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A mean field game approach to equilibrium consumption under external habit formation

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  • Bo, Lijun
  • Wang, Shihua
  • Yu, Xiang

Abstract

This paper studies the equilibrium consumption under external habit formation in a large population of agents. We first formulate problems under two types of conventional habit formation preferences, namely linear and multiplicative external habit formation, in a mean field game framework. In a log-normal market model with the asset specialization, we characterize one mean field equilibrium in analytical form in each problem, allowing us to understand some quantitative properties of the equilibrium strategy and conclude some financial implications caused by consumption habits from a mean-field perspective. In each problem with n agents, we construct an approximate Nash equilibrium for the n-player game using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained.

Suggested Citation

  • Bo, Lijun & Wang, Shihua & Yu, Xiang, 2024. "A mean field game approach to equilibrium consumption under external habit formation," Stochastic Processes and their Applications, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:spapps:v:178:y:2024:i:c:s0304414924001674
    DOI: 10.1016/j.spa.2024.104461
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    References listed on IDEAS

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    1. Shuoqing Deng & Xun Li & Huyên Pham & Xiang Yu, 2022. "Optimal consumption with reference to past spending maximum," Finance and Stochastics, Springer, vol. 26(2), pages 217-266, April.
    2. van Bilsen, Servaas & Bovenberg, A. Lans & Laeven, Roger J. A., 2020. "Consumption and Portfolio Choice under Internal Multiplicative Habit Formation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 55(7), pages 2334-2371, November.
    3. John Y. Campbell & John Cochrane, 1999. "Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior," Journal of Political Economy, University of Chicago Press, vol. 107(2), pages 205-251, April.
    4. Holger Kraft & Claus Munk & Frank Thomas Seifried & Sebastian Wagner, 2017. "Consumption habits and humps," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(2), pages 305-330, August.
    5. Xun Li & Xiang Yu & Qinyi Zhang, 2021. "Optimal consumption with loss aversion and reference to past spending maximum," Papers 2108.02648, arXiv.org, revised Mar 2024.
    6. Jerome B. Detemple & Fernando Zapatero, 1992. "Optimal Consumption‐Portfolio Policies With Habit Formation1," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 251-274, October.
    7. Daniel Lacker & Thaleia Zariphopoulou, 2019. "Mean field and n‐agent games for optimal investment under relative performance criteria," Mathematical Finance, Wiley Blackwell, vol. 29(4), pages 1003-1038, October.
    8. Constantinides, George M, 1990. "Habit Formation: A Resolution of the Equity Premium Puzzle," Journal of Political Economy, University of Chicago Press, vol. 98(3), pages 519-543, June.
    9. Abel, Andrew B, 1990. "Asset Prices under Habit Formation and Catching Up with the Joneses," American Economic Review, American Economic Association, vol. 80(2), pages 38-42, May.
    10. Shuoqing Deng & Xun Li & Huyen Pham & Xiang Yu, 2020. "Optimal Consumption with Reference to Past Spending Maximum," Papers 2006.07223, arXiv.org, revised Mar 2022.
    11. Shuoqing Deng & Xun Li & Huyên Pham & Xiang Yu, 2022. "Optimal consumption with reference to past spending maximum," Post-Print hal-03947571, HAL.
    12. Gilles-Edouard Espinosa & Nizar Touzi, 2015. "Optimal Investment Under Relative Performance Concerns," Mathematical Finance, Wiley Blackwell, vol. 25(2), pages 221-257, April.
    13. Guanxing Fu & Chao Zhou, 2023. "Mean field portfolio games," Finance and Stochastics, Springer, vol. 27(1), pages 189-231, January.
    14. Detemple, Jerome B & Zapatero, Fernando, 1991. "Asset Prices in an Exchange Economy with Habit Formation," Econometrica, Econometric Society, vol. 59(6), pages 1633-1657, November.
    15. Mark Schroder & Costis Skiadas, 2002. "An Isomorphism Between Asset Pricing Models With and Without Linear Habit Formation," The Review of Financial Studies, Society for Financial Studies, vol. 15(4), pages 1189-1221.
    16. Xiang Yu, 2011. "Utility maximization with addictive consumption habit formation in incomplete semimartingale markets," Papers 1112.2940, arXiv.org, revised May 2015.
    17. Goncalo dos Reis & Vadim Platonov, 2020. "Forward utility and market adjustments in relative investment-consumption games of many players," Papers 2012.01235, arXiv.org, revised Mar 2022.
    18. Guanxing Fu, 2023. "Mean field portfolio games with consumption," Mathematics and Financial Economics, Springer, volume 17, number 4, January.
    19. Carroll, Christopher D., 2000. "Solving consumption models with multiplicative habits," Economics Letters, Elsevier, vol. 68(1), pages 67-77, July.
    20. Abel, Andrew B., 1999. "Risk premia and term premia in general equilibrium," Journal of Monetary Economics, Elsevier, vol. 43(1), pages 3-33, February.
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