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Gamma Discounting 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 economists. 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 economist 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 consumption-equivalentnet benefits: changes induced by a project, in terms of "ex-
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