# Hyperbolic discounting

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In behavioral economics, **hyperbolic discounting** refers to the empirical finding that people generally prefer smaller, sooner payoffs to larger, later payoffs when the smaller payoffs would be imminent. However, when the same payoffs are distant in time,^{[1]} people tend to prefer the larger outcome, even though the time lag from the smaller to the larger would be the same as before.

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## [edit] History

The phenomenon of hyperbolic discounting is implicit in Richard Herrnstein's "matching law," the discovery that most subjects allocate their time or effort between two non-exclusive, ongoing sources of reward (concurrent variable interval schedules) in direct proportion to the rate and size of rewards from the two sources, and in inverse proportion to their delays. That is, subjects' choices "match" these parameters.

After the report of this effect in the case of delay (Chung and Herrnstein, 1967), George Ainslie pointed out that in a single choice between a larger, later and a smaller, sooner reward, inverse proportionality to delay would be described by a plot of value by delay that had a hyperbolic shape, and that this shape should produce a reversal of preference from the larger, later to the smaller, sooner reward for no other reason but that the delays to the two rewards got shorter. He demonstrated the predicted reversal in pigeons^{[vague]} (Ainslie, 1974).

A large number of subsequent experiments have confirmed that spontaneous preferences by both human and nonhuman subjects follow a hyperbolic curve rather than the conventional, "exponential" curve that would produce consistent choice over time (Green et al., 1994; Kirby, 1997). For instance, when offered the choice between $50 now and $100 a year from now, many people will choose the immediate $50. However, given the choice between $50 in five years or $100 in six years almost everyone will choose $100 in six years, even though that is the same choice seen at five years' greater distance.

Notice that whether discounting future gains is logically correct or not, and at what rate such gains should be discounted, depends greatly on circumstances. Many examples exist in the financial world, for example, where it is logically reasonable to assume that there is an implicit risk that the reward will not be available at the future date, and furthermore that this risk increases with time. Consider: Paying $50 for your dinner today or delaying payment for sixty years but paying $100,000. In this case the restaurateur would be reasonable to discount the promised future value as there is significant risk that it might not be paid (possibly due to your death, his death, etc).

In cases where both alternatives are fairly certain to occur if chosen this pattern of discounting is dynamically inconsistent, and therefore inconsistent with standard models of rational choice, since the rate of discount between time *t* and *t+1* will be low at time *t-1*, when *t* is the near future, but high at time *t* when *t* is the present and time *t+1* the near future. Nevertheless, it appears to be descriptively accurate.^{[vague]}

## [edit] Applications

More recently these observations about discount functions have been used to study saving for retirement, borrowing on credit cards, and procrastination. However, hyperbolic discounting has been most frequently used to explain addiction.

Hyperbolic discounting has been found to relate to real-world examples of self control. Indeed, a variety of studies have used measures of hyperbolic discounting to find that drug dependent individuals discount delayed consequences more than matched nondependent controls, suggesting that extreme delay discounting is a fundamental behavioral process in drug dependence (e.g., Bickel & Johnson, 2003; Madden et al., 1997; Vuchinich & Simpson, 1998). Some evidence suggests pathological gamblers also discount delayed outcomes at higher rates than matched controls (e.g., Petry & Casarella, 1999). Whether high rates of hyperbolic discounting precede addictions or vice-versa is currently unknown, although some studies have reported that high-rate discounting rats are more likely to consume alcohol (e.g., Poulos et al., 1995) and cocaine (Perry et al., 2005) than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (gambling) outcomes more satisfying (Madden et al., 2007).

## [edit] Mathematical model

The functional equation for hyperbolic discounting is as follows:

where *f*(*D*) is the discount factor that multiplies the value of the reward, *D* is the delay in the reward, and *k* is a parameter governing the degree of discounting.

The degree of discounting is vitally important in describing hyperbolic discounting, especially in the discounting of specific rewards such as money. The discounting of monetary rewards varies across age groups due to the varying rate of *k* (Green, Frye, and Myerson, 1994). *k* depends on a variety of factors, including the species being observed, age, experience, and the amount of time needed to consume the reward (Lowenstein and Prelec, 1992; Raineri and Rachlin, 1993).

## [edit] Quasi-hyperbolic approximation

The "quasi-hyperbolic" discount function, which approximates the hyperbolic discount function above, is given (in discrete time) by

*f*(0) = 1, and *f*(*D*) = β * δ^{D},

where β and δ are constants between 0 and 1; and again *D* is the delay in the reward, and *f*(*D*) is the discount factor. The condition *f*(0) = 1 is stating that rewards taken at the present time are not discounted.

Quasi-hyperbolic time preferences are also referred to as "present-biased" or "beta-delta" preferences. They retain much of the analytical tractability of exponential discounting while capturing the key qualitative feature of discounting with true hyperbolas.

While originally introduced only as an approximation of hyperbolic discounting, there is now some evidence from neurophysiology that it is in fact a more accurate model of human behavior.^{[2]}

## [edit] See also

## [edit] References

### [edit] Footnotes

**^**Laibson, David, 1997. "Golden Eggs and Hyperbolic Discounting," The Quarterly Journal of Economics, MIT Press, vol. 112(2), pages 443-77, May.**^**S. McClure et al., "Separate neural systems evaluate immediate and delayed monetary rewards,"*Science*306 (2004), 503-7, cited in Jon Elster,*Explaining Social Behavior*, 118 and 123

### [edit] General references

- Ainslie, G. W. (1974) Impulse control in pigeons.
*Journal of the Experimental Analysis of Behavior*21,485-489. - Ainslie, G. W. (1975) Specious reward: A behavioral theory of impulsiveness and impulsive control.
*Psychological Bulletin*, 82, 463-496. - Ainslie, G. (1992)
*Picoeconomics: The Strategic Interaction of Successive Motivational States Within the Person*. Cambridge. Cambridge University Press. - Ainslie, G. (2001)
*Breakdown of Will*Cambridge, Cambridge University Press, ISBN 978-0521596947 - Bickel, W. K., & Johnson, M. W. (2003). Delay discounting: A fundamental behavioral process of drug dependence. In G. Loewenstein, D. Read & R. F. Baumeister (Eds.), Time and Decision. New York: Russell Sage Foundation.
- Chung, S. H. and Herrnstein, R. J. (1967). Choice and delay of Reinforcement.
*Journal of the Experimental Analysis of Behavior, 10*67-64. - Green, L., Fry, A. F., and Myerson, J. (1994). Discounting of delayed rewards: A life span comparison.
*Psychological Science, 5*, 33-36. - Kirby, K. N. (1997) Bidding on the future: Evidence against normative discounting of delayed rewards.
*Journal of Experimental Psychology: General*126, 54-70. - Loewenstein, G. and Prelec, D. (1992).
*Choices Over Time*New York, Russell Sage Foundation - Madden, G. J., Petry, N. M., Bickel, W. K., and Badger, G. J. (1997). Impulsive and self-control choices in opiate-dependent patients and non-drug-using control participants: Drug and monetary rewards.
*Experimental and Clinical Psychopharmacology, 5*, 256-262. - Madden, G. J., Ewan, E. E., & Lagorio, C. H. (2007). Toward an animal model of gambling: Delay discounting and the allure of unpredictable outcomes.
*Journal of Gambling Studies, 23*, 63-83. - Perry, J. L., Larson, E. B., German, J. P., Madden, G. J., and Carroll, M. E. (2005). Impulsivity (delay discounting) as a predictor of acquisition of i.v. cocaine self-administration in female rats.
*Psychopharmacology, 178*, 193-201. - Petry, N. M., and Casarella, T. (1999). Excessive discounting of delayed rewards in substance abusers with gambling problems.
*Drug and Alcohol Dependence, 56*, 25-32. - Poulos, C. X., Le, A. D., and Parker, J. L. (1995). Impulsivity predicts individual susceptibility to high levels of alcohol self administration.
*Behavioral Pharmacology, 6*, 810-814. - Vuchinich, R. E., and Simpson, C. A. (1998). Hyperbolic temporal discounting in social drinkers and problem drinkers.
*Experimental and Clinical Psychopharmacology, 6*, 292-305. - Rachlin, H. (2000).
*The Science of Self-Control*Cambridge;London: Harvard University Press - Raineri,A., and Rachlin, H. (1993). The effect of temporal constraints on the value of money and other commodities.
*Journal of Behavioral Decision-Making, 6,*77-94.