Statistical mechanics of money, income and wealth: foundations and applications

1. 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

2. 1 1′ = 1 +  2′ = 2 −  2 Boltzmann-Gibbs probability distribution of energy Collisions between atoms Conservation of energy: 1 + 2 = 1′ + 2′ Detailed balance in equilibrium: 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 thetime-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.

3. 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.

4. Statistical mechanics of money Transactions between agents 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. m1 m1′ = m1 + m m2′ = m2 − m m2 • 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: W(m1m2m1′m2′)P(m1)P(m2)=W(m1′m2′m1m2)P(m1′)P(m2′) • 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 moneymhas the Boltzmann-Gibbs exponential form P(m)  exp(−m/T), where T = m is the money temperature,independent of model rules.

5. Computer simulation of money redistribution Entropy increases, then saturates The stationary distribution of money m is exponential: P(m)  e−m/T

6. 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.

7. 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 Qis 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 profitGwith respect to Kand 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.

8. 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 balancemi: 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.

9. 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 frequency1/s. The stationary probability distribution is not exponential. The subsidy reduces the low-money 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.

10. 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.

11. For a stationary distribution, tP = 0 and Diffusion model for income kinetics Suppose income changes by small amounts r over time t. ThenP(r,t) satisfies the Fokker-Planck equation for 0<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.

12. Money (energy) Low T1, developing countries High T2, developed countries T1 < T2 Products Thermal machine in the world economy In general, different countries have different temperatures T, which makes possible to construct a thermal machine: 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 differenceT2–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

13. Conclusions • The analogy in conservation laws between energy in physics and money in economics results in the universal exponential (“thermal”) Boltzmann-Gibbsprobability 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 non-exponential and non-universal, depending on model details. The time-reversal 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.