# Methodology

The assessment of the effects of the Health Check regulation by the introduction of SPS regionalization on European farms is conducted by analysing, in addition to economic performance and farming system, the change in the farm’s strategy as a consequence of the new policy and market scenario.

We propose the adoption of a model that integrates the Positive Mathematical Programming (PMP) approach — which represents the characteristics of the farms and simulates the effects of the agricultural policy measures — with a cluster analysis technique able to group farms characterized by the same production strategy and economic characteristics.

## The PMP model

In its classical form, PMP as presented by Paris and Howitt (1998) is an articulated method consisting of three different phases, each of which is geared at obtaining additional information on the behaviour of the farm so as to be able to simulate its behaviour when maximizing total gross margin (Paris and Howitt, 1998; Arfini and Paris, 2000). The PMP method has been widely used in the simulation of alternative policy and market scenarios, utilising micro technical-economic data relative both to individual farms and to average farms that are representative of a region or a sector (Arfini *et al., *2005). The success of the method is largely attributable to the relatively low requirement for information on farm activities and, first and foremost, to the possibility to use databases, such as the FADN (Arfini *et al.,* 2005).

Notwithstanding the numerous studies that have adopted the PMP approach using FADN data, the methodology nonetheless comes up against a limitation consisting of the lack of specific production costs per farm enterprise. The lack of this information poses a problem during the calibration phase of the model, when the estimation of the cost function requires a non-negative marginal cost for all enterprises of a single holding (Paris and Arfini, 2000).

This problem is dealt with in this analysis by resorting to an approach that utilises dual optimality conditions directly in the estimation phase of the non-linear function. The approach represents an extension of the Heckelei proposal (2002), according to which the first phase of the classical PMP method can be avoided by imposing first-order conditions directly in the cost function estimation phase by introducing the value of the rented land, given from the market, as a dual value. This procedure requires the use of information external to the FADN dataset and provided by experts or by regional investigations. The main disadvantage of this procedure is that the external data are not always homogeneous with the characteristics of the farms covered in the FADN sample. For several reasons, the rental value of the land may change within the region, and the dual price of the land may also be quite different for different farm types according to their production sector (milk or cereals), their size, their level of specialization and the specific characteristics of each farm holder. In sum, the value of the rented land is not easy to collect and can lead to miss-specification of PMP models.

Moreover, as a guide to the correct estimation of the explicit production cost per crop, we propose the consideration of the information relative to the total variable costs available in the European FADN archive. This innovation becomes particularly important as it enables us to perform analyses utilising the EU data base without having to resort to parameters that are exogenous to the model.

According to this new approach, the PMP model falls into two phases: 1) the estimation of specific accounting variable costs per crop through the reconstruction of a non-linear function of the total variable cost observed for each individual farm of the FADN sample; and 2) the calibration of the observed production situation through the solution of a farm gross margin maximization problem, in the objective function of which the cost function estimated in the previous phase is considered.

The first phase is defined by an estimation model of a quadratic cost function in which the squares of errors are minimised:

By means of the model [1]-[9], a non-linear cost function can be estimated using the explicit information on the total farm variable costs (TVC) available in the FADN database. The restrictions [2] and [3] define the relationship between marginal costs derived from a linear function and marginal costs derived from a quadratic cost function. c+к defines the sum of the accounting variable costs and the differential marginal costs. The latter are implicit in the decision-making process of the entrepreneur and are not accounted for in the holding’s accounts. Both components are variables endogenous to the minimization problem. To guarantee consistency between the estimate of the total specific costs and those effectively recorded by the farm accounting system, the constraint [4] ensures that the total estimated explicit cost should not be greater than the total variable cost observed in the FADN database. Equation [5] defines that the costs estimated by the model by the non-linear cost function must at least equal to the value of the total variable cost (TVC) measured. In order to guarantee consistency between the estimation process and the optimal conditions, restriction [6] introduces the traditional condition of economic equilibrium, where total marginal costs must be greater than or equal to marginal revenues. The total marginal costs also consider the use cost of the factors of production defined by the product of the technical coefficients matrix A' and the shadow price of the restricting factors y; while the marginal revenues are defined by the sum of the product selling prices, p, and any existing coupled subsidies. The additional constraint [7] defines the optimal condition, where the value of the primal function must correspond exactly to the value of the objective function of the dual problem. In order to ensure that the matrix of the quadratic cost function is symmetric positive semi-definite, the model adopts Cholesky’s decomposition method, according to which a matrix that respects the conditions stated is the result of the product of a triangular matrix, a diagonal matrix and the transpose of the first triangular matrix [8]. The estimated matrices are presented in Annex 4.1. Last but not least, restriction [9] establishes that the sum of the errors, u, must be equivalent to zero.

The cost function estimated with the model [1]-[9] may be used in a model of maximization of total gross margin, ignoring the calibration restrictions imposed during the first phase of the classical PMP approach. In this case, the dual relations entered in the preceding cost estimation model guarantee the reproduction of the situation observed. The model therefore appears as follows:

The model [10]-[12] precisely calibrates the farming system observed, thanks to the non-linear cost function entered in the objective function, which preserves the (economic) information on the levels of production effectively attained. The matrix *Q* estimated is

reconstructed using Cholesky’s decomposition: Q = R'R = LDL'. Constraint [11] represents the restriction on the structural capacity of the farm, while the relation [12] enables us to obtain information on the hectares of land (or number of animals) associated with each process j. Once the initial situation has been calibrated through the maximization of the corporate gross margin, it is possible to introduce variations in the public aid mechanisms and/or in the market price levels in order to evaluate the reaction of the farm to the changed environmental conditions. The reaction of the farm production plan takes into account the information used during the estimation phase of the cost function, where it is possible to identify a true matrix of the firm’s choices, i.e. Q.

This PMP model can be used in two different contexts: 1) the estimation of the explicit variable accounting cost (c) related to each activity whose data are collected by the FADN, and 2) the estimation of the total variable cost per crop perceived by the farmer (c + к). This latter provides an information set useful for evaluating farm behaviour by means of the definition of a new profit function.

An additional element to consider is given by the introduction of full decoupling, and the related SPS, in the model. This aspect is given by a specific constraint that links *ex ante* the entitlement value — per unit — to the number of entitlements. Only the eligible area represented by eligible crops can benefit from the decoupled payment.

Equation [13] ensures that the variable related to the admissible area *ham _{n}* should be less than or equal to the number of entitlements in each farm,

*hdir*, where

_{n}*n*represents the

*n*farm

^{th}*(n=1,...,N).*The second constraint [14] means that the land admissible to the payment,

*ham*, plus the land admissible but not payable because not linked to the number of entitlements owned,

_{n}*hamd*, must be less than or equal to the total land attributed to the eligible farm crops. Obviously, only the variable

_{n}*ham*is present in the objective function.

_{n}In the case of regionalization, the structure of the constraints does not change, but rather the value of each entitlement, which will be equal for all the farms belonging to the same region. Moreover, the *j* admissible activities cover the whole farm surface.