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{{Refimprove|date=June 2010}}
Given two similar rewards, humans show a preference for one that arrives sooner rather than later. Humans are said to ''discount'' the value of the later reward, by a factor that increases with the length of the delay. In [[behavioral economics]], '''hyperbolic discounting''' is a particular mathematical model thought to approximate this discounting process; that is, it models how humans actually make such valuations. Hyperbolic discounting is sharply different in form from [[exponential discounting]], a [[rationality|rational]] function used in [[finance]] used in the analysis of [[intertemporal choice|choice over time]]. Hyperbolic discounting has been observed in humans and animals.

In hyperbolic discounting, valuations fall very rapidly for small delay periods, but then fall slowly for longer delay periods. This contrasts with exponential discounting, in which valuation falls by a constant factor per unit delay, regardless of the total length of the delay. The standard experiment used to reveal a test subject's hyperbolic discounting curve is to compare short-term preferences with long-term preferences. For instance: "Would you prefer a dollar today or three dollars tomorrow?" or "Would you prefer a dollar in one year or three dollars in one year and one day?" Typically, subjects will take less money today versus tomorrow, but will gladly wait one extra day in a year in order to receive more money.<ref>Thaler, R.H. (1981) Some Empirical Evidence on Dynamic Inconsistency. ''Economic Letters'' 8, 201-07.</ref>

In studies of pigeons, for example the pigeon is given two buttons: button A provides a small amount of food quickly while button B provides more seed but after a delay. The bird then experiments for a while and settles on preferring A or B.<ref name="Ainslie1974">[[George Ainslie (psychologist)|Ainslie, G.W.]] (1974) Impulse control in pigeons. ''Journal of the Experimental Analysis of Behavior'' 21, 485-489.</ref>

Subjects using hyperbolic discounting reveal a strong tendency to make choices that are inconsistent over time. In other words, they make choices today that their future self would prefer not to make, despite using the same reasoning. This dynamic inconsistency<ref>Laibson, David, 1997. "Golden Eggs and Hyperbolic Discounting," The Quarterly Journal of Economics, MIT Press, vol. 112(2), pages 443-77, May.</ref> happens because hyperbolic discounts value future rewards much more than exponential discounting.

==Observations==
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,<ref>Chung, S.H. and Herrnstein, R.J. (1967). Choice and delay of Reinforcement. ''Journal of the Experimental Analysis of Behavior, 10'' 67-64.</ref> [[George Ainslie (psychologist)|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 function|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|date=March 2009}}<ref name="Ainslie1974"/>

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 discounting|exponential]]" curve that would produce consistent choice over time.<ref name="Green et al">Green, L., Fry, A.F., and Myerson, J. (1994). Discounting of delayed rewards: A life span comparison. ''Psychological Science, 5'', 33-36.</ref><ref>Kirby, K.N. (1997) Bidding on the future: Evidence against normative discounting of delayed rewards. ''Journal of Experimental Psychology: General'' 126, 54-70.</ref> 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.

Hyperbolic discounting has also 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.<ref>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.</ref><ref>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.</ref><ref>Vuchinich, R.E., and Simpson, C.A. (1998). Hyperbolic temporal discounting in social drinkers and problem drinkers. ''Experimental and Clinical Psychopharmacology, 6'', 292-305.</ref> Some evidence suggests pathological gamblers also discount delayed outcomes at higher rates than matched controls.<ref>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.</ref> 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<ref>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.</ref> and cocaine<ref>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.</ref> than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (gambling) outcomes more satisfying.<ref>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.</ref>

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 discount rate.<ref name="Green et al"/> The rate depends on a variety of factors, including the species being observed, age, experience, and the amount of time needed to consume the reward.<ref>Loewenstein, G. and Prelec, D. (1992). ''Choices Over Time'' New York, Russell Sage Foundation</ref><ref>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.</ref>

==Mathematical model==
[[File:Hyperbolic vs. exponential discount factors.svg|thumb|300px|right|Comparison of the discount factors of hyperbolic and exponential discounting. In both cases, <math>k=1</math>. Hyperbolic discounting is shown to ''over-value'' future assets compared to exponential discounting.]]
Hyperbolic discounting is mathematically described as:
:<math>f_H(D)=\frac{1}{1+kD}\,</math>

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. This is compared with the formula for exponential discounting:
:<math>f_E(D)=e^{-kD}\,</math>

===Quasi-hyperbolic approximation===
The "quasi-hyperbolic" discount function, which approximates the hyperbolic discount function above, is given (in [[discrete time]]) by 
:<math>f_{QH}(0)=1\,</math>, and
:<math>f_{QH}(D)=\beta \times \delta^D\,</math>,

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 preferences|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.

==Explanations==

===Uncertain risks===
Notice that whether discounting future gains is rational 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 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).

Uncertainty of this type can be quantified with [[Bayesian probability|Bayesian analysis]].<ref name="sozou1998">{{Cite doi|10.1098/rspb.1998.0534}}</ref> For example, suppose that the probability for the reward to be available after time ''t'' is, for known hazard rate λ
:<math>P(R_t|\lambda) = \exp(-\lambda t)\,</math>

but the rate is unknown to the decision maker. If the [[prior probability]] distribution of λ is
:<math>p(\lambda) = \exp(-\lambda/k)/k\,</math>

then, the decision maker will expect that the probability of the reward after time ''t'' is
:<math>P(R_t) = \int_0^\infty P(R_t|\lambda) p(\lambda) d\lambda = \frac{1}{1 + k t}\,</math>

which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for λ.<ref name="sozou1998"/>

==Applications==
More recently these observations about [[discount function]]s have been used to study saving for retirement, borrowing on credit cards, and [[procrastination]]. However, hyperbolic discounting has been most frequently used to explain [[substance dependence|addiction]].

==See also==
* [[Time value of money]]
* [[Time preference]]
* [[Intertemporal choice]]
* [[Deferred gratification]]

==Footnotes==
<references/>

==Further reading==
* 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
* Musau, A. (2009): Modeling Alternatives to Exponential Discounting, MPRA Paper 16416, University Library of Munich, Germany.
* Rachlin, H. (2000). ''The Science of Self-Control'' Cambridge;London: Harvard University Press⏎
In [[economics]], '''hyperbolic discounting''' is a time-''inconsistent'' model of [[discounting]].

Given two similar rewards, humans show a preference for one that arrives sooner rather than later. Humans are said to ''discount'' the value of the later reward, by a factor that increases with the length of the delay. This process is traditionally modeled in form of [[exponential discounting]], a time-''consistent'' model of discounting. A large number of studies have since demonstrated that the constant [[discount rate]] assumed in exponential discounting is systematically being violated.<ref>{{cite journal |last=Frederick |first=Shane |first2=George |last2=Loewenstein |first3=Ted |last3=O'Donoghue |year=2002 |title=Time Discounting and Time Preference: A Critical Review |journal=[[Journal of Economic Literature]] |volume=40 |issue=2 |pages=351–401 |doi=10.1257/002205102320161311 }}</ref>  Hyperbolic discounting is a particular mathematical model devised as an improvement over exponential discounting.  Hyperbolic discounting has been observed in humans and animals.

In hyperbolic discounting, valuations fall very rapidly for small delay periods, but then fall slowly for longer delay periods. This contrasts with exponential discounting, in which valuation falls by a constant factor per unit delay, regardless of the total length of the delay. The standard experiment used to reveal a test subject's hyperbolic discounting curve is to compare short-term preferences with long-term preferences. For instance: "Would you prefer a dollar today or three dollars tomorrow?" or "Would you prefer a dollar in one year or three dollars in one year and one day?" For certain range of offerings, a significant fraction of subjects will take the lesser amount today, but will gladly wait one extra day in a year in order to receive the higher amount instead.<ref>{{cite journal |last=Thaler |first=R. H. |year=1981 |title=Some Empirical Evidence on Dynamic Inconsistency |journal=Economic Letters |volume=8 |issue=3 |pages=201–207 |doi=10.1016/0165-1765(81)90067-7 }}</ref>  Individuals with such preferences are described as "[[present-biased preferences|present-biased]]".

Individuals using hyperbolic discounting reveal a strong tendency to make choices that are inconsistent over time—they make choices today that their future self would prefer not to make, despite using the same reasoning. This [[dynamic inconsistency]] happens because the value of future rewards is much lower under hyperbolic discounting than under exponential discounting.<ref name="Laibson1997QJE">{{cite journal |authorlink=David Laibson |last=Laibson |first=David |year=1997 |title=Golden Eggs and Hyperbolic Discounting |journal=[[Quarterly Journal of Economics]] |volume=112 |issue=2 |pages=443–477 |doi=10.1162/003355397555253 }}</ref>

==Observations==
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,<ref>{{cite journal |last=Chung |first=S. H. |last2=Herrnstein |first2=R. J. |year=1967 |title=Choice and delay of Reinforcement |journal=Journal of the Experimental Analysis of Behavior |volume=10 |issue=1 |pages=67–74 |doi=10.1901/jeab.1967.10-67 }}</ref> [[George Ainslie (psychologist)|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 function|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|date=March 2009}}<ref name="Ainslie1974">{{cite journal |authorlink=George Ainslie (psychologist) |last=Ainslie |first=G. W. |year=1974 |title=Impulse control in pigeons |journal=Journal of the Experimental Analysis of Behavior |volume=21 |issue=3 |pages=485–489 |doi=10.1901/jeab.1974.21-485 }}</ref>

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 discounting|exponential]]" curve that would produce consistent choice over time.<ref name="Green et al">{{cite journal |last=Green |first=L. |last2=Fry |first2=A. F. |last3=Myerson |first3=J. |year=1994 |title=Discounting of delayed rewards: A life span comparison |journal=Psychological Science |volume=5 |issue=1 |pages=33–36 |doi=10.1111/j.1467-9280.1994.tb00610.x }}</ref><ref>{{cite journal |last=Kirby |first=K. N. |year=1997 |title=Bidding on the future: Evidence against normative discounting of delayed rewards |journal=Journal of Experimental Psychology: General |volume=126 |issue=1 |pages=54–70 |doi=10.1037/0096-3445.126.1.54 }}</ref> 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.

Hyperbolic discounting has also 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.<ref>{{cite book |last=Bickel |first=W. K. |last2=Johnson |first2=M. W. |year=2003 |chapter=Delay discounting: A fundamental behavioral process of drug dependence |editor1-first=G. |editor1-last=Loewenstein |editor2-first=D. |editor2-last=Read |editor3-first=R. F. |editor3-last=Baumeister |title=Time and Decision |location=New York |publisher=Russell Sage Foundation |isbn=0-87154-549-7 }}</ref><ref>{{cite journal |last=Madden |first=G. J. |last2=Petry |first2=N. M. |last3=Bickel |first3=W. K. |last4=Badger |first4=G. J. |year=1997 |title=Impulsive and self-control choices in opiate-dependent patients and non-drug-using control participants: Drug and monetary rewards |journal=Experimental and Clinical Psychopharmacology |volume=5 |issue= |pages=256–262 |doi= |pmid=9260073 }}</ref><ref>{{cite journal |last=Vuchinich |first=R. E. |last2=Simpson |first2=C. A. |year=1998 |title=Hyperbolic temporal discounting in social drinkers and problem drinkers |journal=Experimental and Clinical Psychopharmacology |volume=6 |issue=3 |pages=292–305 |doi=10.1037/1064-1297.6.3.292 }}</ref> Some evidence suggests pathological gamblers also discount delayed outcomes at higher rates than matched controls.<ref>{{cite journal |last=Petry |first=N. M. |last2=Casarella |first2=T. |year=1999 |title=Excessive discounting of delayed rewards in substance abusers with gambling problems |journal=Drug and Alcohol Dependence |volume=56 |issue=1 |pages=25–32 |doi=10.1016/S0376-8716(99)00010-1 }}</ref> 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<ref>{{cite journal |last=Poulos |first=C. X. |last2=Le |first2=A. D. |last3=Parker |first3=J. L. |year=1995 |title=Impulsivity predicts individual susceptibility to high levels of alcohol self administration |journal=Behavioral Pharmacology |volume=6 |issue=8 |pages=810–814 |doi=10.1097/00008877-199512000-00006 }}</ref> and cocaine<ref>{{cite journal |last=Perry |first=J. L. |last2=Larson |first2=E. B. |last3=German |first3=J. P. |last4=Madden |first4=G. J. |last5=Carroll |first5=M. E. |year=2005 |title=Impulsivity (delay discounting) as a predictor of acquisition of i.v. cocaine self-administration in female rats |journal=Psychopharmacology |volume=178 |issue=2–3 |pages=193–201 |doi=10.1007/s00213-004-1994-4 |pmid=15338104}}</ref> than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (gambling) outcomes more satisfying.<ref>{{cite journal |last=Madden |first=G. J. |last2=Ewan |first2=E. E. |last3=Lagorio |first3=C. H. |year=2007 |title=Toward an animal model of gambling: Delay discounting and the allure of unpredictable outcomes |journal=Journal of Gambling Studies |volume=23 |issue=1 |pages=63–83 |doi=10.1007/s10899-006-9041-5 }}</ref>

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 discount rate.<ref name="Green et al"/> The rate depends on a variety of factors, including the species being observed, age, experience, and the amount of time needed to consume the reward.<ref>{{cite book |last=Loewenstein |first=G. |authorlink2=Drazen Prelec |last2=Prelec |first2=D. |year=1992 |title=Choices Over Time |location=New York |publisher=Russell Sage Foundation |isbn=0-87154-558-6 }}</ref><ref>{{cite journal |last=Raineri |first=A. |last2=Rachlin |first2=H. |year=1993 |title=The effect of temporal constraints on the value of money and other commodities |journal=Journal of Behavioral Decision-Making |volume=6 |issue=2 |pages=77–94 |doi=10.1002/bdm.3960060202 }}</ref>

==Mathematical model==
[[File:Hyperbolic vs. exponential discount factors.svg|thumb|300px|right|Comparison of the discount factors of hyperbolic and exponential discounting. In both cases, <math>k=1</math>. Hyperbolic discounting is shown to ''over-value'' future assets compared to exponential discounting.]]
Hyperbolic discounting is mathematically described as:
:<math>f_H(D)=\frac{1}{1+kD}\,</math>

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. This is compared with the formula for exponential discounting:
:<math>f_E(D)=e^{-kD}\,</math>

====Simple derivation====

If <math>f(n)=2^{-n}\,</math> is an exponential discounting function and <math>g(n)=\frac{1}{1+n}\,</math> a hyperbolic function (with n the amount of weeks), then the exponential discounting a week later from "now" (n=0) is <math>\frac{f(1)}{f(0)}=\frac{1}{2}\,</math>, and the exponential discounting a week from week n is <math>\frac{f(n+1)}{f(n)}=\frac{1}{2}\,</math>, which means they are the same. For g(n), <math>\frac{g(1)}{g(0)}=\frac{1}{2}\,</math>, which is the same as for f, while <math>\frac{g(n+1)}{g(n)}=1-\frac{1}{n+2}\,</math>. From this one can see that the two types of discounting are the same "now", but when n is much greater than 1, for instance 52 (one year), <math>\frac{g(n+1)}{g(n)}\,</math> will tend to go to 1, so that the hyperbolic discounting of a week in the far future is virtually zero, while the exponential is still 1/2.

===Quasi-hyperbolic approximation===
The "quasi-hyperbolic" discount function, proposed by Laibson (1997),<ref name="Laibson1997QJE"/> approximates the hyperbolic discount function above in [[discrete time]] by

:<math>f_{QH}(0)=1, \, </math>

and

:<math>f_{QH}(D)=\beta \times \delta^D,\,</math>

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 "beta-delta" preferences. They retain much of the analytical tractability of [[exponential discounting]] while capturing the key qualitative feature of discounting with true hyperbolas.

===Present Value of an Standard Annuity===

The present value of a series of equal annual cash flows in arrears discounted hyperbolically:

:<math>V = P \frac{ln(1+kD)}{k}\,</math> 

Where V is the present value, P is the annual cash flow, D is the number of annual payments and k is the factor governing the discounting.

===Present Value of an Growth Annuity===

The present value of a series of annual cash flows growing at a constant rate, g, in arrears discounted hyperbolically:

:<math>V = \frac{k exp(-g/k +g(D+1/k)} {(-g/k +g(D+1/k)}\ ,</math> 

Where V is the present value, P is the annual cash flow, D is the number of annual payments and k is the factor governing the discounting.

==Explanations==
===Uncertain risks===
Notice that whether discounting future gains is rational 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 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 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 (e.g. due to the death of the restaurateur or the diner).

Uncertainty of this type can be quantified with [[Bayesian probability|Bayesian analysis]].<ref name="sozou1998">{{Cite doi|10.1098/rspb.1998.0534}}</ref> For example, suppose that the probability for the reward to be available after time ''t'' is, for known hazard rate λ
:<math>P(R_t|\lambda) = \exp(-\lambda t)\,</math>

but the rate is unknown to the decision maker. If the [[prior probability]] distribution of λ is
:<math>p(\lambda) = \exp(-\lambda/k)/k\,</math>

then, the decision maker will expect that the probability of the reward after time ''t'' is
:<math>P(R_t) = \int_0^\infty P(R_t|\lambda) p(\lambda) d\lambda = \frac{1}{1 + k t}\,</math>

which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for λ.<ref name="sozou1998"/>

==Applications==
More recently these observations about [[discount function]]s have been used to study [[Retirement#Saving_for_retirement|saving for retirement]], borrowing on [[credit card]]s, and [[procrastination]]. 
It has frequently been used to explain [[substance dependence|addiction]].<ref>{{cite journal |last=O'Donoghue |first=T. |first2=M. |last2=Rabin |year=1999 |title=Doing it now or later |journal=[[The American Economic Review]] |volume=89 |pages=103–124 }}</ref><ref>{{cite journal |last=O'Donoghue |first=T. |first2=M. |last2=Rabin |year=2000 |title=The economics of immediate gratification |journal=[[Journal of Behavioral Decision Making]] |volume=13 |pages=233–250}}</ref>
Hyperbolic discounting has also been offered as an explanation of the divergence between privacy attitudes and behaviour.<ref>{{cite book |last1=Acquisti |first1=Alessandro  |last2=Grossklags |first2=Jens  |editor1-first=J. |editor1-last=Camp |editor2-first=R. |editor2-last=Lewis |chapter=Losses, Gains, and Hyperbolic Discounting: Privacy Attitudes and Privacy Behavior |title=The Economics of Information Security|year=2004 |publisher=Kluwer |pages=179–186}}</ref>

==See also==
* [[Time value of money]]
* [[Time preference]]
* [[Intertemporal choice]]
* [[Deferred gratification]]
* [[Akrasia]]

==References==
{{Reflist|30em}}

==Further reading==
*{{cite journal |last=Ainslie |first=G. W. |year=1975 |title=Specious reward: A behavioral theory of impulsiveness and impulsive control |journal=Psychological Bulletin |volume=82 |issue=4 |pages=463–496 |doi=10.1037/h0076860 |pmid=1099599}}
*{{cite book |last=Ainslie |first=G. |year=1992 |title=Picoeconomics: The Strategic Interaction of Successive Motivational States Within the Person |location=Cambridge |publisher=Cambridge University Press |isbn= }}
*{{cite book |last=Ainslie |first=G. |year=2001 |title=Breakdown of Will |location=Cambridge |publisher=Cambridge University Press |isbn=978-0-521-59694-7 }}
*{{cite book |last=Rachlin |first=H. |year=2000 |title=The Science of Self-Control |location=Cambridge; London |publisher=Harvard University Press |isbn= }}

{{DEFAULTSORT:Hyperbolic Discounting}}
[[Category:Cognitive biases]]
[[Category:Behavioral finance]]

[[ja:双曲割引]]⏎
[[pl:Hiperboliczne obniżenie wartości]]