Dynamics of Money and Income Distributions

1. Dynamics of Money and Income Distributions Przemyslaw Repetowicz, Peter Richmond and Stefan Hutzler Department of Physics, Trinity College Dublin 2, Ireland repetowp@tcd.ie richmond@tcd.ie Econophysics of wealth distributions Kolkata March 2005

2. Summary • Income distributions and Pareto’s law • Empirical data • Agent based models • Lotka Volterra type • Analogies with molecular gas • Single agent collisions • Multi point collisions • Non Markovian models • Continuous time random walks • Non linear collision dynamics Econophysics of wealth distributions Kolkata March 2005

3. Vilfredo Pareto Vilfredo Pareto born Paris 1848 to Italian aristocratic family.  Following father, studied classics, then engineering at Polytechnic Institute, Turin. Here he acquired proficiency in mathematics and basic ideas about mechanical equilibrium that characterized his contributions to economics. Graduated top of class in 1870. Took job as director of Rome Railway Company.  In 1874, became managing director of iron and steel concern, Società Ferriere d'Italia in Florence. Appointed to Chair of Economics, University of Lausanne in 1894. In Cours, proposed law of income distribution - in all countries and times: Vilfredo Pareto, 1848-1923 Econophysics of wealth distributions Kolkata March 2005

4. High and Low incomes(USA 2000)WJ Reed & BD Hughed Phys Rev E66 2002 067103 • “Law only applies to incomes a little above the minimum. The form of the curve in the immediate neighbourhood of this minimum income is still undetermined, for statistics do not furnish us with sufficient information for its determination.” • New Theories of Economics, J Pol Econ 5 (1897) 485-502 Econophysics of wealth distributions Kolkata March 2005

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13. The New Theories of Economics, J Pol Econ 5 (1897) 485-502 • “This law of the distribution of wealth which has so lately been discovered may some day be of use in the study of different races of men. In this respect…..it can be compared to Kepler’s law in astronomy; we still lack a theory that may make this law rational in the way in which the theory of universal gravitation has made Kepler’s law rational.” Econophysics of wealth distributions Kolkata March 2005

14. Multiplicative random processes • Robert Gibrat • Les inegalites economiques, Paris Librairie du Sirey, 1931 Econophysics of wealth distributions Kolkata March 2005

15. Multiplicative random processes • Gibrat • Levy & Solomon • Int J Mod Phys C7 1996 65-72 • Solomon & Richmond • Int J Mod Phys C12 2001 1-11 Econophysics of wealth distributions Kolkata March 2005

16. Lotka-Volterra models • Basic idea, N agents, wealth m • Random multiplicative wealth and wealth redistribution • Mean field Σmi /N→<m> Rescale wealth xi→mi/<m> • Equations decouple Econophysics of wealth distributions Kolkata March 2005

17. 2 7 1 7 5 6 5 6 1 1 4 3 2 4 3 2 0 5 6 1 6 7 1 3 2 4 2 3 4 5 3 4 Econophysics of wealth distributions Kolkata March 2005

18. B Mandelbrot • “There is a great temptation to consider the exchanges of money which occur in economic interaction as analogous to the exchanges of energy which occur in physics shocks between molecules” • The Pareto Levy law and the distribution of income, International Economic review 1 (1960) 79-106 Econophysics of wealth distributions Kolkata March 2005

19. Money in gas like marketsF Slanina Phys Rev E69 (2004) 46102-1-7 molecules → agents Scattering → money exchange β fraction exchanged ε average profit Econophysics of wealth distributions Kolkata March 2005

20. Stationary solution Econophysics of wealth distributions Kolkata March 2005

21. Money in gas like markets with random exchange Money from interaction Speculation/ competition 0≤ ε≤1 Fraction saved Econophysics of wealth distributions Kolkata March 2005

22. Conjecture of Patriarki, Chatterjee and Chakrabarti Econophysics of wealth distributions Kolkata March 2005

23. Repetowicz, Hutzler and Richmond cond-mat/040771; Physica A (submitted) Econophysics of wealth distributions Kolkata March 2005

24. Stationary solution: All agents save in identical manner λi= λ Result only depends on one free parameter: λ Econophysics of wealth distributions Kolkata March 2005

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26. Stationary solution for random λ Econophysics of wealth distributions Kolkata March 2005

27. Uniform distribution of λ Econophysics of wealth distributions Kolkata March 2005

28. Gaussian distribution of λ Econophysics of wealth distributions Kolkata March 2005

29. Agents with memory Non Markovian Econophysics of wealth distributions Kolkata March 2005

30. 4 agent 2 time distribution function Derivative with respect to α is <0 for α<1 and >0 for α>1 Hence 2 solutions for α Econophysics of wealth distributions Kolkata March 2005

31. α=1 Econophysics of wealth distributions Kolkata March 2005

32. Continuous time random walk? Econophysics of wealth distributions Kolkata March 2005

33. Meerschaert Econophysics of wealth distributions Kolkata March 2005

34. The New Theories of Economics, J Pol Econ 5 (1897) 485-502 • “The laws of the distribution of wealth evidently depend on the nature of man and on the economic organisation of society. We might derive these laws by deductive reasoning, taking as a starting point the data of the nature of man and of the economic organisation of society. Will this work sometime be completed?” Econophysics of wealth distributions Kolkata March 2005

35. END Econophysics of wealth distributions Kolkata March 2005