Trap/Behavioral Economics/No. 0478
Hyperbolic Discounting
Hyperbolic discounting is a pattern in which the rate used to discount future rewards falls as the wait grows. In behavioral economics, George Ainslie and James Mazur linked it to preference reversals: people can favor a larger, later reward, then switch as a smaller one nears.
- Evidence
- Mixed evidence
- Read
- 6 min
- Links
- 9 connections
01You've seen this when…
- in life
On Sunday afternoon, you plan to get to bed early all week. At 10 p.m. on Monday, another episode feels more appealing than tomorrow’s rested morning.
- at work
A client offers a smaller fee this Friday or a larger fee next month. You choose next month. On Friday, with the smaller payment ready to collect, you ask to change.
- out in the world
At a public budgeting workshop, you favor larger rebates next quarter over smaller checks next month. When the smaller checks become available, you want yours immediately.
02The idea
A month feels like a substantial wait when the reward is available today. The same extra month can feel minor when both options are a year away. Hyperbolic discounting describes this pattern: the rate at which waiting reduces a reward’s present value falls as the delay grows.
A commonly used formula is V = A / (1 + kD). Here, V is the reward’s value now, A is its value without delay, D is the wait, and k controls impatience. A higher k means waiting reduces value more sharply.
The important consequence is a possible preference reversal. From a distance, a larger, later reward wins. As both dates approach, the smaller, sooner reward gains value faster and can take the lead. The amounts and the gap between payment dates remain unchanged.
For comparison, exponential discounting reduces value by the same proportion for each additional unit of time. Under stable conditions, moving both rewards equally closer preserves their ranking. Hyperbolic discounting allows that ranking to switch.
This is one particular model within temporal discounting. Choosing an immediate reward by itself gives too little information to identify the curve.
03Why it happens
The curve describes behavior more securely than it explains its psychological cause. Several processes can produce or strengthen decreasing impatience:
- People experience time unevenly. The difference between today and tomorrow can feel larger than the difference between day 100 and day 101. Models of subjective time can produce hyperbolic-looking choices.
- Nearby rewards become vivid. A payment available this afternoon is easy to imagine using. Attention and anticipation can increase its pull as the date approaches.
- Waiting introduces uncertainty. A delayed payment may arrive late or fail to arrive. Beliefs about that risk can contribute to the observed discounting pattern.
- The choice setting changes what people notice. Reward size, how dates are described, and whether payment is credible all affect choices.
These explanations can overlap. A fitted hyperbolic curve does not establish which process caused the behavior. Researchers also find differences across reward types, tasks and individuals.
04A worked example
Imagine a freelancer choosing between $100 in 60 days and $140 in 90 days. The client allows revisions until day 60. Assume both payments are guaranteed and the freelancer’s financial needs stay unchanged.
For illustration, use the simple hyperbolic formula with k set to 0.02 per day and value proportional to dollars:
- The $100 payment has a present value of $45.45: 100 / (1 + 0.02 × 60).
- The $140 payment has a present value of $50: 140 / (1 + 0.02 × 90).
The freelancer chooses $140.
What it looks like Two months later, the freelancer changes course and collects $100 immediately. The earlier willingness to wait has evaporated.
What’s actually going on On day 60, the same formula values the immediate $100 at $100 and the $140, now 30 days away, at $87.50. The larger reward won when both payments were distant. The smaller reward wins when it reaches the present. The payment gap stays at 30 days throughout.
What would have helped If the freelancer wanted the earlier choice to hold, agreeing to a fixed payment schedule would remove the later opportunity to switch. That is a commitment device. The parameter here is hypothetical; it demonstrates how the reversal can arise.
05How to spot it
These are clues. To identify a hyperbolic pattern, compare choices at several delays. Check whether income needs, payment risk or new information changed along the way.
06What to do instead
- Make advance decisions easier to keep. Arrange an automatic savings transfer or select a payout schedule before the smaller reward becomes available. A commitment device can be soft and reversible when the stakes are uncertain.
- Compare both options from a distance. Write down what you would choose if both rewards arrived six months later. Then record today’s choice. A difference exposes the role of proximity without proving that either choice is mistaken.
- Bring some benefit forward. If a useful activity pays off much later, pair it with something enjoyable now. Temptation bundling, such as saving a favorite audiobook for walks, gives the activity an earlier reward.
- Prepare for the predictable decision point. Use an implementation intention: if the early-payment offer arrives, first check the payment schedule and the reason for choosing it.
Leave room for changed circumstances. A commitment that prevents an impulsive switch can also block a sensible response to an emergency. Match its strength to the cost of being locked in.
07When it isn’t hyperbolic discounting
Taking less money sooner can be sensible. Rent may be due, borrowing may be expensive, or the later payment may be unreliable. Those conditions belong in the decision. A reversal after learning that a client might default gives little evidence about a person’s discount function.
Present bias is the broader tendency to give special weight to immediate outcomes. Quasi-hyperbolic discounting models that with an extra drop in value between now and any future date, followed by exponential discounting among future dates. Hyperbolic discounting predicts decreasing impatience across delays, including comparisons where both rewards remain in the future.
A single now-versus-later choice cannot distinguish these models. Even a preference reversal can have several explanations. Identifying the mathematical form requires choices across multiple delays and assumptions about reward value, risk and payment credibility.
The opening bedtime scene is a familiar warning sign, but sleep, entertainment and money need not follow the same discount curve.
08Roots
At Harvard, Richard Herrnstein studied pigeons pecking two keys that delivered food at different rates. His 1961 paper addressed how animals divide their behavior between competing sources of reward. Their relative pecking rates tracked relative reinforcement rates. This became the matching law, an important precursor to later accounts of delayed reward.
George Ainslie took up the puzzle of impulse control: an animal or person can favor a larger, later reward in advance and abandon it as a smaller reward approaches. His 1975 synthesis connected that reversal to a sharply curved relationship between delay and reward value. It also explained why arranging a restriction ahead of time could help preserve an earlier preference.
James Mazur made the curve easier to measure. In his 1987 work, he adjusted the delay on one reward until pigeons chose between alternatives about equally. These points of indifference supported a simple hyperbolic equation that became widely used in delay-discounting research.
The idea traveled from animal-learning experiments into behavioral economics, where researchers examined saving, consumption and self-control. Its central contribution was a way to explain how stable rewards can produce changing choices simply because time passes.
09How solid is this?
Decreasing impatience and preference reversals are well documented in human and animal choice studies. Hyperbolic curves fit many datasets, but the exact form varies across tasks and rewards; immediate-versus-delayed impatience alone does not establish this model.
10Connections
- Often confused with Present Bias, Quasi-Hyperbolic Discounting
- Countered by Commitment Device, Temptation Bundling, Implementation Intentions
- Part ofTemporal Discounting
- See alsoPreference Reversal, Time Consistency, Sophisticated vs. Naive Present Bias
11Origin and sources
Richard Herrnstein’s matching-law research (1961) was a precursor. George Ainslie developed the preference-reversal account in 1975; James Mazur presented a widely used hyperbolic delay function in 1987.
- [1]Herrnstein, R. J. (1961). Relative and absolute strength of response as a function of frequency of reinforcement. Journal of the Experimental Analysis of Behavior, 4(3), 267–272.
- [2]Ainslie, G. (1975). Specious reward: A behavioral theory of impulsiveness and impulse control. Psychological Bulletin, 82(4), 463–496.
- [3]Mazur, J. E. (1987). An adjusting procedure for studying delayed reinforcement. In M. L. Commons, J. E. Mazur, J. A. Nevin, & H. Rachlin (Eds.), Quantitative analyses of behavior: Vol. 5. The effect of delay and of intervening events on reinforcement value (pp. 55–73). Lawrence Erlbaum Associates.
- [4]Loewenstein, G., & Prelec, D. (1992). Anomalies in Intertemporal Choice: Evidence and an Interpretation. The Quarterly Journal of Economics, 107(2), 573–597.
- [5]Frederick, S., Loewenstein, G., & O'Donoghue, T. (2002). Time Discounting and Time Preference: A Critical Review. Journal of Economic Literature, 40(2), 351–401.
Suggest an edit· Updated 2026-10-02