Dynamics of Money and Income Distributions

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Dynamics of Money and Income Distributions

• Przemyslaw Repetowicz,
• Peter Richmond
• and
• Stefan Hutzler
• Department of Physics, Trinity College Dublin 2,
  Ireland
• repetowp_at_tcd.ie
• richmond_at_tcd.ie

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Summary

• Income distributions and Paretos 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

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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
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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

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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 Keplers 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 Keplers law
  rational.

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Multiplicative random processes

• Robert Gibrat
• Les inegalites economiques, Paris Librairie du
  Sirey, 1931

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Multiplicative random processes

• Gibrat
• Levy Solomon
• Int J Mod Phys C7 1996 65-72
• Solomon Richmond
• Int J Mod Phys C12 2001 1-11

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Lotka-Volterra models

• Basic idea, N agents, wealth m
• Random multiplicative wealth and wealth
  redistribution
• Mean field Smi /N?ltmgt Rescale wealth xi ?mi/ltmgt
• Equations decouple

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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

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Money in gas like marketsF Slanina Phys Rev E69
(2004) 46102-1-7
molecules ? agents Scattering ? money exchange ß
fraction exchanged e average profit
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Stationary solution
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Money in gas like markets with random exchange
Money from interaction Speculation/ competition
0 e 1
Fraction saved
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Conjecture of Patriarki, Chatterjee and
Chakrabarti
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Repetowicz, Hutzler and Richmond
cond-mat/040771 Physica A (submitted)
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Stationary solution All agents save in identical
manner ?i ?
Result only depends on one free parameter ?
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Stationary solution for random ?
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Uniform distribution of ?
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Gaussian distribution of ?
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Agents with memory Non Markovian
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4 agent 2 time distribution function
Derivative with respect to a is lt0 for alt1 and gt0
for agt1 Hence 2 solutions for a
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a1
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Continuous time random walk?
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Meerschaert
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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?

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