Beta-Delta or Delta-Tau? A Reformulation of Quasi-Hyperbolic Discounting

Abstract

This paper introduces the index [Formula: see text] as a measure of time inconsistency and vulnerability to self-control problems in the quasi-hyperbolic, beta-delta ([Formula: see text] discounting model. We provide a preference foundation for [Formula: see text] and, consequently, a revealed preference definition of failed self-control. The [Formula: see text] index is independent of utility and has an intuitive interpretation as the maximum number of future selves who can disagree with the current self with respect to uniform deviations from an intertemporal plan. The index is also computable for continuous discount functions after an appropriate mapping of functions onto the ([Formula: see text] family. The [Formula: see text] index thus provides a common yardstick for comparing temporal inconsistency across different functional forms. This paper was accepted by Manel Baucells, behavioral economics and decision analysis.