the optimal carbon sequestration in agricultural soils: do the dynamics of the physical process...

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Journal ofEconomic Dynamics & Control

Journal of Economic Dynamics & Control 32 (2008) 3847– 3865

0165-18

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The optimal carbon sequestration in agricultural soils:Do the dynamics of the physical process matter?

Lionel Ragot a,b, Katheline Schubert a,c,�

a CES, Universite Paris 1 Pantheon-Sorbonne, Franceb EQUIPPE, Universites de Lille, Francec Paris School of Economics, Universite Paris 1 Pantheon-Sorbonne, 106-112 boulevard de l’Hopital, 75013 Paris, France

a r t i c l e i n f o

Article history:

Received 25 April 2007

Accepted 26 March 2008Available online 22 April 2008

JEL classification:

C61

H23

Q01

Q15

Keywords:

Environment

Agriculture

Carbon sequestration

Kyoto Protocol

Optimal control

89/$ - see front matter & 2008 Elsevier B.V

016/j.jedc.2008.03.007

esponding author at: C.E.S., 106-112 boule

144 07 82 31.

ail address: katheline.schubert@univ-paris

a b s t r a c t

The Kyoto Protocol, which came into force in February 2005,

allows countries to resort to ‘supplementary activities’, consisting

particularly in carbon sequestration in agricultural soils. Existing

papers studying the optimal carbon sequestration recognize the

importance of the temporality of sequestration, but overlook the

fact that it is an asymmetric dynamic process. This paper takes

explicitly into account the temporality of sequestration. Its first

contribution lies in the modelling of the asymmetry of the

sequestration/de-sequestration process at a micro level, and of its

consequences at a macro level. Its second contribution is

empirical. We compute numerically the optimal path of seques-

tration/de-sequestration for specific damage and cost functions,

and a calibration that mimics roughly the world conditions. We

show that with these assumptions sequestration must be

permanent, and that the error made when sequestration is

supposed immediate can be very significant.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The Kyoto Protocol, ratified by the European countries in 2002 and in force from February 2005,allows countries to resort to ‘supplementary activities’, consisting in the sequestration of carbon inforests and in agricultural soils (Articles 3.3 and 3.4). The emissions trapped by such voluntary

. All rights reserved.

vard de l’Hopital, 75647 Paris Cedex 13, France. Tel.: +33144 07 82 26;

1.fr (K. Schubert).

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mailto:[email protected]

ARTICLE IN PRESSL. Ragot, K. Schubert / Journal of Economic Dynamics & Control 32 (2008) 3847–38653848

activities, set up after 1990, can be deduced from the emissions of greenhouse gas. They can be theresult of afforestation projects (Article 3.3) or of changes of practices in the agricultural and forestrysectors (Article 3.4). As far as this last Article is concerned, the list of eligible activities proposed bythe IPCC (2000) gives appreciable opportunities of reduction of emissions to countries possessingsignificant surfaces of agricultural land. For example, gross French emissions of greenhouse gas wereestimated at 148 MtC in 2000. France emits low levels of GHG per capita and will encounterdifficulties in further reducing its emissions, in part because of the importance of French nuclearenergy generating capacity. Given the area of land devoted to agriculture, the prospects opened byArticle 3.4 may be of interest for the French policy of greenhouse gas mitigation. The French NationalInstitute for Agricultural Research has estimated the potential additional carbon storage for the next20 years, between 1 and 3 MtC/year, for the whole of mainland France (INRA, 2002). This potential isequivalent to 1%–2% of annual French greenhouse gas emissions, a large proportion of the effortsrequired to comply with the commitments to the Kyoto Protocol. At the European level, this potentialis estimated at 1.5%–1.7% of the EU-15 anthropogenic CO2 emissions during the first commitmentperiod (European Climate Change Programme, 2003).

The Bonn Agreement (COP6bis) in July 2001 clarifies the implementation of Article 3.4: eligibleactivities in agriculture comprise ‘cropland management’, ‘grazing land management’ and‘revegetation’, provided that these activities have occurred since 1990 and are human-induced.Carbon sequestration can occur either through a reduction in soil disturbance (zero tillage or reducedtillage, set-aside land, growth of perennial cropsy) or through an increase of the carbon input to thesoil (animal manure, sewage sludge, composty). Switching from conventional arable agriculture toother land-uses with higher carbon input or reduced disturbance can also increase the soil carbonstock (conversion of arable land to grassland or woodland, organic farmingy).

Lal et al. (1998) provide estimates of the carbon sequestration potential of agriculturalmanagement options in the USA. A few studies present estimations of agricultural soil carbonsequestration potentials for EU-15 (see European Climate Change Programme, 2003), and one studydoes the same for France (INRA, 2002). They all show that soil carbon sequestration is a non-linearprocess. Increases in soil carbon are often greatest soon after a land use or land management changeis implemented. There is also a sink saturation effect: as the soil reaches a new equilibrium, the rateof change decreases, so that after 20 to 100 years a new equilibrium is reached and no further changetakes place. Moreover, by changing agricultural management or land-use, soil carbon is lost morerapidly than it accumulates (Smith et al., 1996; INRA, 2002). Carbon de-sequestration is far fasterthan sequestration or, to put it differently, carbon storage in agricultural soils takes far more timethan carbon release (the unit of measure of the storage time is tens of years, the one of release isyears, see INRA, 2002).

The aim of this paper is to study the optimal policy of carbon sequestration in agricultural soils.While many technical papers try to quantify the potential of carbon sequestration, very few studythe optimal path of sequestration. To the best of our knowledge, the paper by Feng et al. (2002) is theonly one. But their dynamic representation of the process is limited by the assumption that thesequestration potential of a unit of land on which a change in land use or land management takesplace is instantaneously obtained. Technical papers recognize the importance of the temporality ofsequestration, and we take here this temporality explicitly into account. Moreover, we also take intoaccount the fact that sequestration is an asymmetric dynamic process, which most experts in thefield of agriculture consider determining (INRA, 2002).

We adapt the Feng et al. (2002) model to take into account explicitly the dynamics of thesequestration process. The main characteristics of our model are the following. Economic activitycauses exogenous carbon emissions that accumulate into the atmosphere. Damages are associated tothe atmospheric carbon stock. Carbon sequestration in agricultural soils has a cost, depending onhow much land is devoted to it. When a change of practice occurs on a unit of land in order toenhance its carbon sequestration, it stores its potential gradually; when the unit of land returns tothe usual practice, it releases carbon more rapidly than it has stored it. The total amount of land onwhich a change of practice can take place is bounded from above. In the same way, at each date, theamount of new land that can be used to store carbon is bounded from above, as well as the amount ofland that can go back to the usual practice. This assumption expresses in a simple way the existence

ARTICLE IN PRESSL. Ragot, K. Schubert / Journal of Economic Dynamics & Control 32 (2008) 3847–3865 3849

of adjustment costs and of physical limits to the land-use changes. We suppose that it is impossibleto sequester again on a unit of land that has already been used for sequestration and then went backto the usual practice. The problem is then to find at each date the optimal amount of land newlydevoted to sequestration or de-sequestration, subject to the evolution of the carbon and land stocksand of the constraints. We characterize analytically the different possible solutions. Then, wecalibrate the model and make numerical simulations that allow us to compare our solutions to theones obtained when the dynamics and the asymmetry of the process are not taken into account.

Section 2 presents the modelling of the sequestration/de-sequestration process. Section 3 isdevoted to the exposition and the analytical resolution of the dynamic optimization problem takingthe temporality and the asymmetry of this process into consideration. In Section 4, numericalsimulations allow us to exhibit the optimal path of carbon sequestration, for specific damage andcost of sequestration functions, and to compare it with the path obtained when sequestration and de-sequestration are taken to be immediate. Section 5 briefly concludes.

2. The dynamics of carbon sequestration in agricultural soils

The total amount of agricultural land is supposed to be B40, constant over time and exogenous.Land is homogeneous and produces a unique agricultural good. The different units of land may,however, differ according to the agricultural practices used on them. Two types of practices aredistinguished: the ‘usual’ practice and a ‘sequestering’ practice, allowing land to store more carbon.All units of land are supposed to be initially cultivated with the usual practice.

Physical studies clearly show that additional sequestration is a non-linear process: it is high in thefirst years following the change of practice, then decreases and tends towards zero as the newequilibrium is approached. We adopt here, following the results of Henin and Dupuit (1945), anexponential approximation of the sequestration process.

The maximal amount of carbon that can be stored in the soil of a unit of land is c�. Let c1ðt; zÞ

represent the amount of carbon in the soil at date t for a change of practice taking place at date z, andcn the amount of carbon stored with the usual practice (c�4cnX0). The dynamics of sequestration ina unit of land is then

c1ðt; zÞ ¼ c� � ðc� � cnÞe�sðt�zÞ; tXz, (1)

where s40 is the parameter of speed of the sequestration process.Let us now consider a unit of land returning to the usual practice at date B after having

experienced the first change of practice at date z. c1ðB; zÞ is the amount of carbon that thesequestering practice has allowed the land to store until date B, and c2ðt; B; zÞ the amount of carbonremaining in the soil at date t. We have

c2ðt; B; zÞ ¼ cn þ ðc1ðB; zÞ � cnÞe�s0 ðt�BÞ; tXBXz, (2)

where s040 is the parameter of speed of the de-sequestration process. As the storage process isslower than the release process,1 we have s04s.

If the only possible change of practice is from the usual to a sequestering one (no possibility ofreturn to the usual practice), the aggregate carbon stock stored in agricultural soils at t is defined by

CðtÞ ¼ cnBþ

Z t

0aðzÞðc1ðt; zÞ � cnÞdz, (3)

where aðzÞ is the number of units of land that moved from the usual practice to a sequestering one atdate z.

The fact that the change of practice can take place at different dates on different units of landintroduces an heterogeneity among the units, even if land is initially homogeneous. At date t, all the

1 For all eligible activities studied in the INRA (2002) report the same result is obtained: the speed of de-sequestration is

greater than the speed of sequestration. See Table 2, which provides several examples. So we take s04s as the basic assumption

of our model.


units which have adopted a sequestering practice do not necessarily store the same amount ofcarbon, the amount stored by each of them depending on the date of the change of practice.

Let us suppose now that a sequence usual practice/sequestering practice/usual practice ispossible. There exist for a given unit of land three conceivable configurations at date t:

�
either it has been cultivated with the usual practice from the beginning, and the carbon stored iscn; � or it has experienced a single change of practice at date zot, and the carbon stored is c1ðt; zÞ given
by Eq. (1);
� or it has experienced the two changes of practice, respectively at dates z and B with zoBot, and
the carbon stored is c2ðt; B; zÞ given by Eq. (2).

The determination of the aggregate carbon stock is not straightforward because of theheterogeneity due to the timing of sequestration. It requires to determine which units of land mustcome back first to the usual practice. It is possible to show that the last in– first out principle applies:these units are those which have changed practice last. This is due to the fact that the benefit of areturn to the usual practice on a given unit of land is independent of the carbon stored in its soil (itsimply consists in a cost reduction), while the social damage due to this return is all the smaller sincethis carbon stock is low. It is natural then to return to the usual practice on units of land classifiedaccording to the carbon they store, those which storage is the lowest first.

To show this formally, let Dðt; B; zÞ be the difference at t in the amount of carbon stored in the soilfor two units of land, the first one having experienced a single change of practice at z and the secondone two changes, at z and B4z. We have Dðt; B; zÞ ¼ c1ðt; zÞ � c2ðt; B; zÞ (cf. Fig. 1), and we see easily that

qDðt; B; zÞqz

¼ �ðc� � cnÞse�sðt�zÞð1� e�ðs0�sÞðt�BÞÞ.

qDðt; B; zÞ=qz is strictly negative since s04s. Then, minimizing this difference requires to have z as nearB as possible, which means that the units of land that must come back first to a usual practice arethose which have just experienced their first change of practice (last in– first out).

Let us now consider two units of land, the first one having adopted the sequestering practice attime z and the second one still using the usual practice. Does it make sense to decide at B4z torelease carbon on unit 1 by returning to the usual practice, and to begin storing carbon on unit 2 byadopting the sequestering practice?

The economic benefit of returning to the usual practice on unit 1, which only consists in a costreduction, is exactly offset by the cost of adopting the sequestering practice of unit 2. Theenvironmental social damage of the double switch is positive at a given date t if the net carbonrelease is greater than the net carbon storage. Net carbon release at t is (see Fig. 2) c1ðt; zÞ � c2ðt; B; zÞ,

Fig. 1. Carbon storage and release (1).

ARTICLE IN PRESS

Fig. 2. Carbon storage and release (2).

L. Ragot, K. Schubert / Journal of Economic Dynamics & Control 32 (2008) 3847–3865 3851

while net carbon storage is c1ðt; BÞ � cn. The difference between the two is:

ðc1ðt; zÞ � c2ðt; B; zÞÞ � ðc1ðt; BÞ � cnÞ ¼ ðc� � cnÞð1� e�sðB�zÞÞðe�sðt�BÞ � e�s0 ðt�BÞÞ.

It is unambiguously positive as we know that s04s. So it does not make sense to begin storing carbonon a new unit of land while simultaneously releasing carbon on another one.

Let T 0 be the date at which carbon release begins at a macro level. The last in– first out principleallows us to define the date at which sequestration that stops in a given unit of land at date B hasbegun, denoted wðBÞ, byZ T 0

wðBÞaðtÞdt ¼

Z B

T 0bðtÞdt, (4)

where bðtÞ is the number of units of land that go back to the usual practice at date t. The left-hand-side is the total number of units having adopted the sequestering practice between wðBÞ and T 0 (in);the right-hand-side is the total number of units having returned to the usual practice between T 0 andB (out); the definition of date wðBÞ ensures that the two are equal: all units having begun to sequesterafter wðBÞ have already returned to the usual practice at B.

The last in–first out principle and the fact that simultaneous decisions of carbon sequestrationand de-sequestration cannot occur allow us to compute the aggregate carbon stock stored inagricultural soils:

CðtÞ ¼ cnBþ

Z T 0

0aðzÞðc1ðt; zÞ � cnÞdz�

Z t

T 0bðzÞDðt; z; wðzÞÞdz. (5)

Notice that we do not consider situations in which a unit of land would experience three or moresuccessive changes of practice. With this assumption, we exclude the possibility of optimal cycles.Such cycles would have been very unusual, because the problem does not reduce to a succession ofidentical patterns of sequestration/de-sequestration. In fact, at the beginning of the firstsequestration stage, land is homogeneous and all units store the same basic amount of carbon cn.But at the end of the first de-sequestration stage (that could become the beginning of the secondsequestration stage), land is not homogeneous any more: units store different amounts of carbon,depending on their history. It does not seem possible to take that heterogeneity into accountproperly.

To sum up, the aggregate carbon stock sequestered in agricultural soils is (with ec ¼ c� � cn):

CðtÞ ¼ cnBþ ecZ t

0aðzÞð1� e�sðt�zÞÞdz 8tpT 0;

CðtÞ ¼ cnBþ ecZ T 0

0aðzÞð1� e�sðt�zÞÞdz� ecZ t

T 0bðzÞðð1� e�sðt�wðzÞÞÞ

�ð1� e�sðz�wðzÞÞÞe�s0 ðt�zÞÞdz 8t4T 0;

8>>>>>><>>>>>>:(6)


and the corresponding flow of carbon emissions trapped at each date in soil is:

FðtÞ ¼ ecs

Z t

0aðzÞe�sðt�zÞ dz 8tpT 0;

FðtÞ ¼ ecs

Z T 0

0aðzÞe�sðt�zÞ dz� ecZ t

T 0bðzÞðse�sðt�wðzÞÞ

þs0ð1� e�sðz�wðzÞÞÞe�s0 ðt�zÞÞdz 8t4T 0:

8>>>>>><>>>>>>:(7)

Finally, the stock of agricultural land cultivated with the sequestering practice at date t is:

AðtÞ ¼

Z t

0aðzÞdz 8tpT 0;

AðtÞ ¼

Z T 0

0aðzÞdz�

Z t

T 0bðzÞdz 8t4T 0

8>>><>>>: (8)

with 0pAðtÞpB 8t.

3. The optimal solution

3.1. The social planner’s program

The social planner minimizes the total cost associated to the emissions of carbon by the economicactivity, composed of the sum of the damage due to the accumulation of carbon in the atmosphereand of the costs of carbon sequestration in agricultural soils.

The carbon stock in the atmosphere is subject to a natural assimilation process at the constantrate d40; besides, it is augmented by net emissions, equal to exogenous raw emissions E minusemissions removed from the atmosphere by sequestration in agricultural soils FðtÞ. The equation ofaccumulation of the carbon stock is then

_SðtÞ ¼ �dSðtÞ þ E� FðtÞ, (9)

where FðtÞ is given by Eq. (7). Emissions E are considered as exogenous in order to concentrate onsequestration decisions, without complicating the analysis by questions about the trade-off betweensequestration and abatement. The initial carbon stock is given, Sð0Þ ¼ S040.

The damage function is DðSðtÞÞ, with D0ðSðtÞÞ40. We do not make any a priori assumption onD00ðSðtÞÞ: the marginal damage can be decreasing, constant or increasing. The cost of sequestration isKðAðtÞÞ, with K 0ðAðtÞÞ40 and K 00ðAðtÞÞ40. This cost is an opportunity cost, and also a cost associated tothe new equipment that the change of practice requires.

Feng et al. (2002) suppose that because of various technical and economic constraints, thenumber of units of land on which a change of practice can take place at each date is bounded fromabove: aðzÞpa 8z. To avoid unessential technical difficulties (see footnote 2), we make the strongerassumption that a is a discrete variable that can only take the two values 0 and a: the number of unitsof land on which the practice can change is either nil or maximum. For the same reasons, we makethe assumption that the number of units of land that can go back to the usual practice at each date iseither 0 or a : bðtÞ ¼ f0; ag 8t.

The social discount rate is r, that we suppose constant. The date at which carbon release begins totake place is T 0.

The social planner seeks to minimize

V ¼

Z 10

e�rtðDðSðtÞÞ þ KðAðtÞÞÞdt,


subject to

_SðtÞ ¼ �dSðtÞ þ E� FðtÞ;_AðtÞ ¼ aðtÞ � bðtÞ;

FðtÞ ¼ ecs

Z t

0aðzÞe�sðt�zÞ dz 8tpT 0;

FðtÞ ¼ ecs

Z T 0


T 0bðzÞðse�sðt�wðzÞÞ

þs0ð1� e�sðz�wðzÞÞÞe�s0 ðt�zÞÞdz 8t4T 0;

8>>>>>><>>>>>>:Z T 0

wðzÞaðtÞdt ¼

Z z

T 0bðtÞdt;

0pAðtÞpB; aðtÞ ¼ f0; ag and bðtÞ ¼ f0; ag 8t;

Sð0Þ ¼ S0 and Að0Þ ¼ 0 given:

8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>:The flow of carbon stored in agricultural soils at instant t, FðtÞ, is determined differently depending

on the relevant stage: carbon storage (before T 0) or carbon release (after T 0). The problem is then atwo-stage optimal control problem. Besides, the switching time T 0 appears as an argument in theequation of accumulation of the atmospheric carbon stock through the value of FðtÞ. The solution ofthis type of problems has been provided by Tomiyama and Rossana (1989). Moreover, we have anintegral state equation, again through the value of FðtÞ. Kamien and Muller (1976) give necessary andsufficient conditions of optimality in this case. We then solve our problem by adapting Tomiyamaand Rossana (1989) and Kamien and Muller (1976) results to our case.

3.2. After T 0: de-sequestration

The social planner solves an optimization program in infinite horizon, beginning at the date T 0 atwhich de-sequestration begins. The values of the stocks SðT 0Þ and AðT 0ÞpB are taken as given andconstitute the initial conditions of the problem after T 0. By definition of T 0, aðtÞ ¼ 0 8tXT 0.

The social planner’s program is given by

minVþðSðT0Þ;AðT 0ÞÞ ¼

Z 1T 0

e�rtðDðSðtÞÞ þ KðAðtÞÞÞdt;

_SðtÞ ¼ �dSðtÞ þ E� FðtÞ;_AðtÞ ¼ �bðtÞ;

FðtÞ ¼ ecs

Z T 0


T 0bðzÞðse�sðt�wðzÞÞ þ s0ð1� e�sðz�wðzÞÞÞe�s0 ðt�zÞÞdz;Z T 0

wðzÞaðtÞdt ¼

Z z

T 0bðtÞdt;

0pAðtÞpB and bðtÞ ¼ f0; ag 8t;

SðT 0Þ and AðT 0Þ given.

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

(10)

Let l1þ be the (positive) shadow price of the carbon stock S and l2þ the one of the land used forsequestration A. Following Kamien and Muller (1976), the proper way to write the current valueHamiltonian of this type of problems is

HþðtÞ ¼ � e�rtðDðSðtÞÞ þ KðAðtÞÞÞ

� l1þðtÞ �dSðtÞ þ E� ecs

Z T 0

0aðzÞe�sðt�zÞ dz

!þ l2þðtÞbðtÞ

� ecbðtÞ

Z 1t

l1þðzÞðse�sðz�wðtÞÞ þ s0ð1� e�sðt�wðtÞÞÞe�s0 ðz�tÞÞdz. (11)

The second term of this Hamiltonian is the variation of carbon in the atmosphere at date t if norelease takes place after T 0, i.e. if the flow of carbon trapped in the soil is FðtÞ ¼ ecs

R T 0

0 aðzÞe�sðt�zÞ dz,


evaluated at the shadow price of carbon l1þðtÞ. The last term allows to take release into account. Itcan be interpreted as the cost associated to carbon release. It is the sum of the flows of carbonreleased from t to infinity by a unit of land returning to the usual practice at t, evaluated each date atthe current shadow value of carbon, times the number of units returning to the usual practice at t.

The Hamiltonian is linear in the control variable bðtÞ. The coefficient associated to bðtÞ is

bðtÞ ¼ l2þðtÞ � ec Z 1t

l1þðzÞðse�sðz�wðtÞÞ þ s0ð1� e�sðt�wðtÞÞÞe�s0 ðz�tÞÞdz. (12)

It is the difference between the marginal benefit, which is the shadow price of the land used forsequestration l2þ, and the marginal cost associated to carbon release. As long as this coefficient bðtÞ ispositive, the benefit of de-sequestration is greater than its cost and bðtÞ ¼ a; when it becomesnegative, carbon release stops and bðtÞ ¼ 0.2 It may happen that release stops before bðtÞ becomesnegative, if all the units of land that have been used for sequestration are back to the usual practice.Let T 00 be the date at which de-sequestration stops. Between T 0 and T 00, we have _AðtÞ ¼ �a and then

AðtÞ ¼ AðT 0Þ � aðt � T 0Þ; T 0ptpT 00.

As we have just pointed out, two configurations may occur.

�

cou

in o

dat

info

(0o

If b never becomes negative after T 0, carbon release takes place until all land used forsequestration has returned to the usual practice (corner solution). Date T 001 is defined as the date atwhich this happens:

AðT 001Þ ¼ 0 3 T 001 ¼AðT 0Þ

aþ T 0.

�
If there exists a date at which b becomes negative (interior solution), this date is denoted T 002 (withT 0oT 002oT 001). We then have bðT 002Þ ¼ 0 and bðtÞ ¼ 0 and AðtÞ ¼ AðT 002Þ 8tXT 002.
The first order necessary conditions for an interior solution are

qHþðtÞ

qSðtÞ� _l1þðtÞ ¼ 0 3 � e�rtD0ðSðtÞÞ þ dl1þðtÞ � _l1þðtÞ ¼ 0, (13)

qHþðtÞ

qAðtÞ� _l2þðtÞ ¼ 0 3 � e�rtK 0ðAðtÞÞ � _l2þðtÞ ¼ 0, (14)

while the transversality conditions are

limt!1

l1þðtÞSðtÞ ¼ 0, (15)

limt!1

l2þðtÞAðtÞ ¼ 0. (16)

As we shall see below, the long term of this economy is a stationary state, where the carbon stockand the amount of land devoted to sequestration are constants. The transversality conditions canthen be written as

limt!1

l1þðtÞ ¼ 0, (17)

limt!1

l2þðtÞ ¼ 0. (18)

2 Without the assumption that bðtÞ is a discrete variable, we could have had 0obðtÞoa for an interval of values of t, that

ld not have been reduced to a single value. Feng et al. (2002) obtain this kind of solution. This would have been untractable

ur problem. Moreover, we are mainly interested in the timing of sequestration/de-sequestration. In any case, before the

e T 00 at which de-sequestration stops, the amount of land newly releasing carbon at each date is a; after T 00 it becomes 0; the

rmation that would have been added by letting the possibility of an interior solution to appear exactly at date T 00

bðT 00Þoa) does not seem very relevant. So we think that the loss of generality due to our assumption is minor.


Integration of Eqs. (13) and (14) taking into account these last conditions yields:

l1þðtÞ ¼ edt

Z 1t

e�ðrþdÞzD0ðSðzÞÞdz; tXT 0, (19)

l2þðtÞ ¼

Z 1t

e�rzK 0ðAðzÞÞdz; tXT 0. (20)

The shadow price of the carbon stock is then the sum of the discounted (at rate r þ d, the sum of thediscount and the natural assimilation rates) future marginal damages of emissions. The shadow priceof the land used for sequestration is the sum of the discounted marginal costs of the change ofpractice.

3.3. Before T 0: sequestration

The social planner solves an optimization program in finite horizon, considering that the horizonT 0 and the value function of the program after T 0, VþðSðT

0Þ;AðT 0ÞÞ, are given. By definition of T 0, we have

bðtÞ ¼ 0 8tpT 0. The social planner’s program is then given by

min

Z T 0

0e�rtðDðSðtÞÞ þ KðAðtÞÞÞdt þ VþðSðT

0Þ;AðT 0ÞÞ;

_SðtÞ ¼ �dSðtÞ þ E� ecs

Z t

0aðzÞe�sðt�zÞ dz;

_AðtÞ ¼ aðtÞ;

0pAðtÞpB and aðtÞ ¼ f0; ag 8t;

Sð0Þ ¼ S0; Að0Þ ¼ 0 given.

8>>>>>>>>>>><>>>>>>>>>>>:(21)

We note l1� the shadow price of the carbon stock S and l2� the one of the land used forsequestration A. The current value Hamiltonian of this program is then (Kamien and Muller, 1976):

H�ðtÞ ¼ � e�rtðDðSðtÞÞ þ KðAðtÞÞÞ � l1�ðtÞð�dSðtÞ þ EÞ � l2�ðtÞaðtÞ

þ ecsaðtÞ

Z T 0

tl1�ðzÞe

�sðz�tÞ dz. (22)

The Hamiltonian is linear in the control variable aðtÞ. The coefficient associated to aðtÞ in theHamiltonian is

aðtÞ ¼ ecs

Z T 0

tl1�ðzÞe

�sðz�tÞ dz� l2�ðtÞ. (23)

It represents the difference between the marginal benefit of sequestration and its marginal cost.The marginal benefit is the sum of the flows of carbon sequestered until T 0 by an additional unit ofland devoted to sequestration at t, evaluated at the shadow value of the carbon stock l1�. Themarginal cost is the shadow price of the land used for sequestration l2�. As long as this coefficientaðtÞ is positive, the benefit of sequestration is greater than its cost and aðtÞ ¼ a; when the coefficientbecomes negative, aðtÞ ¼ 0. It may happen that sequestration stops while aðtÞ is still positive, if all thelands suitable for sequestration are already used. The date T at which sequestration stops can thenbe:

�
T1 in the case of a corner solution, with AðT1Þ ¼ B and aðT1Þ40; � T2 in the case of an interior solution, with AðT2ÞoB and aðT2Þ ¼ 0.
Besides, between 0 and T (equal to T1 or T2) we have _AðtÞ ¼ a, which implies

AðtÞ ¼at; tpT ;

AðTÞ ¼ aT; t 2 ½T ; T 0�:

((24)


The first order necessary conditions are

qH�ðtÞ

qSðtÞ� _l1�ðtÞ ¼ 0 3 � e�rtD0ðSðtÞÞ þ dl1�ðtÞ � _l1�ðtÞ ¼ 0, (25)

qH�ðtÞ

qAðtÞ� _l2�ðtÞ ¼ 0 3 � e�rtC 0ðAðtÞÞ � _l2�ðtÞ ¼ 0. (26)

Integration of Eqs. (25) and (26) yields:

l1�ðtÞ ¼ edtl1�ð0Þ � edt

Z t

0e�ðrþdÞzD0ðSðzÞÞdz; tpT 0, (27)

l2�ðtÞ ¼ l2�ð0Þ �

Z t

0e�rzK 0ðAðzÞÞdz; tpT 0. (28)

3.4. The links between the two stages

3.4.1. Beginning and end of sequestration on a given unit of land

The first link between the two stages is specific to each unit of land. It is the date wðzÞpT 0 at whichsequestration that stops in a given unit of land at date zXT 0 has begun, determined by the lastin–first out principle according to Eq. (4).

For z 2 ½T 0; T 00�, wðzÞ 2 ½0; T�. Eq. (4) can then be written

a

Z T

wðzÞdt ¼ a

Z z

T 0dt,

which allows us to obtain wðzÞ:

wðzÞ ¼ T þ T 0 � z. (29)

It is now possible to make explicit the equation bðT 00Þ ¼ 0, which will yield the date T 00 at which de-sequestration stops in the second stage.

We can first obtain the carbon stock in Eq. (19), by integration of its equation of accumulation (9):

SðtÞ ¼ e�dðt�T 0 ÞSðT 0Þ þE

dð1� e�dðt�T 0 ÞÞ �

Z t

T 0e�dðt�zÞFðzÞdz, (30)

the flow of carbon sequestered in agricultural soils being given by (using Eqs. (7) and (29)):

FðtÞ ¼ ecae�stðesT � 1Þ � eca

�

e�sðt�TÞð1� e�sðt�T 0 ÞÞ þ ð1� e�s0 ðt�T 0 ÞÞ

�s0

s0 � 2sesðTþT 0 Þe�s0tðeðs

0�2sÞt � eðs0�2sÞT 0 Þ; T 0ptpT 00;

e�sðt�TÞð1� e�sðT 00�T 0 ÞÞ þ e�s0tðes0T 00 � es0T 0 Þ

�s0

s0 � 2sesðTþT 0 Þe�s0tðeðs

0�2sÞT 00 � eðs0�2sÞT 0 Þ; tXT 00:

8>>>>>>>>><>>>>>>>>>:(31)

We obtain SðtÞ 8t 2 ½T 0;1½ as a function of SðT 0Þ, T and T 00 which are still unknown.Secondly, the amount of land devoted to sequestration in Eq. (20) is

AðtÞ ¼aðT þ T 0 � tÞ; T 0ptpT 00;

AðT 00Þ ¼ aðT þ T 0 � T 00Þ; tXT 00:

((32)


Lastly, it is possible to calculate bðT 00Þ, using Eqs. (12), (19), (20) and (29), as a function of T and T 0

which are still unknown. bðT 00Þ ¼ 0 then reads

0 ¼

Z 1T 00

e�rzK 0ðAðzÞÞdz� ec Z 1T 00

Z 1z

e�ðrþdÞtD0ðSðtÞÞdt� �

edzðse�sðz�ðTþT 0�T 00 ÞÞ

þ s0ð1� e�sð2T 00�ðTþT 0 ÞÞÞe�s0 ðz�T 00 ÞÞdz. (33)

In the same way, it is possible to make explicit the equation aðTÞ ¼ 0, which will yield the date T atwhich sequestration stops in the first stage.

We obtain first the carbon stock in Eq. (27) by integration of its equation of accumulation (9):

SðtÞ ¼ e�dtS0 þE

dð1� e�dtÞ �

Z t

0e�dðt�zÞFðzÞdz, (34)

with

FðtÞ ¼ ecað1� e�stÞ; 0ptpT ;

e�stðesT � 1Þ; TptpT 0:

((35)

It yields SðtÞ 8t 2 ½0; T 0�, as a function of T which is still unknown.Secondly, the amount of land devoted to sequestration in Eq. (28) is given by Eq. (24).Lastly aðTÞ ¼ 0 reads, using Eqs. (23), (27) and (28):

0 ¼ ecs

Z T 0

Tl1�ð0Þ �

Z z

0e�ðrþdÞtD0ðSðtÞÞdt

� �edze�sðz�TÞ dz� l2�ð0Þ

þ

Z T

0e�rzK 0ðAðzÞÞdz, (36)

where T 0, l1�ð0Þ and l2�ð0Þ are still unknown. The two initial values of the shadow prices and the dateT 0 separating the two stages of the problem will be given by the matching conditions.

3.4.2. The matching conditions

The second link between the two stages of the problem is constituted of the matching conditions,characterizing how the shadow prices and the value functions behave at the switching time T 0.Tomiyama and Rossana (1989) show that these conditions write, in the case of an interior solution0oT 0o1:

l�1�ðT0Þ ¼ l�1þðT

0Þ, (37)

l�2�ðT0Þ ¼ l�2þðT

0Þ, (38)

H��ðT0Þ þ

Z T 0

0

qH��ðtÞ

qT 0dt ¼H�

þðT0Þ �

Z 1T 0

qH�þðtÞ

qT 0dt. (39)

The first two conditions express the continuity of the shadow prices of carbon and land betweenthe two stages. The last one is more intricate. It shows that the Hamiltonians do not match at theswitching time, due to the dependence of the determinants of the atmospheric carbon accumulationto the switching time.3

In the case of a corner solution the last one of these conditions becomes an inequality. If the left-hand side of the inequality is greater than the right-hand side, the optimal solution is to sequestercarbon in the agricultural soils permanently: T 0 ! 1. If the LHS is smaller than the RHS, the optimalsolution is T 0 ¼ 0 and then T ¼ 0: it is never optimal to begin storing carbon in agricultural soils.

3 Notice that Eq. (39) reduces to the usual condition of equality of the Hamiltonians at the switching time T 0 if the

Hamiltonians are independent of T 0 .


Using Eqs. (27) and (19), Eq. (37) reduces to

l1�ð0Þ ¼

Z 10

e�ðrþdÞzD0ðSðzÞÞdz. (40)

Using Eqs. (28) and (20), Eq. (38) becomes

l2�ð0Þ ¼

Z 10

e�rzK 0ðAðzÞÞdz

¼

Z T

0e�rzK 0ðAðzÞÞdzþ

K 0ðAðTÞÞ

rðe�rT � e�rT 0 Þ

þ

Z T 00

T 0e�rzK 0ðAðzÞÞdzþ

K 0ðAðT 00ÞÞ

re�rT 00 . (41)

Eqs. (22) and (11) allow us to compute the last matching condition (39). After taking into accountthe first two matching conditions, using aðT 0Þ ¼ 0, bðT 0Þ ¼ a and Eq. (29), and simplifying(calculations are tedious but straightforward), Eq. (39) reduces to

0 ¼ l2þðT0Þ � ec Z 1

T 0l1þðzÞðse�sðz�wðT 0 ÞÞ þ s0ð1� e�sðT 0�wðT 0 ÞÞÞe�s0 ðz�T 0 ÞÞdz

þ ecs

Z 1T 0

bðtÞ

a

qwðtÞqT 0

Z 1t

l1þðzÞðse�sðz�wðtÞÞ � s0e�sðt�wðtÞÞe�s0 ðz�tÞÞdz

� �dt,

i.e.

l2þðT0Þ ¼ ec Z 1

T 0l1þðzÞðse�sðz�TÞ þ s0ð1� e�sðT 0�TÞÞe�s0 ðz�T 0 ÞÞdz

� ecsesðTþT 0 Þ

Z 1T 0

bðtÞ

ae�st

Z 1t

l1þðzÞðse�sz � s0eðs0�sÞte�s0zÞdz

� �dt, (42)

l2þðT0Þ being given by Eq. (20) and l1þðzÞ by Eq. (19). This equation allows us to calculate T 0 as a

function of T and T 00.

3.5. The optimal dates

It is now possible to find the optimal dates T , T 0 and T 00, in the different configurations that canoccur. In every configuration, these dates must obviously satisfy the following inequality:

0pTpB

apT 0pT 00pT þ T 0. (43)

Any meaningful combination of the different solutions below can take place.

�
Interior solution:The date T at which sequestration stops is T2oB=a, given by aðTÞ ¼ 0, i.e., according to Eqs. (36),(40) and (41):Z 1
Te�rzK 0ðAðzÞÞdz ¼ ecsesT

Z T 0

T

Z 1z


eðd�sÞz dz. (44)

The date T 00 at which de-sequestration stops is T 002oT þ T 0, given by bðT 00Þ ¼ 0, i.e., according to Eq.(33): Z 1

T 00e�rzK 0ðAðzÞÞdz

¼ ec Z 1T 00

Z 1z


edzðse�sðz�ðTþT 0�T 00 ÞÞ

þ s0ð1� e�sð2T 00�ðTþT 0 ÞÞÞe�s0 ðz�T 00 ÞÞdz. (45)


The date T 0 at which de-sequestration begins is given by Eq. (42), using Eqs. (19) and (20):Z 1T 0

e�rzK 0ðAðzÞÞdz

¼ ec Z 1T 0

edz

Z 1z


ðse�sðz�TÞ þ s0ð1� e�sðT 0�TÞÞe�s0 ðz�T 0 ÞÞdz� ecsesðTþT 0 Þ

�

Z 1T 0

bðtÞ

ae�st

Z 1t

edz

Z 1z


ðse�sz � s0eðs0�sÞte�s0zÞdz

� �dt. (46)

This solution is the one in which benefit and cost considerations prevail over quantitativeconstraints. In the first stage, sequestration stops before saturation of the land storage capacity,because it becomes too costly. In the second stage, de-sequestration stops before all units havereturned to usual practices, because the benefits of an additional release are too small.
� Corner solution after T 0:
The date T 00 at which de-sequestration stops becomes T 001 ¼ T þ T 0. In this solution, all units returnto usual practices in the second stage.
� Corner solution before T 0:
The date T at which sequestration stops becomes T1 ¼ B=a. In the first stage, sequestration islimited by the land storage capacity.
� Corner solution for T 0:
It may happen that it is never optimal to begin to store carbon (T 0 ¼ 0) or that it is never optimalto release carbon in the atmosphere (T 0 ! 1). In these two cases, the third matching condition(42) is no longer verified. As far as its left hand side is nil, the solution T 0 ! 1 prevails if the RHSis negative, and T 0 ¼ 0 if it is positive.

It is not possible to solve explicitly and find the optimal dates in the general case that we studyhere. We will then use below specific functional forms and simulations.

3.6. The stationary state

This economy can admit two different steady states, according to the solution that prevails: nocarbon release (formally, T 0 ! 1 or T 00 ¼ T 0o1) and de-sequestration (T 0o1, T 004T 0). In the twocases, the flow of carbon sequestered and the carbon stock are given by

F� ¼ 0, (47)

S� ¼E

d, (48)

but the stock of land devoted to sequestration is different in the two cases:

A� ¼AðTÞ if T 0 ! 1 or T 00 ¼ T 0;

AðT 00Þ if T 0o1 and T 004T 0:

((49)

Eq. (48) shows that the steady state value of the carbon stock is independent of the choices madeabout carbon sequestration in agricultural soils. Sequestration only affects the dynamics to reach thesteady state.

This result is independent of the characteristics of the dynamic process of sequestration/de-sequestration, and is also obtained by Feng et al. (2002). Its explanation is the following. When a unitof carbon is emitted by the economic activity, it goes either into the atmosphere or into agriculturalsoils. In the first case, it disappears at rate d, due to natural processes of absorption. In the secondcase, it either stays in the soil indefinitely or is released after some time and goes into theatmosphere. Moreover, the amount of carbon that can be released into the atmosphere is infinitewhereas agricultural soils have a finite capacity of storage. Then, in the long run, it is impossible tomaintain a permanent flow of carbon sequestration and the only thing that determines theatmospheric carbon stock is the natural absorption capacity.


The dynamics is characterized by Eq. (9), with limt!1 FðtÞ ¼ 0 (cf. Eqs. (35) and (31)). Because ofthis last property, the dynamic equation is asymptotically autonomous. Following Benaım and Hirsch(1996), we first study the corresponding autonomous equation and then examine the convergingproperties of the asymptotically autonomous equation towards the autonomous one.

The autonomous equation is

_SðtÞ ¼ �dSðtÞ þ E, (50)

and it is obviously stable. Its solution is

SðtÞ ¼ S� þ ðS0 � S�Þe�dt .

A simple look at Eqs. (35) and (31) is sufficient to convince us that FðtÞ decreases at an asymptoticrate e�s when there is no carbon release, and at an asymptotic rate e�s0 when there is some carbonrelease. Then, the solution of the dynamic equation (9) converges to the solution of the autonomousone (50) at an asymptotic rate at least as fast as e�s in the first case, e�s0 in the second one.4 Moreover,since (9) converges asymptotically to (50), both have the same local stability properties, and (9) islocally stable.

4. Numerical analysis

4.1. The optimal dates

We suppose that the costs of sequestration are quadratic and the damage linear:

KðAðtÞÞ ¼x2

AðtÞ2; x40, (51)

DðSðtÞÞ ¼ cSðtÞ; c40. (52)

We obtain immediately from Eqs. (19), (20), (27), (28), (40) and (41) the expressions of the shadowprices:

l1þðtÞ ¼c

r þ de�rt ; tXT 0, (53)

l2þðtÞ ¼ xZ 1

te�rzAðzÞdz; tXT 0, (54)

l1�ðtÞ ¼c

r þ de�rt ; tpT 0, (55)

l2�ðtÞ ¼ xZ 1

te�rzAðzÞdz; tpT 0. (56)

In the case of an interior solution, dates T2, T 002 and T 0 are, respectively, given by Eqs. (44)–(46),which can be written here as

xaT

rþ

xa

r2ðe�rðT 00�TÞ � e�rðT 0�TÞÞ ¼

ecscðr þ dÞðr þ sÞ

ð1� e�ðrþsÞðT 0�TÞÞ, (57)

xaðT þ T 0 � T 00Þ

r¼ecc

r þ ds

r þ se�sð2T 00�ðTþT 0 ÞÞ þ

s0

r þ s0ð1� e�sð2T 00�ðTþT 0 ÞÞÞ

� �, (58)

xaT

r�

xa

r2ð1� e�rðT 00�T 0 ÞÞ ¼

ecs0cðr þ dÞðr þ s0Þ

þecc

r þ ds

r þ s�

s0

r þ s0

� �e�sðT 0�TÞ

� 1�s

r þ 2sð1� e�ðrþ2sÞðT 00�T 0 ÞÞ

� �. (59)

4 The proof is directly derived from the results in Benaım and Hirsch (1996), and thus will be omitted here.


We show in the Appendix that an interior solution cannot exist, and that two different cornersolutions can occur.

�
The first one implies T 00 ¼ T 0, which means that de-sequestration stops exactly at the moment atwhich it begins, or in other words that de-sequestration never occurs, because it is always toocostly. We can then find the value of the optimal T2:
eT2 ¼ecscr

xaðr þ dÞðr þ sÞ. (60)

eT2 must be less than B=a.5 eT2 is all the higher since the benefit in terms of sequestration given bythe change of practice ec is high, the marginal damage c is high, the direct cost of sequestration xand the adjustment cost a are low, and the absorption rate d is low. Moreover, it is possible toshow that eT2 is an increasing function of the interest rate r if and only if sd4r2. eT2 is independentof the value of the speed of de-sequestration s0 (with s04s): in this case, the asymmetry of thesequestration process does not affect the optimal solution.Finally, eT2 is an increasing function of the speed of absorption s, and we have

lims!1

eT2 ¼eccr

xaðr þ dÞ. (61)

When we do not take into account the dynamics of sequestration, as in Feng et al. (2002), this dateis the optimal date at which sequestration stops. We have lims!1

eT2=eT2 ¼ 1þ r=s: the error(overvaluation of the optimal period of sequestration) made by ignoring the dynamics ofsequestration can be very significant. This result does not depend on the magnitude of s0, since norelease occurs, but comes from the fact that sequestration takes time and therefore its benefitscome gradually, which reduces the marginal value of sequestration.
� The second one is T 001 ¼ T2 þ T 0: de-sequestration occurs until all land had returned to the usual
practice (the quantitative constraint is binding). In this case, the optimal dates cannot be obtainedexplicitly.

It is impossible to find analytically which solution is the optimal one. We will compute in thenumerical simulations the total value function V� þ Vþ to discriminate between the two solutions, inthe special case of our calibration.

4.2. Numerical simulations

4.2.1. Calibration

The numerical simulations are performed using the values of the parameters given in Table 1.According to FAO data,6 the total agricultural land in the world is about 5 billion ha, and the total

arable land and land under permanent crop is about 1.5 billion ha. Of this, we suppose that 1 billionha can be used to store carbon (B ¼ 1000 Mha). We also suppose that each year a change of practicecan take place on 10 Mha (a ¼ 10 Mha), which means that the conversion of the total disposable landto carbon sequestration would take 100 years.

The initial stock of carbon in the atmosphere is 370 ppmv (or about 778 GtC), the 2001 level. Themodel is calibrated to obtain a long term stock of 450 ppmv (or about 947.25 GtC). For a rate of decayequal to 0.0125 (which means that the average life of carbon in the atmosphere is 1=0:0125 ¼ 80years), this requires emissions of 450� 0:0125 ¼ 5:625 ppmv.

Table 2 provides several examples, detailed in INRA (2002), about the potential of additional soilcarbon storage (ec) and the speeds of sequestration (s) and de-sequestration (s0).

For example, the additional carbon storage induced by the afforestation of arable land (cerealcrops with conventional working practice) is evaluated to 30 tC/ha. The speed of accumulation which

5 The solution is a corner solution before T 0 , ðT1; T0 ; T 002 ¼ T 0Þ, if and only if ecscr=xaðr þ dÞðr þ sÞXB=a.

6 Source: FAOSTAT database of Land Use Statistics.

ARTICLE IN PRESS

Table 2Parameters for the storage– release process

A B ec s s0

(tC/ha) ðA! BÞ ðB! AÞ

Cereal crops, conventional working practice Cereal crops, no tillage 12 0.02 0.04

Cereal crops, conventional working practice Forest 30 0.0175 0.07

Cereal crops, conventional working practice Permanent grassland 25 0.025 0.07

Source: INRA (2002).

Table 1Calibration

Parameter Value

a 10 Mha/y

B 1000 Mha~c 0.015 GtC/Mha

S0 370 ppmv

E 5.625 ppmv

s 0.02

d 0.0125

r 0.03

c 105

x 1

L. Ragot, K. Schubert / Journal of Economic Dynamics & Control 32 (2008) 3847–38653862

results from afforestation is estimated at a value of s ¼ 0:017, while the release is more rapid, withs0 ¼ 0:07. All potential activities studied in this report lead to the same result: the speed of de-sequestration is greater than the speed of sequestration.

According to the data reported in INRA (2002), a good estimation of the additional carbon that canbe stored in agricultural land in France is 15 tC/ha. Even if this value can be different from onegeographical area to another, we take here ec ¼ 0:015 GtC=Mha. The same study for France allows usto evaluate the speeds of sequestration and de-sequestration: s ¼ 0:02 and s0 ¼ 0:037.

As units for total costs and damage are arbitrary, we take x ¼ 1. We then choose c such that theoptimal date at which sequestration stops in the case s ¼ s0 ¼ 1, given by Eq. (61), is significantlypositive. With c ¼ 105 this date is around 20 years.

Finally, we choose r ¼ 3% per year.

4.2.2. Results

For s ¼ 0:02 and 8s04s, the computation of the total value function shows that the optimalsolution is the first corner solution (T ¼ T 0 ¼ T 00). The optimal solution always consists of storingcarbon in agricultural soils during about 20 years and never releasing it. Optimal sequestration ispermanent.

With the assumption s ¼ s0 ¼ 1, in other terms when the dynamics of sequestration is ignored(case studied by Feng et al., 2002), the optimal sequestration is equivalent to half the physicalpotential sequestration (T ¼ 50 years 3A ¼ 500 Mha). Whereas this optimal effort corresponds tothe fifth of the potential (T ¼ 20 years 3A ¼ 200 Mha) for speeds of sequestration and de-sequestration close to the empirical French ones (s ¼ 0:02 and s0 ¼ 0:03). We see that the error madewhen not taking into account the dynamics of sequestration is very significant, with an errorcoefficient (1þ r=s) of 2.5 for our calibration.

Figs. 3a and b show respectively the flow of carbon emissions trapped and the additional carbonstock stored when we assume that these last values of s and s0 are good proxis for the worldconditions and for these two sequestration efforts (T ¼ 50 and 20).

ARTICLE IN PRESS

Fig. 3. Numerical results. (a) Flow of carbon trapped. (b) Additional carbon stock sequestered.

L. Ragot, K. Schubert / Journal of Economic Dynamics & Control 32 (2008) 3847–3865 3863

The additional carbon stock sequestered converges to a steady level of 7.5 GtC when we do nottake the dynamics of the physical process into consideration in the determination of the optimalpolicy (T ¼ 50), while the optimal long term level is only 3 GtC (T ¼ 20). The omission of the dynamicprocess leads to a very significant over-sequestration.

5. Conclusion

This paper takes explicitly into account the temporality of sequestration. Its first contribution liesin the modelling of the asymmetry of the sequestration/de-sequestration process at a micro level,and of its consequences at a macro level. Its second contribution is empirical. We computenumerically the optimal path of sequestration/de-sequestration for specific damage and costfunctions, and a calibration that mimics roughly the world conditions. We show that with theseassumptions sequestration must be permanent, and that the error made when sequestration issupposed immediate can be very significant.

Interesting extensions would consist of considering factors that could make the optimalsequestration temporary. For instance, technical progress in abatement could induce a decrease ofnet emissions in the course of time, and make sequestration less useful. Or technical progress inusual practices and not in sequestering practices could increase the cost of sequestration.

Acknowledgements

The authors acknowledge financial support from the French Ministry of Ecology and SustainableDevelopment (APR GICC 2002). They gratefully thank three anonymous referees for their helpfulcomments, and the participants to the Workshop on Carbon Sequestration in Agriculture andForestry, Thessaloniki, June 2007.

Appendix

We first show that there is no interior solution for s04s.


Taking the difference between Eqs. (58) and (59) yields

xa

r2ð1� e�rðT 00�T 0 ÞÞ �

xaðT 00 � T 0Þ

r

¼ecc

r þ ds0

r þ s0�

s

r þ s

� �e�sðT 0�TÞ 1� e�2sðT 00�T 0 Þ �

s

r þ 2sð1� e�ðrþ2sÞðT 00�T 0 ÞÞ

� �,

which can be written, with X ¼ T 00 � T 0X0, FðXÞ ¼ 1� e�rX � rX andGðXÞ ¼ 1� e�2sX � s=r þ 2sð1� e�ðrþ2sÞXÞ,

xa

r2FðXÞ ¼

eccrðs0 � sÞ

ðr þ dÞðr þ sÞðr þ s0Þe�sðT 0�TÞGðXÞ.

The LHS of this equality has the same sign as FðXÞ, and the sign of the RHS depends on the sign ofs0 � s and GðXÞ.

We show easily that FðXÞp0 and GðXÞX0 8XX0 : Fð0Þ ¼ 0 and F 0ðXÞ ¼ rðe�rX � 1Þo0 8X40;Gð0Þ ¼0 and G0ðXÞ ¼ se�2sX ð2� e�rXÞ40 8XX0.

When s04s, the only solution is then X ¼ 0 i.e. T 00 ¼ T 0. For s0 ¼ s the equality reduces to FðXÞ ¼ 0,and the only solution is again T 00 ¼ T 0.

For s04s and T 00 ¼ T 0, Eqs. (58) and (59) become identical. The optimal dates T and T 0 are thensolution of the system composed of this last equation and (57):

xaT

r¼

eccs

ðr þ dÞðr þ sÞð1� e�ðrþsÞðT 0�TÞÞ;

xaT

r¼

eccs0

ðr þ dÞðr þ s0Þ1�

rðs0 � sÞ

s0ðr þ sÞe�sðT 0�TÞ

� �:

8>>><>>>:The elimination of xaT=r between these two equations yields

rðs0 � sÞ

r þ s0ð1� e�sðT 0�TÞÞ ¼ �se�ðrþsÞðT 0�TÞ,

which is impossible for T 0 2 ½T ;1½ since the LHS is positive and the RHS strictly negative.We now examine the corner solutions.There can first exist a corner solution before T 0. Then, Eqs. (58) and (59) are valid and imply as

before T 00 ¼ T 0, i.e. no de-sequestration. The date T at which sequestration stops can take two values:T 0 or B=a.

If T ¼ T 0, (58) and (59) become identical and allow us to obtain

~T2 ¼eccsr

xaðr þ dÞðr þ sÞ.

We obtain the same value ~T2 for a corner solution both before and after T 0ðT ¼ T 0 ¼ T 00Þ.This solution is valid as long as ~T2oB=a, i.e. as long as eccsr=xðr þ dÞðr þ sÞoB.If T ¼ B=a, we have

e�sT 0 ¼ e�sðB=aÞ xðr þ dÞðr þ sÞðr þ s0Þeccs0rðs� s0Þ�

s0ðr þ sÞ

rðs� s0Þ

� �,

which requires Boeccs0r=xðr þ dÞðr þ s0Þ. The solutions T ¼ B=a and T ¼ ~T2 can coexist foreccsr=xðr þ dÞðr þ sÞoBoeccs0r=xðr þ dÞðr þ s0Þ. The corner solution before T 0 (T ¼ B=a) and T 00 ¼ T 0

gives the same result.There can also exist a corner solution after T 0, T 001 ¼ T2 þ T 0, but in this case we cannot obtain the

optimal dates analytically.

References

Benaım, M., Hirsch, M.W., 1996. Asymptotic pseudotrajectories and chain recurrent flows with applications. Journal ofDynamics and Differential Equations 8, 141–176.

European Climate Change Programme, 2003. Working group on sinks related to agricultural soils. Final report.


Feng, H., Zhao, J., Kling, C., 2002. The time path and implementation of carbon sequestration. American Journal of AgriculturalEconomics 84, 134–149.

Henin, S., Dupuit, M., 1945. Essai de bilan de la matiere organique du sol. Annales Agronomiques 15, 147–157.INRA, 2002. Increasing Carbon Stocks in French Agricultural Soils? Report for the French Ministry for Ecology and Sustainable

Development, October.IPCC, 2000. Land Use, Land-use Change and Forestry (LULUCF). Cambridge University Press, UKKamien, M.I., Muller, E., 1976. Optimal control with integral state equation. The Review of Economic Studies 43 (3), 469–473.Lal, R., Kimble, J.M., Follet, R.F., Cole, C.V., 1998. The Potential of US Cropland to Sequester Carbon and Mitigate the Greenhouse

Effect. Ann Arbor Press, Chelsea, MI.Smith, P., Smith, J.U., Powlson, D.S., 1996. Soil Organic Matter Network: 1996 Model and Experimental Metadata. GCTE Report

7, GCTE Focus 3, Wallingford, Oxon.Tomiyama, K., Rossana, R., 1989. Two-stage optimal control problems with an explicit switch point dependence: optimality

criteria and an example of delivery lags and investment. Journal of Economic Dynamics and Control 13, 319–337.

the optimal carbon sequestration in agricultural soils: do the dynamics of the physical process...

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