Discounting and Climate Change

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So normal intertemporal economic problems and long-term en- vironmental ..... approach to sustainable development” in Social Choice and Welfare, 1996.
WORKING PAPER SERIES in MONEY, ECONOMICS AND FINANCE

DISCOUNTING AND CLIMATE CHANGE

Geoffrey Heal Columbia Business School

May 1997 PW-97-03

COLUMBIA BUSINESS SCHOOL COLUMBIA UNIVERSITY http:// www.columbia.edu/cu/business/wp/

Discounting and Climate Change Geo¤rey Heal Columbia Business School Climatic Change 37: 335-343, 1997

Abstract Why has discounting been controversial, and why has the controversy been particularly acute in the environmental area? The environmental area forces one to consider long time horizons. In the climate change area, a century is the minimum time horizon to make sense. The same is true about species extinction and biodiversity loss and disposal of nuclear waste. Such time horizons are completely outside the normal range in economic decision-making. Corporations and governments normally look at most decades ahead, rather then centuries. Horizons of this type are not long enough to raise one of the issues central to long-term environmental problems, namely equity between generations. For …ve to …fteen years in the context of business plans, the e¢cient use of money is the central issue: for a century or more, when considering the planet’s life support systems, equity and “sustainability” naturally come to the center of the stage. So normal intertemporal economic problems and long-term environmental problems involve di¤erent issues. It is natural that the methodology for one does not …t the other perfectly.

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Discounting: a perspective

Discounting has always been a source of controversy between economists and those from other disciplines interested in the environment. After all, if you discount at 5% over 100 years, then you are giving a future dollar a present equivalent of only $e¡5 = $6: 7379 £ 10¡3 , which is roughly 2/3 of a cent. It is hard for a non-economist to reconcile this with taking the future at all seriously. Perhaps less well-known is the fact that discounting has also been a source of controversy within the economics profession. Frank Ramsey, the …rst person to think seriously about dynamic economics and author of a 1926 article on long-run planning [13] which led the …eld until at least the 1960s, commented that discounting “is ethically indefensible and arises merely from the weakness of the imagination”. His contemporary Roy Harrod [7] added that it is a “polite expression for rapacity and the conquest of reason by passion”. These are strong words, from people who clearly believe that they are right. Why has discounting been controversial in economics, and why has the controversy been particularly acute in the environmental area? The key point about the environmental area is the following: it often forces one to consider long time horizons. In the climate change area, a century is the minimum time horizon to make sense. Probably it is too short1 . The same is true about species extinction and biodiversity loss and disposal of nuclear waste. Scienti…c processes relating to the environment naturally unfold over this type of horizon. Such time horizons are completely outside the normal range in economic decisionmaking. Corporations and governments normally look at most decades ahead, rather then centuries. The longest “normal” time horizons in economics are those for infrastructure investments such as power stations, where thirty years is a possible horizon. Five to …fteen years is much more common as a planning horizon. Horizons of this type are not long enough to raise one of the issues central to long-term environmental problems, namely equity between generations. For …ve to …fteen years in the context of business plans, the e¢cient use of money is the central issue: for a century or more, when considering the planet’s life support systems, equity and “sustainability” naturally come to the center of the stage (although e¢cient use of capital does not leave the stage). So normal intertemporal economic problems and long-term environmental problems have quite di¤erent time scales and involve di¤erent issues. It is natural that the methodology for one does not …t the other perfectly. In resolving long-term environmental issues, we want to achieve two aims: one is to strike a fair balance between the present and the future, and the other is to use our limited economic resources e¢ciently. The key resource here is some type of capital. How do we use capital e¢ciently? We let a market charge for its use, and so make sure that it is allocated preferentially to those uses where its value is highest. 1

For a more detailed discussion of the inconsistency of economic and scienti…c timescales in the context of global climate change, see Chichilnisky et al. [4].

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This gives rise naturally to a cost of capital, which is one of the things economists sometimes think that they are capturing by discounting. We do need a cost of capital, to make sure we don’t waste it: however green your complexion, you certainly want to use society’s capital as productively as possible. How do we strike a proper balance between present and future? This is a harder question to answer: in the end, this is a matter of ethical judgement, the same category of judgement as we make when we say that a particular distribution of income is acceptable, or is too unequal. This emphatically does not mean that it is just a matter of opinion, and that all opinions are equally good. There are some basic principles that we can establish, and these can provide valuable guides. But in the end, there is no purely scienti…c de…nition of a “proper” balance between present and future. The controversy within economics over discounting arises from disagreements about what is a proper balance, plus perhaps some confusion of the issues of intergenerational equity and e¢cient use of capital and a lack of clarity on how they interact. In the balance of this editorial I look at these issues in turn: specifying what we might mean by fairness to the future, by using capital e¢ciently, and then at how the two interact. Then I use these insights that this gives to comment on the papers by Hasselmann et al. and by Nordhaus, both of this issue.

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Equity

Suppose that society’s well-being in time period t can be represented by a function u (ct ) ; where ct is a vector of goods consumed and u is a real concave function of ct (i.e., increasing but at a decreasing rate). Then the sequence U = u (c1) ; u (c2 ) ; u (c3 ) ; u (c4 ) ; ::: represents the time pro…le of social welfare. Suppose for analytical convenience that this sequence U is in…nite: we avoid end-point problems this way. Some possible Us may have high welfare levels now and low ones in the future: other may do the reverse. Still others may show little variation over time. Of course, not any U is possible: resource endowments and technologies limit the welfare levels that we can achieve. How do we rank di¤erent Us? This is the analytical basis of the problem of …nding the proper balance between present and future. In the discounted utilitarian approach, the default approach in economics, we rank alternative u sequences as follows: 1

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U is ranked above U if and only if

t=1 X t=0

u

³ ´

c1t

t

± >

t=1 X t=0

³ ´

u c2t ± t

(1)

where 0 · ± · 1 is a discount rate. Frank Ramsey and Roy Harrod felt that we must have ± = 1 : in this case the in…nite sums will often not converge, so there have been attempts to extend the ranking implied by (1) to cases in which these sums are not

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de…ned. They have not been terribly successful2 . According to what ethical principles is this an appropriate way to rank alternative welfare sequences? This question was addressed by Koopmans [10], who showed that it has an answer: our preferences or rankings over well-being sequences must satisfy axioms which Koopmans called “stationarity” and “impatience”. Stationarity means that if I rank sequence ³ ´ ³ ´ u (c1) ; u1 c12 ; u1 c13 ; :::: above

³ ´

³ ´

u (c1) ; u2 c22 ; u2 c23 ; :::: then I also rank u1 (c12) ; u1 (c13 ) ; ::: above u2 (c22 ) ; u2 (c23 ) ; :::. In words, if two sequences have the same start, then eliminating that common start and bringing the rest forward does not change my ranking. This is not an innocuous axiom. Independence is less easy to state succinctly: roughly, it means that the relative values that we place on levels of well-being in any two periods do not depend on levels of well-being attained in periods other than those two. Again, this is not an innocuous axiom. Koopmans showed that, given some other technical assumptions, the conditions of independence and stationarity characterize discounted utilitarianism and imply ± < 1 and constant. This is a nice result: it tells us what we must “buy”, ethically speaking, to use the discounted framework. What are the alternatives? Chichilnisky [2] has an interesting one. She notes, quite consistent with the environmental critique, that the discounted utilitarian approach is insensitive to the long-run future. She proposes an alternative set of axioms, the key ones of which are that the way we rank sequences of well-being is sensitive to the present and also sensitive to the long-run future. In a neat analysis, she formalizes these concepts and shows that, as in the Koopmans case plus some technical conditions, they imply that the discounted sum of utilities should be supplemented by a term representing the long-run or sustainable level of well-being. In symbols, we rank Us according to the number ®

t=1 X t=0

u (ct ) ¢ (t) + (1 ¡ ®) lim u (ct) ; 0 < ® < 1 t!1

P

(2)

where ¢ (t) < 1 and ¢ (t) < 1: ¢ (t) is a discount factor: it declines over time, but not necessarily geometrically. This way of ranking turns out to have implications that may be much more future-oriented than the standard approach. Another alternative introduced by Beltratti, Chichilnisky and Heal [1] involves the concept of the “green golden rule”: according to this approach, the optimal policy is that which leads to the highest inde…nitely maintainable (or sustainable) welfare level. This is a very future oriented approach: it is equivalent to maximizing just the second part of Chichilnisky’s criterion: lim u (ct). In Chichilnisky’s terms, this t!1

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This is the work on “overtaking” criteria: for references and details see Heal [8].

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objective is sensitive to the long-run future, but not to the present: the utilitarian is precisely the opposite.

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Individual behavior

Of some interest in this context is a body of empirical evidence on the way individuals rank the present against the future. There is interesting evidence that they use a framework di¤erent in certain salient respects from the standard discounted utilitarian approach. Of course, even if we have a clear picture of how individuals form their judgements about the relative weights of present and future, this does not necessarily have normative implications: we might still feel that relative to some appropriate set of ethical standards they give too little (or too much) weight to the future, and so are an imperfect guide to social policy. However, in a democratic society, individual attitudes towards the present-future trade-o¤ presumably have some informative value about the appropriate social trade-o¤ and have at least an element of normative signi…cance. There is a growing body of empirical evidence3 which suggests that the discount rate which people apply to future projects depends upon, and declines with, the futurity of the project. Over relatively short periods up to perhaps …ve years, they use discount rates which are higher even than many commercial rates - in the region of 15% or in some cases very much more. For projects extending about ten years, the implied discount rates are closer to standard rates - perhaps 10%. As the horizon extends the implied discount rates drops, to in the region of 5% for thirty to …fty years and down to of the order of 2% for one hundred years.

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Logarithmic discounting

This empirically-identi…ed behavior has been termed “hyperbolic discounting”4 and is consistent with a very general set of results from natural sciences which …nd that human responses to a change in a stimulus are non-linear, and are inversely proportional to the existing level of the stimulus. For example, the human response to a change in the intensity of a sound is inversely proportional to the initial sound level: the louder the sound initially, the less we respond to a given increase. This is an example of the Weber-Fechner law, which is formalized in the statement that human response to a change in a stimulus is inversely proportional to the pre-existing stimulus. The empirical results on discounting cited above suggest that something similar is happening in human responses to changes in the futurity of an event: a given change in futurity (e.g., postponement by one year) leads to a smaller response in terms 3

See for example Lowenstein and Thaler [11], the papers in the volume by Lowenstein and Elster [12] or Cropper et. all. [5]. 4 See Henderson and Bateman [9] and references therein.

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of the decrease in weighting, the further the event already is in the future. This is quite natural: postponement by one year from next year to the year after, is clearly quite a di¤erent phenomenon from postponement from …fty to …fty one years hence. The former represents a major change: the latter, a small one. If we accept that the human reaction to postponement of a payo¤ or cost by a given period of time is indeed inversely proportional to its initial distance in the future, then the WeberFechner law can be applied to responses to distance in time, as well as to sound and light intensity, with the result that the discount rate is inversely proportional to distance into the future. Another way of saying this, is that we react to proportional rather than absolute increases in the time distance. Let the discount factor applied to bene…ts or costs at date t be ¢ (t) so that the discount rate is q (t) = ¢1 d¢ . Then dt we can formalize the idea that a given increase in the number of years into the future has an impact on the weight given to the event which is inversely proportional to the initial distance in the future as q (t) =

1 d¢ K = or ¢ (t) = eK log t = tK ¢ dt t

for K a negative constant. A discount factor ¢ (t) = eK log t has the interesting interpretation noted above: the replacement of t by log t implies that we are measuring time by equal proportional increments rather than by equal absolute increments. We react in the same way to a given percentage increase in the number of years hence of an event, rather than to a given absolute increase in its number of years hence. We shall call this “logarithmic discounting”: this is quite consistent with the approach taken in for example acoustics, where in response to the Weber-Fechner law sound intensity is measured in decibels which respond to the logarithm of the energy content of the sound waves, and not to energy content itself.

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E¢ciency

We want to allot resources fairly between generations: we also want, at least other things being equal, to use them e¢ciently. What does this mean? In the case of capital or renewable resources, e¢cient use means satisfying the “Ramsey rule”: in the case of extractive resources, there is an equivalent “Hotelling rule”. The latter is simpler, so let us start with that. Hotelling’s rule simply says that use of an extractive resource should be distributed over time in such a way that the derivative of society’s objective with respect to the level of resource use is the same at all points in time. This is a rather obvious …rst order condition for optimality: but in spite of being obvious, it has some interesting and non-obvious implications. In the utilitarian case, it means that the present value of the resource price must rise over time at the discount rate. 5

In the case of e¢cient use of capital, the issue highlighted by Ramsey’s rule is the following. By postponing a little bit of consumption, we can accumulate slightly more capital, which gives us a stream of bene…ts continuing onto the future. Postponing consumption is a loss: having more capital is a gain. E¢cient use means that the two just balance out. Again, a pretty obvious …rst order condition, with some interesting implications. A key aspect of both these concepts of e¢ciency, is that they depend on society’s objectives, which include its concept of equity. This is explicit in the statement of the Hotelling rule: “use of an extractive resource should be distributed over time in such a way that the derivative of society’s objective with respect to resource use is the same at all points in time”. In the case of the Ramsey rule, it is implicit in the statement that the losses from postponing consumption just balance out the gains from accumulating more capital. Here the losses are now and the gains in the future, so operationally this concept requires a speci…cation of the trade-o¤ between present and future5 . Another key aspect of both concepts, more operational than the previous, is that both are characterized by di¤erential equations. In fact, both are characterized by di¤erential equations similar to the classical Euler-Lagrange equations which emerge from …rst order conditions for optimality in variational problems. There is generally a continuum of paths satisfying such equations, of which only one is optimal: which is actually optimal is determined by initial or terminal conditions.

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Equity + E¢ciency = Optimality

How do we combine equity and e¢ciency, and what are the implications for discounting? As noted, given a de…nition of equity, we can characterize e¢ciency. There are many e¢cient paths: the best of them is optimal, and is the one we would want the economy to follow. What does optimality imply for valuing the future? The easiest way to understand this is too look at what economists call “shadow prices”: the shadow price of a good or service is the partial derivative of the objective function evaluated on an optimal plan with respect to the provision of that good or service. It is the adjoint variable associated with the good or service in question in the variational problem that one solves to …nd the optimal time-path of resource use. This is clearly a measure of the “social value” of the commodity concerned (for more details, see Heal [8]). Some rather important results in intertemporal welfare economics show that if …rms in an economy maximize their pro…ts as measured by the “shadow prices”, then they will be led to choose production and consumption levels identical to those on the optimal path6 . In particular, what we need to look at, is what value these shadow prices place 5 6

For a detailed discussion of these rules, see Dasgupta and Heal [6]. See Chichilnisky and Kalman [3].

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on goods provided far into the future: how the values of goods provided in the future compare with those of the same goods provided today. We can illustrate this for a case which is analytically simple by considering the shadow prices in the stationary states to which the optimal solutions asymptote. I quote some results from a simple model in which welfare or utility is a function of the rate of consumption and of the remaining stock of an exhaustible resource, so that welfare at date t is expressed as u1 (ct ) + u2 (st ), where st is the remaining stock of a resource and ct is the rate at which it is being depleted7 . This is the simplest model in which one can capture the con‡ict between welfare from consumption and welfare from conservation. In this model with the utilitarian objective and discount rate ± 0 the asymptotic social value is given by (using u to denote the derivative of u with respect to its argument) 0 ¸util = u2 (s¤) =± 0

where s¤ is the stationary value of the resource stock in the long run and u2 (s¤) =± is the present value of the stream of bene…ts resulting from an incremental increase in this stock. If we use the Chichilnisky criterion as a maximand, then we have instead ¸chichiln isky = u02 (sb) =± + u02 (sb) (1 ¡ ®) =®

where sb is the stationary stock value in the long run on a path optimal according to Chichilnisky’s criterion. In this case the shadow price is the sum of two terms: one, as before, is the present value of the stream of bene…ts resulting from an incremental increase in this stock. The second represents the contribution of an increase in the stock to the limiting utility level. The second term, although it represents contributions in the far distant future, is not discounted. For small values of the parameter ®; it can clearly be the dominant term: this corresponds to the case when the objective places more weight on the very long run than on the more immediate future. The key point here is that if we are placing su¢cient weight on the future, then the shadow price for a resource may naturally contain an element representing its contribution to future bene…ts that is not discounted, its futurity notwithstanding. This is not equivalent to saying that a zero discount rate is appropriate: we are instead saying that the shadow price of a good re‡ects its contribution to the social objective, and this is the sum of two terms, one a present value and one undiscounted. Suppose instead that our objective is to achieve the maximum sustainable utility, what I described above as the green golden rule. If society is so future-oriented as to wish to support the highest sustainable utility level, then we need correspondingly future-oriented behavior on the parts of agents in the economy. Rather than having …rms maximize pro…ts as measured by shadow prices, we need …rms to seek the highest sustainable pro…ts (i.e., the maximum value of pro…ts that can be maintained for ever) and resource owners to manage their resources so as to yield the highest 7

The model comes from Heal [8].

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sustainable revenues from the resources. In this case, it is no longer appropriate to compute present values8 .

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Conclusions

I have skated over the surface of an extremely complex set of issues, without the space to do full justice to any of them. Someone who wants to work in the area of intertemporal policy evaluation has to understand these issues and have a position on them, and there is no substitute for hard work in getting there: getting these issues wrong is enough to invalidate any policy analysis. Where does this brief overview leave us with respect to the papers by Hasselmann et. al. and Nordhaus? Firstly, and perhaps most importantly, there is nothing sacrosanct about the objective of maximizing a sum of bene…ts over time discounted to the present at a constant discount rate. Accepting this as the right objective depends on accepting a particular axiomatization of intertemporal fairness, one which as been rejected by many people who have looked closely at it. Second, also important, there are well-de…ned and non-arbitrary alternatives. Chichilnisky’s criterion is one. Over …nite time horizons, as in the studies at issue here, one would implement this by maximizing a weighted average of a discounted sum of utilities plus the terminal utility value. Alternatively, one can argue for an extremely future-oriented position and adopt the “green golden rule” as a desideratum. Thirdly, if one does not accept the discounted utilitarian approach as axiomatized by Koopmans, then one does not have to work with a discount rate that is constant over time. The discount rate could fall over time, implying far more weight on future bene…ts than with a discount rate constant at the initial value. I should add that I see no justi…cation for the approach of Hasselmann et. al. in using di¤erent discount rates for di¤erent sectors of the economy: all of the approaches that I have mentioned involve using a discount rate that is uniform over sectors.

References [1] Beltratti, Andrea, Graciela Chichilnisky & Geo¤rey Heal. “The Green Golden Rule”, Working Paper, Columbia Business School 1994, Economics Letters, 1995. [2] Chichilnisky, Graciela. “What is sustainable development?” Paper presented at Stanford Institute for Theoretical Economics, 1993. Published as “An axiomatic approach to sustainable development” in Social Choice and Welfare, 1996. 8

For details, see Chichilnisky and Kalman [3] and Heal [8].

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[3] Chichilnisky, Graciela and Peter Kalman. “Application of Functional Analysis to Models of E¢cient Allocation of Economic Resources”, Journal of Optimization Theory and Applications, Vol. 30, No. 1, January 1980, pp. 19-32. [4] Chichilnisky, Graciela, Vivien Gornitz, Geo¤rey Heal, David Rind and Cynthia Rosenzweig. “A new approach to integrated assessment: building linkages between economic and climate models”. Working paper, Goddard Institute for Space Studies, NASA-Columbia University. [5] Cropper, Maureen L., Sema K. Aydede and Paul R. Portney. “Preferences for Life-Saving Programs: How the Public Discounts Time and Age”. Journal of Risk and Uncertainty, 8: 243-265, 1994. [6] Dasgupta, Partha and Geo¤rey Heal. Economic Theory and Exhaustible Resources, Cambridge University Press, 1979. [7] Harrod, Roy. Towards a Dynamic Economics. Macmillan Press, London, 1948. [8] Heal, Geo¤rey M. Valuing the Future: Economic theory and Sustainability. Manuscript: an early version was delivered as the Lief Johansen Memorial Lectures at the University of Oslo in March 1995, and circulated as a University of Oslo Economics Working Paper. [9] Henderson, Norman and Ian Bateman. “Empirical and public choice evidence for hyperbolic social discount rates and the implications for intergenerational discounting”, Environmental and Resource Economics, 5: 413-423, 1995. [10] Koopmans, Tjalling. “Stationary ordinal utility and impatience”, Econometrica, 28 1960 287-309. [11] Lowenstein, George and Richard Thaler. “Intertemporal Choice”, Journal of Economic Perspectives, 3, 181-193, 1989. [12] Lowenstein, George and Jon Elster (eds.). Choice over Time. Russel Sage Foundation, New York, 1992. [13] Ramsey, Frank. “A Mathematical Theory of Saving”, Economic Journal, 38 (1928), pp 543-559. [14] Thaler, Richard. “Some Empirical Evidence on Dynamic Inconsistency”, Economics Letters, 8, 201-207, 1981.

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