Russell measures in data envelopment analysis

A producer who cannot vary his output enquires for possible reduction of inputs. If reduction is not possible he is efficient, otherwise inefficient. In this case the producer minimizes the total cost of production. Alternatively, if inputs cannot be varied, which is often in short run, the entrepreneur enquires for further output augmentation, if such augmentation is not possible he is efficient, otherwise inefficient. In this situation the implicit assumption is revenue maximization. However, in long run, inputs as well as outputs can be varied simultaneously, where the underlying optimization is profit maximization. If the producer neither reduces inputs nor augments outputs, production is profit efficient, otherwise inefficient.

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Introduction

The theoretical literature on productive efficiency originated with the work of The first attempt to estimate efficiency was found in used linear programming techniques to estimate efficiency in U.S. agriculture. Research on efficiency estimation continued on through the development of Stochastic Frontier Analysis (SFA), (Aigner, Lovell, and Schmidt 1977) [3] . DEA was developed at about the sametime by Each of these techniques has subsequently been extended and developed further (see Kumbhakar and Lovell 2000 for a modern textbook treatment of SFA and Cooper, Seiford, and Zhu 2004 for DEA). The empirical application of these techniques to estimate efficiency and/or productivity has resulted in a multitude of studies in numerous areas. Researchers have employed SFA or DEA (and in some cases both techniques) to analysis of agriculture, utilities, education, health economics, transportation, labour economics, management science, environmental economics, and financial institutions. As an example of the amount of output, present a survey of research on the efficiency of financial institutions with a count of 130 studies across 21 countries.

Russell Measure of Input Technical Efficiency

In a k input, m output technology, Russell's measure of input technical efficiency can be viewed as a linear programming problem.

 Radial measure of technical efficiency: OP OQ  Q is weakly efficient input combination  Input x 1 can be further reduced, so that P can be compared with R that belongs to Eff L (u)  The line segment SR constitutes the efficient subset of Isoq L (u)

To estimate Russell measure of technical efficiency, we solve the Following linear programming problem

Russell Cost Measure of Technical Efficiency

The Farrell input technical efficiency can be given a Cost interpretation.

  

Where the denominator is observed cost. More generally, fork input case, this measure takes the form,

Russell Measure of Input Allocative Efficiency

The Russell measure of input allocative efficiency is defined as the ratio of factor minimal cost to the cost measure of Russell input technical efficiency.

is the factor minimal cost obtained by solving the following optimization problem:

Russell Measure of Overall Technical Efficiency

The Russell measure of overall technical efficiency depends on the input and output vectors, viz., 0

x and 0 u of the decision making unit in focus. Consequently, the measure is defined as,   0 0 , px p u Q However, the above ratio measures overall input cost efficiency in Farrell's measurement.

Thus, Russell measure of overall input technical efficiency can be decomposed into the product of Russell input allocative efficiency and measure of overall technical efficiency.

Russell Output Measures of Productive Efficiencies

 Efficiencies are measured comparing the observed output vectors with efficient output vectors which belong to Eff P(x 0 )  P(x) is output level set  Augmentation of additional outputs are non-radial  The various efficiencies are obtained by appropriately formulating and solving linear programming problems.

i) Russell Output Measure of Technical Efficiency

Let x 0 be the input vector of the DMU whose output efficiency is under evaluation. The Russell output measure of technical efficiency is defined as,

ii) Russell Revenue Measure of Output Technical Efficiency Let u 0 be the output vector of the DMU whose efficiency is under evaluation. If r is the revenue vector, the Russell revenue measure of output technical efficiency is the ratio of potential output revenue to the realized revenue at output price vector r. The measure takes the following form:

 iii) Russell Overall Output Efficiency Measure

The overall output efficiency measure of Russell is the ratio of maximal revenue to observed revenue. It depends on output price vector, observed input and output vectors of the DMU whose efficiency is under evaluation. Maximum revenue can be estimated solving the following optimization problem:

If production technology is piecewise linear, under certain assumptions potential revenue can be obtained by solving the following linear programming problem.

 

 

Russell Output Allocative Efficiency Measure

This measure is viewed as the ratio of potential revenue to the revenue attained at

iii) Russell output pure technical efficiency can be found solving the following optimization problem.

Conclusions

Theoretical consideration of technical efficiency has existed since Koopmans defined it for production possibilities for which it is not possible to increase any output without simultaneously increasing any input, ceteris paribus. The nonparametric approach to efficiency measurement known as Data Envelopment Analysis is based on the index of Farrell, which measures radial reduction in all inputs consistent with observed output. Even after Farrell efficiency is achieved, however, there may exist additional slack in individual inputs, suggesting that the Farrell index does not necessarily measure Koopmans inefficiency. To solve this problem, the non-radial Russell measure was introduced. In this paper we proposed various concepts of Russell measure of technical efficiencies in data envelopment analysis,

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