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arXiv:1008.2179v1 [q-fin.ST] 12 Aug 2010 steBlzanGbsdsrbto feeg 2] It [20]. energy of Boltzmann-Gibbs the is oterve aesctdabove. cited papers refer review please the literature, the to the of a on survey For comprehensive results world. more new the around con- present distribution also consumption energy and of debt, issues transfers. often-contentious and money servation the random on of focus models We we founda- the article, conceptual underlying this the tion In of shortcoming discussion [19]. extended This an present the foundations. by criticized conceptual was models, on mathematical not of details but intricate of analysis and [18]. scientists social and economists recognition Econo- receive the to from [17]. starting money gradually of are theory ideas search distribution Miguel the within the money the counterpart of studied Only recently no [16] virtually literature. Molico [13– has economic articles approach the subsequent popular novel in The and This [9–12] [8]. in 15]. Angle reviewed John is by sociologist progress ear- proposed the much was by and transfers [4–7] lier idea money 2000 This around random econophysicists several of agents. develop result economic can between a money as of very en- spontaneously a distribution random analogy, By of probability result collisions. un- unequal molecular a widely in as a transfers energies ergy develop of spontaneously distribution (“equal”) gas equal identical a that known in well very molecules statisti- is In it society. physics, any cal in population [3]. the among proceedings equality its in printed and [2] 1995 in Kolkata cnmcl n nnilpolm 1.Tetr was conference Stan- term the Eugene The at statistical ley the social, [1]. by to introduced problems first physics financial and statistical economical, of methods mathematical I H OTMN-IB DISTRIBUTION BOLTZMANN-GIBBS THE II. h udmna a feulbimsaitclphysics statistical equilibrium of law fundamental The cnpyisppr yial ou ncalculations on focus typically papers in- economic persistent the is problem social puzzling A cnpyisi nitricpiayfil applying field interdisciplinary an is Econophysics ncneta onain o hsapoc,o h suso distributio issues consumption fo the energy payment on the in approach, for agents results this the new for mode present between foundations these money conceptual In of on 1990s. transfers late random the as since literature econophysics ebifl eiwsaitclmdl o h rbblt dis probability the for models statistical review briefly We Mny tsags”Pn Floyd, Pink gas.” a it’s “Money, eateto hsc,Uiest fMrln,CleePar College Maryland, of University Physics, of Department .INTRODUCTION I. ttsia ehnc fmny et n nryconsumpti energy and debt, money, of mechanics Statistical FENERGY OF yaiso ope Systems Complex of Dynamics akSd fteMoon the of Side Dark Dtd 2Ags 2010) August 12 (Dated: itrM Yakovenko M. Victor in tr,wihi qa oteaeaeeeg e particle: per energy average the to T equal is which ature, gnrlyueul at.Te,tettleeg sthe is parts: energy total the the of two Then, sum into system parts. the unequal) divide us (generally Let follows. as summarized conservation and energy system the of of character statistical ents: Here h xoeta function exponential the stepouto : of product the is e rsbytmi tt ihteenergy the with state a in subsystem or tem ttsta h probability the that states ttsia ytm ihacnevdquantity. conserved a with other to systems apply statistical would (1) distribution exponential the that 5 rudta uhacnevdqatt smoney is quantity conserved Yakovenko a such Dr˘agulescu and that conserved argued economy? a [5] there the Is ap- in for target mechanics. quantity promising statistical a of is it plications so agents, participating of ucin() q 1 a eas eie ymaximizing by derived exponential also be the can entropy is (1) the equations Eq. two (1). these function of solution only xdttlenergy total fixed aacso h gnscag sfollows as change agents the of balances fprilshvn h energy the having particles of a il.Ltu osdra cnmctascinbe- transaction economic an ∆ consider us agents are Let tween They dol- print bills. agents. to e.g., other lar money, receive to “manufacture” to only money permitted give can not and agents from economic money ordinary the Indeed, m h ∼ eiaino q 1 novstetomi ingredi- main two the involves (1) Eq. of derivation A hs eiain r eygnrl ooemyexpect may one so general, very are derivations These h cnm sabgsaitclsse ihmillions with system statistical big a is economy The oteagent the to c ε rudteworld. the around n i sanraiigcntn,and constant, normalizing a is pt offiin fteodro 1. of order the of coefficient a to up , s cnmctascin r modeled are transactions economic ls, I.CNEVTO FMONEY OF CONSERVATION III. oe osrainaddb,and debt, and conservation money f rbto fmnydvlpdi the in developed money of tribution ε 2] n ftesots eiain a be can derivations shortest the of One [20]. ,Mrln 04-11 USA 20742-4111, Maryland k, S od n evcs efocus We services. and r i = and j − m m E o oegoso evcs h money the services, or goods some for ε P i j = P j = → → k hnteagent the When . ( P ε N ε = ) 1 k m k m + N ln( P i ′ j ′ e c k = ( ε = ε N ε 2 ε ffidn hsclsys- physical a finding of ) − k hra h probability the whereas , m m k P k where , ε/T /N . i ( j ε − ∆ + = ) . ftesse o a for system the of ) ∆ on m, m. P T N ( k ε stetemper- the is i 1 stenumber the is ) ε asmoney pays P sgvnby given is ( ε 2 .The ). (1) (2) m . 2

The total amount of money of the two agents before and Another potential problem with conservation of money after transaction remains the same is debt, which will be discussed in Sec. V. Most of econo- ′ ′ physics models, such as [4–6], and some economic models mi + mj = mi + mj , (3) [16, 17] do not permit debt. Thus, money balances of the agents cannot drop below zero: mi ≥ 0 for all i. Trans- i.e., there is a local conservation law for money. The action (2) takes place only when an agent has enough transfer of money (2) is analogous to the transfer of en- money to pay the : mi ≥ ∆m. An agent with ergy in molecular collisions, and Eq. (3) is analogous to m = 0 cannot buy goods from other agents, but can conservation of energy. Conservative models of this kind i receive money for delivering goods or services to them. are also studied in some economic literature [16, 17]. In a big statistical ensemble of agents, monetary trans- We should emphasize that the transfer of money from actions (2) take place between many different agents. one agent to another represents payment for goods and Although purposeful and rational for individual agents, services in a economy. However, Eq. (2) only these transactions can be treated as effectively random keeps track of money flow, but does not keep track of for the whole ensemble. This is similar to statistical what goods and are delivered. One reason for physics, where each atom follows deterministic equations this is that many goods, e.g., food and other supplies, of motion, but the whole system is effectively random. and most services, e.g., getting a haircut or going to a We are interested in the probability distribution of movie, are not tangible and disappear after consump- money P (m) among the economic agents resulting from tion. Because they are not conserved, and also because the random transfers (2). For this purpose, it is appropri- they are measured in different physical units, it is not ate to make the simplifying macroeconomic idealizations, practical to keep track of them. In contrast, money is as described above, in order to ensure overall stability of measured in the same unit (within a given country with the system and existence of statistical equilibrium. The a single ) and is conserved in local transactions concept of “equilibrium” is a very common idealization in (3), so it is straightforward to keep track of money flow. economic literature, even though the real economy might It is also important to realize that an increase in mate- never be in equilibrium. We extend this concept to a sta- rial production does not result in an automatic increase tistical equilibrium, characterized by a stationary prob- in . The agents can grow apples on trees, ability distribution P (m), in contrast to a mechanical but cannot grow money on trees. equilibrium, where the “forces” of demand and supply Enforcement of the local conservation law (3) is cru- balance each other. cial for successful functioning of money. If the agents were permitted to “manufacture” money, they would be printing money and buying all goods for nothing, which IV. PROBABILITY DISTRIBUTION OF would be a disaster. The purpose of the conservation MONEY law is to ensure that an agent can buy goods from the society only if he or she contributes something useful to Having recognized the principle of local money con- the society and receives monetary payment for these con- servation, Dr˘agulescu and Yakovenko [5] argued that the tributions. Money is an accounting device, and, indeed, distribution of money P (m) should be given by the expo- all accounting systems are based on the conservation law nential Boltzmann-Gibbs function analogous to Eq. (1) (2). The physical medium of money is not essential as − long as the conservation law is enforced. The fiat money P (m)= ce m/Tm . (4) (declared to be money by the ) may be in the form of paper currency, but more often it is represented Here c is a normalizing constant, and Tm is the “money by digits on computer accounts. So, money is just bits of temperature”, which is equal to the average amount of information, and monetary system constitutes an infor- money per agent: T = hmi = M/N, where M is the total mational layer of the economy. Monetary system inter- money, and N is the number of agents. acts with physical system (production and consumption To verify this conjecture, Dr˘agulescu and Yakovenko of material goods), but the two layers cannot be trans- [5] performed agent-based computer simulations of formed into each other because of their different nature. money transfers between agents. Initially all agents were Unlike, ordinary economic agents, a central bank or a given the same amount of money, say, $1000. Then, a central government can inject money into the economy, pair of agents (i, j) was randomly selected, the amount thus changing the total amount of money in the system. ∆m was transferred from one agent to another, and the This process is analogous to an influx of energy into a process was repeated many times. Time evolution of the system from external sources. As long as the rate of probability distribution of money P (m) is shown in com- money influx is slow compared with the relaxation rate puter animation videos [21] and [22]. After a transitory in the economy, the system remains in a quasi-stationary period, money distribution converges to the stationary statistical equilibrium with slowly changing parameters. form shown in Fig. 1. As expected, the distribution is This situation is analogous to slow heating of a kettle, well fitted by the exponential function (4). where the kettle has a well defined, but slowly increasing, In the simplest model [5], the transferred amount was temperature at any moment of time. fixed to a constant ∆m = $1. Computer animation [21] 3

N=500, M=5*105, time=4*105. contrast, plenty of data are available on income distribu- 18 tion from the tax agencies. Quantitative analysis of such 16 data for the USA [9] shows that the population consists 〈m〉, T of two distinct social classes. Income distribution follows 14 3 the exponential law for the lower class (about 97% of pop- 12 ulation) and the power law for the upper class (about 3% 2 of population). Although social classes have been known 10 since , it is interesting that they can be rec-

log P(m) 1 8 ognized by fitting the empirical data with simple math-

Probability, P(m) 0 ematical functions. The computer scientist Ian Wright 6 0 1000 2000 3000 Money, m [23, 24] has demonstrated emergence of two classes in 4 sophisticated agent-based simulations of initially equal

2 agents. This work is further developed in the book [25], integrating economics, computer science, and physics. 0 0 1000 2000 3000 4000 5000 6000 Money, m V. MODELS OF DEBT FIG. 1: Histogram and points: Stationary probability dis- tribution of money P (m) obtained in the random transfers model [5]. Solid curves: Fits to the exponential distribution Now let us discuss how the results change when debt is (4). Vertical line: The initial distribution of money. permitted. From the standpoint of individual economic agents, debt may be considered as negative money. When an agent borrows money from a bank [26], the cash bal- ance of the agent (positive money) increases, but the shows that the initial distribution of money first broad- agent also acquires a debt obligation (negative money), ens to a symmetric Gaussian curve, typical for a diffusion so the total balance (net worth) of the agent remains process. Then, the distribution starts to pile up around the same. Thus, the act of money borrowing still sat- the m = 0 state, which acts as the impenetrable bound- isfies a generalized conservation law of the total money ary, because of the condition m ≥ 0. As a result, P (m) (net worth), which is now defined as the algebraic sum becomes skewed (asymmetric) and eventually reaches the of positive (cash M) and negative (debt D) contribu- stationary exponential shape, as shown in Fig. 1. The tions: M − D = Mb, where Mb is the original amount boundary at m = 0 is analogous to the ground-state en- of money in the system, the monetary base [27]. When ergy in . Without this boundary con- an agent needs to buy a product at a price ∆m exceed- dition, the probability distribution of money would not ing his money balance mi, the agent is now permitted reach a stationary state. Computer animations [21, 22] to borrow the difference from a bank. After the trans- also show how the entropy of money distribution, defined action, the new balance of the agent becomes negative: ′ as S/N = − Pk P (mk) ln P (mk), grows from the initial mi = mi − ∆m< 0. The local conservation law (2) and value S = 0, where all agents have the same money, to (3) is still satisfied, but now it involves negative values the maximal at the statistical equilibrium. of m. Thus, the consequence of debt is not a violation of Dr˘agulescu and Yakovenko [5] studied different ad- the conservation law, but a modification of the bound- ditive rules for ∆m. Other papers studied multiplica- ary condition by permitting negative balances mi < 0, so tive rules, such as the proportional rule ∆m = γmi m = 0 is not the ground state any more. [4, 8], the propensity [6], and negotiable price [16]. If the computer simulation with ∆m = $1 is repeated These models produce Gamma-like distributions, as well without any restrictions on the debt of the agents, the as a power-law tail for a random distribution of saving probability distribution of money P (m) never stabilizes, propensities. Despite some mathematical differences, all and the system never reaches a stationary state. As time these models demonstrate spontaneous development of goes on, P (m) keeps spreading in a Gaussian manner a highly unequal probability distribution of money as a unlimitedly toward m = +∞ and m = −∞. Because of result of random money transfers between the agents. the generalized conservation law, the first moment of the Many papers use the term “wealth” instead of “money”. algebraically defined money m remains constant hmi = We believe that these terms have different meanings and Mb/N. It means that some agents become richer with should not be used interchangeably [5, 9]. positive balances m > 0 at the expense of other agents It would be very interesting to compare these theoret- going further into debt with negative balances m< 0. ical results with empirical data on money distribution. Common sense, as well as the experience with the cur- Unfortunately, it is very difficult to obtain such data. rent financial crisis, tell us that an can- The probability distribution of balances on deposit ac- not be stable if unlimited debt is permitted [28]. In this counts in a big enough bank would be a reasonable ap- case, the agents can buy goods without producing any- proximation for money distribution among the popula- thing in exchange by simply going into unlimited debt. tion. However, such data are not publicly available. In Arguably, the current financial crisis is caused the enor- 4

is not effective in stabilizing debt in the system, because

it applies only to deposits from general public, but not

40 from corporations. Moreover, there are alternative in-

) struments of debt, including derivatives and various un- -3 ) -3 10 regulated “financial innovations”. As a result, the total 30 debt is not limited in practice and can reach catastrophic

1 proportions. Here we briefly discuss several models with

20 log(P(m)) (1x10 non-stationary debt. Dr˘agulescu and Yakovenko [5] stud-

0.1 ied a model with different rates for deposits into

-50 0 50 100 150 and loans from a bank. Computer simulations show that,

10 Monetary Wealth,m depending on the choice of parameters, the total amount of money in circulation either increases or decreases in

0 Probability,P(m)(1x10 time. The interest amplifies the destabilizing effect of debt, because positive balances become even more posi- -50 0 50 100 150 tive and negative even more negative. A macroeconomic

Monetary Wealth,m model studied by the economist [30, 31] ex- hibits a debt-induced breakdown, where all economic ac- tivity stops under the burden of heavy debt and cannot FIG. 2: The stationary distribution of money [29] for the re- be restarted without a “debt moratorium”. The inter- quired reserve ratio R = 0.8. The distribution is exponential est rates were fixed in these models and not adjusted for both positive and negative money with different “temper- self-consistently. Cockshott and Cottrell [32] proposed atures” T+ and T−, as shown in the inset on log-linear scale. a mechanism, where the interest rates are set to cover probabilistic withdrawals of deposits from a bank. In an agent-based simulation, they found that money supply mous debt accumulation in the system, triggered by sub- increases first, and then the economy crashes under the prime mortgages and financial derivatives based on them. weight of accumulated debt. Detailed discussion of the current economic situation is not a subject of this paper. Going back to the idealized model of money transfers, one would need to impose a Bankruptcy is a mechanism for debt stabilization. It modified boundary condition in order to prevent unlim- erases the debt of an agent (the negative money) and re- ited growth of debt and to ensure overall stability of the sets the balance to zero. However, somebody else (a bank system. Dr˘agulescu and Yakovenko [5] considered a sim- or a lender) counted this debt as a positive asset, which ple model where the maximal debt of each agent is limited also becomes erased. In the language of physics, creation to m . In this model, P (m) again has the exponential of debt is analogous to particle-antiparticle generation d (creation of positive and negative money), whereas can- shape, but with the new boundary condition at m = −md and the higher money temperature T = m +M /N. By cellation of debt corresponds to particle-antiparticle an- d d b nihilation (annihilation of positive and negative money). allowing agents to go into debt up to md, we increase the amount of money available to each agent by m . The former and latter dominate during economic booms d and busts and represents monetary expansion and con- Xi, Ding, and Wang [29] considered a more realistic traction. Bankruptcy is the crucial mechanism for stabi- boundary condition, where a constraint is imposed on lization of money distribution, but it is often overlooked the total debt of all agents in the system. This is ac- by the economists. Interest rates are meaningless without complished via the required reserve ratio R [27]. Banks a mechanism specifying when bankruptcy is triggered. are required by law to set aside a fraction R of the money deposited into bank accounts, whereas the re- maining fraction 1 − R can be lent. If the initial amount Numerous failed attempts were made to create alter- of money in the system (the money base) is Mb, then, native community money from scratch. In such a sys- with repeated lending and borrowing, the total amount tem, when an agent provides goods or services to another of positive money available to the agents increases to agent, their accounts are credited with positive and neg- M = Mb/R, where the factor 1/R is called the money ative tokens, as in Eq. (2). However, because the initial multiplier [27]. This is how “banks create money”. This global money balance is zero in this case, the probability extra money comes from the increase in the total debt distribution of money P (m) is symmetric with respect to D = Mb/R − Mb. Given the two constraints on M and positive and negative m. Unless a boundary condition is D, we expect to find the exponential distributions of pos- imposed on the lower side, P (m) never stabilizes. Some itive and negative money characterized by two different agents accumulate unlimited negative balances by con- temperatures: T+ = Mb/RN and T− = Mb(1 − R)/RN. suming and not contributing anything This is exactly what was found in computer simulations in return, thus undermining the system. A capitalist so- [29] shown in Fig. 2. ciety imposes a lower bound on money balances, whereas However, in the real economy, the reserve requirement a socialist one may consider an upper bound [33]. 5

World distribution of energy consumption ponential distribution [9]. In the Lorenz plot for 1990, 1 one can notice a kink or a knee indicated by the ar- 0.9 IND 1990 row, where the slope of the curve changes appreciably. 2000 This point represents the boundary between developed 0.8 2005 Exponential and developing countries. Mexico, Brazil, China, and In- 0.7 KEN function dia are below this point, whereas Britain, France, Japan, Australia, Russia, and USA are above. Thus, the differ- 0.6 CUB ence between developed and developing countries lies in BRA 0.5 CHN the degree of energy consumption and utilization, rather than in the more ephemeral monetary measures, such as 0.4 BRA dollar income per capita. However, the Lorenz curve for 2005 is closer to the exponential curve, and the kink is 0.3 MEX less pronounced. It means that the energy consumption 0.2 ISR JPN Fraction of the world population Average GBR inequality and the gap between developed and develop- JPN GBR 2.2 kW FRA AUS RUS ing countries have decreased, as also confirmed by the 0.1 FRA AUS USA USA decrease in the Gini coefficient G listed in Fig. 4. We at-

0 1 2 3 4 5 6 7 8 9 10 11 tribute this result to the rapid globalization of the world Kilowatts/Person economy in the last 20 years. Ultimately, the energy con- sumption distribution in a well-mixed globalized world FIG. 3: Cumulative distribution functions of the energy con- economy is expected to be exponential and not equal. sumption per capita around the world for 1990, 2000, and The energy/ecology and financial/economic crises are 2005. The solid curve is the exponential function. the biggest challenges faced by the mankind today. There is an urgent need to find ways for a manageable and VI. PROBABILITY DISTRIBUTION OF realistic transition from the current breakneck growth- ENERGY CONSUMPTION oriented economy, powered by the ever-expanding use of fossil fuels, to a stable and sustainable society, based on renewable energy and balance with the Nature. Un- While money is the informational side of the economy, doubtedly, both money and energy will be the key factors material standards of living are controlled by the phys- shaping up the future of human civilization. ical layer. They are primarily determined by the level of energy consumption and are widely different around the globe. Using the data from the World Resources Inequality in the global energy consumption Institute, Banerjee and Yakovenko [34] found that the 1 USA probability distribution of energy consumption per capita around the world approximately follows the exponential 0.9 1990 G=0.56 2000 G=0.54 law, as shown in Fig. 3. The limited energy resources 0.8 2005 G=0.51 in the world (predominantly fossil fuels) are partitioned Exponential RUS 0.7 among the world population. As in Sec. II, maximization distribution G=0.5 FRA RUS of the entropy with the constraint results in the exponen- AUS 0.6 tial distribution of energy consumption. GBR The world average energy consumption per capita is 0.5 FRA GBR about 2.2 kW, compared with 10 kW in USA and 0.6 kW ISR 0.4 in India [34]. However, if India and other countries were ISR CHN to adopt the same energy consumption level per capita as 0.3 in USA, there would not be enough energy resources in MEX BRA 0.2 CHN the world to do that. The global energy consumption in- BRA

equality results from the constraint on energy resources. Fraction of the world energy consumption 0.1 IND Fig. 4 shows the Lorenz curves for the global energy IND 0 consumption in 1990, 2000, and 2005 [34]. The hori- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 zontal axis in Fig. 4 represents the cumulative popula- Fraction of the world population tion ranked by the energy consumption per capita, and the vertical axis represents the cumulative energy con- FIG. 4: Lorenz curves for the energy consumption per capita sumption of this population. The black solid line is the around the world in 1990, 2000, and 2005, compared with the theoretical curve y = x + (1 − x) ln(1 − x) for an ex- Lorenz curve for the exponential distribution.

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