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

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

Lionel Ragot a,b, Katheline Schua CES, Universite Paris 1 Pantheon-Sorbonne, France

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forests and in agricultural soils (Articles 3.3 and 3.4). The emissions trapped by such voluntary

ARTICLE IN PRESS

Contents lists available at ScienceDirect

Journal ofEconomic Dynamics & Control

Journal of Economic Dynamics & Control 32 (2008) 384738650165-1889/$ - see front matter & 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.jedc.2008.03.007

Corresponding author at: C.E.S., 106-112 boulevard de lHopital, 75647 Paris Cedex 13, France. Tel.: +33144078226;fax: +33144078231.

E-mail address: katheline.schubert@univ-paris1.fr (K. Schubert).The Kyoto Protocol, ratied 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 in1. IntroductionEnvironment

Agriculture

Carbon sequestration

Kyoto Protocol

Optimal control

empirical. We compute numerically the optimal path of seques-

tration/de-sequestration for specic 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 signicant.

& 2008 Elsevier B.V. All rights reserved.b EQUIPPE, Universites de Lille, Francec Paris School of Economics, Universite Paris 1

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 classication:

C61

H23

Q01

Q15

Keywords:bert a,c,

eon-Sorbonne, 106-112 boulevard de lHopital, 75013 Paris, France

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 rst

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 isjournal homepage: www.elsevier.com/locate/jedc

ARTICLE IN PRESSactivities, 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 possessingsignicant surfaces of agricultural land. For example, gross French emissions of greenhouse gas wereestimated at 148MtC in 2000. France emits low levels of GHG per capita and will encounterdifculties 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 3MtC/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 rst commitmentperiod (European Climate Change Programme, 2003).

The Bonn Agreement (COP6bis) in July 2001 claries the implementation of Article 3.4: eligibleactivities in agriculture comprise cropland management, grazing land management andrevegetation, 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 theeld 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 of

L. Ragot, K. Schubert / Journal of Economic Dynamics & Control 32 (2008) 384738653848land that can go back to the usual practice. This assumption expresses in a simple way the existence

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 back

2 n 1 n

0

ARTICLE IN PRESSL. Ragot, K. Schubert / Journal of Economic Dynamics & Control 32 (2008) 38473865 3849where s40 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 dened by

Ct cnBZ t0azc1t; z cndz, (3)

where az 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 assumptionto the usual practice. The problem is then to nd 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 specic 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 briey 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 therst 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 c1t; zrepresent 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 (c4cnX0). The dynamics of sequestration ina unit of land is then

c1t; z c c cnestz; 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 havingexperienced the rst change of practice at date z. c1B; z is the amount of carbon that thesequestering practice has allowed the land to store until date B, and c2t; B; z the amount of carbonremaining in the soil at date t. We have

c t; B; z c c B; z c es0 tB; tXBXz, (2)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 congurations 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 c1t; z givenby Eq. (1);

or it has experienced the two changes of practice, respectively at dates z and B with zoBot, andthe carbon stored is c2t; B; z 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 rst to the usual practice. It is possible to show that the last in rst out principle applies:

ARTICLE IN PRESSL. Ragot, K. Schubert / Journal of Economic Dynamics & Control 32 (2008) 384738653850these units are those which have changed practice last. This is due to the fact that the benet 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 classiedaccording to the carbon they store, those which storage is the lowest rst.

To show this formally, let Dt; B; z be the difference at t in the amount of carbon stored in the soilfor two units of land, the rst one having experienced a single change of practice at z and the secondone two changes, at z and B4z. We have Dt; B; z c1t; z c2t; B; z (cf. Fig. 1), and we see easily that

qDt; B; zqz

c cnsestz1 es0stB.

qDt; B; z=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 rst to a usual practice arethose which have just experienced their rst change of practice (last in rst out).

Let us now consider two units of land, the rst 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 benet 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) c1t; z c2t; B; z,Fig. 1. Carbon storage and release (1).

ARTICLE IN PRESSwhile net carbon storage is c1t; B cn. The difference between the two is:

c1t; z c2t; B; z c1t; B cn c cn1 esBzestB es0 tB.

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 rst out principleallows us to dene the date at which sequestration...

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