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Positive Mathematical Programming for Agricultural and Environmental Policy Analysis: Review and Practice

Bruno Henry de Frahan^{1}, Jeroen Buysse^{2}, Philippe Polomé^{1}, Bruno Fernagut^{3}, Olivier Harmignie^{1}, Ludwig Lauwers^{3}, Guido Van Huylenbroeck^{2} and Jef Van Meensel^{3}

1 Université catholique de Louvain
2 Ghent University
3 Centre for Agricultural Economics, Brussels

Abstract: Positive mathematical programming (PMP) has renewed the interest in mathematical modeling of agricultural and environmental policies. This chapter explains first the main advantages and disadvantages of the PMP approach, followed by a presentation of an individual farmbased sector model, called SEPALE. The farmbased approach allows the introduction of differences in individual farm structures in the PMP modeling framework. Furthermore, a farmlevel model gives the possibility of identifying the impacts according to various farm characteristics. Simulations of possible alternatives to the implementation of the CAP Mid Term Reform illustrate the value of such a model. This chapter concludes with some topics for further research to resolve some of the PMP limitations.
Key words: positive mathematical programming, Common Agricultural Policy, agroenvironmental policy analysis
1.Introduction
There is a renewed interest in mathematical programming (MP) to model economic behaviour. This originates from a combination of factors. First, the emergence in the late 1980's of the positive mathematical programming (PMP) has brought an appealing breath of positivism in the determination of the optimising function parameters. This method formalised later by Howitt (1995a) makes it indeed possible to calibrate MP models exactly. Second, as a result of the former, PMP has provided a more flexible and realistic simulation behaviour of MP models avoiding unlikely abrupt discontinuities in the simulation solutions. Third, the increasing need to model and simulate behavioural functions under numerous technical, economic, policy and, more recently, environmental conditions has strengthened the recourse to MP models. Fourth, in an environment of oftenlimited amount of adequate information and data to treat complex decisions, MP models are nevertheless able to handle decision problems which econometrics cannot. With the increasing number of available databases assembled from data collected at the regional, territorial, farm and even land plot levels, the construction of MP models is now possible at a more disaggregated level of decision making. This allows the analysis of agricultural, environmental and land use policy in accordance with local conditions. This renewed interest in MP modelling for analysing agricultural and environmental policies has generated numerous applications as well as extensions at different investigation levels of which several are reported in Heckelei and Britz (2005).
This chapter concentrates on PMP and its recent developments as a tool for policy analysis and on the practical elaboration in Belgium. It is organised as follows: The next section shows how PMP renovates the calibration of mathematical programming models. Section 3 explains how its weaknesses have generated various developments including extensions bridging mathematical programming and econometrics. Exploiting together the advantages of mathematical programming and econometric approaches leads of a new field of empirical investigation that we would like to name econometric programming. The fourth section shows how the Belgian regional mathematical programming model SEPALE tackles some of the PMP weaknesses and adopts a calibration method able to exploit the richness of the European Union's Farm Accountancy Data Network (FADN) in representing the economic behaviour of a collection of farmers. The application shows how current reforms of the Common Agricultural Policy (CAP) are treated and simulated in SEPALE. The fifth section discusses some issues left to PMP. The last section concludes with a summary of the advantages and limitations of PMP for agricultural and environmental policy analysis.
PMP is a method to calibrate mathematical programming models to observed behaviours during a reference period by using the information provided by the dual variables of the calibration constraints (Howitt 1995a, Paris and Howitt 1998). The dual information is used to calibrate a nonlinear objective function such that the observed activity levels are reproduced for the reference period but without the calibration constraints. The term "positive" that qualifies this method implies that, like in econometrics, the parameters of the nonlinear objective function are derived from an economic behaviour assumed to be rational given all the observed and nonobserved conditions that generates the observed activity levels. The main difference with econometrics is that PMP does not require a series of observations to reveal the economic behaviour, which as a drawback deprives PMP from inference and validation tests.
Formalised by Howitt (1995a), PMP follows a procedure in three steps. The first step consists in writing a MP model as usual but adding to the set of limiting resource constraints a set of calibration constraints that bound the activities to the observed levels of the reference period. Taking the case of maximising gross margins with upper bounded calibration constraints, we write the initial model as in Paris and Howitt (1998):
Maximise Z = p' x  c' x (1)
subject to: A x b [] (1a)
x x_{o} + [] (1b)
x 0 (1c)
where:
Z scalar of the objective function value,
p (n x 1) vector of product prices,
x (n x 1) nonnegative vector of production activity levels,
c (n x 1) vector of accounting costs per unit of activity,
A (m x n) matrix of coefficients in resource constraints,
b (m x 1) vector of available resource levels,
x_{o} (n x 1) nonnegative vector of observed activity levels,
(n x 1) vector of small positive numbers for preventing linear dependency between the structural constraints (1a) and the calibration constraints (1b),
(m x 1) vector of duals associated with the allocable resource constraints,
(n x 1) vector of duals associated with the calibration constraints.
Assuming that all activity levels are strictly positive and all allocable resource constraints are binding at the optimal solution, the firstorder conditions of model (1) provide the following dual values as in Heckelei and Wolff (2003):
^{p} = p^{p}  c^{p}  A^{p}' (2)
^{m} = 0 (3)
= (A^{m}')^{1} (p^{m}  c^{m}) (4)
The vector x is partitioned into [(n – m) x 1] vector of preferable activities x^{p} constrained by the calibration constraints (1b) and (m x 1) vector of marginal activities x^{m} constrained by the allocable resource constraints (1a). The other vectors , p and c and the matrix A are partitioned accordingly.
Howitt (1995a) and Paris and Howitt (1998) interpret the dual variable vector associated with the calibration constraints as capturing any type of model misspecification, data errors, aggregate bias, risk behaviour and price expectations. In the perspective of calibrating a nonlinear decreasing yield function as in Howitt (1995a), this dual vector represents the difference between the activity average and marginal value products. In the alternative perspective of calibrating a nonlinear increasing cost function as in Paris and Howitt (1998), this dual vector is interpreted as a differential marginal cost vector that together with the activity accounting cost vector c reveals the actual variable marginal cost of supplying the observed activity vector x_{o}.
The second step of PMP consists in using these duals to calibrate the parameters of the nonlinear objective function. A usual case considers calibrating the parameters of a variable cost function C^{v} that has the typical multioutput quadratic functional form, however, holding constant variable input prices at the observed market level as follows:
C^{v}(x) = d' x + x' Q x / 2 (5)
where:
d (n x 1) vector of parameters of the cost function,
Q (n x n) symmetric, positive (semi) definite matrix with typical element q_{ii'} for activities i and i’.
Other functional forms are possible. The generalized Leontief and the weightedentropy variable cost function (Paris and Howitt, 1998) and the constant elasticity of substitution (CES) production function (Howitt, 1995b) in addition to the constant elasticity of transformation production function (Graindorge et al., 2001) have also been used.
The variable marginal cost vector MC^{v} of this typical cost function is set equal to the sum of the accounting cost vector c and the differential marginal cost vector as follows:
MC^{v} = C^{v}(x)_{xo}' = d + Q x_{o} = c + (6)
where:
C^{v}(x) is a (1 x n) gradient vector of first derivatives of C^{v}(x) for x = x_{o}.
To solve this system of n equations for [n + n(n + 1)/2] parameters and, thus, overcome the underdetermination of the system, PMP modellers rely on various solutions that are reviewed in the next section.
The third step of PMP uses the calibrated nonlinear objective function in a nonlinear programming problem similar to the original one except for the calibration constraints. This calibrated nonlinear model is consistent with the choice of the nonlinear activity yield or cost function derived in the preceding step and exactly reproduces observed activity levels and original duals of the limiting resource constraints. The following PMP model is ready for simulation.
Maximise Z = p' x  ' x  x' x / 2 (7)
subject to: A x b [] (7a)
x 0 (7b)
where the vector and matrix are the calibrated parameters of the nonlinear objective function.
Assuming again that all optimal activity levels are strictly positive and allocable resource constraints are all binding at the optimal solution, the firstorder conditions of model (7) provide the following dual values of the resource constraints as in Heckelei and Wolff (2003):
= (A ^{1} A')^{ 1} (A ^{1} (p  )  b) (8)
This calibration approach can be applied at the farm, regional and sector levels. When accounting data of a sample of F farms are available such as from the FADN, F PMP models can be defined for each farm of the sample. Simulations can then be performed on these individual PMP models and simulation results may be aggregated as shown in the application below.
