13
Production, Manufacturing and Logistics Allocative efficiency of technically inefficient production units Peter Bogetoft a, * , Rolf Fa ¨re b , Børge Obel c a Department of Economics, Royal Agricultural University (KVL), 23 Rolighedsvej, DK-1958 Frederiksberg C, Denmark b Department of Economics, Oregon State University, 303 Ballard Extension Hall, Corvallis, 97331 OR, USA c Department of Organization and Management, University of Southern Denmark, Campusvej 55, DK 5230 Odense M, Denmark Received 9 May 2003; accepted 14 May 2004 Available online 3 August 2004 Abstract We discuss how to measure allocative efficiency without presuming technical efficiency. This is relevant when it is easier to introduce reallocations than improvements of technical efficiency. We compare the approach to the traditional one of assuming technical efficiency before measuring allocative efficiency. In particular, we develop necessary and suf- ficient conditions in the technology to ensure consistent measures, we suggest alternative interpretations of the approaches, and we relate them to motivational and organizational change perspectives. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Technical efficiency; Allocative efficiency; Homothetic technologies; Organizational slack; Organizational change strategies 1. Introduction In the production economic literature, the over- all inefficiency of a production plan is usually decomposed into technical and allocative efficiency in the following manner: first, the plan is projected onto the efficient frontier and some measure of the gains in moving from the original to the projected plan is called the technical efficiency. Next, it is determined which plan is optimal with the given prices. The gains in moving from the projected to the optimal plan is called allocative efficiency. One can say that allocative efficiency is treated as the residual when evaluating the overall perform- ance––first it is determined what can be explained by technical inefficiency. The rest is called alloca- tive inefficiency. The usual justifications for this approach is that the rate of substitution between the production factors is only well-defined at the frontier. We believe however that it is relevant to con- sider the other order of decomposition––i.e., to first evaluate the allocative efficiency and then the technical efficiency. In particular, it is relevant 0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.05.010 * Corresponding author. Tel.: +45 35282282; fax: +45 35282295. E-mail addresses: [email protected] (P. Bogetoft), rolf.fare@or- st.edu (R. Fa ¨re), [email protected] (B. Obel). European Journal of Operational Research 168 (2006) 450–462 www.elsevier.com/locate/ejor

Allocative efficiency of technically inefficient production units

Embed Size (px)

Citation preview

Page 1: Allocative efficiency of technically inefficient production units

European Journal of Operational Research 168 (2006) 450–462

www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Allocative efficiency of technically inefficient production units

Peter Bogetoft a,*, Rolf Fare b, Børge Obel c

a Department of Economics, Royal Agricultural University (KVL), 23 Rolighedsvej, DK-1958 Frederiksberg C, Denmarkb Department of Economics, Oregon State University, 303 Ballard Extension Hall, Corvallis, 97331 OR, USA

c Department of Organization and Management, University of Southern Denmark, Campusvej 55, DK 5230 Odense M, Denmark

Received 9 May 2003; accepted 14 May 2004Available online 3 August 2004

Abstract

We discuss how to measure allocative efficiency without presuming technical efficiency. This is relevant when it iseasier to introduce reallocations than improvements of technical efficiency. We compare the approach to the traditionalone of assuming technical efficiency before measuring allocative efficiency. In particular, we develop necessary and suf-ficient conditions in the technology to ensure consistent measures, we suggest alternative interpretations of theapproaches, and we relate them to motivational and organizational change perspectives.� 2004 Elsevier B.V. All rights reserved.

Keywords: Technical efficiency; Allocative efficiency; Homothetic technologies; Organizational slack; Organizational change strategies

1. Introduction

In the production economic literature, the over-all inefficiency of a production plan is usuallydecomposed into technical and allocative efficiencyin the following manner: first, the plan is projectedonto the efficient frontier and some measure of thegains in moving from the original to the projectedplan is called the technical efficiency. Next, it is

0377-2217/$ - see front matter � 2004 Elsevier B.V. All rights reservdoi:10.1016/j.ejor.2004.05.010

* Corresponding author. Tel.: +45 35282282; fax: +4535282295.

E-mail addresses: [email protected] (P. Bogetoft), [email protected] (R. Fare), [email protected] (B. Obel).

determined which plan is optimal with the givenprices. The gains in moving from the projected tothe optimal plan is called allocative efficiency.One can say that allocative efficiency is treated asthe residual when evaluating the overall perform-ance––first it is determined what can be explainedby technical inefficiency. The rest is called alloca-tive inefficiency. The usual justifications for thisapproach is that the rate of substitution betweenthe production factors is only well-defined at thefrontier.We believe however that it is relevant to con-

sider the other order of decomposition––i.e., tofirst evaluate the allocative efficiency and thenthe technical efficiency. In particular, it is relevant

ed.

Page 2: Allocative efficiency of technically inefficient production units

P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462 451

to examine how such a decomposition could beundertaken and in which cases the two approacheslead to similar results.First, from a conceptual point of view, we sug-

gest that the interpretations associated with thenotions of technical and allocative efficiency aremore ambiguous if the size of the effects dependon the order of decomposition. In a hierarchicalorganization composed of several productionunits, for example, technical inefficiency is inter-preted as what could be gained by intra-unitchanges while allocative inefficiency is interpretedas what can be gained by inter-unit changes or––more correctly––by interaction with a perfect mar-ket. Yet if the latter presumes the former, the twoconcepts are interdependent and not well-definedon their own. In particular, if we change our meas-ure of technical efficiency, say because certainreductions in the resources used may be more nat-ural to consider given the power structure in theorganization, we would also change the allocativeefficiency. This means that we can substitute be-tween what we diagnose as technical and allocativeproblems, which in turn make the interpretationssomewhat unclear. From a theoretical perspective,therefore, it is interesting to determine underwhich technological assumptions the decomposi-tion is unique.Secondly, from the point of view of (micro-eco-

nomic) productivity analysis, the allocative aspectsmay be the most natural ones to focus on. This lit-erature usually treats the production units as blackboxes or at least in very stylized manners. Thefocus of micro-economics is more on the interac-tion of firms over market and less on the processesthat take place inside the firms. It may therefore bemore appropriate to measure what can be accom-plished by reallocating resources between the blackbox units and via markets instead of what can beaccomplished by changes inside the black box pro-duction units. It seems that the latter is the field ofprocess engineers and organizational specialistswhile the former is the field of economists ingeneral.Thirdly, from a managerial and organizational

point of view, it is relevant to explore alternativeroutes to improve efficiency. Technical efficiencyis ‘‘to do things right’’ and allocative efficiency is

‘‘to do the right things’’. It may therefore beworthwhile to improve allocative efficiency of tech-nically inefficient units. The responsibilities ofimproving technical and allocative efficiency maybe associated with different groups in a firm andit may be important to evaluate the performanceof one group without assuming optimal behaviorof the others. Likewise it may be easier to motivateone group rather than another. One can argue thatit may be easier to reallocate resources within ahierarchy or via markets, than to actually changethe production procedures (including the culture,power configuration, incentive structure etc.) usedin the individual production units. Even in thosecase, where the same individuals are responsiblefor both types of efficiency, it may be easier tolearn and motivate using the right procedures,i.e. to improve technical efficiency, once the goalsand priorities are set right, i.e. allocative efficiencyis ensured. Either way, we need methods of esti-mating the potential gains from improving alloca-tive efficiency without presuming that technicalefficiency has already been accomplished.This paper is organized as follows. In Section

2, we formalize some key concepts about the tech-nology. In Section 3, we recall the traditionalapproach to allocative efficiency and suggest areverse approach that does not presume techni-cal efficiency. Some theoretical results on theconsistency of the approaches under different tech-nological assumptions are provided in Section 4.Alternative formulations of the two measuresof allocative efficiency are given in Section 5.Organizational interpretations are discussed inSection 6 and final remarks are provided inSection 7.

2. Technology

We consider the case where a production unithas used inputs x 2 Rp

þ to produce outputsy 2 Rq

þ. The technology T is given by

T ¼ fðx; yÞ 2 Rpþ � Rq

þjx can produce yg;

and assumed to satisfy the following standardassumptions (e.g. Fare and Primont, 1995):

Page 3: Allocative efficiency of technically inefficient production units

452 P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462

T is closed. Inputs and outputs are freely disposable:(x, y)2T, x 0 P x, y 0

6 y) (x 0, y 0)2T. T is convex. fyjðx; yÞ 2 T g is bounded for each x 2 Rp

þ.

The assumptions imposed here are not the min-imal under which most of the following analysescan be undertaken. Thus for example, we could re-lax the assumption of T being convex into theassumption that production and consumption sets,fyjx; yÞ 2 Tg, x 2 Rp

þ, and fxjðx; yÞ 2 Tg, y 2 Rqþ,

are convex (cf. the analysis of such productionmodels in Bogeoft, 1996; Bogetoft et al., 2000and Petersen, 1990). We stick to the more classicalassumptions here since they allow us to easilydrawn on a series of previously proved dualityresults.Classical measures of the distance between the

production plan (x, y) and the frontier of T is givenby Shephard�s input and output distance functions

Diðy; xÞ ¼ supkfk > 0jðx=k; yÞ 2 Tg;

D0ðx; yÞ ¼ infhfh > 0jðx; y=hÞ 2 Tg;

which measure the largest radial contraction of theinput vector and the largest radial expansion of theoutput vector that are feasible in T, (cf. Shephard,1953, 1970). Note that each of these functions givea complete characterization of the technology Tsince

Diðy; xÞP 1() ðx; yÞ 2 T ;

D0ðx; yÞ6 1() ðx; yÞ 2 T :

We are going to stage most of our discussions inthe input space. We note however that a paralleldiscussion is possible in the output space or evenin the full input–output space.The input requirement set L(y) is defined as the

set of inputs that can produce output y,

LðyÞ ¼ fx 2 Rpþjðx; yÞ 2 Tg;

and the input isoquant IsoqL(y) is the sub-set here-of where it is not possible to save on all inputs

IsoqLðyÞ ¼ fx 2 LðyÞje < 1) ex 62 LðyÞg:

The cost function C(y, x) is defined as the minimalcost of producing y when input prices are x 2 Rp

þ:

Cðy;xÞ ¼ minxfx � xjx 2 LðyÞg:

Note that by the assumed convexity of T and here-by L(y), there is a dual relationship between thecosts and the input distance function in the sensethat (see Fare and Primont, 1995, p. 48)

Cðy;xÞ ¼ minx

xxDiðy; xÞ

� �;

Diðy; xÞ ¼ infx

xxCðy;xÞ

� �:

Two properties of the technology will ease thedevelopment of alternative decompositions. Wesay that the technology T is input homothetic ifthe input sets for different output vectors are ‘‘par-allel’’, i.e.

LðyÞ ¼ HðyÞ � Lð1Þ;for some H(Æ) :Rq!R. Equivalently, this meansthat the cost function can be written as

Cðy;xÞ ¼ HðyÞ � CðxÞ;where Cð�Þ : Rp ! R, or that the input distancefunction equals

Diðy; xÞ ¼ DðxÞ=HðyÞ;for some Dð�Þ : Rp ! R.Similarly, we say that the technology T is input

ray-homothetic if the input sets for different outputvectors with the same direction are ‘‘parallel’’

LðyÞ ¼ GðyÞGðy=kykÞ � L

ykyk

� �;

where k � k denote the Euclidean norm in Rqþ and

G(Æ) :Rq!R. Equivalently this means that the costfunction can be written as

Cðy;xÞ ¼ GðyÞGðy=kykÞ � C

ykyk ;x

� �or that the distance function equals

Diðy; xÞ ¼Gðy=kykÞGðyÞ � Di

ykyk ; x

� �:

In adjusting inputs and outputs to obtain effi-ciency, input homotheticity does not place any

Page 4: Allocative efficiency of technically inefficient production units

P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462 453

restriction on the ‘‘freedom’’ of adjusting inputproportions, and this ‘‘freedom’’ should remainthe same for proportional adjustment of outputs.In the short run, proportional adjustments of out-puts are a ‘‘natural’’ way of conceiving inputchanges. It is, in fact, a rational response in a prin-cipal agent setting with asymmetric informationabout the cost drivers (cf. Bogetoft, 2000). Whena principal seeks to redirect the outputs from aproducing agent but do not know the relative costsof the different outputs, any non-proportionaladjustments of the outputs give the agent a strate-gic advantage as he can always claim that the leastreduced (or mostly expanded) outputs are the pri-mary cost drives.

3. Efficiency

In the following, we evaluate the productionplan (x, y) relative to the technology T and inputprices x. The efficiency indices therefore dependon (x, y), T and x. To simplify the exposition,however, we do not show this dependence in ournotation.The overall efficiency OE of production plan

(x, y) in the technology T when input prices arex is defined as the minimal costs of producing y

relative to the actual costs xx, i.e.,

OE ¼ Cðy;xÞxx

:

Thus, if OE=0.8, it means that 20% of actual costscould have been saved by improving efficiency. Allefficiency measures in this paper are weakly lessthan 1 and the smaller they are, the larger the inef-ficiency and savings potentials.The standard Farrell approach views the over-

all (in)efficiency as originating from two sources,viz.

technical (in)efficiency, which corresponds to aproportional reduction: x!x/Di (y,x) and

allocative (in)efficiency, which corresponds tothe an adjustment to the cost minimal inputcombination:x=Diðy; xÞ ! x�ðy;xÞ 2 argminx02LðyÞxx0.

This leads to the decomposition of the overallefficiency into technical efficiency TE and allocativeefficiency AE as follows:

OE ¼ Cðy;xÞxx

¼ 1

Diðy; xÞ� Cðy;xÞx x

Diðy;xÞ

� � ¼ TE �AE:

We propose here to supplement this standard ap-proach by what we will call a reverse Farrell ap-

proach. In this, we first correct for allocativeefficiency and next for technical efficiency:

allocative (in)efficiency, which corresponds toan adjustment of the input to a cost minimalinput combination:

x ! x�ðy;xÞ 2 arg minx02LðyÞ

xx0

where y is a reference output vector, cf. below,such that x 2 IsoqLðyÞ, and

technical (in)efficiency, which corresponds to aproportional reduction:

x�ðy;xÞ ! x�ðy;xÞ=Diðy; x�ðy;xÞÞ:

Taking this perspective, we get the following re-versed allocative efficiency AE* and reversed techni-cal efficiency TE* measures

AE�ðyÞ ¼ Cðy;xÞxx

;

TE�ðyÞ ¼ 1

Diðy; x�ðy;xÞÞ :

Both approaches, the standard Farrell and the re-versed Farrell decompositions are illustrated inFig. 1.The input requirement sets L(y) and LðyÞ are

illustrated in Fig. 1 with the initial input vector xinterior to L(y) but on the isoquant of LðyÞ. Thetwo paths of ‘‘reducing’’ x are indicated. In the tra-ditional Farrell approach, we reduce x from C to Busing the input distance function. Next we find theoverall efficient x*(y, x) atH. The overall efficiencyis measured as C to A. The second path, the re-versed Farrell approach, first moves x to its alloca-tive efficient point on LðyÞ, x�ðy;xÞ at E. Next, itapplies the input distance function removingtechnical inefficiency by moving to the point

Page 5: Allocative efficiency of technically inefficient production units

A

B

C

F

E

D

G

H

O x1

x2

*

*

OB OETE AE

OC OFOA OD

AE TEOB OE

= =

= =

x

( , )i

x

D y x* ˆ( , )x y ω

*( , )x y ω*

*

ˆ( , )

( , )i

x y

D y x

ω

ˆ( )L y( )L y

Fig. 1. Two decompositions into allocative and technicalefficiency.

454 P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462

x�ðy;xÞ=Diðy; x�ðy;xÞÞ at D on the isoquant ofL(y). Observe that this point may not be overallefficient in the sense of minimizing the cost of pro-ducing y, i.e. it may not coincide with H. We willexplore when D and H differ in the Section 4,and use a second order allocative inefficiency fac-tor to capture the possible differences in Section 5.In our new decomposition, there is some free-

dom in the choice of the reference output vectory, at which to evaluate the allocative efficiency.So far we have only assumed that the referencevector y is such that x 2 IsoqLðyÞ. With this out-put, x is at the frontier, viz. the isoquantIsoqLðyÞ, and allocative efficiency can be meas-ured. We now discuss this and some alternativeassumptions. In general, we use bY to denote anon-empty set of reasonable reference vectors.There is no obvious ‘‘best’’ way to choose y. In-

deed, this is precisely what makes the calculationof allocative efficiency without having technicalefficiency difficult. Using a sensitivity or lack ofinformation (ignorance) argument, one can there-fore presume that the reversed allocative and tech-nical efficiency concepts should give reasonableresults, i.e. consistent results along the lines de-fined below, for all y restricted only by the generalrequirement:

x 2 IsoqLðyÞ: ðA1ÞWe denote this set bY ðA1Þ, i.e. bY ðA1Þ ¼ fy 2Rq

þjx 2 IsoqLðyÞg, and use similar notation forother such sets.Another possibility is to assume that we can use

any y such that

x 2 IsoqLðyÞ and y P y; ðA2Þ

in the reversed evaluations. One way to rationalizethis assumption is in terms of the economics oforganizations. One can assume that the produc-tion unit has produced y, but that it ‘‘consumed’’y � y P 0 as slack, either directly by actually pro-ducing y and taking out y � y for off-the-recorduses or indirectly, by supplying less effort thanwhat is needed to actually produce y (cf. also Sec-tion 5).Of course, (A2) can be refined in a number of

ways by making more precise conjectures aboutwhich of the production plans y P y the productiveunit has actually chosen––or been targeting at. Anatural refinement in this case is to assume thatthe production unit has consumed outputs propor-tionally. This leads to the Farrell selection y ¼ ky,where k is set to ensure

x 2 IsoqLðkyÞ: ðA3ÞNote that whichever assumption we impose on thechoice of y, we want bY to be non-empty. This is anexistence condition.To clarify, we note that the distinction between

the three sets of reference vectors cannot be illus-trated in Fig. 1 since this depicts production ininput space, and the value of y and y cannot beseen here. The idea of the three sets however is thatthey represent different hypotheses of what outputvector may have been underlying the input vectorin an efficient production plan ðx; yÞ. This is easiestto understand if one thinks of the inefficiency as aresults of some of the underlying outputs y beinglost (eaten) before they are actually registered iny. It is natural to assume therefore that y P y asin (A2). If we furthermore assume that the con-sumption of outputs are proportional to the pro-duction levels, we get y ¼ ky as in (A3). If we donot want to be restricted to any of these interpre-tations, we simply require that y can be an arbi-trary output vector such that ðx; yÞ is efficient as

Page 6: Allocative efficiency of technically inefficient production units

P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462 455

in (A1). We shall return to these interpretations inSection 5.

4. Consistency properties

In this section, we show when the choice ofcomparison basis y according to the principles(A1), (A2) and (A3) leads to reversed allocativeand technical efficiencies with desirable properties.We commence by discussing some desirable con-sistency properties.One requirement that is reasonable in general is

the weak consistency requirement:

OE ¼ AE�ðyÞ � TE�ðyÞ 8y 2 bY :This requirement says that we want our measuresof technical and allocative efficiency to give a (mul-tiplicative) decomposition of the overall efficiency.We note that this is a trivial requirement in the tra-ditional Farrell approach where the allocative effi-ciency is essentially defined as a residual. It is nottrivial in the new approach, however, becausenone of the new components are residuals.The weak consistency requirement is a guaran-

tee that the results are not too peculiar in the sensethat the combined effect exceeds or falls short ofthe overall efficiency OE. As such, it is a mildrequirement.Depending on the technology and the choice of

the reference output vector y, there may still be lotsof room for what to assign to the allocative and tothe technical efficiency components––i.e., to sub-stitute between the two effects. Also, the order ofdecomposition may still be important. As arguedin the introduction, such possibilities of varyingthe results make the interpretation and applicationof the technical and allocative efficiency conceptsambiguous. A natural refinement of the weak con-sistency requirement is therefore to require that thetraditional and the reversed Farrell decomposi-tions lead to the same measures of allocative andtechnical efficiency. We call this the strong consist-

ency requirement:

AE ¼ AE�ðyÞ and TE ¼ TE�ðyÞ 8y 2 bY :With strong consistency, the measures of technicaland allocative efficiency depend neither on the

order of decomposition nor do they involve anyarbitrariness in the substitution between them.This allows the most clear and unambiguous inter-pretations of the concepts. In a recent paper,McDonald (1996) provided an example whichshowed that the decomposition of Farrel�s (1957)measure of technical efficiency into scale and con-gestion was order dependent, i.e., the order inwhich the two components are computed has im-pact on their size. Fare and Grosskopf (2000)showed that this result is related to a partial orderof the three input sets involved in the computation.In this paper, we do not have the partial orderproblem.Although the strong consistency requirement is

in general stronger than the weak consistencyrequirement, we shall show below that they do infact impose the same regularities on the technol-ogy for reasonable choices of the bY set.Initially, we note that one part of the weak con-

sistency requirement is trivial. The saving poten-tials calculated by the reversed measures cannever exceed the actual saving potentials. We re-cord this as a lemma.

Lemma 1. We have

OE6AE�ðyÞ � TE�ðyÞ 8y 2 bY ðA1Þ:Proof. As above, let x (y, x) be a cost minimalinput vector capable of producing y when inputprices are x. We then have

AE�ðyÞ � TE�ðyÞ ¼ Cðy;xÞxx

� 1

Diðy; x�ðy;xÞÞ

¼ xx�ðy;xÞ=Diðy; x�ðy;xÞÞxx

Pxx�ðy;xÞ

xx¼ OE

because x�ðy;xÞ=Diðy; x�ðy;xÞÞ may not be a costminimizer for y. h

The proof of Lemma 1 suggests that in orderfor the reversed Farrell measures to be weakly con-sistent, we basically need the cost minimizer for yand y to be proportional. Depending on the struc-ture of bY , this means that we need some type of

Page 7: Allocative efficiency of technically inefficient production units

456 P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462

homothetic technology to get consistent measures.We formalize this in the next propositions. Weemphasize that this should not be interpreted tosay that homotheticity is necessary for the reversedmeasure to make sense. Consistency is one of sev-eral attractive properties of a measurement ap-proach. What the next proposition shows is thatto get consistency, we need homotheticity and thatin this case, the standard and the reversed decom-positions coincide.

Proposition 1. With arbitrary choice of reference

output, y 2 bY ðA1Þ, the following properties are

equivalent:

1. Weak consistency:

8ðx; y;xÞ 2 T � Rpþ : OE ¼ AE�ðyÞ � TE�ðyÞ

8y 2 bY ðA1Þ;2. Strong consistency:

8ðx; y;xÞ 2 T � Rpþ : AE ¼ AE�ðyÞ and

TE ¼ TE�ðyÞ 8y 2 bY ðA1Þ;3. T is input homothetic.

Proof. First, assume that the weak consistencyrequirement holds for an arbitrary y 2 Rq

þ suchthat x 2 IsoqLðyÞ. From the proof of Lemma 1,we therefore have

x�ðy;xÞ ¼ x�ðy;xÞ=Diðy; x�ðy;xÞÞ:

If there are multiple cost minimizers, we can atleast choose x*(y, x) such that this holds. Thisshows that the expansion path is a ray when wemove between any y and y, for which there existsan x such that (x, y)2T and x 2 IsoqðyÞ.In turn, this implies that T is input homothetic:

Let y1 and y2 be arbitrary feasible output vectors.Then, by the disposability of inputs, there exists anx such that (x, y1)2T and x2 Isoq(y2) or (x, y2)2T

and x2 Isoq(y1). Either way, we arrive at x*(y1, x)

and x*(y2, x) being proportional and therefore thatT is input homothetic.Next assume that T is input homothetic. Then

the evaluations are also strongly consistent since

AE�ðyÞ ¼ Cðy;xÞxx

¼ Cðy;xÞDiðy; xÞxx

¼ HðyÞCðxÞDðxÞxxHðyÞ ¼ HðyÞCðxÞDðxÞ

xxHðyÞ

¼ Cðy;xÞxx=Diðy; xÞ

¼ AE

and

TE�ðyÞ ¼ 1

Diðy; x�ðy;xÞÞ ¼1

Diðy; xÞ¼ TE

since x�ðy;xÞ and x both belong to LðyÞ.Lastly, given strong consistency, we immedi-

ately get weak consistency as the traditionalFarrell measures gives a decomposition of OE:

OE ¼ AE � TE ¼ AE�ðyÞ � TE�ðyÞ:

We have hereby proved (1)) (3)) (2)) (1) andtherefore the desired equivalences. h

According to Proposition 1, we need an inputhomothetic technology to ensure the consistencyof the new decomposition for arbitrary choice ofthe output level y (with x 2 IsoqLðyÞ).From an organizational point of view, it is eas-

ier to justify (A2) than (A1) (cf. the interpretationthat some of the underlying outputs are lost). It isinteresting to note therefore that restricting thechoice of output reference to y 2 bY ðA2Þ does notaffect the conclusions from Proposition 1, includ-ing the need for a homothetic technology to ensurethat the reversed evaluations are consistent. We re-cord this as a corollary.

Corollary 1. If we restrict the reference output to

exceed the actual output, y 2 bY ðA2Þ, Proposition 1

is still valid.

Proof. As in Proposition 1, we may show that(1)) (3)) (2)) (1). The proofs of the last twoimplications are the same as in the proof of Prop-osition 1. To show the remaining first implication,assume that the weak consistency requirementholds for an arbitrary y 2 bY ðA2Þ, i.e. an arbitraryy 2 Rq

þ such that x 2 IsoqLðyÞ and y P y. Fromthe proof of Lemma 1, we therefore have

x�ðy;xÞ ¼ x�ðy;xÞ=Diðy; x�ðy;xÞÞ:

Page 8: Allocative efficiency of technically inefficient production units

P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462 457

If there are multiple cost minimizers, we can choosex*(y, x) such that this holds. This shows that theexpansion path is a ray when we move betweenany y and y P y for which there exists an x suchthat (x, y)2T and x 2 IsoqðyÞ. In turn, this impliesthat T is input homothetic: Let namely y1 and y2 bearbitrary feasible output vectors and letx12 IsoqL(y1) and x22 IsoqL(y2). Then, by the dis-posability of inputs, both x1 and x2 can producey ¼ minfy1; y2g where minf�g is the coordinate-wise minimum. Letting x=x1 and y ¼ y1 we get

x�ðy;xÞ ¼ x�ðy1;xÞ=Diðy; x�ðy1;xÞÞ;and letting x=x2 and y ¼ y2 we get

x�ðy;xÞ ¼ x�ðy2;xÞ=Diðy; x�ðy2;xÞÞ:Therefore the cost minimizers of arbitrary outputvectors y1 and y2 are proportional:

x�ðy1;xÞ ¼ Diðy; x�ðy1;xÞÞDiðy; x�ðy2;xÞÞ � x

�ðy2;xÞ;

which shows that T is input homothetic asdesired. h

If we refine the choice of reference vector intoy 2 bY ðA3Þ, i.e., we assume that the reference vec-tor is found by proportional expansion of the ac-tual y, we get the following necessary andsufficient conditions for consistency.

Proposition 2. If we restrict the reference output to

be proportional to the observed output, y 2 bY ðA3Þ,the following properties are equivalent:

1. Weak consistency:

8ðx; y;xÞ 2 T � Rpþ : OE ¼ AE�ðyÞ � TE�ðyÞ

8y 2 bY ðA3Þ:2. Strong consistency:

8ðx; y;xÞ 2 T � Rpþ : AE ¼ AE�ðyÞ

and TE ¼ TE�ðyÞ 8y 2 bY ðA3Þ:3. T is input ray-homothetic.

Proof. First, assume that the weak consistencyrequirement holds for (any) y ¼ ky 2 Rq

þ such thatx 2 IsoqLðkyÞ. From the proof of Lemma 1, wetherefore have

x�ðy;xÞ ¼ x�ðky;xÞ=Diðy; x�ðky;xÞÞ:If there are multiple cost minimizers, we canchoose x*(y, x) such that this holds. This showsthat the expansion path is a ray when we move be-tween any y and ky for which there exists an x suchthat (x, y)2T and x 2 IsoqðkyÞ. In turn, this im-plies that T is input ray-homothetic.Next assume that T is input ray-homothetic.

Then the evaluations are also strongly consistentsince

AE�ðyÞ ¼ Cðky;xÞxx

¼GðkyÞ

Gðky=kkykÞCky

kkyk ;x� �

xx

¼GðkyÞ

Gðy=kykÞCy

kyk ;x� �

xx¼

GðyÞGðy=kykÞC

ykyk ;x

� �xx GðyÞ

GðkyÞ

¼ Cðy;xÞxx Diðky;xÞ

Diðy;xÞ

¼ Cðy;xÞxx=Diðy; xÞ

¼ AE;

and

TE�ðyÞ ¼ 1

Diðy; x�ðky;xÞÞ¼ 1

Diðy; xÞ¼ TE

since x�ðky;xÞ and x both belong to IsoqðkyÞ andtherefore have the same distance to Isoq(y).Lastly, given strong consistency, weak consist-

ency is trivial since the traditional Farrell measuresgives a decomposition of OE:

OE ¼ AE � TE ¼ AE�ðyÞ � TE�ðyÞ:

We have hereby proved (1)) (3)) (2))(1) andtherefore the desired equivalences. h

According to Proposition 2, we need an inputray-homothetic technology to ensure the consist-ency of the new decomposition, if we choosekðxÞy (with x 2 IsoqLðkðxÞyÞ) as the basis for eval-uating the allocative efficiency.To intuitively clarify the relationship between

Propositions 1 and 2, we recall that the technolog-ical assumption of homotheticity is stronger thanthat of ray-homotheticity since the former pre-sume ‘‘parallel’’ input isoquants for arbitrary out-put vectors while the latter only compares inputisoquants for proportional output vectors. Like-wise, the consistency requirements are strongerin Proposition 1 (and Corollary 1) than in

Page 9: Allocative efficiency of technically inefficient production units

458 P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462

Proposition 2 since the consistencies must hold forlarger sets of possible reference outputs y in theformer rather than the latter case.When the technology does not satisfy the hom-

otheticity requirements in Propositions 1 and 2,the reversed measures do not fully capture the sav-ing potentials. Indeed, we know from Lemma 1that there will––in these cases––typically be somepossibility to gain from reallocation after the sec-ond step elimination of technical inefficiency. Thatis, if we do not restrict the technology and if weinitially seek to improve performance by a reallo-cation and next by an (Farrell) improvement oftechnical efficiency, we may be left with some ‘‘sec-ond order’’ allocative inefficiency:

AAE�ðyÞ ¼ xx�ðy;xÞxx�ðy;xÞ=Diðy; x�ðy;xÞÞ ;

resulting from the fact that a proportional (Far-rell) reduction in x�ðy;xÞ may not be the cost min-imizer for the production of y (cf. the proof ofLemma 1). This leads to a three factor decomposi-tion of the overall efficiency:

OE ¼ AE�ðyÞ � TE�ðyÞ �AAE�ðyÞ 8y 2 bY ðA1Þor

OE ¼ Cðy;xÞxx

� 1

Diðy; x�ðy;xÞÞ

� xx�ðy;xÞxx�ðy;xÞ=Diðy; x�ðy;xÞÞ 8y 2 bY ðA1Þ;

where the first allocative effect, AE*, does not pre-sume technical efficiency and may therefore be rel-atively easy to generate while the latter allocativeterm, AAE*, does presume technical adjustments.In Fig. 1, AAE* corresponds to OG/OD.

5. Input or output slack

An important rationale for the reversed decom-position is that it may be easier to reallocate re-sources than to improve technical efficiency.Following the concept of X-efficiency by Lei-

benstien (1966, 1978), there are two obvioussources of technical inefficiencies.One is inadequate decision making. An ineffi-

cient production unit uses sub-optimal decision

making procedures and production practices. Thisis the typical explanation found in the productivityanalysis literature. It implies that performance canbe improved by learning the procedures and prac-tices of the efficient units.The other possible source of inefficiency is re-

lated to the conflicts of interest and asymmetricinformation in a decentralized organization. Theinefficient unit has excessive on-the-job consump-tion of resources and does not motivate sufficient(non-measured) effort to save on the inputs or ex-pand the output. It implies that performance canbe improved by changing the incentive schemes.This perspective has been advocated in the produc-tivity analysis literature by for example Agrellet al. (2002) and Bogetoft (1994, 1997, 2000).Allocative efficiency measures the improve-

ments that can be accomplished by reallocationsof inputs. The two measures, the traditional andthe reverse allocative efficiencies AE and AE*, pro-vide potentially different estimates. This is not sur-prising given their definition as part of twodifferent change strategies. We will discuss thesechange sequences from an organization perspec-tive at some length in the next section. Here, weprovide alternative interpretations of the twomeasures.The new interpretations offered in this section

do not rely on different adjustment sequences.Rather, they rely on dual assumptions about thesources of the technical inefficiency.The traditional measure of allocative efficiency

presumes excessive consumption of inputs orsub-optimal input handling procedures as thesource of technical inefficiency. To see this, notethat AE can be rewritten as

AE ¼ Cðy;xÞxx=Diðy; xÞ

¼ minx0

xx0

xx=Diðy; xÞ

����x0 2 LðyÞ� �

¼ minx0

xx0Diðy; xÞxx

����x0 2 LðyÞ� �

¼ minx00

xx00

xx

����x00 2 Diðy; xÞLðyÞ� �

:

One interpretation of this is that we seek to reducethe cost of producing y but we realize that we actu-ally need inputs in a factor Di (y,x) in excess ofwhat is truly needed. That is, if we allocate x00 to

Page 10: Allocative efficiency of technically inefficient production units

P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462 459

the production process, we realize that only x00/Di(y,x) is actually strictly needed for production.The rest:

x00 1� 1

Diðy; xÞ

� �is lost due to direct or indirect consumption of in-put for off-the-record activities. Note that in thisinterpretation, we do not presume that technicalinefficiencies have been eliminated as a first step.Rather we assume that the technical inefficienciesare on the input side and will continue at the samegeneral level as before.The other measure, the reverse allocative effi-

ciency, involves excessive consumption of outputor sub-optimal output handling procedures. Tosee this, and for ease of comparison, let us considerthe case where the reference output is chosen asy ¼ ky. In this case AE* can be rewritten as

AE� ¼ Cðy=D0ðx; yÞ;xÞxx

¼ minx0

xx0

xx

����x0 2 Lðy=D0ðx; yÞÞ� �

:

One interpretation of this is that we seek to reducethe cost of producing y but we realize that to actu-ally get y, we need to produce y/D0(x, y). The ex-cess output

y1

D0ðx; yÞ� 1

� �is lost due to direct or indirect consumption of out-put for off-the-record purposes. Again, we do notpresume that technical inefficiencies have beeneliminated. Rather, we assume that the technicalinefficiencies are on the output side and will con-tinue at the same general level as before.In some cases, it makes sense to assume that

some inputs are actually used for unintended pur-poses. Simple examples could be time spent mak-ing private phone calls or cars used for privatetrips. In other cases, it makes more sense to as-sume that outputs are actually used off-the-recordas when a baker eats his own donuts or a sharecrop farmer under-reports his produce. Also, insome cases, it may be reasonable to assume thatthe input handling is sub-optimal as when a fac-tory gets costly breakdowns due to inadequate

maintenance or that the output handling is inade-quate as when fresh products deteriorates due toinsufficient sales. In such cases, the likely sourceof inefficiency can guide our choice between thetwo measurement procedures. If slack is mainlyon the output side, the new approach is preferable.In other cases, both (or neither) perspectives maybe relevant. Indeed, it requires a more elaboratebehavioral model of the inefficiency generationprocess to make a strictly formal distinction ofthe two perspectives. It is beyond the scope of thiswork to do so. Papers along the lines of Bogetoftand Hougaard (2003) may however be inspira-tional.

6. Organizational change processes

The two propositions have some interestingorganizational and managerial interpretations.Technical efficiency defined as producing the

most outputs from a bundle of inputs or con-versely using the least inputs to produce a givenbundle of outputs is related to the concept oforganizational slack (cf. Cyert and March, 1963).They define organizational slack as ‘‘the differencebetween total resources and total necessary pay-ments’’. Organizational slack can be interpretedas the existence of excess resources to producegiven outputs. Thus technical efficiency is relatedto the broader definition of efficiency––doing thething you do in the best way.Allocative efficiency, also called behavioral effi-

ciency by Fare et al. (1984), captures the organiza-tion�s ability to choose the inputs in the ‘‘correct’’proportions given the weights of the inputs. Theseweights may represent factor prices or organiza-tional priorities. In the more general case withmultiple inputs and multiple outputs this means,e.g. to find the correct proportions of the inputsand outputs given the prices on the inputs and goalpriorities on the outputs. Output priorities maycome from the vision and mission statements ofthe organization as a part of the strategic plan-ning. Interpreted in this way allocative efficiencymay relate to organizational effectiveness––doingthe right thing. Thus overall efficiency is to dothe right thing the right way.

Page 11: Allocative efficiency of technically inefficient production units

460 P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462

Assume now that the organization wants to im-prove efficiency in a situation where the organiza-tion experiences both technical and allocativeinefficiencies. Which organizational change process

could be applied to obtain overall efficiency? Over-all efficiency may be obtained by simultaneouslyobtaining technical and allocative efficiency (mov-ing directly from C to H in Fig. 1), or by a sequen-tial approach where the organization first becomestechnical efficient (moving from C to B) and thenbecomes allocative efficient by moving along thecurve L(y) till reaching H where overall efficiencyis obtained. The reverse procedure where theorganization first obtains allocative efficiency(moving from C to E) and then becomes technicalefficient (moving from E to D) may in general notlead to overall efficiency. Proposition 1, however,shows that if the technology T is input homothetic,then both of the above sequential procedures willlead to overall efficiency. The assumption thatthe technology is input homothetic means thatthe input proportions required for a given levelsof output do not depend on the output levels(LðyÞ being parallel to L(y)). Many productioncontexts can with as a reasonable approximationbe modelled using homothetic technologies.From a motivational and organizational change

point of view, it may be advantageous to choose aparticular path to obtain overall efficiency if theprocess has to be carried out sequentially over aperiod of time. It may invoke less resistance tochange to first adapt the resources to do the rightthings, and subsequently to improve technical effi-ciency––to do things the right way. First optimiz-ing doing the wrong things and then moving todo the right things, may discourage the employees,particularly if some of the inputs are related tohuman resources.It is worth noting that both procedures will in-

duce non-monotone adjustment of some resources.In the traditional approach, some resources willfirst be laid off and then reemployed while in thereverse procedure, some resources will be tempo-rarily hired and then laid off again. To see this,note that the technical inefficiency adjustment al-ways implies a reduction in all inputs, while thereallocation involves a reduction in some andexpansion in other inputs. Now, combining this

with the order of adjustments in the two ap-proaches gives the suggested non-monotonicities.Which of these non-monotonicities are preferablewill depend on the context, including the extentof reversed resource usages and the factors that be-come subject to these adjustments.Technical efficiency can not only be obtained by

reducing the inputs but also by increasing the out-puts. For many organizations both choices arepossible. If that is the case, an increase in outputsis most likely to occur if the allocative efficiency isachieved before the technical efficiency. The ‘‘tra-ditional’’ procedure: firstly reducing the inputs,then undertaking a substitution of inputs to movetowards allocative efficiency, followed by an in-crease in inputs and outputs, may motivationalbe difficult. The ‘‘reverse’’ procedure: a movementtowards the allocatively efficient point for the in-tended higher output followed by an expansionof the actual outputs, may be motivational sim-pler. Also, it only consists of two steps.Technical efficiency and allocative efficiency

may also be related to the concept of single and

double loop learning discussed in Agryris andSchon (1978). Single loop learning assumes thatgoals are stable and focus on the means by whichexisting goals are pursued. Double loop learninginvolves changing the goals as well as developingnew rules and methods of decision-making. Inlearning terms, the choice between the two proce-dures of this paper to obtain overall efficiency isthus the choice of first doing single loop learningand then double loop learning or the reverse se-quence. Again the natural choice is the change ofthe overall procedures (double loop learning),and then to perform those procedures in the bestpossible way (single loop learning).The idea that certain changes are easier than

others are found in the productivity analysis liter-ature as well. The possibility of an organization tocontrol inputs or outputs are commonly used toguide the choice between input and output basedmeasures. Also, the existence of non-discretionaryparts of the input or output vectors are used tomodify the Farrell measures by only looking forproportional improvements in the discretionarydimensions (cf. Charnes et al., 1995). The changeperspective of this paper is a natural refinement

Page 12: Allocative efficiency of technically inefficient production units

P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462 461

of these somewhat extreme cases. We essentiallysuggest that the possibility of changing certain var-iables may not be fixed exogenously, but be endog-enous and depend on the sequence of change.It is worth emphasizing also that from an

broader organizational and economic point ofview, slack is not necessarily wasted. In the organ-izational literature there have been numerous dis-cussions of the use of organizational slack as abuffer for uncertainty, as a means of de-couplingactivities and hereby diminishing the informationflow and coordination among sub-units and as anecessity for providing resources for innovation(cf. e.g. Galbraith, 1974; Stabler and Sydow,2002). 1 In the economic and bureaucracy litera-ture there have been many other explanationswhy some degree of technical inefficiency may berational. Inefficiency is typically present in an opti-mal second best solution to an incentive problem.Inefficiency can be part of the (fringe) compensa-

tion paid to stakeholders, e.g. the employees, theowners, the local community etc., and it may actu-ally be the cheapest way to provide such compen-sation. Technical inefficiency may reflect firmpreferences that deviate from simple short runprofit maximization or it may reflect special mar-ket conditions. A classical example of the alterna-tive preference hypothesis is a bureaucracy wherethe size of the budget and a large staff in itselfmay have value (cf. Niskanen, 1971; Williamson,1964). So there may be gains from technical ineffi-ciency per se. This idea is pursued in Bogetoft andHougaard (2003), where we show how to use ob-served inefficiencies to reveal the preferences fordifferent types of slack and we discuss how thiscan improve planning, incentive provision andproductivity measurement. If we accept the ideathat technical inefficiency is potentially valuablein a broader organizational and economic context,the reverse procedure suggested here is particularlyrelevant. The reverse allocative efficiency measure

1 Additionally, Charnes et al. (1989) hypothesized thatorganizations, that were subject to repeated performanceevaluations (of the DEA type) would benefit from developingcertain types of slack. However they did not find a systematictendency for slack accumulation in their study of vehiclemaintenance units in the US Air Force.

estimates the gains from reallocation without pre-suming that the––in a broader perspective––poten-tially useful technical inefficiency is eliminated. 2

7. Final remarks

In this paper we have discussed ways of measur-ing allocative efficiency without presuming techni-cal efficiency. In the traditional Farrell approach,the technical efficiency is evaluated first and theallocative efficiency next. We introduced a reversedFarrell approach, in which the allocative efficiencyis evaluated before technical adjustments areintroduced.We identified necessary and sufficient techno-

logical regularities for the two approaches to giveconsistent measures. For natural choices of theoutput reference, we need input homothetic or atleast input ray-homothetic technologies, for thetwo approaches to be equivalent.We also linked the procedures to several organ-

izational perspectives. Firstly, if the source of tech-nical inefficiency can naturally be related to theinput or output sides, this may guide the choice be-tween the traditional and the reversed measures.The new measure is particular relevant when out-put slack is the source of the technical inefficiency.Secondly, technical inefficiency may not just bewasted in a broader organizational interpretation.It may be a way to compensate employees, it maygive a buffer against uncertainty or it may provideroom for experimentation and innovation to men-tion a few more prominent examples. In such cases,it is useful to be able to measure possible gains fromreallocations that do not presume that the (possiblyattractive) technical inefficiency is eliminated.Thirdly, and most significantly, the two procedureshave quite different properties from a organiza-tional change perspective. The traditional and thereversed Farrell approach delineate two adapta-tion processes to become overall efficient. Thepropositions give technological conditions under

2 In some cases, one can make similar arguments torationalize allocative inefficiency, namely if the prices do notfully reflect the broader organizational and economic values ofthe different resources.

Page 13: Allocative efficiency of technically inefficient production units

462 P. Bogetoft et al. / European Journal of Operational Research 168 (2006) 450–462

which the manager can choose between them with-out affecting the expected production economicconsequences. Thus the manager can choose theprocedure that seems best from an organizational,motivational and learning point of view. We havealso argued that in many cases the reverse proce-dure to obtain overall efficiency may be advanta-geous from these perspectives. In particular, itmay be easier to motivate doing things right oncethe organization is targeted on the right things.The research in this paper can be extended in

many ways. From a technical perspective, it is rel-evant to develop the reversed output oriented Far-rell approach to complement the input measuresanalyzed here. Also, it would be interesting to de-velop non-parametric tests of ray input (output)homotheticity for the general multiple inputs mul-tiple outputs case, and to develop parallel theoriesusing translation homothetic technologies. Frommore of an organizational perspective, it is rele-vant to develop more explicit models of the ineffi-ciency generation process and to use these to guidethe choice of change strategies and productivitymeasures.

Acknowledgement

Useful comments from two referees and the edi-tor are gratefully acknowledged.

References

Agrell, P., Bogetoft, P., Tind, J., 2002. Incentive plans forproductive efficiency, innovation and learning. InternationalJournal of Production Economics 78, 1–11.

Agryris, C., Schon, D., 1978. Organizational Learning.Addison-Wesley, London.

Bogetoft, P., 1994. Incentive efficient production frontiers: Anagency perspective on DEA. Management Science 40, 959–968.

Bogeoft, P., 1996. DEA on relaxed convexity assumptions.Management Science 42, 457–465.

Bogetoft, P., 1997. DEA-based yardstick competition: Theoptimality of best practice regulation. Annals of OperationsResearch 73, 277–298.

Bogetoft, P., 2000. DEA and activity planning under asym-metric information. Journal of Productivity Analysis,7–48.

Bogetoft, P., Hougaard, J.L., 2003. Rational inefficiency.Journal of Productivity Analysis 20, 243–271.

Bogetoft, P., Tama, J.M., Tind, J., 2000. Convex input andoutput projections of nonconvex production possibility sets.Management Science 46, 858–869.

Charnes, A., Cooper, W.W., Lewin, A.Y., Seiford, L.M.(Eds.), 1995. Data Envelopment Analysis: Theory, Meth-odology, and Application. Kluwer Academic Publishers.

Charnes, A., Clarke, R.L., Cooper, W.W., 1989. An approachto testing for organizational slack via Banker�s gametheoretic DEA formulations. Research in Governmentaland Nonprofit Accounting 5, 221–229.

Cyert, R.M., March, J.G., 1963. A Behavioral Theory of theFirm. Prentice Hall, Englewood Cliffs, NJ.

Fare, R., Grosskopf, S., 2000. Research note: Decomposingtechnical efficiency with care. Management Science 46, 167–168.

Fare, R., Grosskopf, S., Lovell, C.A.K., 1984. The Measure-ment of Efficiency of Production. Kluwer AcademicPublishers.

Fare, R., Primont, D., 1995. Multi-Output Production andDuality: Theory and Applications. Kluwer Academic Pub-lishers, Boston.

Farrell, M.J., 1957. The measurement of productive efficiency.Journal of the Royal Statistical Society, Series A, General120 (3), 253–281.

Galbraith, J.R., 1974. Organizational design: An informationprocessing view. Interfaces 4, 28–36.

Leibenstien, H., 1966. Allocative efficiency vs. �X-efficiency�.The American Economic Review 56, 392–415.

Leibenstein, H., 1978. X-inefficiency Xists: Reply to an Xorcist.The American Economic Review 68, 203–211.

McDonald, J., 1996. Note: A problem with the decompositionof technical inefficiency into scale and congestion compo-nents. Management Science 42, 473–474.

Niskanen, W.A., 1971. Bureaucracy and Representative Gov-ernment. Aldine Press.

Petersen, N.C., 1990. Data envelopment analysis on arelaxed set of assumptions. Management Science 36, 305–314.

Shephard, R.W., 1953. Cost and Production Functions. Princ-eton University Press, Princeton, NJ.

Shephard, R.W., 1970. Theory of Cost and Production Func-tions. Princeton University Press, Princeton, NJ.

Stabler, U., Sydow, J., 2002. Organizational adaptive capacity:A structuration perspective. Journal of ManagementEnquiry 11 (4), 408–424.

Williamson, O., 1964. The Economics of Discretionary Behav-ior: Managerial Objectives in a Theory of the Firm.Prentice-Hall, Englewood Cliffs, NJ.