Developments of PMP

68 the support values applied in the GME estimation. One way to deal with inappropriate modelling of farm reaction to changes, when the problem is ill-posed, consists in the use of exogenous information, such as supply elasticities. Heckelei and Britz (2000) extended the use of GME in a PMP framework when multiple observations are available. Multiple observations give information on the second order derivatives of the cost function and if the problem is well-posed arbitrary curvature of the cost function is avoided. When multiple observations are available and the problem is well-posed either GME or Least Squares (LS) can be implemented in a PMP framework to estimate the non-linear cost function (Paris and Arfini, 2000).

When all the parameters of the quadratic matrix are estimated either by GME or by LS, the Cholesky factorisation is applied in order to guarantee the symmetry and positive semi-definiteness of the Q matrix. The Cholesky factorisation decomposes the quadratic matrix into a lower triangular matrix and a diagonal matrix such that:

= '

Q LDL (3.12)

where L is a unit lower triangular matrix and D is a diagonal matrix whose elements are restricted to be non-negative. The estimation of the parameters of a mathematical programming model by the traditional econometrics techniques such as LS and GME on multiple observations are still a few (Heckelei et al., 2012).

Once the non-linear cost function has been estimated, the third step of the standard PMP uses this function to recover a calibrated non-linear programming model (equations 3.13-3.15) which reproduces exactly the base period level of primal and dual solutions without the calibration constraints.

max ' ' 1 '

π =p x d x− −2x Qx (3.13)

subject to Ax≤b ( )y (3.14)

x≥0 (3.15)

69 approach. Paris and Arfini work (2000) dealt with the problem of zero activity levels in some farms of an homogenous sample. They solved the self-selection problem by adding to the n farm LP models of the first step an additional model for an artificial farm and through this they calibrated a frontier cost function. The artificial farm resulted by summing the resources and the crop activity levels of all farms in the sample.

Paris (2001) proposed the Symmetric Positive Equilibrium Problem (SPEP) as a way to avoid the linear technology of the fixed input and to make the demand and supply of fixed input responsive to output levels and input price changes. The SPEP model contains symmetric primal and dual constraints in the first step, it recovers a total cost function which includes the cost of quasi-fixed inputs in the second step, and in the third step it results in a set of input demand, output supply, marginal cost and marginal revenue equations.

One of the most important modifications to the original PMP approach was proposed by Heckelei and Wolff (2003). The authors argued that the standard PMP approach leads to inconsistent parameters estimates of the calibrated cost function when multiple observations are used. They showed that, when multiple observations are available, the dual values of the resource constraints of the first step of PMP are different from the values of the third step non-linear model, which is assumed to be the ‘true’ model. As the dual values of the first step are used to recover the non-linear cost function and these values are shown to be distorted, they concluded that the estimates of the cost function parameters are biased. Howitt (2005) showed that for a single observation the dual values are the same in the first phase model and in the third phase model; when multiple observations are available, he proposed a two-step LP calibration in order to avoid the inconsistency. The alternative calibration procedure proposed by Heckelei and Wolff consists of skipping the first step of PMP and employing directly the first order conditions of the desired programming model to estimate simultaneously the non-linear cost function and the dual values. Let’s consider the desired programming model taking this form:

max ' ' 1 '

π =g x d x− −2x Qx (3.16)

subject to Ax≤b ( )y (3.17)

x≥0 (3.18)

where d and Q need to be estimated and gis the per unit of activity gross margin vector.

70 Assuming that the observed activity levels deviate from the optimum choices by a small stochastic errore, with mean zero and standard deviationσ , and that all the resource constraints are binding, the optimality first order conditions are:

( ) ' 0

− − − − =

g d Q x e y A (3.19)

A x e( − )=b (3.20)

The two optimality conditions (3.19) and (3.20) are used to estimate the parameters of non-linear cost function by either GME or LS techniques.

The work of Heckelei and Wolff represents a remarkable attempt to join mathematical programming model with econometric techniques within the new framework of EMP. De Frahan et al. (2007) showed the implementation of the direct estimation of the optimality conditions of the desired model to calibrate an agricultural model, SEPALE, composed by a collection of farm-level mathematical programming models. The authors did not include any resource constraints in the model, but the resources are considered tradable, thus entering the cost function as variable inputs. Another empirical application of the approach proposed by Heckelei and Wolff is represented by the work of Buysse et al. (2007a), which analysed the impact of the 2003 sugar reform in the EU by applying a PMP model to a sample of Belgian farmers. The authors applied directly GME estimation to the first order conditions of a farm level model with a quadratic cost function, assuming the tradability of land among farmers.

A further extension to the standard PMP approach is represented by the model proposed by Arfini and Donati (2008). Paris in his book ‘Economic Foundations of Symmetric Programming’ (2011: 397-404) refers to this model as an ingenious answer to two problems arising from the standard PMP. The first problem is the tautology problem raised by the presence of calibration constraints in the first phase of PMP, while the second issue concerns the non- obvious connection between the - LP independent problems of the first phase and the recovery of a quadratic matrix of the cost function common to all farms in the second phase. The idea of Arfini and Donati is to merge the first linear phase with the second non-linear phase of the PMP and to estimate simultaneously the parameters of the non-non-linear cost function, the shadow price of resources and the differential marginal costs. The theoretical representation of the model for the n^thfarm is:

71

( )

1 1

min 1 ' ' '

2

N N

n n n n n n n n

n n

b y

= =

 + + − 

 



∑

∑

subject to A'_n y +_n r_n ≥p _n (3.22)

r_n =Qx_n +u _n (3.23)

, 0

n n

y r ≥ (3.24)

where x is the vector of observed activity levels, _n b is the land available on the farm, _n A is '_n the matrix of technical coefficients, p is the vector of output prices,_n y is the shadow price of _n land, r is the activity marginal cost vector, Q is the quadratic term of the non linear cost _n function, which is common to all farms, while u is the vector of farm deviations from the _n cost function . The model merges the first linear phase with the second non-linear phase by minimising an objective function subject to a set of constraints. The objective function has two components: the square of the farm deviations from the average cost function and the differences between the primal and dual objective function of an LP problem which should be equal to zero. The constraint (3.22) is the dual constraint of the LP problem and it indicates the traditional economic equilibrium stating that marginal cost must be larger or equal to marginal revenue while the constraint (3.23) represents the link between the LP problem and the quadratic problem. Paris provided the Karush-Kunh-Tucker (KKT) conditions for the resolution of this model and he showed the ability of the model to reproduce the base year activity levels without explicit calibration constraints, which are nevertheless implicit in the setup of the problem.

Arfini and Donati (2011) provided an empirical application of a similar model to analyse the effects of some Health Check CAP reform proposals on farm samples in three different European regions. The main contribution of this model compared to the 2008 version consists in the estimation of the specific variable accounting costs, c , together with the estimation of the non linear cost function, while the shadow value of land is approximated by some exogenous rental prices. In order to estimate c correctly an additional constraint is added to the model which bounds the total estimated accounting cost to be lower or equal to the total variable costs included in the FADN database. Another constraint bounds the total estimated non-linear cost function to be larger or equal to the total variable costs included in the FADN database.

72