A mathematical theory for wealth distribution

  • Bertram Düring
  • Daniel Matthes
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


We review a qualitative mathematical theory of kinetic models for wealth distribution in simple market economies. This theory is a unified approach that covers a wide class of such models which have been proposed in the recent literature on econophysics. Based on the analysis of the underlying homogeneous Boltzmann equation, a qualitative description of the evolution of wealth in the large-time regime is obtained. In particular, the most important features of the steady wealth distribution are classified, namely the fatness of the Pareto tail and the tails’ dynamical stability. Most of the applied methods are borrowed from the kinetic theory of rarefied gases. A concise description of the moment hierarchy and suitable metrics for probability measures are employed as key tools.


Boltzmann Equation Wealth Distribution Trade Rule Total Wealth Homogeneous Boltzmann Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Bertram Düring is supported by the Deutsche Forschungsgemeinschaft, grant JU 359/6 (Forschergruppe 518). Daniel Matthes is supported by the Deutsche Forschungsgemeinschaft, grant JU 359/7.


  1. 1.
    L. Amoroso, Ricerche intorno alla curva dei redditi. Ann. Mat. Pura Appl.Ser. 4 21(1925), 123–159.Google Scholar
  2. 2.
    J. Angle, The surplus theory of social stratification and the size distribution of personal wealth. Social Forces 65(2) (1986), 293–326.Google Scholar
  3. 3.
    L. Baringhaus and R. Grübel, On a class of characterization problems for random convex combination. Ann. Inst. Statist. Math. 49(3) (1997), 555–567.Google Scholar
  4. 4.
    F. Bassetti, L. Ladelli and D. Matthes. Central limit theorem for a class of one-dimensional kinetic equations, preprint, 2009.Google Scholar
  5. 5.
    B. Basu, B.K. Chakrabarti, S.R. Chakravarty and K. Gangopadhyay, Econophysics & Economics of Games, Social Choices and Quantitative Techniques, New Economic Windows Series, Springer, Milan, 2010.Google Scholar
  6. 6.
    A.V. Bobylev, The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules. Dokl. Akad. Nauk SSSR 225(1975), 1041–1044.Google Scholar
  7. 7.
    A.V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules. Sov. Sci. Rev. C 7(1988), 111–233.Google Scholar
  8. 8.
    J.-F. Bouchaud and M. Mézard, Wealth condensation in a simple model of economy. Physica A 282(2000), 536–545.Google Scholar
  9. 9.
    Z. Burda, D. Johnston, J. Jurkiewicz, M. Kamiński, M.A. Nowak, G. Papp and I. Zahed, Wealth condensation in Pareto macroeconomies. Phys. Rev. E 65(2002), 026102.Google Scholar
  10. 10.
    J.A. Carrillo, S. Cordier, G. Toscani, Over-populated tails for conservative-in-the-mean inelastic Maxwell models. Discr. Cont. Dynamical Syst. A 24(2009), 59–81.Google Scholar
  11. 11.
    J.A. Carrillo and G. Toscani, Contractive probability metrics ans asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma 6(7) (2007) 75–198.Google Scholar
  12. 12.
    A. Chakraborti and B.K. Chakrabarti, Statistical mechanics of money: how saving propensity affects its distributions. Eur. Phys. J. B. 17(2000), 167–170.Google Scholar
  13. 13.
    B.K. Chakrabarti, A. Chakraborti and A. Chatterjee, Econophysics and Sociophysics: Trends and Perspectives, Wiley VCH, Berlin, 2006.Google Scholar
  14. 14.
    A. Chatterjee and B.K. Chakrabarti, Ideal-gas-like market models with savings: Quenched and annealed cases. Physica A 382(2007a), 36–41.Google Scholar
  15. 15.
    A. Chatterjee and B.K. Chakrabarti, Kinetic exchange models for income and wealth distributions. Eur. Phys. J. B 60(2007b), 135–149.Google Scholar
  16. 16.
    A. Chatterjee, B.K. Chakrabarti and S.S. Manna, Pareto law in a kinetic model of market with random saving propensity. Physica A 335(2004), 155–163.Google Scholar
  17. 17.
    A. Chatterjee, B.K. Chakrabarti and R.B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution. Phys. Rev. E 72(2005), 026126.Google Scholar
  18. 18.
    A. Chatterjee, S. Yarlagadda and B.K. Chakrabarti, Econophysics of Wealth DistributionsNew Economic Windows Series, Springer, Milan, 2005.Google Scholar
  19. 19.
    A. Chakraborti, Distribution of money in model markets of economy. Int. J. Mod. Phys. C 13(10) (2002), 1315–1321.Google Scholar
  20. 20.
    V. Comincioli, L. Della Croce and G. Toscani, A Boltzmann-like equation for choice formation. Kinetic and related Models 2(2009), 135–149.Google Scholar
  21. 21.
    S. Cordier, L. Pareschi and C. Piatecki, Mesoscopic modelling of financial markets. J. Stat. Phys. 134(1) (2009), 161–184.Google Scholar
  22. 22.
    S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy. J. Stat. Phys. 120(2005), 253–277.Google Scholar
  23. 23.
    R. D’Addario, Intorno alla curva dei redditi di Amoroso. Riv. Italiana Statist. Econ. Finanza, 4(1) (1932).Google Scholar
  24. 24.
    A. Drǎgulescu and V.M. Yakovenko, Statistical mechanics of money. Eur. Phys. Jour. B 17(2000), 723–729.Google Scholar
  25. 25.
    B. Düring, P.A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. Lond. A 465(2009), 3687–3708.Google Scholar
  26. 26.
    B. Düring and G. Toscani, Hydrodynamics from kinetic models of conservative economies. Physica A 384(2007), 493–506.Google Scholar
  27. 27.
    B. Düring and G. Toscani, International and domestic trading and wealth distribution. Comm. Math. Sci. 6(4) (2008), 1043–1058.Google Scholar
  28. 28.
    B. Düring, D. Matthes and G. Toscani, Exponential and algebraic relaxation in kinetic models for wealth distribution. In: “WASCOM 2007” - Proceedings of the 14th Conference on Waves and Stability in Continuous Media, N. Manganaro et al. (eds.), pp. 228–238, World Sci. Publ., Hackensack, NJ, 2008.Google Scholar
  29. 29.
    B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: a comparison of approaches. Phys. Rev. E 78(5) (2008), 056103.Google Scholar
  30. 30.
    B. Düring, D. Matthes and G. Toscani, A Boltzmann-type approach to the formation of wealth distribution curves. Riv. Mat. Univ. Parma 1(8) (2009), 199–261.Google Scholar
  31. 31.
    E. Gabetta, P.A. Markowich and A. Unterreiter, A note on the entropy production of the radiative transfer equation. Appl. Math. Lett. 12(4) (1999), 111–116.Google Scholar
  32. 32.
    E. Gabetta, G. Toscani and B. Wennberg, The Tanaka functional and exponential convergence for non-cut-off molecules. Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994). Transport Theory Stat. Phys. 25(3–5) (1996), 543–554.Google Scholar
  33. 33.
    U. Garibaldi, E. Scalas and P. Viarengo, Statistical equilibrium in simple exchange games II. The redistribution game. Eur. Phys. J. B 60(2) (2007), 241–246.Google Scholar
  34. 34.
    K. Gupta, Money exchange model and a general outlook. Physica A 359(2006), 634–640.Google Scholar
  35. 35.
    S. Ispolatov, P.L. Krapivsky and S. Redner, Wealth distributions in asset exchange models. Eur. Phys. J. B 2(1998), 267–276.Google Scholar
  36. 36.
    B. Mandelbrot, The Pareto-Lacute{e}vy law and the distribution of income. Int. Econ. Rev. 1(1960), 79–106.Google Scholar
  37. 37.
    D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies. J. Stat. Phys. 130(2008), 1087–1117.Google Scholar
  38. 38.
    D. Matthes and G. Toscani, Analysis of a model for wealth redistribution. Kinetic Rel. Mod. 1(2008), 1–22.Google Scholar
  39. 39.
    D. Matthes and G. Toscani. Propagation of Sobolev regularity for a class of random kinetic models on the real line, preprint, 2009.Google Scholar
  40. 40.
    P.K. Mohanty, Generic features of the wealth distribution in an ideal-gas-like market. Phys. Rev. E 74(1) (2006), 011117.Google Scholar
  41. 41.
    M. Patriarca, A. Chakraborti, E. Heinsalu and G. Germano, Relaxation in statistical many-agent economy models. Eur. Phys. J. B 57(2007), 219–224.Google Scholar
  42. 42.
    M. Patriarca, A. Chakraborti and K. Kaski, A statistical model with a standard Gamma distribution. Phys. Rev. E 70(2004), 016104.Google Scholar
  43. 43.
    L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models. J. Stat. Phys. 124(2–4) (2006), 747–779.Google Scholar
  44. 44.
    V. Pareto, Cours d’Économie Politique, Lausanne and Paris, 1897.Google Scholar
  45. 45.
    M. Patriarca, A. Chakraborti and G. Germano, Influence of saving propensity on the power-law tail of the wealth distribution. Physica A 369(2006), 723–736.Google Scholar
  46. 46.
    P. Repetowicz, S. Hutzler and P. Richmond, Dynamics of money and income distributions. Physica A 356(2005), 641–654.Google Scholar
  47. 47.
    H. Takayasu, Application of Econophysics, Springer, Tokyo, 2004.Google Scholar
  48. 48.
    H. Takayasu, Practical fruits of econophysics, Springer, Tokyo, 2005.Google Scholar
  49. 49.
    G. Toscani, Kinetic models of opinion formation. Commun. Math. Sci. 4(2006), 481–496.Google Scholar
  50. 50.
    G. Toscani, Wealth redistribution in conservative linear kinetic models with taxation. Europhys. Lett. 88(2009), 10007.Google Scholar
  51. 51.
    C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Mathematical Society, Providence, 2003.Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institut für Analysis und Scientific ComputingTechnische Universität WienWienAustria

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