Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints
In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely inc...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/22/3536 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850226424515919872 |
|---|---|
| author | Geonwoo Kim Junkee Jeon |
| author_facet | Geonwoo Kim Junkee Jeon |
| author_sort | Geonwoo Kim |
| collection | DOAJ |
| description | In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely increasing their income and borrowing against future income, we additionally consider a non-negative wealth constraint that prohibits borrowing based on future income. This utility maximization problem is a mixture of stochastic control, where the agent chooses consumption and investment, and singular control, where the agent chooses a non-decreasing income process. To solve this non-trivial and challenging problem, we derive the Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint using the dynamic programming principle (DPP). Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy. |
| format | Article |
| id | doaj-art-3bf8e87f006645ddb495c65064d7521d |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-3bf8e87f006645ddb495c65064d7521d2025-08-20T02:05:04ZengMDPI AGMathematics2227-73902024-11-011222353610.3390/math12223536Optimal Consumption and Investment with Income Adjustment and Borrowing ConstraintsGeonwoo 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 KoreaIn this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely increasing their income and borrowing against future income, we additionally consider a non-negative wealth constraint that prohibits borrowing based on future income. This utility maximization problem is a mixture of stochastic control, where the agent chooses consumption and investment, and singular control, where the agent chooses a non-decreasing income process. To solve this non-trivial and challenging problem, we derive the Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint using the dynamic programming principle (DPP). Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy.https://www.mdpi.com/2227-7390/12/22/3536income adjustmentconsumption and investmentHJB equationsingular controlfree boundary problemlinearization |
| spellingShingle | Geonwoo Kim Junkee Jeon Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints Mathematics income adjustment consumption and investment HJB equation singular control free boundary problem linearization |
| title | Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints |
| title_full | Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints |
| title_fullStr | Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints |
| title_full_unstemmed | Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints |
| title_short | Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints |
| title_sort | optimal consumption and investment with income adjustment and borrowing constraints |
| topic | income adjustment consumption and investment HJB equation singular control free boundary problem linearization |
| url | https://www.mdpi.com/2227-7390/12/22/3536 |
| work_keys_str_mv | AT geonwookim optimalconsumptionandinvestmentwithincomeadjustmentandborrowingconstraints AT junkeejeon optimalconsumptionandinvestmentwithincomeadjustmentandborrowingconstraints |