Characterizing the amount and speed of discounting procedures

This paper introduces the concept of categorizing the amount and speed of a discounting procedure in order to generate well-characterized families of procedures for use in social project evaluation. Exponential discounting isolates the concepts of amount and speed into a single parameter that must be disaggregated in order to characterize nonconstant rate procedures. The inverse of the present value of a unit stream of benefits provides a natural measure of the amount a procedure discounts the future. We propose geometrical- and time horizon-based measures of how rapidly a procedure acquires its ultimate present value, and we prove these values are the same. This equivalency provides an unambiguous measure of the speed of discounting, with values between 0 (slow) and 2 (fast). Exponential discounting has a speed of 1. A commonly proposed approach to aggregating individual discounting procedures into a social one for project evaluation averages the individual functions. We point to a serious shortcoming with this approach and propose an alternative for which the amount and time horizon of the social procedure are the average values of the amounts and time horizons of the individual procedures. We further show that this social procedure will in general be slower than the average of the individual procedures' speeds. We then characterize three families of two-parameter discounting procedures - hyperbolic, gamma, and Weibull - in terms of their discount functions, discount rate functions, amounts, speeds, and time horizons. (The appendix characterizes additional families, including the quasi-hyperbolic procedure.) A one-parameter version of hyperbolic discounting, d(t)=(1+rt)-2, has amount r and speed 0. We argue that this zero-speed hyperbolic procedure is a good candidate for use in social project evaluation, although additional empirical work is needed to fully justify a one-parameter simplification of the more general procedure.