Hyperbolic discounting
Encyclopedia
In behavioral 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. 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. Individuals with such preferences are described as "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.
'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, 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.
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. 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. Some evidence suggests pathological gamblers also discount delayed outcomes at higher rates than matched controls. 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 and cocaine than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (gambling) outcomes more satisfying.
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. 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.
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:
by
, and
,
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.
Uncertainty of this type can be quantified with Bayesian analysis
. For example, suppose that the probability for the reward to be available after time t is, for known hazard rate λ
but the rate is unknown to the decision maker. If the prior probability
distribution of λ is
then, the decision maker will expect that the probability of the reward after time t is
which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for λ.
s have been used to study saving for retirement, borrowing on credit card
s, and procrastination
. However, hyperbolic discounting has been most frequently used to explain addiction
.
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
Discount rate
The discount rate can mean*an interest rate a central bank charges depository institutions that borrow reserves from it, for example for the use of the Federal Reserve's discount window....
assumed in exponential discounting is systematically being violated. 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. Individuals with such preferences are described as "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
Dynamic inconsistency
In economics, dynamic inconsistency, or time inconsistency, describes a situation where a decision-maker's preferences change over time in such a way that what is preferred at one point in time is inconsistent with what is preferred at another point in time...
happens because the value of future rewards is much lower under hyperbolic discounting than under exponential discounting.
Observations
The phenomenon of hyperbolic discounting is implicit in Richard HerrnsteinRichard Herrnstein
Richard J. Herrnstein was an American researcher in animal learning in the Skinnerian tradition. He was one of the founders of quantitative analysis of behavior....
's "matching law
Matching law
In operant conditioning, the matching law is a quantitative relationship that holds between the relative rates of response and the relative rates of reinforcement in concurrent schedules of reinforcement...
," 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, George Ainslie
George Ainslie (psychologist)
George W. Ainslie is an American psychiatrist, psychologist and behavioral economist.Unusually for a psychiatrist, Ainslie undertook experimental animal research in operant conditioning, under the guidance of Howard Rachlin...
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
Hyperbolic function
In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...
, 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.
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. 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. Some evidence suggests pathological gamblers also discount delayed outcomes at higher rates than matched controls. 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 and cocaine than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (gambling) outcomes more satisfying.
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. 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.
Mathematical model
Hyperbolic discounting is mathematically described as: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:
Quasi-hyperbolic approximation
The "quasi-hyperbolic" discount function, proposed by Laibson (1997), approximates the hyperbolic discount function above in discrete timeDiscrete time
Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...
by
, and,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.
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 analysis
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
. For example, suppose that the probability for the reward to be available after time t is, for known hazard rate λ
but the rate is unknown to the decision maker. If the prior probability
Prior probability
In Bayesian statistical inference, a prior probability distribution, often called simply the prior, of an uncertain quantity p is the probability distribution that would express one's uncertainty about p before the "data"...
distribution of λ is
then, the decision maker will expect that the probability of the reward after time t is
which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for λ.
Present self vs. future self
Instant gratification pleases the present self while delayed rewards benefit the future self. If time-dependent decisions are viewed as competition between the present self and a future self, many apparent paradoxical decisions make much more sense.Applications
More recently these observations about discount functionDiscount function
A discount function is used in economic models to describe the weights placed on rewards received at different points in time. For example, if time is discrete and utility is time-separable, with the discount function...
s have been used to study saving for retirement, borrowing on credit card
Credit card
A credit card is a small plastic card issued to users as a system of payment. It allows its holder to buy goods and services based on the holder's promise to pay for these goods and services...
s, and procrastination
Procrastination
In psychology, procrastination refers to the act of replacing high-priority actions with tasks of low-priority, and thus putting off important tasks to a later time...
. However, hyperbolic discounting has been most frequently used to explain addiction
Substance dependence
The section about substance dependence in the Diagnostic and Statistical Manual of Mental Disorders does not use the word addiction at all. It explains:...
.
See also
- Time value of moneyTime value of moneyThe time value of money is the value of money figuring in a given amount of interest earned over a given amount of time. The time value of money is the central concept in finance theory....
- Time preferenceTime preferenceIn economics, time preference pertains to how large a premium a consumer places on enjoyment nearer in time over more remote enjoyment....
- Intertemporal choiceIntertemporal choiceIntertemporal choice is the study of the relative value people assign to two or more payoffs at different points in time. Most choices require decision-makers to trade-off costs and benefits at different points in time. These decisions maybe about savings, work effort, education, nutrition,...
- Deferred gratificationDeferred gratificationDeferred gratification and delayed gratification denote a person’s ability to wait in order to obtain something that he or she wants. This intellectual attribute is also called impulse control, will power, self control, and “low” time preference, in economics...