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

Ever since Richard Stone (1954) first estimated a system of demand equations derived explicitly from consumer theory, there has been a continuing search for alternative specifications and functional forms. Many models have been proposed, but perhaps the most important in current use, apart from the original linear expenditure system, are the Rotterdam model (see Henri Theil, 1965, 1976; Anton Barten) and the translog model (see Laurits Christensen, Dale Jorgenson, and Lawrence Lau; Jorgenson and Lau). Both of these models have been extensively estimated and have, in addition, been used to test the homogeneity and symmetry restrictions of demand theory. In this paper, we propose and estimate a new model which is of comparable generality to the Rotterdam and translog models but which has considerable advantages over both. Our model, which we call the Almost Ideal Demand System (AIDS), gives an arbitrary first-order approximation to any demand system; it satisfies the axioms of choice exactly; it aggregates perfectly over consumers without invoking parallel linear Engel curves; it has a functional form which is consistent with known household-budget data; it is simple to estimate, largely avoiding the need for non-linear estimation; and it can be used to test the restrictions of homogeneity and symmetry through linear restrictions on fixed parameters. Although many of these desirable properties are possessed by one or other of the Rotterdam or translog models, neither possesses all of them simultaneously. In Section I of the paper, we discuss the theoretical specification of the AIDS and justify the claims in the previous paragraph. In Section II, the model is estimated on postwar British data and we use our results to test the homogeneity and symmetry restrictions. Our results are consistent with earlier findings in that both sets of restrictions are decisively rejected. We also find that imposition of homogeneity generates positive serial correlation in the errors of those equations which reject the restrictions most strongly; this suggests that the now standard rejection of homogeneity in demand analysis may be due to insufficient attention to the dynamic aspects of consumer behavior. Finally, in Section III, we offer a summary and conclusions. We believe that the results of this paper suggest that the AIDS is to be recommended as a vehicle for testing, extending, and improving conventional demand analysis. This does not imply that the system, particularly in its simple static form, is to be regarded as a fully satisfactory explanation of consumers' behavior. Indeed, by proposing a demand system which is superior to its predecessors, we hope to be able to reveal more clearly the problems and potential solutions associated with the usual approach.

/pdf/an-almost-ideal-demand-system-se3smx57kd.pdf

An Almost Ideal Demand System

The economic theory of index numbers and the measurement of input, output, and productivity

Exact and superlative index numbers

1. The Process Production Function.- 2. Heuristic Principle of Minimum Costs.- 3. The Producer's Minimum Cost Function.- 4. Dual Determination of Production Function From Cost Function.- 5. Geometric Interpretation of the Duality Between Cost and Production Function.- 6. Constraints on the Factors of Production.- 7. Homothetic Production Functions.- 8. The Cobb-Douglas Production Function.- 9. The Problem of Aggregation.- 10. The Dynamics of Monopoly.

Ronald William Shephard

Papers

Cost and production functions