Gamma Discounting

By MARTIN L. WEITZMAN*

By incorporating the probability distributiondirectly into the analysis, this paper proposes a new theoretical approach to resolving the perennial dilemma of being uncertain about what discount rate to use in cost-benefit analysis. A numerical example is constructedfrom the results of a survey based on the opinions of 2,160 . The mainfinding is that even if every individual believes in a constant discount rate, the wide spread of opinion on what it should be makes the effective social discount rate decline significantly over time. Implications and ramifications of this proposed "gamma-discounting"approach are discussed. (JEL H43)

The concept of a "discountrate" is central to Economic opinion is divided on a number of economic analysis, as it allows effects occurring fundamentalaspects, including what is the ap- at different future times to be compared by propriatevalue of an uncertainfuture "marginal converting each future dollar amount into product of capital," what are the relevant effi- equivalentpresent dollars. Because of this cen- ciency distortions and how are they possibly trality, the choice of an appropriatediscount magnifiedby public-sectorprojects, how should rate is one of the most criticalproblems in all of large long-termpublic investmentsbe placed in economics. And yet, to be perfectly honest, a the framework of the capital-asset-pricing great deal of uncertainty beclouds this very model, how are we to view intergenerational issue. transferswhen future generations are not pres- The problem of the unsure discount rate has ently represented,how do we account for equity long bedeviled benefit-cost analysis but has ac- and other distributionaleffects in aggregating quired renewed relevance lately because econ- costs, benefits, and discount rates over individ- omists are now being asked to analyze uals or across countries or over time-and so environmentalprojects or activities the effects forth, and so on. While economists may differ in of which will be spread out over hundreds of their projections of distant future real interest years, and the evaluation of which is therefore rates (which depend, after all, on the ultimately extremely sensitive to the discount rate being unpredictable rate of technological progress), used. Prominent examples include: global cli- this is just the tip of the iceberg. More funda- mate change, radioactivewaste disposal, loss of mental disagreements below the surface con- biodiversity, thinning of stratospheric ozone, cern basic interpretationsof welfare economics, groundwaterpollution, minerals depletion, and including the role of governmentsand even the many others.I ethical foundations of intergenerational dis- The most critical single problem with dis- counting. All these, and many more, consider- counting future benefits and costs is that no ations are fundamentallymatters of judgement consensus now exists, or for thatmatter has ever or opinion, on which fully informed and fully existed, about what actualrate of interestto use. rationalindividuals might be expected to differ. To some extent the economics profession deals with this disturbing state of affairs by * Departmentof Economics, HarvardUniversity, Cam- looking the other way. First of all, the profes- bridge, MA 01238. I am gratefulto RobertDorfman, David sion maintains,outsiders tend to exaggerate the Laibson, and RobertM. Solow for their valuable comments. actual differences among economists-there This work was supportedin part by a grant from the Na- being an implicit premise here that a properly tional Science Foundation. l For a recent discussion with further references, see appointedcommittee of "experts"might be able William D. Nordhaus(1994) or Paul R. Portneyand John P. to thrash out their differences in favor of a Weyant (1999). consensus social discount rate. Secondly, every 260 VOL.91 NO. 1 WEITZMAN:GAMMA DISCOUNTING 261

good knows that the proper proce- bility distributionis a one-parameterfamily de- dure is to perform a sensitivity analysis using scribing the waiting time until breakdownof a several plausible discount-ratevalues. Then, the single-component system, while the more gen- logic continues, hopefully the right policy or eral gamma probability distribution can be project will be relatively robust to the discount viewed as a two-parameterfamily representing rate being used for the evaluations. the waiting time until breakdownof a multiple- The real situation,as we all know in our heart componentsystem, where the second parameter of hearts, is very different. There does not now specifies the number of components in the exist within the economics profession, nor has system.) there ever existed, anything remotely resem- The empirical part of this paper attempts to bling a consensus, even-or, perhaps one determine, from a survey of the opinions of should say, especially-among the "experts"on 2,160 economists, the two defining parameters this subject. (Actually, with very little exagger- of the appropriategamma distribution.It turns ation or cynicism, an "expert"here might be out empiricallythat the second, nonexponential, defined as an economist who knows the litera- parameterof the gamma distributionplays, or at ture well enough to be able to justify any rea- least should play, an extremely significant role sonable social discount rate by some internally in actual long-term discounting. consistent story.2) Furthermore,and also con- A numerical example is constructed, which trary to the party line, expected net present suggests the practicaldesirability of thinking in discounted values for long-term projects are terms of a future subdividedinto approximately notoriously hypersensitive to the interest rate five subperiods. Named in order, these are, being used to make the evaluation. In a great roughly: the Immediate Future (1 to 5 years many practicalanalyses of public projects, per- hence), the Near Future (6 to 25 years hence), haps a majority,the most significantuncertainty the Medium Future (26 to 75 years hence), the of all concerns the discount rate itself. Distant Future(76 to 300 years hence), and the This paper sets forth a radically different Far-Distant Future (more than 300 years approachto the issue of what interestrate to use hence). The numerical example suggests using for discountingthe future, startingwith the rec- the following approximationof within-period ognition that the discount-rateproblem is rooted marginal discount rates for long-term public in fundamental differences of opinion, which projects: ImmediateFuture about 4 percent per are unlikely to go away soon. No committee of annum; Near Future about 3 percent; Medium "experts"will likely be able to resolve the di- Future about 2 percent; Distant Future about 1 lemma-no matter how deeply they delve into percent;and Far-DistantFuture about 0 percent. it. Therefore,it follows, we should be operating The paper concludes by discussing some the- from within a frameworkthat incorporatesthe oretical ramificationsand practicalimplications irreducibleuncertainty about discount rates di- of gamma discounting for global warming. rectly into our benefit-cost methodology. The paper proposes an operational method- I. The Setting for the Problem ological frameworkto resolve the discount-rate dilemma, which is centered on the concept of I begin this section by trying to avoid the bias "gammadiscounting." Viewed from within this that comes from being too close to see the forest framework, the more familiar exponential dis- for the trees. Let us stand back here to view the counting is considered to be a special case of landscape as seen through the eyes of a hypo- gamma discounting in the same spirit that an thetical decision maker,here called the "WNO" exponentialprobability distribution can be seen (standing for "Wise Neutral Observer" or as a special case of the more general gamma "WorldNations Official").The WNO is respon- probabilitydistribution. (An exponentialproba- sible for evaluating various projects or propos- als to mitigate the possible effects of global climate change. 2 By this criterion, after conducting the survey of this The WNO recognizes at once that a fun- paper I now rate myself as perhaps the world's leading damentalist, full-blown, fully disaggregated, "expert"on the social discount rate. general-equilibrium-styleanalysis of the impact 262 THE AMERICANECONOMIC REVIEW MARCH2001

of environmentalprojects is undoable, essen- The {Z(t) I from equation (2) constitutes si- tially because of analytical intractability and multaneously the fundamentaloutput of Panel informationaloverload. Instead, the WNO fol- No. 1 and also the fundamentalinput of Panel lows most economists in opting for a practi- No. 2. This paper also takes {Z(t) } as given, cal compromise, which might be called the and henceforthis focused almost exclusively on "pragmatic-decomposition"approach to inter- the work of the second panel. temporal evaluation. The WNO decides to Panel No. 2 is assigned the task of recom- "pragmaticallydecompose" the evaluation of mending the proper time-dependent weights various projects by moving them through two {A(t) } to be used for aggregatingthe net ben- specially appointedpanels of expert appraisers. efits from different times {Z(t)} into a single Panel No. 1 makes period-by-periodestimates overall index of present-discounted expected of the aggregatenet value of relevantincome flow -equivalentnet benefits: changes induced by a project, in terms of "ex- pected consumption-equivalent"net real dollars for each period,and then deliversthese estimates 00 to the WNO. Such a task amounts to a truly (3) JA(t)Z(t) dt. formidableassignment, which is worthyof a more 0 complete treatment.Yet this paperwili not elab- oratefurther the work of Panel No. 1-except to describe how each panel member's individual Loosely speaking, then, the "principle of opinion is aggregatedinto an overall expected- pragmaticdecomposition" reduces the problem net-benefitestimate. to evaluatinga linear expression of the form (3), Panel No. 1 consists of m experts on evalu- and then furthersubdivides the remaining task ating global warming. Expert i (1 ' i ' m) into two reasonably distinct subtasks. Panel consults model i, where a "model" is under- No. 1 provides the time series of overall ex- stood here in the broadestpossible generic sense pected net benefits {Z(t) 1. Panel No. 2 provides of "some mechanism"for predicting outcomes the correspondingtime-evaluation aggregation of projects to mitigate the possible effects of weights {A(t) 1. The WNO realizes full well global climate change. Suppose that expert i (or that this procedureyields at best an approximate model i) predicts a prototypeproject will yield answer for what, in its full generality, consti- expected consumption-equivalent net benefit tutes an extraordinarilycomplex web of inter- (i.e., expected consumption-equivalent gross related questions.3 Nevertheless, the "principle benefits minus costs) at time t of: of pragmatic decomposition" seems like a sat- isfactory practical compromise to the WNO- (1) Zi (t). especially when compared with the leading alternative,which is basically to curse the dark- From the perspective of the WNO, each ex- ness by bemoaning the analytical and informa- pert serving on Panel No. 1 is viewed as being tional intractabilityof the problem. The WNO of approximately equal standing in terms of opts instead for lighting a small candle to illu- objective credentials or any other observable minate a few modest guidelines for rough an- marker of professional legitimacy. Therefore, swers to some importantquestions. implicitly using a diffuse uniform prior, what Panel No. 2 consists of n experts on "the the WNO takes as the best statisticalestimate of discount-ratequestion." Expert j, (1 ' j ' n), the appropriateoverall expected consumption- consults model j, where here a "model"is un- equivalent net benefit at time t is simply the derstood in the broadestpossible generic sense average response of "some mechanism" for generating time-

E Zi(t) 3At best, the combining of A(t) and Z(t) in the linear form (3) can be rigorouslyjustified only when the underly- ing { } are small relative to the world economy and (2) Z(t)- Zi(t) m independentlydistributed relative to the underlying {Aj(t) I . VOL. 91 NO. I WEITZMAN: GAMMA DISCOUNTING 263

500

4500

400 -

U) 350- 0 300-

250 0 () 200- E 150 z 100

50

Real Discount Rate (percent)

FIGURE 1. ACTUAL (HISTOGRAM) AND FITTED (CURVE) FREQUENcy DISTIRBUTIONS evaluation weights. Suppose that expert j, or that furtherpractical progress will not be made model j, recommends the time-evaluation ag- without specifying more precisely the exact gregation weights: question to be asked each of the n experts serving on the second panel, and without plac- (4) Ai (t). ing somewhatmore structureon the distribution of their various answers. Like the first panel, then, members of this second panel also hold differing opinions-in II. The Model this case about the discount-ratequestion. And, for this second panel also, the WNO has a Imagine that a large number of economists diffuse prior on the credibility of each mem- have been asked to name the real-interest-rate ber's opinion. Only, here it is far less clear to value they favor for discounting long-term en- the WNO how to combine these different opin- vironmental projects. Suppose the replies are ions into a reasonable overall discounting distributed approximately as the histogram function. shown in Figure 1. What might such an exercise So far as the WNO can comprehend, social suggest? discounting weights apparentlyrepresent opin- First of all, the large number of responses ions more than anything else. The social dis- might suggest that the question is not altogether count rate here seems not so much to be a thing unfamiliar.It seems as if many economists are that economists observe, even in principle,but it accustomed to thinking in terms of exponential appears rather more to represent a subjective discounting, but they exhibit a fair amount of judgement which they believe in to varying de- variation in the preferredvalue of the constant grees. At this point, the WNO wisely realizes discount rate that an individual economist is 264 THE AMERICANECONOMIC REVIEW MARCH2001 more or less willing to recommend. To the because it reduces the WNO's problem to pin- WNO, desperate to render the problem more ning down the parametervalues of a and P3. manageable,all of this suggests postulatingthe Imposing the gamma distributionhere is a stra- following enormously simplifying assumption. tegic assumptionthat will deliver enormous an- alytical tractabilityat very little cost in terms of ASSUMPTION 1 [Reduced-FormSpecification actual empirical restrictiveness. #1]: The individual time-evaluation aggrega- What is the expected value today of an extra tion weights {Aj(t)} take the traditional expo- expected dollar at time t? It should be the ex- nential form pected present discounted value of a dollar at time t, weighted by the "probabilityof correct- (5) Ai(t) = e Xji. ness" or the "probabilityof actuality"of the rate at which it is being discounted: Secondly, the histogrampattern suggests the general shape of a gamma probabilitydistribu- tion, like the smooth curve superimposed in Figure 1. Ever the empirical opportunist, the 0 WNO accordingly makes the following as- sumption. The functionA(t) may be called the effective ASSUMPTION2 [Reduced-FormSpecification discount function for time t. Note that the WNO #2]: The { xj I in (5) are distributedas the re- is essentially taking a probability-weightedav- alizations of a randomvariable x, whose prob- erage (over all types of persons in all states of ability density function f(x), defined for all mind) of discountfunctions of the members of positive x, is of the gammaform Panel No. 2. The key insight here is that what should be probability-averagedat various times is not discount rates, but discountfunctions. (6) f(x) = F( ) -Ie-,x The correspondingmarginal or instantaneous effective discount rate at time t is defined to be with positive parameters a and ,3 to be esti- matedfrom the data.4 In words, reduced-form specification #1 (8) R(t)- A(t) means that the average panel member knows about, and typically does not feel acutely un- comfortablewith, the approximationof constant For the gamma probability distribution (6), exponentialdiscounting. The primarydisagree- the effective discount function defined by (7) is ment among panel members is over the appro- priatevalue of the as-if-constantdiscount rate. a Reduced-form specification #2 then restricts (9) A(t) = () J x e-(P+tOx dx. the empiricallyobserved discount-ratevalues to be distributed"as if' being realizations of the two-parametergamma family. Equation(6) rep- resents an openly opportunistic specification, Now it is a fact of calculus that

- 4 I assume some familiaritywith the gamma distribution, (10) F ya-le-by dy a which is typically describedin any comprehensivebook on probabilitytheory. See, e.g., Morris H. DeGroot (1970). 5To the extent that some panel members may believe in declining discount rates, the basic conclusions of this paper By substitutingthe values a = a and b will only be strengthened.In this spirit, the main conclusion might be stated as follows: Even if everyone believes in a ,e + t into formula (10), and then using the constant discount rate, the effective discount rate declines resulting expression, equation (9) can be trans- strongly over time. formed into VOL. 91 NO. 1 WEITZMAN:GAMMA DISCOUNTING 265

Equation (18) can readily be translatedinto an equivalent formulationthat makes clear the exact sense in which gamma discounting may Now the mean and varianceof (6) are defined be viewed as a generalization of exponential in the usual way discounting. Let N be a fixed parameterstand- ing for the number of discrete times at which interestpayments are to be compoundedwithin (12) /Jfxf(x) dx any time interval. If t represents the relevant interest flow rate, then, with N equally spaced compoundingoperations in the interval [0, t], a dollar on deposit at time zero grows into and

(19) B(t) t1 + N (13) a2 (X - ,t)2f(x) dx.

dollars at time t. Making use again of (10), it can be proved for But now notice, comparing (18) with (19), the gamma distributionthat that

a 1 (14) (20) A() B(t) '

and whenever we interpretN to be defined by

a (15) 2 (21) N

Because / and a-have economically intuitive Exponential discounting can thus be viewed meanings, while a and ,B have no economic as a special case of gamma discounting appro- significance per se, it will be more appropriate priate to a limiting situation where the number in this paper to work with the inverse of (14) of compounding operations N is very large, and (15), namely which by (21) can be interpretedto be essen- tially the same thing as having o- be very small. 2 As economists, we are perhapsunaccustomed (16) a -2 to thinking in terms of a situation where N is held constant in the familiar equation (19), while t is varied. But there is nothing inherently and amiss in the image of a world acting like a bank that limits the "interest paid on interest" by restrictingitself to just N compounding opera- (17) tions performedwithin any time interval [0, t]. Such a "gammabank" goes by the rule that if you deposit a dollar at time zero and then close After substitutingfrom (16) and (17), equa- out your account at time t, the withdrawalvalue tion (11) can be rewrittenas of your savings will be compoundedexactly N times in the interval [0, t], irrespective of the withdrawal time t. In this interpretationthe 1 amount of each interest payment ,utlN credited (18) A(2) 2 (1 ? to2Ipt)p~1' per dollar on deposit varies linearly with 266 THE AMERICANECONOMIC REVIEW MARCH2001 withdrawal time t, while the total number of context of this paper, I think it suffices to think such interest compounding operations N is of formula (18) [or (19)-(21)] as defining a viewed as being fixed beforehand. table of technocratic time-dependent weights, Returningto the primaryformulation (18), by which give future-dollarequivalence values for equation (8) the implied effective discount rate making one-time irreversible investment deci- is sions. The possibility of being able later to revisit and revise time-inconsistent investment choices introduces a set of complicated issues (22) R(t) 1 + tcr2/I that are better treated separately. Equations (18) and (22) represent the theo- retical or conceptual centelpiece of the paper, It was already noted that exponential dis- and provide a starting point for the empirical counting correspondshere to the special limit- estimationof {A(t) } and {R(t) }. In the form of ing case of (18) and (22) when the parameteroc (18) and (22) we have obtaineda simple, useful, approaches zero. Note also from (18) that, for two-parameterclosed-form expression that gen- any fixed positive value of o-, the effective dis- eralizes exponential discounting to cover a sit- count rate starts off at its mean value ,u when uation of uncertaindiscount rates. t = 0, but declines monotonicallytowards zero over time.6 The effective discount rate declines III. The "ProfessionallyConsidered Gut with the passage of time because increasingly Feeling"of Economists greater relative present-value weight is being placed on the states with the lower rates. In We next attemptto push the analysis one step comparison, the higher-discount-ratestates are furtherby deriving actual numerical estimates relatively less important over time, because for the declining effective discount-ratesched- their present value has been more drastically ules appropriateto evaluating long-term envi- shrunkby the power of compound discounting ronmentalprojects, like mitigatingthe effects of at the higher rates. global climate change. Building on equations What are the policy implications of time- (18) and (22), what essentially remains is to fix varying discount rates? It depends, ultimately, empirically the two parametersy and o-. on what is being assumedabout how knowledge This section proceeds as if two surveys were of future interest rates is revealed sequentially. conducted. The first survey represents a large- If the expert panel remains equally divided as scale unscreened sampling of the opinions of time unfolds, because each resolution of inter- over 2,000 professionalPh.D.-level economists. est-rateuncertainty is balancedby a new source The second survey effectively comprises a of uncertainty,there is a potential problem of named subsample of 50 representative blue- time inconsistency because, for sequential ribbon "leading economists." decision-makingproblems, nonexponentialdis- The full text is reproducedin the Appendix, counting may lead to investment plans that do but the operativepart of the questionnaireasked not fully cohere over time.7 the subjects to reply with their "professionally I think it is fair to characterizethis paper as considered gut feeling" to the following re- suggesting empirically that long-run time in- quest: consistency may run deeper and be more ge- neric than was previously suspected. However, Taking all relevantconsiderations into ac- the implicationsof time inconsistency per se are count, what real interestrate do you think properly the subject of another paper. In the should be used to discount over time the (expected) benefits and (expected) costs of projectsbeing proposed to mitigate the possible effects of global climate change? 6 This result justifying hyperbolic discounting is not re- stricted to the gamma form, but holds generally for any The questionnairewas sent by e-mail to about probability distribution. See Weitzman (1998) for more details. 2,800 Ph.D.-level economists. Efforts were 7 For a further exposition of this problem, see, e.g., made to ensure that the sample represented a MaureenL. Cropperand David Laibson (1999). balance of backgrounds,fields, and viewpoints. VOL. 91 NO. 1 WEITZMAN:GAMMA DISCOUNTING 267

Additionally,I tried to include all those whom I derstand simplistic answers to complicated consider to be the very best economists around, questions. irrespective of backgroundor area. Over 2,100 Overall, the most common objection from usable replies were received from economists in respondentswas their complaint that they were some 48 countries, representing every major "not an expert in this area" and consequently field of economics. "have no idea" what to answer. (The most ex- I hasten to add that about one in eight respon- treme position of this sort held that economists dents objected to some aspect or anotherof the should not be asked to express an opinion at all; question they were being asked. What respon- ratherit should be politicians or somebody else dents did not like varied considerably. Some better qualified to reflect public opinion.) Once wanted to be able to give two different interest again, like the tunnel-visionedjudge seeking to rates- one for developed and a second for un- avoid a hung jury, I assuredrespondents of this derdevelopedcountries. For others, the lumping type that there were no real experts out there, together of developed and underdeveloped only folks like themselves with honest differ- countries was all right, but they wanted differ- ences of opinion-some of whom just happento ent rates to be applied separately to costs and be more articulatethan others in crafting stories benefits, or to different projects, or to different to justify their answers. I would then typically parts of differentprojects, or to something else. try to prevail upon these concerned "non- Another set of objections centered on the experts" to give their "professionally consid- need to take explicit account of distributional ered gut feeling" for the imperfect world in considerations,with some of these respondents which we live, effectively asking them as seemingly wanting me to supply them with an Ph.D.-level economists to integrate out of the appropriateset of income-distributionwelfare process their subjective uncertaintiesconcern- weights. Still another group of disgruntled ing whatever they "have no idea" about.8 participants wanted, in effect, to be supplied For "extreme"responses of less than 0.5 per- with the appropriate"beta" coefficient for the cent or greater than 12 percent, I typically re- particular investment project in mind, since, quested confirmationand asked for some brief according to them, it is impossible to speak rationalization. If the respondent confirmed meaningfully about a general discount rate "for with any remotely plausible story (and held a projectsto mitigate the effects of global climate Ph.D. degree in economics from a recognizable change"without specifying more explicitly how graduateschool), the response was recorded. the payoffs are supposed to be correlatedwith At 2,160 replies from economists in 48 coun- the performance of some index of alternative tries, representingall majorfields of economics, investments, like the value of the world's stock the coverage was comprehensiveand the overall markets. response rate was excellent. But, the reader Many of these objections struckme as being should be warned again, approximately 12 legitimate, at least in principle. The various types of complaints actually tended to have relatively little in common with each other, 8 For those very few respondents who volunteered to aside from the shared bottom-line feeling that reporta range of discount-ratevalues, over which paramet- ric analysis could be performed, I typically reacted by the question as it stood was problematical,or acknowledging that they were, of course, correct in princi- perhaps even meaningless. It seems as if what ple, but I asked them nevertheless to pinpoint a single sometimes bothered one economist to an ex- number as if a simplemindedpolicy maker could only use pression of indignation about the survey typi- one value. Had I thought in the first place that a large cally failed to arouse another at all. For almost number of respondents might cooperate analogously, by revealing a distributionof their beliefs, I could have de- every case of a raised objection, I acted like signed the survey question along such lines from the begin- the sitting judge who wants badly to avoid a ning; however, as was confirmed by the difficulties I hung jury-I tried to make these sophisticated- subsequently experienced in getting some economists to sounding jurist-respondentsgive an actual nu- comply with revealing just a single number, I feared that requests for any more information, even for a range of merical answer for the imperfect world where values, might have met with refusals to cooperate at all, we happen to live, which is dominatedby sim- which could have introduced a significant element of re- pleminded people, like me, who can only un- sponse bias. 268 THE AMERICANECONOMIC REVIEW MARCH2001

TABLE 1-DISTRIBUTION OF RESPONSES rametervalues of ,u and o- as the sample, is the smooth curve drawn in Figure 1. (For visual Discount rate comparison, both distributions are scaled to (Rounded to nearest whole percentage) Number of responses have the same area, namely n = 2,160.) Even without digressing into fancy nonlinear statis- -3 1 -2 1 tics by attempting here to define formally -1 1 "goodness of fit," I think that the resemblance 0 46 in Figure 1 between the empirical histogram 1 236 and the theoreticalcurve is, intuitively,a "suffi- 2 454 ciently reasonablematch" to allow, for the prac- 3 427 4 362 tical purposesof this paper,the very considerable 5 227 analyticalconvenience of the gammaform. Thus, 6 136 it may be noted in passing,the statisticalmethod- 7 71 ology of this study is simple in the extreme. 8 44 Any one particulareconomist may feel some- 9 28 10 44 what unsure about what answer to give, but the 11 15 enormous size of the sample allows there to be 12 25 revealed a great deal of underlying statistical 13 12 regularity in the aggregate responses. We are 14 5 15 8 lookingin Table 1 at a highly skeweddistribution, 16 3 showing a lot of variation,with a mean of about 17 2 4 percent,a medianof 3 percent,and whose mode 18 3 is 2 percent. Note the pronounced "round-off 19 1 20 4 effect" for "familiar"large integer rates, like 25 2 10 percent,12 percent,15 percent,or 20 percent. 26 1 I also conducted an "as-if" second survey, 27 1 which is just a broken-out subsample of 50 Total responses = 2,160 high-profilenamed economists, selected to con- stitute a hypothetical "balanced blue-ribbon panel" of expert opinion. In alphabeticalorder, the 50 leading economists chosen to constitute percentof the replies might legitimatelybe clas- the "as-if" expert panel are: sified in the categoryof "answeringunder duress" or "responding under objection." My opinion was, and remains, that it is far better to have a sample relatively free of response bias than to ; ; Robert rely on self-styled "experts"who, in my expe- Anthony ATKINSON; BARRO; WilliamBAUMOL; ; David rience, are sometimes bluffing and, in any case, BRADFORD; John CAMPBELL;David may have all kinds of agendas hidden behind CARD;William CLINE; ; their willingness to thrustforward "their" num- ; Jacques DREZE; Martin ber. Accordingly, I invariably prodded unsure FELDSTEIN; Stanley FISCHER; Roger respondents,sometimes to the point of hector- GORDON; Robert HALL; Arnold ing, to come up with some "best estimate" HARBERGER;Jerry HAUSMAN; Dale rather than to leave the profession with the JORGENSON; Lawrence KOTLIKOFF; results of a survey based upon a biased sample. David KREPS;Paul KRUGMAN;Jean-J. The sample mean is t = 3.96 percent, with a LAFFONT; Hayne LELAND; Robert LUCAS; Karl-G. MALER; Burton standarddeviation a- = 2.94 percent. In Table MALKIEL; Alan MANNE; Daniel 1 all 2,160 responses are listed, rounded off to McFADDEN; Robert MERTON; James the nearest whole integer, and the frequency MIRRLEES; Kevin MURPHY; William distributionis depicted as the histogramof Fig- NORDHAUS; Pierre PESTIEAU; Paul ure 1. The corresponding gamma probability PORTNEY; James POTERBA; Stephen distribution,which is defined by the same pa- ROSS; Agnar SANDMO; Thomas VOL. 91 NO. 1 WEITZMAN:GAMMA DISCOUNTING 269

SCBELLING;Richard SCHMALENSEE; discount-rateschedule declining hyperbolically Myron SCHOLES;; Robert to its asymptote of zero at a "convergence ve- SIIILER; JohnSHOVEN; ; locity" given by formula (26). Joseph STIGLITZ;Lawrence SUMMERS; It is importantto understandwhy the effec- JamesTOBIN; W. Kip VISCUSI. tive discount rate declines more rapidly when the underlyingspread of professional opinion is I cannot see how it might be said fairly that broader.For the same mean, a greatervariance the 50 members of this are not distin- group of opinion signifies that more professionals be- are of guished economists or ignorant the issues lieve in the lower-rate scenario, which is the involved-any more thanit could be arguedthat scenariothat predominates over time in present- the full sample of 2,160 names is not suffi- value calculations. Of course, more profession- to uncover the ciently comprehensive underly- als then also believe in the higher-ratescenario, ing probabilitydistribution. but their present-value weight becomes rela- The aggregate response for the above-listed tively less significant over time, due to the "balancedblue-ribbon panel" of expert opinion power of compound discounting at the higher is: y = 4.09 percent; a = 3.07 percent.Because rate. (In a sense, the high-ratebelievers discount the mean and standarddeviation for the entire away the relevance of their own scenario, leav- sample of 2,160 professional economists coin- ing the future ultimately to the low-rate believ- cides so with the mean and standard closely ers.) Summing the two effects, a greaterspread deviation for the subsample of 50 blue-ribbon of opinion hastens the day when the low-rate "leading-expert"economists, I take advantage scenario predominatesin present-valuecalcula- of the opportunityhere to average the two re- tions, thus causing the effective discount rate to sults and round off to the nearest one-tenth of decline more rapidly.9 one Henceforth in this will percent. paper, I Equation(22) can readily be invertedto yield work with the numerical values: the expression

(23) u= 4 percent per annum R (27) t(R) v and

(24) o- 3 percent per annum. where the velocity parameter v is defined by (26). IV. Gamma Discounting for Global Warming With the numerical values implied by (23), (24), and (26), formula (27) can be used to From equation (22), the schedule of effective determinethe time cutoff values corresponding discount rates is a simple rectangularhyperbola to R = 3.5 percent,R = 2.5 percent,R = 1.5 that is easy to calculate. What might be called percent, and R = 0.5 percent. After rounding its elasticity of discount-ratedecline is off, this numericalexercise then suggests a very rough periodizationof the future along the ap- tR(t) 1 proximate guidelines expressed in Table 2. (25) -R(t) 1 Table 2 speaks for itself in presenting the principal policy implication of this paper. The vt existing discount-rate uncertainty generates a sliding-scale effective discount-rate schedule, where the coefficient whose decline over time is significant enough

o-2 v (26) 9 I am unsure how to characterizethe most general class of probabilitydistributions for which higher o-implies lower R(t). I conjecture the condition holds for the class of all can be interpretedas a "convergence velocity" distributionsfor which o- parameterizesa mean-preserving parameter.One can then speak of the effective spread, but at this stage cannot provide a rigorous proof. 270 THE AMERICAN ECONOMIC REVIEW MARCH 2001

TABLE2-APPROXIMATE RECOMMENDEDSLIDING-SCALE DISCOUNT RATES

Marginal discount rate Time period Name (Percent)

Within years 1 to 5 hence Immediate Future 4 Within years 6 to 25 hence Near Future 3 Within years 26 to 75 hence Medium Future 2 Within years 76 to 300 hence Distant Future 1 Within years more than 300 hence Far-Distant Future 0

that this feature ought to be incorporatedinto point-input continuous-outputinvestment, then any discounting of long-term environmental it would be less than 2 percent per annum. projects. With numerical values (23) and (24), What is this paper then implying about the the velocity parameter(26) is sufficiently large optimalform of a long-term project under un- to mandatethe explicit use of a time-dependent certainty(such as taking measuresto ameliorate discount formula, such as (18), (22), for many global warming)? The full answer is compli- practical applicationsof benefit-cost analysis. cated and conditional on the exact time profile The downward-sloping time profile of the of the investment,because of the downwardtilt discount-rateschedule displayed in Table 2 is of the effective discount rate as a function of sufficiently steep to make the equivalent "as-if- time. But the decline is sufficiently steep to constant" discount rate much lower than the suggest, at some considerable risk of oversim- average discount rate of 4 percent. To see this plification, the following sort of generalization. effect clearly in a specific example, suppose that Other things being equal, the basic proposition an investment project is of the familiar point- should make itself felt by biasing the choice of input continuous-outputform. Then the equiv- policy instruments(such as taxes vs. tradeable alent "as-if-constant"discount rate is permits) and levels of desired stringency (such as greenhousegas emission targets)as if toward what is optimal for a low-interest-ratescenario (28) r= because, other things being equal, that kind of JA(t) dt scenario will weigh more heavily in the ex- pected present-discounted difference between benefits and costs.10

Substitutingfor A(t) from (18) and carrying V. Conclusion out the integrationin (28) yields The very wide spreadof professional opinion on discount rates means that society should be (29) r using effective discount rates that decline from a mean value of, say, around 4 percent per annumfor the immediatefuture down to around With the values ,u 4 percent and o- = 3 zero for the far-distantfuture. Furthermore,the percentbeing assumed in (23) and (24), expres- decline in effective social discount rates is suf- sion (29) becomes ficiently pronounced, and comes on line early

(30) r = 1.75 percent per annum. 10 This seems like a roughly accuratedescription of some of the main findings in William A. Pizer (1999). Please note The numerical value (30) therefore implies I am not assertinghere that everythingcan be reducedto an equivalent constant interest rate. It is only that if we force the following. If we were forced here to choose ourselves to think in such oversimplified terms, which is a single constant-equivalentdiscount rate r to wrong to begin with, then the best compromiseis to think in representthe entire sliding-scale schedule for a terms of a low rate. VOL. 91 NO. 1 WEITZMAN:GAMMA DISCOUNTING 271

enough, to warrant inclusion of this sliding- representsa vast oversimplification,but you can scale featurein any serious benefit-costanalysis pretendthat this is the only language that policy of long-term environmentalprojects, like activ- makers understand.) ities to mitigate the effects of global climate Naturally, I would prefer to have your best change. answer be a seriously-considered and well- thought-outestimate. However, an even higher APPENDIX:TEXT OF QUESTIONNAIRE priorityin the study must be placed on attaining extremely high response rates, which are re- Dear XXX: quired here for statistical accuracy by the un- I would very much appreciateenlisting a few derlying experimental design. Therefore, in moments of your time in the cause of introduc- order to eliminate response bias from the sam- ing some economic analysis into the current ple sub-groupthat you represent,I will settle for policy debates about global warming. what might be called your "professionallycon- As part of an empirical study, I need your sidered gut feeling." best point estimate of the appropriatereal dis- I know that you probably don't consider count rate to be used for evaluating environ- yourself to be an expert in this area, but I want mental projectsover a long time horizon. (What your best opinion anyway. Please respond to I am after here is the relevant interest rate for this request by returne-mail, even if your pre- discounting real-dollarchanges in future goods ferred numberis a back-of-the-envelopeguess- and services-as opposed to the rate of pure timate-or even if it just comes off the top of time preference on utility.) your head. No explanationis required. Your response will be held in strictest confi- Thank you for your help. dence. Only highly aggregatedstatistics will be Marty Weitzman reported.No one else will find out what number [email protected] you gave me. Try to imagine, then, that an international REFERENCES organization, such as the U.N. or the World Bank, has commissioned some well-done, Cropper, Maureen L. and Laibson, David. "The comprehensive,and highly-professionalstudies Implications of Hyperbolic Discounting for on the possible quantitativeimpacts of global Project Evaluation,"in Paul R. Portney and climate change. Suppose these studies have John P. Weyant, eds., Discounting and inter- done a credible job of converting costs and generational equity. Washington, DC: Re- benefits from any given year into expected- sources for the Future, 1999, pp. 163-72. consumption-equivalentreal dollarsfor thatyear. DeGroot, Morris H. Optimal statistical deci- Now these organizationswant your advice-and sions. New York: McGraw-Hill, 1970. are possibly going to take it seriously-about Nordhaus, William D. Managing the global what discountrate to use for aggregatingtogether commons: The economics of climate change. the expected-consumption-equivalentreal dollars Cambridge,MA: MIT Press, 1994. fromdifferent years, in orderto calculatethe over- Pizer, WilliamA. "The Optimal Choice of Cli- all net present discountedvalue of a proposed mate Change Policy in the Presence of Un- project. certainty."Resource and Energy Economics, You are being asked the following question. August 1999, 21(3-4), pp. 255-87. "Taking all relevant considerations into ac- Portney, Paul R. and Weyant, John P., eds. Dis- count, what real interest rate do you think countingand intergenerationalequity. Wash- should be used to discount over time the (ex- ington, DC: Resources for the Future, 1999. pected) benefitsand (expected) costs of projects Weitzman, Martin L. "Why the Far-Distant Fu- being proposed to mitigate the possible effects ture Should Be Discounted at Its Lowest of global climate change?" Possible Rate." Journal of Environmental Your answer should be a single number. (I Economics and Management, November realize that a single constant discount rate here 1998, 36(3), pp. 201-08.