M.-oI')/cf.CogIIitiOII1997,25 (5),715-723

Rate of temporal discounting decreaseswith amount of reward

The present, subjective value of a delayed reward is a decreasing function of the duration of thedelay. This phenomenon is termed temporal discounting. To detem1ine whether the amount of thereward influences the rate of temporal discounting, we had subj~ choose between immediate anddelayed hypothetical rewards of different amounts ($100, $2,000, $25,000, and $100,000 delayed re-wards). As predicted by psychological models of the choice process, hyperbolic functions describedthe decrease in the subjective value of the delayed reward as the time until its receipt was increased(R2S from .86 to .99). Moreover, hyperbolic functions consistently provided more accurate descrip-tions of the data than did exponential functions predicted by an economic model of discounted util-ity. Rate of discounting decreased in a negatively accelerated fashion as the amount of the delayedreward increased, leveling off by approximately $25,000. These findings are interpreted in the con-text of different psychological models of choice, and implications for procedures to enhance self-control are discussed

Temporal discounting refers to the fact that the greaterthe delay to a future reward, the less its present. subjectivevalue. For example, most people would prefer $1,000 nowover the same amount in 6 months. This is presumablybecause the subjective value of the delayed $1,000 isless; that is, people discount its value. Although $1,000now would obviously be preferable to $800 now, peoplemay discount delayed rewards to such a degree that theymight well prefer $800 now to the $1,000 in 6 months.

Temporal discounting bas been used to explain the pref-erence reversal phenomenon. Although an individual maychoose $800 now to $1,000 in 6 months, the same indi-vidual might well reverse his or her preference when anequal delay is added prior to the receipt of the amounts.That is, were a year to be added to the delay before the re-ceipt of the amounts, the same individual might now pre-fer the $1,000 in one and a half years to the $800 in oneyear. Such reversals pose problems for the standard ec0-nomic model of intertemporal choice (Koopmans, 1960;Samuelson, 1937) because they violate the stationarityassumption of discounted utility theory (Strotz, 1956; fora review, see Loewenstein, 1992). According to this as-sumption, preference should remain invariant when thedelays to both amounts are increased or decreased equally.

The preference-reversal phenomenon is important foran understanding of impulsivity and self-control. Impul-sivity can be defined as the choice of a smaller, more im-mediate reward over a larger, but more delayed reward;

A preliminary report of these findings was presented at the 1994meetina of the Psychonomic Society in St. Louis. E. McFadden is nowat the University of Connecticut. Correspondence concerning thismanuscript should be addressed to L. Green or J. Myerson, Depart-ment of Psychology, Washington University, Campus Box 112S,St. Louis, MO 63130 (e-mail: Igreen@artsci.wustl.edu).

LEONARD GREEN, JOEL MYERSON, and EDWARD McFADDENWashington University, St. Louis, Mis8OUri

self-control would be exemplified by the choice of thelarger, more delayed reward over the smaller, less de-layed reward (e.g., Green, 1982; Logue, 1988; Rachlin,1974; Rachlin & Green, 1972). The occurrence ofpref-erence reversals (in which the smaller reward is preferredwhen the delays to both rewards are brief but the largerr~ard is preferred when the delays to both rewards arelong) suggests that impulsivity may be a characteristic ofthe choice situation as much as a characteristic of the in-dividual making the choice. That is, the same individualwho acts impulsively when the smaller reward is avail-able immediately may nevertheless show self-controland choose the larger reward when both rewards are de-layed, even though the larger reward is mote delayedthan the smaller.

The form of the function describing the change insubjective value over time is critical in explaining thepreference-reversal phenomenon and related aspects ofself-COntrol.1\vo forms of temporal discounting ftmctionshave been proposed, an exponential model and a hyperbolicmodel (e.g., Ainslie, 1975, 1992; Loewenstein, 1992; Mazur,1987; Samuelson, 1937), along with variations on theseforms (Green, Fry, & Myerson, 1994; Loewenstein & Pre-lec, 1992; Myerson & Green, 1995). The exponentialmodel posits that people make their decisions about thevalue of future rewards on the basis of the assumption of aconstant hazard rate (i.e., people assume that the probabil-ity that something will happen to prevent receipt of a re-ward does not change over time). Conceptualized in thesame terms, the hyperbolic model implies that the hazardrate decreases over time. Thus, the two models imply al-ternative representations of the risks involved in waiting forfuture outcomes (Green & Myerson, 1996).

The exponential function, favored by economists, isgiven by the following equation:

715 Copyright 1997 Psychonomic Society, Inc.

GREEN, MYERSON, AND McFADDEN716

V(A)=Ae ~~~, (I)

where V(A) is the subjective value of a future reward ofamount A, D is the delay to its receipt, and kA representsthe parameter describing the rate of decrease in its value.The hyperbolic function, favored by psychologists, takesthe following form:

V(A) = A/(l + kAD). (2)

Fit to the same data, the hyperbola shows an initiallysteeper decrease in subjective value than the exponen-tial but a more gradual decrease in value as the delay in-creases, reflecting the different risk assumptions of thetwo models.

If it is assumed that the rate of temporal discounting(indexed by the parameter kA) is independent of theamount of the reward, then the exponential model (Equa-tion 1) leads to the prediction that the relative value oftwo delayed rewards will remain the same if the delaysto both rewards are increased or decreased equally. Thus,according to the exponential model, the preference-reversal phenomenon should not occur. In contrast,given the same assumption (i.e., kA is independent ofamount), the hyperbolic model (Equation 2) predicts thatpreference may reverse with equal changes in the delaysto both rewards (Ainslie, 1992; Mazur, 1987).

Recent findings suggest that larger rewards are dis-counted less steeply than smaller rewards (Benzion, Rap-oport, & YIgil, 1989; Myerson & Green, 1995; Raineri &Rachlin, 1993; Thaler, 1981). Hc;,NeVer, these studies haveexamined either few amounts or delays, or a narrowrange of amounts or delays, or have compared discount-ing rates using different groups of participants at the dif-ferent amounts. If the amount dependency of discountingrate is confirmed, this will have important theoreticalconsequences. As Green and Myerson (1993) haveshown, if the Qbserved amount dependency is incorpo-rated into the models so that kA decreases with amountof reward, then both models, the exponential as well asthe hyperbolic, predict the preference-reversal phenom-enon (see Green & Myerson, 1993, for a fuller discus-sion oftbis issue).

Given the importance of this issue, with its funda-mental implications for models of decision making, itbecomes necessary to rigorously test the hypothesis of

amount-dependent discounting suggested by previousresearch. Moreover, if the decrease in discounting ratewith increased amount is validated, then the details ofthe relationship between discounting rate and amount ~

need careful examination. Thus, an important question ~

becomes whether there is a point beyond which further:increases in amount have no additional effect on dis- ;counting rate, and if so, where that point is located. No 'published data address this question. ~

Therefore, the present study was designed to investi- I

gate more fully the effect of amount of reward on tem- :poral discounting. Specifically, subjects were exposed to Ichoices involving four different delayed amounts over a twide range of delays in order to evaluate the relationship r

between amount and discounting rate. In addition. al-though current psychological evidence favors a hyper-bolic model, most economic theorizing remains based onthe exponential model. Consequently, the present studyalso compared the fit of the hyperbolic and e1tponentialmodels at each amount. By evaluating the e1tponentialand hyperbolic models at a number of delayed amounts.the present study is intended to provide a more completeand rigorous assessment of the two models.

Jl(A.) - A..-tAD.

MElHOD

s.bjedITwenty-four Wuhington Univenity undeIJraduate students (20

women and 4 men) between the ages of 18 and 23 years (M = 20.6years) were paid for their voluntary participation in tbis study.

ProcedUftThe subjects were tested individually for about 50 min in a quiet

room containing a table and two chairs. The procedwe, similar tothat used in ~n et al. (1994), Rquired participants to make a se-ries of choices reaarding hypothetical amounts of money, oneavailable immediately and one available after a specified delay.The delayed and immediate amounts were presented oa 4 x 6 in.index cards. The subjects made a series of choices ~n a fixed-amount reward (e.a., SI00) that could be obtained after a delay(e.g., 3 months) and an immediately obtainable reward dlat variedin amount (e.g., from SI to $99). For example. the subject wasgiven a choice between $100 in 3 months or S85 now. The subjectindicated his/her preference by pointing to one of the ~ cards (ei-ther the delayed, faxed-amount reward or the immediate reward).

The subjects were read the following instructions prior to test-ing:

The purpose of die present study is to compare your preference be-tween dilrerent IDIOUnts of money available at difFerent points in time.In this study you will be asked 10 make a series of choicet between hy-

pothetical monetary alternatives. As you can see. there are two sets ofcards in front of you. The cards on your left wi II offer you an amountof InMey 10 be paid right now. This amOllftt will vary from card 10

card. Durinathis study. die amount on the cuds 10 your ript will beeither SIOO, S2,OOO, S25.000. or S 100.000, but its payment will be de-

layed.Pleue look at the example cards at mil time. It will be your job to

dIOoIe ~ the two cards preIenIed and ~ point 10 the card youwould prefer. It is important 10 bile your decision on only the twocards presented on each trial. Do not base your decision on leIs ofcards previously Ieen or ones you expect 10 see. There are IKI corrector iIXOI'rwct choicel. We - int8rellel1 iD the option you would ~.

You will be Ii"8n five .-actiee tria" before you blain, and 1 willturn the cards for you. You willae! plenty ofpraClice so do not worry

If you feel that you do DOt undentand completely at this time. Durinathe prlC!iC8 pleue uk me as many questions as you like because oncePf8CtiCI is over 1 can no Ionpr - your questions.

The seven delays used at each of the four fixed-amount rewards(i.e., SIOO, $2,000, $25,000, and $100,000) were 3 months, 6months, I year, 3 years, 5 years, 10 years, and 20 years. Theimmediate-reward cards depicted 24 values ranging from 1% 1099% of the delayed fixed-amount reward to which they were com-pared (e.g., when the fixed amount was $100, the immediate re-wards were $1, S2.50, $5. $7.50. S10, SI5, S20, S25, S30. S35. $40,S45, S50, S60, S65, $70, S75, S80, S85, S90, S92.50, S95, 597.50.and 599). For each fixed amount at each delay, the immediate re-wards were presented in either ascendina or descending order.When immediate rewards were presented in ascendina order. theamount of the immediate reward judged equal in subjective valueto the delayed reward was calculated as the averaae of the first im-mediate amount preferred over the delayed reward and the preced-

ing immediate amount. When immediate rewards were presentedin deacending order, the amount of the immediate reward judgedequal in subjective value to the delayed reward was calculated asthe average of the last immediate amount preferred over the de-layed reward and the next immediate amount.

The orders of presentation of the immediate amount and the de-layed amounts were both counterbalanced. Half of the participantswere offered the fixed-amount, delayed rewards from $100 up to$100,000; the other half were offered the fixed-amount rewards inthe reverse order. For the briefest delay at each fixed amount, halfof the subjects were given the immediate amounts in ascendingorder and half in descending order; at the next delay for this fixedamount, the half that had just been given the immediate amountsin ascending order now received them in descending order, andvice versa. Tbis alternation continued until all seven delays hadbeen tested at a given rIXed amount, at which point the next fIXed-amount condition began and the immediate rewards were pre-sented in the opposite order to that at the last fixed amount.

RESULTS

Table I gives the amount of the immediate rewardjudged equal in subjective value to the delayed reward,expressed as a proportion of the amount of the delayedreward. Data are group medians. The median was usedbecause the distributions of subjective equivalence pointsare usually negatively skewed when delays are brief andpositively skewed when delays are long (Rachlin, Raineri,& Cross, 1991). The skew is a consequence of the limitsimposed on subjects' choices; that is, the subjectivevalue of a delayed reward is less than its nominal valueand greater than zero.

Two results are apparent from Table I. First, temporaldiscounting occurred at each of the four amounts: Read-ing down the columns, one sees that the amount of theimmediate reward (expressed as a proportion of the de-layed amount) that is subjectively equivalent to the de-layed reward decreases systematically with delay for eachof the four delayed amounts. Second, the degree of dis-counting decreased as the amount of the delayed rewardincreased: Reading across the rows, one sees that theamount of the immediate reward that is subjectivelyequivalent to the delayed reward (expressed as a propor-tion of the delayed amount) increases systematicallyfrom $100 to $25,000, with no further change between$25,000 to $100,000.

Temporal discounting functions (e.g., Equations I and2) attempt to describe decreases in subjective value such

Table IAmount oC the Immediate Reward Judged Equal In SubjectiveValue to the Delayed Reward. Expressed as a Proportion oCtile

Amount oCtile Delayed Reward-Amount

Delay $ I ()() $2,000 $25,000 1100,000- - - 3 months .83 .95 .97 .98

6 months .7& .91 .95 ."I year .68 .89 .91 .933 years .60 .73 .78 .73S years .40 .sS .68 .6510 years .35 .38 .58 .sO20 yeaN .2Q .lJ .30 .30

AMOUNT-DEPENDENT TEMPORAL DISCOUNTING 711

as one sees reading down the columns in Table 1. Fig-ure 1 shows temporal discounting functions for the fourdelayed-reward amounts. For each delay, the data pointrepresents the group median of the immediate rewardsjudged equal in value to the delayed reward. The best fit-ting exponential (Equation 1) and hyperbolic (Equa-tion 2) functions are represented by dashed and solidcurves, respectively. Also shown is the proportion ofvariance accounted for (R2) by each equation. For eachof the four amounts, the proportion of variance ac-counted for by the hyperbolic was larger than that ac-counted for by the exponential function. This was be-cause, as may be seen in the figure, the exponentialtypically overestimated the subjective value of the de-layed reward at briefer delays, and underestimated thesubjective value at longer delays.

For both exponential and hyperbolic functions, thevalue of the discounting parameter k decreased with theamount of the delayed reward, leveling off by approxi-mately $25,000. Focusing on the hyperbolic model, whichprovided the better fit to the data, the same pattern wasalso observed when the hyperbolic function was fit toeach subject's data separately (see Table 2). This may beseen in Figure 2, which shows the k value of the hyper-bolic function fit to the median values and also the me-dian of the individual kvalues, both of which decrease asamount increases from $100 to $25,000. The pattern ofdecreases in k with increases in amount for the groupmedian subjective values was confirmed statisticallyusing nonlinear regression to compare the discount pa-rameters for larger and smaller amounts [$100 vs. $2,000,t(12) = 2.88, p < .05; $2,000 vs. $25,000, t(12) z 5.57,P < .01; $25,000 vs. $100,000, t(12)-1.91, n.s.].1

It may be recalled that for each combination of theseven delays and the four delayed amounts, half of thesubjects were studied with the immediate rewards pre-sented in order of increasing amount until preference re-versed and half were studied with the immediate rewardspresented in order of decreasing amount until preferencereversed. In addition to analyzing the data by combiningthe subjectively equivalent values from the ascendingand descending series, as shown in Figure 1, we also an-alyzed the data by taking the median subjectively equiv-alent values from the ascending and descending seriesseparately. As may be seen in Figure 3, there were no sys-tematic differences between the ascending and descend-ing series with respect to the amounts of the immediatereward judged subjectively equivalent to the delayed re-wards. This was confirmed statistically by nonlinear re-gression, which revealed no significant differences betweenthe discount parameters for ascending and descendingseries at any of the four delayed amounts [$100, t(12) <1.0; $2,000, t(12) < 1.0; $25,000, t(12) ~ 2.03; $100,000,t(12) < 1.0].

Figure 4 presents data from 4 individual subjects (the1st, 8th, 16th, and 24th subjects studied). For each sub-ject at each delayed amount, the subjective value of thedelayed reward at each of the seven delays is shown,along with the best fitting hyperbolic function (Equa-

718 GREEN, MYERSON, AND McFADDEN

-~-w::>-J

~w>~uw-,m:>(/)

FIau~ 1. PRseOt, subjectiw value of the deIa)ied ~nI as a function of the time untllCS receipt.Data for the four delayed amCMInCS an shown in separate panels. Data poinD ~~t the medianamount of immediate ~rd judled equal In subjective value to the delayed ~rd. SoUd and dashedcurws "Pfaent the belt fittinl hyperbolic and exponential dJscounting functions, ~.

tion 2). Although there is some variability, the same pat-tern of decreases in discounting rate as amount increasesobserved at the group level typically holds at the indi-vidual level as well.

The present findings demonstrate that the rate atwhich the value of a future reward is discounted dependson the amount of the reward. More specifically, the rateof discounting decreased in a negatively accelerated fash-ion as the amount of the reward was increased from $100to $25,000. Increasing the amount from $25,000 to$100,000 produced no further decrase in discounting rate.Given arnount-dependent discounting, the exponentialmodel (like the hyperbolic model) predicts that prefer-ence between smaller, sooner rewards and larger, laterrewards will depend on the time at which a choice ismade. Thus, preference reversals in and of themselvescannot be taken as evidence for the hyperbolic andagainst the exponential model. Nevertheless, the presentresults clearly show that a hyperbola provides a betterdescription of the relation between subjective value and

$2,0002,(XX)

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DELAY (months)

delay than does an exponential decay function across awide range of amounts (from $100 to $100,000). In fact,the hyperbolic model accounted for more than 98% ofthe variance at the three largest amounts.

Data from studies using a variety of different proce-dures with real and hypothetical outcomes in human andanimal subjects converge to show that the nature of theprocesses involved in decisions about immediate and de-layed rewards is well captured by a hyperbolic discount-ing model. Although the present results were obtainedasking subjects about their choices between hypotheti-cal rewards, there are good reasons to believe that choicesregarding hypothetical rewards are reflective of the pro-cesses involved in choosing between real rewards. It hasbeen shown that when animals choose between delayedfood rewards, preference decreases hyperbolically withdelay (Chung & Herrnstein, 1967). With humans, Kirbyand Marakovic (1995) have shown that a hyperbolic modeldescribed discounting of small, real monetary rewardsbetter than an exponential model.

The present findings confirm that the rate of discount-ing, as measured by the k parameter in the hyperbolicmodel, decreases with the amount of a delayed reward.

DISCUSSION

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AMOUNTFiaule 1. Rate of discounting (k) as a funcdon or the amount of the

del8)'ed reward. Both the estimates or k based on no of the hyperbolicmodel to tht group median subjective values and the median of theestimates based on fits to the data from Individual subjects areshown.

AMOUNT-DEPENDENT TEMPORAL DISCOUNTING 719

Although Kirby and Marakovic (1995) failed to rmd anamount effect, this discrepancy is likely due to the smallrange of the amounts studied (S14.75 to $28.50), as an-other study by these authors (Kirby &; Marakovic, 1996),which used a wider range (S30 to S85), did find a de-crease in discounting rate with increases in amount. Inaddition, the present results demonstrate that the rate ofdiscounting levels off somewhere in the neighborhood of$25,000. Raineri and Rachlin (1993) reported a decreasein discounting rate with three delayed amounts ($100,$10,000, and $ I million) but failed to observe such a lev-eling off. However, their results are not inconsistent withours. Rather, taken together with the present results, theyaid in localizing the point at which the rate of discount-ing levels off (i.e., between $10,000 and 525,000).

Various explanations have been offered to account forthe fact that rate of discounting decreases as the amountof the delayed reward increases. For example, it has beenproposed that people maintain two different mental ac-counts for large and small amounts, analogous to a "sav-ings" account and a "windfall" account (Loewenstein &;Thaler, 1989). This metaphor may have an intuitive ap-peal. However, at least three different mental accountswould be required to explain the three, significantly dif-ferent discounting rates that were observed in the currentstudy. If, as seems likely, even smaller values (e.g., under$10) are discounted yet more steeply, then it would seem

720 GREEN, MYERSON, AND MCFADDEN

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F1gure 3. Present, subjective wlue of the delayed reward as a runction of the time until Its receipt Datam the mur delayed amounts are shown In sepante panels. Solid and dashed cUn'es rep~t hyperbolicdisoounliDl funcdons flt to the median subjective V81ues for the ascending and descend!. series, rape<:-

dvely.

more parsimonious to postulate a continuous relation be-tween amount and discounting rate rather than four, oreven more, discrete mental accounts.

Another approach to the problem of amount-dependentdiscounting has been suggested by Raineri and Rachlin(1993) as part of an effort to deal with the roles of con-sumption rate and duration in decision making. They as-sume that consumption takes time and that larger re-wards take longer to consume than smaller rewards. Theyassume further that consumption rate increases withamount in a negatively accelerated manner. For example,a S I 0,000 reward might be spent at a faster rate than aSIOO reward but not 100 times faster. In the context oftheir model, these assumptions give rise to an interactionbetween amount and delay, such that larger rewards arediscounted at a lower rate.

Myerson and Green (1995) have suggested two possi-ble explanations for the effect of amount on discountingrate. According to one hypothesis, noted in the introduc-tion, hyperbolic discounting occurs because decisionsare based on the implicit assumption that the likelihood

$100 $2,000

$25,000

80 . 000

DELAY (months)

that something will happen to prevent receipt of a futurereward (i.e., the hazard rate) decreases over the waitingperiod. We have termed this hypothesis the expected-value model. From the perspective of the expected-valuemodel, the phenomenon of amount-dependent discount-ing implies that larger rewards are associated with lowerhazard rates than smaller rewards.

Hyperbolic discounting occurs in both animals andhumans, suggesting a common evolutionary heritage, andit may be useful to consider the expected-value model inthis context. That is, amount-dependent discounting mayrepresent an adaptation to differences in the risks in-volved in foraging for larger and smaller rewards. Theserisks may be either ecological or cognitive (Green & My-erson, 1996). For example, for a foraging animal, the eco-logical risk associated with waiting for a larger rewardmay be lower because, even if a competitor gets to itfirst, it is more likely that some of the reward will be left,whereas with a smaller reward, it is less likely that anywould remain should a competitor find it. The cognitiverisk reflects the fact that waiting for a reward involves

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the possibility of forgetting when and where it may beobtained, and large rewards may be more salient, that is,more memorable.

According to another hypothesis, which we have termedthe repeated-choice model, individuals make an implicitassumption regarding how often they are likely to havean opportunity to choose between a particular pair of al-ternatives, and the discounting rate is determined by thetime betWeen choice opportunities (Myerson & Green,1995). Larger amounts are discounted less steeply becauseindividuals assume that choice opportunities involvingsuch amounts are rarer than choice opportunities involv-ing smaller amounts. Both models appear to be moreeasily testable than metaphorical hypotheses such as sep-arate mental accounts. For example, the expected-valuemodel might be evaluated by manipulating risks (e.g., byincreasing salience while holding amount constant) andthe repeated-choice model might be evaluated by ex-plicitly varying the rate at which choice opportunities arepresented (see Rachlin, Logue, Gibbon. & Frankel, 1986.for a similar approach to testing a model of choice in-

TEMPORAL DISCOUNTING 721AMOUNT-DEPENDENT

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vol ving repeated gambles). Given both the accumulatingevidence for the hyperbolic form of the discounting func-tion and the present findings regarding the range and r0-bustness of the effect of amount on discounting, futureexperiments should be aimed at distinguishing amongthe several theoretical explanations for amount-dependenthyperbolic discounting (e.g., mental accounts, consump-tion rates, expected value, and repeated choice).

Figure 5 illustrates how self-control depends on thedifference between the rates at which large and small re-wards are discounted. The heights of the bars representthe amounts of the two rewards; the solid curves repre-sent discounting functions (i.e., subjective value as a func-tion of time) for the large and small rewards, assuming thatboth are discounted at the same rate. The solid circle rep-resents the crossover point: Before this point (i.e., attimes to the left of the solid circle), the subjective valueof the larger reward is greater than that of the smaller re-ward; after this point, the subjective value of the smallerreward is greater. Thus, if one chooses between the tworewards at time T I' shortly before the smaller, more im-

GREEN. MYERSON. AND McFADDEN722

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FIau~ 5. Subjectiw wlue of ~ ~ ~rds 8S . funcdon ofthe time at which the choke between them b made. The heiKhts of theban ~ment the 8mounts of the two ~ the solid curves rep-ramt temporal ~dDg funcdons 8S8UmIna that bodt ~rdsare discounted at the 18me I'8te, .nd the dashed cune ~ments thetempol'81 diKoundDi fuDCtIoII iJr the larler reward 8S8UmIna that ItIs diICounted - steeply than the smaller IoewanI. The ~ Rpre-sent points at which the two I'eWard5 a~ equiV8lent in subjectivevalue, the solid drde under the 8S8Umpdon or amount-'odependentdiscounting and the open drde under the 8S5umption of amount-dependent dlscountina. T I and T z Rpresent two choice points dls-cussed In the text.

mediate reward could be received, the smaller rewardwill be preferred (i.e., it has the greater subjective value).However, if one chooses at time T 2' so that there is alonger interval until the smaller reward could be received,then the larger, more delayed reward will be preferred.The latter choice is often considered to reflect greater~lf-control relative to the more "impulsive" choice of thesmaller, more immediate reward.

However, large and small rewards are not discountedat the same rate, as the present results demonstrate. Thelower discounting rate associated with a larger reward isillustrated in Figure 5 by the dashed curve, and the re-sulting crossover point is represented by the open circle.Notice that when the discounting rate for the larger re-ward is less, the crossover point moves to the right, thusreducing the time interval over which "impulsive" be-havior (i.e., the choice of the smaller over the larger re-ward) is observed. It follows that any manipulation thatwould further decrease the discounting rate for the largerreward would produce additional reductions in the like-lihood of , 'impulsive" behavior.

Understanding the detenninants of discounting rateand their role in the decision-making process is impor-tant in part because such understanding may suggest waysto enhance self-control. That is, different models suggestthat different variables will influence choice betweenlong-term and short-term interests, as represented by de-layed. larger and more immediate, but smaller rewards,respectively. For example, the expected-value model im-plies that increasing confidence in the certainty offuturerewards will decrease the rate at which they are discounted,thus increasing the likelihood that they will be chosen

over smaller. more immediate rewards. Similarly. therepeated-choice model implies that increasing percep-tion of the rarity of choice opportunities would also en-hance self-control.

Finally, regardless of the theoretical mechanism un-derlying temporal discounting. increasing the perceivedvalue of the delayed reward should result in greater self-control, in the sense of extending the range of delaysover which individuals will choose the larger. delayed re-ward over a more immediate but smaller reward. How-ever, the present results show that although the rate ofdiscounting decreases with amount of reward. it levelsoff at large amounts. This leveling off suggests that therewill be limits to the effectiveness of efforts to increaseself-control by increasing the subjective value of delayedrewards. Consequently, the effectiveness of a variety ofapproaches. including those suggested by the expected-value and repeated-choice models. needs to be explored.

REFERENCES

AlNJUI, G. (1975). Specious rwward: A behavioral theory of impul-sivea8lland impulse control. PS)'Chological Bullnill. 81. 463-496.

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723AMOUNT-DEPENDENT TEMPORAL DISCOUNTING

NOI'E

I. To lesl for pos8ible differences in discounting rates, we fillhefunction

V-I/{I +(k+a.4.k)D],

where V is the 8ubjective value of a delayed reward expressed u a pr0-portion of its nominal amount, k is the discount perimeter for the larpramount, 4.k is die difference between the dilCOUnt perameten for die~er and smaller amounts, and a is an indicator variable that equals 0fur the Iaraer amount and I for the smaller amount To test for the 8ia-nificance of a difference in discountina rate, 4.k is divided by its esti-mated uymptOtic ltaftdard deviation to obtain a va1\W of t with /I - Pdeireel of freedom, where /I is lhe number of data points (i,e., 14, or 7per amount) and p iJ the number of parameters (i.e., 2) in the model(Gallant, I 987}.

(ManUICript received January 26, 1996;revision ICCepted for publicatioa AUiUJt 10, 1996.)