In a closed economic system, money is conserved. Thus, by analogy with
energy, the equilibrium probability distribution of money must follow the
exponential Gibbs law characterized by an effective temperature equal to the
average amount of money per economic agent. We demonstrate how the Gibbs
distribution emerges in computer simulations of economic models. Then we
consider a thermal machine, in which the difference of temperatures allows one
to extract a monetary profit. We also discuss the role of debt, and models with
broken time-reversal symmetry for which the Gibbs law does not hold.Comment: 7 pages, 5 figures, RevTeX. V.4: final version accepted to Eur. Phys.
J. B: few stylistic revisions and additional reference
We present the data on wealth and income distributions in the United Kingdom, as well as on the income distributions in the individual states of the USA. In all of these data, we ÿnd that the great majority of population is described by an exponential distribution, whereas the high-end tail follows a power law. The distributions are characterized by a dimensional scale analogous to temperature. The values of temperature are determined for the UK and the USA, as well as for the individual states of the USA.
Using tax and census data, we demonstrate that the distribution of individual
income in the USA is exponential. Our calculated Lorenz curve without fitting
parameters and Gini coefficient 1/2 agree well with the data. From the
individual income distribution, we derive the distribution function of income
for families with two earners and show that it also agrees well with the data.
The family data for the period 1947-1994 fit the Lorenz curve and Gini
coefficient 3/8=0.375 calculated for two-earners families.Comment: 4 pages, including 5 figures. Uses Springer Verlag style classes for
Eur. Phys. J. B (included). Submitted to the proceedings of APFA2 conference.
V.2: minor stylistic improvement
We study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker-Planck equation exactly and, after integrating out the variance, find an analytic formula for the timedependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow-Jones index for the time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log-returns with a time-dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow-Jones data for 1982-2001 follow the scaling function for seven orders of magnitude.
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