module_for_agricultural_supply_at_regional_level
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module_for_agricultural_supply_at_regional_level [2022/11/07 10:23] – external edit 127.0.0.1 | module_for_agricultural_supply_at_regional_level [2023/09/08 12:11] (current) – [Price depending crop yields and input coefficients] massfeller | ||
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\end{equation} | \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:// | ||
+ | )) and SUPREMA ((https:// | ||
+ | |||
+ | 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), | ||
+ | )), 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, | ||
+ | |||
+ | 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/ | ||
+ | |||
+ | 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, | ||
+ | |||
+ | === 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, | ||
+ | |||
+ | == Gamma density == | ||
+ | |||
+ | Since each transition is non-negative, | ||
+ | |||
+ | {{: | ||
+ | |||
+ | Figure 1: Gamma density graph for mode=1 and various standard deviations. “acc”=" | ||
+ | |||
+ | 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, | ||
+ | |||
+ | $${\max_{T_{\text{lk}}}{\log{\prod_{\text{lk}}^{}{f\left( T_{\text{lk}}|\alpha_{\text{lk}}, | ||
+ | |||
+ | $$\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\; | ||
+ | |||
+ | $$\text{LU}_{k} - \sum_{i}^{}{\text{shar}e_{\text{ki}}\text{LEV}L_{i}} = 0$$ | ||
+ | |||
+ | The last equation is needed to convert land use in UNFCCC classification to land use in CAPRI classification, | ||
+ | |||
+ | $$\ \left( \alpha_{\text{lk}} - 1 \right)T_{\text{lk}}^{- 1} - \beta_{\text{lk}} + \tau_{k}^{} + \tau_{l}^{\text{initial}} = 0$$ | ||
+ | |||
+ | 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< | ||
+ | |||
+ | $$G = Γ*h = \sum_{i, | ||
+ | |||
+ | Where Γ collects the annual GHG effects that correspond to the total GHG effects divided by the time horizon G / h. These annual effects may be calculated as based on average annual transitions and annual effects for the remaining class as follows: | ||
+ | |||
+ | $$Γ= \sum_{i, | ||
+ | + \sum_{i}^{}{ε_{\text{ii}}^{}\text{L}}_{ii}^{} $$ | ||
+ | |||
+ | Where Λ< | ||
+ | |||
+ | Using these average annual transitions for true (off-diagonal) LUC we may compute the final classes as follows: | ||
+ | |||
+ | $$ \text{LU}_{k, | ||
+ | |||
+ | 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, | ||
+ | |||
+ | $$\text{LU}_{k} - \sum_{i}^{}{\text{shar}e_{\text{ki}}\text{LEV}L_{i}} = 0$$ | ||
+ | |||
+ | $$\text{LU}_{k} - \sum_{l}^{}T_{\text{lk}} = 0\; | ||
+ | |||
+ | $$\text{LU}_{l}^{\text{initial}} - \sum_{k}^{}T_{\text{lk}} = 0\; | ||
+ | |||
+ | $$\ \left( \alpha_{\text{lk}} - 1 \right)T_{\text{lk}}^{- 1} - \beta_{\text{lk}} + \tau_{k}^{} + \tau_{l}^{\text{initial}} = 0$$ | ||
+ | |||
+ | The parameters of the inner model **α** and **β// | ||
+ | |||
+ | There are a few methodological and numerical challenges to overcome. In particular, we need to (i) analytically derive $\mathbf{\eta}\left( \mathbf{c}, | ||
+ | |||
+ | $$\sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} = 0$$ | ||
+ | |||
+ | 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, | ||
+ | |||
+ | The first order conditions, and the implicit function, become | ||
+ | |||
+ | $$F\left( LEVL, | ||
+ | \frac{\partial\mathcal{L}}{\partial LEVL_{i}} = & r_{i} - c_{i} - \sum_{j}^{}{D_{\text{ij}}\text{LEV}L_{j}} - \lambda & = 0 \\ | ||
+ | \frac{\partial\mathcal{L}}{\partial\lambda} = & \sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} & = 0 \\ | ||
+ | \end{bmatrix}$$ | ||
+ | |||
+ | 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/ | ||
+ | )) 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 $r_{i}$) | ||
+ | |||
+ | $$\left\lbrack \frac{\partial LEVL}{\partial r} \right\rbrack = - \left\lbrack D_{LEVL, | ||
+ | |||
+ | $$\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$$ | ||
+ | |||
+ | Carrying out the differentiation specifically for land rent // | ||
+ | |||
+ | $$\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 \\ | ||
+ | 0 \\ | ||
+ | \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$$ | ||
+ | |||
+ | In the estimation, we assumed that the prior elasticity matrix is the mode of a density where each entry were independently distributed. Furthermore, | ||
+ | |||
+ | $$\max_{\eta, | ||
+ | |||
+ | subject to | ||
+ | |||
+ | $$\left\lbrack \frac{\partial LEVL_{i}}{\partial r_{j}} \right\rbrack = - \begin{bmatrix} | ||
+ | \left\lbrack {- D}_{\text{ij}} \right\rbrack & - 1 \\ | ||
+ | - 1' & 0 \\ | ||
+ | \end{bmatrix}^{- 1}\begin{bmatrix} | ||
+ | I \\ | ||
+ | 0 \\ | ||
+ | \end{bmatrix}$$ | ||
+ | |||
+ | $$\left\lbrack \eta_{\text{ij}} \right\rbrack = \left\lbrack \frac{\partial LEVL_{i}}{\partial r_{j}} \right\rbrack\left\lbrack \frac{r_{j}}{\text{LEV}L_{i}} \right\rbrack$$ | ||
+ | |||
+ | $$\begin{matrix} | ||
+ | & r_{i} - c_{i} - \sum_{j}^{}{D_{\text{ij}}\text{LEV}L_{j}} - \lambda & = 0 \\ | ||
+ | & \sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} & = 0 \\ | ||
+ | \end{matrix}$$ | ||
+ | |||
+ | 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 // | ||
+ | |||
+ | ==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”: | ||
+ | |||
+ | $$\text{LEVL} = \left\{ \text{ARAC}, | ||
+ | |||
+ | Where | ||
+ | |||
+ | ARAC = arable crops | ||
+ | |||
+ | FRUN = perennial crops | ||
+ | |||
+ | GRAS = permanent grassland | ||
+ | |||
+ | FORE = forest | ||
+ | |||
+ | ARTIF = artificial surfaces (settlements, | ||
+ | |||
+ | OLND = other land | ||
+ | |||
+ | The above categories are matching reasonably well with the definitions in JSA. A mismatch exists in the classification of paddy (part of ARAC in CAPRI but in the perennial group in JSA) and terrestrial wetlands (part of OLND in CAPRI and a separate category in JSA). Inland waters are considered exogenous in CAPRI and hence not included in the above set LEVL. | ||
+ | |||
+ | For carbon accounting we need to identify the six LU classes from IPCC recommendations and official UNFCCC reporting: | ||
+ | |||
+ | $$LU = \left\{ \text{CROP}, | ||
+ | |||
+ | 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> | ||
+ | |||
+ | 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< | ||
+ | |||
+ | 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, | ||
+ | |||
+ | $$\text{OLND}^{0} = {\text{OLND}G}^{0} + {\text{OLND}W}^{0} + {\text{OLND}R}^{0}$$ | ||
+ | |||
+ | 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 | ||
+ | |||
+ | ^// | ||
+ | |WETLND | ||
+ | |RESLND | ||
+ | |||
+ | 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, | ||
+ | |||
+ | where 0 ≤ // | ||
+ | |||
+ | ===Technical implementation=== | ||
+ | |||
+ | The key equations corresponding to the approach explained above are collected in file supply_model.gms or the included files supply/ | ||
+ | |||
+ | // | ||
+ | |||
+ | 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”, | ||
+ | |||
+ | Furthermore, | ||
+ | |||
+ | 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, | ||
+ | |||
+ | === 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 | ||
+ | |||
+ | 1) Additional CAPRI results on land use results related to the complete area coverage, mappings to UNFCCC area categories and their transitions; | ||
+ | |||
+ | 2) The carbon effects linked to these land transitions. | ||
+ | |||
+ | Furthermore, | ||
+ | |||
+ | 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** | ||
+ | |GWPA |Agricultural emissions | ||
+ | |CH4ENT | ||
+ | |CH4MAN | ||
+ | |CH4RIC | ||
+ | |N2OMAN | ||
+ | |N2OAPP | ||
+ | |N2OGRA | ||
+ | |N2OSYN | ||
+ | |N2OCRO | ||
+ | |N2OAMM | ||
+ | |N2OLEA | ||
+ | |N2OHIS | ||
+ | |GLUC |Emissions related to indirect land use changes | ||
+ | |CO2BIO | ||
+ | |CO2SOI | ||
+ | |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 | ||
+ | |N2OBUR | ||
+ | |N2OSOI | ||
+ | |GPRD |Emissions related to the production of non-agricultural inputs to agriculture | ||
+ | |N2OPRD | ||
+ | |O2PRD | ||
module_for_agricultural_supply_at_regional_level.txt · Last modified: 2023/09/08 12:11 by massfeller