Smoothness of the Value Function for Optimal Consumption Model with Consumption-Wealth Utility and Borrowing Constraint

This paper considers an optimal consumption-investment problem for an investor whose instantaneous utility depends on both consumption and wealth. The investor faces a constraint that the investment amount in the risky asset does not exceed an exogenous function of the wealth. We prove that the value function is second-order smooth, and the optimal consumption-investment policy is provided in a feedback form. Moreover, when the risky investment amount is bounded by a fixed constant, we show that under certain conditions, the constraint is binding if and only if an endogenous threshold bounds the portfolio wealth. Our results encompass many well-developed portfolio choice models and imply new applications.