## Abstract

The usual and well-established methods, explained and used in most of the previous chapters, for deriving a specific overall economic inefficiency decomposition associated with a given technical efficiency measure (multiplicative or additive), which we refer to as *the traditional approaches*, rely on the same “modus operandi”; i.e., they are based on dual relationships where allocative efficiency plays a fundamental role. Allocative inefficiency is obtained as the residual from a Fenchel-Mahler *inequality* that shows that the normalized economic inefficiency for a specific firm is greater or equal to its technical inefficiency. Accounting for allocative efficiency allows the closure of the inequality and enables a decomposition of economic efficiency considering technical and price (allocative) criteria, with the value of allocative efficiency clearly depending upon the chosen technical efficiency measure. Researchers have been using these traditional methods for at least half a century, and we take stock of the existing contributions and current state of the art in the previous chapters.

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## Notes

- 1.
- 2.
For a fairly complicated case, revise the case of the graph Russell measure developed in Chap. 7.

- 3.
In this respect, see the illustrative paper of Zofío et al. (2013).

- 4.
In most cases, the mathematical program used is linear or linearizable.

- 5.
This “normalizing factor,” which is expressed in the same monetary units as the corresponding profit inefficiency, was suggested by economist Marc Nerlove (1965), in order to avoid the dependence of profit inefficiency on proportional price changes. In fact, the normalized profit inefficiencies associated with the different efficiency measures we have been dealing with are pure numbers, derived from the corresponding profit inefficiencies but always independent of proportional price changes.

- 6.
The mentioned inequality is a typical Fenchel-Mahler inequality, derived by resorting to duality results—see each of the preceding chapters devoted to different non-multiplicative efficiency measures, including distance functions, to understand how different the normalization factors are.

- 7.
It is possible that multiple solutions may exist for piecewise linear technologies such as (13.2). This situation arises if several firms maximize profit. We consider this possibility later on in the chapter.

- 8.
In what follows, we write

*EM*instead of*EM(G)*, since we will denote oriented measures by*EM(I)*and*EM(O).* - 9.
When evaluating profit, a graph measure is the first choice in preference to any oriented measure because it provides more degrees of freedom in relation to the nonzero input and output slack values. The projection \( \left({\hat{x}}_{oEM},{\hat{y}}_{oEM}\right) \) of any firm (

*x*_{o},*y*_{o}), obtained by means of the specific*graph efficiency measure (EM)*, is always a frontier point. There are two types of frontier points: Firstly, the strongly efficient points that cannot be reduced for any of the inputs or augmented for any of the outputs without leaving the production possibility set*T*and, secondly, the weakly efficient points, which are not Pareto-Koopmans efficient, by admitting a reduction in some inputs or an expansion in some outputs within*T*. For instance, the variable returns to scale (*VRS*) proportional*DDF*generates projections that can be weakly efficient or strongly efficient, depending on the geometrical location with respect to the efficient frontier of the firm being rated. Only the efficiency measures that are known as strong efficiency measures, such as the ERG = SBM, guarantee that all the projections belong to the strongly efficient subset of the technology,*∂*^{S}(*T*). See the indication property (E1) in Sect. 2.2 of Chap. 2. - 10.
If (

*x*_{o},*y*_{o}) is an efficient point, then \( \left({\hat{x}}_{oEM},{\hat{y}}_{oEM}\right)=\left({x}_o,{y}_o\right) \). The last equality can also be valid for certain weakly efficient firms and may depend on the efficiency measure used. In any case, the new profit decomposition we develop is valid for any firm. - 11.
Note that it is verified that \( \left(\left(p\cdot {\hat{y}}_{oEM}-w\cdot {\hat{x}}_{oEM}\right)-\left({py}_o-{wx}_o\right)\right) \) = \( \left(p\cdot \left({\hat{y}}_{oEM}-{y}_o\right)-w\cdot \left({\hat{x}}_{oEM}-{x}_o\right)\right) \) = \( \left(p\cdot {s}_{oEM}^{+\ast }+w\cdot {s}_{oEM}^{-\ast}\right) \), where the last equality follows from \( \left({\hat{x}}_{oEM}-{x}_o\right)=-{s}_{oEM}^{-\ast}\le 0 \). We may also refer to the technological profit gap as

*the technological gap at market prices*. - 12.
When the projection does not belong to the initial sample of firms, we call it a point, which we know is a convex linear combination of firms.

- 13.
This subset includes all the inefficient units and, depending on the efficiency measure used, may represent some or all of the weakly efficient firms.

- 14.
The second one certainly includes all the efficient units and all or some of the weakly efficient firms.

- 15.
Our general direct approach also deals with efficiency measures that give rise to multiple projections but requires selecting one of them in order to propose the corresponding normalized profit inefficiency decomposition, as we explain later on.

- 16.
Since

*TI*_{EM}(*x*_{o},*y*_{o}) > 0, (*x*_{o},*y*_{o}) cannot maximize profit. - 17.
This factor is usually named as the

*normalization factor*, as we make clear later on. - 18.
The second right-hand side term is 0

_{$}if, and only if, the projection is a profit-maximizing point. - 19.
A second geometrical intuitive justification is provided by the fact that the first right-hand side term is the difference between the

*profit inefficiency at firm*(*x*_{o},*y*_{o}) and the*profit inefficiency at*\( \left({\hat{x}}_{oEM},{\hat{y}}_{oEM}\right) \). - 20.
The intuitive geometrical interpretation of the profit allocative term has its roots in the seminal work of Farrell (1957).

- 21.
The projection always satisfies that \( {TI}_{EM}\left({\hat{x}}_{oEM},{\hat{y}}_{oEM}\right)=0 \).

- 22.
See Corollary 13.1.

- 23.
Clearly

*k*_{o$}⋅*TI*_{EM}(*x*_{o},*y*_{o}) =*k*_{o$}⋅ 0 = 0_{$}, with*k*_{o}being any positive number. We prefer to fix the value of*k*_{o}for each firm (*x*_{o},*y*_{o}) when it can be determinant for achieving some desirable property, such as the comparison property, or when we want to compare different decomposition approaches, as described in the next subsection. - 24.
This proposal guarantees that at least the closest inefficient firm has the same allocative inefficiency as its projection.

- 25.
As explained in Chap. 7, it corresponds also to the projection with the highest profit.

- 26.
For each firm that is projected onto itself, select the fourth option described before.

- 27.
The traditional approaches do not usually consider this type of decomposition.

- 28.
This connects with Chap. 12 devoted to the

*Reverse DDF*, which assigns a specific*graph DDF*to any*graph efficiency measure*, satisfying the requirement expressed above. Hence, the*reverse DDF*constitutes the bridge between the graph efficiency measure that does not satisfy the essential property and the*graph DDF*that complies with it. - 29.
Strictly speaking we should have written the normalization factor as

*NF*_{DDF(G)}(*x*,*y*,*g*_{x},*g*_{y},*w*,*p*), but considering the general notation introduced in Sect. 13.2.3.1, we prefer to write a reduced expression. The same applies to the expression*TI*_{DDF(G)}(*x*,*y*). - 30.
It is quite obvious that two

*DDF inefficiency scores*are*comparable*as long as the corresponding*DVs*give rise to the same*profit*, i.e., \( {wg}_{x_j}+{pg}_{y_j}=k \), for all*j*. - 31.
“Slack” means the technical inefficiency detected in any of the inputs and outputs of a firm.

- 32.
When using the general direct approach as we do, to search for a projection with a reduced profit inefficiency is equivalent to getting a reduced allocative profit inefficiency.

- 33.
The reader must be aware that when using a VRS DEA piecewise linear frontier, many profit-maximizing points may exist, for instance, all the points that belong to a certain frontier facet. In the following section dealing with the flexible reverse approach, we account for this possibility and introduce a mathematical program that finds the closest L

_{1}projection of the firm under evaluation to the profit-maximizing benchmarks. - 34.
When we talk about (13.12), we mean to resort to the corresponding equality for decomposing profit inefficiency but adapted to the new situation, where the new projection \( \left({\hat{x}}_{oSR},{\hat{y}}_{oSR}\right) \) is in general different from the old one \( \left({\hat{x}}_{oEM},{\hat{y}}_{oEM}\right) \), obtained through a certain efficiency measure (

*EM*). Let us also observe that minimizing the allocative profit inefficiency is equivalent to minimizing the allocative inefficiency. - 35.
Based on the general direct approach, it is obvious that minimizing the allocative profit inefficiency of any firm is equivalent to maximizing the profit of its frontier projection.

- 36.
The nature of model (13.19) classifies the units directly as efficient or non-efficient.

- 37.
The mentioned article refers to the DDFs, while our approach here is based on the additive slack model.

- 38.
Each output and each input has its own quantity measure.

- 39.
For instance, the units of measurement of each component of

*p*, the output prices in the numerator, are \( \left(\frac{\$}{q_1^{+}},\dots, \frac{\$}{q_N^{+}}\right) \), while in the denominator the corresponding products have as units \( \frac{\$}{q_n^{+}}\times {q}_n^{+}=\$,n=1,\dots, N \). Hence, its output ratios have as units \( \left(\frac{\$}{q_n^{+}}/\$\right)=\frac{1}{q_n^{+}},n=1,\dots, N \). - 40.
- 41.
The only firm that cannot be directly tested in our Table is (10,5) because its projection does not belong to our sample of firms but does belong to the efficient facet defined by the firms B = (6,10) and D = (12,12). In this case, the convex combination that relates the benchmark (10,34/3) with the two mentioned firms is (10,34/3) = 1/3(6,10) + 2/3(12,12), which allows us to express the allocative inefficiency of the mentioned benchmark as a convex combinations of the two considered firms. We leave this exercise to the reader.

- 42.
The subset of non-efficient firms includes both

*the inefficient firms and*the firms that belong to the frontier but are not efficient, also called*weakly efficient firms*. - 43.
The relation between the set of firms

*E*and*N*can equivalently be expressed resorting to the corresponding subindexes*J*_{E}and*J*_{N}. - 44.
Strictly speaking, the optimal solution of program (13.34) is point \( \left(\sum \limits_{j=1}^J{\lambda}_j^{\ast }{x}_j,\sum \limits_{j=1}^J{\lambda}_j^{\ast }{y}_j\right) \). But since its input side is able to generate its output side, it is also able to generate a diminished output side as

*y*_{o}. Within a DEA framework, even in the low-dimensional two input-one output space, we can find multiple optimal solutions, depending on the relative position of the hyperplane*w*_{1}*x*_{1}+*w*_{2}*x*_{2}=*C*(*y*_{o}, (*w*_{1},*w*_{2})) with respect to the polyhedral frontier of the input set within the plane*y*=*y*_{o}. - 45.
Since the initial information needed is only the projection of the firm being rated, the general direct approach for decomposing cost inefficiency has the potential of being applied in quite different settings, as we will show later on when introducing the reverse approaches.

- 46.
All the terms of (13.36) are expressed in monetary units, e.g., in dollars ($).

- 47.
In principle, we can fix

*k*_{o$}= 1_{$}for all firms with*TI*_{EM(I)}(*x*_{o},*y*_{o}) = 0 or consider other possibilities as shown in Example 13.2. - 48.
The firms that satisfy \( \left({x}_o,{y}_o\right)=\left({\hat{x}}_o,{y}_o\right) \) are called the non-inefficient firms.

- 49.
This proposal guarantees that at least the closest inefficient firm has the same allocative inefficiency as its projection.

- 50.
The

*ERG = SBM*, presented in Chap. 7, is likely to generate multiple solutions. See the mentioned chapter where we explain how to calculate them. - 51.
Alternative projections have a common feature: all of them have associated the same valued technical inefficiency.

- 52.
The best cost projection means the projection that has the lowest cost inefficiency among the alternative projections. We just need to calculate one of these projections.

- 53.
In case two or more projections lead to the same smallest cost inefficiency, we can select any of them or consider a second criterion, as suggested before for the case of profit inefficiency.

- 54.
For each firm that is projected onto itself, select one of the three options described before.

- 55.
This input-oriented measure is a particular case of the

*ERG = SBM*, defined by Pastor et al. (1999) for a graph- input or output- representation of the technology. - 56.
Since the measure we are using is a strong efficient measure, in order to classify the efficiency of firms, we can rely on the—simplest—additive model or, alternatively, draw a picture in the two-input plane to see geometrically the position of the firms (see Fig. 13.3).

- 57.
Let us point out that the normalizing factor of the traditional approach for this particular efficiency measure does only depend on the input components of the firm being rated and on market prices, which shows that in case of multiple projections any of them will give rise to the same normalized cost decomposition.

- 58.
This connects with Chap. 12 devoted to the

*reverse DDF*, which assigns a specific input-oriented DDF to any input-oriented efficiency measure, provided the efficiency measure assigns a single projection to each firm. - 59.
*A classical projection of a firm*means a projection that belongs to the efficient frontier which is obtained by nonincreasing the input and by nonreducing the output values of the mentioned firm. - 60.
When using the general direct approach as we do, seeking a projection with the lowest possible cost inefficiency, equates to obtaining the least allocative cost inefficiency of the projection.

- 61.
The projection, according to footnote 1, is\( \left(\sum \limits_{j=1}^J{\lambda}_j^{\ast }{x}_{mj},{y}_o\right)\in L\left({y}_o\right) \).

- 62.
The proofs are a consequence of model (13.44) being a weighted additive inefficiency model. The list of four properties listed above are similar to the list of properties first published by Cooper et al. (1999) for certifying a well-defined efficiency measure (see also Sect. 2.2 in Chap. 2). In our case, the associated efficiency measure

*EM*_{SR}(*x*_{o},*y*_{o}) would be equal to 1 −*TI*_{SR}(*x*_{o},*y*_{o}). - 63.
Let us observe that any optimal solution of (13.56) may not belong to

*P*(*x*_{o}), simply because the optimal inputs \( \sum \limits_{j=1}^J{\lambda}_j^{\ast }{x}_j \) are less than*x*_{o}. In that case, the point \( \left({x}_o,\sum \limits_{j=1}^J{\lambda}_j^{\ast }{y}_j\right) \) that obviously belongs to*T*also belongs to*P*(*x*_{o}) and is also a revenue-maximizing point. - 64.
Even in the low-dimensional one input-two output space, we can find multiple optimal solutions, depending on the relative position of the hyperplane

*p*_{1}*y*_{1}+*p*_{2}*y*_{2}=*R*(*x*_{o}, (*p*_{1},*p*_{2})) with respect to the polyhedral frontier of the output set. - 65.
- 66.
In case the efficiency measure used gives rise to multiple equivalent projections at least for one firm of our finite sample, the method has to be refined in order to preserve the uniqueness of the decomposition.

- 67.
*k*_{o}represents any positive real number that we need to fix for those firms satisfying \( \left({x}_o,{y}_o\right)=\left({x}_o,{\hat{y}}_{oEM(O)}\right) \). - 68.
The

*ERG = SBM*, presented in Chap. 7, is likely to generate multiple solutions. See the already mentioned chapter where we present the DEA program that allows its calculation. - 69.
Alternative projections have a common feature: all of them have associated the same technical inefficiency.

- 70.
The best revenue projection means the projection that hast the lowest revenue inefficiency among the alternative projections. We just need to calculate one of these projections.

- 71.
In case two or more projections lead to the same smallest revenue inefficiency, we can select any of them, since both give rise to the same decomposition. However, we may consider a second proximity criterion, as we have suggested in the profit case.

- 72.
This output-oriented measure overlaps with the output-oriented Russell measure, as shown in Pastor et al. (1999).

- 73.
Since the measure we are using is a strongly efficient measure, we can use the additive model to identify the efficient firms.

- 74.
Since the normalizing factor in the traditional approach is independent of the projection, we can choose any of them for the two mentioned firms with double projections. Moreover, for the general direct approach, in this case, the same conclusion is valid, since both possible benchmarks are revenue-maximizing firms. Hence, the uniqueness of the revenue inefficiency decompositions is guaranteed in this case for both approaches, being independent of the assigned projection. We add an asterisk to the projection of each of these two units in Table 13.9 to indicate that a second projection is possible.

- 75.
This property is always satisfied by the general direct approach, as we show below.

- 76.
See subsection 13.2.5 for an interesting introduction to the two reverse approaches in the case of profit, which also applies to the cases of cost and revenue.

- 77.
When the projection is derived based on the optimal slacks obtained through model (13.63), we add an asterisk to its outputs.

- 78.
This is the reason for maintaining the same terminology in the SR approach.

- 79.
The proofs are a consequence of model (13.67) being a weighted additive inefficiency model. The list of four properties listed above is similar to those published by Cooper et al. (1999) for sanctioning a well-defined efficiency measure (see also Sect. 2.2 in Chap. 2). In our case, the associated efficiency measure

*EM*_{SR(O)}(*x*_{o},*y*_{o}) would be equal to(1 −*TI*_{SR(O)}(*x*_{o},*y*_{o})). - 80.
We can expect almost always a strict second inequality as a consequence of the definition of the common denominator.

- 81.
An output-efficient firm is a firm for which none of its outputs can be increased without leaving

*P*(*x*_{o}). - 82.
We refer the reader to Sect. 2.6.1 in Chap. 2 for the installation of the “Benchmarking Economic Efficiency” Julia package. All Jupyter notebooks implementing the different economic models in this book can be downloaded from the reference site: http://www.benchmarkingeconomicefficiency.com

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Pastor, J.T., Aparicio, J., Zofío, J.L. (2022). A Unifying Framework for Decomposing Economic Inefficiency: The General Direct Approach and the Reverse Approaches. In: Benchmarking Economic Efficiency. International Series in Operations Research & Management Science, vol 315. Springer, Cham. https://doi.org/10.1007/978-3-030-84397-7_13

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