Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon
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 allocate...
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MDPI AG
2025-03-01
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| Series: | Mathematics |
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| author | Geonwoo Kim Junkee Jeon |
| author_facet | Geonwoo Kim Junkee Jeon |
| author_sort | Geonwoo Kim |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a8ea3b33dae24c9890a0a781f75dcc46 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-a8ea3b33dae24c9890a0a781f75dcc462025-08-20T01:49:05ZengMDPI AGMathematics2227-73902025-03-0113698910.3390/math13060989Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite HorizonGeonwoo Kim0Junkee Jeon1School of Natural Sciences, Seoul National University of Science and Technology, Seoul 01811, Republic of KoreaDepartment of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of KoreaWe 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.https://www.mdpi.com/2227-7390/13/6/989optimal consumptionlabor–leisure choiceportfolio selectionborrowing constraintsfree boundary problem |
| spellingShingle | Geonwoo Kim Junkee Jeon Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon Mathematics optimal consumption labor–leisure choice portfolio selection borrowing constraints free boundary problem |
| title | Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon |
| title_full | Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon |
| title_fullStr | Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon |
| title_full_unstemmed | Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon |
| title_short | Optimal Consumption, Leisure, and Investment with Partial Borrowing Constraints over a Finite Horizon |
| title_sort | optimal consumption leisure and investment with partial borrowing constraints over a finite horizon |
| topic | optimal consumption labor–leisure choice portfolio selection borrowing constraints free boundary problem |
| url | https://www.mdpi.com/2227-7390/13/6/989 |
| work_keys_str_mv | AT geonwookim optimalconsumptionleisureandinvestmentwithpartialborrowingconstraintsoverafinitehorizon AT junkeejeon optimalconsumptionleisureandinvestmentwithpartialborrowingconstraintsoverafinitehorizon |