Optimal saving over time with self-control costs: A continuous time extension of the dual-self theory of savings

This paper presents a continuous time extension of the dual-self theory of savings by Fudenberg and Levine (2006). The original model, developed in discrete and infinite time, provides valuable theoretical insights but lacks applicability for realistic savings behaviors, which typically assume finite horizons. By extending the model into continuous time, this study addresses these limitations, allowing for finite time frames and better alignment with conventional lifetime savings models. The continuous time approach incorporates critical behavioral factors such as impatience and self-control costs, producing a transparent decomposition: impatience (r) governs the intertemporal shape of the saving path — especially the timing and steepness of the terminal decline — while impulsiveness/self-control costs (γ) act mainly as a vertical shift in saving levels over the life cycle. Furthermore, the model offers guidance for empirical discipline. These results yield implications for personal finance and long-term financial planning by clarifying how impatience and self-control shape saving decisions over the life cycle.