Hyperbolic Discounting
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In
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
, hyperbolic discounting is a time-''inconsistent'' model of
delay discounting In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a Production (economics), good or some cash at an earlier date compared with ...
. It is one of the cornerstones of
behavioral economics Behavioral economics studies the effects of psychological, cognitive, emotional, cultural and social factors on the decisions of individuals or institutions, such as how those decisions vary from those implied by classical economic theory. ...
and its brain-basis is actively being studied by
neuroeconomics Neuroeconomics is an interdisciplinary field that seeks to explain human decision-making, the ability to process multiple alternatives and to follow through on a plan of action. It studies how economic behavior can shape our understanding of the ...
researchers. According to the discounted utility approach,
intertemporal choice Intertemporal choice is the process by which people make decisions about what and how much to do at various points in time, when choices at one time influence the possibilities available at other points in time. These choices are influenced by the ...
s are no different from other choices, except that some consequences are delayed and hence must be anticipated and discounted (i.e., reweighted to take into account the delay). 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 the financial world, this process is normally modeled in the form of
exponential discounting In economics exponential discounting is a specific form of the discount function, used in the analysis of choice over time (with or without uncertainty). Formally, exponential discounting occurs when total utility is given by :U(\_^)=\sum_^\del ...
, a time-''consistent'' model of discounting. Many psychological studies have since demonstrated deviations in instinctive preference from the constant discount rate assumed in exponential discounting. Hyperbolic discounting is an alternative mathematical model that agrees more closely with these findings. According to hyperbolic discounting, valuations fall relatively rapidly for earlier delay periods (as in, from now to one week), but then fall more slowly for longer delay periods (for instance, more than a few days). For example, in an early study subjects said they would be indifferent between receiving $15 immediately or $30 after 3 months, $60 after 1 year, or $100 after 3 years. These indifferences reflect annual discount rates that declined from 277% to 139% to 63% as delays got longer. This contrasts with exponential discounting, in which valuation falls by a constant factor per unit delay and the discount rate stays the same. 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?" It has been claimed that 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". The most important consequence of hyperbolic discounting is that it creates temporary preferences for small rewards that occur sooner over larger, later ones. 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 have made, despite knowing the same information. This
dynamic inconsistency In economics, dynamic inconsistency or time inconsistency is a situation in which a decision-maker's preferences change over time in such a way that a preference can become inconsistent at another point in time. This can be thought of as there be ...
happens because hyperbolas distort the relative value of options with a fixed difference in delays in proportion to how far the choice-maker is from those options.


Observations

The phenomenon of hyperbolic discounting is implicit in
Richard Herrnstein Richard Julius Herrnstein (May 20, 1930 – September 13, 1994) was an American psychologist at Harvard University. He was an active researcher in animal learning in the B. F. Skinner, Skinnerian tradition. Herrnstein was the Edgar Pierce Profess ...
's "
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. For example, if two response alternatives A ...
", which states that when dividing their time or effort between two non-exclusive, ongoing sources of reward, most subjects allocate 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 when the smaller, sooner reward is preferred, this preference can be reversed by increasing both rewards' delays by the same absolute amount. Ainslie's research showed that a substantial number of subjects reported that they would prefer $50 immediately rather than $100 in six months, but would NOT prefer $50 in 3 months rather than $100 in nine months, even though this was the same choice seen at 3 months’ greater distance. More significantly, those subjects who said they preferred $50 in 3 months to $100 in 9 months said they would NOT prefer $50 in 12 months to $100 in 18 months—again, the same pair of options at a different distance—showing that the preference-reversal effect did not depend on the excitement of getting an immediate reward.Ainslie, George and Haendel, V. (1983) The motives of the will. in E. Gottheil, K. Druley, T. Skodola, H. Waxman (eds.),''Etiology Aspects of Alcohol and Drug Abuse'', Springfield, Ill.: Charles C. Thomas, pp. 119-140. Nor does it depend on human culture; the first preference reversal findings were in rats and pigeons. Many subsequent experiments have confirmed that spontaneous preferences by both human and nonhuman subjects follow a
hyperbolic curve In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
rather than the conventional,
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...
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 discounters are more likely to consume
alcohol Alcohol most commonly refers to: * Alcohol (chemistry), an organic compound in which a hydroxyl group is bound to a carbon atom * Alcohol (drug), an intoxicant found in alcoholic drinks Alcohol may also refer to: Chemicals * Ethanol, one of sev ...
and
cocaine Cocaine (from , from , ultimately from Quechuan languages, Quechua: ''kúka'') is a central nervous system (CNS) stimulant mainly recreational drug use, used recreationally for its euphoria, euphoric effects. It is primarily obtained from t ...
than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (
gambling Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three el ...
) 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


Step-by-step explanation

Suppose that in a study, participants are offered the choice between taking ''x'' dollars immediately or taking ''y'' dollars ''n'' days later. Suppose further that one participant in that study employs exponential discounting and another employs hyperbolic discounting. Both participants know that they can invest the money they receive today in a savings plan that gives them an interest of ''r''. Both of them realize that they should take ''x'' dollars immediately if the future value of the savings plan will yield more than ''y'' dollars ''n'' days later. Each participant correctly understands the fundamental question being asked: "For any given value of ''y'' dollars and ''n'' days, what is the minimum amount ''x'' of dollars, that I should be willing to accept? In other words, how many dollars would I need to invest today to get ''y'' dollars ''n'' days from now?" Each will take ''x'' dollars if ''x'' is greater than the answer that they calculate, and each will take ''y'' dollars ''n'' days from now if ''x'' is smaller than that answer. However, the methods that they use to calculate that amount and the answers that they get will be different, and only the exponential discounter will use the correct method and get a reliably correct result: * The exponential discounter thinks "The savings plan adds to its value, in each day, ''r'' percent of the value that it had the previous day. So every day it multiplies its value once by (100% + ''r''%). So if I hold the investment for ''n'' days, its value will have multiplied itself by this amount ''n'' times, making that value (100% + ''r''%)''n'' of what it was at the start – that is, (1 + ''r'')''n'' times what it was at the start. So to figure out how much I would need to start with today to get ''y'' dollars ''n'' days from now, I need to divide ''y'' dollars by + ''r'''n''." * The hyperbolic discounter, however, thinks "The savings plan adds to its value, in each day, ''r'' percent. Therefore, after ''n'' days, it adds to its value ''r×n'' percent
here lies the hyperbolic discounter's error Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a TV ...
So to figure out how much I would need to start with today to get ''y'' dollars ''n'' days from now, I need to divide ''y'' dollars by + ''n''×''r''" As ''n'' becomes very large, the value of (1 + ''r'')''n'' becomes much larger than the value of + ''n''×''r'' with the effect that the value of ''y'' / (1 + ''r'')''n'' becomes much smaller than the value of ''y''/ '1 + n×r'' Therefore, the minimum value of ''x'' (the number of dollars in the immediate choice) that suffices to be greater than that amount will be much smaller than the hyperbolic discounter thinks, with the result that they will perceive ''x''-values in the range from ''y''/(1 + ''r'' )''n'' to ''y''/ '1 + n×r''inclusive as being too small and, as a result, irrationally turn those alternatives down when they are in fact the better investment.


Formal model

Hyperbolic discounting is mathematically described as :g(D)=\frac\, where ''g''(''D'') is the
discount factor Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee.See "Time Value", "Discount", "Discount Yield", "Compound Interest", "Efficient ...
that multiplies the value of the reward, ''D'' is the delay in the reward, and ''k'' is a parameter governing the degree of discounting (for example, the
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
). This is compared with the formula for exponential discounting: :f(D)=e^\,


Comparison

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


Quasi-hyperbolic approximation

The "quasi-hyperbolic" discount function (sometimes called "beta-delta discounting"), proposed by Laibson (1997), approximates the hyperbolic discount function above in
discrete time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
by :f(D)=\begin 1 \quad D = 0\\ \beta \delta^D \quad D = 1, 2, 3, ... \end where ''β'' and ''δ'' are constants between 0 and 1; and ''D'' is the delay in the reward, but now it takes only integer values. The condition ''f''(0) = 1 states that rewards taken at the present time are not discounted. Quasi-hyperbolic discounting retain much of the analytical tractability of
exponential discounting In economics exponential discounting is a specific form of the discount function, used in the analysis of choice over time (with or without uncertainty). Formally, exponential discounting occurs when total utility is given by :U(\_^)=\sum_^\del ...
while capturing the key qualitative feature of hyperbolic discounting.


Explanations


Uncertain risks

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 analysis Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
. For example, suppose that the probability for the reward to be available after time ''t'' is, for known hazard rate λ, :P(R_t, \lambda) = \exp(-\lambda t),\, but the rate is unknown to the decision maker. If the
prior probability In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...
distribution of λ is :p(\lambda) = \exp(-\lambda/k)/k,\, then the decision maker will expect that the probability of the reward after time ''t'' is :P(R_t) = \int_0^\infty P(R_t, \lambda) p(\lambda) d\lambda = \frac,\, which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for λ.


Applications

More recently these observations about
discount 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 f(t) having a negative fi ...
s have been used to study saving for retirement, personal income to drug addiction., borrowing on
credit card A credit card is a payment card issued to users (cardholders) to enable the cardholder to pay a merchant for goods and services based on the cardholder's accrued debt (i.e., promise to the card issuer to pay them for the amounts plus the o ...
s, and
procrastination Procrastination is the action of unnecessarily and voluntarily delaying or postponing something despite knowing that there will be negative consequences for doing so. The word has originated from the Latin word ''procrastinatus'', which itself evo ...
. It has frequently been used to explain
addiction Addiction is a neuropsychological disorder characterized by a persistent and intense urge to engage in certain behaviors, one of which is the usage of a drug, despite substantial harm and other negative consequences. Repetitive drug use o ...
. Hyperbolic discounting has also been offered as an explanation of the divergence between privacy attitudes and behaviour.


Present values of annuities


Present value of a standard annuity

The present value of a series of equal annual cash flows in arrears discounted hyperbolically is :V = P \frac,\, 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.


Criticism

Several alternative explanations of non-exponential discounting have been proposed. An article from 2003 noted that this pattern might be better explained by a
similarity heuristic Similarity may refer to: In mathematics and computing * Similarity (geometry), the property of sharing the same shape * Matrix similarity, a relation between matrices * Similarity measure, a function that quantifies the similarity of two objects * ...
than by hyperbolic discounting. Subjects have also reported changing relative preferences as they see more details of what they are choosing—a “temporal construal” effect. A study by Daniel Read introduces "subadditive discounting": the fact that discounting over a delay increases if the delay is divided into smaller intervals. This hypothesis may explain the main finding of many studies in support of hyperbolic discounting—the observation that impatience declines with time–while also accounting for observations not predicted by hyperbolic discounting. However, although these observations depart from exponential discounting, they do not entail preference reversal as time from the choice to the earlier reward increases. Arousal of appetite or emotion does sometimes lead to preference reversal, and this has been the most widely accepted alternative to a simply hyperbolic function:
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
or quasi-hyperbolic discounting fuses exponential curves with an arousal bump as a visceral reward becomes imminent. Such cases are obviously important, but still do not account for cases where either both or neither choice is made during arousal. The most obvious objection to hyperbolic discounting is that many or most people learn to choose consistently over time in most situations. Similarly, a 2014 paper criticized the existing studies for mostly using data collected from university students and being too quick to conclude that the hyperbolic model of discounting is correct. Human experiments have frequently reported wide between-subject variations. If overcoming the tendency to temporary preference takes learning, the next obvious task for experimenters is to test theories of how and when this learning occurs (e.g. Ainslie, 2012).


See also

*
Akrasia Akrasia (; Greek , "lacking command" or "weakness", occasionally transliterated as acrasia or Anglicised as acrasy or acracy) is a lack of self-control, or acting against one's better judgment. The adjectival form is "akratic". Classical approa ...
*
Deferred gratification Delayed gratification, or deferred gratification, is the resistance to the temptation of an immediate pleasure in the hope of obtaining a valuable and long-lasting reward in the long-term. In other words, delayed gratification describes the pro ...
*
Intertemporal choice Intertemporal choice is the process by which people make decisions about what and how much to do at various points in time, when choices at one time influence the possibilities available at other points in time. These choices are influenced by the ...
*
Temporal motivation theory Temporal motivation theory (TMT) is an integrative motivational theory developed by Piers Steel and Cornelius J. König, the theory emphasizes time as a critical and motivational factor. The argument for a broad, integrative theory stems from the ...
*
Time preference In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good or some cash at an earlier date compared with receiving it at a later ...
*
Time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The t ...


References


Further reading

* * * * {{DEFAULTSORT:Hyperbolic Discounting zh-yue:雙曲線折扣 Cognitive biases Behavioral finance Intertemporal economics