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

Geonwoo Kim and Junkee Jeon ()
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Geonwoo Kim: School of Natural Sciences, Seoul National University of Science and Technology, Seoul 01811, Republic of Korea
Junkee Jeon: Department of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea

Mathematics, 2025, vol. 13, issue 6, 1-13

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.

Keywords: optimal consumption; labor–leisure choice; portfolio selection; borrowing constraints; free boundary problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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