WPS 235
POLICY RESEARCH WORKING PAPER   2635
Optimal Use  of Carbon                                    Does temporary sequestration
of carbon dioxide have a
Sequestration  in  a  Global                              place in a comprehensive
Climate  Change  Strategy                                 policy for mitigating climate
change? Does it "buy time'
for technical change in the
Is there a Wooden Bridge                                  energy sector?
to a Clean Energy Future?
Franck Lecocq
Kenneth Chomitz
The World Bank
Development Research Group
Infrastructure and Environment
July 2001



| POLICY RESEARCH WORKING PAPER 2635
Summary findings
Carbon sequestration aims at raising the amount of            which both sequestration and abatement can be used to
carbon sequestered in biomass and in soils. Whether it        mitigate climate change.
should be part of a global climate mitigation strategy,         They confirm that permanent sequestration, if feasible,
however, remains controversial.                               can be an overall part of a climate mitigation strategy.
One of the key issues is that, contrary to emission         When permanence can be guaranteed, sequestration is
abatement, carbon sequestration might not be                  equivalent to fossil-fuel emissions abatement.
permanent. But some argue that even temporary                   The optimal use of temporary sequestration, on the
sequestration is beneficial as it delays climate change       other hand, depends mostly on marginal damages of
impacts and "buys" time for technical change in the           climate change. Temporary sequestration projects
energy sector.                                                starting now, in particular, are not attractive if marginal
To rigorously assess these arguments, Lecocq and            damages of climate change at current concentration
Chomitz build an intertemporal optimization model in          levels are assumed to be low.
This paper-a product of Infrastructure and Environment, Development Research Group-is part of a larger effort in the
group to assess policies for mitigating climate change. Copies of the paper are available free from the World Bank, 1818
H Street NW, Washington, DC 20433. Please contact Viktor Soukhanov, room MC2-523, telephone 202-473-5721, fax
202-522-3230, email address vsoukhanov(tworldbank.org. Policy Research Working Papers are also posted on the Web
at http://econ.worldbank.org. The authors may be contacted at flecocq@a worldbank.org or kchomitzCaiworldbank.org. July
2001. (27 pages)
The Policy Research Working Paper Series disseminates the fRndings of Dork in progress to encorage the exchange of ideas about
development issuies. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The
papers carry the names of the authors and should be cited accordinglv. The findings, interpretations, and conclusions expressed in this
paper are entirely those of the authors. They do not necessarily represent the vietv of the WXorld Bank, its Executive Directors, or the
countries they represent.
Produced by the Policv Research Dissemination Center



Optimal Use of Carbon Sequestration in a Global Climate Change
Strategy: Is there a Wooden Bridge to a Clean Energy Future?
Franck Lecocq*, Kenneth Chomitz
July 2001
World Bank, Development Economic Research Group, Infrastructure and Environment
1818 H St NW Washington DC 20433 USA
1. Introduction
Does temporary sequestration of CO2 have a place in a comprehensive policy for mitigating climate
change? Three arguments have been advanced in favor of sequestration. First, some proportion of
"temporary" sequestration may prove permanent (Chomitz, 2000). This applies both for plantations,
which may become financially sustainable after initial barriers such as establishment costs have been
overcome, and for deforestation prevention projects in areas where pressure on forest turns out to be only
temporary'. Second, temporary sequestration may succeed in deferring climatic damages, and this
deferral is argued to be a benefit. Third, some argue that temporary sequestration "buys time" for
technical change: in other words, it bridges from a period when energy abatement is expensive to a future
Corresponding author: flecocq@worldbank.org, Tel: +1-202-473-1231, Fax: +1-202-522-3230. The findings,
interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent
the view of the World Bank, its Executive Directors, of the countries they represent.
1 Examples of Europe, the US, Puerto Rico or Malaysia suggest that there is a "high tide" of forest clearance at moderate
levels of development, which recedes as farmers are pulled from forest margins to urban jobs (such "deforestation Kuznets
curves" can be found for example in Vincent et al., 1997). Worldwide, land-use projections show a declining rate of
agricultural expansion over the next 30 years (FAO, 2000a). As a result, temporary (30 year) protection might end up as de
facto permanent.



era when alternative energy sources are cheap (Schwarze and Niles, 2000, Noble and Scholes, 2001). The
present paper aims at rigorously assessing these arguments.
Temporary sequestration delays emissions with regard to the baseline. The key issue is therefore to
know whether energy abatement and permanent sequestration can be reorganized in such a way that the
total costs of the climate policy - damages plus abatement costs - diminish enough to offset the costs of
the temporary sequestration project. To answer this question, we build a long term climate policy
optimization model in which the planner has both the possibility to abate fossil-fuel emissions and to
sequester carbon, either temporarily or permanently (section 2). In this model, we consider only
deforestation prevention projects, which appear to be inexpensive and to offer potentially important side
benefits such as biodiversity conservation, watershed protection and improvement of rural livelihoods
when alternatives to deforestation are offered (Chomitz, 2000).
In section 3, we first apply the model to the class of permanent deforestation prevention projects. We
demonstrate that sequestration and emissions reduction from fossil fuel combustion should then be
considered similarly -marginal costs should be equated - provided spatial externalities are negligible. In
section 4, we turn to temporary deforestation prevention projects, and show these are not cost-effective in
the short and medium run unless marginal damages of climate change are high enough. We next combine
both possibilities in a model where the planner has the choice to keep deforestation prevention running or
to terminate it at each point of time. Both analytically (section 5) and numerically (section 6), marginal
damages at low concentration levels prove again critical for the optimal sequestration patterns section.
We demonstrate the results obtained for deforestation prevention can be generalized to plantations in
section 7.
2. A climate policy optimization model with emission abatement and deforestation
prevention
Let a planner optimize long term world climate policy so as to minimize total costs of climate change,
i.e. the sum of mitigation costs and climate change damages. To reach this aim, the planner can either
reduce fossil-fuel emissions or use carbon sequestration.
2.1.   Fossil-fuel emissions and abatement
Let time be discretely indexed by t (O<t<T). Carbon dioxide emissions from combustion of fossil-fuels
in the business as usual scenario are denoted et. At each period of time, the planner can decide to abate a
quantity a, of these emissions, incurring costs Ct(at). We assume C, to be twice differentiable convex
functions, differing among time because of autonomous technical change. We also assume that C,(O)=O,
which means we do not consider potential "no regret" abatement potential. By construction, abatement at
is positive (1) and less than baseline emissions (2).
0�at                                                                (1)
a, < et                                                              (2)
2



2.2.    Land-use emissions and deforestation prevention
Carbon sequestration aims at raising the amount of carbon sequestered in biomass and in soils with
regard to the baseline. This encompasses a very wide range of activities, including establishment of
timber plantations, encouragement of agroforestry and silvopastoral activities, improved farming
practices, and prevention of deforestation (Watson et al., 2000). As announced in the introduction, we
focus for the time being on the last item which may be cheap in some areas, potentially offers large
cobenefits and is controversial on permanence grounds. Deforestation prevention also presents the
advantage of limiting the analytical complexity of the model. We will come back to the general
sequestration case in section 7.
Let us denote ft the reduction, relative to the baseline, in the area deforested during period t. By
definition, this area is positive (3) and cannot exceed baseline deforestation for this period d, (4).
f Ž0                                                                           (3)
ft < dt                                                                        (4)
For simplicity we assume a constant factor of proportionality Ti (in tons of carbon per hectare) between
ft and emissions reductions, though the actual factor will vary according to the biomass density of the
forest and the land use that replaces it. Finally, we attribute all the avoided emissions to the period where
the protection took place2.
We denote CL, the cost function of deforestation prevention at period t. This function represents
payments compensating landowners for the foregone benefits of shifting to an alternative land-use3. We
assume CL1 are convex and twice differentiable functions which pass through the origin.
2.3.    Carbon cycle and climate change damages
Let us denote elt the baseline anthropogenic emissions from land-use4. Total anthropogenic emissions
in the atmosphere Et are then given by (5) below, where -rft refers to the reduction in emissions due to
deforestation prevention with regard to the baseline5'6.
Et = et + elt - at - rl ft                                                     (5)
We use the simplified carbon cycle model designed by Nordhaus (1992). The model is represented in
equation 6 below, where Mt denotes the atmospheric CO2 concentration over preindustrial equilibrium
Lo. and where Mo is exogenously given. Parameter ,B can be interpreted as the fraction of CO2 emissions
2 If periods are at least one year long (5 years in our numerical simulations), we can consider this assumption to be
roughly valid, in first order approximation. Bolin et al. (2000) indeed assume that, on average, carbon release is complete
two years after deforestation. For deforestation in the Amazon, Houghton et al. (2000) estimate that 20% of biomass is
burned, 70% is left as slash, 8% removed for products and 2% converted to elemental carbon through burning, with
respective rate of decays of O.lyf', O.lyf' and 0.00lyf' for the last three. Assuming equal carbon contents, these figures
imply that 28% of carbon is released one year after deforestation, 52% five years after and 71% ten years after.
Conceptually these payments finance not only plot-level opportunity costs but sectoral interventions such as
agricultural intensification that ensure that there is no leakage.
4 We do not impose any particular relationship between elt and d, as sequestration may be included in the baseline, and
as the amount of deforestation the planner is able to avoid may be less important than the baseline deforestation for the
considered year
5 In this introductory section, we present the simplest case where deforestation prevention is permanent. Changes in
equation (5) when deforestation prevention is temporary will be introduced in sections 4 and 5.
6 In this first model, we assume that once an hectare is cleared, no subsequent regrowth occurs.
3



which is reabsorbed immediately within the ocean. The remainder is stocked in the atmospheric reservoir
and slowly leaks back to sinks over time at rate 6.
Mt+,= Mt +   Et -6 Mt                                                        (6)
Nordhaus' carbon cycle model is easy to parameterize and to embed in an optimization model. But it
underestimates both short term and long term concentrations (Joos et al., 1999)7. Nevertheless, we claim
that our analytical results would still remain valid using more complex carbon cycle representations. We
will indeed see that they depend primarily from the fact - accepted by climate scientists - that the flux of
carbon from the atmosphere to the ocean is positively correlated with atmospheric CO2 concentration.
This is reflected, though crudely, in Nordhaus' model through paramneter 6.
In order to limit the analytical complexity of our model, we assume that climate change damages in
year t depend on spot concentration Mt through a twice differentiable function Dt such that damages at
current concentration are zero. This is a restrictive hypothesis: current damages may indeed depend not
only on the spot, but on the whole path of atmospheric CO2 concentration. Mean surface temperature, for
instance, depends on concentration history (Svirezhev et al., 1999). It can be demonstrated, however, that
using temperature instead of concentration would not modify our qualitative results.
2.4.    Terminal condition and planner's objective function
The choice of horizon T plays a central role in the model's behavior, as damages beyond this point are
not factored in the analysis. To minimize the effects of this choice, we make the following assumptions.
First, we assume that, when concentration returns to current levels or below, damages are zero, and
marginal damages are less than or equal to current marginal damages. Second, we assume that in the
baseline, anthropogenic CO2 emissions return back to zero at some point in the future as the result of the
diffusion of carbon-free technologies. According to the carbon-cycle (6), baseline concentration will
therefore go back to current levels at some subsequent time8. Third, we choose for horizon T precisely
that later date. By doing so, we ensure that, whatever the abatement path, concentration is below current
level at horizon T: no damage beyond this date is considered.
If some carbon is stored through sequestration programs, however, it may still be released after
horizon T, hence resulting in concentration increase and further damages. We therefore assume that any
ton of carbon sequestered up to period T- I - the last period whose emissions count for concentration MT -
is in fact sequestered forever. The costs for T-1, CLT.1(fT-1), include the costs of permanent sequestration
from this point onward.
The planner's objective function is then (7) below, in which 0 is the annual discount factor9.
T
Min (a,ft) E [Ct(at) + CLt(ft) + Dt(M,)] Ot                                  (7)
t=0
Joos et al identify two main shortcomings of the model. First, the instantaneous reabsorption of a fraction 3 of
emissions is in fact much slower. Second, all the CO2 emitted "in excess" since the industrial revolution will not be
absorbed in the ocean: the law of chemistry indeed states it should be split between atmospheric and oceanic reservoirs.
Ultimately however, its sequestration in soils and in the deep layers of the ocean will lead to a complete removal, but on a
much longer time horizon than Nordhaus' model indicates (Kattenberg et aL, 1996).
8 The fact that our carbon cycle underestimates carbon resilience in the atmosphere implies that this time is not too
faraway in the future.
9 If p is the annual discount rate, then 0 =
I+p
4



3. Permanent sequestration: equivalent to energy abatement, as long as past decisions do
not impact on present costs
In this section, we assume the planner is able to secure contracts guaranteeing that the land will remain
forested until horizon T. With an up-front payment, the planner therefore compensates landowners for the
whole stream of foregone revenues from period t to period T and at the same time avoids leakage by
intensifying agriculture, ranching or silviculture somewhere else from period t to period T. At each period
of time, we assume that because of the heterogeneity in costs of protection there is a supply curve for
averted deforestation CLt(ft). This represents the minimum possible cost of reducing the current flow of
deforestation during period t by ft hectares relative to the baseline.
We must then distinguish two cases depending on whether these costs depend only on current
prevention ft or also on past decisions f,l-, ft2, ... , fo. For the first hypothesis to be valid, both
compensations to landowners and agriculture intensification must be independent from period to period.
For example, opportunities to prevent deforestation might arise in different regions at each period of time.
An expanding frontier of low-productivity semi-subsistence farming might be another example. As the
frontier passes, forest land is converted to slash and burn agriculture, then abandoned to incomplete
regrowth. In this model, each period presents an independent new supply curve for deforestation
prevention.
On the other hand, global land scarcity may induce CL, functions to depend on both present and past
protection decisions. This might also result from intensification programs where the costs of an unit of
productivity gain rise with the quantity of gains already realized. In section 3.1, we start with the
assumption that each CLt() is a function only of current deforestation prevention ft. In section 3.2 we will
examine the case of increasing costs for preventing deforestation as a function of cumulative past
10
protection
In both cases, sequestration is permanent, so carbon is never released. Equation (5) remains unchanged.
3.1.   When past decisions do not matter, marginal abatement and sequestration costs are
equalized
Detailed derivation of first order conditions for model (1)-(7) are provided in Annex 1. The key result
is that, along the optimal path, as long as maximum abatement is not required (constraint 2 not binding),
and as long as the supply of "protectable" forest is not exhausted (constraint 4 not binding), the marginal
costs of abating and sequestering carbon should be equal (8).
CLt(ft) = r Ct(at)                                                   (8)
In this case, sequestration and abatement are perfect substitutes from a climate point of view.
3.2.   When past decisions matter, a slightly more complex equilibrium
Let us now assume that sequestration costs depend on both the incremental surface put under
protection ft and the total protected areas Ft-, = fo + fi + ... + ft-,, with partial derivative of CLt with regard
to f and F positive. Again, marginal abatement costs are equal to marginal sequestration costs. But in this
case, setting one additional hectare under protection entails costs not only at present period (first term in
'0 The same might actually be true in the energy sector.
5



the right-hand side of equation 9 below), but also at each subsequent periods (second term). As a result,
the marginal sequestration cost - aCL,/fft - is now less than marginal abatement costs.
aCLp           T aCL
n.QC(at) =      (fq,Fq-i)+  t X (f Fq q)0(
Does it mean more or less sequestration? If we assume that, at each period, the marginal sequestration
costs remain the same (that is the partial derivatives of CL, with regard to f,) as in the preceding rnodel,
then sequestration becomes overall more expensive when history is taken into account. As a result, (i) the
balance between abatement and sequestration shifts towards abatement, and (ii) as action overall becomes
more expensive, the global damage/cost equilibrium shifts towards less emission reduction. Both effects
tend to reduce the surface of forest which is put under protection.
4. Temporary sequestration: where marginal damages play a critical role
Let us turn to temporary sequestration. We now assume that sequestration projects are not permanent,
but defer emissions for exactly T years. Anthropogenic emissions are now given by equations (10) and
(11) below, which replace (5) in previous model to take carbon release at the end of each sequestration
project into account. All the other equations remain unchanged, except that sequestration costs Cl,,(f1) -
for simplicity's sake, we assume that the total surface of forest put under protection does not impact on
protection prices - now represent the costs of preventing deforestation for a fixed period of t years.
Et=et+elt-at-1 ft             for 0t�t-I                                 (10)
Et =et +elt - a,-9 ft + 9ft-,    forc St ST                              (11)
Derivation of first order conditions for model (1,2,3,4,6,7,10,11) is given in annex 2. To facilitate the
exposition of the results, let us introduce the spot shadow price of atmospheric carbon that we denote X,.
Along the optimal path, Xt is equal to the discounted sum of the flow of incremental damages the release
of one additional unit of carbon in the atmosphere would cause.
Along the optimal path, it is therefore not surprising that marginal abatement costs are equal to the
shadow price of atmospheric carbon, at least as long as some abatement is necessa7 (constraint 1 not
binding) or all the abatement opportunities are not exhausted (constraint 2 not binding) 1
Ct(at) = Xt 3                                                            (12)
The same is not true, however, for temporary sequestration. Such projects indeed simply postpone
emissions by X periods, and therefore offset only part of the stream of climate change damages caused by
the emission of one additional ton of carbon. A given temporary sequestration project will be undertaken
only if the shadow price of carbon - and therefore the cost of abatement - is lower in discounted terms at
period t+t than it is at period t. And the optimum is reached when, at each period of time, the planner is
indifferent between abating one unit of fossil-fuel emissions now or bundling together a x-period
sequestration project starting now and a fossil-fuel emission reduction project in T periods'2.
CL4f,) = (Xt - Xt+,O?) Q D      for t S T-t                              (13)
Factor ,B in equation (12) accounts for the fact that only a fraction D of emissions is not immediately reabsorbed in
Nordhaus' carbon cycle model.
12 Equation (13) is of course valid as long as constraints (3) and (4) are not binding. The additional il multiplier
translates shadow prices (in $ per tons of carbon) in deforestation prevention costs (in $ per hectare).
6



This rule has two interesting consequences:
* First, if the shadow price of atmospheric carbon rises at a rate higher than the rate of discount
between periods t and t+'r, then no temporary sequestration project starting from period t is cost-
effective.
* Second, even if the price of atmospheric carbon rises at a rate lower than the rate of discount,
temporary sequestration is cost-effective if and only if its costs are lower than the atmospheric carbon
price difference.
The key issue for temporary sequestration is therefore to understand the trajectory of the implicit price
of atmospheric carbon along the optimal path. We make this detour in sub-section 4. 1, then conclude our
analysis of the optimal use of temporary sequestration in 4.2 and discuss optimal value for T in 4.3.
4.1.   High or low marginal damages? Two paths for the shadow price of carbon
Rigorous computation of the optimal dynamics of the shadow price of atmospheric carbon Xt is given
in annex 1. We propose here an intuitive derivation of the result.
Let us reason backward. At penultimate period T-1, the costs associated with one additional unit of
carbon in the atmosphere are only the incremental damages triggered by this unit at period T (as T is the
horizon of the analysis). The shadow value of carbon at period T- 1 is therefore given by:
XTI = 0 DT(MT)                                                       (14)
Given the choice of horizon T, we know that concentration at horizon T MT is less than or equal to
current concentration, and that marginal damages of climate change are less than or equal to current
marginal damages.
Let us now go one period backward. The release of one more unit of carbon in the atmosphere in
period T-2 will trigger damages at period T-1, and damages at period T. However, because of the "leak"
term a in the carbon cycle model, releasing one more unit of carbon in the atmosphere at period T-2
actually triggers less incremental damages at period T than releasing the same amount at period T-1. A
fraction (1-8) of this carbon has been reabsorbed by the oceans or the soil. The shadow price of carbon at
period T-2 is therefore given by:
XT-2 = 0 DT'1(MT-1) + (1-8) 02 DT(MT) = 0 DT-1(MT-1) + (1-5) 0 2T-1  (15)
This reasoning can be repeated, to yield the general form of the shadow price of carbon along the
optimal path:
T
Xt=  E 0q-t(lo)q-t-'Dq'(Mq)                                          (16)
q=t+1
It will in fact prove easier to use the backward induction equation:
Xtl = (1-8) 0 t + 0 D'(Mt)     for l <t<T-l                          (17)
What does (17) imply for the evolution of Xt along the optimal path? First, as the discounted shadow
price of carbon is less at period T than it was at initial period13 (14), we know that Xt will ultimately
decrease. But for the first periods, equation ( 17) simply states that X, can cannot rise at a rate higher than
the rate of discount plus 8, the rate of "depreciation" of atmospheric carbon. Consequently, the discounted
13 Indeed, we have XT-I = 6 DT(MT). By construction of T, we guarantee that MT < MO and DT(MT) < Do(MO). At the same
time, we know from equation (17) that X0 > 6 Ds(MI) 2 0 Do(Mo). Thus XT-I < B0 and XT,T-1 < Xo.
7



price of atmospheric carbon can follow either one of the different paths illustrated below: rise and
decrease, with possibly several oscillations before finally decreasing (Figure 1), or decrease from thle start
(Figure 2).
Which path we follow depends on two elements: the marginal damages of climate change, especially at
low concentration levels, which drives the variations of Xt in equation (17), and the global equilibrium
between climate change damages, abatement costs and baseline emissions, which drives the value of ko.
We will see numerically that the first path appears to be more plausible than the second (section 6). At
this analytical level, however, all we can say is that if marginal climate change damages are zero at low
concentration levels, we are in the first scenario regardless of the initial value of X0.
T  Time                                           T  Tinme
Figure 1: Optimal discounted price of atmospheric carbon Figure 2:Optimal discountedprice of atmospheric carbon with
with low marginal abatement costs at low concentration levels. high marginal abatement costs at low concentration levels.
4.2.   Temporary sequestration is cost-effective only when current marginal damages are high
enough
We can now come back to rule (13) and examine under which conditions temporary sequestration
occurs. If the shadow price of carbon rises at a rate higher than the rate of discount between periods t and
t+T, then it is not cost-effective to undertake any T-period sequestration projects. They would indeed
bridge the gap to a period where, along the optimal path, the marginal abatement cost is in fact higher
(even in discounted terms) than it is at period t. The preceding discussion informs us that such a situation
occurs when marginal damages at low concentration levels are low or nil.
On the contrary, if the shadow price of carbon rises at a rate lower than the rate of discount between
periods t and t+-r, then some of the temporary sequestration projects are undertaken up to the point where
the costs of sequestration just bridge the difference (in discounted terms again) between present and
future shadow prices of carbon. When this difference becomes too large, all the available deforestation
prevention projects are undertaken (that is ft=d,).
We have seen that the shadow price of atmospheric carbon will eventually decrease along an optimal
path. We are therefore sure that, beyond some point in the future, the shadow price of carbon will rise at a
rate lower than the rate of discount, and therefore that, beyond this point, some r-period long temporary
sequestration will become cost-effective. However, depending on the shape of the damage function and
on the emission baseline, this point might come at the very first period or on the very far in the future.
8



4.3.   On the optimal length of deforestation prevention
In the above analysis, the length of temporary deforestation prevention X was fixed. Let us now discuss
the optimal value of this parameter. To this aim, let us consider a l-period temporary sequestration project
worthwhile undertaking along the optimal climate mitigation path. This means that the cost of this project
are more than offset by the benefits in terms of discounted shadow prices of carbon. We denote A the
difference.
Does it make sense to extend protection by one additional period? The project obviously becomes more
expensive. But on the other hand, the difference of discounted shadow prices of carbon between present
period and termination periods also grows. The key issue is therefore to know whether the net gain A plus
this incremental difference still remains positive. Two cases are therefore possible:
* If the costs of extending protection become high enough not only to offset further gains in terms of
shadow price of carbon, but also to offset the initial benefit of the project (A), then it becomes useless
to further extend protection. This equilibrium is defined precisely as the point where the planner is
indifferent between abating now, or bundling sequestration and future abatement.
*  On the other hand, extending protection might even add to A (if, for instance, beyond some
threshold, extending deforestation prevention further becomes nearly cost-free). In this case, the
optimal length of the project is in fact until horizon T, that is permanent sequestration.
We now go one step further and assume the planner can choose the length of forest protection, in a
model where the quantity of forest already put under protection matters.
5. Optimal deforestation prevention for carbon sequestration
5.1.   Deforestation prevention for a single body of forest: the model
In the two previous sections, only one type of sequestration, either temporary or permanent, was
admissible at the same time. We now combine both options. We assume a unique forest which shrinks by
an incremental dt hectares at each period in the baseline scenario. At each period, however, the planner
can compensate potential deforesters and save f, hectares out of the dt, adding to a total "belt" of Ft
hectares (equation 18) around what would have remained standing at period t in the baseline. To account
for the possibility of temporary sequestration, the incremental area protected at one given time may be
negative, which implies that constraint (3) is replaced by (19) below.
t
Ft= Efq                                                            (18)
q=0
Ft Ž0                                                               (19)
We model deforestation as follows. At period t, given the structure of the transportation system and the
general demand for agricultural land, we assume an implicit revenue from agriculture or ranching can be
associated with each point of land. If, at the beginning of a given period, this implicit revenue is positive
for some forested patch of land, this surface will be deforested during this period so that, at the end of the
period, the deforestation frontier reaches the point where the implicit revenue from agriculture is zero.
From one period to the next, however, rising demand for agricultural land as well as improvements in the
transportation network shifts this function upward, triggering further deforestation.
9



The planner's sequestration policy is a combination of compensations to "would be deforesters" and of
measures aimed at intensifying agriculture and ranching elsewhere in the country. However, given the
above model of deforestation, we assume that at each period of time the hectares the planner is able to
protect from deforestation are the ones which are closest to the baseline frontier, where distance from
markets and roads make agriculture or ranching the least attractive. This policy thus "holds back"
deforestation frontier from where it would be otherwise'4 (see Figures 3a, b and c).
Contrary to the preceding two cases, however, where the planner could secure long term contracts, we
assume here he cannot prevent deforestation from more than one period at a time. If he stops paying at
one point of time, a rapid clearing of the total protected area F, thus occurs. In other words, the policies he
puts in place have to be renewed at each period, and do not have long term effects. Under these
assumptions, costs CLt now depend on the total area protected F, and not on the incremental ft.
The objective function therefore becomes:
T
Min (at ft) E[Ct(a) + CLt(F,) + Dt(M,)] Ot                               (20)
t=o
5.2.   Two patterns of deforestation prevention
Derivation of first order conditions for problem (1,2,3,5,6,18,19,20) is presented in annex 3. The key
result is that optimal sequestration may follow two different optimal paths depending on the marginal
climate change damages at low concentration levels.
*  When marginal damages are high, sequestration starts at the first period and temporary
sequestration between consecutive periods is possible
* When marginal damages are low at low concentration levels, sequestration follows a "bang-bang"
path: no protection at first, and then full protection of all remaining forest (and temporary
sequestration between consecutive periods is not possible during first periods)
When marginal damages are high, sequestration starts at the first period and temporary
sequestration between consecutive periods is possible
We know from the above discussion that, when marginal damages are sufficiently high, the shadow
price of carbon rises at a rate lower than the discount rate from the very first period onward (the dynamics
of shadow price XA are indeed the same in this model as they were in the two previous sections). At each
period, the planner has an interest in making at least some temporary deforestation, which thus begins
from the start.
What is the optimal deforestation prevention path? Using the same reasoning as in section 4.3, we see
that forest should be protected between the first and second periods up to the point where marginal costs
of one period protection exactly compensate the diminution of discounted shadow price of carbon during
this interval. Hence the value of fo (and therefore of FO). The same reasoning can be applied at each
period, hence yielding the successive values of Ft through equation (21) below.
CL(Ft) = (X, - t,+IO) l p   for 0 < t < T-1                             (21)
14 It is important to note that there is no guarantee that individual plots will be preserved. Indeed, even if one hectare is
protected at first period, rising revenue from agriculture at this spot make it more interesting to let it cut down at second
period to protect another hectare further out on the frontier.
10



Surface
deforested
f fetive                               f+=d+          I=Ft+dt
No clearing
between t and
t+l
t          t+l                     Time
Surface
deforested
baleli.       t~~~~~~~~~t i[
eff/ective                       Some clearing
between t and
t           t+l                      Time
Surface
deforested
baseline  .-                 4                    -Ft
,,a....... _FuUl clearing
between t and
effetive
K                                                 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~p.
t           t+l                      Time
Figures 3a, b and c: Three possible decisions regarding deforestation at period t+l . First, the planner extends the area already protected (F)
by an amount equivalent to the area which should be deforested under the baseline (d,+,). The forest frontier remains unchanged, and there is no
release of carbon between periods I and t+I (3a). In 3b, the planner simply keeps the total amount offorest under protection constant (f,.,=O).
There are emissions as the planner swaps areas previously protected, but too expensive to keep protected now, with areas closer to the baseline
frontier. 3c is the extreme case where the planner abandons allprotection (f +, =-Fd: the whole surface under protection at period t (Fd plus the
area which had to be cleared in the baseline (dt,) are cleared
11



In other words, at each point of time, the marginal cost of permanent sequestration equals the shadow
price of carbon (the shadow price of carbon at this period is by definition zero at last period). These
properties are valid as long as both sequestration and abatement supplies are not exhausted (i.e. as long as
constraints 2 and 4 remain not binding).
T
E CLq(Fq) Oq-t = X,  [3  for 0 < t < T-1                               (22)
q=t
When marginal damages are low at low concentration levels, sequestration follows a "bang-bang"
path: no protection at first, and then full protection of all remaining forest (and temporary
sequestration between consecutive periods is not possible during first periods)
When marginal damages are low at low concentration levels, we have seen that the price of
atmospheric carbon rises at a rate higher than the rate of discount during the first periods (equation 13).
Consequently, deforestation prevention projects which start and end within this time interval are
unattractive. They would postpone emissions to a point where the discounted shadow price of carbon is
higher than it was at the start of the project. As a result, Fo, Fl, and all subsequent F, in the time interval
during which the discounted shadow price of carbon is above its initial value should be zero (i).
But on the other hand, we know that the discounted shadow price of carbon will eventually become
lower than it was at period 0. Consequently, some deforestation prevention projects starting during the
first periods, but ending beyond this time interval are cost-effective. As a result, some of the Fo, F1, and
subsequent F, in the time interval during which the discounted shadow price of carbon is above its initial
value should be strictly positive (ii).
There is an obvious contradiction between (i) and (ii). The reason for that is that, along any
sequestration path (Ft), and henceforth along the optimal one (Ft), the discounted marginal cost of
preventing the emission of one more ton of carbon till horizon T is by construction decreasing with time.
On the other hand, the discounted shadow price of carbon is rising during the first periods. Contrary to the
preceding case (equation 22), there is therefore no way both quantities can be equated at all points in time.
As a result, the optimal use of sequestration is a "bang-bang" solution. Before a "turning point", there
is no sequestration at all (ft=O), and after this point, all the remaining forest standing is protected (that is
f,=dt) (see annex 3 for a rigorous derivation of this result). The marginal cost of permanent sequestration
along the optimal path is higher than the discounted shadow price of carbon before the "turning point"
and lower afterwards. Only at the "turning point" period itself do we observe an equalization of both
quantities in the model of equation (22).
To examine which path is actually chosen, and disentangle the role of the different parameters, we now
turn on to numerical simulations of the above model.
12



6. Numerical simulations
To gain insights on how the shape of climate change damages affects the cost-effectiveness of
sequestration, especially during the first periods of the simulation which are the most relevant for
decision-making, we make numerical simulations of the model of deforestation prevention developed in
the last section using the GAMs software.
6.1.   Model calibration
Fossil-fuel baseline emissions and abatement costs
We assume business-as usual world fossil-fuel emissions grow at the rates of the IPCC IS92a scenario
between 2000 and 2100 (Leggett et al., 1992, figures from Morita and Lee, 1997) with 2000 figures
updated using latest IEA data (2000). Beyond 2100, emissions are assumed to decrease linearly and stop
completely in 2200. With this anthropogenic forcing, carbon cycle (6) predicts atmospheric carbon
dioxide concentration would eventually decrease back to current level in the last quarter of the 24th
century. We therefore set our time horizon T in 2400, dividing time in 5-year periods between 2000 and
2200, and in 25-year periods afterwards.
World marginal abatement costs are assumed to be zero at the origin - we do not take "no regret"
potential into account - and to rise quadratically with abatement. The function is parameterized assuming
a carbon free backstop technology initially available at $900 per ton of carbon. We assume autonomous
technical change cuts this price by half every 50 years (that is by 1.38% annually).
The discount rate is set at 5%.
Land-use CO emissions and sequestration costs
Anthropogenic land-use CO2 emissions derive essentially from tropical deforestation. IPCC estimates
this flux at an average 1.7±0.8 GtC.yr-1 between 1980 and 1989, and 1.6±0.8 GtC.yr'1 between 1990 and
1999 (Bolin et al., 2000), with global deforestation rates estimated 15.1 Mha.yf 1 between 1970 and 1990
(Dixon et al., 1994). Most models forecast from very limited deforestation to fairly important forest
regrowth starting beyond 2050 as world population tends to stabilization, agriculture becomes more
productive, and economic growth provides better job opportunities in the secondary or tertiary sectors.
But the short and medium term transition is much more controversial: some models indeed project a
smooth decline of deforestation over this period, while other forecast a sharp rise followed by a steep
decrease (see Nakicenovic et al., 2000). Recent FAO data for the 1990-2000 decade see, to point towards
smooth decline (FAO, 2000b). In our baseline scenario, we will assume a median trend of a constant
deforestation rate of 8 Mha.yr-l over the next five decades, followed by no deforestation at all beyond this
point.
The quantity of carbon released through deforestation in the tropics depends on the subsequent land-
use. Dixon et al. (1994) assume deforestation prevention saves, on average, 181 tC.ha&'. Sampson et al.
(2000) find cropping after slash and bum of tropical forest release between 152 tC.ha-1 and 224 tC.ha7'
(with a mean value of 184 tC.ha-1). We choose here r = 200 tC.ha' , which, combined with our land-use
scenario, leads to land-use C02 emissions of 1.6 GtC.yr  over the 2000-2050 period.
To model deforestation prevention policies, we assume that at each period, households clear land to
establish pasture and cropping up to the point where the revenue they can expect from these activities (all
13



costs included) drops to zero. In our baseline land-use scenario, exogenous drivers such as total demand
for food cause the zero expected revenue line further and further away in the forest, thus triggering steady
deforestation over the 2000-2050 period. However, we assume that the international community can
divert some of these households from deforesting by a combination of compensations to households and
complementary measures (for instance agriculture intensification) to avoid leakage..
Assuming that the least suitable land for agriculture are protected first at each point of time, the
foregone revenue per hectare is assumed to be increasing with the amount of forest put under protection
with regard to the baseline. At each point of time, the maximum total area that can be put under protection
is obviously the total area that would have been deforested in the baseline between 2000 and the end of
the considered period. We will, however, assume that only half of total world deforestation - 4 MtC .yr ' -
can be addressed through carbon-related policies. We choose quadratic CL, functions, with CLt(O)=O
(costs when no forest is protected are zero), CL4(0)=0 (marginal cost of first hectare protected is zero) and
CLI( E dq), i.e., the marginal annual cost of full protection, starts at $100/ha, rising at a 3% rate until 2050
q=O
and stabilizing afterwards.
Carbon cycle and climate change damages
We calibrate the carbon cycle using values from Nordhaus (1992): ,3=0.38, 5=0.082, preindustrial CO2
concentration of 274 ppm. We estimate 2000 atmospheric CO2 concentration at 370 ppm on the basis of
Keeling and Whorf (2000).
It has been pointed out that widely used "smooth" damage functions - usually polynomial of degree 1
to 3 (Tol and Fankhauser, 1998) - do not correctly capture the possibility of catastrophic damages which
could stem out either of highly non linear climate dynamics, such as a slowdown of the thermohaline
circulation (Broecker, 1997), increased frequency of El Niuo events, or of highly non linear responses to
smooth climate change, such as sudden migratory pressure when agricultural supply reaches a threshold
in the subtropics (Parry et al., 1999) or rapid changes in the opinion's concerns about the importance of
the issue such as in the "mad cow" crisis in Europe (Streets and Glantz, 2000). To take these possibilities
into account, threshold-type damage functions have also been proposed.
In the following simulations, we therefore use both aggregated damage function:
n polynomial damage function, where damages are zero at current concentration and rise linearly (we
will also test quadratic and cubic functions) with concentration. As a benchmark figure, we assume
that double preindustrial concentration level in 2100 would lead to a loss of 1% of world gross
product damage at that time (that is 250 trillion dollars under scenario IS92a economic gTowth
assumption).
*  Threshold damage functions, where damages are assumed to be zero until a given ceiling is reached,
and then rise sharply. For simplicity's sake, and to make the link with the cost-efficiency approach,
we assume damages rise to infinity when the concentrations exceed the threshold. We first consider
the 550 ppm target, but we will also examine the 450 ppm alternative.
14



6.2.    Numerical insights
Threshold damagefunctions
With threshold damage functions, marginal damages are assumed to be zero when concentration is
lower than the threshold. In this case, we find the expected "bang-bang" behavior of sequestration use:
no forest is protected at first, and then, beyond some "turning point", all remaining standing forest is
protected. In our central scenario, the "turning point" is relatively distant in the future (2040) and the
surface of forest preserved permanently is only 39 Mha out of the 200 that could have been protected in
total (19%). Strictly speaking, this is actually not permanent sequestration: these 39 Mha area are in fact
deforested before horizon T when the discounted shadow price of carbon ultimately decreases. But this
happens so late, typically beyond 2160, that for all practical purposes we can consider sequestration
starting from the "turning point" to be permanent.
In Table 1 below, we explore several alternative scenarios to test the sensitivity of the "turning point"
to the different assumptions of the model. First, the position of the threshold appears to have a strong
impact on the "turning point". Forest protection starts in 2025 when the threshold is 500 ppm, and in 2005
when it is set at 450 ppm. These results stem from the fact that the lower the threshold is, the higher
abatement costs need to be, and thus the (relatively) cheaper sequestration becomes.
In comparison, abatement costs in the fossil-fuel sector and discount rate appear to have less impact.
First, even in a highly technology-optimistic scenario - double speed of autonomous technical change -
the "turning point" is just postponed by 5 years (scenario b). Second, lowering the discount rate to 3%,
which means the implicit "pure time preference" of the planner is close to zero, only brings the "turning
point" closer by 5 years.
Similarly, even very optimistic assumptions about future deforestation prevention costs have a limited
impact on the "turning point" and thus on the amount of forest which is ultimately protected. In
scenario c, we indeed assume deforestation prevention becomes nearly free after 2050 ($10 per hectare
and per year at most). As a result, the turning point comes back from 2040 to 2030 and the surface under
protection doubles.
Threshold   discount rate   Abatement and   Turning point  Surface
sequestration                protected from
costs                      deforestation
550         5%              a             2040          39 Mha
500         5%              a             2025          89 Mha
450         5%              a             2005          170 Mha
550         3%              a             2035          41 Mha
550         5%              b             2045          19 Mha
550         5%              c             2030          66 Mha
Tablel: Date at which deforestation prevention begins and surface ultimately protected from deforestation with threshold
damages. a: abatement and sequestration costs assumptions made in the text, b: backstop technology price divided by four in 50
years instead of two; c: sequestration virtuallyfree beyond 2050 ($10/halyr).
We have demonstrated that at the root of this "bang-bang" dynamics is the fact that discounted
marginal abatement costs and discounted marginal costs of permanent sequestration cannot be equalized
over more than one period. We verify this property numerically. Figure 4 shows the discounted shadow
price of carbon (continuous line) and the discounted marginal costs of permanent sequestration (black
squares). We see that the former increases with time (at rate 5) over the 2000-2060 period while the latter,
15



decreases. Before 2040 ("turning point"), sequestration costs more at the margin, and there is no
sequestration at all. But after 2040, the reverse becomes true, and all the remaining land is protected.
1.25   -    I|- Fossil Fuel
---------  I  0 Seluestration
] 075
0 .7 5   -   - - -   - - - - - - - -   - - - -   - - - - - - - - -   - - - -
0
2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2050 2055 2080
Figure 4: Discounted marginal fossil-fuel (in continuous line) and sequestration (with black squares) abatement costs with
threshold damage function.
Polynomial damagefunctions
We start examining polynomial damage functions with the linear case. In this case, marginal damages
are positive from the start. And as expected, we now observe a much smoother deforestation pattern:
prevention starts as early as in 2000, and gradually increases, some deforestation still takes place for a
few decades. This is shown in Figure 5, where the continuous bold line represents the cumulative
deforested area in the business-as-usual scenario, and the line with black squares cumulative deforestation
with linear damages. We see that some forest is protected as early as in 2000, but cumulative deforested
area still rises during 20 years along the optimal path: this means that some clearing still takes place, or,
in other words, that some carbon is temporary sequestration between periods (about 4 Mha per period).
200  -- -   ---   --     -     --   - -- --   --   - - - -   -  --   - --~ --~~~- -----------------~~   ~~ 
1 7 5  - - - - - - - - - -   -  ..   - - - - -   - -   - - - - - - - - - - - - -
X                              13~~~~~~~~~~~aseline 
E  1 5 0  - -   - - -   - - -   - - -   --                   - - - - - - - -
v 125                               . .    /
S                       /        ~~~~~~~~~~Dema ae*t SSOppm = QSX%GWP
1  0 0  - --   - - - - - -- - - - -   -- -    -   - - - -- -
X   75  - -   -- -                                       - - - -- - - - - - -
5 0  -   - - - - -            - - - - -   - - - - - - -   - - -   - - -   - - - - - -
2000  2005  2010  2015  2020  2025  2030  2035  2040  2045  2050
Figure 5: Cumulative deforestation in the baseline scenario (bold line), with linear damages amounting to 0. 5% of Gross
World Product ifconcentration reaches 550 ppm in 21 00 (black triangles), and to I1% of Gross World Product (black squares).
16



Figure 6 shows the discounted marginal permanent sequestration costs (black squares) and the
marginal fossil fuel abatement costs (continuous line). It confirms that, with linear damage function,
marginal damages at low concentration levels are now high enough to trigger decreasing discounted
marginal abatement costs. Consequently, marginal (permanent) sequestration costs and marginal
abatement costs can now be equated (cf. equation 22), at least until 2020. After 2020 however, we reach
the point where the required marginal cost of abatement in the fossil-fuel sector is so great that it exceeds
the marginal cost of protecting (forever) all the remaining standing forest. Marginal abatement and
sequestration costs thus cannot be equated anymore.
We find this general pattern to be robust to the variation of several parameters (damage levels, discount
rates or marginal abatement costs). The surface of forest which is ultimately protected, however, varies
significantly with those (as can be seen, for example, in Figure 5 above, where the line with square
triangles represent the optimal deforestation path when expected damages when concentration reach 550
ppm are divided by two).
e 52- | s sea~IueV1abn
-  _--- -    - ----- -  ----------------  ----- - -----   I -------- - -- -- -- -- -- --
2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2050 2055 2060
Figure 6: Discounted marginal fossil-fuel (in continuous line) and sequestration (with black squares) abatement costs with
linear damage function.
If we move toward damage functions of higher polynomial degrees, we observe a shift towards the
"bang-bang" solution. This evolution comes from the fact that, the higher the exponent, the lower
marginal damages become at low concentration levels. Figure 7 below illustrates this mechanism. It
shows the discounted shadow price of carbon with linear, quadratic and cubic damage functions
respectively (the amount of damage at 550 ppm is adjusted to 0.5%, 1% and 1.5% respectively so as to
ensure initial shadow prices are of the same order of magnitude). We see that the discounted the
discounted shadow price of carbon decreases from the start in the linear case. On the contrary, it first rises
and then curves back with cubic damage functions. The quadratic case is in between, the dynamics of the
shadow price of carbon strongly dependent on the value of the other parameters of the model.
17



*   4                                 --- - - - --   - - - -
3          -
s     >;         .    ~~~~~~cubic  
--- --- --   -  - -- --    uad ati              -- --
g 6       > <       i ~~~~~~~~~~~linear .
0                                                   ,
2000  2005  2010  2015  2020  2025  2030  2035  2040  2045  2050
Figure 7: Discounted abatement costs with linear (square), quadratic (triangles) and cubic (crosses) damage functions.
Damages at 550 ppm have been set to 0.5%, 0.9% and 1.5% of World Gross Product respectively so as to ensure similar initial
abatement costs.
To sum up these numerical findings, let us take a different angle and assume we have the possibility to
undertake a 30-year long sequestration project, knowing that it may not be extended. Under which
circumstances should it be undertaken? Previous results show us that:
*   If one believes there is limited or no damage at low  concentration level (for instance with
"threshold" or cubic damage functions), then this 30-year long sequestration project is not cost-
effective for climate change mitigation, at any positive cost
* On the contrary, if one believes in high marginal damages at low concentration levels (for instance
with linear damage function) then the project should be undertaken provided its cost are lower than
the gains in terms of price of atmospheric carbon (about $4.5 per tons of carbon in the case displayed
on Figure 7).
*   If one believes in quadratic damage functions, then the project may be undertaken, again provided
its costs are less than the gains in terms of price of atmospheric carbon (less than $2 per tons of
carbon in Figure 7). In this limit case however, the cost-effectiveness of the sequestration project is
likely to be strongly dependent on the value of other parameters.
If there was some even some slight chance that the project were to be permanent, then it might still be
cost-effective even in the first and third cases. The computation of the expected costs and benefits are,
however, beyond the scope of this paper.
18



7. The afforestation case
In the previous sections, we have focused on deforestation prevention. We now demonstrate that our
results still hold when considering afforestation or reforestation activities. In 7.1, we demonstrate that any
afforestation activity can be regarded as a combination of elementary projects which are equivalent, as far
as emissions is concerned, to deforestation prevention activities. We then exploit this analogy to derive
the optimal path of afforestation or reforestation within a global mitigation strategy.
7.1.   Plantations as a combination of elementary sequestration projects
We consider the case of a tree plantation over a previously non forested area. We assume that this
plantation is managed so as to reach as rapidly as possible an equilibrium state where there is an equal
number of trees in each age vintage. From this point onward, the forest net emissions are zero as logging
is exactly compensated by regrowth. This plantation thus sequesters carbon only during the interim period
before the equilibrium is established. Let us denote p the length of this transition in periods, and 'lo,   --,
Ilp the incremental quantity of carbon which is sequestered at each period between establishment and
equilibrium. We denote i the sum of all these coefficients, which represents the carbon density of the
plantation at equilibrium.
Let to be the period at which the plantation is established, and t, be the period at which it is abandoned
(we assume the trees are then logged, and all the carbon released in the atmosphere, except if t,=T, in
which case carbon remains sequestered forever). For simplicity's sake, we also assume that the plantation
is abandoned after equilibrium has been reached, i.e. that period t, is posterior to to+p.
The important point here is that the above afforestation project can be regarded as a combination of
p+l elementary projects. The first one postpones the emission of 'lo tons of carbon from period to to
period ti. The second postpones the emission of 11  tons of carbon from to+ I to t1, etc. And postpones the
emission of rp tons of carbon from to+p to tl. With regard to the climate, each of these elementary is
equivalent to a deforestation prevention project.
As a result, we see that the rules governing the optimal use of afforestation in a global mitigation
strategy are similar to those governing deforestation prevention"5. These rules are detailed in 7.2 below.
7.2.   Insights on the optimal plantation path in a climate strategy
First, if the plantations are permanent (i.e. tl=T, a case similar to the one studied in section 3), the
optimal use of afforestation is given by (23). The difference with (8) stems from the fact that in the
afforestation case, carbon is captured progressively over periods (while in the deforestation prevention
case, all the carbon is already there in the trees). Consequently, the weighted sum of future shadow prices
of carbon must now be considered.
p
CLI(ft) =    l11lq XqO4                                                     (23)
q=O
The only difference comes from the combination effect. We do not consider the shadow prices of atmospheric carbon
at periods to and t, anymore, but the difference between the shadow price at period t, and a weighted average of the
shadow prices over periods to, to+l, ... , to+p, with weights precisely equal to the ratio of r1t/r1 times the discount factor
(except if the plantation is cleared before reaching equilibrium, where we consider the weighted average of shadow prices
over interval to,tl).
19



Second, if plantations are temporary (i.e. tc<'T, a case similar to the one studied in section 4), marginal
sequestration follows rule (24) below. With regard to (13) above, we see that, at equal carbon density per
hectare Ti, marginal plantation costs should be higher than marginal deforestation prevention costs ii the
shadow price of carbon rises at a rate higher than the rate of discount between periods t and t+p, hence
with low marginal damages at low concentration levels. On the other hand, marginal plantation costs
should be lower than marginal deforestation prevention costs if marginal damages at low concentration
levels are highl6.
p
CLt(ft) = 3 [  Tlq Xq oq _ Tl AOT-t]                                        (24)
q=O
Let us third consider the optimalforestation path of a unique body offorest where the planner should
pay at each period for protecting the plantations, at a price positively correlated with the total afforested
area F1 (case similar to the one studied in section 5). Again, we obtain two polar sequestration patterns.
When the shadow price of atmospheric carbon rises at a rate higher than the rate of discount, optimal
plantation is "bang-bang". No forest is planted at first, then, beyond some turning point, plantation starts.
On the other hand, if the price of atrnospheric carbon rises at a rate lower than the rate of discount, then
plantation starts from the start17.
16 A difference with deforestation prevention is that, while one cannot prevent prevention on more hectares than it wc,uld
occur in the baseline, there is a priori no such physical limit to the surface afforested. However, if there is no more need
for a constraint similar to 4, the spatial limitation on the extent of plantations is now embedded into the sequestration cost
functions CL,.
17 It might be argued, however, that the model of section 5 should not be transposed to plantation. Contrary to
deforestation prevention, plantations may indeed generate internal revenues and become sustainable in the long run (the
role of the planner being simply to help starting the project by funding establishment costs for instance). This may actually
be true also for deforestation prevention. For example, households could be provided with alternative job opportunities
which generate revenues and become sustainable in the long run.
Nevertheless, beyond some limit, it appears difficult to discard the possibility that global land scarcity might trigger
increasing land-use competition and therefore raise sequestration costs. To put it another way, behind any model which
assumes sequestration costs depend only on the incremental surface forested stands the implicit but necessary assumption
that the total supply of land for sequestration available over all periods is below some kind of acceptability limit beyond
which competition for land-use would become too fierce.
20



8. Conclusion
How should the risk of non permanence affect decision about carbon sequestration? The long-term
climate policy optimization model we have developed in this paper yields three main insights on this
question:
* When permanence can be guaranteed, sequestration is equivalent to abatement from fossil fuel
combustion. Where it is impossible to guarantee permanence of individual projects, it may be
possible to apply the same reasoning to portfolios of projects, adjusted for the expected proportion of
nonpermanent projects.
*  Explicit contracts for temporary sequestration, on the other hand, make sense only when the
rationale for mitigation is reduction of damages, and current marginal damages of climate change are
thought to be significant: sequestration is then cost-effective because it postpone damages. In this
case, there's a rationale to begin temporary sequestration contracts immediately, and they will
probably be renewed indefinitely.
*  Such temporary sequestration contracts also make sense when the rationale for mitigation is
keeping concentration below some threshold, but only when concentrations are approaching that
threshold. Here the sequestration project serves to bridge the "hump" of high energy abatement costs.
Under this rationale, sequestration follows a "bang-bang" optimal dynamics, and begins immediately
only if the goal is to stabilize at very low concentration levels.
Our analysis thus limits the scope of the "buying time" argument for temporary sequestration. We see
in particular that the key parameter for the dynamics of sequestration is the shape of the climate change
damage function, and not the expectations about technical change. One would have to be extremely
optimistic about diffusion of non-carbon energy technologies over the next few decades to justify current
sequestration as a bridge to these technologies.
However, this analysis does not rule out the usefulness of temporary sequestration projects on broader
grounds. First, in the Kyoto context, it may make sense to bundle temporary sequestration projects with
follow-on permanent abatement projects, depending on the expected-price path of carbon - which will
depend on the rules governing each successive commitment period - and the acceptability of this proposal
under the Protocol (Chomitz 2000). Secondly, temporary sequestration projects may be justifiable if they
have a nonzero chance of bringing about permanent change, e.g. by diffusing more carbon-absorbing land
management practices. Third, one could also imagine policies that substituted cheap temporary
sequestration for more expensive energy-based abatement, and invested the savings in research and
development on carbon mitigation technologies. A model where technical change is endogenous would be
necessary to examine this question. Last, if we drop our assumption that the planner operates with full
knowledge of future baselines, climate change impacts and technologies, temporary sequestration projects
might have a value in saving time to gain information. These are areas for future research.
Acknowledgements
The authors would like to thank Maureen Cropper, Alain Karsenty, Timothy Thomas, Vincent Gitz,
and Philippe Ambrosi for valuable comments on this paper. They have also greatly benefited from
discussions with Gunnar Eskeland. Remaining errors are of course the authors' sole responsibility.
21



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23



Annex 1: Optimal climate policy with permanent sequestration
The Lagrangian for problem (1)-(7) is given below. Kt, a,, Xt, it and X, are Lagrange multipliers
associated with constraints (1), (2), (3), (4) and (6) respectively. Xt can be interpreted as the spot shadow
price of atmospheric carbon'8.
T                              T-l
L = - E [CQ(at) + CL,(f,) + D,(M,)] Ot + E t Ot 0[M+,-(l-8) Mt-, (e+elt-at-ii ft)]
t-_0                            t=O
T          T              T           T
+ E ic. 0' at + 2; At 0'(et-at) + E, Xt Ot ft + E 4t Ot (dt-ft)          (al)
t-O       tV=             t7=0       t=O
First order conditions for problem are as follows. First, Lagrange multipliers Kt, (Y,, X, and ,t associated
with inequality constraints are positive or zero, and strictly positive only if the corresponding constraint is
binding. Second, partial derivatives of the Lagrangian with respect to all variables are zero.
Let us start by fossil-fuel abatement. Equation (a2) states that along the optimal path, when the
abatement is strictly positive but does not offset all the emissions, i.e. when Langrange multipliers Kt and
a, are zero, marginal abatement costs are equal to the shadow price of carbon scaled down by parameter ,B
to take into account instantaneous carbon uptake. There is of course no incentive to abate during the last
period T (a3)]9. We come back to this border effect below.
aL
aL _ 0     =>     Ct(a,) = St 4 + (Kt - at)       for 0 < t <T-1             (a2)
aL
dL _ o            CT(aT) = KT - UT    =>    aT =                             (a3)
aaT
Marginal sequestration costs follow exactly the same conditions (a4-a5). The average carbon content
per hectare l in equation (a4) simply translates costs of carbon (in monetary unit per quantity of carbon)
into costs of land (in monetary unit per surface of land). Eliminating Xt in equations (a2) and (a4) and
assuming none of the constraints is binding yield equalization of marginal abatement costs and marginal
sequestration costs (8).
OL
df- =      =0     CLt(ft) = Xt Ti D + (Xt - e)    for O < t: <T- I           (a4)
EL=               CLT(fT) = XT -         'T fT = 0                           (a5)
The dynamics of model (1)-(7) is therefore governed by the variations of the shadow  price of
atmospheric carbon Xt given in (a6) and (a7) below. By backward induction, equations (a6) and (a7)
define a unique trajectory of carbon price, which in turns define a unique trajectory of abatement and
sequestration.
" It would have been possible in theory to eliminate concentration from the objective function by replacing M, with its
value, a function of ao, ..., at, and fo,  f, l. We avoid this fastidious operation by considering M, (I<t�T) as variables
constrained by the carbon cycle (6).
t9 By definition of Lagrange multipliers, KCT aT = 0 and OTŽO. If aT were strictly positive, then KT would be zero. By
equation (a3), we would then have negative marginal abatement costs, which is impossible by definition of functions C.
Consequently, aT=O.
24



aM =° =>   Xt-I6)  1- l6   > St-l = 0 (l-o) )4 + 0 Dt(Mt) (1 < t <T-1) (a6)
aL =0           XT- = DT(MT) f                                       (a7)
As indicated in 2.4, we choose T so that concentration at this period is back to current level even in the
baseline emission scenario. Any abatement prior to this date will only make concentration come back to
its current level more rapidly. As we also assumed marginal damages would be lower than current
marginal damages, we obtain that the discounted shadow price of carbon is (far) lower at period T- I than
it is today, i.e. OT-1 XT.I < X0 (see footnote 13). Consequently, the discounted shadow price of carbon will
eventually start decreasing at some future point of time.
Annex 2: Optimal climate policy with temporary sequestration
Using the same notations as in annex 1, the Lagrangian for problem (1,2,3,4,6,7,10,11) is:
T                            r-1
L = - E [Ct(at) + CLt(ft) + Dt(Mj)] Ot + E At Ot [Mt+±- (1-6) Mt-t (et+elt-at-i ft)]
t-o                         t=o0
T-1
+ O X' et [Mt+1-(l -6) Mt-I3 (et+elt-at-iq ft+l ft.,)]
T         T             T         T
+  Ect Ot at + I cat 0t(et-at) + E Xt Ot ft + E it O (dt-ft)         (a8)
P=O      P=O           t=O        t=O
First order conditions are identical to the ones obtained in annex 1 above, with the exception of partial
derivatives with regard to ft, which now assume three different expressions depending on the value of t.
Like (a5) above, equation (al 1) simply translates the fact that there is no incentive to act at the last period.
For the X preceding periods, sequestration appears "permanent" for the planner, as carbon would only be
released beyond horizon T. The first order equation governing sequestration for these last periods (al 0) is
therefore identical to the one obtained in the permanent sequestration case (a4).
aL                          % L(t =(    t+, 0 ) il D + (Xt - 4t)   for 0 < t <T-r- I    (a9)
aft
-=0 o           Cft) = Xt n P + (xt - t0           for T-c < t <T-1   (alO)
aL
-=0            CLT(fT) = XT - T    > fT = °                        (al 1)
The results given in section 4 follow. Indeed, for all periods before T-c, which constitute most of the
periods if we assume T to be large and r to be small with regard to T, the optimal use of sequestration
depend on the difference between the discounted value of carbon when it is sequestered and when it is
released (equation a9). Precisely, if Xt - 2t+,0T < 0 - i.e. if the shadow price of carbon grows at a rate
higher than the rate of discount between periods t and t+-- - then X, must become strictly positive to
ensure that marginal sequestration costs do not become negative. Which means, by definition of Lagrange
multiplier X, that fe=°: no sequestration is undertaken. On the contrary, if X , - X,+.0 > 0 - i.e. if the shadow
25



value of atmospheric carbon grows at a rate lower than the rate of discount between t and t+r - then ft
becomes positive: temporary sequestration is used along the optimal path. This distinction is
straightforward: temporary sequestering carbon is indeed cost-effective if and only if, at the margin, the
value of one ton emitted later is lower (in discounted terms) than the value of one ton emitted now.
Annex 3: Optimal climate policy with both permanent and temporary sequestration
The Lagrangian for problem (1,2,3,5,6,18,19,20) is given below. Lagrange multipliers Xt are associated
with constraints (19).
T               t                 T-1
L =-  [Ct(a,) + CL,( Efq) + D,(Mt)] Ot + E Xt Ot [Mt,4-(I-8) M-P (et+elta,-rq ft)]
t=o            q=o                 t-=o
T          T              T        t      T
+ E K, Ot at + E u, 0t(et-at) +     Xt 0*.  fq + E  et Et (dt-ft)          (al2)
t'=O      t=O             t        =0    t=O
First order conditions are again identical to the ones obtained in the permanent sequestration case, with
exception of the derivatives relative to the amount of land preserved from deforestation ft.
aL         ~~T          q-= tT        T    o-
aft= 0   =>     E CLq(Fq) P3-   B      + [Y Xq Oqtjt  for 0 < t <T-1    (al3)
q-qt                     q t
aL =0   =>   CLT(fT) = XT - tT            fT = 0                           (a14)
afT
The dynamics of sequestration is governed by equation (al3) which states that, when constraints (3)
and (4) are not binding, the marginal cost of protecting one additional hectare of forest at each period
from period t to horizon T - that is the marginal cost of permanent sequestration - should be equal to the
shadow price of carbon (weighted by ,B and by the carbon per hectare ratio 1). Again, the dynamics of
sequestration depends on whether the shadow price of carbon grows at a rate higher or lower than the
discount rate, which can be seen by "differentiating" equation (al 3).
CLt(F,) = (X - 0 X,+1) ri D + %t + [E,+I 0 - it]                           (al5)
What does this equation imply for the evolution of sequestration? Let us first assume that 2t grows at a
rate strictly higher than the rate of discount between periods t and t+1 (i.e. that kt - 0 Xt+1 < 0). Under this
assumption, equation (al5) requires the term Xt + [4,1 0 - i] to be strictly positive, which occurs in two
cases:
(i)  i,1=O and Xt>, that is when no forest is protected at all at period t (F,=O, which implies Xt>O20) and
when not all forest is protected at period t+1 (f,+1<dt+1, which implies 4,i=°).
(ii)  t,+.0 - 4t>0, that is when all possible forest is protected at period t+l (f,+i=d,+1, which implies
4t+l=0). Note that in this case, we have obviously Xt±l=O, which implies that for equation (al5) to
hold at time t+l, it is necessary that ,t+2 0 - 4t+1>0.
Rules (i) and (ii) define the dynamics of At for the period where the shadow price of carbon grow s at a
rate strictly higher than the rate of discount. Three evolutions are indeed possible:
20 The fact that no forest at all is protected at period t obviously implies that the incremental amount of forest protected
at period t is zero (ft=O). Hence f,<d, and t,=O.
26



*  no sequestration at all along this period (ft=O all along the period): we are always in case (i).
*  some sequestration at first period (fo>O) and full sequestration from period I onward: we are in
case (ii) from the start.
*  or no sequestration until period t (ft=O for all t<t), and full after period t (ft=dt for all t>t): we are in
case (i) until period t- 1 and in case (ii) afterwards.
Now when the shadow price of carbon starts growing less rapidly than the rate of discount (which we
know it will eventually), i.e. when Xt - 0 X,+1 > 0, it is necessary that there is some sequestration for
equation (al 5) to hold (if indeed F, were zero, the left hand side of the equation would be zero, while the
right hand side would be strictly positive). Whether there is full sequestration or not, i.e. whether
constraint (19) is binding depends on the supply of forest, of the marginal sequestration costs, and on the
speed at which discounted price of atmospheric carbon diminishes.
27






Policy Research Working Paper Series
Contact
Title                            Author                   Date              for paper
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Aart Kraay                                 37471
WPS2616 Reforming Land and Real Estate     Ahmed Galal             June 2001          D. Dietrich
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Externalities in the Production  Jesko Hentschel                            33902
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Fragmented Markets: The Case of    Rinku Murgai                             33716
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A Preliminary Assessment         Dimitri Vittas                             37030
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Recovered Role of Education Systems                                         39744
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