A General Mathematical Theory of Discounting

A mathematical formalism is developed for the existence of unique invariants associated with wide classes of observed discounting behavior. These invariants are ‘exponential discount rate spectra,’ derived from the theory of completely monotone functions. Exponential discounting, the empirically important case of hyperbolic discounting, and so-called sub-additive discounting are each special cases of the general theory. This formalism is interpreted at both the individual and social levels. Almost every discount rate spectrum yields a discount function that is ‘hyperbolic’ with respect to some exponential. Such hyperbolic discount functions may not be integrable, and the implications of nonintegrability for intertemporal valuation are assessed. In general, non-stationary spectra lead to discount functions that are not completely monotone. The same is true of discount rate spectra that are not proper measures. This formalism unifies theories of non-constant discounting, declining discount rates, ‘gamma’ discounting, and related notions.