Optimal Job Choice, Consumption, and Investment Under Subsistence-Consumption Constraints

This paper extends a benchmark reversible job-choice framework by imposing a subsistence-consumption constraint. The agent chooses consumption, portfolio allocation, and one of two jobs in continuous time. The first job provides low income and high leisure, whereas the second job provides high income and low leisure. We focus on the case in which the coefficient of relative risk aversion satisfies γ>1, so that the transformed risk-aversion parameter also satisfies γ1>1. Consumption must satisfy the lower bound ct≥c̲ under both jobs. The constraint changes both the natural solvency boundary and the job-switching rule. In the dual problem, each job generates a subsistence-adjusted reward whose form depends on whether the consumption floor is binding. The optimal job is selected by the upper envelope of the two dual rewards. We prove that, when γ>1, the resulting dual switching function has a unique positive zero. We also characterize the location of this zero explicitly in terms of the income gap Y1−Y0 and the subsistence level c̲. Hence the optimal job policy remains a one-threshold rule: the agent chooses the high-income job at low wealth and the high-leisure job at high wealth. The consumption floor shifts this boundary and implies that near the solvency boundary the agent necessarily consumes at the subsistence level and works in the high-income job. We provide a complete closed-form representation of the dual value function in all possible regimes.