J Risk Uncertain (2018) 56:259–287 https://doi.org/10.1007/s11166-018-9281-7
Estimating representations of time preferences and models of probabilistic intertemporal choice on experimental data Pavlo R. Blavatskyy 1 & Hela Maafi 2
Published online: 26 June 2018 # Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract We here estimate a number of alternatives to discounted-utility theory, such as quasi-hyperbolic discounting, generalized hyperbolic discounting, and rank-dependent discounted utility with three different models of probabilistic choice. The data come from a controlled laboratory experiment designed to reveal individual time preferences in two rounds of 100 binary-choice problems. Rank-dependent discounted utility and its special case—the maximization of present discounted value—turn out to be the best-fitting theory (for about two-thirds of all subjects). For a great majority of subjects (72%), the representation of time preferences in Luce’s choice model provides the best fit. Keywords Intertemporal choice . Time preference . Discounted utility . Quasi-hyperbolic discounting . Rank-dependent discounted utility JEL Classification D90
1 Introduction Intertemporal choice involves outcomes that are received at different points in time. A trade-off between smaller outcomes received in the more immediate future and larger We are grateful to the editor Kip Viscusi and one anonymous referee, as well as the participants of D-TEA 2017 (Paris) and BRIC 2017 (London) for helpful comments.
* Pavlo R. Blavatskyy
[email protected] Hela Maafi
[email protected]
1
Montpellier Business School, Montpellier Research in Management, 2300, Avenue des Moulins, 34185 Montpellier Cedex 4, France
2
Université Paris 8, 2 Rue de la Liberté, 93200 Saint-Denis, France
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outcomes at a more distant date can be found in virtually every economic activity. Should one pursue higher education and build up human capital or enter the labor market and build up practical skills? Should the government invest in long-term infrastructure projects or boost current consumption? Should a manager invest in the building of new capacity or boost marketing and outsource production to overseas suppliers? The way these types of trade-offs are resolved ultimately affects our consumption, well-being, health and life satisfaction. The quest for a descriptive model of choice over time that accurately describes how people resolve intertemporal payoffs is therefore of paramount importance. Samuelson (1937) proposed one of the best-known models of intertemporal choice, in which time preferences are represented by discounted utility with constant (or exponential) discounting. This model, known as discounted-utility theory (cf. Section 4.3 below), gained momentum in economics after Koopmans (1960) produced its behavioral characterization (axiomatization). Despite its normative appeal, the underlying discounted-utility theory imposes rather strong consistency requirements on revealed intertemporal choices. For instance, Thaler (1981, p. 202) argued that some people may prefer one apple today over two apples tomorrow but, at the same time, prefer two apples in one year plus one day over one apple in one year and thus violate the time-consistency requirement. The descriptive limitations of discounted-utility theory (such as the commondifference effect, see e.g., Loewenstein and Prelec 1992, p. 574) have stimulated the development of numerous alternative representations of time preferences, including quasi-hyperbolic discounting (Phelps and Pollak 1968), generalized hyperbolic discounting (Loewenstein and Prelec 1992), liminal discounting (Pan et al. 2013) and rank-dependent discounted utility (Blavatskyy 2016). The aim of this paper is to compare the goodness of fit of these various representations of time preferences in experimental data. Our paper is similar in spirit to Hey and Orme (1994), who compared numerous theories of decision making under risk according to their goodness of fit in experimental data. We collect experimental data on repeated intertemporal choice. The elicitation of time preferences from repeated choices enables us to disentangle true preferences from noise, random errors and/or preference imprecision. Data on repeated intertemporal choice can be used to evaluate the relative goodness of fit of not only (deterministic) representations of time preferences but also models of probabilistic intertemporal choice. Examples of the latter include the model of Fechner (1860) of homoscedastic random errors and the choice model of Luce (1959). In many empirical applications of time preferences, a deterministic theory is embedded in a model of probabilistic choice. For example, Tanaka et al. (2010) embed discounted utility in a Fechner (1860)-like model of random errors, while Andersen et al. (2008) and Meier and Sprenger (2015) embed respectively discounted utility and quasi-hyperbolic discounted utility in the choice model of Luce (1959). The use of a probabilistic intertemporal choice model is rarely debated in these applications. As such, the descriptive superiority of a particular probabilistic intertemporal choice model, which is not well documented in the current literature, should be explored. To this end, we designed an experiment to establish which probabilistic intertemporal-choice theory best describes subject behavior. We specifically consider the case of three distinct outcomes received at three distinct points in time, producing 94 possible binary-choice problems (excluding any instances of first-order
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temporal dominance)1: each subject is confronted with the complete set of these 94 problems. This experimental design is thus different from the existing literature, where the experimenter typically picks a much smaller set of the possible binary-choice questions. Such a sample of binary-choice problems may not be the best way to compare the goodness of fit of different theories. For example, discounted-utility theory is known to be violated in questions that test the common-difference effect (Loewenstein and Prelec 1992, p. 574). An experimenter who selects too many (few) such questions will then overestimate (underestimate) the goodness of fit of discounted-utility theory. Our experimental design deliberately employs streams of intertemporal outcomes with a small number of distinct outcomes (three or fewer) and a small number of distinct points in time (three or fewer). This facilitates the non-parametric estimation of representations of time preferences when the subjective parameters are defined over the outcome space (such as in discounted-utility theory). It also facilitates the nonparametric estimation of representations of time preferences when subjective parameters are defined on the time line (such as theories with a subjective discount function). We embed representations of time preferences in three different models of probabilistic choice: the Fechner (1860) model of homoscedastic random errors, the Luce (1959) choice model and the Wilcox (2008) contextual-utility model with heteroscedastic random errors. We show that representations of time preferences with constant discounting (e.g., discounted-utility theory and its special cases—the maximization of present discounted value or cumulative payoffs) generally fit the data better when combined with the Fechner (1860) model or Wilcox (2008) contextual utility. In contrast, representations of time preferences with non-constant discounting (e.g., additively-separable utility or rank-dependent discounted utility) fit the data better when combined with the Luce (1959) choice model. Across all three models of probabilistic choice, the two best-fitting representations of time preferences consistently turn out to be rank-dependent discounted utility and its special case—the maximization of present discounted value. Somewhat surprisingly, the representation of time preferences with additively-separable utility (which includes such popular theories as quasi-hyperbolic discounting and generalized hyperbolic discounting) best fits the revealed choice pattern of only a small minority of subjects. For a great majority of subjects (72%), the representations of time preferences embedded in Luce’s choice model fit the data significantly better than those embedded in the Fechner or Wilcox models. These results provide valuable insights for practitioners (which model to use to describe choices over time) and promising avenues for future research (which models involve assumptions that are too restrictive and that can be dropped/generalized). The remainder of the paper is organized as follows. Section 2 describes our experimental design, subject pool and experimental method, and Section 3 presents some descriptive statistics of the data we collect, such as rates of inconsistency and violations of first-order temporal dominance. Section 4 describes the various representations of time preferences that we consider in this paper. Section 5 then presents the estimation procedure and results for three different econometric models of discrete choice. Last, Section 6 concludes. 1
One option temporally dominates another if, at any moment of time, the cumulative payoff of the first option up to that moment is at least as high as that of the other option. See Section 2.1 for more details.
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2 The experiment In order to examine the descriptive superiority of intertemporal-choice theories and probabilistic-choice theories, an experiment was conducted in which subjects were asked to make 100 binary choices on two different occasions. In what follows, we describe the binary-choice questions that we use and the organization of the experiment. 2.1 Design To render the non-parametric estimation of subjective utility functions easier we use only three monetary payoffs: 0, 10 and 20 Euros. Analogously for the subjective discount functions, we use only three points in time: the present, a delay of one month and a delay of two months. With these three payoffs and points in time, we have 27 choice alternatives (also known as programs, e.g. Koopmans 1960). Thus, a typical choice alternative can be written as {g1, g2, g3}, where g1 ∊ {€0, €10, €20} denotes the instantaneous outcome, and g2 ∊ {€0, €10, €20} and g3 ∊ {€0, €10, €20} the outcomes received with delays of one and two months respectively. With these 27 choice alternatives we can construct a total of 351 binary-choice questions. A typical binary-choice problem is between two alternatives (with g = l, r): a left alternative {l1, l2, l3}, and a right alternative {r1, r2, r3}. One alternative {l1, l2, l3} will first-order temporally dominate another {r1, r2, r3} whenever l1 ≥ r1, l1 + l2 ≥ r1 + r2 and l1 + l2 + l3 ≥ r1 + r2 + r3, with at least one of these inequalities being strict (e.g., Bøhren and Hansen 1980, Section III, p. 48).2 In 257 of the 351 binary-choice questions mentioned above, one alternative first-order temporally dominates the other; in the remaining 94 questions there is no first-order temporal dominance. We use all 94 of these questions in our experiment (cf. questions 7–100 in Table 3 in the appendix). At the same time, we did not ask the subjects to answer all of the 257 binarychoice questions where one alternative first-order temporally dominates the other. Binarychoice decisions without temporal dominance require more cognitive effort than Btrivial^ questions with temporal dominance. We conjectured that many if not all subjects would choose the dominant alternative were one available. We thus only selected six out of the 257 temporal-dominance questions in our experiment (cf. questions 1–6 in Table 3 in the appendix). This does produce some choice data under temporal dominance without overloading the subjects with too many Btrivial^ questions (which might increase noise, random errors or imprecision in responses). All 100 of the binary-choice questions in the experiment appear in Table 3 in the appendix. 2.2 The subjects To control for the transaction costs generally implied by intertemporal-choice experiments, subjects were recruited on the campus of École Polytechnique, France. Subjects studying or working on campus used to receive internal mail in the mailbox of their
2
In other words, one alternative first-order temporally dominates another if it can be obtained from the other via a sequence of increased and/or bought-forward payments.
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dormitory or office. Subjects were recruited from the BLaboratoire d’Economie Expérimentale de Saclay^ pool. As École Polytechnique students are a particular sample (the École Polytechnique is an elite graduate engineering school where students are heavily selected according to their academic ability, and are intensively trained), we also recruited subjects working on campus or visiting students, who more closely resemble traditional experimental subjects. Seventy-five subjects (46 males and 29 female) participated in the experiment. Around half of the subjects (52%) are engineers and half non-engineers. Subjects were aged between 19 and 59, with a mean of 29.84. We ran four sessions with no subject participating in more than one session. 2.3 Organization The experiment was computerized. In each session, the experimenter checked identities once all subjects had arrived, and assigned them randomly to a separate individual computer. The experimenter then distributed written instructions and read them out loud. The instructions described the nature of the alternatives and payment. Subjects were told that there were no right or wrong answers. They were also told that they could ask any question about the experiment, but that once the experiment had started they would not be allowed to communicate with each other but could talk to the experimenter in private. The experiment lasted for almost one hour, including the instruction and payment phases. 2.4 Procedure and implementation The experiment was divided into three parts. The first consisted of the hundred paired choices in Table 3. In the second part, subjects were presented with a distractor task. In the third part, subjects repeated the hundred paired choices of Part 1. To illustrate how the paired choices were framed and presented to subjects, Fig. 1 shows a screenshot of choice problem No. 91. Subjects were provided with a calendar showing the payment dates. Each option was presented in a table with a description of the amounts paid at the three different dates. If the subject preferred the right option, she clicked on it and that option was then highlighted in red (see Fig. 2). The subject had to confirm her choice before moving on to the next choice. If this question was the one selected for payment at end of the experiment, and if the subject chose the right (left) option, she would receive €20 (€20) immediately, €0 (€10) in one month and €20 (€0) in two months. Distractor part: To separate the two intertemporal-choice parts, we added a distractor part with a number of unrelated questions. We opted for the cognitive reflection test, CRT (Frederick 2005). In this task, subjects answer three questions (see Fig. 3), each of which has an incorrect but impulsive answer and a correct but reflective answer. To illustrate, consider the first question in Fig. 3. The first answer that comes to mind is B10 cents^ but this intuitive answer is incorrect since the difference between $1 and $0.10 is not $1 but $0.90. If we think reflectively about this question, we obtain the correct answer of 5 cents. Frederick (2005) shows that the CRT score is correlated with intertemporal behavior: subjects with intuitive answers (a low CRT score) are less patient than those with reflective answers (a high CRT score). Our choice of the CRT was not in fact motivated
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Fig. 1 Screenshot of choice problem No. 91 as seen by a subject before making choice
by this connection; we rather think that the CRT is a good distractor candidate as we wanted a short task which looks different from the paired choices in Table 3. In addition, the CRT is cognitively demanding, which guarantees a clear break between
Fig. 2 Screenshot of choice problem No. 91 as seen by a subject after making choice
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Fig. 3 Cognitive reflection task (CRT)
the two blocks of intertemporal-choice problems. Figure 4 illustrates how the distractor questions were presented to subjects. The distractor part was incentivized, with subjects receiving €0.50 per correct answer. 2.5 Payment Subjects were informed of the payment procedure during the instruction phase. They were told that they would receive an immediate payment of g1 at the end of the experiment (t = 0), an amount g2 in one month (t = 1) and an amount g3 in two months (t = 2), with (g1, g2, g3) ∈(€0, €10, €20),3 with g = l, r. To control for future physical transaction costs, subjects were given two envelopes (one white and one brown) and were told that they would receive their future payment through the internal mailbox: the white envelope containing the earlier future payment (g2) in one month and the brown envelope with the later future payment (g3) in two months. Inside each envelope, there is a letter reminding the subject of participation in the experiment one (two) month(s) ago and that there were earlier and later future payments (the payment figures were however left blank). During the instruction phase, the experimenter asked subjects to wait until the end of the experiment to note their internal mail address on the corresponding envelopes. If both future payments were positive (g2 > 0 and g3 > 0) subjects addressed both envelopes to themselves, otherwise they only needed to put their address on the envelope containing the non-zero gain at t > 0. Subjects chose between receiving their payments by cash or check. They reported this preference during the payment phase at the end of the experiment.3 We did not provide subjects with show-up fees for two reasons. First, to keep the future payment as similar as possible to the immediate payment in this kind of design, we need to give subjects the same show-up fee at each date. Since future payments are made through internal mail, so that subjects do not come back physically to receive them, the future show-up fee has little sense. Second, the average gain in our experiment is relatively high compared to standard experimental earnings on the campus that might include a show-up fee. At the end of the experiment, one paired-choice question was randomly selected by the computer for payment. The final screen showed the randomly-selected question, the subject’s choice, the three amounts of the chosen option, and the bonus payment from the distractor questions. Once they had received their payment information, subjects 3
The large majority of subjects preferred cash.
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Fig. 4 Screenshot of the distractor questions
addressed the envelope(s) to themselves and wrote their future payment(s) in the reminder letter in the corresponding envelope. Subjects were paid in private in a separate office to which they took their envelopes. They first announced whether they preferred cash or check. They then received the immediate payment, and the experimenter put the earlier future payment, g2, in the white envelope and the later future payment, g3, in the brown envelope. Subjects earned €37.75 on average, with a minimum of €20 and a maximum of €51.50.
3 Descriptive statistics of the collected data set 3.1 Consistency Each subject was presented with 100 binary-choice questions twice. We can thus calculate for each subject how many times he or she chose a different alternative in the second round. Figure 5 shows the histogram of inconsistency rates among all 75 subjects for questions 7–100, in which neither alternative first-order temporally dominates the other. The mean inconsistency rate observed in the questions without dominance is 6.40% with a median of 4.26%. The lowest inconsistency rate was 0% (five subjects had perfectly-consistent choice patterns) and the highest was 40.43% (one subject chose inconsistently in 38 out of 94 questions). The most consistent questions were 7, 39, 51, 71 and 82, for which no-one switched their choices in the second round; the most inconsistent question was 85, where 27 out of 75 subjects (36%) switched.
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Fig. 5 Inconsistency rates in questions without dominance
Overall, this inconsistency in decisions over time is less pronounced than that previously observed in the literature for decisions under risk (the inconsistency rate for risk is around 30%). This finding is consistent with Laury et al. (2012), who find more noise in decisions under risk than in those over time. 3.2 First-order temporal dominance We here consider the violation of first-order temporal dominance. One option temporally dominates another if, at all points in time, the cumulative payoff of the first up to that time is at least as high as the cumulative payoff of the second (Blavatskyy 2018, p. 364). As the questions were repeated, the analysis here concerns the violation of temporal dominance in the 12 decisions made by each subject in questions 1–6. Figure 6 shows the histogram of violations of first-order temporal dominance in the first six questions in Table 3. We can see that 32 out of 75 of subjects (43%) never violated temporal dominance, and another 14 subjects (19%) did so only once. The remaining 29 subjects (39%) violated temporal dominance two or more times. Overall, first-order temporal dominance was definitely violated more frequently than we expected ex ante (the average violation rate was 13.9%). There appears to be remarkable heterogeneity in our subject population with respect to violations of temporal dominance. On the one hand, as noted above, 32 out of 75 subjects (43%) never violated temporal dominance; on the other hand, six subjects (8%) were responsible for 35% of all violations of first-order temporal dominance. Our subsequent analysis found that most of these subjects behaved as if they were maximizing the cumulative payoff. This simple heuristic (cf. Section 4.1 below) predicts that a decision maker is exactly indifferent in five out of the six questions involving first-order temporal dominance. Subjects who maximize cumulative payoff might then reveal frequent violations of temporal dominance in our experiment.
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Fig. 6 Violations of first-order temporal dominance in questions 1–6
4 The estimated representations of time preferences 4.1 Maximization of cumulative payoff: CP We first of all consider the simple decision rule (heuristic) of maximizing the cumulative payoff across the three time periods (without discounting). In this case, the utility of an alternative {l1, l2, l3} with outcome l1∊{€0, €10, €20} today, outcome l2∊{€0, €10, €20} in one month and outcome l3∊{€0, €10, €20} in two months is given by U ðl 1 ; l 2 ; l 3 Þ ¼ l 1 þ l 2 þ l 3 The criterion of the maximization of the cumulative payoffs per se does not involve any subjective parameters. 4.2 Maximization of present discounted value: PDV We next consider a decision maker who maximizes the present discounted value of the stream of intertemporal (monetary) outcomes. In this case, the utility of {l1, l2, l3} is U ðl 1 ; l 2 ; l 3 Þ ¼ l 1 þ dl2 þ d 2 l 3 where d∊(0,1) is a subjective discount factor. The maximization of present discounted value thus involves one subjective parameter, the discount factor d. Note that the
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maximization of the cumulative payoff is a special case of the maximization of present discounted value with d = 1. 4.3 Discounted-utility theory (or constant discounting): DUT In discounted-utility theory, also known as constant (or exponential) discounting, a choice alternative {l1, l2, l3} yields utility of U ðl 1 ; l 2 ; l 3 Þ ¼ uðl 1 Þ þ duðl 2 Þ þ d 2 uðl 3 Þ where d∊(0,1) is a subjective discount factor and u : {€0, €10, €20} → ℝ is a subjective utility function. The utility function is unique up to a positive affine transformation, i.e. it can be normalized for any two arbitrary outcomes without loss of generality. We here use the normalization u(€0) = 0 and u(€20) = 1. Discounted-utility theory thus involves two subjective parameters: the discount factor d and utility u(€10)∊(0,1). Note that the maximization of present discounted value is a special case of discounted-utility theory with the parameter restriction u(€10) = 0.5, and the maximization of the cumulative payoff is a special case with two parameter restrictions: d = 1 and u(€10) = 0.5. 4.4 Additively-separable utility: ASU The additively-separable utility of a choice alternative {l1, l2, l3} is given by U ðl 1 ; l 2 ; l 3 Þ ¼ uðl 1 Þ þ duðl 2 Þ þ bduðl 3 Þ where d∊(0,1) denotes a subjective discount factor, u : {€0, €10, €20} → ℝ is a subjective utility function (unique up to a positive affine transformation), and b∊(0,1) is another discount factor. Additively-separable utility represents a large class of time preferences including inter alia quasi-hyperbolic discounting (Phelps and Pollak 1968), generalized hyperbolic discounting (Loewenstein and Prelec 1992) and liminal discounting (Pan et al. 2013). We refer to such models as additively-separable utility because their utility function is additively separable across time periods. We normalize the utility function so that u(€0) = 0 and u(€20) = 1. This leaves additively-separable utility with three subjective parameters to be estimated: the discount factors b and d and utility u(€10)∊(0,1). Note that discounted utility is a special case of additively-separable utility with one parameter restriction: b = d. Consequently, the maximization of present discounted value is a special case of additively-separable utility with two parameter restrictions (u(€10) = 0.5 and b = d), and the maximization of cumulative payoff is a special case of additively-separable utility with three parameter restrictions: d = b = 1 and u(€10) = 0.5. 4.5 Rank-dependent discounted utility: RDDU The rank-dependent discounted utility (Blavatskyy 2016) of a choice alternative {l1, l2, l3} is U ðl 1 ; l 2 ; l 3 Þ ¼ uðl 1 Þ þ d ½uðl 1 þ l 2 Þ−uðl 1 Þ þ bd ½uðl 1 þ l 2 þ l 3 Þ−uðl 1 þ l 2 Þ
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where d, b∊(0,1) are subjective discount factors and u : {€0, €10, €20, €30, €40, €50} → ℝ is a subjective utility function that is unique up to a positive affine transformation. We normalize this utility function so that u(€0) = 0 and u(€50) = 1. Rank-dependent discounted utility thus has six subjective parameters: two discount factors b and d and four utilities u(€10), u(€20), u(€30) and u(€40)∊(0,1). Note that the maximization of present discounted value is a special case of rankdependent discounted utility with five parameter restrictions: u(€10) = 0.2, u(€20) = 0.4, u(€30) = 0.6, u(€40) = 0.8 and b = d. Consequently, the maximization of cumulative payoff is a special case of rank-dependent discounted utility with six parameter restrictions: d = b = 1 and u(€10) = 0.2, u(€20) = 0.4, u(€30) = 0.6 and u(€40) = 0.8. Neither discounted-utility theory nor additively-separable utility are special cases of rank-dependent discounted utility.
5 Econometric estimation 5.1 Estimation procedure for the Fechner model of homoscedastic random errors We embed each (deterministic) representation of time preferences from the previous section into the Fechner (1860) model of homoscedastic random errors. This model was applied to the estimation of time preferences by inter alia Chabris et al. (2008, p. 248), Ida and Goto (2009, p. 1174, formula 1), Tanaka et al. (2010, p. 567, Eq. 1) and Toubia et al. (2013, Section 2.1, p. 617). In this model, the left alternative {l1, l2, l3} is chosen over the right alternative {r1, r2, r3} with probability F 0;s ðU ðl 1 ; l 2 ; l 3 Þ−U ðr1 ; r2 ; r3 ÞÞ
ð1Þ
where F0,s(.) denotes the cumulative distribution function of the normal distribution4 with zero mean and standard deviation s, and U(.) is the utility function of the different representations of time preferences (as described in Section 4). The estimation is carried out individually for each subject as follows. We begin by estimating the simplest decision rule—the maximization of cumulative payoffs. We divide all outcomes by 20 so that the lowest outcome remains zero and the highest outcome becomes one (the middle outcome becomes 0.5). We aggregate the decisions from both rounds into one data set. Specifically, let A1 (A2) denote the choice vectors in the first (second) round, with the convention that zero denotes the choice of the left alternative and one the right alternative. The log-likelihood function is then given by: 0 0 0 0 LL ¼ ∑A1 log 1−PL;R þ A2 log 1−PL;R þ ½1−A1 log PL;R þ ½1−A2 log PL;R where PL,R is the vector of probabilities with which left alternatives are chosen over right alternatives (formula (1) describes this probability for one binary-choice problem).
4 The Fechner model of homoscedastic errors does not require F0,s(.) to be the cumulative distribution function of the normal distribution. It could be any monotonically increasing function, e.g., the cumulative distribution function of the logistic distribution, which is known as a softmax choice rule.
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The maximization of cumulative payoffs per se does not involve any subjective parameters. We thus only need to estimate the standard deviation s of the random errors (using 0.5 as a starting value). We estimate this parameter to maximize log-likelihood LL. Second, we estimate the criterion of the maximization of present discounted value embedded into the Fechner model of random errors. This model has two subjective parameters: the discount factor d and the standard deviation of random errors s. We again estimate these two parameters to maximize LL. We impose an automatic penalty LL = −200 when d is below zero or above one (which is equivalent to a constraint search). We use d = 1 and the best-fitting parameter ^s estimated for the criterion of maximization of cumulative payoffs as the starting values in the maximum-likelihood estimation.5 We use a standard likelihood-ratio test to establish whether a less-restrictive model a fits observed choices significantly better than the more-restrictive model b that is obtained from model a by imposing k parameter restrictions. Formally, let LLa (resp. LLb) denote the estimated log-likelihood for model a without any parameter restrictions (resp. for model b when the parameters are restricted). Thus, LLb has k fewer parameters than LLa. Under the hypothesis that the k restrictions are satisfied, the test statistic Λ = 2(LLa − LLb) is distributed chi-squared with k degrees of freedom. We can see whether the maximization of present discounted value fits a particular subject’s choice pattern significantly better than the maximization of cumulative payoff from (for each subject) a standard likelihood-ratio test. If the full model fits significantly better at the 5% significance level (using the critical value of the chi-squared distribution with one degree of freedom), we set the maximization of present discounted value as the current best-fitting model; if not, the best-fitting model is the maximization of cumulative payoff. Third, we estimate the two discounted-utility parameters and the standard deviation of the random errors, s, to maximize LL. We again have a penalty LL = −200 for a discount factor d below zero or above one, and a non-monotonic utility function (i.e. the utility of the middle outcome u(€10) is below zero or above one). We take u(€10) = 0.5 and the best-fitting parameters d^ and ^s estimated for the maximization of the present discounted value criterion as the starting values in the maximum-likelihood estimation. We run a standard likelihood-ratio test (again, for each subject) to see whether discounted-utility theory fits choices significantly better than either the maximization of cumulative payoff or present discounted value (whichever was previously identified as the best-fitting model). If the new model fits significantly better at the 5% significance level,6 we set discounted-utility theory as the best-fitting model; if not, the best-fitting model remains unchanged. Fourth, we estimate the three subjective parameters of additively-separable utility and the standard deviation of random errors, s, to maximize LL. The same penalty LL = −200 applies when the discount factor d is below zero or above one, and for a non5
Using multiple starting values produced the same results as using the best-fitting parameter for the nested model. 6 We use the critical value of the chi-squared distribution with two degrees of freedom in the test against the maximization of cumulative payoff, and with one degree of freedom in that against the maximization of present discounted value.
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^ Þ and ^ uð€10 monotonic utility function. We use b ¼ d^ and the best-fitting parameters d, ^s estimated for discounted utility as the starting values in the maximum-likelihood estimation of additively-separable utility. The likelihood-ratio test shows for each subject whether additively-separable utility fits choices significantly better than the current best-fitting model. If additivelyseparable utility fits significantly better at the 5% significance level,7 we set it as the best-fitting model and if not leave the latter unchanged. Fifth, we estimate the parameters of rank-dependent discounted utility. The same automatic penalty LL = −200 for discount factors d below zero or above one applies. We use b ¼ d^ and the best-fitting parameters d^ and ^s estimated for the maximization of present discount value, as well as u(€10) = 0.2, u(€20) = 0.4, u(€30) = 0.6, u(€40) = 0.8 as the starting values in the maximum-likelihood estimation of rank-dependent discounted utility. As before, we test whether RDDU fits significantly better than the previous models with fewer parameters. As PDV and CP are special cases of RDDU, but DUT and ASU are not, the log-likelihood ratio cannot discriminate among all these models. We thus proceed as follows. If the current best-fitting descriptive model is PDVor CP, which are special cases of RDDU, then we use the standard likelihood-ratio test as above. If RDDU fits significantly better at the 5% significance level8 than PDV or CP, we set RDDU as the best-fitting model and if not the latter remains unchanged (either PDV or CP). However, if the current best-fitting model is DUT or ASU, which are not nested in RDDU, then we use the Vuong likelihood-ratio test for overlapping models (see Vuong 1989; Appendix A.2 in Loomes et al. 2002). We use the Akaike information criterion to penalize the rank-dependent discounted utility for its extra parameters.9 If RDDU fits significantly better at the 5% significance level then we set it as the best-fitting model; if not, the latter remains DUT or ASU. 5.2 Estimation results for the Fechner model of homoscedastic random errors The median values of the individual estimated parameters (across all subjects) in the theories of decision over time, discussed in Section 4, appear in Table 1. We use a signed-rank test to test the null hypotheses that the subjective discount factors, d and b, are equal to one; we also test the null hypothesis that the utility function is linear. Table 1 shows that the subjective discount factor, d, is significantly below one in all of the intertemporal choice theories under consideration. The discount factor, b, is significantly below one in ASU but not in RDDU. Regarding the utility function in the DUT and ASU specifications, the median of the individual estimates of u(10) is not significantly different from 0.5, suggesting that the utility function of the median agent is linear. These results are in line with previous findings in the literature under discounted 7
We use the critical value of the chi-squared distribution with three degrees of freedom in the test against the maximization of cumulative payoff, with two degrees of freedom in that against the maximization of present discounted value, and with one degree of freedom in that against discounted-utility theory. 8 We use the critical value of the chi-squared distribution with six degrees of freedom in the test against maximization of cumulative payoff, and with five degrees of freedom in that against the maximization of present discounted value. 9 The AIC ranks the different models by penalizing the extra parameters of the less-restrictive models. ^ with L ^ being the maximized log-likelihood for a particular model and k the Formally, the AIC is 2k−2lnL, number of model parameters. The lower the value of AIC, the better the model explains the observed behavior.
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Table 1 Median values of the individual estimated parameters Model of inter-temporal choice Maximization of present discounted value
Discountedutility theory
Additivelyseparable utility
Rank-dependent discounted utility
0.899***
0.901***
0.938***
0.878***
0.878***
0.984
0.50
0.10***
Random error Fechner
d b u(10)a
0.50
u(20)b
0.28***
u(30)b
0.54***
u (40) b Luce
d
0.77** 0.893***
0.889***
0.895*** 0.870***
0.882***
0.50
0.50
0.21
b u(10)a u(20)b
0.36***
u(30)b
0.56***
u (40) b Wilcox
0.809***
d
0.77*** 0.897***
0.899***
0.927*** 0.878***
0.900***
0.50
0.50
0.19
b u(10)a
0.905***
u(20)b
0.39
u(30)b
0.60
u (40) b
0.81
a
Signed-rank test of the null hypothesis u(10) = 0.5 for PDV, DU and ASU. For RDDU, we test the null hypothesis u(10) = 0.2. Significance: * = 10%, ** = 5% and *** = 1%
b
Signed-rank test of the null hypotheses u(20) = 0.4, u(30) = 0.6, and u(40) = 0.8
utility (see, for instance, Andreoni and Sprenger 2012). For RDDU, the median estimate suggests a convex utility function.10 We now turn to the question of the best-fitting theory of intertemporal choice. Our ranking analysis of the five models of intertemporal choice shows that for 26 out of 75 subjects (34%) the best-fitting model is the maximization of present discounted value. Another 24 subjects (32%) behave as if they maximize rank-dependent discounted utility. Quite a large group of subjects (15; 20%) behaves as if they use the simple heuristic of maximizing the cumulative payoff (without any discounting). Somewhat surprisingly, the most widely-used representation of time preferences—discountedutility theory—is the best fit for the choices of only eight subjects (11%). Additivelyseparable utility (which includes inter alia a popular model of quasi-hyperbolic
10
When we focus the analysis on the subjects whose preferences are best-fitted by RDDU, this convexity is less pronounced.
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15, 20%
Maximizaon of cumulave payoff Maximizaon of present discounted value
24, 32%
Discounted-ulity theory
26, 34% 2, 3%
8, 11%
Addively-separable ulity Rank-dependent discounted ulity
Fig. 7 Pie chart of the best-fitting representations of time preferences in the subject population (Fechner model of random errors, 5% significance level)
0
2
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8
10
discounting) is the best fit for only two subjects (3%). Figure 7 below shows the pie chart of the distribution of the best-fitting time preferences in our subject population. There is considerable heterogeneity between subjects, even when their preferences reflect the same theory of intertemporal choice. Due to the richness of our data, we cannot show all the individual results here. We thus choose to discuss briefly the individual results for the two best-fitting theories. Figure 8 shows the histogram of the estimated best-fitting (monthly) discount factors for the 26 subjects whose choices were best described by the maximization of the present discounted value. The average estimated discount factor among here is 0.888 (with the highest value being 0.907 and the lowest 0.867). For the 24 subjects whose choices were best described by rankdependent discounted utility, the average estimated discount factor is 0.766. Figure 9 shows the variability in the estimated best-fitting utility functions for DUT and RDDU. Figure 9a. shows the estimated best fitting utility functions from discounted
0.90
0.89
0.88
0.87
0.86
Fig. 8 Number of subjects whose choices are best described by the maximization of present discounted value with their discount factor shown on the horizontal axis (Fechner model of random errors)
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Euros € Euros € 0
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(a) Discounted utility
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Euros € 0
10
20
30
40
50
(b) Rank-Dependent Discounted Utility
Fig. 9 Estimated best-fitting utility functions for subjects whose revealed choices were best described by DUT or RDDU (Fechner model of random errors)
utility theory for those eight subjects whose choices were best described by discounted utility. For four subjects the best fitting utility function is convex and for the other four it is concave. The average estimated utility of €10 among these 8 subjects is 0.4996 (under the normalization u(€0) = 0 and u(€20) = 1). 5.3 The Luce (1959) choice model A probabilistic extension of deterministic representations of time preferences may be consistent with some behavioral patterns that are not consistent with the original theories. For example, Thaler (1981, p. 202) argued that an individual could choose one apple today over two apples tomorrow but, at the same time, prefer two apples in one year and one day to one apple in one year. Discounted-utility theory cannot explain these choices. However, discounted utility embedded in the Fechner model of probabilistic choice can rationalize some such instances of switching (Blavatskyy 2017, section 4, p. 144). Embedding representations of time preferences in the Fechner (1860) model of homoscedastic random errors may then favor theories with constant discounting (such as discounted utility and the maximization of present discounted value and cumulative payoff) at the expense of theories with non-constant discounting (such as additively-separable utility and rank-dependent discounted utility). We thus consider here an alternative model of probabilistic choice proposed by Luce (1959). This model has been used to estimate time preferences by inter alia Andersen et al. (2008, p. 599, Equation 9) and Meier and Sprenger (Meier and Sprenger 2015, p. 276, Eq. 1). In this model a left alternative {l1, l2, l3} is chosen over a right alternative {r1, r2, r3} with probability U ðl 1 ; l 2 ; l 3 Þμ U ðl 1 ; l 2 ; l 3 Þμ þ U ðr1 ; r2 ; r3 Þμ where μ > 0 denotes a noise parameter. When the parameter μ is close to zero, choices are made almost at random (with the probabilities being close to 50%–50%). With large values of μ, the stream with higher utility is almost certainly chosen. The estimation procedure for Luce’s choice model is the same as that already described for the Fechner (1860) model of homoscedastic random errors.
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4, 5%
Maximizaon of cumulave payoff 19, 25%
34, 46%
Maximizaon of present discounted value Discounted-ulity theory
11, 15%
Addively-separable ulity Rank-dependent discounted ulity
7, 9%
Fig. 10 Pie chart of the best-fitting representations of time preferences in the subject population (Luce’s choice model, 5% significance level)
The second part of Table 1 lists the median value of the individual estimated parameters in the theories of intertemporal choice embedded in Luce’s choice model: both the discount factor d and b are significantly below one for all the intertemporalchoice theories considered. We also note that the estimated subjective discount factors are significantly lower under the time-preference models embedded in Luce’s choice model than in the Fechner model of homoscedastic random errors.11 The pie chart in Fig. 10 shows the distribution of the best-fitting representations of time preferences in our subject population with Luce’s choice model. For 34 out of 75 subjects (46%) the best-fitting model is rank-dependent discounted utility, and another quarter of subjects (19) behave as if they maximize present discounted value. Comparing the results from Luce’s choice model to those in the Fechner model of random errors, we see that the fraction of subjects who behave as if they maximize additively-separable utility (cumulative payoff) is markedly higher (lower) with Luce’s choice model. We also observe considerable heterogeneity between subjects regarding the estimated intertemporal-choice parameters. Figure 11 shows the histogram of the estimated bestfitting (monthly) discount factors for the 19 subjects whose choices were best described by the maximization of present discounted value. The average estimated discount factor here is 0.874 (with the highest value being 0.936 and the lowest 0.824). Figure 12 shows the estimated best fitting utility functions for those 34 subjects, whose choices were best described by rank-dependent discounted utility (under the normalization u(€0) = 0 and u(€50) = 1). The best-fitting utility function is convex for five subjects, concave for one subject, and of mixed shape for the great majority (28 subjects). In particular, 18 subjects have an inverted S-shaped utility function (i.e. utility is first convex and then concave). The average estimated utilities are u(€10) = 0.216, u(€20) = 0.345, u(€30) = 0.496 and u(€40) = 0.680. Figure 13 shows the estimated best fitting discount functions12 for these same 34 subjects. The estimated best-fitting discount function is not monotonic for three 11 We use a sign-rank test to test, for each specification of time preferences, the null hypothesis that the subjective discount factor d (b) is the same under the Luce and Fechner models. The sign-rank test rejects the null hypothesis at the 1% level for four comparisons, and at the 5 and 10% levels for one comparison. 12 By convention, the present is not discounted (the discount factor is one). The delay of one month is discounted with a factor d and that of two months with a factor bd.
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0.92
0.90
0.88
0.86
0.84
0.82
Fig. 11 Number of subjects whose choices are best described by the maximization of present discounted value with their discount factor on the horizontal axis (Luce’s choice model)
subjects. However, the discount function of the rank-dependent discounted utility maximizers is convex for almost 62% of subjects (21 out of 34), but concave for 38% (13) of subjects. The concavity of the discount factor (i.e. increasing impatience) has recently been noted in the literature (Abdellaoui et al. 2013; Cavagnaro et al. 2016). The average discount factor is 0.798 for a delay of one month and 0.679 for a delay of two months. 5.4 The Wilcox contextual-utility model with heteroscedastic random errors Wilcox (2008, p. 211, Section 2.5; 2011, p.96, Section 5) proposed a contextual-utility model for choice under risk that, in principle, could also be applied to intertemporal 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
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Fig. 12 Estimated best fitting utility functions for the 34 rank-dependent discounted utility maximizers (Luce’s choice model)
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delay of 1 month (d)
delay of 2 months (b*d)
Fig. 13 Estimated best fitting discount functions for the 34 rank-dependent discounted utility maximizers (Luce’s choice model)
choice. The contextual-utility model of probabilistic choice is effectively a heteroscedastic Fechner model of random errors with the standard deviation of the random errors being proportional to the difference in utilities of the best and worst possible outcomes. The estimation procedure for the Wilcox contextual-utility model used in this section is the same as that for the Fechner (1860) model of random errors, apart from the following: the probability with which the left alternative {l1, l2, l3} is chosen over the right alternative {r1, r2, r3} is now F 0;s
U ðl 1 ; l 2 ; l 3 Þ−U ðr1 ; r2 ; r3 Þ U ðM ; 0; 0Þ−U ðm; 0; 0Þ
where M ≡ max {l1, l2, l3, r1, r2, r3} denotes the best-possible outcome and m ≡ min {l1, l2, l3, r1, r2, r3} the worst-possible outcome in the context of a specific binary-choice problem (i.e. among outcomes l1, l2, l3, r1, r2, r3). The estimation results for the Wilcox contextual-utility model are qualitatively similar to those in the Fechner (1860) model. Figure 14 shows the pie chart of the best-fitting representations of time preferences in our subject population using the Wilcox model. For 26 out of 75 subjects (34%) the best-fitting model is rankdependent discounted utility, the choices of 23 subjects (31%) are most consistent with the maximization of present discounted value, and those of 14 (19%) with the maximization cumulative payoff (without any discounting). The two least-descriptive representations of time preferences are discounted-utility theory and additively-separable utility (that fit best for nine and three subjects respectively). There is again heterogeneity between subjects. Figure 15 shows the histogram of the estimated best-fitting (monthly) discount factors for the 23 subjects whose choices were best described by the maximization of present discounted value. The average estimated discount factor among these 23 subjects is 0.889 (the highest value is 0.906 and the lowest 0.877). The estimated best-fitting utility functions and discount functions for the 26 subjects whose choices were best described by rank-dependent discounted utility appear in Figs.
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14, 19%
Maximizaon of cumulave payoff Maximizaon of present discounted value
26, 34%
Discounted-ulity theory 23, 31% 3, 4%
Addively-separable ulity Rank-dependent discounted ulity
9, 12%
Fig. 14 Pie chart of the best-fitting representations of time preferences in the subject population (Wilcox contextual utility, 5% significance level)
16 and 17. The majority of these subjects have a convex discount function and a mixedshape utility function. 5.5 The relative goodness of fit of the different models of probabilistic choice
0
2
Number of subjects 6 4
8
10
The comparison of the five representations of time preferences using the three models of stochastic choice produces a variety of conclusions. Table 2 shows that RDDU does pretty well whatever the specification of errors. We can also see that DUT and its special case PDV do a good job in explaining the behavior of a significant proportion of subjects. The results in Table 2 also show some differences between the three models of stochastic choice. It is then tempting to try to identify the representation of stochastic choice that best fits intertemporal preferences. We hence compare the goodness of fit of the Fechner, Luce and Wilcox models of stochastic choice. To do this, we use the
0.90
0.89
0.88
0.87
Fig. 15 Number of subjects whose choices are best described by the maximization of present discounted value with their discount factor on the horizontal axis (Wilcox choice model)
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0 0
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Fig. 16 The estimated best-fitting utility functions for the 26 rank-dependent discounted utility maximizers (Wilcox choice model)
Vuong (1989) likelihood-ratio test for strictly non-nested models. For each subject, we test whether the best-fitting representation of time preferences embedded in the homoscedastic Fechner (1860) model fits significantly better (at the 5% significance level) than that embedded in the Luce (1959) choice model. Note that these two best fits may not necessarily be the same. We use the Akaike information criterion to penalize a more general representation of time preferences for its extra parameters whenever the best fits were not the same for different models of probabilistic choice. Analogously, we also test for each subject the best fit in the homoscedastic Fechner (1860) model against that in the Wilcox (2008, 2011) contextual-utility model, and the best fit in the Luce (1959) choice model against that in the Wilcox contextual-utility model. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 present moment
delay of 1 month (d)
delay of 2 months (b*d)
Fig. 17 The estimated best-fitting discount functions for the 26 rank-dependent discounted utility maximizers (Wilcox choice model)
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Table 2 The likelihood-ratio test and Vuong likelihood-ratio tests of the superiority of intertemporal choice models Preference Functional
Specification of error Fechner
Luce
Wilcox
CP
15
4
14
PDV
26
19
23
DUT
8
11
9
AS
2
7
3
24
34
26
RDDU
For 54 subjects (72%), the best-fitting representation of time preferences embedded in the Luce (1959) choice model fitted significantly better than those in the other two models of probabilistic choice. For four subjects (5%), it was the Wilcox contextualutility model that performed the best. For the remaining 17 subjects (23%) none of the three models of probabilistic choice clearly dominated the others. There is thus clear evidence in favor of the Luce (1959) choice model. We can also compare the goodness of fit of the various representations of time preferences in Section 4 using the best-fitting model of probabilistic choice. We identify the best-fitting representation of time preferences using Luce’s choice model (as described in Section 5.3) for the 54 subjects (72%) for whom this model best fit. We carry out an analogous exercise for the four subjects (5%) for whom the Wilcox contextual-utility model was the best fit. For the remaining 17 subjects there is no clear best-fitting model of probabilistic choice. However, for 10 out of these 17 subjects, the best-fitting representation of time preferences turned out to be the same no matter which model of probabilistic choice is used. Only for seven subjects did the best-fitting representation of time preferences depend on the model of stochastic choice. In this case, we cannot pin down a unique best-fitting representation and we identify two best-fitting representations of time preferences.
1, 1%
4, 5%
Maximizaon of cumulave payoff
5, 7%
Maximizaon of present discounted value 17, 23%
38, 51%
9, 12%
1, 1%
Discounted-ulity theory Discounted-ulity theory or addively-separable ulity Rank-dependent discounted ulity Addively-separable ulity or rank-dependent discounted ulity
Fig. 18 Pie chart of the best-fitting time preferences in the subject population (best-fitting model of probabilistic choice, 5% significance level)
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1, 1%
2, 3%
2, 3%
1, 1%
2, 2%
3, 4% CP+Fechner DUT+Fechner 15, 20%
15, 20%
CP+Luce PDV+Luce DUT+Luce ASU+Luce RDDU+Luce PDV+Wilcox
20, 27%
14, 19%
ASU+Wilcox RDDU+Wilcox
Fig. 19 Model selection based on the Akaike information criterion
Figure 18 depicts the best-fitting time preferences conditional on the best-fitting model of stochastic choice. For 38 out of 75 subjects (51%) the best-fitting representation of time preferences is rank-dependent discounted utility, and for another five subjects (7%) rank-dependent discounted utility ties with additively-separable utility. Seventeen out of 75 subjects (23%) behave as if they maximized present discounted value. Thus, rank-dependent discounted utility and its special case—the maximization of present discounted value—remain the two best-fitting representations of time preferences when we impose each subject’s best-fitting econometric model of random errors. The two worst-fitting representations are the maximization of cumulative payoff and additively-separable utility. 5.6 Model selection based on the Akaike information criterion In this sub-section, we compute the Akaike information criterion for each representation of time preferences described in Section 4 embedded into three models of probabilistic choice. Model selection based on the Akaike information criterion alone avoids sequential model comparison and it does not require different tests for nested and non-nested/overlapping models. However, model selection based on the Akaike information criterion alone does not allow us to assess whether the preferred model13 fits the data significantly better than other models at the required significance level. For each of 75 individual choice patterns, we compute the Akaike information criterion of 15 possible combinations of five representations of time preferences described in Section 4 with three models of probabilistic choice (Fechner (1860) model, Luce (1959) choice model and Wilcox (2008, 2011) contextual-utility model). Figure 19 presents the pie chart of the combinations with the lowest value of the Akaike information criterion. For 66 out of 75 subjects (88%) the preferred model of probabilistic choice is the Luce (1959) choice model. For 22 out of 75 subjects (29%) the 13
I.e., the one with the minimum value of the Akaike information criterion.
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preferred representation of time preferences is additively separable utility. For 17 out of 75 subjects (23%) the preferred representation of time preferences is rank-dependent discounted utility. For 16 out of 75 subjects (21%) the preferred representation of time preferences is maximization of present discounted value.
6 Conclusion The empirical violations of constant (exponential) discounting were behind the development of numerous generalizations of discounted-utility theory. This paper has investigated the relative goodness of fit of these representations of time preferences with newly-collected experimental data. We find that rank-dependent expected utility theory and the maximization of present discounted value are the two best fitting representations of time preferences. This result is robust across three different models of probabilistic choice. On the other hand, a rather general class of additively-separable utility representations of time preferences (which includes quasi-hyperbolic and generalized hyperbolic discounting) fits best only for relatively few subjects: between 3 and 9%, depending on the model of probabilistic choice. For a great majority of subjects (72%), the representation of time preferences embedded in Luce’s choice model provides a significantly better fit than those embedded in the Fechner or Wilcox models. These results identify promising avenues for future research on intertemporal choice.
Appendix Table 3 Questions used in the experiment Question
Left option Now (€)
Right option
In one month (€)
In two months (€)
Now (€)
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Table 3 (continued) Question
Left option Now (€)
Right option
In one month (€)
In two months (€)
Now (€)
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Table 3 (continued) Question
Left option
Right option
Now (€)
In one month (€)
In two months (€)
Now (€)
In one month (€)
In two months (€)
55
10
10
20
10
20
56
10
10
20
20
0
0
57
10
10
20
20
0
10
58
10
10
20
20
10
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59
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20
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20
20
60
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20
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10
10
20
61
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20
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20
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62
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20
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20
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63
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20
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64
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20
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20
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65
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20
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20
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66
10
20
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20
67
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68
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20
20
10
0
72
10
20
20
20
10
10
73
10
20
20
20
20
0
74
20
0
0
0
10
20
75
20
0
0
0
20
10
76
20
0
0
0
20
20
77
20
0
0
10
0
20
78
20
0
0
10
10
10
79
20
0
0
10
10
20
80
20
0
0
10
20
0
81
20
0
0
10
20
10
82
20
0
0
10
20
20
83
20
0
10
0
20
20
84
20
0
10
10
10
20
85
20
0
10
10
20
0
86
20
0
10
10
20
10
87
20
0
10
10
20
20
88
20
0
20
10
20
0
89
20
0
20
10
20
10
90
20
0
20
10
20
20
91
20
0
20
20
10
0
92
20
10
0
0
20
20
93
20
10
0
10
10
20
94
20
10
0
10
20
10
0
286
J Risk Uncertain (2018) 56:259–287
Table 3 (continued) Question
Left option Now (€)
In one month (€)
95
20
10
96
20
10
97
20
98
20
Right option In two months (€)
Now (€)
In one month (€)
In two months (€)
0
10
20
20
0
20
0
20
10
10
10
20
20
10
20
20
20
0
99
20
20
0
10
20
20
100
20
20
0
20
10
20
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