Time Preference or Computational Bias? --A Critique on Matching Task in Discount rate Measurement Xing Zhang 1 Abstract: This research makes the traditional method of subjective discount rate estimation-matching task open to doubt. Matching task indicates that the people’s discount rate is not as constant as what the neoclassical economists assumes, and it is hyperbolic i.e. the subjective discount rate is decreasing with the time. However, the experimental method is problematic due to its ignorance of people’s computational bias when they face the compound interest increasing problems. We believe that the computational bias can lead the estimated discount rate decreasing as well as the time preference that people are impatient in the short run and more patient in the long run. In our experiment, we proved this hypothesis and found that ruling out the factor of time preference, the rate of growth estimated from the subjects still decreases with the time. Therefore, the conclusion that the hyperbolic discount rate acquired from the matching task is due to time preference is too hasty. Keywords: experimental economics; computational bias; time preference; matching task; hyperbolic discounting

1. Introduction Nowadays, we cannot emphasize the importance of hyperbolic discounting models more in the modern intertemporal choice theory. Due to its tremendous applications, we think a comprehensive survey of literature will seem to be clichés. In our research we do not intend to prove whether the discounting pattern is hyperbolic, exponential, or trigonometric. What we want to discuss is the reliability the hyperbolic time preference explanation for the experimental evidence. There are two types of the most common experimental method which are used by economists and psychologists to elicit discount rates in the lab. The first type is called choice task: participants are asked to choose between a smaller sooner reward and a larger later reward. The most typical results from this method is that people prefer 100 1 Xing Zhang is a research assistant in Economics Lab of Jinan University. Thanks to Chen Tao, Ariel Rubinstein, Sun Huojun, Zhang Yaohui for their comments. I am indebted to Lu Yunfeng. Without his thoughtful guidance, insights and assistance, to accomplish this research is certainly impossible. Author can be contacted via Email: [email protected]

units immediately to 110 units tomorrow but prefer 110 units in 31 days to 100 units. In many papers, this phenomenon is considered as “Preference Reversal”. The other type is called matching task which is originated from Thaler (1981): participants are asked to indicate the X units that should occur at time T1 that would make this outcome indifferent to Y units at time T2. By the indifference point acquired from the experiment, the experimenter can estimate a discount rate. In Thaler (1981), subjects were asked to indicate the amount of money they would require in one month/one year/ten years to make them indifference to receive 15$ immediately. On average, the discount rate elicited from this experiment is correspondingly 345% in one month, 120% in one year, and 19%.in ten years. What’s more, in both Thaler (1981) and Benzion (1989), the researcher reported that large dollar amounts suffer less proportional discounting that do small ones, which is as know as ““Magnitude Effect””.

Based on a large amount of experiment results using these two method (you can refer to Fredrick et. al (2002) to see how large the volume is!), many behavioral economists announce that people have such time preference that they (we) are impatient in the short run and more patient in the long run. Inspired by Kahneman and Tversky (1979), Loewenstein et al (1992) first propose a model that can account for many anomalous results in the empirical studies using match task. They assume that discount function and value function are separable. The discount function is a generalized hyperbola: φ(t)= 1 /(1 + ατ )

−γ / α

, α,γ>0

Then the discount function is compatible with the results in the experiment that discount rate is decreasing with the time. They also adopted a function which is convex in log-log coordinates to predict the “Magnitude Effect”. And other version of

hyperbolic discounting models such as Laibson (1997) and O’Donoghue et al (1999) are becoming the popular intertemporal economics analytic tools. However, do hyperbolic discounting models provide a convincing explanation for the experiment results? Before our formal analysis, we would like to discuss the logic.

2. Logical Argument Economists are obsessed to build causal relationship between two different things, however, in some cases, the causal relationship is hard to prove or disprove. Therefore, perhaps one simple way to testify the causal relationship is to rule out other explanations. Hyperbolic time preference provides a plausible explanation for the experiment results. Some economists indicate that it is such time preference that leads to the “Preference Reversal” and decreasing discount rate and these two anomalies are solid evidence for such time preference. While it is tempting, after the controversial and profound research of Rubinstein (2003), the causal relationship is open to doubt. Like the discussion above, Rubinstein provides an alternative explanation that the theory of similarity can lead “preference reversal” in intertemporal choice as well as “present bias” in hyperbolic discounting preference. In his approach, decision makers use simplifying procedure by applying similarity relations when they face to multidimensional choices. In choice task, when participants are asked to make a choice between 100 units immediately and 110 units tomorrow, the time dimension is the decisive factor, because the time” immediately” is quite different from the time” tomorrow” , but in the money dimension, the 100 units and 110 units is much more similar. On the other hand, when participants are asked to make a choice between 100 units in 30 days and 110 units in 31 days, the money dimension is the decisive factor, because “in 30 days” and “in 31 days” are so similar that the participants only compare the amount of money they can acquire. Indeed the theory of similarity is a good alternative explanation of the decreasing discount rate acquired from the experimental method of choice task. However, the conclusion” the procedural approach is in fact superior to hyperbolic discounting in explaining the experimental results” in Rubinstein’s article is premature. Because in matching task (i.e. you will be indifferent between 15$ now and ____ $ in one month/one year/ ten years), participants do not make choices, the theory of similarity is not applicable any more. However, the hyperbolic discounting explanation to the results from matching task is still questionable by our approach-Computational Bias. In next section we will argue that it entirely possible that the computational bias can explain the results in matching task as well as hyperbolic discounting.

3. Preference or Bias?-An experimental argument 1. The procedure of experiment

As the old story goes, a too proud king learns a valuable lesson when he readily grants his wise man a special request: one grain of rice on the first square of a chessboard on the first day, two grains on the second square on the second day, four grains on the third square on the third day and so on. After several days the counting of rice grains gives way to weighing, then the weighing gives way to counting sackfuls, then to wagonfuls. The king soon realizes that there is not enough rice in the entire world to fulfill the wise man’s request. This tale involves people are not sensible facing exponential growth problems. In fact, discount rate is more directly related to interest rate in the financial market and it is some kind of rate of growth(i.e. you put some money in the bank, and the money will grow larger and larger at a rate). Considering the matching task, when people are asked “you will be indifferent between 15$ now and ____ $ in one month/one year/ ten years”, some people will think that if 15$ are deposited in the bank, how much they will be in in one month/one year/ ten years? If there is computational bias and underestimate the consequence of compound interest systematically, the discount rate (rate of growth) calculated by experimenters will decrease with time. In Thaler (1981), Benzion et. al (1989), Fredrick (1999) and so on, experimenters did not provide calculators to the subjects, we suspect that computational bias is inevitable and the pattern of bias is compatible with hyperbolic discounting to explain the experimental results. With such doubt, we designed a simple experiment (quiz) to test whether the pattern of computational bias is the same as the “hyperbolic discounting”: 80 undergraduate students are included in the experiment. Half of them are sophomores in applied mathematics major, and the others are in accounting and finance major whose GPA are higher than 3.5 in their third year. Because all of them are well trained in computation, by using such subject group, we could acquire the by-product whether the intensive training in computation can help people to avoid computational bias. They were asked to answer questions through paper questionnaires. The question is like this: Suppose you are a sale manager of a company, and the sales of your company are increasing by 1% per month. If the sales of this month are 40 wan 2 . Please estimate: The month sales in six month will be _____________wan The month sales in one year will be______________ wan The month sales in two years will be_____________ wan The month sales in three years will be____________ wan The month sales in five years will be_____________ wan The month sales in ten years will be______________wan There are four scenarios in the questionnaires: the sales of this month are 40 wan, 200 wan, 1000 wan, 5000 wan (the amount is use by Benzion et. al (1989)). In order to avoid “Framing Effect” (Kahneman & Tversky, 1971), the different amounts appeared in questionnaires randomly. We gave each participant a fine pan as reward 2

“Wan” is a Chinese money units. 1wan≈1333$

for their cooperation in experiment. Because it is the problem about growth, we think the answers that are the same in different time are lack of careful consideration of the question, and we delete their questionnaires. For instance, in 40 wan scenarios, if the subject estimates the month sales in two years are the same as the month sales in three years, we exclude her questionnaire in our analysis. For this reason, 21 questionnaires are excluded. 2. Results Let R denote the rate of growth calculated from subjects’ estimation; t denotes the time which is from 0.5 year to 10 years; N denotes sales of this month; F denotes sales in the future which are estimated by subjects. Then we have N (1 + R) t = F , and

R=t

F −1 N

By this formula, the rate of growth can be calculated from subjects’ answers. Table 1 shows descriptives inferred from subjects reply. The means of R in different periods and amounts are showed in Figure1. Intuitively, two results come into mind at a glance of the figure: “hyperbolic” rate of growth and “magnitude effect”! Except amount 5000, all the rates of growth in different amount decrease with time from 1 year. This is understandable. Although it is a quiz about estimation with total intuition, subjects still can “calculate” the amount in the very short run. Even the figure of amount 5000 is not satisfactory as other figure, the trend of the line also decreases with time. In addition, the so-called “magnitude effect” is also apparent in the figure: the line of amount 40 is higher than that of other. Therefore, if subjects treat discount rate as one kind of rate of growth, the computational bias is also a sound explanation for the results of matching task. time 0.5 year 1 year 2 years 3 years 5 years 10 years

40 .0369393 (.0517534) .0390046 (.0398359) .0358246 (.0308273) .0331230 (.0250772) .0265063 (.0183053) .0189941 (.0116478)

200 .0268422 (.0356731) .0301165 (.0296550) .0279125 (.0229285) .0273686 (.0203271) .0221683 (.0147866) .0163012 (.0099358)

1000 .0320049 (.0323165) .0383788 (.0354225) .0330744 (.0283453) .0314033 (.0243469) .0255178 (.0185600) .0175178 (.0115730)

5000 .0273847 (.0556486) .0337159 (.0515514) .0288876 (.0399438) .0297895 (.0332665) .0233725 (.0218649) .0166532 (.0142328)

Table 1 Means and Standard Deviations of Inferred Rate

.040 .036 .032 AVERAGE_40 AVERAGE_1000 AVERAGE_5000 AVERAGE_200

.028 .024 .020 .016 0

1

2

3

4

5

6

7

8

9

10 11

T Figure1 3. Significance test

In order to prove our findings are statistically reliable rather than illusion, we make two tests: first, test of “hyperbolic” growth rate; second, test of “magnitude effect”. The time effect is significant (p<0.05) using one-way ANVOVA except in amount 5000. In order to provide the pairwise comparisons between the rate in different time, we adopt the least significant difference test which is a series of t test The multiple comparisons are presented in Table 2. This statistical method was used in Benzion (1989).

Table 2 Multiple Comparisons

(I) Time 0.5

1

2

3 5

sum 40 sum 200 Mean Mean (J) Difference Difference Sig. Sig. Time (I-J) (I-J) 1 -.0020653 .730* -.0032742 .456 2 .0011146 .852 -.0010702 .807 3 .0038163 .523 -.0005264 .905 5 .0104329 .082 .0046739 .288 10 .0179452 .003 .0105410 .017 2 .0031800 .595 .0022040 .616 3 .0058816 .325 .0027478 .532 5 .0124983 .037 .0079482 .071 10 .0200105 .001 .0138153 .002 3 .0027016 .651 .0005438 .901 5 .0093183 .120 .0057442 .191 10 .0168305 .005 .0116113 .009 5 .0066166 .269 .0052003 .237 10 .0141288 .019 .0110674 .012 10 .007512 .209 .0058670 .182 * The mean difference is significant at the .05 level

sum 1000 Mean Difference (I-J) -.0063738 -.0010694 .0006016 .0064871 .0144871 .0022040 .0027478 .0079482 .0138153 .0005438 .0057442 .0116113 .0052003 .0110674 .0058670

Sig. .190 .826 .901 .182 .003 .616 .532 .071 .002 .901 .191 .009 .237 .012 .182

sum 5000 Mean Difference (I-J) -.0063312 -.0015028 -.0024048 .0040120 .0107315 .0048283 .0039263 .0103434 .0170627 -.0009019 .0055150 .0122343 .0064170 .0131363 .0067193

To test the significance of magnitude effect, we adopt Wilcoxon Signed Ranks Test. Because the answers in different amounts in a questionnaire are made by the same subject, they are not independent. Therefore, the Mann-Whitney U test is not applicable and we test the difference between different amounts in the same period. In this paper, we only show the significance of difference between amount 40 and amount 5000 in different time period 3 which is presented in Table 3 Table 3 Test Statistics* Z Asymp. Sig. (2-tailed)

In 0.5 year

In 1 year

Wilcoxon Signed Ranks Test In 2 years

In 3 years

In 5 years

In 10 years

-2.667

-2.348

-3.295

-2.536

-2.676

-2.626

.008

.019

.001

.011

.007

.009

*Based on positive ranks that amount 5000< amount 40

3

There are 36 matches for testing. Although difference of other matches are not as significant as the difference between amount 40 and 5000, there are still a large proportion of the matches are significantly different i.e. there are magnitude effect between other matches. If the readers are interested in the whole results of the test and data of the experiment, please send email to the author, who is glad to share them with others.

Sig. .379 .834 .738 .577 .136 .502 .585 .151 .018 .900 .443 .089 .372 .068 .350

Comparing the amount 40 and 5000, we can see the significant “magnitude effect”. Based on these evidences, we have to take leave to doubt the time preference explanation of the decreasing discount rate, because the computational bias can get same experimental results too.

4. Discussion Recently, more and more experimenters use additional sophisticated experimental techniques to acquire “true” discount rate. Kirby (1997) adopted a second price mechanism and Benhabib et.al (2006) employed a version of Becker-DeGroot-Marschak mechanism to give the subjects incentives to elicit their real discount rate. In essence, their method is one kind of matching task which intends to elicit the indifferent point. In order to defend our finding, we will make two arguments: first of all, although under such mechanisms, it is dominant strategy for subjects to report their indifferent amount, unfortunately, the evidence which illustrate that people will behave following dominant strategy under these two mechanisms is quite rare 4 . On the other hand, the evidence of inconsistence between theory and experimental results is tremendous and solid. Therefore, it is naïve to believe the dominant strategy trick will be effective for the subjects. Secondly, can monetary incentive eliminate bias? Hardly! In Thaler(1986), the author noted that the assertion that systematic mistakes will always disappear if the stakes are large enough should be recognized for what it is-an assertion unsupported by any data. In this case, the so called true values are underestimated by the subjects, so the values elicited by this method are biased in nature. From the perspective of positivist, our research is totally useless, because hyperbolic time preference provide a good prediction for the results of matching task and it is intuitively more appealing than the computational bias explanation in application in economics analysis. However, we must pay attention to the decision-making processes, otherwise we will draw very absurd conclusions if we solely rely on the paradigm of “revealed preference” and “as if” representations (see Benhabib, 2007). Turn back to the scenario of the experiment and to see how ridiculous it is if we believe “revealed preference”. The scenario of the questionnaire is about manager who has to estimate the future sales. If we take the answer seriously as a kind of preference, the manager is overconfident because the all the rates of growth are higher than 1 % given in questions. What is more, the manager is more optimistic in the short run than in the long run i.e. the confidence about the future is “hyperbolic”. However, the scenario is definitely a quiz to test subjects’ computational skills and has nothing to do with manager’s confidence about the future. In order to confirm our findings and conclusion, we should do far more things, because in this research we did not built up the causal relationship between the 4

Basic experimental results about second price auction can be found in Kagel (1995). And for skeptic research on BDM mechanism, refer to Harrison (1992).

decreasing discount rate and computational bias. Therefore, we could check that ruling out the computational bias, what kind of pattern will be in the discount rate measurement experiment. And new experimental techniques should be adopted to elicit the decision process.

References Benhabib, Jess, Alberto Bisin, and Andrew Schotter 2006. Present-Bias, Quasi-Hyperbolic Discounting, and Fixed Costs, NYU working paper Benhabib, Jess, Alberto Bisin, 2006, Choice and Process, Theory Ahead of Measurement, NYU working paper Benzion, Uri; Amnon Rapoport and Joseph Yagil. 1989. Discount Rrates Inferred from Decisions: An experimental study, Management Science.35, pp.270-84 Frederick, Shane 1999. Discounting, Time preference, and Identity, Ph.D. thesis, Department of Social and Decision Science, Carnegie Mellon University Frederick Shane., Loewenstein George and O’Donoghue Ted, 2002. Time Discounting and Time Preference: A critical review. Journal of Economic Literature pp 351-401 Harrison, G. (1992): Theory and Misbehavior of First-Price Auctions: Reply, 82, 1426-43. Kagel, John, 1995, Auctions: A Survey of Experimental Research, in The Handbook of Experimental Economics, John H. Kagel and Alvin E. Roth, editors, Princeton University Press Kahneman, Daniel and Amos Tversky. 1979. “Prospect theory: An analysis of decision under risk,” Econometrica 47, pp. 263-92 Kirby, Kris 1997, Bidding on the Future: Evidence against Normative Discounting of Delayed Rewards, Journal of Experimental Psychology. 126, pp.54-70 Laibson, David. 1997. Golden egg and hyperbolic discounting. Quarterly Journal of Economics. 112, pp.443-77 Loewenstein, George and Richard Thaler. Intertemporal Choice Journal of Economic Perspectives 3(4) (1989): pp. 181-193 Loewenstein, George and Drazen Prelec. 1992 “ Anomalies in intertemporal choice: evidence and an interpretation,” Quarterly Journal of Economics. 107:2 pp.573-97 O’Donoghue, Ted and Matthew Rabin, 1999 ”Doing it now or later,” American Economics Review 89.1pp.103-24 Rubinstein, Ariel, 2003 Economics and psychology? The Case of Hyperbolic Discounting, International Economics Review, Vol. 44, No.4, pp.1207-1216 Thaler, Richard H. 1981. Some empirical evidence on dynamic inconsistency Economics Letter 8, pp.201-07 Thaler, Richard H. 1986. “The Psychology and Economics Conference Handbook: Comments on Simon, on Einhorn and Hogarth, and on Tversky and Kahneman” The Journal of Business, Vol. 59, No. 4, Part 2: The Behavioral Foundations of Economic Theory (Oct., 1986), pp. S279-S284

Appendix I

Data from the experiment Amount 40 Median Mean

45 55 74 94 130 300

52.06966 71.31525 125.4153 195.7102 360.3797 1069.924

Amount 1000 Median Mean

1100 1300 1750 2500 3500 6000

1232.775 1733.683 2981.167 5381.733 14025.73 37751.33

Stdev

24.61133 48.31853 131.306 215.9695 511.2065 2015.288

Stdev

253.9649 969.1676 3991.849 10986.97 41626.49 105287.5

Amount 200 Median Mean

215 250 300 400 560 1000

239.4492 305.4915 458.6695 720.3051 1232.085 3319.661

Amount 5000 Median Mean

5400 6100 7500 9300 12000 25000

6252.898 9858.119 32592.78 65629 108338.9 954835.1

Stdev

69.49908 159.878 368.3367 791.2211 1850.279 6081.641

Stdev

4030.368 15689.98 157934.5 318158.8 512648.3 6393390