Bleichrodt, H., \Potter van Loon\, \RogierJ.D.\ and Prelec, D. (2022). Beta-Delta or Delta-Tau? A Reformulation of Quasi-Hyperbolic Discounting Management Science, 68(8):6326--6335.

This paper introduces the index τ = ln β=ln δ as a measure of time inconsistency and vulnerability to self-control problems in the quasi-hyperbolic, beta-delta (β, δ) discounting model. We provide a preference foundation for τ and, consequently, a revealed preference definition of failed self-control. The τ 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 (β, δ) family. The τ index thus provides a common yardstick for comparing temporal inconsistency across different functional forms.