Quirino Paris

An Analysis of Ill-Posed Production Problems Using Maximum Entropy

Production economics problems are often ill-posed. This means that the number of parameters to be estimated is greater than the number of observations. In this article we show how to recover flexible cost functions from very limited data sets using a maximum entropy approach. We also argue that there exists a continuum of analysis between mathematical programming and traditional econometric techniques which is based solely upon the available information. The limiting case of a multi-output cost function recovered using only a single observation of a farmers allocation decisions can be easily extended to handle more than one observation. Copyright 1998, Oxford University Press.

The von Liebig Hypothesis

Using a celebrated sample of corn response to nitrogen and phosphorus collected by Heady and Pesek, it is shown that a von Liebig model with Mitscherlich regimes is the best interpreter of the experimental data. This assertion is based upon the results of nonnested hypothesis tests among five alternatives. An extension of the test to switching regression models reinforces the superiority of the von Liebig hypothesis for this sample. Cost functions, dual to von Liebig production functions, have continuous derivatives with respect to the parameters and may be easier to estimate when input prices are available.

Testing a von Liebig Crop Response Function against Polynomial Specifications

Crop responses to fertilizers traditionally have been specified as polynomial functions. Recently, criticisms were raised against such specifications because they force substitution between nutrients and overestimate the optimal fertilizer quantity. With those criticisms, an alternative crop response function was presented in the form of a minimum function which equates the realized output to the production potential associated with a limiting input. In this paper a nonnested test is performed to discriminate between the two rival specifications. The results of this test reject the hypothesis that a crop response is of the polynomial type, while they do not reject the hypothesis that it is of the minimum and plateau type.

A New Analytical Framework for the Fertilization Problem

A novel control model for technical and economic analysis of fertilizer recommendations is based on Liebigs Law of the Minimum, Mitscherlichs relative yield theory, and the notions of yield plateau and soil fertility carry-over. The empirical model consists of a multistage separable programming specification which maximizes the discounted stream of net revenues subject to crop response and fertility carry-over functions. The model is applied to the wheat-soybean cropping system in Southern Brazil. It is found that, while the optimal fertility target levels are within a small range of those determined by Brazilian agronomists, their maintenance recommendations could be substantially improved.

Estimation of von Liebig Response Functions

Two distinct but related approaches for the estimation of von Liebig response functions are presented. The first approach is based upon a two-phase, ordinary least squares procedure combined with bootstrapping for computing the standard errors of the estimates. The second approach generates maximum likelihood estimates of the parameters and can be implemented according to two different parameterizations of the model. Application of the procedures to a sample of experimental data suggests a satisfactory degree of conformity of the results. A test of normality of the disturbance term based upon the estimated residuals failed to reject the null hypothesis, further supporting the maximum likelihood approach.

Revenue and Cost Uncertainty, Generalized Mean-Variance, and the Linear Complementarity Problem

Symmetric quadratic programming and the linear complementarity problem are presented in the context of a novel interpretation of risk programming. The novelty admits stochastic supplies of limiting inputs, and yet it preserves the linearity of the structural constraints. This formulation is, therefore, a generalization of the traditional mean-variance approach originated with Markowitzs portfolio analysis and extended by Freund to farm planning under uncertain revenues. The specification admits also nonzero covariances between revenues and costs of limiting inputs and allows the computation of risk coefficients associated with a companion chance-constrained problem.

Symmetric Positive Equilibrium Problem: A Framework for Rationalizing Economic Behavior with Limited Information

The PMP methodology is extended to deal with many economic units, self-selection arising from heterogeneous economic behavior, and multiple limiting inputs. The novel structure for the analysis of this type of problems acquires the name “symmetric positive equilibrium problem.” This methodology is articulated in three phases that parallel those one of positive mathematical programming. The recovery of a cost function that calibrates the base-year decisions of each economic unit is realized within a maximum entropy framework. The methodology is illustrated by the analysis of a sample of Italian farms that operate under the complex policy mandates of the European Union. Copyright 2001, Oxford University Press.

A Programming Model with Stochastic Technology and Prices: The Case of Australian Agriculture

Risk associated with stochastic input-output coefficients and prices is modeled in a programming specification based upon mean-variance analysis and chance-constrained programming. The algorithm which exploits the features of MINOS, a commercially available computer program, is developed and applied to Australian agriculture. Convergence of the algorithm is discussed. With the data used, the thirty runs of this problem achieved a fast convergence with no more than five cycles. The model was used to derive supply response functions of export commodities of Australian agriculture. The relevant information was generated via parametric variations of prices selected according to a suitable experimental design.

Production Efficiency in the von Liebig Model

The objective of this paper is to revisit the von Liebig hypothesis by reexamining five samples of experimental data and by applying to it recent advances in Bayesian techniques. The samples were published by Hexem and Heady as described in a further section. Prior to outlining the estimation strategy, we discuss the intuition underlying our approach and, briefly, the literature on which it is based. We present an algorithm for the basic von Liebig formulation and demonstrate its application using simulated data (table 1). We then discuss the modifications needed to the basic model that facilitate estimation of a von Liebig frontier and we demonstrate the extended algorithm using simulated data (table 2). We then explore, empirically, the relationships between limiting water and nitrogen in the Hexem and Heady corn samples and compare the results between the two formulations (table 3). Finally, some conclusions and suggestions for further research are offered.

A Sure Bet on Symmetry

This paper argues that assertions appearing in the literature about alleged failure of duality to recover all the desired information about the case of multiple outputs with fixed allocatable inputs are incorrect. The paper shows that through the use of shadow prices of fixed inputs, it is possible to apply the familiar Euler-Legendre transformation and recover all the desired information. The principal avenue for achieving these results is the process of decoupling the primal from the dual problem.