Energy Economics 42 (2014) 43–49

Contents lists available at ScienceDirect

Energy Economics

j ourna l homepage: www.e lsev ie r .com/ locate /eneco

Implications of a lowered damage trajectory for mitigation in acontinuous-time stochastic model☆

Jon StrandDevelopment Research Group, Environment and Energy Team, The World Bank, Washington DC 20433, USADepartment of Economics, University of Oslo, Norway

☆ I thankNils Framstad, BårdHarstad,Michael Keen, Robfor their helpful comments to the previous versions. The vthose of the author and not necessarily those of theWorldber countries.

E-mail address: jstrand1@worldbank.org.1 For a more precise definition of adaptation, and the d

anticipatory adaptation (discussed below), see Wickpediresults in welfare improvements, which is frequently theby which these welfare gains are maximized.

2 See Tol (1998, 2005a), Tol et al. (1998), FankhauseHallegatte et al. (2011), Lecocq and Shalizi (2007), and Swider discussion on interrelationships between mitig(2005b, 2007), and IPCC (2007).

0140-9883/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.eneco.2013.11.006

a b s t r a c t

a r t i c l e i n f o

Article history:Received 13 October 2011Received in revised form 5 November 2013Accepted 8 November 2013Available online 20 November 2013

JEL classification:Q54Q58H23C61

Keywords:MitigationAdaptationClimate damagesUncertaintyOption values

We provide counterexamples to the idea that mitigation of greenhouse gas emissions, and adaptation to climatechange, are always substitutes. We consider optimal mitigation policy when climate damages follow a geometricBrownian motion process with positive drift and mitigation is lumpy. Climate damages can be affected by adapta-tion in twomainways: 1) reduced proportionately for given climate impact; or 2) their growth path down-shifted.In either case expectation and variance of the climate damage are both reduced by adaptation. In case 1, the vari-ance effect (which leads to more rapid mitigation as the option value of waiting is reduced) may dominate overthe expectation effect (which reducesmitigation), thus on balance increasingmitigationwhen damages are reduced.Mitigation and adaptation are then complements. A family of functions relating climate damage to adaptation costin this way includes the Cobb–Douglas specification. In case 2, mitigation and adaptation are always substitutes.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Two qualitatively distinct activities are induced by climate change:mitigation of greenhouse gases (GHGs) to limit emissions and future cli-mate change; and adaptation to lessen the negative impact on humansocieties of any given climate change.1 These activities are usuallyassumed to be substitutes. More mitigation (leading to lower GHGemissions) implies less climate change, and thus presumably less needto adapt. Conversely, more adaptation leads to less damage from anygiven climate change, which might reduce the “need” to mitigate.2

For the effect of increasedmitigation on the need for adaptation, such arelationship appears obvious and uncontroversial. In this paper we are

ert Pindyck andMichael Tomaniews expressed in this paper areBank, itsmanagement, ormem-

istinction between reactive anda (2013). When climate changecase, adaptation is the process

r et al. (1999), Burton (1997),halizi and Lecocq (2009). For aation and adaptation, see Tol

ghts reserved.

however concerned with reverse relationships. We ask, and try to an-swer, the following question: Given that the process for future damagesresulting from given climate change will be dampened or down-shifted asa result of increased adaptation, what implication does this have for opti-mal mitigation? The very purpose of adaptation to climate change is toreduce the damages to human societies for any given mitigation level.Thus the two relationships may seem similar.

A main difference is however that, while much or most adaptationactivity can come later, in response to given climate change, mitigationis always anticipatory, aiming to pre-empt anticipated but uncertain cli-mate change.3 The uncertainty part is here crucial. When adaptationactivity is increased or strengthened, expected climate damages arereduced. This in turn reduces the “need” to mitigate, “everything elseequal”. A key issue here is however that “everything” is not equalwhen adaptation activity is increased. Adaptation namely causes theentire climate damage process to be dampened. This reduces not only

3 Adaptation tends to come “later” thanmitigation because climate damages are the re-sult of accumulated emissions, and much adaptation is reactive, in response to the overalldamage level. Any promised future adaptation level must then be credible at the timemit-igation activity is carried out. In our model below this issue is essentially resolved, in thesimplest possible way, by considering only (ex-post) optimal adaptation activity, in re-sponse to given public preferences. We however do not consider the joint optimal set ofmitigation and adaptation actions, which may require formal commitments to future ad-aptation actions, known at the mitigation stage.

4 See Dixit and Pindyck (1994); and for specific environmental applications, Fisher(2000) and Heal and Kriström (2002).

44 J. Strand / Energy Economics 42 (2014) 43–49

expected climate damages, but also theuncertainty about future climateimpacts. As shown in the following discussions, this may in some caseslead to increased mitigation. The intuitive reason is that under uncer-tainty there can be an option value of waiting to mitigate, which istypically greater with more uncertainty. Reduced uncertainty reducesthis option value. This effect by itself leads to earlier or more aggressivemitigation, as the benefit of waiting (to obtain more information) isreduced. The main issue here is whether it is possible to find caseswhere the effect of less uncertainty about future climate impacts,which increases mitigation, more than counteracts the effect of theabsolute level of adaptation being higher, which reduces it.

The contribution of this paper is to apply a widely appliedmodellingframework, developed by Pindyck (2000, 2002), to the issue posedabove. As Pindyck does, we model marginal environmental damages(fromone additional unit of emissions) as a geometric Brownianmotionprocess with positive drift; see also Balikcioglu et al. (2011), Framstad(2011), and Framstad and Strand (2013) for recent related applications.Expected damages from a given concentration of greenhouse gases(GHGs) are in this framework assumed to increase over time, mirroringresults from various integrated assessment models. Actual damages areuncertain. The geometric Brownian motion assumption of Pindyckimplies that the damage level is continuously changing, and that uncer-tainty (represented by the standard deviation of the stochastic processfor marginal damages) increases in proportion to the damage level.

Based on this analytical foundation, we study two alternative casesof adaptation. Both imply that the process for (unmitigated) damagesis altered by adaptation policies that reduce the damages due to climatechange for a given emissions level.

In the first case, studied in Section 3, the stochastic process for dam-ages is “dampened” by adaptation, meaning that the entire process fordamages is shifted down by a constant factor of proportionality at anygiven point in time. We find it natural to call such adaptation activity“reactive”.We also show that such a reactive adaptive response is optimalgiven a particular class of adaptation cost functions, which includes theCobb–Douglas specification. There is then no cumulative effect on climatecosts of adaptation: all gains are instantaneous. Such reactive adaptationreduces damages in all states, but by more, absolutely speaking, in “bad”states (when damages are high) than in “good” states (when damagesare lower). Importantly also, the standard deviation on the stochastic pro-cess for damages is shifted down proportionately by adaptation.

A relevant concrete example of this type of adaptation is better oper-ation of a weather reporting and warning system against extremeevents (which could becomemore frequent and damagingwith greaterclimate change), combinedwith greater efforts to adapt in the short runto adverse climate developments, locally and regionally. Another exam-ple would be better fine-tuning and alertness in adapting suitableproduction techniques and crop types to variation in temperature andrainfall, combined with more “robust” production technologies, inagriculture and other sectors which depend directly on weather andclimate.

In the second case, studied in Section 4, adaptation activity down-shifts themore general process bywhich climate affects human societies,by reducing the constant positive drift of the stochastic process forclimate damages. In this case there is a cumulative dampening effect ofadaptation on damages, as the expected rate of increase for damages pertimeunit is reduced. Unlike in thefirst case, there is in this case no imme-diate reduction in the uncertainty of the damage process; although theprocess variancewill also here be reduced over time due to its geometricstructure together with the lowered trajectory for the level of damages.We call such an adaptation strategy “anticipatory” (as it serves to reducethe overall level of damages and its growth rate, but not directly its var-iance, ahead of actually occurring states). This case is perhapsmost easilythought of as representingmajor infrastructure investments that help tosteer the economy onto a more robust course, to meet and thus reducefuture expected climate damages, but with less implications for the var-iability of damages over time. Other examples of such anticipatory

adaptation might be taking economic activity out of “harm's way”, i.e.,zones vulnerable to extreme weather events; or developments thatmake the economy less sensitive to climate variables, includingdiminishing agriculture's relative importance in the economy.

Pindyck (2000, 2002) and Framstad and Strand (2013) have shownthat when future climate damages are more uncertain, investments in“lumpy”mitigation activity involving large sunk costs (such as replacingcoal-fired power plants with nuclear or renewable energy) are reduced,as the option value of waiting to invest is increased.4 Under uncertainty,future damages may in the end turn out to be small; for such cases, theinitial investment will be wasted. The main point is that greater uncer-tainty increases the probability of very small damages, as well as theprobability of very large damages. Waiting implies retaining the optionto “wait and see” whether damages will actually be small, so that miti-gation today would essentially be wasted. Greater uncertainty thus dis-courages mitigation today.

We study the implications of each type of “adaptation” for the timingof mitigation policy, which in our model takes a simple, discrete andlumpy, form.We assume that authoritiesmaymake amajor investmentin new (energy or other) infrastructure investment that, after havingbeen undertaken, removes all greenhouse gases from the respectiveeconomic activity. The main policy issue in question in the paper isthen when, if ever, this mitigation investment should be made.

In the first of these two cases, treated in Section 3, the level of dam-ages is reduced by a constant factor of proportionality. This reduces bothexpectation and variance of damages, and the two shifts have opposingeffects on the propensity to mitigate. We show that, for some paramet-ric cases, the reduced variance effect (which leads to soonermitigation)can dominate over the reduced expecteddamages effect (which leads topostponed mitigation). The “propensity to mitigate” is then increasedwhen damages are reduced, in the sense that the value for the stochasticvariable triggeringmitigation action is reduced, so that mitigation is ex-ecuted sooner.

We also indicate, in Subsection 3.2, how the specified down-shift ofthe damage function can constitute an optimal level of (“reactive”)adaptation, given that the relationship between adaptation cost andthe value of the adaptation outcomebelongs to a particular class of func-tions (so that only the ratio of adaptation costs to damages matters),and where, we argue, such functional forms can constitute realisticexamples.

In the second, “anticipatory adaptation” case, treated in Section 4,adaptation affects the positive drift of the damage process for given mitiga-tion, but does not directly dampen the stochastic component of dam-ages. Results are then more standard: down-shifting the systematicpart of the damage function then always leads to postponed mitigation(and less ex ante expected mitigation by any given point in time). Ahigher value of the continuous random variable is then necessary totrigger the discrete action to mitigate.

In Section 5 we briefly discuss a case where only the variance on thestochastic process is reduced by adaptation. Results are then unambigu-ous: more adaptation reduces mitigation. It is however more difficult tofind realistic examples of such adaptation activity.

The final section, Section 6, further discusses the two former casesand their realistic real-world interpretations. We seek to explain “adap-tation policy” under case 1 as “reactive” (ex post) adaptation (carriedout by the public or private sector). Adaptation activity in case 2 is, weargue, instead “anticipatory”.

2. Optimal mitigation

This section closely follows Pindyck (2000). DefineM(t) as the stockof greenhouse gases (GHGs) in the atmosphere, and E(t) as a flow var-iable that controls this stock. In the following equation, E is taken as the

45J. Strand / Energy Economics 42 (2014) 43–49

rate of emissions of greenhouse gases, and can take a given number ofnonnegative values. Assume that M is (deterministically) given from

dM tð Þ ¼ E tð Þ−δM tð Þ½ �dt; ð1Þ

where δ is the rate of decay of the greenhouse gas stock. Associatedwiththe stock variable M there is a flow variable of (negative) benefits, B,given by

B M tð Þ; θ tð Þð Þ ¼ −μ tð Þθ tð ÞM tð Þ: ð2Þ

θ is a multiplicative parameter governing the marginal cost to societyfrom increasing the stock of GHGs, in the case of no adaptation activity(μ(t) ≡ 1). We assume that this parameter is stochastic and follows ageometric Brownian motion process. The process can be representedon differential form as follows:

dθ ¼ αθdtþ σθdz; ð3Þ

where α is a (positive) drift parameter (indicating how climate dam-ages are changed systematically by time), and σ is the standard devia-tion of the process.5,6 Considering a starting point 0, at t = T thelogarithm of θ has expectation (α − σ2/2)T and variance σ2T (whileθ itself has expectation αT).7

This process can be affected in threeways, one of which is associatedwith mitigation policy, and the two others with “adaptation policy”.First, as in Pindyck (2000), the mitigation control parameter E can beshifted down to a new and permanently lower level, through (perma-nent) increased mitigation, by incurring a cost K at the shift time.

The two other types of policy intervention are new relative to thePindyck model. The first is represented by downward shifts in μ, inEq. (2), which is assumed to take values less than or equal to unity,and where μ ≡ 1 absent policy interventions. The level of damages ex-perienced by human societies, for given “climate impact” θ, is thenshifted down proportionately by policy. Such a shift reduces both theexpectation and variance of the damage path. The second policy inter-vention is to (permanently) affect the drift rateα for the stochastic pro-cess for damages. In the following discussions we assume, alternatively,that either of the two latter policies is applied together with mitigationpolicy, throughout considered as endogenous and optimal in responseto given (credibly anticipated) future adaptation policy.

Section 3 focuses on shifts in μ, initially considered as exogenous. InSubsection 3.2, we show that a (given and constant) shift in μ can bederived endogenously as part of an optimal adaptation policy, undercertain assumptions about the adaptation cost function. Section 4 dis-cusses exogenous shifts in α.

The policy objective W (at time 0) is

W ¼ Exp t ¼ 0ð ÞZ∞0

−μ tð Þθ tð ÞM tð Þð Þe−rtdt−K E1ð Þe−rT1

8<:

9=;; ð4Þ

which denotes welfare (or the negative of cost) associated with climatechange, r being the (constant) discount rate. T1 is the (uncertain, andendogenous) time at which the mitigation policy will be implemented,modelled as a one-off decision. Following Pindyck (2000), mitigation

5 See Dixit and Pindyck (1994), chapter 3, for a heuristic discussion of the formula andits applications. Formore rigorous presentations see Harrison (1985), Karatzas and Shreve(1988), and Øksendal (2007, page 131).

6 This formulation, following Pindyck (2000), implies that marginal damages are a sys-tematic function of time only, and not e.g. of cumulative emissions. This could have twojustifications: that the global economy subject to damage (andwith no adaptation) growsat the expected rateα; and/or that growth in damages are a function of past emissions and(approximately) manifested in this way. More realistically, the process for marginal dam-ages should be a function also of cumulated emissions. This will however complicate sub-stantially and must be left for future analysis.

7 See Dixit and Pindyck (1994), chapters 3–4; and Borodin and Salminen (2002).

costs take the simple form K = kE. We take expectations at time zeroto indicate that T1 is uncertain at t = 0. The shift in μ is assumed to beconstant, so that adaptation activity of the type studied in Section 3,leads to a proportional downward shift in the damage level. Adaptationcosts are not specified in Eq. (4). One interpretation of this is that theutility value represented by the integral in Eq. (4) is the calculated netof any adaptation costs. Subsection 3.2 extends this analysis by explicit-ly considering adaptation costs.

The time of (possible) mitigation, T1, for given anticipated future ad-aptation, can now be found as an optimal stopping problem.8 Adoptinga dynamic programming approach, we denote the value functions (4)for thepre-mitigation andpost-mitigation states, byWN andWA respec-tively. Since the cost of mitigation is linear in E0 − E1 (denoting theamount of mitigation achieved), the optimal policy is to reduce E to 0given thatmitigation takes place. Consequently, without loss of general-ity we assume that the amount of mitigation is E0, and that mitigationcost K equals kE0.

WN and WA need to satisfy the Bellman equations (Pindyck (2000),Eqs. (5)–(6)):

rWN ¼ −μθMþ E0−δMð ÞWMN þ αμθWθ

N þ 12μ2σ2θ2Wθθ

N ð5Þ

rWA ¼ −μθM−δMWMA þ αμθWθ

A þ 12μ2σ2θ2Wθθ

A ð6Þ

where subscripts denote the first- and second-order derivatives of thevalue functions. These must be solved simultaneously subject to theboundary conditions

WN 0;Mð Þ ¼ 0 ð7Þ

WN θ�;Mð Þ ¼ WA θ�;Mð Þ−kE0 ð8Þ

WθN θ�;Mð Þ ¼ Wθ

A θ�;Mð Þ: ð9Þ

By Eq. (7), the value (or climate cost) functionmust be zero with nocurrent climate damages (and, with a geometric Brownian motion,never any climate damages). Eq. (8) defines the “optimal stoppingpoint” θ* for θ (for mitigation action), by the indifference between ac-tion and non-action. Eq. (9) is the “smooth-pasting condition”: the de-rivative of W must be continuous at the point of mitigation action, θ*.

The solutions to Eqs. (5)–(6) are

WN ¼ A μθð Þγ− μθMr þ δ−α

− E0μθr−αð Þ rþ δ−αð Þ ð10Þ

WA ¼ − μθMrþ δ−α

ð11Þ

where A is a constant, and γ is the positive root of the quadratic equa-tion

12μ2σ2γ γ−1ð Þ þ μαγ−r ¼ 0; ð12Þ

with solution

γ ¼ 12− α

μσ2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

αμσ2 −

12

� �2þ 2rμ2σ2

sN1: ð13Þ

8 This section closely follows the presentation in section 3 of Pindyck (2000).

46 J. Strand / Energy Economics 42 (2014) 43–49

A, and the critical value θ* of θ, beyond which mitigation action istaken, are determined from Eqs. (8) and (9):

A ¼ γ−1k

� �γ−1r−αð Þ rþ δ−αð Þγ½ �−γE0 ð14Þ

θ� ¼ 1μ

γγ−1

k r−αð Þ rþ δ−αð Þ: ð15Þ

Under certainty, the critical value of θ triggeringmitigationwould bedetermined from

θ � � ¼ 1μk r−αð Þ rþ δ−αð Þ: ð16Þ

Increased adaptation (reduced μ) increases θ** proportionately to1/μ, implying that it has a strong delaying effect on mitigation. This isintuitive: adaptation “takes over” most of the role of mitigation,making the latter less relevant and thus less urgent. ComparingEqs. (15) and (16), uncertainty increases θ* relative to θ**, by a factorγ ∕ (γ − 1) N 1 as γ N 1. In the set of cases where θ lies in the range(θ**, θ*), there is then mitigation under certainty, but not underuncertainty.

3. Increased adaptation 1: shifts in μ

3.1. Exogenous adaptation

We now introduce “adaptation” policy in the form of shifting μdownward, to study its effects on optimal mitigation under uncertainty(for cases where γ takes a finite positive value greater than unity).Under certainty this is trivial: from Eq. (16), μθ** is a constant, andlower μ leads to a proportional increase in θ**. This in turn leads to post-poned mitigation.

Under uncertainty, Γ = γ ∕ [μ(γ − 1)] is the relevant measure.Note that γ ∕ (γ − 1) drops in γ (when, as here γ N 1). If γ increaseswhen μ drops, the option value of waiting is then also reduced. Evenstronger, when γ ∕ (γ − 1) is reduced more than μ, θ* drops when μdrops: increased adaptation then leads to earlier mitigation.

Taking the derivative of γ with respect to μ in Eq. (13) yields:

dγdμ

¼ αμσ2 −

αμσ2 −

12

� �2þ 2rμ2σ2

� �−12 α

μσ2 −12

� �α

μ2σ2 þ4r

μ2σ2

� �: ð17Þ

Consider a small change in μ starting from μ = 1 (no initial adapta-tion). Eq. (17) can then be written as

dγdμ

¼ ασ2 1− α

σ2 −12

� �2þ 2rσ2

� �−12 α

σ2 −12þ 4r

α

� �" #: ð18Þ

Here dγ/dμ b 0 in a wide range of circumstances. A more crucialquestion for us is whether the increase in γ can be sufficiently greatwhen μ drops, so that γ ∕ (γ − 1) drops by more than the initial dropin μ. We find

dγ

γ−1

� �dμ

¼ − 1γ−1

� �2 dγdμ

; ð19Þ

where dγ/dμ is found from Eq. (18). Thuswhen γ is close to unity and atthe same time dγ/dμ b 0, γ ∕ [μ(γ − 1)] dropswhen μ drops. Increasedadaptation will then lead to “more mitigation”, in the sense that the(one-off) mitigation investment is made earlier.

3.2. Optimal adaptation

Can the indicated adaptation response to given damages, for thisversion of the model, be optimal? To evaluate this issue, consider acase where adaptation at any given time is reactive in response tothe level of current damages (so that adaptation cannot be viewedas anticipatory). We will search for a specification for adaptationcost as a function of effective adaptation, in the precise sense of ourcase 1. A key feature of this is that damages are dampened by thechosen level of adaptation activity, so as to always be reduced pro-portionately to the initial damage level. One class of adaptationvalue and cost functions that fits to these assumptions is wheregains from and costs of adaptation are both proportional to the initiallevel of damages; and thus the function of net gains from adaptationactivity depends only on the ratio of adaptation costs to such gains. Asuitable net value function (for the value, net of adaptation costs, ofincurring the adaptation cost A given an instantaneous climate dam-age level D) must then take the following form:

V ¼ FAD

� �D−A: ð20Þ

F is here a positive-valued function with the first- and second orderderivatives F′ N 0, and F″ b 0, andwhere F(0) = 0, F(∞) = 1. Maximiz-ing V with respect to A leads to the first-order condition

dVdA

¼ F0−1 ¼ 0⇔F0 ¼ 1: ð21Þ

Eq. (21) yields a unique solution for A/D, call it λ. Net current climatedamages, net of adaptation gains and costs can then be written as fol-lows:

ND ¼ D−FAD

� �Dþ A ¼ D 1−F

AD

� �þ AD

� �¼ D 1−F λð Þ þ λ½ �≡ μD;ð22Þ

where λ is a fixed parameter, positive but less than one. This type of ad-aptation is “reactive”, in response to the particular level of damage atany given time.

Consider next how adaptation activity, when adaptation activitynow is endogenous and determined by Eq. (21), affects the basic ob-jective function (4), and thus the solution to the overall optimizationproblem. In this particular case, however, we can now simply assumethat ND(t) represents the new (“corrected”) net climate cost at anytime t, given the basic (non-corrected) cost D(t), representing thecurrent climate damage term (t)M(t) in Eq. (4). This implies thatwe can simply interpret 1 − F(λ) + λ as the (fixed) parameter μin Eq. (4).

This provides a simple and very straightforward interpretation ofoptimally and endogenously adapted net climate damages. Optimal ad-aptation leads in this particular parametric case to net damages beingreduced by a constant factor of proportionality, relative to gross dam-ages D, in the same way as was presumed in deriving the optimal stop-ping solution (15) for θ⁎.

One such case is where the F function takes the Cobb–Douglas form,so that:

V ¼ D BAD

� �β−A

D

� �: ð20aÞ

where B is a positive constant, and β is the (constant) scale elasticity ofthe F function. In this particular case the optimal solution for A/D = λ isgiven by

λ ¼ Bβð Þ 11−β; ð23Þ

47J. Strand / Energy Economics 42 (2014) 43–49

where λ must lie between zero and unity.9 We find the followingexpression for ND:

ND ¼ D 1−FAD

� �þ AD

� �¼ D 1− Bβð Þβ β−1ð Þ þ Bβð Þβ−1

h i¼ D 1−Bλβ þ λ

� �≡ μD: ð22aÞ

Can a relationship such as Eq. (20) between adaptation costs and cli-mate damages be relevant in practice? This is an empirical question towhich this (analytical) presentation has no clear or obvious answer. Ashort answer is that the Cobb–Douglas production function specifica-tion has proven highly robust in a variety of economic settings, andthus potentially also here.

4. Increased adaptation 2: shift in the drift parameter α

Our second case of “adaptation policy” involves down-shifting of thedrift parameter α in the stochastic process for θ. An interpretation ofsuch policy is (more efficient) “anticipatory mitigation”, insofar as θ asoriginally formulated describes the process by which climate is altered.An alternative interpretation could however be that society, for givenclimate as represented by the (non-shifted) stochastic damage processθ, is affected by climate in the same way as if θ were down-shifted,and made more resilient in the face of climate change. This also givesroom for an interpretation in terms of “anticipatory adaptation” or “cli-mate proofing”.10

To study this case analytically, as above we suppress direct adapta-tion costs. This could as before be taken to imply that adaptation costsare implicit in Eq. (4). Alternatively, adaptation costs are simply exoge-nous, and that adaptation becomes more “efficient”. A third interpreta-tion is that adaptation costs represent long-lasting investments toprepare for future climate change, incurred “up-front” and ahead ofthe period of analysis here.11

An immediate issue in this case is that when α is reduced, the ex-pected time until mitigation, τ, increases for any given level of θ*. Theformula for this expected time is

E τ; θ�; θ0ð Þ ¼log

θ�θ0

� �

α−σ2

2

; ð24Þ

assuming that α N σ2/2. The expected time until mitigation, E(τ), isfinite with probability one if and only if α N σ2/2.12 E(τ) is then a nega-tive function of α so that reduced α leads to higher E(τ). Thus, when alower α is interpreted as “better adaptation” or “climate-proofing”(considered as a possible interpretation below), this delays mitigationas it reduces the “perceived need” to mitigate. This is a feature thatwas not present in case 1.

We are next interested in how θ* is affected by changes in α. Noteagain that a lower α implies that damages are reduced at all futurepoints of time. The cumulative damaging effect of current emissions isthen at the same time reduced. This feature tends to increase the

9 Thus, we must also have 0 b Bβ b 1.10 For a further discussion of “climate proofing” see Asian Development Bank (2005).11 We may then assume that the particular mitigation investments in question here,discussed in Section 2, are too small to affect the overall profitability of such up-front in-vestments; this is particularly relevantwhen such investments aremade jointly for severalcountries.12 Note that even when α b σ2/2, τ is finite with a positive probability which is increas-ing inα − σ2/2, applying Dynkin's (1965) formula; and infinite with the complementaryprobability; thus E(τ) is infinite: see also Borodin and Salminen (2002), and Øksendal(2007). Thus τ is finite in “many” cases, and with probability mass that approaches 1 asα − σ2/2 approaches zero from below.

optimal value of θ. Butwe also nowfind a negative effect on the varianceand thus on the option value of waiting for any given θ. Another majordifference from case 1 above is that the expected time it takes to reacha given θ now increases (as the process for θ is down-shifted).

The derivative of θ* with respect to α is now found as

dθ�dα

¼ − kγ−1ð Þ2

r−αð Þ r−α þ δð Þσ2

dγdα

þ γ γ−1ð Þ 2 r−αð Þ þ δð Þ

; ð25Þ

where

dγdα

¼ − 1σ2 1− α

σ2 −12

� �ασ2 −

12

� �2þ 2rσ2

� �−12

( ): ð26Þ

The first main term in the curled bracket in Eq. (25) can beinterpreted as the effect of reduced uncertainty (variance) of damageswhen α is reduced. The second term can be interpreted as the effectof reduced expected damages. As in Section 3, the variance termworks to increase θ*, and the expectation term works to reduce it.

Is it here possible to find cases where dθ*/dα N 0? The first mainterm in the curled bracket in Eq. (25) (the “variance term”) wouldthen need to dominate the second term (the “expectation term”). It ishowever easy to verify that this can never be the case here: for allvalid parameter combinations, dθ*/dα b 0. Thus, the relationship be-tween mitigation and adaptation is now “traditional”: more adaptationreduces the “need” for mitigation in the sense that the value for the sto-chastic variable θ, beyond which mitigation is triggered, is increasedwhen α drops (“more climate proofing” takes place). There is a doublesuch effect, since the expected time it takes to first reach any givenlevel of θ is now longer, from Eq. (24).

Finally, will any adaptation or cost function produce this type ofadaptation response? One obvious candidate, alluded to earlier in thesection, is an investment sunk early on, which leads to permanent ef-fects directly on the growth trajectory and how this trajectory is affectedby climate change. Such an investment would constitute “anticipatoryadaptation”, but might in some contexts perhaps be indistinguishablefrom mitigation policy.

5. Increased adaptation 3: reduction in damage uncertainty only

Wewill in this section briefly discuss a third option; namely, reduc-tions in the variance, σ2, of the stochastic process for damages withoutchanging the expectation, μ. It is easily derived, from Eq. (13), that θ*is then reduced. The only effect is now on the option value of waitingto mitigate, which drops and spurs more rapid projects for phasingout emissions. Mitigation and adaptation are then complements, andmore clearly so than for the case in Section 3. It may however seemmore difficult than for cases 1 and 2, to find realistic adaptation behav-iour yielding such an outcome. It would need to be anticipatory (sinceex post adaptation tends to reduce actual damages which is here notthe case on the average); and through projects that only affect riskand not expected return. The identification and analysis of such possibleactivity and projects are left for future work. Clearly, however, the exis-tence of such caseswould strengthen the case for outcome types soughthere, where adaptation and mitigation are complementary policies tomeet climate change.13

13 Note that, in the context of our model, projects that reduce only variance and not ex-pectation arewelfare reducing under risk neutrality of the decision maker; in particular asthe possibility of good outcomes is reduced. Under sufficient degree of risk aversion, suchprojects would be welfare enhancing.

48 J. Strand / Energy Economics 42 (2014) 43–49

6. Conclusions and discussion

This paper has studied a simple modification of Pindyck's (2000)well-known model of GHG mitigation, where climate damages de-velop over time according to a geometric Brownian motion processwith positive drift, and mitigation consists of “lumpy” investmentprojects. Within this framework we have interpreted “adaptation”in two main alternative ways, and studied implications of exogenouschanges in adaptation policy for optimal mitigation policy. A keyquestion is whether policies to reduce GHG emissions (mitigation), andpolicies to reduce damages caused by climate change (“adaptation”),can be complements; so that “more adaptation” goes hand in hand with“more mitigation”.

Wefind that this can occur under thefirst type of adaptation studied,in Section 3, which we call “reactive adaptation”. In this case damagescaused by climate change are down-shifted proportionately by adapta-tion. Damages are then reduced, which leads to less need formitigation.But it also reduces the option value of waiting to mitigate, under uncer-tainty about future climate damages, which leads to more rapid mitiga-tion. Under climate uncertainty, climate damages could worsen; butthey could also lessen. The latter possibility leads to a greater incentiveto wait for possible climatic improvements, which are more prevalentwhen uncertainty is greater. Perhaps surprisingly, we find cases wherethis uncertainty effect dominates over the reduced expected damageseffect, so thatmitigation activity is, on balance, increasedwhenmore fu-ture adaptation is anticipated. In Section 3 we also show that such pro-portional dampening of climate damages can result from optimaladaptation, with a concrete example in terms of a Cobb–Douglas “dam-age dampening function”.

In case 2, studied in Section 4, adaptation down-shifts the drift rate ofthe climate damage process. Results are then more traditional: “moreadaptation” always reduces mitigation, in two ways: the time for miti-gation action is delayed for a given value of the stochastic parametergoverning climate damages; and the damage level triggeringmitigationis higher. We have argued that this might correspond to (partial) “cli-mate-proofing” whereby society's ability to withstand climate changesis improved.14 It is here also easier to visualize the impacts that “cli-mate-proofing” may have; perhaps, as a cumulative process whereby“layers” of climate-proofing gradually add to the overall resiliency ofsociety.

Amain difference between cases 1 and 2 is that, in case 1, adaptationaffects both the level and variance on damages, directly and immediately.In case 2, damages and their variance are affected later (as the future dy-namic process for damages is down-shifted via the drift). Evenmore im-portantly, the option value of waiting to mitigate is affected muchmoredirectly, and by more, in case 1.

We also briefly, in Section 5, considered a third case where adap-tation policy reduces only the variance σ2 on the stochastic processfor climate damages. This leads to unambiguously increased mitiga-tion as the only effect is to reduce the option value of waiting tomitigate. It is however not obvious how such a model can be justifiedin practice. It would represent anticipatory adaptation that only im-proves “robustness” only, and does not address climate damages assuch; such policies would seem make sense only when societyexhibits a high degree of risk aversion toward future climatedevelopments.

Can we say more about the possibility for jointly increased adapta-tion and mitigation in case 1? One then needs to look closer atthe parameters for the solution from Eqs. (18)–(19): the drop inγ ∕ (1 − γ) (or increase in γ) needs to be sufficiently great that θ*dropswhen μdrops. To exemplify, considerα/σ2 = ½(when, following

14 For more extensive discussions of similar “climate-proofing” see Asian DevelopmentBank (2005).

Dixit and Pindyck (1994), Et = 0[log θ(τ)] = log θ(0); even thoughEt = 0[θ(τ)] = θ(0) + (ατ)). We find in this particular case:

dγ

γ−1

� �dμ

¼ rα

� �12−1

� �−2 rα

� �12−1

4

� �: ð27Þ

r N α is here a requirement for θ* to be finite. Whenever α is then“not small” relative to r, the expression on the right-hand side ofEq. (20) is greater than one; which is required for a drop in μ to leadto a drop in θ*. The more precise condition is found as α ≥ (4/25)r(by setting the right-hand side of Eq. (27) equal to unity, and solvingthis as a quadratic equation in

ffiffiffiffiffiffiffiffir=α

p). This seems as a not very restric-

tive condition.To exemplify further, set r = 0.05 (so that future climate damages

are discounted at a rate of 5% per year; a rather high discount rate inthe climate policy context).15 The required constraint on the drift pa-rameter is α ≥ 0.008: the “systematic increase” in climate damagesmust be at least 0.8% per year. This is arguably a moderate value;much of the integrated assessment literature points to higher values,1–3%.16 It then appears that, under a wide range of reasonable circum-stances, assumptions are fulfilled that leads to complementarity of mit-igation and adaptation, assuming only that our basic model of optimalreactive adaptation is correct.

Consider a specific, potentially realistic, example where climatedamages increase at 2% annually, so that α = 0.02. Assume that rtakes a lower (albeit still reasonable) long-run value of 0.03, and thatstill α/σ2 = ½ (which we can safely use as an example; the degree offuture uncertainty about further climate damages developments, to beevaluated at the time for a possible future mitigation decision, is itselfconsidered as highly uncertain, and with no agreement on it in thecited literature). Assume that an optimal adaptation action reduces expost damages by 20%; thus μ = 0.8 in the case of optimal ex post adap-tation (while μ = 1with no adaptation). Set θ*(μ = 1) and θ*(μ = 0.8)as the respective levels for the climate damage variable, at which miti-gation action will be triggered. In this case we find, fromEqs. (15)–(16), that [θ*(μ = 0.8)] ∕ [θ*(μ = 1)] = 0.648. Assume alsothat, with no later adaptation action, mitigation action would be trig-gered when the (uncertain) climate damage variable reaches $100 pertonne CO2. Given later adaptation, mitigation action would instead betriggered when the climate damage variable reaches ($100 times0.648=) $64.8 per tonne CO2.

Several strong assumptions behind our analysis can be questioned.First, we assume risk neutrality of decision makers who takemitigationaction. Risk aversion would reduce the option value of waiting; al-though overall effects of risk aversion remain unclear and need furtherscrutiny. Secondly, mitigation is treated as an all-or-nothing decision.Pindyck (2000, 2002) however also considers more gradual mitigationin an otherwise similar context, and shows that it is not crucial formain results that all mitigation costs be incurred up-front. A fixed costcomponent is however crucial, since if not the option value argument,crucial for our results, fails. Thirdly, adaptation is here interpretedsimply as a modification of the rate of change of the climate damageprocess. We would need to verify that such relationships can be empir-ically relevant or (approximately) correct. Surprisingly little is knownabout the shape of such functions. We intend to address such issues infuture research.

15 See e.g. Bijgaart et al. (2013),who in reviewing the literature on integrated assessmentmodels find an average applied value of about 2.5% for the rate of discount.16 See in particular the discussion in Clarke et al. (2007), documenting the research onderived optimal mitigation trajectories from the Climate Change Science Program underthe U.S. Department of Energy.

49J. Strand / Energy Economics 42 (2014) 43–49

References

Asian Development Bank, 2005. Climate Proofing: a Risk Based Approach to Adaptation.Asian Development Bank, Manila.

Balikcioglu, M., Fackler, P.L., Pindyck, R.S., 2011. Solving optimal timing problems inenvironmental economics. Resour. Energy Econ. 33, 761–768.

Bijgaart, I.V.D., Gerlach, R., Korsten, L., Liski, M., 2013. A Simple Formula for the Social Costof Carbon. Nota de Lavoro, 83.2013. Fondazione Eni Enrico Mattei.

Borodin, A.N., Salminen, P., 2002. Handbook of Brownian Motion — Facts and Formulae,2nd ed. Birkhäuser, Basel/Boston/Berlin.

Burton, I., 1997. Vulnerability and adaptive response in the context of climate and climatechange. Climate Change 36, 185–196.

Clarke, L., Edmonds, J., Jacoby, H., Pitcher, H., Reilly, J., Richels, R., 2007. Scenarios of Green-house Gas emissions and Atmospheric Concentrations. Sub-Report 2.1A of Synthesisand Assessment Product 2.1 by the U. S. Climate Change Science Program. U. S.Department of Energy, Washington DC.

Dixit, A.K., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press,Princeton, N.J.

Fankhauser, S., Smith, J.B., Tol, R.S.J., 1999. Weathering climate change: some simple rulesto guide adaptation decisions. Ecol. Econ. 30, 67–78.

Fisher, A.C., 2000. Investment under uncertainty and option value in environmentaleconomics. Resour. Energy Econ. 22, 197–204.

Framstad, N.C., 2011. A remark on R. S. Pindyck: irreversibilities and the timing ofenvironmental policy. Resour. Energy Econ. 33, 756–760.

Framstad, N.C., Strand, J., 2013. Energy intensive infrastructure investments with retrofitsin continuous time: effects of uncertainty on energy use and carbon emissions. WorldBank Policy Research Working Paper, WPS6430, April 2013. The World Bank,Washington, D.C.

Hallegatte, S., Lecocq, F., Perthuis, C., 2011. Designing climate change adaptation policies:an economic framework. World Bank Policy Research Working Paper, WPS5568,February 2011. The World Bank, Washington, D.C.

Harrison, J.M., 1985. Brownian Motion and Stochastic Flow Systems. Wiley, New York.

Heal, G., Kriström, B., 2002. Uncertainty and climate change. Environ. Resour. Econ.22, 3–39.

IPCC, 2007. Inter-relationships between adaptation and mitigation. Chapter 18 in WGIIFourth Assessment Report, Inter-Governmental Panel on Climate Change.

Karatzas, I., Shreve, S.E., 1988. Brownian Motion and Stochastic Calculus. SpringerVerlag, Berlin.

Lecocq, F., Shalizi, Z., 2007. Balancing expenditures on mitigation and adaptation toclimate change: an exploration of issues relevant to developing countries.World Bank Policy Research Working Paper, WPS4299. The World Bank,Washington, D. C.

Øksendal, B., 2007. Stochastic Differential Equations, 4th ed. Springer Verlag, Berlin.Pindyck, R.S., 2000. Irreversibilities and the timing of environmental policy. Resour.

Energy Econ. 22, 223–259.Pindyck, R.S., 2002. Optimal timing problems in environmental economics. J. Econ. Dyn.

Control. 26, 1677–1697.Shalizi, Z., Lecocq, F., 2009. To mitigate or to adapt: is that the question? Observations on

the appropriate response to the climate change challenge to development strategies.World Bank Res. Obs. 25, 295–321.

Tol, R.S.J., 1998. Economic aspects of global environmental models. In: van den Berg, J.C.,Hofkes, M.W. (Eds.), Theory and Implementation of EconomicModels for SustainableDevelopment. Kluwer, Doordrecht, pp. 277–286.

Tol, R.S.J., 2005a. Adaptation andmitigation: trade-offs in substance andmethod. Environ.Sci. Policy 8, 572–578.

Tol, R.S.J., 2005b. Emission abatement versus development as strategies to reducevulnerability to climate change: an application of FUND. Environ. Dev. Econ.10, 615–629.

Tol, R.S.J., 2007. The double trade-off between adaptation and mitigation for sea-levelrise: an application of FUND. Mitig. Adapt. Strateg. Glob. Chang. 12, 741–753.

Tol, R.S.J., Fankhauser, S., Smith, J.B., 1998. The scope for adaptation to climatechange: what can we learn from the impact literature? Glob. Environ. Chang. 8,109–123.

Wickpedia, 2013. Adaptation to Global Warming. http://en.wikipedia.org/wiki/Adaptation_to_global_warming.