MORE TROUBLE FOR COST-BENEFIT ANALYSIS

An Editorial Comment

1. Introduction

Poor cost-benefit analysis. It has been the target of so many recent critiques, es-pecially in its application to the mitigation side of the climate issue, that it canhardly stand. Its detractors point to a series of questions that hammer directly at itsvery foundations. Even if we knew exactly how the future would unfold absent anypolicy intervention, they argue, do we really think that we could measure accuratelyand completely the time series of the cost of removing one ton of carbon fromcurrent emissions? And how could we possibly measure accurately and completelythe time series of benefits that would result? What about the discount rate? Isthere a persuasive case for imposing a positive pure rate of time preference, orshould comparisons across generations depict no impatience to consume even ifthe current generation will be poorer than the next? And finally, how could we, ingood conscience, use a tool that ignores equity so completely that a net dollar inbenefit paid to a rich person is precisely equivalent to a net dollar paid to a poorperson?

These questions are troublesome, of course, but the critics of cost-benefitanalysis are not done. They draw their big ammunition from the recognition thatuncertainty pervades the climate arena. Even if we could measure things accu-rately and completely along one specific future scenario, they continue, how couldwe estimate the distribution of costs and benefits across a range of unknown andunpredictable economic and climate futures? Is there a corresponding distributionof appropriate discount rates over time, and how should we reflect uncertaintyabout time preference to everyone’s satisfaction? Should we not worry about thedistributions of future costs and benefits? Will not the future distribution of incomemake a difference in our evaluations? And where can we turn for distributions ofthe relatively likelihoods of a wide range of ‘not-implausible’ futures?

Given all of these problems, why do researchers continue to conduct cost-benefit analyses of climate issues, particularly if they are trying to provide someinsight into how and when to mitigate greenhouse gas emissions? For the samereason that economists spend so much time examining the properties of perfectlycompetitive markets. Participants in competitive markets decide how much theywant to buy or sell of some good at a price that is determined anonymously by mar-ket powers. Nobody, in short, sets the price; it just magically appears. Economistsdo not study these markets because reality is littered with a plethora of competitive

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markets. Economists study them because competitive equilibria represent bench-marks of economic (i.e., Pareto) efficiency from which nobody can be made betteroff without hurting somebody else (at least until something goes wrong). As aresult, the size and distribution of the economic cost created by any ‘market dis-tortion’ or by any intervention in the name of equity or any other objective can bejudged against this benchmark.

It turns out that maximizing net benefits under the same conditions achievesthe same benchmark of economic efficiency. Absent any inclusion of more pre-cise information, fundamental equity concerns, externalities and/or other ‘marketdistortions’, solutions that set marginal benefits equal to marginal costs achieveexactly equivalent, Pareto efficient outcomes. As a result, allocations that maximizenet benefits can serve as analogous benchmarks against which we can estimate thecost of externalities, the damage caused by distortions, and sometimes the value ofimproved and/or more complete information.

Uncertainty remains a problem, of course, but maximizing expected benefitsminus expected costs can provide a second order efficiency benchmark that showshow we could do as well as possible, ‘on average’ given the circumstances. Theobjective in this context is to achieve an outcome where expected marginal costequals expected marginal benefit; but it is in this context that Richard Tol (2003)has discovered yet another problem for the cost-benefit approach that undercutseven its benchmark function. Armed with his FUND model, he has observed thatthe discounted stream of the marginal benefits derived from reducing emissionsneed not always be finite even if benefits and costs are. It follows that the expectedvalue of the net benefits of mitigation might not always exist.

This short note will produce graphical portraits of how and why this mighthappen in Section 2, but it will turn to confront the ‘So what?’ question directly inSection 3. In the spirit of the television game show ‘Jeopardy’, it is meaningfulto try to articulate the question for which maximizing the discounted value ofexpected net benefits can provide the answer. Concluding remarks will argue thatthe outcome of this search will depend on what we make of the new insight thatTol has provided.

2. Depicting the Convergence Problem

Tol has essentially uncovered a new problem in the application of cost-benefitanalysis to climate change mitigation that focuses our attention on convergencein the computation of the present value of the marginal damage of emissions. It isinstructive to take a moment to see how its sources might be portrayed graphically.Figure 1 begins that process by showing a dynamic portrait of temperature changethat would be associated with the emission of a marginal ton of carbon in the year2000. The specific case portrayed there sets the climate sensitivity associated withan effective doubling of greenhouse gas concentrations at 2.5 degrees. Notwith-

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Figure 1. Typical trajectory of temperature change associated with the emission of one ton of carbonin the year 2000.

standing an underlying reliance on this sensitivity, the effect of a marginal ton ofemissions is never immediate. Indeed, the temperature trajectory shown in Figure 1displays a typical pattern by peaking after some time (about 50 years in this case)and then depreciating toward zero over the course of several centuries; it is takenfrom Newell and Pizer (2001).

Translating this temperature profile into economic damage can now be ac-complished by tracking economic damage as a function of aggregate temperaturechange. Figure 2a offers one trajectory for global damages along an arbitrarily,‘not-implausible’ economic scenario, but comparable and consistent regional por-traits could also be used. Figure 2b presents the correlation between marginaleconomic damage per degree of warming and cumulative temperature change fromwhich this trajectory was derived. Both were produced from a plausible economicscenario with a 1% GDP loss attributed to a benchmark warming of 2.5◦ Centigradeaccording to the DICE-99 construction in which

{1 − �(t)} = {1 − 0.0024T (t)2}−1, (1)

where {1 − �(t)} represent the damage as a proportion of GDP and T (t) indexestemperature change as in Nordhaus and Boyer (2000). Other aggregate and/orregional calibrations could have been used, but the picture would have been es-sentially the same. This typical correlation can also be expressed, as in Figure 2c,in terms of a time series of associated marginal damages per degree of warming bylinking a 2.5◦ warming with a specific year. More to the point, Figure 2c portraysthe transient sensitivity of economic damage to a marginal change in temperaturealong an emissions-cum-economic scenario that reaches 2.5)◦ of warming in 2095.

Figure 3 finally tracks the associated time series of undiscounted marginal dam-ages [denoted MD(t)] caused by one ton of emissions in the year 2000. They are

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Figure 2. (a) Typical trajectory of economic damage along a plausible economic scenario. (b) Typicalcorrelation between marginal economic damage (per degree of temperature change) and temperaturechange derived from the same plausible economic scenario that served as the foundation for (a). (c)Corresponding trajectory of marginal economic damage per degree of temperature change along thesame plausible economic scenario that served as the foundation for (a).

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Figure 3. Trajectory of undiscounted marginal damage associated with the emission of one ton ofcarbon in the year 2000.

expressed as economic costs so that Figure 3 displays the MD(t) for one ton ofcarbon as the future unfolds through the year 2400. These MD(t) values are equalto the simple multiplicative product of the temperature series shown in Figure 1 andmarginal damage series displayed in Figure 2c. It is important to note, of course,that the construction of Figure 3 depends upon the specification of economic andclimate models across many dimensions: carbon cycle parameterization, realizedrates of economic growth, associated rates of greenhouse gas emissions, climatesensitivity, and climate damage functions to name a few. The underlying trajecto-ries depicted in Figures 1 and 2 make this point clear; and, of course, these areincluded in the sources of uncertainty that Tol employed to conduct his MonteCarlo simulations.

We can now illustrate some of the complications highlighted by Tol by explor-ing how the MD(t) series might be discounted over time. Schedule A on Figure 4shows, for example, the stream of discounted marginal damages for a constant 3%discount rate. The fact that we could actually compute the present value of thisstream of damages depicted in Figure 3 with a 3% discount rate is equivalent tostating that the area under Schedule A is finite. Notice that Schedule A convergestoward 0 in the next century; this is a necessary but not sufficient condition. Sched-ule B takes the first step toward complication by moving away from a constantdiscount rate. It reflects, by way of contrast, discounting according to the Ramsey(1928) rule along the economic trajectory that produced Figure 2. In particular,the rate of growth of per capita GDP falls to roughly 1% after 2100. To be morespecific, Schedule B discounts MD(t) according to

r(t) = ρ + θgGDP(t) (2)

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Figure 4. Trajectories of discounted marginal damage associated with the emission of one ton of car-bon in the year 2000. Schedule A shows discounting at a constant 3%. Schedule B shows discountingaccording to the Ramsey rule along the underlying economic scenario. Schedule C displays the effectof economic growth turning negative around 2200. Schedule D suggests the effect of adding equityweighting to the present value calculations.

along the designated economic trajectory where gGDP(t) is the rate of growth ofper capita GDP; it is endogenously determined. Meanwhile, ρ is a pure rate oftime preference; it is set equal to zero. Finally, utility is logarithmic in per capitaGDP so that θ = 1. The Ramsey rule would, in this case, see the discount ratefalling below 1% in the next century. Schedule B in Figure 4 is, as a result, higherthan Schedule A, but the area under Schedule B is probably still finite. Indeed,Schedule B seems to converge toward zero in the distant future.

It is at this point that Tol introduces the several complications that lie at the heartof his new concern. The first simply notes the possibility that economic growthrates could turn negative in extreme cases with high climate sensitivity and severeeconomic sensitivity to climate change. The discount rate, computed according toEquation (2) would turn negative, as a consequence, and the series of discounteddamages would either converge to zero far more slowly (by virtue of natural depre-ciation of the impact of a ton of carbon emissions) or perhaps not at all. It followsthat the critical area that underlies the present value calculation may not be finite.Schedule C in Figure 4 displays such a case, showing the unsettling possibility thatthe series of discounted damage may not even converge to zero. This may not bea very likely future, to be sure. If it had a low but non-zero probability, however,then the expected present value of marginal damages would be undefined.

Tol’s second complication brings regional disaggregation to the mix, but themessage is the same. The global present value of marginal damage along anyfuture scenario could be expressed equally well as the sum of the present value ofregional damages, but the question of convergence would then assume a regional

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perspective. Notationally, the global present value of damages, PV(MD(t)), wouldthen be expressed as

PV[MD(t)] = PV[MD1(t) + · · · + PV[MDN(t)] , (3)

where MDk(t) is the marginal damage series for region k. If any region k displayeda non-convergent series of marginal damages along any future with non-zeroprobability (a case that is much more likely than non-convergence for the globalaggregate), then its component would dominate Equation (3) and guarantee anundefined aggregate value. This, of course, is the very problem discovered by Tol inrun #383 in which water became so scarce in Central Europe, Eastern Europe andthe former Soviet Union that these regional economies collapsed to subsistence.Geographic complication notwithstanding, this case is still portrayed by Sched-ule C in Figure 4. The point, here, is not that accounting for regional differenceswould move the trajectory of discounted marginal damages up or down. Ratherit is that regional differences could make it more likely that aggregate discountedmarginal damages would not converge to zero fast enough to guarantee a finite sum.Incorporating more detailed regional structures would, presumably, only increasethis likelihood even further.

Finally, and crucially, Tol adds equity weights to the mix. Fankhauser etal. (1997) have argued that the appropriate aggregation of regionally computedmonetary present values for marginal damages is

PV[MD(t)] = ω1(t)PV[MD1(t)] + · · · + ωN(t)PV[MDN(t)] , (4)

where the ωk(t) are equity rates set equal to the ratios of global mean per capitaincome to regional per capita income at any time t . Accepting this approach couldexacerbate the convergence problem because the weights assigned to marginaldamages in the poorest countries would climb above unity (their implicit value inEquation (3)) while the weights assigned to the richest countries would fall belowone. If this procedure were not used, a collapsing regional economy would leadto negligible damages measured in dollars. Why does this make sense? Because alogarithmic utility function displays diminishing marginal utility of income so that,for example, 10 dollars worth of damage in a poor country would reduce welfarefar more than 10 dollars worth of damage in a rich country. Adding these weightsto the net benefit calculation could therefore produce aggregate weighted marginaldamage streams that were, as shown in Schedule D of Figure 4, consistently higherthan before. It follows, of course, that it could be even more likely that the criticalpresent value area would not be finite.

3. So What? Why Work with Expected Costs and Benefits?

Is maximizing the expected discounted value of net benefits the correct question tobe posed in the climate context? Even if we are stuck in a ‘once and for all’ policy

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world, why should we choose the one that maximizes discounted expected net ben-efits? Why not review a collection of policies that would maximize the discountedvalue of net benefits along representative scenarios of how the future might un-fold? Subsequent analysis could then look for subsets of similar policy trajectoriesfrom which a ‘robust’ response for multiple possibilities could be selected. Judgingrobustness would be an issue, of course, but the benchmark function of Pareto effi-ciency could certainly be exploited. So, too, could risk analysis or a precautionaryprinciple whose limits were defined by non-economic decisions. Indeed, decisionanalysis teaches us that we should switch from expected utility maximization tosomething more robust (like minimaxing the regret) if uncertainties are ‘too large’.

Application of any such decision framework would uncover extraordinarilyrestrictive mitigation for the cases that would confound an aggregate calculationof expected discounted value. These alternatives would not be chosen for theirrobustness, of course, because they would be atypically severe; but Tol’s analysiscan tell us what to do, instead. The source of their troubling impact on his expectedvalue calculation was a collapse to economic subsistence in one region or another,but why should we expect climate policy to handle all of that by itself? It is wellknown that we need at least two policy instruments if we are trying to achieve twodistinct objectives under uncertainty (see Brainard, 1967) – climate and equity, inthis case. Would it not be possible and preferable, having discovered these possiblefutures, to undertake development aid and knowledge/technology transfers to ad-dress the economic issues? On the climate policy side, this would work to reducethe dispersion in the equity weights of Equation (4) by increasing the rate of growthper capita GDP in critical regions. The outlier policy interventions associated withthese scenarios would diminish, as a result, and move closer to the more robustalternations. The area of present value integral displayed in Figure 3 could beexpected to converge more easily, in other words, so that far less restrictive climate-based interventions would be robust over far more of the range of ‘not-implausible’futures.

We are not, of course, stuck with the problem of once and for all prescribingclimate policies in the first few years of the 21st century that will be in force forthe next one or two hundred years. It would surely be more prudent to use Tol’sapproach to explore the robustness of several representative near-term interventionsacross collections of ‘not-dissimilar’ futures against their associated efficiency so-lutions. We might then be able to (1) enumerate a manageable set of representativeclimate cum economic scenarios, (2) specify robust interventions for each and (3)deduce reasonable hedging strategies for near-term intervention. We could also(4) uncover the climate and economic variables that would be most important indetermining when to contemplate ‘mid-course’ corrections and thereby (5) illu-minate how to make the next round of hedging decisions. Identifying appropriate‘triggering thresholds’ is a difficult problem on the climate side, but Tol’s analysishas already identified something critical on the economic side. Economic declinein a region or country derived from any source is bad and can lead to extreme

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vulnerability to climate policy as well as climate change. The point, quite simply,is that we cannot hold climate policy hostage to the necessity of confronting theseequity issues when there must be a myriad of other remedies.

4. Concluding Remarks

To be fair, Tol was careful to state that his paper was not a policy exercise. We havefocused our attention on policy issues, however, because he did apply expectedcost-benefit techniques to mitigation and mitigation is fundamentally a policy prob-lem. What did we learn despite his protestations? For one thing, we have added onemore element to our list of reasons why it is inappropriate to use the expected valueof discounted net benefits to judge mitigation policy, especially when regional dif-ferences are captured in the analysis. But we have learned more than that. Tol’shighlighting the sources of the technical problems associated with convergencehas suggested where subsequent research might pay the large dividends. Manneand Richels (1992) and Yohe (1996) have already conducted some of the hedgingexercises suggested in Section 3; but Yohe does not use a regional model whileManne and Richels do not use Ramsey discounting. They have not, therefore,confronted the fundamental questions posed here in response to the Tol results.How can a mix of development aid and climate policy be employed to reduce therange of mitigation options across which some hedging might be most productive?And what regional and/or national economic variables might serve in conjunctionwith critical climate variable as leading indicators and triggers for ‘mid-course’corrections in mitigation intervention?

Acknowledgement

Support for this research was received from the National Science Foundation of theUnited States through the Center for Integrated Study of the Human Dimensionsof Global Change at Carnegie Mellon University under SBR-9521914.

References

Brainard, W. D.: 1967, ‘Uncertainty and the Effectiveness of Policy’, Amer. Econ. Rev. 57, 411–425.Fankhauser, S., Tol, R. S. J., and Pearce, D. W.: 1997, ‘The Aggregation of Climate Change Damages:

A Welfare Theoretic Approach’, Environ. Develop. Econom. 3, 59–81.Manne, A. and Richels, R.: 1992, Buying Greenhouse Insurance: The Economic Costs of CO2

Emissions, MIT Press, Cambridge, MA.Newell, R. and Pizer, W.: 2001, Discounting the Benefits of Climate Change Mitigation, Economic

Technical Series, Pew Center on Global Climate Change, Washington, D.C.Nordhaus, W. D. and Boyer, J.: 2000, Warming the World: Economic Models of Climate Change,

MIT Press, Cambridge, MA.

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Ramsey, F. P.: 1928, ‘A Mathematical Model of Saving’, Econom. J. 38, 543–559.Tol, R. S. J.: 2003, ‘Is the Uncertainty About Climate Change too Large for Expected Cost-Benefit

Analysis?’, Clim. Change 56, 265–289.Yohe, G. W.: 1996, ‘Exercises in Hedging against Extreme Consequences of Global Change and the

Expected Value of Information’, Global Environ. Change 6, 87–101.

Department of Economics, GARY W. YOHEWesleyan University,238 Church Street,Middletown, CT 06459,U.S.A.

and

Center for Integrated Study of theHuman Dimensions of Global Change,Carnegie Mellon University,Pittsburgh, PA 15212,U.S.A.