3.2 More Impatient for More Immediate Gratification
3.2.2 Exponential Discounting and Hyperbolic Discounting
You can see how odd hyperbolic discounting is, if you think about interest rates related to bank deposits; the same interest rate would be applied to term deposits with the same maturity (e.g., 1-year maturity), regardless of when the deposit is made (e.g., whether you start it today or 3 months later), unless the interest rate varies due to changes in market conditions. This implies that if you were to behave like a banker, you would apply the same discount rate per period of time to any future amount of money (i.e., to either an amount of money in 1 year or another in 1 year and 3 months).
This type of discounting is called “exponential discounting.” The terminology can be understood if you recall the arithmetic used to compute the initial principal amount of money needed to attain a certain amount of money at a certain future point in time. Consider 1-year compounding. Letrdenote the annual interest rate, a deposit ofAwill be compounded to (1þr)TAinTyears. Conversely, to attainAin Tyears, one must deposit [1/(1þr)]TAthis year. That is, a value inTyears can be converted into the present value by multiplying it by [1/(1þr)]T. The multiplier for
49.1%
30.7%
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Today 1 year later Today 1 year later
Means Medians
Elicited discount rate (%)
Fig. 3.2 Higher discount rates for more immediate future choices. Note: the discount rates are elicited as required interest rates to accept a 1-week delay in receiving USD 10. The differences in means and medians are significant at the 1 % and 5 % significance levels, respectively. Source: The Japan Internet Survey on Preferences Relating to Time and Risk 2010.N¼2,386
discounting, expressed as a function of the delayT, is called a “discount function.”
In the present case, the discount function is given as an exponential function of periodT, [1/(1þr)]T(for a more formal expression, see item (1) in Fig.3.3). The way of discounting is thus referred to as “exponential discounting.”
Figure 3.4 illustrates the relation between discount function values, denoted along the vertical axis, and delays to reward realization, denoted along the hori- zontal axis, in the case of exponential discounting. As a more distant future reward (i.e., a more delayed reward) is discounted more, the discount function is downward sloping. The discount function equals 1 when the delay T is 0, reflecting that a reward that is realized today is not discounted.
The discount rate determines the percentage decrease in the value of the discount function for one unit-period increase in delay. For exponential discounting, the discount rate is constant, and so in Fig. 3.4, the unit-period increases in delay always cause a constant percentage decrease in the discount function value. More specifically, with a constant discount rater, the relative present value of USD 1 with aT-year delay and the same amount with a (Tþτ)-year delay equals (1þr)τ— which depends solely on the difference in delayτ, but not onT. This reflects the property of exponential discounting that any future values, whether immediate or distant, are discounted by the same discount rate, as in the banker’s way of discounting.
In standard economics, so as to simplify the theoretical analysis of consumer behaviors, people have been assumed to be exponential discounters. The assump- tion actually simplifies the discussion; otherwise—that is, if the degree of impa- tience (i.e., the subjective discount rate) which is applied when making a future consumption/saving plan for a year starting from today differs from that which was applied when it was made last year—the two plans would become inconsistent as time passes. By assuming exponential discounting, researchers can concentrate on discussing the issues of national savings, economic growth, and economic devel- opment without being bothered by the cumbersome inconsistency problem.
However, people’s actual discounting does not take such a convenient form; a more immediate future gratification is more intensely discounted. Such dependence
= 1
1 + , > 0
= 1
1 + , > 0
Discounting the amount of the T-period future into the present value :
(1) Exponential discounting
(2) Hyperbolic discounting
Fig. 3.3 Exponential discounting and hyperbolic discounting
of the discount rate on the delay of gratification is known to be well described by specifying the discount function in terms of a hyperbolic function of delay T, 1/(1þaT),a>0, or more generally in its powered form, 1f =ð1þaTÞgb=a, b>0, where the hyperbolic function of variable T is a function that containsT in the denominator (Loewenstein and Prelec 1992). For exponential discounting, the discount function is an exponential function of delay, whereas for hyperbolic discounting, it is given as a hyperbolic function of delay.
To be rigorous, because hyperbolic discounting is defined using a mathematical model, whether or not people’s discounting is hyperbolic should be detected by checking whether their discounting actually satisfies the underlying mathematical formulation. However, the term “hyperbolic discounting” is typically used inter- changeably with “present bias” and “immediacy effect” to express the property wherein more immediate future gratification is discounted at a higher discount rate, without checking whether the resulting discount function is a rigorous hyperbolic discounting function of delay. In this book, I use the term “hyperbolic discounting”
in the broad sense. Incidentally, the term “hyperbolic” is used to describe a person with hyperbolic discounting; a hyperbolic consumer is a consumer who evaluates future gratification with hyperbolic discounting. Similarly, an exponential person is one who follows exponential discounting.
Figure3.5acompares the discount functions under exponential and hyperbolic discounting. As in Fig. 3.4, the discount function under hyperbolic discounting satisfies the basic property that the discount function declines from 1 as the delay increases from 0. The discount function schedule under hyperbolic discounting is more bowed than that under exponential discounting, so that its rate of decline increases as the delay increases. When the delay is short, as it increases, the value of the discount function under hyperbolic discounting declines more steeply than under exponential discounting; meanwhile, when the delay is long, the rate of decline in the discount function value due to a marginal increment of delay is smaller for hyperbolic discounting than for exponential discounting.
Fig. 3.4 The discount function under exponential discounting. The case of a 5 % discount rate
The differences in the discount function schedules of the two discounting cases, illustrated in Fig.3.5a, reflect the difference in how the discount rate relates to the delay. It is to be recalled that the discount rate equals the percentage decline of the discount function due to a unit-period increase in delay. Fig. 3.5b depicts the schedules of the discount rates that correspond to the discount function schedules in Fig.3.5a. It shows that for exponential discounting, the discount rate is constant for any delay, while for hyperbolic discounting, it decreases (increases) as the delay
Exponential discounting Hyperbolic discounting Quasi-hyperbolic discounting
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Exponential discounting Hyperbolic discounting Quasi-hyperbolic discounting (ｂ)
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Fig. 3.5 Exponential discounting, hyperbolic discounting, and quasi-hyperbolic discounting. (a) Delay and discount function value. (b) Delay and discount rates
increases (decreases). That is why the hyperbolic discounting model is used to describe how people tend to be less patient in awaiting more immediate gratification.
Figures 3.5a, b also depict the case of “quasi-hyperbolic” discounting, the simplest model of present bias. Quasi-hyperbolic discounting is a hybrid of expo- nential discounting and hyperbolic discounting; its discount function for aT-period delay is given byβδT, where 0<β1 and 0<δ1. The partδTof the discount function represents the exponential-like discounting in which discounting is inten- sified exponentially as the delay increases. Different from exponential discounting, quasi-hyperbolic discounting discounts future gratification additionally by multi- plyingβirrespective of how far into the future it is. Note thatβdoes not affect the relative values of gratifications in two different points infuturetime (e.g., tomorrow and the day after tomorrow). In contrast, when evaluating the relative values of gratification in the present time and in a future point in time (e.g., today and tomorrow), the future gratification is discounted byβδT—that is, it is discounted more intensively by factorβ. These properties indicate thatpresentgratification is evaluated particularly high relative to a delayed gratification, compared to when a delayed gratification is being evaluated more highly than the same amount of gratification, albeit more delayed. Quasi-hyperbolic discounting describes people’s present bias in this simplest manner. As illustrated in Fig.3.5b, the rate by which to discount a gratification one period ahead in the present time is higher, as in hyperbolic discounting, than that in later periods; meanwhile, the per-period dis- count rate is flat for any future period, as in exponential discounting.