Discounting and relative pricesAbstractIntroductionDiscounting and relative pricesThe case of a constant elasticity of substitutionUncertaintyDevelopment of the discount schedule over timeConclusionAppendix ADerivation of Eq.&newnbsp;10Appendix BDerivation of results in Table&newnbsp;2Derivatives of rDerivatives of RDerivatives of pAppendix CProof that long-run declining U implies R - r < 0 in the long runReferencesDiscounting and relative pricesMichael Hoel & Thomas SternerReceived: 10 March 2006 /Accepted: 15 March 2007 / Published online: 9 May 2007#Springer Science + Business Media B.V. 2007Abstract Environmentalists are often upset at the effect of discounting costs of futureenvironmental damage, e.g., due to climate change. An often-overlooked message is thatwe should discount costs but also take into account the increase in the relative price of theecosystem service endangered. The effect of discounting would thus be counteracted, and ifthe rate of price rise of the item was fast enough, it might even be reversed. The scarcitythat leads to rising relative prices for the environmental good will also have direct effects onthe discount rate itself. The magnitude of these effects depends on properties of theeconomy’s technology and on social preferences. We develop a simple model of theeconomy that illustrates how changes in crucial technology and preference parameters mayaffect both the discount rate and the rate of change of values of environmental goods. Thecombined effect of discounting and the change of values of environmental goods is morelikely to be low – or even negative – the lower the growth rate of environmental quality (orthe larger its decline rate), and the lower the elasticity of substitution betweenenvironmental quality and produced goods.1 IntroductionDuring the past 10–20 years, a vast economic literature has discussed various aspects ofclimate change and climate policies. Part of this literature gives cost–benefit analyses ofvarious proposed measures to reduce greenhouse gases. In analyses of this type, economistsadd up all benefits and costs of any proposed policy. This summation of costs and benefitsincludes the current period and all future periods. In such cost–benefit analyses, future costsClimatic Change (2007) 84:265–280DOI 10.1007/s10584-007-9255-2M. HoelDepartment of Economics, University of Oslo, P.O. Box 1095 0317 Oslo, Norwaye-mail: [email protected]. Sterner (*)Department of Economics, University of Goteborg, Goteborg, Swedene-mail: [email protected] benefits are discounted to present values through the use of a discount (or interest) rate.If some cost or benefit component at a future date t is of the magnitude Vtand the discountrate is r, the present value is 1 þ rðÞtVt.Environmentalists are often upset at the effect of discounting on such benefits and costs.There have been numerous debates – related to the climate problem as well as otherenvironmental problems – in which environmentalists are dismayed to find the cost of someexpected environmental damage is discounted to a present value that is basicallyinsignificant.The logic of discounting is in fact dramatic, but it is correct as long as the assumptionshold. With 5% discounting per year, a cost of $100 in 100 years is worth less than $1 today.There are various arguments in favor of low or nonlinear discount rates, and an extensiveprofessional literature on the subject, but in this article we want to focus on another,separate, factor: that of changes in relative prices.1The cost Vt, for instance, of futureenvironmental damage, should be valued in real future prices. However we only know theprices today. Our best estimate of Vtis Vt¼ V01 þ pðÞt, where p is the expected rise peryear in real price of the relevant good (i.e., the price relative to some general price levelsuch as the consumer price index). For an environmental resource that will become scarcein the future, the relative price will typically rise – and this effect may again be exponential –and thus is potentially just as powerful arithmetically as is discounting. Indeed, combining thetwo formulae we see that the discounted, present, real cost of any given item of damage in thefuture will be V01 þ pðÞt1 þ rðÞt. This general point is well known to economists, but isoften overlooked.The effect of future scarcity of environmental goods is difficult to study empirically fortwo reasons. First, it is difficult because we are speaking of a fairly recent phenomenon, andsecond, because many environmental goods are unpriced nonmarket goods and thus we arespeaking more of a willingness to pay than of a regular price. However, there are marketgoods (partly related to environmental factors) that can illustrate the effect of scarcity onprices. One clear example is that of land and housing markets. We select this examplebecause the stock of housing typically exhibits scarcity. Houses, of course, can bemodernized, which no doubt explains a large part of the rise in price, but the underlyingscarcity of space for housing lots is likely an important explanatory factor why housingshould appreciate faster in densely populated areas. We choose to illustrate this with thehousing market in the United Kingdom, where land scarcity is important. Table 1 showsthat the Greater London area where land is scarcest has a rate of price escalation of 17.8%,compared with 11.5% in Scotland, and with the UK national average of 15.9%. The retailprice index for the same period showed an inflation of 7.4%, far lower than even the lowestprice increase in Table 1. The rate of interest was also much lower; the repo rate of theBank of England for this period was between 3.5 and 7.5%.2The simple, but often overlooked, message is that, in calculating the costs of some futuredamage (say, of excess carbon in the atmosphere), we should discount costs but also takeinto account the increase in the relative price of the ecosystem service endangered(biodiversity, clean air, water, arable land, coastal ecosystems, etc.). Thus the effect ofdiscounting would be counteracted, and if the rate of price rise of the item were fast enoughit might even be reversed. This point is in its simplest form well established in the1Important contributions include Arrow et al. (1996), Cline (1992), Dasgupta (2001), Hasselmann (1999),Heal (1997), Horowitz (1996), Lind (1982), Nordhaus (1997), Portney and Weyant (1999), Schelling (1995),Shogren