WPS5724
Policy Research Working Paper 5724
Implications of a Lowered Damage
Trajectory for Mitigation
in a Continuous-Time Stochastic Model
Jon Strand
The World Bank
Development Research Group
Environment and Energy Team
June 2011
Policy Research Working Paper 5724
Abstract
This paper provides counterexamples to the idea that is down-shifted (“anticipatory adaptation”). In this model
mitigation of greenhouse gases causing climate change, mitigation is a lumpy one-off decision. Policy to reduce
and adaptation to climate change, are always and damages for given emissions is continuous in case 1, but
everywhere substitutes. The author considers optimal may be lumpy in case 2, and reduces both expectation
policy for mitigating greenhouse gas emissions when and variance of damages. Lower expected damages
climate damages follow a geometric Brownian motion promote mitigation, and reduced variance discourages it
process with positive drift, and the trajectory for damages (as the option value of waiting is reduced). In case 1, the
can be down-shifted by adaptive activities, focusing on last effect may dominate. Mitigation then increases when
two main cases: 1) damages are reduced proportionately damages are dampened: mitigation and adaptation are
by adaptation for any given climate impact (“reactive complements. In case 2, mitigation and adaptation are
adaptation”); and 2) the growth path for climate damages always substitutes.
This paper is a product of the Environment and Energy Team, Development Research Group. It is part of a larger effort by
the World Bank to provide open access to its research and make a contribution to development policy discussions around
the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be
contacted at jstrand1@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Implications of a Lowered Damage Trajectory for Mitigation
in a Continuous-Time Stochastic Model1
By
Jon Strand
Development Research Group, Environment and Energy Team
The World Bank
Washington DC 20433, USA
and
Department of Economics
University of Oslo
E-mail: jstrand1@worldbank.org
Key words: Mitigation; adaptation; climate damages; uncertainty; option values.
JEL classification: Q54; Q58; H23; C61.
1
I thank Bård Harstad, Nils Framstad, Michael Keen, Robert Pindyck and Michael Toman for helpful
comments to previous versions. The views expressed in this paper are those of the author and not necessarily
those of the World Bank, its management, Executive Directors or member countries.
1. Introduction
Two qualitatively distinct activities are induced by climate change: mitigation of
greenhouse gases (GHGs) to limit emissions and thus future climate change; and adaptation
to lessen the negative impact on human societies of any given climate change.2 These
activities are usually viewed as substitutes: more mitigation (leading to less GHG emissions)
implies less climate change and thus less need to adapt; conversely, more adaptation reduces
the damage from climate change, thus reducing the “need” to mitigate.3
For the effect of increased mitigation on adaptation, such a view appears robust. In this
paper we are concerned with reverse relationships: Given that the process for future damages
resulting from given climate change is dampened or shifted down, what is the impact on
optimal mitigation? Increased adaptation to climate change is likely to reduce the damages to
human societies for any given mitigation level, thus the two relationships may seem similar.
But while much of adaptation activity is likely to come later, in response to given climate
change, mitigation is always anticipatory, aiming to pre-empt anticipated but uncertain
climate change.
The uncertainty part is here crucial. Two countervailing effects follow from more
mitigation. First and most obviously, expected climate damages are reduced, which reduces
the “need” to mitigate. But when adaptation causes the process describing climate damages to
society to be dampened, uncertainty about future climate impacts (represented by the variance
on the stochastic damage process) is also reduced.
This uncertainty is here modelled as a geometric Brownian motion process with positive
drift. Expected damages from a given (increased) concentration of greenhouse gases (GHGs)
2
When climate change results in welfare improvements, which is frequently the case, adaptation is the process
by which these welfare gains are maximized.
3
See Tol (1998, 2005); Tol, Fankhauser and Smith (1998), Fankhauser, Smith and Tol (1999); Burton (1997).
Ses also the wider discussion on interrelationship between mitigation and adaptation, in IPCC (2007).
2
are then assumed to increase over time. Actual damages are uncertain, and this uncertainty
increases with the level of actual damages for given adaptation.
I study two cases. Both imply that the process for (unmitigated) damages is altered by
policies that reduce climate damage (such policies are, commonly and imprecisely, denoted as
“adaptation”). In the first case, the stochastic process for damages is “dampened”, meaning
that actual damages are shifted down by a constant factor at all points in time. In this case
there is no cumulative effect of the adaptive policy to reduce climate costs: all gains are
instantaneous. In the second case, the process by which climate affects human societies is
downshifted, in the sense that the constant positive drift on this process is dampened. Now
there is a cumulative dampening effect on damages as it is the expected rate of increase for
damages, per time unit, that is reduced. And unlike under case 1, there is no direct and
immediate reduction in the uncertainty of the damage process.
Under both types of dampening of damages, both expectation and variance of climate
damages are reduced. Pindyck (2000, 2002) and others have shown that when future climate
damages are more uncertain, investments in “lumpy” mitigation activity involving large sunk
costs (such as replacing coal-fired power plants with nuclear or renewable energy) are
reduced, as the option value of waiting to invest increases.4. The intuition behind this result is
that, under uncertainty, damages may turn out to be small; for such cases, the initial
investment will be wasted. Waiting implies retaining the option to “wait and see” if damages
will actually be so large that mitigation is worthwhile.
We study implications of each of the two types of “adaptive” measures for the timing of
mitigation policy, which in our model takes a rather primitive, discrete and lumpy, form. We
assume that authorities may make a major investment in new (energy or other) infrastructure
investment that, after having been undertaken, removes all GHG emissions from the
4
See also Fisher (2000), Heal and Kriström (2002).
3
respective economic activity. The policy issue to be decided is when, if ever, this investment
should be made.
In the first “adaptation” case, effects seem counterintuitive at first glance. For my first case
(in section 3), under certain parametric conditions the reduced variance effect on mitigation
timing, of the adaptation measure considered (which leads to sooner mitigation), dominates
over the reduced expected damages effect (which leads to postponed mitigation). This
increases the “propensity to mitigate” when damages are dampened, in the sense that the
value for the stochastic variable, under which mitigation action is triggered, is reduced. In
other words, increased (earlier) mitigation activity follows in cases where damages are
reduced. I also show that the specified down-shift of the damage function may in fact be the
result of optimal (“reactive”) adaptation given that the relationship between adaptive value
and adaptation cost belongs to a particular, but I will argue plausible, class of functions (it is a
function only of the ratio of adaptation costs to damages).
In the second case, treated in section 4, adaptation affects the positive drift of the damage
process for given mitigation, but does not dampen the stochastic component as such. The
results are then more standard: “more adaptation” now leads the decision maker to mitigate
later on average, and a greater value of the continuous random variable is now necessary for
mitigation to be triggered. It is more difficult to find an adaptation value function that fits to
this case; also most reasonably, adaptation must now be “anticipatory”.
The final section 5 discusses these two cases further, in the context of more practical
adaptation policy, and how such activity should most reasonably be interpreted in the two
cases. I suggest that “adaptation policy” under case 1 should most readily be identified with
“reactive” (ex post) adaptation (carried out by the public or private sector). Adaptation
activity under case 2 is anticipatory, or perhaps what may be termed “climate-proofing”,
whereby society is made more “robust” in meeting the challenge of higher global
4
temperatures. In this case adaptation modifies the impacts on society as if they were caused
by lower temperatures.
2. Optimal Mitigation
This section closely follows Pindyck (2000). Define M(t) as the stock of greenhouse gases
(GHGs) in the atmosphere, and E(t) as a flow variable that controls this stock. In the
following, E is taken as the rate of emissions of GHGs, and can take a given number of
nonnegative values. Assume that M is (deterministically) given from
(1) dM ( t ) [ E ( t ) M ( t )]dt ,
where δ is the rate of decay of the GHG stock. Associated with the stock variable M there is a
flow variable of (negative) benefits, B, given by
(2) B(M(t), θ(t)) = -μ(t)θ(t)M(t).
θ is a multiplicative parameter governing the marginal cost to society from increasing the
stock of GHGs. This parameter is stochastic and follows a geometric Brownian motion, on
differential form described by
(3) d dt dz ,
where α is a (positive) drift parameter, and ζ the standard deviation of the process.5
Considering a starting point 0, at t = T the logarithm of θ has expectation (α –ζ2/2)T and
variance ζ2T (while θ itself has expectation αT).6
This process is modified by two types of policy interventions. The first of these is
represented by μ in (2), which is assumed to take values less than or equal to unity. Assuming
α > 0, damages from a given concentration of GHGs in the atmosphere is systematically
drifting upward over time, but with a random component that may make this upward drift
highly uncertain. This process can be affected in three ways, one of which is associated with
5
See Dixit and Pindyck (1994), chapter 3, for a heuristic discussion of the formula and its applications; for more
rigorous presentations see Harrison (1985), Karatzas and Shreve (1988), and Øksendal (2007).
6
See Dixit and Pindyck (1994), chapters 3-4.
5
mitigation policy, and the two others with “adaptation policy”. First, as studied by Pindyck
(2000), the mitigation control parameter E can shift down to a new and permanently lower
level, through (permanent) increased mitigation efforts, by incurring a cost K at the shift time.
The two other types of policy intervention are new here. The first of these implies that the
level of damages experienced by human societies, for given “climate impact” θ, can be
affected by policy, and this is represented by a downward shift in μ. Such shifts work in two
ways: it reduces both expectation and variance of the damage path. The final additional policy
intervention is simply to (permanently) affect the drift rate α for the stochastic process for
damages. In the following we will assume, alternatively, that the two latter policies are
applied, exogenously, together with (endogenous, and optimal) mitigation policy: in section 3
we consider shifts in μ; while in section 4, we assume that α shifts.
The policy objective W (at time 0) is
rt
rT
(4) W E xp ( t 0) ( ( t ) ( t ) M ( t )) e dt K ( E 1 ) e 1
0
denoting welfare (or negative cost) associated with climate change, r being the discount rate.
T1 is the (uncertain, and endogenous) time of mitigation, modelled as a one-off decision. As
in Pindyck (2000), mitigation costs take the simple form K = kE. We take expectations at time
zero to indicate that T1 is uncertain at t = 0. We will in the following also assume that μ is a
constant, so that adaptation activity of the type studied in section 3 below, leads to a
proportional downward shift in the damage level, independent of this level.
We now solve for the time of (possible) mitigation, T1, for given adaptation.7 This is a
classical optimal stopping problem. Adopting a dynamic programming approach, we denote
the value functions (4) for the pre-mitigation and post-mitigation states, by WN and WA
respectively. Since the cost of adoption is linear in E0 – E1 (denoting the amount of mitigation
7
The second step of a full optimisation procedure, left for future research, would be to derive optimal adaptation,
in the form of an optimal value or values of μ (section 3), or an optimal level of α (section 4).
6
achieved), there is nothing to gain by not reducing E to 0 once mitigating. Consequently,
without loss of generality we assume that the amount of mitigation is E0, and that mitigation
cost K equals kE0.
WN and WA must satisfy the Bellman equations (Pindyck (2000), equations (5)-(6)):
1
M ( E 0 M )W M W W
N N N 2 2 N
(5) rW
2
1
M M W M W W
A A N 2 2 N
(6) rW
2
where subscripts denote first- and second-order derivatives of the value functions. These must
be solved simultaneously subject to the boundary conditions
(0, M ) 0
N
(7) W
( *, M ) W ( *, M ) kE 0
N A
(8) W
W ( *, M ) W ( *, M ) .
N A
(9)
(7) states that the value (or climate cost) function must be zero with no current climate
damages (and, with a geometric Brownian motion, never any climate damages). (8) defines
the ”stopping point” θ* for θ (for mitigation), by indifference between action and non-action.
(9) is the “smooth-pasting condition”: the derivative of W must be continuous at the point of
action, θ*.
The solutions to (5)-(6) are
M E 0
A ( )
N
(10) W
r ( r )( r )
M
A
(11) W
r
where A and γ are constants to be determined. γ is the positive root of the quadratic equation
1
( 1) r 0 ,
2 2
(12)
2
leading to the following solution for γ:
7
2
1 1 2r
(13) 2 1.
2 2 2
2 2
The parameter A, and the critical value θ* beyond which mitigation action is taken, are
determined from (8) and (9):
1
1
(14) A [( r )( r ) ] E0
k
1
(15) * k ( r )( r ) .
1
Under full certainty, the critical value of θ triggering mitigation would be determined from
1
(16) ** k ( r )( r ) .
Increased adaptation (reduced μ) increases θ** proportionately to 1/μ, with a strong delaying
effect on mitigation. This is intuitive: adaptation “takes over” the role of mitigation, making
the latter less relevant and thus less urgent. Comparing (15) and (16), uncertainty increases θ*
relative to θ**, by a factor γ/(γ-1) > 1 as γ > 1. In the set of cases where θ lies in the range
(θ**, θ*), there is then mitigation under certainty, but not under uncertainty.
3. Changes in Adaptation Policy 1: Shifts in μ
We now introduce “adaptation” policy in the form of shifting μ downward, 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 a trivial problem: from (16), μθ** is a constant,
and lower μ leads to a proportional increase in θ**; and to mitigation being postponed. Under
uncertainty the situation is more complex as also γ is affected by changes in μ. Γ = γ/[μ(γ-1)]
is then the relevant measure. Note that γ/(γ-1) drops in γ (when, as here γ > 1). If γ increases
when μ drops, the option value of waiting is then also reduced. Even stronger, when γ/(γ-1) is
reduced more than μ, θ* drops when μ drops: increased adaptation then raises mitigation.
8
Taking the derivative of γ with respect to μ in (13):
1
d 1
2
2r 2
1 4r
(17) 2 2 .
d
2
2 2 2 2 2 2 2
2
Consider a small change in μ starting from μ = 1 (no adaptation). (17) can be written
1
d 1
2
2r 2
1 4r
(18) 2 1 2 2 2 .
d
2
2
Here dγ/dμ < 0 in a wide range of circumstances. A more crucial question for us is whether
the increase in γ can be sufficiently great that γ/(γ-1) drops by more than the initial drop in μ.
We find
d( ) 2
1 1 d
(19) ,
d 1 d
where dγ/dμ is found from (18). Thus when γ is close to unity and at the same time dγ/dμ < 0,
γ/[μ(γ-1)] drops when μ drops. Increased adaptation will then lead to “more mitigation”, in the
sense that the (one-off) mitigation investment is incurred earlier.
To exemplify, consider the case α/ζ2 = ½ (when, following Dixit and Pindyck (1994),
Et=0[log θ(η)] = log θ(0); even though Et=0[θ(η)] = θ(0) + (αη)).8 We find:
2
d( ) 1
1
1 r 2 r 2 1
(20)
1 .
d
4
r > 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 of (20) exceeds unity; required for a drop in μ to lead to a
drop in θ*. The more precise condition is found as α ≥ (4/25)r (by setting the right-hand side
8
Note however that this case is in itself not useful for deriving optimal expected times until mitigation
intervention , since in this case, this expected period is infinite; see section 4 below. This case is useful only as a
benchmark and for its simplicity of illustration (and, invoking continuity, by showing that a similar result will
hold also for more interesting cases involving α > ζ 2/2).
9
of (20) equal to unity, and solving this as a quadratic equation in r / ). This seems as a not
very restrictive condition.
Can the indicated adaptation response to given damages, implied by this version of the
model, correspond to optimal adaptation? To answer this question, consider a case where
adaptation is completely reactive, following in response to given damages (so that no
anticipatory adaptation is in question). We seek an adaptation cost function that fits case 1,
where a basic presumption is that damages are dampened by adaptation, proportionately to the
initial level of damages. A class of adaptation value functions that fits to these assumptions is
one where optimal gains from and costs of adaptation are both proportional to the initial level
of damages. A suitable net value function (for the value, net of adaptation costs, of incurring
the adaptation cost A given an instantaneous climate damage level D) has the form
A
(21) V D A ,
D
where β is a positive-valued function with first- and second order derivatives, β’ > 0, β’’ < 0,
β(0) = 0, β(∞) = 1. V is maximized with respect to A, yielding
dV
(22) ' 1 0 ' 1 .
dA
(22) yields a unique solution for β, call it β*, and consequently a unique solution for A/D = γ,
call it γ*. Net climate damages, net of adaptation gains and costs, can now be written D – β*D
+ γ*D = (1-β*+γ*)D, where both β* and γ* constants, and where 0 < β*-γ* < 1. Thus (1-
β*+γ*) is a positive constant less than unity, independent of the damage level D, and
corresponds to the coefficient μ in the analysis in this section. It is clear that this type of
adaptation is “reactive”, carried out in response to the particular level of damage occurring, at
any given time.
When the function β takes the particular Cobb-Douglas form, (21) can be written as
10
A
(23) V D A ,
D
where ε is the elasticity of “adaptive value” with respect to adaptation costs, and where 0 < ε
< 1. In this case,
(24) 1 * * 1 (1 ) 1 1 ,
where μ is a constant. A negative shift in ε is here tantamount to a negative shift in μ.
4. Changes in Adaptation Policy 2: Shift in α
My second case of “adaptation policy” involves down-shifting the drift parameter α of the
stochastic process for θ. The main interpretation of such policy might seem to be one of (more
efficient) mitigation, insofar as θ describes the process by which climate is altered. However,
an interpretation could equally well be that society, for given climate as represented by a basic
(non-shifted) stochastic damage process θ, is affected by climate in the same way as if θ were
down-shifted. One way to view such a shift is that society is made more resilient in the face of
climate change. This also gives room for an interpretation in terms of “adaptation”; and now
most reasonably anticipatory adaptation or “climate proofing”.
An immediate issue for this case is that when α is reduced, the expected time until
mitigation, η, increases for given θ*. The formula for this expected time is9
*
lo g
0
(25) E ( ; *, 0 ) .
2
2
The expected time until mitigation, E(η), is here finite only when α > ζ2/2.10 Given this
inequality, E(η) is a negative function of α so that a reduced α leads to a higher E(η). Thus,
9
See Øksendal (2007), p 131.
11
when a reduced α is interpreted as “better adaptation” or “climate-proofing” (as argued
below), this in itself delays mitigation as it simply reduces the “perceived need” to mitigate.
This is a feature that was not present in our case 1.
We are next interested in how θ* is affected by changes in α. Note again that a lower α
implies that damages are reduced at all future points of time, and thus the cumulative
damaging effect of current emissions reduced. This feature tends to raise the optimal value of
θ. But also here we find a negative effect on the variance and thus the option value of waiting
for any given θ. Another major difference from case a above is that the expected time it takes
to reach a given θ now increases (as the very process for θ is downshifted).
In this case, the derivative of θ* with respect to α is found as
d * k ( r )( r ) d
(26) ( 1)(2( r ) ) ,
d ( 1) d
2 2
where
1
1 2r
2
d 1
2
1
(27) 2 1 2 2 2 .
d 2
2
The first main term in the curled bracket in (26) can be interpreted as the effect of reduced
uncertainty (variance) of damages when α is reduced; while the second term can be
interpreted as the effect of reduced expected damages. As in section 3 the variance term
works to increase θ*, and the expectation term to reduce it.
Can we here find cases where dθ*/dα > 0? The first main term in the curled bracket in (26)
(the “variance term”) would then need to dominate the second term (the “expectation term”).
This is however not the case here: for all valid parameter combinations, dθ*/dα < 0. Thus the
10
Note that even when α < ζ2/2, η is finite with positive probability (that increases in α - ζ2/2), applying
Dynkin’s (1965) formula; and infinite with the complementary probability; thus E(η) is infinite: see also
Øksendal (2007), p 131. Thus η can be finite in “many” cases, and with probability mass that approaches 1 as α -
ζ2/2 approaches zero from below.
12
relationship between mitigation and adaptation is “traditional”: more adaptation reduces the
“need” for mitigation in the sense that the trigger value for the stochastic variable θ, beyond
which mitigation is triggered, is now increased when α drops (“more climate proofing”).
There is a double such effect, since the time it takes to first reach any given level of θ is now
longer in expectation, from (25).
Finally, what if any adaptation or cost function produces this type of adaptation response?
One obvious candidate is an investment sunk early on, which leads to permanent effects
directly on the growth trajectory. Such an investment would constitute “anticipatory
adaptation”, but might in some contexts perhaps be indistinguishable from mitigation policy.
5. Discussion
I have in this paper studied whether policies to reduce GHG emissions (mitigation), and
policies to reduce damages caused by climate change (“adaptation”), are everywhere and in
all circumstances “substitutes” (so that applying more of one policy makes it optimal to apply
less of the other), in the context of a particular model by which climate damages develop
according to a continuous-time Brownian motion process with drift. I present a practical
counter-example to this principle, where policies to reduce GHG emissions (mitigation), and
particular policies to reduce the damages caused by climate change (“adaptation”), are
complements. This possibility exists under uncertainty because “adaptation” broadly can take
two different forms. Complementarity between mitigation and adaptation can be the outcome
when damages caused by climate change are down-shifted proportionately by “reactive
adaptation”. From our model such an argument, and the intuition behind it, fails in some cases
when the “option value” associated with waiting to implement mitigation policy is substantial.
The “option value” is due to uncertainty about future climate damages, and how this
uncertainty evolves over time. Large uncertainty implies that climate (including the damage
13
that it causes) may become worse; but could also improve. The possibility that it can improve
“substantially” from today’s level and into the immediate future is greater, the greater is the
uncertainty of the stochastic underlying variable controlling climate damages. A policy to
reduce this uncertainty reduces the incentive to wait “a little” for possible better times in the
“very near” future, and instead implement the relevant mitigation policy immediately. The
key point we emphasize in our presentation is that this uncertainty effect may dominate over
the effect of reduced expected damages (which, in isolation, leads to mitigation being
postponed). In consequence we have the paradoxical situation that “more adaptation” implies
“more mitigation”: the value for a stochastic damage variable that triggers mitigation policy
shifts down: mitigation is carried out “earlier”.
In analyzing this model, we focus on two distinct interpretations of the role of adaptation
policy, as two different forms that multiplicative downward shifts in net climate damages may
take. In either case, the climate damage trajectory under “business-as-usual” mitigation policy
is assumed to follow a geometric Brownian motion process with constant positive drift. In the
first case, damages resulting from a given climate change are dampened, by “adaptation”.
This is perhaps the most straightforward interpretation of adaptation; it can in principle be
interpreted as either reactive or anticipatory adaptation, but most reasonably as the former.
The policy behind this effect reduces climate damages proportionately in all states. In section
3 we discuss conditions under which the assumed, proportional, dampening of the stochastic
damage process can be a result of optimal (reactive) adaptation. I show that this is the case
when both the degree to which climate damages are mitigated by reactive adaptation, and
adaptation costs, rise in exact proportion to damages. I also provide a concrete example in
terms of a Cobb-Douglas “damage mitigation function”.
In our alternative case, “adaptation” is interpreted as modifying the (damage-related)
climate impact, so that the rate of drift of the stochastic process for damages is shifted down. I
14
then find nothing unusual or surprising. “More adaptation” then always reduces mitigation, in
two distinct ways: the point of time at which mitigation is executed is delayed for given value
of the stochastic parameter governing climate damages; and the value of the stochastic
variable that triggers mitigation is higher. This may, arguably, correspond to “climate-
proofing” whereby society is made better able to withstand given changes in climate. It is here
also easier to visualize the types of impact that our form of “climate-proofing” may have; in
particular, it may, as in our model, be a cumulative process whereby “layers” of climate-
proofing may gradually add to the overall resiliency of society. Again, this view is speculative
and needs confirmation through subsequent research.
Why is there a difference in outcome between our two examples? In the first case, the level
as well as the variance on damages are affected directly and immediately. In the second case
there is only an indirect effect as the stochastic process for damages and its variance are not
affected immediately. Such effects only come later (from the process being down-shifted and
a geometric uncertain component). The option value is affected more directly, and by
relatively more, in the first case than in the second. Intuitively, the short-run reactive type of
adaptation that reduces the (short-run) uncertainty the most. Note here that the option value
concept is essentially one of (very) short-run maximization: should one wait one further
instant to mitigate, or should one do it now. Such tradeoffs are affected much less by our
“climate-proofing” policy, where it is the long run that is on the whole affected.
In terms of policy implications, it is of course desirable that climate damages can be down-
shifted. It may then be useful to know that such a down-shift may, sometimes, make also
more mitigation attractive, so that adaptation and mitigation activity would tend to reinforce
each other. But it is important to stress that this outcome follows from a particular and
perhaps not very general example; such an outcome can thus not be taken for granted.
15
Strong assumptions lie behind this analysis, and these may need relaxation in future work.
First, decision makers (governments) are considered risk neutral. Risk aversion would tend to
reduce the option value of waiting (in particular, highly positive future outcomes are given
less weight); although the overall effect of this remains unclear. Secondly, adaptation and
mitigation are given stylised and abstract interpretations. Mitigation is “one-off”: to mitigate
all GHG emissions. or nothing. It remains to be seen whether more realistic assumptions can
be accommodated without jeopardizing main conclusions. Pindyck (2000, 2002) has
considered partial mitigation in a similar set-up, with similar qualitative results. It neither
appears crucial that all mitigation costs are sunk and paid up-front. A fixed cost component is
crucial; if not, the option value argument, crucial for our results, fails. Thirdly, adaptation is
modelled in very simple ways, as a constant multiplicative modification of the damage path,
or as a modification of the rate of change of the stochastic process causing climate change
damages.
16
References:
Aitchison, J. and Brown, J. A. C., 1957, The Lognormal Distribution. Cambridge, UK:
Cambridge University Press.
Burton, I., 1997, Vulnerability and Adaptive Response in the Context of Climate and Climate
Change. Climate Change, 36, 185-196.
Dixit, A. K. and Pindyck, R. S., 1994, Investment Under Uncertainty. Princeton, N.J.:
Princeton University Press.
Dynkin, E.B. (1965), Markov Processes, Vol I. Berlin: Springer Verlag.
Fankhauser, S., Smith, J. B. and Tol, R. S. J., 1999, Weathering Climate Change: Some
Simple Rules to Guide Adaptation Decisions. Ecological Economics, 30, 67-78.
Fischer, A. C., 2000, Investment Under Uncertainty and Option Value in Environmental
Economics. Resource and Energy Economics, 22, 197-204.
Harrison, J. M., 1985, Brownian Motion and Stochastic Flow Systems. New York: Wiley.
Heal, G. and Kriström, B., 2002, Uncertainty and Climate Change. Environmental and
Resource Economics, 22, 3-39.
IPCC, 2007, Inter-Relationships Between Adaptation and Mitigation. Chapter 18 in WGII
Fourth Assessment Report, Inter-Governmental Panel on Climate Change.
Karatzas, I. and Shreve, S. E., 1988, Brownian Motion and Stochastic Calculus. Berlin:
Springer Verlag.
Øksendal, B., 2007, Stochastic Differential Equations. 4th Edition. Berlin: Springer Verlag.
Pindyck, R. S., 2000, Irreversibilities and the Timing of Environmental Policy. Resource and
Energy Economics, 22, 223-259.
Pindyck, R. S., 2002, Optimal Timing Problems in Environmental Economics. Journal of
Economic Dynamics and Control, 26, 1677-1697.
Tol, R. S. J., 1998, Economic Aspects of Global Environmental Models. In J. C. van den Berg
and M. W. Hofkes (eds.): Theory and Implementation of Economic Models for Sustainable
Development; pp 277-286. Doordrecht: Kluwer.
Tol, R. S. J., 2005, Adaptation and Mitigation: Trade-Offs in Substance and Method.
Environmental Science and Policy, 8, 572-578.
Tol, R. S. J., Fankhauser, S., and Smith, J. B., 1998, The Scope for Adaptation to Climate
Change: What can we Learn from the Impact Literature? Global Environmental Change, 8,
109-123.
17