Journal of Public Economics 85 (2002) 149–166www.elsevier.com/locate/econbaseDiscounting an uncertain futureChristian Gollier´Universite de Toulouse and Institut Universitaire de France,GREMAQ,Place Anatole France,31042Toulouse Cedex,FranceReceived 24 March 1999; received in revised form 16 July 2000; accepted 30 October 2000AbstractThe objective of this paper is to determine the socially optimal discount rate for publicinvestment projects that entail costs and benefits in the very long run. We suppose that thereis an exogenous process for the growth of consumption per capita, which is stochastic. Wefirst evaluate the determinants of the discount rate for a specific horizon when therepresentative agent has a recursive utility. We then explore the influence of the timehorizon in the expected utility model. Under various conditions on preferences, as positiveprudence, decreasing relative risk aversion or decreasing absolute risk aversion, we provethat (1) the fact that growth is uncertain reduces the efficient discount rate at any horizon,and that (2) this discount rate should be smaller for more distant futures. We characterizethe asymptotic value of the discount rate.  2002 Elsevier Science B.V. All rightsreserved.Keywords:Discounting; Uncertain growth; Log-supermodularity; Prudence; Kreps–Porteus preferenceJEL classification:D81; D91; Q25; Q281. IntroductionMuch of life is made of investments. Costly actions are taken today in theprospect of future benefits. In the presence of efficient financial markets, theinvestment decision process is based on the classical concept of the Net PresentValue (NPV). The argument sustaining this decision rule is based on arbitrage.E-mail address:[email protected] (C. Gollier).0047-2727/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved.PII: S0047-2727(01)00079-2150 C.Gollier / Journal of Public Economics85 (2002) 149–166Instead of undertaking the planned investment, one could invest in the financialmarkets. As a consequence, the return of the planned investment should yield atleast the risk free rate. For standard investment projects, this rule is equivalent tohaving a positive NPV. If financial markets are frictionless, the use of the observedrisk free rate to discount public investment projects leads to a socially efficientlevel of investment.The analysis is less easy to perform when benefits and costs of the set of currentpotential actions are expected to last in the long run. The carbon dioxide that oneemits today will not be recycled for a couple of centuries, yielding long term costslike global warming. Some nuclear wastes like plutonium have half-life in the tensof thousands years. Obviously, financial markets are not very helpful to provide aguideline for investing in technologies that prevent this kind of long-lasting risksto occur. Liquid financial instruments with such large durations do not exist. Forthe sake of comparison, US Treasury Bonds have time horizons that do not exceed30 years. We must thus rely on the use of an economic model to value the distantfuture.¨The name of Bohm-Bawerk and Fisher are intimately related to these questions.The first reason offered by these authors is purely psychological. Namely, agentsmay have a pure preference for the present, i.e. they are impatient. The secondreason to discount the future is related to a wealth effect. We expect that thequantity of available consumption goods will increase over time. After all, in thewestern world at least, we experienced an uninterrupted growth during the two lastcenturies. Given decreasing marginal utility of consumption, an investment whichgives one unit of the consumption good in the future in exchange for one unit ofthe consumption good in the present should not be acceptable. Investing for thefuture in a growing economy will increase consumption inequalities over time.Since agents have preferences for the smoothing of consumption over time, thisinvestment should be implemented only if its rate of return is large enough tocompensate for this negative impact on welfare. The larger the growth rate of theeconomy, the larger the socially efficient discount rate.A problem arises with this wealth effect if the growth rate is not known withcertainty. Estimating the growth rate for the coming year is already a difficult task.No doubt, any estimation of growth for the next century/millennium is subject topotentially enormous errors. The history of the western world before the industrialrevolution is full of important economic slumps, as the one due to the invasion ofthe Roman Empire, or the one due to the Black Death during the middle ages. Therecent debate on the notion of a sustainable growth is an illustration of the degreeof uncertainty we face to think about the future of society. Some will argue thatthe effects of the improvements in information technology have yet to be realized,and the world faces a period of more rapid growth. On the contrary, those whoemphasize the effects of natural resource scarcity will see lower growth rates inthe future. Some even suggest a negative growth of the GNP per head in thefuture, due to the deterioration of the environment, population growth anddecreasing returns to scale. They claim that the wealth effect goes the otherC.Gollier / Journal of Public Economics85 (2002) 149–166151direction, so that everything should be made to improve the future. Thisuncertainty at least casts some doubt on the relevance of the wealth effect tojustify the use of a large discount rate. In this paper, we provide an analysis of theeffect of the uncertainty on growth on the socially optimal discount factor.Instrumental to this analysis is the concept of prudence that has been formalizedby Kimball (1990). An agent is prudent if his willingness to save increases in theface of an increase in his future income risk. Technically, an agent is prudent if thethird derivative of his utility function is positive. As shown in this paper, prudencejustifies taking a discount rate that is less than the one that would have beenobtained by assuming a certain growth. The magnitude of the effect depends uponthe degree of prudence and the degree of uncertainty on growth. This analysis isprovided in Section 2. Some numerical simulations are performed in Section 3.In Section 4, we examine the relationship between the time horizon and thesocially optimal discount rate. In order to counterbalance the exponential effect ofdiscounting,