Discussion Paper No. 679

TIME DISCOUNTING:DECLINING IMPATIENCE AND
INTERVAL EFFECT
Yusuke Kinari
Fumio Ohtake
andYoshiro Tsutsui

January 2007

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Discussion Paper No. 679

TIME DISCOUNTING:DECLINING IMPATIENCE AND
INTERVAL EFFECT
Yusuke Kinari
Fumio Ohtake
andYoshiro Tsutsui

January 2007

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Time Discounting: Declining Impatience and Interval EffectYusuke Kinari
Graduate School of Economics, Osaka University
JSPS Research Fellow
Fumio Ohtake
Institute of Social and Economic Research, Osaka University
Yoshiro Tsutsui∗Institute of Social and Economic Research, Osaka University

Correspondence:
Yoshiro Tsutsui,Institute of Social and Economic Research,
Osaka University
6-1 Mihogaoka, Ibaraki,
567-0047 Japan,
Phone: +81-6-6879-8560
Fax: +81-6-6878-2766e-mail: tsutsui@econ.osaka-u.ac.jp∗

An earlier version of this paper was presented in Behavioral Economics Conference and Annual
Meeting of Japanese Economic Association. The authors are grateful to comments by Takanori Ida,
Fumihiko Hiruma, Shinsuke Ikeda, Kazumasa Umeda, and the participants of the conferences.

Abstract
Most studies have not distinguished delay from intervals, so that whether the declining
impatience really holds has been an open question. We conducted an experiment that
explicitly distinguishes them, and confirmed the declining impatience. This implies thatpeople make dynamically inconsistent plans. We also found the interval effect that the
per-period time discount rate decreases with prolonged intervals. We show that the interval
and the magnitude effects are caused, at least partially, because subjects’ choices are
influenced by the differential in reward amount, while Weber’s law solves neither the delay
nor the interval effects.

JEL classification numbers: D81; D90
Keywords: time discount rate; declining impatience; interval effect; subadditivity; Weber’slawIntroduction
Decision-making is time consistent, if per-period time discount rate is constant overtime.1 Many experiments in economics and psychology, however, have found declining
impatience in that per-period time discount rate of the immediate future is higher than that ofthe distant future. Declining impatience implies a reversal of preference and people regret
their plans they have made in the past (Laibson, 1997). Declining impatience has beenformalized as (quasi) hyperbolic time discounting function and has become a standard
decision-making model in behavioral sciences.
However, in many economic experiments on declining impatience so far, two elements,
delay and interval, are mixed. To measure time discount rate in experiments, three timings
should be distinguished—now, earlier option, and later option. Because “now” is fixed, twoquantities define the situation fully; delay represents time difference between now and an
earlier option, and interval represents time difference between earlier and later options.
Although time discount rates may depend on these two factors, most experiments did not
distinguish them. Nevertheless, the researchers interpreted their results indicating that time
discount rates solely depend on the delay, and reported that per-period time discount rate
decreases as the delay increases. However, people’s discounting may depend heavily on theinterval, not on the delay, so that decision-making may be time consistent.
Read (2001) has already pointed out this problem. He divided the total interval (18months) into three subintervals (six months each) and compared discount rates to find“subadditive time discounting” that a product of three discount rates (plus unity) for
subintervals is higher than the discount rate (plus unity) for the total interval. Read (2001)
denied the declining impatience because of lack of evidence when intervals are controlled.

1

This paper does not consider the case of endogenous time discount rate (Uzawa, 1968). If time
discount rate of an individual changes over time, which he or she does not know ex ante, his or her
decision making may be time inconsistent. However, these cases are not objects of this paper.

1

According to Read (2001), who investigated declining impatience by distinguishing thedelay from interval, decision making of human beings is time consistent. Declining
impatience or hyperbolic discounting, which is a standard model in psychology andphysiological neuroscience as well as in behavioral economics, has not indeed had a firm
empirical foundation.
The purpose of this paper is to elucidate whether declining impatience is really abehavioral character of human beings. To achieve this, we requested subjects to choose one of
the two options: an earlier option where one can get a smaller reward sooner; and a later
option where one can get a larger reward, but later. Specifying these options, we control thefollowing three factors: (1) the amount of reward, (2) the interval, and (3) the delay. Declining
impatience is a feature related to (3). As for (1), the magnitude effect that the per-period time
discount rate is smaller for a larger amount of reward has been reported in the literature (e.g.,
Green et al., 1997). On the other hand, a feature related to (2) has seldom been on the focus.
The second aim of this paper is to elucidate “the interval effect” in which the per-period time
discount rate is decreasing to the length of the interval.
As noted above, previous studies have not distinguished the delay from the interval andhave interpreted their results as an outcome based on a change in the delay. In contrast tothem, we explicitly distinguish the delay from the interval, and analyze whether per-period
time discount rate decreases as the delay increases.
With regard to the results of Read (2001), we point out that his analysis has a problem
because even the shortest delay in his experiments is six months, which is too long to analyze
the effect of a change in the delay. The declining impatience should appear at a relatively
immediate future (Ikeda et al., 2005 and Frederick et al., 2001). Therefore, Read’s (2001)
assertion that there is no evidence for the declining impatience needs more examination. In
our experiment, we test much shorter delays such as one day or one week to find the declining2

impatience within eight weeks of delay, with the intervals being controlled.
In addition, we argue that the interval effect defined above also mirror a conspicuous
characteristic of the human being. Specifically, we first show that the interval effect is asufficient condition for the subadditive time discounting. In this sense, the interval effect is amore general concept than the subadditive time discounting. Then, we report that both theinterval effect and the subadditive time discounting are observed in our experiments.
The third aim of this paper is to investigate possible causes of the three anomalies—thedeclining impatience, interval effect, and magnitude effect. Specifically, we examine Weber’s
law that asserts logarithmic time perception and the differential effect hypothesis that we
newly propose.
The rest of this paper is organized as follows: In section 1, we point out problems inprevious studies on time discounting, and present how we address these problems. Section 2
explains our experimental procedure. Section 3 reports experimental results. In section 4, we
check the robustness of our results using a questionnaire survey conducted at the end of theexperiment. In section 5, we inquire the cause of the anomalies on time discounting. Section 6
concludes this paper.

1. Problems Faced in Previous Studies
Many studies by Richards et al. (1999), Pender (1996), Kirby and Marakovic (1995),
Myerson and Green (1995), Benzion et al. (1989), and Thaler (1985) typically ask subjects
how much will they demand if instead of receiving X dollars now, they receive it at time t (inthe future). Varying t and X in their experiments, they drew the conclusion that per-period
time discount rates diminish with delay, since per-period time discount rates from now to t,

R ( 0 , t ), is smaller than that from now to s, R (0, s) , s < t , that is,R(0, s ) ≥ R(0, t ), s < t .

(1)
3

In their deduction, however, they ignore the fact that at the same time, the intervalchanges from s to t. One can interpret inequality (1) as the per-period time discount rate
decreases with increase in the interval.
To solve this problem, we test whether the per-period time discount rate is diminishingwith the delay by comparing discount rates for various delays by controlling the intervals.
However, Read (2001) has already pointed out the above-mentioned problem. He
compared the time discount rate (plus unity) for the total period of 18 months with the
products of three time discount rates (plus unity) for subintervals of six months each, andfound the subadditive time discounting represented by the following inequality (2).

1 + β (0m , 18 m) ≤ (1 + β (0 m,6 m)) × (1 + β (6 m, 12 m)) × (1 + β (12 m,18 m))(2)

where m stands for month(s), and β (a, b) indicates time discount rate for the period fromtime a to time b. That is, if receiving X dollars at time a is indifferent to receiving Y dollars attime b, β (a, b) is (Y − X ) X . Using the per-period time discount rate R(a, b) , defining six
months as a unit period, inequality (2) is rewritten as(1 + R(0 m, 18 m))3 ≤ (1 + R(0 m, 6 m)) × (1 + R(6 m, 12 m)) × (1 + R(12 m, 18 m)) .

(3)

Meanwhile, the interval effect means that the longer the interval, the lower the per-period time
discount rate with the delay and magnitude effects being adjusted. That is,R(a, b) ≤ R( a′, b′), if b − a ≥ b′ − a′, ∀a, b, a′, b′ .

(4)

It is easy to prove that inequality (3) is satisfied if inequality (4) is satisfied. Therefore, theinterval effect is a sufficient condition for the subadditive time discounting.
The interval effect and the subadditive time discounting are independent phenomena
from the declining impatience. Read (2001) elicited a pure effect of change in delay by fixing
the intervals at six months, but founds no evidence of the declining impatience.
His problem is that even the shortest delay is six months, which is too long to find theeffect of a change in the delay. Ikeda et al. (2005) found that a change in the delay longer than
4

one month does not affect time discount rates. Frederick et al. (2002) reported that there is noevidence of declining impatience when the delay is over one year. This indicates thatdeclining impatience appears at the relatively immediate future.
Given these arguments, we examine whether declining impatience is really the case,setting the delay at an adequately short period. Then, we regress per-period time discountrates over the delay, interval, amount of reward, and attributes of subjects to confirm thedeclining impatience, interval effect, and magnitude effect. In addition, we test whether thesubadditive time discounting is observed in various conditions. Read (2001) found thesubadditivity for the case with the interval divided into three six-month subintervals; however,
we examine the cases with a new approach of two or four divisions of shorter intervals.

2.

Experiment

2.1

Subjects
Our experiment was conducted in the morning and afternoon for four days fromFebruary 14 to 17, 2006. The experiment consists of eight sessions in total, using different
subjects. Subjects were 219 students in total who were affiliated to Osaka University. Most of
the experiments on time discounting employ 30 or 40 subjects; however, our experiment waslarge-scale.2
Table 1 provides some information about our subjects. As for age, subjects are 18 to 37
years of age, but most of them are around 20 years old. Among 219 subjects, 26% are female
and 17% are graduate students. Subjects are well-diversified over nine departments of Osaka
University.3

2.2
2
3

Procedure

Benzion et al. (1989) experimented with 282 students, but they did not reward the subjects.
No subjects are from the department of dentistry.

5Each subject is required to choose one of the two options displayed on a computer
screen in front.4 One is option A wherein subjects get smaller reward earlier, and the other is
option B in which subjects get larger rewards later.
As noted in the previous sections, we control three factors: the amount of reward,
interval, and delay. We define the delay as the difference between the present (time 0) and thetime of receipt stated in the option A, and specify eight different delays. We define the intervalas the difference between the time of receipt stated in the options A and B, and specify fourintervals. Pairing these delays and intervals, we specify 15 combinations of receipt timing
shown in Table 2.
We define the amount of reward as the amount stated in the option A; 180 different
amounts are generated randomly from the truncated normal distribution with the mean of
2000; standard deviation of 1000; upper bound of 4000; and lower bound of 500. These 180
amounts are randomly assigned to each question. Twelve rates of return (1%, 2%, 3%, 4%,
5%, 7.5%, 10%, 12.5%, 15%, 25%, 35%, and 50%) are also randomly assigned to 12questions of each combination. The amount of reward stated in the option B is calculated byadding the amount of reward tantamount to the above rates of return to the amount in theoption A.
Thus, there are 15 combinations of receipt timing in the options A and B as shown in
Table 2, and we ask 12 questions for each combination, each of which is specified withdifferent rates of return ranging from 1% to 50%. Consequently, subjects are requested toanswer 180 questions. These 180 questions are determined in advance, and all subjects answer
the same questions. However, the order that the questions are presented to subjects is
randomly determined for each subject.
Subjects are paid 2000 yen in cash for participation in the experiment. At the end of the4

The experiment was programmed and conducted with the software z-Tree (Fischbacher, 1999).

6

experiment, we randomly select one question out of the 180 questions, for which we will
actually pay the reward. Subjects will receive a reward, a gift voucher from amazon.co.jp, on
the date stated in the option they have chosen in this selected question.

2.3

Estimation of Time Discount Rates
We estimate 15 per-period time discount rates per subject, each of which corresponding

to each combination of time of receipt. The procedure of the estimation is as follows. As
noted in the previous subsection, 12 questions are asked for each combination of receipt
timing, changing the rate of return. The answers for these 12 questions, A or B, were sorted inascending order as per the rates of return implied by the options. If a subject chooses theoption A when the rate of return is low and option B when it is high, so that they switch fromthe option A to B only once, then we determine the time discount rate of the subject as anaverage of the two rates of return immediately before and after the switch.
However, some subjects made two or more switches between the two options during
answering of the 12 questions with the same combination of the receipt timing.5 This isprobable because the questions were asked randomly. In such a multi-switching case, weestimate the time discount rates using a logit model. Specifically, the dependent variable is adummy variable that takes unity when a subject chooses B (later option) and zero otherwise,
and is regressed over the 12 rates of return. Then, using the estimates of the coefficient, we
calculate back a rate of return where a probability of choosing B becomes just 0.5, and regard
it as the time discount rate of the subject for this combination of receipt timing. We exclude
the data from our analyses if the estimated time discount rates using the logit model are out of
the 1% to 50% range. Moreover, if a subject consistently chooses A (or B) for 12 questions of

5

1,352 out of 3,285 time discount rates (219 subjects × 15 combinations of receipt timings) fall into
this multi-switching case.

7the same combination of receipt timings, we exclude those observations from our analyses.6
In this paper, we denote the subjects’ time discount rates between time a and b
calculated by the above procedure as β (a, b) , and the corresponding per-period time

discount rate R (a, b) , where the unit period is set at two weeks.7 In Table 3, we show thenumber of observations, the mean and the standard deviation of per-period time discount rates,
and the mean of the amount of reward for each combination of receipt timing.

3. Results
3.1 Declining Impatience
To confirm that the per-period time discount rates are diminishing with the delay, wechoose the combinations in which the delays are different but the intervals are the same, andcompare their discount rates.8 In other words, we select the observations of the same intervaland examine whether there exists a negative correlation between the delay and the time
discount rate within these observations.
Let us first analyze the case where the interval is set at two weeks. Figure 1, which
depicts the relation of the per-period time discount rates (vertical axis) and the delay(horizontal axis), reveals that the per-period time discount rate tends to decline as the delay
increases, keeping the interval at two weeks. While the discount rates are in a range from
0.090 to 0.095 when the delay is within two weeks, they are less than 0.085 on and over afour-week delay, suggesting that the per-period time discount rates shift downward during the6957 out of 3,285 observations are excluded; 625 (43 respectively) observations are excluded because
subjects chose B (A) for all questions; 289 observations are excluded because the time discount rates
estimated by the logit model are not in the range from 1% to 50%.
7
For example, the value of 1 + R (0 day , 4 weeks ) is calculated as the square root of 1 + β (0 day, 4 weeks) ,and the value of 1 + R(0 day, 6 weeks) is calculated as the cube root of 1 + β ( 0 day , 6 weeks ) .
8
We cannot control the amount of reward because it is randomly chosen for each question. As shownin the far-right column of Table 3, however, the average amounts do not differ substantially among 15
combinations of receipt timing. Therefore, possible biases due to the uncontrolled amount may betrivial.

8two- to four-week delay.
To examine its significance, we pick up observations whose intervals are the same andconduct a mean-difference test among groups with different delays. The test results areshown in Table 4, which reveals that the discount rates within a two-week delay are
significantly different from those over a four-week delay. In addition, the discount rates
within a two-week delay are not significantly different from each other, and the discountrates over a four-week delay seldom differ significantly from each other. These results
suggest that the per-period time discount rates with a two-week interval substantiallydecrease during a two- to four-week delay.
Figure 2 depicts the relationship between the per-period time discount rates and thedelay, with the interval being kept at four weeks. Table 5 presents the result of themean-difference tests for the case of a four-week interval. Although there is no significant
difference between the per-period time discount rate with a two-week delay and that with afour-week delay, the per-period time discount rate with zero-day delay is significantly
different from the per-period time discount rates with two- and four-week delays. These
results confirm the declining impatience between the zero-day and two-week delays, for thecase of a four-week interval.
When the interval is fixed at six weeks, the average time discount rate of the zero-day
delay R(0 day, 6 weeks) is 0.066, while that of the two-week delay R (2 weeks, 8 weeks) is
0.044; the latter is significantly lower than the former (t-value is 4.948). This indicates thatthe per-period time discount rates also diminish with the delay in the case of the six-weekinterval. In sum, declining impatience is recognized during the shorter delays irrespective of
the interval.

93.2 Regression Analysis with Panel Data
In the previous subsection, we conducted mean-difference tests keeping the intervals
fixed and found declining impatience. In this subsection, we run a regression to confirm
declining impatience controlling the magnitude effect and subjects’ attributes in addition to
the interval effect. The analysis will elucidate how time discount rates depend on the delay,interval, amount of reward, and attributes of the subjects.
Specifically, we regress the panel data of the per-period time discount rates of 3285
observations (i.e., 219 subjects × 15 combinations of receipt timing) on experimental
conditions (the delay, interval, and amount of reward) and subjects’ attributes (gender, age,and their affiliated department).9 This analysis has an advantage of comprehensive use of theexperimental results.
The following variables of experimental conditions and attributes of subjects are
employed in the estimation: eight dummy variables for delay, taking the zero-day delay as abenchmark; three dummy variables for the interval, taking the two-week interval as abenchmark; and the average amount of reward that is defined as the average of the amount of
reward of 12 questions in the same combination of receipt timing.10 As for subjects’ attributes,
we employ a male dummy variable for gender, age, a dummy variable standing for graduates,and eight dummy variables standing for each department, taking the department ofengineering as the reference group.
The regression results are presented in Table 6. Model 1 is a regression which considers
only experimental conditions, while Model 2 stands for a regression which incorporates
subjects’ attributes into Model 1. The results of Model 1 reveal that although one-day andone-week delays do not affect significantly, the longer delay dummies have significant
9

Subjects’ attributes are based on the questionnaire survey conducted at the end of the experiment.
We use the average of 12 amounts of reward because it is hard to identify the amount of reward
corresponding to the dependent variable, especially in the case where the time discount rate isestimated by a logit model.

1010negative effects. In addition, the coefficients of the dummies keep decreasing until theeight-week delay. These results imply that the declining impatience is observed even when theeffect of the amount of reward as well as the interval is controlled.
All the intervals have significant negative coefficients, supporting the interval effect thatthe per-period time discount rate is diminishing with the interval. Closer inspection reveals
that it may decrease monotonically if we evaluate the coefficients of four- and six-weekintervals as similar.
The average amount of reward has a significantly negative coefficient, indicating themagnitude effect wherein the larger the amount of reward, the lower the per-period time
discount rate.
When the attribute variables are added (see Model 2), the coefficients of the delay andinterval dummies and the average amount of reward are almost unchanged, implying that theabove results are fairly robust.
The male dummy has a significant positive coefficient, indicating male is more
impatient than female, as Ikeda et al. (2005) report. As for the age, Ikeda et al. (2005) reportthat the older subjects tend to have lower per-period time discount rates; however, this age
effect is not confirmed in our experiment. This may be because most of our subjects arearound 20 years old, while the age of their subjects spreads over a wide range. All the
department dummies, taking the department of engineering as the benchmark, did not havesignificant coefficients, although departments of medicine and science are relatively low anddepartments of economics and pharmaceutical science are high. When we set the departmentof economics as the reference group, however, a dummy representing the department of
medicine has a significantly negative coefficient, indicating that students of the department of
medicine are more patient than those of the department of economics.

11

3.3 Subadditive Time Discounting
As noted in section 1, time discounting is subadditive if the interval effect exists.
Therefore, the results of the interval effect presented in the previous subsection imply that thetime discounting should be subadditive. In this subsection, we directly confirm that the time
discounting of our subjects is actually subadditive.
We denote the time discount rate (plus unity) between time a and time b, for the casewhere the period is not divided, as U (a, b). This corresponds to the left-hand side of
inequality (2). Meanwhile, we denote the same time discount rate (plus one), for the casewhere the period is divided into n subintervals, as D (n, a, b). Specifically, D (n, a, b) isdefined as the product of the time discount rates of the subintervals, which corresponds to theright-hand side of inequality (2). In this subsection, we use the time discount rates for various
periods instead of the per-period one for the sake of simplicity of notation.
While Read (2001) divides the total intervals of 18 months and 24 months into threeequal-length subintervals for the test of the subadditivity, we employ various shorter totalintervals of four, six, and eight weeks, and divide them into shorter subintervals such as two
weeks. While Read (2001) examines the case of three divisions, we also test two and four
divisions, as well as three, to see the robustness of the subadditivity.
Table 7, which presents the results on the subadditivity, reveals that the time discountrates in the divided cases are significantly higher than those of the undivided cases
irrespective of the number of divisions. As expected, the subadditive time discounting isobserved in our experiment as in Read (2001). Furthermore, since U (0 day, 8 weeks) < D (2,0 days, 8 weeks) < D (4, 0 days, 8 weeks), we conclude that the subadditivity is stronger as
the number of divisions is larger.
Division is a kind of shortening of an interval, and the interval effect is a sufficientcondition for the subadditivity as explained in section 1. Thus, it is no surprise that we

12

observe the subadditive time discounting, because we have already found the interval effect.
In addition, since the larger number of divisions implies shorter intervals, the result in which
the subadditivity for a larger number of divisions is stronger is also consistent with theinterval effect.11

4. Robustness Check with Survey Data
In this section, we check the robustness of the declining impatience observed in our
experiment by analyzing the results of a questionnaire survey conducted at the end of theexperiment. In our experiment, subjects are requested to answer 180 questions presented in
random order. This randomness of presentation might cause bias. On the other hand, in thequestionnaire survey, subjects are asked two questions. Which one do they prefer: (A) 10,000
yen in two days or (B) X yen in nine days; and the other is between (A) 10,000 yen in 90 daysor (B) Y yen in 97 days. Eight values are assigned for Xs and Ys in these questions.12 Because
both questions differ only in the delay and their interval and amount of reward are the same,
we can test the declining impatience by just comparing the responses to both the questions.
The largest difference from the questions asked in the experiment is that these questions arealigned in ascending order of X and Y, and subjects can read all eight questions before theyanswer. Therefore, subjects’ choice may depend on these eight questions. In addition, thequestionnaire survey is also different from the experiment in that subjects are not paid areward based on their choices.
As in the case of the experiment, we determine time discount rates of the subjects at the11

Of course, this paper does not dismiss the possibility that the subadditivity is caused by some
additional factors to the interval effect. For instance, when the divided and undivided cases arecompared, delay is also different between the cases: while only one delay is involved in the undivided
case, a couple of delays are involved in the divided case. In addition, it might be the case that divisionitself affects the time discount rate. However, this paper does not pursue how the subadditivity isdifferent from the interval effect. To elucidate this point, it is necessary to devise specific experiments,
which will be a future task.
12
Pender (1996) adopts this method in their economic experiment.

13

average of the two rates of return immediately before and after a switch.13 The result of themean-difference test is shown in Table 8, which reveals that the mean of subjects’ per-period
time discount rates in the questionnaire is substantially lower than the corresponding figure in
the experiment shown in Table 3. For example, R(2 days, 9 days) is 0.013 in Table 8, whileR(1 day, 15 days) is 0.094 in Table 3. This discrepancy arises probably because the amount of
reward in the questionnaire is about five times larger than that in the experiment.
Table 8 reveals that the discount rate of a two-day delay is significantly higher than thatof a 90-day delay, implying that the declining impatience is also observed in the answers tosurvey questions. In sum, the results of the declining impatience is robust depending onwhether the questions are presented one by one in random order or simultaneously inascending order, and whether or not a reward is paid depending on their choice.

5. What Causes the Anomalies?
5.1 Traditional Assumption and Time Discounting Anomalies
This paper confirms three anomalies: the declining impatience and interval andmagnitude effects. This section pursues the reason why these anomalies occur.
When subjects consider which is preferable, (A) receive X yen at time s or (B) receive Y
yen at time t, information on the amount of reward in the two options X and Y and that on
timing of receipt in the two options s and t are at hand. Traditional economics, however,assumes that people are rational, so that they have their own per-period time discount rate, R˜ ,a priori, and compare R˜ with the rate of return, R , calculated from the four pieces of
information as
1 (t −s )

⎛Y ⎞
R=⎜ ⎟
⎝X⎠

−1

(5)

13

In this analysis, we regard multi-switching cases as irrational choices and exclude them from theanalysis. There are only a few such cases in the questionnaire survey.

14

to determine their choices. The four pieces of information are condensed into the rate of
return, R, and only R is compared with R˜ to determine the subjects’ decision on the earlier orlater options; no other information has additional effect on their decision. As we observed inprevious sections, such assumption evidently does not match with the reality. Time discountrate, R˜ , of the people is not a constant but is dependent on the experimental conditions ofeach question, X, Y, t, and s. In other words, subjects’ time discount rates, R˜ , depend on the
delay, interval, and amount of reward; and the choice of people is not determined only by therate of return, R, implied in the two options. These facts imply that the traditional assumptionof a rational human being does not apply and therefore are anomalies. In this section, we
examine two hypotheses that may explain these anomalies.

5.2 Differential Effect Hypothesis
In this subsection, we examine a reason why the interval and the amount of reward
affected subjects’ decision-making in addition to the rate of return, R. Specifically, we proposea hypothesis that subjects are affected by a differential in amount (hereafter, differential effecthypothesis) as a source of the interval and magnitude effects and test it. Since the differential
in amount is described as
Y − X = X × [( R + 1) (t −s ) − 1] ,(6)

it increases as the length of the interval and the amount of reward increase. If our subjects
make a decision based on the differential in amount, subjects choose the later option when theinterval becomes longer or the amount of reward gets larger, even though the rate of return
does not change. Consequently, when the differential in amount of reward is large, subjects
report lower discount rate, so that we will find negative coefficients on the intervals and on
the amount of reward when discount rates are regressed over these variables. This is actuallywhat we observed in Table 6. If this differential effect hypothesis is true, we expect that15

coefficients on the interval and the amount become larger when the variable of the differential
is added to this regression.
However, this regression is problematic because, as noted in footnote 10, it is difficult to
identify the amount (therefore, the differentials also) corresponding to the time discount rateestimated by a logit model. To solve this problem, we regress subjects’ binary choices on theearlier or later option over the four experimental conditions, instead of the two-step method inwhich we estimate discount rate at the first step, and then explain it with the experimental
conditions in the second step. The method in this section has an advantage of the efficient useof all the information from 180 choices per subject. More importantly, while the two-stepmethod premises that subjects make a decision based only on their specific per-period time~

discount rate, R , the current method has no need of such a premise that was already foundincorrect.
We regress the dependent variable, a dummy variable that takes unity in case a subject
chooses the later option and zero otherwise, over dummy variables standing for each delay,
dummy variables for each interval, the amount of reward, and logarithm of the per-period rateof return, ln(1 + R) . The estimation results of this specification (Model 3), as well as the
specification (Model 4) that incorporates the differential in amount as an independent variable,
are presented in Table 9.14 The estimation method is a panel logit cum random effect model.
Let us examine the results of Model 3 first. The rate of return has a significantly positivecoefficient, implying that the rate of return is an important factor in decision-making, as
traditional economics assumes. However, the rate of return is not the only factor which solely
determines the intertemporal choice. In addition to the rate of return, the delay has anadditional explanatory power. The dummy variables for the delays, except for those
representing a one-day delay and a one week-delay, have significant positive coefficients and
14

Four variables out of these five explanatory variables determine the remaining one variable.
However, as the relation is nonlinear, colinear problem does not arise.

16

become larger for longer delays. In short, subjects tend to choose the later options as the delay
is larger, even when the rate of return is controlled. This fact is the counterpart to thedeclining impatience reported in Table 6, because the result that subjects choose the later
option as the delay is getting extended indicates that the switch from the earlier option to thelater option occurs at a lower discount rate, as the delay is larger. This in turn implies thatlower per-period time discount rates are found as the delay is extended.
The interval dummies are also significantly positive, implying that subjects increasingly
choose the later options, as the interval extends. This fact corresponds to the interval effectobserved in Table 6.
The coefficient of the amount of reward is also significantly positive, implying that
subjects tend to increasingly choose the later option as the amount of reward increases. The
result corresponds to the magnitude effect observed in Table 6.
Now, let us turn to Model 4 that incorporates the differential in amount as an
explanatory variable. Compared with Model 3, the marginal effects of the interval dummies
change their signs from significantly positive to significantly negative, while those of thedelay dummies are unchanged. The marginal effect of the amount of reward also becomes
smaller to about one-third of the case of Model 3. These results support our hypothesis thatthe interval and the amount of reward affect subjects’ choice via a change in the differential inamount. In other words, the interval and the magnitude effects are caused by the differential
effect at least partially.
How strong is the differential effect? The coefficient of the differential in amount issignificantly positive, which means that subjects increasingly choose the later option as thedifferential increases. Its marginal effect is 1.055 × 10−3. On the other hand, when the
differential is added to the regression, the marginal effect of the rate of return decreases to2.476, about two-fifth of Model 3. These values of the marginal effects imply that a 1%

17

increase in the logarithm of rate of return and a 20-yen increase in the differential in amountwill cause nearly the same increase in the probability of choosing the later option.15 Thus, achange in the differential in amount has a comparable effect on the subjects’ choice as achange in the rate of return.
As possible causes of subadditivity, Read (2001) pointed out a psychological effect offocusing stronger attention to smaller intervals when they are divided (support theory), and abias toward the midpoint in subjective estimates leading to overestimates of small quantities(regression effect). This paper argues that the differential effect may be an additional cause of
the interval effect. Table 9 suggests that once the differential effect is adjusted, the remaining
interval effect is that subjects increasingly choose the earlier option as interval gets longer,
which is the opposite of the total interval effect.

5.3

Weber’s Law
Many psychophysicists have traditionally considered that people perceive external

stimulus (e.g., loudness) after transforming them by a nonlinear function. This is called
Weber’s law. Recent studies further suggest that time perception also follows Weber’s law(Takahashi, 2005, 2006; Dehaene, 2003; Okamoto and Funaki, 2001). In short, relationshipbetween objective and subjective times is nonlinear. Weber’s law gives us another hypothesisthat may explain anomalies on intertemporal choice. Takahashi (2005) argues that hyperbolic
time discounting function is derived from exponential discounting and Weber’s law. In otherwords, if time perception follows Weber’s law, people show declining impatience even if they
discount future reward exponentially. In addition, Takahashi (2006) shows that the interval
effect may be explained by the exponential discounting and Weber’s law. In this section, we
15

The results that use the rate of return R instead of ln (1+R) may better appeal to our intuition. The
estimation results are almost the same as those of Model 3 in Table 9, which imply that a 1% increasein the rate of return and a 16-yen increase in the differential in amount will cause nearly the same
increase in the probability of choosing the later option.

18

empirically investigate whether or not both anomalies are really dissolved by Weber’s law.
Following Takahashi (2005), we define subjective time as logarithm of objective time(logarithmic time perception).

τ ≡ α ln(1+ t) ,(7)

where τ stands for subjective time, while t is objective time, and α is a parameter. Let us
describe how the rate of return is perceived by a subject with logarithmic time perception. The
rate of return of a reward, R, in the experiments is defined as (5), where t and s are objective
time. However, if the subject perceives time as transformed by (7), he or she feels the rate of
return of the reward as R̂ , which is defined ast−s1ln(1 + R) + ln(1 + R)
ln(1 + t ) − ln(1 + s )α.
1≡ A(t , s, R) + ln(1 + R)

ln(1 + Rˆ ) =(8)αThus, the claim by Takahashi (2005, 2006) is examined by testing whether or not the delay
and the interval terms become insignificant, when the rate of return R is replaced with theperceived rate of return R̂ in Model 3 of Table 9. The replacement is actually an addition of
the term A(t , s, R ) of (8) to Model 3, since the second term is nothing but ln(1 + R) . If thelogarithmic time perception model is correct, A(t , s, R) has significant effect on the subject’s
choice.
The estimation results of this regression (Model 5) are presented in the right columns ofTable 9. The first term of (8), which we denote as A(t , s, R ) in Table 9, is not significant at
all; and the coefficient of the second term, ln(1 + R) , is not different from that of Model 3.
The coefficients of the delay and the interval dummies are significant and their values arealmost the same as those of Model 3. In sum, the logarithmic time perception hypothesis does
not solve the anomalies on the delay and interval.
The result of insignificant A(t , s, R) implies that the logarithmic time perception model
19

does not bring about additional explanatory power, suggesting that objective time better
explains the choice of the binary options than the logarithmic time perception hypothesis.
Finally, let us confirm whether the differential effect hypothesis is supported eventhough Weber’s law is considered. In the far-right column, the estimation results that bothA(t , s, R ) and the differential in amount are included in the regression are shown (Model 6).
The results are almost the same as those of Model 4, so that the conclusion of the previous
subsection is retained.

6. Conclusion
This paper examined the effects of the delay and interval on the subjects’ time discountrates, as well as the effect of the differentials in amount of reward on the subjects’ choice,using the result of an experiment on time discount rates and a questionnaire survey conducted
together with the experiment. Thus far, most of the researches did not distinguish the delay
from the interval, so that whether the decreasing impatience really holds has been an open
question. We conducted an experiment that explicitly distinguished the delay from the interval,
and found that per-period time discount rates keep decreasing up to the eight-week delay
when the interval is controlled. This implies that people make dynamic inconsistent plans andshow a preference reversal.
This paper also focused on whether time discount rates depend on the interval between
the two options. Although this problem has seldom been investigated, we found the interval
effect in which the per-period time discount rate decreases as the interval lengthens. This
interval effect is a sufficient condition for subadditive time discounting proposed by Read
(2001), and we confirmed the subadditivity under different conditions. Finally, we found thatthe interval and the magnitude effects are caused, at least partially, because subjects’ choice
are influenced by the differential in reward amount, while Weber’s law does not solve the20

delay nor the interval effects.
While many previous studies propose various functional forms of time discounting, ourresults above suggest that any of them are insufficient to cover the experimental facts.
Hyperbolic and quasi-hyperbolic discounting functions proposed by Loewenstein and Prelec(1992) and Laibson (1997) depend only on the delay, and ignore the interval. Therefore, theycannot explain the interval effect observed in our experiment. Read (2001) proposed afunctional form, which considers the interval effect (subadditivity, in his words) but
disregards declining impatience. However, the results of this paper call for a discounting
function, which properly explains both the declining impatience and the interval effect.

21

References
Benzion, Uri, Amnon Rapoport, and Joseph Yagil. (1989). Discount Rates Inferred from
Decisions: An Experimental Study, Management Science, 35, 270–284.
Dehaene, Stanislas. (2003). The Neural Basis of the Weber–Fechner law: A Logarithmic
Mental Number Line, TRENDS in Cognitive Sciences, 7, 145–7.
Fischbacher, Urs. (1999). Z-Tree: A Toolbox for Readymade Economic Experiments, IEW
Working paper 21, University of Zurich.
Frederick, Shane, George Loewenstein, and Ted O’Donoghue. (2002). Time Discounting andTime Preference: A Critical Review, Journal of Economic Literature, 40, 351–401.
Green, Leonard, Joel Myerson, and Edward McFadden. (1997). Rate of Temporal Discounting
Decreases with Amount of Reward, Memory & Cognition, 25, 715–723.
Ikeda, Shinsuke, Fumio Ohtake, and Yoshiro Tsutsui. (2005). Time Discount Rates: An
Analysis Based on Economic Experiments and Questionnaire Surveys, Institute of Socialand Economic Research Discussion Paper, 638, 1–36 (in Japanese).
Kirby, Kris N., and Nino N. Marakovic. (1995). Modeling Myopic Decisions: Evidence forHyperbolic Delay-Discounting within Subjects and Amounts, Organizational Behavior
and Human Decision Processes, 64, 22–30.
Laibson, David. (1997). Golden Eggs and Hyperbolic Discounting, Quarterly Journal of
Economics, 112, 443–477.
Loewenstein, George, and Drazen Prelec. (1992). Anomalies in Intertemporal Choice:
Evidence and an Interpretation, Quarterly Journal of Economics, 107, 573–597.
Myerson, Joel, and Leonard Green. (1995). Discounting of Delayed Rewards: Models of
Individual Choice, Journal of The Experimental Analysis of Behavior, 64, 263–276.
Okamoto, Hiroshi, and Tomoki Fukai. (2001). Neural Mechanism for a Cognitive Timer,
Physical Review Letters, 86, 3919–22.
Pender, John L. (1996). Discount Rates and Credit Markets: Theory and Evidence from Rural
India, Journal of Development Economics, 50, 257–296.
Read, Daniel. (2001). Is Time-Discounting Hyperbolic or Subadditive? Journal of Risk andUncertainty, 23, 5–32.
Richards, Jerry B., Lan Zhang, Suzanne H. Mitchell, and Harriet De Wit. (1999). Delay or
Probability Discounting in a Model of Impulsive Behavior: Effect of Alcohol, Journal of
The Experimental Analysis of Behavior, 71, 121–143.
Takahashi, Taiki. (2005). Loss of Self-control in Intertemporal Choice may be Attributable to
Logarithmic Time-perception, Medical Hypotheses 65, 691–63.
22

Takahashi, Taiki. (2006). Time-estimation Error Following Weber-Fechner Law May Explain
Subadditive Time-discounting, Medical Hypotheses 67, 1372–74.
Thaler, Richard. (1981). Some Empirical Evidence on Dynamic Inconsistency, Economic
Letters, 8, 201–207.
Uzawa, Hirofumi. (1968). Time Preference, the Consumption Function and Optimum Asset
Holdings, in: J.N. Wolfe, ed., Value Capital and Growth: Papers In Honour of Sir John
Hicks (Chicago: Aldine).

23

Table 1 Attributes of subjects

Number of subjects
Averageage

Departments
Total

Male

FemaleUndergraduates

Graduates

Letters

25

7

18

22

3

21.08

Law

15

9

5

14

1

20.73

Economics

24

16

8

19

5

21.67

Human science

16

4

12

15

1

21.13

Engineering

68

63

5

59

9

20.65

Engineeringscience

34

32

2

26

8

21.62

Science

22

19

3

17

5

22.32

Medicine

12

8

4

7

5

23.58

Pharmaceuticalscience

3

3

0

3

0

19.67

Total

219

161

57

182

37

21.32

24

Table 2 Fifteen combinations of receipt timing in earlier and later options

Delay(timing of receipt in
option A)

Interval

Delay + Interval(timing of receipt
in option B)

0 day

2 weeks

2 weeks

0 day

4 weeks

4 weeks

0 day

6 weeks

6 weeks

0 day

8 weeks

8 weeks

1 day

2 weeks

15 days

1 week2 weeks

3 weeks

2 weeks

2 weeks

4 weeks

2 weeks

4 weeks

6 weeks

2 weeks

6 weeks

8 weeks

4 weeks

2 weeks

6 weeks

4 weeks

4 weeks

8 weeks

6 weeks

2 weeks

8 weeks

8 weeks

2 weeks

10 weeks

10 weeks

2 weeks

12 weeks

12 weeks

2 weeks

14 weeks

25

Table 3 Per-period time discount rates

Delay
Delay + Interval(timing of receipt in (timing of receipt
in option B)
option A)

Number ofobservations

Per-period time
discount rate
Mean

Average amountof reward inStandard
option A
error

0 day

2 weeks

167

0.090

0.091

2263

0 day

4 weeks

170

0.060

0.047

1803

0 day

6 weeks

161

0.066

0.046

1789

0 day

8 weeks

164

0.039

0.027

2238

1 day

15 days

173

0.094

0.074

2181

1 week3 weeks

165

0.091

0.082

2198

2 weeks

4 weeks

157

0.095

0.095

1776

2 weeks

6 weeks

158

0.052

0.044

1814

2 weeks

8 weeks

171

0.044

0.030

1961

4 weeks

6 weeks

149

0.077

0.064

2135

4 weeks

8 weeks

164

0.051

0.044

1903

6 weeks

8 weeks

132

0.081

0.069

2131

8 weeks

10 weeks

135

0.063

0.068

2279

10 weeks

12 weeks

128

0.069

0.079

2155

12 weeks

14 weeks

134

0.070

0.065

2304

Notes: Table shows the number of observations, mean and standard deviation of per-period time
discount rates, and the average of the amount of reward for each combination of receipt timing. The
number of observations is less than the number of subjects, 219, because we exclude the observations
if a subject consistently chooses A (or B) for 12 questions of the same combination, and that theestimated time discount rates using a logit model are out of the 1% to 50% range.

26

Table 4

Test results for the declining impatience for the case that the interval is held at 2
weeks

1 day
0 day

1 week 2 weeks−0.491 −0.110

4 weeks

6 weeks

8 weeks

10 weeks

12 weeks−0.464

1.451

0.961

2.976***

2.079**

2.208**

1 day

-

0.397

−0.0402.235**

1.620*

3.899***

2.794***

3.060***

1 week

--−0.3801.670*

1.135

3.268***

2.291**

2.464**

2 weeks---

1.907*

1.419

3.352***

2.470**

2.622***

4 weeks

-

-

-

-

−0.482

1.862*

0.908

0.938

6 weeks

-

-

-

-

-

2.197**

1.276

1.341

8 weeks

-

-

-

-

-

-

−0.736−0.917

10 weeks

-

-

-

-

-

-

-

−0.082

Notes: Observations whose interval is 2 weeks are picked up and a mean-difference test among groups
with different delays is applied. The figures in each cell are t-values for the mean-difference test
between per-period time discount rates of two groups with different delays shown in the first column
and the first row; ***, **, and * indicate that the values are statistically significant at the 1%, 5% , and
10% levels, respectively.

27

Table 5 Test results for the declining impatience for the case that the interval is held at 4weeks
2 weeks 4 weeks
0 day

1.663*

2 weeks

1.857*
0.167

Notes: Observations whose interval is 4 weeks are picked up and a mean-difference test among groups
with different delays is applied. The figures in each cell are t-values for the mean-difference test
between per-period time discount rates of two groups with different delays shown in the first column
and the first row; * indicates that the values are statistically significant at the 10% level.

28

Table 6 Regression results of per-period time discount rates on experimental conditions andattributes of subjects
Model 1
Independent variables

Model 2Standard
errorsCoefficients

Standard
errors

0.709***

0.147

0.661***

0.152

1 day

0.003

0.004

0.004

0.004

1 week

−0.001

0.004−0.001

0.004

2 weeks−0.014***0.003−0.014***0.003

4 weeks−0.014***0.003−0.014***0.003

6 weeks−0.022***0.005−0.022***0.005

8 weeks−0.032***0.005−0.032***0.005

10 weeks−0.029***0.005−0.029***0.005

12 weeks−0.022***0.005−0.022***0.005

4 weeks−0.036***0.003−0.036***0.003

6 weeks−0.034***0.003−0.034***0.003

8 weeks−0.041***

0.004−0.041***0.004−0.080***0.019−0.081***

0.019

Male dummy

0.028***

0.010

Age of subjects

0.002

0.002

Graduates dummy−0.0050.014

Letters

0.004

0.014

Law

−0.0050.014

Economics

0.018

0.012

Human science

0.007

0.016

Engineering−0.0020.008

Science−0.0130.013

Medicine
Pharmaceuticalscience−0.018

0.016

0.052

0.034

Coefficients

Constant

delay dummies

interval dummies

Average amount of reward

Department dummies

Number of observations
R-squared

2,328

2,316

Within

0.132

0.133

Between

0.140

0.110

Overall
0.067
0.089
Notes: Only the regression results from random effect model are shown because it was not rejectedagainst fixed effect model by Hausman specification test. Dependent variable is the per-period time
discount rates; *** indicates that the values are significant at the 1% level. The number ofobservations is less than the total observations (219 subjects × 15 combinations of receipt timing)
because we exclude the observations that a subject consistently chooses A (or B) for 12 questions of
the same combination, and that the estimated time discount rates using a logit model are out of the 1%
to 50% range.

29

Table 7 Test results on the subadditive time discounting
Number ofTime discount rates

observations

Mean

Standarderror

Minimum

MaximumU(0 day, 8 weeks)

164

1.169

0.127

1.016

1.457

D(4, 0 day, 8 weeks)

107

1.490

0.396

1.057

3.280

U(0 day, 6 weeks)

161

1.218

0.159

1.012

1.482

D(3, 0 day, 6 weeks)

130

1.320

0.277

1.040

2.521

U(2 weeks, 8 weeks)

171

1.142

0.100

1.016

1.465

D(3, 2 weeks, 8 weeks)

113

1.330

0.258

1.045

2.415

U(0 day, 4 weeks)

170

1.126

0.102

1.017

1.473

D(2, 0 day, 4 weeks)

148

1.198

0.180

1.028

1.939

U(2 weeks, 6 weeks)

158

1.108

0.097

1.015

1.473

D(2, 2 weeks, 6 weeks)

137

1.193

0.168

1.027

1.856

U(4 weeks, 8 weeks)

164

1.106

0.095

1.011

1.464

D(2,4 weeks, 8 weeks)

120

1.181

0.129

1.028

1.692

U(0 day, 8 weeks)

164

1.169

0.127

1.016

1.457

D(2, 0 day, 8 weeks)

152

1.246

0.189

1.028

1.901

Difference
in meann=40.320***

n=30.103***

0.188***

n=2
0.071***

0.084***

0.075***

0.076***

Notes: U (a, b) denotes the time discount rate (plus one) of the undivided case between time a to b.
D (n, a, b) denotes the time discount rate (plus one) of the divided case between time a to b; nrepresents the number of subintervals of equal length; *** indicates that the difference in mean issignificant at the 1% level.

30

Table 8

Test results on the declining impatience using the questionnaire survey

Number ofTime discount rates
Difference in meanobservations

Mean

Standard error

R(2 days, 9 days)

193

0.013

0.013

R(90 days, 97 days)

203

0.008

0.010

0.005***

Notes: R(2 days, 9 days) stands for per-period time discount rate with two-day delay and seven-day
interval. R(90 days, 97days) stands for per-period time discount rate with 90-day delay and seven-day
interval. The number of observations is less than the number of subjects, 219, because we exclude theobservations that a subject consistently chose all A or all B and those with multi-switching; ***indicates that the difference in mean is statistically significant at the 1% level.

31

Table 9 Results from the regression with panel data of responses to binary questions
Independent variables
Constant
delay dummies

Interval dummies

1 day
1 week2 weeks
4 weeks
6 weeks
8 weeks
10 weeks
12 weeks
4 weeks
6 weeks
8 weeks

amount of reward×10-4
Differential in amount×10-3
rate of return, ln (1 + R)

Model 3
Coefficients−4.140***−0.165*0.104
0.532***
0.661***
1.043***
1.284***
1.347***
0.981***
0.418***
0.436***
0.576***
6.437***

Model 4

Marginal effects

39.511***−0.025*0.015
0.071***
0.083***
0.113***
0.129***
0.133***
0.108***
0.057***
0.058***
0.072***
0.947***
5.812***

Model 5

Coefficients−2.967***−0.302***0.105
0.495***
0.659***
0.961***
1.239***
1.248***
0.867***−0.393***−0.626***−0.950***2.389***

Marginal effects

7.565***

1.055***

17.750***2.476***−0.046***0.014
0.063***
0.078***
0.100***
0.119***
0.119***
0.093***−0.059***−0.101***−0.169***0.333***

Logarithmic time perception effectA (t , s , R )Number of observations
log likelihood

29,748−10,669.556

29,748−10,182.963

Model 6

Coefficients−4.147***−0.181*0.106
0.536***
0.670***
1.057***
1.301***
1.369***
1.007***
0.420***
0.440***
0.582***
6.437***

Marginal effects

Marginal effects−0.028*0.015
0.071***
0.084***
0.114***
0.130***
0.134***
0.110***
0.057***
0.058***
0.073***
0.949**

Coefficients−2.956***−0.267***0.101
0.487***
0.642***
0.932***
1.201***
1.202***
0.810***−0.398***−0.635***−0.963***2.386***

39.747***5.858***

7.573***
17.219***1.056***
2.401***−0.083−0.012

0.176

0.025

29,748−10,669.571−0.040***0.014
0.062***
0.076***
0.098***
0.116***
0.116***
0.088***−0.060***−0.102***−0.171***0.333***

29,748−10,182.627

Notes: Only the regression results of random effect model are shown because it is not rejected against fixed effect model by Hausman specification test.
Dependent variable is the response to the binary questions; ***, **, and * indicate that the values are statistically significant at the 1%, 5%, and 10% levels,
respectively. The number of observations is less than the total observations (219 subjects × 15 combinations of receipt timing) because we exclude theobservations when a subject consistently chooses A (or B) for 12 questions of the same combination, and that the estimated time discount rates using a logit
model are out of the 1% to 50% range.

32

Per-period time discount rates

0.100

0.090

0.080

0.070

0.060
0 day

1 day

1 week2 weeks

4 weeks

6 weeks

8 weeks

10 weeks 12 weeks

delay

Figure 1

Declining impatience in the case where the interval is kept at 2 weeks.

Note: The average of the per-period time discount rates for each delay designated at the horizontal axis
is plotted. The interval is kept at 2 weeks.

33

Per-period time discount rates

0.062
0.060
0.058
0.056
0.054
0.052
0.050
0 day

2 weeks

4 weeks

delay

Figure 2

Declining impatience in the case where the interval is kept at 4 weeks.

Note: The average of the per-period time discount rates for each delay designated at the horizontal axis
is plotted. The interval is kept at 4 weeks.

34