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

TL;DR

This paper analyzes an optimal consumption-investment problem with a borrowing constraint, demonstrating the smoothness of the value function and deriving the optimal policy, with implications for various portfolio choice models.

Contribution

It establishes the second-order smoothness of the value function and characterizes the optimal policy under general borrowing constraints, extending existing portfolio models.

Findings

Value function is second-order smooth.

Optimal policy expressed in feedback form.

Constraint binding characterized by an endogenous threshold.

Abstract

This paper studies an optimal consumption-investment problem for an investor whose instantaneous utility depends on both consumption and wealth, and the investor faces a general borrowing constraint that the investment amount in the risky asset does not exceed an exogenous function of the wealth. We show that the value function is second-order smooth and present the optimal consumption-investment policy in a feedback form. Moreover, when the risky investment amount is bounded above 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, and we determine the endogenous wealth threshold with the smooth fit condition. Our results encompass several well-developed portfolio choice models and imply new applications.