COMBINING EFFICIENCY AND SCALING EFFECTS IN ACTIVITY ANALYSIS: TOWARDS AN IMPROVED BEST PRACTICE CRITERION

. Efficiency is the main issue in any data envelopment analysis. Realizing output by a minimum of input or reaching a maximum of output by a given input is the credo, scale effects often are only a sort of accessory. In modern economics scale effects play a prominent role, however. What is the right size of a decision making unit (DMU) and how to proceed there. Returns to scale inform a DMU about its hitherto disregard of scale effects and show the way towards its ideal activity size. Combining efficiency aspects and scaling effects leads to a new DEA-best practice criterion of DMUs and gives them a profound orientation of their current position. This combination turns out to be the relation of weighted outputs to weighted inputs – in optimal prices under variable returns to scale (VRS). It is the VRS-productivity. For DMUs with increasing returns to scale the recommended growth path is in accordance with economic rationales, for decreasing returns to scale the recommended shrinking path uncovers severe flaws in VRS-models and needs adjustment. All theoretical considerations are illustrated by little numerical examples. A real world application of 37 Brazilian banks demonstrates the benefits of the new concept.


Introduction
Enterprises transform goods into goods for the purpose of humans' need satisfaction: They transform inputs into outputs.Such transformation processes over a certain time period are called activities, see [17].The intrinsic desire of men to compare and evaluate things makes economists evaluate such transformation processes, either for each isolated activity or comparing similar activities.Here, efficiency comes into play: In a set of activities a selected activity is efficient if its outputs cannot be produced by less inputs, or vice versa its inputs do not permit more outputs, cf.again [17].
For multiple inputs and multiple outputs, from an economic point of view, comparability of activities needs prices of goods, be them market or virtual prices.In a descriptive model of economic activities, prices might be the result of model calculations and hence are called virtual.Such a descriptive model is data envelopment analysis (DEA).It permits measurement of efficiency and productivity of activities.In the model with constant returns to scale (CRS-model), efficiency and productivity coincide.For variable returns to scale (VRS-model), we calculate the efficiencies of activities and their relation to productivity: scale efficiency.For more details on DEA in general, cf.e.g.[4,5,10,11,13,19].
Many researchers in the field of DEA regret weaknesses of the classical CRS-efficiency and VRS-efficiency measures as they do not have any discriminative power beyond efficiency, as discussed e.g. by Despotis [8].All CRS-efficient acitivities show equal assessment and so do all VRS-efficient activities, in their respective models.There are some attempts to overcome this problem by modifying the original activity set and thus assigning super-efficiencies to some activities, see e.g.[14,18].A little satisfying approach, we feel, because it distorts the technology contradicting fundamental axioms of DEA.As to scale efficiency, this measure improves the results of mere CRS-efficiency and/or VRS-efficiency, joining them both.So it shows the difference between global and technical aspects and in parts is a good indicator for an activity's closeness to the point of most productive scale size.Unfortunately, it is unable to discriminate between different activities with equal CRS-projection and VRS-projection, however.So there is a need for a more sophisticated assessment criterion than the classical ones.
The VRS-model suffers from a lack of autonomy.What is VRS-productivity and how does this link efficiency to scale effects?Is there an economically reliable assessment, combining these parameters?Returns to scale (RTS) offer valuable clues to answer these questions.As long as an activity still has great increasing returns to scale (IRS) and hence unused scale effects, its assessment should be downgraded.An activity which has already exploited its scale effects should not be punished.Such reasoning applies for decreasing returns to scale (DRS) as well.
Combining efficiency and RTS yields a concept which conjoins static and dynamic aspects.We call it improved efficiency measure IEM.And for IRS it meets the rationale of economics.For DRS, Dellnitz and Rödder [6] detected misbehaviour of the RTS-concept.Such misbehaviour contaminates IEM and needs amendments which will be presented in this contribution, too.
The paper is organized as follows.After this introduction, we give preliminaries of DEA.In Section 2.1 we sum up the concepts of efficiency, productivity, and (virtual) prices.In Section 2.2 follow returns to scale RTS, scale efficiency SE, and most productive scale size MPSS.Section 2.3 eventually sketches the misbehaviour of RTS in the case of DRS.Section 3 we dedicate to IEM.In Section 3.1 shortcomings of classical efficiency measures are brought up, in Section 3.2 the new IEM is presented and its nature is exemplified by a little example.In Section 4 VRS-productivity is defined (Sect.4.1) and its characteristics are developed (Sect.4.2).As a result of severe flaws in the VRS-model, we propose an NDRS-model instead and develop a respective modified NDRS-IEM.Section 5 shows rankings of DMUs for all efficiency measures reported on so far.Section 6 applies the new findings to 37 Brazilian banks and Section 7 is a resume and a prospect of future research.

Efficiency, productivity, and virtual prices in DEA
The activities of  decision making units (DMUs) are (x  , y  ),  = 1, . . .,; being x  and y  the vectors of  inputs and  outputs, respectively.As is well known, these activities -by means of the axioms completeness, convexity, monotonicity, minimal extrapolation, and expansion -build the possibility set or technology  .Expansion has four different forms: -radial unboundedness U, -radial dilatation D, -radial reduction R, -radial boundedness B.
In the so-called envelopment form, the input-oriented efficiency for each DMU ,  ∈ {1, . . .,}, is calculated by the linear programs The first one often is named CCR after their creators and the last one BCC for the same reason.[12] as well as Dellnitz et al. [7].RTS informs about change of y  to (1 +   )y  when x  changes to (1 + )x  .For the purpose of conserving efficiency, such alterations must satisfy and after reorganisation of terms That is the RTS-equation and  / is the scale elasticity, cf.[11,13,19].
are the loci of altered activities, as long as optimal  *  , u *  , v *  ,  *  remain valid.For this stability issue see [7].For such virtual activities (x, y) the respective RTS-equation reads Obviously, -for  *  > 0 this function decreases with a radial increase of y, -for  *  < 0 this function increases with a radial increase of y.Alternative optima might be present when solving (2.2B) and consequently influence scale elasticities.Equation (2.7) determines the variation of RTS for alternative optima.
As is well-known, is the scale (in)efficiency of DMU , see [4].It informs to what extent the DMU realized scaling effects to improve productivity.For SE  = 1 it is scale-efficient and cannot exploit further scale effects.For  **  =  *  = 1 it is scale-efficient as well as VRS-efficient and CRS-efficient; it has most productive scale size (MPSS), cf.[3].

Misbehaviour of RTS in VRS-technologies
Dellnitz and Rödder [6] show the behaviour of RTS on the efficient boundary of a VRS-technology.We sketch their results as they affect further developments in this paper.
For arbitrary activities (x, y) on the technology's boundary, respective elasticities read Running through boundary activities and calculating u * (y) und  * (y) at a time by means of (2.2B) and (2.7), respectively, will yield elasticities like in Figure 2.
Please verify: -The vertical lines for fix outputs y  are the elasticities for all  −  ≤   ≤  +  like in (2.7).
Table 1.Inputs, outputs, and solutions for 6 DMUs.-For IRS elasticity (y) / is greater than 1 and falls with increasing y till MPSS-activity of DMU 3. The potential for radially improving output/input lessens, in accordance with economic rationales.-For DRS the elasticities are less than 1, increase on facets of  's boundary and collapse for alternative optima, against economic rationale.We would expect an increase of (y) / towards 1 -the elasticity of MPSS -for decreasing y.
Even joining input and output orientations, like in Førsund [11] and the therein studied beam variation equations, does not cure the model's weakness.
After these preliminaries, in Section 3.1 we justify the creation of a new DEA-best practice criterion in VRSmodels, beyond VRS-efficiency.In Section 3.2 the new index will be presented and in Section 3.3 it will be analyzed.

Shortcomings of classical efficiency measures
For measuring efficiency relevant DEA literature cites (i) CRS-efficiency,  (Ad i) CRS-efficiencies for DMUs 1, 2, 3, 4, 5, 6 are 0.417, 0.667, 1.0, 0.667, 0.5, 0.9.The CRS-model shows current productivity but hides possible former and potential future scale effects.This is painful for "little" DMUs: They must compete with "big" DMUs and have little hope to get there.(Ad ii) The VRS-model permits the calculation of scale effects but does not make use therefrom.VRS-efficiencies 1 of DMUs 1, 2, 3, 4, 5 and 0.9 of DMU 6 hide scale effects.Mind the fact that DMU 1 is scored equivalent to DMU 3, even if far away from MPSS.Here, VRS-efficiency is a bad indicator for a DMU's performance.(Ad iii) Scale efficiency does not heal the problems.This measure rates DMUs 3 and 6 equally but does not identify different VRS-efficiencies nor CRS-efficiencies.
The following section is an attempt to cure these deficits.

The improved efficiency measure IEM
Combining elasticity -as a measure of not yet realized scaling potential -and efficiency is a promising attempt to improve the assessment of a DMU.Definition 3.2.For a DMU  with activity x  , y  , VRS-efficiency  *  and elasticity  /, its IEM is the fraction  -DMU 5 has VRS-efficiency 1 and IEM varies in [2.000, ∞].
The intervals of IEM result from alternative optima and hence from ambiguity of RTS, as was outlined in Section 2.2.Furthermore, the values reassure intended effects.DMU 1 has lower IEM than DMU 2 due to greater RTS.DMU 2 has a lower index than DMU 3. DMU 6 loses because of its VRS-efficiency 0.9.High IEMs for DMU 4 and 5 result from elasticities less than 1.They even outperform DMU 3.This irritating fact needs further research and will be picked up later.
Visualization of IEMs might be helpful and so we show it in the next section.We observe the following behaviour of IEM:

IEM for IRS and DRS
-For IRS the new IEM 1 () / is less than 1 and increases when approximating MPSS-activity of DMU 3. Realizing scale effects improves IEM.
-For DRS, IEM is greater than 1, falls on the boundary's facets and shows an abrupt increase at points of alternative optima.This behaviour is economically irrational.For DRS, IEM should be ≤1 and increase when approximating MPSS.
Intervals of IEM need getting used to.To overcome this problem, we make them point estimates.The next section introduces VRS-productivity.This concept helps illuminating economical irregularities for DRS in VRS-models.

From IEM to VRS-productivity
From Definitions 3.2 and 3.3, IEM is the relation between VRS-efficiency and elasticity, combining a static and a dynamic aspect in DEA.Interesting enough, this idea can be put further towards VRS-productivity. Dividing by yields such productivity.We put this result as a theorem.
is productivity in optimal VRS-prices.Productivity as an economic basic concept should fulfil basic principles, see again our observations in Section 2.1: It should be ≤ 1, avoiding "free lunch".It should be in line with quantity-based productivity.The next section is dedicated to such questions.
This meets economic rationale.
(2b) DRS,  *  < 0. For a non-efficient DMU  ( *  < 1), DEA-theory recommends input-projection.And this even whilst VRS-productivity exceeds 1.The inadequate free lunch grows.(1b) and (2b) vividly demonstrate misleading recommendations of DEA-theory in VRS-models.This severe flaw is reinforced when studying quantity-based vs. VRS-productivity.Figure 5 shows quantity-based productivities y /x -dashed line -and VRS-productivities -solid line.As expected, y /x increases until activity of DMU 3 and then decreases.First output increment exceeds input increment and then falls below.The solid line is known from Figure 4. Its zigzaging form foils quantity structure.It increases from y = 4 −  to y = 4 +  for a sufficiently small  > 0, in spite of decreasing quantity-based productivity.VRS-productivity does not follow economic rationale. Resuming: (3a) NDRS,  * ≥ 0. VRS-productivity is in accordance with quantity-based productivity.(3b) DRS,  * < 0. VRS-productivity is inconsistent with quantity-based productivity.
All observed economical irritations in VRS-model are due to negative values of the scale variable .They should be eliminated to make DEA a coherent instrument for evaluating DMUs.This is what the next section is about.

IEM in NDRS-models
To overcome problems related on in the last section, we consider the NDRS-model, which is briefly repeated here.
The course of IEMs now is in line with quantity-based productivity, see again our remarks on this issue in Section 4.2.  3 shows respective results complemented by the following comments:

Rankings of 6 DMUs by different efficiencies
- *  have little explanatory power, only DMU 6 is inefficient.Scale effects are not perceptible.-Scale efficiencies rank DMU 3 on par with DMU 6, an obvious flaw of this criterion.
-ḡ hide improvement potentials for all DMUs.
-IEM  for IRS outperform ḡ but for DRS are inapt.
-NDRS-IEM show all desired effects.For "little" DMUs they conjoin VRS-efficiencies and improvement potentials and for "big" DMUs they cure deficits of VRS-IEM.

Assessment of Brazilian banks
Efficiency and productivity assessments are of great importance, especially in the banking sector due to the difficult interest rate environment.In many applications of DEA -like in banking and finance -divisability of goods (loans, financial instruments) are evident.Scaling effects are present in the VRS-model, of course, and such an effect is the driving force for consolidation processes in the banking sector in order to improve a bank's efficiency or productivity.
In Henriques et al. [15] the authors relate on a DEA-study of Brazilian banks and focus on the so-called intermediation approach.They investigate the banks' efficiencies in the period from 2012 to 2016, using three inputs (fixed assets, total deposits, and personell expenses) and one output (total loans).Table 4 shows data of the fiscal year 2016.Table 5 then provides classical efficiencies and RTS-classification.4 DMUs have MPSS, 10 IRS, and a predominant number of 23 DMUs are oversized showing DRS.The latter being a typical phenomenon in almost all countries' banking systems; cf.[20].The CRS-efficiencies range between 0.027 and 1, VRS-efficiencies range between 0.09 and 1.This highlights the differences between the banks' intermediation strenghts.Scale efficiencies can be seen as "distances" from MPSS.
Table 6 compares classical with new indices developed in this paper.Eye-catching are some entries > 1 for VRS-IEM.They confirm the flaw of VRS-productivity for DRS, see Section 4.2.This defect is cured by NDRS-IEM.Here, (former) IRS-DMUs are downgraded due to unrealized scale effects, former MPSS-activities remain MPSS, and finally former DRS-banks lose efficiency.Table 7 shows all rankings.Determining best practice DMUs is one of the main objectives in DEA.Respective rankings are based on classical indices, such as CRSefficiency, VRS-efficiency and scale efficiency.As demonstrated in this paper, they all suffer from certain flaws, resulting in erroneous best practice DMUs.To overcome these flaws, we have defined new indices combining efficiency and scale effects.These are the two components a responsible management should take into account in order to successfully pilot the decision making unit.
To support managers and policymakers, we show a path that is economically sound from a bank's reality, adapting the procedure of Rödder et al. [21].For this purpose, we pick up the bank Semear (inefficient and operating under IRS) and show its improvements via this management tool.The iterative algorithm consists of two improvement vehicles: (1) RTS-based activity scalings like in equation (2.5).
(2) A further input reduction in each planning period (iteration), confirmed as practicable by the management.We opted for five iterations including RTS-based activity scaling and a further input reduction of five percent in each period.For the bank Semear, the results of this procedure are: -VRS-efficiency ( * 3 ) improves from 0.913 to 0.961.-scale efficiency (SE 3 ) increases from 0.872 to 0.999.-NDRS-IEM 3 consequently upgrades from 0.707 to approx.0.961.We notice that already after five planning periods, the bank Semear reaches an economically valuable position.IEM is an appropriate instrument to control a DMU's activity design.

Resume and prospect of further research
Scale effects have a cardinal impact on modern economies.Marketers seek growth to increase productivity: more output per input.Sometimes this ambition causes overdimensionality and consequently causes a drawback due to, e.g.inefficient control, transports, and personnel structure.Here, economic rationale would command reduction of activities.Data envelopment analysis evaluates activities of marketers and -especially in the VRSmodel -recommends each decision making unit to proceed against its most productive scale size.Returns to scale or production elasticity are respective indicators which might help the units to find the right size.
Data envelopment analysis provides such indicators but unfortunately, they play an ancillary role in the theory.So in the present paper, we show how to combine two objectives: the right way towards efficiency and the way to the right scale size.To this end, classical efficiency becomes improved efficiency.Interestingly
Figure 3 in addition to the 6 DMUs shows the CRS-technology.

Figure 2 in
Figure 2 in Section 2.3 showed all elasticities on the boundary of VRS-technology  .Due to efficiency 1, IEMs are just reciprocal to these values, see Figure 4. Figure 4 (left) shows all IEMs and Figure 4 (right) the IEM-intervals for respective DMUs, including the dashed line for DMU 6.We observe the following behaviour of IEM:

Definition 3 . 3 .
For a DMU  with VRS-efficiency  *  and -CRS make PIEM  = 3 yields benevolent point estimates.In other words: DMU  with IRS gets a high and with DRS gets a low IEM -close to the ideal  * /1.

Table 4 .
Inputs and outputs of 37 Brazilian banks.