Skip to main content
SpringerLink
Log in

The Household Behavior Modeling Based on Mean Field Games Approach

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

We present a modification of the Ramsey model that describes the consumer behavior of the households. We assume that the salary of the households is a stochastic process, defined by the stochastic differential equation (SDE). The impact of the large amount of the households can be modelled by a mean field term. This leads to a Kolmogorov–Fokker–Planck equation, evolving forward in time that describes the evolution of the probability density function of the households. Considering a Hamilton–Jacobi–Bellman equation, evolving backwards in time that describes the optimal strategy of the households behavior, we obtain a Mean Field Game problem. We present a self-similar solution of the Hamilton–Jacobi–Bellman equation and introduce the numerical solution of the Kolmogorov–Fokker–Planck equation.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

REFERENCES

  1. F. Ramsey, ‘‘A mathematical theory of savings,’’ Econ. J. 152 (38), 543–559 (1928).

    Article  Google Scholar 

  2. A. Rudeva and A. Shananin, ‘‘Control synthesis in a modified Ramsey model with a liquidity constraint,’’ J. Differ. Equat. 45, 1835–1839 (2009).

    Article  MathSciNet  Google Scholar 

  3. I. Gimaltdinov, ‘‘Research of the demand for consumer loans and money,’’ J. Math. Model. 24, 84–98 (2012).

    MathSciNet  MATH  Google Scholar 

  4. J.-M. Lasry and P.-L. Lions, ‘‘Jeux á champ moyen. I. Le cas stationnaire,’’ C. R. Math. Acad. Sci. 343, 619–625 (2006).

    Article  MathSciNet  Google Scholar 

  5. J.-M. Lasry and P.-L. Lions, ‘‘Mean field games,’’ Jpn. J. Math. 2, 229–260 (2007).

    Article  MathSciNet  Google Scholar 

  6. O. Guéant, J.-M. Lasry and P.-L. Lions, ‘‘Mean field games and applications,’’ in Paris-Princeton Lectures on Mathematical Finance 2010, Lect. Notes Math. 2003, 205–266 (2011).

    Article  Google Scholar 

  7. M. Huang, P. E. Caines and R. P. Malhamé, ‘‘The NCE (mean field) principle with locality dependent cost interactions,’’ IEEE Trans. Autom. Control 55, 2799–2805 (2010).

    Article  MathSciNet  Google Scholar 

  8. A. Shananin, M. Tarasenko, and N. Trusov, ‘‘Consumer loan demand modeling of households in Russia,’’ Comput. Math. Math. Phys. 61 (6) (2021, in press).

  9. Russia Longitudinal Monitoring Survey, RLMS-HSE (Higher School Economics, Demoscope, Carolina Population Center Univ. North Carolina, Inst. Sociol. RAS).

  10. B. Øksendal, Stochastic Differential Equations: An Introduction with Applications (Springer, Berlin, 2003).

  11. A. Lachapelle, J. Salomon, and G. Turinici, ‘‘Computation of Mean Field equilibria in economics,’’ Math. Models Methods Appl. Sci. 20, 567–588 (2010).

    Article  MathSciNet  Google Scholar 

  12. N. Trusov, ‘‘Numerical solution of Mean Field Games problems with turnpike effect,’’ Lobachevskii J. Math. 41 (4), 559–573 (2020).

    Article  MathSciNet  Google Scholar 

  13. Fed. State Stat. Service of the RF, Russian Federation Household Budget Survey (HBS), 2015. https://obdx.gks.ru. Accessed 2020.

Download references

Funding

The work has been supported by Russian Foundation for Basic Research (grant no. 20-07-00285).

Author information

Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991, Moscow, Russia

    A. A. Shananin & N. V. Trusov

  2. Moscow Institute of Physics and Technology (National Research University), 141701, Dolgoprudny, Moscow oblast, Russia

    A. A. Shananin & N. V. Trusov

  3. Federal Research Center ‘‘Computer Science and Control’’ of Russian Academy of Sciences, 119333, Moscow, Russia

    A. A. Shananin & N. V. Trusov

  4. Peoples’ Friendship University of Russia (RUDN University), 117198, Moscow, Russia

    A. A. Shananin

Authors
  1. A. A. Shananin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. V. Trusov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. A. Shananin or N. V. Trusov.

Additional information

(Submitted by A. V. Lapin)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shananin, A.A., Trusov, N.V. The Household Behavior Modeling Based on Mean Field Games Approach. Lobachevskii J Math 42, 1738–1752 (2021). https://doi.org/10.1134/S1995080221070209

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080221070209

Keywords:

Search

Navigation