GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.19

Time Discounting:
The Delay Effect and Procrastinating Behavior

Shunichiro Sasaki
Shiyu Xie
Shinsuke Ikeda
Jie Qin
Yoshiro Tsutsui

November 2008

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Time Discounting:
The Delay Effect and Procrastinating Behavior∗
Shunichiro Sasakia(Nagoya University of Commerce and Business)
Shiyu Xieb
(Fudan University)
Shinsuke Ikedac(Oasaka University)
Jie Qind(Ritsumeikan University)
Yoshiro Tsutsuie(Osaka University)∗

We are grateful to Fumio Ohtake, Junmin Wan, and Sun Bibo for their support andcomments. This research was financially supported by the 21st Century COE program at
Osaka University and Grant-in-Aid for Scientific Research No. 17203025 of the Ministryof Education, Culture, Sports, Science and Technology in Japan.
aFaculty of Economics, Nagoya University of Commerce and Business, 4-4 Sagamine,
Komenoki-cho, Nisshin, Aichi 470-0193, Japan; e-mail: ssasaki@nucba.ac.jp, Phone:+81-561-73-2111, Fax: +81-561-73-1202b
Department of World Economics, Fudan University, 220 Handan Road, Shanghai 200433,
P. R. China; e-mail: syxie@fudan.edu.cn
cInstitute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki,
Osaka 567-0047, Japan; e-mail: ikeda@iser.osaka-u.ac.jpdFaculty of Economics, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga
525-8577, Japan; e-mail: jieqinjp@yahoo.co.jp
eCorresponding author, Graduate School of Economics, Osaka University, 1-7
Machikaneyama, Toyonaka, 560-0043 Japan, e-mail: tsutsui@econ.osaka-u.ac.jp

Abstract
The delay effect, that people discount the near future more than the distant future, has notbeen verified rigorously. An experiment conducted by us in China confirms that, by
separating the delay from the interval, the delay effect exists only within a short delay. The
results are reliable, because the rewards paid were very large, in order to elicit the subjects’true preferences. The interval and magnitude effects are also confirmed. Finally, subjects’procrastinating behavior, as reported in the questionnaire conducted at the end of theexperiment, is explained by the time discount rates and the degree of the delay effectrevealed in the experiment.

JEL classification numbers: C91; D91
Keywords: discount rate; economic experiment in China; delay effect; naïfs andsophisticates; procrastination

1.

Introduction

This study examines an economic experiment conducted in China to explore how
people discount future values. Although traditional economics has relied on theassumption of “exponential discounting,” i.e., that the time discount rate is constantover the time horizon, many experimental studies have found that the opposite is true:that is, people do not discount future values exponentially (Ainslie 1975, Benzion et al.
1989, Kirby 1997, Kirby and Marakovic 1995, and Thaler 1981). This anomaly of
discounting, that it is not constant over the time horizon, is called the delay effect,
diminishing impatience, or hyperbolic discounting. In addition, it is known that discountrates depend on the value to be discounted: the smaller the value, the more intensively it
is discounted, and this is called “the magnitude effect” (Benzion et al. 1989, Kirby andMarakovic 1995, Kirby 1997, and Thaler 1981).
However, most studies of the delay effect face a problem. In experiments thus far,subjects are typically asked which of two options is more preferable: option A, in which
the subjects will receive a reward in the near future, or option B, in which the subjects
will receive a (usually larger) reward in the distant future. All options shown in thequestion are neatly identified with four experimental conditions: date s and amount X ofreceipt in option A; and date t (t > s) and amount Y (usually larger than X) in option B.
Associated with these conditions, we call s the delay and the difference between the twopoints of time, t − s , the interval h. In what follows, we call the phenomenon that
discount rates depend on the delay “the delay effect,” on the interval “the interval
effect,” and on the reward amount “the magnitude effect.”Thus, the two concepts of time, the delay s and the interval h, should be
distinguished as the factors that characterize intertemporal choices. The delay s is,1among other things, the focal point, because if time discounting depends on the delay, aplan made by such a person will be time-inconsistent. Nonetheless, most studies so far
have not separated the delay from the interval, and subjects are asked to specify theamount of money that they would require, at some future date, to make them indifferent
to receiving some amount now. In these studies, since the receipt time of the earlier
option is fixed at now, their results should be interpreted as an investigation of theinterval effect rather than the delay effect or, at least, as a mixture of these two effects.
Read (2001) was the first to point out this problem. He conducted an experimentthat separated the delay from the interval. He observed a form of the interval effect, but
did not confirm the delay effect.1 However, although the delay effect exists even withquite a short delay (Frederick et al. 2002), the shortest delay used by Read was six
months. Thus, whether subjects show the delay effect within a six-month delay remains
an open question. Given these arguments, the primary aim of this study is to reexamine
whether the delay effect is in fact found when the delay and the interval are explicitly
distinguished. Our experiment differs from Read (2001) in that we investigated thedelay effect using a much shorter delay than his.
The second aim is to explain the procrastinating behavior of the subjects by theirtime discount rates and the degree of the delay effect. Although the delay effect is
expected to cause procrastinating behaviors, because it means a high discount rate in thenear future, it subsequently makes people regret that they procrastinated. Therefore,
1

Recently, Read and Roelofsma (2003) and Read et al. (2005) have reported that the delay effect isfound in experiments in which subjects are asked questions in ways that differ from those in Read
(2001). Read and Roelofsma (2003) utilized the matching method, which asks subjects about theamount at a specified date, which makes them indifferent to a specified option. This is different fromthe choice method used in Read (2001) and in the current study, which asks subjects to choose thebetter of two specified options. On the other hand, Read et al. (2005) focused on how the timing of
two options is described and specified the timing with calendar dates. This method is different from
Read (2001) and from the current paper, which specified the length of the delay.
2

after repeating this serious regret, they would stop procrastinating. Thus, the relation
between the delay effect and procrastinating behavior is not straightforward. In thisstudy, we investigate the relation based on the concept of naïfs and sophisticates, asdeveloped by O’Donoghue and Rabin (1999).
One merit of our study is that, in our experiments, we give the Chinese subjects
strong incentives, by offering a high reward to induce them to reveal their true
preference. This study examines whether real payment to all subjects is indispensablefor eliciting the true preference when some subjects are really paid a large reward.
The remainder of this paper is organized as follows. In the next section, we
explain our experimental design. In section 3, we report the results of the experimentand examine whether the delay, the interval, and the magnitude effects are confirmed bymeans of the test of the same mean, and a regression analysis. In section 4, we examine
whether the discount rate and the degree of the delay effect of the subjects can explain
their procrastinating behavior. In section 5, we present our conclusion.

2.

Design of the experiment

We try to measure subjective time discounting by asking subjects which of the two
options they prefer: option A, to receive a reward X with a shorter delay s, and option B,
to receive a (usually larger) reward Y, with a longer delay t. From these conditions, therate of compensation for a postponement R is defined as

1 Y−X, and the interval h is
X t−sdefined as t − s .
We specified four values for s (2 days, 1 month, 90 days, 10 months), three values
for h (7 days, 3 months, 12 months), and three values for the reward amount (240, 2800,
and 800000 yuan). We also controlled for whether the reward specified in the pay-off
3tables would really be paid, which we denote as P = (YES, NO). Combining these four
conditions, we specified twelve sets of conditions (s, h, X, P), as shown in Table 1. Thus,
the experiment consists of twelve rounds, each of which corresponds to one set of
conditions.
For each set of conditions, we asked about 32 pairs of options, each of which is
specified with a value of R such that the value of the reward of the later option is Y.
Thirty-two pairs of options are arranged in ascending order of R, which varies from avery low value of −20% to a very high value of 300% (annual rate), in a sheet that wecall the “payoff table.”2 The subjects are expected to choose the earlier receipt A when
R is small and the later receipt B when R becomes sufficiently large. Then, we define
the time discount rate TDi,j for the condition j = (s, h, X, P) and the subject i with thevalue of R at which the subject switched from A to B.3
The experiment was conducted on March 11, 2005 at Fudan University,
Shanghai.4 The subjects were 29 undergraduate students recruited from the Department
of World Economics at Fudan University. They were either 20 or 21 years old, and 25of the 29 subjects (86%) were female. The reward was paid by the following procedure.
At the end of the experiment, starting from round one, we randomly selected one pair ofoptions to be paid from the payoff table. Then we randomly selected one subject as arecipient of the reward amount on the date specified in the option that the subject had
chosen.5 The highest reward presented in the payoff tables was 3920 yuan (800000

2

One payoff table is presented in Appendix.
If a subject chooses A (B) for all 32 pairs, we regard the highest (lowest) value of R in the payoff
table as TDij. If a subject switches multiple times, we discard the data because the subject hadprobably misunderstood the question.
4
We conducted an experiment on risk aversion before the experiment on the time discount rate, on
the same day. For the experiment on risk aversion, see Sasaki et al. (2008).
5
Since this payment procedure was explained in the instructions and was rehearsed twice, the34yuan would not be paid), which was a relatively large amount, considering that theannual household income of the subjects was 20−40 thousand yuan. This helped to givethe subjects an incentive to express their true preferences. Including a 120 yuan (about
US$14) participation fee, the average payoff for both the experiments, on time
discounting and risk aversion, was 788 yuan (about US$106). After we selected thewinners, the subjects were required to complete a detailed questionnaire requesting theirpreferences, opinions, and attributes.

3.

Results of the experiment

3.1 Time discount rates elicited by the experiment
In the bottom row of Table 1, we present the average and standard error of the discountrates, which reveal that the discount rates are distributed over a wide range from 1.0%
to 44% (in the annual rate), depending on the experimental conditions. This implies thatthe conventional assumption of exponential discounting is not supported by the data. In
other words, there exist anomalies that are probably associated with the delay, interval,
and reward amount. In this section, we examine the delay, interval, and magnitude
effects. We also investigate whether the results depend on the promise of the paymentof a reward.

3.2

Examination of the three anomalies in time discounting by means of tests of thesame mean
In our framework, the discount rates may differ depending on four experimental
conditions: the delay s, the interval h, the reward amount X, and whether the payment issubjects were expected to understand clearly that what they received would depend on their choice.
5

promised P. In this subsection, in each round we hold three of the four conditions
constant and vary the fourth condition, and we test whether the mean discount ratevaries.

The delay effectTo test the delay effect, we compare the rounds in which the interval and amount are thesame but the delay differs.6 They are as follows:
1) Round 2 (30 days, 90 days, 240 yuan) vs. Round 4 (10 months, 90 days, 240yuan)2) Round 5 (2 days, 7 days, 2800 yuan) vs. Round 6 (90 days, 7 days, 2800 yuan)
3) Round 7 (1 month, 3 months, 2800 yuan) vs. Round 9 (10 months, 3 months,
2800 yuan)
4) Round 10 (1 month, 3 months, 800000 yuan) vs. Round 12 (10 months, 3months, 800000 yuan)
The results of the test of the same mean are presented in Table 2. In all the
combinations, the null hypothesis of the same mean is not rejected. Thus, the delay
effect is not observed by the test of the same mean.

The interval effect
For the interval effect, three comparison tests, round 2 vs. 3, round 7 vs. 8, and round 10
vs. 11, are possible, and the test results are presented in Table 3. When the reward
amount is set at 240 or 2800 yuan, the discount rates are significantly higher in the
shorter intervals than in the longer intervals, and this confirms the interval effect.
6

Whether the payment would be made or not is also controlled.
6However, when the reward amount is set at 800000 yuan, the null of the same mean is
not rejected. This is probably because people will choose sufficiently low discount rates
for an extremely large amount such as 800000 yuan, so that a difference in the otherconditions no longer changes the discount rates. The result of the interval effect isgenerally consistent with the subadditivity found by Read (2001).
The magnitude effect
The results of the tests of the magnitude effect are presented in Table 4. In the top panel,
we show three test results comparing the reward of 240 and 2800 yuan. Resultscomparing the reward of 2800 and 800000 yuan are presented in the middle panel, andthose comparing the reward of 240 and 800000 yuan are presented in the bottom panel.
In all the combinations, the time discount rate is significantly higher for the smaller
reward amount. Thus, the magnitude effect, in which the discount rates are inversely
related to the reward amount, is significantly observed. This is consistent with theresults of previous studies (Benzion et al. 1989, Kirby 1997, Kirby and Marakovic 1997,
and Thaler 1981).
The payment effectFinally, we examine whether the promise of payment affects the discount rates. Rounds
1 and 7 are designed to have the same condition (s, h, X), and they differ only in that the
reward is actually paid in round 5 but not in round 1. The test result of the same mean ispresented in Table 5, which reveals that the null of the same mean is not rejected at all.
As noted above, the reward of 2800 yuan was relatively large in China, and this result
therefore has a strong implication: whether the reward is actually paid to the subjects
has nothing to do with the time discount rates they report, as long as some of the7subjects will really be paid a large reward. This provides grounds for the validity of ourexperimental result with respect to the following: 1) although we did not pay rewards toall subjects except the winner of each round, the results as a whole are trustworthy, and2) the results of rounds 10–12, where the reward of 80000 yuan is only imaginary, arealso reliable.

3.3 Regression analysis for an investigation of three anomalies
In the last subsection, we did not confirm the delay effect, but we did find the interval
and magnitude effects. This is consistent with Read’s (2001) conclusion, and it implies
that the conclusion of many previous studies that argued for the delay effect is false,
because they did not separate the delay from the interval. However, we should not be
hasty, because the comparison tests in the previous subsection may suffer from the
problem of a small sample, and this may lead to a small power to reject the null of thesame mean. Looking at Table 2 carefully, we see that the rounds of the 2-days delay
reports a discount rate of 30.6%, which is substantially higher than 22% in round 6, forthe 90-days delay. Although the p-value is only 32%, this may be a result of the small
sample of 29. To avoid this weakness, we pool all the samples of 348, and conduct a
regression analysis using them.
The regression equation basically controlled all the experimental conditions: thedelay s, the interval h, the reward amount X, and whether the reward is actually paid P.
In particular, we adopt the following variables. For the variables standing for the delay,
we use dummy variables HOR2 j , HOR30 j , and HOR90 j , which is equal to unity if

the delay is 2 days, 1 month, and 3 months in round j, respectively, and 0 otherwise (108months). For the amount of the reward, we use dummy variables AMT 240 j and
AMT 2800 j , which is equal to unity if the amount of reward is 240 and 2800 yuan,
respectively, in round j , and 0 otherwise (800000 yuan). For the interval, we use avariable INT j , which is equal to the length of the interval in round j . Finally, we use adummy variable PAYj , which is equal to unity if the reward will be paid in round j ,and 0 otherwise. Thus, the regression equation isln TDij = α 0 + α1HOR2 j + α 2 HOR30 j + α 3 HOR90 j + α 4 INTj + α 5 AMT 240 j+ α 6 AMT 2800 j + α 7 PAYj + ε i , j(1)where ln TDi , j stands for the logarithm of subject i’s time discount rate in round j.
The results of the regression are presented in Table 6. For the variables of thedelay dummies, the estimated coefficient of HOR2 j is significantly positive, but thoseof HOR30 j and HOR90 j are insignificant, indicating that the delay effect operates
only in 2-days delay, but not in the rounds where the delay is more than 30 days. This
non-linear property of the delay effect is close to the quasi-hyperbolic discount model,
in that people discount much of the future value between the present and the very near
future, whereas they discount it moderately after that period (Laibson 1997). The result
does not contradict Read (2001), who reported that the delay effect is not observed in
the case of 6-months and 8-months delays, which are both substantially longer thanours.
Let us briefly examine the other anomalies. The estimated coefficients of
AMT 240 j and AMT 2800 j are significantly positive, and that of INT j is significantly
negative. These results confirm the magnitude and interval effects, as we have already9confirmed by the test of the same mean. The estimated coefficient of PAYj is
insignificant, which is also consistent with the results in the previous subsection. Thus,
when a part of the subjects are randomly chosen, and are actually paid a large reward
amount, the subjects answer honestly those questions in which the reward is imaginary.
Therefore, all of their responses in our experiment are reliable.
Next, we consider the attributes of the subjects, gender, and household income.
In particular, we add a dummy variable, MALE, which is equal to unity for malesubjects and zero otherwise, and a dummy variable HINCOME, which is equal to unity
if the subject’s annual household income is over 140 thousand yuan and zero otherwise.
Five subjects are classified as high income and 18 as low income. Since 7 subjects who
did not report household income were excluded, the number of observations is 23 forthis regression. The estimates are presented in the right column of Table 6. Bothcoefficients of MALE and HINCOME are negative, but they are not significant at all.
Thus, for our subjects, the discount rates are independent of these attributes. On theother hand, the three anomalies are clearly confirmed with this specification. The resultthat PAY has no effect also remains unchanged.

3.4 Robustness check on our time discounting data
We next examine the reliability of the discount rates elicited in the experiment. In thequestionnaire that we asked subjects to complete at the end of the experiment, questions
similar to those in the experiment were included. In particular, we asked the following
five questions, each of which has a counterpart “round” in the experiment.
QTD5, which corresponds to round 5: s = 2 days, h = 7 days, but X = 800 yuan
instead of 2800 yuan.
10

QTD6, which corresponds to round 6: s = 90 days, h = 7 days, but X = 800 yuan
instead of 2800 yuan.
QTD3&8, which corresponds to rounds 3 and 8: s = 1 month, h = 12 months, but X =800 yuan instead of 240 and 2800 yuan.
QTD11, which corresponds to round 11: s = 1 month, h = 12 months, and X =800000 yuan.
We examine whether or not these four sets of data elicited by the questionnaire
are consistent with the discount rates obtained from the experiment. The mean andstandard deviation of the logarithm of the discount rates elicited by the questionnaire(QTD) and the experiment (ETD) are presented together with the experimental
conditions, in the upper and middle panels of Table 7, respectively. Comparing QTD5with the corresponding ETD5, and QTD6 with ETD6, the former is higher than the latter,
whereas the discount rate of QTD3&8 is lower than ETD3 and higher than ETD8. Thus,
the discount rates with a larger reward amount are consistently lower. As we noted
above, the corresponding pairs are different only in the reward amount, and the resultscorrespond to the magnitude effect that we found in section 3.2. Thus, although the testof the same mean rejects the null of the same mean, the mean may be the same after weadjust the magnitude effect.
To examine whether this is the case, we estimate the following equation (2),which differs from equation (1) only in that the logarithm of the reward amount is used
here, instead of the amount dummies.
ln TDij = α 0 + α1HOR2 j + α 2 HOR30 j + α 3 HOR90 j + α 4 INTj + α 5 AMTj
+ α 7 PAYj + ε i , j(2)where AMT is equal to either log240, log2800, or log800000. Then, the predicted value

11

of the discount rates from the experiment, ADJTD, is calculated by the following
equation (3). For example, in the case of round 5, if the amount were set at 800 yuan
instead of the actual 2800 yuan and the payment was not promised, the discount ratewould beADJTDi ,5 ≡ ln TDi ,5 + αˆ 5 (log 800 − log 2800) − αˆ 7 .

(3)

The mean and standard deviation of the values thus calculated are presented inPanel C of Table 7. The ADJTD values move closer to the discount rates from thequestionnaire, as expected. Although the statistical test (Panel D of Table 7) rejects thenull of the same mean between QTD5 and ADJTD5, and between QTD6 and ADJTD6,
the null that QTD3&8 and ADJTD3 are the same is not rejected at the 30% level, andthe null associated with QTD3&8 and ADJTD8 is not rejected at the 5% level. Thus,
although the discount rates elicited by the questionnaire and the experiment are notexactly the same, in some cases they are close enough.
The consistency between the discount rates elicited by the experiment and by thequestionnaire is confirmed by another analysis. Table 8 presents the correlation
coefficient between these discount rates and reveals that the correlation between thepairs of experimental conditions that are similar is relatively higher (highlighted inyellow) than that of the other pairs, implying that the subjects respond consistently tosimilar questions.
These results indicate that the time discount rates elicited from the experimentare generally consistent with those elicited from the questionnaire, and that ourexperimental data are reliable.

4.

Time discounting and procrastinating behavior

124.1 Analytical framework
People are sometimes unable to behave as they had originally planned. Instead, theytend to postpone discomfort and enjoy pleasure earlier. In this section, we investigate
whether the subjects’ time preference, as revealed in the experiment, can explain theirprocrastinating behavior.
As an example of an unpleasant task in which people tend to procrastinate, we
adopt homework assignments in childhood. In the questionnaire distributed at the end of
the experiment, we asked when they did their assignments:“Thinking about when you were a child and you were given an assignment inschool, when did you usually do the assignment?” (Circle ONE number)
1. Got it done right away
2. Tended to get it done early, before the due date
3. Worked on it daily up until the due date
4. Tended to get it done toward the end
5. Got it done at the last minute

From their answers, we define HWK i as the number that subject i chose. A larger valueof HWK i means a stronger propensity to procrastinate.
Our hypotheses are as follows. First, higher time discounting will lead toprocrastinating behavior. Second, the delay effect may reinforce procrastinating
behavior. A stronger delay effect means higher time discounting in the near future
and/or lower discounting in the remote future. Therefore, controlling the average
discount rate in the regression, a stronger delay effect implies higher time discounting in
the near future, so that it seems to bring about procrastinating behavior. However, thecase may not be so simple. The delay effect causes a preference reversal, so that people

13

subject to the delay effect often procrastinate, and then regret that they had
procrastinated. Thus, after repeatedly suffering bitter regret, they may devise a way toprevent procrastination, in order to avoid having to regret a miserable outcome. If this isso, the stronger delay effect no longer results in greater procrastination.7
Following O’Donoghue and Rabin (1999), let us categorize individuals whoshow the delay effect into two types: naïfs and sophisiticates. Naïfs mistakenly believethat they will behave as they originally planned, and so they delay imminent costs andenjoy immediate rewards. On the other hand, sophisticates know that they will delay anunpleasant task until the date of the execution that was originally planned comes andthat such procrastination may lead to a disaster in the end; therefore, they may
undertake an unpleasant task now (Proposition 2 of O’Donoghue and Rabin, 1999).
Given these arguments, we speculate that (i) if the subjects are naïfs, the delay
effect will enhance their procrastination; (ii) however, if they are sophisticates, thedelay effect may not promote procrastination clearly; it may even suppress
procrastination.
Thus, our regression equation is
HWK i = β + γ TDi + δ DLYi + ui ,i = 1, ... , 29 .

(4)

We expect γ > 0 . However the sign of δ is not known a priori. It depends on whether
the subjects are naïfs or sophisticates.

4.2 Definition of variables used in the regression

7

Here we assume that slight procrastination does not bring about sufficient regret to stop further
procrastination, even by sophisticates.
14

By construction, the relation should be estimated with individual data, so that we need
to use the time discount rate and the degree of the delay effect for each person. Thus,
the time discount rate of subject i, AVTDi , is defined as the average of the logarithmsof the discount rates reported by subject i in round j, LNTDij : that is,
12

AVTDi = ∑ LNTDij / 12 .

(5)

j =1

However, the simple average of the time discount rates elicited in different
conditions may suffer from a problem. For example, we saw in Table 1 that the time
discount rate in the case of large reward amount of 800000 yuan is much smaller than in
the case of smaller reward amounts. Therefore, the difference in the discount rate
between subjects in the case of the large reward amount is substantially smaller than in
the case of the small reward amount. Thus, a simple average would underestimate thecontribution of the discount rates in the case of the large reward amount. To be freefrom this bias, we first standardize logarithmic discount rates in each round to obtain
STDTDij = (LNTD ij − μ j

σj) , where μ j and σ j are the mean and standard deviation oflogarithmic time discount rates in round j over the subjects. Then, the standardized
time discount rate for each subject is defined as
12

AVSTDTD i = ∑ STDTD ij / 12 .

(6)

j =1

On the other hand, we define the degree of the delay effect of subject i, DLY ,as the time discount rate in the case of 2-days delay subtracted from that of 90-daysdelay, that is,15

DLYi = LNTDi 5 − LNTDi 6 ,(7)because, as we confirmed in section 3.2, the delay effect exists only between thesedelays. Nonetheless, to check the robustness of our conclusions, we report the results
using STDDLY, which is defined as the average of the differences in standardized time
discount rates, that is,
STDDLY i = [( STDTD i 2 − STDTD i 4 ) + ( STDTD i 5 − STDTD i 6 )+ ( STDTD i 7 − STDTD i 9 ) + ( STDTD i10 − STDTD i12 )] / 4 .

4.3

(8)

Estimation results on procrastination

Equation (4) is estimated by ordered probit because the dependent variable takes on
integers from one to five, and a larger value represents stronger procrastination. The
results using DLYi as the degree of the delay effect are presented in the upper panel ofTable 9. In the left columns, HWK i is regressed only over the time discount rate, which
reveals that the higher discount rates enhance the procrastination.8 Whether or not thediscount rates are standardized does not change the results. The middle column of thetable shows the estimates in the case in which HWK i is regressed only over the degreeof the delay effect, DLYi . Although DLYi has a negative coefficient, it is not
significant at all. In the columns on the right, HWK i is regressed over both the time
discount rate and the degree of the delay effect, confirming the above results: although
the variables representing the time discount rate are significant at the 1% or 5% level,
the coefficient of the degree of the delay effect DLYi is negative, but not significant at
all.
8Note that the estimated coefficients do not represent the marginal effect on the value of the HWK.
They represent the effect on the continuous latent variable corresponding to the HWK.
16

In the rightmost column of Table 9, we show the results using STDDLY insteadof DLY . The time discount rates have similar coefficients, but the coefficient of
STDDLY is significantly negative. In either case, the results that the coefficients of thedegree of the delay effect are not positive suggest that our subjects may besophisticates.
One might suspect, however, that our results merely failed to reflect the reality,
and are thus not reliable. To address this concern, we conduct an additional analysis.
Following the model of naïfs and sophisticates in O’Donoghue and Rabin (1999), andassuming that slight procrastination does not bring about regret strong enough toprevent further procrastination, even by sophisticates, we examine a hypothesis thatconsists of the following elements (see Table 10):1) Among the naïfs, a higher delay effect brings about stronger procrastination(right column of Table 10).
2) Among the sophisticates with a high delay effect, the effect of the delay
effect is not known a priori: the magnitude of the delay effect may not affectprocrastination at all or it may even affect it negatively (left-upper cell of
Table 10).
3) Among the sophisticates with a low delay effect, a higher delay effect may
or may not bring about stronger procrastination (left-bottom cell of Table
10).
However, the hypothesis is not refutable, because we have not defined who are naïfs
and who are sophisticates independently of the degree of the delay effect. It is natural toassume that sophisticates are more intellectual and introspective persons. Thus, in thisstudy, we assume that all of our subjects are sophisticates, since the students of Fudan

17University are undoubtedly the top elite in China.
To substantiate our interpretation, we define a variable representing the degree ofthe delay effect of the higher group, HDEi, and a variable representing that of the lower
group, LDEi, as follows:⎧ DLYi
HDEi ≡ ⎨⎩ 0

if DLYi ≥ DLY
otherwise⎧ DLYi
LDEi ≡ ⎨⎩ 0

if DLYi < DLY
otherwise

(9)

,

(10)

where DLY stands for a critical value dividing the subjects into two groups that will be
specified below. The regression equation now becomes
HWK i = β + γTDi + δ H HDEi + δ L LDEi + ν i , i = 1, ... , 29 .

(11)

Our hypothesis is δ H ≤ δ L ≥ 0 . On the contrary, if our subjects are all naïfs,then δ H = δ L ≥ 0 will be obtained (see the middle panel of Table 10).
To determine the critical value DLY , let us take a look at the histogram of DLY
in Figure 1. It reveals that there are two spikes, at 0 and 0.74. We do not have a clear
idea of how large the critical value that separates naïfs from sophisticates is. Thus, we
arbitrarily adopt 0.7, 0.9, and 1.0, each of which gives divisions of the samples into(high group, low group) = (12, 17), (6, 23), (4, 25), respectively. The estimation resultsof equation (11) are presented in the upper panel of Table 10. When the critical value isset at 0.7 (the left column), the coefficient of the degree of the delay effect of the highgroup, HDE, is negative, and that of the low group, LDE, is positive, as expected.
However, both of them are not significant. When we separate the groups at the value of18

0.9, the coefficient of the high group becomes smaller and significant at the 10% level(see the middle column). When the critical value is set at 1.0, the coefficient of the highgroup again becomes smaller, with the p-value of 5.7% (the right column). The
coefficient of the low group becomes larger, and its p-value also becomes smaller as thecritical value is raised, although it does not become significant. Meanwhile, the time
discount rate, AVTD, is significant at the 5% level in all cases. These results provide
some evidence for the naïfs-sophisticates interpretation of our results. In sum, we obtainδ H < 0 and δ L = 0 .
In the lower panel of Table 11, we present the results using the average of thedegree of the delay effect, STDDLY, instead of the degree of the delay effect within the
short delays, DLY. The histogram of STDDLY shown in Figure 2 reveals that 15 out of
29 subjects do not report the delay effect, if this variable is taken as the indicator of thedelay effect. Thus, we first stratify all the subjects who report negative STDDLY as the
low-delay-effect group and all others as the high-delay-effect group, and define thedegree of the delay effect of the high group, HSTDDLY, and that of the low group,
LSTDDLY. The estimates in this case are presented in the left column, and they revealthat the high group (HSTDDLY) has a significant negative coefficient and the low group(LSTDDLY) has a positive coefficient, which is almost significant at the 10% level.
Thus, we again obtain δ H < 0 and δ L = 0 . When we raise the critical value to 0.1, thedifference between these coefficients becomes smaller, which suggests that the criticalvalue should be smaller.9 All the results in Table 11 are consistent with our speculation,i.e., the naïfs-sophisticates model.

9

However, we should be careful about the interpretation of the magnitude, because the value is not
the marginal effect.
19However, we should note that our analysis does not offer abundant evidence forthe naïfs-sophisticates model. This study has, at best, found that the model seems tohold for a specific sample of the students at Fudan University, while many studies,including Hiruma and Ikeda (2007) and Tsutsui et al. (2007) have found a positive
correlation between the degree of the delay effect and the loan amount. The
naïfs-sophisticates model interprets this as the former results being obtained forsophisticates and the latter results for naïfs, but this interpretation may seem arbitrary.
To confirm that the model describes the reality, it is necessary to find a negative
correlation between the degree of the delay effect and procrastination in more cases. In
addition, although our presumption that our subjects are all sophisticates may bereasonable because they are all students of the top university, it is still necessary to
examine our hypothesis presented in Table 10, using broader data consisting of bothnaïfs and sophisticates.

5.

Conclusion

This study investigated the time discounting of human beings by an economic
experiment. Economic experiments are often termed as unreliable because the subjects
recruited do not have sufficient incentives to report their true preference. Our
experiment is immune to such criticism because the highest reward shown to thesubjects is close to their monthly household income.
This study achieved two goals. First, the delay effect people discount the near
future more than the remote future is substantiated by explicitly separating the delay

20

from the interval.10 Most previous studies have not separated these two experimental
conditions, while Read (2001) identified these two conditions and reported that thedelay effect was not found. Setting shorter delays than Read, we substantiated the delay
effect for a delay of less than 90 days. This finding is important because the delay effectinduces people to make time-inconsistent plans and then to regret their decision.
Another contribution of this study is its explanation of the subjects’procrastinating behavior by their time discount rates and the degree of the delay effect.
Higher time discounting always promote procrastination, but the delay effect involves
two competing consequences for procrastination: it has a positive effects when it is notstrong, but it reduces procrastination when it is strong. Our results are consistent with
the naïfs-sophisticates model developed by O’Donoghue and Rabin (1999).

10

At the same time, we confirmed the interval effect, in which subjects discount the future more in
shorter intervals than in longer intervals, and the magnitude effect, in which they discount the future
more for smaller amounts of money.
21

References
Ainslie, George (1975), Specious Reward: A Behavioral Theory of Impulsiveness andImpulse Control, Psychological Bulletin, 82 (4), 463-96.
Benzion, Uri, Amnon Rapaport, and Joseph Yagil (1989), Discount Rates Inferred from
Decisions: An Experimental Study, Management Science, 35, 270-84.
Frederick, Shane, George Loewenstein and Ted O’Donoghue (2002), Time Discounting
and Time Preference: A Critical Review, Journal of Economic Literature, 40,
351-401.
Hiruma, Fumihiko and Shinsuke Ikeda (2007) “A Study on Time Discounting Based onExperiments,” Review of Monetary and Financial Studies, 25, 12-25. (in Japanese)
Kirby, Kris N. (1997), Bidding on the Future: Evidence Against Normative Discountingof Delayed Rewards, Journal of Experimental Psychology: General, 126, 54-70.
Kirby, Kris N. and Nino N. Marakovic (1995), Modeling Myopic Decisions: Evidence
for Hyperbolic Delay-Discounting with 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, 443-77.
O’Donoghue, Ted and Matthew Rabin (1999), Doing It Now or Later, American
Economic Review, 89 (1), 103-24.
Read, Daniel (2001), Is Time Discounting Hyperbolic or Subadditive?, Journal of Risk
and Uncertainty, 23, 5-32.
Read, Daniel, and Peter H. M. P. Roelofsma (2003), Subadditive versus Hyperbolic
Discounting: A Comparison of Choice and Matching, Organizational Behavior andHuman Decision Process, 91, 140-153.
Read, Daniel, Shane Frederick, Burcu Orsel, and Juwaria Rahman (2005), Four Score
and Seven Years from Now: The Date/Delay Effect in Temporal Discounting,Management Science, 51, 1326-35.
Sasaki, Shinichiro, Shiyu Xie, Fumio Ohtake, Jie Qin, and Yoshiro Tsustui (2008),
Experiments on Risk Attitude: the Case of Chinese Students, China Economic
Review, 19 (2), 245-59.
Thaler, Richard H. (1981), Some Empirical Evidence on Dynamic Inconsistency,

22

Economic Letters, 8, 201-7.
Tsutsui, Yoshiro, Fumihiko Hiruma, Fumio Ohtake, and Sinsuke Ikeda (2007) “An
Analysis of the Impact of Interest Rate Ceilings to the Consumer Credit:
Behavioral Economic Approach,” Gendai Finance, 22, 25-73. (in Japanese)

23

Table 1. Experimental conditions and mean discount rates
Round
1
2
3
Delay (s)1 month
1 month
1 monthInterval (h)3 months
3 months
1 yearRewards (X)
2800
240
240
Payment (P)
No
Yes
Yes
Mean discount rate
9.586
40.03423.793Standard deviation
11.22840.49820.304

4
10 months
3 months
240
Yes
43.82846.475

5
2 days7 days2800
Yes
30.63849.328

6
90 days
7 days2800
Yes
22.12137.841

Round
Delay (s)Interval (h)Rewards (X)Pay (P)
Mean discount rate
Standard deviation

10
1 month
3 months
800000
No
1.103
2.033

11
1 month
1 year
800000
No
1.491
3.525

12
10 months
3 months
800000
No
0.959
1.612

7
1 month
3 months
2800
Yes
8.276
6.866

8
1 month
1 year
2800
Yes
5.819
5.105

9
10 months
3 months
2800
Yes
9.034
10.98724

Table 2. Discount rates’ and delays
Round
2
4
Delay (s)1 month300
Interval (h)3 months 3 months
Rewards (X)240
240
Payment (P)Yes
Yes
Observations
29
29
Mean discount40.03443.828rate
95% upper bound 24.630
26.14995% lower bound 55.439
61.506
P-value
0.617

52 days7 days
2800
Yes
29

6
90 days
7 days
2800
Yes
29

7
1 month
3 months
2800
Yes
29

9
300
3 months
2800
Yes
29

30.63822.121

8.276

9.034

1.103

0.959

5.664
4.855
10.88713.214
0.455

0.329
1.876

0.346
1.573

11.8757.727
49.40136.514
0.317

25

10
12
1 month 10 months
3 months 3 months
800000800000
No
No
29
29

0.638

Table 3. Discount rates and intervals
Round
2
3
7
8
Delay (s)1 month 1 month 1 month
1 monthInterval (h)3 months
1 year3 months
1 yearRewards (X)
240
240
2800
2800
Payment (P)
Yes
Yes
Yes
Yes
Observations
29
29
29
29
Mean discount40.03423.793
8.276
5.819
rate
95% upper bound24.63016.0705.664
3.877
95% lower bound55.43931.516
10.8877.761
P-value
0.003
0.003

26

10
1 month
3 months
800000No
29

11
1 month
1 year
800000
No
29

1.103

1.491

0.329
1.876

0.151
2.832
0.538

Table 4. Discount rates and reward amounts
2
7
3
8
Round
30
30
30
30
Delay (s)
90
90
365
365
Interval (h)
240
2800
240
2800
Rewards (X)
Yes
Yes
Yes
Yes
Payment (P)
29
29
29
29
Observations
Mean discount40.0348.276
23.793
5.819
rate
24.63
5.664
16.07
3.877
95% upper bound10.88731.5167.761
95% lower bound 55.439
0.000
0.000
P-value
Round
Delay (s)Interval (h)Rewards (X)Payment (P)Observations
Mean discountrate
95% upper bound
95% lower boundP-value
Round
Delay (s)Interval (h)Rewards (X)Payment (P)Observations
Mean discountrate
95% upper bound
95% lower boundP-value

4

9

300
90
240
Yes
29
43.828

300
90
2800
Yes
29
9.034

26.1494.855
61.50613.214
0.000

7

10

8

11

9

12

30
90
2800
Yes
29
8.276

30
90
800000
No
29
1.103

30
365
2800
Yes
29
5.819

30
365
800000
No
29
1.491

300
90
2800
Yes
29
9.034

300
90
800000
No
29
0.959

5.664
0.329
10.887
1.876
0.000

3.877
7.761

0.151
2.832

4.855
0.346
13.214
1.573
0.001

0.001

2

10

3

11

4

12

30
90
240
Yes
29
40.03430
90800000
No
29
1.103

30
365
240
Yes
29
23.79330
365800000
No
29
1.491

300
90
240
Yes
29
43.828300
90800000
No
29
0.959

24.63
0.329
55.439
1.876
0.000

16.07
0.151
31.516
2.832
0.000

27

26.1490.346
61.5061.573
0.000

Table 5. Discount rates and the payment of the reward
Round
Delay (s)Interval (h)Rewards (X)Payment (P)Observations
Mean discount rate
95% upper bound
95% lower boundP-value

1
7
30
30
90
90
2800
2800
No
Yes
29
29
9.586
8.276
5.315
5.664
13.857
10.8870.461

28

Table 6. Result of the regression analysis on three anomalies
Estimated
Estimated
P-valueP-valuecoefficient
coefficient
Constant−1.0740.000−1.070
0.000
HOR2
0.765
0.004
0.733
0.010
HOR30
0.077
0.651
0.198
0.276
HOR90
0.336
0.203
0.297
0.295
AMT240
4.413
0.000
4.518
0.000
AMT2800
2.941
0.000
3.113
0.000
log(AMT)
INT−0.0010.018−0.002
0.012
PAY
0.001
0.997
−0.0290.918
MALE−0.016
0.971
HINCOME−0.1480.720
Number of
348
276
observations
Number of29
23
subjects
0.62840.6525
R2
Note: regression equation is

Estimatedcoefficient
6.067
0.572
0.072
0.249

P-value−0.526−0.001
0.064

0.000
0.053
0.712
0.400

0.000
0.044
0.829

348
29
0.616

ln TDij = α 0 + α1HOR2 j + α 2 HOR30 j + α 3 HOR90 j + α 4 INTj + α 5 AMT 240 j + α 6 AMT 2800 j + α 7 PAYj + ε i , j

29Table 7.

Comparisons of discount rates in the questionnaire and the experiment

Panel A: Questionnaire
Delay (s)Interval (h)Rewards (X)Payment (P)QTD: Mean
Standard Deviation
Standard error
95% upper bound
95% lower bound

QTD5

QTD6

QTD3&8QTD11

2 days7 days
800
No
3.809
1.294
0.249
3.297
4.321

90 days
7 days
800
No
3.677
1.278
0.242
3.181
4.173

1 month
1 year
800
No
2.271
0.765
0.145
1.974
2.568

1 month
1 year
800000
No
0.004
1.145
0.213
−0.432
0.439

ETD5

ETD6

ETD3

ETD8

ETD11

2 days7 days
2800
Yes
2.622
1.386
0.257
2.095
3.149

90 days
7 days
2800
Yes
2.194
1.468
0.273
1.635
2.752

1 month
1 year
240
Yes
2.816
0.892
0.166
2.477
3.156

1 month
1 year
2800
Yes
1.392
0.962
0.179
1.026
1.758

1 month
1 year
800000No−1.4251.991
0.370−2.182−0.667ADJTD5ADJTD6ADJTD3ADJTD82 days7 days
2800
Yes
2.789
1.386
0.257
2.262
3.316

90 days
7 days
2800
Yes
2.789
1.468
0.273
2.231
3.347

1 month
1 year
240
Yes
2.119
0.892
0.166
1.780
2.458

1 month
1 year
2800
Yes
1.988
0.962
0.179
1.622
2.354

Panel B: Experiment
Delay (s)Interval (h)Rewards (X)Payment (P)ETD: Mean
Standard Deviation
Standard error
95% upper bound
95% lower boundPanel C: Adjusted discount rates
Delay (s)Interval (h)Rewards (X)Payment (P)ADJTD: Mean
Standard deviation
Standard error
95% upper bound
95% lower boundNotes: QTD and ETD are the logarithm of the discount rates elicited by the questionnaire
and the experiment, respectively. Adjusted discount rates are calculated from the regression
shown in the rightmost column of Table 6.

30

Panel D: Mean comparison test of the discount rate in the questionnaire and the experimentCompared group
Number of Observations

QTD5
29

P-value
Comparison
Number of Observations
P-value

ADJTD5QTD6ADJTD6

QTD3&8ADJTD3

27

29

28

29

28

0.000
QTD3&8
29

0.001

0.303

ADJTD8

QTD11

ETD11

29

29

29

0.061

0.002

31

Table 8.

Correlation between QTD and ETD
QTD5

QTD5

1.000

QTD6

0.901
(0.000)0.545(0.003)−0.067(0.738)0.700(0.000)0.733(0.000)0.340(0.082)0.256(0.198)−0.144(0.475)

QTD3&8
QTD11
ETD5
ETD6
ETD3
ETD8
ETD11

QTD6

QTD3&8

QTD11

ETD5

ETD6

ETD3

ETD8

ETD11

1.000
0.417
(0.027)−0.149(0.450)0.580(0.001)0.751(0.000)0.234(0.232)0.088(0.655)−0.232(0.235)

1.000
0.174
(0.375)0.345(0.072)0.414(0.029)0.354(0.065)0.557(0.002)0.056(0.775)

1.000
0.102
(0.598)0.014(0.941)0.058(0.765)0.282(0.138)0.838(0.000)

1.000
0.697
(0.000)0.297(0.117)0.333(0.077)−0.062(0.751)

1.000
0.375
(0.045)0.297(0.127)−0.171(0.374)

1.000
0.536
(0.003)0.067(0.730)

1.000
0.296
(0.120)1.000

Notes: QTD and ETD are the logarithm of the discount rates elicited by the questionnaire and the experiment, respectively.
The cells highlighted in yellow indicate the correlation between the pairs of experimental conditions that are similar.
P-values are in parentheses.

32

Table 9. Time discounting and procrastination
Estimatedcoefficient

P-value
0.487
0.009

Estimatedcoefficient
1.394

P-value
0.000

0.774

0.030

Estimatedcoefficient
1.278

P-value
0.000−0.0330.855

Estimatedcoefficient

P-value

0.357
0.803

0.459
0.008

−0.0700.706

Estimatedcoefficient

P-value

Estimatedcoefficient

P-value

1.413

0.000

1.699

0.000

0.775
−0.036

0.030
0.842

1.535

0.001
0.008
0.000
0.000

Constant
AVTD
AVSTDTDDLY
STDDLY

0.331
0.794

μ2
μ3
μ4

1.402

0.000

1.345

0.000

1.223

0.000

1.418

0.000

1.351

0.000

−1.692
1.720

2.053

0.000

1.952

0.000

1.759

0.000

2.065

0.000

1.956

0.000

2.405

Log likelihood

R2
Observations

3.171
0.000
−38.636
0.228
29

3.018
0.000
−39.821
0.157
29

2.746
0.000
−42.204
0.001
29

3.178
0.000
−38.564
0.232
29

3.019
0.000
−39.801
0.158
29

3.502
0.000
−36.0170.370
29Note: μ 2 to μ 4 are estimates of boundary values, where the smallest boundary μ0 is set to –infinity, the largest boundary μ5 is set to infinity. μ1 is
normalized to zero, for estimating the constant. Note that the estimated coefficient is not the marginal effect.

33

Table 10. Expected sign of correlation between the degree of delay effect and the degree of procrastination δ H and δ L
degree ofdelay effectcorrelation

high

δ HS

low

δ LS

sophisticatesnaïfs

sign of correlation

correlation

sign of correlation

δ HN

+

−

or

0

or

0

or +

+

δ LN

Note: δ represents the correlation between the degree of delay effect and the degree of procrastination.
Superscripts S and N stand for sophisticates and naïfs, respectively.
Subscripts H and L means high degree and low degree of procrastination, respectively.

Comparison of the correlation between high and low
type of person

sophisticatesnaïfsδ HS ≤ δ LS

δ HN = δ LN

Comparison of the correlation between sophisticates and naïfs
degree of delay effect

high

low

δ HS < δ HN

δ LS ≤ δ LN

34

0

or +

Table 11. Procrastinating behavior of Naïfs and SophisticatesConstant
AVTD
HDLY
LDLY

μ2
μ3
μ4
Log likelihood
R2
Number of observations
Critical value of DLY(High group, Low group)

Constant
AVSTDTD
HSTDDLY
LSTDDLYμ2
μ3
μ4
Log likelihood
R2
Number of observations
Critical value of STDDLY(High group, Low group)

P-value

Estimatedcoefficient

P-value

1.694
0.963
–0.505
0.294
1.535

0.000
0.010
0.090
0.231
0.000

1.717
0.944
−0.597
0.308
1.600

0.000
0.012
0.057
0.198
0.000

2.168

0.000

2.248

0.000

P-value

Estimatedcoefficient

1.628
0.828
−0.330
0.220
1.409

0.000
0.022
0.247
0.402
0.000

2.016

0.000

Estimatedcoefficient

3.130
0.000
−38.848
0.215
29
0.7
(12, 17)

3.282
0.000
−37.554
0.289
29
0.9
(6, 23)P-value

Estimatedcoefficient

P-value

3.493
2.138
−4.242
0.836
2.801

0.023
0.024
0.027
0.101
0.034

2.026
1.434
−2.375−0.749
1.791

0.001
0.004
0.021
0.551
0.000

3.413

0.012

2.461

0.000

Estimatedcoefficient

5.129
0.002
−13.294
0.723
29
0.0
(14, 15)

3.363
0.000
−36.9470.322
29
1(4, 25)

3.560
0.000
−35.637
0.390
29
0.1
(11, 18)

Note: Refer to the footnote of Table 9.
35

Figure 1. Histogram of the degree of the delay effect, DLY

98number of subjects

7
6
5
4
3
2
1
0
-4

-3

-2

-1
0
1
2
degree of the delay effect (DLY)

36

3

4

Figure 2. Histogram of the standardized degree of the delay effect, STDDLY

14

number of subjects

12
10
8
6
4
2
0
-１～-0.51-0.5～-0.01
0～0.490.5～1.0
standardized degree of the delay effect (STDDLY)37

Appendix: Payoff table used in the experiment (Round 1)① Payoff table (2,800.00 yuan, 1 month vs. 4 months, reward is not actually paid)

Number of theoption pair
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

Option A (yuan)(receipt at 1
month later)
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00
2,800.00

Option B(yuan)(receipt at 4
month later)

Rate of return(annual rate)−10%2,730.00−5%
2,765.04−4%2,772.00−3%
2,779.04−2%2,786.00−1%2,793.040%2,800.00
1%2,807.04
2%2,814.00
3%2,821.04
4%2,828.00
5%
2,835.04
6%2,842.00
7%2,849.048%2,856.00
9%2,863.0410%2,870.00
11%
2,877.04
12%
2,884.00
13%
2,891.04
14%
2,898.00
15%
2,905.04
16%
2,912.00
17%
2,919.0418%
2,926.00
19%
2,933.0420%
2,940.00
25%
2,975.0430%
3,010.00
40%
3,080.00
50%
3,150.00
60%
3,220.00

Notes: 1. Please circle A or B, whichever you prefer.
2. If you circled a wrong option, correct the answer with a double line.

38

Answers
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A

B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B