```
csgy = vsog(ii,ik)*zcc25(ii)
  *(zcc61(ii)*zcc07(ii)
s +zcc62(ii)*zcc18(ii)
s +zcc63(ii)*zcc19(ii)
s +zcc64(ii)*zcc20(ii)
s +zcc65(ii)*zcc21(ii)
s +zcc69(ii)*zcc22(ii)
s +zcc67(ii)*zcc23(ii)
s +zcc70(ii)*zcc24(ii))*0.67*prgch
endif
```

c Descr: CH4 - Enteric fermentation

c Source: Gilles Le Moguedec

c Descr: CH4 - Enteric fermentation (zcc50=2.)

c Source: Equations 4.14 Good Practices

c Comme: 55.65 MJ/kg CH4 entre en dur
  if (im.le.16.and.im.le.21) then
    if (im.eq.animalt(ii,15)) then
      if (zcc50.eq.1.) then
        csgy = fmethane(3,1,ii,ij,ik)*prgch
      endif
      if (zcc50.eq.2.) then
        csgy=brag(ii,ik)*smsag(ii,ik)*(4.185/55650.)*zcc51(ii)*prgch
      endif
    endif
  endif
  return
end

c:F: gesy() : ges
function gesy(im,in)
  gesy=0.
  if ((im.ge.1.and.im.le.22).and.(in.eq.im)) gesy=1.
  return
end

By default parameter values are null, and positive values are displayed below.

zcc01 : 0.01250000 :: zcc17 : 0.00700000 :: zcc36 : 0.01250000
zcc03 : 0.10000000 :: zcc26 : 0.01500000 :: zcc40 : 0.01250000
zcc06 : 0.02000000 :: zcc27 : 0.01500000 :: zcc46 : 36.00000000
zcc08 : 0.01000000 :: zcc28 : 0.01500000 :: zcc47 : 2.00000000
zcc09 : 0.02500000 :: zcc29 : 0.35000000 :: zcc48 : 2.00000000
zcc10 : 0.30000000 :: zcc30 : 0.90000000 :: zcc50 : 2.00000000
zcc11 : 0.02117377 :: zcc31 : 0.01000000 :: zcc52 : 1.00000000
zcc12 : 0.02117377 :: zcc32 : 0.10000000 :: zcc53 : 1.00000000
zcc13 : 0.00103546 :: zcc33 : 0.10000000 :: zcc54 : 1.00000000
zcc15 : 0.00100000 :: zcc34 : 0.01250000 :: zcc55 : 1.00000000
zcc16 : 0.00100000 :: zcc35 : 0.10000000 :: zcc56 : 1.00000000

### 4.2.6 Nitrogen

#### Implementation of the N block and call to N to yield functions

The nitrogen block operates when the related flag parameter is activated (`endrg=1`, 0. by default). The part referring to manure is included when the related flag parameter is activated (`mgesd=1`, 0. by default). All parameters, activities and constraints relating the N block are listed above (see Sections 4.2.1 and 4.2.2).

When this block is not activated, nitrogen comes only through fertilizer costs. When the block is activated, the basic elements refer to the nitrogen requirement of crops and to the nitrogen
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Stylized effects of point (on the left) by response function (on the right) replacement on the optimal yield (*yj*), allocated area (*aj*) and production (*qj*) related to the *j* crop when its price (*pj*) -and only this price- varies. The reference optimally calibrated *LP* solution is materialized by dotted lines.</figcaption></figure>
content of mineral and organic fertilizers. By default, parameters refer to mineral fertilizer prices and nitrogen content ratio of fertilizers. When manure accounts for nitrogen intakes and impacts the nitrogen balance for plants, manure-to-animal relations are parametrized.

In the case of crops for which farm-dependent nitrogen to yield functions are available, the market price ratio supplies the value of required nitrogen and its corresponding yield. These values optimize the plot gross margin and the farm gross margin when a part of crop and a part of fertilizer are marketed. In the case of on-farm use and non-market goods, shadow prices have to replace market prices (see Section 4.3 for discussion and quoted papers for demonstration).

Parameters related to these dose response functions are provided by external data (lisscal.don file called by intern.f). Organic nitrogen accounts for fertilizing with values expressed in mineral nitrogen equivalence and delivered by additional data (included in the parameter file, mgg.par). Optimal yields and nitrogen inputs in generic equivalent terms are calculated for any crop and farm group occurring in the lisscal.don. This is calculated through a subroutine called by the FORTRAN intern.f file.

Urban organic residues may be added in the process, as explained in section 9.2.1. Related parameters and primal and dual variables are integrated in the description of the block. When no iterative AROPAj runs are realized, with the aim of assessing the “long term” impact of regular spreading of organic matter, manure and UOR as well are viewed as “short term” substitution sources of mineral fertilizer. Two categories of urban wastes are now referenced in the model: green waste compost (GWC) and urban sewage sludge (USS).

## The comprehensive MP block

The bio-technical and economical parts detailed here refer to N, E and P blocks, which were described in Section 4.2 on page 27.

The technical side accounts for:

* the nitrogen balance required by each crop

* the organic nitrogen source related to livestock (per animal category)

* the organic nitrogen balance per UOR source

* the nitrogen balance referring to all organic sources (manure and UOR), in terms of equivalent mineral nitrogen

* the balance of organic nitrogen spread per crop

* the N pollutant balance per pollutant category (see section 4.2.6 on page 81)

* the average organic spreading computation aiming at implementing the long term impact of organic matter spreading

The economic side accounts for:

* organic spreading limit per unit of crop area

* implementation of various other environmental policies (regarding GHG and N pollutants)

It must be stated that several LP activities referring to nitrogen are defined at the farm level, while the yields and nitrogen inputs mentioned above refer to calculation per unit of area. This type of procedure allows us to maintain a basic linear structure for the model.

The primal and dual activities involved in the nitrogen block are listed below:
The process of calculation of shadow prices is an iterative process that entails solving the model and re-estimating the shadow prices until a converging solution is found. This methodology extends and enhances the work done by Jayet et al. (2017a), which was previously developed for crop production entirely used on-farm as feedingstuff for livestock (e.g. cereal).

In this case, the optimal solution leads to a substituting of market prices with shadow prices in the yield and nitrogen calculations. These shadow prices, which are required in order to calculate the yield and nitrogen prices, are provided by the linear programming model. The iterative process consists in using at each iteration shadow prices calculated in the previous iteration.

This two-step process starts with the given market prices, yields, and N-inputs that are computed at the plot level using the first order conditions. Then, these yields and N-inputs enter the linear programming model, which, once solved, provides the shadow prices (equal to market prices when optimal production is at least partly marketed). In the case of grass productions, where no market price exists, fodder prices are set at 0.000001 at the first iteration. The addition of multiple response functions and the complex interaction between farming and breeding activities might lead linear program solutions to be unstable. Finally, the full application to the *AROPAj* model appears to be satisfactory in that convergence or weak amplitude oscillation is obtained. Nevertheless, that needs a greater number of iterations than required when only marketed crop response functions were implemented.
<table><thead><tr><th></th><th>primal</th><th>dual</th><th>nitrogen block description</th><th>(unit)</th></tr></thead><tbody><tr><td>po(j,k,o)</td><td></td><td></td><td>UOR quantity of type o used by crop j</td><td>(t/ha)</td></tr><tr><td>qen(j,k)</td><td></td><td></td><td>quantity of regular fertilizer used on j-crop area</td><td>(t/ha)</td></tr><tr><td>q(i,j,k,o)</td><td></td><td></td><td>quantity of manure type o / i-herd sourced</td><td>(t m3)</td></tr><tr><td>qon(j,k)</td><td></td><td></td><td>useful N from manure & UOR spread on area j</td><td>(tN/ha)</td></tr><tr><td>qonlt(k)</td><td></td><td></td><td>org. N average quantity brought on long term per</td><td>(tN/ha)</td></tr><tr><td>xo(o,i,k)</td><td></td><td></td><td>exceeding unused organic N type o i-anim. sourced</td><td>(tN)</td></tr><tr><td>xtk(j,k)</td><td></td><td></td><td>aggregated j-crop area</td><td>(ha)</td></tr><tr><td>ani(i,k)</td><td></td><td></td><td>animal number referring to category i</td><td>(heads)</td></tr><tr><td>epcdt(n)</td><td></td><td></td><td>balance N-pollutant emission (3 cat.: N20 NO3 NH3)</td><td>(tN)</td></tr><tr><td>apnlt(k)</td><td></td><td></td><td>calculation of average per ha organic spreading</td><td>(tN)</td></tr><tr><td>ba(j,k)</td><td></td><td></td><td>N requirement to reach given yield</td><td>(tN)</td></tr><tr><td>cnvol</td><td></td><td></td><td>sum of mineral fertilizer inputs</td><td>(tfert)</td></tr><tr><td>cnorg</td><td></td><td></td><td>sum of organic N intakes in terms of equiv. min. N</td><td>(tN)</td></tr><tr><td>ei(o,i,k)</td><td></td><td></td><td>o-type manure provided by i-herd</td><td>(tRM)</td></tr><tr><td>epro(o)</td><td></td><td></td><td>use of UOR disposal</td><td>(tN)</td></tr><tr><td>eqj(j,k)</td><td></td><td></td><td>balance spreading - source of available N</td><td>(tN)</td></tr><tr><td>epn(m)</td><td></td><td></td><td>balances of N pollut. emis. originated by crops</td><td>(N)</td></tr></tbody></table>

Nitrogen mineral fertilizers impact the objective function through the qen(j,k), and there is the possibility for UOR pricing (negative in case of subsidizing). In addition, environmental policy tools can easily be introduced, such as taxes either referring to inputs (or outputs) or to N-losses (epcdt(n)).

The sub-matrix is displayed on Table 15.

<table><caption>**Table 15.** Sub-matrix referring to technical parameters and LP variables operating for nitrogen.</caption><caption>The term "yexjk(j,k)" is the optimal amount of mineral N equivalent per hectare in case of N-to-yield function<br>We use the functions: qjmy(o,i,j,k) = fned(o,i,k)*coo(o,i,j,k)*(1-volat(o,k))<br>qjoy(o,j,k) = fnedo(o,k)*cpro(o,j,k)*(1-volat(o,k))</caption><thead><tr><th></th><th>q(i,j,k,0)</th><th>qonlt(k)</th><th>qon(j,k)</th><th>qen(j,k)</th><th>xtk(j,k)</th><th>ani(i,k,1)</th><th>po(j,k,o)</th><th>xo(o,i,k)</th><th>zf(k,1)</th><th>RHS</th></tr></thead><tbody><tr><td>obj</td><td></td><td></td><td></td><td>if(mgesd+endrg&gt;1)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>-prxnc(nj,nk)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>*(1+tegkk(k))</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>-fnesc(nj,nk)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>*pgesn</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>else gjk(j,k)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ei(o,i,k)</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>&lt; | 0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>eqj(j,k)</td><td>-</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>&lt; | 0</td></tr><tr><td></td><td>qjmy(o,i,j,k)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>epro(o)</td><td>fned(o,i,k)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>&lt; | 0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ba(j,k)</td><td></td><td></td><td>-1</td><td>-fnesc(j,k)</td><td>benjk(j,k)</td><td></td><td></td><td></td><td></td><td>&lt; | 0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>*rjk(j,k)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>-aqolt(k)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>or</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>yexjk(j,k)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>-qengf(j,k)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>*fnesf(j,k)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>-aqolt(k)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>epa(j,k)</td><td>qjmy(o,i,j,k)</td><td>1</td><td></td><td></td><td>-yxdej(j,k)</td><td></td><td></td><td></td><td></td><td>&lt; | 0</td></tr><tr><td>apnlt(k)</td><td>-apolt(o,k)</td><td>saukl(k,1)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>&lt; | 0</td></tr><tr><td></td><td>or 0 if o&gt;10</td><td>*popul(k,1)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ems()</td><td></td><td></td><td></td><td></td><td>see the GHG block</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
# 5 Typology and farming system clustering

## 5.1 Clustering methodology

### Key-points:

- Regional representativity

- From FADN farm types to AROPAj farm groups

- Metrics for clustering

The information in the FADN database consists of surveys of individual farm samples. These individual farms are grouped by a clustering method to form a group of similar farms that present similar characteristics and thus can form a modeling unit for the AROPAj model: the farm type (also denominated by farm group).

The FADN data are presented at the regional level without any more localization information. Consequently, the farm types that are created belong to a specific region, that approximately corresponds to the NUTS-2 level according to the European classification (Nomenclature of Units for Territorial Statistics). For instance, in France the NUTS-2 classification corresponds to the Administrative Units.

<table><caption>**Table 16.** Organizational form under the FADN classification (simplified version when 2 digits).</caption><thead><tr><td>Code</td><td>Type of farm</td></tr></thead><tbody><tr><td>13</td><td>Cereals, Oleaginous, protein crops</td></tr><tr><td>14</td><td>Other field crops</td></tr><tr><td>2</td><td>Horticulture</td></tr><tr><td>3</td><td>Viticulture</td></tr><tr><td>411</td><td>Bovines milk</td></tr><tr><td>412</td><td>Bovines meat</td></tr><tr><td>42</td><td>Bovines livestock for meat</td></tr><tr><td>43</td><td>Bovines for milk and meat</td></tr><tr><td>44</td><td>Ovines, caprines and other herbivorous</td></tr><tr><td>50</td><td>Grain eater livestock</td></tr><tr><td>72</td><td>Livestock (mainly grain eater)</td></tr><tr><td>60</td><td>Various crops</td></tr><tr><td>71</td><td>Livestock (mainly herbivorous)</td></tr><tr><td>81</td><td>Field crops and herbivorous</td></tr><tr><td>82</td><td>Other livestock/field crop combinations</td></tr></tbody></table>

Within these regional units, the individually sampled farms are aggregated by a clustering method under the condition that each farm type must gather at least 15 different farms (to keep the information fully anonymous). The farms are grouped according to different key variables: organisational form, altitude, irrigation and economic size. Organisational form (FADN variable A18, also called OTEX or FT) refers to a FADN classification of the farms depending on the type of agricultural production, from field crops farms to intensive livestock farming. Table 16 presents the various values that this variable takes (note that horticulture and viticulture are not included in AROPAj). There are three values that correspond to altitude zone (FADN variable A41): 1 for farms below 300 meters, 2 for farms between 300 and 600 meters, and 3 for farms above 600
Calculation of nitrogen pollutant emissions

There is no default evaluation of nitrogen losses in the *AROPAj* model. I must be noted that *N*₂O emission can also be estimated within the *GHG* emission block, so as long as the estimates are *IPCC* sourced. Related *LP* variables are activities (*epcdt*) and constraints (*epn*). Emissions refer to crop areas when no *N*-to-yield function applies ; otherwise they refer to spread *N* inputs (*qen(j,k)*, *q(i,j,k,o)*, *po(j,k,o)*).

When nitrogen-to-yield functions are available and the complete nitrogen block is activated in
the model, it is possible to assess N-losses in three forms, namely nitrous oxide (N₂O), ammonia
(NH₃) and nitrate (NO₃). Such assessment capacity is due to the use of the STICS crop model
simulations and regression calculation (see Section 9.1.2). All that is needed to make a nitrogen-to-
loss function operational is the ability to give the function through AROPAj parameters, knowing
that the function may take any analytical form or point form.

The linear form is based on usual linear regression estimates when *N* crop intake is within the usual range of values ([50,400] kgN/ha). When the related nitrogen to yield function is estimated with the *STICS* model, six additional parameters for each crop and each farm group are needed. There is still no default emission function for the subset of farm groups and crops not referring to *N*-to-yield functions. But *EU* agricultural regions are well covered by these functions in terms of area as well as production (see Section 12.6). This is particularly true in regions intersecting the Seine river basin (see Section 12.3).

4.2.7 Water

The water block is operating in the model since 2017. Basically it runs like the nitrogen block when related to input quantities and crop yields. Water-activities, in the sense of *AROPAj*, refer to irrigation. When the climatic dimension matters, natural resources (rainfalls) are taken into consideration through parameters of yield functions.

Given the multiple interpretation issues that often arise when a concept is used across different
scientific disciplines, it is important to clarify the notion of irrigation water demand. The irrigation
water demand estimated by AROPAj corresponds to the classical economic concept of demand in
response to a "price." More specifically, considering the crop prices and the price of water (or more
commonly in this context, the cost of access to water) at the plot level when the studied crop benefits
from a yield function, the model calculates the optimal yield and optimal water demand required
to achieve this yield (nitrogen demand follows the same pattern). In agronomy, the focus is often
on the maximum yield achievable under given technical and pedoclimatic conditions, with water
demand—the water requirement—becoming the quantity necessary to achieve this yield. When the
available water does not satisfy the demand determined by its cost of access, actual consumption
adjusts to supply. The classical interpretation states that the farmer's marginal valuation of water
is greater than its cost. This point will be fundamental when considering the pressure exerted on
available water (competition between uses, climate change, protection of reserves).

4.2.8 Biogas

Work in progress ...
meters. The irrigation variable was calculated as the ratio of irrigated crop areas divided by the total amount of crop areas. The economic size (FADN variable A27) is an integer between 1 and 10 corresponding to the size of the sampled farm.

A good clustering method is a method that maximizes the similarity within groups and the distance between different groups. We thus chose to perform the typical agglomerative hierarchical clustering method on these four variables. In order to do so, one must be able to calculate a distance between the farms. Each farm was placed on a four dimensional space, and the distance between two farms A and B could then be calculated according to the classical Euler formula:

$$Distance_{AB} = \sqrt{\sum_{i=1}^{N} (xA_{i} - xB_{i})^{2}} \qquad (1)$$

where N is the number of dimensions of the space (4 in our case) and $xA_i$ and $xB_i$ being the coordinates of the farms A and B in the dimension i. The calculation of the distance (a measure of similarity) is a subjective choice that belongs to the modeler, who can emphasize the importance of the variables of his choice.

Another important point is the transformation of the variables to a numerical equivalent that will be used to place the farms on each dimension. Table 17 presents the different variables that were used for the clustering and their arbitrary given value. We decided to put more weight to the organisational form since we intended to group the farms according to their type of agricultural production. The scale goes from 0 to 1200 with 0 being the more extensive (field crops) and 1200 being the more intensive (swine and poultry) type of farm. Irrigation and economic size were considered at a similar weight (scale ranging from 0 to 100) while altitude was considered as the less important variable (scale ranging from 10 to 30).

<table><caption>**Table 17.** Variables used by the hierarchical clustering procedure to create farm-types at the European Union level.</caption><thead><tr><th colspan="2">Organisational form</th><th colspan="2">Economic class</th><th colspan="2">Irrigation</th><th colspan="2">Altitude</th></tr><tr><th>Code</th><th>Value</th><th>Code</th><th>Value</th><th>Irrigation ratio (%)</th><th>Value</th><th>Code</th><th>Value</th></tr></thead><tbody><tr><td>13</td><td>0</td><td>1</td><td>10</td><td>0</td><td>0</td><td>1</td><td>10</td></tr><tr><td>14</td><td>100</td><td>2</td><td>20</td><td>10</td><td>10</td><td>2</td><td>20</td></tr><tr><td>60</td><td>200</td><td>3</td><td>30</td><td>20</td><td>20</td><td>3</td><td>30</td></tr><tr><td>82</td><td>300</td><td>4</td><td>40</td><td>30</td><td>30</td><td>4</td><td>40</td></tr><tr><td>81</td><td>400</td><td>5</td><td>50</td><td>40</td><td>40</td><td>5</td><td>50</td></tr><tr><td>71</td><td>500</td><td>6</td><td>60</td><td>50</td><td>50</td><td>6</td><td>60</td></tr><tr><td>411</td><td>600</td><td>7</td><td>70</td><td>60</td><td>60</td><td>7</td><td>70</td></tr><tr><td>412</td><td>700</td><td>8</td><td>80</td><td>70</td><td>70</td><td>8</td><td>80</td></tr><tr><td>42</td><td>800</td><td>9</td><td>90</td><td>80</td><td>80</td><td>9</td><td>90</td></tr><tr><td>43</td><td>900</td><td>10</td><td>100</td><td>90</td><td>90</td><td>10</td><td>100</td></tr><tr><td>44</td><td>1000</td><td></td><td></td><td>100</td><td>100</td><td></td><td></td></tr><tr><td>50</td><td>1100</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>72</td><td>1200</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

We now have the information to perform the clustering. The algorithm used was the following:

* **for** each region:

- Place all farms on the four dimensional space. Each farm forms a unique farm-type at first.
### 4.2.9 Cover crops

Work in progress ...

## 4.3 Non linearities within the LP framework

Key-points:

- Non linearities within the *AROPAj* approach

- Advantage of keeping the *LP* framework

- Operational proxy solution

- Going further with loop computation

First, non linearities, *scrigto sensu*, come from the use of integer (and usually binary) variables. This is mainly in the case of *CAP* tool implementation, when threshold conditions enter the policy. The solving process is carried out by solvers proposed by the software (e.g. *MIP* modules used by *GAMS*).

The major case to which we refer here is the substitution of *AROPAj* parameters by the so-called dose-response functions. This case occurs when a nitrogen-to-yield point is replaced by a nitrogen-to-yield function. When this function is linear, the *MP* farm group model is quadratic instead of linear. The problem is better solved when the function is monotonously increasing and quasi-concave (see Section 9.1.1, and Figure 5 which illustrates effects of replacing constant yields by response functions).

This ensures that there is a unique solution that depends primarily on the nitrogen and crop price ratio. Some refinements are required when the farm group's optimal solution is made of non-marketed crop output (or of non-marketed nitrogen input). In other words, this occurs when the entire crop product feeds the livestock or when livestock manure is sufficient to supply the crop with the nitrogen required. In this case, market prices have to be replaced by relevant shadow prices which are themselves a part of the optimal solution.

The process of calculation of shadow prices is an iterative process that entails solving the model and re-estimating the shadow prices until a converging solution is found. This methodology extends and enhances the work done by Jayet et al. (2017a), which was previously developed for crop production entirely used on-farm as feedingstuff for livestock (e.g. cereal).

In this case, the optimal solution leads to a substituting of market prices with shadow prices in the yield and nitrogen calculations. These shadow prices, which are required in order to calculate the yield and nitrogen prices, are provided by the linear programming model. The iterative process consists in using at each iteration shadow prices calculated in the previous iteration.

This two-step process starts with the given market prices, yields, and N-inputs that are computed at the plot level using the first order conditions. Then, these yields and N-inputs enter the linear programming model, which, once solved, provides the shadow prices (equal to market prices when optimal production is at least partly marketed). In the case of grass productions, where no market price exists, fodder prices are set at 0.000001 at the first iteration. The addition of multiple response functions and the complex interaction between farming and breeding activities might lead linear program solutions to be unstable. Finally, the full application to the *AROPAj*
- while there exists some farm types with less than 15 individual farms:

* group the farm-type with the lowest number of individual farms to the closest corresponding farm-type using equation (1).

- recalculate the position of each farm-type in the space (i.e. the mean of the values of each farm belonging to the farm-type, for each dimension).

• repeat until each region is completed.

This algorithm was implemented with the *R GNU* software for the whole European Union. The types of output constructed are presented in Table 18 and Figures 6 and 7 for the French region Picardie, as an example. We can see that farm-types 10 and 11 present similar farms (same organisational form) but different economic size classes. Farm-type 14 gathers farms of different organizational forms because too few were surveyed to create a unique farm type.

<table><caption>**Table 18.** Typology created for the french region “Picardie”.</caption><thead><tr><th>farm-type</th><th>sampled farms</th><th>OTEX</th><th>mean economic class</th><th>mean altitude</th><th>ratio irrigation</th></tr></thead><tbody><tr><td>10</td><td>50</td><td>13</td><td>7.98</td><td>1.08</td><td>0.00</td></tr><tr><td>11</td><td>61</td><td>13</td><td>9.11</td><td>1.00</td><td>0.00</td></tr><tr><td>12</td><td>21</td><td>14</td><td>8.00</td><td>1.00</td><td>0.00</td></tr><tr><td>13</td><td>45</td><td>14-60</td><td>9.31</td><td>1.00</td><td>7.88</td></tr><tr><td>14</td><td>35</td><td>43-50-81-411-412</td><td>8.29</td><td>1.17</td><td>0.00</td></tr><tr><td>15</td><td>50</td><td>81</td><td>9.14</td><td>1.00</td><td>0.00</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Figure 6.** The 6 farm-types produced by the clustering method for the French region “Picardie”.</figcaption></figure>

It has to be noted that the *AROPAj* farm type numbering is automatically processed by the algorithm. The *FADN* region index determines the order of farm clustering per country. When the
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Stylized effects of point (on the left) by response function (on the right) replacement on the optimal yield (*yj*), allocated area (*aj*) and production (*qj*) related to the *j* crop when its price (*pj*) -and only this price- varies. The reference optimally calibrated *LP* solution is materialized by dotted lines.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 7.** Types of crops grown by the 6 farm-types of the region “Picardie”.</figcaption></figure>

number becomes close to 99, the country is divided in two (or more) subcountries, and the farm type numbering begins again at 1 for this “new” country. For the use of the model, the *AROPAj* shell will automatically manage the calculation directories per sub-country and per farm type.

## 5.2 Farm groups throughout the *AROPAj* versions

Key-point:

*AROPAj* farm groups over the model versions

The *FADN* sample clustering into farm groups follows the evolution of the European Union size on the one hand, and the clustering criteria on the other hand. The key gap comes when the typology accounts for irrigation information and when we change the weights associated to the different criteria in the clustering process (with the *V5* versions).

Table 19 delivers the typology, in terms of farm group number, at the *EU* country level for currently running versions of the model. Tables 20 and 21 deliver respective associated farm numbers and utilized agricultural areas, again per country. It must be noted that this data relates both on *AROPAj* activities and *FADN* farm types in the *AROPAj* typology. Some agricultural activities are not represented in the model (e.g. greenhouse productions and wine production, among a few others). Moreover, some *FADN* farm types may have *AROPAj* activities but nevertheless be excluded from the model. Tests are undertaken to gather information about missing areas with *AROPAj* activities: missing areas in the major part of countries are less than 2\ and Greece where missing areas are close to 8\.

Let us remark that even if the *FADN* is a noteworthy database and even if real agriculture may strongly change in some countries after their entrance in the *EU*, the figures in Table 21 show that farm sampling may be deficient in a few Member States. Thus, part of the *AROPAj* model’s “aggregation bias” is due to the *FADN*. We have to keep this in mind before attempting to identify bias in the *AROPAj* calibrating process (see Section 7).

Croatia's entry into the EU in 2013 is taken into account in the 2013-2017 FADN years. Table 22 provides farm group numbers for all FADN years related to the V5 version of the model.
model appears to be satisfactory in that convergence or weak amplitude oscillation is obtained.
Nevertheless, that needs a greater number of iterations than required when only marketed crop
response functions were implemented.
<table><caption>**Table 19.** Farm group number per EU country throughout the different AROPAj versions.</caption><thead><tr><th>country<br>(FADN)</th><th>V1<br>(1997)</th><th>V2<br>(2002)</th><th>V3<br>(2004)</th><th>V4<br>(2009)</th><th>V5<br>(2007)</th><th>V5<br>(2008)</th><th>V5<br>(2009)</th><th>V5<br>(2010)</th><th>V5<br>(2011)</th><th>V5*<br>(2012)</th></tr></thead><tbody><tr><td>bel</td><td>15</td><td>27</td><td>24</td><td>22</td><td>23</td><td>25</td><td>26</td><td>30</td><td>29</td><td>30</td></tr><tr><td>bgr</td><td></td><td></td><td></td><td>29</td><td>45</td><td>49</td><td>48</td><td>61</td><td>61</td><td>65</td></tr><tr><td>cyp</td><td></td><td></td><td>9</td><td>5</td><td>5</td><td>8</td><td>9</td><td>10</td><td>11</td><td>11</td></tr><tr><td>cze</td><td></td><td></td><td>23</td><td>22</td><td>42</td><td>44</td><td>45</td><td>44</td><td>47</td><td>47</td></tr><tr><td>dan</td><td>10</td><td>22</td><td>21</td><td>21</td><td>31</td><td>29</td><td>29</td><td>33</td><td>34</td><td>32</td></tr><tr><td>deu</td><td>99</td><td>144</td><td>138</td><td>172</td><td>180</td><td>195</td><td>215</td><td>229</td><td>215</td><td>218</td></tr><tr><td>ell</td><td>45</td><td>54</td><td>59</td><td>56</td><td>86</td><td>90</td><td>93</td><td>80</td><td>90</td><td>106</td></tr><tr><td>esp</td><td>99</td><td>155</td><td>150</td><td>134</td><td>190</td><td>178</td><td>182</td><td>185</td><td>182</td><td>200</td></tr><tr><td>est</td><td></td><td></td><td>11</td><td>13</td><td>16</td><td>18</td><td>17</td><td>22</td><td>22</td><td>23</td></tr><tr><td>fra</td><td>131</td><td>157</td><td>144</td><td>145</td><td>164</td><td>159</td><td>157</td><td>164</td><td>169</td><td>170</td></tr><tr><td>gbr</td><td>51</td><td>59</td><td>64</td><td>66</td><td>74</td><td>73</td><td>74</td><td>74</td><td>71</td><td>78</td></tr><tr><td>hun</td><td></td><td></td><td>45</td><td>49</td><td>54</td><td>53</td><td>57</td><td>53</td><td>54</td><td>48</td></tr><tr><td>irl</td><td>8</td><td>22</td><td>23</td><td>16</td><td>20</td><td>19</td><td>18</td><td>17</td><td>20</td><td>18</td></tr><tr><td>ita</td><td>161</td><td>278</td><td>207</td><td>154</td><td>323</td><td>249</td><td>233</td><td>239</td><td>238</td><td>256</td></tr><tr><td>ltu</td><td></td><td></td><td>22</td><td>25</td><td>30</td><td>28</td><td>30</td><td>30</td><td>30</td><td>31</td></tr><tr><td>lux</td><td>5</td><td>13</td><td>11</td><td>10</td><td>7</td><td>7</td><td>7</td><td>7</td><td>8</td><td>8</td></tr><tr><td>lva</td><td></td><td></td><td>19</td><td>26</td><td>31</td><td>31</td><td>24</td><td>25</td><td>27</td><td>26</td></tr><tr><td>mlt</td><td></td><td></td><td></td><td>6</td><td>9</td><td>11</td><td>11</td><td>11</td><td>10</td><td>11</td></tr><tr><td>ned</td><td>11</td><td>19</td><td>18</td><td>16</td><td>24</td><td>22</td><td>24</td><td>25</td><td>24</td><td>23</td></tr><tr><td>ost</td><td>23</td><td>38</td><td>39</td><td>41</td><td>43</td><td>43</td><td>41</td><td>41</td><td>43</td><td>45</td></tr><tr><td>pol</td><td></td><td></td><td>176</td><td>195</td><td>222</td><td>222</td><td>215</td><td>200</td><td>205</td><td>209</td></tr><tr><td>por</td><td>43</td><td>37</td><td>32</td><td>36</td><td>51</td><td>48</td><td>46</td><td>46</td><td>47</td><td>48</td></tr><tr><td>rou</td><td></td><td></td><td></td><td>65</td><td>38</td><td>55</td><td>102</td><td>178</td><td>171</td><td>180</td></tr><tr><td>suo</td><td>18</td><td>20</td><td>21</td><td>26</td><td>24</td><td>24</td><td>26</td><td>22</td><td>26</td><td>26</td></tr><tr><td>sve</td><td>15</td><td>29</td><td>27</td><td>30</td><td>25</td><td>28</td><td>30</td><td>30</td><td>30</td><td>30</td></tr><tr><td>svk</td><td></td><td></td><td>13</td><td>13</td><td>19</td><td>18</td><td>18</td><td>18</td><td>18</td><td>18</td></tr><tr><td>svn</td><td></td><td></td><td>11</td><td>21</td><td>22</td><td>21</td><td>25</td><td>27</td><td>28</td><td>36</td></tr><tr><td>EU</td><td>734</td><td>1074</td><td>1307</td><td>1414</td><td>1798</td><td>1747</td><td>1802</td><td>1901</td><td>1910</td><td>1993</td></tr></tbody></table>

<footnote>* Including overseas regions</footnote>
# 5 Typology and farming system clustering

## 5.1 Clustering methodology

### Key-points:

- Regional representativity

- From FADN farm types to AROPAj farm groups

- Metrics for clustering

The information in the FADN database consists of surveys of individual farm samples. These individual farms are grouped by a clustering method to form a group of similar farms that present similar characteristics and thus can form a modeling unit for the AROPAj model: the farm type (also denominated by farm group).

The FADN data are presented at the regional level without any more localization information. Consequently, the farm types that are created belong to a specific region, that approximately corresponds to the NUTS-2 level according to the European classification (Nomenclature of Units for Territorial Statistics). For instance, in France the NUTS-2 classification corresponds to the Administrative Units.

<table><caption>**Table 16.** Organizational form under the FADN classification (simplified version when 2 digits).</caption><thead><tr><td>Code</td><td>Type of farm</td></tr></thead><tbody><tr><td>13</td><td>Cereals, Oleaginous, protein crops</td></tr><tr><td>14</td><td>Other field crops</td></tr><tr><td>2</td><td>Horticulture</td></tr><tr><td>3</td><td>Viticulture</td></tr><tr><td>411</td><td>Bovines milk</td></tr><tr><td>412</td><td>Bovines meat</td></tr><tr><td>42</td><td>Bovines livestock for meat</td></tr><tr><td>43</td><td>Bovines for milk and meat</td></tr><tr><td>44</td><td>Ovines, caprines and other herbivorous</td></tr><tr><td>50</td><td>Grain eater livestock</td></tr><tr><td>72</td><td>Livestock (mainly grain eater)</td></tr><tr><td>60</td><td>Various crops</td></tr><tr><td>71</td><td>Livestock (mainly herbivorous)</td></tr><tr><td>81</td><td>Field crops and herbivorous</td></tr><tr><td>82</td><td>Other livestock/field crop combinations</td></tr></tbody></table>

Within these regional units, the individually sampled farms are aggregated by a clustering method under the condition that each farm type must gather at least 15 different farms (to keep the information fully anonymous). The farms are grouped according to different key variables: organisational form, altitude, irrigation and economic size. Organisational form (FADN variable A18, also called OTEX or FT) refers to a FADN classification of the farms depending on the type of agricultural production, from field crops farms to intensive livestock farming. Table 16 presents the various values that this variable takes (note that horticulture and viticulture are not included in AROPAj). There are three values that correspond to altitude zone (FADN variable A41): 1 for farms below 300 meters, 2 for farms between 300 and 600 meters, and 3 for farms above 600
<table><caption>**Table 20.** Farm number represented per EU country throughout the different AROPAj versions (thousands).</caption><thead><tr><th>country<br>(FADN)</th><th>V1<br>(1997)</th><th>V2<br>(2002)</th><th>V3<br>(2004)</th><th>V4<br>(2009)</th><th>V5<br>(2007)</th><th>V5<br>(2008)</th><th>V5<br>(2009)</th></tr></thead><tbody><tr><td>bel</td><td>36</td><td>34</td><td>30</td><td>27</td><td>27</td><td>27</td><td>27</td></tr><tr><td>bgr</td><td></td><td></td><td></td><td>78</td><td>91</td><td>93</td><td>102</td></tr><tr><td>cyp</td><td></td><td></td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>cze</td><td></td><td></td><td>12</td><td>13</td><td>13</td><td>13</td><td>13</td></tr><tr><td>dan</td><td>48</td><td>44</td><td>37</td><td>31</td><td>31</td><td>31</td><td>31</td></tr><tr><td>deu</td><td>256</td><td>205</td><td>200</td><td>164</td><td>163</td><td>163</td><td>164</td></tr><tr><td>ell</td><td>273</td><td>198</td><td>209</td><td>201</td><td>271</td><td>276</td><td>261</td></tr><tr><td>esp</td><td>302</td><td>330</td><td>275</td><td>281</td><td>343</td><td>276</td><td>281</td></tr><tr><td>est</td><td></td><td></td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td></tr><tr><td>fra</td><td>320</td><td>314</td><td>299</td><td>287</td><td>286</td><td>287</td><td>287</td></tr><tr><td>gbr</td><td>124</td><td>118</td><td>94</td><td>85</td><td>85</td><td>85</td><td>85</td></tr><tr><td>hun</td><td></td><td></td><td>67</td><td>65</td><td>62</td><td>62</td><td>65</td></tr><tr><td>irl</td><td>129</td><td>77</td><td>69</td><td>56</td><td>107</td><td>106</td><td>106</td></tr><tr><td>ita</td><td>597</td><td>374</td><td>354</td><td>328</td><td>382</td><td>376</td><td>378</td></tr><tr><td>ltu</td><td></td><td></td><td>30</td><td>39</td><td>39</td><td>39</td><td>39</td></tr><tr><td>lux</td><td>2</td><td>2</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>lva</td><td></td><td></td><td>20</td><td>22</td><td>22</td><td>22</td><td>22</td></tr><tr><td>mlt</td><td></td><td></td><td></td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>ned</td><td>65</td><td>51</td><td>42</td><td>46</td><td>47</td><td>47</td><td>47</td></tr><tr><td>ost</td><td>79</td><td>74</td><td>73</td><td>63</td><td>63</td><td>63</td><td>63</td></tr><tr><td>pol</td><td></td><td></td><td>438</td><td>450</td><td>703</td><td>697</td><td>704</td></tr><tr><td>por</td><td>209</td><td>79</td><td>72</td><td>67</td><td>66</td><td>68</td><td>67</td></tr><tr><td>rou</td><td></td><td></td><td></td><td>245</td><td>815</td><td>834</td><td>835</td></tr><tr><td>suo</td><td>49</td><td>43</td><td>42</td><td>39</td><td>38</td><td>38</td><td>39</td></tr><tr><td>sve</td><td>38</td><td>39</td><td>31</td><td>26</td><td>26</td><td>26</td><td>26</td></tr><tr><td>svk</td><td></td><td></td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>svn</td><td></td><td></td><td>33</td><td>38</td><td>36</td><td>38</td><td>38</td></tr><tr><td>EU</td><td>2527</td><td>1981</td><td>2446</td><td>2672</td><td>3738</td><td>3688</td><td>3700</td></tr></tbody></table>
meters. The irrigation variable was calculated as the ratio of irrigated crop areas divided by the total amount of crop areas. The economic size (FADN variable A27) is an integer between 1 and 10 corresponding to the size of the sampled farm.

A good clustering method is a method that maximizes the similarity within groups and the distance between different groups. We thus chose to perform the typical agglomerative hierarchical clustering method on these four variables. In order to do so, one must be able to calculate a distance between the farms. Each farm was placed on a four dimensional space, and the distance between two farms A and B could then be calculated according to the classical Euler formula:

$$Distance_{AB} = \sqrt{\sum_{i=1}^{N} (xA_{i} - xB_{i})^{2}} \quad (1)$$

where N is the number of dimensions of the space (4 in our case) and $xA_i$ and $xB_i$ being the coordinates of the farms A and B in the dimension i. The calculation of the distance (a measure of similarity) is a subjective choice that belongs to the modeler, who can emphasize the importance of the variables of his choice.

Another important point is the transformation of the variables to a numerical equivalent that will be used to place the farms on each dimension. Table 17 presents the different variables that were used for the clustering and their arbitrary given value. We decided to put more weight to the organisational form since we intended to group the farms according to their type of agricultural production. The scale goes from 0 to 1200 with 0 being the more extensive (field crops) and 1200 being the more intensive (swine and poultry) type of farm. Irrigation and economic size were considered at a similar weight (scale ranging from 0 to 100) while altitude was considered as the less important variable (scale ranging from 10 to 30).

<table><caption>**Table 17.** Variables used by the hierarchical clustering procedure to create farm-types at the European Union level.</caption><thead><tr><th colspan="2">Organisational form</th><th colspan="2">Economic class</th><th colspan="2">Irrigation</th><th colspan="2">Altitude</th></tr><tr><th>Code</th><th>Value</th><th>Code</th><th>Value</th><th>Irrigation ratio (%)</th><th>Value</th><th>Code</th><th>Value</th></tr></thead><tbody><tr><td>13</td><td>0</td><td>1</td><td>10</td><td>0</td><td>0</td><td>1</td><td>10</td></tr><tr><td>14</td><td>100</td><td>2</td><td>20</td><td>10</td><td>10</td><td>2</td><td>20</td></tr><tr><td>60</td><td>200</td><td>3</td><td>30</td><td>20</td><td>20</td><td>3</td><td>30</td></tr><tr><td>82</td><td>300</td><td>4</td><td>40</td><td>30</td><td>30</td><td>4</td><td>40</td></tr><tr><td>81</td><td>400</td><td>5</td><td>50</td><td>40</td><td>40</td><td>5</td><td>50</td></tr><tr><td>71</td><td>500</td><td>6</td><td>60</td><td>50</td><td>50</td><td>6</td><td>60</td></tr><tr><td>411</td><td>600</td><td>7</td><td>70</td><td>60</td><td>60</td><td>7</td><td>70</td></tr><tr><td>412</td><td>700</td><td>8</td><td>80</td><td>70</td><td>70</td><td>8</td><td>80</td></tr><tr><td>42</td><td>800</td><td>9</td><td>90</td><td>80</td><td>80</td><td>9</td><td>90</td></tr><tr><td>43</td><td>900</td><td>10</td><td>100</td><td>90</td><td>90</td><td>10</td><td>100</td></tr><tr><td>44</td><td>1000</td><td></td><td></td><td>100</td><td>100</td><td></td><td></td></tr><tr><td>50</td><td>1100</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>72</td><td>1200</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

We now have the information to perform the clustering. The algorithm used was the following:

* **for** each region:

- Place all farms on the four dimensional space. Each farm forms a unique farm-type at first.
<table><caption>**Table 21.** Utilized agricultural area represented per EU country throughout the different AROPAj versions (thousands of hectares).</caption><thead><tr><th>country<br>(FADN)</th><th>V1<br>(1997)</th><th>V2<br>(2002)</th><th>V3<br>(2004)</th><th>V4<br>(2009)</th><th>V5<br>(2007)</th><th>V5<br>(2008)</th><th>V5<br>(2009)</th></tr></thead><tbody><tr><td>bel</td><td>1355</td><td>1432</td><td>1339</td><td>1393</td><td>1333</td><td>1380</td><td>1411</td></tr><tr><td>bgr</td><td></td><td></td><td></td><td>2797</td><td>2752</td><td>2781</td><td>2985</td></tr><tr><td>cyp</td><td></td><td></td><td>137</td><td>110</td><td>107</td><td>125</td><td>112</td></tr><tr><td>cze</td><td></td><td></td><td>3430</td><td>2987</td><td>3277</td><td>3153</td><td>3222</td></tr><tr><td>dan</td><td>2349</td><td>2437</td><td>2397</td><td>2489</td><td>2486</td><td>2521</td><td>2590</td></tr><tr><td>deu</td><td>14736</td><td>14728</td><td>15414</td><td>15145</td><td>14932</td><td>15306</td><td>15411</td></tr><tr><td>ell</td><td>1418</td><td>1220</td><td>1651</td><td>2053</td><td>2357</td><td>2443</td><td>2423</td></tr><tr><td>esp</td><td>11816</td><td>11329</td><td>10939</td><td>12903</td><td>13566</td><td>13008</td><td>13253</td></tr><tr><td>est</td><td></td><td></td><td>669</td><td>2249</td><td>837</td><td>869</td><td>886</td></tr><tr><td>fra</td><td>22972</td><td>24042</td><td>24321</td><td>24778</td><td>25065</td><td>25208</td><td>25118</td></tr><tr><td>gbr</td><td>11360</td><td>11647</td><td>10337</td><td>10290</td><td>10327</td><td>10322</td><td>10606</td></tr><tr><td>hun</td><td></td><td></td><td>3477</td><td>3656</td><td>3724</td><td>3727</td><td>3737</td></tr><tr><td>irl</td><td>4014</td><td>3198</td><td>3087</td><td>3003</td><td>3992</td><td>4072</td><td>4195</td></tr><tr><td>ita</td><td>8295</td><td>7552</td><td>8125</td><td>7145</td><td>9109</td><td>7848</td><td>7982</td></tr><tr><td>ltu</td><td></td><td></td><td>1470</td><td>1813</td><td>1907</td><td>1889</td><td>1869</td></tr><tr><td>lux</td><td>104</td><td>130</td><td>124</td><td>125</td><td>125</td><td>125</td><td>127</td></tr><tr><td>lva</td><td></td><td></td><td>1000</td><td>1159</td><td>1142</td><td>1157</td><td>1178</td></tr><tr><td>mlt</td><td></td><td></td><td></td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>ned</td><td>1867</td><td>1511</td><td>1234</td><td>1648</td><td>1716</td><td>1728</td><td>1718</td></tr><tr><td>ost</td><td>1882</td><td>1788</td><td>1853</td><td>1771</td><td>1780</td><td>1807</td><td>1791</td></tr><tr><td>pol</td><td></td><td></td><td>7706</td><td>9823</td><td>12594</td><td>13254</td><td>13374</td></tr><tr><td>por</td><td>2253</td><td>1990</td><td>1716</td><td>1971</td><td>2032</td><td>2267</td><td>2124</td></tr><tr><td>rou</td><td></td><td></td><td></td><td>4527</td><td>7917</td><td>10100</td><td>10182</td></tr><tr><td>suo</td><td>1765</td><td>1889</td><td>1978</td><td>2046</td><td>2024</td><td>2084</td><td>2110</td></tr><tr><td>sve</td><td>3228</td><td>2639</td><td>2560</td><td>2406</td><td>2482</td><td>2472</td><td>2496</td></tr><tr><td>svk</td><td></td><td></td><td>1761</td><td>1724</td><td>1925</td><td>1899</td><td>1854</td></tr><tr><td>svn</td><td></td><td></td><td>430</td><td>434</td><td>437</td><td>434</td><td>438</td></tr><tr><td>EU</td><td>89415</td><td>87532</td><td>107153</td><td>120448</td><td>129943</td><td>131980</td><td>133193</td></tr></tbody></table>
- while there exists some farm types with less than 15 individual farms:

* group the farm-type with the lowest number of individual farms to the closest corresponding farm-type using equation (1).

- recalculate the position of each farm-type in the space (i.e. the mean of the values of each farm belonging to the farm-type, for each dimension).

• repeat until each region is completed.

This algorithm was implemented with the *R* GNU software for the whole European Union. The types of output constructed are presented in Table 18 and Figures 6 and 7 for the French region Picardie, as an example. We can see that farm-types 10 and 11 present similar farms (same organisational form) but different economic size classes. Farm-type 14 gathers farms of different organizational forms because too few were surveyed to create a unique farm type.

<table><caption>**Table 18.** Typology created for the french region “Picardie”.</caption><thead><tr><th>farm-type</th><th>sampled farms</th><th>OTEX</th><th>mean economic class</th><th>mean altitude</th><th>ratio irrigation</th></tr></thead><tbody><tr><td>10</td><td>50</td><td>13</td><td>7.98</td><td>1.08</td><td>0.00</td></tr><tr><td>11</td><td>61</td><td>13</td><td>9.11</td><td>1.00</td><td>0.00</td></tr><tr><td>12</td><td>21</td><td>14</td><td>8.00</td><td>1.00</td><td>0.00</td></tr><tr><td>13</td><td>45</td><td>14-60</td><td>9.31</td><td>1.00</td><td>7.88</td></tr><tr><td>14</td><td>35</td><td>43-50-81-411-412</td><td>8.29</td><td>1.17</td><td>0.00</td></tr><tr><td>15</td><td>50</td><td>81</td><td>9.14</td><td>1.00</td><td>0.00</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Figure 6.** The 6 farm-types produced by the clustering method for the French region “Picardie”.</figcaption></figure>

It has to be noted that the *AROPAj* farm type numbering is automatically processed by the algorithm. The *FADN* region index determines the order of farm clustering per country. When the
<table><caption>**Table 22.** Farm group number per EU country throughout the different years of the AROPAj-V5 version.</caption><thead><tr><th>country<br>(FADN)</th><th>V5<br>(2007)</th><th>V5<br>(2008)</th><th>V5<br>(2009)</th><th>V5<br>(2010)</th><th>V5<br>(2011)</th><th>V5*<br>(2012)</th><th>V5*<br>(2013)</th><th>V5*<br>(2014)</th><th>V5*<br>(2015)</th><th>V5*<br>(2016)</th><th>V5*<br>(2017)</th><th>V5*<br>(2018)</th></tr></thead><tbody><tr><td>bel</td><td>23</td><td>25</td><td>26</td><td>30</td><td>29</td><td>30</td><td>31</td><td>33</td><td>30</td><td>25</td><td>29</td><td>29</td></tr><tr><td>bgr</td><td>45</td><td>49</td><td>48</td><td>61</td><td>61</td><td>65</td><td>67</td><td>70</td><td>68</td><td>69</td><td>67</td><td>73</td></tr><tr><td>cyp</td><td>5</td><td>8</td><td>9</td><td>10</td><td>11</td><td>11</td><td>10</td><td>8</td><td>8</td><td>9</td><td>8</td><td>8</td></tr><tr><td>cze</td><td>42</td><td>44</td><td>45</td><td>44</td><td>47</td><td>47</td><td>43</td><td>44</td><td>44</td><td>45</td><td>41</td><td>41</td></tr><tr><td>dan</td><td>31</td><td>29</td><td>29</td><td>33</td><td>34</td><td>32</td><td>31</td><td>39</td><td>37</td><td>41</td><td>40</td><td>40</td></tr><tr><td>deu</td><td>180</td><td>195</td><td>215</td><td>229</td><td>215</td><td>218</td><td>224</td><td>223</td><td>215</td><td>214</td><td>215</td><td>217</td></tr><tr><td>ell</td><td>86</td><td>90</td><td>93</td><td>80</td><td>90</td><td>106</td><td>116</td><td>98</td><td>103</td><td>99</td><td>98</td><td>89</td></tr><tr><td>esp</td><td>190</td><td>178</td><td>182</td><td>185</td><td>182</td><td>200</td><td>193</td><td>193</td><td>191</td><td>199</td><td>189</td><td>191</td></tr><tr><td>est</td><td>16</td><td>18</td><td>17</td><td>22</td><td>22</td><td>23</td><td>18</td><td>20</td><td>21</td><td>20</td><td>22</td><td>19</td></tr><tr><td>fra</td><td>164</td><td>159</td><td>157</td><td>164</td><td>169</td><td>170</td><td>179</td><td>172</td><td>175</td><td>167</td><td>165</td><td>166</td></tr><tr><td>gbr</td><td>74</td><td>73</td><td>74</td><td>74</td><td>71</td><td>78</td><td>73</td><td>78</td><td>74</td><td>78</td><td>82</td><td>81</td></tr><tr><td>hrv</td><td></td><td></td><td></td><td></td><td></td><td></td><td>31</td><td>37</td><td>35</td><td>33</td><td>36</td><td>36</td></tr><tr><td>hun</td><td>54</td><td>53</td><td>57</td><td>53</td><td>54</td><td>48</td><td>48</td><td>49</td><td>48</td><td>50</td><td>48</td><td>45</td></tr><tr><td>irl</td><td>20</td><td>19</td><td>18</td><td>17</td><td>20</td><td>18</td><td>17</td><td>17</td><td>15</td><td>18</td><td>13</td><td>13</td></tr><tr><td>ita</td><td>323</td><td>249</td><td>233</td><td>239</td><td>238</td><td>256</td><td>243</td><td>225</td><td>200</td><td>206</td><td>228</td><td>235</td></tr><tr><td>ltu</td><td>30</td><td>28</td><td>30</td><td>30</td><td>30</td><td>31</td><td>26</td><td>30</td><td>27</td><td>29</td><td>30</td><td>31</td></tr><tr><td>lux</td><td>7</td><td>7</td><td>7</td><td>7</td><td>8</td><td>8</td><td>7</td><td>8</td><td>7</td><td>10</td><td>8</td><td>10</td></tr><tr><td>lva</td><td>31</td><td>31</td><td>24</td><td>25</td><td>27</td><td>26</td><td>26</td><td>24</td><td>26</td><td>25</td><td>26</td><td>26</td></tr><tr><td>mlt</td><td>9</td><td>11</td><td>11</td><td>11</td><td>10</td><td>11</td><td>10</td><td>11</td><td>10</td><td>11</td><td>11</td><td>10</td></tr><tr><td>ned</td><td>24</td><td>22</td><td>24</td><td>25</td><td>24</td><td>23</td><td>26</td><td>26</td><td>26</td><td>28</td><td>27</td><td>26</td></tr><tr><td>ost</td><td>43</td><td>43</td><td>41</td><td>41</td><td>43</td><td>45</td><td>41</td><td>44</td><td>43</td><td>40</td><td>39</td><td>41</td></tr><tr><td>pol</td><td>222</td><td>222</td><td>215</td><td>200</td><td>205</td><td>209</td><td>210</td><td>172</td><td>168</td><td>174</td><td>174</td><td>211</td></tr><tr><td>por</td><td>51</td><td>48</td><td>46</td><td>46</td><td>47</td><td>48</td><td>45</td><td>42</td><td>43</td><td>40</td><td>35</td><td>40</td></tr><tr><td>rou</td><td>38</td><td>55</td><td>102</td><td>178</td><td>171</td><td>180</td><td>189</td><td>116</td><td>142</td><td>180</td><td>181</td><td>157</td></tr><tr><td>suo</td><td>24</td><td>24</td><td>26</td><td>22</td><td>26</td><td>26</td><td>22</td><td>21</td><td>23</td><td>24</td><td>24</td><td>19</td></tr><tr><td>sve</td><td>25</td><td>28</td><td>30</td><td>30</td><td>30</td><td>30</td><td>32</td><td>31</td><td>32</td><td>34</td><td>30</td><td>31</td></tr><tr><td>svk</td><td>19</td><td>18</td><td>18</td><td>18</td><td>18</td><td>18</td><td>17</td><td>21</td><td>16</td><td>16</td><td>19</td><td>16</td></tr><tr><td>svn</td><td>22</td><td>21</td><td>25</td><td>27</td><td>28</td><td>36</td><td>31</td><td>29</td><td>25</td><td>26</td><td>31</td><td>27</td></tr></tbody><tfoot><tr><td>EU</td><td>1798</td><td>1747</td><td>1802</td><td>1901</td><td>1910</td><td>1993</td><td>2006</td><td>1881</td><td>1852</td><td>1910</td><td>1916</td><td>1928</td></tr></tfoot></table>

* Including overseas regions
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Types of crops grown by the 6 farm-types of the region “Picardie”.</figcaption></figure>

number becomes close to 99, the country is divided in two (or more) subcountries, and the farm type numbering begins again at 1 for this “new” country. For the use of the model, the *AROPAj* shell will automatically manage the calculation directories per sub-country and per farm type.

## 5.2 Farm groups throughout the *AROPAj* versions

Key-point:

*AROPAj* farm groups over the model versions

The *FADN* sample clustering into farm groups follows the evolution of the European Union size on the one hand, and the clustering criteria on the other hand. The key gap comes when the typology accounts for irrigation information and when we change the weights associated to the different criteria in the clustering process (with the *V5* versions).

Table 19 delivers the typology, in terms of farm group number, at the EU country level for currently running versions of the model. Tables 20 and 21 deliver respective associated farm numbers and utilized agricultural areas, again per country. It must be noted that this data relates both on *AROPAj* activities and *FADN* farm types in the *AROPAj* typology. Some agricultural activities are not represented in the model (e.g. greenhouse productions and wine production, among a few others). Moreover, some *FADN* farm types may have *AROPAj* activities but nevertheless be excluded from the model. Tests are undertaken to gather information about missing areas with *AROPAj* activities: missing areas in the major part of countries are less than 2\ and Greece where missing areas are close to 8\.

Let us remark that even if the *FADN* is a noteworthy database and even if real agriculture may strongly change in some countries after their entrance in the EU, the figures in Table 21 show that farm sampling may be deficient in a few Member States. Thus, part of the *AROPAj* model’s “aggregation bias” is due to the *FADN*. We have to keep this in mind before attempting to identify bias in the *AROPAj* calibrating process (see Section 7).

Croatia's entry into the EU in 2013 is taken into account in the 2013-2017 FADN years. Table 22 provides farm group numbers for all FADN years related to the V5 version of the model.
# 6 Parameter estimation

## 6.1 Variable costs

**Key-points:**

* Insights about new cost estimates

* Application of General Maximum Entropy

* Individual estimates

### 6.1.1 Estimation of activity variable costs

One of the main problems related to the use of the FADN database in farm programming models designed for policy analysis is the aggregated nature of cost data. This means that costs for various types of inputs are reported at the farm level but their distribution among the farm various activities is not available to the analyst.

Least Squares (LS) was historically the first method used for such cost disaggregation (Errington, 1989). However, LS may possibly return negative or non-significant cost coefficients that cannot be considered as valid estimates. Moreover, LS is suitable for estimating regional cost coefficients since multiple regression with cross-sectional data can only yield estimates of the average cost of input use for each activity across the sample: denoting activities by $i = 1, 2, ..., I$ and farms by $n = 1, 2, ..., N$, the estimating equation for the average cost per activity for any given input is written as $Y_n = \sum_i X_{ni}b_i + u_n$, where $Y_n$ is the aggregate cost paid by farm $n$ for the said input, $X_{ni}$ stands for the level of activity $i$ in the $n$-th farm, $b_i$ is the average cost of the input for activity $i$ and $u_n$ is the error term. However, the cost coefficients in the case of farm models should not correspond to average values but should instead reflect the differences in production costs among the same activities in different farms. Cost disaggregation for each farm in the sample therefore requires the estimation of different $\beta_{ni}$ coefficients so that the budget equation $Y_n = \sum_i X_{ni}\beta_{ni}$ is satisfied for any given input (i.e. $\beta_{ni} \neq b_i$). This means that for databases like FADN, the estimation of activity- and farm-specific costs constitutes an ill-posed problem with negative degrees of freedom because the number of the coefficients to be estimated ($I \times N$) is greater than the number of observations ($N$).

Since the problem of possible negative coefficients and ill-posedness cannot be handled by traditional econometric techniques, the literature suggests the use of information-theoretic statistical approaches like entropy and Bayesian methods that make use of prior information based on agro-nomic data, expert opinion, or other sources (e.g. Louhichi et al., 2012; Léon et al., 1999). Of similar reasoning to the above is the use of goal programming, as implemented by Kampas et al. (2012) for the estimation of activity costs in FADN farms in Thessaly, Greece, using information on costs obtained by a farm survey.

Although the use of exogenous information provides an answer to the problem of activity cost disaggregation, there still remains the problem of a source of homogeneous data at the EU level because available information, especially when it relates to agronomic practices, is usually limited to only a finite number of crops and geographical locales. In this section we present a method that is based on a two-step procedure which combines regression analysis with the Generalized
<table><caption>**Table 19.** Farm group number per EU country throughout the different AROPAj versions.</caption><thead><tr><th>country<br>(FADN)</th><th>V1<br>(1997)</th><th>V2<br>(2002)</th><th>V3<br>(2004)</th><th>V4<br>(2009)</th><th>V5<br>(2007)</th><th>V5<br>(2008)</th><th>V5<br>(2009)</th><th>V5<br>(2010)</th><th>V5<br>(2011)</th><th>V5*<br>(2012)</th></tr></thead><tbody><tr><td>bel</td><td>15</td><td>27</td><td>24</td><td>22</td><td>23</td><td>25</td><td>26</td><td>30</td><td>29</td><td>30</td></tr><tr><td>bgr</td><td></td><td></td><td></td><td>29</td><td>45</td><td>49</td><td>48</td><td>61</td><td>61</td><td>65</td></tr><tr><td>cyp</td><td></td><td></td><td>9</td><td>5</td><td>5</td><td>8</td><td>9</td><td>10</td><td>11</td><td>11</td></tr><tr><td>cze</td><td></td><td></td><td>23</td><td>22</td><td>42</td><td>44</td><td>45</td><td>44</td><td>47</td><td>47</td></tr><tr><td>dan</td><td>10</td><td>22</td><td>21</td><td>21</td><td>31</td><td>29</td><td>29</td><td>33</td><td>34</td><td>32</td></tr><tr><td>deu</td><td>99</td><td>144</td><td>138</td><td>172</td><td>180</td><td>195</td><td>215</td><td>229</td><td>215</td><td>218</td></tr><tr><td>ell</td><td>45</td><td>54</td><td>59</td><td>56</td><td>86</td><td>90</td><td>93</td><td>80</td><td>90</td><td>106</td></tr><tr><td>esp</td><td>99</td><td>155</td><td>150</td><td>134</td><td>190</td><td>178</td><td>182</td><td>185</td><td>182</td><td>200</td></tr><tr><td>est</td><td></td><td></td><td>11</td><td>13</td><td>16</td><td>18</td><td>17</td><td>22</td><td>22</td><td>23</td></tr><tr><td>fra</td><td>131</td><td>157</td><td>144</td><td>145</td><td>164</td><td>159</td><td>157</td><td>164</td><td>169</td><td>170</td></tr><tr><td>gbr</td><td>51</td><td>59</td><td>64</td><td>66</td><td>74</td><td>73</td><td>74</td><td>74</td><td>71</td><td>78</td></tr><tr><td>hun</td><td></td><td></td><td>45</td><td>49</td><td>54</td><td>53</td><td>57</td><td>53</td><td>54</td><td>48</td></tr><tr><td>irl</td><td>8</td><td>22</td><td>23</td><td>16</td><td>20</td><td>19</td><td>18</td><td>17</td><td>20</td><td>18</td></tr><tr><td>ita</td><td>161</td><td>278</td><td>207</td><td>154</td><td>323</td><td>249</td><td>233</td><td>239</td><td>238</td><td>256</td></tr><tr><td>ltu</td><td></td><td></td><td>22</td><td>25</td><td>30</td><td>28</td><td>30</td><td>30</td><td>30</td><td>31</td></tr><tr><td>lux</td><td>5</td><td>13</td><td>11</td><td>10</td><td>7</td><td>7</td><td>7</td><td>7</td><td>8</td><td>8</td></tr><tr><td>lva</td><td></td><td></td><td>19</td><td>26</td><td>31</td><td>31</td><td>24</td><td>25</td><td>27</td><td>26</td></tr><tr><td>mlt</td><td></td><td></td><td></td><td>6</td><td>9</td><td>11</td><td>11</td><td>11</td><td>10</td><td>11</td></tr><tr><td>ned</td><td>11</td><td>19</td><td>18</td><td>16</td><td>24</td><td>22</td><td>24</td><td>25</td><td>24</td><td>23</td></tr><tr><td>ost</td><td>23</td><td>38</td><td>39</td><td>41</td><td>43</td><td>43</td><td>41</td><td>41</td><td>43</td><td>45</td></tr><tr><td>pol</td><td></td><td></td><td>176</td><td>195</td><td>222</td><td>222</td><td>215</td><td>200</td><td>205</td><td>209</td></tr><tr><td>por</td><td>43</td><td>37</td><td>32</td><td>36</td><td>51</td><td>48</td><td>46</td><td>46</td><td>47</td><td>48</td></tr><tr><td>rou</td><td></td><td></td><td></td><td>65</td><td>38</td><td>55</td><td>102</td><td>178</td><td>171</td><td>180</td></tr><tr><td>suo</td><td>18</td><td>20</td><td>21</td><td>26</td><td>24</td><td>24</td><td>26</td><td>22</td><td>26</td><td>26</td></tr><tr><td>sve</td><td>15</td><td>29</td><td>27</td><td>30</td><td>25</td><td>28</td><td>30</td><td>30</td><td>30</td><td>30</td></tr><tr><td>svk</td><td></td><td></td><td>13</td><td>13</td><td>19</td><td>18</td><td>18</td><td>18</td><td>18</td><td>18</td></tr><tr><td>svn</td><td></td><td></td><td>11</td><td>21</td><td>22</td><td>21</td><td>25</td><td>27</td><td>28</td><td>36</td></tr><tr><td>EU</td><td>734</td><td>1074</td><td>1307</td><td>1414</td><td>1798</td><td>1747</td><td>1802</td><td>1901</td><td>1910</td><td>1993</td></tr></tbody></table>

<footnote>* Including overseas regions</footnote>
Maximum Entropy estimator (GME). Specifically, we remain within the entropy framework of for the disaggregation of costs, as suggested by the literature, but we draw prior information from an initial regression exercise that aims at providing an approximate estimate of variable costs per activity at the regional or higher geographical scales.

Applying the GME estimator in the linear model $Y_n = \sum_i x_{ni} \beta_{ni}$ involves finding an entropy maximizing probability distribution over a vector of support values for every $\beta_{ni}$ coefficient, so that each coefficient is re-defined as the expected value of its corresponding unknown probability distribution. The support vectors are specified in the feasible range of the unknown parameters and are defined a priori by the researcher. In this sense they constitute the most subjective aspect of the entropy estimation procedure. In fact, the selection of the support values gives rise to two issues. The first is the choice of the central value of the support interval and the second is the width of the interval, i.e., the width of the interval around the central value. Especially the latter can greatly affect the estimation procedure because the coefficients are not invariant to the chosen support intervals (Caputo & Paris, 2008); narrow bounds can greatly increase the mean square error of the GME estimator (Golan et al., 1997), or even render the entropy maximization program infeasible.

The first step for estimating $\beta_{ni}$ with GME is to construct an appropriate vector of support values. For this purpose, we estimate the equation $Y_n = \sum_i x_{ni}b_i + u_n$ in order to obtain the $\hat{b}_i$ coefficient and the residual $e_n$. The underlying hypothesis is that for a given cost type $t$, the coefficient $\beta_{ni}$ for activity $i$ in farm $n$ should be located in an interval around the average cost $\hat{b}_i$ and the interval must also contain part of the residuals $e_n$. We can therefore redefine the unknown parameter as $\beta_{ni} = \alpha_{ni} + v_{ni}$, where $\alpha_{ni}$ is a parameter derived from the distribution of $\hat{b}_i$, whereas $v_{ni}$ represents an error term that can be defined with respect to $e_n$. Although $\alpha_{ni}$ is directly related to the distribution of $\beta_{ni}$, and thus its meaning is straightforward, the use of an error term merits justification. There are two reasons that require the definition of $\beta_{ni}$ as the sum of two separate parameters, one of which is an “error” term. The first reason is the aggregation of explanatory variables (production activities) into general categories. For instance, some FADN activities are already aggregates of other individual activities which cannot be modelled separately for different technical or sampling reasons. Typical examples are the various age categories in livestock production, flowers, seed crops, fodder root crops and all subcategories indexed as “other…” (e.g., K128: other cereals and K330: other protein crops). The second reason that justifies the inclusion of error terms concerns the feasibility of the maximum entropy problem: as explained by Léon et al. (1999), the role of the disturbances is to “loosen” the model constraints so that the model returns an optimal solution. This implies that there should be a large enough error support interval in order to compensate for deviations between observed aggregate input costs and estimated cost allocations.

The goal is now to find support intervals for $\alpha_{ni}$ and $v_{ni}$. The support interval for $\alpha_{ni}$ can be constructed using the “3σ rule”, that is, three standard errors (s.e.) around the LS estimate: $\hat{b}_i - 3 \text{ s.e.} (\hat{b}_i) \le \alpha_{ni} \le \hat{b}_i + 3 \text{ s.e.} (\hat{b}_i)$. This method requires that (i) the $\hat{b}_i$ coefficient is positive and (ii) the resulting support interval does not contain any negative values. However, this may not always be possible with least square techniques because the analyst can neither impose non-negativity constraints on the estimation procedure nor ensure that the estimated coefficients are statistically significant, so that the interval strictly lies in $\mathbb{R}_+$.<fnref n="6" /> A simple ad-hoc solution when

<footnote n="6">⁶A higher variance will probably lead to a negative lower bound with the “3σ” rule.</footnote>
<table><caption>**Table 20.** Farm number represented per EU country throughout the different AROPAj versions (thousands).</caption><thead><tr><th>country<br>(FADN)</th><th>V1<br>(1997)</th><th>V2<br>(2002)</th><th>V3<br>(2004)</th><th>V4<br>(2009)</th><th>V5<br>(2007)</th><th>V5<br>(2008)</th><th>V5<br>(2009)</th></tr></thead><tbody><tr><td>bel</td><td>36</td><td>34</td><td>30</td><td>27</td><td>27</td><td>27</td><td>27</td></tr><tr><td>bgr</td><td></td><td></td><td></td><td>78</td><td>91</td><td>93</td><td>102</td></tr><tr><td>cyp</td><td></td><td></td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>cze</td><td></td><td></td><td>12</td><td>13</td><td>13</td><td>13</td><td>13</td></tr><tr><td>dan</td><td>48</td><td>44</td><td>37</td><td>31</td><td>31</td><td>31</td><td>31</td></tr><tr><td>deu</td><td>256</td><td>205</td><td>200</td><td>164</td><td>163</td><td>163</td><td>164</td></tr><tr><td>ell</td><td>273</td><td>198</td><td>209</td><td>201</td><td>271</td><td>276</td><td>261</td></tr><tr><td>esp</td><td>302</td><td>330</td><td>275</td><td>281</td><td>343</td><td>276</td><td>281</td></tr><tr><td>est</td><td></td><td></td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td></tr><tr><td>fra</td><td>320</td><td>314</td><td>299</td><td>287</td><td>286</td><td>287</td><td>287</td></tr><tr><td>gbr</td><td>124</td><td>118</td><td>94</td><td>85</td><td>85</td><td>85</td><td>85</td></tr><tr><td>hun</td><td></td><td></td><td>67</td><td>65</td><td>62</td><td>62</td><td>65</td></tr><tr><td>irl</td><td>129</td><td>77</td><td>69</td><td>56</td><td>107</td><td>106</td><td>106</td></tr><tr><td>ita</td><td>597</td><td>374</td><td>354</td><td>328</td><td>382</td><td>376</td><td>378</td></tr><tr><td>ltu</td><td></td><td></td><td>30</td><td>39</td><td>39</td><td>39</td><td>39</td></tr><tr><td>lux</td><td>2</td><td>2</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>lva</td><td></td><td></td><td>20</td><td>22</td><td>22</td><td>22</td><td>22</td></tr><tr><td>mlt</td><td></td><td></td><td></td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>ned</td><td>65</td><td>51</td><td>42</td><td>46</td><td>47</td><td>47</td><td>47</td></tr><tr><td>ost</td><td>79</td><td>74</td><td>73</td><td>63</td><td>63</td><td>63</td><td>63</td></tr><tr><td>pol</td><td></td><td></td><td>438</td><td>450</td><td>703</td><td>697</td><td>704</td></tr><tr><td>por</td><td>209</td><td>79</td><td>72</td><td>67</td><td>66</td><td>68</td><td>67</td></tr><tr><td>rou</td><td></td><td></td><td></td><td>245</td><td>815</td><td>834</td><td>835</td></tr><tr><td>suo</td><td>49</td><td>43</td><td>42</td><td>39</td><td>38</td><td>38</td><td>39</td></tr><tr><td>sve</td><td>38</td><td>39</td><td>31</td><td>26</td><td>26</td><td>26</td><td>26</td></tr><tr><td>svk</td><td></td><td></td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>svn</td><td></td><td></td><td>33</td><td>38</td><td>36</td><td>38</td><td>38</td></tr><tr><td>EU</td><td>2527</td><td>1981</td><td>2446</td><td>2672</td><td>3738</td><td>3688</td><td>3700</td></tr></tbody></table>
$\hat{b}_i$ is not statistically significant would be to re-define the lower bound of the support interval as $\hat{b}_i$ - s.e. ($\hat{b}_i$). However, if this still results in negative interval values, or if $\hat{b}_i$ is negative, then the lower bound should be set to zero.

On the other hand, the support interval for $v_{ni}$ can be defined with respect to the residuals of the initial regression, $e_n$. More precisely, $e_n$ needs to be distributed among the various activities of the same farm. We will therefore assume that $-|e_n| \le v_{ni} \le |e_n|$. This interval is large enough to allow for an optimal solution to the entropy model and prevent any infeasibility problems.

For the formulation of the GME problem we will denote the unknown probability distribution corresponding to $\alpha_{ni}$ by $h_{ni} = [h_{ni1}, h_{ni2}, \dots, h_{nIK}]$ and the distribution for every $v_{ni}$ term by $w_{ni} = [w_{ni1}, w_{ni2}, \dots, w_{nIK}]$.

Similarly, the support vectors for $\alpha_{ni}$ will be denoted by $z_{ni} = [z_{ni1}, z_{ni2}, \dots, z_{nIK}]$ and for $v_{ni}$ by $v_{ni} = [v_{ni1}, v_{ni2}, \dots, v_{nIK}]$. Since entropy is a concave function of the non-negative probability vectors $h_i$ and $w_i$, estimating the allocation of any type of input cost across every activity $i$ can be expressed in the form of the following optimization program which has a unique interior solution:

$$ 
\begin{array}{ll}
\max_{h_{nik}, w_{nik} \ge 0} & H = - \sum_{n,i,k} h_{nik} \ln(h_{nik}) - \sum_{n,i,k} w_{nik} \ln(w_{nik}) \\
\text{s.t.} & y_n = \sum_i \left[ x_{ni} \left( \sum_k z_{nik} h_{nik} + \sum_k v_{nik} w_{nik} \right) \right] \\
& \sum_k h_{nik} = 1, \quad \sum_k w_{nik} = 1 \\
& \forall n \in N, \quad \forall i \in I, \quad \forall k \in K
\end{array}
$$

Assuming that $h_{nik}^*$ and $w_{nik}^*$ are the solutions to the maximum entropy problem, the unknown parameters are calculated as:

$$ \alpha_{ni} = \sum_k z_{nik} h_{nik}^* \quad \text{and} \quad v_{ni} = \sum_k v_{nik} w_{nik}^* $$

Note that we propose an initial regression exercise that uses all observations (farms) in each EU country surveyed by FADN. An objection to this choice would be based on the expected heterogeneity in activity costs among the various regions due to different technologies which are adapted to the local physical conditions. Although this heterogeneity can possibly be captured with the use of dummy variables, the number of additional variables introduced in the model (No. of activities × No. of regions) would make this procedure too cumbersome. Another solution would be to perform regressions for every region, but at the cost of less observations for each activity-cost combination and the increased risk of not statistically significant coefficients.

### 6.1.2 Implementation details for cost estimating

#### 6.1.2.1 Agricultural activities and cost data
The agricultural and animal production activities in AROPAj constitute only a subset of the activities that appear in FADN. A detailed presentation of all activities considered in the cost estimation exercise, along with the corresponding FADN variable codes is given in Table 23. Note that FADN includes several sub-categories for the various animal types, for example, cattle by age or by production orientation. Although AROPAj also uses various livestock categories in order to better represent feeding requirements in its constraint
<table><caption>**Table 21.** Utilized agricultural area represented per EU country throughout the different AROPAj versions (thousands of hectares).</caption><thead><tr><th>country<br>(FADN)</th><th>V1<br>(1997)</th><th>V2<br>(2002)</th><th>V3<br>(2004)</th><th>V4<br>(2009)</th><th>V5<br>(2007)</th><th>V5<br>(2008)</th><th>V5<br>(2009)</th></tr></thead><tbody><tr><td>bel</td><td>1355</td><td>1432</td><td>1339</td><td>1393</td><td>1333</td><td>1380</td><td>1411</td></tr><tr><td>bgr</td><td></td><td></td><td></td><td>2797</td><td>2752</td><td>2781</td><td>2985</td></tr><tr><td>cyp</td><td></td><td></td><td>137</td><td>110</td><td>107</td><td>125</td><td>112</td></tr><tr><td>cze</td><td></td><td></td><td>3430</td><td>2987</td><td>3277</td><td>3153</td><td>3222</td></tr><tr><td>dan</td><td>2349</td><td>2437</td><td>2397</td><td>2489</td><td>2486</td><td>2521</td><td>2590</td></tr><tr><td>deu</td><td>14736</td><td>14728</td><td>15414</td><td>15145</td><td>14932</td><td>15306</td><td>15411</td></tr><tr><td>ell</td><td>1418</td><td>1220</td><td>1651</td><td>2053</td><td>2357</td><td>2443</td><td>2423</td></tr><tr><td>esp</td><td>11816</td><td>11329</td><td>10939</td><td>12903</td><td>13566</td><td>13008</td><td>13253</td></tr><tr><td>est</td><td></td><td></td><td>669</td><td>2249</td><td>837</td><td>869</td><td>886</td></tr><tr><td>fra</td><td>22972</td><td>24042</td><td>24321</td><td>24778</td><td>25065</td><td>25208</td><td>25118</td></tr><tr><td>gbr</td><td>11360</td><td>11647</td><td>10337</td><td>10290</td><td>10327</td><td>10322</td><td>10606</td></tr><tr><td>hun</td><td></td><td></td><td>3477</td><td>3656</td><td>3724</td><td>3727</td><td>3737</td></tr><tr><td>irl</td><td>4014</td><td>3198</td><td>3087</td><td>3003</td><td>3992</td><td>4072</td><td>4195</td></tr><tr><td>ita</td><td>8295</td><td>7552</td><td>8125</td><td>7145</td><td>9109</td><td>7848</td><td>7982</td></tr><tr><td>ltu</td><td></td><td></td><td>1470</td><td>1813</td><td>1907</td><td>1889</td><td>1869</td></tr><tr><td>lux</td><td>104</td><td>130</td><td>124</td><td>125</td><td>125</td><td>125</td><td>127</td></tr><tr><td>lva</td><td></td><td></td><td>1000</td><td>1159</td><td>1142</td><td>1157</td><td>1178</td></tr><tr><td>mlt</td><td></td><td></td><td></td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>ned</td><td>1867</td><td>1511</td><td>1234</td><td>1648</td><td>1716</td><td>1728</td><td>1718</td></tr><tr><td>ost</td><td>1882</td><td>1788</td><td>1853</td><td>1771</td><td>1780</td><td>1807</td><td>1791</td></tr><tr><td>pol</td><td></td><td></td><td>7706</td><td>9823</td><td>12594</td><td>13254</td><td>13374</td></tr><tr><td>por</td><td>2253</td><td>1990</td><td>1716</td><td>1971</td><td>2032</td><td>2267</td><td>2124</td></tr><tr><td>rou</td><td></td><td></td><td></td><td>4527</td><td>7917</td><td>10100</td><td>10182</td></tr><tr><td>suo</td><td>1765</td><td>1889</td><td>1978</td><td>2046</td><td>2024</td><td>2084</td><td>2110</td></tr><tr><td>sve</td><td>3228</td><td>2639</td><td>2560</td><td>2406</td><td>2482</td><td>2472</td><td>2496</td></tr><tr><td>svk</td><td></td><td></td><td>1761</td><td>1724</td><td>1925</td><td>1899</td><td>1854</td></tr><tr><td>svn</td><td></td><td></td><td>430</td><td>434</td><td>437</td><td>434</td><td>438</td></tr><tr><td>EU</td><td>89415</td><td>87532</td><td>107153</td><td>120448</td><td>129943</td><td>131980</td><td>133193</td></tr></tbody></table>
set, for the purpose of cost estimation we have grouped all animal activities.

A special note should be made for irrigated agricultural activities. More precisely, FADN includes data on irrigated agricultural land only for the main crop types (FADN variables K1XX), whereas all K3XX variables are actually sub-categories of the K1XX ones. For example, cotton and fibre flax belong to the general category “Other industrial crops—K135”. Although FADN reports total land for K135 and the part of it which is irrigated (K135AA and K135IRAA respectively according to FADN nomenclature, with K135AA ≥ K135IRAA), it only reports total land for cotton and fibre flax. In order to estimate the irrigated areas for these two crops, we assumed that the irrigated shares of a general category is the same as the crops that comprise it. Hence, if K135IRAA/K135AA = λ (with λ ≤ 1), then λ is also the irrigated share for both cotton and fibre flax. Finally, for completeness we have created a “virtual” activity (a26 and r26) that includes permanent crops, vines, orchards and nurseries which are not represented in AROPAj but nevertheless affect farm expenses.

<table><caption>**Table 23.** Activities considered and related FADN variable codes.</caption><thead><tr><th rowspan="2">Activities</th><th rowspan="2">FADN variable</th><th colspan="2">Notation</th></tr><tr><th>Generic</th><th>Irrigated</th></tr></thead><tbody><tr><td><strong>Agricultural Activities</strong></td><td></td><td></td><td></td></tr><tr><td>Common wheat and spelt</td><td>K120</td><td>a1</td><td>r1</td></tr><tr><td>Durum wheat</td><td>K121</td><td>a2</td><td>r2</td></tr><tr><td>Rye</td><td>K122</td><td>a3</td><td></td></tr><tr><td>Barley</td><td>K123</td><td>a4</td><td>r4</td></tr><tr><td>Oats</td><td>K124</td><td>a5</td><td></td></tr><tr><td>Grain maize</td><td>K126</td><td>a6</td><td>r6</td></tr><tr><td>Rice</td><td>K127</td><td>a7</td><td>r7</td></tr><tr><td>Other cereals</td><td>K125+K128</td><td>a8</td><td>r8</td></tr><tr><td>Dry pulses</td><td>K129</td><td>a9</td><td></td></tr><tr><td>Potatoes</td><td>K130</td><td>a10</td><td>r10</td></tr><tr><td>Sugar beet</td><td>K131</td><td>a11</td><td>r11</td></tr><tr><td>Oilseed rape</td><td>K331*</td><td>a12</td><td>r12</td></tr><tr><td>Sunflower</td><td>K332*</td><td>a13</td><td>r13</td></tr><tr><td>Soya</td><td>K333*</td><td>a14</td><td>r14</td></tr><tr><td>Tobacco</td><td>K134</td><td>a15</td><td>r15</td></tr><tr><td>Cotton</td><td>K347*</td><td>a16</td><td>r16</td></tr><tr><td>Fibre flax</td><td>K373*</td><td>a17</td><td>r17</td></tr><tr><td>Fodder roots and brassicas</td><td>K144</td><td>a18</td><td></td></tr><tr><td>Fodder maize</td><td>K326*</td><td>a19</td><td>r19</td></tr><tr><td>Other fodder plants</td><td>K327*</td><td>a20</td><td>r20</td></tr><tr><td>Other silage cereals</td><td>K328*</td><td>a21</td><td>r21</td></tr><tr><td>Fallow and set aside</td><td>K146</td><td>a22</td><td></td></tr><tr><td>Temporary grass</td><td>K147</td><td>a23</td><td></td></tr><tr><td>Permanent pasture</td><td>K150</td><td>a24</td><td></td></tr><tr><td>Open ground field scale vegetables</td><td>K136</td><td>a25</td><td>r25</td></tr><tr><td>Sum of activities not in AROPAj</td><td>**</td><td>a26</td><td>r26</td></tr><tr><td><strong>Animal Activities</strong></td><td></td><td></td><td></td></tr><tr><td>Cows</td><td>D23LU+...+D32LU</td><td>a27</td><td></td></tr><tr><td>Sheep and goats</td><td>D38LU+...+D41LU</td><td>a28</td><td></td></tr><tr><td>Pigs</td><td>D43LU+...+D46LU</td><td>a29</td><td></td></tr><tr><td>Chicken and other poultry</td><td>D47LU+...+D49LU</td><td>a30</td><td></td></tr></tbody></table>

On the other hand, production costs reported in FADN include the value of all non-capital inputs used in crop and animal production, that is, the “consumption” of productive resources that correspond to farm output during the accounting year. The following are not part of production costs: (1) Costs that do not correspond to the accounting year production; (2) changes in stocks

<footnote>*See text for more details on the calculation of irrigated areas.</footnote>

<footnote>**See text for more details on the activities not in AROPAj.</footnote>
<table><caption>**Table 22.** Farm group number per EU country throughout the different years of the AROPAj-V5 version.</caption><thead><tr><th>country<br>(FADN)</th><th>V5<br>(2007)</th><th>V5<br>(2008)</th><th>V5<br>(2009)</th><th>V5<br>(2010)</th><th>V5<br>(2011)</th><th>V5*<br>(2012)</th><th>V5*<br>(2013)</th><th>V5*<br>(2014)</th><th>V5*<br>(2015)</th><th>V5*<br>(2016)</th><th>V5*<br>(2017)</th><th>V5*<br>(2018)</th></tr></thead><tbody><tr><td>bel</td><td>23</td><td>25</td><td>26</td><td>30</td><td>29</td><td>30</td><td>31</td><td>33</td><td>30</td><td>25</td><td>29</td><td>29</td></tr><tr><td>bgr</td><td>45</td><td>49</td><td>48</td><td>61</td><td>61</td><td>65</td><td>67</td><td>70</td><td>68</td><td>69</td><td>67</td><td>73</td></tr><tr><td>cyp</td><td>5</td><td>8</td><td>9</td><td>10</td><td>11</td><td>11</td><td>10</td><td>8</td><td>8</td><td>9</td><td>8</td><td>8</td></tr><tr><td>cze</td><td>42</td><td>44</td><td>45</td><td>44</td><td>47</td><td>47</td><td>43</td><td>44</td><td>44</td><td>45</td><td>41</td><td>41</td></tr><tr><td>dan</td><td>31</td><td>29</td><td>29</td><td>33</td><td>34</td><td>32</td><td>31</td><td>39</td><td>37</td><td>41</td><td>40</td><td>40</td></tr><tr><td>deu</td><td>180</td><td>195</td><td>215</td><td>229</td><td>215</td><td>218</td><td>224</td><td>223</td><td>215</td><td>214</td><td>215</td><td>217</td></tr><tr><td>ell</td><td>86</td><td>90</td><td>93</td><td>80</td><td>90</td><td>106</td><td>116</td><td>98</td><td>103</td><td>99</td><td>98</td><td>89</td></tr><tr><td>esp</td><td>190</td><td>178</td><td>182</td><td>185</td><td>182</td><td>200</td><td>193</td><td>193</td><td>191</td><td>199</td><td>189</td><td>191</td></tr><tr><td>est</td><td>16</td><td>18</td><td>17</td><td>22</td><td>22</td><td>23</td><td>18</td><td>20</td><td>21</td><td>20</td><td>22</td><td>19</td></tr><tr><td>fra</td><td>164</td><td>159</td><td>157</td><td>164</td><td>169</td><td>170</td><td>179</td><td>172</td><td>175</td><td>167</td><td>165</td><td>166</td></tr><tr><td>gbr</td><td>74</td><td>73</td><td>74</td><td>74</td><td>71</td><td>78</td><td>73</td><td>78</td><td>74</td><td>78</td><td>82</td><td>81</td></tr><tr><td>hrv</td><td></td><td></td><td></td><td></td><td></td><td></td><td>31</td><td>37</td><td>35</td><td>33</td><td>36</td><td>36</td></tr><tr><td>hun</td><td>54</td><td>53</td><td>57</td><td>53</td><td>54</td><td>48</td><td>48</td><td>49</td><td>48</td><td>50</td><td>48</td><td>45</td></tr><tr><td>irl</td><td>20</td><td>19</td><td>18</td><td>17</td><td>20</td><td>18</td><td>17</td><td>17</td><td>15</td><td>18</td><td>13</td><td>13</td></tr><tr><td>ita</td><td>323</td><td>249</td><td>233</td><td>239</td><td>238</td><td>256</td><td>243</td><td>225</td><td>200</td><td>206</td><td>228</td><td>235</td></tr><tr><td>ltu</td><td>30</td><td>28</td><td>30</td><td>30</td><td>30</td><td>31</td><td>26</td><td>30</td><td>27</td><td>29</td><td>30</td><td>31</td></tr><tr><td>lux</td><td>7</td><td>7</td><td>7</td><td>7</td><td>8</td><td>8</td><td>7</td><td>8</td><td>7</td><td>10</td><td>8</td><td>10</td></tr><tr><td>lva</td><td>31</td><td>31</td><td>24</td><td>25</td><td>27</td><td>26</td><td>26</td><td>24</td><td>26</td><td>25</td><td>26</td><td>26</td></tr><tr><td>mlt</td><td>9</td><td>11</td><td>11</td><td>11</td><td>10</td><td>11</td><td>10</td><td>11</td><td>10</td><td>11</td><td>11</td><td>10</td></tr><tr><td>ned</td><td>24</td><td>22</td><td>24</td><td>25</td><td>24</td><td>23</td><td>26</td><td>26</td><td>26</td><td>28</td><td>27</td><td>26</td></tr><tr><td>ost</td><td>43</td><td>43</td><td>41</td><td>41</td><td>43</td><td>45</td><td>41</td><td>44</td><td>43</td><td>40</td><td>39</td><td>41</td></tr><tr><td>pol</td><td>222</td><td>222</td><td>215</td><td>200</td><td>205</td><td>209</td><td>210</td><td>172</td><td>168</td><td>174</td><td>174</td><td>211</td></tr><tr><td>por</td><td>51</td><td>48</td><td>46</td><td>46</td><td>47</td><td>48</td><td>45</td><td>42</td><td>43</td><td>40</td><td>35</td><td>40</td></tr><tr><td>rou</td><td>38</td><td>55</td><td>102</td><td>178</td><td>171</td><td>180</td><td>189</td><td>116</td><td>142</td><td>180</td><td>181</td><td>157</td></tr><tr><td>suo</td><td>24</td><td>24</td><td>26</td><td>22</td><td>26</td><td>26</td><td>22</td><td>21</td><td>23</td><td>24</td><td>24</td><td>19</td></tr><tr><td>sve</td><td>25</td><td>28</td><td>30</td><td>30</td><td>30</td><td>30</td><td>32</td><td>31</td><td>32</td><td>34</td><td>30</td><td>31</td></tr><tr><td>svk</td><td>19</td><td>18</td><td>18</td><td>18</td><td>18</td><td>18</td><td>17</td><td>21</td><td>16</td><td>16</td><td>19</td><td>16</td></tr><tr><td>svn</td><td>22</td><td>21</td><td>25</td><td>27</td><td>28</td><td>36</td><td>31</td><td>29</td><td>25</td><td>26</td><td>31</td><td>27</td></tr></tbody><tfoot><tr><td>EU</td><td>1798</td><td>1747</td><td>1802</td><td>1901</td><td>1910</td><td>1993</td><td>2006</td><td>1881</td><td>1852</td><td>1910</td><td>1916</td><td>1928</td></tr></tfoot></table>

<footnote>* Including overseas regions</footnote>
of inputs, including costs accruing to growing crops; (3) cost of inputs used to increase, repair or replace the holding's fixed assets which are considered as "Investment"; (4) consumption of capital assets ("Depreciation") and/or acquisition of capital; (5) costs of production for private consumption, e.g. inputs for kitchen gardens; (6) expenditure on cost items compensated during the accounting year, e.g. tractor repairs covered by an insurance policy the same year; (7) charges on land, quotas and other rights, and buildings, but not short-term renting and other charges; (8) interests paid.

According to FADN nomenclature, production costs can be divided into four general categories, namely animal-specific costs, crop-specific costs, labour and machinery costs, and farming overheads. As illustrated in Table 24, we have aggregated specific FADN variables in order to produce 15 types of costs that need to be estimated per farm and activity.

<table><caption>Table 24. Production costs considered and related FADN variable codes.</caption><thead><tr><th>Cost Notation</th><th>FADN variable</th><th>Description</th></tr></thead><tbody><tr><td colspan="3"><strong>Animal-specific</strong></td></tr><tr><td rowspan="3">c1</td><td>F64</td><td>Concentrated feedingstuffs for grazing livestock (farm)</td></tr><tr><td>F65</td><td>Coarse fodder for grazing livestock (farm)</td></tr><tr><td>F68</td><td>Feedingstuffs for grazing livestock (market)</td></tr><tr><td rowspan="2">c2</td><td>F66</td><td>Feedingstuffs for pigs (farm)</td></tr><tr><td>F69</td><td>Feedingstuffs for pigs (market)</td></tr><tr><td rowspan="2">c3</td><td>F67</td><td>Feedingstuffs for poultry & other small animals (farm)</td></tr><tr><td>F70</td><td>Feedingstuffs for poultry & other small animals (market)</td></tr><tr><td>c4</td><td>F71</td><td>Other specific livestock costs (market)</td></tr><tr><td colspan="3"><strong>Labour and machinery</strong></td></tr><tr><td>c5</td><td>F59</td><td>Wages paid and social security costs</td></tr><tr><td>c6</td><td>F60</td><td>Contract work and machinery hire</td></tr><tr><td rowspan="2">c7</td><td>F61</td><td>Current upkeep of machinery and equipment</td></tr><tr><td>F63</td><td>Car expenses</td></tr><tr><td>c8</td><td>F62</td><td>Motor fuels and lubricants</td></tr><tr><td colspan="3"><strong>Specific crop costs</strong></td></tr><tr><td rowspan="2">c9</td><td>F72</td><td>Seeds and seedlings purchased</td></tr><tr><td>F73</td><td>Seeds and seedlings produced and used on the farm</td></tr><tr><td>c10</td><td>F74</td><td>Fertilisers and soil improvers</td></tr><tr><td>c11</td><td>F75</td><td>Crop protection products</td></tr><tr><td rowspan="2">c12</td><td>F76</td><td>Other specific crop costs</td></tr><tr><td>F77</td><td>Specific forestry costs</td></tr><tr><td colspan="3"><strong>Farming overheads</strong></td></tr><tr><td>c13</td><td>F79</td><td>Electricity</td></tr><tr><td>c14</td><td>F81</td><td>Water</td></tr><tr><td rowspan="4">c15</td><td>F78</td><td>Upkeep of land improvements and buildings</td></tr><tr><td>F80</td><td>Heating fuels</td></tr><tr><td>F82</td><td>Insurance</td></tr><tr><td>F83</td><td>Taxes and other dues</td></tr><tr><td>F84</td><td>Other farming overheads</td></tr></tbody></table>

**6.1.2.2 Regression analysis** As explained previously, the first step of the cost disaggregation procedure involves the estimation of an average cost per production activity ($\hat{b}_i$) by means of regression analysis. For this purpose, a different model was built for each cost category (15 in total), depending on the activities which were most likely to involve a certain type of cost. For
# 6 Parameter estimation

## 6.1 Variable costs

Key-points:

* Insights about new cost estimates

* Application of General Maximum Entropy

* Individual estimates

### 6.1.1 Estimation of activity variable costs

One of the main problems related to the use of the FADN database in farm programming models designed for policy analysis is the aggregated nature of cost data. This means that costs for various types of inputs are reported at the farm level but their distribution among the farm various activities is not available to the analyst.

Least Squares (LS) was historically the first method used for such cost disaggregation (Errington, 1989). However, LS may possibly return negative or non-significant cost coefficients that cannot be considered as valid estimates. Moreover, LS is suitable for estimating regional cost coefficients since multiple regression with cross-sectional data can only yield estimates of the average cost of input use for each activity across the sample: denoting activities by $i = 1, 2, ..., I$ and farms by $n = 1, 2, ..., N$, the estimating equation for the average cost per activity for any given input is written as $Y_n = \sum_i X_{ni}b_i + u_n$, where $Y_n$ is the aggregate cost paid by farm $n$ for the said input, $X_{ni}$ stands for the level of activity $i$ in the $n$-th farm, $b_i$ is the average cost of the input for activity $i$ and $u_n$ is the error term. However, the cost coefficients in the case of farm models should not correspond to average values but should instead reflect the differences in production costs among the same activities in different farms. Cost disaggregation for each farm in the sample therefore requires the estimation of different $\beta_{ni}$ coefficients so that the budget equation $Y_n = \sum_i X_{ni}\beta_{ni}$ is satisfied for any given input (i.e. $\beta_{ni} \neq b_i$). This means that for databases like FADN, the estimation of activity- and farm-specific costs constitutes an ill-posed problem with negative degrees of freedom because the number of the coefficients to be estimated ($I \times N$) is greater than the number of observations ($N$).

Since the problem of possible negative coefficients and ill-posedness cannot be handled by traditional econometric techniques, the literature suggests the use of information-theoretic statistical approaches like entropy and Bayesian methods that make use of prior information based on agro-nomic data, expert opinion, or other sources (e.g. Louhichi et al., 2012; Léon et al., 1999). Of similar reasoning to the above is the use of goal programming, as implemented by Kampas et al. (2012) for the estimation of activity costs in FADN farms in Thessaly, Greece, using information on costs obtained by a farm survey.

Although the use of exogenous information provides an answer to the problem of activity cost disaggregation, there still remains the problem of a source of homogeneous data at the EU level because available information, especially when it relates to agronomic practices, is usually limited to only a finite number of crops and geographical locales. In this section we present a method that is based on a two-step procedure which combines regression analysis with the Generalized
example, labour and machinery costs should correspond to all activities, whereas, animal specific costs were regressed only against the corresponding animal activities and crop-specific costs against agricultural activities. For c8 (motor fuels and lubricants), all agricultural areas were separated into irrigated and non-irrigated ones. The latter (which will henceforth denoted by 'd') were calculated for each farm by subtracting 'r' areas from their respective 'a' values: $dX = aX - rX$. The reason for this distinction is that irrigated activities may exhibit higher costs than non irrigated ones. Therefore in these regression models we used both r and d entries as independent variables, instead of an aggregate 'a' activity. The same rationale was employed for the c13 (electricity) and c14 (water) costs, since they were regressed against irrigated activities and animals only. For the remaining models we used the total agricultural areas as independent variables (a entries), since we assume that these costs per hectare should not differ between irrigated and non irrigated activities. Finally, agricultural activities that do not involve marketable outputs were not used in certain models. For example, activity a22 (fallow and set aside) was de-associated from cost types c9 (seeds and seedlings), c10 (fertilizers and soil improvers), and c11 (crop protection products). Table 25 presents the explanatory variables (activities) used in the estimation of the different cost types.

<table><caption>**Table 25.** Production costs considered and related FADN variable codes.</caption><thead><tr><th>Cost types</th><th>Agricultural activities</th><th>Animal activities</th></tr></thead><tbody><tr><td>c1</td><td></td><td>a27, a28</td></tr><tr><td>c2</td><td></td><td>a29</td></tr><tr><td>c3</td><td></td><td>a30</td></tr><tr><td>c4</td><td></td><td>a27, a28, a29, a30</td></tr><tr><td>c5</td><td>a1, ..., a26</td><td>a27, a28, a29, a30</td></tr><tr><td>c6</td><td>a1, ..., a26</td><td>a27, a28, a29, a30</td></tr><tr><td>c7</td><td>a1, ..., a26</td><td>a27, a28, a29, a30</td></tr><tr><td>c8</td><td>d1, ..., d26, r1, ..., r26</td><td>a27, a28, a29, a30</td></tr><tr><td>c9</td><td>a1, ..., a21, a23, ..., a26</td><td></td></tr><tr><td>c10</td><td>a1, ..., a21, a23, ..., a26</td><td></td></tr><tr><td>c11</td><td>a1, ..., a21, a25, a26</td><td></td></tr><tr><td>c12</td><td>a1, ..., a21, a25, a26</td><td></td></tr><tr><td>c13</td><td>r1, ..., r26</td><td>a27, a28, a29, a30</td></tr><tr><td>c14</td><td>r1, ..., r26</td><td>a27, a28, a29, a30</td></tr><tr><td>c15</td><td>a1, ..., a26</td><td>a27, a28, a29, a30</td></tr></tbody></table>

Since the goal of the initial regressions is to aid in the construction of the support intervals for the GME model and not to accurately estimate the average cost coefficients, for simplicity's sake at this stage, we chose to perform equation by equation OLS estimation with no constant term. The reason is that a constant term would also have to be distributed to different activities, like the residuals, in each farm in order for the budget constraint to hold. Future work can use WLS estimation in order to also account for the representation weights attributed to each farm in the FADN sample.

### 6.1.2.3 The maximum entropy program
The outputs from each model (cost type) were the $\hat{b}_i$ coefficients, the standard errors and the residuals for each farm-observation. These results were used as data inputs for the second step of the cost disaggregation procedure that involves the use of a GME model for each farm included in the FADN sample. One fundamental differentiation from the initial regressions, was that for the entropy model, we explicitly distinguished between irrigated
Maximum Entropy estimator (GME). Specifically, we remain within the entropy framework of for the disaggregation of costs, as suggested by the literature, but we draw prior information from an initial regression exercise that aims at providing an approximate estimate of variable costs per activity at the regional or higher geographical scales.

Applying the GME estimator in the linear model $Y_n = \sum_i x_{ni} \beta_{ni}$ involves finding an entropy maximizing probability distribution over a vector of support values for every $\beta_{ni}$ coefficient, so that each coefficient is re-defined as the expected value of its corresponding unknown probability distribution. The support vectors are specified in the feasible range of the unknown parameters and are defined a priori by the researcher. In this sense they constitute the most subjective aspect of the entropy estimation procedure. In fact, the selection of the support values gives rise to two issues. The first is the choice of the central value of the support interval and the second is the width of the interval, i.e., the width of the interval around the central value. Especially the latter can greatly affect the estimation procedure because the coefficients are not invariant to the chosen support intervals (Caputo & Paris, 2008); narrow bounds can greatly increase the mean square error of the GME estimator (Golan et al., 1997), or even render the entropy maximization program infeasible.

The first step for estimating $\beta_{ni}$ with GME is to construct an appropriate vector of support values. For this purpose, we estimate the equation $Y_n = \sum_i x_{ni}b_i + u_n$ in order to obtain the $\hat{b}_i$ coefficient and the residual $e_n$. The underlying hypothesis is that for a given cost type $t$, the coefficient $\beta_{ni}$ for activity $i$ in farm $n$ should be located in an interval around the average cost $\hat{b}_i$ and the interval must also contain part of the residuals $e_n$. We can therefore redefine the unknown parameter as $\beta_{ni} = \alpha_{ni} + v_{ni}$, where $\alpha_{ni}$ is a parameter derived from the distribution of $\hat{b}_i$, whereas $v_{ni}$ represents an error term that can be defined with respect to $e_n$. Although $\alpha_{ni}$ is directly related to the distribution of $\beta_{ni}$, and thus its meaning is straightforward, the use of an error term merits justification. There are two reasons that require the definition of $\beta_{ni}$ as the sum of two separate parameters, one of which is an “error” term. The first reason is the aggregation of explanatory variables (production activities) into general categories. For instance, some FADN activities are already aggregates of other individual activities which cannot be modelled separately for different technical or sampling reasons. Typical examples are the various age categories in livestock production, flowers, seed crops, fodder root crops and all subcategories indexed as “other…” (e.g., K128: other cereals and K330: other protein crops). The second reason that justifies the inclusion of error terms concerns the feasibility of the maximum entropy problem: as explained by Léon et al. (1999), the role of the disturbances is to “loosen” the model constraints so that the model returns an optimal solution. This implies that there should be a large enough error support interval in order to compensate for deviations between observed aggregate input costs and estimated cost allocations.

The goal is now to find support intervals for $\alpha_{ni}$ and $v_{ni}$. The support interval for $\alpha_{ni}$ can be constructed using the “3σ rule”, that is, three standard errors (s.e.) around the LS estimate: $\hat{b}_i - 3 \text{ s.e.} (\hat{b}_i) \le \alpha_{ni} \le \hat{b}_i + 3 \text{ s.e.} (\hat{b}_i)$. This method requires that (i) the $\hat{b}_i$ coefficient is positive and (ii) the resulting support interval does not contain any negative values. However, this may not always be possible with least square techniques because the analyst can neither impose non-negativity constraints on the estimation procedure nor ensure that the estimated coefficients are statistically significant, so that the interval strictly lies in $\mathbb{R}_+$.<fnref n="6" /> A simple ad-hoc solution when

<footnote n="6">⁶A higher variance will probably lead to a negative lower bound with the “3σ” rule.</footnote>
and non irrigated activities for all cost types. Since the explanatory variables for most regression models were total crop areas ('a'), some data manipulation was required in order to render the regression results consistent with the GME model. Hence we assumed that the $\hat{b}_i$ coefficients and the standard errors for all cost types were the same for both irrigated and non irrigated crops in the same farm, which finally lead to the same support intervals for the two crop variants. Additionally, as explained before, negative coefficients were set to zero, which meant that the support interval for $\beta_{ni}$ also had a lower bound of zero.

For the maximum entropy program, we introduced additional constraints for c8 (motor fuels and lubricants), c13 (electricity) and c14 (water) in order to reflect possible differences between the two crop variants if they are both recorded in the same farm. More precisely, c13 and c14 were not considered at all for non irrigated crops (as is also shown in Table 25), whereas c8 for irrigated crops were set to be equal or greater than non irrigated ones. Especially for c8 (motor fuels and lubricants), we assumed that any additional costs were the result of using a petrol-based irrigation system. Thus the existence of electricity expenses in the same farm meant that c8 costs were the equal between irrigated and non irrigated crops. Finally, farms with only one recorded activity were excluded from the GME model because the repartition of costs is straightforward. The model was solved within a “loop” command in GAMS which, since entropy was maximized independently for each farm, lead to a number of sub-models equal to the number of remaining farms. A matrix of farm-activity correspondences was created for each remaining farm based on FADN data. Hence each model maximized entropy for farms with more than one activity and for those costs that are actually recorded. This constraint was deemed necessary for feasibility reasons when solving the model with the GAMS programming language.

Since the main decision units in AROPAj are farm-types and not individual farms, the results of the entropy model, i.e., the input costs per hectare for each farm, were transformed into average values for every farm-type. Hence, we have implicitly assumed that all costs are the same for the irrigated and non irrigated variant of the same activity and are numerically equal to the average of costs in the examined farm-type. An exception was made for fuel costs (c8) which were distinguished between irrigated (c8ir) and non irrigated activities (c8nir) because irrigation with no electricity costs implies the use of oil-based water pumps. Therefore, when non zero electricity (c13) expenses were recorded for the irrigated activity then c8 was averaged out for both the irrigated and non irrigated activity variants so that c8ir = c8nir. In contrast, for c13 = 0, we required that c8ir > c8nir. Since the two activity variants at the farm-type level come from different individual farms, a numerical problem that arises is that the previous inequality may not always hold. In such case, and when c13 = 0, the estimated c8ir was assumed to be equal to c8nir, plus their difference:

$$ \hat{c}_{8ir} = c_{8nir} + (c_{8nir} - c_{8ir}) = 2c_{8nir} - c_{8ir} $$

## 6.2 Parameters estimating for farm groups

**Key-point:**

Estimating farm groups' parameters
$\hat{b}_i$ is not statistically significant would be to re-define the lower bound of the support interval as $\hat{b}_i$ - s.e. ($\hat{b}_i$). However, if this still results in negative interval values, or if $\hat{b}_i$ is negative, then the lower bound should be set to zero.

On the other hand, the support interval for $v_{ni}$ can be defined with respect to the residuals of the initial regression, $e_n$. More precisely, $e_n$ needs to be distributed among the various activities of the same farm. We will therefore assume that $-|e_n| \le v_{ni} \le |e_n|$. This interval is large enough to allow for an optimal solution to the entropy model and prevent any infeasibility problems.

For the formulation of the GME problem we will denote the unknown probability distribution corresponding to $\alpha_{ni}$ by $h_{ni} = [h_{ni1}, h_{ni2}, \dots, h_{niK}]$ and the distribution for every $v_{ni}$ term by $w_{ni} = [w_{ni1}, w_{ni2}, \dots, w_{niK}]$.

Similarly, the support vectors for $\alpha_{ni}$ will be denoted by $z_{ni} = [z_{ni1}, z_{ni2}, \dots, z_{niK}]$ and for $v_{ni}$ by $v_{ni} = [v_{ni1}, v_{ni2}, \dots, v_{niK}]$. Since entropy is a concave function of the non-negative probability vectors $h_i$ and $w_i$, estimating the allocation of any type of input cost across every activity $i$ can be expressed in the form of the following optimization program which has a unique interior solution:

$$
\begin{array}{ll}
\max_{h_{nik}, w_{nik} \ge 0} & H = - \sum_{n,i,k} h_{nik} \ln(h_{nik}) - \sum_{n,i,k} w_{nik} \ln(w_{nik}) \\
\text{s.t.} & y_n = \sum_i \left[ x_{ni} \left( \sum_k z_{nik} h_{nik} + \sum_k v_{nik} w_{nik} \right) \right] \\
& \sum_k h_{nik} = 1, \quad \sum_k w_{nik} = 1 \\
& \forall n \in N, \quad \forall i \in I, \quad \forall k \in K
\end{array}
$$

Assuming that $h_{nik}^*$ and $w_{nik}^*$ are the solutions to the maximum entropy problem, the unknown parameters are calculated as:

$$ \alpha_{ni} = \sum_{k} z_{nik} h_{nik}^{*} \quad \text{and} \quad \nu_{ni} = \sum_{k} v_{nik} w_{nik}^{*} $$

Note that we propose an initial regression exercise that uses all observations (farms) in each EU country surveyed by FADN. An objection to this choice would be based on the expected heterogeneity in activity costs among the various regions due to different technologies which are adapted to the local physical conditions. Although this heterogeneity can possibly be captured with the use of dummy variables, the number of additional variables introduced in the model (No. of activities × No. of regions) would make this procedure too cumbersome. Another solution would be to perform regressions for every region, but at the cost of less observations for each activity-cost combination and the increased risk of not statistically significant coefficients.

## 6.1.2 Implementation details for cost estimating

### 6.1.2.1 Agricultural activities and cost data
The agricultural and animal production activities in AROPAj constitute only a subset of the activities that appear in FADN. A detailed presentation of all activities considered in the cost estimation exercise, along with the corresponding FADN variable codes is given in Table 23. Note that FADN includes several sub-categories for the various animal types, for example, cattle by age or by production orientation. Although AROPAj also uses various livestock categories in order to better represent feeding requirements in its constraint
The majority of *AROPAj* parameters are provided by estimations from *FADN* data. The previous section describes the calculation process required to deal with the insufficiently detailed information per crop. After that, it is easy to estimate costs at the farm group level thanks to the provision of (estimated) individual costs for the entire *FADN* sample. We did that separately for fertilizer costs, other variables costs for all *AROPAj* versions, and for irrigation costs for the *V5* version.

Other parameters are provided with two methods. Some are calculated by the means of simple linear regression or estimated average. This is done through *R*-code programs which can be automatically managed per series of parameter categories for all countries. Parameter categories of this part refer to:

* crop yields

* areas (requested by the *AROPAj* calibrating process)

* herds

* on-farm re-used crop products (useful for improving the *AROPAj* calibrating process)

* variable costs (in case of non-operating *GME* estimator)

* farm numbers per crop or per crop category

* ratios between crops or fodders, and categories of crops or fodders

* milk productions and quotas, sugar beet quotas

<figure><img src="image_9.png" /><figcaption>**Figure 8.** Fertilizer cost (1000€/ha)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 9.** Yield (t/ha)</figcaption></figure>

Distribution of fertilizer costs (left) and yields (right), for soft wheat, among the EU *AROPAj* farm groups.

From expert sources, we gather other important information such as:

* IPCC guidelines sourced recommendation

* energy and protein compounds of concentrates and on-farm sourced feed-stuff

* animal requirements in energy, proteins and dry matter, and limits or thresholds regarding animal feeding
set, for the purpose of cost estimation we have grouped all animal activities.

A special note should be made for irrigated agricultural activities. More precisely, FADN includes data on irrigated agricultural land only for the main crop types (FADN variables K1XX), whereas all K3XX variables are actually sub-categories of the K1XX ones. For example, cotton and fibre flax belong to the general category “Other industrial crops—K135”. Although FADN reports total land for K135 and the part of it which is irrigated (K135AA and K135IRAA respectively according to FADN nomenclature, with K135AA ≥ K135IRAA), it only reports total land for cotton and fibre flax. In order to estimate the irrigated areas for these two crops, we assumed that the irrigated shares of a general category is the same as the crops that comprise it. Hence, if K135IRAA/K135AA = λ (with λ ≤ 1), then λ is also the irrigated share for both cotton and fibre flax. Finally, for completeness we have created a “virtual” activity (a26 and r26) that includes permanent crops, vines, orchards and nurseries which are not represented in AROPAj but nevertheless affect farm expenses.

<table><caption>**Table 23.** Activities considered and related FADN variable codes.</caption><thead><tr><th rowspan="2">Activities</th><th rowspan="2">FADN variable</th><th colspan="2">Notation</th></tr><tr><th>Generic</th><th>Irrigated</th></tr></thead><tbody><tr><td colspan="4"><strong>Agricultural Activities</strong></td></tr><tr><td>Common wheat and spelt</td><td>K120</td><td>a1</td><td>r1</td></tr><tr><td>Durum wheat</td><td>K121</td><td>a2</td><td>r2</td></tr><tr><td>Rye</td><td>K122</td><td>a3</td><td></td></tr><tr><td>Barley</td><td>K123</td><td>a4</td><td>r4</td></tr><tr><td>Oats</td><td>K124</td><td>a5</td><td></td></tr><tr><td>Grain maize</td><td>K126</td><td>a6</td><td>r6</td></tr><tr><td>Rice</td><td>K127</td><td>a7</td><td>r7</td></tr><tr><td>Other cereals</td><td>K125+K128</td><td>a8</td><td>r8</td></tr><tr><td>Dry pulses</td><td>K129</td><td>a9</td><td></td></tr><tr><td>Potatoes</td><td>K130</td><td>a10</td><td>r10</td></tr><tr><td>Sugar beet</td><td>K131</td><td>a11</td><td>r11</td></tr><tr><td>Oilseed rape</td><td>K331*</td><td>a12</td><td>r12</td></tr><tr><td>Sunflower</td><td>K332*</td><td>a13</td><td>r13</td></tr><tr><td>Soya</td><td>K333*</td><td>a14</td><td>r14</td></tr><tr><td>Tobacco</td><td>K134</td><td>a15</td><td>r15</td></tr><tr><td>Cotton</td><td>K347*</td><td>a16</td><td>r16</td></tr><tr><td>Fibre flax</td><td>K373*</td><td>a17</td><td>r17</td></tr><tr><td>Fodder roots and brassicas</td><td>K144</td><td>a18</td><td></td></tr><tr><td>Fodder maize</td><td>K326*</td><td>a19</td><td>r19</td></tr><tr><td>Other fodder plants</td><td>K327*</td><td>a20</td><td>r20</td></tr><tr><td>Other silage cereals</td><td>K328*</td><td>a21</td><td>r21</td></tr><tr><td>Fallow and set aside</td><td>K146</td><td>a22</td><td></td></tr><tr><td>Temporary grass</td><td>K147</td><td>a23</td><td></td></tr><tr><td>Permanent pasture</td><td>K150</td><td>a24</td><td></td></tr><tr><td>Open ground field scale vegetables</td><td>K136</td><td>a25</td><td>r25</td></tr><tr><td>Sum of activities not in AROPAj</td><td>**</td><td>a26</td><td>r26</td></tr><tr><td colspan="4"><strong>Animal Activities</strong></td></tr><tr><td>Cows</td><td>D23LU+...+D32LU</td><td>a27</td><td></td></tr><tr><td>Sheep and goats</td><td>D38LU+...+D41LU</td><td>a28</td><td></td></tr><tr><td>Pigs</td><td>D43LU+...+D46LU</td><td>a29</td><td></td></tr><tr><td>Chicken and other poultry</td><td>D47LU+...+D49LU</td><td>a30</td><td></td></tr></tbody></table>

On the other hand, production costs reported in FADN include the value of all non-capital inputs used in crop and animal production, that is, the “consumption” of productive resources that correspond to farm output during the accounting year. The following are not part of production costs: (1) Costs that do not correspond to the accounting year production; (2) changes in stocks

<footnote>*See text for more details on the calculation of irrigated areas.</footnote>

<footnote>**See text for more details on the activities not in AROPAj.</footnote>
* by default characteristics of fertilizers

* by default characteristics of manure

* by default value of ratios within the bovine herd

* CAP tool values linked to (more or less) stylized CAP tool implementation

Finally, in some cases, parameter values can be provided by the *AROPAj* model itself, when
outputs become parameters. This is the case for the following parameters:

* baseline GHG emissions (to be quotas)

* baseline cereals, seed and protein crops (to be CAP supported areas)

* shadow prices (to replace market prices in the case of endogenous computation of yields)

Figures 8 and 9 illustrate the parameter heterogeneity in case of fertilizer costs and yields related to soft wheat, present in 1642 among the 1815 farm groups related to the V5 version and the FADN-2009. The figures show a bi-modal distribution.
of inputs, including costs accruing to growing crops; (3) cost of inputs used to increase, repair or replace the holding's fixed assets which are considered as "Investment"; (4) consumption of capital assets ("Depreciation") and/or acquisition of capital; (5) costs of production for private consumption, e.g. inputs for kitchen gardens; (6) expenditure on cost items compensated during the accounting year, e.g. tractor repairs covered by an insurance policy the same year; (7) charges on land, quotas and other rights, and buildings, but not short-term renting and other charges; (8) interests paid.

According to FADN nomenclature, production costs can be divided into four general categories, namely animal-specific costs, crop-specific costs, labour and machinery costs, and farming overheads. As illustrated in Table 24, we have aggregated specific FADN variables in order to produce 15 types of costs that need to be estimated per farm and activity.

<table><caption>Table 24. Production costs considered and related FADN variable codes.</caption><thead><tr><th>Cost Notation</th><th>FADN variable</th><th>Description</th></tr></thead><tbody><tr><td colspan="3"><strong>Animal-specific</strong></td></tr><tr><td rowspan="3">c1</td><td>F64</td><td>Concentrated feedingstuffs for grazing livestock (farm)</td></tr><tr><td>F65</td><td>Coarse fodder for grazing livestock (farm)</td></tr><tr><td>F68</td><td>Feedingstuffs for grazing livestock (market)</td></tr><tr><td rowspan="2">c2</td><td>F66</td><td>Feedingstuffs for pigs (farm)</td></tr><tr><td>F69</td><td>Feedingstuffs for pigs (market)</td></tr><tr><td rowspan="2">c3</td><td>F67</td><td>Feedingstuffs for poultry & other small animals (farm)</td></tr><tr><td>F70</td><td>Feedingstuffs for poultry & other small animals (market)</td></tr><tr><td>c4</td><td>F71</td><td>Other specific livestock costs (market)</td></tr><tr><td colspan="3"><strong>Labour and machinery</strong></td></tr><tr><td>c5</td><td>F59</td><td>Wages paid and social security costs</td></tr><tr><td>c6</td><td>F60</td><td>Contract work and machinery hire</td></tr><tr><td rowspan="2">c7</td><td>F61</td><td>Current upkeep of machinery and equipment</td></tr><tr><td>F63</td><td>Car expenses</td></tr><tr><td>c8</td><td>F62</td><td>Motor fuels and lubricants</td></tr><tr><td colspan="3"><strong>Specific crop costs</strong></td></tr><tr><td rowspan="2">c9</td><td>F72</td><td>Seeds and seedlings purchased</td></tr><tr><td>F73</td><td>Seeds and seedlings produced and used on the farm</td></tr><tr><td>c10</td><td>F74</td><td>Fertilisers and soil improvers</td></tr><tr><td>c11</td><td>F75</td><td>Crop protection products</td></tr><tr><td rowspan="2">c12</td><td>F76</td><td>Other specific crop costs</td></tr><tr><td>F77</td><td>Specific forestry costs</td></tr><tr><td colspan="3"><strong>Farming overheads</strong></td></tr><tr><td>c13</td><td>F79</td><td>Electricity</td></tr><tr><td>c14</td><td>F81</td><td>Water</td></tr><tr><td rowspan="5">c15</td><td>F78</td><td>Upkeep of land improvements and buildings</td></tr><tr><td>F80</td><td>Heating fuels</td></tr><tr><td>F82</td><td>Insurance</td></tr><tr><td>F83</td><td>Taxes and other dues</td></tr><tr><td>F84</td><td>Other farming overheads</td></tr></tbody></table>

### 6.1.2.2 Regression analysis
As explained previously, the first step of the cost disaggregation procedure involves the estimation of an average cost per production activity ($\hat{b}_i$) by means of regression analysis. For this purpose, a different model was built for each cost category (15 in total), depending on the activities which were most likely to involve a certain type of cost. For
# 7 Calibration of the *AROPAj* model

**Key-points:**

* Finding a *LP* solution at the farm group level

* Improving all time-reference *LPs* through parameter re-estimates

* Applying the algorithm, and running time

The first problem we face after prior estimation of parameters is to obtain a *LP* solution. To do so, we check parameters' consistency with regards to a series of potential problems due to missing information. For instance, at the farm group level, the *FADN* may provide milk quota and milk production but no milk price; it may provide crop area but no crop production neither crop price. This poses a problem even when a *LP* solution exists, and even more when there is no *LP* solution. A series of “expert” checking points are processed in the earlier steps of the *AROPAj* building. They are based on fulfilling farm group data sets with nested calculations: given a parameter (e.g. a yield, a price), in case of missing farm group information, we use regional average values; in case of missing region information, we use country-level average values; in case of missing country information, we retain the default value. In practice, default values were used in very few cases, related to bovine activities (uncomplete livestock categories) in small countries (Cyprus).

The second problem arises when we compare computed optimal activities and actual activities, once we have a *LP* solution derived from prior estimates of parameters. The *FADN* provides a wide range of parameters, some of them being related to *LP* command variables, and, at the same time, the *FADN* has to be supplemented by exogenous technical information (e.g. energy and protein contents of feed intakes, feed requirements, fertilizer characteristics, livestock life cycle, agronomic cropping limits in terms of land allocation).

The calibration process consists of re-estimating a subset composed of previously estimated parameters (crop ratios) and of non-*FADN* sourced parameters (feed requirements, livestock life cycle). In practice, more than one hundred parameters per farm group are re-estimated in this way. The calibration criterion is a sum of weighted squared deviations, each of them being the difference between *FADN* observation and the associated optimal *LP* variable. The set of variables is composed of crop and fodder areas (hectares), livestock categories (head units) and feed intakes (metric tons). Table 26 lists the variables which contribute to the calibration criterion and fully describes the criterion calculation.

Figure 10 illustrates the geometric interpretation of the calibration phase, which can be viewed as a rotational and translational movement of constraints in the primal variable space.

The calibration algorithm is based on sequential calculations, combining Monte-Carlo and gradient methods. In practice, randomization of calibration parameters alternating with “local” gradient based improvement of the criterion is generated from a large number of *LP* runs for each of *AROPAj* farm groups. All operating *AROPAj* versions are based on more than one thousand farm groups. For one farm group, the number of runs ranges from 1000 to 1500 (*AROPAj* versions *V2* and *V3*) or to 2000 (versions *V4* and *V5*). The calibration process is stopped either when the run number limit is reached (i.e. pre-defined number for Monte-Carlo and gradient phases), or when additional change of parameters' value does not improve the criterion during a phase.
example, labour and machinery costs should correspond to all activities, whereas, animal specific costs were regressed only against the corresponding animal activities and crop-specific costs against agricultural activities. For c8 (motor fuels and lubricants), all agricultural areas were separated into irrigated and non-irrigated ones. The latter (which will henceforth denoted by 'd') were calculated for each farm by subtracting 'r' areas from their respective 'a' values: $dX = aX - rX$. The reason for this distinction is that irrigated activities may exhibit higher costs than non irrigated ones. Therefore in these regression models we used both r and d entries as independent variables, instead of an aggregate 'a' activity. The same rationale was employed for the c13 (electricity) and c14 (water) costs, since they were regressed against irrigated activities and animals only. For the remaining models we used the total agricultural areas as independent variables (a entries), since we assume that these costs per hectare should not differ between irrigated and non irrigated activities. Finally, agricultural activities that do not involve marketable outputs were not used in certain models. For example, activity a22 (fallow and set aside) was de-associated from cost types c9 (seeds and seedlings), c10 (fertilizers and soil improvers), and c11 (crop protection products). Table 25 presents the explanatory variables (activities) used in the estimation of the different cost types.

<table><caption>**Table 25.** Production costs considered and related FADN variable codes.</caption><thead><tr><th>Cost types</th><th>Agricultural activities</th><th>Animal activities</th></tr></thead><tbody><tr><td>c1</td><td></td><td>a27, a28</td></tr><tr><td>c2</td><td></td><td>a29</td></tr><tr><td>c3</td><td></td><td>a30</td></tr><tr><td>c4</td><td></td><td>a27, a28, a29, a30</td></tr><tr><td>c5</td><td>a1, ..., a26</td><td>a27, a28, a29, a30</td></tr><tr><td>c6</td><td>a1, ..., a26</td><td>a27, a28, a29, a30</td></tr><tr><td>c7</td><td>a1, ..., a26</td><td>a27, a28, a29, a30</td></tr><tr><td>c8</td><td>d1, ..., d26, r1, ..., r26</td><td>a27, a28, a29, a30</td></tr><tr><td>c9</td><td>a1, ..., a21, a23, ..., a26</td><td></td></tr><tr><td>c10</td><td>a1, ..., a21, a23, ..., a26</td><td></td></tr><tr><td>c11</td><td>a1, ..., a21, a25, a26</td><td></td></tr><tr><td>c12</td><td>a1, ..., a21, a25, a26</td><td></td></tr><tr><td>c13</td><td>r1, ..., r26</td><td>a27, a28, a29, a30</td></tr><tr><td>c14</td><td>r1, ..., r26</td><td>a27, a28, a29, a30</td></tr><tr><td>c15</td><td>a1, ..., a26</td><td>a27, a28, a29, a30</td></tr></tbody></table>

Since the goal of the initial regressions is to aid in the construction of the support intervals for the GME model and not to accurately estimate the average cost coefficients, for simplicity's sake at this stage, we chose to perform equation by equation OLS estimation with no constant term. The reason is that a constant term would also have to be distributed to different activities, like the residuals, in each farm in order for the budget constraint to hold. Future work can use WLS estimation in order to also account for the representation weights attributed to each farm in the FADN sample.

### 6.1.2.3 The maximum entropy program
The outputs from each model (cost type) were the $\hat{b}_i$ coefficients, the standard errors and the residuals for each farm-observation. These results were used as data inputs for the second step of the cost disaggregation procedure that involves the use of a GME model for each farm included in the FADN sample. One fundamental differentiation from the initial regressions, was that for the entropy model, we explicitly distinguished between irrigated
<figure><img src="image_11.png" /><figcaption>**Figure 10.** Geometric overview of the calibrating process on a two-dimension subspace: the target fixed point is marked by the black circle, the *LP* point maximizing the gross margin *p*·*x* is marked by the grey circle, the feasible solution has to belong to the colored area bounded by the *LP* constraints. Calibrating consists on rotational and translational movements aiming to bring the two points closer.</figcaption></figure>
and non irrigated activities for all cost types. Since the explanatory variables for most regression models were total crop areas ('a'), some data manipulation was required in order to render the regression results consistent with the GME model. Hence we assumed that the $\hat{b}_i$ coefficients and the standard errors for all cost types were the same for both irrigated and non irrigated crops in the same farm, which finally lead to the same support intervals for the two crop variants. Additionally, as explained before, negative coefficients were set to zero, which meant that the support interval for $\beta_{ni}$ also had a lower bound of zero.

For the maximum entropy program, we introduced additional constraints for c8 (motor fuels and lubricants), c13 (electricity) and c14 (water) in order to reflect possible differences between the two crop variants if they are both recorded in the same farm. More precisely, c13 and c14 were not considered at all for non irrigated crops (as is also shown in Table 25), whereas c8 for irrigated crops were set to be equal or greater than non irrigated ones. Especially for c8 (motor fuels and lubricants), we assumed that any additional costs were the result of using a petrol-based irrigation system. Thus the existence of electricity expenses in the same farm meant that c8 costs were the equal between irrigated and non irrigated crops. Finally, farms with only one recorded activity were excluded from the GME model because the repartition of costs is straightforward. The model was solved within a “loop” command in GAMS which, since entropy was maximized independently for each farm, lead to a number of sub-models equal to the number of remaining farms. A matrix of farm-activity correspondences was created for each remaining farm based on FADN data. Hence each model maximized entropy for farms with more than one activity and for those costs that are actually recorded. This constraint was deemed necessary for feasibility reasons when solving the model with the GAMS programming language.

Since the main decision units in AROPAj are farm-types and not individual farms, the results of the entropy model, i.e., the input costs per hectare for each farm, were transformed into average values for every farm-type. Hence, we have implicitly assumed that all costs are the same for the irrigated and non irrigated variant of the same activity and are numerically equal to the average of costs in the examined farm-type. An exception was made for fuel costs (c8) which were distinguished between irrigated (c8ir) and non irrigated activities (c8nir) because irrigation with no electricity costs implies the use of oil-based water pumps. Therefore, when non zero electricity (c13) expenses were recorded for the irrigated activity then c8 was averaged out for both the irrigated and non irrigated activity variants so that c8ir = c8nir. In contrast, for c13 = 0, we required that c8ir > c8nir. Since the two activity variants at the farm-type level come from different individual farms, a numerical problem that arises is that the previous inequality may not always hold. In such case, and when c13 = 0, the estimated c8ir was assumed to be equal to c8nir, plus their difference:

$$ \hat{c}_{8ir} = c_{8nir} + (c_{8nir} - c_{8ir}) = 2c_{8nir} - c_{8ir} $$

## 6.2 Parameters estimating for farm groups

**Key-point:**

Estimating farm groups' parameters
<table><caption>**Table 26.** Detailed expression of the calibrating criterion to be maximized *c* ($\delta$ refers to any additional deviation, *w* is 1 by default); each deviation enters the criterion such as $c \to c - 1/w \delta^2$.</caption><thead><tr><th>variable</th><th>additional weight 1/w</th><th>(&delta; unit)</th></tr></thead><tbody><tr><td>each of 8 cereals</td><td></td><td>(ha)</td></tr><tr><td>cereals together</td><td></td><td>(ha)</td></tr><tr><td>each of 3 oilseeds</td><td></td><td>(ha)</td></tr><tr><td>each of 2 proteins crops</td><td></td><td>(ha)</td></tr><tr><td>sugarbeet</td><td></td><td>(ha)</td></tr><tr><td>potatoes</td><td></td><td>(ha)</td></tr><tr><td>each of other crops</td><td></td><td>(ha)</td></tr><tr><td>fallow</td><td></td><td>(ha)</td></tr><tr><td>CAP set aside</td><td></td><td>(ha)</td></tr><tr><td>each of fodders</td><td></td><td>(ha)</td></tr><tr><td>grasslands</td><td></td><td>(ha)</td></tr><tr><td>fallow & set-aside</td><td></td><td>(ha)</td></tr><tr><td>bovine less than 1 year</td><td>16</td><td>(Lh)</td></tr><tr><td>one-year bovine male</td><td>4</td><td>(Lh)</td></tr><tr><td>one-year bovine female</td><td>4</td><td>(Lh)</td></tr><tr><td>male more than 2 years</td><td></td><td>(Lh)</td></tr><tr><td>heifers and cows</td><td></td><td>(Lh)</td></tr><tr><td>other animals</td><td></td><td>(Lh)</td></tr><tr><td>cow life cycle -5</td><td>4</td><td>(1)</td></tr><tr><td>each of feed categories</td><td>25</td><td>(t)</td></tr></tbody></table>

Results provided by the versions V2, V3 and V4 based on linear cost estimates are illustrated on Figures 11, 12, and 13.

The calculations were carried out on a high-performance workstation for the FADN years 2007 up to 2012, initially. Results are illustrated on the figures 14, 15, 16, 17, 18, 19. The operations were renewed for the FADN years 2013-2018, after the FADN data were modified by DG-AGRI of the EC. Corresponding illustrations are provided in figures 20, 21, 22, 23, 24.

Calibration may be considered as acceptable when the criterion $log(-c)$ is lower than 0, it is correct when $log(-c)$ is lower than -2.3 and very good when $log(-c)$ is lower than -4.6. We have to insist on the difficulty of comparing the calibrated farm group distribution involving different AROPAj versions: these versions usually refer to different stages of the European Union building, and recently added countries usually strongly differ in farm types. Typically, East Germany and other countries of Central and Eastern Europe had very large farming structures which did not exist in the founding EU countries. Finally, the calibration criterion was refined after the V2 version.

The computation run time used to be a real difficulty. In the early 2000s, several weeks were required to calibrate from 1000 (V2) to 1400 (V4) farm groups. Now, when new iterations are added, the 1800 farm groups can be calibrated in less than a full day.
The majority of *AROPAj* parameters are provided by estimations from *FADN* data. The previous section describes the calculation process required to deal with the insufficiently detailed information per crop. After that, it is easy to estimate costs at the farm group level thanks to the provision of (estimated) individual costs for the entire *FADN* sample. We did that separately for fertilizer costs, other variables costs for all *AROPAj* versions, and for irrigation costs for the *V5* version.

Other parameters are provided with two methods. Some are calculated by the means of simple linear regression or estimated average. This is done through *R*-code programs which can be automatically managed per series of parameter categories for all countries. Parameter categories of this part refer to:

* crop yields

* areas (requested by the *AROPAj* calibrating process)

* herds

* on-farm re-used crop products (useful for improving the *AROPAj* calibrating process)

* variable costs (in case of non-operating *GME* estimator)

* farm numbers per crop or per crop category

* ratios between crops or fodders, and categories of crops or fodders

* milk productions and quotas, sugar beet quotas

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Fertilizer cost (1000€/ha)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Yield (t/ha)</figcaption></figure>

Distribution of fertilizer costs (left) and yields (right), for soft wheat, among the EU *AROPAj* farm groups.

From expert sources, we gather other important information such as:

* IPCC guidelines sourced recommendation

* energy and protein compounds of concentrates and on-farm sourced feed-stuff

* animal requirements in energy, proteins and dry matter, and limits or thresholds regarding animal feeding
<figure><img src="image_12.png" /><figcaption>Figure 11. "V2" (FADN-2001)</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 13. "V4" (FADN-2009)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12. "V3" (FADN-2004)</figcaption></figure>

Distribution set related to the calibrating criterion among farm groups for different FADN years - V2, V3, V4 versions.
* by default characteristics of fertilizers

* by default characteristics of manure

* by default value of ratios within the bovine herd

* CAP tool values linked to (more or less) stylized CAP tool implementation

Finally, in some cases, parameter values can be provided by the *AROPAj* model itself, when
outputs become parameters. This is the case for the following parameters:

* baseline GHG emissions (to be quotas)

* baseline cereals, seed and protein crops (to be CAP supported areas)

* shadow prices (to replace market prices in the case of endogenous computation of yields)

Figures 8 and 9 illustrate the parameter heterogeneity in case of fertilizer costs and yields related to soft wheat, present in 1642 among the 1815 farm groups related to the V5 version and the FADN-2009. The figures show a bi-modal distribution.
<figure><img src="image_15.png" /><figcaption>Figure 14. "V5_2007"</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 16. "V5_2009"</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 18. "V5_2011"</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 15. "V5_2008"</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 17. "V5_2010"</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 19. "V5_2012"</figcaption></figure>

Distribution set related to the calibrating criterion among EU-27 farm groups for 6 different FADN years (2007 up to 2012) - AROPAj version related to GME cost estimates.
# 7 Calibration of the *AROPAj* model

**Key-points:**

* Finding a *LP* solution at the farm group level

* Improving all time-reference *LPs* through parameter re-estimates

* Applying the algorithm, and running time

The first problem we face after prior estimation of parameters is to obtain a *LP* solution. To do so, we check parameters' consistency with regards to a series of potential problems due to missing information. For instance, at the farm group level, the *FADN* may provide milk quota and milk production but no milk price; it may provide crop area but no crop production neither crop price. This poses a problem even when a *LP* solution exists, and even more when there is no *LP* solution. A series of “expert” checking points are processed in the earlier steps of the *AROPAj* building. They are based on fulfilling farm group data sets with nested calculations: given a parameter (e.g. a yield, a price), in case of missing farm group information, we use regional average values; in case of missing region information, we use country-level average values; in case of missing country information, we retain the default value. In practice, default values were used in very few cases, related to bovine activities (uncomplete livestock categories) in small countries (Cyprus).

The second problem arises when we compare computed optimal activities and actual activities, once we have a *LP* solution derived from prior estimates of parameters. The *FADN* provides a wide range of parameters, some of them being related to *LP* command variables, and, at the same time, the *FADN* has to be supplemented by exogenous technical information (e.g. energy and protein contents of feed intakes, feed requirements, fertilizer characteristics, livestock life cycle, agronomic cropping limits in terms of land allocation).

The calibration process consists of re-estimating a subset composed of previously estimated parameters (crop ratios) and of non-*FADN* sourced parameters (feed requirements, livestock life cycle). In practice, more than one hundred parameters per farm group are re-estimated in this way. The calibration criterion is a sum of weighted squared deviations, each of them being the difference between *FADN* observation and the associated optimal *LP* variable. The set of variables is composed of crop and fodder areas (hectares), livestock categories (head units) and feed intakes (metric tons). Table 26 lists the variables which contribute to the calibration criterion and fully describes the criterion calculation.

Figure 10 illustrates the geometric interpretation of the calibration phase, which can be viewed as a rotational and translational movement of constraints in the primal variable space.

The calibration algorithm is based on sequential calculations, combining Monte-Carlo and gradient methods. In practice, randomization of calibration parameters alternating with “local” gradient based improvement of the criterion is generated from a large number of *LP* runs for each of *AROPAj* farm groups. All operating *AROPAj* versions are based on more than one thousand farm groups. For one farm group, the number of runs ranges from 1000 to 1500 (*AROPAj* versions *V2* and *V3*) or to 2000 (versions *V4* and *V5*). The calibration process is stopped either when the run number limit is reached (i.e. pre-defined number for Monte-Carlo and gradient phases), or when additional change of parameters' value does not improve the criterion during a phase.
<figure><img src="image_21.png" /><figcaption>Figure 20. "V5_2013"</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Figure 22. "V5_2015"</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 24. "V5_2017"</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Figure 21. "V5_2014"</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Figure 23. "V5_2016"</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Figure 25. "V5_2018"</figcaption></figure>

Distribution set related to the calibrating criterion among EU-28 farm groups for 6 different FADN years (2013 up to 2018) - AROPAj version related to GME cost estimates.
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Geometric overview of the calibrating process on a two-dimension subspace: the target fixed point is marked by the black circle, the *LP* point maximizing the gross margin *p*·*x* is marked by the grey circle, the feasible solution has to belong to the colored area bounded by the *LP* constraints. Calibrating consists on rotational and translational movements aiming to bring the two points closer.</figcaption></figure>
# 8 Spatialization of AROPAj outputs

**Key-points:**

* Obtaining spatialized probabilities for crop location

* Calling spatial econometrics and the Multinomial logit approach

* Generalized cross entropy aiming at refining the probabilities

The proposed spatial disaggregation approach (Chakir, 2009) shows how land use data and biophysical data, together available at a fine resolution level, can be combined to disaggregate regional data and provide more complete information. The proposed method is based on a two-step process. First, the estimation of a disaggregated econometric model (multinomial logit - MNL) of land use with Land Use/Cover Area frame Statistical Survey - LUCAS data in combination with data on soil, climate, altitude and land cover (CORINE Land Cover - CLC). Second, using the estimates from the first stage as priors and refining them through a cross entropy method, we disaggregate the FADN data at a fine level (100m pixel resolution).

## 8.1 Disaggregation approach in spatial econometrics devoted to crop location

The proposed disaggregation approach is a two-step procedure. In the first step, we estimate a land use model. In the second step, using estimations from the first step as "priors", we disaggregate the aggregated regional data available at the regional administrative level with the Generalized Cross-Entropy method (GCE).

### 8.1.1 The land use model

The MNL model estimates the probability outcome associated with each category of land use depending on a set of explanatory variables. The probability of observing the land use c at parcel l can be expressed as:

$$p_{lc} = \frac{\exp(\beta'_{c}x_{lc})}{\sum_{c=1}^{C} \exp(\beta'_{c}x_{lc})} \qquad (2)$$

where C is the number of possible land use categories, β'c is a vector of parameters to be estimated for land use c, and xlc are explanatory variables associated with land use c and parcel l.

The log-likelihood function of a sample of L parcels is given by:

$$\ln(L(\beta)) = \sum_{l=1}^{L} \sum_{c=1}^{C} ln p_{lc}^{y_{lc}} \qquad (3)$$

where ylc is a dummy variable such that ylc = 1 if land use c is observed at location l and ylc = 0 otherwise.

### 8.1.2 The disaggregation model: the Generalized Cross Entropy approach

The objective here is to utilize land use estimation results to find a disaggregated spatial allocation of regional agricultural land use data. We used the Generalized Cross Entropy (GCE)
<table><caption>**Table 26.** Detailed expression of the calibrating criterion to be maximized *c* ($\delta$ refers to any additional deviation, *w* is 1 by default); each deviation enters the criterion such as $c \to c - 1/w \delta^2$.</caption><thead><tr><th>variable</th><th>additional weight 1/w</th><th>(&delta; unit)</th></tr></thead><tbody><tr><td>each of 8 cereals</td><td></td><td>(ha)</td></tr><tr><td>cereals together</td><td></td><td>(ha)</td></tr><tr><td>each of 3 oilseeds</td><td></td><td>(ha)</td></tr><tr><td>each of 2 proteins crops</td><td></td><td>(ha)</td></tr><tr><td>sugarbeet</td><td></td><td>(ha)</td></tr><tr><td>potatoes</td><td></td><td>(ha)</td></tr><tr><td>each of other crops</td><td></td><td>(ha)</td></tr><tr><td>fallow</td><td></td><td>(ha)</td></tr><tr><td>CAP set aside</td><td></td><td>(ha)</td></tr><tr><td>each of fodders</td><td></td><td>(ha)</td></tr><tr><td>grasslands</td><td></td><td>(ha)</td></tr><tr><td>fallow & set-aside</td><td></td><td>(ha)</td></tr><tr><td>bovine less than 1 year</td><td>16</td><td>(Lh)</td></tr><tr><td>one-year bovine male</td><td>4</td><td>(Lh)</td></tr><tr><td>one-year bovine female</td><td>4</td><td>(Lh)</td></tr><tr><td>male more than 2 years</td><td></td><td>(Lh)</td></tr><tr><td>heifers and cows</td><td></td><td>(Lh)</td></tr><tr><td>other animals</td><td></td><td>(Lh)</td></tr><tr><td>cow life cycle -5</td><td>4</td><td>(1)</td></tr><tr><td>each of feed categories</td><td>25</td><td>(t)</td></tr></tbody></table>

Results provided by the versions V2, V3 and V4 based on linear cost estimates are illustrated on Figures 11, 12, and 13.

The calculations were carried out on a high-performance workstation for the FADN years 2007 up to 2012, initially. Results are illustrated on the figures 14, 15, 16, 17, 18, 19. The operations were renewed for the FADN years 2013-2018, after the FADN data were modified by DG-AGRI of the EC. Corresponding illustrations are provided in figures 20, 21, 22, 23, 24.

Calibration may be considered as acceptable when the criterion $log(-c)$ is lower than 0, it is correct when $log(-c)$ is lower than -2.3 and very good when $log(-c)$ is lower than -4.6. We have to insist on the difficulty of comparing the calibrated farm group distribution involving different AROPAj versions: these versions usually refer to different stages of the European Union building, and recently added countries usually strongly differ in farm types. Typically, East Germany and other countries of Central and Eastern Europe had very large farming structures which did not exist in the founding EU countries. Finally, the calibration criterion was refined after the V2 version.

The computation run time used to be a real difficulty. In the early 2000s, several weeks were required to calibrate from 1000 (V2) to 1400 (V4) farm groups. Now, when new iterations are added, the 1800 farm groups can be calibrated in less than a full day.
approach (Golan et al., 1996) to carry out the disaggregation procedure. The entropy concept in information theory was originally proposed by Shannon (1948) as a measure of uncertainty or missing information. Suppose there are C possible outcomes for some event and the probability distribution ($p_1, p_2, ..., p_C$) can be used to explain outcomes. Shannon's information entropy is defined as:

$$H(p_1, p_2, ..., p_C) = - \sum_{c=1}^{C} p_c \ln(p_c). \qquad (4)$$

To estimate the unknown probabilities $p_c$, Jaynes (1957) proposed maximizing entropy and subject to sample-moment information and then adding up constraints on the probabilities. This approach is based on the idea that the frequency that maximizes entropy is a reasonable estimate of the true distribution when we lack other information. If we have any additional information from previous experiments or observations, we can use those *priors* to alter this estimate.

Good (1963) introduced the notion of Cross Entropy<fnref n="7" /> (CE). Unlike the ME approach, where we maximize uncertainty implied by the probabilities, in the CE framework we minimize the CE, which is a measure of discrepancy between the posterior probabilities p and their priors q. This yields the minimization problem:

$$Min_p CE(p, q) = \sum_{c=1}^{C} p_c \ln(p_c/q_c) \qquad (5)$$

subject to all relevant adding up constraints on probabilities, where $p = p_1, p_2, ..., p_C$ and $q = q_1, q_2, ..., q_C$.

The GCE approach is applied here to estimate agricultural land use choices at a disaggregated level taking into account the observed aggregated regional level data. The estimation of the land use model by the Multinomial logit provided us with some prior knowledge about the land share of each crop j in each pixel i. This corresponds to the prior probabilities $\hat{\pi}_{ij}$ that will enable us to estimate probabilities $p_{ij}$ to observe crop j in pixel i by solving the non-linear optimisation program:

$$\min_{p} CE(p_{ij}, \varepsilon) = \sum_{i=1}^{I} \sum_{j=1}^{C} p_{ij} \ln(p_{ij}/\hat{\pi}_{ij}) + \sum_{j=1}^{C} \sum_{n=1}^{N} \varepsilon_{jn} \ln(\varepsilon_{jn}) \qquad (6)$$

subject to:

$$\sum_{i=1}^{I} p_{ij} \times s_i + \varepsilon_j = S_j^R, \forall j = 1, ..., C. \qquad (7)$$

$$\sum_{j=1}^{C} p_{ij} = 1, \forall i = 1, ..., I \text{ and } p_{ij} \in [0, 1]. \qquad (8)$$

$$\sum_{n=1}^{N} \varepsilon_{jn} = 1, \forall j = 1, ..., C \text{ and } \varepsilon_{jn} \in [0, 1]. \qquad (9)$$

where $s_i$ is the area of pixel i and $S_j^R$ is the area allocated to crop j at the regional level. Following the GCE framework, the error terms $\varepsilon_j$ associated with the constraints (7) are reparameterized in terms of probabilities such that $\varepsilon_j = \sum_{n=1}^{N} \zeta_n \varepsilon_{jn}$ where $\zeta_1, ..., \zeta_N$ with $N \ge 2$ is the support set associated with probabilities $\varepsilon_{j1}, ..., \varepsilon_{jN}$.

<footnote n="7">⁷The Cross entropy, or Relative Entropy, is also known as the Kullback-Leibler distance or divergence. This distance is not, in general, symmetric</footnote>
<figure><img src="image_13.png" /><figcaption>**Figure 11.** "V2" (FADN-2001)</figcaption></figure>

Criterion distribution (c)

<figure><img src="image_14.png" /><figcaption>**Figure 13.** "V4" (FADN-2009)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 12.** "V3" (FADN-2004)</figcaption></figure>

Distribution set related to the calibrating criterion among farm groups for different FADN years - V2, V3, V4 versions.
Equation (7) is the data compatibility constraint, which ensures that the predicted land allocated to a specific crop *j* across all pixels is equal to the observed area allocated at the regional level to crop *j*. Equation (8) is the adding-up constraint that ensures that the area allocated to all crops in a given pixel *i* is exactly equal to the area of pixel *i*. Equation (9) ensures that $ε_{j1},...,ε_{jN}$ is a probability distribution. The objective of the GCE problem (6)-(9) is to recover the unknown probability distributions $p_{ij}$ and $ε_{jn}$. This is a standard GCE problem; due to its convexity, there will be unique solution.

## 8.2 Method and process implementation

**Key-points:**

* Obtaining spatialized contribution of farm groups to land based activities
* Algorithm implementation

### 8.2.1 Crop location estimates at a fine resolution level

Data utilized for the spatialization block refers to raw physical data, as defined in section 3.2.

For the spatialization of V5 of AROPAj, we use R, GDAL, POSTGIS and GAMS scripts and procedures.

#### Data treatment

The spatialization procedure combines a large variety of information. The starting point is the LUCAS database which contains points of observations on crops (land uses) and land cover. For each LUCAS point data on soils, climate, relief and land cover (CLC) is extracted (see Figure 26). In the same sense, CLC raster data is intersected with the other layers. In order to do so, rasters devoted to agricultural activities treated in the AROPAj model are extracted and polygonized through GDAL programs. Afterwards, the obtained polygons are intersected with the polygons containing the other raw information.

#### Aggregation of crops into crop classes

Crops observed in the FADN data are grouped in different classes depending on their representativeness in the FADN region. When a given crop surpasses a certain threshold (for instance, 5% of agricultural area in the FADN sample), it is modeled independently. Otherwise, it is grouped with other crops. The method is summarized in Table 27.

The same classification of the crops is applied to the LUCAS points before the estimation of the MNL model.

#### MNL and estimation of priors

In order to establish the MNL model formula a dedicated R script is used (package VGAM). The algorithm defines a list of variables (that actually vary in the region) and their corresponding interaction terms. Three loops over this list are then executed. In each loop, each variable (or interaction terms) is added to the MNL model formula. The score of the model is defined as the
<figure><img src="image_16.png" /><figcaption>Figure 14. "V5_2007"</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 16. "V5_2009"</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 18. "V5_2011"</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 15. "V5_2008"</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 17. "V5_2010"</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Figure 19. "V5_2012"</figcaption></figure>

Distribution set related to the calibrating criterion among EU-27 farm groups for 6 different FADN years (2007 up to 2012) - AROPAj version related to GME cost estimates.
<table><caption>**Table 27.** Method of grouping crops depending on area ratio limits.</caption><thead><tr><th>Crop</th><th>Crop &gt;5% (Cat.1)</th><th>Crop &lt;5% (Cat.2)</th><th>Cat.2 &lt; 5% (Cat.3)</th></tr></thead><tbody><tr><td>Cereals</td><td></td><td></td><td></td></tr><tr><td>Wheat (common)</td><td>Wheat (common)</td><td>Cereals</td><td>Cereals</td></tr><tr><td>Wheat (durum)</td><td>Wheat (durum)</td><td>Cereals 2</td><td>Cereals</td></tr><tr><td>Barley</td><td>Barley</td><td>Cereals 2</td><td>Cereals</td></tr><tr><td>Oats</td><td>Cereals 2</td><td>Cereals 2</td><td>Cereals</td></tr><tr><td>Rye</td><td>Cereals 1</td><td>Cereals 1</td><td>Cereals</td></tr><tr><td>Other cereals</td><td>Cereals 1</td><td>Cereals 1</td><td>Cereals</td></tr><tr><td>Maize</td><td>Maize</td><td>Cereals 2</td><td>Cereals</td></tr><tr><td>Rice</td><td>Rice</td><td>Cereals</td><td>Cereals</td></tr><tr><td>Industrial crops</td><td></td><td></td><td></td></tr><tr><td>Dry pulses</td><td>LEG 1</td><td>IND</td><td>IND</td></tr><tr><td>Potatoes</td><td>PDT</td><td>IND</td><td>IND</td></tr><tr><td>Sugar beet</td><td>BTS</td><td>IND</td><td>IND</td></tr><tr><td>Tobacco</td><td>TAB</td><td>IND</td><td>IND</td></tr><tr><td>Cotton</td><td>COT</td><td>IND</td><td>IND</td></tr><tr><td>Flax fiber</td><td>LIN</td><td>IND</td><td>IND</td></tr><tr><td>Other industrial crops</td><td>AIN</td><td>IND</td><td>IND</td></tr><tr><td>Oilseed rape</td><td>CLZ</td><td>PR1</td><td>PRO</td></tr><tr><td>Sunflower</td><td>TOR</td><td>PR1</td><td>PRO</td></tr><tr><td>Soya</td><td>SOJ</td><td>PRO</td><td>PRO</td></tr><tr><td>Fodder crops</td><td>FOU</td><td>PPR</td><td>PPR</td></tr><tr><td>Fallows and set aside</td><td>FRIC</td><td>PPR</td><td>PPR</td></tr><tr><td>Temporary grass</td><td>FRIC</td><td>PPR</td><td>PPR</td></tr><tr><td>Other arable crops</td><td>AUC</td><td>AUC</td><td>AUC</td></tr><tr><td>Permanent pasture</td><td>PPR</td><td>PPR</td><td>PPR</td></tr></tbody></table>
<figure><img src="image_22.png" /><figcaption>Figure 20. "V5_2013"</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 22. "V5_2015"</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Figure 24. "V5_2017"</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Figure 21. "V5_2014"</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Figure 23. "V5_2016"</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Figure 25. "V5_2018"</figcaption></figure>

Distribution set related to the calibrating criterion among EU-28 farm groups for 6 different FADN years (2013 up to 2018) - AROPAj version related to GME cost estimates.
<figure><img src="image_27.png" /><figcaption>**Figure 26.** Intersecting and overlapping of information layers requested by *AROPAj* spatialization procedures. Source: Cantelaube et al. (2012).</figcaption></figure>

percent of correctly predicted observations. If the score of the model with the additional variable is at least as good as the best one already obtained, then the variable is retained in the formula. When all three loops are completed the formula with highest score and lowest number of variables is selected. Based on this formula, predictions of the probability of presence of the different land uses are estimated and exported as priors for the following *GCE* procedure.

## GCE

In this step the predicted crop areas in the *MNL* phase are compared with the crop areas observed in the *FADN* database (as ratios) and the priors are refined (see Section 8.1.2)).

### 8.2.2 Estimation of probabilities for farm types

In *V5* of *AROPAj* the irrigation plays a central part in the model. In order to account better for it in the spatialization we use additional information from the *LUCAS* database, namely the presence of irrigation for agricultural needs at a given point. We first define a 10 km x 10 km regular grid. For each cell of the grid we verify the presence of an irrigating point in the *LUCAS* data. If there are such points we assume that the cell has a greater potential to accommodate agricultural activities subject to irrigation. The polygons of the grid are then intersected with those resulting from the polygonization of the *CLC* database and its intersection with the other data layers.

We describe below how to distribute farm group contribution to agricultural activities across all pixels over a geographical area.
# 8 Spatialization of AROPAj outputs

## Key-points:

* Obtaining spatialized probabilities for crop location

* Calling spatial econometrics and the Multinomial logit approach

* Generalized cross entropy aiming at refining the probabilities

The proposed spatial disaggregation approach (Chakir, 2009) shows how land use data and biophysical data, together available at a fine resolution level, can be combined to disaggregate regional data and provide more complete information. The proposed method is based on a two-step process. First, the estimation of a disaggregated econometric model (multinomial logit - MNL) of land use with Land Use/Cover Area frame Statistical Survey - LUCAS data in combination with data on soil, climate, altitude and land cover (CORINE Land Cover - CLC). Second, using the estimates from the first stage as priors and refining them through a cross entropy method, we disaggregate the FADN data at a fine level (100m pixel resolution).

## 8.1 Disaggregation approach in spatial econometrics devoted to crop location

The proposed disaggregation approach is a two-step procedure. In the first step, we estimate a land use model. In the second step, using estimations from the first step as "priors", we disaggregate the aggregated regional data available at the regional administrative level with the Generalized Cross-Entropy method (GCE).

### 8.1.1 The land use model

The MNL model estimates the probability outcome associated with each category of land use depending on a set of explanatory variables. The probability of observing the land use c at parcel l can be expressed as:

$$p_{lc} = \frac{\exp(\beta'_{c}x_{lc})}{\sum_{c=1}^{C} \exp(\beta'_{c}x_{lc})} \qquad (2)$$

where C is the number of possible land use categories, β'c is a vector of parameters to be estimated for land use c, and xlc are explanatory variables associated with land use c and parcel l.

The log-likelihood function of a sample of L parcels is given by:

$$\ln(L(\beta)) = \sum_{l=1}^{L} \sum_{c=1}^{C} ln p_{lc}^{y_{lc}} \qquad (3)$$

where ylc is a dummy variable such that ylc = 1 if land use c is observed at location l and ylc = 0 otherwise.

### 8.1.2 The disaggregation model: the Generalized Cross Entropy approach

The objective here is to utilize land use estimation results to find a disaggregated spatial allocation of regional agricultural land use data. We used the Generalized Cross Entropy (GCE)
## Prior information

We consider that the FADN region is the basic aggregation unit, and all requested information are region-based.

At the region scale, we have to distinguish:

* the geographical area

* the agricultural area estimated through general spatial statistics and related to AROPAj activities

* the AROPAj area (based on FADN estimates)

We denote by:

* i : the pixel

* j : the crop

* k : the farm group

* a : the elevation class (let us recall that the value of 1 refers to [0,300m[, the value of 2 to [300m,600m[, the value of 3 to altitudes greater than or equal to 600m

* *irr* vs *nir* respectively for irrigated vs non irrigated areas (when available)

We assume to be informed on:

* the location probability to find the agricultural activity *j* on the pixel *i*, denoted by $P_{ij}$, $\forall i, \forall j : P_{ij} > 0, \forall i : \sum_j P_{ij} = 1$; $P_{ij}$ is provided by MNL and cross entropy estimating process

* the AROPAj area devoted to the *j* crop by the *k* farm group, denoted by $S_{jk}$ (prior estimate provided by FADN), and the *k*'s utilised agricultural area $L_k = \sum_j S_{jk}$

* possibly informed share of the $S_{jk}$ area between irrigated ($S_{jk}^{irr}$) and non irrigated ($S_{jk}^{nir}$) areas ($S_{jk}^{irr} + S_{jk}^{nir} = S_{jk}$)

* the share of *k'* utilized agricultural area (UAA) of *a* elevation class $U_{ak}$ (in accordance with $\sum_a U_{ak} = 1$);

We consider that any *i* pixel is elevation class homogeneous ("a"⇒"i")

At the regional level, we assume to know:

* the total amount of irrigated areas relating to AROPAj activities (denoted by $\mathbb{I}$)

* the total amount of pixels of elevation *a* covered by AROPAj activities (denoted by $\mathbb{A}(a)$)

* the total amount of areas covered by AROPAj activities (denoted by $\mathbb{M}$)

The proposed estimate of the contribution of the *k* farm group to agricultural activities (among AROPAj ones) follows:

$$T_{ik} = \sum_{j} P_{ij} \left( \alpha \frac{S_{jk}}{\sum_{k'} S_{jk'}} + \beta \frac{S_{jk}^{irr}}{\sum_{k'} S_{jk'}^{irr}} \mathbb{I}_{\mathbb{I}(i)} \right) + \sum_{a} \gamma_a U_{ak} \mathbb{I}_{\mathbb{A}(a)}(i) \quad (10)$$

using the indicator functions $\mathbb{1}_{\mathbb{A}(a)}(i) = 1$ if pixel *i* elevation is *a*, else the value is 0, and $\mathbb{1}_{\mathbb{I}}(i) = 1$ if pixel *i* is irrigated, else 0.

Some *a priori* constraints have to be checked:
approach (Golan et al., 1996) to carry out the disaggregation procedure. The entropy concept in information theory was originally proposed by Shannon (1948) as a measure of uncertainty or missing information. Suppose there are C possible outcomes for some event and the probability distribution ($p_1, p_2, ..., p_C$) can be used to explain outcomes. Shannon's information entropy is defined as:

$$H(p_1, p_2, ..., p_C) = - \sum_{c=1}^{C} p_c \ln(p_c). \qquad (4)$$

To estimate the unknown probabilities $p_c$, Jaynes (1957) proposed maximizing entropy and subject to sample-moment information and then adding up constraints on the probabilities. This approach is based on the idea that the frequency that maximizes entropy is a reasonable estimate of the true distribution when we lack other information. If we have any additional information from previous experiments or observations, we can use those priors to alter this estimate.

Good (1963) introduced the notion of Cross Entropy<fnref n="7" /> (CE). Unlike the ME approach, where we maximize uncertainty implied by the probabilities, in the CE framework we minimize the CE, which is a measure of discrepancy between the posterior probabilities p and their priors q. This yields the minimization problem:

$$Min_p CE(p, q) = \sum_{c=1}^{C} p_c \ln(p_c/q_c) \qquad (5)$$

subject to all relevant adding up constraints on probabilities, where $p = p_1, p_2, ..., p_C$ and $q = q_1, q_2, ..., q_C$.

The GCE approach is applied here to estimate agricultural land use choices at a disaggregated level taking into account the observed aggregated regional level data. The estimation of the land use model by the Multinomial logit provided us with some prior knowledge about the land share of each crop j in each pixel i. This corresponds to the prior probabilities $\hat{\pi}_{ij}$ that will enable us to estimate probabilities $p_{ij}$ to observe crop j in pixel i by solving the non-linear optimisation program:

$$\min_{p} CE(p_{ij}, \varepsilon) = \sum_{i=1}^{I} \sum_{j=1}^{C} p_{ij} \ln(p_{ij}/\hat{\pi}_{ij}) + \sum_{j=1}^{C} \sum_{n=1}^{N} \varepsilon_{jn} \ln(\varepsilon_{jn}) \qquad (6)$$

subject to:

$$\sum_{i=1}^{I} p_{ij} \times s_i + \varepsilon_j = S_j^R, \forall j = 1, ..., C. \qquad (7)$$

$$\sum_{j=1}^{C} p_{ij} = 1, \forall i = 1, ..., I \text{ and } p_{ij} \in [0, 1]. \qquad (8)$$

$$\sum_{n=1}^{N} \varepsilon_{jn} = 1, \forall j = 1, ..., C \text{ and } \varepsilon_{jn} \in [0, 1]. \qquad (9)$$

where $s_i$ is the area of pixel i and $S_j^R$ is the area allocated to crop j at the regional level. Following the GCE framework, the error terms $\varepsilon_j$ associated with the constraints (7) are reparameterized in terms of probabilities such that $\varepsilon_j = \sum_{n=1}^{N} \zeta_n \varepsilon_{jn}$ where $\zeta_1, ..., \zeta_N$ with $N \ge 2$ is the support set associated with probabilities $\varepsilon_{j1}, ..., \varepsilon_{jN}$.

<footnote n="7">⁷The Cross entropy, or Relative Entropy, is also known as the Kullback-Leibler distance or divergence. This distance is not, in general, symmetric</footnote>
* $T_{ik}$ has to be a probability set such that $\forall i : \sum_k T_{ik} = 1$, and consequently $\alpha + \beta + \sum_a \gamma_a = 1$

* When the region is homogeneous in elevation, the $\gamma_a$ parameters have to be equal to 0

* When there is no information upon irrigation, the $\beta$ parameter has to be equal to 0

At this stage, there is no reason to consider any uplet $\{\alpha, \beta, \gamma\}, \alpha + \beta + \gamma = 1$, as a mean to provide the best estimate of probabilities, in the sense the linear form of term combination is questionable (linearity means that the parameter uplet does not depend on the $i$ pixel). But we have a real probability set, for which we just need to set four of the five parameters $\alpha, \beta, \gamma_a, a \in \{1, 2, 3\}$. Additional information may be useful to do that.

Let us denote the set of irrigated pixels by $\mathbb{I}$ (amount $\|\mathbb{I}\|$ in hectares), and the set of $a$ elevation by $\mathbb{A}(a)$ (amount $\|\mathbb{A}(a)\|$ in hectares). First, irrigation is one of the characteristics of the $i$ pixel (through the LUCAS database). Secondly, the $i$ elevation is assumed to be known (through the DEM database). Three equations describe these informations and relate them to AROPAj-sourced information:

$$ \sum_i 1 = M = \frac{1}{\rho} \sum_{jk} S_{jk} \quad (11) $$

$$ \sum_i \mathbb{1}_{\mathbb{I}}(i) = \|\mathbb{I}\| = \frac{1 + \delta_{irr}}{\rho} \sum_{jk} S_{jk}^{irr} \quad (12) $$

$$ \forall a : \sum_i \mathbb{1}_{\mathbb{A}(a)}(i) = \|\mathbb{A}(a)\| = \frac{1 + \delta_a}{\rho} \sum_k U_{ak} \sum_j S_{jk} \quad (13) $$

where $\rho$ reflects the AROPAj representativity in area of the AROPAj-activities in the region and the $\delta$ refer to distortions due to the fact that AROPAj may differ in farming system representativity accounting for irrigation or for elevation. The $\rho$ parameter is determined by the known amount of CLC pixels dedicated to AROPAj-like activities.

Combining the definition of the $T_{ik}$ and the previous relations and respectively accounting for irrigation calculation (whatever is the elevation) and elevation calculation (whatever is the irrigation) in terms of area balance, we obtain:

$$ \sum_i \mathbb{1}_{\mathbb{I}}(i) = (\alpha + \beta) \sum_i \mathbb{1}_{\mathbb{I}}(i) + \sum_a \gamma_a \sum_i \mathbb{1}_{\mathbb{A}(a)}(i) \mathbb{1}_{\mathbb{I}}(i) \\ \forall a : \sum_i \mathbb{1}_{\mathbb{A}(a)}(i) = (\alpha + \gamma_a) \sum_i \mathbb{1}_{\mathbb{A}(a)}(i) + \beta \sum_i \mathbb{1}_{\mathbb{A}(a)}(i) \mathbb{1}_{\mathbb{I}}(i) $$

leading us to the equations:

$$ (1 - \alpha - \beta) \|\mathbb{I}\| = \sum_a \gamma_a \sum_i \mathbb{1}_{\mathbb{A}(a)}(i) \mathbb{1}_{\mathbb{I}}(i) \quad (14) $$

$$ \forall a : (1 - \alpha - \gamma_a) \|\mathbb{A}(a)\| = \beta \sum_i \mathbb{1}_{\mathbb{A}(a)}(i) \mathbb{1}_{\mathbb{I}}(i) \quad (15) $$

We can be comfortable with these formulas in particular cases such as when there is no irrigation in the region, or when only one elevation class exists in the region. We aim at providing an acceptable solution in $\{\alpha, \beta, \gamma\}$ in any case, taking account of amounts of areas intersecting irrigation and elevation class (we denote $\|\mathbb{I}\mathbb{A}(a)\| = \sum_i \mathbb{1}_{\mathbb{A}(a)}(i)\mathbb{1}_{\mathbb{I}}(i)$ possibly estimated through overlapping of LUCAS and DEM databases).
Equation (7) is the data compatibility constraint, which ensures that the predicted land allocated to a specific crop *j* across all pixels is equal to the observed area allocated at the regional level to crop *j*. Equation (8) is the adding-up constraint that ensures that the area allocated to all crops in a given pixel *i* is exactly equal to the area of pixel *i*. Equation (9) ensures that $ε_{j1},...,ε_{jN}$ is a probability distribution. The objective of the GCE problem (6)-(9) is to recover the unknown probability distributions $p_{ij}$ and $ε_{jn}$. This is a standard GCE problem; due to its convexity, there will be unique solution.

## 8.2 Method and process implementation

**Key-points:**

- Obtaining spatialized contribution of farm groups to land based activities
- Algorithm implementation

### 8.2.1 Crop location estimates at a fine resolution level

Data utilized for the spatialization block refers to raw physical data, as defined in section ??.

For the spatialization of V5 of AROPAj, we use R, GDAL, POSTGIS and GAMS scripts and procedures.

#### Data treatment

The spatialization procedure combines a large variety of information. The starting point is the LUCAS database which contains points of observations on crops (land uses) and land cover. For each LUCAS point data on soils, climate, relief and land cover (CLC) is extracted (see Figure 26). In the same sense, CLC raster data is intersected with the other layers. In order to do so, rasters devoted to agricultural activities treated in the AROPAj model are extracted and polygonized through GDAL programs. Afterwards, the obtained polygons are intersected with the polygons containing the other raw information.

#### Aggregation of crops into crop classes

Crops observed in the FADN data are grouped in different classes depending on their representativeness in the FADN region. When a given crop surpasses a certain threshold (for instance, 5% of agricultural area in the FADN sample), it is modeled independently. Otherwise, it is grouped with other crops. The method is summarized in Table 27.

The same classification of the crops is applied to the LUCAS points before the estimation of the MNL model.

#### MNL and estimation of priors

In order to establish the MNL model formula a dedicated R script is used (package VGAM). The algorithm defines a list of variables (that actually vary in the region) and their corresponding interaction terms. Three loops over this list are then executed. In each loop, each variable (or interaction terms) is added to the MNL model formula. The score of the model is defined as the
**No irrigation** When $\forall i \mathbb{I}(i) = 0$ (implying $\mathbb{I} = 0$), necessarily $\beta = 0$ (thanks to 15 and $\alpha + \beta + \gamma = 1$). There remains to set $\gamma_a$ values. We suggest to retain positive values increasing with absolute values of distorsion $\delta_a$ such as $\gamma_a = 1 - \frac{M-\mathbb{A}(a)}{M}e^{-|\delta_a|}$. Let us note that the subcase in which the region is homogeneous in elevation leads to $\gamma_a = 0$, thanks to Equations 11 and 13. This is compatible with the case below.

**Homogeneous elevation class** When the region is homogeneous in elevation class, there is nothing to loose to set $\gamma = 0$. In case of irrigation areas positively estimated by the FADN-AROPAj source, we suggest to retain a value reflecting the distorsion in irrigation between FADN and DEM sources for estimation, i.e. $\beta = 1 - \frac{M-\mathbb{I}}{M}e^{-|\delta_{irr}|}$. Let us note that when the part of irrigated area is close to 0 or to 1, the $\beta$ value is close to 0. This is a consequence of Equation 12 which implies that $\delta_{irr}$ tends to 0 when $\mathbb{I}$ is close to $M$).

**Mixed cases with areas splitted in irrigation and in elevation** Let us consider the case where irrigation information exists in FADN (correlatively AROPAj $S_{jk}^{irr}$ are estimated). Let us consider the system defined by Equation 14, Equations 15 (as many as a elevation classes) and $\alpha + \beta + \sum_a \gamma_a = 1$ which is linear in the weights parameters $\alpha, \beta$, and $\gamma_a$. Let us note that $\sum_a ||\mathbb{A}(a)|| = ||\mathbb{I}||$, due to the fact that the a areas do not intersect. With some re-arrangements of the system, we get that $\beta ||\mathbb{I}|| = \beta \sum_a (||\mathbb{A}(a)||)^2 / ||\mathbb{A}(a)||$. There is generally no solution except $\beta = 0$, which means that weights of our probability set (Equation 10) is based on redundant information (irrigated areas on the one hand and irrigated areas intersecting elevation classes on the other hand). Moreover, that leads to keep equal the $\gamma$ values, i.e. $\forall a: \gamma_a = \gamma$. We suggest to overcome this parameter under-determination by retaining the parameters as follows:

$$ \gamma = 1 - \alpha = 1 - \frac{M-\mathbb{I}}{M}e^{-|\delta_{irr}|} $$

In other words, we rank first irrigation before elevation, which is nevertheless taken into account through the part of k farm group area of elevation a ($U_{ak}$). The probability set becomes:

$$ T_{ik} = \alpha \sum_j P_{ij} \frac{S_{jk}}{\sum_{k'} S_{jk'}} + \gamma \sum_a \frac{U_{ak}}{\sum_{k'} U_{ak'}} \mathbb{I}_{A(a)}(i) \quad (16) $$

The distorsion term $\delta_{irr}$ comes from Equations 11 and 12: $\delta_{irr} = 1 - \frac{M \sum_{jk}^{irr}}{\mathbb{I} \sum_{jk}^{irr}}$.

## 8.3 Application

User-friendly application is anchored to the general AROPAj-menu (“miraj”, see Appendix A), aiming first at managing the compilation of databases, precisely regarding:

1. FADN region list

2. CLC for Europe raster

3. Soils

4. Elevation

5. Slope

6. year-dependent climate
<table><caption>**Table 27.** Method of grouping crops depending on area ratio limits.</caption><thead><tr><th>Crop</th><th>Crop &gt;5% (Cat.1)</th><th>Crop &lt;5% (Cat.2)</th><th>Cat.2 &lt; 5% (Cat.3)</th></tr></thead><tbody><tr><td>Cereals</td><td></td><td></td><td></td></tr><tr><td>Wheat (common)</td><td>Wheat (common)</td><td>Cereals</td><td>Cereals</td></tr><tr><td>Wheat (durum)</td><td>Wheat (durum)</td><td>Cereals 2</td><td>Cereals</td></tr><tr><td>Barley</td><td>Barley</td><td>Cereals 2</td><td>Cereals</td></tr><tr><td>Oats</td><td>Cereals 2</td><td>Cereals 2</td><td>Cereals</td></tr><tr><td>Rye</td><td>Cereals 1</td><td>Cereals 1</td><td>Cereals</td></tr><tr><td>Other cereals</td><td>Cereals 1</td><td>Cereals 1</td><td>Cereals</td></tr><tr><td>Maize</td><td>Maize</td><td>Cereals 2</td><td>Cereals</td></tr><tr><td>Rice</td><td>Rice</td><td>Cereals</td><td>Cereals</td></tr><tr><td>Industrial crops</td><td></td><td></td><td></td></tr><tr><td>Dry pulses</td><td>LEG 1</td><td>IND</td><td>IND</td></tr><tr><td>Potatoes</td><td>PDT</td><td>IND</td><td>IND</td></tr><tr><td>Sugar beet</td><td>BTS</td><td>IND</td><td>IND</td></tr><tr><td>Tobacco</td><td>TAB</td><td>IND</td><td>IND</td></tr><tr><td>Cotton</td><td>COT</td><td>IND</td><td>IND</td></tr><tr><td>Flax fiber</td><td>LIN</td><td>IND</td><td>IND</td></tr><tr><td>Other industrial crops</td><td>AIN</td><td>IND</td><td>IND</td></tr><tr><td>Oilseed rape</td><td>CLZ</td><td>PR1</td><td>PRO</td></tr><tr><td>Sunflower</td><td>TOR</td><td>PR1</td><td>PRO</td></tr><tr><td>Soya</td><td>SOJ</td><td>PRO</td><td>PRO</td></tr><tr><td>Fodder crops</td><td>FOU</td><td>PPR</td><td>PPR</td></tr><tr><td>Fallows and set aside</td><td>FRIC</td><td>PPR</td><td>PPR</td></tr><tr><td>Temporary grass</td><td>FRIC</td><td>PPR</td><td>PPR</td></tr><tr><td>Other arable crops</td><td>AUC</td><td>AUC</td><td>AUC</td></tr><tr><td>Permanent pasture</td><td>PPR</td><td>PPR</td><td>PPR</td></tr></tbody></table>
7. LUCAS

8. intersecting CLC and countries involved in the application

9. FADN-based AROPAj-dependent information related to farm group's area per crop group

The next steps of the application follow the algorithm described in the previous section, performing the calculations of crop location's priors and posteriors, and the geo-referenced probability sets of farm groups.

We implemented the calculation process for the *V5 AROPAj* version applied to most of EU-27 FADN regions and for the 6 years 2007-2012. This is illustrated for France and the 2009 *FADN* on Figures 27 providing the location probabilities of two of the farm groups in the Bourgogne region.

<figure><img src="image_28.png" /><figcaption>**Figure 27.** Location probabilities of farm groups “GT37” and “GT39” in the Bourgogne region (2009- *V5 AROPAj* version).</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>**Figure 26.** Intersecting and overlapping of information layers requested by *AROPAj* spatialization procedures. Source: Cantelaube et al. (2012).</figcaption></figure>

percent of correctly predicted observations. If the score of the model with the additional variable is at least as good as the best one already obtained, then the variable is retained in the formula. When all three loops are completed the formula with highest score and lowest number of variables is selected. Based on this formula, predictions of the probability of presence of the different land uses are estimated and exported as priors for the following *GCE* procedure.

## GCE

In this step the predicted crop areas in the *MNL* phase are compared with the crop areas observed in the *FADN* database (as ratios) and the priors are refined (see Section 8.1.2)).

### 8.2.2 Estimation of probabilities for farm types

In *V5* of *AROPAj* the irrigation plays a central part in the model. In order to account better for it in the spatialization we use additional information from the *LUCAS* database, namely the presence of irrigation for agricultural needs at a given point. We first define a 10 km x 10 km regular grid. For each cell of the grid we verify the presence of an irrigating point in the *LUCAS* data. If there are such points we assume that the cell has a greater potential to accommodate agricultural activities subject to irrigation. The polygons of the grid are then intersected with those resulting from the polygonization of the *CLC* database and its intersection with the other data layers.

We describe below how to distribute farm group contribution to agricultural activities across all pixels over a geographical area.
# 9 Models linkage

The interest of linking the *AROPAj* economic model with biophysical models relies on the possibility of accounting for the impact of technical as well as environmental and physical variables (i.e. soil and climate) on the agricultural production. Moreover, linking *AROPAj* to biophysical models is important for understanding the environmental externalities. Regarding crop production, the general rationale is to introduce in the model a relation between crop production inputs (e.g. fertilizer applications) and crop production outputs (e. g. yields). The possible connecting points between the economic and the biophysical models are the farmer practices found in the accountings, such as irrigation, application of pesticides, and fertilizers. Among those practices, we chose nitrogen (*N*) fertilization because of (i) its key role in environmental policies in relation to non-point source pollution and greenhouse gas emissions, (ii) its traceability in accounts through fertilizers costs, and (iii) the wealth of usable information for our model on *N* effects on yield.

## 9.1 STICS

### Key-points:

* Introducing crop models and replacing constant yields and inputs by nitrogen to yield functions
* The integrated *ArTiX* database and applications

Economic modeling benefits from accessing biophysical information on the systems they represent. In the case of agricultural production, crop models are often associated with economic models, with the information being presented in a simplified form to fit the economic model's constraints. An important variable is the yield of the crop. This depends on environmental factors, for instance climate, as well as farmers' decisions (fertilizer and water use, technical management etc...). In the case of *AROPAj*, the generic crop model *STICS* (Brisson et al., 1998, 2003) was used to generate data linking the yield of major crops with the technical choices made at the farm level.

The genericity of the *STICS* crop model was one of the main reasons for choosing it among all the existing crop models. The potential crop model linked to *AROPAj* needed to simulate yield as well as environmental variables (such as nitrate fluxes) for all of the main crops simulated by the economic model across the *EU* territory. The *STICS* model simulates crop growth, development, and environmental impacts for the main European crops, in the various pedo-climatic conditions of the *EU*. Indeed, the main processes simulated by *STICS* are crop growth and development as well as water and nitrogen soil balances. A of this information is dynamically driven by data such as daily weather data and information regarding soil characteristics and management. The model is organized in several modules: three modules are dedicated to the simulation of plant aerial part (*LAI*, biomass and yield), three to the soil simulation (water and nitrogen balances, and root growth), and the last one additional module dedicated to the management of interactions between cropping techniques and soil-plant system (i.e. water or fertilizer supplies, or climatic environment of the plant). *STICS* is a deterministic model that simulates the soil-plant-weather system in supposedly homogenous physical and technical conditions, which corresponds to each crop of each
## Prior information

We consider that the FADN region is the basic aggregation unit, and all requested information are region-based.

At the region scale, we have to distinguish:

* the geographical area

* the agricultural area estimated through general spatial statistics and related to AROPAj activities

* the AROPAj area (based on FADN estimates)

We denote by:

* i : the pixel

* j : the crop

* k : the farm group

* a : the elevation class (let us recall that the value of 1 refers to [0,300m[, the value of 2 to [300m,600m[, the value of 3 to altitudes greater than or equal to 600m

* irr vs nir respectively for irrigated vs non irrigated areas (when available)

We assume to be informed on:

* the location probability to find the agricultural activity j on the pixel i, denoted by $P_{ij}$, $\forall i, \forall j : P_{ij} > 0, \forall i : \sum_j P_{ij} = 1$; $P_{ij}$ is provided by MNL and cross entropy estimating process

* the AROPAj area devoted to the j crop by the k farm group, denoted by $S_{jk}$ (prior estimate provided by FADN), and the k's utilised agricultural area $L_k = \sum_j S_{jk}$

* possibly informed share of the $S_{jk}$ area between irrigated ($S_{jk}^{irr}$) and non irrigated ($S_{jk}^{nir}$) areas ($S_{jk}^{irr} + S_{jk}^{nir} = S_{jk}$)

* the share of k' utilized agricultural area (UAA) of a elevation class $U_{ak}$ (in accordance with $\sum_a U_{ak} = 1$);

We consider that any i pixel is elevation class homogeneous ("a"⇒"i")

At the regional level, we assume to know:

* the total amount of irrigated areas relating to AROPAj activities (denoted by $\mathbb{I}$)

* the total amount of pixels of elevation a covered by AROPAj activities (denoted by $\mathbb{A}(a)$)

* the total amount of areas covered by AROPAj activities (denoted by $\mathbb{M}$)

The proposed estimate of the contribution of the k farm group to agricultural activities (among AROPAj ones) follows:

$$T_{ik} = \sum_{j} P_{ij} \left( \alpha \frac{S_{jk}}{\sum_{k'} S_{jk'}} + \beta \frac{S_{jk}^{irr}}{\sum_{k'} S_{jk'}^{irr}} \mathbb{1}_{\mathbb{I}(i)} \right) + \sum_{a} \gamma_a U_{ak} \mathbb{1}_{\mathbb{A}(a)}(i) \quad (10)$$

using the indicator functions $\mathbb{1}_{\mathbb{A}(a)}(i) = 1$ if pixel i elevation is a, else the value is 0, and $\mathbb{1}_{\mathbb{I}}(i) = 1$ if pixel i is irrigated, else 0.

Some a priori constraints have to be checked:
European farm type of the AROPAj model. Regarding the nitrogen sources, the STICS model was also able to account for both chemical fertilizers and on-farm produced manures.

### 9.1.1 N to yield functions

We decided to provide AROPAj with response functions $y(N)$ that describes crop yield level $y$ as a function of nitrogen input $N$. With such a response function, the production process is summarized in a simple function that focuses on the main limiting production factor ($N$) while also implicitly considering the other ones. No general consensus appears in the literature on the ideal shape of such a response curve (Makowski et al. (1999) cited in Godard et al. (2008)). However, the use of concave and strictly increasing curve is consistent from both an economic perspective (requirements for the estimation of a mathematical optimum), and an agronomic perspective (consistent with expected shape, and defined by easily interpretable parameters). Hence, we selected an exponential function, expressed as the following:

$$y(N) = B - (B - A)e^{-\tau N}$$

where $y$ is crop yield, $N$ is the nitrogen fertilizer amount, $A$, $B$, and $\tau$ are, respectively, the minimal yield, maximal yield and sensitivity to $N$ input.

To build up the nitrogen response curve, a two-step procedure was set up and repeated for each crop of each farm type. In the first step, STICS was run for each crop of each farm type using inputs derived from available databases and expert knowledge (see sections 3.2.2, 3.2.4 and 3.3.1). Each case was thus first characterized by weather conditions and by several combinations of soil and agronomic choices. To avoid too many combinations, the most representative soils and techniques were selected for each case studied: namely, 5 soil types, 3 sowing dates or earliness groups, 2 preceding crops and, given the crop, rainfall or irrigated option were selected for one case studied. This procedure yielded a set of 30 or 60 likely $N$-response curves.

In the second step, economic data (i.e. yields and price ratios extracted by AROPAj from the FADN, and from price databases) were used to select the relevant response curve so that a single $N$-response curve emerges with the required biophysical and economic properties. To do so, we first selected the curves for which the reference yield of the crop, given by the FADN database, was reached. Second, we selected the curve for which the slope of the tangent, at the point of the curve corresponding to the above mentioned reference yield value, was the closest to the price ratio (price of nitrogen/price of crop).

In the process, special attention was paid to livestock farms. For these farms, effluents from livestock are spread in addition to (or instead of) chemical nitrogen fertilizer. In this case, the STICS model needs additional data to simulate the appropriate $N$-response curves: first, a curve for the combined organic sources of nitrogen and second a curve of yield against chemical nitrogen only. Indeed, this additional organic nitrogen affects the crop yield but does not have to be represented on the X-axis of a curve showing yield response to chemical nitrogen fertilizer. Consequently, potential organic nitrogen available in each livestock farm type was estimated from the FADN. Used as STICS inputs (complementary $N$ source apart from fertilizer $N$), it enabled one to choose the optimal STICS input set in step 2. Another run of the STICS model, using the selected set of inputs (soil, agronomic choices) then generated the response curve to chemical fertilizer nitrogen
* $T_{ik}$ has to be a probability set such that $\forall i : \sum_k T_{ik} = 1$, and consequently $\alpha + \beta + \sum_a \gamma_a = 1$

* When the region is homogeneous in elevation, the $\gamma_a$ parameters have to be equal to 0

* When there is no information upon irrigation, the $\beta$ parameter has to be equal to 0

At this stage, there is no reason to consider any uplet $\{\alpha, \beta, \gamma\}, \alpha + \beta + \gamma = 1$, as a mean to provide the best estimate of probabilities, in the sense the linear form of term combination is questionable (linearity means that the parameter uplet does not depend on the $i$ pixel). But we have a real probability set, for which we just need to set four of the five parameters $\alpha, \beta, \gamma_a, a \in \{1, 2, 3\}$. Additional information may be useful to do that.

Let us denote the set of irrigated pixels by $\mathbb{I}$ (amount $\|\mathbb{I}\|$ in hectares), and the set of $a$ elevation by $\mathbb{A}(a)$ (amount $\|\mathbb{A}(a)\|$ in hectares). First, irrigation is one of the characteristics of the $i$ pixel (through the LUCAS database). Secondly, the $i$ elevation is assumed to be known (through the DEM database). Three equations describe these informations and relate them to AROPAj-sourced information:

$$ \sum_{i} 1 = M = \frac{1}{\rho} \sum_{jk} S_{jk} \quad (11) $$

$$ \sum_{i} 1_{\mathbb{I}}(i) = \|\mathbb{I}\| = \frac{1 + \delta_{irr}}{\rho} \sum_{jk} S_{jk}^{irr} \quad (12) $$

$$ \forall a : \sum_{i} 1_{\mathbb{A}(a)}(i) = \|\mathbb{A}(a)\| = \frac{1 + \delta_a}{\rho} \sum_{k} U_{ak} \sum_{j} S_{jk} \quad (13) $$

where $\rho$ reflects the AROPAj representativity in area of the AROPAj-activities in the region and the $\delta$ refer to distortions due to the fact that AROPAj may differ in farming system representativity accounting for irrigation or for elevation. The $\rho$ parameter is determined by the known amount of CLC pixels dedicated to AROPAj-like activities.

Combining the definition of the $T_{ik}$ and the previous relations and respectively accounting for irrigation calculation (whatever is the elevation) and elevation calculation (whatever is the irrigation) in terms of area balance, we obtain:

$$ \sum_{i} 1_{\mathbb{I}}(i) = (\alpha + \beta) \sum_{i} 1_{\mathbb{I}}(i) + \sum_{a} \gamma_a \sum_{i} 1_{\mathbb{A}(a)}(i) 1_{\mathbb{I}}(i) \\ \forall a : \sum_{i} 1_{\mathbb{A}(a)}(i) = (\alpha + \gamma_a) \sum_{i} 1_{\mathbb{A}(a)}(i) + \beta \sum_{i} 1_{\mathbb{A}(a)}(i) 1_{\mathbb{I}}(i) $$

leading us to the equations:

$$ (1 - \alpha - \beta) \|\mathbb{I}\| = \sum_{a} \gamma_a \sum_{i} 1_{\mathbb{A}(a)}(i) 1_{\mathbb{I}}(i) \quad (14) $$

$$ \forall a : (1 - \alpha - \gamma_a) \|\mathbb{A}(a)\| = \beta \sum_{i} 1_{\mathbb{A}(a)}(i) 1_{\mathbb{I}}(i) \quad (15) $$

We can be comfortable with these formulas in particular cases such as when there is no irrigation in the region, or when only one elevation class exists in the region. We aim at providing an acceptable solution in $\{\alpha, \beta, \gamma\}$ in any case, taking account of amounts of areas intersecting irrigation and elevation class (we denote $\|\mathbb{I}\mathbb{A}(a)\| = \sum_{i} 1_{\mathbb{A}(a)}(i) 1_{\mathbb{I}}(i)$ possibly estimated through overlapping of LUCAS and DEM databases).
itself. The theoretical framework used for the inclusion of different nitrogen sources in the N-response equation was based on an equivalence rule allowing us to compare any N source to a reference chemical nitrogen fertilizer. Once the exponential shape had been selected for the N-response curve, the minimal and maximal yields were assumed to be independent of the N source, and N sources were then distinguished by the “rate of increase” of the function. Practically, the N-response curve was expressed using the following form:

$$y(\{N_i\}) = B - (B - A)e^{-\sum_i \tau_i N_i}$$

where y is the yield for a given crop on a given farm type, B and A represent the maximal and minimal yield of that crop, respectively, and $\tau_i$ represents the rate of increase of the yield response function to a nitrogen source i (e.g. manure, slurry etc.) the quantity of which is $N_i$.

### 9.1.2 N to emission functions

In addition to the amount of fertilizers, the STICS crop model provides us with the leaching associated to each type of soil and crop together with the climatic conditions and input application date. We derive the leaching functions from STICS outputs when nitrogen input varies within a reasonable range. Linear regression applied to STICS outputs allows us to relate N-losses and fertilizer amounts (through affine functions). Consequently, the emission function for pollutant i, related to crop j and farm group k, depends on the N-input through relation (17); $\alpha$ and $\beta$ are estimated for a given plant on a given farm group in the form:

$$[e_i]_{j,k}(N) = [\alpha_i]_{j,k} N + [\beta_i]_{j,k}; \quad i = \{NO_3, N_2O, NH_3\}. \qquad (17)$$

It should be noted that we focus only on the N-losses at the soil-root level (i.e. $NO_3$) and on two gas emissions ($N_2O$ and $NH_3$). The time lag existing between the N-fertilizer application and the actual pollution is not taken into account.

The spatialization method provides the contribution of each farm group on each cell of the grid to the regional (FADN) agricultural activities and enables AROPAj outputs to be distributed over France. Once the AROPAj farm groups have been distributed, we aggregate the results in terms of emissions among other model outputs.

For instance, Table (28) represents the emissions for $NH_3$, $N_2O$ and $NH_3$ for the main crops analyzed in AROPAj in France

### 9.1.3 Artix

We couple the AROPAj model to the STICS crop model in order to estimate input-output relationships (see above sections) for particular crops and the economic agents considered in the AROPAj model. The combination of an economic agent and a crop is hereafter referred to as a **crop-agent case**. For a specific crop-agent case, we estimate the input-output relationships from several simulations of the STICS crop model, while using various sources of data defining its physical (soil, altitude, slope, climate) and technical (such as sowing date, precocity group of the cultivar used, use of irrigation or not, fertilization calendar, preceding crop etc.) environments. This process thus requires the manipulation of various types of geo-referenced data as inputs to