CHAPTER 5 ANALYTICAL FOUNDATIONS AND MEASUREMENT OF FRONTIER
5.3 Measurement techniques of efficiency in banking
5.3.1 Data Envelopment Analysis (DEA)
DEA was introduced by Charnes, Cooper, and Rhodes (1978), hereinafter referred to as CCR, for measuring efficiency. The original model proposed by CCR in measuring the efficiency of a decision-making unit (DMU) was in the context of an input-oriented measure of technical efficiency with a production frontier exhibiting constant returns to scale (CRS).55
The objective of DEA is to construct a non-parametric envelopment frontier over data points representing output/input ratios (adjusted by output and input weights) such that all observed points lie on or below the production frontier. Although it does not require any functional form to be specified, it must satisfy the general assumptions about the production technology discussed earlier, including feasible input-output combinations, convexity of the production possibility set, and free disposability of both inputs and outputs.
The DEA frontier is thus constructed as the piecewise linear combinations that connect the set of the best-practice observations (of the output/input ratios), yielding a convex production possibilities set. Efficiency scores are then derived by measuring how far an observation is positioned from the ‘envelope’ or frontier.
We briefly illustrate the formulation of the original DEA model (CCR, 1978) as discussed in detail in Coelli (1995) below.
Suppose we have N firms who produce M outputs from J inputs. For the i-th firm, we represent the input and output vectors by ¤¥ and¦¥, respectively. The §' x ¨ input matrix, R and the 'x ¨ output matrix, U, represent the data of all N firms. For each firm, we obtain a measure of the ratio of all outputs over all inputs, such as M&z /©′: where M is an ' 1 vector of output weights and © is a § ' 1vector of input weights.
A linear programming (LP) problem which seeks to find optimal weights for M and © such that the efficiency measure of the i-th firm is maximised, subject to the constraint that all efficiency measures must be less than or equal to one is specified as follows:
Maximise (M&z /©′: ), (5.19)
subject to M&z /©′: ≤ 1
55
The term DMU was used to include firms as well as non-market agencies like schools, hospitals, courts, etcetera with lack of market prices for their outputs.
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M, © ≥ 0
This formulation has an infinite number of solutions and so the constraint ©: = 1is imposed, so we have: 56
Maximise (Š&z )57 (5.20)
subject to ©′: = 1
Š&z − ©′: ≤ 0, q = 1,2, … , ¨,
Š, © ≥ 0
Using duality in linear programming, an equivalent envelopment form of this problem can be derived as follows:
iPitiso-,‚F, (5.21)
sM®qo.¡ ¡l − z + X8 ≥ 0, F: − '8 ≥ 0,
8 ≥ 0
where F is a scalar to be estimated and 8 is an ¨ ' 1 vector of constants. The linear programming problem is solved N times, once for each firm in the sample, and a value of F is then obtained for each firm.
The parameter F is the estimation of technical efficiency (TE) for the i-th firm, with 0 ≤ F ≤
1 and where θ =1 indicates a point on the frontier and hence a technically efficient firm. The vector 8 defines the projected point ('8, X8) against which the efficient score of the i-th firm is derived. The projected point is a linear combination of all the observed data points in the sample. The projected points obtained by each LP together form the feasible surface of the sample, known as the production technology (or frontier).
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This means that if (M∗, ©∗) is a solution, then ( M∗, ©∗)is another solution
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The original CCR DEA model was modified and extended by Banker, Charnes, and Cooper (1984) hereinafter referred to as BCC, by relaxing the restrictive assumption of CRS and assuming variable returns to scale (VRS).
An adjustment is made to equation (5.21) by the inclusion of a convexity constraint ƒ1′8 (where ƒ1 is an I x 1 vector of ones) to give the BCC VRS DEA model as:
iPitiso-,‚F (5.22)
sM®qo.¡ ¡l − z + X8 ≥ 0 F: − '8 ≥ 0
ƒ1&8 = 1
8 ≥ 0
The CCR–BCC DEA models have been used extensively and have also undergone various modifications and extensions to incorporate computation of other efficiency measures, including output-oriented efficiency, allocative efficiency, cost efficiency and profit efficiency.
For instance in the case of cost efficiency score computation using DEA, the following cost minimisation problem is solved:
iPitiso‚,°∗ " ′:∗,
(5.23)
sM®qo.¡ ¡l − z + X8 ≥ 0 :∗− '8 ≥ 0
ƒ1&8 = 1
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where
"
is a N x 1 vector of input prices for the i-th firm and:
∗ (computed by the LP) is the cost-minimising vector of input quantities for the i-th firm, given the input prices"
and output levelsz.58The modifications and extensions of the DEA notwithstanding, the underlying philosophy and fundamental characteristics of the DEA framework in constructing the frontier using non- parametric techniques and attributing all the deviation from the frontier as measures of inefficiencies are retained.
To conclude our review of the DEA technique, we note that the main merits attributable to the DEA framework are its simplicity, as well as the fact that it does not require any functional specification of the production process or any distributional assumptions regarding inefficiency. Accordingly, it avoids problems associated with model sensitivity and functional instability. However, DEA has a number of drawbacks. The major drawback lies in its deterministic nature that rules out the existence of noise and other measurement errors and attributes all deviations from the frontier to inefficiency. Second, and as a consequence of the first point, DEA is particularly sensitive to the presence of outliers in the data. Third, it precludes the possibility of performing direct statistical tests on the results. Fourth, the assumption of homogeneity of firms implicit in DEAs is often untenable.