Clim ate P olicy when the Distan t FutureM a tters: C atastro phic ev ents with hyperbolicdiscoun tingLarry Karp∗Yacov Tsur♦November 10, 2006AbstractWe study climate change policy with hyperbolic discounting un-der the risk of catastrophic events. Using a binary action model, wecompare the set of Markov Perfect Equilibria (MPE) to the optimalpolicy under (time consistent) commitment. For some initial condi-tions there are multiple MPE that may involve either excessive orinsufficient stabilization efforts relative to the policy under commit-ment. Numerical analysis shows that standard Integrat ed Assessm entModels may have significantly understated the optimal level of near-term effort to reduce GHG emissions. Even if the free-rider problemamongst contemporaneous decision-makers were solved, the coordina-tion problem amongst different generations complicates the design ofoptimal policy.Keywords: abrupt climate c hang e, ev ent uncertain ty, catastroph ic risk,hyperbolic discoun ting , Markov Perfect Equilibria∗Department of Agricultural and Resource Economics, 207 Giannini Hall, Universityof California, Berkeley CA 94720 email:[email protected]♦Department of Agricultural Economics and Management, The Hebrew University ofJerusalem, P.O. Box 12 Rehovot 76100, Israel ([email protected]).1 IntroductionIn tegra ted Assessm ent Models (IAM s) — n u m erical models that combine cli-mate and economic modules — t ypically endorse modest near-term efforts toreduce green hou se gas (G HG ) emissions (Nordhaus and Boyer 2000, IPCC2001, Manne 2003, Edmonds and Sands 2003). These models have been in-flu ential in guiding policy recomm enda tion s because of their empirical foun-dation. (See, for example, the open letters from econom ists to Presiden t Bushin the appendix to Griffin (2003), and the c hapter on the Ky oto Protocol inBarrett (2003).) Alth ough few of these models formally incorporate abruptcatastrophic events, discoun tin g cau ses th ese even ts to h ave little effect onoptimal climate policy (Mastrandrea and Schneider 2001). Lo w probabilityev ents (those with lo w ha zard rates) are not likely to occur un til the distan tfuture, and discoun ting at a non-negligible rate makes the distan t future al-most irrelevant to policy today. The conclu sion that low -pr obab ility catas-trophic eve nts are relatively unimportant for clim at e cha ng e policy prob ab lysounds strange to the represen tative citizen (or climate scien tist).We study catastro phic threats with hyperbolic discoun ting to provideanoth er view of the im portance of lo w proba bility climat e ev ents. Our mainpolicy conclusion is that standard IAMs may significantly understate theoptim al reduction in GHG emissions. In reaching this conclusion, w e showhow to solve a new ty pe of optimal con trol/dynamic game model, and w eexplain its relation to more familiar dynamic coordinatio n problems.A low probability ev ent occurs infrequently, but its proba bility of occur-rence o v er a few cen turies is non-negligible. For example, the thermohalinecircuit (THC) (the ocean curren ts that con tribute to the moderate climatein Western Europe) has sh u t down or rev ersed m ore than on ce during thegeological record (Broec ker 1997). Suppose that suc h an “event” reducesthe ann ual flow of utilit y (or income) by 1 util (the value-at-risk). Let theconstant hazard rate be h, so that the probability of the event occurring dur-ing the next year, conditional on it not hav ing y et occurred, is 1 − exp(−h).1W hen the future is discounted at a constant annual rate r, the expectedpresent value of the value-at-risk is1h+r. Suppose that “stabilization” elim -inates the hazard, at a flow c ost (perhaps caused b y a perpetual reductionof econom ic activity) of x(100%), leading to a present value of1−xr.Themaximal flow cost, x∗, that societ y would pay for stabilization equates theset wo expressions, leading to x∗=hh+r. If the annual discount rate is 5%(r = .05) and the probability of the even t occurring within a century is 5%(equivalen t to the annu al hazard h =5.129 × 10−4), then x∗=0.01015.Inthis case, societ y wou ld be willing to sacrifice about 1% of the v alue-at-riskto eliminate the threat.1Und er const ant discounting, societ y will not devote substantial resourcesto reduce a risk unless the hazard rate and discoun t rate are of a similarorder of magnitude. For low probability even ts, this similarity requiresa discount rate near zero. However, a negligible constan t discoun t raterequires excessiv e saving by current generation s, and it is inconsisten t withempirical estimates of time preferences (Frederick et al. 2002).A positiv e social discount rate requires a positiv e pure rate of time pref-erence and/o r the anticip atio n that the future growth rate of incom e willbe positive. Althoug h both of these condition s plausibly hold for the nearterm, they ma y not hold for the dista nt future. We ma y feel closer to ourc h ild ren than to our unborn grandc hildr en, but it is less likely that we caremuch more about the 10thfuture generation than about the 11th.Thereisalso less basis for assuming that incom e will be higher in the 11ththan inthe 10thfuture generation. The longest-term bonds mature within 30 yearsand there are no financial instrumen ts that reflect (very) long-term discoun trates.There is a large literature dealing with the presen t valuation of futurewelfare, motivated b y considerations of inter-generational equit y and sustain-1Inthecaseofclimatechange,whereinertiais important, current actions could alterfuture but not current risk. By assuming that the policy has an immediate effect on thehazard, this example overstates the amount that society would be willing to spend.2abilit y (Solo w 1974, Hartwic k 1977, Chic hilnisky 1996, Arrow 1999, Asheimand Buc hh olz 2004) and uncertaint y (Weitzman 2001, Gollier 2002, Das-gupta and M askin 2005). An importan t strand of this literature uses h y-perbolic discoun ting, where the discount rate falls ov er time. This modelof discounting is consisten t with the theoretical and empirical argumen tsfor a non-negligible near-term discount rate, and the ethical argumen ts forgiving positive w eig ht to the distant future. Cr op per and Laibson (1999)discuss hy perbolic discounting in the context of climate chan ge, and Mas-trandrea and Sc hneider (2001) use hyperbolic discounting with a variationof the DICE model (a