i. introduction · fabio maccheroni, yusufcan masatlioglu, jawwad noor, ben polak, al roth, michael...

AXIOMATIZATION AND MEASUREMENT OFQUASI-HYPERBOLIC DISCOUNTING*

Jose Luis Montiel Olea and Tomasz Strzalecki

This article provides an axiomatic characterization of quasi-hyperbolic dis-counting and a more general class of semi-hyperbolic preferences. We imposeconsistency restrictions directly on the intertemporal trade-offs by relying onwhat we call ‘‘annuity compensations.’’ Our axiomatization leads naturally toan experimental design that disentangles discounting from the elasticity ofintertemporal substitution. In a pilot experiment we use the partial identifica-tion approach to estimate bounds for the distributions of discount factors in thesubject pool. Consistent with previous studies, we find evidence for both presentand future bias. JEL Codes: C10, C99, D03, D90.

I. Introduction

Understanding how agents trade off costs and benefits thatoccur at different periods of time is a fundamental issue in eco-nomics. The leading paradigm used for the analysis of intertem-poral choice has been the exponential (or geometric) discountingmodel introduced by Samuelson (1937) and characterized axio-matically by Koopmans (1960).

The two main properties of this utility representation aretime separability and stationarity. Time separability meansthat the marginal rate of substitution between any two time per-iods is independent of the consumption levels in other periods,which rules out habit formation and related phenomena.Stationarity means that the marginal rate of substitutionbetween any two consecutive periods is the same.

The present bias is a well-documented failure of stationaritywhere the marginal rate of substitution between consumption inperiods 0 and 1 is smaller than the marginal rate of substitution

*We thank Gary Charness, David Dillenberger, Armin Falk, DrewFudenberg, Simon Grant, John Horton, David Laibson, Morgan McClellon,Fabio Maccheroni, Yusufcan Masatlioglu, Jawwad Noor, Ben Polak, Al Roth,Michael Richter, and Larry Samuelson, as well as the audiences at Harvard,NYU, and Yale, for their helpful comments. We thank Jose Castillo, MorganMcClellon, and Daniel Pollmann for expert research assistance. The usual dis-claimer applies. Strzalecki gratefully acknowledges support by the SloanFoundation and the NSF grant SES-1123729 and the NSF CAREER grant SES-1255062.

! The Author(s) 2014. Published by Oxford University Press, on behalf of President andFellows of Harvard College. All rights reserved. For Permissions, please email: [email protected] Quarterly Journal of Economics (2014), 1449–1499. doi:10.1093/qje/qju017.Advance Access publication on May 12, 2014.

1449

by guest on September 17, 2014

http://qje.oxfordjournals.org/D

ownloaded from

http://qje.oxfordjournals.org/

between periods 1 and 2. For example, the following preferencepattern is indicative of present bias.

ð1, 0, 0, 0, . . .Þ � ð0, 2, 0, 0, . . .Þð1aÞ

and

ð0, 1, 0, 0, 0, . . .Þ � ð0, 0, 2, 0, 0, . . .Þ,ð1bÞ

where symbols � and � refer to the preference over consumptionstreams expressed at the beginning of time before receiving anypayoffs.

This article is concerned with a very widely applied model ofpresent bias, the quasi-hyperbolic discounting model, which wasfirst applied to study individual choice by Laibson (1997).1 A con-sumption stream (x0, x1, x2, . . . ) is evaluated by

Vðx0, x1, x2, . . .Þ ¼ uðx0Þ þ ��X1t¼1

�t�1uðxtÞ,

where u is the flow utility function, � 2 ð0, 1Þ is the long-run dis-count factor, and � 2 ð0, 1� is the short-run discount factor thatcaptures the strength of the present bias; �= 1 corresponds to thestandard discounted utility model.

Quasi-hyperbolic discounting retains the property of time sep-arability but relaxes stationarity. However, the departure fromstationarity is minimal: stationarity is satisfied from period t = 1onward; this property is referred to as quasi-stationarity. Furtherrelaxations of stationarity have been proposed, for instance, thegeneralized hyperbolic discounting of Loewenstein and Prelec(1992).2 Our approach is not directly applicable to those models;however, both our axiomatization and experimental design extendto a class of semi-hyperbolic preferences, which approximates anytime-separable preference.

Present bias may lead to violations of dynamic consistencywhen choices at later points in time are also part of the analysis;

1. This formalism wasoriginally proposed by Phelps andPollak (1968) to studyintergenerational discounting. See also Zeckhauser and Fels (1968), published asFels and Zeckhauser (2008).

2. Experimental studies (see, e.g., Abdellaoui, Attema, and Bleichrodt 2010;Van der Pol and Cairns 2011) find that generalized hyperbolic discounting fits thedata better than the quasi-hyperbolic model. However, quasi-hyperbolic discount-ing is being used in many economic models, as quasi-stationarity greatly simplifiesthe analysis.

QUARTERLY JOURNAL OF ECONOMICS1450



ownloaded from


this has led to many different ways of modeling dynamic choice.3

Because our results uncover the shape of ‘‘time 0’’ preferenceswithout taking a stance on how they change, they can informany of these models.

I.A. Axiomatic Characterization

The customary method of measuring the strength of the pre-sent bias focuses directly on the trade-off between consumptionlevels in periods 0 and 1; see, for example, Thaler (1981). Thevalue of � can be revealed by varying consumption in period 1to obtain indifference to a fixed level of consumption in time 0.However, this inference relies on parametric assumptions aboutthe utility function u and is subject to many experimental con-founds; see, for example, McClure et al. (2007), Chabris, Laibson,and Schuldt (2008), and Noor (2009, 2011) among others. Hayashi(2003) and Andersen et al. (2008) employ a conceptually relatedmethod that uses probability mixtures to elicit the trade-offs.However, this method relies on the expected utility assumptionand in addition the assumption that risk aversion is inverselyproportional to the elasticity of intertemporal substitution(EIS). The method that our axiomatization is building on usesonly two fixed consumption levels, but instead varies the timehorizon.4 In the quasi-hyperbolic discounting model the subject-ive distance between periods 0 and 1 (measured by ��) is largerthan the subjective distance between periods 1 and 2 (measuredby �), which is the reason behind the preference pattern (1a)–(1b).We uncover the parameter � by increasing the second distanceenough to make it subjectively equal to the former. The delayneeded to exactly match the two distances is directly related tothe value of �. For example, if �= �, then the gap between periods0 and 1 (��) is equal to the gap between periods 1 and 3 (�2). In thiscase, the following preference pattern obtains:

ðx, y, 0, 0, . . .Þ � ðz, w, 0, 0, . . .Þð2aÞ

3. For example, sophistication and naivete (Strotz 1955), partial sophistica-tion (O’Donoghue and Rabin 2001), costly self-control and dual-self models (Thalerand Shefrin 1981; Gul and Pesendorfer 2001, 2004; Fudenberg and Levine 2006).

4. A related but distinct method of standard sequences that relies on continu-ous time was used by Loewenstein and Prelec (1992) and Attema et al. (2010).

QUASI-HYPERBOLIC DISCOUNTING 1451



ownloaded from


if and only if

ð0, x, 0, y, 0, 0, . . .Þ � ð0, z, 0, w, 0, 0, . . .Þ:ð2bÞ

Since we are working in discrete time, for certain values of �there may not exist a corresponding delay that would providean exact compensation. However, it is always possible to compen-sate the agent with an annuity instead of a single payoff.For example, consider the case of �= �+ �2. In this case thesimple 2-annuity provides an exact compensation:

ðx, y, 0, 0, . . .Þ � ðz, w, 0, 0, . . .Þ

if and only if

ð0, x, 0, y, y, 0, . . .Þ � ð0, z, 0, w, w, 0, . . .Þ:

We show that for any � there exists an annuity that provides anexact compensation.

I.B. Experimental Design

Our idea of using annuity compensations to measure impa-tience leads naturally to a new experimental design. Though inmany cases the annuity needed for exact compensation will bevery complicated, we do not insist on point-identifying the valueof �. Instead, we take a simple annuity and delay it appropriatelyuntil the agent switches from ‘‘patient’’ to ‘‘impatient’’ choice. Forinstance, consider the following switch.

ð1, 0, 2, 2, 0, 0, 0, . . .Þ � ð2, 0, 1, 1, 0, 0, 0, . . .Þð3aÞ

and

ð1, 0, 0, 2, 2, 0, 0, . . .Þ � ð2, 0, 0, 1, 1, 0, 0, . . .Þ:ð3bÞ

In comparison (3a), the agent makes the patient choice becausethe annuity compensation (receiving the payoff twice in a row)comes relatively soon. On the other hand, in comparison (3b), theagent makes the impatient choice because the annuity compen-sation is delayed. The more patient the agent, the later sheswitches from patient to impatient choice.

We use a multiple price list (MPL) in which we vary the delay ofthe annuity. The switch point from early to late rewards yields two-sided bounds on � as a function of �. We then use the same method toelicit the value of �. The width of the bounds on these parameterscan be controlled by the length of the annuity. In our pilot study we




ownloaded from


used the simplest 2-annuity. In Section III.B we derive two-sidedbounds on the discount factors � and � given the agent’s switchpoint. In that section and in Appendix II we show how to use theindividual bounds to partially identify the distribution of � and � inthe population. Our results are consistent with the recent experi-mental studies on discounting, though we treat our pilot with somecaution given its online nature and lack of incentives. The partialidentification methodology we develop may be useful to experimen-talists using the MPL paradigm, independently of the particularpreference parameters being studied.

The key aspect of our measurement method is that it disen-tangles discounting (as measured by � and �) from the EIS (asmeasured by u). This is because we are varying the time horizonof rewards instead of varying the rewards themselves (we onlyuse two fixed nonzero rewards). Thus, for any given � the switchpoint is independent of the utility function u. This is important onconceptual grounds, as impatience and EIS are separate prefer-ence parameters. By disentangling these distinct aspects of pref-erences, we provide a direct measurement method focusingpurely on impatience.5

This facilitates comparisons across different rewards. It mayalso be useful in light of a recent debate about fungibility of re-wards (see, e.g., Chabris, Laibson, and Schuldt 2008; Andreoniand Sprenger 2012; Augenblick, Niederle, and Sprenger 2013).It is often argued that observing choices over monetary payoffs isnot helpful in uncovering the true underlying preferences, asthose are defined on consumption, not money. Because moneycan be borrowed and saved, observing choices over payoff streamsis informative about the shape of subjects’ budget sets, but not theshape of their indifference curves. Thus, we should expect differ-ent patterns of choice between monetary and primary rewards.Because our method makes such comparisons easier, we hopethat it can be used to shed some light on this issue.

The rest of the article is organized as follows. Section II pre-sents the axioms and the representation theorems. Section IIIpresents our method of experimental parameter measurement,

5. Recent experimental work has used risk preferences as a proxy for the elas-ticity of intertemporal substitution. However, even though these two parametersare tied together in the standard model of discounted expected utility, they areconceptually distinct (see, e.g., Epstein and Zin 1989), and there are reasons tobelieve they are empirically different, so one may not be a good proxy for the other.




ownloaded from


as well as the results of a pilot study. Section IV extends ourresults to semi-hyperbolic discounting. Appendix I containsproofs and additional theoretical results. Appendix II containsthe details of our partial identification approach. Appendix IIIcontains additional analyses of the data and robustness checks.

II. Axiomatic Characterization

II.A. Preliminaries

Let C be the set of possible consumption levels, formally aconnected and separable topological space. The set C could bemonetary payoffs, but also any other divisible good, such asjuice (McClure et al. 2007), effort (Augenblick, Niederle, andSprenger 2013), or level of noise (Casari and Dragone 2010).Let T :¼ f0, 1, 2, . . .g be the set of time periods. Consumptionstreams are members of CT . A consumption stream x is constantif x = (c, c, . . . ) for some c 2 C. For any c 2 C we slightly abuse thenotation by denoting the corresponding constant stream by cas well. For any a, b, c 2 C and x 2 CT the streams ax, abx, andabcx denote ða, x0, x1, . . .Þ, ða, b, x0, x1, . . .Þ, and (a, b, c, x0, x1, . . . ),respectively.

For any T and x, y define xTy ¼ ðx0, x1, . . . , xT, yTþ1, yTþ2, . . .Þ.A consumption stream x is ultimately constant if x ¼ xTc forsome T and c 2 C. For any T let XT denote the set of ultim-ately constant streams of length T. Any XT is homeomorphicto CTþ1. Consider a preference % defined on a subset F of CT

that contains all ultimately constant streams. This preferencerepresents the choices that the decision maker makes at thebeginning of time before any payoffs are realized. We focus onpreferences that have a quasi-hyperbolic discounting representa-tion over the set of streams with finite discounted utility.

DEFINITION. A preference % on F has a quasi-hyperbolic discount-ing representation if and only if there exists a nonconstant andcontinuous function u : C! R and parameters � 2 ð0, 1� and� 2 ð0, 1Þ such that % is represented by the mapping:

x � uðx0Þ þ �X1t¼1

�tuðxtÞ:

As mentioned before, the parameter � can be thought of as ameasure of the present bias. The parameter � represents the size




ownloaded from


of the subjective distance between periods 0 and 1. As we will see,this parameter has a clear behavioral interpretation in our axiomsystem, and it will become explicit in what sense � is capturingthe subjective distance between periods 0 and 1.

II.B. Axioms

Our axiomatic characterization involves two steps. First, bymodifying the classic axiomatizations of the discounted utilitymodel, we obtain a representation of the form:

x � uðx0Þ þX1t¼1

�tvðxtÞð4Þ

for some nonconstant and continuous u, v : X ! R and 0<�< 1.Second, we impose our main axiom to conclude that v(c) =�u(c) forsome � 2 ð0, 1�.

Our axiomatization of the representation (4) builds on theclassic work of Koopmans (1960, 1972), recently extended byBleichrodt, Rohde, and Wakker (2008). The first axiom isstandard.

AXIOM 1 (WEAK ORDER). % is complete and transitive.

The second axiom, sensitivity, guarantees that preferencesare sensitive to payoffs in periods t = 0 and t = 1 (sensitivity topayoffs in subsequent periods follows from the quasi-stationarityaxiom, to be discussed later). Sensitivity is a very natural require-ment, to be expected of any class of preferences in the environ-ment we are studying.

AXIOM 2 (SENSITIVITY). There exist e, c, c0 2 C and x 2 F such thatcx � c0x and ecx � ec0x.

The third axiom, initial separability, involves conditions thatensure the separability of preferences across time. (These condi-tions are imposed only on the few initial time periods, but extendbeyond them as a consequence of the quasi-stationarity axiom.)Time separability is a necessary consequence of any additive rep-resentation of preferences and is not specific to quasi-hyperbolicdiscounting.

AXIOM 3 (INITIAL SEPARABILITY). For all a, b, c, d, e, e0 2 C and allz, z0 2 F we have

(i) abz � cdz if and only if abz0 � cdz0,




ownloaded from


(ii) eabz � ecdz if and only if eabz0 � ecdz0,(iii) ex � ey if and only if e0x � e0y.

The standard geometric discounting preferences satisfy a re-quirement of stationarity, which says that the trade-offs made atdifferent points in time are resolved in the same way. Formally,stationarity means that cx � cy if and only if x � y for any con-sumption level c 2 C and streams x, y 2 F . However, as discussedin Section I, the requirement of stationarity is not satisfied byquasi-hyperbolic discounting preferences; in fact, it is the viola-tion of stationarity that is often taken to be synonymous withquasi-hyperbolic discounting. Nevertheless, quasi-hyperbolic dis-counting preferences possess strong stationarity-like properties,since the preferences starting from period 1 onward are geometricdiscounting.

AXIOM 4 (QUASI-STATIONARITY). For all e, c 2 C and all x, y 2 F ,ecx � ecy if and only if ex � ey.

The last three axioms, introduced by Bleichrodt, Rohde, andWakker (2008), are used instead of stronger infinite-dimensionalcontinuity requirements. They are of technical nature, as are allcontinuity-like requirements. However, constant equivalenceand tail continuity have simple interpretations in terms ofchoice behavior.

AXIOM 5 (CONSTANT-EQUIVALENCE). For all x 2 F there exists c 2 Csuch that x � c.

AXIOM 6 (FINITE CONTINUITY). For any T, the restriction of � to XT

satisfies continuity, that is, for any x 2 XT the setsfy 2 XT : y � xg and fy 2 XT : y � xg are open.

AXIOM 7 (TAIL-CONTINUITY). For any c 2 C and any x 2 F if x � c,then there exists � such that for all T � �, xTc � c; if x � c,then there exists � such that for all T � �, xTc � c:

THEOREM 1. The preference % satisfies Axioms 1–7 if and only if itis represented by equation (4) for some nonconstant and con-tinuous u, v : X ! R and 0<�< 1.

Note that the representation obtained in Theorem 1 is a gen-eralization of the quasi-hyperbolic model. The main two featuresof this representation are the intertemporal separability ofconsumption and the standard stationary behavior that fol-lows period 1 (captured by the quasi-stationarity axiom).




ownloaded from


The restriction that specifies representation (4) to the quasi-hyperbolic class imposes a strong relationship between the utilityfunctions u and v. Not only do they have to represent the sameordering over the consumption space C, they also must preservethe same cardinal ranking, that is, u and v relate to each otherthrough a positive affine transformation u = �v (the additive con-stant can be omitted without loss of generality). To capture thisrestriction behaviorally, we express it in terms of the willingnessto make trade-offs between time periods.

We now present three different ways of restricting equation(4) to the quasi-hyperbolic model. It is important to observe thatan axiom that requires the preference relation % to exhibit pref-erence pattern (1) is necessary but not sufficient to pin down the�� model: present bias may arise as an immediate consequence ofdifferent preference intensity—as captured by differences in uand v. Therefore, in the context of representation (4), presentbias could be explained without relying on the �� structure. Theadditional axioms that we propose shed light on what it exactlymeans, in terms of consumption behavior, to have different short-term discount factors and a common utility index.

II.C. The Annuity Compensation Axiom

First, we present an axiom that ensures � is larger than half.We impose this requirement to be able to construct a ‘‘future com-pensation scheme’’ that exactly offsets the lengthening of the firsttime period caused by �. If � is less than half, then there will bevalues of � which we cannot compensate for exactly.6

AXIOM 8 (�� 0.5). If ðc, a, a, . . .Þ � ðc, b, b, . . .Þ for some a, b, c 2 X,then

ðc, b, a, a, . . .Þ % ðc, a, b, b . . .Þ:

In the context of representation (4) the long-run patience (�)can be easily measured. Fix two elements a, b 2 C such that a ispreferred to b. Axiom 8 uncovers the strength of patience by get-ting information about the following trade-off. Consider first aconsumption stream that pays a tomorrow and b forever after.Consider now a second consumption stream in which the order ofthe alternatives is reversed. An agent who decides to postpone

6. Since in most calibrations � is close to 1 for any reasonable length of the timeperiod, we view this step as innocuous.




ownloaded from


higher utility (by choosing b first) reveals a certain degree ofpatience. Under representation (4) the patient choice reveals avalue of �� 0.5.

THEOREM 2. Suppose % is as in Theorem 1. It satisfies Axiom 8 ifand only �� 0.5.

As discussed in Section I, our main axiom relies on the idea ofincreasing the distance between future payoffs to compensate forthe lengthening of the time horizon caused by �. For example, if�= �, then the trade-off between periods 0 and 1 is the same as thetrade-off between periods 1 and 3. Similarly, if �= �t, then thetrade-off between periods 0 and 1 is the same as the trade-offbetween periods 1 and t + 2. Because we are working in discretetime, there exist values of � such that �t+1<�<�t for some t, sothat the exact compensation by one payoff is not possible.However, due to time separability, it is possible to compensatethe agent by an annuity. Lemma 1 in the Appendix shows that aslong as �� 0.5, any value of � can be represented by a sum of thepowers of � with coefficients 0 or 1.7 The set M is the collection ofpowers with nonzero coefficients; formally, let M denote a subsetof f2, 3, . . .g � T . We refer to M as an annuity. Our main axiomguarantees that the annuity M is independent of the consumptionlevels used to elicit the trade-offs.

AXIOM 9 (ANNUITY COMPENSATION). There exists an annuity M suchthat for all a, b, c, d, e

a if t ¼ 0b if t ¼ 1

e otherwise

0@

1A �

c if t ¼ 0d if t ¼ 1

e otherwise

0@

1A

if and only if

a if t ¼ 1b if t 2M

e otherwise

0@

1A �

c if t ¼ 1d if t 2M

e otherwise

0@

1A:

The main result of our article is the following theorem.

7. A similar technique was used in repeated games, see, for example, Sorin(1986) and Fudenberg and Maskin (1991). We thank Drew Fudenberg for thesereferences. See also Kochov (2013), who uses results from number theory to cali-brate the discount factor in the geometric discounting model.




ownloaded from


THEOREM 3. A preference % satisfies Axioms 1–9 if and only if hasa quasi-hyperbolic discounting representation with �� 0.5.In this case, � ¼

Pt2M �

t�2.

II.D. Alternative Axioms

The annuity compensation axiom ensures that v is cardinallyequivalent to u. From the formal logic viewpoint, however, thecompensation axiom involves an existential quantifier. This sec-tion complements our analysis by considering two alternate waysof ensuring the cardinal equivalence: a form of the trade-off con-sistency axiom and a form of the independence axiom.

Both axioms need to be complemented with an axiom thatguarantees that �< 1. The following axiom yields just that.

AXIOM 10 (PRESENT BIAS). For any a, b, c, d, e 2 C, a � c

ðe, a, b, e . . .Þ � ðe, c, d, e, . . .Þ¼)ða, b, e, . . .Þ % ðc, d, e, . . .Þ:

This axiom says that if two distant consumption streams areindifferent, one ‘‘impatient’’ (involving a bigger prize at t = 1, fol-lowed by a smaller at t = 2) and one ‘‘patient’’ (involving a smallerprize at t = 1, followed by a bigger at t = 2), then pushing both ofthem forward will skew the preference toward the impatientchoice.

For both approaches, fix a consumption level e 2 C (forexample in the context of monetary prizes, e could be $0). Forany pair of consumption levels a, b 2 C let (a, b) denote the con-sumption stream (a, b, b, b, . . . ).

1. Trade-off Consistency Axiom.

AXIOM 11 (TRADE-OFF CONSISTENCY). For any a, b, c, d, e1, e2 2 C,

If ðb, e2Þ % ða, e1Þ, ðc, e1Þ % ðd, e2Þ, and ðe3, aÞ � ðe4, bÞ, then ðe3, cÞ % ðe4, dÞ;

and

If ðe2, bÞ % ðe1, aÞ, ðe1, cÞ % ðe2, dÞ, and ða, e3Þ � ðb, e4Þ, then ðc, e3Þ % ðd, e4Þ:

The intuition behind the first requirement of axiom is as fol-lows (the second requirement is analogous and ensures that thetime periods are being treated symmetrically). The first premiseis that the ‘‘utility difference’’ between b and a offsets the utilitydifference between e1 and e2. The second premise is that the




ownloaded from


utility difference between e1 and e2 offsets the utility differ-ence between d and c. These two taken together imply that theutility difference between b and a is bigger than the utility dif-ference between d and c. Thus, if the utility difference between e3

and e4 exactly offsets the utility difference between b and a, itmust be big enough to offset the utility difference between dand c.

THEOREM 4. The preference % satisfies Axioms 1–7 and 11 if andonly if there exists a nonconstant and continuous functionu : C! R and parameters �> 0 and � 2 ð0, 1Þ such that % isrepresented by the mapping:


�tuðxtÞ:

Moreover, it satisfies Axiom 10 if and only if � 1, that is, %has the quasi-hyperbolic discounting representation.

2. Independence Axiom. By continuity (Axioms 6 and 7) forany a, b 2 C there exists a consumption level c that satisfies(c, c)� (a, b). Let c(a, b) denote the set of such consumptionlevels. Note that we are not imposing any monotonicity assump-tions on preferences (the set C could be multidimensional), andfor this reason the set c(a, b) may not be a singleton. However,since all of its members are indifferent to each other, it is safe toassume in the expressions below that c(a, b) is an arbitrarilychosen element of that set.

AXIOM 12 (INDEPENDENCE). For any a, a0, a00, b, b0, b00 2 C ifða, bÞ % ða0, b0Þ, then

ðcða, a00Þ, cðb, b00ÞÞ % ðcða0, a00Þ, cðb0, b00ÞÞ

and

ðcða00, aÞ, cðb00, bÞÞ % ðcða00, a0Þ, cðb00, b0ÞÞ:

The intuition behind the first requirement of the axiom is asfollows (the second requirement is analogous and ensures thatthe time periods are being treated symmetrically). For anyða, bÞ, ða00, b00Þ the stream given by ðcða, a00Þ, cðb, b00ÞÞ is a ‘‘subjectivemixture’’ of bets (a, b) and ða00, b00Þ. The axiom requires that if one




ownloaded from


consumption stream is preferred to another, then mixing eachstream with a third stream preserves the preference.8

The next axiom, is a version of Savage’s P3. It ensures thatpreferences in each time period are ordinally the same.

AXIOM 13 (MONOTONICITY). For any a, b, e 2 C, then

b % a()ðb, eÞ % ða, eÞ and ðe, bÞ % ðe, aÞ:

THEOREM 5. The preference % satisfies Axioms 1–7 and 12–13 ifand only if there exists a nonconstant and continuous func-tion u : C! R and parameters �> 0 and � 2 ð0, 1Þ such that%is represented by the mapping:


�tuðxtÞ:


II.E. Related Theoretical Literature

A large part of the theoretical literature on time preferencesuses the choice domain of dated rewards, where preferences aredefined on C T , that is, only one payoff is made. On this domainFishburn and Rubinstein (1982) axiomatized exponential dis-counting. By assuming that T ¼ Rþ, that is, that time is continu-ous, Loewenstein and Prelec (1992) axiomatized a generalizedmodel of hyperbolic discounting, where preferences are repre-sented by Vðx, tÞ ¼ ð1þ �tÞ�

��uðxÞ. Recently, Attema et al. (2010)

generalized this method and obtained an axiomatization of quasi-hyperbolic discounting, among other models.

The foregoing results share a common problem: the domainof dated rewards is not rich enough to enable the measurement ofthe levels of discount factors. Even in the exponential discountingmodel the value of � can be chosen arbitrarily, as long as it belongsto the interval (0, 1), see, for example, Theorem 2 of Fishburn andRubinstein (1982); see also the recent results of Noor (2011).

8. We thank Simon Grant for suggesting this type of axiom. A similar approachalong the lines of Nakamura (1990) is considered in the Appendix.




ownloaded from


The richer domain of consumption streams that we employ in thisarticle allows us to elicit more complex trade-offs between timeperiods and to pin down the value of all discount factors.

The continuous time approach can be problematic for yet an-other reason. It relies on extracting a sequence of time periods ofequal subjective length, a so-called standard sequence.9 Since thetime intervals in a standard sequence are of equal subjectivelength, their objective duration is unequal and has to be un-covered by eliciting indifferences. In contrast, our method usestime intervals of objectively equal length and does not rely onsuch elicitation.

Finally, an axiomatization of quasi-hyperbolic discountingusing a different approach was obtained by Hayashi (2003). Hestudied preferences over an extended domain that includes lot-teries over consumption streams. He used the lottery mixtures tocalibrate the value of �. His axiomatization and measurementrely heavily on the assumption of expected utility, which is re-jected by the bulk of experimental evidence. Moreover, in hismodel the same utility function u measures both risk aversionand the intertemporal elasticity of substitution; however, thesetwo features of preferences are conceptually unrelated (see, e.g.,Kreps and Porteus 1978; Epstein and Zin 1989) and are shown tobe different in empirical calibrations. Another limitation of hispaper is that his axioms are not suggestive of a measurementmethod of the relation between the short-run and long-discountfactor.

III. Experimental Design and a Pilot Study

In this section we use the idea of annuity compensations thatunderlies our axiomatization and provide a preference elicitationdesign. The method provides two-sided bounds for �i and �i foreach subject i. Because there is a natural heterogeneity of pref-erences in the population, we are interested not only in averagevalues, but in the whole distribution. We use these bounds topartially identify the cumulative distribution functions of �i and

9. The standard sequence method was originally applied to eliciting subjectivebeliefs by Ramsey (1931) and later by Luce and Tukey (1964). Interestingly, thesimilarity between beliefs and discounting was already anticipated by Ramsey:‘‘the degree of belief is like a time interval; it has no precise meaning unless wespecify how it is to be measured.’’




ownloaded from


�i in the population. Our method works independently of the util-ity function, so no functional form assumptions have to be madeand no curvatures have to be estimated. We first discuss thedesign and then report results of a pilot experiment.

III.A. Design

The proposed experiment provides a direct test of stationar-ity; moreover, under the assumption that agent i’s preferencesbelong to the quasi-hyperbolic class, our experimental designyields two-sided bounds on the discount factors �i and �i.

10 Thesize of the bounds depends on the choice of the annuity M. We usethe simplest annuity composed of just two consecutive payoffs;however, tighter measurements are possible. The individualbounds are used to partially identify the (marginal) distributionsof preference parameters �i and �i in the population. All thedetails concerning the partial identification of the marginal dis-tributions are provided in Appendix II.

As mentioned before, the experiment does not rely on anyassumptions about the curvature of the utility function ui. Infact, whether the prizes are monetary is immaterial; the only as-sumption that the researcher has to make is that there exist twoprizes a and b, where b is more preferred than a (it doesn’t matterby how much). As a consequence, the experimental design can beused to study how the nature of the prize (e.g., money, effort,consumption good, addictive good) affects impatience, a featurenot shared by experiments based on varying the amount of mon-etary payoff.

The questionnaire consists of two MPLs.11 In each list, everyquestion is a choice between two consumption plans: A (impatientchoice) and B (patient choice); see, for example, Figures I and II.Each option in the first list involves an immediate payoff followedby a two-period annuity that pays off the same outcome in periodst and t + 1; the second list is a repetition of the first with all payoffsdelayed by one period. Under the assumption of quasi-hyperbolicdiscounting, the agent has only one switch point in each list, that

10. In principle, all our axioms are testable, so that assumption could be verifiedas well.

11. MPLs have been used to elicit discount factors for some time now. For ex-ample, Coller and Williams (1999) and Harrison, Lau, and Williams (2002) usethem under the assumption of linear utility and geometric discounting.Andreoni, Kuhn, and Sprenger (2013) use them under the assumption of CRRAutility.




ownloaded from


FIG

UR

EI

Fir

stP

rice

Lis

t




ownloaded from


FIG

UR

EII

Sec

ond

Pri

ceL

ist




ownloaded from


is, she answers B for questions 1, . . . , k and A for questionsk + 1, . . . , n (where n is the total number of questions in the list).12

III.B. Parameter Bounds

Since the second list does not involve immediate payoffs, theobserved switch point in this list (denoted si,2) yields bounds onthe discount factor �i. For example, suppose that in the list de-picted in Figure II subject i chose B in the first five questions andA in all subsequent questions, so that si,2 = 6. Then,

�i�iuið1Þ þ �i�25i uið2Þ þ �i�

26i uið2Þ � �i�iuið2Þ þ �i�

25i uið1Þ þ �i�

26i uið1Þ

�i�iuið1Þ þ �i�37i uið2Þ þ �i�

38i uið2Þ �i�iuið2Þ þ �i�


38i uið1Þ,

where u(2) is the utility of two ice cream cones and u(1) isthe utility of one cone. If u(2)>u(1) this is equivalent to�36

i þ �37i 1 �24

i þ �25i , so approximately

0:972 �i 0:981:

Therefore, the probability of the event fi j si, 2 ¼ 6g provides alower bound for the probability of the event fi j0:972 �i 0:981g. Appendix II derives upper and lower bounds for themarginal distribution of �i based on the switch point si,2.

Note that if the switch points in the first and second list aredifferent, stationarity is violated and we obtain bounds on �i. Forexample, suppose that in the first price list the subject answeredB in the first three questions and A in all subsequent questions, sothat si,1 = 4. We have

uið1Þ þ �i�6i uið2Þ þ �i�

7i uið2Þ � uið2Þ þ �i�


7i uið1Þ

uið1Þ þ �i�12i uið2Þ þ �i�

13i uið2Þ uið2Þ þ �i�


13i uið1Þ,

or equivalently, si,1 = 4 implies

1

�6i þ �

7i

�i 1

�12i þ �

13i

,

so using the bounds for �i just derived from the second list weconclude that si,1 = 4 and si,2 = 6 imply

0:565 �i 0:712:

12. In fact, the switch point is unique under any time-separable model a laRamsey (1931) with a representation

P1t¼0 DtuðctÞ, where Dtþ1 < Dt, for example

the generalized hyperbolic discounting model of Loewenstein and Prelec (1992).




ownloaded from


Appendix II derives upper and lower bounds for the marginaldistribution of �i based on the switch points si,1 and si,2.

III.C. Implementation of the Pilot Experiment

To illustrate our design, we implemented a pilot study usingan online platform and hypothetical rewards. Though compara-tive studies show that there tends to be little difference betweenchoices with hypothetical and real consequences in discountingtasks (Johnson and Bickel 2002) and that online markets providegood quality data and replicate many lab studies (Horton, Rand,and Zeckhauser 2011), we treat our results with caution andthink of this study as a proof of concept before a thorough incen-tivized laboratory or field experiment can be implemented.13 Weuse two kinds of hypothetical rewards: money and ice cream. Wehave a total of 1,277 participants each with a unique IP address;639 subjects answered the money questionnaire and 640 the icecream questionnaire (548 participants answered both).

The experiment was conducted using Amazon’s MechanicalTurk (AMT), an online labor market. Immediate and convenientaccess to a large and diverse subject pool is usually emphasized asone of the main advantages of the online environment; see, forexample, Mason and Suri (2012). One of the common concernsoften raised by online experiments is that both low wages and thelack of face-to-face detailed instructions to participants mightlead to low-quality answers. However, Mason and Watts (2010),Mason and Suri (2012), and Marge, Banerjee, and Rudnicky(2010) present evidence of little to no effect of wage on the qualityof answers, at least for some kinds of tasks. In our study we paid$5 per completed questionnaire. The average duration of eachquestionnaire was five minutes. Hence, we paid approximately$60 per hour: a significantly larger reward than the reservationwage of $1.38 per hour reported in Mason and Suri (2012) forAMT workers.

The lack of face-to-face detailed instructions is oftenaddressed by creating additional questions to verify subjects’understanding of the experiment (Paolacci, Chandler, andIpeirotis 2010). To address these concerns, we have two questionsat the beginning of the questionnaire that check participants’

13. Hypothetical rewards may offer some benefits compared to real rewardsbecause they eliminate the need for using front-end delays so the ‘‘presentmoment’’ in the lab is indeed present.




ownloaded from


understanding. Out of the 638 (639) participants in the moneytreatment, a subsample of 502 (503) subjects was selected basedon ‘‘monotonicity’’ and ‘‘understanding’’ initial checks; see theOnline Appendix.

We also perform two additional robustness checks: we studyresponse times and we vary worker qualifications. These exer-cises are described in the Online Appendix.

An important consideration when using the MPL paradigmare multiple switch points. As noted in Section III.A, any agentwith a time-separable impatient preference has a unique switchpoint. Three hundred thirty-six out of the 502 subjects in themoney treatment and 444 out of the 503 subjects in the icecream treatment have a unique switch point. We focus only onthose subjects, disregarding the multiple switchers.

We note that there is an important share of ‘‘never switchers’’in our sample; that is, subjects who always chose the patient (orimpatient) prospect in both price lists. Since never switchers arecompatible with both �i 1 and �i� 1, they directly affect thewidth of our bounds for the cumulative distribution function(c.d.f.) of �. We did not disregard never switchers, as we haveno principled way of doing so: their response times were not sig-nificantly faster than those of the subjects that exhibited a switchpoint and the fraction of such subjects was independent of theworker qualifications (for details see the Online Appendix).In small-scale pilot tests with shorter time horizons, even moresubjects were never switching, which is what prompted us to uselonger time horizons.14 We are hopeful that the number of neverswitchers will decrease in the lab and/or with real incentives,which would allow for more practical time horizons.15

III.D. Results of the Experiment

As discussed in Section III.B, for each such subject, we obtaintwo-sided bounds on �i; we use these bounds to partially identifythe distribution of � in the population. To represent the aggre-gate distribution of � in our subject population we graph two

14. Dohmen et al.’s (2012) experiment shows that the elicited preferences candepend on the time horizon. The dependence can be so strong that it leads tointransitives.

15. However, we note that similar behavior was obtained in the lab with realincentives by Andreoni and Sprenger (2012), where in a convex time budget taskroughly 70% of responses were corner solutions and 37% of subjects never choseinterior solutions.




ownloaded from

http://qje.oxfordjournals.org/lookup/suppl/doi:10.1093/qje/qju017/-/DC1




nondecreasing functions, each corresponding to one of the ends ofthe interval. The true c.d.f. must lie in between them. Figure IIIpresents the c.d.f. bounds for the two treatments; the true c.d.f.must lie in the gray area between the dashed line (upper bound)and the solid line (lower bound).

We now turn to �. As discussed in Section III.B, for eachsubject we obtain two-sided bounds on �i using his answers inthe first price list and bounds on his �i obtained above. We usethe same method of aggregating these bounds as above. Figure IVpresents the c.d.f. bounds for the two treatments; once again, thetrue c.d.f. must lie in the gray area. We reiterate that obtainingtighter bounds on the distribution of � is possible by using annu-ity compensation schemes longer than the simple two-periodannuity that we adopted here for simplicity.

The distribution of parameter values seems consistent withresults in the literature. The next section makes detailed com-parisons. A noticeable feature of the data is the high proportion ofsubjects with �> 1, that is, displaying a future bias. This has beendocumented by other researches as well; for example, Read(2001), Gigliotti and Sopher (2003), Scholten and Read (2006),Sayman and Onculer (2009), Attema et al. (2010), Cohen,Tallon, and Vergnaud (2011), Takeuchi (2011), Andreoni andSprenger (2012), and Halevy (2012).

III.E. Relation to the Experimental Literature

There is a large body of research on estimation of time pref-erences using laboratory experiments. The picture that seems toemerge is that little present bias is observed in studies usingmoney as rewards, whereas it emerges strongly in studies usingprimary rewards. For example, Andreoni and Sprenger (2012)introduce the convex time budget procedure to jointly estimatethe parameters of the �-� model with constant relative risk aver-sion (CRRA) utility. They find averages values of � between 0.74and 0.8, and only 16.7% of their subjects exhibit diminishing im-patience. The null hypothesis of exponential discounting, �= 1, isrejected against the one-sided alternative of future bias, �>1.Andreoni, Kuhn, and Sprenger (2013) compare the convex timebudget procedure and what they call dual marginal price lists inthe context of the CRRA discounted utility model. Even thoughthey find substantial difference in curvature estimates arisingfrom the two methodologies, they find similar time preference




ownloaded from


0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

F(δ

)

Set Estimators

Upper BoundLower BoundIdentified Set

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

F(δ

)

Set Estimators


(a)

(b)

FIGURE III

Bounds for the c.d.f. of �: a) Money b) Ice cream




ownloaded from


0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β

)

Bounds for the c.d.f. of β:Set Estimators


0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β

)Bounds for the c.d.f. of β:

Set Estimators


(a)

(b)

FIGURE IV

Bounds for the c.d.f. of �: a) Money b) Ice cream




ownloaded from


parameters. The reported estimates of yearly � are around 0.7.They again find very little evidence of quasi-hyperbolic discount-ing. Using risk aversion as proxy for the EIS Andersen et al.(2008) find that 72% of their subjects are exponential and 28%are hyperbolic.

Another line of work relies on a parameter-free measurementof utility. Using hypothetical rewards and allowing for differen-tial discounting of gains and losses Abdellaoui, Attema, andBleichrodt (2010) show that generalized hyperbolic discountingfits the data better than exponential discounting and quasi-hyperbolic discounting, where the median values of � are closeto 1. In an innovative experiment Halevy (2012) elicits dynamicchoices to study the present bias, as well as time consistency andtime invariance of preferences. Since we only focus on time 0preferences, only his results on the present bias are relevant tous. He finds that 60% of his subjects have stationary preferences,17% display present bias, and 23% display future bias.

On the other hand, the present bias is strong in studies usingprimary rewards. For example, McClure et al. (2007) use fruitjuice and water as rewards and find that on average �& 0.52.Augenblick, Niederle, and Sprenger (2013) compare preferenceover monetary rewards and effort. Using parametric specifica-tions for both utility functions, they show little present biasfor money, but existing present bias for effort: they find thatfor money the average �& 0.98 but ranges between 0.87 and0.9 for effort (depending on the task). Using health outcomes asrewards, Van der Pol and Cairns (2011) find significant violationsof stationarity (however, their results point in the directionof generalized hyperbolic, rather than quasi-hyperbolicdiscounting).

Turning to our experiment, the results of our money treat-ment are consistent with those already mentioned, that is, thepresent bias is not prevalent: at least 10% of subjects have �< 1and at least 30% of subjects have �>1. Our second treatmentused a primary reward—ice cream—in the hope of obtaining adifferential effect. However, the effect is weak: at least 10% ofsubjects have �< 1 and at least 10% of subjects have �> 1. Thisis consistent with the average � being lower for primary rewards.A possible explanation of the weakness of the effect is that hypo-thetical rewards may lead subjects to conceptualize money andice cream similarly. A larger difference would more likely be seenin a study using real incentives.




ownloaded from


IV. Semi-hyperbolic Preferences

As mentioned earlier, other models of the present bias relaxstationarity beyond the first time period. The most general modelthat maintains time separability is one where

Vðx0, x1, . . .Þ ¼X1t¼0

DtuðxtÞ,

where 1 ¼ D0 > D1 > � � � > 0. For these preferences to be definedon constant consumption streams the condition

P1t¼0 Dt <1 has

to be satisfied. We call this class time-separable preferences (TSP).An example of TSP is the generalized hyperbolic discountingmodel of Loewenstein and Prelec (1992) where Dt ¼ ð1þ �tÞ�

��

and � > �.Consider the subclass of semi-hyperbolic preferences, where

D1, . . . , DT are unrestricted and for some � 2 ð0, 1Þ, Dtþ1

Dt¼ � for

all t>T. This class does not impose any restrictions on thediscount factors for a finite time horizon and assumes thatthey are exponential thereafter. Notice that if the time horizonis finite, this implies that semi-hyperbolic preferences coincidewith TSP. We now show that with infinite horizon semi-hyper-bolic preferences approximate any TSP for bounded consumptionstreams. We say that a stream x ¼ ðx0, x1, . . .Þ is bounded when-ever there exist c, c 2 C such that c �� xt �

� c for all t. The restric-tion to bounded plans may be a problem in models whereeconomic growth is unbounded, but seems realistic in experimen-tal settings.

THEOREM 6. For any V that belongs to the TSP class there exists asequence Vn of semi-hyperbolic preferences such thatVnðxÞ ! VðxÞ for all bounded x. Moreover, the convergenceis uniform on any set of equi-bounded consumption streams.Furthermore, this implies that: (i) if x %n y for all n suffi-ciently large, then x %y and (ii) if x � y then for all n largeenough x �n y.

To extend our axiomatization to semi-hyperbolic prefer-ences, quasi-stationarity, initial separability, and annuity com-pensation need to be modified. Quasi-stationarity needs to berelaxed to hold starting from period T. Initial separabilityneeds to be be imposed for periods t ¼ 0, 1, . . . T instead ofjust 0, 1, 2 (this property was implied by initial separability




ownloaded from


together with quasi-stationarity, but the latter axiom is nowweaker, so it has to be assumed directly). Annuity compensa-tion becomes:

AXIOM 14 (EXTENDED ANNUITY COMPENSATION). For each � ¼ 0,1, . . . T there exists an annuity M such that for all a, b, c, d, e

a if t ¼ �b if t ¼ � þ 1e otherwise

0@

1A �

c if t ¼ �d if t ¼ � þ 1e otherwise

0@

1A

if and only if

a if t ¼ T þ 1b if t 2Me otherwise

0@

1A �

c if t ¼ T þ 1d if t 2Me otherwise

0@

1A:

Finally, to understand how to extend our experimentaldesign to semi-hyperbolic preferences, consider the following gen-eralization of quasi-hyperbolic discounting, the �-�-� preferences,where

Vðx0, x1, . . .Þ ¼ uðx0Þ þ ��

�uðx1Þ þ ��

X1t¼2

�t�2uðxtÞ

�:

The elicitation of � is from an MPL like in Figure II, where thefirst payoff is in two years instead of one year. The elicitation of �is from an MPL like in Figure II. The elicitation of � is from anMPL like in Figure I. The practicality of this approach depends onhow well the semi-hyperbolic preferences approximate theobserved preferences for reasonable time horizons. This is anempirical question beyond the scope of this article.

V. Conclusion

This article axiomatizes the class of quasi-hyperbolic dis-counting and provides a measurement technique to elicit thepreference parameters. Both methods extend to what we callsemi-hyperbolic preferences. Both methods are applications ofthe same basic idea: calibrating the discount factors using annu-ities. In the axiomatization we are looking for an exact compen-sation, whereas in the experiment we use a multiple price list toget two-sided bounds. The advantage of this method is that it




ownloaded from


disentangles discounting from the EIS and hence facilitates com-parisons of impatience across rewards. To illustrate our experi-mental design we run an online pilot experiment using the �-�model. We show how to partially identify the distribution ofdiscount factors in the population.

Appendix I: Proofs

Proof of Theorem 1

Necessity of the axioms is straightforward. For sufficiency,we follow a sequence of steps.

Step 1. The initial separability axiom guarantees that the setsf0, 1g, f1, 2g, and f1, 2, . . . , g are independent. To show that for allt ¼ 2, . . . the sets ft, tþ 1g are independent fix x, y, z, z0 2 F andsuppose that

ðz0, z1, . . . , zt�1, xt, xtþ1, ztþ1, . . .Þ � ðz0, z1, . . . , zt�1, yt, ytþ1, ztþ1, . . .Þ:

Apply quasi-stationarity t – 1 times to obtain

ðz0, xt, xtþ1, ztþ1, . . .Þ � ðz0, yt, ytþ1, ztþ1, . . .Þ:

By part (ii) of initial separability, conclude that

ðz0, xt, xtþ1, z0tþ1, . . .Þ � ðz0, yt, ytþ1, z0tþ1, . . .Þ:

By part (iii) of initial separability, conclude that

ðz00, xt, xtþ1, z0tþ1, . . .Þ � ðz00, yt, ytþ1, z0tþ1, . . .Þ:

Apply quasi-stationarity t – 1 times to obtain

ðz00, z01, . . . , z0t�1, xt, xtþ1, z0tþ1, . . .Þ � ðz00, z01, . . . , z0t�1, yt, ytþ1, z0tþ1, . . .Þ:

The proof of the independence of ft, tþ 1, . . .g for t ¼ 2, . . . isanalogous.

Step 2. Show that any period t is sensitive. To see that,observe that by sensitivity of the period t = 1 there exists x 2 Fand c, c0 2 C such that

ðx0, c, xtþ1, xtþ2, . . .Þ � ðx0, c0, xtþ1, xtþ2, . . .Þ:




ownloaded from


By quasi-stationarity, applied t – 1 times conclude that

ðx0, x1, . . . , xt�1, c, xtþ1, xtþ2, . . .Þ � ðx0, x1, . . . , xt�1, c0, xtþ1, xtþ2, . . .Þ:

Step 3. Additive representation on XT. Fix T � 1 and fix e 2 C.Weak order, finite continuity, and steps 1 and 2 imply that (bytheorem 1 of Gorman 1968, together with Vind 1971) the restric-tion of � to XT is represented by

ðx0, x1, . . . , xT, c, c, . . .Þ�XT

t¼0

vt, TðxtÞ þ RTðcÞ;

for some nonconstant and continuous maps vt,T and RT from Cto R. By the uniqueness of additive representations, the abovefunctions can be chosen to satisfy

vt, TðeÞ ¼ RTðeÞ ¼ 0:ð5Þ

Step 4. Since any XT � XTþ1, there are two additive represen-tations of � on XT:

ðx0, x1, . . . , xT , c, c, . . .Þ�XT

t¼0

vt, TðxtÞ þRTðcÞ

and

ðx0, x1, . . . , xT , c, c, . . .Þ�XT

t¼0

vt, Tþ1ðxtÞ þ vTþ1, Tþ1ðxtÞ þRTþ1ðcÞ:

By the uniqueness of additive representations and thenormalization (5), the above functions must satisfy vt, Tþ1ðcÞ ¼�Tþ1vt, TðcÞ for t ¼ 0, 1, . . . , T � 1 and vTþ1, Tþ1ðcÞ þ RTþ1ðcÞ ¼�Tþ1RTðcÞ for some �Tþ1 > 0. By the uniqueness of additive repre-sentations the representations can be normalized so that�Tþ1 ¼ 1. Let vt denote the common function vt, T. With this nota-tion, we obtain

vTþ1ðcÞ þ RTþ1ðcÞ ¼ RTðcÞ:ð6Þ

Step 5. By quasi-stationarity, for any T � 1 the two additiverepresentations of � on XT:

ðe, x0, x1, . . . , xT�1, c, c, . . .Þ� v0ðeÞ þXT

t¼1

vtðxt�1Þ þ RTðcÞ




ownloaded from


and

ðe, x0, x1, . . . , xT�1, c, c, . . .Þ� v0ðeÞ þXT

t¼1

vtþ1ðxt�1Þ þRTþ1ðcÞ;

represent the same preference. By the uniqueness of additiverepresentations, and the normalization (5), there exists �T > 0such that for all t ¼ 1, 2, . . . , vtþ1ðcÞ ¼ �TvtðcÞ for all c 2 C andRTþ1ðcÞ ¼ �TRTðcÞ. Note, that �T is independent of T, sincethe functions v and R are independent of T; let � denote thiscommon value.

Step 6. Define u :¼ v0, v :¼ ��1v1 and R :¼ ��2R1. With thisnotation, equation (6) is �Tþ1vðcÞ þ �Tþ2RðcÞ ¼ �Tþ1RðcÞ for allc 2 C. Observe that � ¼ 1 implies that v is a constant func-tion, which is a contradiction; hence, � 6¼ 1. Thus,RðcÞ ¼ 1

1�� vðcÞ for all c 2 C. Thus, the preference on XT isrepresented by

ðx0, x1, . . . , xT, c, c, . . .Þ� uðx0Þ þXT

t¼1

�tvðxtÞ þ�Tþ1

1� �vðcÞ:

To rule out � > 1 note that because v is nonconstant, there exista, b 2 C such that vðaÞ > vðbÞ. Then, since �þ �2

1�� < 0 it followsthat uðaÞ þ �vðbÞ þ �2

1�� vðbÞ > uðaÞ þ �vðaÞ þ �2

1�� vðaÞ, so eb � a.However, by tail continuity there exists T such that ðebÞTa � a,which implies that

uðaÞ þ ð�þ � � � þ �TÞvðbÞ þ�Tþ1

1� �vðaÞ

> uðaÞ þ ð�þ � � � þ �TÞvðaÞ þ�Tþ1

1� �vðaÞ:

Thus, ð�þ � � � þ �TÞðvðbÞ � vðaÞÞ > 0, which contradictsvðaÞ > vðbÞ and � > 0. Thus, � < 1 and U(x) represents � on XT

for any T.

Step 7. Fix x 2 F . By constant equivalence, there existsc 2 C with x � c. Suppose there exists a 2 C such that c � a.Then by tail continuity there exists � such that for all




ownloaded from


T � �, xTa � a, which by step 6 implies that UðxTaÞ > UðaÞ. Thisimplies that

9�8T��uðx0Þ þXT

t¼1


1� �vðaÞ > uðaÞ þ

XT

t¼1

�tvðaÞ þ�Tþ1

1� �vðaÞ

9�8T��

XT

t¼1

½�tvðxtÞ � �tvðaÞ� > ½uðaÞ � uðx0Þ�

9� infT��

XT

t¼1

½�tvðxtÞ � �tvðaÞ� � ½uðaÞ � uðx0Þ�

sup�

infT��

XT

t¼1

½�tvðxtÞ � �tvðaÞ� � ½uðaÞ � uðx0Þ�,

which means that lim infTPT

t¼1þ½�tvðxtÞ � �

tvðaÞ� � ½uðaÞ � uðx0Þ�.Because the sequence

PTt¼1 �

tvðaÞ converges, it follows that

uðx0Þ þ lim infT

XT

t¼1

�tvðxtÞ � uðaÞ þ limT

XT

t¼1

�tvðaÞ ¼ UðaÞ:

Since this is true for all a � c, by connectedness of C and conti-nuity of u and v it follows that

uðx0Þ þ lim infT

XT

t¼1

�tvðxtÞ � UðcÞ:ð7Þ

On the other hand, suppose that a % c for all a 2 C. Then, byconstant equivalence for all T there exists b 2 C such thatxTc � b. This implies that xTc % c. Thus,

8Tuðx0Þ þXT

t¼1


1� �vðcÞ � uðcÞ þ

XT

t¼1

�tvðcÞ þ�Tþ1

1� �vðcÞ

8T

XT

t¼1

�tvðxtÞ �XT

t¼1

�tvðcÞ � uðcÞ � uðx0Þ

lim infT

XT

t¼1

�tvðxtÞ �XT

t¼1

�tvðcÞ � uðcÞ � uðx0Þ:

Since the sequencePT

t¼1 �tvðcÞ converges, equation (7) follows.




ownloaded from


An analogous argument implies that lim supTPTt¼0 �

tvðxtÞ UðcÞ, which establishes the existence of the limitof the partial sums and the representation. «

Proof of Theorem 2

We have

ðe, b, a, . . .Þ % ðe, a, b, . . .Þ

iff

uðeÞ þ �vðbÞ þ�2

1� �vðaÞ � uðeÞ þ �vðaÞ þ

�2

1� �vðbÞ

iff

vðbÞ þ�

1� �vðaÞ � vðaÞ þ

�

1� �vðbÞ

iff

½vðbÞ � vðaÞ�1� 2�

1� �� 0

iff

1� 2� 0 «

Proof of Theorem 3

The following lemma is key in the proof of Theorem 3.

LEMMA 1. For any � 2 ½0:5, 1� and any � 2 ð0, 1� there exists asequence f�tgt of elements in {0, 1} such that � ¼

P1t¼0 �t�

t.

Proof. Let d0 :¼ 0 and �0 :¼ 0 and define the sequences fdtg

and f�tg by

dtþ1 :¼dt þ �

tþ1 if dt þ �tþ1 �

dt otherwise:

�

and

�tþ1 :¼1 if dt þ �

tþ1 �0 otherwise:

�




ownloaded from


Since the sequence fdng is increasing and bounded from aboveby �, it must converge; let d :¼ lim dt. It follows thatd ¼

P1t¼0 �t�

t. Suppose that d < �. It follows that �t ¼ 1 foralmost all t; otherwise there would exist arbitrarily large t with�t ¼ 0, and since �t < �� d for some such t that would contradictthe construction of the sequence fdtg. Let T :¼ maxft : �t ¼ 0g. We

have d ¼ dT�1 þ�Tþ1

1�� . Since � � 0:5, it follows that �T �Tþ1

1��,

so dT�1 þ �T �, which contradicts the construction of the

sequence fdtg. «

The necessity of Axioms 1–9 follows from Theorems 1 and 2 andLemma 1. Suppose that Axioms 1–9 hold. By Theorems 1 and 2the preference is represented by equation (4) with � � 0:5.Normalize u and v so that there exists e 2 C withuðeÞ ¼ vðeÞ ¼ 0. Let M be as in Axiom 9. Define � :¼

Pt2M �

t�1.Axiom 9 implies that for all a, b, c, d 2 C

uðaÞ þ �vðbÞ > uðcÞ þ �vðdÞ

if and only if

vðaÞ þ �vðbÞ > vðcÞ þ �vðdÞ:

By the uniqueness of the additive representations, thereexists � > 0 and �1, �2 2 R such that vðeÞ ¼ �uðeÞ þ �1 and�vðeÞ ¼ ��vðeÞ þ �2 for all e 2 C. By the abovenormalization, �1 ¼ �2 ¼ 0. Hence, vðeÞ ¼ �uðeÞ for all e 2 C and� ¼

Pt2M �

t�2. «

Proof of Theorem 4

The necessity of Axioms 1–7 and 10 is straightforward.For Axiom 11, if ðb, e2Þ %ða, e1Þ, ðc, e1Þ %ðd, e2Þ and ðe3, aÞ � ðe4, bÞ,it follows that:

uðbÞ þ�

1� ��uðe2Þ � uðaÞ þ

�

1� ��uðe1Þð8Þ

uðcÞ þ�

1� ��uðe1Þ � uðdÞ þ

�

1� ��uðe2Þð9Þ

uðe3Þ þ�

1� ��uðaÞ ¼ uðe4Þ þ

�

1� ��uðbÞ:ð10Þ




ownloaded from


Equations (8)–(9) imply uðbÞ � uðaÞ � uðdÞ � uðcÞ. Suppose thatthe implication of Axiom 11 does not hold, so thatðe4, dÞ � ðe3, cÞ. Then

uðe4Þ þ�

1� ��uðdÞ > uðe3Þ þ

�

1� ��uðcÞ:ð11Þ

Since 0 < �, 0 < � < 1, equations (10)–(11) imply uðdÞ � uðcÞ >uðbÞ � uðaÞ. A contradiction. By analogy, the second condition ofAxiom 11 is also necessary. Therefore, Axiom 11 is satisfied by therepresentation in Theorem 4.

Now we prove sufficiency. From Theorem 1 it follows that %admits the representation in equation (4). Define the binary rela-tion %� over the elements of C2 as follows:

ðb, cÞ %� ða, dÞ

() there exists e1, e2, e3, e4 2 C such that

ðb, e2Þ % ða, e1Þ and ðc, e1Þ % ðd, e2Þ and ðe3, aÞ � ðe4, bÞ:ð12Þ

We break the proof of sufficiency into four steps.

Step 1. First, we argue that %� admits the following additiverepresentation:

ðb, cÞ %� ða, dÞ () uðbÞ þ uðcÞ � uðaÞ þ uðdÞ

Using the definition of %� and the representation (4) of %, itfollows that ðb, cÞ %�ða, dÞ implies the existence of elementse1, e2 2 C such that:

uðaÞ þ�

1� �vðe1Þ uðbÞ þ

�

1� �vðe2Þ

and

uðdÞ þ�

1� �vðe2Þ uðcÞ þ

�

1� �vðe1Þ:

Therefore uðbÞ þ uðcÞ � uðaÞ þ uðdÞ.Now, suppose uðbÞ þ uðcÞ � uðaÞ þ uðdÞ. We consider the

following six cases and we show that Condition 12 is satisfied.

(i) uðbÞ � uðaÞ, uðcÞ � uðdÞ, vðaÞ � vðbÞ: Set e ¼ e1 ¼ e2 for anye 2 C, and choose e3, e4 to satisfy uðe3Þ þ

�1�� vðaÞ ¼

uðe4Þ þ�

1�� vðbÞ. Then, Condition (12) is satisfied.




ownloaded from


(ii) uðbÞ � uðaÞ, uðcÞ � uðdÞ, vðaÞ < vðbÞ: Set e ¼ e1 ¼ e2 forany e 2 C and choose e3, e4 to have uðe3Þ þ

�1�� vðaÞ ¼

uðe4Þ þ�

1�� vðbÞ. Again, condition (12) is satisfied andðb, cÞ %� ða, dÞ.

(iii) uðbÞ � uðaÞ, uðcÞ < uðdÞ, vðaÞ � vðbÞ: Note that uðbÞ � uðaÞ �uðdÞ � uðcÞ > 0. Find e1, e2 to satisfy: �

1�� ½vðe1Þ � vðe2Þ� ¼

uðdÞ � uðcÞ > 0. Set e ¼ e3, e4 to get indifference.(iv) uðbÞ � uðaÞ, uðcÞ < uðdÞ, vðaÞ < vðbÞ: Do the same as above.(v) uðbÞ < uðaÞ, uðcÞ � uðdÞ, vðaÞ � vðbÞ: Find e1, e2 to satisfy:

�1�� ½vðe1Þ � vðe2Þ� ¼ uðbÞ � uðaÞ < 0. Note that

0 ¼ uðbÞ � uðaÞ ��

1� �½vðe1Þ � vðe2Þ�

� uðdÞ � uðcÞ ��

1� �½vðe1Þ � vðe2Þ�:

(vi) uðbÞ < uðaÞ, uðcÞ � uðdÞ, vðaÞ < vðbÞ: Do the same as above.

In any event uðbÞ þ uðcÞ � uðaÞ þ uðdÞ implies ðb, cÞ %� ða, dÞ.Therefore, the preference relation %� admits an additive repre-sentation in terms of u.

Step 2. The preference relation %� also admits a representa-tion in terms of the index v:

ðb, cÞ %�ða, dÞ () vðbÞ þ vðcÞ � vðaÞ þ vðdÞ

Using the definition of %� and Axiom 11 it follows that

uðe3Þ þ�

1� �vðaÞ ¼ uðe4Þ þ

�

1� �vðbÞ

and

uðe3Þ þ�

1� �vðcÞ � uðe4Þ þ

�

1� �vðdÞ,

which implies vðbÞ þ vðcÞ � vðaÞ þ vðdÞ. Now, for the other direction,we proceed as in step 1. Suppose vðbÞ þ vðcÞ � vðaÞ þ vðdÞ.Proceeding exactly as before, there are elements e1, e2, e3, e4

such that ðe2, bÞ % ðe1, aÞ, ðe1, cÞ % ðe2, dÞ and ða, e3Þ � ðb, e4Þ.By Axiom 11, it follows that ðc, e3Þ % ðb, e4Þ. And therefore,uðbÞþ uðcÞ�uðaÞ þ uðdÞ. Therefore, ðb, cÞ%� ða, dÞ() vðbÞ þ vðcÞ �vðaÞ þ vðdÞ.




ownloaded from


Step 3. Since the preference relation %� admits two differentadditive representations, it follows that the two utility indexesare related through a monotone affine transformation. This is,there exists � > 0 and � such that for all a 2 C:

vðaÞ ¼ �uðaÞ þ �

We conclude that % is represented by the mapping


�tuðxtÞð13Þ

with � > 0.

Step 4. Take a, c 2 C such that uðaÞ > uðcÞ. The existence ofsuch an element follows from the sensitivity axiom. Choose b, d tosatisfy:

uðaÞ þ �uðbÞ ¼ uðcÞ þ �uðdÞ:

Axiom 10 implies that

uðaÞ þ ��uðbÞ � uðcÞ þ ��uðdÞ:

The two inequalities imply � 1.

Proof of Theorem 5

REMARK 1. Both Ghirardato and Marinacci (2001) and Nakamura(1990) study Choquet preferences, so their axioms have co-monotonicity requirements. To have simpler statements andto avoid introducing the concept of co-monotonicity in themain text, we use stronger axioms that hold for all, not neces-sarily co-monotone acts, but the co-monotone versions ofthose axioms could be used (are equivalent in the presenceof other axioms).

The necessity of the axioms is straightforward. For suffi-ciency, we rely on the work of Ghirardato and Marinacci (2001).Note that their axiom B1 follows from our Axioms 1 and 2. Theiraxioms B2 and B3 follow from our Axiom 13. Their axiom S1follows from the fact that by Theorem 1 the functions u and vare continuous. Finally their axiom S2 follows from our Axiom12. Thus, by their lemma 31 there exists � 2 ð0, 1Þ and w : C! R




ownloaded from


such that ða, bÞ��wðaÞ þ ð1� �ÞwðbÞ represents %. By unique-ness of additive representations, w is a positive affine transfor-mation of u. Step iv in the proof of Theorem 5 concludes theproof. «

Nakamura’s Axiom. An alternative to Theorem 5 is thefollowing:

AXIOM 15. (Nakamura’s A6) For a, b, c, d 2 C such that b % a,d % c, d % b and c % a:

ðcða, bÞ, cðc, dÞÞ � ðcða, cÞ, cðb, dÞÞ

and

ðcðc, dÞ, cða, bÞÞ � ðcðc, aÞ, cðd, bÞÞ:

THEOREM 7. The preference % satisfies Axioms 1–7 and 13–15 ifand only if there exists a nonconstant and continuous func-tion u : C! R and parameters � > 0 and � 2 ð0, 1Þ such that% is represented by the mapping


�tuðxtÞ:


Proof. The necessity of Axioms 1–7, 10, and 13 is straightfor-ward. For Axiom 15, take a, b, c, d 2 C as in the statement of theaxiom and note that:

cða, bÞ c1, uðc1Þ þ�

1� ��uðc1Þ ¼ uðaÞ þ

�

1� ��uðbÞð14aÞ

cðc, dÞ c2, uðc2Þ þ�

1� ��uðc2Þ ¼ uðcÞ þ

�

1� ��uðdÞð14bÞ

and also,

cða, cÞ c3, uðc3Þ þ�

1� ��uðc3Þ ¼ uðaÞ þ

�

1� ��uðcÞð15aÞ

cðb, dÞ c4, uðc4Þ þ�

1� ��uðc4Þ ¼ uðbÞ þ

�

1� ��uðdÞ:ð15bÞ




ownloaded from


Therefore, using equations (14a)–(14b),

1þ ��

1� �

� �uðc1Þ þ

�

1� �uðc2Þ

� �¼ uðaÞ þ

�

1� ��uðbÞ

þ�

1� ��uðcÞ þ

��

1� ��

�2

uðdÞ

ð16Þ

and using equations (15a)–(15b),

1þ ��

1� �

� �uðc3Þ þ

�

1� �uðc4Þ

� �¼ uðaÞ þ

�

1� ��uðcÞ

þ�

1� ��uðbÞ þ

��

1� ��

�2

uðdÞ:

ð17Þ

So, ðc1, c2Þ � ðc3, c4Þ. The second implication of Axiom 15 followsby analogy.

For sufficiency of the axioms we rely on the proof of Lemma 3(Proposition 1) in Nakamura (1990).16 The argument goes as fol-lows. Consider the restriction of % to elements of the form (a, b),with a, b 2 C and b % a. Denote it by % R. The proof of Theorem 1implies lemma 2 (parts 1 and 2) of Nakamura (1990), withS ¼ ðs1, s2Þ, A ¼ s1, � u and �

1�� v. Our Axioms 13 and 15coincide exactly with A3 and A6 in Nakamura (1990) whenS ¼ ðs1, s2Þ. Therefore, Lemma 3 implies there is a real valuedfunction r(x) such that:

ða, bÞ %R ðc, dÞ () �rðaÞ þ ð1� �ÞrðbÞ � �rðcÞ þ ð1� �ÞrðdÞ,

where r is defined (Nakamura 1990, p. 356) as �ðcÞ� for all c 2 C and� ¼ 1

ð1þ��Þ, with �� such that ðcÞ ¼ ��ðcÞ þ ��, �� > 0. Hence, itfollows that for every c 2 C, �

1�� vðcÞ ¼ ��uðcÞ þ ��. If we set � ¼ 1

��,then we get uðcÞ ¼ �

1�� uðcÞ þ �. The representation (4) becomes:


�tuðxtÞ, � > 0:

Step 4 in the proof of Theorem 5 concludes the proof. «

16. Nakamura’s results are used explicitly by Chew and Karni (1994) and impli-citly by Ghirardato and Marinacci (2001).




ownloaded from


Proof of Theorem 6

Suppose that V is defined by the utility function u : C! R

and the sequence 1 ¼ D0 > D1 > � � � such thatP1

t¼0 Dt <1. LetVn be a semi-hyperbolic preference defined by the same utilityfunction and Dn

t ¼ Dt for t ¼ 0, 1, . . . , nþ 1 and Dnt ¼ Dnþ1�

t�n

for t > nþ 1, where � ¼ D1.For each n define the functions WnðxÞ ¼

Pnt¼0 DtuðxtÞ,

RnðxÞ ¼P1

t¼nþ1 DtuðxtÞ, and EnðxÞ ¼ Dnþ1P1

t¼nþ1 �t�n�1uðxtÞ.

Notice that VðxÞ ¼WnðxÞ þRnðxÞ for any n since the value of thesum is independent of n. Also, VnðxÞ ¼ WnðxÞ þ EnðxÞ for all n.Since the stream x is bounded, all these terms are well defined,and moreover the terms EnðxÞ and RnðxÞ converge to 0. Notice thatthis also implies that Dnþ1 ! 0. Suppose that there exist u < usuch that u uðxtÞ u for all t and define M :¼ maxfjuj, jujg. Wehave:

jVðxÞ � VnðxÞj ¼ jWnðxÞ þRnðxÞ �WnðxÞ � EnðxÞj ¼ jRnðxÞ � EnðxÞj

jRnðxÞj þ jEnðxÞj M

� X1t¼nþ1

Dt þDnþ1

X1t¼nþ1

�t�n

�! 0:

This also proves uniform convergence over all x within u, u.Finally, notice that if x %n y for n large enough, then

VnðxÞ � VnðyÞ for large n, so by the foregoing result VðxÞ � VðyÞ.Moreover, if for some > 0 we have VðxÞ � VðyÞ > then sinceVnðxÞ ! VðxÞ and VnðyÞ ! VðyÞ, we have limn½VnðxÞ � VnðyÞ� � ,so x �n y for n sufficiently large. «

Appendix II: Empirical Results

Econometric Analysis

Each agent i answers seven questions in each of the twoprice lists. We summarize each agent’s set of answers by the‘‘switch point’’ in each list; that is, we report the number of thefirst question (1 to 7) in which the agent chooses the impatientprospect A. If agent i always chooses the patient prospect B, wesay that the switch point has a numerical value of 8. As notedbefore, under the assumption of quasi-hyperbolic discountingthe agent has at most one switch point in each list, that is,she answers B for questions 1, . . . , k and A for questionskþ 1, . . . , 7.




ownloaded from


Let ðsi, 1, si, 2Þ denote the switch points of agent i in price list 1and 2, respectively. The objective of the econometric analysis inthis article is to estimate the marginal distributions of ð�i,�iÞ inthe population based on a sample of switch points for agentsi ¼ 1, . . . I. In the following subsections we argue that our experi-mental design allows us to partially identify the marginal distri-butions of �i and �i.

Data and Distributions of Switch Points. Our initial sampleconsists of two groups of subjects. The Money Group (M) has 639subjects that answered the ‘‘money’’ questionnaire. The IceCream Group (IC) has 640 subjects that answered the ‘‘icecream’’ questionnaire. We associate subjects with an IP addressand we verify that there is no IP repetition inside the group.Consequently, we do not allow for a single IP address to answerthe same questionnaire more than once.

We select a subsample of 336 subjects from the M group and444 subjects from the IC group. The selection is based on threecriteria (monotonicity, understanding, and consistency)described in the Online Appendix. For the selected sample, wefocus on the distributions of switch points. These distributionsare described in Figure A.I.

Our objective is to map the joint empirical distribution ofswitch points in Figure A.I into estimated lower and upperbounds for the marginal distributions of �i and �i.

Marginal Distribution of �i. Partial identification: For� 2 ½0, 1Þ, let Fð�Þ denote the measure of the set of quasi-hyperbolicagents in the MTurk population (denoted P) with parameter�i �. That is:

Fð�Þ ¼ fi 2 P j �i �g:

We argue now that Fð�Þ is partially identified by the switchpoints in the second price list. Let ��ðjÞ be the value of the discountfactor that makes any agent i indifferent between options A and Bin question j of the second price list, j ¼ 1 . . . 7. Note that ��ðjÞ isdefined by the equation:

�iuiðxÞ þ �i��ðjÞtjuiðyÞ þ �i�

�ðjÞtjþ1uiðyÞ

¼ �iuiðyÞ þ �i��ðjÞtjuiðxÞ þ �i�

�ðjÞtjþ1uiðxÞ,




ownloaded from



where tj ¼ f1, 3, 6, 12, 24, 36, 60g. If uiðxÞ > uiðyÞ the latter holds ifand only if:

1 ¼ ��ðjÞtj þ ��ðjÞtjþ1,ð18Þ

which has only one real solution in ½0, 1Þ. The collection ofintervals

½��ð0Þ, ��ð1ÞÞ, ½��ð1Þ, ��ð2ÞÞ . . . ½��ð7Þ, ��ð1ÞÞ

is a partition of ½0, 1Þ (with ��ð0Þ 0 and ��ð8Þ 1).

12

34

56

78

12

34

56

78

0

0.05

0.1

0.15

0.2

Switch Point Price List 21:Very Impatient

Empirical Distribution of Switch Points336 subjectsMoney−Year

Switch Point Price List 11: Very Impatient

(a)

FIGURE A.I

Distribution of Switch Points in the Sample




ownloaded from


PROPOSITION 1. For j ¼ 1 . . . 7

fi 2 P j si, 2 jg Fð��ðjÞÞ fi 2 P j si, 2 jþ 1g:

Proof. Note that

fi 2 P j si, 2 jg ¼ fi 2 P j i chooses A in question jg

fi 2 P j �tj

i þ �tjþ1i 1 ¼ ��ðjÞtj þ ��ðjÞtjþ1

g

¼ fi 2 P j �i ��ðjÞg

¼ Fð��ðjÞÞ:

12

34

56

78

12

34

56

78

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Switch Point Price List 21:Very Impatient

Empirical Distribution of Switch Points444 subjects

IC−Year

Switch Point Price List 11: Very Impatient

(b)

FIGURE A.I

Continued




ownloaded from


Likewise,

Fð��ðjÞÞ fi 2 P j �i < ��ðjþ 1Þg

¼ fi 2 P j �tjþ1

i þ �tjþ1þ1i < ��ðjþ 1Þtjþ1 þ ��ðjþ 1Þtjþ1þ1

¼ 1g

fi 2 P j si, 2 jþ 1g «

COROLLARY. For any � 2 ½��ðjÞ, ��ðjþ 1ÞÞ, j ¼ 1, . . . 7,

Fð�Þ fi 2 P j si, 2 jg Fð�Þ fi 2 P j si, 2 jþ 1g Fð�Þ:

Proof. For the lower bound, the weak monotonicity of thec.d.f. implies

Fð�Þ � Fð��ðjÞÞ

� fi 2 P j si, 2 jg ðby Proposition 1Þ:

For the upper bound:

Fð�Þ fi 2 P j �i < ��ðjþ 1Þg

fi 2 P j si, 2 jþ 1g «

Hence, the marginal distribution of �i is partially identified by theswitch points si,2.

Estimation and inference: lower and upper bounds. Our infer-ence problem falls in the set-up considered by Imbens and Manski(2004) and Stoye (2009): a real-valued parameter, Fð�Þ, is partiallyidentified by an interval whose upper and lower bounds may beestimated from sample data. Given the results in Proposition 1 andits corollary, we consider the following estimators for the lower andupper bounds of Fð�Þ. For any � 2 ½��ðjÞ, ��ðjþ 1Þ�:

Fð�Þ 1

I

XI

i¼1

1fsi, 2 jg

and

^Fð�Þ 1

I

XI

i¼1

1fsi, 2 jþ 1g

¼ Fð�Þ þ1

I

XI

i¼1

fsi, 2 ¼ jþ 1g:




ownloaded from


If the preference parameters ð�i, �iÞ are independent drawsfrom the distribution , then the weak law of large numbersimplies that:

Fð�Þ!p

Fð�Þ and ^Fð�Þ!p

Fð�Þ:

To construct confidence bands for the partially identifiedparameter we use Imbens and Manski (2004)’s approach asdescribed in Stoye (2009, p. 1301). For each � we consider a con-fidence set for the parameter Fð��ð1ÞÞ Fð�Þ Fð��ð7ÞÞ of theform:

CI�

�Fð�Þ �

c��lffiffiffiIp , Fð�Þ þ

c��uffiffiffiIp

�ð19Þ

where

�l ¼�Fð�Þð1� Fð�ÞÞ

1=2and �u ¼

� ^Fð�Þð1� ^Fð�ÞÞ1=2

and c� satisfies

�ðc� þ

ffiffiffiIp

�

maxf�l, �ugÞ ��ð�c�Þ ¼ 1� �,

� ¼^Fð�Þ � Fð�Þ ¼

1

I

XI

i¼1

fsi, 2 ¼ jþ 1g:

Figure A.II shows the estimated upper and lower boundsand the (point-wise) confidence sets for Fð�Þ. Each of thejumps of bounds for the c.d.f. occurs at the (real) roots of theequations

1 ¼ ��ðjÞtj þ ��ðjÞtjþ1,

where tj corresponds to the delay of the rewards in the secondprice list. So, based on our experimental design, the seven jumpsfor the bounds of the c.d.f. occur at:

��ð1Þ ¼ 0:6180, ��ð2Þ ¼ 0:8192, ��ð3Þ ¼ 0:8987, ��ð4Þ ¼ 0:9460

��ð5Þ ¼ 0:9721, ��ð6Þ ¼ 0:9812, ��ð7Þ ¼ 0:9886:

Marginal Distribution of �i. For � � 0, let Gð�Þ denote themeasure of the set of quasi-hyperbolic agents in the MTurk




ownloaded from


0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

F(δ

)

Set Estimators and Confidence Sets

Upper BoundLower BoundIdentified SetConfidence Bands

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

F(δ

)

Set Estimators and Confidence Sets


(a)

(b)

FIGURE A.II

Bounds for Fð�Þ: a) Money b) Ice cream




ownloaded from


population (denoted P) with parameter 0 �i �. That is:

Gð�Þ fi 2 Pj�i �g:

We now show that the switch points in the first andsecond price lists allows us to partially identify the marginaldistribution Gð�Þ. Let ��ðjÞ be the solution to equation (18). Forj ¼ 1, 2 . . . 7 and k ¼ 1, 2, . . . 7, define:

��ðj, kÞ 1

��ðjÞtk þ ��ðjÞtkþ1,

where tk ¼ f1, 3, 6, 12, 24, 36, 60g. Note that tk represents the firstfuture payment date in questions A and B of price list 1. Define:

nðj j�Þ max n ��ðj, nÞ � ��

ð20Þ

nðj j�Þ min n � < ��ðjþ 1, nÞ �

:�

ð21Þ

Let nð0 j�Þ 0 for all �. We start by proving the following result:

LEMMA 2. For j ¼ 0, . . . 7, � � 0, let

Bðj,�Þ ¼ fi 2 P j 0 �i �, si, 2 ¼ jþ 1g:

i 2 P si, 1 nðj j�Þ, si, 2 ¼ jþ 1 ��

� Bðj,�Þ

� i 2 P j si, 1 nðj j�Þ, si, 2 ¼ jþ 1��

ð22Þ

Proof. We establish the lower bound first. The result holdsfor vacuously for j = 0. So, suppose j> 0. Note that si, 2 ¼ jþ 1implies two things. First, the switch point in the second pricelist did not occur at j < jþ 1. Therefore,

1 �tj

i þ �tjþ1i ,

where tj corresponds to the first future payment date in questionj of price list 2. By definition of ��ðjÞ, the latter implies

��ðjÞtj þ ��ðjÞtjþ1 �

tj

i þ �tjþ1i ,

which implies �i � ��ðjÞ. Second, at question j + 1 the switch

occurs. Hence:

��ðjþ 1Þtjþ1 þ ��ðjþ 1Þtjþ1þ1¼ 1 � �

tjþ1

i þ �tjþ1þ1i :




ownloaded from


Consequently, ��ðjþ 1Þ � �i. We conclude that for any i such thatsi, 2 ¼ jþ 1:

�i 2 ½��ðjÞ, ��ðjþ 1Þ�:ð23Þ

In addition, let k0 nðj j�Þ. Note that for a quasi-hyperbolic agentsi, 1 ¼ k implies

�i 1

�tk

i þ �tkþ1i

1

�tnðj j�Þ

i þ �tnðjj�Þþ1

i

¼ ��ðj, nðj j�ÞÞ:ð24Þ

Hence si, 1 nðj j�Þ and s1, 2 ¼ jþ 1 imply equations (23) and (24).Equation (20) implies

0 �i ��ðj, nðj j�ÞÞ �,

and we conclude

fi 2 P j si, 1 nðj j�Þ, si, 2 ¼ jþ 1g � Bðj,�Þ:

Now we establish the upper bound. Suppose i 2 Bðj,�Þ. Then ibelongs to

Bðj,�Þ i 2 P 0 �i � < ��ðjþ 1, nðjþ 1 j�Þ, s1, 2 ¼ jþ 1 �

:�

Since

�i < ��ðjþ 1, nðjþ 1 j�Þ ¼1

��ðjþ 1Þtnðjþ1 j �Þ þ ��ðjþ 1Þtnðjþ1 j�Þþ1

1

�tnðjþ1 j�Þ

i þ �tnðjþ1 j �Þþ1

i

the switch in price list 1 occurred at most at period nðjþ 1 j�Þ.Therefore, si, 1 nðjþ 1 j�Þ. «

We use the previous lemma to partially identify Gð�Þ.

PROPOSITION 2 (BOUNDS FOR Gð�Þ). For j ¼ 0, . . . 7:

(i)P7

j¼0 fi 2 P j si, 1 nðj j�Þ, si, 2 ¼ jþ 1g Gð�Þ

(ii) Gð�Þ P7

j¼0 fi 2 P j si, 1 nðj j�Þ, si, 2 ¼ jþ 1g.

Proof. First we establish the lower bound. By Lemma 2, foreach j ¼ 0, . . . 7:�

i 2 P j si, 1 nðj j�Þ, si, 2 ¼ jþ 1�� Bðj,�Þ




ownloaded from


Therefore,

[7j¼0

ni 2 P j si, 1 nðj j�Þ , si, 2 ¼ jþ 1

o�[7j¼0

Bðj,�Þ

¼[7j¼0

ni 2 Pj0 �i �

�ðj, nðjj�Þ, si, 2 ¼ jþ 1o

�[7j¼0

ni 2 Pj0 �i �, si, 2 ¼ jþ 1

o

¼�i 2 Pj0 �i �

�:

Hence,

�[7j¼0

fi 2 P j si, 1 nðj j�Þ, si, 2 ¼ jþ 1g

�[7j¼0

ni 2 P j 0 �i �

o

¼ Gð�Þ:

Now we establish the upper bound. From Lemma 2:

ni 2 Pj0 �i �, si, 2 ¼ jþ 1

o

is a subset of

ni 2 Pjsi, 1 nðjj�Þ, si, 2 ¼ jþ 1

o:

The result then follows. «

Estimation and inference: lower and upper bounds. Based onProposition 2, the estimators for the upper and lower bounds ofthe population are given by:

(i)P7

j¼01I

PIi¼1 1


o

(ii)P7

j¼01I

PIi¼1 1


o,

which can be written as:

(i) Gð�Þ ¼ 1I

PIi¼1 1

ni 2 Pj

S7j¼0 ðsi, 1 nðjj�Þ, si, 2 ¼ jþ 1Þ

o

(ii) ^Gð�Þ ¼ 1I

PIi¼1 1

ni 2 Pj

S7j¼0 ðsi, 1 nðjj�Þ, si, 2 ¼ jþ 1Þ

o.




ownloaded from


0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β

)



0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β

F(β

)



(a)

(b)

FIGURE A.III

Bounds for Gð�Þ: a) Money b) Ice cream




ownloaded from


Imbens and Manski (2004)’s approach is used to build a con-fidence set for the parameter Gð�Þ:

CI�

�Gð�Þ �

c��lffiffiffiIp ,

^Gð�Þ þc��uffiffiffi

Ip

�;ð25Þ

where

�l ¼

�Gð�Þð1� Gð�ÞÞ

�1=2

and �u ¼

�^Gð�Þ 1� ^Gð�ÞÞ

�1=2

;

and c� satisfies

�

�c� þ

ffiffiffiIp

�

maxf�l, �ug

��ð�c�Þ ¼ 1� �,

� ¼^Gð�Þ � Gð�Þ:

Figure A.III reports the estimates for the lower and upperbounds along with a 95% confidence set for the partially identifiedparameter Gð�Þ.

New York University

Harvard University

Supplementary Material

An Online Appendix for this article can be found at QJEonline (qje.oxfordjournals.org).

References

Abdellaoui, Mohammed, Arthur E. Attema, and Han Bleichrodt, ‘‘IntertemporalTradeoffs for Gains and Losses: An Experimental Measurement ofDiscounted Utility,’’ Economic Journal, 120 (2010), 845–866.

Andersen, Steffen, Glenn W. Harrison, Morten I. Lau, and E. Elisabet Rutstrom,‘‘Eliciting Risk and Time Preferences,’’ Econometrica, 76 (2008), 583–618.

Andreoni, James, and Charles Sprenger, ‘‘Estimating Time Preferences fromConvex Budgets,’’ American Economic Review, 102 (2012), 3333–3356.

Andreoni, James, Michael A. Kuhn, and Charles Sprenger, ‘‘On Measuring TimePreferences,’’ Working paper, UC San Diego, 2013.

Attema, A. E., H. Bleichrodt, K. I. M. Rohde, and P. P. Wakker, ‘‘Time-TradeoffSequences for Analyzing Discounting and Time Inconsistency,’’ ManagementScience, 56 (2010), 2015–2030.

Augenblick, Ned, Muriel Niederle, and Charles Sprenger, ‘‘Working over Time:Dynamic Inconsistency in Real Effort Tasks,’’ NBER Working Paper, 2013.

Bleichrodt, H., K. I. M. Rohde, and P. P. Wakker, ‘‘Koopmans’ ConstantDiscounting for Intertemporal Choice: A Simplification and aGeneralization,’’ Journal of Mathematical Psychology, 52 (2008), 341–347.




ownloaded from



Casari, M., and D. Dragone, ‘‘On Negative Time Preferences,’’ Economics Letters,111 (2010), 37–39.

Chabris, Christopher, David Laibson, and Jonathan Schuldt, ‘‘IntertemporalChoice,’’ in Palgrave Dictionary of Economics (London: Palgrave, 2008).

Chew, H. C., and E. Karni, ‘‘Choquet Expected Utility with a Finite State Space:Commutativity and Act-Independence,’’ Journal of Economic Theory, 62(1994), 469–479.

Cohen, Michele, Jean-Marc Tallon, and Jean-Christophe Vergnaud, ‘‘AnExperimental Investigation of Imprecision Attitude and its Relation withRisk Attitude and Impatience,’’ Theory and Decision, 71 (2011), 81–109.

Coller, Maribeth, and Melonie B. Williams, ‘‘Eliciting Individual Discount Rates,’’Experimental Economics, 2 (1999), 107–127.

Dohmen, Thomas J., Armin Falk, David Huffman, and Uwe Sunde, ‘‘InterpretingTime Horizon Effects in Inter-Temporal Choice,’’ CESifo Working PaperSeries No. 3750, 2012.

Epstein, Larry G., and S. E. Zin, ‘‘Substitution, Risk Aversion, and the TemporalBehavior of Consumption and Asset Returns: A Theoretical Framework,’’Econometrica, 57 (1989), 937–969.

Fels, S., and R. Zeckhauser, ‘‘Perfect and Total Altruism across the Generations,’’Journal of Risk and Uncertainty, 37 (2008), 187–197.

Fishburn, P. C., and A. Rubinstein, ‘‘Time Preference,’’ International EconomicReview, 23 (1982), 677–694.

Fudenberg, Drew, and David K. Levine, ‘‘A Dual-Self Model of Impulse Control,’’American Economic Review, 96 (2006), 1449–1476.

Fudenberg, Drew, and Eric Maskin, ‘‘On the Dispensability of PublicRandomization in Discounted Repeated Games,’’ Journal of EconomicTheory, 53 (1991), 428–438.

Ghirardato, P., and M. Marinacci, ‘‘Risk, Ambiguity, and the Separation of Utilityand Beliefs,’’ Mathematics of Operations Research, 26 (2001), 864–890.

Gigliotti, Gary, and Barry Sopher, ‘‘Analysis of Intertemporal Choice: A NewFramework and Experimental Results,’’ Theory and Decision, 55 (2003),209–233.

Gorman, W. M., ‘‘The Structure of Utility Functions,’’ Review of Economic Studies,35 (1968), 367–390.

Gul, Faruk, and Wolfgang Pesendorfer, ‘‘Temptation and Self-Control,’’Econometrica, 69 (2001), 1403–1435.

———, ‘‘Self-Control and the Theory of Consumption,’’ Econometrica, 72 (2004),119–158.

Halevy, Yoram, ‘‘Time Consistency: Stationarity and Time Invariance,’’ MicroTheory Working Papers, 2012.

Harrison, Glenn W., Morten I. Lau, and Melonie B. Williams, ‘‘EstimatingIndividual Discount Rates in Denmark: A Field Experiment,’’ AmericanEconomic Review, 92 (2002), 1606–1617.

Hayashi, T., ‘‘Quasi-Stationary Cardinal Utility and Present Bias,’’ Journal ofEconomic Theory, 112 (2003), 343–352.

Horton, John J., David G. Rand, and Richard J. Zeckhauser, ‘‘The OnlineLaboratory: Conducting Experiments in a Real Labor Market,’’Experimental Economics, 14 (2011), 399–425.

Imbens, Guido W., and Charles F. Manski, ‘‘Confidence Intervals for PartiallyIdentified Parameters,’’ Econometrica, 72 (2004), 1845–1857.

Johnson, Matthew W., and Warren K. Bickel, ‘‘Within-Subject Comparison of Realand Hypothetical Money Rewards in Delay Discounting,’’ Journal of theExperimental Analysis of Behavior, 77 (2002), 129–146.

Kochov, Asen, ‘‘Geometric Discounting in Discrete, Infinite-Horizon ChoiceProblems,’’ Mimeo, 2013.

Koopmans, T. C., ‘‘Stationary Ordinal Utility and Impatience,’’ Econometrica, 28(1960), 287–309.

———, ‘‘Representation of Preference Orderings over Time, in Decision andOrganization, C. B. McGuire, and Roy Radner eds. (Amsterdam: North-Holland, 1972).

Kreps, D. M., and E. L. Porteus, ‘‘Temporal Resolution of Uncertainty andDynamic Choice Theory,’’ Econometrica, 46 (1978), 185–200.




ownloaded from


Laibson, D., ‘‘Golden Eggs and Hyperbolic Discounting,’’ Quarterly Journal ofEconomics, 112 (1997), 443–477.

Loewenstein, G., and D. Prelec, ‘‘Anomalies in Intertemporal Choice: Evidenceand an Interpretation,’’ Quarterly Journal of Economics, 107 (1992),573–597.

Luce, R. D., and J. W. Tukey, ‘‘Simultaneous Conjoint Measurement: A New Typeof Fundamental Measurement,’’ Journal of Mathematical Psychology (1964),1–27.

Marge, Matthew, Satanjeev Banerjee, and Alexander I. Rudnicky, ‘‘Using theAmazon Mechanical Turk for Transcription of Spoken Language,’’ CarnegieMellon University Computer Science Department Paper 1350, 2010.

Mason, Winter, and Duncan J. Watts, ‘‘Financial Incentives and the Performanceof Crowds,’’ ACM SigKDD Explorations Newsletter, 11 (2010), 100–108.

Mason, Winter, and Siddharth Suri, ‘‘Conducting Behavioral Research onAmazon’s Mechanical Turk,’’ Behavior Research Methods, 44 (2012), 1–23.

McClure, S. M., K. M. Ericson, D. I. Laibson, G. Loewenstein, and J. D. Cohen,‘‘Time Discounting for Primary Rewards,’’ Journal of Neuroscience, 27 (2007),5796–5804.

Nakamura, Y., ‘‘Subjective Expected Utility with Non-additive Probabilities onFinite State Spaces,’’ Journal of Economic Theory, 51 (1990), 346–366.

Noor, J., ‘‘Hyperbolic Discounting and the Standard Model: Eliciting DiscountFunctions,’’ Journal of Economic Theory, 144 (2009), 2077–2083.

———, ‘‘Temptation and Revealed Preference,’’ Econometrica, 79 (2011), 601–644.O’Donoghue, Ted, and Matthew Rabin, ‘‘Choice and Procrastination,’’ Quarterly

Journal of Economics, 116 (2001), 121–160.Paolacci, Gabriele, Jesse Chandler, and Panagiotis Ipeirotis, ‘‘Running

Experiments on Amazon Mechanical Turk,’’ Judgment and DecisionMaking, 5 (2010), 411–419.

Phelps, E. S., and R. A. Pollak, ‘‘On Second-Best National Saving and Game-Equilibrium Growth,’’ Review of Economic Studies, 35 (1968), 185–199.

Ramsey, F., ‘‘Truth and Probability, in Ramsey, The Foundations of Mathematicsand Other Logical Essays, R. B. Braithwaite, ed. (London: Kegan Paul,Trench, Trubner, 1931), pp. 156–198.

Read, Daniel, ‘‘Is Time-Discounting Hyperbolic or Subadditive?,’’ Journal of Riskand Uncertainty, 23 (2001), 5–32.

Samuelson, P. A., ‘‘A Note on Measurement of Utility,’’ Review of EconomicStudies, 4 (1937), 155–161.

Sayman, Serdar, and Ayse Onculer, ‘‘An Investigation of Time Inconsistency,’’Management Science, 55 (2009), 470–482.

Scholten, Marc, and Daniel Read, ‘‘Discounting by Intervals: A Generalized Modelof Intertemporal Choice,’’ Management Science, 52 (2006), 1424–1436.

Sorin, S., ‘‘On Repeated Games with Complete Information,’’ Mathematics ofOperations Research, 11 (1986), 147–160.

Stoye, Jorg, ‘‘More on Confidence Intervals for Partially Identified Parameters,’’Econometrica, 77 (2009), 1299–1315.

Strotz, Robert Henry, ‘‘Myopia and Inconsistency in Dynamic UtilityMaximization,’’ Review of Economic Studies, 23 (1955), 165–180.

Takeuchi, Kan, ‘‘Non-parametric Test of Time Consistency: Present Bias andFuture Bias,’’ Games and Economic Behavior, 71 (2011), 456–478.

Thaler, R., ‘‘Some Empirical Evidence on Dynamic Inconsistency,’’ EconomicsLetters, 8 (1981), 201–207.

Thaler, Richard H., and Hersh M. Shefrin, ‘‘An Economic Theory of Self-Control,’’Journal of Political Economy, 89 (1981), 392–406.

Van der Pol, Marjon, and John Cairns, ‘‘Descriptive Validity of AlternativeIntertemporal Models for Health Outcomes: An Axiomatic Test,’’ HealthEconomics, 20 (2011), 770–782.

Vind, K., ‘‘Note on ‘The Structure of Utility Functions’,’’ Review of EconomicStudies, 38 (1971), 113.

Zeckhauser, R., and S. Fels, ‘‘Discounting for Proximity with Perfect and TotalAltruism,’’ Harvard Institute of Economic Research, Discussion Paper 50,1968.




ownloaded from


This page intentionally left blank



ownloaded from


i. introduction · fabio maccheroni, yusufcan masatlioglu, jawwad noor, ben polak, al roth, michael...

Documents