Statistical mechanics of money, income and wealthVictor Yakovenko 1 Distributions of Money, Income, and Wealth: The Social Inequality Data Victor M. Yakovenko

1Statistical mechanics of money, income and wealthVictor Yakovenko

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

http://www2.physics.umd.edu/~yakovenk/econophysics.html

Publications

• European Physical Journal B 17, 723 (2000), cond-mat/0001432• European Physical Journal B 20, 585 (2001), cond-mat/0008305• Physica A 299, 213 (2001), cond-mat/0103544• Modeling of Complex Systems: Seventh Granada Lectures, AIP Conference Proceeding 661, 180 (2003), cond-mat/0211175• Europhysics Letters 69, 304 (2005), cond-mat/0406385• Physica A 370, 54 (2006), physics/0601176• Springer Encyclopedia of Complexity and System Science, arXiv:0709.3662


Boltzmann-Gibbs versus Pareto distribution

Ludwig Boltzmann (1844-1906) Vilfredo Pareto (1848-1923)

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


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:w121’2’P(1) P(2) = w1’2’12P(1′) P(2′)

Collisions between atoms

1

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:w121’2’P(m1) P(m2) = w1’2’12P(m1′) P(m2′)

Economic transactions between agents

m1

m2

m1′ = m1 + m

m2′ = m2 − m

Conservation of money:m1 + m2 = m1′+ m2′

money


Computer simulation of money redistribution

The stationarydistribution ofmoney m isexponential:P(m) e−m/T



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!


Probability distribution of individual income

US Censusdata 1996 –histogram andpoints A

PSID: Panel Study of Income Dynamics, 1992(U. Michigan) –points B

Distribution of income ris exponential:P(r) e−r/T


Probability distribution of individual income

IRS data 1997 – main panel andpoints A, 1993 –points B

Cumulativedistribution of income ris exponential:

( )

' ( ')

exp( / )r

C r

dr P r

r T


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*


Income distribution in the USA, 1983-2001

Very robust exponential law for the great majority of population

No changein the shape ofthe distribution– only changeof temperature T

(income / temperature)


Income distribution in the USA, 1983-2001

The rescaled exponential part does not change,

but the power-law part changes significantly.

(income / average income T)


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 2000.

• 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.


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

,

/1

R TN Tf

R R N

where N is the total number of people


Time evolution of income temperature

The nominal average income T doubled:20 k$ 1983 40 k$ 2001,but it is mostlyinflation.


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)


Diffusion model for income kineticsSuppose income changes by small amounts r over time t. Then P(r,t) satisfies the Fokker-Planck equation for 0<r<:

2, , .

2

rP rAP BP A B

t r r t t

For a stationary distribution, tP = 0 and .BP APr

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 = br2. 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.


Additive and multiplicative income diffusion

0

0arctan

1 / 2201 ( / )

( )

r r

T r

a b

C e

r rP r

If the additive and multiplicative diffusion processes are present simultaneously, then A= A0+ar and B= B0+br2 = b(r0

2+r2). The stationary solution of the FP equation is

It interpolates between the exponential and the power-law distributions and has 3 parameters:

T = B0/A0 – the temperature of the exponential part

= 1+a/b – the power-law exponent of the upper tail

r0 – the crossover income between the lower and upper parts.

Yakovenko (2007) arXiv:0709.3662, Fiaschi and Marsili (2007) Web site preprint


A measure of inequality,the Gini coefficient is G =Area(diagonal line - Lorenz curve)Area(Triangle beneath diagonal)

Lorenz curves and income inequalityLorenz curve (0<r<):

0( ') '

rx r P r dr

0' ( ') ' '

ry r r P r dr r

With a tail, the Lorenz curve is

where f is the tail income, andGini coefficient is G=(1+f)/2.

(1 )[ (1 ) ln(1 )]

( 1),

y f x x x

f x

For exponential distribution, G=1/2 and the Lorenz curve is

(1 ) ln(1 )y x x x


Income distribution for two-earner families

2

1 1

0

exp( /

' ",

' '

)

'r

r r r

P r P r

T

dr

P r

r r

r

The average family income is 2T. The most probable family income is T.


No correlation in the incomes of spouses

Every family is represented by two points (r1, r2) and (r2, r1).

The absence of significant clustering of points (along the diagonal) indicates that the incomes r1 and r2 are approximately uncorrelated.


Lorenz curve and Gini coefficient for familiesLorenz curve is calculated for families P2(r)r exp(-r/T).The calculated Gini coefficient for families is G=3/8=37.5%

Maximum entropy (the 2nd 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.


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)


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.


Wealth distribution in the United Kingdom

For UK in 1996,T = 60 k£

Pareto index = 1.9

Fraction of wealth in the tailf = 16%


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 2000.

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.


Income distribution in the United Kingdom

For UK in 1998,T = 12 k£= 20 k$

Pareto index = 2.1

For USAin 1998,T = 36 k$


Thermal machine in the world economyIn general, different countries have different temperatures T, which makes possible to construct a thermal machine:

High T2,developed countries

Low T1,developing countries

Money (energy)

Products

T1 < T2

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 T1 and sold to a high-temperature country at a high price T2.

The temperature difference T2–T1 is the profit of an intermediary.

In full equilibrium, T2=T1 No profit “Thermal death” of economy

Money (energy) flows from high T2 to low T1 (the 2nd law of thermodynamics – entropy always increases) Trade deficit

Documents

Statistical mechanics of money, income and wealthVictor Yakovenko 1 Distributions of Money, Income, and Wealth: The Social Inequality Data Victor M. Yakovenko