The Amount and Speed of Discounting1

This paper introduces the concepts of amount and speed of a discounting procedure. Exponential discounting sequesters both concepts into a single parameter that needs to 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 discounting procedure acquires its ultimate present value, and we prove these to be the same. This provides an unambiguous measure of the speed of discounting, a measure whose values lie 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 averages the individual discount functions. We point to serious shortcoming with this approach and propose an alternative that, for logarithmic utility, is market based and for which the amount and time horizon of the social procedure are the averages of the amounts and time horizons of the individual procedures. We further show that the social procedure will in general be slower than the average of the speeds of the individual procedures. We then characterize three families of discounting procedures in terms of their discount functions, their discount rate functions, their amounts, their speeds and their time horizons. A one parameter hyperbolic discounting procedure, d(t) = (1 + rt)−2, has amount r and speed 0, and we argue that this zero-speed hyperbolic is well suited for social project evaluation. JEL codes: D51, D90, H43 and Q51