In this paper, we address the optimal consumption, investment, and labor-leisure decision of an economic agent who does not allow a decrease in consumption. The agent has a Cobb-Douglas utility function dependent on consumption and leisure. Additionally, the agent's income is determined proportionally to the working time, which is the remaining time after enjoying leisure from the total available time for labor and leisure. Since consumption is a non-decreasing process, the agent's utility maximization problem is formulated as a two-dimensional stochastic and singular control problem. To tackle this non-trivial problem, we apply the dual-martingale approach to derive a dual problem represented as a two-dimensional pure singular control problem. By utilizing the relationship between the non-decreasing process and a collection of stopping times, we obtain the explicit-form solution for the dual problem. Finally, by establishing the duality theorem, we also derive the explicit-form optimal strategies.
An et al. (2025) studied this question.