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--- scenario_simulation [2020/05/01 08:58] – matsz
+++ scenario_simulation [2023/09/08 12:07] (current) – [Land use, land use change and forestry (LULUCF)] massfeller
@@ Line 25: / Line 25: @@
 **Figure 13: Link of modules in CAPRI**
-{{::figure13.png?600|Source: CAPRI Modelling System. Note: the number of regions is outdated in December 2011}}
+{{::figure_13.png?600|Source: CAPRI Modelling System. Note: the number of regions is outdated in December 2011}}
 =====Module for agricultural supply at regional level=====
@@ Line 350: / Line 350: @@
 \end{align}
-The scaling factor to map from the legal quota legalquotA (as the B quota has been eliminated in the sugar reform, it holds that \(q^A = q^{A+B}) \)to the behavioural quota qA depends on the projected sugar beet sales quantity in the calibration point \(NETTRD_{SUGB}^{cal})\ : For a country with a high over quota production (say 40%) we would obtain a scaling factor of 1.31, such that this producer will behave like a moderate C-sugar producer: responsive to both the C-beet prices as well as to the quota beet price (and the legal quotas). Without this scaling factor, producers with significant over quota p   roduction, like France and Germany, would not show any sizeable response to a 10% cut of either the legal quotas or the quota price (at empirically observed coefficients of variation). As it is likely that the profitability of ethanol beets benefit from cross-subsidisation from the quota beets such a zero responsiveness was considered implausible.
+The scaling factor to map from the legal quota legalquotA (as the B quota has been eliminated in the sugar reform, it holds that \(q^A = q^{A+B} \) )to the behavioural quota qA depends on the projected sugar beet sales quantity in the calibration point \( NETTRD_{SUGB}^{cal} \) : For a country with a high over quota production (say 40%) we would obtain a scaling factor of 1.31, such that this producer will behave like a moderate C-sugar producer: responsive to both the C-beet prices as well as to the quota beet price (and the legal quotas). Without this scaling factor, producers with significant over quota p   roduction, like France and Germany, would not show any sizeable response to a 10% cut of either the legal quotas or the quota price (at empirically observed coefficients of variation). As it is likely that the profitability of ethanol beets benefit from cross-subsidisation from the quota beets such a zero responsiveness was considered implausible.
 ===Update note===
@@ Line 356: / Line 357: @@
 A number of recent developments are not covered in the previous exposition of supply model equations
-  -A series of projects have added a distinction of rainfed and irrigated varieties of most crop activities which is the core of the so-called “CAPRI-water” version of the system((A more complete presentation is given in [[https://ec.europa.eu/jrc/en/publication/eur-scientific-and-technical-research-reports/extension-capri-model-irrigation-sub-module]].)).
+   - A series of projects have added a distinction of rainfed and irrigated varieties of most crop activities which is the core of the so-called “CAPRI-water” version of the system
-  -Several projects have added endogenous GHG mitigation options((These are most completely included in the “trunk” version of the CAPRI system. For details, see, for example, [[http://publications.jrc.ec.europa.eu/repository/bitstream/JRC101396/jrc101396_ecampa2_final_report.pdf]].))
+  - Several projects have added endogenous GHG mitigation options
-  -Several new equations serve to explicitly represent environmental constraints deriving from the Nitrates Directive and the NEC directive((These are most completely included in the “trunk” version of the CAPRI system but developments are still ongoing.)).
+  - Several new equations serve to explicitly represent environmental constraints deriving from the Nitrates Directive and the NEC directive
-  -A complete area balance monitoring the land use changes according to the six UNFCCC land use types (cropland, grassland, forest land, wetland, settlements, residual land) has been introduced for carbon accounting
 ====Calibration of the regional programming models====
@@ Line 392: / Line 392: @@
 Y_{j,t}=Y_{j,t-1}^{[\epsilon_jlog \frac{p_o,t-1}{p_t}]}
 \end{equation}
+==== LULUCF in the supply model of CAPRI ====
+=== Introduction ===
+This technical paper explains how the most aggregate level of the CAPRI area allocation in the context of the supply models has been re-specified in the TRUSTEE ((https://www.trustee-project.eu/
+)) and SUPREMA ((https://www.suprema-project.eu/)) projects and subsequently adopted in the CAPRI trunk. The former specification for land supply and transformation functions focused on agricultural land use and the transformation of agricultural land between arable land and grass land.
+During the subsequent period, CAPRI was increasingly adapted to analyses of greenhouse gas (GHG) emission studies. Examples include CAPRI-ECC, GGELS, ECAMPA-X, AgCLim50-X, (European Commission, Joint Research Centre), ClipByFood (Swedish Energy Board), SUPREMA (H2020). This vein of research is very likely to gain in importance in the future.
+In order to improve land related climate gas modelling within CAPRI, it was deemed appropriate to (1) extend the land use modelled to //all// available land in the EU (i.e. not only agriculture), and (2) to explicitly model //transitions// between land use classes. The pioneering work was carried out within the TRUSTEE project((https://www.trustee-project.eu/
+)), but as always, an operational version emerged only after integrating efforts by researchers in several projects working at various institutions. Within the SUPREMA project another important change in the depiction of land use change was made: the Markov chain approach was replaced by prespecifying the total land transitions as average transitions per year times the projection. This paper focusses on the theory applied while data and technical implementation are only briefly covered.
+=== A simple theory of land supply ===
+Recall the dual methodological changes attempted in this paper:
+  - Extend land use modelling to the entire land area, and
+  - Explicitly model transitions between each pair of land uses
+In order to keep things as simple as possible, we opted for a theory where the decision of how much land to allocate to each use is independent of the explicit transitions between classes. This separation of decisions is simplifying the theoretical derivations, but also seem to have some support in theory: land use transitions show a good deal of stability over time. We would like to remind sceptics of this assumption that the converse is not implied: land transitions are certainly strongly depending on the land use requirements.
+The land supply and transformation model developed here is a bilevel optimization model. At the higher level (sometimes termed the //outer problem//), the land owner decides how much land to allocate to each aggregate land use based on the rents earned in each use and a set of parameters capturing the costs required in order to ensure that the land is available to the intended use. At the lower level (sometimes termed the //inner problem//), the transitions between land classes are modelled, with the condition that the total land needs of the outer problem are satisfied. The inner problem is modelled as a stochastic process involving no explicit economic model.
+For the outer problem, i.e. the land owner’s problem, we propose a quadratic objective function that maximizes the sum of land rents minus a dual cost function. The parameters of the dual cost function were specified in two steps:
+  - A matrix of land supply elasticities was estimated (by TRUSTEE partner Jean Saveur Ay, CESEAR, Dijon (JSA). This estimation might be updated in future work or replaced with other sources for elasticities.
+  - The parameters of the dual cost function are specified so that the supply behaviour replicates the estimated elasticities as closely as possible while exactly replicating observed/estimated land use and land rents.
+The model is somewhat complicated by the fact that land use classes in CAPRI are defined somewhat differently compared to the UNFCCC accounting and also in the land transition data set. Therefore, some of the land classes used in the land transitions are different from the ones used in the land supply model. In particular, “Other land”, “Wetlands” and “Pasture” are differently defined. To reconcile the differences, we assumed constant shares of the intersections of the different sets, as explained below.
+=== Inner model – transitions ===
+A vector of supply of land of various types could result from a wide range of different transitions. The inner model determines the matrix of land transitions that is “most likely”. The concept of “most likely” is formalized by assuming a joint density function for the land transitions, based on the historically observed transitions. The model then is to find the transition matrix that maximizes the joint density function.
+== Gamma density ==
+Since each transition is non-negative, but in principle unlimited upwards, we opted for a gamma density function, that has the support $\lbrack 0,\infty\rbrack$. For those that cannot immediately recall what the gamma density function looks like, and as entertainment for those that can, Figure 1 shows the graph of the density function for different parameters, all derived from an assumed mode of “1” and different assumed ratios “mode/standard deviations” (that we called “acc” for “accuracy” in the figure).
+{{:wiki:gamma_dens_land.png?nolink|}}
+Figure 1: Gamma density graph for mode=1 and various standard deviations. “acc”="mode/standard deviation".
+Let $i$ denote land use classes in CAPRI definition, whereas //l// and //k// are land uses in UNFCCC classification. Let $\text{LU}_{k}$ be total land use after transitions and $\text{LU}_{l}^{\text{initial}}$ be land use before transitions. Furthermore, let $T_{\text{lk}}$ denote the transition of land from use $l$ to use $k$. Noting that it is simpler and fully equivalent to maximize a sum of logged densities than a product of densities, the likelihood maximization problem can be written (with //f// being the gamma density function)
+$${\max_{T_{\text{lk}}}{\log{\prod_{\text{lk}}^{}{f\left( T_{\text{lk}}|\alpha_{\text{lk}},\beta_{\text{lk}} \right)}}}}{= \max_{T_{\text{lk}}}{\sum_{\text{lk}}^{}{\log{f\left( T_{\text{lk}}|\alpha_{\text{lk}},\beta_{\text{lk}} \right)}}}}$$
+$$\Rightarrow \max_{T_{\text{lk}}}\sum_{\text{lk}}^{}\left\lbrack \left( \alpha_{\text{lk}} - 1 \right)\log T_{\text{lk}} - \beta_{\text{lk}}T_{\text{lk}} \right\rbrack$$
+subject to
+$$\text{LU}_{k} - \sum_{l}^{}T_{\text{lk}} = 0 \; \left\lbrack \tau_{k} \right\rbrack$$
+$$\text{LU}_{l}^{\text{initial}} - \sum_{k}^{}T_{\text{lk}} = 0\;\left\lbrack \tau_{l}^{\text{initial}} \right\rbrack$$
+$$\text{LU}_{k} - \sum_{i}^{}{\text{shar}e_{\text{ki}}\text{LEV}L_{i}} = 0$$
+The last equation is needed to convert land use in UNFCCC classification to land use in CAPRI classification, using a fixed linear transformation matrix $\text{shar}e_{\text{ki}}$. This discrepancy between land class accounts will be expanded on in a subsequent section. Forming the Lagrangian function and taking the derivatives with respect to land transitions gives the following first-order optimality conditions:
+$$\ \left( \alpha_{\text{lk}} - 1 \right)T_{\text{lk}}^{- 1} - \beta_{\text{lk}} + \tau_{k}^{} + \tau_{l}^{\text{initial}} = 0$$
+The parameters $\alpha$ and $\beta$ of the gamma density function were computed by assuming that (i) the observed transitions are the mode of the density, and (ii) the standard deviation equals the mode. Then the parameters are obtained by solving the following quadratic system:
+$$\text{mode} = \frac{\alpha - 1}{\beta}$$
+$$\text{variance} = \frac{\alpha}{\beta^{2}}$$
+== Annual transitions via Marcov chain in basic model ==
+The implementation in CAPRI differs from the above general framework in that it explicitly identifies the //annual// transitions in year t $T_{\text{lk}}^{t}$ from the initial $\text{LU}_{l}^{\text{initial}}$ land use to the final land use $\text{LU}_{k}$. This is necessary to identify the annual carbon effects occurring only in the final year in order to add them to the current GHG emissions, say from mineral fertiliser application in the final simulation year. If the initial year is the base year = 2008 and projection is for 2030, then the carbon effects related to the change from the 2008 $\text{LU}_{l}^{\text{initial}}$ to the final land use $\text{LU}_{k}$ (=$T_{\text{lk}}$in the above notation, without time index) refer to a period of 22 years that cannot reasonably be aggregated with the “running” non-CO2 effects from the final year 2030. Furthermore the historical time series used to determine the mode of the gamma density for the transitions also refer to annual transitions.
+Initially the problem to link total to annual transitions has been solved by assuming a linear time path from the initial to the final period, but this was criticised as being an inconsistent time path (by FW). Ultimately the time path has been computed therefore in the supply model in line with a static Markov chain with constant probabilities $P_{\text{lk}}$ such that both land use $\text{LU}_{l}^{t}$ as well as transitions $T_{\text{lk}}^{t}$ in absolute ha require a time index (e_luOverTime in supply_model.gms).
+$$\text{LU}_{k}^{t} - \sum_{l}^{}{P_{\text{lk}}\text{LU}_{l}^{t - 1}} = 0\ ,\ t = \{ 1,\ldots s\}$$
+Where $\text{LU}_{k}^{s}$ is the final land use in the simulation year s and $\text{LU}_{k}^{0} = \text{LU}_{k}^{\text{iniital}}$ is the initial land use. The transitions in ha in any year may be recovered from previous years land use and the annual (and constant) transition probabilities (e_LUCfromMatrix in supply_model.gms).
+$$T_{\text{lk}}^{t} = P_{\text{lk}}*\text{LU}_{l}^{t - 1}$$
+The absolute transitions may enter the carbon accounting (ignored here) and if we substitute the last period’s transitions we are back to the condition for consistent land balancing in the final period from above:
+$$\text{LU}_{k}^{s} = \sum_{l}^{}{P_{\text{lk}}\text{LU}_{l}^{s - 1}} = \sum_{l}^{}T_{\text{lk}}^{s}$$
+When using the transition probabilities in the consistency condition for initial land use we obtain
+$$\text{LU}_{l}^{\text{initial}} - \sum_{k}^{}T_{\text{lk}}^{1} = 0$$
+$$\Longleftrightarrow \text{LU}_{l}^{\text{initial}} = \sum_{k}^{}{P_{\text{lk}}^{}\text{LU}}_{l}^{\text{iniital}}$$
+$$\Leftrightarrow 1 = \sum_{k}^{}P_{\text{lk}}$$
+So the simple condition is that probabilities have to add up to one (e_addUpTransMatrix in supply_model.gms).
+== Annual transitions if SUPREMA is active ==
+As the use of the Marcov-chain approach allows the annual transitions to be explicit model variables that could be used to compute annual carbon effects but leads to computational limitations especially in the market model a new approach was developed under SUPREMA (i.e. if %supremaSup% == on) by re-specifying the total land transitions as average transitions per year times the projection horizon and by considering for the remaining class without land use change (on the diagonal of the land transition matrix) only the annual carbon effects per ha, relevant for the case of gains via forest management.
+The new accounting in the CAPRI global supply model may be explained as follows, starting from a calculation of the total GHG effects G over horizon h = t-s from total land transitions L<sub>lk</sub> and carbon effects per ha for the whole period e<sub>lk</sub>:
+$$G = Γ*h = \sum_{i,l}^{}{e_{\text{il}}^{}\text{L}}_{il}^{}$$
+Where Γ collects the annual GHG effects that correspond to the total GHG effects divided by the time horizon G / h. These annual effects may be calculated as based on average annual transitions and annual effects for the remaining class as follows:
+$$Γ= \sum_{i,l}^{}{e_{\text{il}}^{}\text{L}}_{il}^{}/h = \sum_{i≠l}^{}{e_{\text{il}}^{}\text{Λ}}_{il}^{}
++ \sum_{i}^{}{ε_{\text{ii}}^{}\text{L}}_{ii}^{} $$
+Where Λ<sub>il</sub> = L<sub>il</sub> / h is the average land use change per year and ε<sub>ii</sub> is the annual carbon effect on a remaining class (relevant might be an annual increase due to growing forests while this will be zero for most effects based on IPCC default assumptions).
+Using these average annual transitions for true (off-diagonal) LUC we may compute the final classes as follows:
+$$ \text{LU}_{k,t} = \sum_{l}^{}{L_{\text{lk}}^{}\text{}}_{}^{} = \sum_{l≠k}^{}{Λ_{\text{lk}}^{}*h +\text{LU}_{kk}\text{}}_{}^{}$$
+While adding up of shares (or probabilities) of LUC from class I to k over all receiving classes k continues to hold as stated above. It should be highlighted that the land use accounting implemented under SUPREMA avoids the need to explicitly trace the annual transitions in the form of a Markov chain and thereby economised on equations and variables.
+=== Outer model – land supply ===
+The outer problem is defined as a maximization of the sum of land rents minus a quadratic cost term, subject to the first order optimality conditions of the inner problem:
+$$\max{\sum_{i}^{}{\text{LEV}L_{i}r_{i}} - \sum_{i}^{}{\text{LEV}L_{i}c_{i}} - \frac{1}{2}\sum_{\text{ij}}^{}{\text{LEV}L_{i}D_{\text{ij}}\text{LEV}L_{j}}}$$
+subject to,
+$$\text{LU}_{k} - \sum_{i}^{}{\text{shar}e_{\text{ki}}\text{LEV}L_{i}} = 0$$
+$$\text{LU}_{k} - \sum_{l}^{}T_{\text{lk}} = 0\;\left\lbrack \tau_{k} \right\rbrack$$
+$$\text{LU}_{l}^{\text{initial}} - \sum_{k}^{}T_{\text{lk}} = 0\;\left\lbrack \tau_{l}^{\text{initial}} \right\rbrack$$
+$$\ \left( \alpha_{\text{lk}} - 1 \right)T_{\text{lk}}^{- 1} - \beta_{\text{lk}} + \tau_{k}^{} + \tau_{l}^{\text{initial}} = 0$$
+The parameters of the inner model **α** and **β// //**may be determined as explained in the previous sections. For the outer model, we need to define the parameters **c** and **D**. We have a single data point of land use and land rent for each land use class. Since we have, for $N$ land classes, $N + N(N - 1)/2$ parameters, but only $N$ price-quantity pairs (one data point for each land class). This means that without any additional information, we could e.g. calibrate the model exactly by computing the **c** parameter, but have no information left for defining **D**//.// However, we have at our disposal prior estimates of the regional matrices of land supply elasticities that may be used to define prior densities for the elasticity matrix implied jointly by the **c** and **D** parameters and the inner problem. Another way of expressing this is that we compute a meta parameter matrix $\mathbf{\eta}\left( \mathbf{c},\mathbf{D},\mathbf{L}\mathbf{U}^{\text{initial}} \right)$ that is a function of the real parameters, and use the prior elasticity matrix as a prior for this meta parameter. If cast in this way, the problem becomes a Bayesian econometric estimation.
+There are a few methodological and numerical challenges to overcome. In particular, we need to (i) analytically derive $\mathbf{\eta}\left( \mathbf{c},\mathbf{D},\mathbf{L}\mathbf{U}^{\text{initial}} \right)$, and (ii) ensure that the resulting model has the appropriate curvature to ensure a unique interior solution – anything else would result in a rather useless model. We start by simplifying the problem by observing that all the constraints (the first order conditions of the inner problem) can be replaced with an ordinary land constraint:
+$$\sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} = 0$$
+Note that the second sum is a constant. This simplification is based on the observation that the land transitions don’t appear in the objective function of the outer problem, so that all solutions to the inner problems are equivalent from the perspective of the outer problem, and that any land use vector that preserves the initial land endowment is a feasible solution to the inner problem.
+Next, we formulate the first order condition (FOC) of the modified outer problem to obtain land use as an implicit function of the parameters, $F\left( LEVL,c,D,LU^{\text{initial}},r \right) = 0$. We can then use the implicit function theorem to compute the derivative of land supply $\text{LEV}L_{i}$ with respect to land rent $r_{j}$, which in turn can be used to define the elasticity matrix $\mathbf{\eta}$.
+The first order conditions, and the implicit function, become
+$$F\left( LEVL,\lambda,c,D,LU^{\text{initial}},r \right) = \begin{bmatrix}
+\frac{\partial\mathcal{L}}{\partial LEVL_{i}} = & r_{i} - c_{i} - \sum_{j}^{}{D_{\text{ij}}\text{LEV}L_{j}} - \lambda & = 0 \\
+\frac{\partial\mathcal{L}}{\partial\lambda} = & \sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} & = 0 \\
+\end{bmatrix}$$
+In order to apply the implicit function theorem((Recall that the implicit function theorem states that if F(x,p) = 0, then dx/dp = -[dF/dx]<sup>-1</sup>[dF/dp]
+)) we need to differentiate the FOC once w.r.t. the variables $\text{LEV}L_{i}$ and $\lambda$ and once with respect to the parameter of interest, $r_{j}$, invert the former and take the negative of the matrix product. If (currently) irrelevant parameter are omitted, the following matrix of $(N + 1) \times (N + 1)$ is obtained (the “+1” is the uninteresting derivative of total land rent $\lambda$ with respect to individual land class rent $r_{i}$)
+$$\left\lbrack \frac{\partial LEVL}{\partial r} \right\rbrack = - \left\lbrack D_{LEVL,\lambda}F(LEVL,\lambda,r) \right\rbrack^{- 1}D_{r}F(LEVL,\lambda,r)$$
+$$\begin{bmatrix}
+\frac{\partial LEVL}{\partial r} \\
+\frac{\partial\lambda}{\partial r} \\
+\end{bmatrix} = - \begin{bmatrix}
+\frac{\partial F}{\partial LEVL} & \frac{\partial F}{\partial\lambda} \\
+\end{bmatrix}\left\lbrack \frac{\partial F}{\partial r} \right\rbrack$$
+Carrying out the differentiation specifically for land rent //r<sub>j</sub>//, we obtain:
+$$\begin{bmatrix}
+\frac{\partial LEVL_{i}}{\partial r_{j}} \\
+\frac{\partial\lambda}{\partial r_{j}} \\
+\end{bmatrix} = - \begin{bmatrix}
+\left\lbrack {- D}_{\text{ij}} \right\rbrack & - 1 \\
+ - 1' & 0 \\
+\end{bmatrix}^{- 1}\begin{bmatrix}
+I \\
+\\
+\end{bmatrix}$$
+Discarding the last row of the resulting $(N + 1) \times N$ matrix finally lets us compute the elasticity as
+$$\left\lbrack \eta_{\text{ij}} \right\rbrack = \left\lbrack \frac{\partial LEVL_{i}}{\partial r_{j}} \right\rbrack\left\lbrack \frac{r_{j}}{\text{LEV}L_{i}} \right\rbrack$$
+In the estimation, we assumed that the prior elasticity matrix is the mode of a density where each entry were independently distributed. Furthermore, the off-diagonal or any diagonal elements with negative priors were normally distributed, whereas the diagonal elements with positive priors (as required for a well-behaved curvature) were gamma distributed. For the standard deviation of elasticities we used either information from the prior estimates or some fall-back assumptions on standard deviations relative to the mode of elasticities. Denoting the prior elasticities with $e_{\text{ij}}$, we solved the following optimization problem, where parameters $\alpha$ and $\beta$ were already estimates as explained in the sections on the inner problem.
+$$\max_{\eta,c,D}{\sum_{ij \in normal(i,j)}^{}{- {\frac{1}{s_{\text{ij}}^{2}}\left( \eta_{\text{ij}} - e_{\text{ij}}^{\text{jsa}} \right)}^{2}} + \sum_{ij \in gamma(i,j)}^{}\left\lbrack \left( \alpha_{\text{ij}} - 1 \right)\log\eta_{\text{ij}} - \beta_{\text{ij}}\eta_{\text{ij}} \right\rbrack}$$
+subject to
+$$\left\lbrack \frac{\partial LEVL_{i}}{\partial r_{j}} \right\rbrack = - \begin{bmatrix}
+\left\lbrack {- D}_{\text{ij}} \right\rbrack & - 1 \\
+ - 1' & 0 \\
+\end{bmatrix}^{- 1}\begin{bmatrix}
+I \\
+\\
+\end{bmatrix}$$
+$$\left\lbrack \eta_{\text{ij}} \right\rbrack = \left\lbrack \frac{\partial LEVL_{i}}{\partial r_{j}} \right\rbrack\left\lbrack \frac{r_{j}}{\text{LEV}L_{i}} \right\rbrack$$
+$$\begin{matrix}
+ & r_{i} - c_{i} - \sum_{j}^{}{D_{\text{ij}}\text{LEV}L_{j}} - \lambda & = 0 \\
+ & \sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} & = 0 \\
+\end{matrix}$$
+and the curvature constraint using a stricter variant of the Cholesky factorization
+$$D_{\text{ij}}\left( 1 - \delta I_{\text{ij}} \right) = \sum_{k}^{}{U_{\text{ki}}U_{\text{kj}}}$$
+where $\delta$ is a small positive number and $I_{\text{ij}}$ entries of the identity matrix such that the factor $(1 - \delta I_{\text{ij}})$ shrinks the diagonal of the D-matrix, ensuring //strict// positive definiteness instead of //semi-//definiteness. We used $\delta = 0.05$. Furthermore, the Lagrange multiplier of the total land constraint, $\lambda$, was fixed at the weighted average of the rents $r_{i}$, i.e. $\lambda = \frac{\sum_{i}^{}{\text{LEV}L_{i}r_{i}}}{\sum_{i}^{}{\text{LEV}L_{i}}}$. Without the latter assumption, the parameters c and D are not uniquely identified.
+==Prior elasticities and area mappings==
+The empirical evidence obtained in the TRUSTEE project applied to prior elasticities for land categories based on Corine Land Cover (CLC) data. These categories are also covered in the CAPRI database based on various sources (see the database section in the CAPRI documentation):
+The introduction has mentioned already three systems of area categories that need to be distinguished. The first one is the set of area aggregates with good coverage in statistics that has been investigated recently by JS Ay (2016), in the following “JSA”:
+$$\text{LEVL} = \left\{ \text{ARAC},\ \text{FRUN},\ \text{GRAS},\ \text{FORE},\ \text{ARTIF},\text{OLND} \right\}$$
+Where
+ARAC = arable crops
+FRUN = perennial crops
+GRAS = permanent grassland
+FORE = forest
+ARTIF = artificial surfaces (settlements, traffic or industrial)
+OLND = other land
+The above categories are matching reasonably well with the definitions in JSA. A mismatch exists in the classification of paddy (part of ARAC in CAPRI but in the perennial group in JSA) and terrestrial wetlands (part of OLND in CAPRI and a separate category in JSA). Inland waters are considered exogenous in CAPRI and hence not included in the above set LEVL.
+For carbon accounting we need to identify the six LU classes from IPCC recommendations and official UNFCCC reporting:
+$$LU = \left\{ \text{CROP},\ \text{GRS}\text{LND},\ \text{FORE},\ \text{ARTIF},WETLND,RESLND \right\}$$
+which is typically indexed below with “l” or “k” ∈ LU and where
+CROP = crop land (= sum of arable crops and perennial crops)
+GRSLND = grassland in IPCC definition (includes some shrub land and other “nature land”, hence GRSLND>GRAS)
+WETLND = wetland (includes inland waters but also terrestrial wetlands)
+RESLND = residual land is that part of OLND not allocated to grassland or wetland, hence RESLND<OLND
+FORE = forest
+ARTIF = artificial surfaces
+In the CAPRI database, in particular for its technical base year, we have estimated an allocation of other land OLND into its components attributable to the UNFCCC classes GRSLND,WETLND, and RESLND:
+$$\text{OLND}^{0} = {\text{OLND}G}^{0} + {\text{OLND}W}^{0} + {\text{OLND}R}^{0}$$
+Lacking better options to make the link between sets LEVL (activity level aggregates) and LU (UNFCCC classes, technically in CAPRI code: set “LUclass”) we will assume that these shares are fixed and may estimate the “mixed” LU areas from activity level aggregates as follows
+^//GRSLND//^=^//GRAS + OLND · OLNDG<sup>0</sup>/OLND<sup>0</sup>//^
+|WETLND    |=|//INLW + OLND · OLNDW<sup>0</sup>/OLND<sup>0</sup>//|
+|RESLND    |=|//OLND · OLNDR<sup>0</sup>/OLND<sup>0</sup>//       |
+which means that the mapping from set LEVL to set LU only uses some fixed shares of LEVL areas that are mapped to a certain LU:
+$$LU_k=\sum_i{\text{share}_{\text{i,k}}\text{LEVL}_i}$$
+where 0 ≤ //share<sub>i,k</sub>// ≤ 1.
+===Technical implementation===
+The key equations corresponding to the approach explained above are collected in file supply_model.gms or the included files supply/declare_calibration_models_for_luc.gms and supply/declare_calibration_models_for_land_supply.gms. The declarations of parameters, variables, equations, models and even some sets only used in the calibration given in these files are included by the “supply_model.gms” only if “BASELINE==ON” or if it was a CAPREG base year task that was carried out. Loading of priors, initialisation of parameters and variables for the calibration as well as the organisation of solve attempts are handled in new sections of file “cal_land_nests.gms”, in turn called by the gams file “prep_cal.gms”. This implies that the land supply and land use change calibrations were inserted before the ordinary calibration of the supply models.
+//SupremaSup// should be active together with //trustee_land// to have smoother adjustments which may be set via the CAPRI GUI. In order to store the results of the calibration in a compact way that is compatible with the existing code, the existing parameter files “pmppar_XX.gdx” was used. The parameters of the land supply functions, called “c” and “D” above, were stored on two parameters “p_pmpCnstLandTypes” and “p_pmpQuadLandTypes”. As a new symbol (p_pmpCnstLandTypes) is introduced in an existing file, the first run of CAPRI after setting %trustee_land%==on may give errors if the file exists already but has been used with the previous land supply specification before. In this case it helps to delete or rename the old pmppar files.
+At this point, it should also be explained that rents for non-agricultural land types were entirely based on assumptions (a certain ratio to agricultural rents). As there were no plans to run scenarios with modified non-agricultural rents, these land rents //r// used in calibration for those land types were subtracted from the “c-paramter”, so that it is implicitly stored in p_pmpCnstLandTypes and enters the objective function through the PMP terms. This requires changes if the rents shall be modified or if non-agricultural production shall be included in some simplified form.
+Furthermore, the class Inland Waters (INLW) was given a special treatment: it is supposed to be entirely exogenous. For this purpose the special acronym “exogenousLandSupply” was introduced, and stored on the p_pmpCnstLandTypes and used to trigger an equation “e_exogenousLand “ in the supply model setting the variable to a constant. In that way, the fixity of INLW (or any land type, should it happen) is stored in the pmp terms and cannot be “forgotten”.
+More detailed explanations on the technical implementation are covered elsewhere, for example in the “Training material” included in the EcAMPA-4 deliverable D5.
+Concerning the improvements made under SUPREMA from a technical perspective, the changes are merged to the trunk. The approach is controlled by globals in capmod\set_global_variables.gms. If the global variable %supremaSup% == on, the yearly transition rate p_lucAnnualFac_sup is calculated. If it is %supremaSup% == off, the old approach using the Marcov chain is used with the respective variable v_luYearly. The FOC-approach to calculate LUC as described above is standard and independent from if the global variable supremaSup is on or off.
+=== Emission Equations ===
+Under EcAMPA 3 and partly in earlier projects (inter alia EcAMPA 2) new modelling outputs have been developed for indicators without matching reporting infrastructure helping users to organise the additional information. This applied for example to
+) Additional CAPRI results on land use results related to the complete area coverage, mappings to UNFCCC area categories and their transitions;
+) The carbon effects linked to these land transitions.
+Furthermore, additional non-CO2 and CO2 related mitigation measures had been included under EcAMPA 3.
+The scenarios including the emission equations are only run if %ghgabatement% == on, otherwise emissions are only calculated and not simulated.
+The following emission equations have been implemented:
+^**Code**          ^**Description**                                                                                                ^
+|GWPA              |Agricultural emissions                                                                                         |
+|CH4ENT            |Methane emissions from enteric fermentation                                                                    |
+|CH4MAN            |Methane emissions from manure management                                                                       |
+|CH4RIC            |Methane emissions from rice production                                                                         |
+|N2OMAN            |Direct nitrous oxide emissions stemming from manure management (only housing and storage)                      |
+|N2OAPP            |Direct nitrous oxide emissions stemming from manure application on soils except grazings per animal activity   |
+|N2OGRA            |Direct nitrous oxide emissions stemming from manure managment on grazings                                      |
+|N2OSYN            |Direct nitrous oxide emissions from anorganic fertilizer application                                           |
+|N2OCRO            |Direct nitrous oxide emissions from crop residues                                                              |
+|N2OAMM            |Indirect nitrous oxide emissions from ammonia volatilisation                                                   |
+|N2OLEA            |Indirect nitrous oxide emissions from leaching                                                                 |
+|N2OHIS            |Direct nitrous oxide emissions from cultivation of histosols                                                   |
+|GLUC              |Emissions related to indirect land use changes                                                                 |
+|CO2BIO            |Carbon dioxide emissions from land use change due to losses of carbon in biomass and litter                    |
+|CO2SOI            |Carbon dioxide emissions from land use change due to soil carbon losses                                        |
+|CO2HIS\\ \\ CH4HIS|Carbon dioxide emissions from the cultivation of histosols\\ \\ Methane emissions from cultivation of histosols|
+|CO2LIM\\ \\ CO2BUR|Carbon dioxide emissions from limestone and dolomit\\ \\ Carbon dioxide emissions from burning                 |
+|CH4BUR            |Methane emissions from burning                                                                                 |
+|N2OBUR            |Nitrous oxide emissions from burning                                                                           |
+|N2OSOI            |N2O emissions from land use change due to soil carbon losses                                                   |
+|GPRD              |Emissions related to the production of non-agricultural inputs to agriculture                                  |
+|N2OPRD            |Nitrous oxide emissions during fertilizer production                                                           |
+|O2PRD             |Carbon Dioxide emissions during fertilizer production                                                          |
 =====Premium module=====
@@ Line 451: / Line 767: @@
 **Figure 14: Example of technical implementation of a premium scheme**
-{{:figure14.png?600|Source: CAPRI Modelling System. Note: The parameter PPDATA_E is now called p_premDataE.}}
+{{:figure_14.png?600|Source: CAPRI Modelling System. Note: The parameter PPDATA_E is now called p_premDataE.}}
 The sets of payments, exemplified by DPGRCU in the figure, and the activity groups, exemplified by PGGRCU and PGPROT are defined in the file policy/policy_sets.gms. Since this is a “static” GAMS file used in any simulation, it contains the gross list of all policies that currently can be simulated, including legacy ones. In order to work efficiently with the acronyms which define the application types, these are converted to numerical attributes as shown below (//‘policy/policy.gms’)//:
@@ Line 472: / Line 788: @@
 **Figure 15: General flow of logic of CAPRI model as regards premiums**
-{{::fiugre15.png?600|Source: own illustration}}
+{{::figure_15.png?600|Source: own illustration}}
 Generally, all attributes for a premium scheme are mapped down in space, e.g. from EU27 to EU 27 member states, from countries to NUTS1 regions inside the country, from there to the NUTS2 regions inside the NUTS1, and from NUTS2 regions to the farm types in a NUTS2 region (see //‘policy/policy.gms’//), e.g.
@@ Line 561: / Line 877: @@
 **Figure 16: General way of SFP implementation in CAPRI**
-{{::figure16.png?600|Source: own illustration}}
+{{::figure_16.png?600|Source: own illustration}}
 In opposite to the reforms until Agenda 2000, there are hence in most cases not longer premium rates or individual ceilings in hectares found in legal texts. Rather, these are calculated by the model itself from the decoupled part of the “old” Mac Sharry and Agenda 2000 premiums which introduces additional complexity in the model code.
@@ Line 679: / Line 995: @@
 **Figure 17: The logic of the CAP 2014-2020 reference policy as implemented in the CAPRI policy module**
-{{:figure17.png?600|Source: own illustration}}
+{{:figure_17.png?600|Source: own illustration}}
 ===Tradable Single Premium Scheme entitlements===
@@ Line 815: / Line 1131: @@
 Finally, an extensification effect to the AE payments is introduced using the possibility to make technological variants differently eligible.
-{{:code_p179_2.png?600}} \\
+{{:code_p179_2.png?600}}
 {{:code_p180.png?600}}
@@ Line 842: / Line 1159: @@
 {{:code_p182.png?600}}
-Currently, the following budget categories are supported (see ‘//sets.gms//’ and ‘//policy\policy_sets.gms//’):
+Currently, the following budget categories are supported (see ‘//sets.gms//’ and ‘//policy/policy_sets.gms//’):
 {{:code_p182_2.png?600}}
@@ Line 921: / Line 1238: @@
 **Figure 18: Land supply curve examples**
-{{::figure18.png?600|Source: own calculations}}
+{{::figure_18.png?600|Source: own calculations}}
 In order to parameterize the land demand function, information about yield and supply elasticities is used. The marginal reaction of land to a marginal change in one of the prices is defined as the total supply effect minus the yield effect:
@@ Line 960: / Line 1277: @@
 The disadvantage of the behavioural functions above is the fact that they might generate non-positive values. That situation might be interpreted as a combination of prices where the marginal costs exceed marginal revenues. Accordingly, a fudging function is applied for supply, feed and (see below) processing demand which ensures strictly positive quantities. That fudging function is highly non-linear, and therefore only switched on on demand.
+====Land use, land use change and forestry (LULUCF) ====
+===LULUCF in the basic model ===
+Before SUPREMA LULUCF and area-based carbon accounting were not depicted in the global market model. Land demand was conceptually derived from maximising farmers profit. Land supply was represented with a function that links supply to agricultural land rents with an elasticity. Non-agricultural land use that complements farm land to give the total region area was disaggregated into forestry, built up areas (urban or “artificial” land) and a remaining “other land” category. There was neither a mapping of land use categories in the market model to the UNFCCC categories, nor a modelling of the transition matrix accompanied by a very limited product-based carbon accounting which was not in line with IPCC.
+The pre-SUPREMA specification may be described as follows.
+Agricultural outputs i (barley, wheat, beef ...) have land requirements LV<sub>i</sub> derived from production of these outputs (via yields that respond to prices according to yield elasticities). Adding up all land requirements gives total agricultural land (LT<sub>ag</sub>).
+$${LT}_{ag} = \sum_{i}^{}{{LV}_{i}(\mathbf{P},R_{ag})}$$
+Land demand depends on a vector of prices and the agricultural and rent R<sub>ag</sub> (treated separately). Total agricultural land is just one of several land types (l) that play a role:
+l = {ag, tc, pc, fd, no, fr, ur, ot, iw}, where
+ag = total agricultural land
+tc = temporary (non-fodder) crops
+pc = permanent crops
+fd = temporary fodder, permanent grassland and fallow land
+no = non-agricultural land
+fr = forest land
+ur = settlements, industrial, built up md any other artificial areas
+ot = other land
+iw = inland waters (exogenous)
+Matching with land demand there is a land supply function for total agricultural land
+$${LT}_{ag} = \alpha{R_{ag}}^{\beta}$$
+Given agricultural land and an exogenous region area as well as exogenous inland waters permits to compute total non-agricultural land residually:
+$${LT}_{no} = T - {LT}_{ag} - {LT}_{iw}$$
+This total non-agricultural land (beyond inland waters) is currently allocated to non-agricultural land types n = { fr, ur, ot} according to the shares of “intermediate” areas for non-agricultural land types:
+$${LT}_{n} = {LT}_{no}*{\widehat{LT}}_{n}/\sum_{n}^{}{\widehat{LT}}_{n}$$
+The “intermediate” areas in turn result from the change in the non-agricultural area against the baseline, considering elasticities that reflect the responsiveness of land types to imbalances:
+$${{\widehat{LT}}_{n} = LT}_{n}^{0}*\left( \frac{{LT}_{no}^{}}{{LT}_{no}^{0}} \right)^{\gamma_{n}}$$
+The concept of elasticities of land types to imbalances expresses the expectation that any disequilibrium in the land balance is very unlikely to be removed by changes in settlement area, and probably only to a small extent by changes in forest land and therefore most of all by changes in other land category.
+In spite of full consistency, we have changed this specification under SUPREMA, as 1) it is difficult to reconcile with welfare accounting, 2) it turned out that the scaling mechanism may dominate the planned responsiveness of land types, 3) the asymmetric specifications for supply of agricultural and non-agricultural land is ad-hoc and intransparent and 4) it does not link to standard empirical parameter estimation.
+== Land transitions via Gamma density and Marcov chain  ==
+The area-based carbon modelling and accounting requires the land transition matrix describing how an initial allocation of land uses (either from the base year or from an intermediate simulation year) is transformed into the currently simulated one. The transition matrix may be expressed in terms of absolute areas L<sub>jk</sub> changing from land use LU<sub>j</sub> in the initial year s into another land use LU<sub>k</sub> in the final year t or in terms of a transition matrix sh<sub>jk</sub> giving the share (probability in a Markov chain) of initial land use LU<sub>j</sub> converted into the final LU<sub>k</sub> over the whole horizon of (t-s):
+$${LU}_{k,t} = \sum_{j}^{}{{sh}_{jk}{LU}_{j,s} = \sum_{j}^{}L_{jk}}$$
+Where the shares (probabilities) have to add up to one:
+$$1 = \sum_{k}^{}{{sh}_{jk},\ \forall j}$$
+The total areas converted from initial land use j into final land use k over the horizon (t-s) are denoted L<sub>jk</sub> above. For those land transitions we would expect that the pattern of changes resembles that observed in the past, at least if total land uses LU<sub>j</sub> change similarly as in the past. This expectation corresponds to the most likely land transitions maximising a Gamma density, giving for each transition a corresponding FOC:
+$$\ \left( \lambda_{jk} - 1 \right)L_{jk}^{- 1} - \mu_{jk} + \tau_{k}^{} + \tau_{j}^{initial} = 0$$
+Where λ<sub>jk</sub> and μ<sub>jk</sub> are parameters related to the mode (determined from the database or baseline projection) and standard deviation (assumed = 1) of the Gamma density. The variables τ<sub>k</sub> and τ<sub>j</sub> are shadow values paired with the final year land use accounting from transition probabilities and the adding up condition for probabilities.
+The original specification for land transitions as used in the CAPRI supply models involve 6x(t-s) = 120 equations for a 20 years time horizon to represent a Markov chain of annual land transitions for each region. The advantage of this specification was that annual transitions were explicit model variables that could be used to compute annual carbon effects. These were hence comparable to annual non-CO2 effects related to agricultural production and could be added therefore to obtain total GHG effects from the LULUCF sector and non-CO2 GHGs. Having annual land transitions in each regional was also acceptable from a computational viewpoint for the relatively small regional supply models of CAPRI (about 1500 equations).
+However, in the global market model all regions (about 80 with agents like farmers, consumers or landowners) have to be solved simultaneously such that the additional equations and variables for the extended land use modelling and carbon accounting (addressed in the following section) could increase solution time beyond critical limits. Given that the standard market model already includes about 80000 equations the above framework was adjusted to give the land transitions in //one// step for the change from the initial years to the final year t, while still considering that we need annual carbon effects for comparability with the annual non-CO2 emissions. This has been achieved
+  * by re-specifying the total land transitions as average transitions per year times the projection horizon and
+  * by considering for the remaining class without land use change (on the diagonal of the land transition matrix) only the annual carbon effects per ha, relevant for the case of gains via forest management.
+=== LULUCF and carbon accounting if SUPREMA is active ===
+Within the SUPREMA project two major changes were made:
+First, integration across spatial scales was improved: a) the land activity and LULUCF representation was extended to non-European countries and b) the product-based carbon accounting was replaced by an area-based carbon accounting.
+Second, the methodological approach was changed including a) statistical estimation of land use changes assuming a gamma density as in the supply model, b) re-specification of the total land transitions as average transitions per year times the projection horizon as in the supply model (replacement of the Markov chain approach) and c) representation of the disaggregated land supply in the market model through multinomial logit form to. These changes in the SUPREMA project allow for a more symmetric land use representation and carbon accounting between the supply models for European NUTS2 regions and in the global market model of CAPRI.
+== Multinomial logit function ==
+Under SUPREMA we have introduced a multinomial logit form for land supply of all major endogenous land types f = g = h = m = {ag, fr, ur, ot}. This approach is conceptually fully in line with land supply in the regional supply models. In this way we have integrated and replaced the above separate treatment of land supply for agricultural and non-agricultural land:
+$${LT}_{m} = {SH}_{m}*T$$
+where the area share of land type m is
+$${SH}_{g} = \frac{\exp\left( \delta_{g0} + \sum_{f}^{}{\delta_{gf}R_{f}} \right)}{\sum_{m}^{}{\exp\left( \delta_{m0} + \sum_{f}^{}{\delta_{mf}R_{f}} \right)}}$$
+and the elasticity of share g (and due to constant region area also land type LT<sub>g</sub>) with respect to rent R<sub>h</sub> may be derived as
+$$\varepsilon_{gh} = \frac{\partial{SH}_{g}}{\partial R_{h}}\frac{R_{h}}{{SH}_{g}} = R_{h}\left( \delta_{gh} - \sum_{m}^{}{\delta_{mh}{SH}_{m}} \right)$$
+which permits to make use of the same empirical information (on elasticities of agricultural land supply) and assumptions (on the ranking of responsiveness of non-agricultural areas) that have been used so far in the pre-SUPREMA version. For this purpose, a calibration problem has been set up that minimises weighted squared differences to the starting values by modifying parameters δ<sub>mh</sub>. Due to its symmetric treatment of all major land uses the system also includes supply elasticities for non-agricultural areas. In “standard” scenarios these are unlikely to play a relevant role. The key parameters are the “cross-rent” elasticities of non-agricultural areas with respect to agricultural rents as these are steering now which non-agricultural areas are increasing if agricultural area declines and vice versa. However, in the context of global carbon price scenarios also the supply elasticities of non-agricultural areas play an important role even though we do not introduce assumptions on changing prices of urban land, forest land or other land. This is because a global carbon price creates an endogenous mark-up for land rental prices of non-agricultural areas that reflect the value of the carbon effects from changing land use. In particular the rental price of forest land R<sub>fr</sub> is strongly reduced and might become negative in scenarios with high carbon prices.
+== Spatial extension ==
+Under SUPREMA the land use categories of the market model are mapped to the UNFCCC categories. The mapping of market model land types LT<sub>l</sub> to UNFCCC land use LU<sub>k</sub> will rely on the most recent historical shares ϕ<sub>kl</sub> of UNFCCC land use k in CAPRI land type l:
+$${LU}_{k} = \sum_{l}^{}{\varphi_{kl}{LT}_{l}}$$
+These shares are trivially zero or one in case that certain land types like “temporary non-fodder crops” (tc) and permanent crops (pc) are exclusively mapped to one UNFCCC category (cropland). The remainder to total cropland derives from temporary fodder and fallow land which is a fraction of total fodder area with the remainder being (productive) permanent grassland. The allocation of “other land” (ot) to grassland (ϕ<sub>glot</sub>), wetland (ϕ<sub>wlot</sub>) and residual land (ϕ<sub>rlot</sub>) may occur as in the European database but required that some CAPRI code in use for the supply models was transferred into the context of the extended global market model.
+== Land transitions as average transitions per year times the projection horizon  ==
+The new accounting in the CAPRI global market model may be explained as follows, starting from a calculation of the total GHG effects G over horizon h = t-s from total land transitions L<sub>jk</sub> and carbon effects per ha for the whole period e<sub>jk</sub>:
+$$G = \Gamma \bullet h = \sum_{i,j}^{}{e_{ij}L_{ij}}$$
+Where Γ collects the annual GHG effects that correspond to the total GHG effects divided by the time horizon G / h. These annual effects may be calculated as based on average annual transitions and annual effects for the remaining class as follows:
+$
+$$$\Gamma = \sum_{i,j}^{}{e_{ij}L_{ij}/h} = \sum_{i \neq j}^{}{e_{ij}\Lambda_{ij}} + \sum_{i}^{}{\varepsilon_{ii}L_{ii}}$$
+Where Λ<sub>ij</sub> = L<sub>ij</sub> / h is the average land use change per year and ε<sub>ii</sub> is the annual carbon effect on a remaining class (relevant might be an annual increase due to growing forests while this will be zero for most effects based on IPCC default assumptions).
+Using these average annual transitions for true (off-diagonal) LUC we may compute the final classes as follows:
+$${LU}_{k,t} = \sum_{j}^{}L_{jk} = \sum_{j \neq k}^{}\Lambda_{jk} \bullet h + L_{kk}$$
+While adding up of shares (or probabilities) of LUC from class I to k over all receiving classes k continues to hold as stated above. It should be highlighted that the land use accounting implemented under SUPREMA avoids the need to explicitly trace the annual transitions in the form of a Markov chain and thereby economised on equations and variables. In this form LUC by CAPRI region and the associated accounting of carbon effects turned out computationally feasible even though the number of equations in the global market model increased from about 78000 to about 83000. Apart from feasibility the format above also permitted to retain the typical CAPRI accounting identity that some total “quantity” (“GROF”) should be computable as the effects “per activity” times activity levels. It was therefore also adopted in the CAPRI supply models.
+=== Technical aspects ===
+Concerning the improvements made under SUPREMA from a technical perspective, the changes are merged to the trunk. The approach is controlled by globals in capmod\set_global_variables.gms. If the global variable %supremaMrk% == on, the yearly transition rate p_lucAnnualFac_sup is calculated, land activity is expaned to non-european countries and the multinomial logit form approach is used to model land supply responsiveness. If it is %supremaMrk% == off, the old approach using the Marcov chain is used with the respective variable v_luYearly (for details on Marcov chain approach see [[module_for_agricultural_supply_at_regional_level|supply model description]]). The FOC-approach to calculate LUC as described above is standard and independent from if the global variable supremaMrk is on or off.
+=== Carbon accounting ===
+A last recent change concerns the transfer of the existing carbon accounting equations from the supply model to the global market model. These equations run if the global variable %supremaMrk% == on. More information on the equations can be found in the [[module_for_agricultural_supply_at_regional_level|description of the supply model]]. The equations concerned are, indicated with their present “CAPRI names” in the supply models and plus “Mrk” in the market model:
+Table 3. Equations concerning mitigation modelling in CAPRI
+^__Supply model__^__Market model__  ^
+|GWPCO2BIO_      |GWPCO2BIOMrk_(RMS)|
+|GWPCO2SOI_      |GWPCO2SOIMrk_(RMS)|
+|GWPN2OSOI_      |GWPN2OSOIMrk_(RMS)|
+|GWPCO2BUR_      |GWPCO2BURMrk_(RMS)|
+|GWPCH4BUR_      |GWPCH4BURMrk_(RMS)|
+|GWPN2OBUR_      |GWPN2OBURMrk_(RMS)|
+|GWPCO2HIS_      |GWPCO2HISMrk_(RMS)|
+|GWPCH42HIS_     |GWPCH4HISMrk_(RMS)|
+As most CAPRI regions combine only a small number of climate zones and we may assume for an illustrative calculation three (out of 9). In this case we would have 22 additional equations for carbon accounting per region and 1738 equations in total (on top of those for land use modelling mentioned above), confirming the order of magnitude for the additional equations that was mentioned above.
+For the technical coefficients we could rely on the FAO data compiled for the implementation of LULUCF accounting in the supply models. Here they served often only a fall-back solution in case that some European dataset was missing, but for the global market model the FAO data are often the only source of data readily available.
+There is one new element required for the planned implementation of a carbon tax deriving from land use and land use changes: In the supply models the tax is simply added as a cost element in the existing income accounting for the regional farms to make it effective. In the global market model there is no explicit income accounting. The carbon tax levied so far on non-CO2 emissions has been translated therefore into a tax on outputs, depending on the product based non-CO2 emission factors of outputs that may be changed implicitly. For LULUCF an explicit tax on land use has been introduced. This required to treat land demand and land supply for temporary and permanent crops and fodder as separate qualities with distinct rental prices for market clearing.
 ====Behavioural equations for final demand====
@@ Line 1100: / Line 1571: @@
 |Other fruits	|  3	|  3  |
 |Sugar| 	12  |	12  |
-|All other products|	8	|  10  | \\Source: own calculations
+|All other products|	8	|  10  |
+Source: own calculations
 There are some specific settings, such as a value of 2 for rice and the EU15, 2.5 respectively 5 for Japan to account for its specific tariff system, as well as some lower values for EU’s Mediterrean partner countries.
@@ Line 1233: / Line 1704: @@
 **Figure 20: Witzke et al. calibration, two-goods case**
-{{::figure20.png?600|Source: Witzhe et al 2005}}
+{{::figure_20.png?600|Source: Witzhe et al 2005}}
 The additional commitment parameter involves another degree of freedom that needs to be eliminated with additional information. During the calibration this is provided by the expected imports from region 2 at the second hypothetical set of relative prices. Following the dual approach, the lower Armington nest is represented with Armington share-equations and with equations for the composite price indexes:
@@ Line 1249: / Line 1720: @@
 **Figure 21: GUI Option for the non-homothetic Armington system**
-{{::figure21.png?600}}
+{{::figure_21.png?600}}
 The calibration of the non-homothetic Armington demand system does not require a full re-calibration of the complete CAPRI modelling system; it can be found under the workstep “Run scenario”, task: “Run scenario with market model”. Technically, the calibration modifies the simini/sim_ini.gdx file that contains the necessary starting parameters for a CAPRI simulation: if the above GUI option is selected then the existing sim_ini.gdx file is automatically deleted and the create_sim_ini CAPRI module is called. This is all steered in the main capmod.gms file by a specific GAMS setglobal variable called modArmington:
@@ Line 1487: / Line 1958: @@
 \begin{equation}
-intd_{i,r} = (intk_{i,r}+intp__{i,r}) errf \left ( \left( uvae_{i,r} - pmrk_{i,r} + \gamma_{i,r}^p\right ) / stddev_{i,r} \right )
+intd_{i,r} = (intk_{i,r}+intp_{i,r}) errf \left ( \left( uvae_{i,r} - pmrk_{i,r} + \gamma_{i,r}^p\right ) / stddev_{i,r} \right )
 \end{equation}