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

3.Further PMP developments

A second development of the PMP methodology concerns the integration of risk. For example, Paris (1997) uses a von Neumann-Morgenstern expected utility approach assuming a normal distribution of output prices and a constant absolute risk aversion.

A third development is the inclusion of greater competitiveness among close competitive activities whose requirements for limiting resources are more similar than with other activities. Rohm and Dabbert (2003) represent these close competitive activities as variant activities from their generic activities and add to the first PMP step calibration constraints for these variant activities that are less restrictive than their counterparts for their generic activities.

A fourth development to overcome criticisms that have been raised against the use of a linear technology in limiting resources and the zero-marginal product for one of the calibrating constraints is the expansion of the PMP framework into a Symmetric Positive Equilibrium Problem (SPEP). Paris (2001) and Paris and Howitt (2001) express the first step of this new structure as an equilibrium problem consisting of symmetric primal and dual constraints and the third step as an equilibrium problem between demand and supply functions of inputs, on the one hand, and between marginal cost and marginal revenue of the output activities, on the other hand. For these authors, the key novelty of this new framework is rendering the availability of limiting inputs responsive to output levels and input price changes. Britz et al. (2003), however, address several conceptual concerns with respect to the SPEP methodology and question the economic interpretation of the final model ready for simulations.

Other shortcomings comprise the under-determination of the system, the unequal treatment of the marginal and preferable activities and the first phase estimation bias. They are treated in the following three sub-sections.

3.1The under-determination problem

q_ii = (c_i + _i)/ x_io and d_i = 0 for all i = 1, …n,

or equal to the accounting cost vector c, which leads to:

q_ii = _i / x_io and d_i = c_i for all i = 1, …n.

Another calibration rule called the average cost approach equates the accounting cost vector c to the average cost vector of the quadratic cost function, which leads to:

q_ii = 2 _i / x_io and d_i = c_i - _i for all i = 1, …n.

Exogenous supply elasticities _ii are also used to derive the parameters of the quadratic cost function as in Helming et al. (2001):

q_ii = p_io / _ii x_io and d_i = c_i + _i - q_ii x_io for all i = 1, …n.

All these specifications would exactly calibrate the initial model as long as equations (6) are verified, but lead to different simulation responses to external changes.

A subsequent development from Paris and Howitt (1998) to calibrate the marginal cost function is to exploit the maximum entropy estimator to determine all the [n + n(n + 1)/2] elements of the vector d and matrix Q using the Cholesky factorisation of this matrix Q to guarantee that the calibrated matrix Q is actually symmetric positive semi-definite.¹ This estimator in combination with PMP enables to calibrate a quadratic variable cost function accommodating complementarity and competitiveness among activities still based on a single observation but using a priori information on support bounds. Nevertheless, as argued in Heckelei and Britz (2000), the simulation behaviours of the resulting calibrated model would be still arbitrary because heavily dominated by the supports.

Heckelei and Britz (2000) exploit the suggestion from Paris and Howitt (1998) to use the maximum entropy estimator to determine these parameters on the basis of additional observations of the same farm or region in a view to collect information on second order derivatives. They estimate the parameters of the vector d and matrix Q on the basis of cross sectional vectors of marginal costs and the use of the Cholesky decomposition of the matrix of the second order derivatives as additional constraints. They obtain a greater successful ex-post validation than using the standard "single observation" maximum entropy approach. This cross sectional procedure is an interesting response to the lack of empirical validation for models that are calibrated on a single reference period. It is used to calibrate the cost functions of the regional activity supplies of the Common Agricultural Policy Regional Impact (CAPRI) modelling system (Heckelei and Britz, 2001).