Download Abatement vs. Adaptation and the Prospects for International

EQUILIBRIUM CLIMATE POLICY: ABATEMENT VS. ADAPTATION,
AND THE PROSPECTS FOR INTERNATIONAL COOPERATION
Martin Farnham and Peter W. Kennedy
Department of Economics
University of Victoria
April 2010
We study the relationship between the availability of adaptation as a defense
against climate change, and the prospects for international cooperation to reduce
emissions. We examine a model of the global economy in which countries differ
according to income, and each country can undertake a mix of abatement and
adaptation. Damage from climate change is proportional to the amount of
undefended economic activity, because climate change is a public factor in
production. We begin with a setting in which adaptation is not available. We
show that the non-cooperative equilibrium in this setting yields an inverted Ushaped relationship between income and emissions across countries (a crosssection environmental Kuznets curve). We then compare the equilibrium with the
first-best solution, and derive a condition on global income variance under which
the first-best solution Pareto-dominates the equilibrium. We then introduce the
possibility of adaptation. We derive the first-best solution, and then characterize
the equilibrium relationship between national income and the equilibrium mix of
abatement and adaptation as a function of the relative cost of these two actions.
We show that the availability of adaptation can make low-income countries worse
off in equilibrium than when adaptation is not available, because global emissions
are higher when countries substitute into adaptation and away from abatement.
Countries that are negatively affected by equilibrium adaptation in this way have
more incentive to reach a cooperative agreement over abatement than when
adaptation is not available, and this means that the availability of adaptation
actually enhances the prospects for cooperation. In particular, we show that the set
of conditions under which the global first-best solution Pareto-dominates the noncooperative Nash equilibrium is enlarged by the availability of adaptation.
1. INTRODUCTION
The international debate on global climate policy to date has focused primarily on
reducing emissions. Yet adaptation will surely also play a key role in any optimal
policy response to climate change. The availability of adaptation – as a partial
substitute for abatement – also has important implications for negotiations over an
international agreement to reduce emissions. In particular, the threat point for any
country in those negotiations depends on the viability of adaptation as an
alternative to abatement. Our purpose here is to investigate this relationship
between adaptation and the prospects for international cooperation on abatement.
One might be inclined initially to think that the availability of adaptation
necessarily reduces the prospects for cooperation over abatement. Adaptation is
for the most part a private good while abatement is a public good. Failure to reach
a cooperative agreement over contributions to that public good would appear to
become less important to all players if a viable alternative – via a private good – is
available. However, this simple logic breaks down in a setting with
heterogeneous countries. In such a setting, the availability of adaptation can make
some countries worse off in equilibrium than when adaptation is not available,
because global emissions are higher when countries substitute into adaptation and
away from abatement. We show that this negative impact of adaptation on some
countries can occur even when all countries have equal access to adaptation.
Countries that are negatively affected by equilibrium adaptation in this way have
more incentive to reach a cooperative agreement over abatement than when
adaptation is not available, and this means that the availability of adaptation
actually enhances the prospects for cooperation. In particular, we show that the set
of conditions under which the global first-best solution Pareto-dominates the noncooperative Nash equilibrium is enlarged by the availability of adaptation.
This result hinges on two key features of our model: strict convexity of
abatement costs; and a positive relationship between climate-related damage and
the total value of economic activity affected. In the non-cooperative equilibrium
in our model, large economies (as measured by GDP) adopt cleaner technologies
than do smaller ones because larger economies suffer greater damage, in absolute
terms, from any given amount of climate change. Large countries therefore
internalize more of the global damage from their own emissions, and as a
consequence, adopt cleaner technologies. This means that high-GDP countries
are positioned higher up the marginal abatement cost curve than are lower-GDP
countries. This in turn means that any given substitution out of abatement and into
adaptation yields larger cost savings for high-GDP countries than for lower-GDP
countries. In contrast, all countries are exposed to the same increase in global
emissions stemming from substitution into adaptation. Thus, the net impact of
adaptation in equilibrium can be positive for high-GDP countries but negative for
1
lower-GDP countries. This makes a cooperative agreement to reduce emissions
more attractive to lower-GDP countries than when adaptation is not available.
Our paper adds to a growing body of literature on the relationship between
abatement and adaptation.1 A key concern expressed in much of the policyoriented elements of that literature is that any enhanced emphasis on adaptation in
practice will distract effort away from abatement, and hence further widen the
divide between large and small countries in terms of reaching agreement on an
abatement treaty, and in terms of net damages suffered (see for example, UNDP
(2007)). Our paper is an attempt to construct a formal theoretical framework in
which to consider these issues. The only other paper we know that attempts to
build such a framework is Barrett (2008b).
Barrett studies the same key question that we do: whether the availability
of adaptation – as an imperfect substitute for abatement – helps or hinders
collective action on abatement. He considers a setting with identical countries in
which abatement and adaptation are both binary actions, and focuses on a region
of the parameter space where, in the Nash equilibrium, every country adapts and
no country abates, and in the first-best solution, every country abates and no
country adapts. He then examines the treaty equilibria in a game where countries
first choose whether or not to participate in an abatement treaty, and then choose
whether or not to adapt individually. He shows that the availability of adaptation
raises the equilibrium number of treaty participants because adaptation raises the
payoff to every country when cooperation fails. It is therefore necessary for more
countries to participate if cooperative abatement is to be more attractive than nonparticipation. He then extends consideration to a setting in which a subset of the
countries (“poor” countries) can neither abate nor adapt, and shows that the
availability of adaptation to the other countries (“rich” countries) does not
necessarily make the former group worse off, because of the positive impact of
adaptation on the treaty equilibrium outcome.
While Barrett’s model is more stark and stylized than our own – there is
no income heterogeneity across countries, and all policy decisions are binary – he
considers a richer set of questions with respect to cooperative treaties than we do;
we restrict consideration to the question of Pareto-dominance. In this respect we
see our paper and his as usefully complementary.
The rest of our paper proceeds as follows. In section 2 we present our
model. In section 3 we examine a setting in which adaptation is not available, as a
1
Much of the theoretical work in this area to date has focused on the implications of adaptation for
optimal climate policy in a dynamic setting (particularly in the presence of uncertainty); see for
example, McKibbin and Wilcoxen (2004), Ingham et. al. (2007), Agrawala et. al. (2009), and
Felgenhauer and Bruin (2009). While clearly important for the policy problem in practice, we
abstract entirely from dynamics and uncertainty in this paper so as to focus on the strategic
interaction between countries in a tractable and transparent way.
2
benchmark against which to consider the implications of adaptation. We first
solve for the first-best solution, and then characterize the non-cooperative
equilibrium and show that it yields a cross-section environmental Kuznets curve;
that is, there is an inverted U-shaped relationship between GDP and emissions.
We then derive a condition under which the first-best solution Pareto-dominates
the equilibrium. We show that the variance across countries with respect to GDP
plays a key role in that condition. In section 4 we introduce adaptation. We derive
the first-best solution and examine the optimal mix of adaptation and abatement.
In section 5 we characterize the non-cooperative equilibrium, and show that the
availability of adaptation to all countries can make low-GDP countries worse off.
We then derive the Pareto-domination condition when adaptation is available, and
show that it is weaker than when adaptation is not available. In section 6 we
conclude with some summary remarks and a brief discussion of key limitations of
our analysis. An appendix contains proofs of all propositions.
2. THE MODEL
Let
denote aggregate production in country i (as measured by GDP). We
will henceforth refer to
as the income of country i, but it is important to note
that we do not mean per capita income; we revisit this distinction in section 6.
Global average income is
(1)
where n is the number of countries, and total global income is
variance in income across countries is given by
. The
(2)
Emissions from country i are a function of its economic output, as
measured by , and the technology used to produce that output. In particular, let
denote the technology employed by country i. Then its emissions are
(3)
3
Note that
measures the emissions-intensity of production in country i. In
addition, if we define baseline emissions to be
choice
, then
, corresponding to technology
measures abatement from that baseline, and
measures the percentage reduction in emissions from that
baseline. It will sometimes help to think of
in these terms when interpreting
our results.
The cost of producing output
using technology is
(4)
This quadratic specification means that unit production cost is increasing and
strictly convex in the cleanliness of the technology used. This is a key property of
our model. The specification in (4) also implies that there are no scale economies
in production per se: for any given production technology, unit production cost is
the same across all countries. Differentiating
with respect to
yields
the marginal cost of cleaner technology adoption (MCCTA) for country i, equal to
. Note that MCCTA is increasing in and .
Aggregate global emissions are given by
(5)
In the absence of adaptation, damage to country i from global emissions is
,
where
is the “damage parameter”. Thus, absolute damage to country i is
proportional to its economic size. This approach to modeling damage from
climate change comes from Hutchinson and Kennedy (2010).2 Their argument is
that climate change is a damaging public factor, whose impact (in the absence of
adaptation) is likely to be more or less proportional to the value of the economic
activity affected. For example, violent storm surges will cause more economic
damage to heavily developed coastlines than to less developed ones (think of how
hurricane costs have risen over time as wealth has grown); drought-induced crop
losses will be proportionately higher for large planted acreages than for small
ones; weather-induced electricity outages will impose costs that are typically
2
Hutchinson and Kennedy (2010) focus on the relationship between income growth and
equilibrium dynamics for a global pollutant when damage is strictly convex in emissions; there is
no possibility of adaptation. In contrast, here we focus on the role of adaptation, but in a simpler
setting in which damage is linear in emissions and incomes do not grow over time.
4
increasing in the value of the economic activity disrupted. This public-factor
approach to modeling damage reflects the fundamental importance of climate –
and climate-induced weather – to most aspects of economic activity. Note that the
strict proportionality assumption means that damage per unit of output is
independent of economic size, and so all countries suffer the same damage
relative to the size of their economies.3
Our damage function also assumes that damage is proportional to global
emissions. While this specification is common in the literature, it understates the
complexity of the climate change problem, in two respects. First, the physical
evidence indicates that climate change is a lagged function of the atmospheric
stock of carbon. Our model is a static one, so in our setting the stock and flow of
emissions are identical, but ideally the climate change problem should be modeled
as a stock-dynamic one.4 Second, damage in reality may be strictly convex in the
atmospheric stock of carbon, at least at levels beyond potential “tipping points”.
Modifying our model to impose strict convexity in global emissions is relatively
straightforward. The key change is that the marginal damage function for any
individual country is no longer independent of other-country emissions, and so the
equilibrium in the no-adaptation setting is no longer characterized by simple
dominant strategies, as it is in the game we solve. However, the additional
strategic interaction that strictly convex damage adds to the game without
adaptation arises even without strict convexity once adaptation is available. We
see this as an interesting result in itself since it highlights the strategic complexity
that adaptation adds to the global climate change game.
Our treatment of adaptation is simple but tractable. Adaptation in practice
will take many forms, including the adoption of different agricultural crops, the
construction of dykes and sea-walls, the relocation of populations, and a variety of
other defensive measures.5 Here we model adaptation as defensive action to
protect some fraction
of the economy from the damaging impact of
climate change. The “undefended” residual fraction of the economy remains
3
In reality,
will differ across countries according to their vulnerability to climate change – as
determined by geographical factors and the composition of their economic activity – but to keep
the analysis tractable, and to focus on the direct importance of income differences, we do not
allow countries to differ on this dimension. We revisit this issue in section 6.
4
Stock-dynamic models with strategic interaction (such as Levhari and Mirman (1980) in the
context of fisheries) are often intractable when they involve heterogeneous agents, and typically
do not yield transparent analytical results. We believe our simple static model does yield some
useful insights but we cannot confidently claim that the same key results would necessarily
emerge from a properly specified dynamic model. Extending the model to a dynamic setting is a
task for future work.
5
Some defensive measures – such as geoengineering – may have significant spillover effects on
other countries, possibly negative; see Barrett (2008a). Here we restrict attention to purely private
defensive measures.
5
subject to damage. In particular, if country i undertakes adaptation
then
the damage it suffers from climate change is
.
The cost of adaptation for country i is strictly convex (and for simplicity,
quadratic) in the coverage of adaptation, and proportional to the magnitude of
economic activity that must be adapted:
(6)
where
is a parameter reflecting the cost of adaptation relative to cleaner
technology adoption. This will be a key parameter in the analysis that follows.6
3. A WORLD WITHOUT ADAPTATION
We begin by studying the properties of this economy when adaptation is not
available. This facilitates a simple and transparent exposition of some key features
of the economy, and allows us to construct a benchmark against which the impact
of adaptation can be assessed. We first derive the first-best solution, and then
characterize the non-cooperative equilibrium.
3.1 FIRST-BEST ABATEMENT
The first-best global abatement policy is derived as the solution to a planning
problem in which total global cost – equal to the sum of production cost and
damage for each country, aggregated across countries – is minimized via the
choice of production technologies:
(7)
Let
denote the first-best technology for country i. The first-order conditions for
an interior solution to (7) are
6
In reality,
might differ across countries, depending on the composition of their economies.
Moreover, there may exist economies of scale in adaptation (such as in the development of new
seed varieties) that give higher income countries an advantage over lower income countries. The
extent to which these economies of scale might be so significant as to extend to the national level
is unclear, but it is an issue we revisit in section 6.
6
(8)
These are standard conditions equating the marginal cost and marginal benefit of
using a cleaner technology. The LHS of (8) is the MCCTA for country i. The
RHS of (8) is the marginal global benefit of using a cleaner technology in country
i. This marginal global benefit comprises two additive terms: the first term is the
marginal benefit that accrues to country i; the second term is the marginal benefit
bestowed on other countries. Solving (8) yields the first-best technology for
country i. This is described in Proposition 1.
PROPOSITION 1. If adaptation is not available, then the first-best technology
for country i is
(a)
(b)
if
; and
otherwise.
This first-best solution has two noteworthy properties. First, the optimal
technology is the same for all countries. This result arises because production cost
is proportional to output.7 Second, in the interior solution, given by part (a), the
first-best technology is increasing in aggregate global income. This result stems
from the fact that damage from emissions is increasing in income, because
climate change is a damaging public factor. A higher global income therefore
calls for cleaner technology.8 Note that if global income is high enough (or if
emissions are damaging enough, as measured by ) then first-best requires zero
emissions for all countries (
). While this corner solution is a theoretical
possibility in the model, we henceforth restrict attention to a setting in which the
first-best solution is weakly interior. This is guaranteed by Assumption 1.
7
In the appendix we provide a generalized version of Proposition 1 and show that if there are
economies of scale in production then the first-best technology for country i is increasing in its
own income. Conversely, if there are diseconomies of scale in production then the first-best
technology for country i is declining in its own income.
8
It should be noted that the distribution of global income across countries is irrelevant here only
because damage is strictly proportional to income. If the increasing relationship between damage
and income is not strictly proportional then the distribution of income matters. See equation (A2)
in the Appendix with
.
7
ASSUMPTION 1.
.
When Assumption 1 holds, first-best emissions for country i are decreasing in
aggregate global income, but increasing in its own income:
(9)
Thus, first-best emissions are highest for the highest-income countries. Summing
across countries in (9) yields first-best global emissions
(10)
Note from (10) that global emissions are quadratic in global income, Y. Thus, an
inverted U-shaped relationship – an environmental Kuznets curve (EKC) – arises
between global income and global emissions. This EKC reflects the competing
forces of a scale effect and a technique effect, as first described at the individual
country level by Grossman and Krueger (1991), and modeled formally in a traderelated context by Copeland and Taylor (1994). The scale effect stems from the
positive relationship between output and emissions for a given production
technology: emissions grow as output grows. The technique effect reflects the
optimal adoption of cleaner technology as income grows. The source of the
technique effect in Copeland and Taylor is preference-based: citizens demand
higher environmental quality as income grows because environmental quality is a
normal good. This translates into greater market and regulatory pressure for the
adoption of cleaner technologies. In contrast, the technique effect in our model
stems from climate change as a public factor: total damage is increasing in the
amount of economic activity exposed to the changing climate, and so too
therefore is the benefit from reducing emissions.9
It is important to note that the EKC in (10) involves a relationship between
global emissions and global income. In contrast, the relationship between income
and emissions across countries, for a given total global income, does not exhibit
an inverted U-shape; recall from (9) that this relationship is linear. The first-best
technology choice for any single country depends only on global income (rather
9
See Hutchinson and Kennedy (2010) for a discussion of how the technique effect arising from a
public factor approach to climate change relates to other models of the technique effect. See Brock
and Taylor (2005) for a comprehensive treatment of existing models. Carson (2010) provides a
good recent treatment of theoretical and empirical issues relating to EKC hypotheses.
8
than its own income) because the planner is concerned with global damage, which
is proportional to global income. The make-up of global income across countries
has no bearing on global damage, so individual country incomes are not relevant
for the first-best technology choice. We will see below that the equilibrium
outcome is quite different.
3.2 EQUILIBRIUM ABATEMENT
We now turn to the non-cooperative equilibrium in a world without adaptation.
The policy problem for country i is to set
to minimize its total domestic cost,
which is equal to the sum of domestic production cost and domestic damage:
(11)
where
denotes aggregate emissions from all countries other than
country i. The associated equilibrium abatement technology choice is described in
Proposition 2.10
PROPOSITION 2. If adaptation is not available and Assumption 1 holds,11 then
(a) the equilibrium technology for country i is
(12)
,
(b) equilibrium emissions for country i are
(13)
, and
(c) equilibrium global emissions are
(14)
10
This equilibrium is characterized by dominant strategies (because damage is linear in global
emissions). This will no longer be true once we introduce adaptation.
11
If Assumption 1 does not hold (that is, if the first-best solution is not interior) then the
equilibrium abatement choice may also not be interior for some high income countries. In that
case, the first-best action and equilibrium action may coincide for some high income countries.
9
This equilibrium has a number of noteworthy properties. First, the
equilibrium technology for any country is increasing in its domestic income. A
high-income country has more incentive than a low-income country to reduce
emissions because damage from emissions is proportional to income. This
relationship between technology and individual country income does not arise in
the first-best solution because in the planning problem it is global damage – and
its relationship to global income – that matters. This difference in accounting for
transboundary damage also means that in equilibrium each country chooses a
technology that is more polluting than first-best.
Second, in contrast to the first-best solution, the equilibrium exhibits a
cross-section EKC: an inverted U-shaped relationship between income and
emissions across countries. This EKC reflects the same competing scale and
technique effects described earlier, but here these effects act at the level of an
individual country. Each country is concerned only with domestic damage, and
this depends only on its domestic income – not global income – for any given
level of global emissions. Thus, higher-income countries adopt cleaner
technologies, and this technique effect dominates the scale effect for countries
with sufficiently high incomes, causing a turning point in the cross-section EKC.
All countries nonetheless produce excessive emissions relative to first-best.
Third, there are two important effects on aggregate emissions in
equilibrium, relative to the first-best solution. The first of these is the usual
transboundary externality distortion. This is evident from comparing the
bracketed term in (14) with equation (10), which describes first-best aggregate
emissions. The two expressions differ only by the presence of n in the
denominator of the former; if all countries are identical then a larger number of
countries means a larger externality distortion. This is the usual “public bad”
distortion.
The second effect on aggregate emissions relates to the fact that countries
are not identical. The term outside the brackets in (14) tells us that equilibrium
aggregate emissions are declining in
, the variance across national incomes.
This result arises because a high-income country internalizes a greater fraction of
global damage than would a collection of low-income countries with the same
aggregate income. For example, consider a setting where a single large economy
accounts for 99% of global income. That country incurs 99% of the global
damage associated with its own emissions, because damage is proportional to
income, and so it internalizes most of that global damage when choosing its
technology. In contrast, in a collection of small economies, each country
internalizes only a small fraction of the global damage caused by its emissions,
and this larger collective externality translates into higher global emissions. Note
that variance across countries plays no role in the first-best solution because
10
global damage is fully internalized – by construction – for all countries, regardless
of their size.
It is important to note that the “variance effect” in (14) cannot be so large
as to outweigh the “public bad” effect, and thereby lead to equilibrium aggregate
emissions that are lower than first-best. This is intuitively clear, but it is not
obvious from a cursory comparison of (14) with (10). The key to this comparison
is that
has an upper bound. In particular, for any distribution with strictly
positive support and mean , the upper bound on its variance is
. It is
straightforward to show that equilibrium aggregate emissions must therefore
exceed first-best emissions for any
.
This relationship between the mean and variance of a distribution with
strictly positive support will be important in understanding some other results that
follow, and so it is worth highlighting the result at this point for future reference.
It will be most convenient to state the result in its converse form, as follows.
LEMMA 1. If
then
.
Having characterized both the first-best solution and the non-cooperative
equilibrium, we can now turn to the question of Pareto dominance.
3.3 PARETO DOMINANCE
A key corollary of Propositions 1 and 2 is that the gap between first-best and
equilibrium technology is declining in income; that is, lower-income countries are
further from first-best than are higher-income countries. This result is central to
the political difficulty of negotiating an international climate change treaty. For
example, the distributional consequences of requiring a larger percentage
emissions reduction – relative to the equilibrium baseline – from smaller
economies than from larger ones are clearly problematic. More generally,
heterogeneity across countries means that the first-best solution may not Paretodominate the equilibrium, in which case a cooperative agreement to achieve firstbest must be supported with compensating transfers. This necessarily makes a
cooperative agreement more difficult to implement. Proposition 3 describes a
necessary and sufficient condition for Pareto dominance in this economy.
PROPOSITION 3. If adaptation is not available and Assumption 1 holds, then
the first-best solution Pareto-dominates the non-cooperative equilibrium if and
only if
11
(15)
The intuition behind this result is as follows. Consider the net gain from
the first-best solution over the equilibrium outcome for country i. This net gain is
given by
(16)
The first term inside the square brackets measures the benefit per unit of output
from lower global emissions; this is the same for all countries. The second term
inside the square brackets measures the increase in production cost, per unit of
output, associated with a switch to the first-best technology. This term is not the
same for all countries, because the gap between the first-best and equilibrium
technologies is greater for lower-income countries (recall that
is increasing in
income but
is not). This means that
is increasing in income.
Now consider the link to income variance. Recall that a higher variance
means a smaller gap between equilibrium global emissions and first-best global
emissions, because a higher-income country internalizes a greater share of global
damage than does a collection of lower-income countries. Thus, the benefit term
in (16), given by
, is declining in income variance. If income variance
is high enough then the net gain to the lowest-income country will be negative,
and the first-best solution will not Pareto-dominate the equilibrium.12
In the following sections we investigate how the potential for Paretodominance is affected by the availability of adaptation.
4. A WORLD WITH ADAPTATION: FIRST-BEST POLICY
The first-best solution when adaptation is available solves a planning problem in
which total global cost – equal to the sum of production cost, adaptation cost, and
damage for each country, aggregated across countries – is minimized via the
choice of production technologies and adaptation actions:
12
Conversely, note from (15) that Pareto dominance is more likely if there are many countries (n
large), because the free-rider problem is worse when there are many countries, or if the global
economy is large ( large), because the positive externality associated with abatement is larger in
that case. Both of these forces make the equilibrium outcome less attractive for all countries
relative to the first-best solution.
12
(17)
The solution to this problem is described in Proposition 4.
PROPOSITION 4. Let
adaptation is available.
(a) If
and
denote the first-best policy for country i when
, then
(18)
, and
(19)
(b) If
and
, then
and
(c) If
and
; and
, then
and
This first-best solution is summarized in Figure 1. The figure partitions the
parameter space into critical regions corresponding to the possible first-best
solutions.13 In regions A1 and A2 (both shaded), the first-best solution is weakly
interior –
and
– and described by Proposition 4(a). In
regions B and C (both unshaded), the solution involves a corner, as described by
parts (b) and (c) of Proposition 4 respectively. Note that in all cases, the solution
is identical across countries.
13
Note that in Figure 1, the domain of Y has a strictly positive lower bound, by Lemma 1.
13
Figure 1: First-Best Policy
The key difference between regions A1 and C on one hand, and regions
A2 and B on the other, is whether the marginal cost of adaptation is less than or
greater than the MCCTA at equal values of the two actions (that is,
or
, respectively). Not surprisingly, the characteristics of the first-best solution
are very different in these two areas of the parameter space. Consider each case in
turn.
(i)
: Regions A1 and C
In these regions, the marginal cost of adaptation is lower than the MCCTA at
equal values of the two actions (
). The first-best policy mix therefore places
more emphasis on adaptation than on abatement via cleaner technology adoption.
Figure 2 illustrates the profiles of
and
against in region A1, for a given
value of
. These profiles depict first-best technology and adaptation choices
(for a country of any size) as a function of global income, . The most notable
feature of this first-best policy is that technology is not monotonic in . The firstbest technology is initially increasing in Y but eventually begins to fall for
sufficiently large, and drops to zero at the boundary with region C. Conversely,
14
adaptation rises monotonically with
until the boundary with region C is
reached, at which point the global economy is fully defended against climate
change.
Figure 2: First-Best Policy in Region A1
These properties of the first-best solution reflect the low cost of adaptation
in this region of the parameter space. A higher global income means there is more
global economic activity to protect from climate change, and this makes
adaptation more valuable. It also initially makes cleaner technology adoption
more valuable – since damage is proportional to emissions – and so both actions
initially rise as
rises. However, as adaptation becomes increasingly complete,
the benefits of fighting climate change via abatement eventually begin to fall, and
so the optimal technology begins to decline as well. At the boundary of regions
A1 and C – and throughout the entire region C – adaptation is complete (
),
and hence there is no point at all to abatement; thus,
in region C.
15
(ii)
: Regions A2 and B
In these regions, the marginal cost of adaptation is higher than MCCTA at equal
values of the two actions (since
); the optimal policy mix therefore favors
cleaner technology adoption over adaptation. Figure 3 illustrates the profiles of
and
against in region A2, for a given value of
. These profiles are
the opposite of those in region A1: the technology becomes monotonically cleaner
as Y rises, while adaptation initially rises with before eventually falling to zero
(at the boundary with region B).
Figure 3: First-Best Policy in Region A2
The intuition behind this policy is simply the reverse of that underlying the
policy profile in region A1. A larger (and hence more valuable) global economy
warrants greater protection from climate change via adaptation, and greater effort
to reduce emissions, which are proportional to . Thus, both actions initially rise
with . However, since adaptation is more costly than cleaner technology
adoption (
), the policy mix favors abatement. As technology becomes
cleaner – and emissions fall – the value of adaptation eventually declines, and
16
optimal adaptation declines with it. As the boundary with region B is reached,
abatement becomes complete (
) and adaptation has no value at all; thus,
in region B.
(iii) The
Boundary
Cleaner technology adoption and adaptation are equally costly in the knife-edge
case where
. Thus,
at any
when
. Moreover,
and
are both continuous in
for any
. However, the optimal policy is
discontinuous in at
for any
because the global cost function is
not convex in this range. Thus, when
, the optimal policy jumps
discontinuously at
from one corner solution to the other. At
, both
corner solutions yield the same total social cost, and a planner would be
indifferent between them.
The corner solutions identified in Proposition 4 are unlikely scenarios in
reality. Accordingly, we henceforth restrict attention to that part of the parameter
space in which both
and
are weakly interior (regions A1 and A2 in Figure
1) by invoking the following assumption.
ASSUMPTION 2.
and
Note that Assumption 2 subsumes Assumption 1 from Section 3. We now turn to
the non-cooperative equilibrium.
5. EQUILIBRIUM ADAPTATION AND ABATEMENT
The policy problem for country i is to set
and
to minimize its total domestic
cost, as measured by the sum of domestic production cost, domestic adaptation
cost, and domestic damage:
(20)
where
denotes aggregate emissions from all countries other than
country i. The solution to this problem yields best response functions for the
17
technology choice and adaptation.14 The parameter restrictions in Assumption 2
guarantee that these best response functions solve for an interior and stable
equilibrium. The key properties of that equilibrium are described in Propositions 5
and 6.
PROPOSITION 5. If Assumption 2 holds then the equilibrium technology and
adaptation choices for country i are
(21)
(22)
respectively, and
(23)
is the associated level of emissions for that country.
This equilibrium has a number key properties. First, technologies are less
clean for all countries relative to a setting without adaptation, and the difference is
increasing in domestic income.15 Thus, the cross-section EKC between income
and emissions is shifted up and skewed to the right by the availability of
adaptation. This impact of adaptation on the EKC is illustrated in Figure 4. The
reason for this impact is straightforward. Adaptation reduces the damage from
emissions, and so reduces the marginal benefit of cleaner technology adoption;
that is, adaptation and cleaner technology are substitutes in the protection of
economic activity. This reduction in the marginal benefit of cleaner technology is
larger for higher-income economies because damage is proportional to economic
activity.
14
The equilibrium is no longer characterized by dominant strategies when adaptation is available.
The best response functions for
and a are decreasing and increasing, respectively, in
. See
(A16) and (A17) in the Appendix.
15
Note that
is increasing in (by Assumption 1 and Lemma 1), and taking the limit of
as
yields
.
18
Figure 4: The Impact of Adaptation on the Equilibrium EKC
Second, equilibrium adaptation is not a function of income. This reflects
the fact that adaptation cost is proportional to income, and so too is the benefit
from adaptation. In contrast, technology adoption cost is proportional to income,
but the benefit from technology adoption is not, because emissions are a public
bad. For example, a very small economy could reduce its own emissions by 99%
and see no significant reduction in global emissions, or in the damage it suffers
from those emissions, because its contribution to global emissions is so small in
the first place. This asymmetry between adaptation and abatement means that the
relative importance of adaptation in the equilibrium policy mix – as measured by
– is decreasing in income. That is, low-income countries favor adaptation
over cleaner technology in their policy mix relative to the policy mix chosen by
higher-income countries. In short, a small country has little power to reduce
global emissions, so its best policy is to defend itself against the damaging impact
of those emissions.
Third, higher income variance leads to cleaner technologies, and less
adaptation, for all countries:
is increasing in
;
is decreasing in
. The
reason is as follows. A high-income country internalizes a greater share of global
19
damage than does a collection of lower-income countries with the same total
income. Thus, higher income variance leads to lower global emissions. This in
turn causes all countries to shift away from adaptation and towards cleaner
technology adoption in their policy mix. (Note that this increased emphasis on
cleaner technology is smallest for low-income countries, so the cross-section EKC
is shifted down and skewed to the left by an increase in income variance).
We next examine how the cost of adaptation affects overall domestic costs
in equilibrium. This is of central importance for the issue of Pareto-dominance.
PROPOSITION 6. Suppose Assumption 2 holds, and let
equilibrium total domestic cost for country i.
(a) If
, where
(24)
then
(b) If
(i) for
,
is increasing in
.
then
, where
(25)
,
is increasing in
(ii) for
denote
,
; and
is increasing in
(26)
and decreasing in
for
, where
,
for
.
Proposition 6 states that a lower marginal cost of adaptation (a smaller )
benefits all countries in equilibrium if income variance is relatively small.
However, if income variance is large, a lower marginal cost of adaptation benefits
relatively high-income countries but makes relatively low-income countries worse
off unless is small. The high-variance scenario – case (b) in Proposition 6 – is
20
illustrated in Figure 5. The figure depicts equilibrium total domestic cost, as a
share of domestic income, for three different countries, where
.
16
(Depicting cost relative to income normalizes the vertical scale). Country 1 has
income
; a lower marginal cost of adaptation benefits this country
unambiguously. In contrast, countries 2 and 3 have incomes below ; these
countries benefit from a lower marginal cost of adaptation only if that marginal
cost is small. The availability of adaptation, but at high cost, makes these
countries worse off than when no adaptation is available at all (where
).
Figure 5: Lower Adaptation Cost: Low-Income Countries Lose
The intuition behind this result is as follows. A lower marginal cost of
adaptation causes all countries to substitute into adaptation and away from cleaner
technology. This exposes all countries to higher global emissions. The impact of
higher global emissions is the same for all countries in terms of damage per unit
16
Note also that cost as a share of domestic income is the same for all countries at
There is a corner solution, with
, at this value of .
.
21
of output. However, the substitution away from cleaner technology yields a larger
reduction in production costs, on a per unit basis, for high-income countries than
for lower-income countries, because high-income countries are positioned higher
up the MCCTA curve in equilibrium. High-income countries therefore gain more
from any given move away from cleaner technology than do lower-income
countries. Thus, the equilibrium substitution into adaptation, which high-income
countries make in response to lower adaptation costs, can make lower-income
countries worse off. This adverse impact on lower-income countries is rooted in
the fact that abatement is a public good, while adaptation is a private good.
We now turn to a comparison of the equilibrium with the first-best
solution. The key elements of this comparison are described in Proposition 7.
PROPOSITION 7. Suppose Assumption 2 holds. Then relative to the first-best
solution, the equilibrium involves
(a) dirtier abatement technology for all countries:
;
(b) more adaptation by all countries:
;
(c) higher emissions from all countries:
(d) higher global emissions:
; and
.
These results reflect the same transboundary externality that underlies
Proposition 3, which describes a world without adaptation. Technologies are too
dirty (and emissions are too high) relative to the first-best solution because much
of the benefit from abatement is external to the country taking the action. In
contrast, the benefits of adaptation are fully internalized because adaptation is a
private good. Thus, the equilibrium policy is skewed towards too much adaptation
and too little abatement.
The magnitude of this policy distortion depends critically on the size of .
Recall Figure 1. It characterizes the first-best solution in terms of
and .
Figure 6 reproduces Figure 1 – focusing on regions A1 and A2 from that figure,
where the first-best solution is interior – and partitions the parameter space
according to the characteristics of the non-cooperative equilibrium. Region A1 in
Figure 6 corresponds exactly to that same region in Figure 1. Regions A2-1 and
A2-2 split region A2 from Figure 1 into two distinct sub-regions, divided by a
threshold value of
labeled
in Figure 6. The equilibrium is interior in all of
these regions but its characteristics – especially in relation to the first-best
solution – differ markedly across them. Consider each region in turn.
22
Figure 6: Equilibrium Policy
(i) Region A1
Recall from Figure 2 that the first-best solution in this region involves a relative
emphasis on adaptation and an inverted U-shaped relationship between abatement
action and global income. The equilibrium policies in this region follow a similar
pattern, but with too much adaptation and too little abatement relative to the firstbest solution. Figure 7 plots the equilibrium policies for the average-income
country against global income. These are labeled
and
for technology and
adaptation respectively. The first-best policies, labeled
and
, are also
depicted for comparison purposes.
It is important to be clear about what Figure 7 tells us. It illustrates how a
country with domestic income equal to
(the global average income) would
behave as Y rises. At any given Y, a country with income less than
would
choose a dirtier technology than , while a country with income greater than
would choose a cleaner technology. In contrast, equilibrium adaptation is
the same across countries in any given global economy, so in Figure 7 applies
to all countries. Similarly,
and
in Figure 7 apply to all countries.
23
Figure 7: Equilibrium Policy in Region A1
(ii) Region A2-1
The first-best solution in all of region A2 involves a relative emphasis on
abatement (recall Figure 3). In stark contrast, the equilibrium policies in region
A2-1 follow the same pattern as those in region A1: a relative emphasis on
adaptation. Figure 8 depicts the technology and adaptation policies for the
average-income country, for parameter values corresponding to region A2. The
first-best policies are plotted for comparison purposes.
Note that the equilibrium polices in this region have a completely different
profile from the first-best policies. The distortion associated with the emissions
externality reverses the relative importance of abatement and adaptation as global
income grows. In this setting, first-best policy for a very wealthy global economy
calls for a near-exclusive emphasis on abatement. In contrast, equilibrium policy
makes adaptation the priority.
24
Figure 8: Equilibrium Policy in Region A2-1
(iii) Region A2-2
In this region the equilibrium mix of abatement and adaptation more closely
resembles the first-best solution (see Figure 9). While technologies are too dirty
and there is too much adaptation, the relative importance of the two actions is
more consistent with the first-best policy.
Consider now the role of income variance in the patterns evident in
Figures 7 – 9. The critical threshold that separates between regions A2-1 and A22 in Figure 6 is
(27)
Note that a higher value of
expands region A2-1, in which the relative
importance of climate policy actions is reversed vis-à-vis the first-best solution.
Consider the determinants of . First,
is increasing in n. This simply reflects
25
the fact that free-riding gets worse as the number of participants in the public
good provision problem grows, and hence there is greater emphasis on adaptation
in equilibrium. Second, and more interesting, is the role of
, the variance
across incomes. Recall that a high-income country, whose emissions account for
a relatively large share of global emissions, internalizes more of the damage from
its own emissions than does a small country. A higher variance – a meanpreserving spread of the income distribution – pushes more countries into the
upper income tail of the distribution, where more internalization takes place. It
also pushes more countries into the lower tail, but these countries are small
contributors to global emissions anyway. Thus, the region in which the externality
is most severe – region A2-1 – shrinks as variance rises.
Figure 9: Equilibrium Policy in Region A2-2
26
5.3 PARETO DOMINANCE
We now turn to the question of Pareto dominance, and how the availability of
adaptation might affect prospects for reaching a first-best cooperative agreement.
PROPOSITION 8. Suppose Assumption 2 holds. When adaptation is available,
the first-best solution Pareto-dominates the non-cooperative equilibrium if and
only if
(a)
(b)
; or
and
Condition (a) in Proposition 8 is the same as that in Proposition 3 (in a
world without adaptation): if income variance is sufficiently small then the firstbest solution Pareto-dominates the non-cooperative equilibrium. However, in a
world with adaptation, Pareto dominance obtains even with higher income
variance, if the cost of that adaptation is not too high. This result is illustrated in
Figure 10. The figure depicts the difference between equilibrium domestic total
cost and first-best domestic total cost, denoted
, as a fraction of
domestic income, for two different values of marginal adaptation cost, where
. Income variance is held constant at a value that does not satisfy
condition (a) in Proposition 8. At
, the lowest-income countries are worseoff under the first-best solution, and the net loss – as a fraction of income – is
decreasing in income. Thus, Pareto dominance does not obtain. In contrast, at
, all countries are better-off under the first-best solution (though the gains
still favor the highest-income countries).
In some respects, Proposition 8 seems counter-intuitive. In particular, one
might expect a priori that the availability of a private substitute for a public good
would reduce the benefits of cooperating over that public good. This is true if
countries are identical. In that special case, the difference between domestic cost
under the equilibrium and domestic cost under the first-best solution declines
monotonically as the marginal cost of adaptation falls. Nonetheless, when
countries are homogeneous, the first-best solution always Pareto-dominates the
equilibrium, so there is always some incentive to cooperate towards the first-best
solution even if the benefits are small. A world with heterogeneous countries is
quite different. The availability of adaptation in that setting can hurt lower-income
27
countries in equilibrium (unless
is small), but benefits those countries in the
first-best solution. As a consequence, low-income countries have more to gain
from cooperation when adaptation is available than when adaptation is not
available.
Figure 10: Marginal Adaptation Cost and the Gains from First-Best Cooperation
6. CONCLUSION
Our central result is that the availability of adaptation weakens the condition
under which the first-best solution Pareto-dominates the non-cooperative
equilibrium. However, this is a good news-bad news story. On one hand, the
weaker Pareto dominance condition means that a first-best cooperative agreement
may be easier to reach. On the other hand, the mechanism through which
adaptation has this effect on the prospects for cooperation is not good news for
low-income countries. These countries may be made worse off by the availability
of adaptation in equilibrium because global emissions are higher when adaptation
provides a viable alternative to abatement.
28
We believe these results, and the modeling behind them, effectively
capture some of the concerns that have been raised in the policy community
regarding adaptation and its potential to further disadvantage poor countries
relative to rich ones [UNDP (2007)]. However, our modeling framework by no
means captures all of those concerns. First among the limitations of our model is a
lack of distinction between income and per-capita income. There are two
relatively straightforward ways in which this distinction could be introduced into
our model. One approach would be to allow the cost of adaptation to be
increasing in the number of people in the economy, in addition to the size of the
economy itself. This is most obviously true with respect to the relocation of
people as a defensive measure against climate change.
A second approach would be to introduce a preference-based link between
damage and per-capita income, reflecting the notion that environmental quality is
a normal good. As noted earlier, this linkage motivates the technique effect in
Copeland and Taylor (1994). We do not see this as an alternative to the public
factor model linking income and damage, but as complementary to it. Moreover,
while some adverse effects of climate change clearly fit the normal good model –
shortened ski seasons, for example – there are others that perhaps do not, such as
drought-induced damage to basic cereal crops.
A more satisfactory approach to this issue might be to formalize a
relationship between per-capita income and the composition of the economy. In
particular, countries with low per-capita incomes typically depend more heavily
on agriculture than do richer countries, and this sector is more susceptible to
damage from climate change than is manufacturing or services. Incorporating this
source of variation across countries requires a richer model than the one we have
examined here but doing so would clearly add to its scope.
Among the other limitations of our model, the one we see as perhaps most
important is the lack of dynamics. The stock pollutant nature of the climate
change problem, coupled with inertial lags in the climate system, mean that
damage today is a function of past emissions. This means that abatement and
adaptation are not substitutes in terms of avoiding immediate damage, and the
optimal mix of these two actions therefore depends on the discount rate, and on
the degree to which costs imposed on future generations can be internalized.
Adding these dynamic elements to our model is a task for future work.
29
APPENDIX
PROOF OF PROPOSITION 1
We prove a generalized version of the result reported in the text. Let production
cost for country i be given by
, where
, and let damage to country i be
given by
, where
and
. Then total global cost is
(A1)
The first-best solution solves
(A2)
Taking the ratio of any two of the conditions in (A2) yields
(A3)
It follows that
(i) for
(economies of scale in production):
(ii) for
if and only if
(diseconomies of scale in production):
;
if and only if
; and
(iii) for
Setting
:
.
in (A2) yields Proposition 1 in the text.
PROOF OF PROPOSITION 2
The first-order condition for (11) is
(A4)
This solves for the dominant strategy reported in (a). Part (b) then follows from
(3) in the text. Summing across i in (13) yields
30
(A5)
.
It is straightforward to show that second-order conditions are satisfied.
PROOF OF LEMMA 1
If
then
. Since
. But
. Thus,
, it follows that
or
.
PROOF OF PROPOSITION 3
The net gain for country i from the first-best solution over the equilibrium is
(A6)
Substituting for the first-best and equilibrium values from Propositions 1 and 2
respectively yields the result.
PROOF OF PROPOSITION 4
The first-order conditions for and
are, respectively,
(A7)
(A8)
From (A8) we obtain
(A9)
31
Thus, adaptation is identical across countries. Substituting a from (A9) for
in
(A7), and rearranging, we obtain
(A10)
,
where
. Thus, technologies are identical across countries. Substituting
from (A10) for
in (A9), and rearranging, we obtain
(A11)
Solving for
yields
(A12)
Finally, substituting
for a in (A10) then yields
(A13)
Solving for the conditions under which
and
yields the three
parts of Proposition 4. It is straightforward to show that second-order conditions
are satisfied.
PROOF OF PROPOSITION 5
The first-order conditions for and
are, respectively,
(A14)
(A15)
Solving (A14) and (A15) yields best-response functions for
respectively,
and
. These are,
32
(A16)
(A17)
From (A16) we can obtain the best-response function in terms of emissions:
(A18)
Rearranging, and noting that
, we can sum across i to obtain
(A19)
Making this substitution into (A18), and summing across i again, yields
equilibrium aggregate emissions
(A20)
Setting
in (A18), we can solve for the equilibrium
:
(A21)
Then from (A21) we can obtain
(A22)
Substituting
for
in (A14) then yields
(A23)
33
It is straightforward to show that when Assumption 2 is satisfied, these solutions
are weakly interior, second-order conditions hold, and the equilibrium is stable.
PROOF OF PROPOSITION 6
Equilibrium domestic cost for country is
(A24)
Making the substitutions for , and from (A22), (A23) and (A20)
respectively, and differentiating with respect to , yields the results in the
Proposition.
PROOF OF PROPOSITION 7
This is a straightforward corollary of Propositions 4 and 5.
PROOF OF PROPOSITION 8
The net gain for country i from the first-best solution over the equilibrium is
(A25)
Substituting for the first-best and equilibrium values yields the result.
REFERENCES
Agrawala, S., F. Bosello, C. Carraro and E. De Cian (2009), “Adaptation,
Mitigation and Innovation: A Comprehensive Approach to Climate Policy”,
Working Paper No.25, Department of Economics, Ca Foscari University of
Venice.
Barrett, S. (2008a), “The Incredible Economics of Geoengineering”, Environmental and Resource Economics, 39, 45–54.
Barrett, S. (2008b), “Dikes versus Windmills: Climate Treaties and Adaptation”,
unpublished manuscript, Johns Hopkins University, School of Advanced
International Studies.
Brock, W.A., and M.S. Taylor (2005), “Economic Growth and the Environment:
A Review of Theory and Empirics”, in Handbook of Economic Growth,
Volume 1, by P. Aghion and S. N. Durlauf (eds.), North-Holland, Amsterdam.
34
Carson, R.T. (2010), “The Environmental Kuznets Curve: Seeking Empirical
Regularity and Theoretical Structure”, Review of Environmental Economics
and Policy, 4, 3-23.
Copeland, B. and M.S. Taylor (1994), “North-South Trade and the Environment”,
Quarterly Journal of Economics, 109, 755-787.
Felgenhauer, T. and de Bruin, K. (2007), “The Optimal Paths of Climate Change
Mitigation and Adaptation under Certainty and Uncertainty”, International
Journal of Global Warming, 1, 66-88.
Grossman, G.M. and A.B. Krueger (1991), “Environmental Impacts of a North
American Free Trade Agreement”, NBER Working Paper 3914, National
Bureau of Economic Research, Cambridge, MA.
Hutchison, E. and P.W. Kennedy (2010), “Income Growth and Equilibrium
Dynamics for a Global Pollutant”, Department of Economics, University of
Victoria.
Ingham, A., J. Ma and A. Ulph (2007), “Climate Change, Mitigation and
Adaptation with Uncertainty and Learning”, Energy Policy, 35, 5354–5369.
Levhari, D. and L.J. Mirman (1980), “The Great Fish War: An Example Using a
Dynamic Cournot-Nash Solution”, The Bell Journal of Economics, 322-334.
McKibbin, W.J. and P.J. Wilcoxen (2004), “Climate Policy And Uncertainty:
The Roles Of Adaptation Versus Mitigation”, Brookings Discussion Papers In
International Economics, No. 161.
United Nations Development Programme (2007), Fighting Climate Change:
Human Solidarity in a Divided World, Human Development Report
2007/2008. New York: UNDP.
35