Measuring the efficiency of decision making units

In management contexts, mathematical programming is usually used to evaluate a collection of possible alternative courses of action en route to selecting one which is best. In this capacity, mathematical programming serves as a planning aid to management. Data Envelopment Analysis reverses this role and employs mathematical programming to obtain ex post facto evaluations of the relative efficiency of management accomplishments, however they may have been planned or executed. Mathematical programming is thereby extended for use as a tool for control and evaluation of past accomplishments as well as a tool to aid in planning future activities. The CCR ratio form introduced by Charnes, Cooper and Rhodes, as part of their Data Envelopment Analysis approach, comprehends both technical and scale inefficiencies via the optimal value of the ratio form, as obtained directly from the data without requiring a priori specification of weights and/or explicit delineation of assumed functional forms of relations between inputs and outputs. A separation into technical and scale efficiencies is accomplished by the methods developed in this paper without altering the latter conditions for use of DEA directly on observational data. Technical inefficiencies are identified with failures to achieve best possible output levels and/or usage of excessive amounts of inputs. Methods for identifying and correcting the magnitudes of these inefficiencies, as supplied in prior work, are illustrated. In the present paper, a new separate variable is introduced which makes it possible to determine whether operations were conducted in regions of increasing, constant or decreasing returns to scale in multiple input and multiple output situations. The results are discussed and related not only to classical single output economics but also to more modern versions of economics which are identified with "contestable market theories."

http://www.utdallas.edu/~ryoung/phdseminar/BCC1984.pdf

Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis

http://people.stern.nyu.edu/wgreene/FrontierModeling/Reference-Papers/Aigner-Lovell-Schmidt-JE1977-ALS.pdf

Formulation and estimation of stochastic frontier production function models

Efficiency Estimation from Cobb-Douglas Production Functions with Composed Error

This paper analyzes productivity growth in seventeen OECD countries over the period 1979-88. A nonparametric programming method (activity analysis) is used to compute Malmquist productivity indexes. These are decomposed into two component measures, namely, technical change and efficiency change. The authors find that U.S. productivity growth is slightly higher than average, all of which is due to technical change. Japan's productivity growth is the highest in the sample with almost half due to efficiency change. Copyright 1994 by American Economic Association.

Productivity growth, technical progress and efficiency change in industrial countries

The Measurement of Productive Efficiency

The Measurement of Production Efficiency

AN earlier paper by one of us (Farrell, 1957) developed, inter alia, a method for estimating efficient production functions from observations of the inputs and outputs of a number of individual production units The method consisted essentially in plotting these observations as points in a space of a suitable number of dimensions, forming the convex closure of this set of points and taking the appropriate part of the surface of this convex closure as the estimate of the efficient production function In the case where n inputs are used to produce a single output under conditions of constant returns to scale, if output is represented by X and the inputs by xl, x2, , Xw then each firm can be represented as a point in n-dimensional space with coordinates (xl/X, x2/X, , x/X) This set of points is then augmented by the points at infinity on each of the n axes, and the convex closure of the augmented set is formed The surface of this convex closure, omitting the facet at infinity, is our estimate of the efficient production function The earlier paper was primarily concerned with this simple version of the method Not only was it used for the basic exposition but a large part of the paper was devoted to applying it to observations on American agricultural production The paper did, however, discuss at a formal level the relaxation of the assumptions of single product and constant returns to scale, and in Section 23 showed how the method could be generalized to the case of m outputs X1, X2, , Xm by treating each observation as a point in n + m-dimensional space with coordinates (X,, X2, , Xm, x1, x2, , xn) A broadly similar procedure which would accommodate the possibility of decreasing returns to scale was indicated in Section 24 Underlying the use of the method in all these cases is the assumption that the efficient production function to be estimated is convex In no case does this assumption hold necessarily, but it is one frequently made in economics, and will probably hold in most actual situations Certainly there is nothing in the nature of multiple outputs or decreasing returns to scale that is inconsistent with the convexity assumption This happy state of affairs ceases to obtain as soon as we consider a situation with increasing returns to scale, because this essentially involves a non-convex production function; indeed they are in practice probably the most important example of nonconvex production functions The estimation of efficient production functions under

Estimating Efficient Production Functions Under Increasing Returns to Scale

The object of this paper is to examine how far the allocation of resources brought about by s perfectly competitive market remains optimal when indifference maps and production functions cease to be convex. It has recently been asserted that the allocative efficiency of a competitive system breaks down under these circumstances. If this true, it would constitute s serious limitation, for, as I shall argue. In the real world the relevant functions are often not convex. However, we shall see that the traditional assumptions of convexity are by no means essential to the optimality of competitive markets and that the assertions to the contrary are based on an elementary fallacy. I shall examine in some detail the behavior of competitive markets under conditions of certainty.

The Convexity Assumption in the Theory of Competitive Markets

There is an extensive literature concerning the conditions necessary for the existence of a social welfare function (see e.g. [1]) or a social choice function (see e.g. [4]) in the absence of 'any explicit principle of " Liberalism"; to this discussion we attempt no addition. However, Sen [3] has shown that the conditions become even more restrictive if such a principle is added in a particular form, and Gibbard [2], in discussing this and related problems, has uncovered yet more severe conditions. We are concerned in Sections 1 to 5 to analyse the way in which these difficulties arise and to argue that the way in which " Liberalism " is inserted into the problem has much to answer for. In Section 6 we propose a radically different way of incorporating " Liberalism "-an arguably more plausible waywhich does not lead to these difficulties. There is a set S of possible social states, each of which is " a complete description of society, including every individual's place in it ". Each individual i, i = 1, 2, ..., M, has a preference ordering Ri over S; an ordering is by definition complete, reflexive and transitive. The symbol Ri is used (without ambiguity) both to denote the ordering and in propositions like " xRiy " to denote pairwise preferences. The related symbols Pi and Ii are similarly used. A collective choice rule C is a function determining, for each M-tuple (R1, R2, ..., RM) of orderings on S a social preference relation R on S. (Again, R, P, and I are also used to denote pairwise preferences.) The properties of R are not specified, but for R to be either an ordering or a choice function it is necessary that it be acyclic. The authors quoted above show that, if " Liberalism " is introduced in a particular way, cycles of the form " x1Px2, x2Px3, ..., xnPx1 " can be generated, violating acyclicity. We define individual i as decisive between two social states x and y by the two implications

M. J. Farrell

Papers

The Measurement of Productive Efficiency

The Measurement of Production Efficiency

Estimating Efficient Production Functions Under Increasing Returns to Scale

The Convexity Assumption in the Theory of Competitive Markets

Liberalism in the Theory of Social Choice