Time-Sharing Experiments for the Social Sciences. Exponential-Growth Bias: Theory and Experiments. For Review Only

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1 Exponential-Growth Bias: Theory and Experiments Journal: Time-Sharing Experiments for the Social Sciences Manuscript ID: TESS-0.R Manuscript Type: Original Article Specialty Area: Economics

2 Page of Time-Sharing Experiments for the Social Sciences Exponential-Growth Bias: Theory and Experiments Introduction In recent years the average personal savings rate in the U.S. has been low, between % and % during This is far below what many believe to be appropriate to maintain retirement consumption (Munnell et al., 00). In addition, people are substantially indebted. Conditional on credit card ownership, the average revolving credit card debt is $,0 (00 Survey of Consumer Finances). We posit that a contributing factor to both of these facts is that people downwardly bias their estimates of exponential growth due to a failure to compound interest, a version of exponential-growth bias (Stango and Zinman, 00a). They consequently make many predictable mistakes regarding their personal finances, including systematic misestimation of interest rates. Such a bias could lead people to make costly errors in savings, borrowing, and investment decisions. A person who exhibits exponential-growth bias will underestimate the cost of holding debt and will underestimate the benefit of investment. This could potentially be a large behavioral cause of the middle-class s low savings rate for retirement, and the use of debt at extremely high interest rates among lower-income households. Payday loans, for example, seem to exploit this bias by masking compounding interest to achieve a typical APR of 0% (Skiba and Tobacman, 0). Our main goal is to determine whether people systematically misestimate exponential growth and to measure the magnitude of this mistake. Our primary hypothesis is that people do not adequately account for compounding and estimate exponential growth to be somewhere in between linear growth and the correct exponential growth. In addition, we will explore for which segments of the population and for which types of financial questions the bias is most severe. We hope to address this potential heterogeneity by using a representative panel of participants. Model While the tendency for people to under-appreciate compound growth tendency has been known in the psychology literature for some time, no adequate model currently exists. We propose a simple

3 Page of one-parameter model, with several distinct advantages over earlier attempts. In the simplest setting of a constant growth rate, we model subjective estimates of exponential growth as given by: V t (i) =V 0 ( + it ) (t ) ; œ [0, ] () We believe this model captures several key features of exponential-growth bias in the most parsimonious form. First, it embeds the correct exponential function and a linear growth function at the extreme values of its single parameter. That is, =0corresponds to the correct exponential growth function, while =corresponds to simple linear growth. While there may be other biases or errors that cause subjects to ignore time horizons, we focus here on exponential-growth bias and note that our model fully allows other such factors to be incorporated where appropriate. We therefore also predict that multiple periods are necessary to generate an error: a single period of growth is correctly estimated for any degree of bias. Finally, we note that this model satisfies a criterion we call no long-run exponential equivalence. An extension of the model allows for exponential-growth bias in the presence of variable rates in such a way that it reduces to () for the case where į is constant over time. By formally stating a model of exponential-growth bias, we achieve three important goals. First, the model enables a precise definition of the bias. This in turn enables us to generate specific hypotheses about the e ect of the bias in di erent questions, and thus conduct a more powerful experimental test. Finally, it generates specific predictions about which real-world situations will be the most costly to people with the bias. For example, the cost of borrowing through a frequently- In particular, we are attempting here to develop a portable extension of existing models (c.f. Rabin (00)). Suppose subjective beliefs are given by  V t. We say that these beliefs demonstrate long-run exponential equivalence if there exists a value s such that lim tæœ [  V t /(s ( + i) t )] =. That is, as the number of periods grows large,  V approaches the correct exponential growth function with a starting value of s. This criterion generates a testable restriction on the model (and is one we expect to verify). Equally importantly, it helps distinguish our parametrization from, for example, a simple weighted average of exponential and linear growth (which would fail the no long-run exponential equivalence criterion) Our preferred approach is to re-cast the variable interest rate as a constant rate and then apply equation (). Intuitively, people may do exactly this simplification when faced with a complex setting. Formally, suppose a biased agent faces (possibly changing) period-specific interest rates į =< i,i,...i T >. Then the agent s perceived principal balance at time T is given by:  Vt (į) = V t (M (į)). wherem (į) is the generalized power mean of į, given by: M (į) = C T Tÿ s= i g( ) s D /g( ) g :[0, ] æ [0, ], g(0) = 0, g() = ()

4 Page of Time-Sharing Experiments for the Social Sciences compounding payday loan would be under-estimated to a greater degree than would be the cost of borrowing on an APR-based credit card. TESS Proposal Subjects will face a sequence of monetarily incentivized binary decisions where they must choose one of two assets. The value of one asset will be deterministic while the value of the second asset will be based on chance (see Appendix C). The value of each asset will not be clearly indicated but rather will require computation. Questions will be drawn from four di erent financial domains. The first domain is a simple choice between a growing asset and a fixed amount. The second is between an amortized investment and a fixed amount. The third domain is a choice between two growing assets that begin with di erent principals, and the fourth domain is a choice between an asset that grows with a fixed interest rate and an asset that grows with a deterministically fluctuating interest rate. All questions are listed in Appendix D. Subjects will be experimentally exposed to an intervention (Appendix C) designed to increase the salience of compounding in exponential growth. We hypothesize that we will identify a significant degree of exponential growth bias (H0), with a considerable degree of heterogeneity explained by demographic characteristics (H). Moreover, exposure to our intervention will decrease the extent of exponential growth bias in all domains (H) and will do so to a greater extent for individuals and demographic groups that exhibit greater bias (H). We use a x design. Subjects will rotate through the four domains but the starting point will be randomly determined for each subject. Half of the subjects will be in the control and half will be in the treatment. The treated will be given the graphical intervention for the latter three of the four domains. This allows for the estimation of both a between-subject and within-subject e ect of the graphical intervention on bias. Because a significant contribution of this paper is the application of economic methodology, we enforce incentive-compatibility by randomly selecting a question for each subject for an additional payment. The subject s payment will depend on her choice in that decision, and by the randomly determined payo. The subjects choices will reveal their subjective This research is funded by the Fletcher Jones Foundation. We have external funding to pay subjects for their decisions.

5 Page of valuations of the two assets.. Request and Justification for Survey Lengths and Sample Size We are requesting N=0 subjects and survey lengths per subject. The survey lengths comprise our experimental questions (Appendix D) and six-question post-experiment survey (Appendix E). In addition we request the Financial Services Profile: ppfs000 ppfs0, and ppfs0. Our request for N=0 subjects is based on our findings in our pilot study (see below). Our power calculations indicate that this sample will provide adequate power to precisely estimate the distribution of exponential-growth bias, and to detect di erences in the distribution as a result of exposure to our experimental intervention. For example, if the intervention lowers the average value of in Domain by half the distance to that of the relatively easier Domain, our desired sample will provide the desired power of (- ) = 0.. Similarly, our request for experimental survey lengths will allow us to obtain responses per subject within each Domain. This will enable us to adequately test the hypothesis of exponential-growth bias against competing alternatives (such as random error) by testing both the within-domain and between-domain reliability of our model. Pilot Results The proposed study design builds o of an online pilot run on a student sample during May 0. The subjects were all assigned to the equivalent of the control groups in the current proposal, and were paid for both their participation and responses in the form of Amazon.com gift certificates. All subjects answered eight questions in each of four domains, yielding thirty-two responses per subject in total. Because both the correct answer and the cost of error varied considerably across questions, it is most useful to consider the results in terms of the implied degrees of exponentialgrowth bias. That is, for each response and across responses at the subject-domain or subject level one can calculate what value of œ [0, ] will minimize the di erence between the predicted response and the actual response. The distribution of these values at the individual response level is shown in Figure. The results of the pilot indicate that there is considerable heterogeneity in the extent of exponential

6 Page of Time-Sharing Experiments for the Social Sciences growth bias. Roughly half of subjects were completely biased ( =), while a smaller mass were completely unbiased ( =0). A significant mass also located at intermediate levels of bias. Furthermore, the extent of bias was significantly greater in the more di cult domains. More than half of subjects displayed no bias in Domain, which required only a single calculation and would be relatively simple to so with just the aid of a calculator. In contrast, nearly three-quarters of subjects located at =for Domain, which requires the most calculations to solve. However, there was clearly a common driver of a subject s responses: a factor analysis of subject s average in each domain yielded a just a single factor with an eigenvalue exceeding, and which received positive loadings from all four domains. This is consistent with our hypothesis that there is an individual-level degree of exponential-growth bias. References Benzion, Uri, Alon Granot, and Joseph Yagil, The Valuation of the Exponential Function and Implications for Derived Interest Rates, Economic Letters,,, 0. Cole, Shawn and Gauri Kartini Shastry, Smart Money: The E ect of Education, Cognitive Ability, and Financial Literacy on Financial Market Participation, February 00. Working Paper. Eisenstein, Eric M. and Stephen J. Hoch, Intuitive Compounding: Perspective, and Expertise, December 00. WorkingPaper. Framing, Temporal Jones, Gregory V., Polynomial Perception of Exponential Growth, Perception and Psychophysics,, (), 00. Keren, Gideon, Cultural Di erences in the Misperception of Exponential Growth, Perception and Psychophysics,, (),. Laibson, David, Golden Eggs and Hyperbolic Discounting, Quarterly Journal of Economics,. Lusardi, Annamaria and Olivia S. Mitchell, How Ordinary Consumers Make Complex Economic Decisions: Financial Literacy and Retirement Readiness, September 00. Working Paper. and Peter Tufano, Debt Literacy, Financial Incentives, and Overindebtedness, April 00. Working Paper. Munnell, Alicia, Anthony Webb, and Luke Delorme, Retirements at Risk: A New National Retirement Index, June 00. Working Paper. Rabin, Matthew, Improving Theory with Experiments, Improving Experiments with Theory, November Skiba, Paige Marta and Jeremy Tobacman, Do Payday Loans Cause Bankruptcy?, 0. Vanderbilt Law School Working Paper -.

7 Page of Stango, Victor and Jonathan Zinman, Exponential Growth Bias and Household Finance, Journal of Finance, December00, (), 0. and, Fuzzy Math, Disclosure Regulation and Credit Market Outcomes: Evidence from Truth-in-Lending Reform, June 00. Working Paper. Timmers, Hans and Willem A. Wagenaar, Inverse Statistics and Misperception of Exponential Growth, Perception and Psychophysics,, (),. Wagenaar, William A. and Sabato D. Sagaria, Misperception of Exponential Growth, Perception and Psychophysics,, (),.

8 Page of Time-Sharing Experiments for the Social Sciences A Protocol Results Figure : Distribution of Subject-Question EGB Domain Domain Domain Domain Notes: Calibrated distribution of exponential-growth bias using individual question answers from online pilot. The outcome variable is the value of œ [0, ] which minimizes the distance to a subject s actual answer subject to () and () above. When =0there is no bias and when =exponential growth is perceived as linear growth.

9 Page of B Study Design Table : Recruitment Subjects Initial Domain Salience Treatment in Initial Domain C 00 Random No No T 0 No Yes T 0 No Yes T 0 No Yes T 0 No Yes Salience Treatment in Subsequent Domains

10 Page of Time-Sharing Experiments for the Social Sciences C Elicitation Task Asset A has an initial value of $00 and grows at an interest rate of 0% in odd periods (starting with period ) and 0% in even periods No-Priming Protocol: Would you prefer: Logged in as 00. Logout Asset B has an initial value of $00 and grows at an interest rate of X% in all periods Your payment for this question will be the value of your chosen asset after: 0 periods. Please indicate the smallest value of X for which you would prefer the asset on the right: (Question ) Next Priming Protocol:

11 Page 0 of D Diagnostic Questions Domain : Simple Savings. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $00 and grows at an interest rate of 0% each period. Asset B: has an initial value of $X, and does not grow.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $00 and grows at an interest rate of % each period. Asset B: has an initial value of $X, and does not grow.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $0 and grows at an interest rate of % each period. Asset B: has an initial value of $X, and does not grow.. Your payment for this question will equal the value of your chosen asset at the end of periods. Asset A: has an initial value of $0 and grows at an interest rate of % each period. Asset B: has an initial value of $X, and does not grow.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $00 and grows at an interest rate of % each period. Asset B: has an initial value of $X, and does not grow.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $00 and shrinks at an interest rate of % each period. Asset B: has an initial value of $X, and does not shrink.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $00 and shrinks at an interest rate of % each period. Asset B: has an initial value of $X, and does not shrink.. Your payment for this question will equal the value of your chosen asset at the end of periods. Asset A: has an initial value of $000 and shrinks at an interest rate of % each period. Asset B: has an initial value of $X, and does not shrink. Domain : Defined Benefit vs. Risk-Free Defined Contribution. Asset A: you contribute $0 at the beginning of every period. Your contributions earn % interest every period, and you will receive your contributions and all interest earned at the end of period 0. Asset B: you also contribute $0 each period, but you will instead earn a fixed amount $X at the end of period 0 (and do not receive your contributions).. Asset A: you contribute $ at the beginning of every period. Your contributions earn.% interest every period, and you will receive your contributions and all interest earned at the end of period 0. Asset B: you also contribute $ each period, but you will instead earn a fixed amount $X at the end of period 0 (and do not receive your contributions).. Asset A: you contribute $ at the beginning of every period. Your contributions earn 0% interest every period, and you will receive your contributions and all interest earned at the end of period 0. Asset B: you also contribute $ each period, but you will instead earn a fixed amount $X at the 0

12 Page of Time-Sharing Experiments for the Social Sciences end of period 0 (and do not receive your contributions).. Asset A: you contribute $0 at the beginning of every period. Your contributions earn % interest every period, and you will receive your contributions and all interest earned at the end of period 0. Asset B: you also contribute $0 each period, but you will instead earn a fixed amount $X at the end of period 0 (and do not receive your contributions).. Asset A: you contribute $0 at the beginning of every period. Your contributions earn % interest every period, and you will receive your contributions and all interest earned at the end of period. Asset B: you also contribute $0 each period, but you will instead earn a fixed amount $X at the end of period (and do not receive your contributions).. Asset A: you contribute $0 at the beginning of every period. Your contributions earn % interest every period, and you will receive your contributions and all interest earned at the end of period 0. Asset B: you also contribute $0 each period, but you will instead earn a fixed amount $X at the end of period 0 (and do not receive your contributions).. Asset A: you contribute $ at the beginning of every period. Your contributions earn % interest every period, and you will receive your contributions and all interest earned at the end of period. Asset B: you also contribute $ each period, but you will instead earn a fixed amount $X at the end of period (and do not receive your contributions).. Asset A: you contribute $ at the beginning of every period. Your contributions earn % interest every period, and you will receive your contributions and all interest earned at the end of period 0. Asset B: you also contribute $ each period, but you will instead earn a fixed amount $X at the end of period 0 (and do not receive your contributions). Domain : Savings Vehicle Choice. Your payment for this question will equal the value of your chosen asset at the end of periods. Asset A: has an initial value of $00, and grows at an interest rate of % each period. Asset B: has an initial value equal to $X, and grows at an interest rate of % each period.. Your payment for this question will equal the value of your chosen asset at the end of periods. Asset A: has an initial value of $0, and grows at an interest rate of % each period. Asset B: has an initial value equal to $X, and grows at an interest rate of 0% each period.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $0, and grows at an interest rate of 0% each period. Asset B: has an initial value equal to $X, and grows at an interest rate of % each period.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $00, and grows at an interest rate of 0% each period. Asset B: has an initial value equal to $X, and grows at an interest rate of % each period.. Your payment for this question will equal the value of your chosen asset at the end of periods. Asset A: has an initial value of $0, and grows at an interest rate of % each period. Asset B: has an initial value equal to $X, and grows at an interest rate of % each period.

13 Page of Your payment for this question will equal the value of your chosen asset at the end of periods. Asset A: has an initial value of $0, and grows at an interest rate of % each period. Asset B: has an initial value equal to $X, and grows at an interest rate of % each period.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $0, and grows at an interest rate of % each period. Asset B: has an initial value equal to $X, and grows at an interest rate of 0% each period.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $00, and grows at an interest rate of % each period. Asset B: has an initial value equal to $X, and grows at an interest rate of % each period. Domain : Consolidation. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $ and grows at an interest rate of 0% in odd periods (starting with period ) and 0% in even periods. Asset B: has an initial value of $ and grows at an interest rate of X% in all periods.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $ and grows at an interest rate of 0% in odd periods (starting with period ) and 0% in even periods. Asset B: has an initial value of $ and grows at an interest rate of X% in all periods.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $0 and grows at an interest rate of 0% in odd periods (starting with period ) and 0% in even periods. Asset B: has an initial value of $0 and grows at an interest rate of X% in all periods.. Your payment for this question will equal the value of your chosen asset at the end of periods. Asset A: has an initial value of $ and grows at an interest rate of % in odd periods (starting with period ) and 0% in even periods. Asset B: has an initial value of $ and grows at an interest rate of X% in all periods.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $0 and grows at an interest rate of 0% in all periods. Asset B: has an initial value of $0 and grows at an interest rate of -0% in odd periods (starting with period ) and +X% in even periods.. Your payment for this question will equal the value of your chosen asset at the end of periods. Asset A: has an initial value of $00 and grows at an interest rate of 0% in all periods. Asset B: has an initial value of $00 and grows at an interest rate of -0% in odd periods (starting with period ) and +X% in even periods.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $ and grows at an interest rate of 0% in all periods. Asset B: has an initial value of $ and grows at an interest rate of -0% in odd periods (starting with period ) and +X% in even periods.. Your payment for this question will equal the value of your chosen asset at the end of 0 periods. Asset A: has an initial value of $0 and grows at an interest rate of 0% in all periods. Asset B: has an initial value of $0 and grows at an interest rate of -0% in odd periods (starting

14 Page of Time-Sharing Experiments for the Social Sciences with period ) and +X% in even periods. E Post-Experiment Survey. Did you use a calculator or another tool to help answer these questions? If so, what tool(s)?. What sort of mathematics was necessary for the calculations in the study? (Check as many that apply.) a. simple arithmetic (addition, subtraction, multiplication, division) b. advanced arithmetic (exponentiation, logarithmic operations) c. pre-calculus (trigonometric operations) d. calculus and other advanced math. Have you ever filed for Chapter or Chapter Bankruptcy? yes or no. Have you ever used or had any of the following? (Check as many that apply.) a. credit card b. payday loan c. car loan d. mortgage e. second mortgage. Do you get and use financial advice from any of the following? (Check as many that apply.) a. A friend or family member. b. A professional advisor whom I pay. c. Newspapers, magazines, internet, books, and television. d. I do not request nor use financial advice.. For each of the Asset choices you made previously, we have computed the answer that exactly equalizes the value of the two assets. If we were to make a window above and below this amount for each question, how wide do you think it must be so that 0% of your answers fell inside the window? (e.g. if the equalizing answer were 000, a 0% window would be between 00 and 00) [integer response]

15 Page 0 of Appendix A: Graphical Intervention

16 Page of Time-Sharing Experiments for the Social Sciences Appendix B: Instructions Regarding Monetary Incentives Thank you for participating in this experiment on individual decision-making. Several research foundations fund this research. Your decisions in this experiment will affect how much money you receive. Our computer using a random number generator will randomly select one of the questions and implement your choice from that question. That choice will determine your earnings. The computer s selection will be independent of any of your choices. Thus, it is in your interest to choose the option that you think will earn you the most money for each question because any choice may count, and your choices will not affect which question the computer selects. All earnings will be paid by an Amazon.com gift code sent to you by . The questions that follow will make use of an experimental currency. Your earnings will be translated in to real dollars at the end, using an exchange rate of experimental dollars = $ of Amazon.com gift codes. You will face a series of questions in which you will choose between Asset A and Asset B. The values of the assets will be calculated according to the description in the questions. One of these assets will be increasing in a variable X, and you will be asked to state the minimum value of X (cutoff) for which you would be indifferent to obtaining Asset A or Asset B. For the question that is chosen to count, the value of X will be randomly determined by the computer. Your stated value will have no effect on the random number and how it is determined. If the value of X is equal to or above your stated value (X cutoff) then you will get Asset B. If the value of X is less than your stated value (X < cutoff) then you will get asset A. Example : Asset A: is worth $ Asset B: is worth $X+ Maria states a cutoff of. Then the computer randomly determines X. This random determination is unaffected by Maria s choice of cutoff. If X then Maria gets Asset B. If X < then Maria gets A. Suppose the computer determines that X=. Then because is higher than Maria s cutoff of, she gets Asset B. With this computer draw, it is worth $+$ = $. If the computer determined that X=/ then since this is below Maria s cutoff of, Maria would get Asset A, and leave the experiment with $. Notice that you are best off truthfully stating the cutoff that makes you indifferent between the two assets. There is no advantage to strategic behavior. Your cutoff determines which asset you will receive, but does not affect its value. Consider example that illustrates this point. Example : Asset A: is worth $ Asset B: is worth $*X Saul is best off stating a minimum value of X (cutoff) as. If Saul submitted a lower minimum value, for example, and the computer randomly determines X=. then since the X is above the cutoff Saul gets Asset B, but it is only worth $.0. However, if he had chosen a cutoff of, he

17 Page of would have gotten Asset A and left the experiment with $. Likewise, if he had chosen any cutoff greater than he would do no better. He would still get Asset A worth $. Now consider what would happen if Saul submits a cutoff greater than. If Saul submits as his cutoff, for example, and the computer randomly determines X=, then since his cutoff his above X, Saul gets Asset A, but it is only worth $. If he had instead chosen a cutoff of, he would have gotten asset B and left the experiment with $. Likewise, if he had chosen any cutoff lower than he would do no better. He would still get asset B worth $. Since each round has an equal likelihood of being selected, you should approach each round with equal care and consideration. Example : Asset A: is worth $0 Asset B: is initially worth $X and doubles every period. You will receive your chosen asset at the end of one period. Please indicate the smallest value of X for which you would prefer Asset B: [They must answer the correct value, X=, to continue to the experiment. Incorrect answers prompt a request to try again.]