## Abstract

In this chapter, we summarize the analytical framework found in the book by presenting the main concepts in an intuitive and accessible way while relying on supporting graphical illustrations to ease comprehension. Arguably, the measurement of economic efficiency dates back to the seminal paper by Farrell (1957), who introduced the definition, decomposition, and measurement of *overall* (economic) efficiency, which he named productive efficiency. The model is based on the cost function and input-oriented technical efficiency, and we initiate our presentations with this approach to guide and illustrate the different concepts underlying the measurement of economic efficiency. Throughout the text, we use economic efficiency as a measure to compare best and actual economic performance reserving the term productive efficiency for the technical dimension of the analysis, which may include aspects related to alternative characterizations of returns to scale (e.g., constant or variable returns to scale technical efficiency resulting in scale efficiency), disposability (e.g., strong and weak disposability of inputs and outputs), etc.

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## Notes

- 1.
For the origins of Data Envelopment Analysis, we refer the reader to Førsund and Sarafoglou (2002).

- 2.
ten Raa (2008) compares the Farrell and Debreu concepts of economic efficiency at length, expressing disapproval given that he perceives both as being casually equated. He claims that Debreu’s coefficient of resource allocation encompasses both Farrell’s technical efficiency and his allocative efficiency measures while frees the latter from prices. Debreu’s coefficient “calculates the resource costs not of a given consumption bundle, but of an (intelligently chosen) Pareto equivalent allocation. (And the prices are not given, but support the allocation).”

- 3.
General notation: For a vector

*v*of dimension*D*, \( v\in {\mathrm{\mathbb{R}}}_{++}^D \) means that each element of*v*is positive; \( v\in {\mathrm{\mathbb{R}}}_{+}^D \) means that each element of*v*is nonnegative;*v*> 0_{D}means*v*≥ 0_{D}but*v*≠ 0_{D}; and, finally, 0_{D}denotes a zero vector of dimension*D*. Given the two vectors*v*and*u*,*v*≤*u*⇔*v*_{d}≤*u*_{d},*d*∈ {1, …,*D*},*v*<*u*⇔*v*_{d}<*u*_{d},*d*∈ {1,…,*D*}, and*v*<^{*}*u*⇔*v*_{d}<*u*_{d}or*v*_{d}=*u*_{d}= 0,*d*∈ {1, …,*D*}. The inner product of two*D*dimensional vectors*v*≡ [*v*_{1}, ...,*v*_{D}] and*u*≡ [*u*_{1}, ...,*u*_{D}] is denoted as \( v\cdot u\equiv {\sum}_{d=1}^D{v}_d{u}_d \). - 4.
In empirical studies approximating the technology through DEA, a

*global*CRS characterization is assumed for computational convenience when relevant definitions, such as the profitability efficiency index, require these returns to scale, and therefore, their associated distance functions are defined with respect to that benchmark technology. Nevertheless, as remarked in Sects. 2.3.3 and 2.4.3, this requirement requires only the existence of*local*CRS. - 5.
- 6.
- 7.
Russell and Schworm (2018:17) make an alternative distinction between “path-based” indices and “slack-based” indices. However, these categorizations are not equivalent to the multiplicative and additive definition of the technical efficiency measures.

- 8.
- 9.
It is important to remark that whatever the approach for technical efficiency measurement, it is generally assumed that inputs are to be decreased and outputs increased. However, there are instances in economic efficiency measurement that taking advantage of the flexibility of some measures, it is possible to endogenize the direction so as to project inefficient observations directly onto the optimal economic benchmark by increasing inputs or reducing outputs; see Zofio et al. (2013) and Petersen (2018). This proposal questions the decomposition of economic efficiency into technical and allocative components, as the former is normally associated with a subjective choice of orientation. We discuss the possibility of endogenizing the orientation in subsequent chapters.

- 10.
As shown in the remainder of the book, other projections that keep inputs’ proportions (mix) fixed and associated with alternative aggregating functions different from the L

_{2}norm ‖⋅‖ (Euclidean length) are possible. - 11.
However, in the more general case that allows for non-homothetic technologies, Aparicio et al. (2015a, 2015b) and Aparicio and Zofío (2017) show that the standard radial measures a la Farrell would generally measure both technical

*and*allocative efficiencies. We return to these qualifications in the following chapter. - 12.
This is rather counterintuitive since one would rather look for reference benchmarks that are closer to the evaluated firm. However, from an empirical perspective, minimizing the value of the input slacks involves solving substantially more complex programs than the textbook DEA models based on maximization; see Aparicio et al. (2007) and Aparicio et al. (2017c). We comment on this feature when evaluating the properties of technical (in)efficiency measures.

- 13.
The DEA additive model is not unit invariant, and consequently, the sum of slacks in the objective function, expressed in different units, has no apparent meaning. Additionally, the efficient projection of each firm depends on the units of measurement considered for each input and output.

- 14.
The main critic to the RAM, already contained in the original paper, is that it delivers small technical inefficiency values due to the large values of the normalization factors. To improve it, the BAM was proposed 12 years later by Cooper et al. (2011), relying on smaller normalization factors.

- 15.
Some authors define the output-oriented technical efficiency measure as the ratio of optimal to observed output quantities (i.e., as its input-oriented counterpart (2.4)), and therefore, \( {TE}_{R(O)}\left(x,y\right)=\left\Vert \hat{y}\right\Vert /\left\Vert y\right\Vert \ge 1 \). Here, we follow the convention of defining all multiplicative efficiency measures as being bounded by one from above, regardless of the orientation.

- 16.
They refer to this index as the “Farrell Graph Measure of Technical Efficiency”. The measure inherits its name from the hyperbolic path it follows toward the production frontier.

- 17.
Moreover, as shown in Chap. 4 presenting the duality between the profitability function and the generalized distance function introduced by Chavas and Cox (1999), this latter measure corresponds to

*φ*^{*}; i.e., the square root of*TE*_{H(G)}(*x*,*y*). Actually, we could have developed the duality results presented below based on this measure, but it requires exponentiating*φ*^{*}to*α*and (1−*α*) for inputs and outputs, respectively. Hence, for simplicity, in this chapter, we stay with the (square of) hyperbolic efficiency measure. - 18.
Chambers (1988), Morrison (1993), and Beattie et al. (2009) discuss the concepts of returns to scale and elasticity of scale within a standard (neoclassical) presentation of production theory, with emphasis on duality theory and its implications for empirical work. As a result, from a primal perspective, these books rely on (single output) functions rather than adopting a general axiomatic approach based on (multiple output-multiple input) distance functions like that initiated by Shephard (1953, 1970) and followed by Färe and Primont (1995). This is necessary since empirical production studies resort to regression analysis to estimate the technology, where production is the dependent variable. They also discuss duality issues within a parametric context since the definition and econometric estimation of cost, revenue, or profit functions represent a convenient framework for studying multi-output technologies and characterize their properties.

- 19.
Distance functions represent specific aggregating functions for inputs and outputs. Data Envelopment Analysis techniques, in their dual (multiplier) formulations, aggregate inputs and outputs through shadow (optimizing) prices, while economic functions (i.e., cost, revenue, profitability, or profit) rely on observed market prices. Eventually, allocative efficiency can be seen as a disparity between vectors of aggregating weights (shadow and market prices) when evaluating technical efficiency (using shadow prices) and economic efficiency (using market prices).

- 20.
Balk (1998:19) refers to these firms as those exhibiting a technically optimal scale. O’Donnell (2012:260) identifies them as

*mix-invariant optimal scales*, because the radial projection of the firms to the efficient frontier keeps the input and output mixes constant. In the multiple output-multiple input case, the*y*and*x*axes in Figure 2.4 could be seen as aggregated outputs and inputs using specific aggregating functions. The comparison between an observed firm and its projections on the frontier associated with the radial efficiency measures (or Shephard’s input or output distance functions) corresponds to particular aggregating functions of the input and output vectors:*X*(*x*) and*Y*(*y*)—i.e., in these cases equivalent to the use of the L_{2}norm ‖·‖ associated with the Euclidean length. O’Donnell (2012) shows that efficiency measurement can be performed considering alternative aggregating functions (e.g., linear, CES). Consequently, if an alternative aggregating function is selected (e.g., linear functions with specific weight vectors for inputs and outputs:*X*(*x*) =*w*⋅*x*, \( w\in {\mathrm{\mathbb{R}}}_{+}^M \), and*Y*(*y*) =*p*⋅*y*, \( p\in {\mathrm{\mathbb{R}}}_{+}^N \)), then smaller (bigger) aggregate scalar input (output) quantities than those obtained with the radial efficient projections could be obtained. The scalar difference between the aggregate efficient output obtained by using the radial distance function and those obtained with alternative aggregating functions is termed*mix efficiency*by O’Donnell (2012), because the use of these alternative aggregating functions and associated weights results in changes in the input and output mixes. Later on, we recall that this interpretation is actually equivalent to the concept of allocative efficiency when the aggregating input and output functions correspond to the usual cost, revenue, profitability, or profit functions, i.e., those having a meaningful economic significance. - 21.
Resorting to trigonometry, scale efficiency defines equivalently as the ratio of the slope of the ray vector joining the origin and the efficient projection on the variable returns to scale technology to that of observation A, i.e., in Figure 2.4b,

*SE*_{G(H)}(*x*,*y*) = tan*β*/ tan*α*. - 22.
In trigonometric terms, productive efficiency defines as the ratio of the slope of the ray vector joining the origin and observation D to that of observation A, i.e., \( {TE}_{H(G)}^{CRS}\left({x}_{\mathrm{D}},{y}_{\mathrm{D}}\right) \) = slope 0D/slope 0A = tan χ/tan

*α*. - 23.
- 24.
There is only one exception presented in Chap. 13 and related to the so-called reverse approaches, where technical inefficiency is a subsidiary of allocative inefficiency.

- 25.
From an empirical perspective, a key issue when calculating efficiency measures is to ensure that the chosen formulations satisfy the homogeneity conditions. In the case of the graph (hyperbolic) efficiency measure, Cuesta and Zofío (2005) show how the almost homogeneity condition can be imposed on the translog specification. Hence, although Russell and Schworm (2011) state that for graph measures, there is no obvious counterpart to the homogeneity properties of the partially oriented measures, there seems to be some guidance when exploring meaningful counterparts.

- 26.
For the use of negative data with DEA, see also Pastor (1996).

- 27.
Notice that

*L*(*y*) is closed since we are supposing that*T*is closed. However, it is not enough to assure that “inf” can be substituted by “min” in \( \underset{x}{\operatorname{inf}}\left\{w\cdot x\left|x\in L(y)\right.\right\} \). So, hereafter, we assume that the optimization problem associated with the calculation of the cost function*C*(*y*,*w*) always attains its minimum in the set*L*(*y*). There exist several sufficient conditions in the literature which ensure such result. For example, Shephard (1970, p. 223) assumed that the subset of Pareto-efficient points of*L*(*y*) are bounded. Another case is when the technology is a polyhedral set (see Mangasarian, 1994, p. 130). - 28.
Shephard’s lemma allows us to recover the system of demand equations defined by the partial derivatives of the cost function with respect to input prices:

*x*^{C}(*y*,*w*) = ∇_{w}*C*(*y*,*w*). The result requires that the multiple output-multiple input transformation function characterizing the technology is (i) well behaved satisfying all desirable neoclassical properties and regularity conditions, particularly quasi-concavity, which ensures that the associated input production possibility sets are convex, e.g., Madden (1986), and (ii) continuous and twice differentiable. In textbooks, where the technology is represented by the single output production function*y*=*f*(*x*), for any two inputs*k*and*l*with associated market prices*w*_{k}and*w*_{l}, the first-order conditions also imply that the marginal rate of technical substitution of factor*k*for factor*l*must be equal to their price ratio: \( {MRS}_l^k=- dl/ dk={f}_k(x)/{f}_l(x)={w}_k/{w}_l, \) where*f*_{k}(*x*) =*∂f*(*x*)/*∂x*_{k}and*f*_{l}(*x*) =*∂f*(*x*)/*∂x*_{l}are marginal productivities. It is assumed that given our assumptions about the production function, the second-order conditions are verified, and therefore, the sign of the bordered Hessian determinant is negative. - 29.
Equivalent remarks to those previously made for the cost function as to whether

*R*(*x*,*p*) attains a maximum in the set*P*(*x*) can be recalled here. - 30.
On this occasion, under the conditions previously stated, Shephard’s lemma allows us to recover the system of supply equations defined by the partial derivatives of the revenue function with respect to output prices:

*y*^{R}(*x*,*p*) = ∇_{p}*R*(*x*,*p*). Then, for any two outputs*k*and*l*with associated market prices*p*_{k}and*p*_{l}, the first-order conditions also imply that the marginal rate of technical transformation of output*k*for output*l*, defined on a general transformation function*g*(*y*), must be equal to the price ratios: \( {MRT}_l^k=- dl/ dk={g}_k(y)/{g}_l(y)={p}_k/{p}_l, \) where*g*_{k}(*y*) =*∂g*(*y*)/*∂y*_{k}and*g*_{l}(*y*) =*∂g*(*y*)/*∂y*_{l}. Again, given the necessary assumptions about the transformation function, the second-order conditions are also verified. - 31.
Diewert (2014:59) credits Balk (2003:9–10) for introducing the term “profitability.” Georgescu-Roegen (1951:103) also proposed this function to characterize economic behavior and termed it “return-to-dollar.” This function is the inverse of the so-called efficiency ratio, representing one of the most important financial key performance indicators. Grifell-Tatjé and Knox Lovell (2015: Chap. 2) discuss the origins of profitability as an indicator of economic performance.

- 32.
Again, we assume that the optimization problem associated with the calculation of Γ(

*w*,*p*) always attains its maximum in*T*. - 33.
In the standard, single output-multiple input production function,

*y*=*f*(*x*), scale elasticity is defined as follows: \( \varepsilon \left(x,y\right)={\left.\left(\partial \ln f\left(\psi x\right)/\partial \ln \psi \right)\right|}_{\psi =1}={\sum}_{m=1}^M\left(\partial f(x)/\partial {x}_m\right)\cdot \left({x}_m/y\right)={\sum}_{m=1}^M{\varepsilon}_m \). The definition implies proportional changes in the inputs quantities; i.e., the input mix (relative input quantities) remains unchanged. - 34.
We can relate this result to the comparison of productivities in Sect. 2.1. Resorting once again to trigonometry, profitability efficiency defines equivalently as the ratio of the slope of the ray vector joining the origin and firm under evaluation to that of firm A maximizing profitability, i.e., in Figure 2.5a,

*ΓE*(*x*,*y*,*w*,*p*) = tan*χ*/tan*α*. - 35.
“Operating income” is a standard accounting figure that measures the amount of profit realized from a firm’s operations, after deducting from the sales revenue all operating expenses such as wages and the cost of other inputs (e.g., materials, rent and utilities), known as cost of goods sold (COGS), and depreciation. Another relevant measure of a company’s operating performance widely used by analysts is earnings before interest, tax, depreciation, and amortization (EBITDA). This measure allows evaluating a firm’s performance without having to factor in financing decisions, accounting decisions, or tax environments.

- 36.
Again, we assume that the optimization problem associated with the calculation of Π(

*w*,*p*) always attains its maximum in*T*. - 37.
See Rockafellar (1972:112) or Chambers (1988:306) for a general exposition of Minkowski’s theorem. On the relevance of the convexity assumption for duality theory, we refer the reader to Kuosmanen (2003), who concludes its necessity for the decomposition of economic efficiency. For a recent overview of non-convexity in production and economic (cost) functions, see Briec et al. (2021).

- 38.
Expression (2.27) corresponds to the standard cost-minimizing program presented in every microeconomics textbook, where the technological restriction

*x*∈*L*(*y*) is substituted by single output production function*y*=*f*(*x*). - 39.
From a parametric perspective, duality theory assumes that (i) the production function is well behaved satisfying all desirable neoclassical properties and regularity conditions, particularly quasi-concavity, which ensures that the associated input production possibility sets are convex, e.g., Madden (1986), and (ii) it is continuous and twice differentiable. In the case of single output production functions,

*y*=*f*(*x*), the quasi-concavity assumption, ensuring that the input isoquants are convex, is satisfied by the most common functional forms—e.g., Cobb-Douglas and CES. Although, in this book, we do not consider the parametric approach to measure and decompose economic efficiency (given the existing difficulties to impose the desired properties on the system of demand equations that characterize the optima and the challenges it poses for econometric estimation), the regularity and differentiability conditions of the production function pass on the distance functions defined in subsequent chapters—see Blackorby and Diewert (1979). - 40.
Considering the single output cost function

*C*(*y*,*v*),*v*_{k}and*v*_{l}are the (shadow) prices that, applying Shephard’s lemma, precisely correspond to minus the ratio of the given input combination that, in consequence, is optimal by minimizing the cost of producing output amount*y*: i.e.,*C*_{k}(*y*,*v*)/*C*_{l}(*y*,*v*) = \( -\left({x}_k^C\left(y,v\right)/{x}_l^C\left(y,v\right)\right) \), where*C*_{k}(*y*,*v*) =*∂C*(*y*,*v*)/*∂v*_{k}and*C*_{l}(*y*,*v*) =*∂C*(*y*,*v*)/*∂v*_{l}are the marginal costs associated with the input prices. Correspondingly, given the shadow prices, \( {x}_k^C\left(y,v\right) \) and \( {x}_k^C\left(y,v\right) \) are the optimal input quantities whose marginal rate of technical substitution equals their ratio: \( {MRS}_l^k=- dl/ dk={f}_k\left({x}_k^C\left(y,v\right)\right)/{f}_l\left({x}_l^C\left(y,v\right)\right)={v}_k/{v}_l \). - 41.
Since Farrell radial input measure (2.4) can be expressed as

*TE*_{R(I)}(*x*,*y*) = {*θ*^{∗}:*θ*^{∗}*x*∈*L*(*y*)} ⇒\( w\cdot {\theta}^{\ast }x=w\cdot \hat{x}\ge C\left(y,w\right) \) - 42.
This duality can be expressed equivalently as \( C\left(y,w\right)=\underset{x}{\min}\kern0.5em \left\{w\cdot x:{\theta}^{\ast}\le 1\right\} \), if and only if \( {\theta}^{\ast }=1/\left(\underset{w}{\min}\left\{w\cdot x:C\left(y,w\right)\ge 1\right\}\right) \). Here, we follow Färe and Primont (1995) who present this result in terms of Shephard’s (1953, 1970) input distance function rather than its inverse, corresponding to Farrell’s input technical efficiency measure. Chapter 3 revisits the decomposition of cost efficiency based on the input distance function.

- 43.
- 44.
Considering the single input revenue function

*R*(*x*,*μ*),*μ*_{k}and*μ*_{l}are the (shadow) prices that, applying Shephard’s lemma, precisely correspond to minus the ratio of the given output combination that, in consequence, is optimal by maximizing the revenue using input amount*x*: i.e.,*R*_{k}(*y*,*μ*)/*R*_{l}(*y*,*μ*) = \( -\left({y}_k^R\left(x,\mu \right)/{y}_l^R\left(x,\mu \right)\right) \), where*R*_{k}(*y*,*μ*) =*∂R*(*y*,*μ*)/*∂μ*_{k}and*R*_{l}(*y*,*μ*) =*∂R*(*y*,*μ*)/*∂μ*_{l}are the marginal revenues corresponding to each output price. Correspondingly, given the shadow prices, \( {y}_k^R\left(x,\mu \right) \) and \( {y}_l^R\left(x,\mu \right) \) are the optimal output quantities whose marginal rate of technical transformation equals their ratio: \( {MRT}_l^k=- dl/ dk={g}_k(y)/{g}_l(y)={\mu}_k/{\mu}_l \). - 45.
The Farrell radial output measure (2.4) can be expressed as

*TE*_{R(O)}(*x*,*y*) = min {*ϕ*^{∗}:*y*/*ϕ*^{∗}∈*P*(*x*)} ⇒ \( p\cdot y/{\phi}^{\ast }=p\cdot \hat{y}\le R\left(x,p\right) \). - 46.
This duality can be expressed equivalently as \( R\left(x,p\right)=\underset{y}{\max}\kern0.5em \left\{p\cdot y:{\phi}^{\ast}\le 1\right\} \) ⇔ \( {\phi}^{\ast }=\underset{p}{\max}\left\{p\cdot y:R\left(x,p\right)\le 1\right\} \)—see Färe and Primont (1995).

- 47.
- 48.
This constitutes McFadden’s (1978, p. 22)

*envelopment technology*but, on this occasion, is defined on the constant returns to scale subset of the technology, relevant for profitability maximization (rather than for the case of the cost function and its dual input set developed by McFadden). Hence, it follows that since*T*^{CRS}⊆*T*⇒*∂*^{S}(*T*^{CRS}) ⊆*∂*^{S}(*T*). - 49.
This is in contrast to the nonparametric Data Envelopment Analysis (DEA) techniques discussed in Sect. 2.4 and used in the empirical applications throughout the book, where, for convenience, the benchmark technology is characterized through global constant returns to scale, rather than allowing for variables returns to scale, which nevertheless allows identifying the reference hyperplanes (faces) consistent with local CRS.

- 50.
It is assumed once again that the profitability function is continuous and twice differentiable.

- 51.
Similar considerations to those made for the cost and revenue shadow prices can be made with respect to their profitability counterparts. The link between these functions and their first-order conditions is straightforward since the profitability function can be equivalently expressed either as \( \Gamma \left(w,p\right)=\underset{y}{\max}\left\{p\cdot y/C\left(y,w\right):\left(x,y\right)\in {T}^{CRS}\right\} \) or \( \Gamma \left(w,p\right)=\underset{x}{\max}\left\{R\left(x,p\right)/w\cdot x:\left(x,y\right)\in {T}^{CRS}\right\} \). However, relevant to duality theory, it has been shown that the first-order conditions characterize local constant returns to scale.

- 52.
- 53.
This duality can be expressed equivalently as \( \Gamma \left(w,p\right)=\underset{x,y}{\max}\kern0.5em \left\{\Gamma \left(w,p\right):{\left({\varphi}^{CRS\ast}\right)}^2\le 1\kern0.5em \right\} \), if and only if \( \kern0.5em {\left({\varphi}^{CRS\ast}\right)}^2=\underset{w,p}{\max}\left\{\left(p\cdot y/w\cdot x\right):\Gamma \left(w,p\right)\le 1\right\} \).

- 54.
Here,

*X*(*x*,*ν*) and*Y*(*y*,*μ*) are aggregator functions that are nonnegative, nondecreasing, and linearly homogeneous, i.e., satisfy constant returns to scale. - 55.
It is once again assumed that a maximum is achieved and that the profit function is continuous and twice differentiable.

- 56.
For example, the hyperbolic measure (2.10) cannot be recovered from the profit function in the form of a Fenchel-Mahler inequality.

- 57.
For (

*x*,*y*) ∉*T*, we define \( \sum \limits_{m=1}^M{s}_m^{-\ast }+\sum \limits_{n=1}^N{s}_n^{+\ast } \) = −∞, since the associated optimization problem is unfeasible. - 58.
This is a particular case of the weighted additive duality presented in Chap. 6, where the weights are set equal to one and, therefore, the slacks are measured in the units of the original observed data.

- 59.
Following these authors, we comment on the output-revenue decomposition, and for clarity in the exposition, adopt our notation.

- 60.
Alternatively, normalizing profit inefficiency by observed cost yields the alternative decomposition Π

*E*(*x*,*y*,*w*,*p*) = [*R*/*C*] + [Π*I*(*x*,*y*,*w*,*p*)]/*C*, which can be related to Farrell’s radial input measure. Also, combining observed revenue and cost, Π*E*(*x*,*y*,*w*,*p*) = [*R*(*x*,*p*)/*R*] + [*C*−*C*(*y*,*w*)]/*C*, where the first summand is the inverse of revenue efficiency in (2.20). - 61.
It is rather curious that the first DEA radial models due to Charnes et al. (1978) were designed based on engineering reasoning.

- 62.
- 63.
A recent review of the available general purpose and dedicated software options for efficiency and productivity analysis can be found in Daraio et al. (2019).

- 64.
An early example of these difficulties when decomposing economic efficiency into technical and allocative factors was the so-called Greene problem. Formal analysis of economic, technical, and allocative inefficiencies requires estimation of both a cost (revenue) and profit function along with its corresponding system of demand and supply share equations. This model complies with the neoclassical assumptions by which the latter system is consistent with Shephard’s lemma in the case of the cost (revenue) functions and Hotelling’s lemma for the profit function. Therefore, deviations from the optimality conditions in any input or output dimension translate into higher costs or lower profits. The difficulties emerge because the error terms capturing technical and allocative inefficiencies in the different equations of the model are not independently distributed, resulting in biased and inconsistent estimators. Moreover, it is necessary to disentangle technical and allocative efficiencies from the error terms in the cost or profit functions and their share equations because they interact among themselves. These errors present complex and nonlinear specifications, and under the usual distributional assumptions, the likelihood function necessary for estimation does not have a closed-form expression. See Kumbhakar et al. (2015) for recent single and multiple equations cross-sectional models of profit efficiency based on the primal approaches, overcoming some of these difficulties and using seemingly unrelated regression (SUR) and maximum-likelihood estimation methods.

- 65.
Färe and Grosskopf (1996; Chap. 2) prove that the DEA technology satisfies the usual axioms.

- 66.
- 67.
A facet can be full dimensional, in which case it corresponds to a face, or not. For instance, in a three-dimensional space, under VRS, the full-dimensional facet corresponds to a subset of a plane defined by three extreme efficient points, while a non-full-dimensional facet corresponds to a subset of a line defined by two extreme efficient points.

- 68.
- 69.
This is the also the case for the multiplicative approach based on the Farrell graph measure (2.10), identifying whether the firm belongs to the weakly efficient set (2.3). For a dedicated discussion on radial measures based on Shephard’s (1953, 1970) input and output distance functions; see Chap. 3. Chapter 4 is devoted to the case of the generalized distance function, extending the Farrell graph approach corresponding to the hyperbolic efficiency measure.

- 70.
This trade-off has prompted research on the general problem of transforming any weak DEA (in)efficiency measure into a strong DEA (in)efficiency, e.g., Fukuyama and Weber (2009) and Pastor and Aparicio (2010). Pastor et al. (2016) show that any DEA model that projects inefficiency observations onto the weakly efficient frontier, rather than onto the strongly efficient frontier, can be related to a reversed directional distance function, RDDF. They propose a two-stage process that combines a given efficiency measure (e.g., radial), which offers a first-stage projection for each observation on the weakly efficient frontier, with a second stage based on the additive model that projects each first-stage projection onto the strongly efficient frontier, ending up with a strongly efficient benchmark. Relating each inefficient observation with that final second-stage benchmark through the corresponding RDDF results in a comprehensive DDF (in)efficiency measure that combines radial and non-radial inefficiencies into a single scalar. In Chap. 11, we present the decomposition of economic efficiency based on the RDDF.

- 71.
Some alternatives to approximate the value of the hyperbolic efficiency measures have been recently proposed in the literature. Färe et al. (2016) devise a method that relates its value to that of the additive directional distance function. This results in an algorithm that, relying on the dual (multiplier) formulation of the DDF and the quadratic formula, allows for the estimation of the HDF through linear programming techniques. Recently, Halická and Trnovská (2019) reformulate the hyperbolic model into a semidefinite programming framework, opening the way to solving it with reliable and efficient interior point algorithms, as well as establishing the primal-dual correspondence.

- 72.
Ipopt, short for “Interior Point Optimizer,” is a software library for large-scale nonlinear optimization of continuous systems. At the time of writing, the latest stable release is 3.14.4, from Sep. 20, 2021. See https://github.com/coin-or/Ipopt. A list of commercial and free solvers compatible with the Julia (JuMP) environment can be found in http://www.juliaopt.org/JuMP.jl/stable/installation/

- 73.
Relying on semidefinite programming Halická and Trnovská (2019: 415) introduce the dual counterpart of the hyperbolic envelopment formulation under CRS and VRS.

- 74.
Nevertheless, the computational effort of solving the envelopment problems grows in proportion to powers of the number DMUs,

*J*. As the number of firms is considerably larger than the number of inputs and outputs (*N*+*M*), it takes longer and requires more memory to solve the envelopment problems. We contend that except for simulation analyses and the use of recursive statistical methods such as bootstrapping, nowadays processing power allows calculation of either method without computational burdens. - 75.
Julia is available for download with accompanying documentation from ‘The Julia Programming Language’ at https://julialang.org

- 76.
- 77.
- 78.
To install the Julia Language Support for Visual Studio Code visit:

https://marketplace.visualstudio.com/items?itemName=julialang.language-julia

- 79.
We are grateful to these authors for sharing the data. The same dataset has been used to illustrate the decompositions of total factor productivity change using quantities-only and price-based indices by Balk and Zofío (2018), as well as symmetric decompositions of cost variation by Balk and Zofío (2020).

- 80.
All Jupyter Notebooks implementing the different economic models in this book can be downloaded from the reference site: http://www.benchmarkingeconomicefficiency.com

- 81.
Also, we note that in the second stage of the profitability model calculating the slacks, the optimizer may flag infeasibility results before stopping at the last iteration. The reason is that for this model, nonlinear optimizers are called upon to solve the generalized distance function (or hyperbolic efficiency measure). The precision of the solution will depend on the level of tolerance. Since, for the calculation of slacks, firms are projected to the efficient frontier, small deviations would result, effectively, in a super efficiency model, for which the standard additive formulation cannot find a solution, as some slacks would adopt infinitesimal negative values, thereby violating the nonnegativity constraints on the slacks.

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Pastor, J.T., Aparicio, J., Zofío, J.L. (2022). Conceptual Background: Firms’ Objectives, Decision Variables, and Economic Efficiency. 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_2

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