Abstract
We propose and test a new method for eliciting curvature-controlled discount rates that are invariant to the form of the utility function. Our method uses a single elicitation task and obtains individual discount rates without knowledge of risk attitude or parametric assumptions about the form of the utility function. We compare our method to a double elicitation technique in which the utility function and discount rate are jointly estimated. Our experiment shows that these methods yield consistent estimates of the discount rate, which is reassuring given the wide range of estimates in the literature. We find little evidence of probability weighting, but in a second experiment, we observe that discount rates are sensitive to the length of the front-end delay, suggesting present bias. When the front-end delay is at least two weeks, we estimate average discount rates to be 11.3 and 12.2% in the two experiments.
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Notes
See Frederick et al. (2002) for a review of the literature.
Note that our approach does not require that Eq. 6 holds for all combinations of p t and p t + τ . We can cover a broad range of discount rates (0–347%) by varying p between 0.5 and .65, thus allowing us to avoid extreme probabilities where probability weighting has been shown to be most severe.
We chose the parameters for our first experiment to satisfy this assumption: all payments are made either three or twelve weeks in the future.
Longer time horizons introduce the possibility of background consumption changing over time, which when not accounted for will bias discount rate estimates. See Noor (2009) for a discussion.
In subject instructions and decision sheets, the Sooner option is referred to as Option A and the Later option as Option B.
The actual decision table presented to subjects is shown in the Appendix.
The instructions and decision sheets for all tasks in both experiments are available at http://www.excen.gsu.edu/swarthout/LMS/.
As in Task P, in subject instructions and decision sheets these were labeled Option A and Option B.
Our pilot experiment had sessions with and without the AEIR information. Excluding this information did not appear to have substantial effect on the pilot results. Ultimately we decided to exclude the AEIR from the P-Task because there is no naturally-occurring counterpart for expressing probabilities as interest rates. Coller and Williams (1999) first introduced the AEIR information to the discount rate task and found that including the AEIR led to lower discount rate estimates and residual variance.
These are the same payment levels used by Holt and Laury (2002) in their highest payment treatment.
This equates the expected value of the payment in the standard and binary-lottery discount rate tasks, as the baseline probability of payment was only 50% in Task P, but 100 percent in the two other tasks. Randomly determining whether a subject receives payment is used for consistency with earlier discount rate studies, including Coller and Williams (1999) and Andersen et al. (2008).
It is common to exclude multi-switchers from analyses that focus on a single switch point. An advantage of the maximum likelihood estimation that follows is that we can add error terms to the model and include all decisions.
In a pilot experiment with the P Task that did not include a 0% interest rate choice (Morgan 2009), 45.7% of participants choose Option B for all decisions implying a discount rate of less than 2.02%. The pilot experiments differed in other important dimensions (including lower stakes, shorter time horizon, AEIR provided in some sessions, and conducted at different university); yet, the pilot results are broadly consistent with the findings we report here.
At the time the experiment was conducted, interest rates were quite low. For example, Bankrate.com’s weekly survey of banks conducted October 21, 2009, found a yield for one-year CD of only 0.92%. http://www.bankrate.com/finance/cd/national-cd-rate-averages8-134136.aspx
Both the data and estimation routines are available at http://www.excen.gsu.edu/swarthout/LMS/.
In comparing Andreoni and Sprenger (2012) to our results, note that in their model ω has the opposite sign to ours.
The instructions and decision sheets for all tasks in both experiments are available at http://www.excen.gsu.edu/swarthout/LMS/.
As with Table 6, subjects who switch more than once are excluded from this tabulation.
d_NF is equal to 1 if an observation is from the No FED treatment, 0 otherwise; d_SF is equal to 1 if an observation is from the Short FED treatment, 0 otherwise; d_SH is equal to 1 if an observation is from the Short Horizon treatment, 0 otherwise. We now express the discount rate as \(\delta = \delta^B + \delta^{NF} \times \textbf{d\_NF} + \delta^{SF} \times \textbf{d\_SF} + \delta^{SH} \times \textbf{d\_SH}\).
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Acknowledgements
We benefitted greatly from discussions with Erica Von Nessen and thank her for piloting the instructions during her dissertation (Morgan 2009). We thank seminar participants at Emory University, University of South Carolina, and Xiamen University, as well as conference participants at the 2009 Economic Science Association North American Regional Meeting, the 2010 Foundations and Applications of Utility, Risk and Decision Theory International Conference, the 2010 Southern Economic Association Annual Meeting, and the 2012 International Meeting on Experimental and Behavioural Economics. We especially want to acknowledge and thank Glenn Harrison for his timely and detailed comments on earlier drafts. All errors remain our own.
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Appendix: Experiment 1 Task P instructions and decision sheet
Appendix: Experiment 1 Task P instructions and decision sheet
(The instructions and decision sheets for all tasks in both experiments are available at http://www.excen.gsu.edu/swarthout/LMS/.)
Each person in this room has the chance to earn a large sum of money from this part of the experiment. The amount of money you earn in this part of the experiment will depend both on your choices and also on chance. We will first explain to you the choices that you will be making, and then how your earnings are determined.
1.1 Your choices
In this part of the experiment, you will be asked to make a series of choices between Option A and Option B. On the lab monitors, you can see three 10-sided dice: a red die, a white die, and a blue die.
Your earnings in this part of the experiment depend, in part, on the outcome from rolling these three dice. We will throw the dice, and then record the outcomes as a three digit number:
For example, if we threw a “6” on the Red Die, a “4” on the White Die, and a “0” on the Blue Die, the number recorded would be 640. On the other hand, if we threw a “0” on the Red Die, “6” on the White Die and a “4” on the Blue Die, the number recorded would be 064. Notice that there are one thousand possible outcomes of the drawing. They can be listed as:
Each of these numbers has a 1-in-1000 chance of being thrown (if three zeros are thrown, this will represent an outcome of 1000).
Please look at the Decision Table. Each Decision row gives you a choice between Option A and Option B. As an example, look at the row for Decision 2. Option A pays $200 in three weeks if the number thrown on the die is between 001 to 500. Thus, there are 500-in-1000 chances of receiving $200 if you choose Option A. This represents a 50% chance of winning. Now look at Option B for Decision 2. Option B pays $200 in 12 weeks if the number thrown on the die is between 001 to 501. There are 501-in-1000 chances (50.1%) that the number drawn will be in this range. As you move down the table, Option A is always the same, but with Option B the chances of winning increase.
Looking down the rows of the Decision Table you can see that you have 20 Decisions to make. In each of these Decisions, Option A represents a chance to earn $200 in three weeks if the number thrown on the die is between 001 and 500. In each of these Decisions, Option B represents a chance to earn $200 in 12 weeks, with the chance of earning this money increasing as you move down the rows of the table. For example, in Decision 16, there is a 500-in-1000 chance of earning $200 in 3 weeks if you choose Option A and a 536-in-1000 chance of earning $200 in 12 weeks if you choose Option B.
You will make twenty choices. For each Decision row, you will have to choose between Option A and Option B. You may choose A for some Decision rows and B for other rows, and you may change your choice and make them in any order. So now please look at the boxes on the right side of the record sheet. You will have to circle your choice of A or B for each of the twenty Decisions.
Even though you will make twenty Decisions, only one of these will end up possibly affecting your earnings, but you will not know in advance which Decision will be used. Each Decision has an equal chance of being used for payment.
1.2 How you will be paid
For each Decision row, choose whether you prefer A or B for that row. After you have made all 20 Decisions, we will throw a black 20-sided die. You can see this die on the lab monitors. The number that is thrown on the 20-sided die will determine which one of your 20 Decisions will count for payment. For example, if we throw a 12, we will look at Decision Row 12 to see if you chose Option A or Option B. We will not look at any of your other Decisions when we determine your earnings for this part of the experiment.
Next, we will throw the three colored 10-sided dice to determine whether the monetary outcome is $200, as we described above. Continuing with the example for Decision 12, the table below illustrates the outcomes that would occur depending on the 3-digit die throw and whether you chose Option A or Option B.
Die throw | Outcome if you choose Option A | Outcome if you choose Option B |
|---|---|---|
001–500 | $200 in 3 weeks | $200 in 12 weeks |
501–518 | $0 | $200 in 12 weeks |
519–000 | $0 | $0 |
Suppose instead that when we threw the 20-sided die we had determined that Decision 3 was the one that counts for payment. In this case, the table below illustrates the outcomes that would occur depending on the 3-digit die throw and whether you chose Option A or Option B.
Die throw | Outcome if you choose Option A | Outcome if you choose Option B |
|---|---|---|
001–500 | $200 in 3 weeks | $200 in 12 weeks |
501–502 | $0 | $200 in 12 weeks |
503–000 | $0 | $0 |
After this, we will throw a black 10-sided die for the final payment phase. If the outcome is between 2 and 9, then none of your choices in this part of the experiment will count for payment. If the outcome is a 0 or 1, then your choices and the earlier rolls will determine your earnings as described above.
Note that we will roll the dice individually for each person, and so each person is equally likely to receive payment in the final payment phase of the experiment. More than one person may be selected for payment, and each person selected will be paid the amount determined by their choices and die rolls as described above.
If you earn $200 in this task, we will give you a certificate for $200 redeemable in cash here in the Andrew Young School Building in three weeks or 12 weeks, depending on the option you choose. If you do earn this money, then at the end of the experiment we will give you more detailed instructions for redeeming your certificate and the location in this building where you will go to redeem your certificate.
1.3 Summary
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1.
You will choose Option A or Option B for each of the 20 rows of the Decision Table.
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2.
We will throw a black 20-sided die to determine which ONE of these Decisions will count.
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3.
We will look at the choice you made in this Decision, and then throw the three 10-sided dice (one red, one blue, and one white).
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4.
We will look at the three-digit-number that comes from the dice roll and also at your choice for this one Decision to determine the monetary outcome: receive $0, receive $200 in three weeks, or receive $200 in 12 weeks.
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5.
Finally, we will throw a 10-sided die to determine whether or not this outcome will be paid. If we throw a 2-9 then you will not be paid for your Decision. If we throw a 0 or 1, then you will receive payment according to the outcome in step #4. If the outcome is for you to receive $200, then you will receive a certificate for $200 that may be redeemed in three or 12 weeks (depending on whether you chose Option A or Option B).
We will next go through a simple demonstration of this task before you make your choices. This demonstration is to help you better understand the task and will not count for money. Please listen and watch the demonstration before making your own choices.
If you have any questions, please raise your hand and one of us will come to your desk to answer it.
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Laury, S.K., McInnes, M.M. & Todd Swarthout, J. Avoiding the curves: Direct elicitation of time preferences. J Risk Uncertain 44, 181–217 (2012). https://doi.org/10.1007/s11166-012-9144-6
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DOI: https://doi.org/10.1007/s11166-012-9144-6