Time Discounting and Time Preference: A Critical Review PDF

Journal of Economic Literature Vol. XL (June 2002), pp. 351–401 Frederick, Loewenstein, and O’Donoghue: Time Journal of Economic Discounting...

Journal of Economic Literature Vol. XL (June 2002), pp. 351–401 Frederick, Loewenstein, and O’Donoghue: Time Journal of Economic Discounting Literature, Vol. XL (June 2002) Time Discounting and Time Preference: A Critical Review S HANE FREDERICK , GEORGE LOEWENSTEIN , and T ED O’DONOGHUE 1 1. Introduction wealth of nations, the Scottish economist John Rae was examining the sociologi- I NTERTEMPORAL CHOICES —decisions cal and psychological determinants of involving tradeoffs among costs and these choices. In section 2, we briefly benefits occurring at different times— review the perspectives on intertempo- are important and ubiquitous. Such deci- ral choice of Rae and nineteenth- and sions not only affect one’s health, wealth, early twentieth-century economists, and and happiness, but, may also, as Adam describe how these early perspectives Smith first recognized, determine the interpreted intertemporal choice as economic prosperity of nations. In this the joint product of many conflicting paper, we review empirical research on psychological motives. intertemporal choice, and present an All of this changed when Paul Sam- overview of recent theoretical formula- uelson proposed the discounted-utility tions that incorporate insights gained (DU) model in 1937. Despite Samuel- from this research. son’s manifest reservations about the Economists’ attention to intertempo- normative and descriptive validity of ral choice began early in the history of the formulation he had proposed, the the discipline. Not long after Adam DU model was accepted almost in- Smith called attention to the impor- stantly, not only as a valid normative tance of intertemporal choice for the standard for public policies (e.g., in cost- benefit analyses), but as a descriptively 1 Frederick: Sloan School of Management, Mas- accurate representation of actual behav- sachusetts Institute of Technology. Loewenstein: ior. A central assumption of the DU Department of Social and Decision Sciences, Carnegie Mellon University. O’Donoghue: De- model is that all of the disparate mo- partment of Economics, Cornell University. We tives underlying intertemporal choice can thank Colin Camerer, David Laibson, John be condensed into a single parameter— McMillan, Drazen Prelec, Daniel Read, Nachum Sicherman, Duncan Simester, and three anony- the discount rate. In section 3 we exam- mous referees for useful comments. We thank ine this and many other assumptions Cara Barber, Rosa Blackwood, Mandar Oak, and underlying the DU model. We do not Rosa Stipanovic for research assistance. For finan- cial support, Frederick and Loewenstein thank the present an axiomatic derivation of the Integrated Study of the Human Dimensions of model, but instead focus on those Global Change at Carnegie Mellon University features that highlight the implicit (NSF Grant SBR-9521914), and O’Donoghue thanks the National Science Foundation (Award psychological assumptions underlying SES-0078796). the model. 351 352 Journal of Economic Literature, Vol. XL (June 2002) Samuelson’s reservations about the rate stems partly from differences in descriptive validity of the DU model elicitation procedures. But it also stems were justified. Section 4 reviews from the faulty assumption that the var- the growing list of “DU anomalies”— ied considerations that are relevant in patterns of choice that are inconsistent intertemporal choices apply equally to with the model’s theoretical predic- different choices and thus that they can tions. Virtually every assumption under- all be sensibly represented by a single lying the DU model has been tested discount rate. and found to be descriptively invalid in Throughout the paper, we stress the at least some situations. Moreover, as importance of distinguishing among the we discuss at the end of the section, varied considerations that underlie in- these anomalies are not anomalies in the tertemporal choices. We distinguish sense that they are regarded as errors time discounting from time preference. by the people who commit them. Unlike We use the term time discounting many of the better-known expected- broadly to encompass any reason for utility anomalies, the DU anomalies do caring less about a future consequence, not necessarily violate any standard or including factors that diminish the ex- principle that people believe they pected utility generated by a future should uphold. consequence, such as uncertainty or The insights about intertemporal changing tastes. We use the term time choice gleaned from this empirical re- preference to refer, more specifically, to search have led to the proposal of nu- the preference for immediate utility merous alternative theoretical models, over delayed utility. In section 7, we which we review in section 5. Some of push this theme further, by examining these modify the discount function, per- whether time preference itself might mitting, for example, declining discount consist of distinct psychological traits rates or “hyperbolic discounting.” Oth- that can be separately analyzed. Section ers introduce additional arguments into 8 concludes. the utility function, such as the utility of anticipation. Still others depart from 2. Historical Origins of the Discounted the DU model more radically, by in- Utility Model cluding, for instance, systematic mis- predictions of future utility. Many of The historical developments that cul- these new theories revive psychological minated in the formulation of the DU considerations discussed by Rae and model help to explain the model’s limi- other early economists that were extin- tations. Each of the major figures in the guished with the adoption of the DU development of the DU model—John model and its expression of intertem- Rae, Eugen von Böhm-Bawerk, Irving poral preferences in terms of a single Fisher, and Paul Samuelson—built parameter. upon the theoretical framework of his In section 6, we review attempts to predecessors, drawing on little more estimate discount rates. While the DU than introspection and personal obser- model assumes that people are charac- vation. When the DU model eventually terized by a single discount rate, this became entrenched as the dominant literature reveals spectacular variation theoretical framework for modeling in- across (and even within) studies. The tertemporal choice, it was due largely to failure of this research to converge to- its simplicity and its resemblance to the ward any agreed-upon average discount familiar compound interest formula,  Frederick, Loewenstein, and O’Donoghue: Time Discounting 353 and not as a result of empirical research ting into the vortex of extravagant fashion, demonstrating its validity. live economically. War and pestilence have always waste and luxury, among the other evils Intertemporal choice became firmly that follow in their train. (Rae 1834, p. 57) established as a distinct topic in 1834, with John Rae’s publication of The So- A second factor that limited the ef- ciological Theory of Capital. Like Adam fective desire of accumulation was the Smith, Rae sought to determine why excitement produced by the prospect of wealth differed among nations. Smith immediate consumption, and the con- had argued that national wealth was de- comitant discomfort of deferring such termined by the amount of labor allo- available gratifications: cated to the production of capital, but Such pleasures as may now be enjoyed gener- Rae recognized that this account was in- ally awaken a passion strongly prompting to complete because it failed to explain the partaking of them. The actual presence of the determinants of this allocation. In the immediate object of desire in the mind by Rae’s view, the missing element was exciting the attention, seems to rouse all the faculties, as it were to fix their view on it, and “the effective desire of accumulation”—a leads them to a very lively conception of the psychological factor that differed across enjoyments which it offers to their instant countries and determined a society’s possession. (Rae 1834, p. 120) level of saving and investment. Along with inventing the topic of in- Among the four factors that Rae iden- tertemporal choice, Rae also produced tified as the joint determinants of time the first in-depth discussion of the psy- preference, one can glimpse two funda- chological motives underlying inter- mentally different views. One, which was temporal choice. Rae believed that later championed by William S. Jevons intertemporal-choice behavior was the (1888) and his son, Herbert S. Jevons joint product of factors that either pro- (1905), assumes that people care only moted or limited the effective desire of about their immediate utility, and ex- accumulation. The two main factors plains farsighted behavior by postulat- that promoted the effective desire of ing utility from the anticipation of accumulation were the bequest motive future consumption. On this view, de- (“the prevalence throughout the society ferral of gratification will occur only if of the social and benevolent affections,” it produces an increase in “anticipal” p. 58) and the propensity to exercise utility that more than compensates for self-restraint (“the extent of the intel- the decrease in immediate consumption lectual powers, and the consequent utility. The second perspective assumes prevalence of habits of reflection, and equal treatment of present and future prudence, in the minds of the mem- (zero discounting) as the natural base- bers of society,” p. 58). One limiting line for behavior, and attributes the factor was the uncertainty of human overweighting of the present to the life: miseries produced by the self-denial When engaged in safe occupations, and living required to delay gratification. N. W. in healthy countries, men are much more apt Senior, the best-known advocate of this to be frugal, than in unhealthy, or hazardous “abstinence” perspective, wrote, “To occupations, and in climates pernicious to hu- abstain from the enjoyment which is in man life. Sailors and soldiers are prodigals. our power, or to seek distant rather In the West Indies, New Orleans, the East Indies, the expenditure of the inhabitants is than immediate results, are among the profuse. The same people, coming to reside most painful exertions of the human in the healthy parts of Europe, and not get- will” (Senior 1836, p. 60). 354 Journal of Economic Literature, Vol. XL (June 2002) The anticipatory-utility and absti- Böhm-Bawerk’s analysis of time pref- nence perspectives share the idea that erence, like those of his predecessors, intertemporal tradeoffs depend on im- was heavily psychological, and much of mediate feelings—in one case, the im- his voluminous treatise, Capital and mediate pleasure of anticipation, and in Interest, was devoted to discussions of the other, the immediate discomfort of the psychological constituents of time self-denial. The two perspectives, how- preference. However, whereas the early ever, explain variability in intertemporal- views of Rae, Senior, and Jevons ex- choice behavior in different ways. The plained intertemporal choices in terms anticipatory-utility perspective attrib- of motives that are uniquely associated utes variations in intertemporal-choice with time, Böhm-Bawerk began model- behavior to differences in people’s ing intertemporal choice in the same abilities to imagine the future and to terms as other economic tradeoffs—as a differences in situations that promote “technical” decision about allocating re- or inhibit such mental images. The ab- sources (to oneself) over different points stinence perspective, on the other hand, in time, much as one would allocate explains variations in intertemporal- resources between any two competing choice behavior on the basis of individ- interests, such as housing and food. ual and situational differences in the Böhm-Bawerk’s treatment of inter- psychological discomfort associated with temporal choice as an allocation of con- self-denial. In this view, one should sumption among time periods was for- observe high rates of time discounting malized a decade later by the American by people who find it painful to delay economist Irving Fisher (1930). Fisher gratification, and in situations in which plotted the intertemporal consumption deferral is generally painful—e.g., when decision on a two-good indifference one is, as Rae worded it, in the “actual diagram, with consumption in the cur- presence of the immediate object of rent year on the abscissa, and consump- desire.” tion in the following year on the ordi- Eugen von Böhm-Bawerk, the next nate. This representation made clear major figure in the development of the that a person’s observed (marginal) economic perspective on intertemporal rate of time preference—the marginal choice, added a new motive to the list rate of substitution at her chosen con- proposed by Rae, Jevons, and Senior, sumption bundle—depends on two arguing that humans suffer from a considerations: time preference and di- systematic tendency to underestimate minishing marginal utility. Many econo- future wants: mists have subsequently expressed dis- comfort with using the term “time It may be that we possess inadequate power to imagine and to abstract, or that we are not preference” to include the effects of dif- willing to put forth the necessary effort, but ferential marginal utility arising from in any event we limn a more or less incom- unequal consumption levels between plete picture of our future wants and espe- time periods (see in particular Mancur cially of the remotely distant ones. And Olson and Martin Bailey 1981). In then there are all those wants that never come to mind at all. (Böhm-Bawerk 1889, pp. Fisher’s formulation, pure time prefer- 268–69) 2 ence can be interpreted as the marginal 2 In a frequently cited passage from The Eco- nomics of Welfare, Arthur Pigou (1920) proposed telescopic faculty is defective, and we, therefore, a similar account of time preference, suggesting see future pleasures, as it were, on a diminished that it results from a type of cognitive illusion: “our scale.”  Frederick, Loewenstein, and O’Donoghue: Time Discounting 355 rate of substitution on the diagonal, in Samuelson’s simplified model, all the where consumption is equal in both psychological concerns discussed over the periods. previous century were compressed into Fisher’s writings, like those of his a single parameter, the discount rate. predecessors, included extensive discus- The DU model specifies a decision sions of the psychological determinants maker’s intertemporal preferences over of time preference. Like Böhm-Bawerk, consumption profiles (c t,...,c T). Under he differentiated “objective factors,” the usual assumptions (completeness, such as projected future wealth and transitivity, and continuity), such pref- risk, from “personal factors.” Fisher’s erences can be represented by an in- list of personal factors included the four tertemporal utility function U t(c t,...,c T). described by Rae, “foresight” (the abil- The DU model goes further, by as- ity to imagine future wants—the inverse suming that a person’s intertemporal of the deficit that Böhm-Bawerk postu- utility function can be described by the lated), and “fashion,” which Fisher be- following special functional form: lieved to be “of vast importance... in T−t its influence both on the rate of interest U t(ct,...,cT) = ∑ D(k)u(ct + k) and on the distribution of wealth itself.” k=0 k (Fisher 1930, p. 88):  1  where D(k) =  . The most fitful of the causes at work is prob- 1 + ρ ably fashion. This at the present time acts, In this formulation, u(ct + k) is often inter- on the one hand, to stimulate men to save preted as the person’s cardinal instanta- and become millionaires, and, on the other hand, to stimulate millionaires to live in an neous utility function—her well-being in ostentatious manner. (Fisher 1930, p. 87) period t + k—and D(k) is often inter- preted as the person’s discount func- Hence, in the early part of the twen- tion—the relative weight she attaches, in tieth century, “time preference” was period t, to her well-being in period t + k. viewed as an amalgamation of various ρ represents the individual’s pure rate intertemporal motives. While the DU of time preference (her discount rate), model condenses these motives into the which is meant to reflect the collective discount rate, we will argue that resur- effects of the “psychological” motives recting these distinct motives is crucial discussed in section 2.3 for understanding intertemporal choices. Samuelson did not endorse the DU model as a normative model of in- 3. The Discounted Utility Model tertemporal choice, noting that “any connection between utility as discussed In 1937, Paul Samuelson introduced here and any welfare concept is dis- the DU model in a five-page article avowed” (p. 161). He also made no titled “A Note on Measurement of Util- claims on behalf of its descriptive valid- ity.” Samuelson’s paper was intended to ity, stressing, “It is completely arbitrary offer a generalized model of intertem- to assume that the individual behaves so poral choice that was applicable to mul- as to maximize an integral of the form tiple time periods (Fisher’s graphical envisaged in [the DU model]” (p. 159). indifference-curve analysis was difficult However, despite Samuelson’s manifest to extend to more than two time peri- ods) and to make the point that repre- 3 The continuous-time analogue is U t({c } τ τ ∈[t,T]) = senting intertemporal tradeoffs re- ∫τT= t e − ρ(τ − t)u(cτ). For expositional ease, we shall quired a cardinal measure of utility. But restrict attention to discrete-time throughout. 356 Journal of Economic Literature, Vol. XL (June 2002) reservations, the simplicity and ele- new alternatives by integrating them gance of this formulation was irresist- with her existing plans. To illustrate, ible, and the DU model was rapidly consider a person with an existing con- adopted as the framework of choice for sumption plan (c t,...,c T) who is offered analyzing intertemporal decisions. an intertemporal-choice prospect X, The DU model received a scarcely which might be something like an op- needed further boost to its dominance tion to give up $5000 today to receive as the standard model of intertemporal $10,000 in five years. Integration means choice when Tjalling C. Koopmans that prospect X is not evaluated in isola- (1960) showed that the model could be tion, but in light of how it changes the derived from a superficially plausible person’s aggregate consumption in all set of axioms. Koopmans, like Samuel- future periods. Thus, to evaluate the son, did not argue that the DU model prospect X, the person must choose what was psychologically or normatively her new consumption path (c′t,…,c′T) plausible; his goal was only to show that would be if she were to accept prospect under some well-specified (though ar- X, and should accept the prospect if guably unrealistic) circumstances, in- U t(c′t,…,c′T) > U t(ct,…,cT). dividuals were logically compelled to An alternative way to understand in- possess positive time preference. Pro- tegration is to recognize that intertem- ducers of a product, however, cannot poral prospects alter a person’s budget dictate how the product will be used, set. If the person’s initial endowment is and Koopmans’ central technical mes- E 0, then accepting prospect X would sage was largely lost while his axiom- change her endowment to E 0 ∪ X. Let- atization of the DU model helped to ting B(E) denote the person’s budget cement its popularity and bolster its set given endowment E—i.e., the set of perceived legitimacy. consumption streams that are feasible In the remainder of this section, we given endowment E—the DU model describe some important features of the says that the person should accept DU model as it is commonly used by prospect X if: economists, and briefly comment on the T τ−t  1  normative and positive validity of these max ∑ 1 + ρ u(cτ) assumptions. These features do not rep- (ct,...,cT ) ∈B(E 0 ∪ X) τ= t  resent an axiom system—they are nei- T τ−t ther necessary nor sufficient conditions  1  for the DU model—but are intended > max ∑ 1 + ρ u(cτ). (ct,...,cT ) ∈B(E 0) τ= t  to highlight the implicit psychological assumptions underlying the model. 4 While integration seems normatively compelling, it may be too difficult to 3.1 Integration of New Alternatives actually do. A person may not have with Existing Plans well-formed plans about future con- sumption streams, or be unable (or un- A central assumption in most models willing) to recompute the new optimal of intertemporal choice—including the plan every time she makes an intertem- DU model—is that a person evaluates poral choice. Some of the evidence we 4 There are several different axiom systems for review below supports the plausible the DU model—in addition to Koopmans, see presumption that people evaluate the Peter Fishburn (1970), K. J. Lancaster (1963), results of intertemporal choices inde- Richard F. Meyer (1976), and Fishburn and Ariel Rubinstein (1982). pendently of any expectations they have  Frederick, Loewenstein, and O’Donoghue: Time Discounting 357 regarding consumption in future time pects are not affected by the conse- periods. quences that the prospects share—i.e., that the utility of an experienced out- 3.2 Utility Independence come is unaffected by other outcomes The DU model explicitly assumes that that one might have experienced (but the overall value—or “global utility”— did not). In intertemporal choice, con- of a sequence of outcomes is equal to sumption independence says that pref- the (discounted) sum of the utilities in erences over consumption profiles are each period. Hence, the distribution of not affected by the nature of consump- utility across time makes no difference tion in periods in which consumption is beyond that dictated by discounting, identical in the two profiles—i.e., that which (assuming positive time prefer- an outcome’s utility is unaffected by ence) penalizes utility that is experi- outcomes experienced in prior or future enced later. The assumption of utility periods. For example, consumption in- independence has rarely been discussed dependence says that a person’s prefer- or challenged, but its implications are ence between an Italian and Thai res- far from innocuous. It rules out any taurant tonight should not depend on kind of preference for patterns of utility whether she had Italian last night, nor over time—e.g., a preference for a flat whether she expects to have it tomor- utility profile over a roller-coaster util- row. As the example suggests, and as ity profile with the same discounted Samuelson and Koopmans both recog- utility. 5 nized, there is no compelling rationale for such an assumption. Samuelson 3.3 Consumption Independence (1952, p. 674) noted that, “the amount of wine I drank yesterday and will drink The DU model explicitly assumes that tomorrow can be expected to have ef- a person’s well-being in period t + k is fects upon my today’s indifference independent of her consumption in any slope between wine and milk.” Simi- other period—i.e., that the marginal larly, Koopmans (1960, p. 292) acknowl- rate of substitution between consump- edged that, “One cannot claim a high tion in periods τ and τ′ is independent degree of realism for [the indepen- of consumption in period τ″. dence assumption], because there is no Consumption independence is analo- clear reason why complementarity of gous to, but fundamentally different from, goods could not extend over more than the independence axiom of expected- one time period.” utility theory. In expected-utility the- ory, the independence axiom specifies 3.4 Stationary Instantaneous Utility that preferences over uncertain pros- When applying the DU model to spe- 5 “Utility independence” has meaning only if cific problems, it is often assumed that one literally interprets u(ct + k) as well-being expe- the cardinal instantaneous utility func- rienced in period t + k. We believe that this is, in fact, the common interpretation. For a model that tion u(cτ) is constant across time, so that relaxes the assumption of utility independence, the well-being generated by any activity see Benjamin Hermalin and Alice Isen (2000), is the same in different periods. Most who consider a model in which well-being in period t depends on well-being in period t – 1— economists would acknowledge that sta- i.e., they assume u t = u(c t, u t – 1). See also Daniel tionarity of the instantaneous utility Kahneman, Peter Wakker, and Rakesh Sarin function is not sensible in many situ- (1997) who propose a set of axioms that would justify an assumption of additive separability in ations, because people’s preferences do, instantaneous utility. in fact, change over time in predictable 358 Journal of Economic Literature, Vol. XL (June 2002) and unpredictable ways. Though this period discount rate (ρ n = ρ for all unrealistic assumption is often retained n). 7 for analytical convenience, it becomes less Constant discounting entails an even- defensible as economists gain insight handedness in the way a person evalu- into how tastes change over time (see ates time. It means that delaying or Loewenstein and Angner, forthcoming, accelerating two dated outcomes by a for a discussion of different sources of common amount should not change preference change). 6 preferences between the outcomes—if in period t a person prefers X at τ to Y 3.5 Independence of Discounting at τ + d for some τ, then in period t she from Consumption must prefer X at τ to Y at τ + d for all τ. The assumption of constant discounting The DU model assumes that the dis- permits a person’s time preference to count function is invariant across all be summarized as a single discount forms of consumption. This feature is rate. If constant discounting does not crucial to the notion of time preference. hold, then characterizing one’s time If people discount utility from different preference requires the specification of sources at different rates, then the no- an entire discount function. tion of a unitary time preference is Constant discounting implies that a meaningless. Instead we would need to person’s intertemporal preferences are label time preference according to the time-consistent, which means that later object being delayed—”banana time preferences “confirm” earlier prefer- preference,” “vacation time prefer- ences. Formally, a person’s preferences ence,” and so on. In section 7, we dis- are time-consistent if, for any two con- cuss in more detail the validity of the sumption profiles (ct,...,cT) and (c′t,...,c′T), assumption that the same rate of time with ct = c′t, U t(ct,ct + 1,...,cT) ≥ U t(c′t,c′t + 1, preference applies to all forms of...,c′T) if and only if U t + 1(ct + 1,...,cT) ≥ consumption. U t + 1(c′t + 1,...,c′T). 8 For an interesting dis- 3.6 Constant Discounting and Time cussion that questions the normative va- Consistency lidity of constant discounting, see Martin Albrecht and Martin Weber (1995). Any discount function 1can be written in the form D(k) = Π kn −= 10  1 + ρ  , where ρ n rep-  n  3.7 Diminishing Marginal Utility resents the per-period discount rate and Positive Time Preference for period n—that is, the discount rate While not core features of the DU applied between periods n and n + 1. model, virtually all analyses of intertem- Hence, by assuming that the discount  1  k poral choice assume both diminishing function takes the form D(k) =  1 + ρ  , the DU model assumes a constant per- 7 An alternative but equivalent definition of con- stant discounting is that D(k)/D(k + 1) is indepen- 6 As we discuss in section 5, endogenous prefer- dent of k. ence changes, due to things such as habit forma- 8 Constant discounting implies time-consistent tion or reference dependence, are best understood preferences only under the ancillary assumption in terms of consumption interdependence and not of stationary discounting, for which the dis- nonstationary utility. In some situations, nonsta- count function D(k) is the same in all periods. As a tionarities clearly play an important role in behav- counterexample,  1  k if the period-t discount function ior—e.g., Steven Suranovic, Robert Goldfarb, and is D t(k) =  1 + ρ  while  the 1  k period-t + 1 discount Thomas Leonard (1999), and O’Donoghue and function is D t + 1(k) =  1 + ρ′  for some ρ′ ≠ ρ, then Mathew Rabin (1999a; 2000) discuss the impor- the person exhibits constant discounting at both tance of nonstationarities in the realm of addictive dates t and t + 1, but nonetheless has time- behavior. inconsistent preferences.  Frederick, Loewenstein, and O’Donoghue: Time Discounting 359 marginal utility (that the instantaneous erence was made by Derek Parfit (1971; utility function u(ct) is concave) and posi- 1976; 1982), who contends that there is tive time preference (that the discount rate no enduring self or “I” over time to ρ is positive). 9 These two assumptions which all future utility can be ascribed, create opposing forces in intertemporal and that a diminution in psychological choice: diminishing marginal utility mo- connections gives our descendent fu- tivates a person to spread consumption ture selves the status of other people— over time, while positive time prefer- making that utility less than fully ence motivates a person to concentrate “ours” and giving us a reason to count it consumption in the present. less: 11 Since people do, in fact, spread con- sumption over time, the assumption of We care less about our further future... because we know that less of what we are diminishing marginal utility (or some now—less, say, of our present hopes or plans, other property that has the same effect) loves or ideals—will survive into the further seems strongly justified. The assump- future... [if] what matters holds to a lesser tion of positive time preference, on the degree, it cannot be irrational to care less. other hand, is more questionable. Sev- (Parfit 1971, p. 99) eral researchers have argued for posi- Parfit’s claims are normative, not de- tive time preference on logical grounds scriptive. He is not attempting to ex- (Jack Hirshleifer 1970; Koopmans 1960; plain or predict people’s intertemporal Koopmans, Peter A. Diamond, and choices, but is arguing that conclusions Richard E. Williamson 1964; Olson and about the rationality of time preference Bailey 1981). The gist of their argu- must be grounded in a correct view of ments is that a zero or negative time personal identity. However, if this is the preference, combined with a positive only compelling normative rationale for real rate of return on saving, would time discounting, it would be instruc- command the infinite deferral of all tive to test for a positive relation be- consumption. 10 But this conclusion as- tween observed time discounting and sumes, unrealistically, that individuals changing identity. Frederick (2002) have infinite life-spans and linear (or conducted the only study of this type, weakly concave) utility functions. Never- theless, in econometric analyses of sav- 11 As noted by Frederick (2002), there is much ings and intertemporal substitution, posi- disagreement about the nature of Parfit’s claim. In tive time preference is sometimes treated her review of the philosophical literature, Jennifer Whiting (1986, p. 549) identifies four different in- as an identifying restriction whose vio- terpretations: (1) the strong absolute claim: that it lation is interpreted as evidence of is irrational for someone to care about their future misspecification. welfare, (2) the weak absolute claim: that there is no rational requirement to care about one’s future The most compelling argument sup- welfare, (3) the strong comparative claim: that it is porting the logic of positive time pref- irrational to care more about one’s own future welfare than about the welfare of any other per- 9 Discounting is not inherent to the DU model, son, and (4) the weak comparative claim: that one because the model could be applied with ρ ≤ 0. is not rationally required to care more about their However, the inclusion of ρ in the model strongly future welfare than about the welfare of any other implies that it may take a value other than zero, person. We believe that all of these interpretations and the name discount rate certainly suggests that are too strong, and that Parfit endorses only a it is greater than zero. weaker version of the weak absolute claim. That is, 10 In the context of intergenerational choice, he claims only that one is not rationally required Koopmans (1967) called this result the paradox of to care about one’s future welfare to a degree that the indefinitely postponed splurge. See also Ken- exceeds the degree of psychological connectedness neth J. Arrow (1983), S. Chakravarty (1962), and that obtains between one’s current self and one’s Robert M. Solow (1974). future self. 360 Journal of Economic Literature, Vol. XL (June 2002) and found no relation between mone- larger-later reward (see section 6 for a tary discount rates (as imputed from description of these procedures), the procedures such as “I would be indiffer- implicit discount rate over longer time ent between $100 tomorrow and $____ horizons is lower than the implicit dis- in five years”) and self-perceived stabil- count rate over shorter time horizons. ity of identity (as defined by the follow- For example, Richard Thaler (1981) ing similarity ratings: “Compared to asked subjects to specify the amount of now, how similar were you five years money they would require in [one ago [will you be five years from month/one year/ten years] to make them now]?”), nor did he find any relation indifferent to receiving $15 now. The between such monetary discount rates median responses [$20/$50/$100] imply and the presumed correlates of identity an average (annual) discount rate of stability (e.g., the extent to which peo- 345 percent over a one-month horizon, ple agree with the statement “I am still 120 percent over a one-year horizon, embarrassed by stupid things I did a and 19 percent over a ten-year hori- long time ago”). zon. 12 Other researchers have found a similar pattern (Uri Benzion, Amnon 4. DU Anomalies Rapoport, and Joseph Yagil 1989; Gretchen B. Chapman 1996; Chapman Over the last two decades, empirical and Arthur S. Elstein 1995; John L. research on intertemporal choice has Pender 1996; Daniel A. Redelmeier and documented various inadequacies of the Daniel N. Heller 1993). DU model as a descriptive model of be- Second, when mathematical functions havior. First, empirically observed dis- are explicitly fit to such data, a hyper- count rates are not constant over time, bolic functional form, which imposes but appear to decline—a pattern often declining discount rates, fits the data referred to as hyperbolic discounting. better than the exponential functional Furthermore, even for a given delay, form, which imposes constant discount discount rates vary across different rates (Kris N. Kirby 1997; Kirby and Nino types of intertemporal choices: gains Marakovic 1995; Joel Myerson and Leon- are discounted more than losses, small ard Green 1995; Howard Rachlin, Andres amounts more than large amounts, and Raineri, and David Cross 1991). 13 explicit sequences of multiple outcomes Third, researchers have shown that are discounted differently than outcomes 12 That is, $15 = $20∗(e–(3.45)(1/12)) = $50∗(e–(1.20)(1)) = considered singly. $100∗(e–(0.19)(10)). While most empirical studies re- port average discount rates over a given horizon, it 4.1 Hyperbolic Discounting is sometimes more useful to discuss average “per- period” discount rates. Framed in these terms, The best documented DU anomaly Thaler’s results imply an average (annual) discount is hyperbolic discounting. The term rate of 345 percent between now and one month “hyperbolic discounting” is often used from now, 100 percent between one month from now and one year from now, and 7.7 percent to mean, in our terminology, that a per- between one year from now and ten years son has a declining rate of time prefer- from now. That is, $15 = $20∗(e–(3.45)(1/12)) = ence (in our notation, ρ n is declining in $50∗(e–(3.45)(1/12) e–(1.00)(11/12)) = $100 ∗ (e –(3.45)(1/12) e –(1.00)(11/12)e –(0.077)(9)). n), and we adopt this meaning here. 13 Several hyperbolic functional forms have Several results are usually interpreted been proposed: George Ainslie (1975) suggested as evidence for hyperbolic discounting. the function D(t) = 1/t, Richard Herrnstein (1981) and James Mazur (1987) suggested D(t) = 1/(1 + αt), First, when subjects are asked to com- and George Loewenstein and Drazen Prelec (1992) pare a smaller-sooner reward to a suggested D(t) = 1/(1 + αt) β/α.  Frederick, Loewenstein, and O’Donoghue: Time Discounting 361 preferences between two delayed re- evidence that discount rates continue to wards can reverse in favor of the more decline. In fact, after excluding the stud- proximate reward as the time to both ies with short time horizons, the corre- rewards diminishes—e.g., someone may lation between time horizon and discount prefer $110 in 31 days over $100 in 30 factor is almost exactly zero (–0.0026). days, but also prefer $100 now over Although the collective evidence out- $110 tomorrow. Such “preference re- lined above seems overwhelmingly to versals” have been observed both in support hyperbolic discounting, a re- humans (Green, Nathaniel Fristoe, and cent study by Daniel Read (2001) Myerson 1994; Kirby and Herrnstein points out that the most common type 1995; Andrew Millar and Douglas of evidence—the finding that implicit Navarick 1984; Jay Solnick et al. 1980) discount rates decrease with the time and in pigeons (Ainslie and Herrnstein horizon—could also be explained by 1981; Green et al. 1981). 14 “subadditive discounting,” which means Fourth, the pattern of declining dis- the total amount of discounting over a count rates suggested by the studies temporal interval increases as the inter- above is also evident across studies. In val is more finely partitioned. 16 To dem- section 6, we summarize studies that es- onstrate subadditive discounting and timate discount rates. Figure 1a plots distinguish it from hyperbolic discount- the average estimated discount factor ing, Read elicited discount rates for a two- (= 1/(1 + discount rate)) from each of year (24-month) interval and for its three these studies against the average time constituent intervals, an eight-month horizon for that study. 15 As the regres- interval beginning at the same time, an sion line reflects, the estimated dis- eight-month interval beginning eight count factor increases with the time ho- months later, and an eight-month inter- rizon, which means that the discount val beginning sixteen months later. He rate declines. We note, however, that found that the average discount rate after excluding studies with very short for the 24-month interval was lower than time horizons (one year or less) from the compounded average discount rate the analysis (see figure 1b), there is no over the three eight-month subintervals— a result predicted by subadditive dis- 14 These studies all demonstrate preference re- counting but not predicted by hyper- versals in the synchronic sense—subjects simulta- bolic discounting (or any type of discount neously prefer $100 now over $110 tomorrow and function, for that matter). Moreover, prefer $110 in 31 days over $100 in 30 days, which is consistent with hyperbolic discounting. But there was no evidence that discount rates there seems to be an implicit belief that such pref- declined with time, as the discount erence reversals would also hold in the diachronic rates for the three eight-month inter- sense—that if subjects who currently prefer $110 in 31 days over $100 in 30 days were brought back vals were approximately equal. Similar to the lab thirty days later, they would prefer $100 empirical results were found earlier by at that time over $110 one day later. Under the J. H. Holcomb and P. S. Nelson (1992), assumption of stationary discounting (as discussed in footnote 8), synchronic preference reversals im- ply diachronic preference reversals. To the extent 16 Read’s proposal that discounting is subaddi- that subjects anticipate diachronic reversals and tive is compatible with analogous results in other want to avoid them, evidence of a preference for domains. For example, Amos Tversky and Derek commitment could also be interpreted as evidence Koehler (1994) found that the total probability as- for hyperbolic discounting (we discuss this issue signed to an event increases the more finely the more in section 5.1.1). event is partitioned—e.g., the probability of 15 In some cases, the discount rates were com- “death by accident” is judged to be more likely if puted from the median respondent. In other one separately elicits the probability of “death by cases, the mean discount rate was used. fire,” “death by drowning,” “death by falling,” etc. 362 Journal of Economic Literature, Vol. XL (June 2002) 1.0 1.0 imputed discount factor imputed discount factor 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 5 10 15 0 5 10 15 time horizon (years) time horizon (years) Figure 1a. Discount Factor as a Function of Time Figure 1b. Discount Factor as a Function of Time Horizon (all studies) Horizon (studies with avg. horizons > 1 year) although they did not interpret their 4.2 Other DU Anomalies results the same way. The DU model not only dictates that If Read is correct about subadditive the discount rate should be constant for discounting, its main implication for all time periods; it also assumes that the economic applications may be to provide discount rate should be the same for all an alternative psychological underpin- types of goods and all categories of ning for using a hyperbolic discount intertemporal decisions. There are sev- function, because most intertemporal eral empirical regularities that appear to decisions are based primarily on dis- contradict this assumption, namely: counting from the present. 17 (1) gains are discounted more than losses; (2) small amounts are discounted 17 A few studies have actually found increasing discount rates. Frederick (1999) asked 228 respon- more than large amounts; (3) greater dents to imagine that they worked at a job that discounting is shown to avoid delay consisted of both pleasant work (“good days”) and of a good than to expedite its receipt; unpleasant work (“bad days”) and to equate the attractiveness of having additional good days this (4) in choices over sequences of year or in a future year. On average, respondents outcomes, improving sequences are were indifferent between 20 extra good days this often preferred to declining sequences year, 21 the following year, or 40 in five years, implying a one-year discount rate of 5 percent and though positive time preference dic- a five-year discount rate of 15 percent. A possible tates the opposite; and (5) in choices explanation is that a desire for improvement is over sequences, violations of indepen- evoked more strongly for two successive years (this year and next) than for two separated years dence are pervasive, and people seem (this year and five years hence). Rubinstein (2000) to prefer spreading consumption over asked students in a political science class to choose time in a way that diminishing marginal between the following two payment sequences: utility alone cannot explain. March 1 June 1 Sept 1 Nov 1 A: $997 $997 $997 $997 4.2.1 The “Sign Effect” (gains are April 1 July1 Oct 1 Dec 1 discounted more than losses) B: $1000 $1000 $1000 $1000 Then, two weeks later, he asked them to choose Many studies have concluded that between $997 on November 1 and $1000 on gains are discounted at a higher rate December 1. Fifty-four percent of respondents than losses. For instance, Thaler (1981) preferred $997 in November to $1000 in Decem- ber, but only 34 percent preferred sequence A to sequence B. These two results suggest increasing ments may have masked the differences in the discount rates. To explain them Rubinstein specu- timing of the sequence of dated amounts, while lated that the three more proximate additional ele- making the differences in amounts more salient.  Frederick, Loewenstein, and O’Donoghue: Time Discounting 363 asked subjects to imagine they had re- change in delivery time of an outcome ceived a traffic ticket that could be paid is framed as an acceleration or a delay either now or later and to state how from some temporal reference point. much they would be willing to pay if For example, respondents who didn’t payment could be delayed (by three expect to receive a VCR for another months, one year, or three years). The year would pay an average of $54 to re- discount rates imputed from these an- ceive it immediately, but those who swers were much lower than the discount thought they would receive it immedi- rates imputed from comparable questions ately demanded an average of $126 to about monetary gains. This pattern is delay its receipt by a year. Benzion, prevalent in the literature. Indeed, in many Rapoport, and Yagil (1989) and Shelley studies, a substantial proportion of sub- (1993) replicated Loewenstein’s findings jects prefer to incur a loss immediately for losses as well as gains (respondents rather than delay it (Benzion, Rapoport, demanded more to expedite payment and Yagil 1989; Loewenstein 1987; L. D. than they would pay to delay it). MacKeigan et al. 1993; Walter Mischel, 4.2.4 Preference for Improving Joan Grusec, and John C. Masters 1969; Sequences Redelmeier and Heller 1993; J. Frank Yates and Royce A. Watts 1975). In studies of discounting that involve choices between two outcomes—e.g., X 4.2.2 The “Magnitude Effect” (small at τ vs. Y at τ′—positive discounting is outcomes are discounted more the norm. Research examining prefer- than large ones) ences over sequences of outcomes, how- Most studies that vary outcome size ever, has generally found that people have found that large outcomes are prefer improving sequences to declin- discounted at a lower rate than small ing sequences (for an overview, see ones (Ainslie and Varda Haendel 1983; Ariely and Carmon, in press; Frederick Benzion, Rapoport, and Yagil 1989; Green, and Loewenstein 2002; Loewenstein and Fristoe, and Myerson 1994; Green, Prelec 1993). For example, Loewen- Astrid Fry, and Myerson 1994; Hol- stein and Nachum Sicherman (1991) comb and Nelson 1992; Kirby 1997; found that, for an otherwise identical Kirby and Marakovic 1995; Kirby, job, most subjects prefer an increasing Nancy Petry and Warren Bickel 1999; wage profile to a declining or flat one Loewenstein 1987; Raineri and Rachlin (see also Robert Frank 1993). Christo- 1993; Marjorie K. Shelley 1993; Thaler pher Hsee, Robert P. Abelson, and 1981). In Thaler’s (1981) study, for ex- Peter Salovey (1991) found that an in- ample, respondents were, on average, creasing salary sequence was rated as indifferent between $15 immediately highly as a decreasing sequence that and $60 in a year, $250 immediately conferred much more money. Carol and $350 in a year, and $3000 immedi- Varey and Kahneman (1992) found that ately and $4000 in a year, implying dis- subjects strongly preferred streams of count rates of 139 percent, 34 percent, decreasing discomfort to streams of in- and 29 percent, respectively. creasing discomfort, even when the over- all sum of discomfort over the interval 4.2.3 The “Delay-Speedup” Asymmetry was otherwise identical. Loewenstein Loewenstein (1988) demonstrated and Prelec (1993) found that respon- that imputed discount rates can be dents who chose between sequences of dramatically affected by whether the two or more events (e.g., dinners or 364 Journal of Economic Literature, Vol. XL (June 2002) vacation trips) on consecutive weekends which consumption is identical in the or consecutive months generally pre- two profiles. Thus, anyone preferring ferred to save the better thing for last. profile B to profile A (which share the Chapman (2000) presented respondents fifth period “Eat at Home”) should also with hypothetical sequences of head- prefer profile D to profile C (which ache pain that were matched in terms share the fifth period “Fancy Lobster”). of total pain that either gradually less- As the data reveal, however, many ened or gradually increased with time. respondents violated this prediction, Sequence durations included one hour, preferring the fancy French dinner on one day, one month, one year, five the third weekend, if that was the only years, and twenty years. For all se- fancy dinner in the profile, but prefer- quence durations, the vast majority ring the fancy French dinner on the (from 82 percent to 92 percent) of sub- first weekend if the profile contained jects preferred the sequence of pain another fancy dinner. This result could that lessened over time. (See also W. T. be explained by the simple desire to Ross, Jr. and I. Simonson 1991). spread consumption over time—which, in this context, violates the dubious as- 4.2.5 Violations of Independence sumption of independence that the DU and Preference for Spread model entails. The research on preferences over se- Loewenstein and Prelec (1993) pro- quences also reveals strong violations of vide further evidence of such a prefer- independence. Consider the following ence for spread. Subjects were asked to pair of questions from Loewenstein and imagine that they were given two cou- Prelec (1993): pons for fancy ($100) restaurant din- Imagine that over the next five weekends you must ners, and were asked to indicate when decide how to spend your Saturday nights. From each they would use them, ignoring consid- pair of sequences of dinners below, circle the one you erations such as holidays, birthdays, and would prefer. “Fancy French” refers to a dinner at a fancy French restaurant. “Fancy Lobster” refers to an such. Subjects either were told that exquisite lobster dinner at a four-star restaurant. Ignore “you can use the coupons at any time scheduling considerations (e.g., your current plans). between today and two years from to- first second third fourth fifth day” or were told nothing about any weekend weekend weekend weekend weekend constraints. Subjects in the two-year Option A constraint condition actually scheduled Fancy Eat at Eat at Eat at Eat at [11%] both dinners at a later time than those French home home home home who faced no explicit constraint—they Option B Eat at Eat at Fancy Eat at Eat at [89%] delayed the first dinner for eight weeks home home French home home (rather than three) and the second din- Option C ner for 31 weeks (rather than thirteen). Fancy Eat at Eat at Eat at Fancy [49%] This counterintuitive result can be ex- French home home home Lobster plained in terms of a preference for Option D spread if the explicit two-year interval Eat at Eat at Fancy Eat at Fancy [51%] home home French home Lobster was greater than the implicit time hori- zon of subjects in the unconstrained As discussed in section 3.3, consump- group. tion independence implies that prefer- 4.3 Are These “Anomalies” Mistakes? ences between two consumption pro- files should not be affected by the In other domains of judgment and nature of the consumption in periods in choice, many of the famous “effects”  Frederick, Loewenstein, and O’Donoghue: Time Discounting 365 that have been documented are re- coordinate their responses to conform garded as errors by the people who to DU’s postulates when they evaluated commit them. For example, in the “con- rewards of different sizes, it suggests junction fallacy” discovered by Tversky that they consider the different dis- and Kahneman (1983), many people will— count rates to be normatively appropri- with some reflection—recognize that a ate. Similarly, even after Loewenstein conjunction cannot be more likely than and Sicherman (1991) informed respon- one of its constituents (e.g., that it can’t dents that a decreasing wage profile be more likely for Linda to be a femi- ($27,000, $26,000,... $23,000) would nist bank teller than for her to be (via appropriate saving and investing) “just” a bank teller). In contrast, the permit strictly more consumption in patterns of preferences that are re- every period than the corresponding garded as “anomalies” in the context increasing wage profile with an equiv- of the DU model do not necessarily vio- alent nominal total ($23,000, $24,000, late any standard or principle that peo-... $27,000), respondents still pre- ple believe they should uphold. Even ferred the increasing sequence. Perhaps when the choice pattern is pointed out they suspected that they could not to people, they do not regard them- exercise the required self control to selves as having made a mistake (and maintain their desired consumption probably have not made one!). For sequence, or felt a general leeriness example, there is no compelling logic about the significance of a declining that dictates that one who prefers to wage, either of which could justify delay a French dinner should also pre- that choice. As these examples illus- fer to do so when that French dinner trate, many DU “anomalies” exist as will be closely followed by a lobster “anomalies” only by reference to a model dinner. that was constructed without regard Indeed, it is unclear whether any of to its descriptive validity, and which the DU “anomalies” should be regarded has no compelling normative basis. as mistakes. Frederick and Read (2002) found evidence that the magnitude ef- fect is more pronounced when subjects 5. Alternative Models evaluate both “small” and “large” amounts than when they evaluate either In response to the anomalies just one. Specifically, the difference in the enumerated, and other intertemporal- discount rates between a small amount choice phenomena that are inconsistent ($10) and a large amount ($1000) was with the DU model, a variety of alter- larger when the two judgments were nate theoretical models have been made in close succession than when developed. Some models attempt to they were made separately. Analogous achieve greater descriptive realism by results were obtained for the sign ef- relaxing the assumption of constant fect, as the differences in discount discounting. Other models incorporate rates between gains and losses were additional considerations into the in- slightly larger in a within-subjects stantaneous utility function, such as design, where respondents evaluated the utility from anticipation. Still others delayed gains and delayed losses, than depart from the DU model more in a between-subjects design where radically, by including, for instance, they evaluate only gains or only losses. systematic mispredictions of future Since respondents did not attempt to utility. 366 Journal of Economic Literature, Vol. XL (June 2002) 5.1 Models of Hyperbolic Discounting ing by Jon Elster (1979). It assumes that the per-period discount rate between In the economics literature, R. H. now and the next period is 1 −βδβδ whereas Strotz (1955–56) was the first to con- the per-period discount rate between sider alternatives to exponential dis- any two future periods is 1 −δ δ < 1 −βδβδ. counting, seeing “no reason why an Hence, this (β,δ) formulation assumes a individual should have such a special declining discount rate between this pe- discount function” (p. 172). Moreover, riod and next, but a constant discount Strotz recognized that for any discount rate thereafter. The (β,δ) formulation is function other than exponential, a highly tractable, and captures many of person would have time-inconsistent the qualitative implications of hyperbolic preferences. 18 He proposed two strate- discounting. gies that might be employed by a per- Laibson and his collaborators have son who foresees how her preferences used the (β,δ) formulation to explore will change over time: the “strategy of the implications of hyperbolic discount- precommitment” (wherein she commits ing for consumption-saving behavior. to some plan of action) and the “strat- Hyperbolic discounting leads a person egy of consistent planning” (wherein to consume more than she would like she chooses her behavior ignoring plans from a prior perspective (or, equiva- that she knows her future selves will lently, to under-save). Laibson (1997) not carry out). 19 While Strotz did not explores the role of illiquid assets, such posit any specific alternative functional as housing, as an imperfect commit- forms, he did suggest that “special ment technology, emphasizing how a attention” be given to the case of person could limit overconsumption by declining discount rates. tying up her wealth in illiquid assets. Motivated by the evidence discussed Laibson (1998) explores consumption- in section 4.1, there has been a recent saving decisions in a world without illiq- surge of interest among economists in uid assets (or any other commitment the implications of declining discount technology). These papers describe how rates (beginning with David Laibson hyperbolic discounting might explain 1994, 1997). This literature has used a some stylized empirical facts, such as particularly simple functional form which the excess comovement of income and captures the essence of hyperbolic consumption, the existence of asset-spe- discounting: cific marginal propensities to consume,  1 if h = 0 low levels of precautionary savings, and D(k) =  k βδ if k > 0. the correlation of measured levels of This functional form was first introduced patience with age, income, and wealth. by E. S. Phelps and Pollak (1968) to Laibson, Andrea Repetto, and Jeremy study intergenerational altruism, and was Tobacman (1998), and George-Marios first applied to individual decision mak- Angeletos et al. (2001) calibrate models of consumption-saving decisions, using 18 Strotz implicitly assumes stationary discount- both exponential discounting and (β,δ) ing. hyperbolic discounting. By comparing 19 Building on Strotz’s strategy of consistent planning, some researchers have addressed the simulated data to real-world data, they question of whether there exists a consistent path demonstrate how hyperbolic discount- for general non-exponential discount functions. ing can better explain a variety of See in particular Robert Pollak (1968), Bezalel Peleg and Menahem Yaari (1973), and Steven empirical observations in the consump- Goldman (1980). tion-saving literature. In particular,  Frederick, Loewenstein, and O’Donoghue: Time Discounting 367 Angeletos et al. (2001) describe how O’Donoghue and Rabin (1999a, hyperbolic discounting can explain 2000), Jonathan Gruber and Botond the coexistence of high preretirement Koszegi (2000), and Juan D. Carrillo wealth, low liquid asset holdings (rela- (1999) have applied (β,δ) preferences tive to income levels and illiquid asset to addiction. These researchers de- holdings), and high credit-card debt. scribe how hyperbolic discounting can Carolyn Fischer (1999) and lead people to overconsume harmful O’Donoghue and Rabin (1999c, 2001) addictive products, and examine the have applied (β,δ) preferences to pro- degree of harm caused by such over- crastination, where hyperbolic discount- consumption. Carrillo and Thomas ing leads a person to put off an onerous Mariotti (2000) and Roland Benabou activity more than she would like from a and Jean Tirole (2000) have examined prior perspective. 20 O’Donoghue and how (β,δ) preferences might influence a Rabin (1999c) examine the implications person’s decision to acquire informa- of hyperbolic discounting for contract- tion. If, for example, a person is decid- ing when a principal is concerned with ing whether to embark on a specific combating procrastination by an agent. research agenda, she may have the op- They show how incentive schemes with tion to get feedback from colleagues “deadlines” may be a useful screening about its likely fruitfulness. The stan- device to distinguish efficient delay from dard economic model implies that peo- inefficient procrastination. O’Donoghue ple should always choose to acquire this and Rabin (2001) explore procrastina- information if it is free. However, Car- tion when a person must not only rillo and Mariotti show that hyperbolic choose when to complete a task, but discounting can lead to “strategic igno- also which task to complete. They show rance”—a person with hyperbolic dis- that a person might never carry out a counting who is worried about with- very easy and very good option because drawing from an advantageous course of they continually plan to carry out an action when the costs become imminent even better but more onerous option. might choose not to acquire free infor- For instance, a person might never take mation if doing so increases the risk of half an hour to straighten the shelves in bailing out. her garage because she persistently plans to take an entire day to do a major 5.1.1 Self Awareness cleanup of the entire garage. Extending this logic, they show that providing peo- A person with time-inconsistent pref- ple with new options might make pro- erences may or may not be aware that crastination more likely. If the person’s her preferences will change over time. only option were to straighten the Strotz (1955–56) and Pollak (1968) shelves, she might do it in a timely discussed two extreme alternatives. At manner; but if the person can either one extreme, a person could be com- straighten the shelves or do the major pletely “naïve” and believe that her cleanup, she now may do nothing. future preferences will be identical O’Donoghue and Rabin (1999d) apply to her current preferences. At the this logic to retirement planning. other extreme, a person could be com- pletely “sophisticated” and correctly 20 While not framed in terms of hyperbolic dis- predict how her preferences will counting, George Akerlof’s (1991) model of pro- crastination is formally equivalent to a hyperbolic change over time. While casual observa- model. tion and introspection suggest that 368 Journal of Economic Literature, Vol. XL (June 2002) people lie somewhere in between these than those with evenly spaced dead- two extremes, behavioral evidence re- lines (whether externally imposed or garding the degree of awareness is self-imposed). 21 quite limited. O’Donoghue and Rabin (1999b) ex- One way to identify sophistication is amine how people’s behaviors depend to look for evidence of commitment. on their sophistication about their own Someone who suspects that her prefer- time inconsistency. Some behaviors, such ences will change over time might take as using illiquid assets for commit- steps to eliminate an option that seems ment, require some degree of sophisti- inferior now but might tempt her later. cation. Other behaviors, such as over- For example, someone who currently consumption or procrastination, are prefers $110 in 31 days to $100 in 30 more robust to the degree of aware- days but who suspects that in a month ness, though the degree of misbehavior she will prefer $100 immediately to may depend on the degree of sophisti- $110 tomorrow, might attempt to elimi- cation. To understand such effects, nate the $100 reward from the later O’Donoghue and Rabin (2001) intro- choice set, and thereby bind herself duce a formal model of partial naïveté, now to receive the $110 reward in 31 in which a person is aware that she will days. Real-world examples of commit- have future self-control problems but ment include “Christmas clubs” or “fat underestimates their magnitude. They farms.” show that severe procrastination cannot Perhaps the best empirical demon- occur under complete sophistication, stration of a preference for commit- but can arise even if the person is only ment was conducted by Dan Ariely and a little naïve. For more discussion on Klaus Wertenbroch (2002). In that self-awareness, see O’Donoghue and study, MIT executive-education stud- Rabin (in press). ents had to write three short papers The degree of sophistication versus for a class and were assigned to one naiveté has important implications for of two experimental conditions. In one public policy. If people are sufficiently condition, deadlines for the three pa- sophisticated about their own self- pers were imposed by the instructor control problems, providing commit- and were evenly spaced across the se- ment devices may be beneficial. How- mester. In the other condition, each ever, if people are naïve, policies student was allowed to set her own might be better aimed at either edu- deadlines for each of the three papers. cating people about loss of control In both conditions, the penalty for (making them more sophisticated), or delay was 1 percent per day late, re- providing incentives for people to gardless of whether the deadline was use commitment devices, even if externally or self-imposed. Although they don’t recognize the need for students in the free-choice condition them. could have made all three papers due at the end of the semester, many did, in 21 A similar “natural” experiment was recently fact, choose to impose deadlines on conducted by the Economic and Social Research Council of Great Britain. They recently eliminated themselves, suggesting that they ap- submission deadlines and now accept grant pro- preciated the value of commitment. posals on a “rolling” basis (though they are still Few students chose evenly spaced reviewed only periodically). In response to this policy change, submissions have actually declined deadlines, however, and those who by about 15–20 percent (direct correspondence did not performed worse in the course with Chris Caswill at ESRC).  Frederick, Loewenstein, and O’Donoghue: Time Discounting 369 5.2 Models That Enrich the monotonic consumption profile. The di- Instantaneous Utility Function rection of the effect depends on things such as how much one has already con- Many discounting anomalies, espe- sumed (as reflected in the initial habit cially those in section 4.2, can be un- stock), and, perhaps most importantly, derstood as a misspecification of the whether current consumption increases instantaneous utility function. Similarly, or decreases future utility. many of the confounds we discuss in In recent years, habit-formation mod- section 6 are caused by researchers at- els have been used to analyze a variety tributing to the discount rate aspects of of phenomena. Gary Becker and Kevin preference that are more appropriately Murphy (1988) use a habit-formation considered as arguments in the instan- model to study addictive activities, and taneous utility function. As a result, in particular to examine the effects of alternative models of intertemporal past and future prices on the current choice have been advanced that add ad- consumption of addictive products. 22 ditional arguments, such as utility from Habit formation can help explain asset- anticipation, to the instantaneous utility pricing anomalies such as the equity- function. premium puzzle (Andrew Abel 1990; John 5.2.1 Habit-Formation Models Campbell and John Cochrane 1999; George M. Constantinides 1990). Incor- James Duesenberry (1952) was the porating habit formation into business- first economist to propose the idea of cycle models can improve their ability “habit formation”—that the utility from to explain movements in asset prices current consumption (“tastes”) can be (Urban Jermann 1998; Michele Boldrin, affected by the level of past consump- Lawrence Christiano, and Jonas Fisher tion. This idea was more formally devel- 2001). Some recent papers have shown oped by Pollak (1970) and Harl Ryder that habit formation may help explain and Geoffrey Heal (1973). In habit for- other empirical puzzles in macro- mation models, the period-τ instantane- economics as well. Whereas standard ous utility function takes the form growth models assume that high saving u(cτ;cτ − 1,cτ − 2,...) where ∂2u ⁄ ∂cτ ∂cτ ′ > 0 rates cause high growth, recent evi- for τ′ < τ. For simplicity, most such dence suggests that the causality can models assume that all effects of past run in the opposite direction. Christo- consumption for current utility enter pher Carroll, Jody Overland, and David through a state variable. That is, they Weil (2000) show that, under conditions assume that period-τ instantaneous util- of habit formation, high growth rates ity function takes the form u(cτ;zτ) can cause people to save more. Jeffrey where z τ is a state variable that is in- Fuhrer (2000) shows how habit forma- creasing in past consumption and tion might explain the recent finding ∂2 ⁄ ∂cτ∂zτ > 0. Both Pollak (1970) and that aggregate spending tends to have a Ryder and Heal (1973) assume that zτ is gradual “hump-shaped” response to the exponentially weighted sum of past ∞ 22 For rational-choice models building on consumption, or zτ = ∑ i = 1γ icτ − i. Becker and Murphy’s framework, see Athanasios Although habit formation is often Orphanides and David Zervos (1995), Ruqu Wang said to induce a preference for an in- (1997), and Suranovic, Goldfarb, and Leonard creasing consumption profile, it can, (1999). For addiction models that incorporate hyperbolic discounting, see O’Donoghue and under some circumstances, lead a per- Rabin (1999a, 2000), Gruber and Koszegi (2000), son to prefer a decreasing or even non- and Carrillo (1999). 370 Journal of Economic Literature, Vol. XL (June 2002) various shocks. The key feature of habit asymmetry. They show that if the elas- formation that drives many of these re- ticity of the value function is increasing sults is that, after a shock, consumption in the magnitude of outcomes, people adjustment is sluggish in the short term will discount smaller magnitudes more but not in the long term. than larger magnitudes. Intuitively, the elasticity condition captures the insight 5.2.2 Reference-Point Models that people are responsive to both dif- ferences and ratios of reward amounts. Closely related to, but conceptually It implies that someone who is indiffer- distinct from, habit-formation models ent between, say, $10 now and $20 in a are models of reference-dependent util- year should prefer $200 in a year over ity, which incorporate ideas from pros- $100 now because the larger rewards pect theory (Kahneman and Tversky have a greater difference (and the same 1979; Tversky and Kahneman 1991). ratio). Consequently, even if a person’s According to prospect theory, outcomes time preference is actually constant are evaluated using a value function de- across outcomes, she will be more will- fined over departures from a reference ing to wait for a fixed proportional in- point—in our notation, the period-τ in- crement when rewards are larger, and, stantaneous utility function takes the thus, her imputed discount rate will be form u(c τ, r τ) = v(c τ – r τ). The reference smaller for larger outcomes. Similarly, point, r τ, might depend on past con- if the value function for losses is more sumption, expectations, social compari- elastic than the value function for gains, son, status quo, and such. A second then people will discount gains more feature of prospect theory is that the than losses. Finally, such a model helps value function exhibits loss aversion— explain the delay-speedup asymmetry negative departures from one’s refer- (Loewenstein 1988). Shifting consump- ence consumption level decrease utility tion in any direction is made less desir- by a greater amount than positive de- able by loss aversion, since one loses partures increase it. A third feature of consumption in one period and gains it prospect theory is that the value func- in another. When delaying consump- tion exhibits—diminishing sensitivity for tion, loss aversion reinforces time dis- both gains and losses, which means that counting, creating a powerful aversion the value function is concave over gains to delay. When expediting consumption, and convex over losses. 23 loss aversion opposes time discounting, Loewenstein and Prelec (1992) ap- reducing the desirability of speedup plied a specialized version of such a (and, occasionally, even causing an value function to intertemporal choice aversion to it). to explain the magnitude effect, the Using a reference-dependent model sign effect, and the delay-speedup that assumes loss aversion in consump- 23 Reference-point models sometimes assume tion, David Bowman, Deborah Mine- there is a direct effect of the consumption level or hart, and Rabin (1999) predict that reference level, so that u(cτ,rτ) = v(cτ − rτ) + w(cτ) or “news” about one’s (stochastic) future u(cτ,rτ) = v(cτ − rτ) + w(rτ). Some habit-formation income affects one’s consumption models could be interpreted as reference-point models, where the state variable z τ is the refer- growth differently than the standard ence point. Indeed, many habit-formation models, Permanent Income Hypothesis predicts. such as Pollak (1970) and Constantinides (1990), According to (the log-linear version of) assume instantaneous utility functions of the form u(cτ − zτ), although they typically assume neither the Permanent Income Hypothesis, loss aversion nor diminishing sensitivity. changes in future income should not  Frederick, Loewenstein, and O’Donoghue: Time Discounting 371 affect the rate of consumption growth. Loewenstein describes how utility For example, if a person finds out that from anticipation may play a role in her permanent income will be lower many DU anomalies. Because near-term than she formerly thought, she would consumption delivers only consumption reduce her consumption by, say, 10 per- utility whereas future consumption de- cent in every period, leaving her con- livers both consumption utility and an- sumption growth unchanged. If, how- ticipatory utility, anticipatory utility ever, this person were loss averse in provides a reason to prefer improve- current consumption, she would be un- ment and for getting unpleasant out- willing to reduce this year’s consump- comes over with quickly instead of tion by 10 percent—forcing her to re- delaying them as discounting would duce future consumption by more than predict. It provides a possible explana- 10 percent, and thereby reducing the tion for why people discount different growth rate of her consumption. Two goods at different rates, because utility studies by John Shea (1995a,b) support from anticipation creates a downwa