Specification and estimation of multiple output technologies: A primal approach

This paper addresses specification and estimation of multiple-outputs and multiple-inputs production technology in the presence of technical inefficiency. The primary focus is on the primal formulations. Several competing specifications such as production function, input (output) distance function, input requirement function are considered. We show that all these specifications come from the same transformation function and are algebraically identical. We also show that: (i) unless the transformation function is separable (i.e., outputs are separable from inputs), the input (output) ratios in the input (output) distance function can not be treated as exogenous (uncorrelated with technical inefficiency) resulting inconsistent estimates of the input (output) distance function parameters. (ii) Even if input (output) ratios are exogenous, estimation of the input (output) distance function will result in inconsistent parameter estimates if outputs (inputs) are endogenous. We address endogeneity and instrumental variable issues in details in the context of flexible (translog) functional forms. Estimation of several specifications using both single and system approaches are discussed using Norwegian dairy farming data.     

Estimation of production technology when the objective is to maximize return to the outlay

This paper deals with estimation of production technology where endogeneous choice of input and output variables is explicitly recognized. To address this endogeneity issue, we assume that producers maximize return to the outlay. We start from a flexible (translog) transformation function with a single output and multiple inputs and show how the first-order conditions of maximizing return to the outlay can be used to come up with an ‘estimating equation’ that does not suffer from the econometric endogeneity problem although the output and input variables are chosen endogenously. This is because the regressors in this estimating equation are in ratio forms which are uncorrelated with the error term under the assumption that producers maximize return to the outlay. The analysis is then extended to the multiple outputs and multiple inputs case with technical inefficiency. Although the estimating equations in both single and multiple output cases are neither production nor distance functions, they can be estimated in a straightforward manner using the standard stochastic frontier technique without worrying about endogeneity of the regressors. Thus, we provide a rationale for estimating the technology parameters consistently using an econometric model which requires data on only input and output quantities.     

Estimation of production risk and risk preference function: a nonparametric approach

While estimating parametric production models with risk, one faces two main problems. The first problem is associated with the choice of functional forms on the mean production function and the risk (variance) function. The second problem is associated with the specification of the risk preference function. In a parametric model the researcher chooses some ad hoc functional form on all these. It is obvious that the estimated (i) technology (mean production function), (ii) risk and (iii) risk preference functions are affected by the choice of functional form. In this paper we consider an estimation framework that avoids assuming parametric functions on all three. In particular, this paper deals with nonparametric estimation of the technology, risk and risk preferences of producers when they face uncertainty in production. Uncertainty is modeled in the context of production theory where producers’ maximize expected utility of anticipated profit. A multi-stage nonparametric estimation procedure is used to estimate the production function, the output risk function and the risk preference function. No distributional assumption is made on the random term representing production uncertainty. No functional form is assumed on the underlying utility function. Rice farming data from Philippines are used for an empirical application of the proposed model. Rice farmers are, in general, found to be risk averse; labor is risk decreasing while fertilizer, land and materials are risk increasing. The mean risk premium is about 3% of mean profit.     

Constrained Global Optimization of Expensive Black Box Functions Using Radial Basis Functions

We present a new strategy for the constrained global optimization of expensive black box functions using response surface models. A response surface model is simply a multivariate approximation of a continuous black box function which is used as a surrogate model for optimization in situations where function evaluations are computationally expensive. Prior global optimization methods that utilize response surface models were limited to box-constrained problems, but the new method can easily incorporate general nonlinear constraints. In the proposed method, which we refer to as the Constrained Optimization using Response Surfaces (CORS) Method, the next point for costly function evaluation is chosen to be the one that minimizes the current response surface model subject to the given constraints and to additional constraints that the point be of some distance from previously evaluated points. The distance requirement is allowed to cycle, starting from a high value (global search) and ending with a low value (local search). The purpose of the constraint is to drive the method towards unexplored regions of the domain and to prevent the premature convergence of the method to some point which may not even be a local minimizer of the black box function. The new method can be shown to converge to the global minimizer of any continuous function on a compact set regardless of the response surface model that is used. Finally, we considered two particular implementations of the CORS method which utilize a radial basis function model (CORS-RBF) and applied it on the box-constrained Dixon–Szegö test functions and on a simple nonlinearly constrained test function. The results indicate that the CORS-RBF algorithms are competitive with existing global optimization algorithms for costly functions on the box-constrained test problems. The results also show that the CORS-RBF algorithms are better than other algorithms for constrained global optimization on the nonlinearly constrained test problem.     

Construction of proximal distances over symmetric cones

Abstract

This paper is devoted to the study of proximal distances defined over symmetric cones, which include the non-negative orthant, the second-order cone and the cone of positive semi-definite symmetric matrices. Specifically, our first aim is to provide two ways to build them. For this, we consider two classes of real-valued functions satisfying some assumptions. Then, we show that its corresponding spectrally defined function defines a proximal distance. In addition, we present several examples and some properties of this distance. Taking into account these properties, we analyse the convergence of proximal-type algorithms for solving convex symmetric cone programming (SCP) problems, and we study the asymptotic behaviour of primal central paths associated with a proximal distance. Finally, for linear SCP problems, we provide a relationship between the proximal sequence and the primal central path.     

On methods for converting exhausters of positively homogeneous functions

In the paper we develop a method for converting a (not necessarily uniformly bounded) upper (lower) primal exhauster of a continuous positively homogeneous function to a lower (upper) primal exhauster of the same function. The method is based on representation of a continuous positively homogeneous function as a pointwise supremum (infimum) of an one-parameter monotone family of Lipschitz continuous positively homogeneous functions that is of independent interest.     

Metric Distance Function and Profit: Some Duality Results

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 Hölder 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 Färe appear as special cases of some l _p distance (also called Hölder 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.     

中国能源资本替代的不确定性

在研究能源经济问题的可计算一般均衡模型中，都考虑能源、资本和劳动的相互替代。通过对中国1978～2000年历史数据回归分析[1]发现在常替代弹性(CES)生产函数和Cobb—Douglas(C-D)生产函数中，劳动投入对生产函数几乎没有影响。剔除劳动投入后使用能源和资本作为投入[1]重新估计了中国的生产函数，在新的生产函数中发现能源和资本间的替代具有很高的不确定性。本在[1]的基础上，设定了五个情景，包括常规发展情景(BAU)，节能情景(EC)及减少温室气体排放5％(ER5)、10％(ER10)和15％(ER15)三个情景探索节能和控制温室气体排放对中国经济的影响。研究发现，尽管存在很大的不确定性，中国需要增加大量的资本去替代能源，所以在确保可持续发展所需的能源和控制温室气体排放上中国经济面临着巨大的挑战。     

Profit,Directional Distance Functions,and Nerlovian Efficiency

The directional technology distance function is introduced, given an interpretation as a min-max, and compared with other functional representations of the technology including the Shephard input and output distance functions and the McFadden gauge function. A dual correspondence is developed between the directional technology distance function and the profit function, and it is shown that all previous dual correspondences are special cases of this correspondence. We then show how Nerlovian (profit-based) efficiency measures can be computed using the directional technology distance function.     

A new clustering approach using data envelopment analysis

In this paper, we present a new clustering method that involves data envelopment analysis (DEA). The proposed DEA-based clustering approach employs the piecewise production functions derived from the DEA method to cluster the data with input and output items. Thus, each evaluated decision-making unit (DMU) not only knows the cluster that it belongs to, but also checks the production function type that it confronts. It is important for managerial decision-making where decision-makers are interested in knowing the changes required in combining input resources so it can be classified into a desired cluster/class. In particular, we examine the fundamental CCR model to set up the DEA clustering approach. While this approach has been carried for the CCR model, the proposed approach can be easily extended to other DEA models without loss of generality. Two examples are given to explain the use and effectiveness of the proposed DEA-based clustering method.     

Primal-dual row-action method for convex programming

We present a primal-dual row-action method for the minimization of a convex function subject to general convex constraints. Constraints are used one at a time, no changes are made in the constraint functions and their Jacobian matrix (thus, the row-action nature of the algorithm), and at each iteration a subproblem is solved consisting of minimization of the objective function subject to one or two linear equations. The algorithm generates two sequences: one of them, called primal, converges to the solution of the problem; the other one, called dual, approximates a vector of optimal KKT multipliers for the problem. We prove convergence of the primal sequence for general convex constraints. In the case of linear constraints, we prove that the primal sequence converges at least linearly and obtain as a consequence the convergence of the dual sequence.The research of the first author was partially supported by CNPq Grant No. 301280/86.     

Scale characterizations in a DEA directional technology distance function framework

The present study extends prior research on data envelopment analysis (DEA)-based returns to scale in view of the fact that the standard methods are very restrictive. The extension is made in the context of the directional technology distance function that generalizes the Shephard input and output distance functions. The two main results in this extended setting are as follows. First, we show how Banker's most productive scale size (MPSS) concept is characterized. Second, we obtain scale elasticity measures and examine the properties and the relationships with standard scale elasticity measures.     

How to generate regularly behaved production data? A Monte Carlo experimentation on DEA scale efficiency measurement

Monte Carlo experimentation is a well-known approach used to test the performance of alternative methodologies under different hypotheses. In the frontier analysis framework, whatever the parametric or non-parametric methods tested, experiments to date have been developed assuming single output multi-input production functions. The data generated have mostly assumed a Cobb–Douglas technology. Among other drawbacks, this simple framework does not allow the evaluation of DEA performance on scale efficiency measurement. The aim of this paper is twofold. On the one hand, we show how reliable two-output two-input production data can be generated using a parametric output distance function approach. A variable returns to scale translog technology satisfying regularity conditions is used for this purpose. On the other hand, we evaluate the accuracy of DEA technical and scale efficiency measurement when sample size and output ratios vary. Our Monte Carlo experiment shows that the correlation between true and estimated scale efficiency is dramatically low when DEA analysis is performed with small samples and wide output ratio variations.     

Global convergence of an SQP method without boundedness assumptions on any of the iterative sequences

Usual global convergence results for sequential quadratic programming (SQP) algorithms with linesearch rely on some a priori assumptions about the generated sequences, such as boundedness of the primal sequence and/or of the dual sequence and/or of the sequence of values of a penalty function used in the linesearch procedure. Different convergence statements use different combinations of assumptions, but they all assume boundedness of at least one of the sequences mentioned above. In the given context boundedness assumptions are particularly undesirable, because even for non-pathological and well-behaved problems the associated penalty functions (whose descent is used to produce primal iterates) may not be bounded below for any value of the penalty parameter. Consequently, boundedness assumptions on the iterates are not easily justifiable. By introducing a very simple and computationally cheap safeguard in the linesearch procedure, we prove boundedness of the primal sequence in the case when the feasible set is nonempty, convex, and bounded. If, in addition, the Slater condition holds, we obtain a complete global convergence result without any a priori assumptions on the iterative sequences. The safeguard consists of not accepting a further increase of constraints violation at iterates which are infeasible beyond a chosen threshold, which can always be ensured by the proposed modified SQP linesearch criterion. The author is supported in part by CNPq Grants 301508/2005-4, 490200/2005-2, 550317/2005-8, by PRONEX–Optimization, and by FAPERJ Grant E-26/151.942/2004.     

Pricing with markups in industries with increasing marginal costs

We study a game that models a market in which heterogeneous producers of perfect substitutes make pricing decisions in a first stage, followed by consumers that select a producer that sells at lowest price. As opposed to Cournot or Bertrand competition, producers select prices using a supply function that maps prices to production levels. Solutions of this type of models are normally referred to as supply function equilibria. We consider a market where producers’ convex costs functions are proportional to each other, depending on the efficiency of each particular producer. We provide necessary and sufficient conditions for the existence of an equilibrium that uses simple supply functions that replicate the cost structure. We then specialize the model to monomial cost functions with exponent \(q>0\) , which allows us to reinterpret the simple supply functions as a markup applied to the production cost. We prove that an equilibrium for the markups exists if and only if the number of producers in the market is strictly larger than \(1+q\) , and if an equilibrium exists, it is unique. The main result for monomials is that the equilibrium nearly minimizes the total production cost when the market is competitive. The result holds because when there is enough competition, markups are bounded, thus preventing prices to be significantly distorted from costs. Focusing on the case of linear unit-cost functions on the production quantities, we characterize the equilibrium accurately and refine the previous result to establish an almost tight bound on the worst-case inefficiency of equilibria. Finally, we derive explicitly the producers’ best response for series-parallel networks with linear unit-cost functions, extending our previous result to more general topologies. We prove that a unique equilibrium exists if and only if the network that captures the market structure is 3-edge-connected. For non-series-parallel markets, we provide an example that does not admit an equilibrium on markups.     

Necessary and Sufficient Conditions for Strong Fenchel–Lagrange Duality via a Coupling Conjugation Scheme

Given a general primal problem and its Fenchel–Lagrange dual one, which is obtained by using a conjugation scheme based on coupling functions and the perturbational approach, the aim in this work is to establish conditions under which strong duality can be guaranteed. To this purpose, even convexity and properness are a compulsory requirement over the involved functions in the primal problem. Furthermore, two closedness-type regularity conditions and a characterization for strong duality are derived.     

Mean value cross decomposition applied to integer programming problems

The mean value cross decomposition method for linear programming problems is a modification of ordinary cross decomposition, that eliminates the need for using the Benders or Dantzig-Wolfe master problems. As input to the dual subproblem the average of a part of all known dual solutions of the primal subproblem is used, and as input to the primal subproblem the average of a part of all known primal solutions of the dual subproblem. In this paper we study the lower bounds on the optimal objective function value of (linear) pure integer programming problems obtainable by the application of mean value cross decomposition, and find that this approach can be used to get lower bounds ranging from the bound obtained by the LP-relaxation to the bound obtained by the Lagrangean dual. We examplify by applying the technique to the clustering problem and give some preliminary computational results.     

On the convergence of the entropy-exponential penalty trajectories and generalized proximal point methods in semidefinite optimization

The convergence of primal and dual central paths associated to entropy and exponential functions, respectively, for semidefinite programming problem are studied in this paper. It is proved that the primal path converges to the analytic center of the primal optimal set with respect to the entropy function, the dual path converges to a point in the dual optimal set and the primal-dual path associated to this paths converges to a point in the primal-dual optimal set. As an application, the generalized proximal point method with the Kullback-Leibler distance applied to semidefinite programming problems is considered. The convergence of the primal proximal sequence to the analytic center of the primal optimal set with respect to the entropy function is established and the convergence of a particular weighted dual proximal sequence to a point in the dual optimal set is obtained.     

Directional fuzzy clustering and its application to fuzzy modelling

The paradigm of clustering (unsupervised learning) viewed as a fundamental tool for data analysis has been found useful in fuzzy modelling. While the objective functions guiding the clustering mechanisms are by and large direction-free (namely, they do not distinguish between independent (input) and dependent (output) variables, for most of the models this discrimination becomes of vital importance. The method of directional clustering takes the directionality requirement into account by incorporating the nature of the functional relationships into the objective function guiding the formation of the clusters. The complete clustering algorithm is presented. The role of this method in a two-phase fuzzy identification scheme is also revealed in detail.     

Competitive equilibria in economies with multiple divisible and indivisible commodities and without money

A general equilibrium model is considered with multiple divisible and multiple indivisible commodities. In models with indivisibles it is typically assumed that an indivisible commodity, called money, is present that is needed to transfer the value of amounts of indivisible goods. For economies with divisible and indivisible goods and money and without producers it is well understood in the literature that a general equilibrium exists if the individual demands and supplies for the indivisible goods all belong to the same class of discrete convexity. In this paper we consider economies with multiple divisible and multiple indivisible commodities, but without money as one specific commodity for value transfer. Moreover, we allow for one or more producers that own a nonincreasing returns to scale technology. However, one of the producers has a production technology which is linear in producing divisible goods. In this way the composite of the divisible goods takes over the role of money in the model. Individual endowments being large enough for production together with discrete convexity guarantees the existence of a competitive equilibrium using Kakutani’s fixed point theorem.