CAPRI Online Manual (update)

There are two sources for interactions between activities in simulation experiments: the objective function and constraints. In the current version of CAPRI, the objective function does solve inter-activity terms for groups of arable crops, so that the major interplay is due to constraints. The interaction is best understood by looking at the first order conditions of a programming model including PMP terms:

\begin{equation} Rev_j = Cost_j+ac_j+\sum_k bc_{j,k}Levl_k+\sum_i^m\lambda_ia_{ij} \end{equation}

The left hand side (Rev) shows the marginal revenues, which are typically equal to the fixed prices times the fixed yields plus premiums. The right hand side shows the different elements of the marginal costs. Firstly, the variable or accounting costs (Cost) which are fix as they are based on the Leontief assumption. The term \( (ac_j+\sum_k bc_{j,k}Levl_k) \) shows the marginal non-linear costs, which are increasing with the activity levels. The cross effects are only introduced to let major arable crop groups interact, whereas for fruits & vegetables, permanent crops, grassland and the animal sectors, only diagonal terms are introduced. The methodology for the estimation of these terms is described in Jansson and Heckelei (2011).

The remaining term \( (\sum_i^m\lambda_ia_{ij}) \) captures the marginal costs linked to the use of exhausted resources and is equal to the sum of the shadow prices \lambda multiplied the per unit demand of resource i for activity j; the matrix A being again based on Leontief technology. The shadow values of binding resources hence are the drivers linking the activities.

The land balance plays a central role in the CAPRI supply model. The land shadow price appears as a cost in all crop activities including fodder producing ones, so that animals are indirectly affected as well. The second major link is the availability of not-marketable feeding stuff, and finally, less important, organic fertiliser.

The basic effects are best discussed with a simple example. Assume an increase of a per hectare premium for soft wheat, all other things unchanged.

• What will happen in the model? The increased premium will lead to an imbalance between marginal revenues (= yield times prices plus premium) and marginal costs (=accounting costs, ‘resource use cost’, non-linear costs). In order to close the gap, as marginal revenues are fixed, the area under soft wheat will be increased until marginal costs of producing soft wheat have increased to a point where they are again equal to marginal revenues. As the marginal costs linked to the non-linear cost function \( (ac_j+\sum_k bc_{j,k}Levl_k) \) are increasing in activity levels, increasing the area under soft wheat will hence reduce that gap. At the same time, as the land balance must be kept closed, other crop activities must be reduced. The non-linear cost function will for these crops now provoke a countervailing effect: reducing the activity levels of competing crops will lead to lower costs for these crops. With marginal revenues (Rev) and accounting costs (Cost) fixed, that will require the shadow price  of the land balance to increase.
• What will be the impact on animal activities? Again, the shadow price of the land balance will be crucial. For activities producing non-marketable feed, marginal revenues are not defined as prices times yields, but as internal feed value times prices. The internal feed value is determined as the substitution value of non-marketable fodder against other feeding stuff, and depends on their nutrient content and further feed restrictions. Increasing the shadow price of land will hence either require decreasing other costs in producing fodder or increasing the internal marginal revenues. In other words, a high shadow price of land renders non-marketable fodder less competitive compared to other feeding stuff. As feed costs are – however very slightly – increasing in quantities fed per head, feed costs for animals will increase. But as there are several requirement constraints involved, some feeding stuff may increase and other decrease. Clearly, the higher the share of non-marketable fodder in the mix for a certain animal type, the higher the effect. As marginal feed costs will increase, and marginal revenues for the animal process are not changing, other marginal costs in animal production need to be reduced, and again the non-linear cost function will be the crucial part, as the marginal cost related to it will decrease if herd sizes drop.

To summarize the supply response, increasing premiums for a crop will hence increase the cropping share of that crop, reduce the share of other crops, increase the shadow price of land, lead to less fodder production, higher fodder costs and thus reduced herd size of animals.

• What will be the impacts covered by the market? The changes in hectares will lead to increased supply of the crop with the higher premium and less supply of all other crops at given prices, i.e. one upward and many downward shifts of the supply curves. Equally, supply curves for animal products will shift downwards. On the other hand, some feed demand curve will shift as well, some upward, other downward. These shifts will move the market module away from the former fixed points where market balances were closed. For the crop product with the increased premiums, increased supply plus some changes in feed will most probably lead to lower prices, whereas prices of other crops will most probably increase. That will require new adjustments during the next iteration where the supply models are solved, with to a certain extent countervailing effects.

Table 25: Overview on a regional aggregate programming model

The definition of the supply model can be found in ‘supply\supply_model.gms’

The feed block ensures that the requirements of the animal processes in terms of feed energy and protein are met and links these to the markets and crop production decisions.

\begin{equation} \overline{AREQ}_{r,act,req} \overline{DAYS}_{r,act,req}= \sum_{feed} FEDNG_{r,act,feed} \overline{REQCNT}_{r,act,feed} \end{equation}

The left hand side captures the daily animal requirements (AREQ) for each region r, animal activity act and requirement AREQ multiplied with the days (DAYS) the animal is in the production process. Both are parameters fixed during the solution of the modelling system. The right hand side ensures that the requirement content of the actual feed mix represented by the feeding (FEDNG) of certain type of feed to the animals multiplied with the requirement content (REQCNT) in the regions covers these nutritional demands. Requirements and contents are specified in the feed calibration while production days are determined in the “COCO1” module. Total feed use (FEDUSE) in a region is defined as the feeding per head multiplied with the activity level (LEVL) for the animal activities:

\begin{equation} FEDUSE_{r,feed} = \sum_{aact} LEVL_{r,aact} FEDNG_{r,aact,feed} \end{equation}

Total feed use might be either produced regionally in the case of fodder assumed not tradable (grass, fodder root crops, silage maize, other fodder from arable land), or bought from the market at fixed prices.

The model distinguishes arable and grassland and comprises thus two land balances:

\begin{equation} \overline{LEVL}_{r,"arab"} \le \sum_{arab} LEVL_{r,arab} \end{equation}

\begin{equation} \overline{LEVL}_{r,"gras"} \le LEVL_{r,"grae"} + LEVL_{r,"grai"} \end{equation}

Both land balances might become slack if marginal returns to land drops to zero. For arable land, idling land not in set-aside (activity FALL) is a further explicit activity. For the grassland, the model distinguishes two types with different yields (GRAE: grassland extensive, GRAI: grassland intensive) so that idling grassland can be expressed of an average lower production intensity of grassland by changing the mix between the two intensities.

The model comprises a land use module with two major components:

1. Imperfect substitution between arable and grass lands depending on returns to the two types of agricultural land uses.
2. A land supply curve which determines the land available to agriculture as a function to the returns to land.

There are hence two further equations:

\begin{equation} \overline{LEVL}_{r,"uaar"} = \overline{LEVL}_{r,"arab"} +\overline{LEVL}_{r,"gras"} \end{equation}

And a further one which prevents numerical problems with the terms relating to land supply in the objective function

\begin{equation} \overline{LEVL}_{r,"uaar"} = 0.999 \overline{LEVL}_{r,"asym"} \end{equation}

Where “asym” is the land asymptote, i.e. the maximal amount of economically usable agricultural area in a region when the agricultural land rent goes towards infinity. For an application where the land market is used see Renwick et al. (2013).

Set aside policies have changed frequently during CAP reforms. The recent specification is covered in the context of the premium modelling in Section 5.3 . The obligatory set-aside restriction introduced by the McSharry reform 1992 and valid until the implementation of the Luxembourg compromise of June 2003 has been explicitly modelled through this equation:

\begin{align} \begin{split} &LEVL_{r,"iset"} + LEVL_{r,"gset"} + LEVL_{r,"tset"} \\ &=\sum_{arab} LEVL_{r,arab}\left(1-NONS_{r,arab}\right ) \frac {1/100SETR_{r,arab}}{1- 1/100 SETR_{r,arab}} \end{split} \end{align}

LEVL_{r,“iset”}As seen from above, the model distinguishes between three types of obligatory set-aside: idling (ISET), for grass land use (GSET) and for forestation purposes (TSET). The share of so-called non-food production exempt from set-aside (NONS) for each activity and region is fixed and given.

The equation above is replaced for years where the Luxembourg compromise of June 2003 is implemented by a Member State, where the level of obligatory set-aside is fixed instead to the historical obligations.

For certain years of the McSharry reform, the total share of set-aside – be it obligatory or voluntary – on a list of certain crops was not allowed to exceed a certain ceiling. That restriction is captured by the following equation:

\begin{align} \begin{split} &LEVL_{r,"iset"} + LEVL_{r,"gset"} + LEVL_{r,"tset"}+ LEVL_{r,"vset"} \\ & \le \sum_{arab \wedge SETF_{r,arab}} LEVL_{r,arab}/\overline{MXSETA} \end{split} \end{align}

As of CAPRI Stable Release 2.1, the fertilizer allocation was modified, and this section of the documentation updated. Notation has changed compared with previous versions of the model and documentation. Here, we represent the equations in more general mathematical notation, avoiding the long GAMS code names of the source code, in order to save space.

We distinguish the three macro-nutrients N, P and K. The supply and uptake of those nutrients are modelled in a uniform way, save for the fact that there is fixation and atmospheric deposition only of N.

Each crop has a requirement per hectare, calculated based on the yield. Yields are exogenous from the vantage point of the producer, but there are alternative technologies available for each cropping activity, and a separable, i.e. handled outside of the optimization model, relation between prices and optimal yields.

From the basic nutrient requirement we first deduct the rate of biological fixation (only for nitrogen and selected crops). The remainder is inflated by a (calibrated) factor and additive term of over-fertilization, and then scaled with a soil-specific factor (only for nitrogen), to arrive at the total amount of nutrients that need to be supplied to the crop. This is the left hand side of Equation 96 .

Nutrient supply, shown on the right handside, comes from mineral fertilizer, manure, crop residues and atmospheric deposition. Mineral fertilizer may have ammonia losses during application. For manure, there are both losses and inefficiencies. When manure is applied to crops, there is an efficiency factor applied to the nutrient content (denoted by ϕ_(r,“excr”, n) ), corresponding to the Fertilizer Value (FV) of manure relative to mineral fertilizer. The efficiency factor is a key parameter of interest in simulations carried out in some studies. Crop residues can be re-distributed among crop groups for annual arable crops but not for grassland and permanent crops, where it stays with the crop that produced it. For crop residues there is both a loss rate and a fertilizer value.

\begin{align} \begin{split} &\sum_{i \in I \, j,k} \left[ levl_{rik} \left ( ret_{rni} (1-biofix_{rni}) \lambda_{rnik}^{prop} + \lambda_{rni}^{const} \right ) soil_{rn} \, yf_{rnik} \right ] \\ & = fmine_{rni}(1-loss_{rn}) +fexcr_{rni} \phi_{r,excr,n} +(1-isPerm_j)fcres_{rni}(1-loss_{rn})\phi_{r,cres,n}\\ & + isPerm_j \sum_{ i \in I \, j,k} levl_{rik}res_{rni}(techf_{rink}+1)(1-loss_{rn})\phi_r,cres,n \\ & \forall r,n,j \end{split} \end{align}

Indices:

\(r\) = region
\(i\) = crop
\(j\) = crop group
\(k\) = technological crop option (high/low yield)
\(n\) = nutrient (N/P/K)
\(isPerm_j\) = indicates that crop group \(j\) contains permanent crops

Endogenous choice variables:

\(levl_{rik}\) = Area (ha) of each crop \(i\) and technology \(k\) in region \(r\).
\(fmine_{rnj}\) = Application of mineral fertilizer \(n\) to crop group \(j\) in region \(r\).
\(fexcr_{rnj}\) = Application of manure \(n\) to crop group \(j\) in region \(r\).
\(fcrex_{rnj}\) = Allocation of crop residue \(n\) to crop group \(j\) in region \(r\).

Parameters: \(ret_{rni}\) = Retention (uptake) of nutrients by the crop
\(res_{rni}\) = Crop residues output
\(biofix_{rni}\) = Biological fixation, share (only for N and selected crops)
\(λ_{rnik}^{prop} \) = Over-fertilization factor, calibrated
\(λ_{rni}^{const} \) = Over-fertilization term, calibrated
\(soil_{rn}\) = Soil factor
\(yf_{rnik}\) = Yield factor for technologies
\(loss_{rn}\) = Loss rate
\(ϕ_{r",excr" ,n}\) = Nutrient availability ratio for manure
\(ϕ_{r,"cres" ,n}\) = Nutrient availability ratio for crop residues

The reader may have noted that there is no loss rate for manure in the Equation 96 . CAPRI does contain such loss rates, but they are specific for each animal type and therefore happens on the manure supply side of the regional manure balance (see section on input allocation).

The model contains three types of manure: N-manure, P-manure and K-manure. From an agricultural point of view this may seem odd. It might be more intuitive to think of one type of manure per animal category. The motivation is to keep the system simple and flexible. With the present representation, where each animal category supplies N, P, and K-manure, the number of manure classes can be limited and yet the unique mix of nutrients from each animal category can be defined.

The supply of each manure type is collected in a “pool” for each regional farm model, i.e. for each NUTS2 region. Regions within a member state may trade manure, subject to a cost. The supply in the pool plus the traded quantities has to be distributed to the crops in the region, i.e. there is an equality-restriction in place. This is handled in the equations “FertDistExcr_” and “ManureNPK_”. Note that fertilizer flows are measured in tons, for the sake of scaling, whereas other total quantities in CAPRI are measured in 1000 tons. Hence the factors 1000 and 0.001.

\begin{equation} \sum_j fexcr_{rnj}= 1000 v\_ManureNPK_{rn} \end{equation}

\begin{equation} v\_ManureNPK_{rn} + \sum_s T_{rs}nutshr_{rn} = 0.001 \sum_{i\in Anim_j,k} levl_{rik}o_{rnik}(1-loss_{rin}) \quad \forall r, n \end{equation}