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On Steady Distributions of Kinetic Models of Conservative Economies

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Abstract

We analyze the large-time behavior of various kinetic models for the redistribution of wealth in simple market economies introduced in the pertinent literature in recent years. As specific examples, we study models with fixed saving propensity introduced by Chakraborti and Chakrabarti (Eur. Phys. J. B 17:167–170, 2000), as well as models involving both exchange between agents and speculative trading as considered by Cordier et al. (J. Stat. Phys. 120:253–277, 2005) We derive a sufficient criterion under which a unique non-trivial stationary state exists, and provide criteria under which these steady states do or do not possess a Pareto tail. In particular, we prove the absence of Pareto tails in pointwise conservative models, like the one in (Eur. Phys. J. B 17:167–170, 2000), while models with speculative trades introduced in (J. Stat. Phys. 120:253–277, 2005) develop fat tails if the market is “risky enough”. The results are derived by a Fourier-based technique first developed for the Maxwell-Boltzmann equation (Gabetta et al. in J. Stat. Phys. 81:901–934, 1995; Bisi et al. in J. Stat. Phys. 118(1–2):301–331, 2005; Pareschi and Toscani in J. Stat. Phys. 124(2–4):747–779, 2006) and from a recursive relation which allows to calculate arbitrary moments of the stationary state.

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References

  1. Bisi, M., Carrillo, J.A., Toscani, G.: Contractive Metrics for a Boltzmann equation for granular gases: Diffusive equilibria. J. Stat. Phys. 118(1–2), 301–331 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. 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  MATH  MathSciNet  Google Scholar 

  3. Bobylev, A.V.: The theory of the spatially Uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev. C 7, 112–229 (1988)

    MathSciNet  Google Scholar 

  4. Bobylev, A.V., Carrillo, J.A., Gamba, I.: On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Stat. Phys. 98, 743–773 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bobylev, A.V., Cercignani, C.: Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions. J. Stat. Phys. 110, 333–375 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bouchaud, J.P., Mézard, M.: Wealth condensation in a simple model of economy. Physica A 282, 536–545 (2000)

    Article  ADS  Google Scholar 

  7. Carrillo, J.A., Toscani, G.: Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma 6, 75–198 (2007)

    MathSciNet  Google Scholar 

  8. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York (1994)

    MATH  Google Scholar 

  9. Chakraborti, A.: Distributions of money in models of market economy. Int. J. Mod. Phys. C 13, 1315–1321 (2002)

    Article  ADS  Google Scholar 

  10. 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 

  11. 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  ADS  MathSciNet  Google Scholar 

  12. Chakraborti, A., Germano, G., Heinsalu, E., Patriarca, M.: Relaxation in statistical many-agent economy models. Eur. Phys. J. B 57, 219–224 (2007)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Das, A., Yarlagadda, S.: Analytic treatment of a trading market model. Phys. Scr. T 106, 39–40 (2003)

    Article  ADS  Google Scholar 

  15. Desvillettes, L., Furioli, G., Terraneo, E.: Propagation of Gevrey regularity for solutions of Boltzmann equation for Maxwellian molecules. Trans. Am. Math. Soc. (in press)

  16. Di Matteo, T., Aste, T., Hyde, S.T.: In: Mallamace, F., Stanley, H.E. (eds.) The Physics of Complex Systems (New Advances and Perspectives). IOS Press, Amsterdam (2004)

    Google Scholar 

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

    Article  ADS  Google Scholar 

  18. Drǎgulescu, A., Yakovenko, V.M.: Exponential and power law probability distribution of wealth and income in the United Kingdom and the United States. Physica A 299, 213–221 (2001)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  20. Ernst, M.H., Brito, R.: High energy tails for inelastic Maxwell models. Europhys. Lett. 43, 497–502 (2002)

    Google Scholar 

  21. Ernst, M.H., Brito, R.: Scaling solutions of inelastic Boltzmann equation with over-populated high energy tails. J. Stat. Phys. 109, 407–432 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. 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  MATH  ADS  MathSciNet  Google Scholar 

  23. Gamba, I.M., Rjasanow, S., Wagner, W.: Direct simulation of the uniformly heated granular Boltzmann equation. Math. Comput. Model. 42, 683–700 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gosse, L., Toscani, G.: Identification of asymptotic decay to self-similarity for one-dimensional filtration equations. SIAM J. Numer. Anal. 43(6), 2590–2606 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. 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 

  27. Kac, M.: Probability and Related Topics in the Physical Sciences. Interscience Publishers, New York (1959)

    Google Scholar 

  28. Levy, M., Solomon, S.: New evidence for the power-law distribution of wealth. Physica A 242, 90–94 (1997)

    Article  ADS  Google Scholar 

  29. Levy, H., Levy, M., Solomon, S.: Microscopic Simulations of Financial Markets. Academic Press, New York (2000)

    Google Scholar 

  30. Malcai, O., Biham, O., Solomon, S., Richmond, P.: Theoretical analysis and simulations of the generalized Lotka–Volterra model. Phys. Rev. E 66, 031102 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  31. Mandelbrot, B.: The Pareto–Lévy law and the distribution of income. Int. Econ. Rev. 1, 79–106 (1960)

    Article  MATH  Google Scholar 

  32. Matthes, D., Toscani, G.: Analysis of a model for wealth redistribution. Preprint (2007)

  33. Mohanty, P.K.: Generic features of the wealth distribution in an ideal-gas-like market. Phys. Rev. E 74(1), 011117 (2006)

    Article  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  35. Pareto, V.: Cours d’Economie Politique. Lausanne and Paris (1897)

  36. Patriarca, M., Chakraborti, A., Kaski, K.: Statistical model with a standard Γ distribution. Phys. Rev. E 70, 016104 (2004)

    Article  ADS  Google Scholar 

  37. Pulvirenti, A., Toscani, G.: Asymptotic properties of the inelastic Kac model. J. Stat. Phys. 114, 1453–1480 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Repetowicz, P., Hutzler, S., Richmond, P.: Dynamics of money and income distributions. Physica A 356(2–4), 641–654 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  39. Rjasanow, S.: Monte-Carlo methods for the Boltzmann equation. In: Degond, P., Pareschi, L., Russo, G. (eds.) Modeling and Computational Methods for Kinetic Equations, pp. 81–115. Birkhäuser Boston, Cambridge (2004)

    Google Scholar 

  40. Sinha, S.: The Rich Are Different!: Pareto Law from asymmetric interactions in asset exchange models. In: Chatterjee, A., Chakrabarti, B.K., Yarlagadda, S. (eds.) Econophysics of Wealth Distributions, pp. 177–184. Springer, New York (2005)

    Chapter  Google Scholar 

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

    Article  ADS  Google Scholar 

  42. Solomon, S.: Stochastic Lotka–Volterra systems of competing auto-catalytic agents lead generically to truncated Pareto power wealth distribution, truncated Levy distribution of market returns, clustered volatility, booms and crashes. In: Refenes, A.P.N., Burgess, A.N., Moody, J.E. (eds.) Computational Finance 97. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universität, Staudingerweg 9, 55099, Mainz, Germany

    Daniel Matthes

  2. Department of Mathematics, University of Pavia, via Ferrata 1, 27100, Pavia, Italy

    Giuseppe Toscani

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  1. Daniel Matthes
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  2. Giuseppe Toscani
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Correspondence to Giuseppe Toscani.

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Matthes, D., Toscani, G. On Steady Distributions of Kinetic Models of Conservative Economies. J Stat Phys 130, 1087–1117 (2008). https://doi.org/10.1007/s10955-007-9462-2

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