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Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon

Author

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  • Geonwoo Kim

    (School of Natural Sciences, Seoul National University of Science and Technology, Seoul 01811, Republic of Korea
    These authors contributed equally to this work.)

  • Junkee Jeon

    (Department of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea
    These authors contributed equally to this work.)

Abstract

We study an optimal consumption, leisure, and investment problem over a finite horizon in a continuous-time financial market with partial borrowing constraints. The agent derives utility from consumption and leisure, with preferences represented by a Cobb–Douglas utility function. The agent allocates time between work and leisure, earning wage income based on working hours. A key feature of our model is a partial borrowing constraint that limits the agent’s debt capacity to a fraction of the present value of their maximum future labor income. We employ the dual-martingale approach to derive the optimal consumption, leisure, and investment strategies. The problem reduces to solving a variational inequality with a free boundary, which we analyze using analytical and numerical methods. We provide an integral equation representation of the free boundary and solve it numerically via a recursive integration method. Our results highlight the impact of the borrowing constraint on the agent’s optimal decisions and the interplay between labor supply, consumption, and portfolio choice.

Suggested Citation

  • Geonwoo Kim & Junkee Jeon, 2025. "Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon," Mathematics, MDPI, vol. 13(6), pages 1-13, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:6:p:989-:d:1614552
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    References listed on IDEAS

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    1. Jing-Zhi Huang & Marti G. Subrahmanyam & G. George Yu, 1999. "Pricing And Hedging American Options: A Recursive Integration Method," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 8, pages 219-239, World Scientific Publishing Co. Pte. Ltd..
    2. Nicole El Karoui & Monique Jeanblanc-Picqué, 1998. "Optimization of consumption with labor income," Finance and Stochastics, Springer, vol. 2(4), pages 409-440.
    3. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    4. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
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