In order to study the adaptation of farmers’ practices in response to the implementation of the 2003 CAP reform, a mathematical programming model was built. This method allows us to identify the effects of the decoupling on the production system (i.e. the allocation of land areas to different crops, the level of intensification, environmental impact, etc.). An econometric model would not meet this objective because in that type of model there is no change of farmer’s practice; the structure is constant. With the mathematical programming method, the model can stop certain activities or increase others.

Bio-economic model: a farm-level approach

We built a bio-economic model which takes into account the farmer’s response to price variation and several technical and biological elements in order to represent as accurately as possible the functioning of a dairy farm. Mathematical Programming is a technique which enables us to represent the farm functioning in reaction to a set of constraints. It is an appropriate technique because its assumptions correspond to those of classic microeconomics: rationality and the optimising nature of the agent (Hazell and Norton, 1986). This method allows us to study threshold effects and to calculate dual values of inputs (marginal yields). Farm-level modelling enables simultaneous consideration of production, price and policy information.

Any model derived from mathematical optimisation has three basic elements (Matthews etal., 2006): 1) an objective function, which minimises or maximises a function of the set of activity levels; 2) a description of the activities within the system, with coefficients representing their productive responses; and 3) a set of constraints that define the operational conditions and the limits of the model and its activities. Given the objective function, the solution procedure determines the optimal solution considering all activities and restrictions simultaneously.

The model optimises the farm plan, which represents the quantities of different outputs produced and inputs used. The economic results follow from those quantities and their prices. The model is used to estimate the effects of institutional, technical and price changes on the farm plan, economic results and intensification indicators.

Many studies have demonstrated that farmers typically behave in a risk-averse way (Hardaker et al., 2004). As such, farmers often prefer farm plans that provide a satisfactory level of security even if this means sacrificing some income. For the farmer, the main issue raised by variability of price and production is how to respond tactically and dynamically to opportunities or threats in order to generate additional income or to avoid losses. Moreover, during the years 2007, 2008 and 2009, prices of agricultural commodities were subject to strong variations so that we had to take the farmer’s sensitivity to price volatility. For example, the price of milk paid to the producers nearly doubled through 2007, from EUR 240/tonne to EUR 380/tonne before strongly decreasing to EUR 220/tonne in April 2009. Since the beginning of 2010, milk price seems to be on an increasing trend. Prices of cereals such as wheat have followed the same fluctuations. Cereals play a special role in dairy farming because they can be both input and output.

Lambert and McCarl (1985) present a mathematical programming formulation that allows identification of the expected utility function. Their approach, which does not require an assumption of normally distributed income (unlike the E-V, MOTAD and Target MOTAD methods), can accommodate the assumption that the utility function is monotonically increasing and concave (risk-averse). Patten et al. (1988) reformulated this approach as Utility Efficient Programming (UEP). Moreover, Zuhair et al. (1992) show that the negative exponential utility function (with Constant Absolute Risk Aversion, CARA) can better predict farmers’ behaviour than cubic and quadratic functions. The CARA function is a reasonable approximation to the real but unknown utility function: the coefficient of absolute risk variation can be validly applied to consequences in terms of losses and gains for variations in annual income. The UEP method enables the model to take into account asymmetric price distribution: the skewness becomes an element of decision as well as the variation amplitude. Thus, the model maximizes the expected utility of the income as follows:

Maximize: E[U] = p U(k, r), r varying [1]

with: Uk = 1 - exp(-ra x Zk) where Z is the net farm income for state k, and r is a non-negative parameter representing the coefficient of absolute risk aversion:

where X is a parameter reflecting variation in risk preference, and r_{max} and r_{min} are upper and lower bounds of the coefficient of absolute risk aversion (r_{a}).

In a more detailed form, the income Z is defined by:

• The main part of the income Z is given by milk revenues: the milk quantity multiplied with Ta the total number of animal of type a (dairy cows, heifers, calves and young bulls); mYa the milk yield (litre/day) per animal by mP the milk price (EUR/litre).

• There is then the meat revenue with aS_{a} the number of animals sold, aW_{a} the animals’ average carcass weight (kg) and aP_{a} the meat price (EUR/kg). At the end of the lactation, cull cows are sold and benefit from the female slaughter premium (SP_{a}) and young bulls benefit from the special premium for bovine male (SPBM_{a}).

• Then we take out livestock costs as: cfQ_{conc},_{p},_{a} the quantity of concentrate feed ingested (kg/day/animal), cfP_{conc} the concentrate feed price (EUR/kg per type of concentrate conc); I_{a} the specific inputs for animals (artificial insemination, medicines, herd book, minerals).

• We add the crop revenue as: X_{c} the cultivated area (ha) for each type of crop c (wheat, maize (corn), rapeseed, pea, maize silage, pasture, hay and grass silage); Y_{c} the crop yield (kg/ha); cP_{c} the crop price (EUR/kg); I_{c} the specific crop inputs (seed, treatments and harvesting); nQ_{c} the nitrogen quantity (kg/ha); and nP the nitrogen price (EUR/kg).

• Finally we consider the fixed costs FC (electricity, water, mechanisation, buildings, rent for land, insurance, taxes and other fixed costs). These fixed costs are specific to each type of farming.

The central element in the Linear Programming model is the dairy cow. The model represents the operation of a dairy farm for a one-year period. The classical duration of lactation is 305 days, followed by 60 days of drying off. The year is divided into four seasons of 91.25 days. The fecundity rate is lower for the most productive cows, thus decreasing the number of calves per cow per year. Regarding the progeny, it is assumed that, according to the intensification level of the type of farming, 25% to 35% of the dairy cows are replaced per year by heifers raised on the farm. Concerning female calves which are not assigned to replace cows, the model can choose between: 1) selling the calves at the age of 8 days; and 2) keeping the calves until two years old and then selling to the slaughterhouse (with the female slaughter premium).

Regarding plant production, the forage crops produced in France are mainly maize silage, grass silage, hay and pasture. All farmers aim for forage self-sufficiency; the purchase and/or sale of forage are not considered because these are activities linked to exceptional events (e.g. drought or exceptional harvest) in these areas. Farmers must comply with the set-aside requirement in order to benefit from the crop premium: we use a binary variable which is 0 if the farmer does not set aside land, and 1 if he does. It is assumed that the cereals are sold at harvest time, i.e. no crop storage except for wheat used to feed the cows.

Thornton and Herrero (2001) show a wide variety of separate crop and livestock models, but the nature of crop-livestock interactions, and their importance in farming systems, makes their integration difficult. That is why, in order to precisely describe the operation of a dairy farm, this model considers four important characteristics: 1) the seasonality of labour and grass production, 2) the response of crop yield to nitrogen use, 3) the non-linearity of milk yield per cow, and 4) the interaction between crop and animal production.

Four periods p (spring, summer, autumn and winter) are distinguished in the model. It allows for seasonal specification of grass production and grassland use (Berentsen et al., 2000). Seasonal variations enable us to integrate differences in the growth potential of grass during the growing season as well as the evolution of the nutrient content of grass. Moreover, we introduce seasonal labour constraints by allocating labour needs to each activity according to the work peaks (harvesting and calving). It is assumed that the farmer and his family/associates execute all the work, and thus there is no option to hire temporary labour. The model is more able to reflect temporal conditions thanks to the addition of these parameters.

The global working time per period (with Wt_{a},_{p} the working time per animal; Wt_{cp} the working time per ha of crop; FL is the fixed labour) has to be lower than the labour availability per period (AL_{p} the available labour for each annual work unit (AWU)).

Crop yield depends on the quantities of nitrogen used. Godard et al. (2008) formulated an exponential function, which satisfies economic requirements for attaining a mathematical optimum (the yield curve has to be concave and strictly increasing) and is consistent with its expected agronomic shape and with parameters with an agronomic interpretation.

where Y_{c} is yield for each crop, and Ymin_{c} and Ymax_{c} are respectively the minimal and maximal yield (different according to the type of farming and its level of intensification); ti represents the rate of increase in the yield response function to a nitrogen source i (e.g. manure, slurry, chemical nitrogen) the quantity of which is N_{i}. This enables us to take the increasing price of nitrogen into account and also the flow of organic nitrogen (such as manure) on the farm (Manos et al., 2007).

In order to give more flexibility to the model, milk production per cow is not fixed. Farmers have the possibility to choose the milk yield per animal in a range of 1 000 litres below the dairy cow’s genetic potential. It is also possible for farmers to produce beyond the genetic potential (Brun-Lafleur et al., 2009); in this case, nutritional requirements needed to produce one litre of milk are increased (from 0.44 to 1.2 energy units per litre of milk, and from 48 to 140 units of protein per litre of milk) (Faverdin et al., 2007).

With these three elements, we can very accurately represent the feeding system. The quantity ingested per cow per day is determined by using nutritional requirements in biological unit b (energy and protein), and the composition of forages and concentrate feed in equation 6 (INRA, 2007). The concentrate feeds conc available in the model are soybean meal, rapeseed meal, wheat, production concentrate and milk powder.

For each nutrient unit b and period p:

with: MR_{ab} the maintenance requirement (in energy and protein)

MYa the milk yield (in litre per animal per day)

LR_{ab} the lactation requirement (in energy and protein for one litre of milk)

fnc_{c},_{p},_{b} the forage nutrient content (in energy and protein per kg of forage)

fQ_{c},_{P},_{a} the forage consumption (kg) for each crop c, each period p and each type of animal a

Cfnc_{concpb} the concentrate feed nutrient content (in energy and protein per kg of concentrate)

CfQcon_{c},_{P},_{a} the concentrate feed consumption (in kg per day per concentrate per period per animal)

The global nutritional needs for the herd must not exceed the availability in forage and concentrate feed.. Moreover, the forage consumption (for each type of forage c) has to be lower than the forage production:

subject to:

Consequently, in order to maximise the farm’s income, the model determines the optimum for the following endogenous variables: number of each type of animal (T_{a} and aS_{a} for sale); milk yield per cow (mY_{a} in kg per cow per day); concentrate feed and forage consumption for each type of animal and per period (CfQ_{conc},_{P},_{a} and, fQ_{c},_{P},_{a} in kg per animal per day per season); the crop rotation (Y_{c} in ha); the level of nitrogen fertilisation (nQ_{c}for chemical nitrogen and manure, in kg); and crop yield (Y_{c} in kg per ha).

The model tries to offer the largest choice of technical practice for crop and animal production. That is why we choose to incorporate each “quantity variable” (as ha and kg) as endogenous variables in the model. Thus, the model has access to all possible situations, e.g. the model can choose a full grass diet for a cow which produces 7 000 litres of milk or a full maize diet for the same cow. The model will therefore calculate the optimal quantity of input and output.