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A Kinetic Description of Individual Wealth Growth and Control

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Abstract

The evolution problem of individual wealth growth and control is investigated by applying the kinetic theory. The microscopic variation of individual wealth growth around a universal desired target is analyzed by discussing a suitable value function, which characterizes the internal trading mechanism. Inheritance, capital gifts from others, and capital gains from rising prices are treated as external mechanisms that result in the growth of individual wealth. Under the grazing collision limit, the steady-state solution of the Fokker–Planck type equation is the product of an inverse gamma distribution and a generalized inverse gamma distribution, and exhibits a fat-tailed property. To prevent the excessive growth of individual wealth, we design additive and multiplicative controls, which reduce the possibility of excessive growth of individual wealth.

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References

  1. Albi, G., Pareschi, L., Zanella, M.: Opinion dynamics over complex networks: kinetic modelling and numerical methods. Kinet. Relat. Model 10(1), 1–32 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ajmone Marsan, G., Bellomo, N., Gibelli, L.: Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics. Math. Models Methods Appl. Sci. 26, 1051–1093 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aristov, V.V.: Biological systems as nonequilibrium structures described by kinetic methods. Results Phys. 13, 102232 (2019)

    Article  Google Scholar 

  4. Ballante, E., Bardelli, C., Zanella, M., Figini, S., Toscani, G.: Economic segregation under the action of trading uncertainties. Symmetry 12, 1390 (2020)

    Article  Google Scholar 

  5. Bellomo, N., Burini, D., Dosi, G., Gibelli, L., Knopof, D., Outada, N., Terna, P., Virgillito, M.E.: What is life? A perspective of the mathematical kinetic theory of active particles. Math. Models Methods Appl. Sci. 31(9), 1821–1866 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bisi, M., Spiga, G., Toscani, G.: Kinetic models of conservative economies with wealth redistribution. Commun. Math. Sci. 7, 901–916 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bisi, M.: Some kinetic models for a market economy. Boll. Unione Mat. Ital. 10, 143–158 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boghosian, B.M., Devitt-Lee, A., Johnson, M., Marcq, J.A., Wang, H.: Oligarchy as a phase transition: the effect of wealth-attained advantage in a Fokker–Planck description of asset exchange. Physica A 476, 15–37 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Brugna, C., Toscani, G.: Kinetic models for goods exchange in a multi-agent market. Physica A 499, 362–375 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Chakraborti, A., Chakrabarti, B.K.: Statistical mechanics of money: effects of saving propensity. 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. Cordier, S., Pareschi, L., Toscani, G.: On a kinetic model for a simple market economy. J. Stat. Phys. 120, 253–277 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Cordier, S., Pareschi, L., Piatecki, C.: Mesoscopic modelling of financial markets. J. Stat. Phys. 134(1), 161–184 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Crooks, G.E.: The Amoroso distribution. Statistics (2010). https://arxiv.org/pdf/1005.3274v1.pdf.

  15. Devitt-Lee, A., Wang, H., Li, J., Boghosian, B.: A nonstandard description of wealth concentration in large-scale economies. SIAM J. Appl. Math. 78(2), 996–1008 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dimarco, G., Toscani, G.: Kinetic modeling of alcohol consumption. J. Stat. Phys. 177, 1022–1042 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Dimarco, G., Toscani, G.: Social climbing and Amoroso distribution. Math. Models Methods Appl. Sci. 30(11), 2229–2262 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dolfin, M., Lachowicz, M.: Modeling altruism and selfishness in welfare dynamics: the role of nonlinear interactions. Math. Models Methods Appl. Sci. 24(12), 2361–2381 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dolfin, M., Lachowicz, M.: Modeling opinion dynamics: how the network enhances consensus. Netw. Heterog. Media 4, 877–896 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dolfin, M., Leonida, L., Outada, N.: Modeling human behavior in economics and social science. Phys. Life Rev. 22–23, 1–21 (2017)

    Article  Google Scholar 

  21. Düring, B., Toscani, G.: International and domestic trading and wealth distribution. Commun. Math. Sci. 6, 1043–1058 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Düring, B., Pareschi, L., Toscani, G.: Kinetic models for optimal control of wealth inequalities. Eur. Phys. J. B 91(10), 265 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  23. Furioli, G., Pulvirenti, A., Terraneo, E., Toscani, G.: Fokker–Planck equations in the modelling of socio-economic phenomena. Math. Models Methods Appl. Sci. 27, 115–158 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Furioli, G., Pulvirenti, A., Terraneo, E., Toscani, G.: Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution. Math. Models Methods Appl. Sci. 30, 685–725 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gualandi, S., Toscani, G.: Call center service times are lognormal. A Fokker–Planck description. Math. Models Methods Appl. Sci. 28, 1513–1527 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gualandi, S., Toscani, G.: Human behavior and lognormal distribution. A kinetic description. Math. Models Methods Appl. Sci. 29, 717–753 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gualandi, S., Toscani, G.: The size distribution of cities: a kinetic explanation. Physica A 524, 221–234 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  28. Islam, M.R., McGillivray, M.: Wealth inequality, governance and economic growth. Econ. Model. 88, 1–13 (2020)

    Article  Google Scholar 

  29. Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47(2), 263–292 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  31. Li, J., Boghosian, B.M., Li, C.: The affine wealth model: an agent-based model of asset exchange that allows for negative-wealth agents and its empirical validation. Physica A 516, 423–442 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  33. Pareschi, L., Toscani, G.: Interacting Multiagent Systems. Kinetic Equations & Monte Carlo Methods. Oxford University Press, Oxford (2013)

    MATH  Google Scholar 

  34. Pareschi, L., Toscani, G.: Wealth distribution and collective knowledge. A Boltzmann approach. Philos. Trans. R. Soc. A 372, 1–15 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Piketty, T.: On the long-run evolution of inheritance: France 1820–2050. Q. J. Econ. 126(3), 1071–1131 (2011)

    Article  MATH  Google Scholar 

  36. Piketty, T.: About capital in the twenty-first century. Am. Econ. Rev. 105(5), 1–15 (2015)

    Article  Google Scholar 

  37. Polk, S.L., Boghosian, B.M.: The nonuniversality of wealth distribution tails near wealth condensation criticality. SIAM J. Appl. Math. 81(4), 1717–1741 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Preziosi, L., Toscani, G., Zanella, M.: Control of tumor growth distributions through kinetic methods. J. Theor. Biol. 514(6), 110579 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tobin, J., Golub, S.: Money, Credit and Capital. McGraw-Hill/Irwin, Boston (1998)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Toscani, G., Brugna, C., Demichelis, S.: Kinetic models for the trading of goods. J. Stat. Phys. 151, 549–566 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Toscani, G.: Statistical description of human addiction phenomena. Trails Kinet. Theory 25, 209–226 (2019)

    Article  MathSciNet  Google Scholar 

  43. Trimborn, T., Pareschi, L., Frank, M.: Portfolio optimization and model predictive control: a kinetic approach. Discret. Cont. Dyn. B 24(11), 6209–6238 (2019)

    MathSciNet  MATH  Google Scholar 

  44. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143, 273–307 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics. Elsevier Press, New York (2002)

    Google Scholar 

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 11471263). The authors are very grateful to the reviewers for their helpful and valuable comments, which have led to a meaningful improvement of the paper.

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  1. Xia Zhou and Shaoyong Lai have contributed equally to this work.

Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Xia Zhou & Shaoyong Lai

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  1. Xia Zhou
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Correspondence to Xia Zhou.

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Communicated by Pierpaolo Vivo.

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Zhou, X., Lai, S. A Kinetic Description of Individual Wealth Growth and Control. J Stat Phys 188, 30 (2022). https://doi.org/10.1007/s10955-022-02961-z

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