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Kinetic models for the trading of goods

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  • G. Toscani
  • C. Brugna
  • S. Demichelis

Abstract

In this paper we introduce kinetic equations for the evolution of the probability distribution of two goods among a huge population of agents. The leading idea is to describe the trading of these goods by means of some fundamental rules in price theory, in particular by using Cobb-Douglas utility functions for the binary exchange, and the Edgeworth box for the description of the common exchange area in which utility is increasing for both agents. This leads to a Boltzmann-type equation in which the post-interaction variables depend in a nonlinear way from the pre-interaction ones. Other models will be derived, by suitably linearizing this Boltzmann equation. In presence of uncertainty in the exchanges, it is shown that the solution to some of the linearized kinetic equations develop Pareto tails, where the Pareto index depends on the ratio between the gain and the variance of the uncertainty. In particular, the result holds true for the solution of a drift-diffusion equation of Fokker-Planck type, obtained from the linear Boltzmann equation as the limit of quasi-invariant trades.

Suggested Citation

  • G. Toscani & C. Brugna & S. Demichelis, 2012. "Kinetic models for the trading of goods," Papers 1208.6305, arXiv.org.
  • Handle: RePEc:arx:papers:1208.6305
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    References listed on IDEAS

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    4. David D. Friedman, 1990. "Price Theory: An Intermediate Text," Online economics textbooks, SUNY-Oswego, Department of Economics, number prin13.
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    6. Adrian Dragulescu & Victor M. Yakovenko, 2000. "Statistical mechanics of money," Papers cond-mat/0001432, arXiv.org, revised Aug 2000.
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    8. Gupta, Abhijit Kar, 2006. "Money exchange model and a general outlook," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 359(C), pages 634-640.
    9. Levy, Moshe & Levy, Haim & Solomon, Sorin, 1994. "A microscopic model of the stock market : Cycles, booms, and crashes," Economics Letters, Elsevier, vol. 45(1), pages 103-111, May.
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    11. Arnab Chatterjee & Bikas K. Chakrabarti & S. S. Manna, 2003. "Pareto Law in a Kinetic Model of Market with Random Saving Propensity," Papers cond-mat/0301289, arXiv.org, revised Jan 2004.
    12. Arnab Chatterjee & Bikas K. Chakrabarti & Robin B. Stinchcombe, 2005. "Master equation for a kinetic model of trading market and its analytic solution," Papers cond-mat/0501413, arXiv.org, revised Aug 2005.
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