Consumption and Portfolio Selection with Recursive Utility, Stochastic Income, and Liquidity Constraints

Abstract

This paper studies a continuous-time optimal consumption and portfolio selection problem when an economic agent with recursive utility has stochastic income and liquidity constraints. To tackle this problem, we introduce a transform of the Hamilton-Jacobi-Bellman equation into a free boundary value ODE (ordinary differential equation). This transform is designed to simplify the procedure required for performing a duality method. Then, we characterize the optimal policies by deriving an integral equation with a fixed interval from the free boundary ODE. The integral equation is used to prove the verification theorem and to generate a stable numerical scheme. It is notable that the optimal portfolio depends on the elasticity of intertemporal substitution (EIS) due to liquidity constraints even if the investment opportunity is constant. Furthermore, we perform various analyses to investigate the impact of the borrowing constraints on the optimal policies and the marginal propensity to consume.