Global tail optimality and global preferences consistency under non-decomposable discounting

Abstract

When time preferences in an intertemporal optimization problem are represented by non–exponential discounting, a control plan that is optimal today may no longer be optimal from the perspective of a future self. We examine this issue by revisiting four notions of temporal consistency: local and global tail optimality of a control plan, and local and global preferences consistency of the agent. We introduce decomposability, super–decomposability, and sub–decomposability of the discount function, identifying decomposability as the property that allows Bellman’s optimality principle to extend beyond exponential discounting. Under decomposability, global tail optimality and preferences consistency hold simultaneously. Conversely, under non–decomposable discounting, this connection breaks down and the problem becomes time–inconsistent. We therefore consider the three canonical approaches to time–inconsistent control: precommitment, dynamic optimality (naive), and Nash equilibrium (consistent planning). We apply this theory to a continuous–time consumption–investment problem extending Merton’s [1] model to non–decomposable discounting. We derive closed–form policy rules and evaluate the tail optimality and preferences consistency of the three strategies. We show that, under super–decomposability, precommitment induces a uniformly higher propensity to consume than the dynamically optimal and Nash–equilibrium strategies, whereas the ordering is reversed under sub–decomposability. Under additional regularity conditions, analogous ordering results hold for consumption levels.