The optimal carbon sequestration in agricultural soils: do the dynamics of the physical process matter? Lionel Ragot a;b and Katheline Schubert a;c a CES, Université Paris 1 Panthéon-Sorbonne b EQUIPPE, Universités de Lille c

Paris School of Economics

4th January 2008

The authors acknowledge …nancial support from the French Ministry of Ecology and Sustainable Development (APR GICC 2002). They gratefully thank three anonymous referees for their helpful comments, and the participants to the Workshop on Carbon Sequestration in Agriculture and Forestry, Thessaloniki, June 2007. The usual disclaimers apply.

1

Abstract The Kyoto Protocol, which came in 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 is empirical. We compute numerically the optimal path of sequestration / de-sequestration for speci…c 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 signi…cant. JEL Classi…cation: C61, H23, Q01, Q15 Keywords: Environment, agriculture, carbon sequestration, Kyoto Protocol, optimal control.

2

1

Introduction

The Kyoto Protocol, rati…ed 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 in forests and in agricultural soils (Articles 3.3 and 3.4). The emissions trapped by such voluntary activities, set up after 1990, can be deduced from the emissions of greenhouse gas. They can be the result of a¤orestation projects (Article 3.3) or of changes of practices in the agricultural and forestry sectors (Article 3.4). As far as this last Article is concerned, the list of eligible activities proposed by the IPCC gives appreciable opportunities of reduction of emissions to countries possessing signi…cant surfaces of agricultural land. For example, gross French emissions of greenhouse gas were estimated at 148 MtC in 2000. France emits low levels of GHG per capita and will encounter di¢ culties in further reducing its emissions, in part because of the importance of French nuclear energy generating capacity. Given the area of land devoted to agriculture, the prospects opened by Article 3.4 may be of interest for the French policy of greenhouse gas mitigation. The French National Institute for Agricultural Research has estimated the potential additional carbon storage for the next 20 years, between 1 and 3 MtC/year, for the whole of mainland France (INRA (2002)). This potential is equivalent to 1 to 2% of annual French greenhouse gas emissions, a large proportion of the e¤orts required to comply with the commitments to the Kyoto Protocol. At the European level, this potential is estimated at 1.5 to 1.7% of the EU-15 anthropogenic CO2 emissions during the …rst commitment period (European Climate Change Program (2003)). The Bonn Agreement (COP6bis) in July 2001 clari…es the implementation of Article 3.4: eligible activities in agriculture comprise “cropland management”, “grazing land management” and “revegetation”, provided that these activities have occured since 1990 and are human-induced. Carbon sequestration can occur either through a reduction in soil disturbance (zero tillage or reduced tillage, set-aside land, growth of perennial crops...) or through an increase of the carbon input to the soil (animal manure, sewage sludge, compost...). Switching from conventional arable agriculture to other land-uses with higher carbon input or reduced disturbance can also increase the soil carbon stock (conversion of arable land to grassland or woodland, organic farming...). Lal et al. (1998) provide estimates of the carbon sequestration potential of agricultural management options in the USA. A few studies present estimations of agricultural soil carbon sequestration potentials for EU-15 (see European Climate Change Programme (2003)), and one study does the same for France (INRA (2002)). They all show that soil carbon sequestration is a non-linear process. Increases in soil carbon are often greatest soon after a land use or land management change is implemented. There is also a sink saturation e¤ect: as the soil reaches a new equilibrium, the rate of change decreases, so that after 20 to 100 years a new equilibrium is reached and no further change takes place. Moreover, by changing agricultural management or land-use, soil carbon is lost more rapidly than it accumulates (Smith et al. (1996), INRA (2002)). Carbon de-sequestration is far faster than sequestration or, to put it di¤erently, carbon storage in agricultural soils takes far more time than carbon release (the unit of measure of the storage time is tens of years, the one of release is years, see INRA (2002)).

3

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 study the optimal path of sequestration. To the best of our knowledge, the paper by Feng, Zhao and Kling (2002) is the only one. But their dynamic representation of the process is limited by the assumption that the sequestration potential of a unit of land on which a change in land use or land management takes place is instantaneously obtained. Technical papers recognize the importance of the temporality of sequestration, and we take here this temporality explicitly into account. Moreover, we also take into account the fact that sequestration is an asymmetric dynamic process, which most experts in the …eld of agriculture consider determining (INRA (2002)). We adapt the Feng, Zhao and Kling’s (2002) model to take into account explicitly the dynamics of the sequestration process. The main characteristics of our model are the following. Economic activity causes exogenous carbon emissions that accumulate into the atmosphere. Damages are associated to the atmospheric carbon stock. Carbon sequestration in agricultural soils has a cost, depending on how much land is devoted to it. When a change of practice occurs on a unit of land in order to enhance its carbon sequestration, it stores its potential gradually; when the unit of land returns to the usual practice, it releases carbon more rapidly than it has stored it. The total amount of land on which a change of practice can take place is bounded from above. In the same way, at each date, the amount of new land that can be used to store carbon is bounded from above, as well as the amount of land 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 impossible to sequester again on a unit of land that has already been used for sequestration and then went back to the usual practice. The problem is then to …nd at each date the optimal amount of land newly devoted to sequestration or de-sequestration, subject to the evolution of the carbon and land stocks and of the constraints. We characterize analytically the di¤erent possible solutions. Then, we calibrate the model and make numerical simulations that allow us to compare our solutions to the ones 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 is devoted to the exposition and the analytical resolution of the dynamic optimization problem taking the temporality and the asymmetry of this process into consideration. In section 4, numerical simulations allow us to exhibit the optimal path of carbon sequestration, for speci…c damage and cost of sequestration functions, and to compare it with the path obtained when sequestration and de-sequestration are taken to be immediate. Section 5 brie‡y concludes.

2

The dynamics of carbon sequestration in agricultural soils

The total amount of agricultural land is supposed to be B > 0; constant over time and exogenous. Land is homogeneous and produces a unique agricultural good. The di¤erent units of land may however di¤er according to the agricultural practices used on them. Two types of practices are distinguished: 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.

4

Physical studies clearly show that additional sequestration is a non-linear process: it is high in the …rst years following the change of practice, then decreases and tends towards zero as the new equilibrium is approached. We adopt here, following the results of Hénin and Dupuis (1945), an exponential 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 c1 (t; z) represent the amount of carbon in the soil at date t for a change of practice taking place at date z, and cn the amount of carbon stored with the usual practice (c > cn 0). The dynamics of sequestration in a unit of land is then c1 (t; z) = c (c cn )e s(t z) ; t z; (1) where s > 0 is the parameter of speed of the sequestration process. Let us now consider a unit of land returning to the usual practice at date & after having experienced the …rst change of practice at date z. c1 (&; z) is the amount of carbon that the sequestering practice has allowed the land to store until date &, and c2 (t; &; z) the amount of carbon remaining in the soil at date t: We have 0 c2 (t; &; z) = cn + (c1 (&; z) cn )e s (t &) ; t & z; (2) where s0 > 0 is the parameter of speed of the de-sequestration process. As the storage process is slower than the release process1 , we have s0 > s: If the only possible change of practice is from the usual to a sequestering one (no possibility of return to the usual practice), the aggregate carbon stock stored in agricultural soils at t is de…ned by Z t C(t) = cn B + a(z)(c1 (t; z) cn )dz; (3) 0

where a(z) is the number of units of land that moved from the usual practice to a sequestering one at date z. The fact that the change of practice can take place at di¤erent dates on di¤erent units of land introduces an heterogeneity among the units, even if land is initially homogeneous. At date t; all the units which have adopted a sequestering practice do not necessarily store the same amount of carbon, 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 is possible. There exist for a given unit of land three conceivable con…gurations at date t: either it has been cultivated with the usual practice from the beginning, and the carbon stored is cn ; or it has experienced a single change of practice at date z < t, and the carbon stored is c1 (t; z) given by equation (1); or it has experienced the two changes of practice, respectively at dates z and & with z < & < t; and the carbon stored is c2 (t; &; z) given by equation (2). 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 s0 > s as the basic assumption of our model.

5

6

c1 (t; z)

c1 (&; z) (t; &; z) c2 (t; &; z) &

z

-

t

Figure 1: Carbon storage and release (1) The determination of the aggregate carbon stock is not straigthforward because of the heterogeneity due to the timing of sequestration. It requires to determine which units of land must come back …rst to the usual practice. It is possible to show that the last in – …rst out principle applies: these units are those which have changed practice last. This is due to the fact that the bene…t of a return to the usual practice on a given unit of land is independent of the carbon stored in its soil (it simply consists in a cost reduction), while the social damage due to this return is all the smaller since this carbon stock is low. It is natural then to return to the usual practice on units of land classi…ed according to the carbon they store, those which storage is the lowest …rst. To show this formally, let (t; &; z) be the di¤erence at t in the amount of carbon stored in the soil for two units of land, the …rst one having experienced a single change of practice at z and the second one two changes, at z and & > z: We have (t; &; z) = c1 (t; z) c2 (t; &; z) (cf. …gure 1), and we see easily that @ (t; &; z) 0 = (c cn )se s(t z) 1 e (s s)(t &) : @z @ (t;&;z) @z

is strictly negative since s0 > s: Then, minimizing this di¤erence requires to have z as near & as possible, which means that the units of land that must come back …rst to a usual practice are those 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 at time z and the second one still using the usual practice. Does it make sense to decide at & > z to release carbon on unit 1 by returning to the usual practice, and to begin storing carbon on unit 2 by adopting the sequestering practice? The economic bene…t of returning to the usual practice on unit 1, which only consists in a cost reduction, is exactly o¤set by the cost of adopting the sequestering practice of unit 2. The environmental social damage of the double switch is positive at a given date t if the net carbon release is greater than the net carbon storage. Net carbon release at t is (see …gure 2) c1 (t; z) c2 (t; &; z); while net carbon storage is c1 (t; &) cn : The di¤erence between the two is: (c1 (t; z)

c2 (t; &; z))

(c1 (t; &)

cn ) = (c

cn )(1

e

s(& z)

) e

s(t &)

e

s0 (t &)

:

It is unambiguously positive as we know that s0 > s: So it does not make sense to begin storing carbon on a new unit of land while simultaneously releasing carbon on another one. 6

6

r

c c1 (t; z)

r

c1 (t; &)

r

c2 (t; &; z) cn z

-

& t Figure 2: Carbon storage and release (2)

Let T 0 be the date at which carbon release begins at a macro level. The last in –…rst out principle allows us to de…ne the date at which sequestration that stops in a given unit of land at date & has begun, denoted (&); by Z T0 Z & a( )d = b( )d ; (4) T0

(&)

where b( ) is the number of units of land that go back to the usual practice at date : The left-handside is the total number of units having adopted the sequestering practice between (&) and T 0 (in); the right-hand-side is the total number of units having returned to the usual practice between T 0 and & (out); the de…nition of date (&) ensures that the two are equal: all units having begun to sequester after (&) have already returned to the usual practice at &. The last in –…rst out principle and the fact that simultaneous decisions of carbon sequestration and de-sequestration cannot occur allow us to compute the aggregate carbon stock stored in agricultural soils: C(t) = cn B +

Z

T0

a(z)(c1 (t; z)

cn )dz

Z

t

b(z) (t; z; (z))dz:

(5)

T0

0

Notice that we do not consider situations in which a unit of land would experience three or more successive 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 of identical patterns of sequestration / de-sequestration. In fact, at the beginning of the …rst sequestration stage, land is homogeneous and all units store the same basic amount of carbon cn . But at the end of the …rst de-sequestration stage (that could become the beginning of the second sequestration stage), land is not homogeneous any more: units store di¤erent amounts of carbon, depending on their history. It does not seem possible to take that heterogeneity into account properly.

7

To sum up, the aggregate carbon stock sequestered in agricultural soils is (with e c=c 8 Rt > c 0 a(z)(1 e s(t z) )dz 8t T 0 > < C(t) = cn B + e R T0 C(t) = cn B + e c 0 a(z)(1 e s(t z) )dz > R 0 > t : e c T 0 b(z) (1 e s(t (z)) ) (1 e s(z (z)) )e s (t z) dz 8t > T 0 ;

and the corresponding ‡ow of carbon emissions trapped at each date in soil is: 8 Rt > F (t) = e c s a(z)e s(t z) dz 8t T 0 > < R0T 0 F (t) = e cs 0 a(z)e s(t z) dz > Rt 0 > : e c T 0 b(z) se s(t (z)) + s0 (1 e s(z (z)) )e s (t z) dz 8t > T 0 :

cn ):

Finally, the stock of agricultural land cultivated with the sequestering practice at date t is: ( Rt A(t) = 0 a(z)dz 8t T 0 R T0 Rt 0 A(t) = 0 a(z)dz T 0 b(z)dz 8t > T with 0

3 3.1

A(t)

(6)

(7)

(8)

B 8t:

The optimal solution The social planner’s program

The social planner minimizes the total cost associated to the emissions of carbon by the economic activity, composed of the sum of the damage due to the accumulation of carbon in the atmosphere and of the costs of carbon sequestration in agricultural soils. The carbon stock in the atmosphere is subject to a natural assimilation process at the constant rate > 0; besides, it is augmented by net emissions, equal to exogenous raw emissions E minus emissions removed from the atmosphere by sequestration in agricultural soils F (t). The equation of accumulation of the carbon stock is then _ S(t) =

S(t) + E

F (t);

(9)

where F (t) is given by equation (7). Emissions E are considered as exogenous in order to concentrate on sequestration decisions, without complicating the analysis by questions about the trade-o¤ between sequestration and abatement. The initial carbon stock is given, S(0) = S0 > 0. The damage function is D(S(t)); with D0 (S(t)) > 0: We do not make any a priori assumption on D00 (S(t)): the marginal damage can be decreasing, constant or increasing. The cost of sequestration is K(A(t)); with K 0 (A(t)) > 0 and K 00 (A(t)) > 0: This cost is an opportunity cost, and also a cost associated to the new equipment that the change of practice requires. Feng et al. (2002) suppose that because of various technical and economic constraints, the number of units of land on which a change of practice can take place at each date is bounded from above: a(z) a 8z. To avoid unessential technical di¢ culties (see footnote 2), we make the stronger assumption that a is a discrete variable that can only take the two values 0 and a: the number of units of land on which

8

the practice can change is either nil or maximum. For the same reasons, we make the assumption that the number of units of land that can go back to the usual practice at each date is either 0 or a: b(t) = f0; ag 8t. The social discount rate is r; that we suppose constant. The date at which carbon release begins to take place is T 0 : The social planner seeks to minimize Z 1 e rt (D(S(t)) + K(A(t)))dt; V = 0

subject to 8 > > > > > > > > > > > > > > <

_ S(t) = S(t) + E F (t) _ A(t) = a(t) b(t) 8 Rt > cs 0 a(z)e s(t z) dz > < F (t) = e R T0 F (t) = e cs 0 a(z)e s(t z) dz > Rt > : e c T 0 b(z) se s(t (z)) + s0 (1 R T0 Rz (z) a( )d = T 0 b( )d

8t

> e s(z (z)) )e > > > > > > > > > > > 0 A(t) B, a(t) = f0; ag and b(t) = f0; ag 8t > > : S(0) = S0 and A(0) = 0 given.

s0 (t z)

dz

T0

8t > T 0

The ‡ow of carbon stored in agricultural soils at instant t; F (t); is determined di¤erently depending on the relevant stage: carbon storage (before T 0 ) or carbon release (after T 0 ). The problem is then a two-stage optimal control problem. Besides, the switching time T 0 appears as an argument in the equation of accumulation of the atmospheric carbon stock through the value of F (t). The solution of this type of problems has been provided by Tomiyama and Rossana (1989). Moreover, we have an integral state equation, again through the value of F (t): Kamien and Muller (1976) give necessary and su¢ cient conditions of optimality in this case. We then solve our problem by adapting Tomiyama and Rossana (1985) and Kamien and Muller (1976) results to our case.

3.2

After T 0 : de-sequestration

The social planner solves an optimization program in in…nite horizon, beginning at the date T 0 at which de-sequestration begins. The values of the stocks S(T 0 ) and A(T 0 ) B are taken as given and constitute the initial conditions of the problem after T 0 . By de…nition of T 0 ; a(t) = 0 8t T 0 . The social planner’s program is given by: 8 R 0 ); A(T 0 )) = 1 e rt (D(S(t)) + K(A(t)))dt > min V (S(T + > T0 > > > > > _ > S(t) = S(t) + E F (t) > > > > _ > = b(t) < A(t) R T0 Rt 0 s(t z) dz s(t (z)) + s0 (1 (10) e s(z (z)) )e s (t z) dz F (t) = e c s a(z)e e c 0 b(z) se 0 T > > R R 0 > T z > > > (z) a( )d = T 0 b( )d > > > > 0 A(t) B and b(t) = f0; ag 8t > > > : S(T 0 ) and A(T 0 ) given. 9

Let 1+ be the (positive) shadow price of the carbon stock S and 2+ the one of the land used for sequestration A. Following Kamien and Muller (1976), the proper way to write the current value Hamiltonian of this type of problems is: H+ (t) =

e

rt

(D(S(t)) + K(A(t)))

1+ (t)

e cb(t)

Z

S(t) + E 1

1+ (z)

se

e cs

s(z

Z

T0

a(z)e

s(t z)

+ s0 (1

e

dz

0

(t))

s(t

!

+

(t))

2+ (t)b(t)

)e

s0 (z t)

dz:

(11)

t

The second term of this Hamiltonian is the variation of carbon in the atmosphere at date t if no R T0 release takes place after T 0 ; i.e. if the ‡ow of carbon trapped in the soil is F (t) = e cs 0 a(z)e s(t z) dz; evaluated at the shadow price of carbon 1+ (t): The last term allows to take release into account. It can be interpreted as the cost associated to carbon release. It is the sum of the ‡ows of carbon released from t to in…nity by a unit of land returning to the usual practice at t; evaluated each date at the 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 b(t). The coe¢ cient associated to b(t) is Z 1 0 s(z (t)) (12) (t) = 2+ (t) e c + s0 (1 e s(t (t)) )e s (z t) dz: 1+ (z) se t

It is the di¤erence between the marginal bene…t, which is the shadow price of the land used for sequestration 2+ ; and the marginal cost associated to carbon release. As long as this coe¢ cient (t) is positive, the bene…t of de-sequestration is greater than its cost and b(t) = a; when it becomes negative, carbon release stops and b(t) = 02 . It may happen that release stops before (t) becomes negative, 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 A(t) = a and then A(t) = A(T 0 )

a(t

T 0 ); T 0

t

T 00 :

As we have just pointed out, two con…gurations may occur. If never becomes negative after T 0 ; carbon release takes place until all land used for sequestration has returned to the usual practice (corner solution). Date T100 is de…ned as the date at which this happens: A(T 0 ) + T 0: A(T100 ) = 0 , T100 = a 2

Without the assumption that b(t) is a discrete variable, we could have had 0 < b(t) < a for an interval of values of t; that could not have been reduced to a single value: Feng et al. (2002) obtain this kind of solution. This would have been untractable in our problem. Moreover, we are mainly interested in the timing of sequestration / de-sequestration. In any case, before the date 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 information that would have been added by letting the possibility of an interior solution to appear exactly at date T 00 (0 < b(T 00 ) < a) does not seem very relevant. So we think that the loss of generality due to our assumption is minor.

10

If there exists a date at which becomes negative (interior solution), this date is denoted T200 (with T 0 < T200 < T100 ). We then have (T200 ) = 0 and b(t) = 0 and A(t) = A(T200 ) 8t T200 : The …rst order necessary conditions for an interior solution are: @H+ (t) @S(t) @H+ (t) @A(t)

_ 1+ (t) = 0 ,

e

rt

D0 (S(t)) +

_ 2+ (t) = 0 ,

e

rt

K 0 (A(t))

1+ (t)

_ 1+ (t) = 0;

_ 2+ (t) = 0;

(13) (14)

while the transversality conditions are lim

1+ (t)S(t)

= 0;

(15)

lim

2+ (t)A(t)

= 0:

(16)

t!1 t!1

As we shall see below, the long term of this economy is a stationary state, where the carbon stock and the amount of land devoted to sequestration are constants. The transversality conditions can then be written: lim

1+ (t)

= 0;

(17)

lim

2+ (t)

= 0:

(18)

t!1 t!1

Integration of equations (13) and (14) taking into account these last conditions yields: Z 1 t e (r+ )z D0 (S(z))dz; t T 0 ; 1+ (t) = e Z 1t e rz K 0 (A(z))dz; t T 0 : 2+ (t) =

(19) (20)

t

The shadow price of the carbon stock is then the sum of the discounted (at rate r + ; the sum of the discount and the natural assimilation rates) future marginal damages of emissions. The shadow price of the land used for sequestration is the sum of the discounted marginal costs of the change of practice.

3.3

Before T 0 : sequestration

The social planner solves an optimization program in …nite horizon, considering that the horizon T 0 and the value function of the program after T 0 ; V+ (S(T 0 ); A(T 0 )); are given. By de…nition of T 0 ; we have b(t) = 0 8t T 0 : The social planner’s program is then given by: 8 R T0 > min 0 e rt (D(S(t)) + K(A(t)))dt + V+ (S(T 0 ); A(T 0 )) > > > > Rt > > S(t) _ < cs 0 a(z)e s(t z) dz = S(t) + E e (21) _ A(t) = a(t) > > > > > 0 A(t) B and a(t) = f0; ag 8t > > : S(0) = S0 , A(0) = 0 given. 11

We note 1 the shadow price of the carbon stock S and 2 the one of the land used for sequestration A. The current value Hamiltonian of this program is then (Kamien and Muller (1976)): H (t) =

e

rt

(D(S(t)) + K(A(t))) Z T0 s(z t) +e csa(t) dz: 1 (z)e

1

(t)

S(t) + E

2

(t)a(t) (22)

t

The Hamiltonian is linear in the control variable a(t): The coe¢ cient associated to a(t) in the Hamiltonian is Z T0 s(z t) (t) = e cs dz (23) 1 (z)e 2 (t): t

It represents the di¤erence between the marginal bene…t of sequestration and its marginal cost. The marginal bene…t is the sum of the ‡ows of carbon sequestered until T 0 by an additional unit of land devoted to sequestration at t; evaluated at the shadow value of the carbon stock 1 . The marginal cost is the shadow price of the land used for sequestration 2 : As long as this coe¢ cient (t) is positive, the bene…t of sequestration is greater than its cost and a(t) = a; when the coe¢ cient becomes negative, a(t) = 0. It may happen that sequestration stops while (t) is still positive, if all the lands suitable for sequestration are already used. The date T at which sequestration stops can then be: T1 in the case of a corner solution, with A(T1 ) = B and (T1 ) > 0; T2 in the case of an interior solution, with A(T2 ) < B and (T2 ) = 0: _ Besides, between 0 and T (equal to T1 or T2 ) we have A(t) = a, which implies ( at; t T A(t) = A(T ) = aT; t 2 [T; T 0 ] :

(24)

The …rst order necessary conditions are: @H (t) @S(t) @H (t) @A(t)

_ 1 (t) = 0 ,

e

rt

D0 (S(t)) +

_ 2 (t) = 0 ,

e

rt

C 0 (A(t))

1

(t)

_ 1 (t) = 0;

_ 2 (t) = 0:

(25) (26)

Integration of equations (25) and (26) yields: 1

2

(t) = e (t) =

t

2

1

(0)

(0)

e Z

t

Z

t

e

(r+ )z

D0 (S(z))dz;

t

T 0;

(27)

0

t

e

rz

K 0 (A(z))dz;

0

12

t

T 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 …rst link between the two stages is speci…c to each unit of land. It is the date (z) T 0 at which sequestration that stops in a given unit of land at date z T 0 has begun, determined by the last in – …rst out principle according to equation (4). For z 2 [T 0 ; T 00 ] ; (z) 2 [0; T ] : Equation (4) can then be written Z T Z z d =a d ; a T0

(z)

which allows us to obtain (z): (z) = T + T 0

z:

(29)

It is now possible to make explicit the equation (T 00 ) = 0; which will yield the date T 00 at which de-sequestration stops in the second stage. We can …rst obtain the carbon stock in equation (19), by integration of its equation of accumulation (9): Z t E (t T 0 ) 0 (t T 0 ) e (t z) F (z)dz; (30) S(t) = e S(T ) + 1 e T0

the ‡ow of carbon sequestered in agricultural soils being given by (using equations (7) and (29)): F (t) = e cae st esT 1 e ca ( 0 0 0 0 0 0 0 0 e s(t T ) (1 e s(t T ) ) + (1 e s (t T ) ) s0 s 2s es(T +T ) e s t (e(s 2s)t e(s 0 00 0 0 0 00 0 0 0 0 0 00 e s(t T ) (1 e s(T T ) ) + e s t (es T es T ) s0 s 2s es(T +T ) e s t (e(s 2s)T

(31) 2s)T 0 );

e(s

0

T0

t T 00 2s)T 0 ); t T 00 :

We obtain S(t) 8t 2 [T 0 ; 1[ as a function of S(T 0 ); T and T 00 which are still unknown. Secondly, the amount of land devoted to sequestration in equation (20) is: ( a(T + T 0 t); T 0 t T 00 ; A(t) = A(T 00 ) = a(T + T 0 T 00 ); t T 00 ;

(32)

Lastly, it is possible to calculate (T 00 ); using equations (12), (19), (20) and (29), as a function of T and T 0 which are still unknown. (T 00 ) = 0 then reads

0 =

Z

1

T 00

e c

e

Z

rz

1

T 00

K 0 (A(z))dz Z 1 e (r+ ) D0 (S( ))d

(33) e

z

se

s(z (T +T 0 T 00 ))

+ s0 (1

e

s(2T 00 (T +T 0 ))

)e

s0 (z T 00 )

z

In the same way, it is possible to make explicit the equation (T ) = 0; which will yield the date T at which sequestration stops in the …rst stage. We obtain …rst the carbon stock in equation (27) by integration of its equation of accumulation (9): Z t E 1 e t e (t z) F (z)dz; (34) S(t) = e t S0 + 0

13

dz:

with

(

F (t) = e ca

1 e st ; 0 e st esT 1 ;

t T

T t

(35)

T 0:

It yields S(t) 8t 2 [0; T 0 ] ; as a function of T which is still unknown. Secondly, the amount of land devoted to sequestration in equation (28) is given by equation (24). Lastly (T ) = 0 reads, using equations (23), (27) and (28): Z T Z z Z T0 (r+ ) 0 z s(z T ) e rz K 0 (A(z))dz; (36) e D (S( ))d e e dz 0=e cs 2 (0) + 1 (0) T 0 T;

0

0

where 1 (0) and 2 (0) are still unknown. The two initial values of the shadow prices and the 0 date T 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 solution 0 < T 0 < 1:

Z

1

(T 0 ) =

1+ (T

0

)

2

(T 0 ) =

2+ (T

0

)

T0

(37) Z

(38) 1

@H+ (t) @H (t) dt = H+ (T 0 ) dt: (39) 0 @T @T 0 T0 0 The …rst two conditions express the continuity of the shadow prices of carbon and land between the two stages. The last one is more intricate. It shows that the Hamiltonians do not match at the switching time, due to the dependence of the determinants of the atmospheric carbon accumulation to the switching time3 . 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 sequester carbon in the agricultural soils permanently: T 0 ! 1: If the LHS is smaller than the RHS, the optimal solution is T 0 = 0 and then T = 0: it is never optimal to begin storing carbon in agricultural soils. Using equations (27) and (19), equation (37) reduces to Z 1 e (r+ )z D0 (S(z))dz: (40) 1 (0) = H (T 0 ) +

0

Using equations (28) and (20), equation (38) becomes

2

(0) =

Z

1

e

rz

e

rz

K 0 (A(z))dz

0

=

Z

0

+

T

Z

T 00

e

K 0 (A(z))dz + rz

K 0 (A(T )) e r

K 0 (A(z))dz +

T0

rT

K 0 (A(T 00 )) e r

e rT 00

:

rT 0

(41)

Notice that equation (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 . 3

14

Equations (22) and (11) allow us to compute the last matching condition (39). After taking into account the …rst two matching conditions, using a(T 0 ) = 0; b(T 0 ) = a and equation (29), and simplifying (calculations are tedious but straightforward), equation (39) reduces to Z 1 0 0 0 0 0 s(z (T 0 )) 0 = 2+ (T ) e c + s0 (1 e s(T (T )) )e s (z T ) dz 1+ (z) se T0 Z 1 Z 1 b(t) @ (t) 0 s(z (t)) s0 e s(t (t)) e s (z t) dz dt; +e cs 1+ (z) se 0 a @T T0 t

i.e.

2+ (T

0

) = e c

Z

1

1+ (z)

T0

e cses(T +T

0 2+ (T ) being given by equation T 0 as a function of T and T 00 :

3.5

0)

s(z T )

se

Z

1

T0

b(t) e a

(20) and

+ s0 (1 e Z 1

st

1+ (z)

s(T 0 T )

1+ (z)

se

)e sz

s0 (z T 0 )

s0 e(s

0

dz s)t

e

s0 z

dz dt;

(42)

t

by equation (19). This equation allows us to calculate

The optimal dates

It is now possible to …nd the optimal dates T; T 0 and T 00 ; in the di¤erent con…gurations that can occur. In every con…guration, these dates must obviously satisfy the following inequality: B T 0 T 00 T + T 0 : a Any meaningful combination of the di¤erent solutions below can take place. 0

T

(43)

Interior solution The date T at which sequestration stops is T2 < Ba ; given by (T ) = 0; i.e., according to equations (36), (40) and (41): Z T0 Z 1 Z 1 e (r+ ) D0 (S( ))d e( s)z dz: (44) e rz K 0 (A(z))dz = e csesT T

T

z

T 00

The date at which de-sequestration stops is T200 < T + T 0 ; given by (T 00 ) = 0; i.e., according to equation (33): Z 1 e rz K 0 (A(z))dz (45) 00 T Z 1 Z 1 0 00 00 0 0 00 e (r+ ) D0 (S( ))d e z se s(z (T +T T )) + s0 (1 e s(2T (T +T )) )e s (z T ) dz: = e c T 00

The date and (20): Z 1

T0

z

at which de-sequestration begins is given by equation (42), using equations (19)

rz

K 0 (A(z))dz Z 1 Z 1 0 0 0 z = e c e e (r+ ) D0 (S( ))d se s(z T ) + s0 (1 e s(T T ) )e s (z T ) dz T0 z Z 1 Z 1 Z 1 b(t) st 0 0 e cses(T +T ) e ez e (r+ ) D0 (S( ))d se sz s0 e(s s)t e a 0 T t z e

(46)

T0

15

s0 z

dz dt:

This solution is the one in which bene…t and cost considerations prevail over quantitative constraints. In the …rst 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 have returned to usual practices, because the bene…ts of an additional release are too small. Corner solution after T 0 The date T 00 at which de-sequestration stops becomes T100 = T + T 0 : In this solution, all units return to usual practices in the second stage. Corner solution before T 0 The date T at which sequestration stops becomes T1 = limited by the land storage capacity.

B a:

In the …rst stage, sequestration is

Corner solution for T 0 It may happen that it is never optimal to begin to store carbon (T 0 = 0) or that it is never optimal to release carbon in the atmosphere (T 0 ! 1). In these two cases, the third matching condition (42) is no longer veri…ed. As far as its left hand side is nil, the solution T 0 ! 1 prevails if the RHS is negative, and T 0 = 0 if it is positive. It is not possible to solve explicitly and …nd the optimal dates in the general case that we study here. We will then use below speci…c functional forms and simulations.

3.6

The stationary state

This economy can admit two di¤erent steady states, according to the solution that prevails: no carbon release (formally, T 0 ! 1 or T 00 = T 0 < 1) and de-sequestration (T 0 < 1, T 00 > T 0 ). In the two cases, the ‡ow of carbon sequestered and the carbon stock are given by:

F S

= 0; E ; =

but the stock of land devoted to sequestration is di¤erent in the two cases: ( A(T ) if T 0 ! 1 or T 00 = T 0 ; A = A(T 00 ) if T 0 < 1 and T 00 > T 0 :

(47) (48)

(49)

Equation (48) shows that the steady state value of the carbon stock is independent of the choices made about carbon sequestration in agricultural soils. Sequestration only a¤ects the dynamics to reach the steady state. This result is independent of the characteristics of the dynamic process of sequestration / desequestration, and is also obtained by Feng et al. (2002). Its explanation is the following. When a unit of carbon is emitted by the economic activity, it goes either into the atmosphere or into 16

agricultural soils. In the …rst case, it disappears at rate ; due to natural processes of absorption. In the second case, it either stays in the soil inde…netely or is released after some time and goes into the atmosphere. Moreover, the amount of carbon that can be released into the atmosphere is in…nite whereas agricultural soils have a …nite capacity of storage. Then, in the long run, it is impossible to maintain a permanent ‡ow of carbon sequestration and the only thing that determines the atmospheric carbon stock is the natural absorption capacity. The dynamics is characterized by equation (9), with limt!1 F (t) = 0 (cf. equations (35) and (31)). Because of this last property, the dynamic equation is asymptotically autonomous. Following Benaïm and Hirsch (1996), we …rst study the corresponding autonomous equation and then examine the converging properties of the asymptotically autonomous equation towards the autonomous one. The autonomous equation is _ (50) S(t) = S(t) + E; and it is obviously stable. Its solution is S(t) = S + (S0

S )e

t

:

A simple look at equations (35) and (31) is su¢ cient to convince us that F (t) decreases at an 0 asymptotic rate e s when there is no carbon release, and at an asymptotic rate e s when there is some carbon release. Then, the solution of the dynamic equation (9) converges to the solution of 0 the autonomous one (50) at an asymptotic rate at least as fast as e s in the …rst case, e s in the second one4 . Moreover, since (9) converges asymptotically to (50), both have the same local stability properties, and (9) is locally stable.

4

Numerical analysis

4.1

The optimal dates

We suppose that the costs of sequestration are quadratic and the damage linear: K(A(t)) = A(t)2 ; 2

> 0;

(51)

D (S(t)) = S(t);

> 0:

(52)

We obtain immediately from equations (19), (20), (27), (28), (40) and (41) the expressions of the shadow prices:

1+ (t)

=

2+ (t) =

r Z+

e 1

e

rt

;

rz

t

T 0;

A(z)dz;

(53) t

T 0:

t

4

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

17

(54)

1

(t) =

2

(t) =

r Z+

rt

e 1

e

;

rz

t

T 0;

A(z)dz;

(55) t

T 0:

(56)

t

In the case of an interior solution, dates T2 , T200 and T 0 are respectively given by equations (44), (45) and (46), which can be written here as aT a + 2 e r r a(T + T 0 r

T 00 )

r(T 00 T )

=

e c r+

e

r(T 0 T )

s e r+s

=

e cs 1 (r + )(r + s)

s(2T 00 (T +T 0 ))

+

s0 (1 r + s0

aT a e cs0 r(T 00 T 0 ) 1 e = r r2 (r + )(r + s0 ) s e c s s0 0 + e s(T T ) 1 1 0 r+ r+s r+s r + 2s

e

e

(r+s)(T 0 T )

e

s(2T 00 (T +T 0 ))

(r+2s)(T 00 T 0 )

;

(57) ) ;

(58)

:

(59)

We show in the Appendix that an interior solution cannot exist, and that two di¤erent corner solutions can occur. The …rst one implies T 00 = T 0 ; which means that de-sequestration stops exactly at the moment at which it begins, or in other words that de-sequestration never occurs, because it is always too costly. We can then …nd the value of the optimal T2 : Te2 =

e cs r : a(r + )(r + s)

(60)

Te2 must be less than Ba 5 . Te2 is all the higher since the bene…t in terms of sequestration given by the change of practice e c is high, the marginal damage is high, the direct cost of sequestration and the adjustment cost a are low, and the absorption rate is low. Moreover, it is possible to show that Te2 is an increasing function of the interest rate r if and only if s > r2 : Te2 is independent of the value of the speed of de-sequestration s0 (with s0 > s): in this case, the asymmetry of the sequestration process does not a¤ect the optimal solution. Finally, Te2 is an increasing function of the speed of absorption s, and we have lim Te2 =

s!1

e c r : a(r + )

(61)

When we do not take into account the dynamics of sequestration, as in Feng et al. (2002), Te2 this date is the optimal date at which sequestration stops. We have lims!1 = 1 + rs : the Te2 error (overvaluation of the optimal period of sequestration) made by ignoring the dynamics of sequestration can be very signi…cant. This result does not depend on the magnitude of s0 ; since no release occurs, but comes from the fact that sequestration takes time and therefore its bene…ts come gradually, which reduces the marginal value of sequestration. 5

The solution is a corner solution before T 0 ; (T1 ; T 0 ; T200 = T 0 ); if and only if

18

c es r a(r+ )(r+s)

B : a

The second one is T100 = T2 + 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 obtained explicitly. It is impossible to …nd analytically which solution is the optimal one. We will compute in the numerical simulations the total value function V + V+ to discriminate between the two solutions, in the special case of our calibration.

4.2 4.2.1

Numerical simulations Calibration

The numerical simulations are performed using the values of the parameters given in table 1. Table 1: Calibration Parameter Value a B c~ S0 E s r

10 Mha/y 1000 Mha 0:015 GtC/Mha 370 ppmv 5:625 ppmv 0:02 0:0125 0:03 105 1

According to FAO data6 , 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 billion ha can be used to store carbon (B = 1000 Mha). We also suppose that each year a change of practice can take place on 10 Mha (a = 10 Mha), which means that the conversion of the total disposable land to carbon sequestration would take 100 years. The initial stock of carbon in the atmosphere is 370 ppmv (or about 778 GtC), the 2001 level. The model is calibrated to obtain a long term stock of 450 ppmv (or about 947.25 GtC). For a rate of decay equal to 0.0125 (which means that the average life of carbon in the atmosphere is 1=0:0125 = 80 years), this requires emissions of 450 0:0125 = 5:625 ppmv. Table 2 provides several examples, detailed in INRA (2002), about the potential of additional soil carbon storage (e c) and the speeds of sequestration (s) and de-sequestration (s0 ). For example, the additional carbon storage induced by the a¤orestation of arable land (cereal crops with conventional working practice) is evaluated to 30 tC/ha. The speed of accumulation which results 6

Source: FAOSTAT database of Land Use Statistics.

19

Table 2: Parameters for the storage–release process A B e c s (tC/ha) (A ! B)

Cereal crops, conventional working practice

Cereal crops, no tillage

Cereal crops, conventional working practice

Forest

Cereal crops, conventional working practice

Permanent grassland

s0 (B ! A)

12

0.02

0.04

30

0.0175

0.07

25

0.025

0.07

source: INRA (2002)

from a¤orestation is estimated at a value of s = 0:017, while the release is more rapid, with s0 = 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 can be stored in agricultural land in France is 15 tC/ha. Even if this value can be di¤erent from one geographical area to another, we take here e c = 0:015 GtC/Mha. The same study for France allows us to evaluate the speeds of sequestration and de-sequestration: s = 0:02 and s0 = 0:037: As units for total costs and damage are arbitrary, we take = 1: We then choose such that 0 the optimal date at which sequestration stops in the case s = s = 1; given by equation (61), is signi…cantly positive. With = 105 this date is around 20 years. Finally, we choose r = 3% per year. 4.2.2

Results

For s = 0:02 and 8s0 > s; the computation of the total value function shows that the optimal solution is the …rst corner solution (T = T 0 = T 00 ). The optimal solution always consists of storing carbon in agricultural soils during about 20 years and never releasing it. Optimal sequestration is permanent. With the assumption s = s0 = 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 physical potential sequestration (T = 50 years , A = 500 Mha). Whereas this optimal e¤ort corresponds to the …fth of the potential (T = 20 years , A = 200 Mha) for speeds of sequestration and desequestration close to the empirical French ones (s = 0:02 and s0 = 0:03): We see that the error made when not taking into account the dynamics of sequestration is very signi…cant, with an error coe¢ cient (1 + rs ) of 2:5 for our calibration. Figures 3 and 4 show respectively the ‡ow of carbon emissions trapped and the additional carbon stock stored when we assume that these last values of s and s0 are good proxis for the world conditions and for these two sequestration e¤orts (T = 50 and T = 20).

20

Figure 4: Additional carbon stock sequestered

Figure 3: Flow of carbon trapped

The additional carbon stock sequestered converges to a steady level of 7.5 GtC when we don’t take the dynamics of the physical process into consideration in the determination of the optimal policy (T = 50), while the optimal long term level is only 3 GtC (T = 20). The omission of the dynamic process leads to a very signi…cant over-sequestration.

5

Conclusion

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 is empirical. We compute numerically the optimal path of sequestration / de-sequestration for speci…c 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 signi…cant. Interesting extensions would consist of considering factors that could make the optimal sequestration temporary. For instance, technical progress in abatement could induce a decrease of net emissions in the course of time, and make sequestration less useful. Or technical progress in usual practices and not in sequestering practices could increase the cost of sequestration.

References Benaïm M. and M.W. Hirsch (1996), “Asymptotic Pseudotrajectories and Chain Recurrent Flows with Applications”, Journal of Dynamics and Di¤ erential Equations, 8, 141–176. European Climate Change Programme (2003), Working Group on Sinks Related to Agricultural Soils - Final report. Feng H, J. Zhao and C. Kling (2002), “The Time Path and Implementation of Carbon Sequestration”, American Journal of Agricultural Economics, 84, 134–149.

21

Hénin S.and M. Dupuit (1945), “Essai de bilan de la matière 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, U.K. Kamien M.I. and E. Muller (1976), “Optimal Control with Integral State Equation”, The Review of Economic Studies, 43(3), 469–473. Lal R., J.M. Kimble, R.F. Follet and C.V. Cole (1998), The Potential of US Cropland to Sequester Carbon and Mitigate the Greenhouse E¤ ect. Ann Arbor Press, Chelsea, MI. Smith P., J.U. Smith and D.S. Powlson (1996), Soil Organic Matter Network: 1996 Model and Experimental Metadata. GCTE Report 7, GCTE Focus 3, Wallingford, Oxon. Tomiyama K. and R. Rossana (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.

Appendix We …rst show that there is no interior solution for s0 > s: Taking the di¤erence between equations (58) and (59) yields

=

a 00 0 1 e r(T T ) 2 r e c s0 s 0 r+ r+s r+s

a(T 00 r e

T 0)

s(T 0 T )

1

e

2s(T 00 T 0 )

s 1 r + 2s

e

(r+2s)(T 00 T 0 )

which can be written, with X = T 00 T 0 0; F (X) = 1 e rX rX and G(X) = 1 s e (r+2s)X ; r+2s 1 e c r(s0 s) a 0 F (X) = e s(T T ) G(X): r2 (r + )(r + s)(r + s0 )

; e

2sX

The LHS of this equality has the same sign as F (X); and the sign of the RHS depends on the sign of s0 s and G(X): We show easily that F (X) 0 and G(X) 0 8X 0 : F (0) = 0 and F 0 (X) = r(e rX 1) < 0 8X > 0; G(0) = 0 and G0 (X) = se 2sX (2 e rX ) > 0 8X 0: When s0 > s; the only solution is then X = 0 i.e. T 00 = T 0 . For s0 = s the equality reduces to F (X) = 0; and the only solution is again T 00 = T 0 . For s0 > s and T 00 = T 0 ; equations (58) and (59) become identical. The optimal dates T and T 0 are then solution of the system composed of this last equation and (57): 8 0 e c s < aT = e (r+s)(T T ) r (r+ )(r+s) 1 :

aT r

=

e c s0 (r+ )(r+s0 )

1

22

r(s0 s) s(T 0 T ) s0 (r+s) e

:

The elimination of

aT r

between these two equations yields r(s0 s) 1 r + s0

e

s(T 0 T )

=

se

(r+s)(T 0 T )

;

which is impossible for T 0 2 [T; 1[ since the LHS is positive and the RHS strictly negative. We now examine the corner solutions. There can …rst exist a corner solution before T 0 : Then, equations (58) and (59) are valid and imply as before T 00 = T 0 ; i.e. no de-sequestration. The date T at which sequestration stops can take two values: T 0 or Ba : If T = T 0 ; (58) and (59) become identical and allow us to obtain T~2 =

e c sr : a(r + )(r + s)

We obtain the same value T~2 for a corner solution both before and after T 0 (T = T 0 = T 00 ). sr < B. This solution is valid as long as T~2 < Ba , i.e. as long as (r+ec )(r+s) B If T = a ; we have e which requires B < e c s0 r (r+ )(r+s0 ) .

sT 0

=e

e c s0 r (r+ )(r+s0 ) .

sB a

(r + )(r + s)(r + s0 ) e c s0 r(s s0 )

The solutions T =

B a

s0 (r + s) r(s s0 )

;

and T = T~2 can coexist for

e c sr (r+ )(r+s)


The corner solution before T 0 (T = Ba ) and T 00 = T 0 gives the same result. There can also exist a corner solution after T 0 , T100 = T2 + T 0 ; but in this case we cannot obtain the optimal dates analytically.

23