Dynamics of Money and Income Distributions Przemyslaw Repetowicz

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 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 1

Summary • Income distributions and Pareto’s law • Empirical data • Agent based models – Lotka Volterra type – Analogies with molecular gas • • • 10/3/2020 Single agent collisions Multi point collisions Non Markovian models Continuous time random walks Non linear collision dynamics Econophysics of wealth distributions Kolkata March 2005 2

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: 10/3/2020 Vilfredo Pareto Econophysics of wealth distributions Kolkata March 2005 Vilfredo Pareto, 1848 -1923 3

High and Low incomes (USA 2000) WJ Reed & BD Hughed Phys Rev E 66 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 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 4

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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 theory of universal gravitation has made Kepler’s law rational. ” 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 13

Multiplicative random processes • Robert Gibrat – Les inegalites economiques, Paris Librairie du Sirey, 1931 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 14

Multiplicative random processes • Gibrat • Levy & Solomon – Int J Mod Phys C 7 1996 65 -72 • Solomon & Richmond – Int J Mod Phys C 12 2001 1 -11 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 15

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 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 16

2 1 7 5 4 5 6 3 3 6 1 4 6 3 4 1 2 5 0 7 2 1 1 5 3 2 6 7 4 3 2 4 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 17

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 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 18

Money in gas like markets F Slanina Phys Rev E 69 (2004) 46102 -1 -7 molecules → agents Scattering → money exchange β fraction exchanged ε average profit 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 19

Stationary solution 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 20

Money in gas like markets with random exchange Money from interaction Fraction saved 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 Speculation/ competition 0≤ ε ≤ 1 21

Conjecture of Patriarki, Chatterjee and Chakrabarti 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 22

Repetowicz, Hutzler and Richmond cond-mat/040771; Physica A (submitted) 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 23

Stationary solution: All agents save in identical manner λi= λ Result only depends on one free parameter: λ 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 24

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Stationary solution for random λ 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 26

Uniform distribution of λ 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 27

Gaussian distribution of λ 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 28

Agents with memory Non Markovian 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 29

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 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 30

α=1 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 31

Continuous time random walk? 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 32

Meerschaert 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 33

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? ” 10/3/2020 Econophysics of wealth distributions Kolkata March 2005 34

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