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Under a technology based on the generalized mean of inputs and outputs with constant returns to scale (CRS), the firms have incentive to merge (in a firm game) in order to improve their technical efficiency. A directional complementarity property in inputs and in outputs is introduced. It is shown that the core of the firm game is non-void whenever the aggregate technology of each coalition exhibits complementarity in outputs and CRS. In the case of $$\alpha $$ -returns to scale, the firms have incentive to merge (improvement of technical efficiency) when there are both directional complementarity in inputs and in outputs.

The renormalization procedure facilitates the detection of periodic and chaotic patterns of many two-dimensional systems of nonlinear equations. We first give a comprehensive presentation of this procedure. We then generalize it to make it more applicable to economic frameworks.

This contribution introduces the so-called quasi-Leontief functions. In the framework and the language of tropical algebras, our quasi-Leontief functions are the additive functions defined on a semimodule with values in the semiring of scalars. This class of functions encompasses as a special case the usual Leontief utility function. We establish the existence of efficient Nash equilibria when the strategy spaces are compact and pathconnected topological semilattices.

The purpose of this contribution is to highlight an underexplored property of the directional distance function, a recently introduced generalization of the Shephard distance function. It diagnoses in detail the economic conditions under which infeasibilities may occur for the case of directional distance functions and explores whether there exist any solutions that remedy the problem in an economically meaningful way. This discussion is linked to determinateness as a property in index theory and is illustrated by analyzing the Luenberger total factor productivity indicator, based upon directional distance functions. This indicator turns out to be impossible to compute under certain weak conditions. A fortiori, the same problems can also occur for less general productivity indicators and indexes.

This paper starts from Fukuyama and Weber's (2004) work on testing for inverse homotheticity to show that the equality of the output gain function and the Zieschang output gain function is not equivalent to inverse homotheticity. Hence, testing for the equality of these gain functions cannot prove that a technology is inverse homothetic. Moreover, it is established that the gain functions are equal if and only if the direct and indirect isoquant and the corresponding efficient set are equal, which does not depend on inverse homotheticity.

This paper presents new developments on the state-contingent theory of production under uncertainty with stochastic prices. Our main purpose is to generalize the usual finite discrete state-contingent production model to infinite dimensional, possibly uncountable spaces which look like a more realistic framework. Usual duality results are established in this general context, shedding some light on the links between risk-neutral probabilities and shadow prices. A direct generalized production risk premium is defined and is shown to be independent of the inputs level when the technology is output translation homothetic. In such a case, the technology exhibits constant absolute riskiness.

The Luenberger productivity indicator has many attractive features to evaluate productivity, technical and efficiency changes. Paralleling Fare et al. (1997), this paper shows that the technical change indicator can be expressed as the sum of a magnitude indicator and a bias indicator that is the sum of two bias indicators input and output oriented respectively. Using a recent concept of ``parallel neutrality'' introduced by Briec et al. (2006), some conditions under which each bias indicator makes no contribution to productivity change are established. Among the key contributions of this paper is a new linear programming model involving a graph translation homotheticity property.

This paper develops the notion of parallel-neutral technical change for inputs and outputs. It is shown that parallel-neutral technical change generalizes existing radial notions of neutral technical change. A taxonomy of the structural consequences of parallel-neutral technical change is developed.

In a recent article, Briec, Kerstens and Vanden Eeckaut (2004) develop a series of nonparametric, deterministic non-convex technologies integrating traditional returns to scale assumptions into the non-convex FDH model. They show, among other things, how the traditional technical input efficiency measure can be analytically derived for these technology specifications. In this paper, we develop a similar approach to calculate output and graph measures of technical efficiency and indicate the general advantage of such solution strategy via enumeration. Furthermore, several analytical formulas are established and some algorithms are proposed relating the three measurement orientations to one another.

Introducing a new difference-based Luenberger-Hicks-Moorsteen productivity indicator, this contribution establishes theoretically its relations with some existing ratio- and difference-based productivity indexes and indicators. The first main result is an approximation proposition stating that the logarithm of the Hicks-Moorsteen productivity index is about equal to the Luenberger-Hicks-Moorsteen productivity indicator. Secondly, we also establish the specific conditions under which the Luenberger-Hicks-Moorsteen indicator equals the recently introduced Luenberger indicator and compare these to the conditions governing the relations between ratio-based Hicks-Moorsteen and Malmquist indices.

The Markowitz portfolio theory (Ref. 1) has stimulated research into the efficiency of portfolio management. This paper studies existing nonparametric efficiency measurement approaches for single-period portfolio selection from a theoretical perspective and generalizes currently used efficiency measures into the full mean-variance space. We introduce the efficiency improvement possibility function (a variation on the shortage function), study its axiomatic properties in the context of the Markowitz efficient frontier, and establish a link to the indirect mean-variance utility function. This framework allows distinguishing between portfolio efficiency and allocative efficiency; furthermore, it permits retrieving information about the revealed risk aversion of investors. The efficiency improvement possibility function provides a more general framework for gauging the efficiency of portfolio management using nonparametric frontier envelopment methods based on quadratic optimization.

Three main objectives are pursued in this paper. First, we intend to analyze the aggregation problem of directional distance functions from a constructive viewpoint. We provide necessary and sufficient conditions concerning the structural properties of the production technology and of the nature of groups of firms. Indeed, exact additive aggregation holds for a linear technology and for a direction solely defined in the output space. Second, since these conditions are somewhat restrictive, we are interested in providing a measure for the aggregation bias through the relationship between industrial and structural technical efficiency. Finally, we show that this aggregation bias is a lower bound for industrial allocative efficiency.

This paper extends the recent work by Frei and Harker on projections onto efficient frontiers (1999) in three ways. First, we provide a formal definition of the production set as the intersection of a finite number of closed halfspaces. We emphasize the necessity of a complete enumeration of the supporting hyperplanes to define the production set properly. We focus on the problem of exhaustive enumeration of the supporting hyperplanes to characterize the production set. Second, we consider the problem of an arbitrary-norm projection on the boundary of the production set. We use the concept of the Holder distance function and we derive the necessary mathematics to calculate distances and projections of inefficient DMUs onto the efficient frontier. Third, we introduce a relevant weighting scheme for inputs and outputs so that the Holder distance function respects the commensurability axiom defined by Russell (1988). Finally, we present an illustration using the same data set as Frei and Harker (1999) to highlight some of the extensions proposed in the paper.

The purpose of this short article is to simplify goodness-of-fit methods to obtain qualitative information about returns to scale for individual observations. Traditional and new goodness-of-fit methods developed for estimating returns to scale on nonparametric deterministic reference technologies are reviewed. Using composition rules for technologies with specific returns to scale assumptions, we show how these goodness-of-fit methods can be simplified in the case of convex technologies (Data Envelopment Analysis (DEA) models).

In this paper, we intend to establish relations between the way efficiency is measured in the literature on efficiency analysis and the notion of distance in topology. To this effect, we are interested particularly in the Holder norm concept, providing a duality result based upon the profit function. Along this line, we prove that the Luenberger shortage function and the directional distance function of Chambers, Chung, and Fare appear as special cases of some l p distance (also called Holder distance), under the assumption that the production set is convex. Under a weaker assumption (convexity of the input correspondence), we derive a duality result based on the cost function, providing several examples in which the functional form of the production set is specified.

The subject of this paper is to study the problem of the minimum distance to the complement of a convex set. Nirenberg has stated a duality theorem treating the minimum norm problem for a convex set. We state a duality result which presents some analogy with the Nirenberg theorem, and we apply this result to polyhedral convex sets. First, we assume that the polyhedral set is expressed as the intersection of some finite collection of m given half-spaces. We show that a global solution is determined by solving m convex programs. If the polyhedral set is expressed as the convex hull of a given finite set of extreme points, we show that a global minimum for a polyhedral norm is obtained by solving a finite number of linear programs.

We have two goals we wish to accomplish in this article. The first is the development of a framework for measuring efficiency in the full input–output space. This approach introduces a graph-type extension of the Farrell measure of technical efficiency. The second is the introduction of a weighting scheme for inputs and outputs, taking account of the particularity of the market summarized by input and output prices.

Luenberger [8] introduced the so-called benefit function that converts preferences into a numerical function that has some cardinal meaning. This measure has a number of remarkable properties and is a powerful tool in analyzing welfare issues ([10], [12], [13], [14]). This paper studies the conditions for a general function to make it a relevant welfare measure. Therefore, we introduce a large class of measures, called generalized benefit functions. The generalized benefit function is derived from the minimization of a convex function over the complement of a convex set. We show this class encompases as a special case the benefit function and is suitable to provide an alternative characterization of preferences. We also make a connection to the expenditure function through Fenchel duality theory and derive a duality result from Lemaire [7] for reverse convex optimization. Finally, we study the relationship between our class of functions and Hicksian compensated demand and we establish a link to the Slutsky matrix.