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
Victor Yakovenko
Statistical mechanics of money, income and wealth
1
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
Victor Yakovenko
Statistical mechanics of money, income and wealth
2
Boltzmann-Gibbs probability distribution of money
energy
Collisions between atoms
1
1′ = 1 + 
Conservation of energy:
1 + 2 = 1′ + 2′
Detailed balance:
w121’2’P(1) P(2) = w1’2’12P(1′)
2′ = 2 − 
2
P(2′)
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.
Boltzmann-Gibbs distribution maximizes entropy S = − P() lnP() under
the constraint of conservation law  P()  = const.
Economic transactions between agents
m1
m1′ = m1 + m
Conservation of money:
m1 + m2 = m1′+ m2′
Detailed balance:
w121’2’P(m1) P(m2) = w1’2’12P(m1′)
m2′ = m2 − m
m2
P(m2′)
Boltzmann-Gibbs probability distribution P(m)  exp(−m/T) of money m,
where T = m is the money temperature.
Victor Yakovenko
Statistical mechanics of money, income and wealth
3
Computer simulation of money redistribution
The stationary
distribution of
money m is
exponential:
P(m)  e−m/T
Victor Yakovenko
Statistical mechanics of money, income and wealth
4
Victor Yakovenko
Statistical mechanics of money, income and wealth
5
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!
Victor Yakovenko
Statistical mechanics of money, income and wealth
6
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
Victor Yakovenko
Statistical mechanics of money, income and wealth
7
Probability distribution of individual income
IRS data 1997 –
main panel and
points A, 1993 –
points B
Cumulative
distribution
of income r
is exponential:
C (r ) 


r
dr 'P (r ')
 exp( r / T )
Victor Yakovenko
Statistical mechanics of money, income and wealth
8
Income distribution in the USA, 1997
Two-class society
Upper Class
• Pareto power law
• 3% of population
• 16% of income
• Income > 120 k$:
investments, capital
r*
Lower Class
• Boltzmann-Gibbs
exponential law
• 97% of population
• 84% of income
• Income < 120 k$:
wages, salaries
“Thermal” bulk and “super-thermal” tail distribution
Victor Yakovenko
Statistical mechanics of money, income and wealth
9
Income distribution in the USA, 1983-2001
No change
in the shape of
the distribution
– only change
of temperature
T
(income / temperature)
Very robust exponential law for the great majority of population
Victor Yakovenko
Statistical mechanics of money, income and wealth
10
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)
Victor Yakovenko
Statistical mechanics of money, income and wealth
11
Time evolution of the tail parameters
The Pareto index  in
C(r)1/r is non-universal.
It changed from 1.7 in 1983
to 1.3 in 2000.
• Pareto tail changes in time
non-monotonously, in line
with the stock market.
• The tail income swelled 5fold from 4% in 1983 to 20%
in 2000.
• It decreased in 2001 with
the crash of the U.S. stock
market.
Victor Yakovenko
Statistical mechanics of money, income and wealth
12
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
f
R  TN
T
 1
,
R
R / N 
Victor Yakovenko
where N is the total number of people
Statistical mechanics of money, income and wealth
13
Time evolution of income temperature
The
nominal
average
income T
doubled:
20 k$ 1983
40 k$ 2001,
but it is
mostly
inflation.
Victor Yakovenko
Statistical mechanics of money, income and wealth
14
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)
Victor Yakovenko
Statistical mechanics of money, income and wealth
15
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<:
P  


  AP   BP   ,
t r 
r

r
A
, B
t
For a stationary distribution, tP = 0 and
 r 
2
2t
.

 BP    AP.
r
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.
Victor Yakovenko
Statistical mechanics of money, income and wealth
16
Additive and multiplicative income diffusion
If the additive and multiplicative diffusion
processes are present simultaneously, then
A= A0+ar and B= B0+br2 = b(r02+r2). The
stationary solution of the FP equation is
P( r ) 
r
r
 0 arctan  
T
 r0 
Ce
2 1 a / 2 b
1( r / r0 )


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
Victor Yakovenko
Statistical mechanics of money, income and wealth
17
Lorenz curves and income inequality
Lorenz curve (0<r<):
x  r    P(r ') dr '
r
0
y  r    r ' P(r ') dr ' r '
r
0
A measure of inequality,
the Gini coefficient is G =
Area(diagonal line - Lorenz curve)
Area(Triangle beneath diagonal)
For exponential distribution,
G=1/2 and the Lorenz curve is
y  x  (1  x ) ln(1  x )
With a tail, the Lorenz curve is
y  (1  f )[ x  (1  x) ln(1  x)]
 f ( x  1),
where f is the tail income, and
Gini coefficient is G=(1+f)/2.
Victor Yakovenko
Statistical mechanics of money, income and wealth
18
Income distribution for two-earner families
r  r ' r ", P2  r  
r
 P  r ' P  r  r 'dr '
1
1
0
 r exp(r / T )
The average family income is 2T. The most probable family income is T.
Victor Yakovenko
Statistical mechanics of money, income and wealth
19
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.
Victor Yakovenko
Statistical mechanics of money, income and wealth
20
Lorenz curve and Gini coefficient for families
Lorenz curve is calculated
for families P2(r)r exp(-r/T).
The calculated Gini
coefficient for families is
G=3/8=37.5%
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.
Maximum entropy (the 2nd
law of thermodynamics) 
equilibrium inequality:
G=1/2 for individuals,
G=3/8 for families.
Victor Yakovenko
Statistical mechanics of money, income and wealth
21
World distribution of Gini coefficient
The data from the World Bank (B. Milanović)
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.
A sharp increase of
G is observed in
E. Europe and
former Soviet
Union (FSU) after
the collapse of
communism – no
equilibrium yet.
Victor Yakovenko
Statistical mechanics of money, income and wealth
22
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$,
probably related to a welfare threshold set by the government.
Victor Yakovenko
Statistical mechanics of money, income and wealth
23
Wealth distribution in the United Kingdom
For UK
in 1996,
T = 60 k£
Pareto
index
 = 1.9
Fraction
of wealth
in the tail
f = 16%
Victor Yakovenko
Statistical mechanics of money, income and wealth
24
Conclusions
 The analogy in conservation laws between energy in physics and money
in economics results in the exponential (“thermal”) Boltzmann-Gibbs
probability distribution of money and income P(r)exp(-r/T) for individuals
and P(r)r exp(-r/T) for two-earner families.
 The tax and census data reveal a two-class structure of the income
distribution in the USA: the exponential (“thermal”) law for the great
majority (97-99%) of population and the Pareto (“superthermal”) power
law for the top 1-3% of population.
 The exponential part of the distribution is very stable and does not change
in time, except for slow increase of temperature T (the average income).
The Pareto tail is not universal and was increasing significantly for the last
20 years with the stock market, until its crash in 2000.
 Stability of the exponential distribution is the consequence of entropy
maximization. This results in the concept of equilibrium inequality in
society: the Gini coefficient G=1/2 for individuals and G=3/8 for families.
These numbers agree well with the data for developed capitalist countries.
Victor Yakovenko
Statistical mechanics of money, income and wealth
25
Income distribution in the United Kingdom
For UK
in 1998,
T = 12 k£
= 20 k$
Pareto
index
 = 2.1
For USA
in 1998,
T = 36 k$
Victor Yakovenko
Statistical mechanics of money, income and wealth
26
Thermal machine in the world economy
In general, different countries have different temperatures T,
which makes possible to construct a thermal machine:
Money (energy)
Low T1,
High T2,
developing
developed
T1 < T2
countries
countries
Products
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.
Money (energy) flows from high T2 to low T1 (the 2nd law of
thermodynamics – entropy always increases)  Trade deficit
In full equilibrium, T2=T1  No profit  “Thermal death” of economy
Victor Yakovenko
Statistical mechanics of money, income and wealth
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