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Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints

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

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.

Suggested Citation

  • Geonwoo Kim & Junkee Jeon, 2024. "Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints," Mathematics, MDPI, vol. 12(22), pages 1-14, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:22:p:3536-:d:1519427
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    References listed on IDEAS

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