Fully Characterizing the Set of Time Consistent Discount Functions

This paper reexamines the time consistency of discount functions as analyzed by Strotz (1956) and Koopmans (1960). Following on an observation by Heal (1998) regarding the time consistency of logarithmic discounting, I fully characterize the class of time consistent—but not necessarily autonomous—discount functions. I use my findings to explain an apparent “paradox” in the use of the gamma discounting method formulated by Weitzman (2001). One conclusion from the analysis is that a policymaker, when valuing intertemporal projects, can select a discount function which exhibits at most two of the following three properties: (1) time consistency, (2) autonomy, and/or (3) declining discount rates. The results of the paper suggest that the discounting debate be reframed to consider the costliness of time inconsistency against the benefits of employing declining discount rates and retaining autonomy. An interesting mathematical result of the paper is that time consistent discounting is equivalent to creating a metric space which is isometric to the real numbers.