An extended Fa
K
re
}
Lovell technical e
$
ciency measure
qW. Briec
*
ManLtre de Confe&rences, Universite& de Rennes 1, CREREG, 11 rue Jean Mace&-BP1997, 35019 Rennes Cedex, France
Received 13 March 1998; accepted 5 February 1999
Abstract
Recently, Chambers et al. (Journal of Economic Theory 70 (1996) 407}419; Working Paper, Southern Illinois University, Carbondale, 1996) have developed the concept of directional distance to measure technical e$ciency. However, there is no guarantee that such a measure must intersect the e$cient subset (also called strong e$cient subset) de"ned by Koopmans (Activity Analysis of Production and Allocation, 1951, Vol. 36, pp. 27}56), that is primarily the Pareto e$cient subset. So, we develop a framework for measuring e$ciency in the full input/output space. Following FaKre and Lovell (Journal of Economic Theory 19 (1978) 150}162), we introduce a graph-type extension of the Russell measure. Moreover, we show that our new measure can be computed by using linear programming. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: E$ciency indices; Graph measure; Technical e$ciency; Production technology; Distance functions
1. Introduction
In recent years, a number of studies on the theor-etical and empirical measurement of technical e$-ciency has been generated by researchers, and two di!erent notions of technical e$ciency have emerged in the economic literature. The"rst notion due to Koopmans [1], de"nes a producer as tech-nically e$cient if a decrease in any input requires a increase of at least one other input. This de"nition is closely related to the Pareto e$ciency concept,
*Corresponding author. Tel: (33) 02 99 84 78 08; fax: (33) 02 99 84 77 90.
E-mail address:walter.briec@univ-rennes1.fr (W. Briec) q
The"rst version of this paper was presented to the 1996 Latin American Meeting of the Econometric Society (Rio de Janeiro, August).
and its great intuitive appeal has led to its adoption by several authors, in particular by FaKre and Lovell [2]. The second notion introduced by Debreu [3] and Farrell [4], is based on radial measures of technical e$ciency. In the input case, the De-breu}Farrell index measures the minimum amount that a vector can be shrunk along a ray while holding output levels constant. This e$ciency in-dex is constructed around a technical component that involves equiproportionate modi"cation of inputs, and this has received a growing interest during the last few years. Following Charnes et al. [5], several empirical papers have implemented the Debreu}Farrell measure. In particular, de-scribing the production set as a piece-wise linear technology, it can be computed by linear program-ming.
Recently, several authors suggested some new development in order to judge adjustments of both input and output quantities simultaneously. In
particular, Chambers et al. [6,7] have introduced the`directional distance functionaas an e$ciency measure, by how much the outputs can be in-creased and the inputs dein-creased. We made inde-pendently the same thing by de"ning the so-called `Farrell proportional distancea [8]. These new measures are derived from a recent concept due to Luenberger [9]: the`shortage functiona.
The aim of this paper is the introduction of a new distance function for measuring technical e$ciency in the full-input}output space. Why do we want another e$ciency measure? First, the above-men-tioned indices are not able to measure the technical e$ciency (also called strong e$ciency) in the full-input}output space, with respect to the criterion introduced by Koopmans [1]. In particular, the directional distance function, and the Farrell pro-portional distance function are constructed around a technical component that has not the#exibility to select a strong e$cient vector on the boundary of the production set. These measures, in the general case, select a vector on the weak e$cient subset, generally smaller. Second, FaKre and Lovell [2] in-troduced the Russell measure that evaluates tech-nical e$ciency, but this has not the ability to judge adjustments of both input and output quantities simultaneously.
From the above considerations, the objective of this paper is to merge the ideas of the directional distance function with the Russell measure in order to form an extended `FaKre}Lovell technical e$-ciency measurea. This selects a strongly e$cient vector and measures technical e$ciency in the full-input}output space. According to FaKre and Lovell we term it: Russell proportional distance function. This measure calls a vector technically e$cient if and only if it belongs to the strong e$cient subset.
From the theoretical standpoint, our measure can be used to compare the production vectors: it characterises the observed e$cient subset, it is weakly monotonic and unit invariant. The Russell proportional distance has another important prop-erty. It is constructed around a technical compon-ent that has the#exibility to take into account the particularity of the market by introducing a weigh-ing scheme. In our opinion this property is interest-ing, because the producer may consider that some
particular inputs and outputs are more important than the others.
The paper is organised as follows: in Section 2, we focus our attention on the shortage function [9] the directional distance function [6,7], and the Far-rell proportional distance function [10]. Moreover, we show that a proportionally modi"ed input}out-put vector does not necessarily belong to the strong e$cient subset of the full-input}output space.
In Section 3, we introduce a graph-type exten-sion of the Russell measure and we state its basic properties. Moreover, among the outcomes of the section is the comparison of the Russell tional distance function and the Farrell propor-tional distance function [10]. In particular, we show that the comparison is in accordance with the result provided by FaKre et al. [11], alternatively in input or output space.
2. Distance function and technical e7ciency
First we de"ne the standard notations used in this paper. LetR`n be the non-negative Euclidean
n-orthant, for z,z@3Rn with n'1 we denote:
z6z@Qzi)z@i∀i3M1,2, nN, z)z@Qz6z@ and zOz@, z(z@Qzi(z@i∀i3M1,2,nN. A pro-duction technology transforming inputs x" (x
1,x2,2,xn)3Rn`into outputsy"(y1,y2,2,yp)3
Rp`can be characterised by the input correspond-ence¸:yP¸(y)LRn`and the output correspond-enceP:xPP(x)LRp`, where¸(y) is the set of all input vectors which yield at leastyandP(x) is the subset of all outputs vectors obtainable fromx. The set of all the input}output vectors which are feas-ible is called the graphTand de"ned as follows:
¹"M(x,y);x3¸(y),y3Rp`N
"M(x,y);y3P(x),x3Rn`N. (1)
We assume that the production set Tsatis"es the following axioms (see [11,12]):
T1: (0, 0)3¹, (0,y)3¹Ny"0,
T2: ¹(x)"M(u,y)3¹;u6xNis bounded∀x3Rn`,
T3: ¹is a closed set,
The following assumption may also be intro-duced:
T5: ¹is a convex set.
T5 postulates the convexity of the production set. This assumption is often used in empirical works and it can be useful to describe the produc-tion set as a convex combinaproduc-tion of positive vectors (see [5,11,13]). In such case, the production set is de"ned as a piece-wise linear technology. More precisely, under convexity and strong dis-posability assumptions, consider m activities (x1,y1), (x2,y2),2, (xm,ym). The production set can now be de"ned as
¹"
G
(u,v)3Rn`p` ;u7+mThe setCcharacterises the returns to scale chosen by the producer. It is possible to characterise con-stant returns to scale (see [5]), various returns to scale [14], non-increasing returns to scale [15,16], non-decreasing returns to scale [17]:
(a) Constant returns to scale [5]:
C:"C
CRS"Rm`. (3)
(b)Various returns to scale[14]:
C:"C
(c)Non-increasing returns to scale[17]:
C:"C
(d)Non-decreasing returns to scale[15,16]:
C:"C
Note that all the above speci"cations of the pro-duction set imply the convexity assumption. It is however, possible to drop the convexity assump-tions by using the FDH methodology by Tulkens and Vanden-Eeckaut [18]. In such case the set Ccan be characterized as follows:
C:"C
In order to judge adjustments of both inputs and output quantities simultaneously, it is possible to model the technology with the graph [13]. A graph measure of technical e$ciency is constructed around a technical component that involves a vari-ation of both input and output: inputs are reduced, while outputs are simultaneously increased until it reaches the boundary of the production set.
Recently, Luenberger [19] introduced a function he terms the bene"t function. The bene"t function, has its roots in a construction by Dupuit [20]) called `relative utilitya. In particular Luenberger introduces the notion of a `shortage functionain his studies of production. Luenberger [19] has seen the `shortage functiona as a shortage of (x,y) to reach the boundary of¹. More recently, Chambers et al. [6,7] have interpreted the distance as an e$ciency measure, by how much output can be increased and input decreased. They term the new function as`directional distancea. This can be con-sidered as a development of what Luenberger calls the`shortage functionaand can be used to evaluate technical e$ciency. The directional distance func-tion in the direcfunc-tion ofg"(!gi,g0) can be de"ned as follows:
De5nition 1. LetT be a production set satisfying T1}T4. Letg"(!gi,g0)3(!Rn`)]Rp`be a vec-tor, the functionDogT:¹PR
`de"ned by
DogT(x,y)"max
d Md*0; (x!dgi,y#dg0)3¹N is called directional distance function in the direc-tion ofg.
and J(b)"Mj3M1,2,pN;bj(o)'0N. Similarly, we denote I(x)"Mi3M1,2,nN;xi'0Nand J(y)" Mj3M1,2,pN;yj'0N. Now, consider the two matrices A and B which are, respectively, n]n
and p]p non-negative diagonal matrices, such thatA"Diag(a(n)) andB"Diag(b(o)). The Farrell proportional distance function is de"ned as follows:
De5nition 2. LetT be a production set satisfying T1}T4. Let (a(n),b(o))3Rn`p` be a price dependant orientation. Assume that (I(x)WI(a))X(J(y)WJ(b)) O0, the functionD(a,b)
T,F :¹PR`de"ned by
D(a,b)
T,F(x,y)"maxd Md*0; ((I!dA)x, (I#dB)y)3¹N
is called oriented Farrell proportional distance function.
From the above considerations, the Farrell pro-portional distance function can be expressed at point (x,y) as a particular directional distance func-tion in the direcfunc-tion ofg"(!Ax,By). This point was suggested by Chambers et al. [6,7].
A di$culty with the measures introduced is that a projected input}output vector does not necessar-ily belong to the e$cient subset of the production set.
For instance, consider the Fig. 1. Clearly the point M is weakly e$cient [11] and the propor-tional distance function is zero. However, M does not satisfy the FaKre}Lovell [2] criterion for tech-nical e$ciency: a decrease of the inputxdoes not require a decrease of the outputy. This means that the Farrell proportional distance function does not characterize the e$cient points. On the contrary, it is clear thatM@is technically e$cient. That is, the Farrell proportional distance function projects an input}output vector onto the weak e$cient subset and not necessarily onto the graph e$cient subset, generally smaller, de"ned by FaKre et al. [11]. The e$cient subsets ofTcan be de"ned as follows:
De5nition 3. LetT be a production set satisfying T1}T4. We have the following de"nitions:
(1) Lety*0, the e$cient subset of¸(y) is de"ned as LK(¸(y))"Mx3¸(y);x@)xNx@N¸(y)N and we haveLK(¸(0))"M0N.
Fig. 1.
(2) If P(x)OM0N the subset of P(x) de"ned by LK(P(x))"My3P(x);y@*yNy@NP(x)Nis called e$cient subset ofP(x), if P(x)"M0N, we have LK(P(0))"M0N.
(3) The subset LK(¹)"M(x,y)3¹; (!x,y)) (!x@,y@)N(!x@,y@)N¹N is called the graph e$cient subset ofT.
Since the notion of technical e$ciency compels a vector to belong to the strong e$cient subset, there is reason to develop a graph measure that projects an input}output vector onto the graph e$cient subset.
3. An extended FaKre+Lovell technical e7ciency
measure
Now, let us de"ne the Russell measure of tech-nical e$ciency, which was introduced by FaKre and Lovell [2]. The Russell measure}theoretical and empirical } evaluates e$ciency alternatively in input or output space (note that FaKre et al. [11] have de"ned a`hyperbolicaRussell graph measure of technical e$ciency). This measure is related with the e$cient subset of ¸(y), and satis"es the weak monotonicity conditions. These properties were established by FaKre and Lovell [2] and Russell [21], respectively.
proportionate increasing in individual outputs in coordinate direction. We denote Card(I(x)) the number of positivex
is, and Card(J(y)) the number
of positive y
js. The Russell measure of technical
e$ciency is de"ned as follows (see [13]):
De5nition 4. Let ¹be a production set satisfying T1}T4.The functionEiR:¹PR
is called Russell input measure of technical e$-ciency.
is called Russell output measure of technical e$-ciency.
Note that contrary to the Russell input measure, the Russell output measure does not completely characterise the e$cient subset for all technologies, in particular if some output is null. Now, we de"ne an`orientationaof measurement of technical e$-ciency. We modify the previous Farrell propor-tional distance, by maximising the modi"cations of inputs and outputs in co-ordinate direction. Let p ando be, respectively, inputs and outputs price vectors as de"ned in Section 2. It is possible to transform inputs and outputs introducing coe$-cients which are price dependant:
f each individual inputx
i, for i"1,2,n, is
pro-portionally decreased by the factor,d
iai(n) with
a
i(n)3[0, 1]. Thus, the input obtained is
(1!d
iai(n))xi;
f each individual outputy
j, forj"1,2,p, is
pro-portionally increased by the factor e
jbj(o) with
b
j(o)3[0, 1]. Thus, the output obtained is
(1#e
jbj(o))yj.
With this notation, it is clear that the oriented Russell proportional distance of (x,y) exists if and only if we have (I(x)WI(a))X(J(y)W(Jb))O0. We will discuss later the relationship between Russell proportional distance function and Russell measure of technical e$ciency. The oriented Rus-sell proportional distance function is de"ned as follows:
De5nition 5. LetT be a production set satisfying assumptions T1}T4. Let (a(n),b(o))3Rn`p` be a price-dependant orientation. and let us denote
r(x,a)"Card(I(x)WI(a)) and s(y,b)"Card(J(y)W
is called the oriented Russell proportional distance function.
Since¹ is closed, and T2 holds the maximum exists and the Russell proportional distance is well de"ned.AandBde"ne an orientation, and accord-ing to this we have input, output or graph-type methodology.
Consider the two-dimensional case (see Fig. 1). In such case, matricesAandBare scalar and thus, we haveA"aandB"b. Ifa"0 and b'0 the Russell proportional distance function measures the maximal proportional increase in output given the technology and the input vectorx. Ifa'0 and b"0, the Russell proportional distance measures the maximal proportional decrease in input given the technology and the output vectory. Ifa'0 and b'0 the Russell proportional distance is a graph-type methodology. It measures the maximum sum of proportionate modi"cations in inputs and out-puts in co-ordinate directions.
Russell proportional distance we have de"ned above. Thus, in order to salvage these properties it is necessary to consider the subset of all the produc-tion units (x,y) such thaty'0. Since in the general case the products of the observed decision-making units are positive, this assumption is not very re-strictive. We denote¹Hthe subset ofTde"ned as
¹H"M(x,y)3¹;y'0N. (8)
Thus,∀(x,y)3¹we denotePH(x) the subset ofP(x) de"ned asPH(x)"My3P(x);y'0N. The properties of the oriented Russell proportional distance func-tion are summarized as follows:
Proposition. Let T be a production set satisfying as-sumptions T1}T4. Assume that (I(x)WI(a))X(J(y)W
J(b))O0, and let D(a,b)
T,R be the Russell proportional
distance function, then
(4)The Russell proportional distance satisxes the weak monotonicity conditions over the subset¹H. We have
∀(x,y), (x@,y@)3¹H, (!x@,y@)6(!x,y)
ND(Ta,,Rb)(x,y))D(Ta,,Rb)(x@,y@).
(5) The Russell proportional distance function is unit invariant.
of the maximisation program computing the Rus-sell proportional distance. Since T2 holds, M(u,v)3Rn`p` ;u6x,v7yNis closed and bounded, The proofs of (b) and (c) are similar.
(2) Assume that j"I!d. Assume that for all
therefore, we have the relationship
D(a,b)
(3) The proof is similar to (2) making the trans-formationh"I#d.
(4) First, we prove the input monotonicity. Let (x@,y)3¹, with x6x@ and assume that X(x,y)" M(d,e)3Rn`p` ; ((I!dA)x, (I#eB)y)3¹N, thus we haveX(x,y)LX(x@,y) and the input monotonicity is proved. Now, assume that y7y@'0, since
y@'0 we have necessarily X(x,y@)M(x,y). Thus, the output monotonicity holds. We deduce im-mediately (4).
(5) LetLbe a (n#p)](n#p) positive diagonal matrices de"ned overRn`psuch that
¸"
A
¸1 0 0 ¸2
B
where¸
1and ¸2are, respectively,n]nand p]p
positive diagonal matrices. We need to prove that
D(a,b)
is the Russell proportional distance de"ned over ¸(¹). (¸
1x,¸2y) is the input}output vector
ob-tained with respect to a change in the units of measurement. Thus we have
SinceLis an isomorphism overRn`p, thus
D(LTa,b,)R(¸
Consequently, the Russell proportional distance function is invariant with respect to a change in the units of measurement.
(6) It is possible to show that
D(Ta,,bF)(x,y)
The fourth result proves that the generalised Russell}FaKre}Lovell measure characterises the observed decision-making units that are strongly e$cient. In the same idea, it is possible to relate the input and output e$cient subsets and the Russell proportional distance. Property (4) shows that the Russell proportional distance function satis"es the weak monotonicity condition. A similar proof is for instance given in Ref. [21]. (5) proves that the proportional distance is invariant with respect to a change in the units of measurement.
The property (6) states a comparison between the Farrell and Russell proportional distances. A sim-ilar result is obtained by FaKre et al. [11] alterna-tively in the input or output space. From properties (2) and (3), it is obvious that (6) is in accordance with the result obtained by FaKre et al.
Now, we focus on the particular relationship between the proportional distance and the Russell input and output measurements of technical e$-ciency. Assume thatb"0. The input set dimension is two. Fig. 2 illustrates the di!erent cases.
Assume that a1"a
2"1, the Russell
propor-tional distance function coincides with the Russell measurement of technical e$ciency and we have the relationship D(Ra,b)"1!EiR. The factors of
production are equiproportionately decreased. If a1(a2the second input is reduced more than the "rst one, and the Russell proportional distance function is oriented more to the direction of the
x
1-axis. Ifa1'a2, the"rst input is reduced more
than the second one, and the Russell proportional distance is oriented more in the direction of the
x
2-axis. Of course, a similar analysis can be
pro-vided in the output case. We will not discuss this point.
We are now able to, from the above results, provide some linear program in order to compute the Russell proportional distance function when the production set is de"ned as a piece-wise linear technology. From relationship (2), we get
max d,e
1
r(xk,a) +
i|I(a)
di# 1
s(yk,b) +
j|J(b)
ej,
(I!dA)xk7+m
i/1
h
ixi,
(I#eB)yk6+m
i/1
h
iyi,
d,e60, h3C, (P)
where C is a polyhedral subset that characterises the returns to scale. Recalling the relationship between the proportional Distance and Russell measurement of technical e$ciency, for some par-ticular values of the parameters, the above linear program is identical to the linear program comput-ing Russell input and output measures (see [11]). Hence, the new methodology presented in this paper generalises the Russell measure introduced by FaKre and Lovell [2].
4. Conclusion
The foregoing discussion is entirely in the spirit of the recent literature on measures of technical e$ciency, discussing radial and non-radial measures. The measure we have introduced above is a generalisation, in the full-input space, of the Russell measure of technical e$ciency, introduce by FaKre}Lovell [2]. From the practical standpoint this measure has the advantage to select a strong e$cient vector onto the frontier of technical
e$ciency. Moreover, it has the #exibility to take into account some managerial criterion, and can be computed by linear programming.
Acknowledgements
I am grateful to C. Blackorby for helpful sugges-tions and I would like to thank two anoymous referees for comments that greatly improved the exposition of the paper.
References
[1] T.C. Koopmans, Analysis of production as an e$cient combination of activities, in: T.C. Koopmans (Ed.), Activ-ity Analysis of Production and Allocation, vol. 36, 1951, pp. 27}56.
[2] R. FaKre, C.A.K. Lovell, Measuring the technical e$ciency of production, Journal of Economic Theory 19 (1978) 150}162.
[3] G. Debreu, The coe$cient of resource utilization, Econo-metrica 19 (1951) 273}292.
[4] M.J. Farrell, The measurement of productive e$ciency, Journal of the Royal Statistical Society 120 (1957) 253}281.
[5] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the e$ciency of decision-making units, European Journal of Operational Research 3 (1978) 429}444.
[6] R. Chambers, Y. Chung, R. FaKre, Bene"t and distance functions, Journal of Economic Theory 70 (1996) 407}419. [7] R. Chambers, Y. Chung, R. FaKre, Pro"t, directional distance functions, and Nerlovian e$ciency, Working Pa-per Southern Illinois University, Carbondale; Journal of Optimization Theory and Application (1996) fothcoming. [8] W. Briec, An extended FaKre}Lovell technical e$ciency measure, Presented at XIV Latin American Meeting of the Econometric Society, 5}12 August 1996, Rio de Janeiro. [9] D.G. Luenberger, Microeconomic Theory, McGraw Hill,
Boston, 1995.
[10] W. Briec, A graph-type extension of Farrell technical e$ciency measure, Journal of Productivity Analysis 8 (1997) 1.
[11] R. FaKre, S. Grosskopf, C.A.K. Lovell, The Measurement of E$ciency of E$ciency of Production, Kluwer Nijho!
Publishing, Dordrecht, 1985.
[12] R.W. Shephard, Theory of Cost and Production Func-tions, Princeton University Press, Princeton, NJ, 1970. [13] R. FaKre, S. Grosskopf, C.A.K. Lovell, Production Frontier,
Cambridge University Press, Cambridge, 1994.
[15] L.M. Seiford, A Bibliography of data envelopment analysis (1978}1989)-DEA Bibliography 5.0 (University of Mass-achusetts, Amherst, MA), Department Industrial Engin-eering and Operations Research, 1989.
[16] L.M. Seiford, Recent development in DEA: The math-ematical programming approach to frontier analysis, Jour-nal of Econometrics 46 (1990) 7}38.
[17] S. Grosskopf, The role of the reference technology in measuring productive e$ciency, The Economic Journal 96 (1986) 499}513.
[18] H. Tulkens, P. Vanden Eeckaut, Non-parametric e$ciency progress and regress-measures for panel data: Method-ological aspects, European Journal of Operational Research 80 (1995) 474}499.
[19] D.G. Luenberger, Bene"t function and duality, Journal of Mathematical Economics 21 (1992) 461}481.
[20] J. Dupuit, De la mesure de l'utiliteH des travaux publics, Anales des Ponts et ChausseHes 8 (1844) 332}375. [21] R.R. Russell, Measures of technical e$ciency, Journal of
