soil organic carbon changes in dynamic land use decision models
TRANSCRIPT
Soil organic carbon changes in dynamic land use decision models
Uwe A. Schneider a,b,*
a Research Unit Sustainability and Global Change, Center for Marine and Atmospheric Research, Hamburg University, Germanyb International Institute for Applied Systems Analysis Laxenburg, Austria
Received 19 April 2005; received in revised form 17 July 2006; accepted 25 July 2006
Available online 20 September 2006
www.elsevier.com/locate/agee
Agriculture, Ecosystems and Environment 119 (2007) 359–367
Abstract
Soil organic carbon can be sequestered through various land management options depending on the soil organic carbon status at the
beginning of a management period. This initial soil organic carbon status results from a certain soil management history in a certain soil
climate regime. Similarly, the prediction of future carbon storage depends on the time path of future soil management. Unfortunately, the
number of possible management trajectories reaches non-computable levels so fast that explicit representations of future management
trajectories are infeasible for most existing empirical land use decision models. Not surprisingly, the impact of different management
trajectories has been ignored.
This article proposes a computationally feasible mathematical programming method for integration of soil state dependent sequestration
rates in dynamic land use decision optimization models. The possible soil organic carbon range is divided in several adjacent carbon states.
For each soil management practice, location, and initial soil organic carbon state, transition probabilities of moving into a different or
remaining in the same state are computed. Subsequently, these probabilities are used in dynamic equations to update the soil organic carbon
level before and after each period. To illustrate the impacts of this Markov chain based method, a case study portraying Australian wheat
farmers is conducted.
# 2006 Elsevier B.V. All rights reserved.
Keywords: Soil organic carbon sequestration; Sink dynamics; Mathematical programming; Land use; Optimization; Agriculture; Markov chain
1. Introduction
Soil organic carbon (SOC) sequestration through
agriculture and forestry has been regarded as a potentially
important option to lower greenhouse gas concentrations in
the atmosphere (Schlesinger, 1999). It has been recognized
that unlike some other greenhouse gas emission abatement
technologies, soil sequestration is very heterogeneous over
location, time, and management. Local variations of soil
management regimes have been studied by West and Post
(2002), Pautsch et al. (2001), Lal et al. (1998), De Cara and
Jayet (2000), and McCarl and Schneider (2001) among
many others. Furthermore, West et al. (2004), Marland et al.
(2004), Lal et al. (1998), Murray et al. (2004) integrate
* Correspondence address: Bundesstrasse 55, 20146 Hamburg, Germany.
Tel.: +49 40 428386593; fax: +49 40 428387009.
E-mail address: [email protected].
0167-8809/$ – see front matter # 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.agee.2006.07.020
changes in SOC sequestration rates over time to address
issues of saturation and non-permanence. Continuous use of
no-till agriculture for example, will eventually lead to
equilibrium, where no additional storage can be achieved
without a management change (Marland et al., 2004).
Direction and magnitude of SOC sequestration rates
depend on the difference between initial SOC and
equilibrium levels. If the initial SOC level is far away from
the equilibrium level, sequestration rates will be relatively
high. Without management or climate changes, these rates
will eventually converge to zero as SOC levels approach the
equilibrium. For mineral soils, studies by West et al. (2004)
imply that most of the organic carbon adjustment takes place
within 20–30 years. Besides, many soil management
strategies can act as a sink or a source depending on the
initial SOC level. For example, consider no-till wheat
production. This often advocated tillage system might
indeed sequester substantial carbon amounts on fields that
U.A. Schneider / Agriculture, Ecosystems and Environment 119 (2007) 359–367360
were intensively tilled during previous years. However, the
same system could emit carbon if it were used on native
grassland soils or drained wetlands.
While the dynamic interaction between SOC levels and
sequestration rates has been observed and verified in many
experimental plots, it has been ignored in land use decision
models. These models examine the adoption of alternative
management decisions within a certain range of soil-climate
regimes and are frequently used to find economic SOC
sequestration potentials under various policy scenarios.
Spatial variations have been integrated in these land use
models through coupling with bio-physical models (Antle
et al., 2002; McCarl and Schneider, 2001; Pautsch et al.,
2001) or through application of IPCC sequestration
coefficients (De Cara and Jayet, 2000; Perez and Britz,
2004).
However, the dynamic nature of carbon sequestration
rates has so far been ignored. Quite a few studies use static
models in the beginning (Antle et al., 2002; Pautsch et al.,
2001; McCarl and Schneider, 2001; De Cara and Jayet,
2000; Perez and Britz, 2004). Static models can only predict
equilibrium changes between the initial and subsequent
management. These models are frequently used to assess
intermediate implications of alternative policies or technol-
ogies. Because of their static nature, they implicitly assume a
single adjustment only. Dynamic models, on the other hand,
mostly employ constant sequestration rates (Sohngen and
Mendelsohn, 2003; Murray et al., 2004). For example, the
same sequestration rate is applied to no-till soils that were
intensively tilled over many years and to soils, which have
not been tilled for some time. In reality, the sequestration
rates should be higher for the first and closer to zero for the
second case.
The relatively simple carbon sequestration modeling
approach adopted in existing land use decision models is
caused by lack of data and computational restriction. Data
wise, it is certainly easier to determine a single, average
coefficient for a given soil management in a given location
than to estimate carbon sequestration as a function of soil,
climate, and land management over time. Suitable functions
include carbon management response curves developed by
West et al. (2004). As for the computation, a decision
optimization with all soil management sequences over time
enumerated explicitly can be very demanding if not
prohibitive.
This study integrates a Markov chain based approach for
the representation of the dynamic interactions between land
use decisions, SOC levels, and sequestration rates within a
mathematical programming framework. This approach is
computationally more feasible and provides the opportunity
to trade computing time against accuracy on a continuous
scale. The paper is structured as follows. The next section
shows the first best implementation of dynamic SOC
sequestration rates within land use decision models. It will
be demonstrated why this theoretically ideal approach
comes at high computational cost. The following section
develops the alternative approach. Subsequently, this
approach is numerically tested with a case study. Finally,
conclusions are given.
2. Methodology
2.1. Land use decision models
SOC sequestration requires certain suitable land manage-
ment practices. To analyze the economic potential of SOC
storage, land use decision models are employed. These
models are tools to understand, guide, and predict land use
decisions under various economic and environmental
conditions. Mathematical programming is frequently used
to determine optimal decisions when carbon sequestration
efforts relate to substantial structural changes in the
agricultural and forestry sector. These changes may include
the introduction of new soil management practices or the
implementation of new governmental policies. The
approach presented here will be equally applicable to farm
level, regional, and sector models, where a land use based
objective is maximized.
Common to all land use optimization models is a
predefined set of alternative land use decision variables and
an objective function. In addition, most models include
various biophysical or economic constraints. Let us denote
alternative land use decisions by a nonnegative variable
block Xt,r,i,u, where t = {1, . . ., T} denotes the set of time
periods, r = {1, . . ., R} the set of explicit geographic regions,
i = {1, . . ., I} the set of soil types, and u = {1, . . ., U} the set
of land use categories. Note that the indexes r, i, and u should
carefully be defined to sufficiently capture the heterogeneity
of landscape and agricultural production technologies.
Different soil types, for example, may distinguish soils that
differ in texture, layer depth, slope, and other physically
stable properties.
The general structure of dynamic land use optimization
models is displayed in equation block (1). The first line
represents the objective function, where the total value V is a
function of all possible land use decisions Xt,r,i,u. This
potentially non-linear function may include both economic
and environmental objectives. Line 2 generalizes restrictions
on all decision variables, where gj denotes j different implicit
functions.
Maximize V ¼ vðXt¼1;r¼1;i¼1;u¼1; . . . ;Xt¼T ;r¼R;i¼I;u¼UÞs:t: g jðXt¼1;r¼1;i¼1;u¼1; . . . ;Xt¼T ;r¼R;i¼I;u¼UÞ � 0
(1)
The main purpose of this paper is to demonstrate how
dynamic carbon sequestration rates can be feasibly modeled
within a land use optimization framework. Without loss of
generality, I use a linear system, where I maximize the sum
of land use profits over all time periods, regions, soil types,
and management systems as shown in Eq. (2). The profit of a
U.A. Schneider / Agriculture, Ecosystems and Environment 119 (2007) 359–367 361
land use activity is the product of an exogenous profit
coefficient vt;r;i;u ¼ vMt;r;i;u þ vC
t;r;i;u, a time period specific
discount factor bt, and the activity level Xt,r,i,u. The vMt;r;i;u
values represent the gross margins of an activity, i.e.
revenues minus variable costs. The second objective value
coefficient vCt;r;i;u denotes carbon emission taxes (negative
values) or carbon sequestration benefits (positive values)
which are obtained by multiplying carbon emission or
sequestration rates by an exogenously given carbon price.
MaxX
t;r;i;u
ðbtvt;r;i;uXt;r;i;uÞ (2)
Furthermore, I include land endowment restrictions reflect-
ing physical limits on available land. These restrictions are
contained in equation block (3). Particularly, the sum of all
land use activities Xt,r,i,u over all land management practices
u is limited by a given natural land endowment Lt,r,i.
Together, Eqs. (2) and (3) yield a simple starting point
for implementing dynamic SOC sequestration rates. Note
that such a linear model specification forces a specialized
solution. Particularly, for each soil type in each region, only
one management would be chosen over the entire area.X
u
Xt;r;i;u � Lt;r;i 8 t; r; i (3)
2.2. Explicit representation of soil management
trajectories
To capture the dynamics of SOC sequestration, a model
has to be able to represent and track all different
management decision paths. If a farmer could choose
between 20 alternative land uses in each period, he would
face 20 decisions in period 1, 202 combinations of past and
current land use decisions in period 2, 203 in period 3 and so
forth. For our simple model, this implies that the land use
decision variables would have to be modified to explicitly
represent decisions from all past periods, i.e. Xt,r,i,u would
change to Xr;i;ud , where ud denotes the index containing all
possible management decision paths. Suppose, we have only
two land use alternatives u = {1,2} and three time periods
t = {1,2,3}. The index ud would then contain all possible
management decision paths, i.e. ud = {111,112,121,
122,211,212,221,222}. For each decision path, specific
carbon and profit net present values would have to be
generated. The objective would then be to maximizePr;i;ud ½ðvM
r;i;ud þ vCr;i;udÞXr;i;ud �, where vM
r;i;ud and vCr;i;ud repre-
sent the net present market and carbon values for a given
management decision path on land in a given region and soil
type, respectively. The land use restriction from above would
slightly change toP
ud Xr;i;ud � Lr;i.
It can easily be seen that this approach is computationally
very expensive for models with numerous management
alternatives and many time periods. Suppose, a dynamic
model had 30 time periods and 20 different land manage-
ment options resulting from combinations of different crop,
tillage, fertilization, and irrigation systems. The total
number of possible decision paths would equal 2030
alternatives. For models with multiple regions, multiple
crops, and many explicit soil types, this number would be
accordingly higher.
2.3. Management dynamics with a Markov chain
In this section, I will demonstrate an alternative approach,
which employs dynamically updated carbon sequestration
rates and which is computationally more feasible. The
principal idea is to integrate past management impacts
indirectly through a Markov chain where SOC is used as a
state variable. High initial values imply a carbon friendly
management history whereas low values indicate an
emission intensive history. State transition probabilities
control how SOC levels change from one period to the next.
Basically, SOC levels affect carbon emission or sequestra-
tion rates and in turn these rates determine how SOC levels
change.
2.3.1. Soil organic carbon states, emissions coefficients,
and transition probabilities
To implement dynamically changing carbon sequestra-
tion rates in the above land use decision model, several
modifications have to be made. First, the SOC level at the
beginning of each management period has to be identified.
To make this a feasible task, we must group the possible
range of SOC values into several discrete states denoted here
by the index o = {1,. . ., O}. We then append this index to the
land use decision variable, which changes from Xt,r,i,u to
Xt,r,i,u,o. Each land use decision is now associated with a
certain SOC state, which by definition corresponds to the
SOC state at the beginning of a management period. Note
that the grouping of the possible SOC range into several
discrete states will cause approximation errors. However,
through an increase in the number of carbon states, the width
of these states will decrease and so will the approximation
losses. This provides an opportunity for the decision analyst
to trade carbon accounting accuracy against computing time
according to preferences and resources.
Second, sequestration coefficients sr,i,u,o need to be
established for all regions, soil types, management alter-
natives, and all discrete SOC states. These rates can be based
on estimated functions or observed data. Biophysical field
process models such as EPIC (Williams et al., 1989) or
CENTURY (Parton et al., 1987) may be a good tool to
simulate a complete and consistent set of necessary
sequestration rates for a large number of alternative
locations, soil types, and alternative management practices.
Third, SOC state transition probabilities must be
computed for all regions, soil types, management alter-
natives, and SOC states. Suppose the SOC level at the
beginning of time period t is at o 2 {1, . . ., O} within the
range [olow, oup]. Then, the SOC level at the end of time
period t must lie somewhere in the interval
U.A. Schneider / Agriculture, Ecosystems and Environment 119 (2007) 359–367362
wr;i;u;o ¼ ½olow þ slowr;i;u;o; o
up þ supr;i;u;o�. Define rr,i,u,o,o 2 [0, l]
as the state transition probability of moving from initial SOC
state o at the beginning of the management period to state o
at the end. Because sequestration rates and the determination
of the range covered by each carbon states are completely
independent, I assume a uniform probability for the initial
carbon level to be anywhere between the lower carbon state
boundary olow and the upper carbon state boundary oup.
Thus, the probability of moving from original SOC state o to
subsequent state o can be simply calculated as the fraction
of the interval w that is part of carbon state o. Five general
cases are possible, which result in different computations
of rr,i,u,o,o. These cases are illustrated in Fig. 1. The corr-
esponding calculation of probabilities is straightforward
and shown below.
Case I : If oup þ supr;i;u;o� oup and olow þ slow
r;i;u;o� olow;
then rr;i;u;o;o ¼oup � ðolow þ slow
r;i;u;oÞwr;i;u;o
Case II : If oup þ supr;i;u;o � oup and olow þ slow
r;i;u;o � olow;
then rr;i;u;o;o ¼ðoup þ sup
r;i;u;oÞ � olow
wr;i;u;o
Case III : If oup þ supr;i;u;o� oup and olow þ slow
r;i;u;o � olow;
then rr;i;u;o;o ¼oup � olow
wr;i;u;o
Case IV : If oup þ supr;i;u;o � oup and olow þ slow
r;i;u;o� olow;
then rr;i;u;o;o ¼ 1
Case V : If oup þ supr;s;u;o;o � olow or olow þ slow
r;s;u;o;o� oup;
then rr;s;u;o;o ¼ 0
To check that the probability computations are correct,
one can verify the validity ofP
orr;i;u;o;o ¼ 1, which must
hold for all regions, soil types, land uses, and initial carbon
levels. In addition, one should be able to use the state
Fig. 1. Possible cases of SOC change from initial SOC state o to subsequent
state o. The hatched rectangle represents the interval of width
wr;i;u;o ¼ ðoup þ supr;i;u;oÞ � ðolow þ slow
r;i;u;oÞ.
transition probability matrix to replicate the observed or
simulated carbon response functions.
2.3.2. Soil organic carbon balance
SOC state transition probabilities are used in Eq. (4) to
balance SOC levels before and after each management
period. Particularly, all land being in SOC state o at time t
must equal the transition probability weighted sum over all
land uses u and initial SOC levels o at the previous time.
These equations illustrate the implicit representation of
soil management trajectories. The summation over
management alternatives u on each side of the equation
implies that the model does not directly link previous with
subsequent management practice. However, this link is
established indirectly through the explicit account of the
SOC state.
X
u
Xt;r;i;u;o ¼X
u;o
ðrr;i;u;o;oXt�1;r;i;u;oÞ 8 t; r; i; o (4)
Eq. (5) is an accounting equation, which computes the
SOC stock as the sum of the carbon stock at the beginning in
period t plus DSt,r,i the amount of carbon added or lost in
period t.
St;r;i ¼ St�1;r;i þ DSt;r;i 8 t; r; i; o (5)
The carbon stock change DSt,r,i can be measured in two
ways. It can be calculated as the sum of sequestration levels
over all land use activities, i.e.P
u;oðsr;i;u;oXt;r;i;u;oÞ, or as the
sum of SOC levels over all SOC changes, i.e.Pu;oðcr;i;u;oXt;r;i;u;oÞ �
Pu;oðcr;i;u;oXt�1;r;i;u;oÞ, where cr,i,u,o
denotes the amount of contained SOC at the end of a period.
Because of the approximation, the two measures will not be
identical. However, as the number of explicit states
increases, the width of each state decreases and the
deviation between the two measures converges towards zero.
2.3.3. Assumptions and extensions
The following assumptions have been used in the Markov
chain based approach described above: (i) a certain land
management history is sufficiently identified by the current
SOC status; (ii) SOC can be treated of equal quality, (iii)
SOC sequestration rates converge monotonically from an
initial value towards zero provided a chosen soil manage-
ment practice is continued forever. Depending on the initial
carbon level, convergence can occur from above or below.
All of these assumptions can be relaxed through additional
soil state indexes. However, such extensions require
additional research, data, and computing power. A brief
discussion on why and how to relax these assumptions
follows below.
Assumption (i) may lead to incorrect results because
different management histories are likely to affect not only
total SOC but also other soil properties such as soil depth,
soil density, soil flora, and soil chemical characteristics. To
capture these impacts, one would need to define additional
U.A. Schneider / Agriculture, Ecosystems and Environment 119 (2007) 359–367 363
Fig. 2. SOC functions for four wheat management regimes in New
South Wales, Australia. Each carbon function is computed as
state indexes for each relevant individual property. If
different soil properties have interdependencies, their
associated state transition probabilities should be computed
jointly. Emission and sequestration rates would have to be
computed for all possible combinations of the involved state
indexes. Assumption (ii) could be relaxed in a similar way.
State index modifications could involve adding age and
stability categories. Finally, assumption (iii) may be
inappropriate as findings by West et al. (2004) suggest that
carbon stock changes after adoption and maintenance of
reduced tillage may not always be monotonic decreasing. To
allow non-monotonic behavior with respect to SOC within
the framework presented here, at least one additional soil
state index, i.e. soil flora activity, is needed. A certain SOC
state would then result in higher or lower net emission rates
depending on the activity state of soil flora.
C(t,a,b,C0) = (�ae(at/100)C0)/(�a � bC0 + be(at/100)C0) where C0 is the car-bon level at t = 0. The parameters a and b are taken from stepwise linear
percentage SOC change equations (100(dC/C)/dt = a + bC) estimated by
Hill (2003) based on CENTURY simulations of the four management
alternatives.
Fig. 3. Soil organic carbon functions reproduced with a Markov chain
process using eight carbon states equally distributed between 5 and 45 t of
carbon per hectare.
3. Carbon policies and wheat management in New
South Wales
In this section, I use simulated data from wheat
management alternatives in New South Wales, Australia
to implement the approach explained in Section 2 and
examine the optimal management path under different
carbon policies, different initial carbon levels, and different
discount rates.
3.1. Wheat management alternatives
The four alternative wheat (Triticum aestivum) manage-
ment practices include (1) conventional tillage based wheat-
fallow (CTWF); (2) no-till wheat-fallow (NTWF); (3) 6-
year wheat- 3-year lucerne (Medicago sativa) rotation
(63WL), and (4) a 3-year wheat- 3-year lucerne rotation
(33WL). SOC sequestration functions for each practice
were computed based on CENTURY simulations by Hill
(2003). In particular, the stepwise linear carbon change
equations of the form 100((dC/C)/dt) = a + bC (Hill, 2003)
were solved for carbon (C) as a function of time, initial SOC
(C0), and the two parameters a and b. Because the resulting
functions implied a very slow convergence in the order of
several centuries, I calibrated all functions using
C(t) = CHill(1/15)t. The calibrated functions are shown in
Fig. 2. The simulated SOC equilibriums equal 6(CTWF),
12(NTWF), 15(63WL), and 40(33WL), all in t C/ha/20 cm.
Thus, the relevant SOC range for these four strategies lies
between 6 and 40 t C/ha/20 cm. Average annual profits are
based on Wylie (1996) and amount to 144(CTWF),
136(NTWF), 116(63WL), and 97(33WL), all in $/ha. Note
that these data show an inverse relationship between market
profits and SOC levels.
Throughout the analysis, I assume that technological and
environmental conditions do not change over time. In
addition, intra rotational differences in sequestration rates
are not provided in the study by Hill (2003) and therefore, are
ignored. The first empirical exercise is to check the
functionality of the Markov chain. As argued above, the
carbon accounting accuracy will increase as the range covered
by one state decreases. On the other hand, a large number of
SOC states will increase the computational cost. The
empirical land management optimization model deployed
here is relatively small as it contains only four alternative
management practices, one region, and one soil type. While
this model can easily be solved with a Markov chain linking
several thousand SOC states, stricter computational con-
straints may arise in more complex models. In light of these
possible restrictions, Fig. 3 shows a reproduction of the
original carbon functions (see Fig. 2) with a Markov chain that
uses only eight carbon states, which were equally distributed
between 2 and 48 t C/ha/20 cm. This reproduction was
obtained by forcing each practice to continue over 50 years
U.A. Schneider / Agriculture, Ecosystems and Environment 119 (2007) 359–367364
first starting from the lower SOC boundary (2 t C/ha/20 cm)
and subsequently from the upper boundary (48 t C/ha/20 cm).
SOC changes from one period to the next were determined
through the carbon state transition probability matrix
(Table 1), which was computed as described in Section
2.3.1. For example, if we had 1 ha of 33WL land in carbon
state SOC1 at time t, we would have 0.23 ha in state SOC2
plus 0.77 ha in state SOC3 at time t + 1. Comparing original
and reproduced carbon functions, we find a relatively good fit
despite the low number of SOC states.
3.2. Optimal wheat management response to carbon
mitigation
After verifying the accuracy of the Markov process, let us
examine how implementing this process affects the optimal
soil management path within the land use decision model.
The optimal path gives the best sequence of soil manage-
ment decisions over time, where ‘‘optimal’’ relates to the
Table 1
SOC state transition probabilities for eight equally distributed states and four
columns = SOC at the end of a period, SOC1 = [2.00, 7.75] t C/ha, . . ., SOC8 =
SOC1 SOC2 SOC3 SOC
Wheat-lucerne 3/3
SOC1 0.23 0.77
SOC2 1
SOC3 0.24 0.63
SOC4
SOC5
SOC6
SOC7
SOC8
Wheat-lucerne 6/3
SOC1 0.3 0.7
SOC2 0.48 0.52
SOC3 0.9 0.1
SOC4 1
SOC5 0.13
SOC6
SOC7
SOC8
No-till wheat-fallow
SOC1 0.81 0.19
SOC2 1
SOC3 0.09 0.91
SOC4 0.31 0.69
SOC5 0.5
SOC6
SOC7
SOC8
Tilled wheat-fallow
SOC1 1
SOC2 0.06 0.94
SOC3 0.66 0.34
SOC4 1
SOC5 0.09 0.91
SOC6 0.53
SOC7
SOC8
specified objective function for the decision model in
question. The model has a profit maximizing objective with
total profit equaling the discounted sum over all time periods
of market profits plus carbon premiums minus carbon taxes.
By design, the only dynamic elements in our model are SOC
sequestration rates and SOC levels. If, on the other hand,
carbon sequestration rates were constant over time, the
optimization would become a sequence of autarkic
decisions. In such a case, the optimal soil management
path would involve adopting a single practice over the entire
planning horizon because the decision problem in all periods
would be identical. With dynamically linked carbon levels
and sequestration rates, however, the carbon premium of a
soil management practice in a certain period depends on the
management decisions taken in all previous periods.
Consequently, we might observe multiple management
changes over time.
To better understand why a certain decision path
becomes optimal under various conditions, let us first
different crop rotations (SOC in rows = SOC at the beginning, SOC in
[42.25, 48.00] t C/ha)
4 SOC5 SOC6 SOC7 SOC8
0.13
0.99 0.01
1
0.57 0.43
1
0.66 0.34
0.87
0.52 0.48
1
0.08 0.92
0.5
0.74 0.26
1
0.04 0.96
0.47
1
0.02 0.98
U.A. Schneider / Agriculture, Ecosystems and Environment 119 (2007) 359–367 365
consider a situation without discounting. In such a situation,
the farmer receives the same carbon revenue whether he
stores carbon early or late. The net carbon payment would
simply be determined by the difference between final and
initial carbon level and by the carbon price. Suppose, the
optimal carbon difference is known and positive implying
that the final SOC level is higher than the initial level.
Suppose further that the time needed to store this additional
amount of carbon is less than the planning horizon. When
would then be the optimal timing for carbon storage: early
or late? Clearly, the optimal timing would involve a
management plan that achieves the required final carbon
level but otherwise maximizes profits from crop production.
Among the four wheat practices considered here, market
profits are higher for carbon emission intensive strategies.
Thus, the optimal timing of SOC storage would be late
because it would allow the farmer to take advantage of
higher profits from emission intensive strategies for a longer
time. This insight is confirmed in Fig. 4, which shows the
optimal SOC path over time for two different initial carbon
level scenarios and five different carbon price scenarios.
Several things can be observed. First, the optimal final
carbon level is roughly the same regardless whether the
Fig. 4. Optimal SOC path for five different carbon prices, zero discount
rate, and a Markov chain with 20 carbon states. Upper panel: low initial
carbon level. Lower panel: high initial carbon level.
initial carbon was high or low. Second, the optimal carbon
path in late periods is similar for a given carbon price
regardless whether the initial carbon level is high or low.
Third, higher carbon prices lead to higher final carbon
levels. Thus, in absence of discounting, the optimal carbon
storage would be at the end of the planning horizon, which
could imply that there is no carbon storage at all because as a
decision maker moves into the future the planning horizon
will move out as well and the carbon storage would be
delayed even further.
While a zero discount rate may yield important insights,
it is not a realistic assumption. Without doubt, farmers value
early income more than late income. Thus, we might expect
a shift in sequestration activities towards earlier periods.
Indeed, as shown in Fig. 5, relatively low discount rate
changes are sufficient to substantially change the optimal
SOC path and the associated soil management systems.
While the optimal final carbon level does not change much
for different discount rates, the path linking initial and final
carbon level moves considerably upwards. Note that a
similar effect as the discount rate effect could be achieved by
changing carbon prices or relative costs between different
management practices over time.
Fig. 5. Optimal SOC path for four different discount rates, a carbon price of
$80/t C, and a Markov chain with 20 carbon states. Upper panel: low initial
carbon level. Lower panel: high initial carbon level.
U.A. Schneider / Agriculture, Ecosystems and Environment 119 (2007) 359–367366
Table 2
Net present value per hectare over 30 periods with 5% discount rate, a Markov process with 20 carbon states
Initial SOC Carbon price Fully dynamic Recursive dynamic Always CTWF Always NTWF Always 63WL Always 33WL
Low 0 2324 2324 2324 2195 1872 1566
Low 15 2362 1992 2344 2276 2051 1992
Low 30 2522 2417 2363 2356 2230 2417
Low 50 3039 2985 2390 2464 2469 2985
Low 80 3910 3837 2429 2625 2827 3837
Low 100 4491 4405 2455 2733 3065 4405
Low 150 5943 5825 2520 3001 3662 5825
Low 200 7396 7244 2585 3270 4258 7244
Middle 15 2090 1740 2070 2034 1816 1740
Middle 30 2083 1915 1817 1874 1760 1915
Middle 50 2244 2148 1478 1659 1684 2148
Middle 80 2631 2497 970 1338 1572 2497
Middle 100 2889 2730 632 1124 1496 2730
Middle 150 3534 3313 �215 588 1308 3313
Middle 200 4179 3895 �1061 52 1120 3895
High 15 1835 1502 1797 1805 1551 1411
High 30 1577 1300 1271 1415 1230 1256
High 50 1349 1090 568 895 802 1049
High 80 1125 739 �486 115 160 739
High 100 981 532 �1188 �405 �268 532
High 150 620 15 �2945 �1704 �1338 15
High 200 259 �502 �4701 �3004 �2408 �502
How does a carbon policy affect profits under different
management pathways? To answer this question, total
profits are computed for eight carbon price levels, three
initial carbon states, and six alternative decision rules
(Table 2). With a fully dynamic optimization algorithm in
place, management decisions in all periods are chosen
simultaneously to maximize the sum of discounted profits
over all periods. Recursive dynamic optimization involves a
repeated maximization of current profits without consider-
ing consequences for future profits. The other four decision
rules consist of choosing the same management practice
over the entire planning horizon. By construct, fully
dynamic optimization always outperforms the other
decision rules. The difference is generally higher for higher
initial carbon levels. Second, for low carbon prices (15 $/
tC) recursive optimization leads to smaller total profits than
adopting conventional tillage over the entire time horizon.
Recursive optimization leads to relatively early carbon
storage because the negative profit consequences of
decreasing future carbon sequestration options are ignored.
Third, the difference between the true dynamic optimiza-
tion and the best constant practice scenario can be
interpreted as the cost of ignoring dynamic SOC sequestra-
tion rates.
4. Conclusions
The dynamic interactions between SOC sequestration
rates, soil management decisions, and SOC levels, while
recognized in the soil science literature, have not been
implemented in most existing land use decision models. A
major obstacle is the high data and computing require-
ments for an explicit representation of alternative land use
sequences in dynamic decision models. The objective of
this study was to develop and test an alternative method,
which integrates the carbon sequestration dynamics at a
computationally more feasible level. The objectives
were met by (i) introducing a SOC state index which
approximates the cumulative effect of past soil manage-
ment decisions, (ii) making land use decisions
and sequestration rates dependent on the current SOC
state, and (iii) integrating Markov chain based SOC state
transition equations in the land use decision model.
Furthermore, the mathematical programming approach
developed here allows the decision analyst to trade
accuracy of SOC dynamics against computational feasi-
bility.
The proposed method was tested in a case study about the
effects of carbon policies on wheat management in New
South Wales, Australia. Case study results imply that the use
of sequestration rates as a function of SOC may notably
improve the prediction of optimal land management
decisions and the prediction of land use based carbon
sequestration quantities. Improved predictions may in turn
improve the design and efficiency of carbon mitigation
policies.
Appendix A. Supplementary data
Supplementary data associated with this article can be
found, in the online version, at doi:10.1016/
j.agee.2006.07.020.
U.A. Schneider / Agriculture, Ecosystems and Environment 119 (2007) 359–367 367
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