Optimal Consumption and Investment with Early Retirement Under a Target Wealth Constraint

We study an infinite-horizon consumption, portfolio, and early-retirement problem for a wage earner who receives labor income while working but bears a constant utility cost of labor. Retirement is irreversible and removes both labor income and work disutility. In addition, retirement is feasible only after the agent has accumulated a prescribed level of financial wealth. We solve the problem under constant relative risk aversion using a dual martingale method. The post-retirement problem reduces to the standard Merton problem, while the pre-retirement problem becomes an optimal stopping problem with a target-induced obstacle. We derive an explicit dual value function, characterize the free boundary, recover the primal value, and obtain closed-form consumption, portfolio, and retirement policies. The solution exhibits a sharp slack–binding dichotomy: small targets do not affect the classical disutility retirement policy, whereas large targets become the effective retirement threshold and reshape both consumption and risky investment before retirement.