Hyperbolic Discounting

Definition

Hyperbolic discounting refers to the tendency for people to increasingly choose a smaller-sooner reward over a larger-later reward as the delay occurs sooner rather than later in time. When offered a larger reward in exchange for waiting a set amount of time, people act less impulsively (i.e., choose to wait) as the rewards happen further in the future. Put another way, people avoid waiting more as the wait nears the present time. Hyperbolic discounting has been applied to a wide range of phenomena. These include lapses in willpower, health outcomes, consumption choices over time, and personal finance decisions.

Background and History

The notion of discounting future rewards relative to immediate pleasure has a long history. People generally want rewards sooner rather than later. Thus, options that delay a reward appear less attractive, and people discount them. The neoclassical view of economics assumes that people discount a future reward by a fixed percentage for each unit of time they must wait. If the discount rate is 10% per year, a person should equally like $100 now and $110 a year from now. As well, the same person should also equally like $100 in 1 year and $110 in 2 years. According to this view (called exponential discounting), the amount people discount a future reward depends only on the length of the wait and a discount rate that is constant across different wait times.

Although exponential discounting has been used widely in economics, a large body of evidence suggests that it does not explain people's choices. People choose as if they discount future rewards at a greater rate when the delay occurs sooner in time. To illustrate, many people prefer $100 now to $110 in a day, but very few people prefer $100 in 30 days to $110 in 31 days. It appears people would rather wait 1 day for $10 if the wait happens a month from now. However, they prefer the opposite if they must wait right now. More generally, the rate at which people discount future rewards declines as the length of the delay increases. This phenomenon has been termed hyperbolic discounting by the psychologist Richard Herrnstein.

There are several reasons why people might rationally choose a smaller reward now over a larger reward later. They may like the sure thing, their preferences could change, or they may have an urgent need such as hunger or paying the rent. Even so, people still seem to show inconsistencies in their choices over time. When choosing between $100 or $110 a day later as in the earlier example, people believe that in a month they will want to wait a day for an extra $10. Yet after a month passes, many of these people will reverse their preferences and now choose the immediate $100 rather than wait a day for an additional $10. In sum, even when facing the same exact choice, people act impulsively in the short term but exhibit greater patience in the long term.

The amount that people discount future rewards has been mathematically represented in several ways. The classical economic view of exponential discounting reduces a future reward by a factor of 1 / (1 + k)t where k is the constant discount rate per time unit and t is the length of the delay. The amount a future reward is discounted depends only on the length of the delay, given a constant discount rate. Alternatively, hyperbolic discounting reduces a future reward by a factor of 1 / (1 + kt)β/α where α and β are greater than zero. The term hyperbolic is used because this formula is the generalized function for a hyperbola. With hyperbolic discounting, the rate of discounting decreases as the delay occurs further in the future. Thus, the amount a future reward is discounted depends on the length of the delay and when the delay occurs. Hyperbolic discounting will generally discount future rewards more than exponential discounting for short delays, yet less than exponential discounting for long delays.

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