Stability of Risk Preference Estimates Over Payoff Horizons

Transcription

1 Stability of Risk Preference Estimates Over Payoff Horizons Alexander Myers Faculty Advisors: Professor Daniel Barbezat Professor Jessica Reyes April 21, 2010 Submitted to the Department of Economics of Amherst College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors. i

2 Acknowledgements I would like to thank Professor Barbezat for introducing me to Behavioral Economics and guiding me through the process of picking a topic and turning it into a thesis. I would also like to thank Professor Reyes for imparting in me a small amount of her discipline and dedication and helping me to organize my time and my thoughts. Thank you to Professor Ishii for convincing me to register for the thesis seminar. In addition, Professors Nicholson and Alpanda gave me readings that were very helpful in learning about risk. I may have spent more time in Jeanne Reinle s office than in the computer lab: Thank you Jeanne for being such a good friend. Lastly, I want to thank Ivan Petkov and Professor Liao for teaching me the requisite econometrics and statistics for this project. ii

3 Abstract Rational risk- averse individuals should exhibit risk neutral behavior if the stakes of a gamble are sufficiently small. In order to explain why they don t, I develop theory that explains this anomalous risk aversion as an effect of sub- optimal consumption smoothing, itself a result of emotional temptation. I conduct an experiment on Amherst College students and elicit their risk preferences using lotteries with payoffs in the present and with payoffs in the future, on the assumption that individuals will have more temptation in the present than in the future. While I find that this assumption is not universally true for everyone, my results support the notion that an individual will be more risk averse when payoffs are in the present than when they are in the future provided that the individual has more temptation in the present than in the future. iii

4 Table Of Contents Title Page Acknowledgements Abstract Table of Contents i ii iii iv Introduction 1 I: Theory... 3 II: Experimental Design III: Econometric Specification.. 20 IV: Experimental Implementation. 26 V: Estimates. 29 VI: Discussion. 36 VII: References.. 39 Appendix A.. 41 Appendix B.. 45 iv

5 Introduction Although Expected Utility Theory (EUT) allows for a concave utility function and thus risk aversion, Arrow (1971) demonstrated that, over sufficiently small stakes, a risk-averse individual would behave as though he were risk-neutral. This is because a small increase in one period s consumption only causes a negligible decrease in the marginal utility of money. 1 If an individual spreads the additional consumption from unanticipated earnings out over his lifetime, he would act as though he were risk-neutral so long as the stakes were small when compared to his lifetime wealth. Laboratory experiments such as Holt & Laury (2002) and Andersen, Harrison, Lau & Rutstrom (2008), (AHLR), however, find strong evidence of risk aversion even over such small stakes. I theorize that these findings are evidence of sub-optimal consumption smoothing whereby an individual fails to maximize his lifetime utility by failing to spread consumption evenly across his lifetime. I further argue that this suboptimal consumption smoothing is a result of short-run temptation. 2 We would expect variation in risk preferences to follow from variation in temptation. I use a dual-self model from Fudenberg & Levine (2006), (FL), to explain the presence of temptation. This model views behavior as the interplay between emotional short-run selves and a rational long-run self. Although the long-run self s behavior conforms to the assumptions of EUT, risk aversion over small stakes arises from the 1 Under EUT, a concave utility function simply indicates that there is diminishing marginal utility of money. This concept is the actual cause of risk aversion for the following reason: Assuming diminishing marginal utility of money, a random event with a 50% chance of winning $100 and a 50% chance of winning $0 does not have the same expected utility as a certain event with a payoff of $50, since the last $50 are not worth as much as the first $50. Such a person would prefer to pay a nonzero risk premium to participate in the certain event, in essence the definition of risk aversion. See Chapter 7 of Nicholson & Snyder (2007) for a more rigorous explanation. 2 Temptation is an irrational desire to consume in a sub-optimal way. If an agent yields to temptation he will regret the decision later because he failed to maximize lifetime utility. 1

6 influence of the short-run selves and their corresponding temptation. The dual-self model, however, is agnostic about how much of an influence the short-run selves will have (Andersen et al., 2008). 3 AHLR write that they generally assume temptation to be higher in the present than in the future. At the same time, a footnote acknowledges an important maintained assumption in our empirical model is that the risk parameter r is constant over time and planning horizons. 4 I view these two assumptions as contradictory, and pose the following hypothesis: If participants experience more temptation in the present than in the future, they will have higher risk aversion when payoffs are in the present than when payoffs are in the future. To test this hypothesis, I run an experiment on 85 Amherst College students. 5 I give participants risk preference exercises developed by Holt & Laury (2002) with payoffs in the present and risk preference exercises with payoffs in the future, allowing me to compare risk preferences between the two periods. I also collect observable characteristics about participants in a post-experiment survey. Following AHLR, I use Maximum Likelihood Estimation to estimate the relative risk aversion parameter most likely to generate the choices observed in these exercises. 3 FL do provide some guidance: If the cost of self-control is high, the long-run self cannot constrain the short-run selves and they will thus have a stronger influence on behavior. If the cost of self-control is low, the short-run self s temptation will be less relevant. See Section I for a more detailed explanation. 4 The footnote continues, Thus we assume the same r applies to utility over the lottery payoffs, which are paid immediately, as to utility over the asset choices, which all have a front-end delay. This assumption is plausible from both an empirical and a theoretical perspective. Andersen, Harrison, Lau, and Rutstrom (2008b) used data from a panel experiment with similar monetary incentives and found some variation in risk attitudes over time, but they did not detect a general tendency for risk attitudes to increase or decrease over a 17-month time span. Theoretically, this allows us to focus on the role of differences in consumption smoothing between the two selves of our agent. Rich as our experimental design is, we do not believe it would be possible to identify differences in the risk parameter as well as differences in consumption smoothing (Andersen et al., 2008). Note that the study they cite confirms that risk preferences remain stable over a time horizon of 17-months, but does not address the stability of risk preferences over any type of planning horizon. 5 Not all 85 participants made it into my experimental sample. The reasons why are discussed in Section II. 2

7 This procedure allows me to define the risk aversion parameter as a linear function of observable characteristics and treatment effects. I obtain estimates of the relative risk aversion parameter for my experimental sample of 0.715, a value that is consistent with the literature and indicates strong risk aversion. In addition, I allow the estimated risk aversion parameter to vary based on observable characteristics and treatment effects, allowing for some variation in estimated risk aversion. My findings support the hypothesis that individuals who have more temptation in the present than in the future will exhibit stronger risk aversion when payoffs are in the present than when they are in the future. Contrary to the assumptions in AHLR, I conclude that the difference in an individual s risk preferences between the two time periods is proportional to the difference in his temptation between the two time periods. Section I contains the theory that supports this thesis. In Section II I explain my experimental design. I elaborate on my estimation procedure in Section III. In Section IV I demonstrate how I was able to implement the experimental design. I also provide summary statistics on my sample and compare its composition to the larger Amherst College population. In Section V I present my results. Additionally, in Section VI I discuss the implications of these results and propose an idea for future research to explore some of the questions raised by this thesis. I Theory CRRA Utility Function I define utility using a modification of the Constant Relative Risk Aversion utility function that allows for background consumption. The utility function I use is, 3

8 (1.1) U(M) = (ω + M)1 r 1 r, where ω is a measure of background consumption in each period, M is an experimental payoff, and r is a parameter that reflects risk preferences. 6 The concavity of equation 1.1 is parametized by the risk preferences parameter r as shown in Figure 1.1. FIGURE 1.1 Plot of CRRA utility function for two values of r. The marginal utility of money is represented in this graph by the slope of the two curves. 7 When r = 0, the graph is linear so the marginal utility of money is constant. When r > 0, the graph is concave down so the marginal utility of money is strictly decreasing. Under EUT, risk aversion arises from the diminishing marginal utility of money or, equivalently, from the concavity of an agent s utility function. For equation 1.1, r > 0 6 The Constant Relative Risk Aversion utility function appears when we set ω equal to zero. A non- zero ω means that the relative risk aversion implied by the utility function will change, and thus is no longer constant. 7 Technically the derivative is the marginal utility of $100 because of the units on the x-axis. The marginal utility of money is thus the derivative divided by

9 indicates risk aversion while r = 0 indicates risk-neutrality. 8 It is generally assumed that individuals are risk-averse (Nicholson & Snyder, 2007, p. 213). 9 Consumption Smoothing The situation is complicated by the addition of multiple time periods. The primary model for equating utility in distinct periods is the Discounted Utility model developed by Samuelson (1937). Samuelson introduced the concept of a discount rate, which allows us to convert utility from one period into its equivalent utility in another period. One of the central assumptions of Discounted Utility is consumption independence, meaning that an agent s utility in one period is independent of his utility in any other period (Frederick, Loewenstein & O Donoghue, 2002). Combining consumption independence with the assumption of risk aversion for individuals yields a prediction of perfect consumption smoothing for rational individuals. 10 We can see this result by assuming that a risk-averse individual starts off with perfect consumption smoothing. If he moves one unit of consumption from any one period t to period s he will suffer a net loss of utility, because the marginal utility gained in period s will be less than the marginal utility lost in period t. 11 Implications of Consumption Smoothing This is not the whole story, however, since we must also look at the effects of a positive discount rate and a real return on savings and how these two concepts nullify each other. While not strictly a condition of the model, the literature strongly assumes that the discount rate is positive, implying diminished utility over time. One argument 8 Similarly, r < 0 indicates an agent that is risk loving. 9 Nicholson & Snyder (2007) write that empirical evidence is consistent with an r value between 2 and Perfect consumption smoothing means that an individual allocates consumption such that the marginal utility of consumption is equal across all periods. 11 This of course is due to the diminishing marginal utility of money implied by risk aversion. 5

10 for a positive discount rate is that a negative or zero value, when combined with a real return on savings, would cause people to defer consumption indefinitely into the future (Frederick et al., 2002). Meyer (1976) writes that the discount rate for money equals the real return on savings so long as an agent can save and borrow at the same interest rate. Indeed, Fisher (1907, p. 128) writes, by buying and selling, the various parties in the community adjust their rates of [time] preference to a common level an implicit rate of interest. If we make the simplifying assumption that there are no transactions costs, the discount rate equals the real return on savings. 12 The discount rate and the real return on savings are opposing forces on intertemporal utility, with the discount rate decreasing future utility and the return on savings increasing it. So long as these two parameters have the same value they cancel each other out. 13 A rational agent that smoothes his consumption will consume the same amount in each period This mirrors an assumption of Discounted Utility that states there is only one discount rate that applies to all sources of utility. 13 Assume that the discount rate = the real return on savings = ρ. If we save some amount s for 1 consumption in the next period, the utility of that consumption is ( 1+ ρ)s. The first 1+ ρ multiplicative term is the discount factor while the second is a simple one period interest calculation. Notice that the two terms cancel each other out. As a result, the utility in the next period equals s, though we actually have (1+ ρ)s available for consumption. This example makes the simplifying assumption that the utility function is linear with respect to consumption, though the same idea applies if we allow the utility function to be concave. 14 The optimization problem is as follows: Max τ t =0 τ t = ρ t ( 1+ ρ) t U(x t ), subject to the constraint that t 1 x 1+ ρ t = the present discounted value of lifetime wealth. The result is x 0 = x 1 = x t. One important conclusion to make is that even if the agent had the present discounted value of his lifetime wealth on hand he would not be dividing it evenly among all periods. τ t = ρ t s t models how much ( ). This series would be money would be set aside in the present for each period in the future s t geometrically decreasing, with real returns on savings allowing him to consume the same amount in each period. 6

11 By introducing another assumption from Samuelson s model, we can show that a rational agent will spread the additional consumption from an unanticipated payoff across all periods. Frederick et al. (2002) refer to this assumption as the integration of new alternatives with existing plans, meaning that an agent reevaluates his entire consumption stream when determining the utility of a payoff. 15 Remember that the utility maximizing consumption stream is one that exhibits perfect consumption smoothing, so that is exactly how a rational agent would reallocate his new endowment. Specifically, the agent will divide the unanticipated earnings up among each period of his lifetime, adding only a marginal amount of consumption to any one period. 16 EUT s Prediction of Risk-neutrality Arrow (1971) would have used this result to conclude that a risk-averse agent will behave as though he were risk-neutral when faced with an unanticipated and risky choice, so long as the risky payoffs were small when compared to lifetime wealth. To see why, consider the example of an agent who consumes $120 every day (ω =120). His utility function for one period is shown in Figure 1.2 with a red hash mark indicating his normal, background consumption, and a black circle indicating his background consumption plus an additional $80 unexpected reward that he received and consumed in this period. The derivatives on the graph show the marginal utility of money evaluated at 15 Note that Frederick et al. also mention that this assumption is not unique to Discounted Utility but is rather quite common in models of inter-temporal choice. 16 If the discount rate were zero, then the agent would divide the earnings up equally among each period. I am, however, interested in consumption smoothing behavior for an agent with a strictly positive discount rate. In this case, the earnings would still be divided amongst the same number of periods, but we would see a geometrically decreasing series representing the amount of money from the reward that is allocated for each period going forward. With interest, he would end up consuming the same amount each period. Thus if there are τ periods, the agent would be consuming strictly more than (ω + reward /τ) in each period. 7

12 $120, for his normal consumption, and at $200, the case in which he consumes the $80 reward immediately. Notice the lower marginal utility of money at the point representing FIGURE 1.2. Utility function of a risk-averse agent with $120 background consumption and an $80 unexpected reward that is consumed immediately. $200 of consumption. We know that if instead of an $80 reward with certainty the agent were offered a 50% chance of an $80 reward and a 50% chance of receiving nothing, this agent would be indifferent between the risky payoff and some amount less than $40 with certainty. 17 If Δx is the increase in a period s consumption, we can say that Δxis $80 for the example in Figure 1.2. If the agent s behavior follows the assumptions of discounted utility and he recalculates his optimal consumption stream upon receiving the $80, however, Δx would be quite close to zero. 18 With such a low difference between the old and new levels of 17 Such a trade indicates risk aversion; this agent is risk-averse in this situation because the marginal utility of the last $40 is lower than the marginal utility of the first $ The exact Δx is determined by the number of periods in a lifetime and the discount rate. Generally, the agent s allocation of extra earnings for period t ( s t )would be t τ 1 s 1+ ρ 0, with s t = $80 t =0. (Note that 8

13 consumption, we can quite accurately calculate the new utility per period with a linear approximation of the utility function. The marginal utility of money is effectively unchanged in each period after readjusting consumption, so the agent will behave as though he were risk-neutral. The presence of risk aversion over small stakes is thus the result of sub-optimal consumption smoothing. An agent will only express risk aversion when he expects to consume the payoffs from a small stakes risk over a small number of periods as opposed to spreading out the extra consumption over his lifetime. Although this violates the axioms of Discounted Utility, there are several alternate models that can explain such behavior. Dual-Self Model The literature views sub-optimal consumption smoothing as a self-control issue whereby an agent yields to temptation. One way of explaining self-control issues is to model a person s actions as the equilibrium decision between two distinct selves. 19 This concept has existed in the background of economics for quite some time. 20 Thaler (1981) this is not the optimization problem but rather the solution to it.) Once again we have a geometrically decreasing series of money set aside in the present. When we include the real return on savings, ρ, there is the same amount of additional consumption in each period. In this case, τ is the number of periods in a lifetime, so each x must be quite small, especially since a period has been defined as one day. This example also allows us to see why small stakes are defined relative to lifetime wealth. In order for there to still be risk aversion after an agent readjusts his lifetime consumption stream, U'(ω + x 0 ) must be noticeable smaller than U'(ω). Since the CRRA utility function has a strictly negative but increasing second derivative, as ω increases, x 0 must be larger yet for this to still be true. We compare to lifetime wealth instead of background consumption because x 0 represents the stakes divided among τ periods: The stakes are thus τ x 0, while τ ω is lifetime wealth. 19 Thaler justified the structure model using a quote by McIntosh (1969), The idea of self-control is paradoxical unless it is assumed that the psyche contains more than one energy system, and that these energy systems have some degree of independence from each other. 20 Ashraf, Camerer, & Loewenstein (2005) write that Adam Smith's The Theory of Moral Sentiments describes man s behavior as being driven by a passionate, emotional self. At the same time, Smith believed that this behavior could be overridden by man s capacity to view his own actions from the point of view of a third party. 9

14 expanded upon this idea and created an economic model of self-control, calling it a dualself model. The two selves are the rational planner, focused on long term, lifetime utility, and the impulsive doer, focused on utility only in the current time period. Temptation comes from the influence of the short-run self because it aims to increase consumption in the present. 21 Thaler found a model for the optimization of a pair of selves with different interests in theories of the firm, where an owner and manager may have divergent incentives. 22 In his paper, he writes that the solution to the problem of self control can come in two forms: Either the far seeing planner can use some sort of a commitment mechanism to change the costs of the doer and cause him to make a choice that is optimal from the planner s perspective, or he can use a set of rules to eliminate the doer s agency. 23 FL expanded Thaler s model into a game theoretic framework. The model presents decision-making as a multi-stage game between a set of short-run selves and a singular long-run self. Both the short-run selves and the long-run self share the same baseline consumption preferences, but they differ in time preferences. As in Thaler, short-run selves are characterized as impulsive and myopic; the long-run self is labeled as patient and rational. The game they propose has the following structure: For each round of the game, the long-run self first has an opportunity to modify the short-run self s preferences at some cost (γ). In other words, the imposition of an incentive shifting cost 21 Temptation in the future is simply the expectation of temptation when the future period becomes the present. 22 As pointed out by a referee, the employee is always able to quit and find another job. The appropriate analog is thus the short-run model of this relationship. 23 Ariely & Wertenbroch (2002) conducted an experiment to test our ability to actually utilize these optimization strategies and found that students performed better when intermediate deadlines were forced upon them rather than when they were able to set by the students or when they were only given a final deadline. 10

15 upon the short-run self is in and of itself costly. In the second stage, the short-run self plays its best response strategy and executes its utility maximizing set of actions, given the constraints imposed upon its behavior by the long-run self. By exerting self-control to change short-run self behavior, the long-run self aims to maximize the utility that results from the set of short-run self actions across all periods. Implications of Dual-Self Model on Behavior I will first explore the implications of various self-control cost functions on the consumptions/savings problem. Absent self-control (γ = ), the short-run self will consume lifetime wealth in the first period. The example mentioned in Thaler, but also applicable to FL s model, is that the round one short-run self will borrow Y y 0 on the perfect capital market and consume all of it. 24 On the other hand, if self-control is costless (γ = 0), the long-run self will fully manipulate the short-run selves such that each short-run self chooses an optimal consumption path to maximize lifetime utility. If selfcontrol had no cost, there would be no anomalies to explain, since the optimal consumption path would be perfect consumption smoothing and thus follow the predictions of EUT. Implications of Dual-Self Model on Risk Aversion Neither of these cases explains risk aversion over small stakes in experiments. If an agent s behavior aims to maximize his lifetime utility, then Arrow s prediction of riskneutrality will apply. The short-run self acting in the absence of self-control will also be risk-neutral over the same magnitude of stakes. Although it is only concerned with the present period, present period consumption in this case would equal lifetime wealth, so 24 Y is lifetime wealth while y 0 is the wealth available in the first period. Y y 0 is thus all future consumption. The perfect capital market is one that has no transactions costs. 11

16 stakes would still need to be of a comparable magnitude to lifetime wealth in order for the agent to display risk aversion. These extremes, however, are an insignificant part of the dual-self model. Self-control is assumed to have a non-zero, finite cost. Behavior is thus a weighted average of these two extreme consumption paths, with the weights determined by the amount of self-control exerted by the long-run self. A higher amount of self-control will cause an agent to behave in a more rational manner. If self-control is to be seen as the aspect of human behavior that pushes an agent towards acting rationally, its opposing force is temptation. For a given amount of selfcontrol, higher temptation means that the short-run self has a stronger influence on behavior and thus that the agent s consumption smoothing is less optimal. Marginal Propensity to Consume FL provide a qualitative description of self-control within the context of the consumption/savings problem. They make a distinction between cash available on hand and other types of wealth, like money in a bank account. The scenario that they use as an example is an agent that is preparing to go out to a nightclub. Stage One of the game is when the agent goes to the bank to withdraw funds. The long-run self is able to apply self-control in this situation by, for example, limiting the amount of cash that is withdrawn. When the agent is in the club later that evening, it is assumed that the shortrun self will spend all available cash, though not all available wealth. The key point here is that the model argues that the marginal propensity to consume (MPC) out of cash is higher than the MPC out of wealth held in other, less accessible asset classes. This is because the short-run self is relatively unrestrained in his ability to spend cash EUT, on the other hand, assumes that there is one MPC for all forms of wealth because it assumes that transactions costs are negligible, so any type of asset can be costlessly converted into a cash equivalent. 12

17 When FL use their model to predict risk aversion in the future, the key determinant is the agent s expectation of his MPC out of the future payoff. If a future, risky payoff will arrive in cash, with a high MPC, the agent will be more risk-averse than if the future risky payoff will be deposited into his bank account. This echoes the view of risk aversion as caused by sub-optimal consumption smoothing, because a high MPC also means the unanticipated payoff will be integrated over fewer periods. Declaration of Hypotheses It is generally assumed that temptation is higher in the present. 26 My first hypothesis is thus: H1: Participants will exhibit more temptation in the present than the future. Temptation causes agents to smooth their consumption in a sub-optimal manner. This, in turn, means that they will experience the diminishing marginal utility of money during their periods of extra consumption, causing them to behave in a risk-averse manner even if the experimental stakes are small. My second hypothesis is thus: H2: If participants experience more temptation in the present than in the future, they will have higher risk aversion in the present than the future. II- Experimental Design This thesis investigates whether risk preferences change as payoffs move from being in the present to being in the future. My experimental procedures borrow heavily from Harrison, Lau, Rutstrom & Sullivan (2005). I use the same risk preferences task 26 This assumption is explicitly mentioned by AHLR when they say, there is evidence that decision makers exhibit a passion for the present when offered choices between monetary amounts today or in the future (Andersen et al., 2008). Furthermore, Thaler (1991) and Akerlof (1991) articulate the problem of self-control as one of dynamically inconsistent decision making, meaning that the discount rate is higher between the current period and the next period in the future than it is between two subsequent future periods. This feature of human behavior is captured by a hyperbolic discount function. 13

18 developed by Holt & Laury (2002). The exercise is given to each participant twice with payoffs in the present and twice with payoffs one month in the future, for a total of four risk preferences exercises. Risk Preferences Exercise The risk associated with a random event is defined by the difference in payoffs between a favorable outcome and an unfavorable one. Risk aversion can be conceptualized as a measure of the premium (risk premium) that an agent would pay to avoid a risky event and replace it with a certain payoff. 27 A risk-averse agent would thus prefer a safe lottery to a riskier one with a higher expected value, so long as the diminished risk is sufficient to account for the loss of expected value. Holt & Laury (2002) created an exercise to test for risk aversion, as follows: Subjects are asked to choose between two lotteries. One of these lotteries, referred to as Option A, is a relatively safe lottery because the two payoff possibilities are relatively close together in value. The other lottery, Option B, is risky because there is a much larger difference between the high and low payoff. The exercise developed by Holt and Laury is able to measure risk preferences by asking subjects to choose between a safe lottery and a risky lottery ten different times. Although the high and low payoffs are the same for each of these ten choices, the probabilities associated with these payoffs change incrementally from one decision to the next. This has the effect of changing the relative expected values of the safe and risky lotteries without changing their relative risk. Table 2.1 on the next page shows the ten lottery choices and also has a column showing the expected values of each lottery. This 27 The risk premium is defined as the difference between the expected value of the risky payoff and the value of the lowest certain payoff that the agent would accept instead. 14

19 column is included here for reference but did not appear in the experimental materials. 28 The first row of the table shows that there is a 10% chance of receiving a high payoff and a 90% chance of receiving a low payoff. The expected value of the each lottery is thus dominated by the value of corresponding low payoff. Since Option A s low payoff is much higher than Option B s low payoff, Option A has a much higher expected value. TABLE 2.1: RISK PREFERENCES EXERCISE WITH EXPECTED VALUES a a Note that the last column was not present in the experimental materials and is only included here to simplify analysis of the exercise. While we would expect an agent to prefer Option A in this first row since it has both a higher expected value and is less risky, we can still look at what it would mean if an 28 In addition, the actual exercise is formatted differently. Please refer to Appendix B to see the exercise as it was presented to experimental participants. 15

20 agent preferred Option B instead. This choice can be represented by the following inequality: (2.1) E[ U(OptionA) ] < E[ U(OptionB) ]. We can then substitute in our utility function. In addition, we can calculate the expected utilities for each lottery since we know the payoffs and their corresponding probabilities. (2.2) (.1) 381 r r ( ) r 1 r <.1 ( ) r 1 r r ( ) r. 29 The left side of the inequality is the expected utility of Option A, while the right side is the expected utility of Option B. If we solve for r, our solution is that r < -1.71, implying that an individual would have to be quite risk loving to prefer Option B in the first row. Notice that each choice by itself gives us quite limited information on the value of r. For any of the individual decisions, when the agent prefers Option B, we know that r must be less than some value. Similarly, if the agent prefers Option A, the inequality is reversed and we know that r must be greater than some value. When we look at the 10 decisions for each exercise together, however, we get a bounded interval that contains the agent s risk preferences. This occurs at the point where an agent first prefers Option B. Lets say that an agent prefers Option A in row 5 and Option B in row 6. The inequality for row 5 lets us know that r > The inequality for row 6, however, tells us that r < 0.41, so we know that 0.14 < r < For this example, I am making the simplifying assumption that background consumption (ω) is zero. Later we will assume a non-zero value for ω that represent s the average Amherst College student s daily consumption. Including a measure of background consumption has the effect of changing the magnitude of implied risk preferences. Furthermore, this example and the calculations contained within are only included to explore the risk preferences exercise and should not be taken as a method used to analyze the data from this experiment. While solving inequalities is useful for calculating the risk preferences of one individual filling out one exercise, this method does not scale particularly well. Please see Section III to see the method that will be used to analyze data on multiple exercises for multiple people. 16

21 Since most people are risk-averse, we expect them to pick Option A in the first row. The last row, on the other hand, doesn t actually have any risk, since there is a 100% chance of the high payoff. Since Option B has a higher payoff with certainty, we expect participants to prefer Option B in this row. 30 Thus, we expect participants to switch to Option B at some point. Since Option B s expected value increases relative to Option A s expected value from one row to the next, we would also expect participants to continue to prefer Option B once they have switched from preferring Option A to preferring Option B. 31 If an agent is risk-neutral ( r = 0), they will first prefer Option B in row 5 since that is the earliest choice where Option B has a higher expected value than Option A. We would expect most individuals to first prefer Option B after row 5, since that is the area of the exercise that indicates risk aversion. Subjects were first given this exercise with the values in Table 1 (A1: $38.00, A2: $30.40 B1: $73.15, B2: $1.90) as well as with a different set of values (A1: $40.00, A2: $16.00 B1: $72.00, B2: $0.80). As argued in AHLR, including an alternate set of payoffs increases the resolution of the experiment because the risk preferences intervals will be slightly different so long as the two sets of payoff values are not simply scaled version of each other. The risk aversion intervals for the payoff values in Table 1 are ( to -1.71, to -0.95, to -0.49, to -0.15, to 0.14, 0.14 to 0.41, 0.41 to 0.68, 0.68 to 0.97, 0.97 to, N/A). 32 The risk aversion intervals for the alternate set of 30 AHLR refer to this row as a test to see if participants understand the exercise. Later I exclude 3 participants from my sample for failing this test and instead preferring Option A or being indifferent between Options A and B in this last row. 31 If an individual does prefer Option B and then switches back to preferring Option A in a later row he is indicating inconsistent preferences. 9 of my participants ultimately indicated inconsistent preferences and were thus excluded from my sample. 32 These intervals assume that ω equals zero. The first interval implies that a subject first preferred B in the first row, the second interval implies that a subject first preferred B in the second row, etc. First choosing 17

22 values are ( to -0.75, to -0.32, to -0.05, to 0.16, 0.16 to 0.34, 0.34 to 0.52, 0.52 to 0.70, 0.70 to 0.91, 0.91 to, N/A). Concerns One concern about using different sets of payoffs is that they may skew participant answers in a certain direction. The intervals for the second set of payoffs, for example, generally reflect a higher measure of risk aversion than the first set of payoffs. Although participant answers should theoretically be determined solely by their risk preferences and not by the payoff values, Harrison, Lau & Rutstrom (2007) find evidence that supports the significance of framing effects in these types of exercises. Adjustments A large concern was that participants would perceive that they were filling out the same exercise, remember their answers, and reproduce them in an effort to avoid the cognitive costs that arise when dealing with lotteries and uncertainty. I dealt with this issue on two fronts. Firstly giving participants the exercise with two payoff sets forced them to actually reevaluate and compare expected utilities. 33 The second strategy was to give participants other tasks in the middle of the experiment that could serve as an intermission. First, participants completed a discount rate task with a similar, 10-row structure. 34 They were then given a short article adapted from Reyes (n.d.). 35 The hope B in the last row reveals nothing about risk preferences because the last row does not actually involve any risk. 33 I chose payoff sets that were not just different, but also looked different, to discourage participants from simply reusing their previous answers. 34 The discount rate exercise was originally developed by Coller & Williams (1999). In this task, subjects are given the choice between $60 in one week and a larger amount in two months and one week. The larger amount is $60 plus two months worth of interest. As you move from one row to the next, the interest rate used to calculate the second quantity increases making the future option more tempting. The point at which a person switches from preferring the early option to preferring the later option is indicative of that person s discount rate. To see what this exercise looks like, see Appendix B. 18

23 was that these two activities would distract participants and cause them to forget their previous answers. Anecdotally, the presence of calculations in the margins and eraser marks indicates that at least some participants did indeed incur the relevant cognitive costs when filling out the exercises for the second time. Post Experiment Survey After participants complete the risk preferences exercise they are given a survey to collect data on their heterogeneous characteristics. I measure participant s expectations about their temptation in both the present and future by asking the following two questions: 1) If you received $70 right now, how long do you think it will take you to spend it? days 2) If you received $70 in the mail in one month, how long do you think it will take you to spend it? days Survey data is appropriate because even if participants are naïve towards what their actual behavior will be, that naïveté will be reflected in their choices on the risk preferences exercises. Comparing each individual s answers to these two questions allows me to evaluate H1, my hypothesis that states participants will have more temptation in the present than in the future. Actual Payoffs Participants were guaranteed a $5 award for participating in the experiment. In addition, they were offered a 10% chance of receiving an additional payoff of up to $ At the end of the experimental session, each participant rolled a 10-sided die. 35 Reyes (n.d.) is about teaching the art of economics research to senior thesis writers. The reading assignment was mostly the introduction to this article, with an additional paragraph describing the economics thesis process at Amherst College 36 Some payoffs were given to participants immediately while others were sent to them via the campus mail, depending on the exercise. 19

24 If a participant rolled a 1, they then rolled a series of dice to determine which exercise and then which row within the exercise their payoff would come from. Once the specific row was randomly determined, they received whichever payoff option they had indicated a preference for. In the case of the risk preference exercises, an additional die was rolled to play out the lottery. III- Econometric Specification I now outline the analytical strategy used to estimate r from the risk preferences exercises outlined in Section II in order to test my hypothesis H2 that participants with more temptation in the present will have more risk aversion in the present. For each row of each exercise, the subject either picked A or B, or in a few cases indicated that they were indifferent between A and B. Rather than directly producing a dataset of risk preferences, each exercise produces a set containing 10 of these discrete choices for each individual. 37 According to Greene (1997), normal regression methods are poorly suited for datasets in which the dependant variable is discrete rather than continuous. Instead, I borrow from AHLR and use Maximum Likelihood Estimation (MLE) to determine which value of risk preferences is most likely to generate the observed data, assuming a specification for the data generating process. A detailed explanation of MLE can be found in Appendix A. Estimating Risk Preferences In order to approximate the data generating process, Holt & Laury (2002) created a specification to model the way that a person decides which lottery they prefer among the two they are choosing between. First, the individual evaluates the utility of each of 37 Since each individual fill out 4 exercises, there are 40 observations per individual in the sample. With 71 individuals in the sample, there are ultimately 2840 observations in my sample. 20

25 the four payoff options, two for the lottery in Option A and two for the lottery in Option B. He then calculates the expected utility of each lottery by multiplying the utility of a payoff by the probability of receiving that payoff. Lastly, the agent compares these two values and selects the lottery with the higher expected utility. 38 This last step is represented by a latent index function. 39 Calculating the utility of each payoff is as simple as evaluating U(ω + M), where ω is the assumed amount of background consumption and M is the payoff. The expected utility of each lottery can be written as (3.1) EU = p i U(ω + M i ) + p j U(ω + M j ), a linear combination of the two payoffs it contains, weighted by their respective probabilities. Latent Index Function The latent index function models an agent s decision-making process after the agent has computed the expected utility of each lottery. When the function is written in the form of (3.2) EU = EU B 1 µ EU A 1 µ + EU B 1 µ, 38 Note that the lottery with the higher expected utility depends on the individual s risk preferences. If an agent is risk loving, they will prefer the riskier lottery even if it has a lower expected utility because they will get additional utility from the potential high payoff. (A risk loving individual exhibits increasing marginal utility of money.) 39 An index function is a function used to compare different values. I refer to a latent index function because my econometric specification assumes that an individual plugs the expected values of the lotteries into the specified function every time they form a preference between two lotteries. Latent simply means present in the unconscious mind but not consciously expressed (cite American heritage medical dictionary via dictionary.com) 21

26 the index EU represents the probability that the agent will prefer Option B. 40 This value will always be between zero and one, because the denominator of the ratio is strictly larger than the numerator. EU A 1 µ and 1 µ EU B are the adjusted expected utilities of lotteries A and B, respectively. Holt & Laury (2002) adjust the expected utilities of the lotteries by raising them to the 1 µ power. This exponent represents stochastic errors on the part of the agent when comparing the expected utilities of the lotteries. First I will explain how equation 3.2 models the decision-making process, and then I will show the effect of these stochastic errors. Since the index EU represents the probability that an agent will prefer Option B, a value 0.50 indicates indifference between Options A and B; 41 this would occur if the expected utility of A equaled the expected utility of B. As the value of EU B 1 µ increase relative to the value of EU 1 µ A, the index also increases in value, indicating a higher probability that the agent will prefer Option B over Option A. The stochastic error term µ amplifies or diminishes the sensitivity of EU to changes in the difference between EU A 1 µ and 1 µ EU B depending on the value of µ. We can see this by dividing the 1 µ numerator and denominator of equation 5 by EU B yielding 1 (3.3) EU = 1+ EU A EU B 1 µ µ The other form of this function would have EU A in the numerator instead, in which case the index term would be the probability that the agent will prefer Option A. 41 This is true because the probability of the agent preferring Option A is 1 minus the probability that the agent prefers Option B. 22

27 As µ, an agent will be indifferent between Options A and B regardless of the expected utility of each lottery. 42 As µ 0, however, the agent will strongly prefer the lottery with the higher expected utility. 43 The maximum likelihood estimation procedure is an iterative process that calculates the probability that a set of candidate values for r and µ ( r ˆ and ˆ µ ) would generate the observed data, assuming that the econometric specification is an accurate representation of the data generating process. 44 Given an ˆ r, a ˆ µ, and a set of payoff values along with their respective probabilities, EU is the probability that the agent would prefer Option B and 1 EU is the probability that the agent would prefer Option A. The joint likelihood for the entire dataset is the product of EU for every observation where the agent preferred Option B, and 1 EU for every observation where the agent preferred Option A. 45 We can represent the joint likelihood function as n (3.4) L(r,µ;y,ω) = ( EU i y i =1) (1 EU i y i = 0), 46 i i n 1 µ EU 42 A As µ, 1/µ 0. The term thus goes to 1. 1/(1+1) = 0.5, indicating indifference. EU B 1 µ EU 43 As µ 0, 1/µ. If EU A > EU A B, the term will go to infinity. Since it is in the EU B denominator of the overall ratio, the probability that the agent will prefer Option B goes to zero. If EU A < EU B, however, the probability that the agent will prefer Option B goes to 1. In either case, the agent will have a 100% chance of picking the lottery with the higher expected utility. If the lotteries have the same expected utility, the agent will be indifferent between them. 44 Likelihood simply means probability. The distinction between the two terms is simply the direction of reasoning. If we know the parameters of a data generating process, probability allows us to predict unknown outcomes. If we instead know the outcomes and would like to calculate the parameters, we refer to likelihood. 45 We know that we can simply take the product of the probability for each observation because each observation is independent from the rest of the observations. 46 In this equation, i is indexing over the n observations in the sample. 23

28 where y i =1 represents an observation where the agent preferred Option B and y i = 0 represents an observation where the agent preferred Option A. We evaluate the loglikelihood because this product can instead be represented by the Riemann sum n (3.5) lnl(r,µ;y,w) = ((ln( EU) y i =1) + (ln(1 EU) y = 0)), i using the same notation for the values of y i as before. While equation 3.5 gives us the log-likelihood for a candidate r ˆ, we are interested in finding the r ˆ that maximizes the log-likelihood. Stata uses an iterative process to maximize lnl, first picking an arbitrary ˆ r and ˆ µ, then calculating the log-likelihood, and then finally computing a direction vector comprised of partial derivatives for both parameters in order to determine the next candidate values to use for the following iteration so that it can hone in on the values that produce the maximum log-likelihood (Steenbergen, 2003). AHLR expanded equation 3.5 to allow participants to express indifference between the two lotteries. If a participant indicated indifference, participants were told that payoffs would be determined by flipping a coin to decide between options A and B. The log-likelihood equation thus becomes (3.6) lnl(r,µ;y,ω) = n ln( EU) y i =1 i ( ) + ( ln(1 EU) y = 0) ln( EU) ln(1 EU) y i = 1 with y i = 1indicating indifference. The last term provides the log-likelihood of an indifference observation by simply averaging the log-likelihood of the agent preferring Option A and the log-likelihood of the agent preferring Option B. 47, 47 The log-likelihood of an indifference observation would be maximized when the probability of preferring either Options A or B equals 50%. We can see this by looking at the likelihood instead of the log- 24