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Journal of Environmental Economics and Management ] (]]]]) ]]]–]]]
www.elsevier.com/locate/jeem
Carbon sequestration or abatement? The effect of rising
carbon prices on the optimal portfolio of greenhouse-gas
mitigation strategies
Klaas van ’t Velda,, Andrew Plantingab
a
Department of Economics and Finance, Department 3985, University of Wyoming, 1000 East University Avenue,
Laramie, WY 82071, USA
b
Department of Agricultural and Resource Economics, Oregon State University, USA
Received 29 August 2003; received in revised form 4 August 2004; accepted 13 September 2004
Abstract
Existing projections of the optimal share of carbon sequestration in an overall portfolio of greenhousegas mitigation strategies almost all assume the carbon price to be constant over time. This paper shows
analytically that if the price instead increases over time—consistent with projections from integrated
assessment models—it becomes optimal to delay certain sequestration projects, whereas the optimal timing
of energy-based abatement projects remains unchanged. As a result, the optimal share of sequestration
falls, and significantly so. Calibrating our analytical model, we find that a modest, 3% rate of price increase
results in about a 60% reduction in the optimal sequestration share relative to constant-price projections.
Numerical simulations based on predicted carbon-price paths from Nordhaus’ RICE01 model indicate
quantitatively similar reductions under an economically efficient scenario, and much larger reductions
(80–100% for up to 80 years) under a scenario that aims to limit the atmospheric CO2 concentration to
double its pre-industrial level.
r 2004 Elsevier Inc. All rights reserved.
Keywords: Climate change; Carbon sequestration; Forestry; Carbon price; Integrated assessment models
Corresponding author. Fax: +307 766 5090.
E-mail address: [email protected] (K. van ’t Veld).
0095-0696/$ - see front matter r 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jeem.2004.09.002
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1. Introduction
The United Nations Framework Convention on Climate Change (UNFCCC), adopted in 1992
by a majority of the world’s nations in response to global concern over human-induced climate
change, provides the institutional structure for ongoing negotiations over an international treaty
on climate change. A central, and often controversial, issue in these negotiations has been the use
of terrestrial carbon sinks (e.g., forests, agricultural soils) to reduce net emissions of carbon
dioxide (CO2 ). The 1997 Kyoto Protocol to the UNFCCC included provisions for industrialized
nations to manage carbon sinks in order to meet specified emissions-reduction targets. Subsequent
negotiations in The Hague in 2000 were to clarify the permit trading and carbon sink provisions of
the Protocol, but ended in failure in large part because of disagreement over how to assign credits
for CO2 reductions from carbon sinks. Agreement was only reached at subsequent meetings in
Bonn and Marrakech in 2001, though this time without the participation of the United States. The
Bush Administration, while withholding support of the Kyoto Protocol, has proposed a series of
climate change initiatives that include additional funding for land conservation to enhance carbon
storage.
Why have carbon sinks been at the center of international negotiations over a climate treaty?
While many factors have come into play, the high level of interest in carbon sinks stems, in part,
from a belief that they offer a relatively low-cost means of offsetting CO2 emissions when
compared to alternative mitigation and abatement strategies. Because CO2 diffuses rapidly in the
atmosphere, the climate-related benefits—but not necessarily other environmental benefits, as we
discuss below—of reducing CO2 emissions are independent of the strategy employed. Consequently, researchers have sought to identify the portfolio of strategies that achieve a given target
for net-emissions reductions at least cost (e.g., [1,18,31,32]).
As part of this effort, a large number of studies have estimated the costs of sequestering carbon
in forests for many regions of the world.1 The basic approach taken in most studies is to estimate
the opportunity cost of taking land out of its current use (e.g., agricultural production), compute
the additional carbon sequestered if the land is converted to forest, and then combine these figures
to yield estimates of the average or marginal cost of sequestering carbon.
In one such study, focusing on the U.S., Stavins [32] estimates a marginal cost schedule for
carbon sequestration and proceeds to compare it with other researchers’ estimates of the marginal
cost schedule for energy-based carbon abatement strategies. He finds that marginal sequestration
costs lie everywhere above marginal abatement costs, but up to a marginal cost level of around
US$50 per ton of carbon (tC), the difference is quite small. This leads him to conclude that
(subject to various caveats) ‘‘sequestration ought to be part of our overall portfolio of greenhouse
strategies in the short term, providing a significant fraction of overall carbon reductions, although
less than from conventional abatement activities’’ (emphasis in the original). A recent review
article by Kauppi et al. in IPCC [10], summarizing a large number of sequestration cost estimates
for various regions of the world, similarly concludes that ‘‘most studies, of all methodologies,
suggest that there are many opportunities for relatively low-cost carbon sequestration through
forestry. Estimates of the private costs of sequestration range from about US$0.10–US$100/tC,
which are modest compared with many of the energy alternatives’’.
1
See Richards and Stokes [29] for a recent survey.
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An assumption made in almost all of the sequestration cost studies is that the price of carbon is
constant over time.2 The main objective of this paper is to show that this assumption has
important implications for the costs of sequestering carbon in forests. In particular, when carbon
prices increase over time (below, we argue why this is a plausible case to consider), the optimal
supply curve of carbon sequestration shifts in relative to when carbon prices are assumed to
remain constant. At first blush, it would seem that rising prices could only enhance incentives for
the conversion of land to forest and, thus, cause the carbon-sequestration supply curve to shift
out. We show that precisely the opposite is true: any expected increase in carbon prices will make
it optimal to delay conversion, thereby reducing the supply of carbon sequestration at any given
price. This implies that, relative to the constant-price case, the share of carbon sequestration in an
optimal portfolio of greenhouse-gas mitigation strategies is smaller.3
This result has a biological origin: it hinges on the fact that rates of carbon sequestration decline
as a forest stand ages. The intuition for the result is easiest to grasp if one assumes that conversion
is always to permanent forest stands, rather than to periodically harvested ones. In this case, the
trees in a given stand are young, and therefore sequester carbon at high rates, only once over the
lifetime of a forestry project. When the price of carbon is constant, this fact is immaterial:
conversion should then always take place either immediately or not at all. When the price of
carbon increases over time, however, delaying conversion becomes optimal for some range of
forestry projects, because having young trees at a later date then implies that higher-valued units
of carbon will be sequestered at a greater rate. In contrast, there is no such reason to delay any
project that offsets emissions at a constant rate over its lifetime, as is typical of energy-based
abatement projects.
In Section 2 below, we provide background for this study by discussing carbon flows in
managed forest stands and by reviewing results from integrated assessment models of the
combined world economy and climate systems. Section 3 presents an analytical model of the
problem of when to optimally undertake carbon sequestration or abatement projects. There, we
establish the paper’s key result, namely that rising prices induce optimal delay of certain
sequestration projects, whereas the optimal timing of abatement projects remains unchanged.
Sections 4 and 5 calibrate the analytical model to explore the implications of this result for the
optimal aggregate supply of both types of projects. The analytical model abstracts from many
real-world complications, however, most notably by assuming that the rates at which
sequestration declines and carbon prices increase are constant over time, and that conversion is
always to permanent forest. In Section 6, we therefore develop a numerical simulation model that
accounts for these complications. The model combines price and discount-rate paths estimated for
various scenarios using Nordhaus’ [20] RICE01 integrated assessment model with data on actual
sequestration profiles for southern pine and spruce-fir forests compiled by Birdsey [2]. In addition,
the numerical model allows for harvesting of forests at endogenously determined intervals. A final
section presents discussion and conclusions.
2
Two important exceptions, discussed in detail below, are Sedjo et al. [30] and Sohngen and Mendelsohn [31].
Others have similarly argued that the use of carbon sequestration should be delayed, though for different reasons.
For example, Herzog et al. [9] discuss a case (see their footnote 2) in which the use of forest carbon sinks should be
delayed to offset future leakages of carbon from ocean or geologic reservoirs.
3
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2. Background
In this section, we discuss carbon flows associated with forest management and review results
on the time path of carbon prices from integrated assessment models. This provides background
material for the analysis in the remainder of the paper.
Trees and other plants in the forest convert CO2 to carbon through photosynthesis. Because
more carbon is typically stored in forests than in lands used for agriculture, the conversion of
agricultural land to forest achieves a net reduction in atmospheric CO2 concentrations.4 After the
establishment of a forest stand, the total volume of carbon in the forest increases, as carbon is
stored in the biomass of trees and understory plants, and is deposited in forest soils and floor litter
through leaf and root abscission. Following the pattern of tree growth, the rate of carbon storage
increases in young stands, but then declines as the stand ages. In an old stand, forest carbon is
roughly constant over time, as old trees die and provide opportunities for new trees to grow.
Cumulative carbon sequestration is shown in Fig. 1 (solid line) for a permanent southern pine
stand in the southeastern U.S. planted on former cropland.5 For this species, the rate of carbon
storage begins to decline at approximately age 20 and is close to zero by age 100.
The harvesting of trees for timber releases carbon back into the atmosphere. Within a decade
following a harvest, most of the carbon in the litter, understory vegetation, and nonmerchantable
portions of trees (e.g., branches) is converted back to CO2 :6 For 30-year-old southern pine, this
amounts to approximately 40% of the total carbon stored in the stand. Additional carbon is
released when the merchantable portion of trees is processed into primary products (e.g., lumber,
paper), when primary products are transformed into end-use products (e.g., housing), and when
end-use products are disposed of. For sawtimber harvests in the southeastern U.S. approximately
60% of the carbon in the merchantable portion of trees has been released by 50 years after the
harvest [25]. The rest of the carbon can remain in end-use products and landfills for a century or
more. Cumulative carbon storage is shown in Fig. 1 (dashed line) for a southern pine stand that is
periodically harvested.
We next turn to the results of integrated assessment models. Table 1 lists a number of wellknown studies that have used such models to compute economically efficient policies to mitigate
climate change. All these studies find that, rather than adopting the draconian measures that
would be needed to stabilize marginal damages from carbon emissions any time soon, marginal
damages should be permitted to increase over time, at rates that vary from 1.5% to 4% per year.
The implication of these findings is that, if one believes that future climate negotiations will seek
to implement an efficient policy, one should expect carbon prices to increase at roughly these
rates.
Economic efficiency is just one goal that future climate negotiations might pursue, however,
and not necessarily the most likely one. As discussed at some length by Nordhaus and Boyer [21],
4
Other strategies for augmenting carbon sinks include modifying the management of existing forests to increase rates
of tree growth, and switching to cropland tillage practices that conserve organic matter in the soil. The latter strategy is
briefly discussed in the concluding section.
5
The figure is similar to Fig. 1 in Newell and Stavins [17], but is based on different data and assumptions about
carbon releases following a harvest.
6
Carbon in the soils, however, is largely unaffected by timber harvesting and thus accumulates over successive
rotations.
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160
Cumulative Carbon
Sequestered (tons/acre)
120
80
40
Permanent
Periodically harvested
Exponential approximation
0
0
20
40
60
80
100
Years
Fig. 1. Cumulative carbon sequestered by a permanent stand of southern pine and by a periodically harvested stand.
Also shown is a stylized approximation, which has flow sequestration declining at a constant exponential rate.
Table 1
Estimated marginal damagesa from CO2 emissions under economically efficient climate-policy scenariosb
Study
1991–2000
2001–2010
2011–2020
2021–2030
Nordhaus [19]
Best guess
Expected value
Cline [4,5]
Peck and Teisberg [24]
Maddison [15]
5.3
12.0
5.8–124
10–12
5.9
6.8
18.0
7.6–154
12–14
8.1
8.6
26.5
9.8–186
14–18
11.1
10.0
n.a.
11.8–221
18–22
14.7
a
In 1990 US$/ton of carbon.
Source: Pearce et al. [23].
b
because of concerns about hard-to-predict, catastrophic impacts of climate change (e.g., shifts in
ocean currents or monsoons, collapsing Antarctic ice sheets resulting in sudden sea-level increases,
melting permafrost resulting in runaway global warming) the public debate on climate policy is
more commonly framed in terms of alternative goals, such as limiting the buildup of CO2 in the
atmosphere, or limiting the increase in global temperatures.7 According to several studies, costeffective implementation of these alternative goals still requires that carbon prices increase over
7
In fact, the UNFCCC specifically identifies ‘‘stabilization of GHG concentrations in the atmosphere at a level that
would prevent dangerous anthropogenic interference with the climate system’’ as its ultimate objective.
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time; in fact, they should increase even faster than if the goal is economic efficiency. Goulder and
Mathai [8] show, for example, that if the goal is to stabilize CO2 concentrations by some target
date in a way that optimally exploits the natural rate of removal of atmospheric CO2 ; then carbon
taxes or credit prices should increase until that date at a rate that slightly exceeds their assumed
5% discount rate. Using a different integrated assessment model, Nordhaus and Boyer [21] arrive
at essentially the same conclusion and find that very similar initial rates of price increase are
optimal if the goal is to keep global temperature increases below 2:5 C:
3. The model
In this section, we start our analysis of optimal carbon sequestration and abatement decisions
by developing an analytical model that incorporates the two key stylized facts discussed in the
previous section: (i) carbon sequestration by forests tends to decline over time, and (ii) carbon
prices are expected to increase over time.
Consider then a public or private decisionmaker choosing at what time T (if ever) to undertake
a project that either sequesters or abates carbon. Either type of project will involve a one-time
investment cost C i incurred immediately at T, and thereafter constant flow opportunity costs ai ;
where the subscript i denotes the type of project. If the project in question is a carbon
sequestration project—e.g., converting an acre of agricultural land to forest—then the up-front
cost C s (s for sequestration) may include the costs of planting trees, and the flow cost as ;
agricultural profits forgone after T. If the project is a carbon abatement project—e.g., converting
an electric power plant from coal to an alternative, less carbon-intensive energy source—then C a
(a for abatement) may include costs of construction, and aa ; possibly higher costs of fuel.
For analytical convenience, all sequestration projects are assumed to involve conversion to
permanent forest, so that trees planted at time T are never harvested. Consistent with stylized fact
(i) above, we assume that trees sequester carbon at a rate qs ðt TÞ that declines over time at a
constant exponential rate ls 40: Thus, qs ðt TÞ ¼ q0s els ðtTÞ ; where q0s is the rate at which newly
planted trees sequester carbon. (The curve labeled ‘‘exponential approximation’’ in Fig. 1 is the
cumulative of this equation for q0s ¼ 4:8 and ls ¼ 0:03:) The decisionmaker receives carbon credits
for each ton sequestered, as and when that sequestration occurs. That is, at time t, the
decisionmaker receives credits at exactly the rate qs ðt TÞ at which a ðt TÞ-year old forest
sequesters carbon.8All abatement projects are assumed to be infinitely-lived,9and to reduce carbon
8
Alternatively, we could assume that the forest is harvested and re-planted (i.e., ‘‘rotated’’) at fixed intervals S, at
which time the decisionmaker receives a stumpage value P for the harvested timber but has to re-incur the planting costs
C s : We could then treat the infinite sequence of rotations as a single ‘‘meta-project’’ with net investment cost
C^ s ¼ C s þ
1
X
i¼1
½eirS ðC s PÞ ¼
C s erS P
:
1 erS
The analysis of this section would go through unchanged, provided we retain the assumption that sequestration by this
sequence of rotations can be roughly approximated by an exponentially declining function. (In Fig. 1, an exponential
curve with parameters q0s ¼ 2:2 and ls ¼ 0:02 would reasonably approximate the cumulative-sequestration profile of a
periodically harvested stand.)
9
Alternatively, we could assume that abatement projects have a fixed, finite lifetime S, after which the decisionmaker
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emissions at a constant rate qa ðt TÞ ¼ q0a ; for which the decisionmaker again receives an
equivalent amount of carbon credits.
Consistent with stylized fact (ii) above, we assume that the market price of carbon credits
increases over time, at a constant exponential rate aX0: The carbon price at any given time t is
therefore pðtÞ ¼ pð0Þeat ; where pð0Þ is the price at time zero, when the carbon market is first
established.10 Lastly, until Section 5 below, we will assume that aor; where r is the constant rate
at which the decisionmaker discounts the future.
Given these assumptions, the decisionmaker’s optimization problem is to choose T so as to
maximize the net present value of the project. Expressed as a function of T and of the rate of price
increase a; this net present value can be written as
Z 1
Z 1
rt
at 0 li ðtTÞ
e pð0Þe qi e
dt ert ai dt erT C i ;
(1)
NPV ðT; aÞ ¼
T
T
where for an abatement project la ¼ 0; while for a sequestration project ls 40: Converting C i to
its annualized equivalent rC i ; and letting ci ai þ rC i denote total annualized costs, we can
simplify Eq. (1) to
pð0Þq0i
ci rT
aT
e e :
(2)
NPV ðT; aÞ ¼
r a þ li
r
To analyze the decision whether or not to undertake the project immediately, at time T ¼ 0; two
critical values of ci are important. One is the cost level up to which immediately undertaking the
project is profitable, in the sense that NPV ðT; aÞX0 at T ¼ 0: Letting ci ðaÞ denote this critical cost
level expressed as a function of a; we have, from solving NPV ð0; aÞ ¼ 0 for ci ;
r
ci ðaÞ ¼
pð0Þq0i :
(3)
r a þ li
The other critical cost level is that up to which immediately undertaking the project is
not just profitable but also optimal, in the sense that NPV ðT; aÞ is maximized with respect to T at
T ¼ 0: Letting ci ðaÞ denote this critical cost level, we have, from solving @NPV ð0; aÞ=@T ¼ 0 for
ci ;11
ra
ci ðaÞ ¼
pð0Þq0i :
(4)
r a þ li
(footnote continued)
has to renew the project by again investing C a : We could then treat the infinite sequence of such projects as a single
meta-project with investment cost
C~ a ¼
1
X
i¼0
eirS C a ¼
Ca
1 erS
and the analysis of this section would go through unchanged.
10
We are making the strong assumption that future generations also will commit to managing climate change through
support of a carbon market. See Lind and Schuler [14] for a discussion of the difficulties of developing institutions to
formulate long-term responses to climate change.
11
It can be shown that NPV ðT; aÞ is strictly quasi-concave in T. The condition @NPV ð0; aÞ=@Tp0 is therefore both
necessary and sufficient for T ¼ 0 to maximize NPV ðT; aÞ subject to TX0: Moreover, because @2 NPV ð0; aÞ=@T@ci 40;
the largest value of ci for which the condition @NPV ð0; aÞ=@Tp0 holds is that at which it holds with equality.
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Consider now first the case where the decisionmaker expects the price to remain constant
forever at its initial value pð0Þ: Evaluating (3) and (4) at a ¼ 0 yields
r
ci ð0Þ ¼ ci ð0Þ ¼
pð0Þq0i ;
(5)
r þ li
showing that the two critical cost levels coincide in this case. The implication is that when a ¼ 0;
any project with sufficiently low costs to make undertaking it profitable—in the sense of yielding
non-negative net present value—should optimally be undertaken immediately in order to
maximize that net present value.12
Consider next the case where the decisionmaker expects prices to increase over time at rate
a40: In this case, it is immediate from comparing (3) and (4) that the critical cost levels ci ðaÞ and
ci ðaÞ no longer coincide. Differentiating ci ðaÞ; we find that
@ci ðaÞ
r
¼
pð0Þq0i 40:
@a
ðr a þ li Þ2
This shows that, not surprisingly, an expectation of higher future carbon prices expands the range
of projects that can profitably be undertaken immediately. In contrast, differentiating ci ðaÞ yields
@ci ðaÞ
li
¼
pð0Þq0i p0;
@a
ðr a þ li Þ2
(6)
where the inequality is strict if li 40: This shows that the range of projects that will optimally be
undertaken immediately remains unchanged in the case of abatement projects, but actually
shrinks in the case of sequestration projects.
4. Aggregate supply when aor
Combined with the evidence presented in Section 2, indicating that carbon prices can indeed be
expected to increase over time under a range of climate-policy scenarios, the results of the
previous section have clear implications for the optimal share of carbon sequestration in an
overall portfolio of greenhouse-gas mitigation strategies. In particular, relative to existing
projections of this share—which typically assume that carbon prices will remain constant over
time—the optimal share will be reduced.
12
By assuming ci to be constant over time, we abstract from possibly optimal delay in undertaking positive-NPV
projects because of anticipated future reductions in fixed costs C i (see Kerr and Newell [11] for a discussion of this
possibility in the context of the U.S. lead phasedown). Any such delay effect is orthogonal to that induced by
anticipated future carbon-price increases, however: incorporating it would merely complicate our exposition, without
adding new insights. The same is true of any delay effect induced by uncertainty around the expected price path, as
analyzed by Dixit and Pindyck [6].
That said, if successive innovations reduce ci so substantially over time as to materially reduce the rate of carbonprice increases (see, e.g., Chakravorty et al. [3] for an argument that advances in photovoltaic technology will turn
global warming into a ‘‘short-run’’ problem), then the delay effect that we identify will be accordingly reduced
(provided of course that the innovations are anticipated). See Section 4.2 below for an analysis of how sensitive the
delay effect is to changes in a:
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In this section, we make a first attempt at quantifying this reduction, whereby we focus initially
on climate-policy scenarios for which aor; leaving the case where aXr (for which we have to
slightly modify the analytical model) to Section 5. Before we can do so, however, we must
introduce a number of additional assumptions.
First, we assume hereafter that both the interest rate r and the carbon-price pðtÞ at any given
time t are socially optimal, i.e., equal, respectively, to the shadow price of capital and to the
shadow cost of releasing an additional ton of carbon into the atmosphere. Given this assumption,
and provided decisionmakers correctly anticipate future carbon-price increases, the privately
optimal aggregate supply of sequestration and abatement will then also be socially optimal.
Second, we assume that both the interest rate r and the rate of carbon-price increases a can be
treated as fixed, even when we consider discrete—and possibly large—changes in aggregate
sequestration supply. This assumption may appear strong, particularly in light of our earlier
discussion of studies suggesting that carbon sequestration should play a significant role in any
socially optimal portfolio of greenhouse strategies. Simulations with Nordhaus and Boyer’s [21]
DICE99 model and Nordhaus’ [20] RICE01 model suggest, however, that the assumption is in
fact quite weak. In the DICE99 model, for example, even halving or doubling13 marginal
abatement costs turns out to have a negligible effect on either the shadow price of capital or the
rate at which the shadow cost of carbon increases over time.14
Third and last, in order to quantify what the optimal delay of individual forestry projects
implies for sequestration supply in aggregate, we need some estimate of how per-acre conversion
costs cs are distributed across all acres of non-forested land. Clearly, the optimal reduction in
initial acreage converted at a40 relative to that at a ¼ 0 (and thereby the reduction in the initial
sequestration supply) will depend on what fraction of those acres have conversion costs in the
range ðcs ðaÞ; cs ð0Þ: For a rough estimate of this fraction, we make use of a carbon sequestration
study by Newell and Stavins [17] based on data from 36 counties in the Mississippi Delta region of
the U.S. The sequestration supply curve estimated by Newell and Stavins (under an assumption of
constant carbon prices) suggests a very close to linear relationship between total acreage
converted and the conversion costs of the marginal acre converted, up to a point where marginal
conversion costs reach a level of about $80/acre.15 It is reasonable to assume that most of the
13
Halving is the more relevant experiment. This is because the DICE99 model’s mitigation cost function is based on
estimates by Nordhaus [18] that suggest a very limited optimal contribution of carbon sequestration to overall
greenhouse-gas mitigation, on the order of 10% or less. Raising this contribution to something closer to the levels
suggested by more recent studies such as Stavins [32] will therefore tend to reduce marginal mitigation costs overall.
14
Under the economically efficient price scenario (which is most relevant to the analysis of this section) even carbonprice levels are hardly affected: it is the optimal emissions-control rate that adjusts, roughly doubling for example when
marginal costs are halved. The reason why this doubling of control rates does not feed back to noticeably lower carbon
prices is that emissions-reduction levels under the optimal scenario are quite low to begin with, particularly in
comparison to the stock of atmospheric carbon. It is this stock that drives the damages from global warming, and
thereby the shadow cost of carbon. In contrast, under the concentration-limit scenario (which is most relevant to the
analysis of Section 5), it is the emissions-control rate that is hardly affected, and carbon-price levels that adjust. The
reason here is that climate policy under this scenario aims to keep the atmospheric CO2 concentration below some limit
at minimal cost. This goal essentially fixes a path of control rates, and the carbon price simply adjusts to equal the
marginal mitigation cost along that path. However, this adjustment takes place in a manner that leaves the rate of
carbon-price increase essentially the same, and it is this rate that drives our results.
15
Expressed, in 1990 US$, as are all prices hereafter in the paper.
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variation in opportunity costs of conversion across different acres comes from variation in the
agricultural potential of land, i.e., from variation in the component as of cs as þ rC s : Below, we
make the simplifying assumptions that (i) all the variation comes from as ; i.e., that the up-front
component C s is constant across all acres, and that (ii) agricultural profits as of the marginal
acre converted are exactly linear in total acreage converted. Finally, because our focus will be on
sequestration projects alone, we simplify notation by dropping subscripts on all variables
except q0s :
4.1. Quantifying optimal sequestration supply for the U.S.
Given constant up-front conversion costs, we have from substituting Eq. (4) into the identity
a c rC that the marginal acre optimally supplied immediately yields agricultural profits
(expressed as a function of a) equal to
ra
(7)
aðaÞ ¼
pð0Þq0s rC
raþl
up to the time of conversion. A linear relationship between marginal agricultural profits and total
acreage converted then implies that the aggregate acreage supply can be written as
ra
0
(8)
pð0Þqs rC
AðaÞ ¼ kaðaÞ ¼ k
raþl
for some constant k40: Newell and Stavins’ estimates, when extrapolated to the U.S. as a whole
in the manner adopted by Stavins [32],16 suggest that k is around 2.8 million.
With this estimate in hand, and using Eq. (8), we can quantify the importance of expectations
about the rate of future price increases a for optimal aggregate acreage supply, given any initial
carbon price pð0Þ: To do this, it will be useful to first consider the special case where the up-front
conversion cost C is essentially zero, as might be true, for example, when forests are not actively
planted but allowed to regenerate naturally, or when planting costs are offset by stumpage
revenues from future harvests (a plausible case that our analytical model cannot capture, but see
footnote 8 above). From (8), optimal acreage supply becomes in this case directly proportional to
the carbon price.
The dashed curve labeled Að0Þ in the left-hand panel of Fig. 2 represents the optimal acreage
supply curve when both a ¼ 0 and C ¼ 0; and when the remaining parameters of our model take
on what we shall treat as benchmark values, namely r ¼ 0:05; l ¼ 0:03; q0s ¼ 4:8; and k ¼ 2:8
million. Substituting these values into (8) yields that acreage supply (expressed in millions of
acres) at any given carbon price pð0Þ is
0 rqs
pð0Þ 8:4 pð0Þ:
(9)
Að0Þ ¼ k
rþl
Also shown in the figure, in the right-hand panel, is the corresponding sequestration supply curve,
labeled Qs ð0Þ: This curve represents the aggregate quantity of sequestration supplied at any given
carbon price pð0Þ; calculated on an annualized basis. That is, rather than considering the actual,
16
See Stavins’ footnote 15.
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50
Qa
50
40
A (0 )
30
20
20
15
10
15.6
10
0
0
54
84
131
Acres (million)
200
Year
40
A(α )
30
Carbon price ($/tC)
Carbon price ($/tC)
40
50
Qs ()
40
30
Qs(0 )
24.4
30
20
20
10
0
10
0
Year
50
11
37 82 103
252
400
615
Sequestration /abatement (MtC)
Fig. 2. Optimal acreage, sequestration, and abatement supply at a ¼ 0 and at a ¼ 0:03:
time-varying rate qs ðtÞ at which each of the Að0Þ acres supplied sequesters carbon, we treat each
acre as if it sequesters carbon at the constant rate rq0s =ðr þ lÞ: This rate is the annualized
equivalent of qs ðtÞ; in that over an infinite time horizon it yields the same present value of tons
sequestered:
Z 1
Z 1
rq0
rt
e pð0Þqs ðtÞ dt ¼
ert pð0Þ s dt:
rþl
0
0
The reason for introducing this annualization is that it facilitates comparison with the aggregate
quantity of carbon abatement supplied at the same price: in a present-value sense, each acre of
land converted can be treated as equivalent to rq0s =ðr þ lÞ 3 tons per year of carbon abatement
projects undertaken.
Multiplying Eq. (9) by rq0s =ðr þ lÞ; we find that aggregate sequestration supply (expressed in
megatons of carbon, or MtC) at any given carbon price pð0Þ equals
0 2
r
rqs
0
qs ¼ k
pð0Þ 25:2 pð0Þ:
(10)
Qs ð0Þ ¼ Að0Þ
rþl
rþl
To put this estimate in perspective, the U.S. annual emissions target for the years 2008–2012
originally agreed to by the Clinton administration under the Kyoto Protocol was about 1500
MtC—93% of total U.S. emissions in 1990—which is about 700 MtC below projections of
business-as-usual emissions in the year 2010. Moreover, Nordhaus’ [20] RICE01 model predicts
that, if the U.S. were to re-join the Kyoto Protocol at its original target, the initial carbon price
would be around $10 per ton. Fig. 2 shows that, at this price, 84 million acres of farmland would
optimally be converted, corresponding to about 14.4% of total U.S. farmland. The resulting
annualized sequestration supplied would be 252 MtC, or close to 40% of the required reduction.17
17
Note our implicit assumption that both agricultural profits in the Mississippi Delta and sequestration rates of
southern pine can be extrapolated to the U.S. as a whole. Stavins [32] makes the same assumption, but estimates the
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The central result of this paper, however, is that this supply is optimal only if the
carbon price remains at $10 forever, i.e., if a ¼ 0: If instead the price increases over time, t
hen, from Eq. (8), the optimal acreage supply will be lower at any given value of the initial
carbon price pð0Þ: As shown in the left-hand panel of Fig. 2(a), at our benchmark
value of a ¼ 0:03; for example, optimal acreage supply is reduced by 36%, to only 54 million
acres at $10.
Moreover, this is not the only effect of increasing prices. Compounding the effect of the
lower acreage supply is that the annualized, or ‘‘abatement-equivalent,’’ value of sequestration by
each acre that still does get supplied is lower as well. Intuitively, if the carbon price increases
over time rather than staying constant, sequestering carbon at declining rate qs ðtÞ becomes less
valuable (both privately and socially) relative to abating carbon at constant rate q0a ; since
relatively more of the sequestration will take place up front, when prices are still low.
Mathematically, since
Z 1
Z 1
ðr aÞq0s
rt
at
;
e pð0Þe qs ðtÞ dt ¼
ert pð0Þeat
raþl
0
0
annualized sequestration by each acre supplied drops to ðr aÞq0s =ðr a þ lÞ 1:92 tons=year: As
shown in the right-hand panel of Fig. 2, the combined effect of both the drop in acreage supply
and the drop in annualized sequestration by each acre is a reduction in optimal sequestration
supply by 59%, to only 103 MtC at $10.
Of course, if the carbon price indeed increases over time, it will be optimal to convert additional
acres in subsequent years. Eventually, therefore, optimal acreage supply at increasing prices may
well catch up with and ultimately exceed the optimal immediate supply under a constant-price
assumption. However, this does not imply that the delay effect identified in Section 3 is of only
transitory importance. The reason is that, although for expositional reasons the analysis of
Section 3 was presented from the vantage point of time 0, when the carbon market starts off at
price pð0Þ; the analysis is easily shown to go through unchanged for any subsequent time t,
conditional on the then-current price pðtÞ: This is true in particular of Eqs. (4) and (6), which
show, respectively, the critical cost level ci ðaÞ up to which it is optimal to supply projects at time 0
and the change in this cost level—negative for sequestration projects but zero for abatement
projects—when a increases. Both equations apply at any subsequent time t as well, but with pðtÞ
substituted for pð0Þ:
Graphically, this implies that when a increases, the acreage and sequestration supply curves
pivot up in their entirety, as illustrated by the curves labeled AðaÞ and Qs ðaÞ in Fig. 2. In contrast,
the abatement supply curve, illustrated by the curve labeled Qa ; stays put. The Qa curve represents
the optimal U.S. supply of carbon abatement as estimated by the RICE01 model. Note that, over
the price range shown in the figure, this abatement supply curve is close to linear, implying that
(footnote continued)
annualized contribution of carbon sequestration at a constant price of $10 to be only about a quarter as large as ours.
The reason is that his estimate of annualized per-acre sequestration is only half as large as ours (around 1.5 tons per
acre) in part because he assumes that forests will be periodically harvested. As a result, his estimate of acreage supply at
$10 is only half as large as well (around 42 million acres) and the resulting total supply of annualized sequestration is
only a quarter as large (around 63 MtC). Note from Eq. (10) that annualized per-acre sequestration enters quadratically
in aggregate sequestration supply.
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13
both optimal abatement and optimal sequestration increase over time roughly in proportion to the
carbon price. As a result, the optimal ratio of sequestration to abatement stays roughly constant
over time.
Fig. 2 shows that at the initial price of $10, optimal abatement supply is 37 MtC, about
3 times smaller than the optimal sequestration supply of 103 MtC, and almost 7 times
smaller than the constant-price projected sequestration supply of 252 MtC.18 Thirty years later, as
the carbon price has increased to pð30Þ ¼ 10e0:0330 $25; optimal sequestration supply has
increased 2.5-fold to 252 MtC, and thereby caught up with constant-price projected supply in year
0. At the same time, however, abatement supply has increased 2.5-fold as well, to 82 MtC. This is
therefore still about 3 times smaller than optimal sequestration supply (and also still almost 7
times smaller than projected sequestration supply if the price were assumed constant at $25
forever).
In sum, even though the optimal levels of sequestration supply may eventually catch up with
constant-price projections of immediate supply in year 0, the optimal share of sequestration
relative to that of abatement in the overall portfolio of greenhouse strategies may never catch
up.19
4.2. Effect of changes in a and l
To see how the reduction in the optimal share of sequestration depends on the key model
parameters a and l; let DQs =Qs ½Qs ð0Þ Qs ðaÞ=Qs ð0Þ denote the proportional reduction in this
contribution at any given price pðtÞ: Since abatement supply Qa conditional on pðtÞ is independent
of a; DQs =Qs also captures the proportional reduction in the optimal sequestration-to-abatement
ratio: DQs =Qs ¼ ½Qs ð0Þ=Qa Qs ðaÞ=Qa =½Qs ð0Þ=Qa : Using Eq. (8) and the fact that Qs ðaÞ ¼
AðaÞðr aÞq0s =ðr a þ lÞ; we have
ðr aÞq0s ðr aÞq0s
pðtÞ rC
raþl raþl
0
:
(11)
DQs =Qs ¼ 1 rq0s
rqs
pðtÞ rC
rþl rþl
18
As noted in the introduction, Stavins [32] estimates optimal sequestration supply at constant prices to be somewhat
smaller than optimal abatement supply, rather than 7 times larger. Two factors explain the large discrepancy with our
estimate. One is Stavins’ lower estimate of annualized per-acre sequestration, which, as explained in footnote 17 above,
reduces his estimate of aggregate sequestration by a factor of 4. The other is the RICE01 model’s higher estimate of
marginal U.S. abatement costs, which exceeds that of the various abatement-cost studies cited by Stavins by a factor of
2.
19
It is worth noting, as an aside, that a 30-year catch-up time for the aggregate supply level may itself constitute a
significant delay if policies to encourage carbon sequestration are seen as a way of buying time for the development of
cheaper carbon abatement technologies. See, for example, the March 2001 testimony at the Senate Committee on
Agriculture, Nutrition and Forestry’s Hearing on Biomass and Environmental Trading by agricultural economist Bruce
McCarl, arguing that carbon sequestration could provide an approximate 20-year bridge for the energy industry to
come to terms with its CO2 emissions and develop a less costly and longer-term system to reduce them. (‘‘Panel eyes
emissions trading plan for farmers, foresters’’, Environment and Energy Daily, April 2, 2001).
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100%
λ= ∞
75%
λ = 0 .03
∆Qs /Q s
59%
50%
λ = 0 .01
25%
0%
0.00
λ=0
0.01
0.02
0.03
0.04
0.05
(= r )
Fig. 3. Proportional reduction in sequestration supply as a function of a; for various values of l:
If up-front conversion costs C are zero, DQs =Qs becomes independent of pðtÞ and the initial
sequestration rate q0s ; reducing to
ra rþl 2
:
DQs =Qs ¼ 1 raþl r
(12)
It follows from (12) that the proportional reduction is zero when l ¼ 0; at all values of a; and that
it goes to 100% when a approaches r, regardless of the value of l:
Fig. 3 plots the right-hand side of (12) as a function of a for various values of l; assuming
r ¼ 0:05: The figure shows that the proportional reduction is increasing in l: Any increase in l
reduces the optimal sequestration supply level Qs conditional on any a; because the present value
of sequestration revenues is lower if the sequestration rate drops more sharply over time. Because
Qs ð0Þ falls proportionally by more than Qs ðaÞ; however, DQs =Qs increases in l:20
If up-front conversion costs are nonzero, DQs =Qs is no longer independent of the carbon price
pðtÞ and initial sequestration rate q0s ; these terms no longer cancel in Eq. (11). However, because
the right-hand side of (11) is increasing in C, the simpler expression (12) can still serve as a lower
bound on the actual proportional supply reduction.
20
This discussion assumes that q0s is independent of l: In this case, both Qs ð0Þ and Qs ðaÞ fall to zero in the limit as l
goes to infinity, and the ratio DQs =Qs is then no longer defined. More plausibly, however, q0s and l are related. If, for
example, high-l tree species tend to mature rapidly, they will tend to have high values of q0s as well. In the simple
(though unrealistic) case where all tree species eventually attain the same cumulative carbon sequestration level V q0s =l; so that we can substitute V l for q0s everywhere, Qs ð0Þ and Qs ðaÞ will both increase in l: That is, supply levels will
be higher for rapidly maturing (i.e., high-l and high-q0s ) species than for slowly maturing ones. Because the V l term will
cancel out in Eq. (11) when C ¼ 0; however, DQs =Qs as given by (12) still increases in l:
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5. Aggregate supply when initially aXr
Although the model developed above constrained a to lie strictly below r, our discussion in Section
2 suggested that a may reasonably be expected to equal or exceed r for many decades, namely if future
climate-change negotiations continue to focus on keeping atmospheric CO2 concentrations or global
temperature increases below limits considered ‘‘safe.’’ Under a concentration-limit scenario considered
by Goulder and Mathai [8], for example, the carbon price optimally increases at a rate 0.8% above the
discount rate for 200 years. Similarly, under concentration- and temperature-limit scenarios
considered by Nordhaus and Boyer [21], the carbon price increases at a rate that exceeds the
discount rate by about the same amount for just over 100 years, after which the rate of price increase
drops quite abruptly to a level well below the discount rate.
An analytically tractable approximation to these scenarios arises if we assume that the price
increases at rate aXr until some future time T and then abruptly stops rising, staying constant
thereafter at pðTÞ ¼ pð0ÞeaT : In this case, the net present value of undertaking an abatement or
sequestration project at any time TpT becomes
Z T
Z 1
rt
at 0 li ðtTÞ
NPV ðT; aÞ ¼
e pð0Þe qi e
dt þ
ert pð0ÞeaT q0i eli ðtTÞ dt
T
T
Z 1
ert ai dt erT C i
T
pð0Þq0
pð0Þq0i
ai
aT
ðraþli ÞðTTÞ
i aT ðrþli ÞðTTÞ
þ
¼
e
1e
e e
C i erT :
r a þ li
r þ li
r
Differentiating with respect to T yields
@NPV ðT; aÞ
ra
¼ pð0ÞeaT q0i
@T
r a þ li
r
ra
aT 0 ðraþli ÞðTTÞ
pð0Þe qi e
þ ai þ rC i erT :
r þ li r a þ li
ð13Þ
For abatement projects, with la ¼ 0; this derivative reduces to that of NPV ðT; aÞ as given by
Eq. (2) in Section 3. The analysis of the case where aor therefore goes through essentially
unchanged when initially aXr:
For sequestration projects, in contrast, with ls 40; there is an important difference with the case
where aor in that, if the target time T lies sufficiently far into the future, the right-hand side of
(13) may be positive at T ¼ 0 even when as þ rC s ¼ 0: If so, optimal sequestration supply will be
zero up to some critical time T ‘ ; i.e., the optimal sequestration supply curve will have a positive
price intercept at pðT ‘ Þ: Setting the right-hand side of (13) equal to zero and solving for T at
as þ rC s ¼ 0; we find this critical time to be
1
r a þ ls r
‘
T T
:
log 1 r a þ ls
r a r þ ls
At r ¼ 0:05; a ¼ 0:058; and T ¼ 100 (values roughly consistent with Nordhaus and Boyer’s [21]
concentration-limit scenario), and at our benchmark value of ls ¼ 0:03; T ‘ evaluates to 55. At
these parameter values, therefore, it would be optimal to delay conversion of any agricultural land
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to at least 55 years after the carbon market is first established, by which time the carbon price will
have increased at least e0:05855 24-fold from its initial value pð0Þ:
Setting the right-hand side of (13) equal to zero and solving for as shows that, once T exceeds
T ‘ ; the marginal acre supplied at price pðTÞ will yield agricultural profits
ra
r
ra
0
as ðaÞ ¼
pðTÞqs þ
(14)
pðTÞq0s eðraþls ÞðTTÞ rC:
r a þ ls
r þ ls r a þ ls
Using again our assumption that A ¼ kas ; it is then straightforward to derive expressions for
aggregate sequestration supply and for the proportional reduction in that supply after T ‘ relative
to projected supply at constant prices. Clearly, by our assumption that the price becomes constant
at T; this proportional reduction goes to zero in the limit as T approaches T:
6. Numerical model
In this section, we report simulation results from a numerical model that relaxes many of the
simplifying assumptions of the analytical model. First, rather than assuming that carbon prices
increase at a constant exponential rate and that the discount rate is constant, we use the RICE01
model to simulate optimal price and discount-rate paths under two climate-policy scenarios. The
‘‘optimal’’ scenario solves for the economically efficient carbon-price path from the year 2005
onwards, while the ‘‘2 CO2 ’’ scenario solves for the path that most cost-effectively prevents the
atmospheric concentration of CO2 from ever exceeding 560 parts per million, double its preindustrial level.
Second, rather than assuming that sequestration declines with forest age at a constant
exponential rate (which ignores in particular the typical initial increase in sequestration just after
planting), we use data by Birdsey [2] on the actual time profiles of sequestration in a southern pine
stand in the southeastern U.S. and a spruce-fir stand in the Lake States region. Southern pine
refers to a collection of commercially important pine species found primarily in the southern U.S.;
the Southeast region would likely play a central role in a national afforestation policy, given
climatic conditions suitable for fast-growing southern pine plantations and the prevalence of
wood-products manufacturing in the region.21 We also examine spruce-fir, the dominant
commercial timber type in the Lake States region, because recent empirical evidence (e.g.,
Plantinga et al. [27]) suggests that sequestration costs are relatively low in this region.
Third, rather than assuming that conversion is always to permanent forest, we allow
landowners to generate revenue from timber harvesting, at endogenously determined intervals. A
new assumption that then becomes relevant is that landowners internalize the full social cost of all
carbon releases resulting from timber harvests. More specifically, we assume that landowners have
to buy carbon credits at the prevailing carbon price to cover all resulting emissions, as and when
those emissions occur.22 This allows us to focus on first-best optimal sequestration paths.
21
The Conservation Reserve Program provides subsidies to farmers to convert cropland to grassland, trees, and other
permanent vegetative cover. Compared to other regions, the Southeast region has by far the highest percentage of land
enrolled in the tree planting category.
22
See Feng et al. [7] and Sohngen and Mendelsohn [31] for discussion of arguably more practical schemes that would
pay landowners an annual carbon rent for each ton that remains sequestered. Expected payments under these schemes
are exactly equivalent to those under our assumed scheme.
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To the landowner’s optimization problem given these modifications, we need some additional
Rt
rðxÞ dx
denote the discount factor at time t, given the now variable
notation. First, let dðtÞ e 0
discount rate function rðtÞ: Also, let T 0 now denote the time of initial conversion to forest, and
T 1 ; T 2 ; T 3 ; . . . the subsequent times at which the forest is rotated. Define Qs ðT i ; T iþ1 Þ R T iþ1
qðt T i Þ dt as the cumulative amount of carbon sequestered by a rotation planted at time T i
Ti
and harvested at time T iþ1 ; and Qm
s ðT i ; T iþ1 Þ as the carbon sequestered by that same rotation in
merchantable wood only. Finally, let g denote the quantity of carbon contained in a volumetric
unit of merchantable wood—the unit in which the stumpage price P is expressed—and let ‘m ðsÞ
denote the fraction of carbon originally sequestered in wood products that is released s years after
harvest. The landowner’s optimization problem is then
Z T iþ1
1 X
dðT i ÞC þ
dðtÞ½pðtÞqðt T i Þ a dt
max NPV ðT 0 ; T 1 ; . . .Þ ¼
Ti
Ti
i¼0
þ
dðT iþ1 ÞPgQm
s ðT i ; T iþ1 Þ
dðT iþ1 ÞpðT iþ1 Þ½Qs ðT i ; T iþ1 Þ Qm
s ðT i ; T iþ1 Þ
Z 1
m
dðtÞpðtÞQs ðT i ; T iþ1 Þ‘m ðt T iþ1 Þ dt :
T iþ1
The first two terms on the right-hand side are familiar from the analytical model: they represent,
respectively, the cost of planting (or, for later rotations, re-planting) forest and flow revenues from
sequestration net of forgone agricultural revenues over the course of each rotation. The last three
terms are new: they represent, respectively, the stumpage revenue received at the time of harvest
and the costs of carbon releases at and after the time of harvest.23
The four panels of Fig. 4 show the results of simulating this model for the two types of forest
considered and the two climate-policy scenarios from the RICE01 model. The interpretation of
the axes and the curves labeled Qa ; Qs ðaÞ; and Qs ð0Þ in the panels is identical to that in the righthand panel of Fig. 2, except that the initial year in which the carbon price first becomes positive is
taken to be the year 2005 rather than some arbitrary year 0.24 In general, the simulation results
confirm those of the analytical model. Optimal sequestration supply given the increasing carbonprice path under both climate-policy scenarios is below projected supply at constant prices, and
more so in the 2 CO2 scenario than in the optimal scenario.25
23
Further detail on the assumptions underlying the simulations and on the solution procedure used is provided in an
appendix available from the authors upon request.
24
In particular, the simulations extrapolate regional conditions to the U.S. as a whole in the same manner as that used
to generate Fig. 2. That is, the distribution of agricultural profits in the Mississippi Delta is extrapolated to all farmland
in the U.S., and growth conditions of, respectively, southern pine forest in the Southeast and spruce-fir forest in the
Lake States are similarly applied to the U.S. as a whole.
25
As a sensitivity check, we have run simulations (not reported here) in which we extrapolate our model of U.S.
sequestration supply in a somewhat crude fashion to the world as a whole, and then iteratively update the RICE01 model’s
price and discount-rate paths to account for feedback effects from sequestration supply to optimal abatement supply and
capital investment. Consistent with our experiments with the DICE99 model, discussed in Section 4, these feedback effects
turn out to be quite small; the major difference with the simulation results reported here is that the level (but importantly not
the rate of increase) of carbon prices is lower in the 2 CO2 scenario when feedback effects are allowed for.
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50
50
2065
2045
Q s ()
2035
Q s (0)
20
Year
Carbon price (1990$ / tC)
30
Qa
40
2055
2025
Carbon price (1990$ / tC)
Qa
40
2065
2055
Q s ( )
2045
Q s (0)
30
2035
20
2025
2015
2015
6 .9
0
2005
37
65
108
200
(a)
300
2005
9 .6
400
0
MtC
37 53
111
200
(b)
250
300
400
MtC
250
200
200
Q s ( )
Qa
2065
50
(c)
100
Qa
2065
50
Q s (0)
2045
Q s (0)
4.4
0 15 41
2085
164 .2
150
Year
Carbon price (1990$ / tC)
100
Qs ( )
Year
Carbon price (1990$ / tC)
2085
150
Year
18
2045
2025
2005
250
500
MtC
4.4
0 15 47
750
(d)
2025
2005
250
500
750
MtC
Fig. 4. Simulated optimal sequestration supply curves for two region/species combinations and two climate-policy
scenarios. (a) Optimal scenario, Southeast pine. (b) Optimal scenario, Lake States spruce-fir. (c) 2 CO2 scenario,
Southeast pine and (d) 2 CO2 scenario, Lake States spruce-fir.
Under the optimal RICE01 scenario, the carbon price starts out in the year 2005 at $9.60/ton
and thereafter increases at a rate that is always below the discount rate; the gap between the two
(analogous to r a in the analytical model) is about 1% initially, and widens to 2% by the year
2050. The discount rate itself falls over time, and in fact does so in the same manner in both
scenarios; starting at about 5%, it drops to around 2% in the year 2250.
As shown in panels (a) and (b), the optimal initial supply of abatement under this scenario is 37
MtC. For both southern pine and spruce-fir, the projected supply of sequestration assuming the
carbon price will stay at $9.60 forever is about three times larger: 108 MtC for southern pine, and
111 MtC for spruce-fir. The optimal supply given the actual, increasing price path is less than
twice that of abatement, however: only 65 MtC for southern pine and 53 MtC for spruce-fir.
Compared to the constant-price estimate of initial sequestration supply, this amounts to
respectively a 40% and 52% reduction. By the year 2025, when under this scenario the carbon
price has roughly doubled to $18.90, optimal sequestration supply has roughly doubled as well,
and thereby caught up with projected initial supply. However, optimal abatement supply has by
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19
then also doubled, implying that the optimal share of carbon sequestration relative to that of
abatement has not caught up at all. This share remains for many decades lower than the initial
constant-price estimate.
Consistent with the predictions from our analytical model of Section 5, the effect of increasing
prices is considerably larger under the 2 CO2 scenario. Under this scenario, the carbon price
starts out in the year 2005 at just $4.40/ton. As mentioned earlier, the price thereafter increases for
over a century at a rate 0.8% above the discount rate, but once the 2 CO2 concentration limit is
reached, around the year 2115, the rate of price increase drops quite abruptly.
As shown in panel (c), the optimal resulting sequestration supply for southern pine is zero
initially, but almost immediately becomes positive. Catch-up with projected constant-price
supply of 41 MtC does not occur until almost 40 years later, however, by which time optimal
abatement supply has increased 6-fold, from 15 to 92 MtC. The optimal supply share is therefore
still only one-half that of abatement at the time of catch-up in levels, and the share remains
roughly one-half long after. This amounts to a proportional reduction of 80% relative to the
constant-price estimate. As for spruce-fir, panel (d) shows that the optimal supply for this type of
forest remains zero for almost 80 years, and then very soon catches up with the projected
constant-price supply of 47 MtC. The optimal supply share stays far below that of carbon
abatement, however.
We comment briefly on several features of our simulation results. First, optimal rotation
lengths turn out to be highly sensitive to the carbon-price level when that price is assumed
constant over time. As in earlier studies by Plantinga and Birdsey [26] and Van Kooten et al. [33],
rotations become infinite at some price levels. More specifically, in the simulations for southern
pine under the optimal RICE01 scenario, the optimal rotation length is 30 years assuming a
constant year-2005 price of $9.60. At the year-2017 price of $14.80, however, the optimal rotation
length is infinite, both when that price is assumed constant and conditional on the actual,
increasing price path. As a result, the equilibrium outcomes of our simulation model from the year
2017 onwards closely match those of our calibrated analytical model from Section 4. Indeed, the
simulation model’s estimated sequestration supply reduction conditional on the year-2017 price is
54%, just slightly below the benchmark estimate of 59% from our analytical model.
Second, the marked difference under the 2 CO2 scenario in the vertical intercepts of the
sequestration supply curves for southern pine and spruce-fir is driven by the much greater
importance of soil sequestration for spruce-fir. As with sequestration by trees, the opportunity to
rapidly sequester carbon in forest soils is available only in the years immediately following
conversion to forest, after which the rate of sequestration falls. For trees, forest rotation can
partially renew this opportunity (though it also results in carbon releases), and this reduces the
incentive to delay initial conversion. Forest rotation has little effect on the declining rate of soil
sequestration, however. As a result, the incentive to delay conversion to forest types such as
spruce-fir, for which soil sequestration represents a greater proportion of total sequestration, is
relatively larger.
Finally, the discontinuous upward jumps in the optimal sequestration supply curves, most
notably under the 2 CO2 scenario, arise for reasons to do with optimal timing of rotations in
anticipation of a sudden future drop in the rate of increase of carbon prices, as occurs under the
2 CO2 scenario around the year 2115. Full explanations of all these results are beyond the scope
of this paper, but are available from the authors upon request.
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7. Conclusions
In this paper, we consider socially optimal carbon sequestration and abatement decisions under
different expectations about future carbon prices. It is shown that if carbon prices increase over
time—consistent with projections from integrated assessment models under various assumptions
about future climate-policy goals—it becomes optimal to delay certain carbon sequestration
projects, whereas the optimal timing of abatement projects remains unchanged. Equivalently, the
social opportunity cost of undertaking these carbon sequestration projects immediately—i.e., of
not delaying them—increases relative to that of carbon abatement projects.
Earlier estimates of the costs of carbon sequestration have almost all been based on an
assumption that prices are constant over time, in which case there is no reason for delay. The
central implication of our paper is that increasing carbon prices reduce, relative to constant-price
projections, the optimal share of carbon sequestration in an overall portfolio of greenhouse-gas
mitigation strategies, and more so, the more rapidly carbon prices are expected to increase. With
rising prices, the absolute amount of carbon sequestration may eventually catch up with and
ultimately exceed the amount under constant prices. However, because the incentive to delay
projects under rising prices is unique to carbon sequestration, the amount of carbon sequestration
at any given point in time relative to that of abatement remains smaller compared to the constantprice case.
This analytical result is of course relevant to climate policy only if it is quantitatively, by some
measure, significant. When we calibrate our analytical model, we find that a 3% rate of increase in
carbon prices results in about a 60% reduction in the optimal share of carbon sequestration
relative to constant-price projections. Simulations with our numerical model, based on predicted
carbon-price paths from the RICE01 model, indicate quantitatively similar reductions under an
economically efficient scenario, and much larger reductions (80–100% for up to 80 years) under a
scenario that aims to limit the atmospheric CO2 concentration to double its pre-industrial level.
Apart from papers already cited, the three papers in the existing literature closest to our own
are Feng et al. [7], Sedjo et al. [30], and Sohngen and Mendelsohn [31]. Feng et al.’s theoretical
study of the optimal timing of carbon sequestration concludes, in seeming contradiction to our
results, that carbon sinks should be utilized ‘‘as early as possible.’’ Closer examination reveals,
however, that Feng et al.’s conclusion applies only to acres with conversion costs c below a value
that is equivalent to cs ðaÞ in our model. In our model, too, such acres should be converted
immediately; the delay effect applies only to acres with costs in the range ðcs ðaÞ; cs ðaÞ: In fact,
because Feng et al. assume that all sequestration takes place instantaneously—i.e., that l is
effectively infinite—the delay effect implicit in their model is maximally large, as can be seen from
Fig. 3.26
Sohngen and Mendelsohn’s simulation study of optimal world-wide sequestration supply over
time is exceptional in that, rather than assuming constant carbon prices, it considers two different,
increasing carbon-price scenarios of the DICE99 model. Roughly consistent with our simulation
26
More precisely, Feng et al.’s model can be interpreted as a limiting case of our model that arises when l goes to
infinity, while q0s is adjusted in the manner discussed in footnote 20 to keep cumulative sequestration constant at V.
Substituting V l for q0s in Eq. (12) and taking the limit as l goes to infinity yields a maximal proportional reduction
(conditional on a) of DQs =Qs ¼ 1 ½ðr aÞ=r2 ; and hence a maximal delay effect.
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results, they find that under the economically efficient scenario the optimal share of sequestration
is only about half that of abatement (measured, however, in cumulative rather than annualized
terms). The study does not note the sensitivity of this share to price expectations, however. This is
because the second, higher-damage scenario considered, while yielding thrice higher price levels,
happens to yield almost identical rates of price increase over time. Not surprisingly (in light of our
analysis), the optimal sequestration share is therefore almost identical as well.27
Using the same modeling framework, Sedjo et al. [30] analyze a larger set of carbon price paths,
including ones with constant and increasing prices. Two of the scenarios can be examined for
evidence of the delay effect analyzed in our paper. The first scenario involves a constant carbon
price of $50 while the second has an initial carbon price of $50 rising at a rate of 2.5% per year
before stabilizing after 60 years. Fig. 1 in Sedjo et al. reports the cumulative carbon sequestration
for each scenario over a 100-year horizon. Our results suggest that cumulative carbon under rising
prices should lag, at least for a while, behind cumulative carbon under constant prices, because
under rising prices less land is initially converted to forest. However, no delay effect is apparent in
Fig. 1: cumulative carbon under the rising-price scenario always exceeds that under the constantprice scenario. This result is difficult to square with our analysis, but may be due to the greater
complexity of Sedjo et al.’s model. For example, forest managers in their model can respond to
carbon incentives not just by lengthening rotations, but also by changing forest management
practices in ways that increase forest growth, and thereby sequestration.
Although we have conducted our analysis under the assumption that carbon sequestration is
the only environmental benefit associated with conversion of land to forest, a number of authors
have drawn attention to other environmental impacts, referred to as ‘‘co-benefits.’’ These include
changes in biodiversity [16], reductions in agricultural externalities [28,12], and effects on water
quality [22]. Our model of Section 4 can be easily modified to include two stylized specifications of
co-benefits. First, suppose that co-benefits are a constant proportion of agricultural profits, which
could occur if their production increases with the fertility of agricultural land. They would then
proportionally increase acreage supply at any given carbon price (in effect increasing the constant
k in Eq. (8)) and thereby increase sequestration supply as well. But they would have no effect on
the relative reduction in sequestration supply that results from rising prices (note that k does not
appear in Eq. (11)). Thus, the key result of our paper is unaltered. Second, suppose that cobenefits are independent of land fertility. They would then offset the annualized conversion costs
in Eq. (8), again increasing both acreage and sequestration supply, but more so under constant
than under rising prices. Thus, co-benefits would increase the absolute reduction in sequestration
supply under rising prices, though the relative reduction (given by Eq. (11)) can be shown to
decline.
A final point worth noting is that, although our study focuses on carbon sequestration in
forests, our key findings should hold also for other approaches to greenhouse-gas mitigation
characterized by declining rates of effectiveness over time. Enhancing carbon sequestration in
agricultural soils through adoption of conservation tillage and other best-management practices
(BMPs) is one such case. Lal and Bruce [13] estimate the worldwide potential contribution of this
27
Interestingly, Sohngen and Mendelsohn, too, find negligible feedback effects from world sequestration supply on
either carbon prices or abatement supply: both fall by less than 1% by 2100 when feedback is incorporated in their
model.
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approach to be about 450–600 MtC per year, but note that ‘‘soil’s capacity to sequester C through
adoption of BMPs may be filled over a 20–50-year period’’ and that the rate of sequestration
usually peaks within 10–15 years. This suggest that, as with sequestration by forests, rising carbon
prices will reduce the share of this approach in an optimal portfolio of mitigation strategies,
though further analysis will be needed to quantify any such effect.
Acknowledgements
The authors thank Matthew Kotchen, Urvashi Narain, and seminar participants at the 2002
AEA meetings and the 2003 NBER Summer Institute Workshop on Public Policy and the
Environment for helpful comments.
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