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...

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Bibliographic Details
Published in:Mathematics (Basel) 2024-11, Vol.12 (22), p.3536
Main Authors: Kim, Geonwoo, Jeon, Junkee
Format: Article
Language:English
Subjects:
Borrowing
Closed form solutions
Constraints
Consumption
consumption and investment
Cost control
Decision making
Dynamic programming
Earnings
Financial planning
Flexibility
free boundary problem
HJB equation
Human capital
Income
income adjustment
Investment analysis
Investments
linearization
Maximization
Optimal control
Optimization
Present value
Securities markets
singular control
Skills
Stochastic processes
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12223536