Notes on discounting.
Use a two-knob power curve, not the old one-knob hyperbola, when you plot how delayed rewards lose value.
01Research in Context
What this study did
Rachlin (2006) wrote a math note. He asked: does one simple curve really map how rewards lose value when they are delayed?
He compared two formulas. One has a single knob. The other has two knobs. He showed why the two-knob power curve hugs real data better.
What they found
The two-parameter power form, kD^S, fit discounting curves more cleanly than the classic hyperbola.
In plain words, you need both a rate knob (k) and a shape knob (s) to trace how steeply value falls.
How this fits with other research
van den Bos et al. (2013) took the same complaint further. They said hyperbolic forms are too weak and sketched a brain-based model. Their paper is the next-generation sequel to Howard’s math fix.
McKerchar et al. (2023) tested the idea in a shopper study. They framed waiting costs as opportunity costs and saw the power-form shape hold up with real adult consumers. The theory moved from blackboard to grocery aisle.
Last et al. (1984) had already spotted power functions inside multiple schedules. Their sensitivity curve and Howard’s discount curve share the same bend, hinting that one math shape may cover both reinforcer value and choice timing.
Why it matters
If you graph client discounting data, stop forcing it into a single-knob hyperbola. Plug the data into a two-parameter power form instead. A tighter curve gives you a clearer picture of impulsivity, which means better-informed interventions for delay tolerance or self-control training.
Want CEUs on This Topic?
The ABA Clubhouse has 60+ free CEUs — live every Wednesday. Ethics, supervision & clinical topics.
Join Free →Open your last discounting Excel file, add a power-function trendline, and compare R² to the hyperbola.
02At a glance
03Original 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
Journal of the experimental analysis of behavior, 2006 · doi:10.1901/jeab.2006.85-05