Transcript The Hot Spots and Transition from d

Statistical mechanics of money, income and
wealth: foundations and applications
Victor M. Yakovenko
Adrian A. Dragulescu and A. Christian Silva
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 Proceedings 661, 180 (2003), cond-mat/0211175
• Europhysics Letters 69, 304 (2005), cond-mat/0406385
Victor Yakovenko
Statistical mechanics of money, income and wealth
1
Boltzmann-Gibbs probability distribution of energy
Collisions between atoms
Conservation of energy:
1 + 2 = 1′ + 2′
1
1′ = 1 + 
2
Detailed balance in equilibrium:
2′ = 2 −  W(121′2′)P(1)P(2)=W(1′2′12)P(1′)P(2′)
Because of time-reversal symmetry in physics, the direct and reverse
transition probabilities are equal: W(121′2′)=W(1′2′12) and
cancel.
Then, the only solution is the Boltzmann-Gibbs exponential probability
distribution P()  exp(−/T) of energy , where T =  is temperature.
The Boltzmann-Gibbs distribution is universal – independent of model
rules, provided a model belongs to the time-reversal symmetry class.
Boltzmann-Gibbs distribution can be also derived by maximizing entropy
S = − P() lnP() under the constraint of conservation law  P()  = const.
Interestingly, entropy maximization does not invoke time-reversal symmetry.
Victor Yakovenko
Statistical mechanics of money, income and wealth
2
Money, Wealth, and Income
Money is conserved in local transactions between agents. Agents can
only receive and give money, but cannot “manufacture” money. In the
sense of conservation law, money is similar to energy in physics. Central
Bank can emit money, but let us start with the case of a closed system with
no external money supply.
Wealth = Money + Other Assets (Property, Material Wealth)
• Material objects (food, consumer goods, houses) are not conserved,
they can be manufactured and consumed or destroyed. The price of
tangible assets (stocks, real estate) fluctuates and changes their value.
• Some transactions do not change wealth: money is exchanged for
tangible assets of equal value. Other transactions do change wealth:
money paid for intangible services, such as entertainment or travel.
• For the lower class with few assets, wealth  money.
For the upper class, wealth  assets (investments in stocks, real estate)
Distribution of wealth is more complicated than distribution of money.
Income is the flux of money: d(Money)/dt = Income – Spending
A kinetic theory is required.
Victor Yakovenko
Statistical mechanics of money, income and wealth
3
Statistical mechanics of money
Transactions between agents
m1
m1′ = m1 + m
m2
m2′ = m2 − m
Agent 2 pays agent 1 money m for some
product or service. We do not keep track
of various products or services, because
they are manufactured and destroyed.
We only keep track of money m.
• Conservation of money: m1 + m2 = m1′+ m2′.
• Effective randomness of transactions. Agents buy and sell products and
services rationally. But, because of enormous multitude of products and
preferences, money transactions look effectively random.
• Detailed balance:
1m2m1′m
2′)P(m
1)P(m2)=W(m
1′m2′m1m
2)P(m1′)P(m2′)
• W(m
Approximate
(not
fundamental)
time-reversal
symmetry:
W(m1,m2m1+m,m2-m)  W(m1+m,m2-mm1,m2)
For example, the price m that an agent pays for a product is independent
(not proportional) of his money balance, if he can afford the price at all.
• Then, the probability distribution of money m has the Boltzmann-Gibbs
exponential form P(m)  exp(−m/T), where T = m is the money temperature,
independent of model rules.
Victor Yakovenko
Statistical mechanics of money, income and wealth
4
Computer simulation of money redistribution
Entropy increases,
then saturates
The stationary
distribution of
money m is
exponential:
P(m)  e−m/T
Victor Yakovenko
Statistical mechanics of money, income and wealth
5
Different rules of exchange
 Exchange a small constant amount m, say $1.
 Exchange a random fraction  of the average money per agent: m=M/N.
 Exchange a random fraction  of the average money of the pair of agents:
m=(mi + mj)/2.
Selection of Winners and Losers
 For a given pair (i,j) of agents, the buyer and seller are selected randomly
every time they interact: i  j  k. Money can flow either way.
 For every pair of agents, the buyer and seller are randomly established once,
before the simulation start. In this case, money flows one way along directed
links between the agents: i  j  k.
For all of these models, computer simulations produced the same exponential
Boltzmann-Gibbs distribution, independent of the model rules – universality.
Victor Yakovenko
Statistical mechanics of money, income and wealth
6
Model with firms
To simulate economy better, we introduced firms in the model.
One agent at a time is randomly selected to be a “firm”:
 borrows capital K from a randomly selected agent and returns it with an interest rK
 hires L other randomly selected agents and pays them wages W
 makes Q items of a product and sells it to Q randomly selected agents at a price R
The net result is a many-body transaction, where
 one agent increases his money by rK
 L agents increase their money by W
 Q agents decrease their money by R
 the firm receives profit G = RQ – LW – rK
Parameters of the model are selected as in economics textbooks:
 The aggregate demand-supply curve for the product is taken to be: R(Q) = V/Q,
where Q is the quantity people would buy at a price R, and =0.5 and V=100.
 The production function of the firm has the conventional Cobb-Douglas form:
Q(L,K) = LK1- with = 0.8.
 We set W = 10. After maximizing profit G with respect to K and L, we find: L=20,
Q=10, R=32, G=68.
The stationary probability distribution of money in this model has the same
exponential Boltzmann-Gibbs form.
Victor Yakovenko
Statistical mechanics of money, income and wealth
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Models with proportionality rules
Many papers study models where transactions are proportional to the current money
balance, e.g., Ispolatov, Krapivsky, Redner, Eur. Phys. J. B 2, 267 (1998).
In this model, the buyer i pays the price proportional to his money balance mi:
m=mi.. This means Bill Gates would be charged 1000 times more for a cup of
coffee than an ordinary person.
The proportionality principle and
the time-reversal symmetry are
incompatible. Direct and reverse
transactions do return to the original
configuration:
[mi,mj]  [(1-)mi,mj+mi] 
[(1-)mi+(mj+mi),(1-)(mj+mi)]
 [mi,mj]
The stationary probability distribution
in this model is not exponential.
Various models violating time-reversal
symmetry generate different
distributions depending on model
details – no universality.
Victor Yakovenko
Statistical mechanics of money, income and wealth
8
Redistribution of money by taxation
Consider a special agent (“government”) that collects a tax on every transaction in
the system. The collected money are equally divided between all agents of the
system, so that each agent receives the subsidy m with the frequency 1/s.
The stationary probability
distribution is not exponential.
The subsidy reduces the lowmoney population. However,
even at the tax rate of 40%, the
deviation from exponential is
not very big, i.e. the policy is
not very efficient.
Here government acts as an anti-entropic “Maxwell’s demon”, pushing the distribution
away from exponential. However, it is very hard to work against entropy.
Victor Yakovenko
Statistical mechanics of money, income and wealth
9
Boltzmann equation and Fokker-Planck equation
Time evolution of probability distribution is described by the Boltzmann equation:
dP(m)/dt = m’,m [ – W(m,m’m+m,m’-m) P(m) P(m’)
+ W(m+m,m’-mm,m’) P(m+m) P(m’-m) ]
When the money transfer m is small, the integral Boltzmann equation
reduces to the differential Fokker-Planck equation. For example, for the $1
exchange model (m=1), we find
dP(m)/dt = [P(m+1) + P(m-1) – 2P(m)] + P(0) [P(m) - P(m-1)]
 d2P(m)/dm2 + P(0) dP(m)/dm
This equation has the exponential stationary solution P(m)  exp(−m/T).
It is known as the barometric distribution of density in atmosphere.
Diffusion tries to spread the distribution, whereas gravity pulls it down, so
the balance is achieved by the exponential distribution.
Victor Yakovenko
Statistical mechanics of money, income and wealth
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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
11
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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Conclusions
 The analogy in conservation laws between energy in physics and money
in economics results in the universal exponential (“thermal”) BoltzmannGibbs probability distribution of money P(m)exp(-m/T) in models with the
time-reversal symmetry, independent of model details.
 If the time-reversal symmetry is violated, then the distribution is nonexponential and non-universal, depending on model details. The timereversal symmetry is typically violated by the proportionality principle.
 Often the time-reversal symmetry is more realistic than the proportionality
symmetry. Time reversal does not mean reverse transactions between
the same agents, but reversal in the money space.
 The Fokker-Planck equation with the additive and multiplicative diffusion
processes can describe the observed two-class distribution of income.
 Difference of income and money temperatures between different countries
can be exploited to make profit in international trade and leads to a
systematic trade deficit for the higher-temperature countries.
Victor Yakovenko
Statistical mechanics of money, income and wealth
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Victor Yakovenko
Statistical mechanics of money, income and wealth
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