The standard PMP approach

PMP is a methodology to calibrate mathematical programming models to the base year observed outcomes by using the dual information provided by the calibration constraints.

Besides the perfect calibration to the base year level of the endogenous variables, PMP avoids the introduction of artificial constraints and it guarantees smooth reactions to changes in parameter values. Although the PMP methodology was already applied in the eighties in agricultural economics research, it was formalised and published for the first time by Howitt in 1995. The basic assumption of PMP is that the observed choices of the decision maker are the optimum choices, which maximise the decision maker’s objective function. Howitt (1995:332) stated that ‘if the model does not calibrate to observed production activities with the full set of general linear constraints that are empirically justified by the model, a necessary condition for profit maximization is that the objective function be nonlinear in at least some of the activities’. The information contained in the dual values of the calibration constraints are used to infer the marginal cost of each activity level and by that to recover a farm non-linear cost function such that the endogenous variables replicate exactly their level in the base year.

The standard PMP approach is a three step procedure (Paris and Howitt, 1998). In the first step a linear or non-linear programming model is specified adding to the set of resource constraints (equation 3.7) a set of calibration constraints (equation 3.8) that bind the activities to the observed levels:

maxπ =p x c x' − ' (3.6)

subject to Ax≤b ( )y (3.7)

x≤ x+ε ( )λ (3.8)

≥

x 0 (3.9)

where π is the profit to be maximised, p and c are the n×1vectors of output prices per unit of activity and accounting variable costs per unit of activity respectively and x is the n×1 vector of endogenous activity levels. In the standard PMP approach, the x vector can be partitioned into a (n−m) 1× vector of preferable activities bounded by the calibration constraints and into a m ×1vector of marginal activities, or less profitable activities, bounded

66 by the resource constraints. Ais the m ×n matrix of technical coefficients which indicates the amount of each resource used per unit of each activity, bis the m×1vector of resource availability, x is the n×1vector of observed activity levels in the reference year and εis a small positive number vector (perturbance term vector) which prevents the linear dependency between the calibration and the resource constraints. The first step results in the dual value vectors yand λassociated to the resource constraints and to the calibration constraints respectively. The dual values vector of the calibration constraints λ represents the differential marginal costs vector, which added to the activity observed accounting costs vector provides the total marginal costs to produce observed activity levels. The value of λ is equal to zero for the marginal activities, as these activities are bounded by the resource constraints. λis not present in the accounting book of the farmer but it is implicitly included in the observed output level and it captures any type of model misspecification, data errors, aggregate bias, risk behaviour and price expectations (De Frahan et al., 2007).

The second step of PMP uses the dual values of the calibration constraints to recover a non-linear cost function. As stated by Heckelei and Britz (2005) “Any type of non-non-linear function with the required properties qualifies for phase 2. For reasons of computational simplicity and lacking strong arguments for other type of functions, a quadratic cost function is often employed.” The quadratic cost function usually applied is a multi-output quadratic functional form without input prices, which are assumed to be fixed at the market level:

( ) ' 1 '

C x =d x+2x Qx (3.10)

where dis the n ×1 linear term vector of the cost function and Qis the quadratic n n× matrix which is symmetric and positive semi-definite to guarantee the convexity property of the cost function.

The parameters of the cost function are recovered by using the information contained in the dual values of the calibration constraints by using the equation:

( ) = + = +

MC x c λ d Qx (3.11)

where M Cis the farm marginal cost function vector. This marginal cost function does not include the opportunity cost of fixed inputs, which are implicitly accounted in the dual value of the resource constraints of the final model.

67 The estimation of d andQ implies the estimation of (n+n*(n+1)/2) parameters where n indicates the number of activity in the base year farm production plan. Originally PMP was proposed to calibrate mathematical programming models with a minimum dataset. The PMP pioneering works provided examples of recovering the farm cost function when only data on a single farm in one year was available. Besides the extreme case of just one observation available, most PMP problems are ill-posed, meaning that the number of parameters to be estimated is larger than the number of observations. To overcome the under-determination problem, PMP modellers adopted some ad hoc restrictions on the parameters (see de Frahan et al., 2007 and Heckelei and Britz, 2005 for an overview). The most common assumption of the first implementations of PMP was setting the off diagonal elements of the Qmatrix equal to zero, implying that the marginal cost of activity is not affected by the level of activity .

Under this assumption, the substitution and complementarity relationships between activities are not considered.

Together with this common hypothesis, some additional restrictions were imposed. The most common ones consisted in setting the linear term d of the marginal cost function equal to zero or equal to the accounting costs vector c; one may also equates the accounting costs vector cto the average costs vector of the quadratic cost function. In all these cases the marginal costs of the less profitable activities, which are bounded by the resource constraints, are constant as their activity shadow values, λ , derived from the first phase of PMP, are equal to zero. An ad hoc solution to get an increasing marginal costs for the marginal activities consisted in retrieving some share of the dual values of the resource constraint and transfer it to the dual values of the calibration constraints. Helming (2005) used the exogenous supply elasticity to recover all the on-diagonal elements of the Q matrix.

Paris and Howitt (1998) proposed the application of the Generalized Maximum Entropy (GME) technique in the second step of PMP in order to recover all the (n+n*(n+1)/2) parameters of the farm total cost function in the case of both ill-posed and well-posed problems. GME also allows the integration of exogenous information, such as supply elasticity, in the estimation phase and it accommodates multiple observations. By the introduction of GME in the mathematical programming framework, the authors started to cross the bridge between econometrics and mathematical programming. However, their work still used only one observation. Heckelei and Britz (2005) argued that the use of a single observation does not provide any information on the curvature of the cost function and hence the farm response to economic changes is largely dependent on the ad hoc restrictions or on

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)