Statistical mechanics of money, income and wealthVictor Yakovenko 1 Distributions of Money, Income, and Wealth: The Social Inequality Data Victor M. Yakovenko with A. A. Dragulescu, A. C. Silva, and A. Banerjee Department of Physics, University of Maryland, College Park, USA Publications European Physical Journal B 17, 723 (2000), cond-mat/ European Physical Journal B 20, 585 (2001), cond-mat/ Physica A 299, 213 (2001), cond-mat/ Modeling of Complex Systems: Seventh Granada Lectures, AIP Conference Proceeding 661, 180 (2003), cond-mat/ Europhysics Letters 69, 304 (2005), cond-mat/ Physica A 370, 54 (2006), physics/ Springer Encyclopedia of Complexity and System Science, arXiv:

Statistical mechanics of money, income and wealthVictor Yakovenko 2 Boltzmann-Gibbs versus Pareto distribution Ludwig Boltzmann ( ) Vilfredo Pareto ( ) Boltzmann-Gibbs probability distribution P(  )  exp(  /T), where  is energy, and T=  is temperature. Pareto probability distribution P(r)  1/r (  +1) of income r. An analogy between the distributions of energy  and money m or income r

Statistical mechanics of money, income and wealthVictor Yakovenko 3 Boltzmann-Gibbs probability distribution of energy Boltzmann-Gibbs probability distribution P(  )  exp(−  /T) of energy , where T =  is temperature. It is universal – independent of model rules, provided the model belongs to the time-reversal symmetry class. Detailed balance: w 12  1’2’ P(  1 ) P(  2 ) = w 1’2’  12 P(  1 ′) P(  2 ′) Collisions between atoms 11 22  1 ′ =  1 +   2 ′ =  2 −  Conservation of energy:  1 +  2 =  1 ′ +  2 ′ Boltzmann-Gibbs distribution maximizes entropy S = −   P(  ) lnP(  ) under the constraint of conservation law   P(  )  = const. Boltzmann-Gibbs probability distribution P(m)  exp(−m/T) of money m, where T =  m  is the money temperature. Detailed balance: w 12  1’2’ P(m 1 ) P(m 2 ) = w 1’2’  12 P(m 1 ′) P(m 2 ′) Economic transactions between agents m1m1 m2m2 m 1 ′ = m 1 +  m m 2 ′ = m 2 −  m Conservation of money: m 1 + m 2 = m 1 ′+ m 2 ′ money

Statistical mechanics of money, income and wealthVictor Yakovenko 4 Computer simulation of money redistribution The stationary distribution of money m is exponential: P(m)  e −m/T

Statistical mechanics of money, income and wealthVictor Yakovenko 5

Statistical mechanics of money, income and wealthVictor Yakovenko 6 Comparing theoretical models of money distribution with the data Where can one find and get access to the data on money distribution among individuals? These data is not routinely collected by statistical agencies... (unlike income distribution data) However, assuming that most people keep most of their data in banks, the instantaneous distribution of bank accounts balances would approximate the money distribution. There are some technical issues: individuals may have multiple accounts in multiple banks, etc. The BIG problem: Banks would not share these data with the public. It would be easy for a big commercial bank to run a computer query and generate the distribution of account balances. Most likely, banks actually do this operation routinely. However, they have no incentives to share the results with the research community, even though the confidentiality of clients would not be compromised. If anybody can help with access to the bankers, please do!

Statistical mechanics of money, income and wealthVictor Yakovenko 7 Probability distribution of individual income US Census data 1996 – histogram and points A PSID: Panel Study of Income Dynamics, 1992 (U. Michigan) – points B Distribution of income r is exponential: P(r)  e −r/T

Statistical mechanics of money, income and wealthVictor Yakovenko 8 Probability distribution of individual income IRS data 1997 – main panel and points A, 1993 – points B Cumulative distribution of income r is exponential:

Statistical mechanics of money, income and wealthVictor Yakovenko 9 Income distribution in the USA, 1997 Two-class society Upper Class Pareto power law 3% of population 16% of income Income > 120 k$: investments, capital Lower Class Boltzmann-Gibbs exponential law 97% of population 84% of income Income < 120 k$: wages, salaries “Thermal” bulk and “super-thermal” tail distribution r*r*

Statistical mechanics of money, income and wealthVictor Yakovenko 10 Income distribution in the USA, Very robust exponential law for the great majority of population No change in the shape of the distribution – only change of temperature T (income / temperature)

Statistical mechanics of money, income and wealthVictor Yakovenko 11 Income distribution in the USA, The rescaled exponential part does not change, but the power-law part changes significantly. (income / average income T)

Statistical mechanics of money, income and wealthVictor Yakovenko 12 Time evolution of the tail parameters Pareto tail changes in time non-monotonously, in line with the stock market. The tail income swelled 5- fold from 4% in 1983 to 20% in It decreased in 2001 with the crash of the U.S. stock market. The Pareto index  in C(r)  1/r  is non-universal. It changed from 1.7 in 1983 to 1.3 in 2000.

Statistical mechanics of money, income and wealthVictor Yakovenko 13 The fraction of total income in the upper tail The average income in the exponential part is T (the “temperature”). The excessive income in the upper tail, as fraction of the total income R, is where N is the total number of people

Statistical mechanics of money, income and wealthVictor Yakovenko 14 Time evolution of income temperature The nominal average income T doubled: 20 k$ k$ 2001, but it is mostly inflation.

Statistical mechanics of money, income and wealthVictor Yakovenko 15 The origin of two classes  Different sources of income: salaries and wages for the lower class, and capital gains and investments for the upper class.  Their income dynamics can be described by additive and multiplicative diffusion, correspondingly.  From the social point of view, these can be the classes of employees and employers.  Emergence of classes from the initially equal agents was recently simulated by Ian Wright "The Social Architecture of Capitalism“ Physica A 346, 589 (2005)

Statistical mechanics of money, income and wealthVictor Yakovenko 16 Diffusion model for income kinetics Suppose income changes by small amounts  r over time  t. Then P(r,t) satisfies the Fokker-Planck equation for 0<r<  : For a stationary distribution,  t P = 0 and For the lower class,  r are independent of r – additive diffusion, so A and B are constants. Then, P(r)  exp(-r/T), where T = B/A, – an exponential distribution. For the upper class,  r  r – multiplicative diffusion, so A = ar and B = br 2. Then, P(r)  1/r  +1, where  = 1+a/b, – a power-law distribution. For the upper class, income does change in percentages, as shown by Fujiwara, Souma, Aoyama, Kaizoji, and Aoki (2003) for the tax data in Japan. For the lower class, the data is not known yet.

Statistical mechanics of money, income and wealthVictor Yakovenko 17 Additive and multiplicative income diffusion If the additive and multiplicative diffusion processes are present simultaneously, then A= A 0 +ar and B= B 0 +br 2 = b(r 0 2 +r 2 ). The stationary solution of the FP equation is It interpolates between the exponential and the power-law distributions and has 3 parameters:  T = B 0 /A 0 – the temperature of the exponential part   = 1+a/b – the power-law exponent of the upper tail  r 0 – the crossover income between the lower and upper parts. Yakovenko (2007) arXiv: , Fiaschi and Marsili (2007) Web site preprint

Statistical mechanics of money, income and wealthVictor Yakovenko 18 A measure of inequality, the Gini coefficient is G = Area(diagonal line - Lorenz curve) Area(Triangle beneath diagonal) Lorenz curves and income inequality Lorenz curve (0<r<  ): With a tail, the Lorenz curve is where f is the tail income, and Gini coefficient is G=(1+f)/2. For exponential distribution, G=1/2 and the Lorenz curve is

Statistical mechanics of money, income and wealthVictor Yakovenko 19 Income distribution for two-earner families The average family income is 2T. The most probable family income is T.

Statistical mechanics of money, income and wealthVictor Yakovenko 20 No correlation in the incomes of spouses Every family is represented by two points (r 1, r 2 ) and (r 2, r 1 ). The absence of significant clustering of points (along the diagonal) indicates that the incomes r 1 and r 2 are approximately uncorrelated.

Statistical mechanics of money, income and wealthVictor Yakovenko 21 Lorenz curve and Gini coefficient for families Lorenz curve is calculated for families P 2 (r)  r exp(-r/T). The calculated Gini coefficient for families is G=3/8=37.5% Maximum entropy (the 2 nd law of thermodynamics)  equilibrium inequality: G=1/2 for individuals, G=3/8 for families. No significant changes in Gini and Lorenz for the last 50 years. The exponential (“thermal”) Boltzmann-Gibbs distribution is very stable, since it maximizes entropy.

Statistical mechanics of money, income and wealthVictor Yakovenko 22 World distribution of Gini coefficient A sharp increase of G is observed in E. Europe and former Soviet Union (FSU) after the collapse of communism – no equilibrium yet. In W. Europe and N. America, G is close to 3/8=37.5%, in agreement with our theory. Other regions have higher G, i.e. higher inequality. The data from the World Bank (B. Milanović)

Statistical mechanics of money, income and wealthVictor Yakovenko 23 Income distribution in Australia The coarse-grained PDF (probability density function) is consistent with a simple exponential fit. The fine-resolution PDF shows a sharp peak around 7.3 kAU$, probably related to a welfare threshold set by the government.

Statistical mechanics of money, income and wealthVictor Yakovenko 24 Wealth distribution in the United Kingdom For UK in 1996, T = 60 k£ Pareto index  = 1.9 Fraction of wealth in the tail f = 16%

Statistical mechanics of money, income and wealthVictor Yakovenko 25 Conclusions  The analogy in conservation laws between energy in physics and money in economics results in the exponential (“thermal”) Boltzmann-Gibbs probability distribution of money and income P(r)  exp(-r/T) for individuals and P(r)  r exp(-r/T) for two-earner families.  The tax and census data reveal a two-class structure of the income distribution in the USA: the exponential (“thermal”) law for the great majority (97-99%) of population and the Pareto (“superthermal”) power law for the top 1-3% of population.  The exponential part of the distribution is very stable and does not change in time, except for slow increase of temperature T (the average income). The Pareto tail is not universal and was increasing significantly for the last 20 years with the stock market, until its crash in  Stability of the exponential distribution is the consequence of entropy maximization. This results in the concept of equilibrium inequality in society: the Gini coefficient G=1/2 for individuals and G=3/8 for families. These numbers agree well with the data for developed capitalist countries.

Statistical mechanics of money, income and wealthVictor Yakovenko 26 Income distribution in the United Kingdom For UK in 1998, T = 12 k£ = 20 k$ Pareto index  = 2.1 For USA in 1998, T = 36 k$

Statistical mechanics of money, income and wealthVictor Yakovenko 27 Thermal machine in the world economy In general, different countries have different temperatures T, which makes possible to construct a thermal machine: High T 2, developed countries Low T 1, developing countries Money (energy) Products T 1 < T 2 Prices are commensurate with the income temperature T (the average income) in a country. Products can be manufactured in a low-temperature country at a low price T 1 and sold to a high-temperature country at a high price T 2. The temperature difference T 2 –T 1 is the profit of an intermediary. In full equilibrium, T 2 =T 1  No profit  “Thermal death” of economy Money (energy) flows from high T 2 to low T 1 (the 2 nd law of thermodynamics – entropy always increases)  Trade deficit