Duality of Shortfall Aversion

Abstract

Shortfall aversion describes a form of consumption habit formation in which an individual's utility depends on consumption relative to the historical consumption peak. This study develops a duality framework for a broad class of utility functions, with and without discounting, avoiding the need for determining the local optimality required by the HJB approaches commonly adopted in the literature. We show that shortfall aversion can instead be incorporated directly into the dual problem through a singular-control mechanism characterized by a variational inequality. Furthermore, we establish that the resulting value function strictly dominates those obtained in the existing literature while remaining attainable under an admissible optimal control. We demonstrate that the suboptimality of existing solutions stems from their myopic treatment of updates to the consumption peak, whereas our duality approach enlarges the admissible class to treat the consumption peak as a separate nondecreasing control and determines its optimal evolution through duality, accounting for future consumption opportunities. These results reveal a previously unrecognized advantage of the dual formulation and provide a more general and flexible framework for consumption optimization under shortfall aversion.