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{{Importartikel}}
'''Hyperbolische Diskontierung''' ist ein Modell der zeitinkonsistenten [[Abzinsung und Aufzinsung|Diskontierung]] aus der [[Wirtschaftswissenschaft]].

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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 [[
hHyperbolic 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 Analysi(contracted; show full)ot;/> 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  ==
[[FileDatei:Hyperbolic vs. exponential discount factors.svg|thumb|300px|right|Comparison of the discount factors of hyperbolic and exponential discounting.]]
Der Diskontierungsfaktor wird durch die Funktion
:<math>f_H(D)=\frac{1}{1+kD}\,</math>

beschrieben. Der subjektive Wert einer in der Zukunft liegenden Belohnung wird dabei durch Multiplikation des Diskontierungsfaktors ''f(D)'' mit dem jetzigen Wert errechnet. ''D'' ist die Verzögerung der Belohnung und ''k'' ein Parameter, der die Stärke der Diskontierung – die Diskontierungsrate (?) – beschreibt. Im Gegensatz dazu lautet die Formel für exponentielle Diskontierung:
:<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 "[[pPresent-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 dDoi|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 [[sSubstance 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 978-0-521-59694-7
* 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

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[[Category:Cognitive biases]]
[[Category:Behavioral finance]]

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