valuing future private and social benefits: the discounted utility model versus hyperbolic...

Valuing future private and social bene®ts: The discountedutility model versus hyperbolic discounting models

John Cairns, Marjon van der Pol *

Health Economics Research Unit, University of Aberdeen, University Medical Buildings, Foresterhill,

Aberdeen AB25 2ZD, UK

Received 2 August 1998; accepted 10 September 1999

Abstract

It is standard practice in economic evaluation and in any economic analysis of future events

to assume the discounted utility model on the part of economic agents. This paper compares

the discounted utility model with three hyperbolic discounting models with respect to private

and social ®nancial bene®ts. The discounting models are ®tted using non-linear regression

techniques to data collected from the general public. Regression analysis is then used to test

the theoretical validity of the models. The main test is whether the period of years for which

the bene®t is to be delayed is a statistically signi®cant predictor of the respective values of the

discounted utility model and the three hyperbolic discounting models. This tests whether these

discounting models are satisfactory representations of the individualsÕ intertemporal prefer-

ences. The results show that there is evidence in favour of hyperbolic discounting models over

the discounted utility model. There were not any statistically signi®cant di�erences in the

discounting models ®tted for the private and social ®nancial bene®ts. The regression results

were very similar for the private and social ®nancial bene®ts. Ó 2000 Elsevier Science B.V. All

rights reserved.

PsycINFO classi®cation: 2340

Journal of Economic Psychology 21 (2000) 191±205www.elsevier.com/locate/joep

* Corresponding author. Tel.: +44 1224 552783; fax: +44 1224 662994.

E-mail address: [email protected] (M. van der Pol).

0167-4870/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 7 - 4 8 7 0 ( 9 9 ) 0 0 0 4 2 - 2

JEL classi®cation: D90

Keywords: Time preference; Hyperbolic discounting

1. Introduction

It is standard practice in economic evaluation and in any economicanalysis of future events to assume the discounted utility model on the part ofeconomic agents. One of the key axioms of the discounted utility model isstationarity. It refers to the assumption that preference between two out-comes depends only on the absolute time interval separating them. So indi-viduals who prefer receiving £100 after 1 month to receiving £110 after 2months should also prefer receiving £100 after 12 months to receiving £110after 13 months. However, in practice preferences between two delayedoutcomes often switch when both delays are incremented by a given constantamount. Loewenstein and Prelec (1992) refer to this as the common di�erencee�ect. The common di�erence e�ect implies that discount rates should de-crease as a function of the time delay over which they are estimated. Hy-perbolic models allow for the common di�erence e�ect.

This paper compares the discounted utility model with three hyperbolicdiscounting models using data on the valuation of future private and social®nancial bene®ts collected from a sample of the general public. There are tworeasons for testing the discounted utility model and hyperbolic discountingmodels, namely: to inform discounting practice in economic evaluation; andimprove understanding of intertemporal behaviour.

Gyrd-Hansen and Sùgaard (1998), writing in the context of valuing futurehealth bene®ts, argue that ``the basic question is whether and to what extentintertemporal preferences revealed from consumption behaviour should formthe basis for social decision making over time''. How important a questionthis is will depend on the extent to which the private and social implied ratesof discount di�er. They might di�er because the factors upon which theydepend may di�er. For example, there is a greater risk of an individual notbeing able to bene®t from higher future consumption than there is of a so-ciety being unable to bene®t. It has long been argued that individuals maynot want their private marginal time preference rates to be used in deter-mining the rate to be used to appraise public projects. One person's decisionto save may provide an external bene®t to other people currently alive, in that

192 J. Cairns, M. van der Pol / Journal of Economic Psychology 21 (2000) 191±205

they bene®t from the prospect that future generations will enjoy additionalconsumption (Sen, 1967). This paper considers whether or not individuals usedi�erent models of time preference for di�erent types of choice, and whetheror not the implied discount rates di�er for private and social choices.

2. Discounting models

Albrecht and Weber (1995) provide a useful framework for comparingdi�erent discounting models. They de®ne decision weights (wt) as follows:

wt � 1

�1� r�a�t�; �1�

where a(t) is the time perception function which indicates how fast time isperceived to pass in the individualÕs mind. A linear a(t) gives the standarddiscounting model:

wt � 1

�1� r�t and a�t� � t: �2�

Concave a(t) yield hyperbolic discounting models. Three such discountingmodels are investigated in this paper. The models are hyperbolic because thediscount factor applied to each period declines as a hyperbolic function oftime. Loewenstein and Prelec (1992) propose a general functional formwhich, using Albrecht and Weber (1995) notation, gives the following deci-sion weights and time perception function:

wt � 1

�1� gt�h=gand a�t� � h ln�1� gt�

g ln�1� r� : �3�

The parameter h measures the speed of the individualÕs time perception. Thehigher h the longer will one time period be perceived to last. If h � 0 the timeperiods are perceived to pass in®nitely fast and the individual is timing in-different. As h!1 time is perceived not to pass at all. A value cannot bederived from future consequences and the discount factors of all periodst > 0 are zero (Albrecht & Weber, 1995). The g parameter determines howmuch the function departs from the traditional model. As g approaches 0, wt

approaches the exponential discount function.

J. Cairns, M. van der Pol / Journal of Economic Psychology 21 (2000) 191±205 193

Loewenstein and PrelecÕs two parameter model encompasses a number ofhyperbolic models which have appeared in the literature. For example, set-ting g � 1 yields the model implied by Harvey (1986):

wt � 1

�1� t�h and a�t� � hln�1� t�ln�1� r� : �4�

Alternatively, setting h=g � 1 yields the model proposed by Mazur (1987):

wt � 1

�1� gt� and a�t� � ln�1� gt�ln�1� r� : �5�

3. Previous empirical studies

Several studies have assumed a hyperbolic discounting function whenexamining intertemporal preferences (Green, Fry & Myerson, 1994; Green,Myerson, Lichtman & Frey, 1996; Green, Myerson & McFadden, 1997;Kirby & Herrnstein, 1995; Kirby & Markovi�c, 1996; Rachlin, Raineri &Cross, 1991; Richards, Mitchell de Wit & Seiden, 1997). Rather fewer studieshave explicitly compared hyperbolic models with the exponential model. InGreen, Fristoe and Myerson (1994) 24 undergraduates were asked to choosebetween pairs of hypothetical sums of money available after di�erent delays.When smaller, more immediate sums were selected over larger, more delayedsums, the addition of a constant delay to both outcomes resulted in reversalsof preference. Exponential and hyperbolic models (Mazur model) were ®ttedto data for 12 of those undergraduates in Myerson and Green (1995). For thewhole group and on an individual basis temporal discounting was betterdescribed by hyperbola-like functions than by exponential decay functions.

Kirby and Marakovi�c (1995) report two experiments with 22 college stu-dents, one using real rewards in a simulated auction and the other usinghypothetical monetary rewards. Subjects were o�ered ®ve di�erent rewardseach with six di�erent delays and they had to indicate the smallest amountthat they would accept immediately in exchange for the delayed rewards.Hyperbolic (Mazur model) and exponential models were ®tted for eachsubject. Both functions ®tted the data very well but the hyperbolic function®tted better for all of the delayed rewards.

Albrecht and Weber (1995) identify time perception functions and periodweights for data from studies by Benzion, Rapoport and Yagil (1989) andShelley (1993). They compare these with those for a hyperbolic model (the


Harvey model) and for the standard discounting model. They found theempirical data are more consistent with a hyperbolic rather than the standarddiscounting model.

Cairns and van der Pol (1997a) analyse data on preferences for future ®-nancial and health bene®ts collected from 473 members of the general public.They compare the standard discounting model with two hyperbolic models:the Loewenstein and Prelec (1992) model and a special case of this modelwhere wt � �1� gt�ÿs

(Rachlin, 1989). They found greater support for thehyperbolic models than for the standard discounting model. This earlier re-search is extended in this paper by comparing the standard model with threehyperbolic models (Loewenstein and Prelec, Mazur, and Harvey). In addi-tion, non-linear least squares regression is used to select the optimal valuesfor g, h and r in these models.

4. Data

Two thousand people were selected randomly from the electoral registerfor the parliamentary constituencies of Aberdeen North and AberdeenSouth. They were sent a postal questionnaire containing six intertemporalchoices: two concerning social lifesaving choices; two concerning private ®-nancial choices; and two concerning social ®nancial choices. The analysis ofthe life-saving data is reported in Cairns and van der Pol (1997b). Each pairof intertemporal choices had a similar structure. Respondents were asked toindicate what future level of bene®t would lead them to be indi�erent betweena speci®ed bene®t to be received one year in the future and the delayed moredistant bene®t. There was a Ôshort runÕ choice and a Ôlong runÕ choice. In thecase of the former the future bene®t was delayed 2, 4, 5, 6 or 8 years, and inthe case of the latter the bene®t was delayed 12, 14, 15, 16, 17, 18 or 19 years.Posing future private ®nancial choices is straightforward and has been donein numerous studies. Most instances of eliciting social preferences have in-volved the saving of future statistical lives. In this study individuals wereinformed that money from outside the community would be made availableto the broader community in which they lived for the purpose of improvingpublic transport. They had to indicate the desirability of receiving di�erentsums at di�erent dates in the future. See Appendix A for examples of theprivate and social ®nancial questions.

The data were originally collected in order to examine di�erences in ex-ponential discount rates between health and ®nancial bene®ts. Although the


limited number of observations per respondent restricts, to some extent, theanalysis of alternative discounting functions it does o�er scope for such aninvestigation.

Data were also collected on the individualÕs age, gender, long-term health,cigarette consumption, educational attainment, and whether they were par-ents of children under 10. Table 1 shows some descriptive statistics of thecharacteristics of the respondents. One hundred and three questionnaireswere returned by the Post O�ce for a variety of reasons and 473 usable re-sponses were received. A further 22 were returned without the time prefer-ence choices being attempted, yielding a response rate of 25%.

5. Methods

Non-linear Least Squares Regression in Limdep (Greene, 1993) is used toselect the optimal values for r, h and g in the discounting models. A non-linear regression model is one for which the ®rst-order conditions for leastsquares estimation of the parameters are non-linear functions of the pa-rameters (Greene, 1993). Thus, non-linearity is de®ned in terms of thetechniques needed to estimate the parameters, not the shape of the regressionfunction. Since there are only two observations per respondent functionalforms cannot be selected on an individual basis. Functional forms aretherefore selected on a group basis. The goodness of ®t is measured by R2.The R2 is compared for the functional forms of the discounted utility modeland the hyperbolic models. A t-test is used to test whether differences be-tween parameter values for private ®nancial and social ®nancial bene®ts arestatistically signi®cant.

Table 1

Descriptive statistics

Variable

Age Mean: 46.9

Range: 18±88

Female 54.2%

Education beyond secondary school 50.0%

Fair or poor health 24.8%

Children under the age of ten 13.6%

Smoker 21.9%


Regression analysis is used to test the theoretical validity of the dis-counting models. Theoretical validity assesses the extent to which the resultsare consistent with economic theory or more generally, a priori expectations.For each individual observation the following parameter values are esti-mated: r in the discounted utility model; h in the Loewenstein and Prelecmodel while assuming a ®xed g; h in the Harvey model; and g in the Mazurmodel. These parameter values are then regressed on a range of variablesincluding: the period in years for which the bene®t is to be delayed (Delay);whether or not the individual was told to assume a zero rate of in¯ation(In¯ation); and the individualÕs characteristics, their age in years (Dummyvariables for the age groups 30±43, 44±63, 64±88), their gender (Female),whether or not they currently smoked cigarettes (Smoker), whether theirlong-term health was fair or poor rather than good (Health), whether or notthey had children under ten years of age (Children), and whether or not theyhad been educated beyond secondary school level (Education). An h valuecannot be estimated for respondents with an implied discount rate of zero. Inorder to provide a truer comparison, these individuals (33) have been left outof the regression estimates for the three discounting models.

The strongest test of theoretical validity is that the coe�cient of Delayshould not be statistically signi®cant from zero if the discounted utility modeland the hyperbolic models where the rs, hs, and gs come from are accuratedescriptions of individualsÕ time preferences. Hypotheses can be made withrespect to the signs of the coef®cients of the other included independentvariables but these tests are of limited value because counter hypotheses arereadily constructed.

It is hypothesised that the impact of age on time preference could di�eracross age groups, for example younger respondents are likely to be disad-vantaged with respect to the terms on which they can borrow while olderrespondents may be more keenly aware of their own mortality and as resultdiscount future bene®ts more heavily. On the other hand younger respon-dents have longer life expectancy and might therefore attach more impor-tance to the distant future.

Smokers may discount the future more heavily since they will be awarethat they have a lower chance than non-smokers of living to enjoy that fu-ture. It is anticipated that more highly educated individuals may discount thefuture less heavily since they may face better borrowing possibilities. Thepossibility of endogeneity bias should be noted with respect to these twovariables in that the direction of causation is not unambiguous. Those in-dividuals who discount the future less heavily will more readily invest in


themselves by acquiring additional education. Similarly those who discountthe future more heavily may be more likely to choose to smoke. Parents withchildren under the age of ten would be expected to place a relatively highvalue on the near future. Similarly those respondents with poor long-termhealth may discount the more distant future relatively heavily.

Another way to test the appropriateness of the discounting models is theRESET test which can be viewed as a test of functional form (Godfrey,1988). With the RESET yt is regressed on xt; y2

t , and y3t . The likelihood ratio

statistic is used to test that the coef®cients of the powers of yt are zero(Godfrey, 1988).

For each type of choice (private and social) there are two observations perrespondent. As a result of this clustering of observations OLS can producesmall standard errors which overestimate the statistical signi®cance of ex-planatory variables. The multilevel model can be expressed as (Goldstein,1995)

yij � a� bxij � lj � eij: �6�In this equation lj and eij are both random quantities. The respondent-levelrandom variable lj are the departures of the jth respondentÕs actual interceptfrom the overall mean value a. This is thus a level two residual. The obser-vation-level random variable eij is the observation-level residual for the ithobservation in the jth respondent and measures random variation acrossobservations. Because the collected data has few level one units (observa-tions) for each level two unit (respondents) restrictive iterative generalizedleast squares (RIGLS) estimation is used (Goldstein, 1995).

6. Results

Table 2 shows the optimal values of r, h and g. The values are very similarfor the private and the social choices. Using a t-test none of the differences inthe parameter values for private and social bene®ts are statistically signi®-cant. The table also shows the goodness of ®t for the different discountingfunctions measured in terms of R2. Comparing the results for the goodness of®t the hyperbolic models all ®t the data better than the discounted utilitymodel. The Loewenstein and Prelec model is the best ®tting model followedby the Mazur model. It should be noted that the best ®tting model, theLoewenstein and Prelec model, is a two-parameter model while the othermodels are one-parameter models.


The multilevel results for the private choices are reported in Table 3 andfor the social choices in Table 4. For the purposes of this paper the coe�cienton the Delay variable is of greatest relevance. With respect to the privatebene®ts, Delay is statistically signi®cant in all cases. This suggests that noneof the discounting models are wholly satisfactory representations of the in-tertemporal preferences of the individuals surveyed. However, the Loewen-stein and Prelec model appears to be the best performing model since the t-value for the coef®cient on Delay is substantially lower than for the otherdiscounting models. With respect to social bene®ts, Delay is statisticallysigni®cant in all but one case. The Loewenstein and Prelec model is clearlythe best performing model. The lower t-value for the coef®cient of Delay inthe Mazur model suggests that this model performs better than the dis-counted utility model and the Harvey model. These results are con®rmed bythe RESET test. The RESET results suggest that the discounted utility modeland Harvey model are misspeci®ed while the RESET scores are insigni®cantfor the Loewenstein and Prelec model and the Mazur model.

The signs of the coe�cients on the other variables are the same for alldiscounting models apart from the variable Health. However, the variableHealth is not statistically signi®cant in any of the cases. The variable Smokeris statistically signi®cant in all cases except in the discounted utility model forsocial choices. As hypothesised smokers tend to discount the future moreheavily. The variable Female is statistically signi®cant in the case of privatechoices. Females tend to discount the future more heavily. Respondents whoare educated beyond secondary school tend to discount the future less heavilyin the case of social choices. This variable is statistically signi®cant only in thediscounted utility model and the Loewenstein and Prelec model. Respondentsaged between 44 and 63 tend to discount the future less heavily. This variableis statistically signi®cant only in the Loewenstein and Prelec model and theHarvey model.

Table 2

The speci®c functional forms and their goodness of ®t

Private ®nancial t-Test

difference

Social ®nancial

Parameters R2 Parameters R2

Discounted utility model r � 0:192 0.37715 1.89 r � 0:203 0.35429

Loewenstein and Prelec g � 0:255; 0.47666 1.27 g � 0:316; 0.49234

h � 0:309 1.05 h � 0:344

Harvey h � 0:606 0.43618 0.68 h � 0:615 0.45943

Mazur g � 0:351 0.47416 1.07 g � 0:365 0.49174


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7. Discussion

This paper compared the discounted utility model with three hyperbolicmodels, namely: the Loewenstein and Prelec model, the Harvey model andthe Mazur model. There is evidence in favour of the hyperbolic dis-counting models over the discounted utility model. First, the hyperbolicdiscounting models have a better goodness of ®t than the discountedutility model. The Loewenstein and Prelec model was the best ®ttingmodel followed by the Mazur model. Second, the regression results sug-gest that the Loewenstein and Prelec model and to some degree the Mazurmodel are better representations of the intertemporal preferences than thediscounted utility model. It should be noted that not all of the hyperbolicmodels investigated in this study are better representations of individualsintertemporal preferences than the discounted utility model. More specif-ically, the Harvey model did not seem to perform better than the dis-counted utility model.

That certain hyperbolic models ®t the data better than the exponentialmodel is indirect evidence that the axiom of stationarity is violated. For adirect test reaching the same conclusion see Cairns and van der Pol (1997a).Generally stationarity has been regarded as important because it avoidsdynamic inconsistency and consequent preference reversal. However, Albr-echt and Weber (1995) argue that stationarity is not always necessary. Spe-ci®cally, in the case of the many decisions which cannot be unmade in thefuture, stationarity is stronger than is required for rationality.

None of the di�erences in the parameter values of the discountingmodels for private bene®ts and social bene®ts are statistically signi®cant.The regression models also appear to be very similar in the case of privatebene®ts and social bene®ts. The main di�erence is that gender is a sta-tistically signi®cant predictor of the parameter values in the discountingmodels for private bene®ts but not in the models for social bene®ts. Also,the Loewenstein and Prelec model is clearly the best performing model forsocial bene®ts while the evidence for this model is less clear for privatebene®ts.

There are several ways in which this research might be taken for-ward. Three potential lines of enquiry are: one, to devise less challengingtime preference choices (thus reducing the scope for respondents to makeerrors); two, to obtain more observations per individual; and three, to allowthe speci®c functional forms of the hyperbolic models to vary across indi-viduals.


Acknowledgements

HERU is funded by the Chief Scientist O�ce of the Scottish ExecutiveHealth Department. This study received ®nancial support from the Nu�eldFoundation and the NHS R&D Health Technology Assessment Pro-gramme (project 94/35/1). The views expressed in this paper are those of theauthors and not necessarily those of the Standing Group on HealthTechnology Assessment, the HTA Commissioning Board, the SGHT Panelmembers, the Department of Health or the Scottish Executive Health De-partment.

Appendix A. Examples of questions

Private ®nancial choices: Imagine that you had a choice between receiving£500 in one yearÕs time, or some other sum 9 years from now. How muchwould you require in nine yearsÕ time, in order to just compensate you for notgetting £500 in one yearÕs time? Circle one of the options or write your ownchoice in the box.

Social ®nancial choices: Suppose that the European Community hasagreed to provide Grampian region with money to be spent on improve-ments in public transport. What is the smallest sum of money to be re-ceived in 7 years time, that you feel would be as good as getting £1 millionin 1 year time? Circle one of the options or write your own choice in thebox.

£500 received 1 year in the future� Sum received 9 years in the future:£700

£1100£1600£2200£3000£3700£4500£6000


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