Skip to main content
SpringerLink
Log in

Kinetic Models for the Trading of Goods

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  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. Baldassarri, A., Marini Bettolo Marconi, U., Puglisi, A.: Kinetic models of inelastic gases. Math. Models Methods Appl. Sci. 12, 965–983 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bassetti, F., Toscani, G.: Explicit equilibria in a kinetic model of gambling. Phys. Rev. E 81, 066115 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  5. Ben-Naim, E., Krapivski, P.: Multiscaling in inelastic collisions. Phys. Rev. E 61, R5–R8 (2000)

    Article  ADS  Google Scholar 

  6. Ben-Naim, E., Krapivski, P.: Nontrivial velocity distributions in inelastic gases. J. Phys. A 35, L147–L152 (2002)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules. Sov. Sci. Rev. C 7, 111–233 (1988)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brugna, C., Toscani, G.: Kinetic models for trading. A numerical approach (work in progress)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  13. Cercignani, C.: The Boltzmann Equation and Its Applications. Springer Series in Applied Mathematical Sciences, vol. 67. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  14. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer Series in Applied Mathematical Sciences, vol. 106. Springer, Berlin (1994)

    MATH  Google Scholar 

  15. Chakraborti, A., Chakrabarti, B.K.: Statistical mechanics of money: how saving propensity affects its distributions. Eur. Phys. J. B 17, 167–170 (2000)

    Article  ADS  Google Scholar 

  16. Chakrabarti, B.K., Chakraborti, A., Chatterjee, A.: Econophysics and Sociophysics: Trends and Perspectives. Wiley, Berlin (2006)

    Book  Google Scholar 

  17. Chatterjee, A., Chakrabarti, B.K.: Kinetic exchange models for income and wealth distributions. Eur. Phys. J. B 60, 135–149 (2007)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  20. Chatterjee, A., Sudhakar, Y., Chakrabarti, B.K.: Econophysics of Wealth Distributions. New Economic Windows Series. Springer, Milan (2005)

    Book  Google Scholar 

  21. Cordier, S., Pareschi, L., Piatecki, C.: Mesoscopic modelling of financial markets. J. Stat. Phys. 134, 161–184 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Cordier, S., Pareschi, L., Toscani, G.: On a kinetic model for a simple market economy. J. Stat. Phys. 120, 253–277 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Drǎgulescu, A., Yakovenko, V.M.: Statistical mechanics of money. Eur. Phys. J. B 17, 723–729 (2000)

    Article  ADS  Google Scholar 

  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)

    Chapter  Google Scholar 

  25. Düring, B., Matthes, D., Toscani, G.: Kinetic equations modelling wealth redistribution: a comparison of approaches. Phys. Rev. E 78, 056103 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  26. Düring, B., Toscani, G.: Hydrodynamics from kinetic models of conservative economies. Physica A 384, 493–506 (2007)

    Article  ADS  Google Scholar 

  27. Edgeworth, F.Y.: Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. Kegan Paul, London (1881)

    Google Scholar 

  28. Friedman, D.D.: Price Theory: An Intermediate Text. South-Western Publishing, Cincinnati (1990)

    Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Gupta, K.: Money exchange model and a general outlook. Physica A 359, 634–640 (2006)

    Article  ADS  Google Scholar 

  31. Hayes, B.: Follow the money. Am. Sci. 90(5), 400–405 (2002)

    MathSciNet  Google Scholar 

  32. Ispolatov, S., Krapivsky, P.L., Redner, S.: Wealth distributions in asset exchange models. Eur. Phys. J. B 2, 267–276 (1998)

    Article  ADS  Google Scholar 

  33. Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263–292 (1979)

    Article  MATH  Google Scholar 

  34. Kahneman, D., Tversky, A.: Choices, Values, and Frames. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  35. Levy, M., Levy, H., Solomon, S.: A microscopic model of the stock market: cycles, booms and crashes. Econ. Lett. 45, 103–111 (1994)

    Article  MATH  Google Scholar 

  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. Lux, T., Marchesi, M.: Volatility clustering in financial markets: a microscopic simulation of interacting agents. Int. J. Theor. Appl. Finance 3, 675–702 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lux, T., Marchesi, M.: Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397(11), 498–500 (1999)

    Article  ADS  Google Scholar 

  39. Maldarella, D., Pareschi, L.: Kinetic models for socio-economic dynamics of speculative markets. Physica A 391, 715–730 (2012)

    Article  ADS  Google Scholar 

  40. Matthes, D., Toscani, G.: On steady distributions of kinetic models of conservative economies. J. Stat. Phys. 130, 1087–1117 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Matthes, D., Toscani, G.: Analysis of a model for wealth redistribution. Kinet. Relat. Models 1, 1–22 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Naldi, G., Pareschi, L., Toscani, G. (eds.): Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Birkhauser, Boston (2010)

    MATH  Google Scholar 

  43. Pareschi, L., Russo, G.: An introduction to Monte Carlo methods for the Boltzmann equation. ESAIM Proc. 10, 35–75 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. Pareschi, L., Toscani, G.: Self-similarity and power-like tails in nonconservative kinetic models. J. Stat. Phys. 124(2–4), 747–779 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Slanina, F.: Inelastically scattering particles and wealth distribution in an open economy. Phys. Rev. E 69, 046102 (2004)

    Article  ADS  Google Scholar 

  47. Takayasu, H.: Application of Econophysics. Springer, Tokyo (2004)

    Book  Google Scholar 

  48. Takayasu, H.: Practical Fruits of Econophysics. Springer, Tokyo (2005)

    Google Scholar 

  49. Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4, 481–496 (2006)

    MathSciNet  MATH  Google Scholar 

  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 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pavia, Pavia, Italy

    Giuseppe Toscani

  2. Department of Physics, University of Pavia, Pavia, Italy

    Carlo Brugna

  3. Department of Economics and Business, University of Pavia, Pavia, Italy

    Stefano Demichelis

Authors
  1. Giuseppe Toscani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlo Brugna
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefano Demichelis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Toscani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Toscani, G., Brugna, C. & Demichelis, S. Kinetic Models for the Trading of Goods. J Stat Phys 151, 549–566 (2013). https://doi.org/10.1007/s10955-012-0653-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-012-0653-0

Keywords

Search

Navigation