Hyperbolic Discounting

Hyperbolic Discounting, Preference Reversals and Present Bias in Health

A hyperbolic discounter is impatient about a delay that starts now but patient about the same delay when it lies in the future, so a smoker can sincerely plan to quit next month and then, when next month arrives, prefer one more month of smoking. The pattern matters in health economics because many health behaviours set an immediate cost against a delayed benefit, and because it shapes how incentive and commitment schemes are designed, even though the NICE manual (PMG36) prescribes constant rates for evaluation. Pure time preference and its measurement are on the time preference page and the present-value formula on the discount rate page.

From constant to declining rates of impatience

Frederick, Loewenstein and O'Donoghue use the term hyperbolic discounting for a declining rate of time preference and call it the best documented anomaly of the discounted utility model. Under exponential discounting the weight on an outcome $t$ periods away is $\delta^{t}$, so the rate between any two adjacent periods is the same however far ahead they lie; a hyperbolic function lets it fall as the delay grows.

In the study by Thaler that Frederick and colleagues describe, median answers implied an average annual discount rate of 345% over a one-month horizon, 120% over one year and 19% over ten years, and when functions are fitted to such data a hyperbolic form fits better than an exponential one.

Hyperbolic and quasi-hyperbolic discount functions

Several hyperbolic forms have been proposed. The choice matters because each implies a different path for the per-period rate. Frederick and colleagues list Ainslie's suggestion $D(t) = 1/t$, the simple hyperbola proposed by Herrnstein and by Mazur, and the generalised hyperbola of Loewenstein and Prelec, shown below.

$$D_{H}(t) = \frac{1}{1 + \alpha t}, \qquad D_{G}(t) = \frac{1}{(1 + \alpha t)^{\beta/\alpha}}$$

where $D(t)$ is the weight on an outcome $t$ periods ahead, $\alpha > 0$ controls how quickly the rate declines with delay and $\beta > 0$ sets the overall level of discounting. The per-period rate implied by the simple hyperbola between $t$ and $t+1$ is $\alpha/(1+\alpha t)$, which falls steadily with $t$. As $\alpha$ approaches zero the generalised hyperbola approaches the exponential form $e^{-\beta t}$, so it nests constant discounting as a limiting case.

The economics literature on declining rates, beginning with Laibson, has used a simpler quasi-hyperbolic form, first introduced by Phelps and Pollak, written as follows.

$$D(0) = 1, \qquad D(t) = \beta \delta^{t} \;\text{for}\; t > 0$$

where $\beta \le 1$ is a one-off discount applied to everything that is not immediate and $\delta$ is the per-period discount factor thereafter. These $\beta$ and $\delta$ differ from the generalised hyperbola's parameters. The per-period rate is $(1 - \beta\delta)/\beta\delta$ between now and the next period and $(1 - \delta)/\delta$ between any two later periods. Frederick and colleagues describe this form as highly tractable and as capturing many of the qualitative implications of hyperbolic discounting. Setting $\beta = 1$ returns exponential discounting, and empirical studies such as Wang and Sloan's read $\beta < 1$ as present bias.

Why plans unravel through time inconsistency and self-awareness

The mechanism is a change in the ranking of two dated outcomes as both draw nearer. A smaller-sooner outcome can lose to a larger-later one when both are distant, because the extra delay is discounted at the low long-run rate, and win when it becomes immediate, because the same delay now meets the high short-run rate. Under exponential discounting the ratio of the two weights depends only on the gap between them, so the ranking never changes.

Laibson states that hyperbolic discount functions induce dynamically inconsistent preferences and so a motive to constrain one's own future choices; his model treats an illiquid asset as an imperfect commitment technology. Strotz, the first economist to consider alternatives to exponential discounting, described two responses for a person who foresees the change, namely precommitment and consistent planning that ignores plans the future self will not carry out. O'Donoghue and Rabin applied the quasi-hyperbolic model to procrastination, putting off an onerous task more than one would like from a prior perspective.

What happens next depends on self-awareness. A naive person believes future preferences will match current ones; a sophisticated person correctly predicts how they will change; and a partially naive person, in the model of O'Donoghue and Rabin, knows that future self-control problems exist but underestimates them. Frederick and colleagues draw the policy implication that commitment devices may help sophisticated people, whereas for naive people policies may do better to educate them about loss of control or to provide incentives to use commitment devices, which ties hyperbolic discounting to self-control.

Present bias in smoking, adherence and screening

Health behaviours are a natural test because their costs and benefits fall at different times. Wang and Sloan note that with smoking the pleasure comes first and the harm later, while with adherence to guidelines for a chronic condition the costs come first; present bias favours the immediate side in both.

For smoking, Bickel, Odum and Madden found that current smokers discounted delayed hypothetical money more than never-smokers and ex-smokers, that delayed cigarettes lost value faster than delayed money, and that the hyperbolic equation fitted better than the exponential equation in 74 of 89 comparisons. Gruber and Köszegi cite evidence of unrealised plans to quit. Among high school seniors who smoked, 56% said they would not be smoking five years later, but only 31% had quit five years on. Modelling smokers as time-inconsistent, they argue that policy should account for the internalities smokers impose on themselves, and in their US calibration estimate that the optimal tax per pack of cigarettes is at least USD 1 higher than under the rational addiction model.

For medication adherence and self-care, Wang and Sloan estimated a dynamic model for older people with diabetes and found substantial present bias ($\beta$ estimated at 0.38) together with complete naivety about it. In their simulation, removing both raised predicted adherence to diabetes care guidelines by 8.5 percentage points from 51.7%, a 16.4% relative increase. They report that an earlier study of mammography by Fang and Wang estimated $\beta$ between 0.51 and 0.79 and rejected both exponential discounting and sophistication, which places screening attendance in the same framework.

Designing incentives and commitment devices

If people overweight the present, the model implies that an incentive has most effect when it changes what happens now. Deposit contracts, in which participants put their own money at risk and recover it if they change behaviour, act as commitment devices, and Halpern and colleagues expected loss aversion to favour them; rewards add a gain that is itself delayed and discounted.

Halpern and colleagues randomised smokers to reward-based programmes worth about USD 800 or to deposit-based programmes that combined a refundable USD 150 deposit with USD 650 in rewards. Acceptance was 90.0% for rewards against 13.7% for deposits, and in intention-to-treat analyses sustained abstinence at six months was higher with rewards (15.7% against 10.2%). In instrumental-variable analyses, among the estimated 13.7% who would accept either type, abstinence was 13.2 percentage points higher with deposits. A final bonus payment was included to offset smokers' tendency to discount future events. An evaluation of such schemes therefore has to model uptake and efficacy separately. Wang and Sloan suggest that, because naive hyperbolic discounters heavily discount future benefits, efforts to raise adherence should reduce the immediate cost of adhering, for example by removing cost sharing for screening and preventive visits.

Worked example of a smoker's plan to quit next month

The figures are illustrative. Each month a smoker chooses between continuing to smoke, worth 10 units of satisfaction in the current month, and quitting, worth 13 units of better health and wellbeing arriving one month later. Evaluated from today, the weights are $D(0)$ and $D(1)$ for this month's choice, and $D(1)$ and $D(2)$ for next month's choice.

1. Compute the weights. Four discount functions are compared, namely exponential with $\delta = 0.99$ and with $\delta = 0.75$ a month, the simple hyperbola with $\alpha = 0.4$, and the quasi-hyperbolic form with $\beta = 0.7$ and $\delta = 0.99$. For the quasi-hyperbolic case the weights are as follows.

$$D(1) = 0.7 \times 0.99 = 0.693, \qquad D(2) = 0.7 \times 0.99^{2} = 0.686$$

where 0.693 and 0.686 are rounded to three decimal places. For the simple hyperbola, $D(1) = 1/1.4 = 0.714$ and $D(2) = 1/1.8 = 0.556$.

2. Compare the options today and for next month.

For the quasi-hyperbolic smoker today, quitting is worth 13 × 0.693 = 9.009, below 10. For next month, smoking is worth 10 × 0.693 = 6.93 and quitting 13 × 0.68607 = 8.92 (using the unrounded weight), so quitting wins.

3. Interpret. Both exponential smokers are consistent, because the ratio of the two weights is the same at every horizon. The patient one quits now and the impatient one never intends to. The hyperbolic smokers plan to quit next month, but next month becomes today and the first column applies again, so a naive smoker abandons the plan month after month. A sophisticated one foresees this; O'Donoghue and Rabin show severe procrastination cannot occur under complete sophistication. A declining rate, not steep discounting, causes it.

4. Price a commitment device. For the quasi-hyperbolic smoker, a penalty felt this month on smoking, such as a deposit forfeited in the same month, needs to exceed 10 − 9.009 = 0.991 units to tip today's choice. A reward paid with next month's health gain is weighted by 0.693, so it needs to exceed 10 ÷ 0.693 − 13 = 1.430 units. The 44% premium comes from timing alone (1 ÷ 0.693 = 1.443); a reward paid this month would need only 0.991.

Hyperbolic discounting in economic evaluation and HTA

Hyperbolic discounting enters health economics in two distinct ways. As a positive model of behaviour, it can inform parameters such as uptake of a commitment scheme or adherence, and explain why a cost-effective preventive service is under-used. As a normative rule for weighting future costs and QALYs in an economic evaluation, it is rarely adopted.

The NICE manual (PMG36, last updated 13 October 2026) states in section 4.5.1 that for the reference case costs and health effects should be discounted at the same rate of 3.5% per year, and section 4.5.3 allows the committee to consider 1.5% a year for both when all of its criteria are met; both are constant rates. Attema, Brouwer and Claxton note that hyperbolic discounting explains individual behaviour better but leads to inconsistencies over time, and that almost all guidelines prescribe constant rates. A reference-case ranking that could reverse a year later with nothing changed would be of little use to a decision maker, which is the practical case for a constant rate.

Limits of the hyperbolic account and common misreadings

The evidence for hyperbolic discounting has limits. Several errors recur in health applications.

1. Reading any declining rate as hyperbolic. Frederick and colleagues report Read's argument that rates falling with horizon could reflect subadditive discounting, where total discounting over an interval grows as it is split more finely; in their own review, excluding studies with horizons of a year or less removes the link between horizon and discount factor.
2. Equating present bias with a high discount rate. In the worked example the steep exponential smoker never reverses; reversal needs a rate that declines with delay.
3. Treating the quasi-hyperbolic form as literal. Gruber and Köszegi describe it as a theoretical device that confines self-control conflict to one period, so the period length must be chosen with that limit in mind.
4. Assuming people know their own bias. Wang and Sloan found complete naivety in their diabetes sample, so commitment devices would need incentives or education to be taken up.

Sources

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