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.

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 farmer's allocation decisions can be easily extended to handle more than one observation. Copyright 1998, Oxford University Press.

This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry. We study the validity of variance inequalities in spaces of non-positive and non-negative Aleksandrov curvature. In this last scenario, we show that variance inequalities hold provided geodesics, emanating from a barycenter, can be extended by a constant factor. We also relate variance inequalities to strong geodesic convexity. While not restricted to this setting, our results are largely discussed in the context of the 2-Wasserstein space.Given a separable and complete metric space (M, d), define P 2 (M ) as the set of Borel probability measures P on M such that). (1.1) When it exists, a barycenter stands as a natural analog of the mean of a (square integrable) probability measure on R d . Alternative notions of mean value include local minimisers [Kar14], p-means [Yok17], exponential barycenters [ÉM91] or convex means [ÉM91]. Extending the notion of mean value to the case of probability measures on spaces M with no Euclidean (or Hilbert) structure has a number of applications ranging from geometry [Stu03] and optimal transport [Vil03, Vil08, San15, CP19] to statistics and data science [Pel05, BLL15, BGKL18, KSS19], and the context of abstract metric spaces provides a unifying framework encompassing many non-standard settings. Properties of barycenters, such as existence and uniqueness, happen to be closely related to geometric characteristics of the space M . These properties are addressed in the context of Riemannian manifolds in [Afs11]. Many interesting examples of metric spaces, however, cannot be described as smooth manifolds because of their singularities or infinite dimensional nature. More general geometrical structures are geodesic metric spaces which include many more examples of interest (precise definitions and necessary background on metric geometry are reported in Appendix A). The barycenter problem has been addressed in this general setting. The scenario where M has non-positive curvature (from here on, curvature bounds are understood in the sense of Aleksandrov) is considered in [Stu03]. More generally, the case of metric spaces with upper bounded curvature is studied in [Yok16] and [Yok17]. The context of spaces M with lower bounded curvature is discussed in [Yok12] and [Oht12].Focus on the case of metric spaces with non-negative curvature may be motivated by the increasing interest for the theory of optimal transport and its applications. Indeed, a space of central importance in this context is the Wasserstein space M = P 2 (R d ), equipped with the Wasserstein metric W 2 , known to be geodesic and with non-negative curvature (see Section 7.3 in [AGS08]). In this framework, the barycenter problem was first studied by [AC11] and has si...

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 ofthis 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.The question of how to generate an adequate food supply remains a pressing concern in developed and developing countries. As a consequence, appropriate fertilization technologies are important cornerstones of any sensible food policy.For the last three decades, the majority of agricultural economists has suggested the use of polynomial functions (quadratic and square root) for representing crop responses to fertilizer nutrients (for example, Baum, Heady, and Blackmore; Hexem and Heady; Woodworth). Proponents of polynomials argued that since the exact mathematical nature of crop response is unknown, approximation by polynomials is preferable in view of their computational simplicity and high fit. During the last decade, however, researchers have pointed out the inappropriateness of polynomial crop response functions (Anderson and Nelson, Lanzer and Paris). These relations consistently overestimate the maximum yield and the optimal fertilizer recommendations; they introduce the appearance of a nonexistent biological substitution between nutrients and, finally, their parameters do not possess agronomic interpretations easily discernible.Among these undesirable properties, the most crucial one is the excessive use of fertilizers derived from recommendations based on polynomials. Such a waste of scarce resources unnecessarily pollutes the environment and is intolerable for developing countries attempting to attain food self-sufficiency by boosting crop production through an increasing use of fertilizers.With the criticisms, a new family of crop response functions was introduced. It is inspired by basic agronomic principles and was called the von Liebig family of response functions in honor of the German chemist who first formulated those principles. Contrary to the polynomial response, a von Liebig function does not allow for nutrient substitution (although it admits interaction) and implies a response surface with a plateau maximum. An empirical study of crop response using a von Liebig specification was presented by Lanzer and Paris. Many of the criticisms of the polynomial response were detailed in that paper. However, no formal statistical test between the two rival specifications was given. This paper will provide such a rigorous test using a new set of data.' The evidence shows that the von Liebi...

A novel control model for technical and economic analysis of fertilizer recommendations is based on Liebig's Law of the Minimum, Mitscherlich's 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.

The development of optimal fertilizer recommendations requires a renewed collaborative effort between agronomists and agricultural economists. The purpose of this study is to emphasize the direction of this interdisciplinary effort in the area of crop response analysis. Using information from two separate experiments [corn (Zea mays L.) response to N and P fertilization and cotton lint (Gossypium hirsutum) response to N and irrigation water], it is shown that the best response model is a von Liebig specification, more commonly known as the law of the minimum. When the selection of a statistical specification is made among models that do not share the parameter space, hypothesis testing cannot be done by means of the familiar F‐test and likelihood ratio. The selection of the von Liebig specification (chosen against square‐root polynomial and Mitscherlich‐Baule models) is, thus, made on the basis of a rigorous statistical analysis using a nonnested hypothesis approach.

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.

This work establishes fast rates of convergence for empirical barycenters over a large class of geodesic spaces with curvature bounds in the sense of Alexandrov. More specifically, we show that parametric rates of convergence are achievable under natural conditions that characterize the bi-extendibility of geodesics emanating from a barycenter. These results largely advance the state-of-the-art on the subject both in terms of rates of convergence and the variety of spaces covered. In particular, our results apply to infinite-dimensional spaces such as the 2-Wasserstein space, where bi-extendibility of geodesics translates into regularity of Kantorovich potentials.