Notes on discounting

Abstract

In general, if a variable can be expressed as a function of its own maximum value, that function may be called a discount function. Delay discounting and probability discounting are commonly studied in psychology, but memory, matching, and economic utility also may be viewed as discounting processes. When they are so viewed, the discount function obtained is hyperbolic in form. In some cases the effective discounting variable is proportional to the physical variable on which it is based. For example, in delay discounting, the physical variable, delay (D), may enter into the hyperbolic equation as kD. In many cases, however, the discounting data are not well described with a single-parameter discount function. A much better fit is obtained when the effective variable is a power function of the physical variable (kDS in the case of delay discounting). This power-function form fits the data of delay, probability, and memory discounting as well as other two-parameter discount functions and is consistent with both the generalized matching law and maximization of a constant-elasticity-of-substitution utility

Fig 1. Social discounting.

Amount of money forgone as a function of social distance from the receiver. The solid curve is the best-fitting simple hyperbolic discount function (Equation 4). The dashed curve is the best-fitting exponential discount function (Equation 6).

Fig 2. Ebbinghaus's forgetting function.

Savings (learning time minus relearning time as a percentage of learning time) as a function of time between learning and relearning. The solid curve is the best-fitting exponentiated hyperbolic discount function (Equation 8).