This paper explores the conditions for local dynamics and the uniqueness of a steady state in the money-in-utility-function, and transaction-costs models with recursive utility. When a money demand depends only on nominal interest rates and a monetary Brock-Gale (BG) condition holds around a steady state, the steady state is locally stable. Even in the case of the general money demand function, the local stability is sometimes possible. Furthermore, when a money demand function exists and the BG condition holds globally, the steady state is unique.