Optimal Job, Consumption, and Portfolio Choice with Multiple Income–Leisure Regimes

Abstract

We study an infinite-horizon consumption, portfolio, and job-choice problem in which an investor may move costlessly and reversibly among N income–leisure regimes. Job i provides constant labor income Yi and leisure Li, with higher-income jobs offering less leisure. Merely listing N jobs does not imply that all of them are ever selected: an intermediate job can lie below the upper envelope of the relevant dual payoffs and therefore be economically redundant. We formulate a full-activity condition as a strict ordering of adjacent dual intersection points and prove that it is necessary and sufficient for every job to be optimal on a nonempty state interval. When the condition fails, an explicit upper-hull reduction removes the inactive jobs and converts the problem into an equivalent model with a smaller active set. In the complete-market benchmark, the martingale method reduces the mixed control problem to a one-dimensional dual resolvent. The active switching thresholds are explicit, the remaining coefficients follow from a finite transfer recursion, and strong duality yields the optimal consumption, portfolio, wealth, and job policies. Relative to the classical two-job model, the N-job formulation identifies which intermediate occupations survive, how wealth-region widths differ, and when a nominal job menu collapses to fewer effective choices. Numerical exercises report a failure-of-full-activity case, sensitivity analysis, a two-job comparison, and a simulated wealth/job path. The paper is deliberately theoretical and frictionless: it provides a transparent benchmark for richer empirical and structural models rather than statistically testing the wealth–leisure mechanism.