Journal of Statistical Physics

, Volume 151, Issue 3–4, pp 549–566 | Cite as

Kinetic Models for the Trading of Goods

  • Giuseppe Toscani
  • Carlo Brugna
  • Stefano Demichelis


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.


Wealth and income distributions Kinetic models Boltzmann equation Fokker-Planck equation 



This work has been done under the activities of the National Group of Mathematical Physics (GNFM). The support of the MIUR project “Variational, functional-analytic, and optimal transport methods for dissipative evolutions and stability problems” is kindly acknowledged.


  1. 1.
    Angle, J.: The surplus theory of social stratification and the size distribution of personal wealth. Soc. Forces 65(2), 293–326 (1986) Google Scholar
  2. 2.
    Baldassarri, A., Marini Bettolo Marconi, U., Puglisi, A.: Kinetic models of inelastic gases. Math. Models Methods Appl. Sci. 12, 965–983 (2002) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bassetti, F., Toscani, G.: Explicit equilibria in a kinetic model of gambling. Phys. Rev. E 81, 066115 (2010) MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Basu, B., Chackabarti, B.K., Chackavarty, S.R., Gangopadhyay, K.: Econophysics & Economics of Games, Social Choices and Quantitative Techniques. New Economic Windows Series. Springer, Milan (2010) MATHCrossRefGoogle Scholar
  5. 5.
    Ben-Naim, E., Krapivski, P.: Multiscaling in inelastic collisions. Phys. Rev. E 61, R5–R8 (2000) ADSCrossRefGoogle Scholar
  6. 6.
    Ben-Naim, E., Krapivski, P.: Nontrivial velocity distributions in inelastic gases. J. Phys. A 35, L147–L152 (2002) MATHCrossRefGoogle Scholar
  7. 7.
    Bisi, M., Carrillo, J.A., Toscani, G.: Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model. J. Stat. Phys. 124(2–4), 625–653 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules. Sov. Sci. Rev. C 7, 111–233 (1988) MathSciNetMATHGoogle Scholar
  9. 9.
    Bobylev, A.V., Carrillo, J.A., Gamba, I.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys. 98, 743–773 (2000). Erratum on: J. Stat. Phys. 103, 1137–1138 (2001) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Brugna, C., Toscani, G.: Kinetic models for trading. A numerical approach (work in progress) Google Scholar
  11. 11.
    Carrillo, J.A., Cordier, S., Toscani, G.: Over-populated tails for conservative-in-the-mean inelastic Maxwell models. Discrete Contin. Dyn. Syst., Ser. A 24(1), 59–81 (2009) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Carrillo, J.A., Toscani, G.: Contractive probability metrics ans asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma Ser. 7 6, 75–198 (2007) MathSciNetGoogle Scholar
  13. 13.
    Cercignani, C.: The Boltzmann Equation and Its Applications. Springer Series in Applied Mathematical Sciences, vol. 67. Springer, Berlin (1988) MATHCrossRefGoogle Scholar
  14. 14.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer Series in Applied Mathematical Sciences, vol. 106. Springer, Berlin (1994) MATHGoogle Scholar
  15. 15.
    Chakraborti, A., Chakrabarti, B.K.: Statistical mechanics of money: how saving propensity affects its distributions. Eur. Phys. J. B 17, 167–170 (2000) ADSCrossRefGoogle Scholar
  16. 16.
    Chakrabarti, B.K., Chakraborti, A., Chatterjee, A.: Econophysics and Sociophysics: Trends and Perspectives. Wiley, Berlin (2006) CrossRefGoogle Scholar
  17. 17.
    Chatterjee, A., Chakrabarti, B.K.: Kinetic exchange models for income and wealth distributions. Eur. Phys. J. B 60, 135–149 (2007) ADSCrossRefGoogle Scholar
  18. 18.
    Chatterjee, A., Chakrabarti, B.K., Manna, S.S.: Pareto law in a kinetic model of market with random saving propensity. Physica A 335, 155–163 (2004) MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Chatterjee, A., Chakrabarti, B.K., Stinchcombe, R.B.: Master equation for a kinetic model of trading market and its analytic solution. Phys. Rev. E 72, 026126 (2005) ADSCrossRefGoogle Scholar
  20. 20.
    Chatterjee, A., Sudhakar, Y., Chakrabarti, B.K.: Econophysics of Wealth Distributions. New Economic Windows Series. Springer, Milan (2005) CrossRefGoogle Scholar
  21. 21.
    Cordier, S., Pareschi, L., Piatecki, C.: Mesoscopic modelling of financial markets. J. Stat. Phys. 134, 161–184 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Cordier, S., Pareschi, L., Toscani, G.: On a kinetic model for a simple market economy. J. Stat. Phys. 120, 253–277 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Drǎgulescu, A., Yakovenko, V.M.: Statistical mechanics of money. Eur. Phys. J. B 17, 723–729 (2000) ADSCrossRefGoogle Scholar
  24. 24.
    Düring, B., Matthes, D., Toscani, G.: Exponential and algebraic relaxation in kinetic models for wealth distribution. In: Manganaro, N., et al. (eds.) “WASCOM 2007”—Proceedings of the 14th Conference on Waves and Stability in Continuous Media, pp. 228–238. World Scientific, Hackensack (2008) CrossRefGoogle Scholar
  25. 25.
    Düring, B., Matthes, D., Toscani, G.: Kinetic equations modelling wealth redistribution: a comparison of approaches. Phys. Rev. E 78, 056103 (2008) MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Düring, B., Toscani, G.: Hydrodynamics from kinetic models of conservative economies. Physica A 384, 493–506 (2007) ADSCrossRefGoogle Scholar
  27. 27.
    Edgeworth, F.Y.: Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. Kegan Paul, London (1881) Google Scholar
  28. 28.
    Friedman, D.D.: Price Theory: An Intermediate Text. South-Western Publishing, Cincinnati (1990) Google Scholar
  29. 29.
    Gabetta, E., Toscani, G., Wennberg, B.: Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81, 901–934 (1995) MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Gupta, K.: Money exchange model and a general outlook. Physica A 359, 634–640 (2006) ADSCrossRefGoogle Scholar
  31. 31.
    Hayes, B.: Follow the money. Am. Sci. 90(5), 400–405 (2002) MathSciNetGoogle Scholar
  32. 32.
    Ispolatov, S., Krapivsky, P.L., Redner, S.: Wealth distributions in asset exchange models. Eur. Phys. J. B 2, 267–276 (1998) ADSCrossRefGoogle Scholar
  33. 33.
    Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263–292 (1979) MATHCrossRefGoogle Scholar
  34. 34.
    Kahneman, D., Tversky, A.: Choices, Values, and Frames. Cambridge University Press, Cambridge (2000) Google Scholar
  35. 35.
    Levy, M., Levy, H., Solomon, S.: A microscopic model of the stock market: cycles, booms and crashes. Econ. Lett. 45, 103–111 (1994) MATHCrossRefGoogle Scholar
  36. 36.
    Levy, M., Levy, H., Solomon, S.: Microscopic Simulation of Financial Markets: From Investor Behaviour to Market Phenomena. Academic Press, San Diego (2000) Google Scholar
  37. 37.
    Lux, T., Marchesi, M.: Volatility clustering in financial markets: a microscopic simulation of interacting agents. Int. J. Theor. Appl. Finance 3, 675–702 (2000) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Lux, T., Marchesi, M.: Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397(11), 498–500 (1999) ADSCrossRefGoogle Scholar
  39. 39.
    Maldarella, D., Pareschi, L.: Kinetic models for socio-economic dynamics of speculative markets. Physica A 391, 715–730 (2012) ADSCrossRefGoogle Scholar
  40. 40.
    Matthes, D., Toscani, G.: On steady distributions of kinetic models of conservative economies. J. Stat. Phys. 130, 1087–1117 (2008) MathSciNetADSMATHCrossRefGoogle Scholar
  41. 41.
    Matthes, D., Toscani, G.: Analysis of a model for wealth redistribution. Kinet. Relat. Models 1, 1–22 (2008) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Naldi, G., Pareschi, L., Toscani, G. (eds.): Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Birkhauser, Boston (2010) MATHGoogle Scholar
  43. 43.
    Pareschi, L., Russo, G.: An introduction to Monte Carlo methods for the Boltzmann equation. ESAIM Proc. 10, 35–75 (2001) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Pareschi, L., Toscani, G.: Self-similarity and power-like tails in nonconservative kinetic models. J. Stat. Phys. 124(2–4), 747–779 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  45. 45.
    Silver, J., Slud, E., Takamoto, K.: Statistical equilibrium wealth distributions in an exchange economy with stochastic preferences. J. Econ. Theory 106, 417–435 (2002) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Slanina, F.: Inelastically scattering particles and wealth distribution in an open economy. Phys. Rev. E 69, 046102 (2004) ADSCrossRefGoogle Scholar
  47. 47.
    Takayasu, H.: Application of Econophysics. Springer, Tokyo (2004) CrossRefGoogle Scholar
  48. 48.
    Takayasu, H.: Practical Fruits of Econophysics. Springer, Tokyo (2005) Google Scholar
  49. 49.
    Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4, 481–496 (2006) MathSciNetMATHGoogle Scholar
  50. 50.
    Yakovenko, V.M.: Statistical mechanics approach to econophysics. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and System Science. Springer, New York (2009) Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Giuseppe Toscani
    • 1
  • Carlo Brugna
    • 2
  • Stefano Demichelis
    • 3
  1. 1.Department of MathematicsUniversity of PaviaPaviaItaly
  2. 2.Department of PhysicsUniversity of PaviaPaviaItaly
  3. 3.Department of Economics and BusinessUniversity of PaviaPaviaItaly

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