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 it at short delay such as less than 8-week delay. This implies that people make dynamically inconsistent plans. We also found the interval effect that the time discount rate decreases with prolonged intervals. We show that the interval and the magnitude effects are caused because intertemporal choice is made partially based on the differential in reward amount, while Weber’s law explains neither the delay nor the interval effects sufficiently.
The original article first appeared in Journal of Risk and Uncertainty 39:87–112, 2009. A newly written addendum has been added to this book chapter.
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Notes
- 1.
This chapter 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 chapter.
- 2.
- 3.
The situation is similar to the Newton–Raphson method of numerical optimization, which sometimes depends on initial value that researchers set.
- 4.
Benzion et al. (1989) experimented with 282 students, but they did not reward the subjects.
- 5.
No subjects are from the department of dentistry.
- 6.
The experiment was programmed and conducted with the software z-Tree (Fischbacher 1999).
- 7.
1,352 out of 3,285 time discount rates (219 subjects × 15 combinations of receipt timings) fall into this multi-switching case.
- 8.
957 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 %.
- 9.
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).
- 10.
We cannot control the amount of reward because it is randomly chosen for each question. As shown in the far-right column of Table 3.3, however, the average amounts do not differ substantially among 15 combinations of receipt timing. Therefore, possible biases due to the uncontrolled amount may be trivial.
- 11.
Subjects’ attributes are based on the questionnaire survey conducted at the end of the experiment.
- 12.
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 is estimated by a logit model.
- 13.
Of course, this chapter 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 are compared, 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 division itself affects the time discount rate. However, this chapter does not pursue how the subadditivity is different from the interval effect. To elucidate this point, it is necessary to devise specific experiments, which will be a future task.
- 14.
Pender (1996) adopts this method in their economic experiment.
- 15.
In this analysis, we regard multi-switching cases as irrational choices and exclude them from the analysis. There are only a few such cases in the questionnaire survey.
- 16.
The estimated coefficient of the logarithmic average amount of reward in Table 3.6 indicates that the time discount rate decreases by about 0.1 when the average amount of the reward changes from 2,000 to 10,000 yen (0.080 × (ln(10000)−ln(2000)). The time discount rates in the experiment come close to the questionnaire survey when the magnitude effect is considered.
- 17.
Read and Scholten (2006) examine a similar idea.
- 18.
We are grateful to an anonymous referee for explaining this with a concrete example.
- 19.
Four variables out of these five explanatory variables determine the remaining one variable. However, as the relation is nonlinear, the collinear problem does not arise.
- 20.
The results that use the rate of return R instead of ln (1 + R) may appeal more to our intuition. The estimation results are almost the same as those of Model 3 in Table 3.9, which imply that a 1 % increase in the rate of return and a 16-yen increase in the differential amount will cause nearly the same increase in the probability of choosing the later option.
- 21.
This is a phenomenon that Scholten and Read (2006) call superadditivity.
- 22.
In these models we added the amount of reward, because Weber’s law as defined in this chapter does not explain the magnitude effect theoretically. However, when we delete the amount of reward, the results are essentially unchanged.
- 23.
Of course, we do not deny the possibility that Weber’s law will solve the anomalies using a different approach from ours. For example, Read and Scholten (2006) apply Weber’s law not only to scaling time but also to scaling the amount of money and examine the validity of the law.
- 24.
When we delete the delay and interval dummies, A(t, s, R) becomes significant, while the differential in amount remains significant.
- 25.
The paper also investigate time horizon, concerning two different points starting either at different points in time or at the same point in time. However, since we criticized the latter method, arguing that declining impatience cannot be separated from the interval effect, we do not report the results that were obtained with this method.
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Acknowledgments
An earlier version of this chapter was presented at the Behavioral Economics Conference and Annual Meeting of Japanese Economic Association. The authors are grateful to comments by Takanori Ida, Fumihiko Hiruma, Shinsuke Ikeda, Yuichi Fukuta, Kazumasa Umeda, and the participants of the conferences. Especially, we would like to thank an anonymous referee of this journal for comments, which resulted in a substantial revision and improvement.
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Addendum: Examination of Elicitation Designs
This addendum has been newly written for this book chapter.
Addendum: Examination of Elicitation Designs
In the text of this chapter, we report that declining impatience is recognized at short delays of less than 8 weeks. As was noted in Sect. 1.2, this is a new finding, because it involves (1) selection of one of two options randomly shown on computer display, (2) asking subjects to choose from specified two options, and (3) specifying the timing by the length of the day instead of by calendar date. Using methods (2) and (3) and a multiple price list (MPL) format that presents whole choices simultaneously instead of (1), Read (2001) found no evidence of declining impatience. However, declining impatience has been observed with a matching method that asks subjects regarding an amount at a specified date, making them indifferent between the specified options (Read and Roelofsma 2003). It is also observed when the timing is specified with calendar dates (Read et al. 2005). These results motivate us to investigate under which method declining impatience is more easily observed. One of the authors of the text (Fumio Ohtake) studied this topic, and wrote about it in Hanaoka et al. (2014). In this addendum we briefly introduce that paper.
Hanaoka et al. (2014) mainly focuses on two designs for elicitation of time discounting.Footnote 25 The first of these is (1) the elicitation format, in which options are presented as a multiple price list (MPL) or a titration. In a titration, a subject is presented with each intertemporal choice, based on a decision tree; according to each chosen outcome, the options presented in the subsequent choice are adjusted to present the remaining options as the range of remaining discount rates becomes narrower. The second is (2) the framing, which concerns whether subjects are asked amounts or delays.
An original nationwide web survey on time and preferences was conducted in 2011, with 4,970 subjects surveyed. Analysis of the responses indicates that using the net comparison approach, declining discounting is observed only for the titration elicitation format within the framing of “ask amount,” whereas the other procedures show no bias. As per the framing, increasing discounting is more likely to be observed when using the framing of “ask delay” than when using “ask amount.”
The paper further investigates whether the elicited discount rates can accurately predict behavior such as debt, credit card borrowing, obesity, smoking, and gambling. A subject is classified as having dynamically inconsistent time preferences if the subject exhibits either declining discounting (present-biased preferences) or increasing discounting (future-biased preferences); a subject is classified as having dynamically consistent time preferences (no-biased preferences) if he/she exhibits constant discounting. Each behavior is regressed over the two dummy variables according to the classification (present-bias and future-bias) and the discount factor.
The estimation reveals a weak correlation between behavior and the values elicited by the MPL; none of the coefficients on present bias and future bias are statistically significant. In sharp contrast, most of the coefficients elicited by the titration are significantly correlated with behavior and show the expected signs. Thus, the discount rate is more correlated with behavior such as debt, credit card borrowing, obesity, smoking, and gambling when elicited with the titration method than with the MPL, when amount is asked. On the other hand, the parameters of present- or future-biased preferences elicited under the framing of “ask delay” fail to capture behavior in both the MPL and the titration, whereas asking amount often produces reasonable outcomes in the titration method.
The results are summarized as follows. (1) Whether subjects exhibit dynamically consistent or inconsistent time preferences depends on which elicitation designs are adopted. (2) Whether the elicited discount rates accurately predict behavior varies across elicitation designs. Specifically, the results suggest that an elicitation design using a titration format and/or asking subjects on amount would provide more relevant measures for predicting behavior than other designs.
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Kinari, Y., Ohtake, F., Tsutsui, Y. (2016). Time Discounting: Declining Impatience and Interval Effect. In: Ikeda, S., Kato, H., Ohtake, F., Tsutsui, Y. (eds) Behavioral Economics of Preferences, Choices, and Happiness. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55402-8_3
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