Optimal risk taking with flexible income

We study the portfolio selection problem of an investor who can optimally exert costly effort for more income. The possibility of generating more income, if necessary, increases the risk-taking appetite of the investor. We find the optimal allocation to the risky security as a proportion of financial wealth and as a proportion of the total wealth, defined as the combination of the financial wealth and the human capital of the investor. When the investor's objective is the maximization of the terminal wealth, we show that the optimal allocation to the risky security is a hump-shaped function of the investment horizon. However, when the investor maximizes utility from intertemporal consumption, the optimal allocation in the risky security is a constant proportion of the total wealth of the investor.