Sunk Decision Points: A Unified Theory of the Endowment Effect and Present Bias

Transcription

1 Sunk Decision Points: A Unified Theory of the Endowment Effect and Present Bias Peter Landry February 8, 2014 Abstract This paper presents a utility-based model in which situational cues compel an agent to consider the associated consumption decision. The premise gives rise to a unified account of the endowment effect and present bias that, unlike prevailing behavioral theories of inconsistent preferences, captures both anomalies (and without extra parameters). Among other features, the model captures evidence that the endowment effect and present bias are often inactive (and that cues, such as exposure to the good, can activate them), as well as key properties of prospective memory e.g. the effect of mere reminders and value in forgetting in that successful recall carries an opportunity cost. Stemming from the hypothesis that the anomalies are linked, the model offers testable new predictions, including a form of crowding out between the endowment effect and present bias. A method to deactivate both anomalies is also proposed, the viability of which fits with related evidence. I thank Attila Ambrus, Peter Arcidiacono, Mike Dalton, Rachel Kranton, and Philipp Sadowski for helpful feedback. Division of the Humanities and Social Sciences, California Institute of Technology, MC , 1200 E. California Blvd, Pasadena, CA plandry@caltech.edu. 1

2 1 Introduction The standard economic model of decision-making is built on the premise that preferences are consistent, yet behavioral researchers drawing heavily on experimental evidence have highlighted compelling challenges to this assumption. One of the most well-known such behavioral anomalies is the endowment effect, i.e. the tendency of those who receive a good to value it more than they otherwise would. 1 The endowment effect is typically captured by models of loss aversion, which depart from consistent consumption preferences by adopting a utility function that depends on a reference point (e.g. current assets) from which losses hurt more than gains help. 2 present bias, i.e. disproportionately steep impatience to delay immediate rewards. 3 A second important behavioral anomaly is Present bias is typically captured by (quasi-)hyperbolic discounting, given as βδ t with present bias factor β < 1, which departs from consistent time preferences by devaluing future utilities at a nonconstant rate. 4,5 While relaxing the explicit assumption of consistent preferences, the loss aversion and hyperbolic discounting models maintain an implicit assumption regarding the decisions that an agent considers. That is, both the standard model and its behavioral counterparts take the underlying decision problem as given. In dynamic settings, the same consumption decision arises once in every period by assumption. Hence, situational factors cannot affect when or whether the decision is considered, nor can different decisions be considered at different times. When interpreting experimental research through this lens, it is presumed that the decisions a subject considers are independent of experimental protocols. For instance, receiving a good or being asked to evaluate an intertemporal tradeoff in- 1 See Thaler (1980), Knetsch (1989), or Kahneman et al. (1990). 2 See Kahneman and Tversky (1979), Tversky and Kahneman (1991), and Kőszegi and Rabin (2006). When referencing the loss aversion model(s), I will implicitly include (and not differentiate between) pure ownership- and newer expectations- based approaches because they are not distinguished by the typical laboratory illustration of the endowment effect, which is the formal setting considered in this paper. 3 For example, the mean annual discount rates inferred from the median willingnesses to delay $250 in Thaler s (1981) study were 219% over one month from the present, yet only 14% over the next 9 years, 11 months. Also see Benzion et al. (1989) or Kirby (1997). 4 See Laibson (1997) and O Donoghue and Rabin (1999). 5 Here and throughout, endowment effect and present bias will refer to the associated behaviors, as opposed to their formal connotations as models based on inconsistent preferences. 2

3 volving the good (i.e. time-preference elicitation) as in the typical experiments that reveal the endowment effect and present bias, respectively are assumed to have no effect on whether the subject considers the decision to consume the good. While maintaining consistent preferences, this paper presents a model in which the decisions we consider can depend on the situations we encounter. Namely, situational cues induce decision points in that they compel the agent to consider the associated consumption decision. 6 A cue also entails a decision opportunity cost, understood as the loss due to the inability to fully engage other possible activities when thinking about the decision of interest. It follows that exposure to a cue increases the effective utility of owning the associated good; or put differently, a good is most valuable to you when you re already thinking about consuming it. The model accounts for both behavioral anomalies of interest in a manner that maps to the typical experiments that reveal them. To start, Section 4 shows how the endowment effect arises when receiving a good is modeled as a cue (inducing a decision point), as the willingness-to-accept (WTA) to sell a received good exceeds a potential buyer s ex-ante willingness-to-pay (WTP). Due to the irreversibility of the induced decision point, simply receiving the good makes the agent invested in the consumption decision. Hence, the endowment effect exists because the decision point is sunk or more precisely, the accompanying decision opportunity cost is sunk for a potential seller but not for a potential buyer, where the WTA-WTP gap quantifies the cost of being stuck having to think about consuming a good that is no longer available for consumption. In Section 5, time-preference elicitation is modeled as a cue, giving rise to present bias. Analogous to receiving a good, simply being asked to choose between an immediate good and a future good makes the agent think about and therefore become invested in consumption at the time of elicitation, while choosing the future good entails being stuck having to think about the foregone immediate good. The form of the apparent present bias β depends on whether 6 While focusing on addiction, Landry (2014) also models cues as decision points. For other representations of cues in economics, which also focus on addiction, see Laibson s (2001) preference-based approach and Bernheim and Rangel s (2004) treatment of cues as triggers of mistaken consumption. 3

4 or not choosing the future good guarantees an accompanying future decision point. If so, β < 1 embodies a fixed-cost present bias (as in Benhabib et al., 2010), which reflects the extra opportunity cost from having to re-think about consumption after elicitation. This fixed-cost present bias also captures the welldocumented magnitude effect whereby individuals exhibit greater patience for high-value than low-value goods. For the case in which it is uncertain whether a decision point will accompany the future good, β < 1 reflects the possibility that the agent will be thinking about something else when the future good becomes available. This characterization fits with the observation that individuals sometimes forget while cues can serve as reminders to carry out a future plan. 7 Related to this, the model accommodates value in forgetting due to the opportunity cost inherent in remembering; this implied tradeoff between forgetting and remembering is consistent with evidence that remembering to perform a given task is associated with reduced performance on a secondary task (Park et al., 1997; Smith, 2003; Smith et al., 2007). Sections 4 and 5 collectively demonstrate that, contrary to the mainstream view, the endowment effect and present bias are both compatible with consistent preferences. In doing so, the model fits with evidence (reviewed in Section 7) that both anomalies can be activated by situational cues, as well as the related mere exposure and mere question effects, whereby exposure to or being asked about a good increases one s valuation of it. 8 Furthermore, unlike prevailing behavioral models, the current framework simultaneously captures both anomalies and without requiring additional parameters. That is, relative to the most strippeddown version of the standard decision-making model, only one new parameter is needed: the natural per-period probability of considering the decision of interest in the absence of a cue. Section 6 characterizes a method to control for sunk decision points. In particular, the theory predicts that both anomalies vanish when their typical experiments are modified to include the ex-ante receipt of an appropriately-timed 7 See Ericson (2011) for evidence on forgetting. For evidence that cues can function as reminders or nudges to carry out the associated behavior, see Karlan et al. (2012). 8 For example, Peck and Shu (2009) and Bushong et al. (2010) document the effect of physical exposure. The mere question effect is covered by Sprott et al. (2006), among others. See Section 7.2 for additional references and discussion. 4

5 auxiliary good (e.g. a replica of the primary good). Receiving the auxiliary good eliminates the endowment effect because it sinks the decision point for all subjects: even potential buyers who do not receive the primary good will assuredly consider the decision to consume it. Hence, WTP for the primary good rises to the level of WTA, which matches evidence that subjects do not exhibit an endowment effect for goods that they already own, as well as its manifestation as a rise in WTP instead of a drop in WTA (Morewedge et al., 2009). Present bias is similarly eliminated by the ex-ante endowment of a future auxiliary good to be received at the time associated with the future good offered in elicitation because it sinks the future decision point. In this case, the consumption decision is not only considered with certainty at elicitation, but also at the time the future good would be acquired (even if the immediate good is chosen). This result extends to a general auxiliary endowment sequence, which fits with the apparent lack of present bias when time preferences are elicited over sequences of outcomes. 9 While carrying a new interpretation of the experimental evidence, the model also features a novel link between the endowment effect and present bias, as both anomalies arise because the agent rationally values a good more when she is already thinking about consuming it. Formally tightening this proposed link, Section 7 demonstrates that in an idealized experimental setting, the two anomalies standard laboratory measures are in fact equal: i.e. β = WTP < 1. This WTA equality is not a coincidence because both measures quantify the effective devaluation of a good when its decision point is not yet sunk. For example, in the endowment effect experiment, a potential buyer in essence devalues the nonendowed good by β relative to a potential seller s valuation even though the good is immediately available. Conversely, in the typical present bias experiment, the future good s value reflects WTP, while the immediate good s value reflects WTA even though the subject is not endowed with it ex-ante. Along these lines, the theory offers some new predictions to test the hypothesis that the endowment effect and present bias share a common basis, such as: (i) hidden versions of each anomaly are generated in the experiments known to produce the other for example, the theory predicts that the typical endowment 9 See Loewenstein and Prelec (1991, 1993). For an elaboration and caveats, see Footnote 45. 5

6 effect experiment activates a hidden present bias that could be uncovered if the experimenter attempts to compare the endowed good s value to that of a future good; and (ii) crowding out between the endowment effect and present bias would occur in an experiment designed to simultaneously detect both in that the combined strength of the two anomalies would be significantly weaker than would be implied by the standard notion that their underlying mechanisms are distinct. 10,11 Lastly, Section 8 shows how the hypothesized artificial separation between the endowment effect and present bias can extend to the field. That is, in formalizing conventional means of field inference, a detection threshold arises that partitions the domains for which each anomaly is observable. In particular, only the endowment effect can be detected for goods with raw value above the threshold in that market prices will reveal a WTA-WTP disparity; below the threshold, only present bias can be detected in that agents will value commitment, as evidenced by a willingness to pay a cost to avoid receiving the good. This result complements the laboratory analysis in addressing why, if they are supposedly variations of an equivalent sunk decision points rationale, the endowment effect and present bias are still generally inferred in different settings. 2 Related Theories Although the current approach is distinguished in its attempt to unify the anomalies, the theory can be related to existing theories that seek to explain one or the other anomaly without the usual behavioral model. To start, Huck et al. (2005) propose an evolutionary basis for the endowment effect, suggesting that it can improve one s bargaining position in bilateral trade. Gintis (2007) also posits an adaptive advantage, suggesting that an endowment effect could have facilitated 10 For instance, suppose a subject is endowed with an immediately available good at the time of time-preference elicitation (and thus has the option to exchange it for a future good that is not in her endowment). In this combined experiment, either WTP WTA or β (or both) would be closer to their classical value of 1 than when they are measured in separate experiments, ceteris paribus. 11 The proposed coexistence between the anomalies is not far off suggestive evidence (reviewed in Section 7.2) that their associated cues may trigger the other anomaly; namely, present bias has been linked to physical contact with a good, while an endowment effect has been linked to merely asking questions about the good. 6

7 enforcement of private property rights in the absence of legal institutions (which are a relatively modern invention). Isoni (2011) and Weaver and Frederick (2012) emphasize the role of reference prices in arguing that the endowment effect fits better with bad deal aversion than loss aversion. Bordola et al. (2012) suggest that the endowment effect can arise because due to the influence of context on the salience of goods attributes, which in turn affects the degree to which these attributes are weighted in choice. Ungureanu (2012) demonstrates how the endowment effect can arise due to locally-limited knowledge of one s own utility function. 12 As for present bias, Sozou (1998) suggests that it can arise without hyperbolic discounting if a Bayesian updater does not know the hazard rate that determines whether future payoffs will be realized. Dasgupta and Maskin (2005) instead attribute present bias to uncertainty with respect to when future payoffs will be realized. Havely (2008) demonstrates how present bias can be explained by nonlinear termination probabilities (which can be interpreted as a mortality rate or as the likelihood a reward will disappear). Ok and Masatlioglu (2007) show how relaxing the assumption of transitive time preferences can give rise to present bias. Rubinstein (2003) argues that a similarity heuristic in evaluating different rewards across time can explain behavioral patterns typically attributed to hyperbolic discounting. Lastly, Robson and Samuelson (2009) offer an evolutionary hypothesis in which present bias emerges amidst aggregate uncertainty in the form of a discount rate that falls with age, due to the imperfect correlation of aggregate shocks to agents survival probabilities. 3 Model 3.1 Benchmark Setup In each period, t = 0, 1, 2,..., a utility-maximizing agent with consistent time and consumption preferences might consider the decision to consume a particu- 12 Also of note, Plott and Zeiler (2007) informally propose (and experimentally test) a few alternatives to loss aversion in explaining experimental illustrations of the endowment effect, such as: other-regarding preferences towards the experimenter (i.e. subjects view their endowment as a gift, making them feel somewhat obligated to keep it), perceived signaling of the good s relative value by way of the fact that the experimenter chose to endow the subject with it, and informational cascades stemming from the observation of other subjects choices. 7

8 lar good. The decision points are defined as the times that the agent considers this decision. For ease of exposition, we assume that the good is perfectly perishable. 13 We also assume (implicitly) that the instantaneous utility from consumption is strictly increasing. 14 Hence, in period t, the agent optimally chooses to consume her full period-t endowment of the good (which may be zero) if and only if t is a decision point. 15 If t is not a decision point, the agent engages in some outside opportunity, e.g. perhaps a different consumption decision is considered. Since the return to this outside opportunity is equivalent to the opportunity cost of a decision point, it will be referred to as the decision opportunity cost. For simplicity, the decision opportunity cost is normalized to 1 (in absolute value). The benchmark refers to the control setting in which the agent is not (yet) subjected to an experiment. For all t in the benchmark, the endowment of the good is normalized to zero and the decision point probability is fixed to some π (0, 1). The benchmark expected lifetime utility is thus given by U = t=0 δ t (1 π) = 1 π 1 δ, where δ (0, 1) is a constant discount factor. 3.2 Alternate Scenarios: Endowments and Cues Now suppose the agent is subjected to alternate, experimental scenarios. Relative to the benchmark, an alternate scenario may involve changes to the endowment stream, to the decision point probabilities, or both. Let e(τ) denote a time-τ endowment of the good with consumption value e > 0 (i.e. consuming the good brings e utils). For a scenario that features an endowment of this form but does not include any other changes to the benchmark, the expected lifetime utility is given by U[e(τ)] = U + πδ τ e. 13 As shown in Appendix A.8, all results are (qualitatively) robust to consideration of durable goods. 14 Equivalently, we could instead model an indivisible good for which its consumption brings positive utility. 15 Thus, we abstract from the possibility that the agent considers consuming the good, but chooses not to. We maintain such assumptions to keep the model as simple and standard as possible. 8

9 This expression implies that the present value of the endowed good, πδ τ e, is its discounted consumption value weighted by the decision point probability π. This weighting by π follows because e(τ) provides e utils in period τ if and only if τ is a decision point (otherwise the good perishes). Exposure to a cue associated with the consumption good will formally describe a situational factor that increases the probability of a decision point relative to the benchmark probability, π. Thus, exposure to a cue precludes the outside opportunity, which reflects the idea that a cue distracts the agent from whatever else she happens to be doing, and compels her to consider the consumption decision associated with the cue. For simplicity, we assume that a cue at t guarantees a decision point at t and leaves all other decision point probabilities unaffected. 16 Also, to avoid ambiguity when translating real-world cues into the model, only situational factors that are exclusively associated with the primary consumption good will induce a decision point; in particular, exposure to a situation that is symmetrically associated with the primary consumption good and with an outside opportunity or with a different good (such as money) will not affect the decision point probability. 17 Absent any accompanying changes to the endowment stream, the expected lifetime utility associated with a single cue at t is U[t] = U (1 π)δ t. (1) Hence the cue s cost, (1 π)δ t, is the discounted decision opportunity cost, weighted by the benchmark probability that t would not have been a decision point in the absence of a cue. Thus 1 π represents the expected decision opportunity cost attributable to a cue, and will be referred to as such from this point forward This assumption isolates the immediate impact of a cue and treats it as a binary event. See Subsection 3.3 for a discussion of this assumption. 17 If a cue is associated with multiple and different goods/activities, (at least) one must be included in the outside opportunity. Since the agent s attention is a limited resource and thus may be partially diverted from the primary good when such combined cues are present, it is not necessarily clear whether their net effect would be to increase or decrease the decision point probability. To abstract from this ambiguity, the model simply assumes that the net effect is zero. 18 As defined here, the expected decision opportunity cost of a cue is expressed relative to and not to be confused with the expected opportunity cost of a decision point in the 9

10 For a scenario that involves a cue at t and an endowment of the good, e(τ), the expected lifetime utility is: 19 { U (1 π)δ t + πδ τ e if t τ U[t; e(τ)] = (2) U (1 π)δ t + δ τ e if t = τ As shown here, cues and endowments will be separated by a semicolon in the argument of U. From equation (2), if the endowment and the cue do not coincide (t τ), the net value of both is simply the sum of their individual values. However, if they do coincide (t = τ), then the weight π on the endowed good s discounted consumption value vanishes, which follows because the cue guarantees a decision point at the time of the endowment. The increase in utility when the endowed good coincides with the cue highlights the inefficiency in thinking about consuming a good and having it available for consumption at different times. Put differently, it is best to possess a good when considering the decision to consume it otherwise the agent may forget (with probability 1 π) to consume the good when it is available. 20, Simplifications in the Cue Representation While it will suffice to illustrate the main properties of the model, the cue representation described in the previous subsection in which a cue at t guarantees a decision point at t and leaves all other decision point probabilities unaffected is limited in its realism. First, this simple cue abstracts from any effect beyond the present, even though cues realistically may affect future decision points. For instance, an individual may remember past cues or a cue may be a catalyst benchmark, which is just π. 19 In general, a scenario can involve sequences of cues and endowments: {t i }, {e j (τ j )}. The corresponding lifetime utility expression is U[{t i }; {e j (τ j )}] = U (1 π) i δti + j:τ j {t i} δτj e j + π j :τ j / {t i} δτ j e j. As in (2), the cost of a cue at t is (1 π)δ t, while the value of an endowment e(τ) is δ τ e if it coincides with a cue and is πδ τ e if it does not. 20 Note that not considering the decision to consume (i.e. thinking about something else) and forgetting to consume are, in essence, equivalent in the current framework, which follows from the feature that the agent consumes e(τ) if and only if τ is a decision point. 21 As addressed in Appendix A.8, this mismatch between the decision point and possession of the good when t τ is still costly for a durable good e(τ), in which forgetting at τ does not preclude its eventual consumption. Rather, if τ is not a decision point, the value of the good is subject to additional discounting, out to the time it is eventually consumed. 10

11 for planning future decision points e.g. setting a mental reminder, or using an external planning device (such as a calendar). 22 Appendix A.9 considers the almost-as-simple case of a lagged decision point, in which a cue at t induces a decision point in t+1, but leaves all other decision point probabilities unaffected; here it is shown that the results derived from the simple cue model extend to the case in which a cue s effect is felt in the next period. 23 The simple cue representation also abstracts from potential uncertainty regarding the induced decision point. likelihood of a decision point without guaranteeing it. Realistically, a cue may only raise the persistence, cues could be ranked in terms of their salience. 24 With uncertainty and In contrast, any two cues must be equally salient in the current setup. Moreover, since the first cue guarantees the decision point, a second and concurrent cue has no additive effect (even though realistically two cues are likely more salient than one, all else equal). Though superfluous for the formal results, consideration of uncertainty and persistence will be helpful to informally address real-world intricacies that the simple cue representation does not readily motivate; uncertainty and persistence also represent potentially promising generalizations of the basic model for future research. 22 Dramatic examples of cue persistence can be seen in the realm of addiction. For instance, when a drug addict attempting to quit cold turkey encounters a cue for the drug, it may induce a prolonged craving, in which the abstinent addict is continually preoccupied with thoughts of drug use. The example also alludes to the notion that the extent to which a cue endures beyond the present can depend on the chosen course of action when the cue is encountered, as in Landry (2014). For instance, choosing not to consume a craved good likely brings a near-term escalation of decision points due to the unsatisfied craving. If the good is durable, the physical presence of unconsumed portions can also invite continued decision points. 23 As another basic way to model persistence, a cue at t could guarantee decision points in t and in t + 1. This case is equivalent to a scaled version of the simple cue model without persistence and with a perishable good. That is, the expected decision opportunity cost would become (1 + δ)(1 π), provided that realized cues do not overlap in that they are separated by at least one period without a cue. This scaling of the decision opportunity cost by (1 + δ) follows because the cue precludes the outside opportunity both today and tomorrow. 24 Suppose πt [π, 1] denotes the decision point probability at t = 0, 1,..., in a scenario that features a single cue at time-zero (without loss of generality). Then t δt (πt π) would be a natural measure of the cue s salience since this sum quantifies the cue s impact, given as the present expected value of the induced decision point sequence. 11

12 4 A Receipt-Induced Endowment Effect 4.1 The Endowment Effect Experiment In the typical endowment effect experiment, some subjects are given (endowed with) a good, while others are not. 25 All subjects must then choose between the good or some amount of money. That is, subjects who initially receive the good are prospective sellers who can either sell the good or keep it, while the remaining subjects are prospective buyers who can either buy the good or keep their money. In this context, willingness-to-pay (WTP) refers to the maximum price that a prospective buyer is willing to pay for the good, while willingness-toaccept (WTA) is the minimum price that a prospective seller is willing to accept to relinquish the good. The endowment effect is revealed by this experiment if WTA exceeds WTP, where a large WTA-WTP disparity (or a small WTP/WTA ratio) signifies a strong endowment effect. 4.2 Receiving the Good as a Cue To formally consider the typical endowment effect experiment, receiving the good is now modeled as a cue. 26 Thus, if the agent receives e(τ), τ will be a decision point. For this experiment and without loss of generality, we will only consider a time-zero endowment of the good, e(0). Let v P (e) denote the agent s WTP for the good, defined as the net value from receiving it; WTA, denoted by v A (e), is then defined as the value of retaining e(0) upon its receipt. WTP and WTA can be interpreted as the prices to a buyer and to a seller, respectively, such that the net return to a transaction is zero. Relative to the benchmark, WTP and WTA 25 See Horowitz and McConnell (2002) for a review of experimental research on the endowment effect. 26 Note that following the criteria given in Section 3.2, the opportunity to buy or sell the good does not constitute a cue. This follows because buying and selling opportunities essentially boil down to a choice between two goods the primary consumption good and a numeraire good (i.e. money) and only situations that are exclusively associated with a single good are modeled as cues. To be sure, the emergence of an endowment effect in this section is robust to generalizations of the exclusivity requirement that allow, for instance, a potential buyer s decision point probability to exceed π. See Appendix A.10 for an elaboration. 12

13 are thus given by: v P (e) = U[0; e(0)] U, v A (e) = U[0; e(0)] U[0]. (3) Here, WTP is the expected utility gain from receiving the good and WTA is the ensuing loss upon relinquishing it. 27 Since relinquishing the good does not change the fact that it had been received, the benchmark utility does not enter WTA; rather, the scenario in which e(0) is relinquished still entails a cue at t=0. Due to the irreversibility of this cue, simply receiving the good makes the agent invested in the associated consumption decision. Quantifying this investment, the first result captures the endowment effect: Proposition 1 [Endowment Effect] v A (e) v P (e) = 1 π > 0. Here and in all subsequent results, the benchmark is the base for comparison unless otherwise specified; all proofs are in the appendix. Proposition 1 establishes a WTA-WTP disparity, equal to the expected decision opportunity cost, 1 π. This endowment effect reflects the cost of being stuck having to consider the decision to consume a good that was immediately relinquished upon its receipt. 28 Thus, by causing the recipient to think about consumption, receiving the good causes its value to rise from its benchmark value put differently, this WTA-WTP disparity exists because the decision point is sunk for a prospective seller, but not for a prospective buyer. 29 Observe if π = 1, as in the standard decision-making model, there would be no WTA-WTP disparity; this lack of an endowment effect for π = 1 follows because the cue (receiving the good) would no longer carry a decision opportunity cost if the decision would have been considered regardless. 27 If the default scenario is something other than the benchmark, i.e. if it involves sequences of cues {t i } and of endowments {e j (τ j )}, the general forms for WTP and WTA are: v P (e) = U[0 {t i }; e(0) {e j (τ j )}] U[{t i }; {e j (τ j )}] and v A (e) = U[0 {t i }; e(0) {e j (τ j )}] U[0 {t i }; {e j (τ j )}] (respectively). 28 In the language of loss aversion, the loss from relinquishing the good exceeds the gain from receiving it because the loss is the good s full consumption value, while the gain is only its net value, i.e. the consumption value minus the expected decision opportunity cost from its receipt. 29 To reiterate, variations on the phrase sunk decision point are used as shorthand to refer to the sunk decision opportunity cost. 13

14 The emergence of an endowment effect in Proposition 1 is robust to notions of cue persistence discussed in Section 3.3. In fact, if we maintain that receiving the good guarantees the present decision point, the endowment effect would strengthen with cue persistence because the total expected decision opportunity cost underlying the WTA-WTP disparity would exceed 1 π if receiving the good sinks additional decision points beyond the present. That is, a seller of the good may continue to reminisce or think about using the good after selling it due to the continued decision points that were effectively sunk when the good was initially received. 30 It similarly follows that, with cue persistence, the endowment effect can exist even for goods that were received yesterday. In contrast, using our simple cue model without persistence, Proposition 1 implies that there will be no endowment effect for goods at the time that they are ripe (or available) for consumption if they were received in the past; e.g. there is no WTA-WTP disparity at τ for an endowment e(τ) if it was received in τ 1. 5 An Elicitation-Induced Present Bias 5.1 The Present Bias Experiment The typical experimental illustration of present bias is based on a time-preference elicitation protocol that involves asking subjects to choose between an immediately available good and a future endowment of the same good, but of a higher quantity or value. From a subject s responses, the analyst makes inferences on the form of the subject s discount function. The total elicited discount from the present to a future time τ is calculated as the ratio of a presently-available good s value to the value of a good at τ such that the subject is indifferent between the two. A present bias is then revealed if the elicited discount function is quasihyperbolic, i.e. if it takes the form βδ τ for τ > 0, where β < 1 is the present bias factor (Laibson, 1997; O Donoghue and Rabin, 1999). 30 The notion that receiving the good could induce future decision points (via cue-induced planning) would likely be important in understanding the endowment effect for mugs the classical endowment effect illustration (Kahneman et al., 1990) in the context of the model, as mugs are generally not used immediately upon receipt. Recall, the robustnesses of Proposition 1 to durable goods and to a basic form of cue persistence are established in Appendices A.8 and A.9, respectively. 14

15 5.2 Elicitation of Time Preferences as a Cue To formally consider the typical present bias experiment, time-preference elicitation is now modeled as a cue. 31 The elicited discount function, denoted by D τ, measures the total discount from elicitation at t = 0 (without loss of generality) to the time of the future good e(τ) that is offered as an option. Thus, D τ is defined as the unique value such that the agent is indifferent between e(τ) and D τ e(0). 32 To compute D τ we still need to know more about the elicitation protocol because when the future good e(τ) is chosen, the period-τ decision point probability remains ambiguous. If the agent receives e(τ) in period τ, then τ must be a decision point since receiving the good is a cue. That said, it is conceivable that τ may not be a decision point even if e(τ) is chosen, e.g.: (i) elicitation may only involve a choice among hypothetical rewards; 33 (ii) the subject may have to remember to claim the future reward; or (iii) a good that is not yet ripe for consumption may be acquired at the time of elicitation (e.g. choosing between one yellow banana and two green bananas). To account for these possibilities, the term non-cueing is used to refer to a good for which its acquisition does not induce a decision point in contrast to a received good. 34 Proposition 2 [Present Bias] D τ = βδ τ, where β < 1 is given by: (i) β = 1 (1 π)/e for received goods, (ii) β = π for non-cueing goods. 31 Note that time-preference elicitation is exclusively associated with a single good, as both of the subject s options are endowments of the primary consumption good. Thus, time-preference elicitation meets the criteria for a cue from Section 3 unlike buying and selling opportunities in the endowment effect experiment, which are symmetrically associated with a numeraire good (i.e. money) in addition to the primary good. See Appendix A.10 for additional discussion on this distinction as it relates to the main results. 32 Put differently, if elicitation reveals that the agent is indifferent between e (0) and e(τ), then D τ = e e. Note that D τ may implicitly depend on e (as well as on π). 33 When hypothetical rewards are involved, it is often unclear whether subjects ought to choose the alternative that they would prefer to have had in their pre-existing endowment, in which case the future good may not be evaluated as if it involves a cue, or to choose the alternative that they would prefer to receive into their endowment at that time. This ambiguity is evident in an excerpt from the sample instructions in Thaler s (1981) classic study: choose between: (A.1) One apple today. (A.2) Two apples tomorrow. 34 That is, if e(τ) is a non-cueing good then choosing it in the elicitation task does not affect the decision point probability at τ. 15

16 From part (i), elicitation induces a present bias for received goods in the form of a fixed-cost present bias, in which the agent appears to discount as if future returns entail a fixed cost from which present returns are exempt. This fixed cost for future returns equals the expected decision opportunity cost and is attributable to the property that time-preference elicitation at t = 0 sinks the decision point for the immediate good, but not for the future good. 35 From part (ii) of Proposition 2, elicitation induces a present bias for non-cueing goods, equal to the benchmark decision point probability. In this case, present bias arises because an accompanying decision point is only guaranteed for the immediate good (since it coincides with elicitation). If the future good is chosen, with probability π the agent will not consider the decision to consume it, in which case the good perishes. 36 Together, parts (i) and (ii) of Proposition 2 show that an elicitation-induced present bias will exist with or without our previous assumption that receiving the good induces a decision point. In both cases, the present bias reflects the extra incentive to own a good when its decision point is already sunk from timepreference elicitation. As with the endowment effect, present bias (in both forms) disappears if π = 1 in that D τ = δ τ. For received goods, this disappearance of present bias reflects that the expected decision opportunity cost from receiving a future good is zero if the future decision point was assured regardless. For non-cueing goods, present bias disappears because the agent will always consider the consumption decision meaning it is certain that the good will not perish since decision points are guaranteed in all periods with π = Magnitude Effect (for Received Goods) Observe from part (i) of Proposition 2 that a received good s consumption value nontrivially enters the expression for β. This dependence implies: 35 To see that the fixed cost equals the expected decision opportunity cost, observe that the time-zero value associated with receiving e(τ) is βδ τ e = δ τ (e (1 π)), while the present value from receiving e(0) is just e. For a formal consideration and evidence of fixed-cost present-bias, see Benhabib and Bisin (2005) and Benhabib et al. (2010). 36 With durable non-cueing goods, the elicited present bias instead reflects the extra discounting from the time that the future good is first available until the time of its eventual consumption (i.e. the first decision point on or after the time the good first becomes available for consumption). See Appendix A.8 for details. 16

17 Corollary 1 [Magnitude Effect] For received goods, D τ increases with e. Hence, the elicited discount function for received goods is increasing in the good s consumption value. This corollary captures the widely-documented magnitude effect an inherent feature of a fixed-cost present-bias whereby individuals exhibit greater patience for large rewards than for small rewards, ceteris paribus. 37 Here, the magnitude effect exists because the expected decision opportunity cost of a future, receipt-induced decision point is comparatively small when the received good s consumption value is large. That is, if the agent chooses the received future good e(τ), the nuisance from having to consider the decision (again) at τ is relatively minor if the future reward e is large. 5.4 Memory and Cue Persistence The current treatment of present bias relates to aspects of memory. To start, the present bias for non-cueing goods reflects the intuitive (and empirically supported) notion that people sometimes forget to carry out their plans or intentions for the future. For example, subjects in Ericson s (2011) experiment had to choose between a smaller payment to be automatically given to them six months later and a larger payment that they had to remember to claim after the same six-month waiting period; roughly half of the subjects who chose the larger payment did not remember to claim it. Furthermore, receiving a future good in essence serves as a reminder to consume it. The the notion that such a reminder works as a nudge to perform the associated behavior likewise matches introspection and evidence. 38 While a reminder may on balance be beneficial (i.e. provided e > 1 π), there can also be value in forgetting because the decision opportunity cost is avoided if 37 In addition to Benhabib et al. s (2010) evidence of a fixed-cost present bias, the magnitude effect is evident from survey data in Thaler (1981), Benzion et al. (1989), and Green et al. (1997). Also see Loewenstein and Prelec (1992) or Frederick et al. (2002) for discussion and additional references. For a recent axiomatization of the magnitude effect, see Noor (2011). 38 Karlan et al. s (2012) field study shows how mere reminders can be effective increasing individuals propensity to save. As the authors note, not only does the hyperbolic discounting model fail to explain the effect of reminders on behavior, the fact that reminders matter also points to a different channel through which inconsistent behavior typically attributed to timeinconsistent preferences can arise. 17

18 the agent does not remember to consider consumption. 39 This implied tradeoff between remembering and forgetting as well as its conception in terms of an opportunity cost fits with laboratory evidence that successfully remembering to perform a given task is associated with reduced performance on a secondary task (e.g. Park et al., 1997; Smith, 2003; Smith et al., 2007). Both forms of present bias in Proposition 2 are robust to cue persistence. 40 Similar to its effect on the WTA-WTP disparity, cue persistence can exacerbate the elicited fixed-cost present bias because the total corresponding opportunity costs are higher if receiving the good has an enduring effect on decision points beyond the present. For non-cueing goods, however, cue persistence would generally weaken present bias because elicitation at t = 0 could then raise the decision point probability at τ above the benchmark value, thereby increasing the likelihood that the agent remembers to consume. Bearing in mind that the present bias for non-cueing goods reflects the probability of remembering to consume, a realistic specification of cue persistence motivated by an understanding of memory would likely enhance the model s predictive power. For example, the likelihood of remembering to consume e(τ) is probably higher if τ = 1 than if τ = 100. In particular, incorporating the (empirically-supported) hyperbolic retention function would give rise to the appearance of the smoother hyperbolic discount function, as opposed to the cruder quasi-hyperbolic form Controlling for Sunk Decision Points In contrast to their usual interpretations as evidence against consistent preferences, the endowment effect and present bias emerged in preceding sections because goods are rationally valued more by those who are already considering 39 Recall from Section 3.1 that the agent optimally chooses to consume a good if and only if she considers the decision to consume it. Hence, remembering to consume and remembering to consider the decision to consume can, in effect, be regarded as equivalent. 40 Save for unreasonably extreme cases. For example, if elicitation is so salient that it guarantees decision points for all t (the agent never again considers any decision besides the decision to consume this particular good), present bias would disappear because, in essence, we would have π = Lee s (2004) estimates from individual retention data favor a hyperbolic memory model over alternative specifications such as an exponential and a quasi-hyperbolic-equivalent retention function. 18

19 the associated consumption decision. While quantifying the influence of such sunk decision points on the valuation of goods may have merit, if the aim is to test preference consistency assumptions, the results suggest that standard experiments may not be suitably controlled. Can we control for sunk decision points? As discussed, if decision points are guaranteed in all periods (π = 1), both anomalies vanish. Although guaranteeing decision points for all t is not a practical experimental control, with our simple cue concept that lacks persistence, decision points would only need to be induced in periods relevant to the valuation problem. Namely, suppose the benchmark is modified such that the agent receives an auxiliary good, denoted by a(τ). Here τ will be the period for which some, but not all participants in the original experiment will have an induced decision point. This modified benchmark will be referred to as the auxiliary-appended benchmark. 6.1 Disappearance of the Endowment Effect The next result shows that the endowment effect from Proposition 1 disappears in the auxiliary-appended benchmark: Proposition 3 [WTA-WTP parity] From the auxiliary-appended benchmark, v A (e) = v P (e), both of which equal v A (e) in benchmark. The WTA-WTP disparity vanishes here because receiving the auxiliary good a(0) guarantees a decision point at t = 0 regardless of whether or not the agent receives the primary good e(0) i.e. receiving the auxiliary good sinks the decision point even for potential buyers of the primary good. Only WTP changes from its benchmark value because, to a recipient of e(0), receiving a(0) as well does not affect the time-zero decision point probability, implying that WTA is unaffected. This disappearance of the endowment effect and its attribution to an increase in WTP (as opposed to a decrease in WTA) are both consistent with Morewedge et al. s (2009) finding that potential buyers of a coffee mug that already own an identical mug are generally willing to pay the price demanded by sellers. Proposition 3 ought to remain valid even if the auxiliary good is a different good that, when received, triggers the same consumption decision as the primary good. Since different goods may be used in a similar fashion, it is plausible 19

20 that such (possibly imperfect) decision-point substitutes exist. This concept fits with Chapman s (1998) finding that the endowment effect is significantly stronger between a writing utensil and a chocolate than between two different types of writing utensils (or between two different types of chocolates). 42 For example, receiving an auxiliary pen would control for the endowment effect from receiving a pencil if both items cause the agent to consider writing. However, the endowment effect should persist if the auxiliary good triggers a different decision altogether e.g. a chocolate cannot serve as a proper control for a pencil because receiving the chocolate would not compel the agent to consider writing. 6.2 Disappearance of Present Bias The same principle that eliminated the endowment effect can be applied to eliminate present bias. That is, an exponential discount function is elicited if it is assured that the agent will receive an auxiliary good at the time associated with the primary future good offered in the elicitation task: 43 Proposition 4 [Exponential Discounting] From the auxiliary-appended benchmark, D τ = δ τ. Receiving the auxiliary good a(τ) eliminates present bias because it sinks the decision point at τ, offsetting the elicitation-induced decision point at t = 0. Proposition 4 holds for both received and non-cueing goods, although it is easier to conceive implementation with received goods for consistency with how the auxiliary good is acquired. For received goods, there is no longer a fixedcost present bias as in part (i) of Proposition 2 because the associated fixed cost at τ (i.e. the expected decision opportunity cost) is paid ex-ante in the 42 That is, individuals endowed with a chocolate tend to be much less reluctant to exchange it for a different type of chocolate than to exchange it for a writing utensil (and vice versa). Similar evidence is provided by Hanemann (1991) and by Shogren et al. (1994) who find that goods with more substitutes in the marketplace feature a smaller WTA-WTP gap than goods with fewer substitutes. 43 Stated differently, if the original procedure compared e (0) to e(τ), then now the pairs {(e +a)(0), a(τ)} and {a(0), (e+a)(τ)} are effectively compared. In the current setting, receiving a(0) for both options is extraneous, but it may be a helpful control in practice to ensure the present cue is as potent as the future cue even when the agent opts for the alternative with the larger consumption value at τ. 20

21 auxiliary-appended benchmark. For non-cueing goods, there is no longer any doubt whether the agent will consider consumption at τ because receiving a(τ) guarantees the decision point; hence the value of e(τ) is no longer weighted by π, which was the elicited β from Proposition 2, part (ii). Although it would have been extraneous, Proposition 4 would still hold if the agent also received an additional, time-zero auxiliary good, a(0). In this case, both alternatives in the elicitation task would effectively reduce to a pair of nonzero endowments one at t = 0 and one at τ that differ only in their per-period consumption values. 44 Hence, Proposition 4 suggests that there would be no present bias when eliciting preferences over sequences of outcomes at prespecified times. This fits with the apparent lack of present bias when discount rates are inferred from preferences over sequences of outcomes (Loewenstein and Prelec, 1991, 1993). 45 Intuitively, present bias would not be apparent here because the periods for which the agent receives some amount of the good and hence the periods for which the agent considers the decision are fixed ex-ante and thus do not depend on the chosen endowment sequence Linking the Endowment Effect and Present Bias So far, several similarities between the endowment effect and present bias have emerged. The results from Sections 4-5 show how both behavioral anomalies can be inferred from consistent preferences in conventional experimental settings. Both anomalies were attributable to the increase in a good s effective value when 44 For example, if utility is linear and the consumption value of the immediate primary good is e, the elicitation task would entail asking the subject to choose between: (i) endowments at t = 0 and at τ worth a + e and a, respectively, or (ii) endowments at t = 0 and at τ worth a and a + e, respectively. 45 In fact, Loewenstein and Prelec document revealed negative time preferences, as improving sequences of outcomes tend to be favored over nonimproving sequences. The simple cue representation cannot account for revealed negative time preferences. That said, if receiving e(τ) has a persistent and positive effect on decision point probabilities after τ where the degree of persistence (i.e. salience beyond its present impact) rises with e, then improving sequences could become more favorable to mitigate the decision opportunity costs in the interim periods when no endowment is received. 46 The logic is the same for sequences that involve more than a pair of endowments. If an auxiliary good is received in all t = 0, 1, 2,..., the disappearance of present bias and also of the endowment effect can be understood in light of the fact that the induced sequence of decision points is, in essence, a reduction to standard discrete-time. 21