Choice of metric in the measurement of relative productive efficiency

i

international journal of

product[on economics ELSEVIER

Int. J. Production Economics 46 47 (1996) 433- 439

Choice of metric in the measurement of relative productive efficiency P.C. R i t c h i e , J.E. R o w c r o f l * University of New Brunswick. P.O. Box 4400, Fredericton A' B.. Canada E3B 53,4

Abstract In this paper different data envelope analysis (DEA) models are compared in terms of their potential to generate a quantitative measure of the degree of inefficiency. The Russell [1] measure in particularly appealing since it is not restricted to simple proportional comparisons. However, its actual value is sensitive to the choice of weights on the different inputs (or outputs) in the objective function. This may be of little significance in a simple binary classification of efficient and inefficient firms. However when evaluating the degree of efficiency, and hence the "distance" of a firm from the efficient envelope, choice of metric becomes critical. A number of alternative metrics are compared, including the "stability index" of Charnes et al. [2] and an alternative approach is suggested using the nature of the linear program itself. The paper concludes with a numerical illustration. Keywords: DEA; Efficiency; Metric; Stability index

1. Introduction The b r o a d measurement of productive efficiency may be conveniently categorised in two ways, parametric and non-parametric, stochastic and deterministic. M u c h of the economics literature has been devoted to the fitting of parametric p r o d u c t i o n functions such as the C o b b - D o u g l a s using statistical techniques to recognize noise in the data. Equivalently cost functions m a y be fitted. Such a production or cost function m a y be used as the basis for c o m p a r i s o n of firms' performance (see for

* Corresponding author: Tel.: (506) 453-4655: fax: (506) 4535102: c-mail: [email protected].

example [3]). The main d r a w b a c k of such a procedure is the reliance on a chosen functional form for the p r o d u c t i o n or cost relationship. Button and W e y m a n - J o n e s [4] note in their survey of efficiency studies that "in all instances, the degree of measured inefficiency is very sensitive to the researcher's m e t h o d of analysis." In its basic form, data envelope analysis IDEA) offers a deterministic n o n - p a r a m e t r i c a p p r o a c h which requires only the assumption of a convex technology, that is, one in which all linear combinations of existing technologies are feasible. While this obviates the need to impose a particular functional form, it also means that each firm is c o m p a r e d only with existing firms, so called "best practice" [5]. If all firms are inefficient in some

0925-5273/96/'$15.00 Copyright ~i~, 1996 Elsevier Science B.V. All rights reserved. SSDI 0 9 2 5 - 5 2 7 3 { 9 5 ) 0 0 1 8 5 - 9

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P . C Ritchie, ~LE. Rowcroft/lnt. ~ Production Economics 46-47 (1996) 433 439

absolute sense, DEA will only reveal firms which are unusually so. A deterministic technique avoids the assumption of particular error distributions and hence excludes the means to distinguish between data noise and inefficiency. Attempts have been made to modify the DEA approach to incorporate disturbance terms (see for example [6]). However, there is a significant increase in complexity and often the introduction of an assumed functional form which offsets much of the appeal of the envelope approach. Nevertheless there is a frequent wish to go beyond the categorisation of firms to attempt to fit an explanatory model using statistical methods. Within the simple DEA model this leads to the use of envelope efficiency as a binary variable in a logit type of model, or the need for actual efficiency scores to incorporate in a probit regression of some kind. Thus the use of DEA in a modelling framework requires the examination of the efficiency scores which the envelope analysis provides. In this sense the non-stochastic form of the analysis can be justified as a (somewhat elaborate) process of data manipulation prior to probability based testing of explanatory models. Before looking directly at efficiency measures we consider some of the various forms of data envelope analysis which have emerged in the literature.

2. Data envelope analysis types DEA models may be classified under five categories according to whether they are (i) radial or non-radial, (ii) input or output, (iii) primal or dual, (iv) linear or non-linear, and (v) with or without exogenous prices. Not all cross-classifications are of interest, but the basic taxonomy is worth a brief examination. (i) Radial and non-radial: Farrell's original model [7] proposed a radial change in an efficient firm's input or output set to bring it to the efficient, or best practice, envelope. The consequent measure corresponds to Shephard's 1-8] input measure. If constant returns to scale prevail throughout then input and output forms yield the same measure and there is little ambiguity about the distance from

the envelope or, equivalently, the degree of inefficiency. The requirement of strictly proportional expansion (of outputs) or contraction (of inputs) was relaxed by Russell [1] at the cost of differences between input and output measures. Russell also implicitly imposed a uniform weighting of all inputs (or outputs) in the objective function. (ii) Input and output: As noted in the preceding subsection, under restrictive conditions, input and output measures may be the same. Where they differ it appears to be the practice in the literature to appeal to the circumstances of the industry. Thus in Brander et al. [9] it is argued that for the airline industry output limitations are likely to be largely market determined and the most plausible approach is to look at an input model to determine inefficiency. Bjurek et al. [10] based their use of an output model on the presence of excess demand in the daycare market. (iii) Primal and dual: Since the DEA analysis is conducted via a mathematical program the model may be expressed either as a primal in terms of input and output quantities or, equivalently, as a dual in terms of the shadow prices of these commodities. For example, Bjurek et al. [10] use a concise form of the dual program to modify the radial model to accommodate inefficiencies of scale. (iv) Linear and non-linear: The most common forms of DEA use a linear programming format. Ffire et al. [11] develop a non-linear programming approach in their graph formulation of the Russell model. While this is an interesting line of enquiry, the computational and manipulative convenience of the linear program recommended its continued investigation. (v) Exogenous prices: To investigate profit and economic as opposed to technical efficiency, prices have been introduced into envelope analysis (see for example [6]). While recognising the importance of price and market implications for the firm, the present analysis focuses on the technical efficiency question separately, and independent of price considerations. Implicitly we assume that all firms in the data set face equivalent (not necessarily constant) price conditions. Given the establishment of technical efficiency measures, this assumption is testable against market data, at least in principle.

P.C. Ritchie. J.E. Rowcrofi/lnt. J. Production Economics 46 47 (1996) 433 439

A more extensive discussion of expansion measures is given in Fiire et al. [-12]. O u r concern in the remainder of the paper is primarily with nonradial input and output models expressed as either primal or dual linear programs without exogenous prices. We turn now to the consideration of efficiency measures in this context.

435

solve the following linear program: max

0

s.t.

ys - / + - Odo = yk,

x s + ). es=

+ Odl = x k,

(1)

],

s,,:.+,2 ,0 >~ 0. 3. M e a s u r e s o f e f f i c i e n c y

In the radial input model, the linear program solves for the minimum value for 0 for which Ox k lies on the industry envelope for yk where x k, yk are the inefficient firm's bundles of inputs and outputs respectively. For the corresponding output model, the program solves for the m a x i m u m q5 such that ~by lies on the envelope for x. 0 and q5 are both scalar, and the efficiency measure follows as 0 or 4) 1. For the Russell measure, O = {01, ... ,0,,} and 4~ = {01 . . . . . ~b,} are vectors and the corresponding objective functions are Y~Oi and 52qSj which are used to derive efficiency scores of 2 O i / m , and n./V~qSi, where m and n are the numbers of inputs and outputs respectively. As noted above, this assigns equal weight to each input (or output) somewhat arbitrarily as well as leading to two measures which will in general be different. For a subsequent Iogit analysis using a simple classification of firms into efficient and inefficient, these considerations are largely unimportant. Input and output measures will render the same classification. However the binary classification ignores much of the information in the data set. Haag et al. [13] note that DEA is not a units invariant model. They propose scaling by component averages. Thus each input and output quantity is standardised by dividing by the industry mean. This provides the obvious benefit of making subsequent measurements independent of units. Secondly Charnes et al. [2] recommend the computation of a stability index for each firm. Efficient firms are scored on the basis of the largest simultaneous perturbation of all inputs and outputs which would permit the firm to be continued to be classified as efficient. For an inefficient firm the authors

(xk, y k refer to the inefficient firm under considera-

tion). Here x, y are input vectors as before, s is a vector of weights, 2 +, 2- are vectors of slacks, do, dl and e are unit vectors of the appropriate size and 0 is a scalar. Thus 0 determines the m a x i m u m simultaneous perturbation of all inputs and outputs which would not cause the firm to be reclassified as efficient, that is to achieve the envelope. In a sense, the term stability index applies most aptly to the efficient firm measure since it attempts to measure the extent to which the firm remains efficient in the face of random (or other) changes in its input-output mix. It is less easy to think of an inefficient firm moving from one efficient position to another as exhibiting stability. More substantially program (1) imposes two stringent constraints which are hard to justify a priori, except on the basis of simplicity. The simultaneous expansion and contraction of outputs and inputs resolves the dilemma of input and output measures but the use of a scalar 0 reduces the problem to an essentially radial one. Both restrictions imply an equal weighting within and between inputs and outputs which have been scaled by component averages. In the next section we turn out to an alternative, non-radial approach.

4. Input and output s h a d o w prices

The Russell input measure may be conveniently derived from the following program: min

O

s.t.

y s >/ yk

Ox k - x s >~ O,

(2)

436

P.C Ritchie, J.E. Rowcrofi/Int. A Production Economics 46-47 (1996) 433 439

where the notation is as before. Similarly, the output measure may be written: rain s.t.

Cyk _ y t <~ O,

(3)

x t <~ X k,

where the t are a set of non-negative firm weights, corresponding to (but not necessarily equal to) the s. Both programs may be modified by incorporating arbitrary sets of non-negative weights = (~1 . . . . . ~m) and fl = ( i l l , ... , f i n ) i n t o the r e s p e c t i v e objective functions which become: rain

Lemma. T h e r e exists a pair (u, v) which solves pro9rams (4D) and (5D). The proof is given in the Appendix. Let (u*, v*) represent a set of shadow prices which are valid for both input and output programs. Through the remaining constraints (u*, v*) define a set of weights

(~,/~). Using these prices the input measure of efficiency is E I-

and the output measure is

~O

V* y k E o -

s.t.

ys ~ yk, O x k - x s >1 O, -diO

>/ -

(4)

Since primal and dual objective functions are equal at optimum, U* XS* u,xk

EI-

fl@

Ugcxk ~) * -

s.t.

(5)

x t <~ x k,

- -

U* X k

(by k - yt <~ O,

-do~<

v*yt* "

1

and max

bl * XS* u,xk '

from primal

u*xkO*

- - /?,/,* - -

from output objective function

- 1, u*xk~

with corresponding dual programs: max

ykv

s.t.

y ' v -- X'U <~ O, x k u -- W <~ ~

v * yk fl~ *

yk v* v*.yk ~:/)*

(4D)

s.t.

from output primal

E0 .

xku

--y'v--x'u>lO, -- ykv -- z >~ fl,

from input objective functions

ykv*

v*yt*

and min

from output dual

(5D)

where u, v are input and output vectors of shadow prices and w, z are slacks on the :~, fl constraints. If :~ and /3 are chosen in each case such that the appropriate slacks are zero, the constraints systems in (4D) and (5D) are identical. Thus a pair (u,v) which is feasible in (4D) must be feasible in (5D).

Thus the ambiguity of different input and output measure is resolved. Using the above equivalences E = EI = Eo may be written variously as: u*xs* E -- - -

u* x k

v*y k

v* yt*

~0"

flop* "

Since the measure is derived from a set of homogeneous constraints, it is unit invariant, and standardisation is unnecessary. It also appears to

P.C. Ritchie, J.E. Rowcrofi/lnt. J. Production Economies 46 47 (1996) 433 439

reconcile the alternative Russell formulations in a theoretically appealing way. In fact, one possible solution is the Farrell radial model in which :~ and fi are scalars and E1 and E o necessarily collapse into a single measure. However, as defined, the measure is not necessarily unique, and following the spirit of H a a g et al. [13] we characterise E as the m i n i m u m attainable value subject to the nonnegativity of u and r [equivalently :~ and fl).

437

The Russell output measure is the solution to the program max

4)1 + 4'2

s.t.

44)1 -- 4tl -- 3t2 -- 2t3 ~< O, 44) 2 --

4tl - t2 - - 2t3 <~ 0,

7tl + 1 2 + 2 t 3 ~ < 7 , 8 t l + 2t2 + t 3

~< 8,

which yields 5.

An illustration

tl = O, 12 = 3, t3 = 2, and 4)~ = 11/4, 4)2 = 7/4 An industry consists in part of three firms each using the same two inputs ( x l , x 2 ) to produce two outputs (Yl,Y2), according to the technology in Table 1. We consider various measures of the efficiency of firm 1 relative to the best practice defined by firms 2 and 3. A simple radial measure is given by the output program: max

4)

s.t.

4~b -- 4tl -- 3t2 -- ts <~ O,

for an efficiency measure of 2(11/4 + 7/4) i = 4/9 or 0.4444, whereas the corresponding input measure gives sl = 0, s2 = 0.8, s3 = 1.6, and O~ = 4/7, 02 = 2/5 for an efficiency measure of 1(4/7 + 2/5)

1%, 5 or

0.4857.

W h a t may be termed the Charnes 'stability' index for firm 1 is the solution to the following program, which adapts p r o g r a m (1) to the present context, using data standardised on c o m p o n e n t means.

44) - 4tl - t 2 - - 2t3 ~< 0, max

0

s.t.

--0 + 1.50sl + 1.13s2 +0.38s3 - 2 1 = 1.50,

7t~ + t2 + 2 / 3 ~ 7,

8t~ + 2t2 + t~ <~ 8,

--0 * 1 . 7 1 s l +0.43s2 +0.86s3 - 2 2 =1.71,

which yields tl = 0 , t 2 = 3 ,

0 + 2.10sl + 0.30s2 + 0.60s3 + t3=2,

4)=7/4

Table I Inputs

2.10,

0 + 2.18sl + 0.55Se + 0.27s3 + )-4 = 2,18,

for an efficiency measure of 4)- t = 4/7 = 0.5714. As noted earlier, a simple input radial p r o g r a m yields the same measure.

Firm

23 =

Outputs

xl

x2

)'1

3'2

1

7

8

4

4

2 3

1 2

2 1

3 1

1 2

which

gives

sl = 0,

s23 = 1.11.

s3 = 2.06,

0 = 0.5342.

Finally, the reconciliation of input and o u t p u t s h a d o w prices suggested in Section 4 proceeds as follows. F r o m (4) and (4D) the input primal and dual are: rain

~1(11 + 3(202

s.t.

4 s 1 -t- 3 s 2 + s 3 /> 4,

4s~ 4 s2 + 2S3 >/4, 70

-- 7&

-

s2 - -

2t3 >/- 0,

8 ( 1 - - 8S 1 - - 2S 2 + S 3 / > 0 ,

P.C. Ritchie, J.E. Rowcroft/Int. ~L Production Economics 46-47 (1996) 433-439

438

-01/>

--i,

--02>

-1

and max

4vl + 4v2

s.t.

4v~ + 4v2 - 7ut - 8u2 ~ O, 3v~ + v2 - u 1 - 2u2 <~ O, v~+2v2-2ul-u2<<.O, 7ul + wl <~ cq, 8U 2 @ W2 ~ 0~2.

F r o m the dual, v~ = 0.6u2 and v2 = Ul + 0.2u2, whence

of a firm's efficiency relative to other firms in the same industry - the so-called comparison with best practice. However, if we wish to develop explanations of firms' behaviour it is desirable to have a quantitative index of efficiency (or rather inefficiency) which can be used as a dependent variable in subsequent statistical investigations• In this paper it was argued that the measures proposed to date are arbitrary to an unacceptable degree and that a more appropriate distance measure should be based on the structure of the underlying programs. This paper is very much a first stage in the development of such a measure and much remains to be done particularly in terms of the sensitivity of particular problems to the use of different measures.

dual O F = 4vl + 4v2 = 4u~ + 3.2 u2 = primal O F = cq01 + c~202 = 7u10~ + 8u202 from the

dual. Thus 0 ~ = 0 . 5 7 , 0 2 = 0 . 4 0 , s ~ = 0 , s z = 0 . 8 and s3 = 1.6. The measure of efficiency is given by EI ~--

usx UXt

--

4ul + 3.2u2 7Ul + 8u 2

Similarly, using (5) and (5D) the output measure yields (])1 = 11/4, 4~2 =- 7/4, tl = 0, t2 = 3, t3 = 2 as expected and the measure of efficiency is given by g O ~

vy 1

4Vl Jr- 4V2

vty

l l v l + 7v2

Using the values for v~,v2 in terms of Ul,U2 confirms immediately that E~ = E o . Noting that u~,u2, v~, v2 must be non-negative, E~ e (0.4000, 0.5714) and Eo e (0.3600, 0.5714) hence the minimum feasible value for E = E~ = Eo is 0.4000. We also note that the upper limit corresponds to the radial measure determined earlier.

6. Conclusion One of the primary virtues of DEA is its relative simplicity as a deterministic non-parametric gauge

Appendix Lemma. There exists a pair (u, v) which solves pro9rams (4D) and (5D). Proof. Note first that if the ~ are determined by the solution the remaining constraints are homogeneous and program (4D) may be considered as the problem of solving for a set of v in terms of a given set of u. Similarly, program (5D) is solved for u in terms of v. N o w let (u*,v*) maximise ykv in (4D). M a p (u*,v*) to the pair (u***,v***) which miniraises xku in (5D). N o w map (u**, v**) to (u**, v**) which maximises ykv in (4D). The mapping (u*, v*) --+ (u***, v***) is continuous by application of Berge's [14] m a x i m u m theorem and it is defined on a non-empty compact convex set of R "+" by nature of the linear program. Therefore by Kakutani's [15] theorem the mapping has a fixed point such that

(u*,v*)

= (u***,v***).

Therefore (u*, v*) is a solution to both programs.

References [1] Russell, R.R., 1985. M e a s u r e s of technical efficiency. J. E c o n o m . Theory, 35: 109-126.

P.C Ritchie, ,LE. Rowcroft/Int. J. Production Economics 46 47 (1996) 433 439

[2] Charnes, A., Haag, S., Jaska, P. and Semple, J., 1992. Sensitivity of efficient classifications in the additive model of data envelopment analysis. Int. J. Systems Sci., 23: 789 798. [3] Cornwell, C., Schmidt, R. and Sickles, R.C., 1990, Production frontiers with cross-section and time series variation in efficiency levels. J. Econometrics, 46: 185-200. [4] Button, K.J. and Weyman-Jones, T.G., 1994. X-efficiency and technical efficiency. Public Choice, August 272--274. [5] Colwell, R.J. and Davis, E.P., 1992. Output and productivity in banking. Scand. J. Econom., 94: Sl11-S129. [6] Aigner, D.J. and Chu, S.F., 1968. On estimating the industry production function. Amer. Econom. Rev., LVlII 4: 826 839. [7] Farrell, M.J., 1957. The measurement of productive efficiency. J. Roy. Statist. Soc. Ser. A, 120: 253-290. [8] Shephard, R.W., 1970, Theory of Cost and Production Functions. Princeton University Press, Princeton. [9] Brander, J.R.G., Ritchie, P.C. and Rowcroft, J.E., 1992. Deregulation and productivity changes in the U.S. airline system. 6th World Conf. on Transport Research, Lyon, June. 10] Bjurek, H., Kjulin, U. and Gustafsson, B., 1992. Efficiency, productivity and determinants of inefficiency at public

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day care centers in Sweden. Scand. J. Econom., 94: S173-S187. F/ire, R., Grosskopf, S. and Lovell, C.A.K., 1985. The Measurement of Efficiency of Production. Kluwer-Nijhoff, Boston. F/ire, R., Grosskopf, S. and Lovell, C.A.K., 1994. Production Frontiers. Cambridge University Press, Cambridge. Haag, S., Jaska, P. and Semple, J., 1992. Assessing the relative efficiency of agricultural production units in the Blackland Prairie, Texas. Appl. Econom., 24:559 565. Berge, C., 1963. Topological Spaces, tr. Patterson. Macmillan, New York. Kakutani. S., 1941. A generalization of Brouwer's fixed point theorem. Duke Math. J., 8: Fare, R., Lovell, C.A.K. and Zieschang, K, 1982. Measuring the technical efficiency of multiple output production technologies, in: W. Eichhorn et al. (Eds.), Quantitative Studies in Production and Prices. Physica-Verlag, Wurzburg. Lovell, C.A.K. and Schmidt, P., 1988. A comparison of alternative approaches to the measurement of productive efficiency, in: A. Dogramaci and R. F/ire (Eds.), Applications of Modern Production Theory. Kluwer, Boston.