Hyperbolic Discounting
What is Hyperbolic Discounting?
Hyperbolic discounting values future rewards with a discount rate that falls as the delay grows, so waiting now costs far more than the same wait later on.
Standard theory uses exponential discounting, where each extra period of delay costs the same fixed percentage, so a one day wait feels equally costly whether it starts today or a year from now. Hyperbolic discounting replaces that with a curve that drops steeply at first and then flattens, so the implied rate of impatience is high over the next week and low between two dates far out. Written as a hyperbola, present value equals the reward divided by one plus k times t, and the per-period rate shrinks as t grows. The practical result is preference reversals: a plan that looked sensible when both options were distant gets overturned once the sooner option is at hand. It is a discount function, not a bias in itself, and present bias is the behavior such a function predicts.
Hyperbolic Discounting: a worked example
Set k = 1 per year and compare $100 now with $120 in one year. Present value of the first is 100 ÷ (1 + 0) = 100 and the second is 120 ÷ (1 + 1) = 60, so the sooner, smaller reward wins easily. Now push both choices five years out. The $100 at five years is worth 100 ÷ (1 + 5) = 16.67, while the $120 at six years is worth 120 ÷ (1 + 6) = 17.14, so the larger, later reward now wins. Same one year gap between the payments, opposite decision, purely because of where the pair sits on the curve.
The mistake students make with hyperbolic discounting
Students often think hyperbolic discounting means people discount the future more than they should. The model says nothing about the right amount of impatience; what makes it hyperbolic is that the rate changes with the delay instead of staying fixed. A second error is assuming exponential discounters are patient. An exponential discounter can be extremely impatient and still never reverse a choice, because a fixed rate keeps rankings stable over time.
Hyperbolic Discounting questions
What is the difference between hyperbolic and exponential discounting?
Exponential discounting keeps impatience constant no matter how far off the reward is, while hyperbolic discounting lets impatience fall as the wait gets longer. Under the exponential version, a choice you make for next year is still the choice you make when next year arrives; under the hyperbolic version it may not be. That instability is the reason the hyperbolic form was introduced.
What is a preference reversal?
A preference reversal is picking the larger, later reward while both options are distant, then switching to the smaller, sooner reward once the sooner one becomes immediate. It is the signature prediction of a discount rate that falls with delay. A constant rate cannot produce one, so observed reversals are the main evidence for hyperbolic models.
Do people really discount hyperbolically?
Experiments with both people and animals fit a falling discount rate better than a constant one, so the hyperbolic shape describes the data more closely. Most applied work still uses the simpler quasi-hyperbolic version, which keeps a constant rate across future periods and adds one extra penalty for anything not immediate. Neither shape is treated as literally true, since both are approximations chosen for fit and tractability.
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