1. INTRODUCTION2. DESCRIPTION OF THE ECONOMY3. THE YIELD CURVE WHEN THERE IS NO RISK OF RECESSIONlabel idnorecession4. THE YIELD CURVE WITH A RISK OF RECESSION IN THE FUTURE5. THE YIELD CURVE WHEN THERE IS A RISK OF A RECESSION IN EACH PERIOD6. CONCLUSIONREFERENCES463⁄0022-0531/02 $35.00© 2002 Elsevier Science (USA)All rights reserved.Journal of Economic Theory 107, 463–473 (2002)doi:10.1006/jeth.2001.2952Time Horizon and the Discount RateChristian Gollier11We are grateful to Claude Henry, Jean-Pierre Florens, Jennifer Gann, Jean-JacquesLaffont, Pierre Lasserre, Jean-Charles Rochet, and Patrick Roger for their helpful comments.Financial support from the European Commission is gratefully acknowledged.University of Toulouse, Toulouse, France22Address correspondence to IDEI, Université des Sciences Sociales, Manufacture desTabacs, 21 allée de Brienne, 31000 Toulouse, [email protected] September 15, 1999; final version received November 8, 2001We consider an economy à la Lucas (1978, Econometrica 46, 1429–1446) with arisk-averse representative agent. The exogenous growth rate of the economyfollows a random walk. We characterize the set of utility functions for which it isefficient to discount more distant cash flows at a lower rate. The benchmark resultis that, when the growth rate is almost surely nonnegative, the yield curve isdecreasing if and only if relative risk aversion is decreasing with wealth. Relaxingthe assumption on the absence of recession requires more restrictions on pref-erences, such as increasing relative prudence. Journal of Economic LiteratureClassification Numbers: D81, D91, Q25, Q28. © 2002 Elsevier Science (USA)Key Words: discounting; uncertain growth; prudence; time horizon.1. INTRODUCTIONThe objective of this paper is to examine the relationship that existsbetween the socially efficient discount rate and the time horizon. We con-sider a simple economy à la Lucas [12] with a representative risk-averseagent facing an exogenous and uncertain growth of his or her consumptionover time. This paternalistic agent wants to maximize the net present valueof the flow of future expected utility. We characterize the set of utilityfunctions for which it is optimal for the representative agent to discountmore distant cash flows at a lower rate. This question has not beenaddressed in the literature so far, except by Cox et al. [2, 3] in the specialcase of power utility functions, in which case the socially efficient discountrate is independent of the time horizon.This paper is motivated by the difficulty in using the standard cost–benefit analysis with a constant discount rate for public investment projectswhose costs and benefits are generated over a long period of time, as is thecase for projects related to mitigating global warming, or for the manage-ment of nuclear wastes. Discounting far distant costs and benefits at thesame rate as for the shorter terms is equivalent to ignoring these long-termeffects. Using discount rates that are decreasing with the time horizonwould reduce the exponential effect of discounting.Before examining the effect of time on the socially efficient discount rate,it is useful to recall the determinants of the level of the efficient discountrate. With a sure positive growth of the economy, we do not want tobenefit overmuch future generations who will enjoy a larger GNP percapita. Under decreasing marginal utility of consumption, one more unit ofconsumption in the future is less valuable than one more unit of consump-tion today. This wealth effect is the standard argument for using a positivediscount rate. But the growth of the economy is affected by random shockswhich should be taken into account in the selection of the discount rate.The effect of a future risk on the willingness to invest for the future is wellknown from Leland [11] and Drèze and Modigliani [4]: if people areprudent, the uncertainty about the growth of incomes should induce themto invest more for the future. This precautionary effect provides an argu-ment to reduce the discount rate. This effect increases with the degree ofprudence, an index introduced by Kimball [9] to measure the propensityto accumulate savings in the face of future risk.We want to determine how the wealth effect interacts with the precau-tionary effect when the time horizon is expanded. We first examine the caseof an economy facing no risk of recession. In that case, we show that thesocially efficient discount rate is decreasing with the time horizon if andonly if relative risk aversion is decreasing. The case of an economy with arisk of a recession is more complex. In such a situation, there are twoconflicting effects of a longer time horizon on the valuation of future cashflows. When the expected growth of the economy is positive, a longer timehorizon raises the expected consumption and the risk affecting this con-sumption. Thus, a longer time horizon yields a positive wealth effect and anegative precautionary effect on the socially efficient discount rate. If theprecautionary effect dominates the wealth effect, one should recommendselecting a smaller discount rate for longer horizons. In this paper, weprovide conditions for this dominance to hold. Without surprise, our con-ditions depend upon properties of the index of prudence. In spite of thissimple intuition, the reader should not expect to get simple necessary andsufficient conditions. The basic problem is to determine how temporal464 CHRISTIAN GOLLIERgrowth risks interact with each other. The recent literature on multiplerisks, as initiated by Pratt and Zeckhauser [14], clearly shows that theseinteractions are complex and that conditions on the third and fourth deri-vatives of the utility function are necessary to yield unambiguous com-parative statics results.This work is related to a recent paper by Weitzman [16] who alsoproves that the discount rate should be decreasing with the time horizon.Weitzman’s conclusion is obtained in a different framework, with riskneutral agents together with a simple early revelation of future uncertainproductivity of capital.2. DESCRIPTION OF THE ECONOMYThere is a paternalistic representative consumer who maximizes the sumof future expected utility discounted at rate d=b−1−1. Parameter d mea-sures the rate of pure preference for the present. It must be constant overtime to guarantee the time consistency of the decision process. The utilityfunction u on consumption is assumed to be