linear scale in the inset of Fig. 6 demonstratesthe exponential Boltzmann-Gibbs law for the lower class,and the straight line on the log-log scale in the main panelillustrates the Pareto power law for the upper class. Theintersection point between the exponential and power-lawfits in Fig. 6 defines the boundary between the lower andupper classes. For 1997, the annual income separatingthe two classes was about 120 k$. About 3% of the pop-ulation belonged to the upper class, and 97% belonged to

10

1 10 100 10000.1%

1%

10%

100%

Adjusted Gross Income, k$

Cum

ulat

ive

perc

ent o

f ret

urns

United States, IRS data for 1997

Pareto

Boltzmann−Gibbs

0 20 40 60 80 100

10%

100%

AGI, k$

FIG. 6 Cumulative probability distribution of tax returns forUSA in 1997 shown on log-log (main panel) and log-linear(inset) scales (Dragulescu and Yakovenko, 2003). Points rep-resent the IRS data, and solid lines are fits to the exponentialand power-law functions.

the lower class. Although the existence of social classeshas been known since Karl Marx, it is interesting thatthey can be straightforwardly identified by fitting theempirical data with simple mathematical functions. Theexponential distribution (9) applies only to the wages ofworkers in the lower class, whereas the upper tail repre-sents capital gains and other profits of the firms owners.

Silva and Yakovenko (2005) studied historical evolu-tion of income distribution in the USA during 1983–2001using the IRS data. The structure of income distribu-tion was found to be qualitatively similar for all years, asshown in Fig. 7. The average income in nominal dollarshas approximately doubled during this time interval. So,the horizontal axis in Fig. 7 shows the normalized incomer/Tr, where the income temperature Tr was obtained byfitting the exponential part of the distribution for eachyear. The values of Tr are shown in Fig. 7. The plots forthe 1980s and 1990s are shifted vertically for clarity. Weobserve that the data points for the lower class collapseon the same exponential curve for all years. This demon-strates that the relative income distribution for the lowerclass is extremely stable and does not change in time, de-spite gradual increase in the average income in nominaldollars. This observation confirms that the lower class isin a statistical “thermal” equilibrium.

On the other hand, Fig. 7 also shows that income dis-tribution of the upper class does not rescale and signifi-cantly changes in time. Banerjee and Yakovenko (2010)found that the exponent α of the power law C(r) ∝ 1/rα

has decreased from 1.8 in 1983 to 1.3 in 2007, as shownin Panel (b) of Fig. 8. This means that the upper tailhas become “fatter”. Another informative parameter isthe fraction f of the total income in the system goingto the upper class (Banerjee and Yakovenko, 2010; Silva

0.1 1 10 1000.01%

0.1%

1%

10%

100%

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

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

Cum

ulat

ive

perc

ent o

f ret

urns

Rescaled adjusted gross income

4.017 40.17 401.70 4017

0.01%

0.1%

1%

10%

100%

Adjusted gross income in 2001 dollars, k$

100%

10% Boltzmann Gibbs

Pareto

1980’s

1990’s

FIG. 7 Cumulative probability distributions of tax returnsfor 1983–2001 plotted on log-log scale versus r/Tr (the an-nual income r normalized by the average income Tr in theexponential part of the distribution) (Silva and Yakovenko,2005). The columns of numbers give the values of Tr for thecorresponding years.

and Yakovenko, 2005):

f =〈r〉 − Tr〈r〉

. (17)

Here 〈r〉 is the average income of the whole population,and the temperature Tr is the average income in the ex-ponential part of the distribution. Eq. (17) gives a well-defined measure of the deviation of the actual incomedistribution from the exponential one and, thus, of thefatness of the upper tail. Panel (c) in Fig. 8 shows his-torical evolution of the parameters 〈r〉, Tr, and f (Baner-jee and Yakovenko, 2010). We observe that Tr has beenincreasing approximately linearly in time (this increasemostly represents inflation). In contrast, 〈r〉 had sharppeaks in 2000 and 2007 coinciding with the heights ofspeculative bubbles in financial markets. The fraction fhas been increasing for the last 20 years and reached max-ima exceeding 20% in the years 2000 and 2007, followedby sharp drops. We conclude that speculative bubblesgreatly increase overall income inequality by increasingthe fraction of income going to the upper class. Whenbubbles collapse, income inequality decreases. Similar re-sults were found for Japan by Aoyama, Souma, and Fu-jiwara (2003); Fujiwara, Souma, Aoyama, Kaizoji, andAoki (2003).

Income inequality can be also characterized by theLorenz curve and the Gini coefficient (Kakwani, 1980).The Lorenz curve is defined in terms of the two coordi-nates x(r) and y(r) depending on a parameter r:

x(r) =

∫ r

0

P (r′) dr′, y(r) =

∫ r0r′P (r′) dr′∫∞

0r′P (r′) dr′

. (18)

The horizontal coordinate x(r) is the fraction of the pop-ulation with income below r, and the vertical coordi-

11

0%

5%

10%

15%

20%

25%

30%

Perc

enta

ge o

f tot

al in

com

e in

tail

Years

1985 1990 1995 2000 20050

10

20

30

40

50

60

YearsAv

erag

e in

com

e in

k$

1.4

1.6

1.8

2

0.5

0.6G

ini

(c)

(b)

(a)

housing bubble

.com bubble

r

T

f

FIG. 8 (a) The Gini coefficient G for income distribution inthe USA in 1983–2009 (connected line), compared with thetheoretical formula G = (1 + f)/2 (open circles). (b) The ex-ponent α of the power-law tail of income distribution. (c) Theaverage income 〈r〉 in the whole system, the average income Trin the lower class (the temperature of the exponential part),and the fraction of income f , Eq. (17), going to the upper tail(Banerjee and Yakovenko, 2010).

nate y(r) is the fraction of the total income this popu-lation accounts for. As r changes from 0 to ∞, x andy change from 0 to 1 and parametrically define a curvein the (x, y) plane. For for an exponential distributionP (r) = c exp(−r/Tr), the Lorenz curve is (Dragulescuand Yakovenko, 2001a)

y = x+ (1− x) ln(1− x). (19)

Fig. 9 shows the Lorenz curves for 1983 and 2000 com-puted from the IRS data (Silva and Yakovenko, 2005).The data points for 1983 (squares) agree reasonably wellwith Eq. (19) for the exponential distribution, which isshown by the upper curve. In contrast, the fraction f ofincome in the upper tail becomes so large in 2000 thatEq. (19) has to be modified as follows (Dragulescu andYakovenko, 2003; Silva and Yakovenko, 2005)

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

The last term in Eq. (20) represents the vertical jump ofthe Lorenz curve at x = 1, where a small percentage ofpopulation in the upper class accounts for a substantialfraction f of the total income. The lower curve in Fig. 9shows that Eq. (20) fits the data points for 2000 (circles)very well.

The deviation of the Lorenz curve from the straightdiagonal line in Fig. 9 is a measure of income inequality.Indeed, if everybody had the same income, the Lorenzcurve would be the diagonal line, because the fraction

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

10

20

30

40

50

60

70

80

90

100%

Cumulative percent of tax returns

Cum

ulat

ive

perc

ent o

f inc

ome

US, IRS data for 1983 and 2000

1983

2000

4%19%

1980 1985 1990 1995 2000 0

0.5

1

Year

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

FIG. 9 Main panel: Lorenz plots for income distribution in1983 and 2000 (Silva and Yakovenko, 2005). The data pointsare from the IRS, and the theoretical curves represent Eq.(20) with the parameter f deduced from Eq. (17). Inset: Theclosed circles are the IRS data for the Gini coefficient G, andthe open circles show the theoretical formula G = (1 + f)/2.

of income would be proportional to the fraction of thepopulation. The standard measure of income inequalityis the Gini coefficient 0 ≤ G ≤ 1, which is defined asthe area between the Lorenz curve and the diagonal line,divided by the area of the triangle beneath the diagonalline (Kakwani, 1980). It was shown by Dragulescu andYakovenko (2001a) that the Gini coefficient is G = 1/2for a purely exponential distribution. Historical evolu-tion of the empirical Gini coefficient is shown in the insetof Fig. 9 and in Panel (a) of Fig. 8. In the first ap-proximation, the values of G are close to the theoreticalvalue 1/2. However, if we take into account the uppertail using Eq. (20), the formula for the Gini coefficientbecomes G = (1 + f)/2 (Silva and Yakovenko, 2005).The inset in Fig. 9 and panel (a) in Fig. 8 show that thisformula gives a very good fit of the IRS data startingfrom 1995 using the values of f deduced from Eq. (17).The values G < 1/2 in the 1980s cannot be captured bythis formula, because the Lorenz data points for 1983 lieslightly above the theoretical curve in Fig. 9. We con-clude that the increase of income inequality after 1995originates completely from the growth of the upper-classincome, whereas income inequality within the lower classremains constant.

So far, we discussed the distribution of individual in-come. An interesting related question is the distributionP2(r) of family income r = r1+r2, where r1 and r2 are theincomes of spouses. If individual incomes are distributedexponentially P (r) ∝ exp(−r/Tr), then

P2(r) =

∫ r

0

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

where c is a normalization constant. Fig. 10 shows thatEq. (21) is in good agreement with the family income dis-

12

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

1%

2%

3%

4%

5%

6%

Annual family income, k$

Prob

abilit

y

United States, Bureau of Census data for 1996

FIG. 10 Histogram: Probability distribution of family in-come for families with two adults, US Census Bureau data(Dragulescu and Yakovenko, 2001a). Solid line: Fit to Eq.(21).

tribution data from the US Census Bureau (Dragulescuand Yakovenko, 2001a). It is assumed in Eq. (21) thatincomes of spouses are uncorrelated. This simple ap-proximation is indeed supported by the scatter plot ofincomes of spouses shown in Dragulescu and Yakovenko(2003). The Gini coefficient for the family income dis-tribution (21) was analytically calculated by Dragulescuand Yakovenko (2001a) as G = 3/8 = 37.5%. Fig.11 shows the Lorenz quintiles and the Gini coefficientfor 1947–1994 plotted from the US Census Bureau data(Dragulescu and Yakovenko, 2001a). The solid line, rep-resenting the Lorenz curve calculated from Eq. (21), isin good agreement with the data. The Gini coefficient,shown in the inset of Fig. 11, is close to the calculatedvalue of 37.5%. Stability of income distribution until thelate 1980s was pointed out by Levy (1987), but a verydifferent explanation of this stability was proposed.

The exponential distribution of wages (9) was derivedassuming random realizations of all possible configura-tions in the absence of specific regulations of the labormarket. As such, it can be taken as an idealized limitingcase for comparison with actual distributions in differ-ent countries. If specific measures are implemented forincome redistribution, then the distribution may deviatefrom the exponential one. An example of such a redistri-bution was studied by Banerjee, Yakovenko, and Di Mat-teo (2006) for Australia, where P (r) has a sharp peak ata certain income stipulated by government policies. Thisis in contrast to the data presented in this section, whichindicate that income distribution in the USA is close tothe idealized one for a labor market without regulation.Numerous income distribution studies for other countriesare cited in the review paper by Yakovenko and Rosser(2009).

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

10

20

30

40

50

60

70

80

90

100%

Cum

ulat

ive

perc

ent o

f fam

ily in

com

e

United States, Bureau of Census data for 1947 1994

Cumulative percent of families

1950 1960 1970 1980 19900

0.20.375

0.60.8

1

Year

Gini coefficient 38

FIG. 11 Main panel: Lorenz plot for family income, calcu-lated from Eq. (21) and compared with the US Census Bu-reau data points for 1947–1994 (Dragulescu and Yakovenko,2001a). Inset: Data points from the US Census Bureau forthe Gini coefficient for families, compared with the theoreticalvalue 3/8=37.5%.

VII. PROBABILITY DISTRIBUTION OF ENERGYCONSUMPTION

So far, we discussed how monetary inequality devel-ops for statistical reasons. Now let us discuss physicalaspects of the economy. Since the beginning of the in-dustrial revolution, rapid technological development ofhuman society has been based on consumption of fossilfuel, such as coal, oil, and gas, accumulated in the Earthfor billions of years. As a result, physical standards ofliving in modern society are primarily determined by thelevel of energy consumption per capita. Now it becomesexceedingly clear that fossil fuel will be exhausted in thenot-too-distant future. Moreover, consumption of fossilfuel releases CO2 into the atmosphere and affects globalclimate. These pressing global problems pose great tech-nological and social challenges (Rezai, Foley, and Taylor,2012).

Energy consumption per capita varies widely aroundthe globe. This heterogeneity is a challenge and a compli-cation for reaching a global consensus on how to deal withthe energy problems. Thus, it is important to understandthe origin of the global inequality in energy consumptionand characterize it quantitatively. Here I approach thisproblem using the method of maximal entropy.

Let us consider an ensemble of economic agents andcharacterize each agent j by the energy consumption εjper unit time. Notice that here εj denotes not energy,but power, which is measured in kiloWatts (kW). Simi-larly to Sec. II, let us introduce the probability densityP (ε), so that P (ε) dε gives the probability to have energyconsumption in the interval from ε to ε + dε. Energyproduction, based on extraction of fossil fuel from the

13

0 1 2 3 4 5 6 7 8 9 10 11

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Kilowatts/Person

Frac

tion

of th

e wo

rld p

opul

atio

n

World distribution of energy consumption

GBR

USA USA

IND

CHN

BRA

BRA

KEN

JPN

FRAAUS RUS

ISR

JPN

MEX

CUB

FRA

GBR

AUS

199020002005Exponentialfunction

Average 2.2 kW

FIG. 12 Cumulative distribution functions C(ε) for the en-ergy consumption per capita around the world for 1990, 2000,and 2005. The solid curve is the exponential function (22)with the parameter Tε = 〈ε〉 = 2.2 kW.

Earth, is a physically limited resource, which is dividedfor consumption among the global population. It wouldbe very improbable to divide this resource equally. Muchmore likely, this resource would be divided according tothe entropy maximization principle, subject to the globalenergy production constraint. Following the same proce-dure as in Sec. II, we arrive to the conclusion that P (ε)should follow the exponential law analogous to Eq. (5)

P (ε) = c e−ε/Tε . (22)

Here c is a normalization constant, and the temperatureTε = 〈ε〉 is the average energy consumption per capita.

Using the data from the World Resources Institute,Banerjee and Yakovenko (2010) constructed the probabil-ity distribution of energy consumption per capita aroundthe world and found that it approximately follows theexponential law, as shown in Fig. 12. The world av-erage energy consumption per capita is 〈ε〉 = 2.2 kW,compared with 10 kW in the USA and 0.6 kW in India(Banerjee and Yakovenko, 2010). However, if India andother developing countries were to adopt the same energyconsumption level per capita as in the USA, there wouldbe not enough energy resources in the world to do that.One can argue that the global energy consumption in-equality results from the constraint on energy resources,and the global monetary inequality adjusts accordinglyto implement this constraint. Since energy is purchasedwith money, a fraction of the world population ends upbeing poor, so that their energy consumption stays lim-ited.

Fig. 13 shows the Lorenz curves for the global energyconsumption per capita in 1990, 2000, and 2005 fromBanerjee and Yakovenko (2010). The black solid lineis the theoretical Lorenz curve (19) for the exponentialdistribution (22). In the first approximation, the empir-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fraction of the world population

Frac

tion

of th

e wo

rld e

nerg

y co

nsum

ptio

n

Inequality in the global energy consumption

ISR

IND

CHNBRA

MEX

GBRFRA

AUS

RUS

USA

CHN

BRA

ISR

GBR

FRARUS

IND

1990 G=0.56 2000 G=0.542005 G=0.51 Exponentialdistribution G=0.5

FIG. 13 Lorenz curves for the energy consumption per capitaaround the world in 1990, 2000, and 2005, compared with theLorenz curve (19) for the exponential distribution.

ical curves are reasonably close to the theoretical curve,but some deviations are clearly visible. On the Lorenzcurve for 1990, there is a kink or a knee indicated by thearrow, where the slope of the curve changes appreciably.This point separates developed and developing countries.Mexico, Brazil, China, and India are below this point,whereas Britain, France, Japan, Australia, Russia, andUSA are above. The slope change of the Lorenz curverepresents a gap in the energy consumption per capitabetween these two groups of countries. Thus, the phys-ical difference between developed and developing coun-tries lies in the degree of energy consumption and utiliza-tion, rather than in more ephemeral monetary measures,such as dollar income per capita. However, the Lorenzcurve for 2005 is closer to the exponential curve, and thekink is less pronounced. It means that the energy con-sumption inequality and the gap between developed anddeveloping countries have decreased, as also confirmedby the decrease in the Gini coefficient G listed in Fig. 13.This result can be attributed to the rapid globalizationand stronger mixing of the world economy in the last 20years. However, the energy consumption distribution ina well-mixed globalized world economy approaches to theexponential one, rather than to an equal distribution.

The inherent inequality of the global energy consump-tion makes it difficult for the countries at the oppositeends of the distribution to agree on consistent measuresto address the energy and climate challenges. While notoffering any immediate solutions, I would like to point outthat renewable energy has different characteristics thanfossil fuel. Because solar and wind energy is typically gen-erated and consumed locally and not transported on theglobal scale, it is not subject to the entropy-maximizingredistribution. Thus, a transition from fossil fuel to re-

14

newable energy gives a hope for achieving a more equalglobal society.

VIII. CONCLUSIONS

Statistical approach and entropy maximization arevery general and powerful methods for studying big en-sembles of various nature, be it physical, biological, eco-nomical, or social. Duncan Foley is one of the pioneers ofapplying these methods in economics (Foley, 1994, 1996,1999). The exponential distribution of wages, predictedby the statistical equilibrium theory of a labor market(Foley, 1996), is supported by empirical data on incomedistribution in the USA for the majority of population.In contrast, the upper tail of income distribution fol-lows a power law and expands dramatically during fi-nancial bubbles, which results in a significant increaseof the overall income inequality. The two-class struc-ture of the American society is apparent in the plots ofincome distribution, where the lower and upper classesare described by the exponential and power-law distri-butions. The entropy maximization method also demon-strates how a highly unequal exponential probability dis-tribution of money among the initially equal agents isgenerated as a results of stochastic monetary transac-tions between the agents. These ideas also apply tothe global inequality of energy consumption per capitaaround the world. The empirical data show convergenceto the predicted exponential distribution in the processof globalization. Global monetary inequality may be aconsequence of the constraint on global energy resources,because it limits energy consumption per capita for alarge fraction of the world population.

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