A closed-form solution for the continuous-time consumption model with endogenous labor income

In this paper, we study the consumption, labor supply, and portfolio decisions of an infinitely lived individual who receives a certain wage rate and income from investment into a risky asset and a risk-free bond. Uncertainty about labor income arises endogenously, because labor supply evolves randomly over time in response to changes in financial wealth. We derive closed-form solutions for optimal consumption, labor supply and investment strategy. We find that deferring the retirement age stimulates optimal consumption over time and discourages optimal labor supply during the working life. We also find explicitly that optimal portfolio allocation becomes more ‘conservative’ when the individual approaches his prescribed retirement age. The effects of risk-aversion coefficients on optimal decisions are examined.