The standard reverse approach for decomposing economic inefficiency

Abstract

The traditional approach decomposing profit inefficiency into the sum of its technical and its allocative components identifies first the frontier projection of each firm based on the exogenous choice of a specific technical measure, e.g., based on slacks, directional, etc. However, in real life situations, firms and organizations are interested in benchmarking themselves against competitors representing the largest feasible profit improvement given market prices. Resorting to the recently defined general direct approach decomposition of profit inefficiency, which decomposes profit loss into the profit technical gap existing between the firm and its frontier projection, and a remaining profit allocative gap, we introduce a decomposition that endogenizes the technical component. This is achieved by securing technical inefficiency reductions that, simultaneously, search to maximize the profit of the projected benchmark. The proposal defines a new measure of technical inefficiency that corresponds to a monetized version of the weighted additive model. We also present a normalized version of the reversed decomposition that is units’ invariant through the definition of a suitable normalization factor.

Keywords:

• Data envelopment analysis
• efficiency
• linear programming

1. Introduction

Bogetoft and Hougaard (2003) proposed years ago a change of paradigm when measuring economic inefficiency. Rather than focusing solely on reducing technical inefficiency per se, these authors indicate that a firm could be interested in directly improving its economic performance, without focusing primarily on the technical dimension. That is, market conditions and economic performance are brought to the forefront of the efficiency evaluation, while the reduction of technical inefficiency is contingent on achieving a higher economic goal, e.g., profit maximization, and therefore can be seen as the result of dynamic or transitory adjustments towards the final economic goal. In fact, they argue that “technical inefficiency can be part of the (fringe) compensation paid to stake holders…It can create loyal employees and thereby reduce costly turnovers in the labour force”. Bogetoft et al. (2006) added one more argument for adopting what they named “the reverse approach”, suggesting that employees and stakeholders must be convinced to recognize that “doing the right things according to the market conditions is the first goal of an organization”.¹ From a conceptual perspective the reverse approach brings the improvement of the economic performance of the firm to the forefront of efficiency analysis, i.e., solving technical and allocative inefficiencies should be linked to the economic goal of the firm, most notably maximum profit.

The reverse approach is in stark contrast to the traditional approach decomposing profit inefficiency, which focus on technical inefficiency and treats allocative inefficiency as a residual. The latter approach consists of the next two steps: first, a certain technical inefficiency measure is exogenously chosen, as well as its associated Fenchel-Mahler inequality, establishing that, for each firm, its profit inefficiency is greater or equal than its technical inefficiency times a positive normalization factor; and second, the mentioned inequality is transformed dividing by the normalization factor so as to get that the resulting normalized economic inefficiency is greater than or equal to its technical inefficiency, deriving then its allocative inefficiency as a residual. The traditional approach for decomposing profit inefficiency was introduced by Chambers et al.’s (1998), who show the duality between the graph directional distance function (DDF) and the profit function. This duality enables the definition of a Fenchel-Mahler inequality where normalized profit inefficiency is greater than or equal to the directional distance function (Luenberger, 1992; Chambers et al.’s, 1996). Despite being the DDF model the most popular choice to decompose economic inefficiency, many authors have adapted this methodology to decompose profit inefficiency based on alternative technical inefficiency measures. For example, Cooper et al. (2011a) show the duality between the weighted additive technical inefficiency measure and the profit function. Depending on the weights this proposal encompasses a number of normalized decompositions of the weighted additive model, including the measure of inefficiency proportions (MIP) introduced by Cooper et al. (1999), the range adjusted measure (RAM) of inefficiency, Cooper et al. (1999), and the bounded adjusted measure (BAM) of inefficiency, Cooper et al. (2011b). In this study, to develop the standard reverse approach, we propose a novel model that considers input and output prices as weights of the additive technical inefficiency measure. Additionally, Aparicio et al. (2016) propose the weighted additive distance function and derive its associated Fenchel-Mahler inequality, while Aparicio et al. (2017) develop the duality that allows decomposing profit inefficiency using the Enhanced Russell Graph Measure introduced by Pastor et al. (1999)—also known as Slack-Based Measure, see Tone (2001). Finally, traditional decompositions of economic inefficiency are possible through the Russell graph measure introduced by Färe and Lovell (1978), who extend the input- and output-oriented counterparts. Halická and Trnovská (2018) present an exact method to calculate the original graph Russell measure through semidefinite programming, and establish its duality with the profit function, which allows its decomposition.

Recently, Pastor et al. (2022) introduced a methodology that provides an alternative to the above traditional approach and does not require to derive duality results and their associated Fenchel-Mahler inequalities. It is based on the natural relationship between each firm, its projection, and the slacks that connect them. This approach is known as the general direct approach because it is valid for any technical inefficiency measure. Once the projection on the frontier associated to a given a measure is identified, the general direct approach decomposes the economic loss associated to profit inefficiency into a profit technical gap, separating observed profit from the profit at the projected benchmark, and a profit allocative gap, which is the profit inefficiency of the projected benchmark with respect to maximum profit. The sum of both profit losses, technical and allocative, yields the difference between maximum profit and observed profit, i.e., the profit inefficiency of the firm under evaluation.

In this paper, inspired by the reverse approach of Bogetoft et al. (2006) and relying on the general direct approach by Pastor et al. (2022), we introduce a new procedure to decompose profit inefficiency that starts defining a new inefficiency measure corresponding to a singular weighted additive model, whose weights are precisely the market prices. This model identifies as projection on the technical frontier the benchmark whose profit is the largest attainable by reducing the input slacks and increasing the output slacks. This procedure is equivalent to find the projection whose allocative inefficiency is the smallest, bringing economic criteria to the forefront of the analysis, i.e., the reverse approach (eventually, if the projected benchmark corresponds to a profit maximizing firm, the remaining allocative inefficiency is zero). The relationship between each firm and its projection exhibiting the largest feasible profit, expressed through an equality, gives rise to a new decomposition based on the general direct approach. That is, it decomposes profit inefficiency into technical and allocative criteria, with the former measuring the profit loss due to technical inefficiency, and the second one is related to the profit inefficiency of the mentioned projection. By dividing the equality by a suitable normalization factor, we end up with the corresponding normalized economic inefficiency decomposition that identifies its second term as its allocative inefficiency, with the novelty that it is directly related to the identified technical projection.

The paper unfolds as follows. In Section 2, we present the traditional approach to decompose profit inefficiency based on the weighted additive technical inefficiency measure. Then, in Section 3 we introduce the new standard reverse profit inefficiency decomposition, based on the equality that relates each firm with its projection through the corresponding slacks. This allows us to derive a new graph technical inefficiency measure and prove its properties. We then present the standard reverse decomposition in monetary terms and, subsequently, considering a suitable normalization factor, introduce its normalized counterpart, which is units’ invariant. Section 4 presents an illustrative numerical example, while Sections 5 and 6 adapt the methodology to the decomposition of cost and revenue inefficiency as particular cases. The last section draws the main conclusions.

2. The traditional approach to decompose profit inefficiency

To ease the understanding of the new methodology we formalize first the background required for the measurement and decomposition of profit inefficiency and present the traditional approach to decompose profit inefficiency based on Fenchel-Mahler inequalities. Data Envelopment Analysis approximates the production technology from observed cross-sectional data, relying on the Activity Analysis approach introduced by Koopmans (1951). Based on the principle of minimum extrapolation, DEA yields the smallest subset of the input-output space ${\mathbb{R}}_{+}^{M+N}$ as an inner approximation containing all observations and satisfying certain technological assumptions: convexity and free disposability of inputs and outputs. DEA generates convex polyhedral technologies (i.e., intersections of finite numbers of half-spaces), consisting of piecewise linear combinations of a finite sample of firms $F=\{(x_j^{},y_j^{})\in {\mathbb{R}}_{+}^{M+N},j\in J\},$ thereby allowing for multiple inputs, $x\in {\mathbb{R}}_{+}^M,$ and outputs, $y\in {\mathbb{R}}_{+}^N.$

In the DEA framework, the production technology $T=\{(x,y)\in {\mathbb{R}}_{+}^{M+N}:x\text{\hspace{0.5em}can\hspace{0.5em}produce\hspace{0.5em}}y\}$ can be approximated as follows: (1) $$T=\{\left(x,y\right):\sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}\le x_m,m=1,\dots ,M;\sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}\ge y_n,n=1,\dots ,N;\sum _{j=1}^J{\lambda }_j=1,{\lambda }_j\ge 0,j=1,\dots ,J\}$$(1)

Where $(x_j,y_j)=(x_{j1},\dots ,x_{\mathit{\text{jM}}},y_{j1},\dots ,y_{\mathit{\text{jN}}})$ denotes the input-output vector of each observation j, $j=1,\dots ,J.$

Since we are interested in profit inefficiency, we assume that the technology is characterized by variables returns to scale, VRS, ensuring that at least one profit maximizing firm exists and belongs to the strongly (Pareto-Koopmans) efficient production possibility set (Koopmans, 1951, p. 60).² The strongly efficient production set corresponds to: (2) $${\partial }^S\left(T\right)=\{\left(x,y\right)\in T:\hspace{0.17em}\left(x\prime ,-y\prime \right)\le \left(x,-y\right),\hspace{0.17em}\left(x\prime ,y\prime \right)\ne \left(x,y\right)\Rightarrow \left(x\prime ,y\prime \right)\notin T\}$$(2)

Let us consider the specific firm j = o, denoted by $(x_o^{},y_o^{}).$ Let us focus on the traditional approach that seeks a projection on the strongly efficient frontier (2)(2) $${\partial }^S\left(T\right)=\{\left(x,y\right)\in T:\hspace{0.17em}\left(x\prime ,-y\prime \right)\le \left(x,-y\right),\hspace{0.17em}\left(x\prime ,y\prime \right)\ne \left(x,y\right)\Rightarrow \left(x\prime ,y\prime \right)\notin T\}$$(2) , denoted by $({\widehat{x}}_o,{\widehat{y}}_o).$ It is assumed that $(-x_o^{},y_o^{})\le (-{\widehat{x}}_o,{\widehat{y}}_o),$ or, equivalently, that both input-slacks $s_o^{-}=(x_o^{}-{\widehat{x}}_o)$ and output-slacks $s_o^{+}=({\widehat{y}}_o-y_o^{})$ are non-negative. Within DEA methods, we can resort to several additive measures to achieve this goal. Here, to consider the reverse approach, we rely on the standard weighted additive technical inefficiency measure (WA) to determine such projection. Following Lovell and Pastor (1995), the corresponding model is (see also Pastor et al., 2012): (3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3)

The vectors ${\rho }^{-}=({\rho }_1^{-},\dots ,{\rho }_M^{-})\in R_{++}^M$ and ${\rho }^{+}=({\rho }_1^{+},\dots ,{\rho }_N^{+})\in R_{++}^N$ contain, respectively, input and output weights representing the relative importance of unit inputs and unit outputs from a technical perspective.³ If $(s_{\mathit{\text{oWA}}}^{-*},s_{\mathit{\text{oWA}}}^{+*},{\lambda }_o^{*})$ is an optimal solution of (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) , then the corresponding projection point is defined as $({\widehat{x}}_{\mathit{\text{oWA}}}^{},{\widehat{y}}_{\mathit{\text{oWA}}}^{})=(x_o-s_{\mathit{\text{oWA}}}^{-*},y_o+s_{\mathit{\text{oWA}}}^{+*}).$ We can determine the efficiency status of the firm under evaluation by looking at the optimal slacks $(s_{\mathit{\text{oWA}}}^{-*},s_{\mathit{\text{oWA}}}^{+*}).$ A firm under evaluation is deemed efficient if, and only if, the optimal slack values are all equal to 0, i.e., $T{I}_{\mathit{\text{WA}}}(x_o,y_o;{\rho }^{-},{\rho }^{+})$ = 0; otherwise, it is technically inefficient, $T{I}_{\mathit{\text{WA}}}(x_o,y_o;{\rho }^{-},{\rho }^{+})$ > 0.

Regarding the economic dimension, let us assume that a common fixed vector of market prices $(w,p)\in {\mathbb{R}}_{++}^{M+N}$ is known. Consequently, the market value of the inputs of firm $(x_o^{},y_o^{}),$ i.e., observed cost, is given by the interior product $w\cdot x_o^{}$ while that corresponding to its outputs, i.e., observed revenue, is $p\cdot y_o^{}.$ We can then define observed profit as revenue less cost, i.e., ${\Pi }_o=p\cdot y_o^{}-w\cdot x_o^{}.$ Since each $x_{\mathit{\text{om}}}^{},m=1,\dots ,M,$ is expressed in its own quantity units, $q_m^{},$ and, similarly, each $y_{\mathit{\text{on}}}^{},n=1,\dots ,N,$ is expressed in its own quantity units $q_n^{},$ we conclude that each input price $w_m$ must be expressed in $\frac{\$}{q_m^{}}$ and each output price $p_n$ in $\frac{\$}{q_n^{}},$ so that the profit generated by firm $(x_o^{},y_o^{})$ is expressed in $. Let us further denote as $\Pi (w,p)$ the maximum profit associated with sample of firms F, $\Pi (w,p)=\underset{x,y}{\mathrm{\text{max}}}\left\{p\cdot y-w\cdot x:(x,y)\in T\right\},$ and, based on it, let us further define the profit inefficiency of firm $(x_o^{},y_o^{})$ as (4) $$\Pi I(x_o^{},y_o^{},w,p)=\Pi (w,p)-(p\cdot y_o^{}-w\cdot x_o^{}).$$(4)

Profit inefficiency can be decomposed considering technical and allocative criteria by making use of the benchmark projection $({\widehat{x}}_{\mathit{\text{oWA}}}^{},{\widehat{y}}_{\mathit{\text{oWA}}}^{})$ obtained through the traditional approach—model (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) . Expressing the firm under evaluation as $(x_o,y_o)=(s_{\mathit{\text{oWA}}}^{-}+{\widehat{x}}_{\mathit{\text{oWA}}},-s_{\mathit{\text{oWA}}}^{+}+{\widehat{y}}_{\mathit{\text{oWA}}})$ and substituting last equality into (4)(4) $$\Pi I(x_o^{},y_o^{},w,p)=\Pi (w,p)-(p\cdot y_o^{}-w\cdot x_o^{}).$$(4) , we obtain the following decomposition: (5) $$\begin{array}{c}\Pi I\left(x_o,y_o,w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o-w\cdot x_o\right)=\Pi \left(w,p\right)-\left(p\cdot \left(-s_{\mathit{\text{oWA}}}^{+}+{\widehat{y}}_{\mathit{\text{oWA}}}\right)-w\cdot \left(s_{\mathit{\text{oWA}}}^{-}+{\widehat{x}}_{\mathit{\text{oWA}}}\right)\right)\\ =p\cdot s_{\mathit{\text{oWA}}}^{+}+w\cdot s_{\mathit{\text{oWA}}}^{-}+\Pi \left(w,p\right)-\left(p\cdot {\widehat{y}}_{\mathit{\text{oWA}}}-w\cdot {\widehat{x}}_{\mathit{\text{oWA}}}\right)\\ =\left(p\cdot s_{\mathit{\text{oWA}}}^{+}+w\cdot s_{\mathit{\text{oWA}}}^{-}\right)+\Pi I\left({\widehat{x}}_{\mathit{\text{oWA}}},{\widehat{y}}_{\mathit{\text{oWA}}},w,p\right).\end{array}$$(5)

In these expressions we identify a term measuring the firm’s technical profit inefficiency: $T\Pi I_{\mathit{\text{WA}}}(x_o,y_o,w,p)$ = $(p\cdot s_{\mathit{\text{oWA}}}^{+}+w\cdot s_{\mathit{\text{oWA}}}^{-})$ and a second term measuring allocative profit inefficiency: $A\Pi I_{\mathit{\text{SR}}}(x_o,y_o,w,p)$ = $\Pi I({\widehat{x}}_{\mathit{\text{oWA}}},{\widehat{y}}_{\mathit{\text{oWA}}},w,p).$ That is, (6) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right).$$(6)

A desirable characteristic of profit inefficiency is that it should satisfy the so-called commensurability property, thereby being units’ invariant. Unfortunately, expression (6)(6) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right).$$(6) does not satisfy this property as it depends on the reference monetary units, i.e., $. Units’ invariance can be achieved by dividing profit inefficiency by a suitable normalization factor. Duality theory relates profit inefficiency and technical inefficiency measures through the so-called Fenchel-Mahler inequality in terms of the corresponding normalization factor. Cooper et al. (2011a) proved the following Fenchel-Mahler inequality between normalized (N) profit inefficiency and the value of the Weighted Additive Measure (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) . (7) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=\underset{\mathrm{(}\text{Normalized}\mathrm{)}\mathrm{ }\text{Profit}\mathrm{ }\text{Inefficiency}\mathrm{(}\Pi I_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{\frac{\Pi I\left(x_o,y_o,w,p\right)}{\mathrm{\text{min}}\left\{\frac{w_1}{{\rho }_1^{-}},\dots ,\frac{w_M}{{\rho }_M^{-}},\frac{p_1}{{\rho }_1^{+}},\dots ,\frac{p_N}{{\rho }_N^{+}}\right\}}}}\ge \underset{\text{Technical}\mathrm{ }\text{Inefficiency}\mathrm{(}T{I}_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{T{I}_{\mathit{\text{WA}}}\left(x_o,y_o,{\rho }^{-},{\rho }^{+}\right)}},$$(7) where $N{F}_{\mathit{\text{WA}}}^{}(x_o,y_o,w,p)=\mathrm{\text{min}}\left\{\frac{w_1}{{\rho }_1^{-}},\dots ,\frac{w_M}{{\rho }_M^{-}},\frac{p_1}{{\rho }_1^{+}},\dots ,\frac{p_N}{{\rho }_N^{+}}\right\}$ is the normalization factor. The normalization of profit inefficiency prompts us to denote it by including $(\tilde{w},\tilde{p}),$ since one can consider that the normalization factor is associated with prices: i.e., $(\tilde{w},\tilde{p})=(w/N{F}_{\mathit{\text{WA}}}^{}(x_o,y_o,w,p),p/N{F}_{\mathit{\text{WA}}}^{}(x_o,y_o,w,p)).$ The decomposition of normalized profit inefficiency is completed by closing the above inequality. In this way, recalling expression (5)(5) $$\begin{array}{c}\Pi I\left(x_o,y_o,w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o-w\cdot x_o\right)=\Pi \left(w,p\right)-\left(p\cdot \left(-s_{\mathit{\text{oWA}}}^{+}+{\widehat{y}}_{\mathit{\text{oWA}}}\right)-w\cdot \left(s_{\mathit{\text{oWA}}}^{-}+{\widehat{x}}_{\mathit{\text{oWA}}}\right)\right)\\ =p\cdot s_{\mathit{\text{oWA}}}^{+}+w\cdot s_{\mathit{\text{oWA}}}^{-}+\Pi \left(w,p\right)-\left(p\cdot {\widehat{y}}_{\mathit{\text{oWA}}}-w\cdot {\widehat{x}}_{\mathit{\text{oWA}}}\right)\\ =\left(p\cdot s_{\mathit{\text{oWA}}}^{+}+w\cdot s_{\mathit{\text{oWA}}}^{-}\right)+\Pi I\left({\widehat{x}}_{\mathit{\text{oWA}}},{\widehat{y}}_{\mathit{\text{oWA}}},w,p\right).\end{array}$$(5) , normalized allocative inefficiency is defined as ${\mathit{\text{NAI}}}_{\mathit{\text{WA}}}(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+})$ = $N\Pi I_{\mathit{\text{WA}}}(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+})-T{I}_{\mathit{\text{WA}}}(x_o,y_o;{\rho }^{-},{\rho }^{+})=$ $\left[\Pi (w,p)-(p\cdot (-s_{\mathit{\text{oWA}}}^{+}+{\widehat{y}}_{\mathit{\text{oWA}}})-w\cdot (s_{\mathit{\text{oWA}}}^{-}+{\widehat{x}}_{\mathit{\text{oWA}}}))-(p\cdot s_{\mathit{\text{oWA}}}^{+}+w\cdot s_{\mathit{\text{oWA}}}^{-})\right]/N{F}_{\mathit{\text{WA}}}^{}(x_o,y_o,w,p)=$ $\left[\Pi (w,p)-(p\cdot {\widehat{y}}_{\mathit{\text{oWA}}}-w\cdot {\widehat{x}}_{\mathit{\text{oWA}}})\right]$/$N{F}_{\mathit{\text{WA}}}^{}(x_o,y_o,w,p).$ Last equality shows that normalized allocative inefficiency is interpreted as the profit loss between maximum profit and profit at the efficient projection: ${\mathit{\text{NAI}}}_{\mathit{\text{WA}}}(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+})$ ≡ $N\Pi I_{\mathit{\text{WA}}}({\widehat{x}}_{\mathit{\text{oWA}}},{\widehat{y}}_{\mathit{\text{oWA}}},\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}).$

Therefore, normalized profit inefficiency can be decomposed into the weighted additive technical inefficiency measure and normalized allocative inefficiency: (8) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)+{\mathit{\text{NAI}}}_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right).$$(8)

It is easy to see that these terms depend on the normalization factor $N{F}_{\mathit{\text{WA}}}^{}(x_o,y_o,w,p)=\mathrm{\text{min}}\left\{\frac{w_1}{{\rho }_1^{-}},\dots ,\frac{w_M}{{\rho }_M^{-}},\frac{p_1}{{\rho }_1^{+}},\dots ,\frac{p_N}{{\rho }_N^{+}}\right\},$ whose economic meaning is dubious, thereby hampering the interpretation of the values of the normalized decomposition.

3. The standard reverse approach to decompose profit inefficiency

Let us now introduce the Standard Reverse method (SR) to decompose profit inefficiency, which does not rely on duality theory and Fenchel-Mahler inequalities. It resorts to the general direct approach, see Pastor et al. (2022), which only needs to know for each firm $(x_o^{},y_o^{})$ its efficient projection. In particular, the reverse approach starts identifying the projection of the firm under evaluation, $({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}}),$ satisfying $(x_o,y_o)=(s_{\mathit{\text{oSR}}}^{-}+{\widehat{x}}_{\mathit{\text{oSR}}},-s_{\mathit{\text{oSR}}}^{+}+{\widehat{y}}_{\mathit{\text{oSR}}}),$ but searching for the maximum profit gains, as shown below. Considering the last equality, together with the profit inefficiency of $(x_o^{},y_o^{})$ and its projection, we now get the next equalities: (9) $$\begin{array}{c}\Pi I\left(x_o,y_o,w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o-w\cdot x_o\right)=\Pi \left(w,p\right)-\left(p\cdot \left(-s_{\mathit{\text{oSR}}}^{+}+{\widehat{y}}_{\mathit{\text{oSR}}}\right)-w\cdot \left(s_{\mathit{\text{oSR}}}^{-}+{\widehat{x}}_{\mathit{\text{oSR}}}\right)\right)\\ =p\cdot s_{\mathit{\text{oSR}}}^{+}+w\cdot s_{\mathit{\text{oSR}}}^{-}+\Pi \left(w,p\right)-\left(p\cdot {\widehat{y}}_{\mathit{\text{oSR}}}-w\cdot {\widehat{x}}_{\mathit{\text{oSR}}}\right)\\ =\left(p\cdot s_{\mathit{\text{oSR}}}^{+}+w\cdot s_{\mathit{\text{oSR}}}^{-}\right)+\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right).\end{array}$$(9)

Under the reverse approach, this expression follows the structure of expression (5)(5) $$\begin{array}{c}\Pi I\left(x_o,y_o,w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o-w\cdot x_o\right)=\Pi \left(w,p\right)-\left(p\cdot \left(-s_{\mathit{\text{oWA}}}^{+}+{\widehat{y}}_{\mathit{\text{oWA}}}\right)-w\cdot \left(s_{\mathit{\text{oWA}}}^{-}+{\widehat{x}}_{\mathit{\text{oWA}}}\right)\right)\\ =p\cdot s_{\mathit{\text{oWA}}}^{+}+w\cdot s_{\mathit{\text{oWA}}}^{-}+\Pi \left(w,p\right)-\left(p\cdot {\widehat{y}}_{\mathit{\text{oWA}}}-w\cdot {\widehat{x}}_{\mathit{\text{oWA}}}\right)\\ =\left(p\cdot s_{\mathit{\text{oWA}}}^{+}+w\cdot s_{\mathit{\text{oWA}}}^{-}\right)+\Pi I\left({\widehat{x}}_{\mathit{\text{oWA}}},{\widehat{y}}_{\mathit{\text{oWA}}},w,p\right).\end{array}$$(5) , by decomposing profit inefficiency in the sum of firms’ technical profit inefficiency: $T\Pi I_{\mathit{\text{SR}}}(x_o,y_o,w,p)$ = $(p\cdot s_{\mathit{\text{oSR}}}^{+}+w\cdot s_{\mathit{\text{oSR}}}^{-})$ and allocative profit inefficiency: $A\Pi I_{\mathit{\text{SR}}}(x_o,y_o,w,p)$ = $\Pi I({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p).$ Therefore, we now propose the following equality decomposing profit inefficiency in monetary terms: (10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10)

Contrary to the traditional decomposition in expression (6)(6) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right).$$(6) , which is based on technical grounds by relying on the exogenous choice of a technical efficiency measure, e.g., the weighted additive measure (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) , the reverse approach (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10) brings economic performance to the forefront of the efficiency analysis. The economic criterion underlying the reverse approach when searching for the best target on the frontier is to aim at the benchmark with the highest possible profit or, alternatively, that minimizes the firm’s allocative inefficiency with respect to maximum profit $\Pi (w,p).$ Since the left-hand side term in (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10) is given for each firm $(x_o^{},y_o^{}),$ our goal is to maximize $T\Pi I_{\mathit{\text{SR}}}(x_o,y_o,w,p).$ Hence, we propose to solve the following linear model measuring the technical profit inefficiency term: (11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11)

By using market prices as weights, i.e., ${\rho }^{-}=(w_1^{},\dots ,w_M^{-})\in R_{++}^M$ and ${\rho }^{+}=(p_1^{},\dots ,p_N^{})\in R_{++}^N,$ this is an economically driven weighted additive model (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) .⁴ Consequently, the objective function is expressed in monetary units. Model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) searches for a target on the production frontier with the maximum feasible profit. In case the optimal solution is $p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}=0_{\$},$ firm $(x_o^{},y_o^{})$ is technically efficient, while if $p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}>0_{\$}$ it is technically inefficient. A relevant feature of the new model is that, as its original weighted additive precursor (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) , it is able to classify each firm as strongly efficient by belonging to (2)(2) $${\partial }^S\left(T\right)=\{\left(x,y\right)\in T:\hspace{0.17em}\left(x\prime ,-y\prime \right)\le \left(x,-y\right),\hspace{0.17em}\left(x\prime ,y\prime \right)\ne \left(x,y\right)\Rightarrow \left(x\prime ,y\prime \right)\notin T\}$$(2) —in case all optimal slacks are equal to 0—or inefficient, otherwise.⁵ The virtue of model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) is that it allows us to define in what follows a new technical inefficiency measure, which we now formally call the standard reverse graph technical inefficiency measure. This new measure is instrumental in the normalized decomposition of profit inefficiency. Consequently, we first prove that it is a true technical inefficiency measure satisfying the usual properties.

3.1. The standard reverse technical inefficiency measure

Let us now elaborate the definition of the new technical inefficiency measure associated with the standard reverse approach and show that it satisfies the usual properties (Cooper et al., 1999). Linear model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) gives us an optimal non-negative objective function value for each firm of our sample, which means that we have exactly J optimal output and input slack values. Assuming that our sample of firms contains at least one inefficient firm, any upper bound for the optimal values of model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) is positive. After solving model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) for all firms, let us consider the upper bound U_SR of the calculated optimal values: (12) $$\hspace{0.17em}\hspace{0.17em}{U}_{\mathit{\text{SR}}}=\mathrm{\text{max}}\left\{p\cdot s_{\mathit{\text{jSR}}}^{+*}+w\cdot s_{\mathit{\text{jSR}}}^{-*},j\in J\right\}>0_{\$}.$$(12)

This value corresponds to at least one of the inefficient firms of our sample and, therefore, $U_{\mathit{\text{SR}}}\ge (p\cdot s_{\mathit{\text{jSR}}}^{+*}+w\cdot s_{\mathit{\text{jSR}}}^{-*}),\forall j\in J.$ Now we are ready to define the standard reverse technical inefficiency measure $T{I}_{\mathit{\text{SR}}}(x_o,y_o).$

Definition 1.

For firm $(x_o,y_o)$ of our finite sample, the standard reverse technical inefficiency measure $T{I}_{\mathit{\text{SR}}}(x_o,y_o)$ is defined as: (13) $$T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right):=\frac{T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=\frac{p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}}{U_{\mathit{\text{SR}}}}.$$(13)

For each firm, there are two remarkable differences between $T\Pi I_{\mathit{\text{SR}}}(x_o,y_o,w,p)$ in (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10) and $T{I}_{\mathit{\text{SR}}}(x_o,y_o)$ as defined in (13)(13) $$T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right):=\frac{T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=\frac{p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}}{U_{\mathit{\text{SR}}}}.$$(13) . First, $T{I}_{\mathit{\text{SR}}}(x_o,y_o)$ is a pure number, since $U_{\mathit{\text{SR}}}$ is expressed in the same monetary units as $p\cdot s_{\mathit{\text{oSR}}}^{+*}$ and $w\cdot s_{\mathit{\text{oSR}}}^{-*};$ and second, $T{I}_{\mathit{\text{SR}}}(x_o,y_o)$ is bounded by 0 and 1, because $(p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*})$ is bounded by 0_$and $U_{\mathit{\text{SR}}}.$⁶ The new introduced technical inefficiency measure $T{I}_{\mathit{\text{SR}}}(x_o,y_o)$ satisfies four basic properties, first introduced by Cooper et al. (1999) for technical efficiency measures, TE = 1−TI. Since in our economic context we are interested in inefficiency measures, we are going to enunciate and proof the corresponding versions for $T{I}_{\mathit{\text{SR}}}(x_o,y_o).$ These properties are indication, strong monotonicity, inefficiency requirement (as called by Sueyoshi & Sekitani, 2009) and commensurability. Regarding strong monotonicity, it is relevant to remark that, in the case of the standard reverse approach, this property is satisfied within the sample, i.e., comparing any two firms j = k, l ∈ J, if firm k dominates firm l in the sense of Pareto-Koopmans, then the inefficiency of the latter is always greater than that of the former, $T{I}_{\mathit{\text{SR}}}(x_l,y_l)$ > $T{I}_{\mathit{\text{SR}}}(x_k,y_k),$ as it uses more of at least one input and/or produces less of at least one output.

Proposition 1.

$T{I}_{\mathit{\text{SR}}}(x_o,y_o)$, the SR technical inefficiency measure, satisfies the next four properties.

P1. Indication (of inefficiency): $T{I}_{\mathit{\text{SR}}}(x_o,y_o)=0$ if, and only if, $(x_o,y_o)$ is a Pareto-efficient firm.

P2. Strong (or strict) monotonicity: $T{I}_{\mathit{\text{SR}}}(x_o,y_o)$ is a strongly monotonic inefficiency measure within the sample.

P3. Inefficiency requirement: $0\le T{I}_{\mathit{\text{SR}}}(x_o,y_o)\le 1.$

P4 Commensurability (or units’ invariance): $T{I}_{\mathit{\text{SR}}}(x_o,y_o)$ is invariant to the units of measurement of inputs and outputs.

Proof.

P1. Every weighted additive model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) with strictly positive weights satisfies the property of indication (see Cooper et al., 1999, and Halická & Trnovská, 2021).

P2. Every weighted additive model with strictly positive weights satisfies the property of strong monotonicity (see Cooper et al., 1999, and Halická & Trnovská, 2021). This implies that the numerator of (13)(13) $$T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right):=\frac{T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=\frac{p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}}{U_{\mathit{\text{SR}}}}.$$(13) satisfies strong monotonicity. Given $J,$ i.e., a particular sample of DMUs, we calculate $\hspace{0.17em}\hspace{0.17em}{U}_{\mathit{\text{SR}}},$ which is a fixed value for the observations belonging to $J.$ Consequently, if $(x_l,y_l)$ and $(x_k,y_k)$ belong to $J,$ $(x_l,y_l)\ne (x_k,y_k),$ with DMU $k$ dominating DMU $l$ in the sense of Pareto-Koopmans, then $p\cdot s_{\mathit{\text{lSR}}}^{+*}+w\cdot s_{\mathit{\text{lSR}}}^{-*}>p\cdot s_{\mathit{\text{kSR}}}^{+*}+w\cdot s_{\mathit{\text{kSR}}}^{-*}.$ This strict inequality allows deriving that $\hspace{0.17em}\frac{p\cdot s_{\mathit{\text{lSR}}}^{+*}+w\cdot s_{\mathit{\text{lSR}}}^{-*}}{U_{\mathit{\text{SR}}}}>\frac{p\cdot s_{\mathit{\text{kSR}}}^{+*}+w\cdot s_{\mathit{\text{kSR}}}^{-*}}{U_{\mathit{\text{SR}}}},$ that is, $T{I}_{\mathit{\text{SR}}}(x_l,y_l)$>$T{I}_{\mathit{\text{SR}}}(x_k,y_k).$

P3. This property is a direct consequence of (12)(12) $$\hspace{0.17em}\hspace{0.17em}{U}_{\mathit{\text{SR}}}=\mathrm{\text{max}}\left\{p\cdot s_{\mathit{\text{jSR}}}^{+*}+w\cdot s_{\mathit{\text{jSR}}}^{-*},j\in J\right\}>0_{\$}.$$(12) and (13)(13) $$T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right):=\frac{T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=\frac{p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}}{U_{\mathit{\text{SR}}}}.$$(13) . Moreover, having shown indication and strong monotonicity, the existence of inefficient firms can be assumed, one of which will exhibit a technical inefficiency score of one through the adoption of the upper bound as normalizing factor, while the presence of at least one efficient firm guarantees that the lower bound is also reached. Hence our new technical inefficiency measure classifies the firms in two disjoint subsets: the subset of efficient firms, for which $T{I}_{\mathit{\text{SR}}}(x_o,y_o)=0,$ and the subset of inefficient firms, for which $T{I}_{\mathit{\text{SR}}}(x_o,y_o)\in \left]0,1\right].$

P4. Since $T{I}_{\mathit{\text{SR}}}(x_o,y_o)=\left(\frac{p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}}{U_{\mathit{\text{SR}}}}\right)$ and the units of measurement of numerator and denominator are the same, it is clear that, following Lovell and Pastor (1995), $T{I}_{\mathit{\text{SR}}}(x_o,y_o)$ is units’ invariant. □

Note that the first two properties, indication and strong monotonicity, derive directly from the weighted additive model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) , i.e., regardless of its transformation by the normalization factor.

3.2. The normalized standard reverse approach to decompose profit inefficiency

After having defined in the preceding section the technical inefficiency associated with the standard profit reverse approach, and based on equality (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10) , we are ready to deduce the corresponding normalized profit inefficiency decomposition. Since equality (13)(13) $$T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right):=\frac{T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=\frac{p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}}{U_{\mathit{\text{SR}}}}.$$(13) can be reformulated as $T{I}_{\mathit{\text{SR}}}(x_o,y_o)\times U_{\mathit{\text{SR}}}=(p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}),$ equality (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10) can be rewritten as follows. (14) $$\Pi I\left(x_o,y_o,w,p\right)=T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right)\times U_{\mathit{\text{SR}}}+\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right).$$(14)

Knowing that $\hspace{0.17em}\hspace{0.17em}{U}_{\mathit{\text{SR}}}>0_{\$},$ we can divide each term of (14)(14) $$\Pi I\left(x_o,y_o,w,p\right)=T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right)\times U_{\mathit{\text{SR}}}+\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right).$$(14) by this value, obtaining the corresponding normalized profit inefficiency decomposition associated with the reverse approach:⁷ (15) $$N\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,\tilde{w},\tilde{p}\right)=\frac{\Pi I\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right)+\frac{\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right)}{U_{\mathit{\text{SR}}}}.$$(15)

Note that the upper bound of the technical profit inefficiencies represents a proper normalization factor in the decomposition of normalized profit inefficiency, $N{F}_{\mathit{\text{SR}}}\equiv U_{\mathit{\text{SR}}}.$ Last equality also shows that allocative inefficiency, corresponding to the last term, is not obtained as a residual derived from Fenchel-Mahler inequalities, but directly from our reverse approach. We also know its definition in terms of the projection of $(x_o,y_o)$ and of the obtained normalization factor; i.e., (16) $${\mathit{\text{NAI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,\tilde{w},\tilde{p}\right)=\frac{\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right)}{U_{\mathit{\text{SR}}}}.$$(16)

4. A numerical example

Let us illustrate the new standard reverse profit inefficiency decomposition and compare the obtained results to those of the traditional approach presented in Section 2. We consider eleven firms in the one input–one output space whose input and output values, denoted A thru K, are presented in the first column of Table 1. The first row specifies the content of each column. The second row replicates the first one introducing the associated mathematical notation used in previous sections, with the only exception of column 2, where the technical status of each firm—efficient or inefficient according to model (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) or (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) , is reported. The third row identifies each column, C1, C2, …, C9, as well as the relation of each column with the preceding ones. The next eleven rows show the results associated with each firm.

Table 1. The traditional profit inefficiency decomposition based on the weighted additive model (Range Adjusted Measure, RAM). (w,p) = (1, 4).

Firm	Model (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3)	WA Project. (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3)	Monetary (6)(6) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right).$$(6)			Normalized^(1) (8)(8) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)+{\mathit{\text{NAI}}}_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right).$$(8)
			Profit Ineff.	Tech. Profit Ineff.	Alloc. Profit Ineff.	Profit Ineff.	Tech. Ineff.	Alloc. Ineff.
$(x_o,y_o)$	Tech. status	$({\widehat{x}}_{\mathit{\text{oWA}}}^{},{\widehat{y}}_{\mathit{\text{oWA}}}^{})$	$\Pi I$	$T\Pi I_{\mathit{\text{oWA}}}$ =$p\cdot s_{\mathit{\text{oWA}}}^{+*}+w\cdot s_{\mathit{\text{oWA}}}^{-*}$	$A\Pi I_{\mathit{\text{WA}}}$	$N\Pi I_{\mathit{\text{WA}}}$	$T{I}_{\mathit{\text{WA}}}$	${\mathit{\text{NAI}}}_{\mathit{\text{WA}}}$
C1	C2	C3	C4	C5	C6= C4-C5	C7= C4/$N{F}_{\mathit{\text{WA}}}^{}$	C8	C9= C7-C8
A = (6,9)	Eff.	(6, 9)	9	0	9	0.75	0	0.75
B = (9,12)	Eff.	(9, 12)	0	0	0	0	0	0
C = (7,10)	Eff.	(7, 10)	6	0	6	0.5	0	0.5
D = (6,4)	Ineff.	(6, 9)	29	20	9	2.417	0.278	2.139
E = (8,8)	Ineff.	(6, 9)	15	6	9	1.250	0.222	1.028
F = (8,9)	Ineff.	(6, 9)	11	2	9	0.917	0.167	0.75
G = (9,9)	Ineff.	(6, 9)	12	3	9	1	0.25	0.75
H = (9,10)	Ineff.	(7, 10)	8	2	6	0.667	0.167	0.5
I = (10,8)	Ineff.	(6, 9)	17	8	9	1.417	0.389	1.028
J = (12,10)	Ineff.	(7, 10)	11	5	6	0.917	0.417	0.5
K = (12,3)	Ineff.	(6, 9)	39	30	9	3.25	0.833	2.417

⁽¹⁾The normalization factor in expression (7(7) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=\underset{\mathrm{(}\text{Normalized}\mathrm{)}\mathrm{ }\text{Profit}\mathrm{ }\text{Inefficiency}\mathrm{(}\Pi I_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{\frac{\Pi I\left(x_o,y_o,w,p\right)}{\mathrm{\text{min}}\left\{\frac{w_1}{{\rho }_1^{-}},\dots ,\frac{w_M}{{\rho }_M^{-}},\frac{p_1}{{\rho }_1^{+}},\dots ,\frac{p_N}{{\rho }_N^{+}}\right\}}}}\ge \underset{\text{Technical}\mathrm{ }\text{Inefficiency}\mathrm{(}T{I}_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{T{I}_{\mathit{\text{WA}}}\left(x_o,y_o,{\rho }^{-},{\rho }^{+}\right)}},$$(7) ) is $\mathrm{\text{min}}\left\{w/{\rho }_{}^{-},p/{\rho }_{}^{+}\right\}$ = $\mathrm{\text{min}}\left\{1/(1/(2\times 6)),4/(1/(2\times 9))\right\}$ = $\mathrm{\text{min}}\left\{12,72\right\}$= 12.

We start reporting the values of the traditional decomposition choosing the weighted additive model (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) to measure technical efficiency. Specifically, we rely on the Range-Adjusted Measure (RAM) proposed by Cooper et al. (1999) as it satisfies the set of desirable properties (P1-P4) previously considered to determine the suitableness of the normalized standard reverse technical inefficiency.⁸ For this measure, the respective input and output weights are $({\rho }_{}^{-},{\rho }_{}^{+})$ = $(1/\left[(M+N)\times R^{-}\right],1/\left[(M+N)\times R^{+}\right]),$ where $R^{-}=(R_1^{-},\dots ,R_M^{-})$ with $R_m^{-}=\underset{1\le j\le J}{\mathrm{\text{max}}}\left\{x_{\mathit{\text{jm}}}\right\}-\underset{1\le j\le J}{\mathrm{\text{min}}}\left\{x_{\mathit{\text{jm}}}\right\},$ $m=1,\dots ,M,$ and $R^{+}=(R_1^{+},\dots ,R_N^{+})$ with $R_n^{+}=\underset{1\le j\le J}{\mathrm{\text{max}}}\left\{y_{\mathit{\text{jn}}}\right\}-\underset{1\le j\le J}{\mathrm{\text{min}}}\left\{y_{\mathit{\text{jn}}}\right\},$ $n=1,\dots ,N.$ Solving model (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3) with these weights we categorize firms as efficient or inefficient, C2, and determine their input-output targets on the strongly efficient frontier (2)(2) $${\partial }^S\left(T\right)=\{\left(x,y\right)\in T:\hspace{0.17em}\left(x\prime ,-y\prime \right)\le \left(x,-y\right),\hspace{0.17em}\left(x\prime ,y\prime \right)\ne \left(x,y\right)\Rightarrow \left(x\prime ,y\prime \right)\notin T\}$$(2) —reported in C3. The first three firms are technically efficient as the objective function values zero. The remaining firms are inefficient obtaining a positive value.

Considering now market prices (w,p) = (1,4), we calculate the profit of each firm: ${\Pi }_o^{}=(p\cdot y_o^{}-w\cdot x_o^{}),$ finding that B maximizes profit, i.e., $\Pi (w,p)$ = 4 × 12 − 1 × 9 = $39. Subsequently, we obtain the profit inefficiency of each firm with respect to maximum profit, $\Pi I(x_o^{},y_o^{},w,p)=\Pi (w,p)-(p\cdot y_o^{}-w\cdot x_o^{}),$ as shown in (C4). The decomposition of profit inefficiency in monetary terms into technical profit inefficiency $T\Pi I_{\mathit{\text{WA}}}$ and allocative profit inefficiency $A\Pi I_{\mathit{\text{WA}}}$ according to expression (6)(6) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right).$$(6) is reported in columns C5 and C6. The first term corresponds to the values of the slacks separating each firm from its projection, $p\cdot s_{\mathit{\text{oWA}}}^{+*}+w\cdot s_{\mathit{\text{oWA}}}^{-*}.$ Consequently, all profit inefficiency is allocative for the technically efficient firms, while the rest of the firms exhibit both technical and allocative profit inefficiencies. The normalized version of profit inefficiency $N\Pi I_{\mathit{\text{WA}}}$ obtained through duality—i.e., the Fenchel-Mahler inequality (7)(7) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=\underset{\mathrm{(}\text{Normalized}\mathrm{)}\mathrm{ }\text{Profit}\mathrm{ }\text{Inefficiency}\mathrm{(}\Pi I_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{\frac{\Pi I\left(x_o,y_o,w,p\right)}{\mathrm{\text{min}}\left\{\frac{w_1}{{\rho }_1^{-}},\dots ,\frac{w_M}{{\rho }_M^{-}},\frac{p_1}{{\rho }_1^{+}},\dots ,\frac{p_N}{{\rho }_N^{+}}\right\}}}}\ge \underset{\text{Technical}\mathrm{ }\text{Inefficiency}\mathrm{(}T{I}_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{T{I}_{\mathit{\text{WA}}}\left(x_o,y_o,{\rho }^{-},{\rho }^{+}\right)}},$$(7) —ensures that these measures are units’ invariant. The normalization factor in this example is $N{F}_{\mathit{\text{WA}}}^{}(x_o,y_o,w,p)$ = $\mathrm{\text{min}}\left\{w/{\rho }_{}^{-},p/{\rho }_{}^{+}\right\}$ = $\mathrm{\text{min}}\left\{1/(1/(2\times 6)),4/(1/(2\times 9))\right\}$ = $\mathrm{\text{min}}\left\{12,72\right\}$ = 12. The normalized values of technical and allocative inefficiencies, $T{I}_{\mathit{\text{WA}}}$ and ${\mathit{\text{NAI}}}_{\mathit{\text{WA}}},$ are presented in columns C8 and C9. While these values do not depend on the units of measurement, their relative values are referred to the normalization factor, whose interpretation lacks economic meaning.

We now report the results of the profit inefficiency decomposition based on the standard reverse approach in Table 2. First, the efficiency status of each firm does not change with respect to the traditional approach because this result depends on the value of the slacks, and not on the choice of specific weights. However, we see that using markets prices as weights in order to search for the technical benchmark with the maximum feasible profit, i.e., that minimizing allocative profit inefficiency according to model (8)(8) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)+{\mathit{\text{NAI}}}_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right).$$(8) , results in different projections. Except for firm D, which is projected to the same reference benchmark using the traditional (RAM) and standard reverse models, all technically inefficient firms identify a different benchmark on the strongly efficient frontier, with the last five being directly projected to the profit maximizing firm B (see column C3).⁹ Consequently, while profit inefficiency in monetary terms $\Pi I$ is the same in both models, see column C4 in Tables 1 and 2, its decomposition following expression (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10) will differ if firms identify alternative benchmarks, see columns C5 and C6. For these firms identifying different benchmarks, allocative profit inefficiency (C6) is now smaller than in the traditional approach: $A\Pi I_{\mathit{\text{WA}}}$ > $A\Pi I_{\mathit{\text{SR}}}.$ Equivalently, technical profit inefficiency, equal to $p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*},$ is larger: $T\Pi I_{\mathit{\text{WA}}}$ < $T\Pi I_{\mathit{\text{SR}}}.$ Moreover, allocative profit inefficiency drops to zero for the last five firms because having firm B as reference implies that, once projected to it, they become allocatively efficient. Accordingly, all profit inefficiency is technical.

Table 2. The standard reverse profit inefficiency decomposition. (w,p) = (1, 4).

Firm	Model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11)	SR Project. (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11)	Monetary (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10)			Normalized (15)(15) $$N\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,\tilde{w},\tilde{p}\right)=\frac{\Pi I\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right)+\frac{\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right)}{U_{\mathit{\text{SR}}}}.$$(15)
			Profit Ineff.	Tech. Profit Ineff.	Alloc. Profit Ineff.	Profit Ineff.	Tech. Ineff.	Alloc. Ineff.
$(x_o,y_o)$	Tech. status	$({\widehat{x}}_{\mathit{\text{oSR}}}^{},{\widehat{y}}_{\mathit{\text{oSR}}}^{})$	$\Pi I$	$T\Pi I_{\mathit{\text{SR}}}$ =$p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}$	$A\Pi I_{\mathit{\text{SR}}}$	$N\Pi I_{\mathit{\text{SR}}}$	$T{I}_{\mathit{\text{SR}}}$	${\mathit{\text{NAI}}}_{\mathit{\text{SR}}}$
C1	C2	C3	C4	C5	C6 = C4-C5	C7 = C4/$N{F}_{\mathit{\text{SR}}}^{}$	C8 = C5/$N{F}_{\mathit{\text{SR}}}^{}$	C9 = C6/$N{F}_{\mathit{\text{SR}}}^{}$
A = (6,9)	Eff.	(6, 9)	9	0	9	0.231	0	0.231
B = (9,12)	Eff.	(9, 12)	0	0	0	0	0	0
C = (7,10)	Eff.	(7, 10)	6	0	6	0.154	0	0.154
D = (6,4)	Ineff.	(6, 9)	29	20	9	0.744	0.513	0.231
E = (8,8)	Ineff.	(8, 11)^(1)	15	12	3	0.385	0.308	0.077
F = (8,9)	Ineff.	(8, 11)^(1)	11	8	3	0.282	0.205	0.077
G = (9,9)	Ineff.	(9, 12)	12	12	0	0.308	0.308	0
H = (9,10)	Ineff.	(9, 12)	8	8	0	0.205	0.205	0
I = (10,8)	Ineff.	(9, 12)	17	17	0	0.436	0.436	0
J = (12,10)	Ineff.	(9, 12)	11	11	0	0.282	0.282	0
K = (12,3)	Ineff.	(9, 12)	39	39	0	1	1	0
–	–	–	–	$N{F}_{\mathit{\text{SR}}}^{}$ = 39 = max C5	–	–	–	–

⁽¹⁾Model (11(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) ) yields (8, 11) as projection, which is a linear combination of firms A and B, with ${\lambda }_{\mathrm{A}}^{*}=0.333$ and ${\lambda }_{\mathrm{B}}^{*}=0.667$ as optimal solutions.

Finally, we proceed to illustrate the decomposition of the normalized version of profit inefficiency $N\Pi I_{\mathit{\text{SR}}}$ according to the standard reverse approach, expression (15)(15) $$N\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,\tilde{w},\tilde{p}\right)=\frac{\Pi I\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right)+\frac{\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right)}{U_{\mathit{\text{SR}}}}.$$(15) , whose values are presented in the last three columns of Table 2 (C7, C8 and C9). To achieve this goal we identify first the normalizing factor equivalent to the maximum value of the technical profit inefficiencies, which in this case corresponds to firm K, i.e., $N{F}_{\mathit{\text{SR}}}\equiv U_{\mathit{\text{SR}}}$= $39. This normalization factor is reported at the bottom of C5. Finally, the normalized decomposition (15)(15) $$N\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,\tilde{w},\tilde{p}\right)=\frac{\Pi I\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right)+\frac{\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right)}{U_{\mathit{\text{SR}}}}.$$(15) is presented in the last three columns. These terms are obtained by dividing the preceding three columns by the common normalization factor. Comparing the normalized values of the traditional and standard reverse approaches we see that the values of the traditional approach are unbounded and harder to interpret. In Table 1, for firm K, normalized profit inefficiency is 3.25, while its technical and allocative components are 0.833 and 2.417, respectively.¹⁰ Also, these values are driven by the choice of technical inefficiency measure, which is at odds with reality when managers are interested in gaining profit efficiency. In the standard reverse approach, normalizing by the upper bound of the technical profit inefficiencies results in a well-defined valuation of the inefficiencies with respect to the worse technical behavior in economic terms, which provides a suitable interpretation. In this example, normalized profit inefficiency is 1, and since firm K is projected to the profit maximizing benchmark, its technical and allocative components are 1 and 0, respectively.

5. The standard reverse cost inefficiency decompositions

The decomposition of cost inefficiency using the reverse approach, monetary and normalized, follows the same philosophy as the above profit inefficiency decompositions. However, the analysis differs in four basic dimensions. The first one is that the analysis is performed in terms of the input set, rather than the technology. The second one is that only market input prices are required. The third one is that the efficient frontier is defined in terms of the input values. And the last one is that considering cost inefficiency instead of profit inefficiency requires to adapt the rules of action. In this case, following the methodology introduced above for the profit model means that, for each firm, an initial linear program will identify its input-efficient projection, by maximizing the price of its input slacks. Then, based on the calculated optimal input slacks for each firm of our sample, we define a new input-oriented technical inefficiency measure, that is crucial for recovering the associated reverse cost inefficiency decomposition as well as its normalized version. As could be expected, the last-mentioned reverse decompositions have also the same four desirable properties introduced for the profit case.

We start considering a VRS input production possibility set generated by the finite sample of firms $L(y_o)=\left\{x:(x,y_o^{})\in T\right\}.$ Last set is based on the observed output quantities $y_o,$ which is the reference non-zero N-dimensional output vector of the unit under evaluation. A common vector of market input prices $w\in {\mathbb{R}}_{++}^M$ is known, delivering the positive input-cost $w\cdot x_j$ of each firm j. The minimum cost of producing $y_o$ is defined as $C(y_o,w)=\mathrm{\text{min}}\left\{w\cdot x:x\in L(y_o)\right\}.$ The cost inefficiency of firm $(x_o^{},y_o^{})$ is the cost excess between observed cost and minimum cost: $\mathit{\text{CI}}(x_o,y_o,w)=w\cdot x_o-C(y_o,w).$ For the same reason argued in the profit case, we assume that there exists at least one inefficient firm. To decompose cost inefficiency we first need to find a projection $({\widehat{x}}_{\mathit{\text{oSR}}},y_o)$ of firm $(x_o^{},y_o^{}),$ that satisfies ${\widehat{x}}_{\mathit{\text{oSR}}}\le x_o^{},$ or, equivalently, with non-negative input-slacks, $s_{\mathit{\text{jSR}}}^{-}=(x_j^{}-{\widehat{x}}_{\mathit{\text{jSR}}}).$ Last equality gives rise to the next one, simply by obtaining the market value of each of its three elements, $w\cdot s_{\mathit{\text{oSR}}}^{-}=w\cdot x_o^{}-w\cdot {\widehat{x}}_{\mathit{\text{oSR}}},$ which in turn can be reordered to give $w\cdot x_j^{}=w\cdot s_{\mathit{\text{oSR}}}^{-}+w\cdot {\widehat{x}}_{\mathit{\text{oSR}}}.$ Based on this result, we are going to formulate the standard reverse cost inefficiency decomposition as an equality, relating the cost inefficiency of firms $(x_o^{},y_o^{})$ and its projection $({\widehat{x}}_{\mathit{\text{oSR}}},y_o)$ as follows: (17) $$\begin{array}{c}\mathit{\text{CI}}\left(x_o,y_o,w\right)=w\cdot x_o^{}-C\left(y_o,w\right)=w\cdot s_{\mathit{\text{oSR}}}^{-}+w\cdot {\widehat{x}}_{\mathit{\text{oSR}}}-C\left(y_o,w\right)\\ =w\cdot s_{\mathit{\text{oSR}}}^{-}+\mathit{\text{CI}}\left({\widehat{x}}_{\mathit{\text{oSR}}},y_o,w\right)={\mathit{\text{TCI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right)+{\mathit{\text{ACI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right).\end{array}$$(17)

Equality (17)(17) $$\begin{array}{c}\mathit{\text{CI}}\left(x_o,y_o,w\right)=w\cdot x_o^{}-C\left(y_o,w\right)=w\cdot s_{\mathit{\text{oSR}}}^{-}+w\cdot {\widehat{x}}_{\mathit{\text{oSR}}}-C\left(y_o,w\right)\\ =w\cdot s_{\mathit{\text{oSR}}}^{-}+\mathit{\text{CI}}\left({\widehat{x}}_{\mathit{\text{oSR}}},y_o,w\right)={\mathit{\text{TCI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right)+{\mathit{\text{ACI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right).\end{array}$$(17) decomposes cost inefficiency in the sum of two terms, which we call as technical cost inefficiency ${\mathit{\text{TCI}}}_{\mathit{\text{SR}}}(x_o,y_o,w)$ and allocative cost inefficiency ${\mathit{\text{ACI}}}_{\mathit{\text{SR}}}(x_o,y_o,w).$ Next, to implement the reverse approach we mirror the profit case, see model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) , and formulate the following input weighted additive model that allows calculating technical cost inefficiency ${\mathit{\text{TCI}}}_{\mathit{\text{SR}}}(x_o,y_o,w)$: (18) $$\begin{array}{cccc}{\mathit{\text{TCI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{omSR}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{o1\mathit{\text{SR}}}^{-},\dots ,s_{\mathit{\text{oMSR}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill \end{array}$$(18)

This model identifies the projection with the lowest feasible cost inefficiency or, alternatively, with the minimum allocative inefficiency with respect to minimum cost $C(y_o,w).$ If model (18)(18) $$\begin{array}{cccc}{\mathit{\text{TCI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{omSR}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{o1\mathit{\text{SR}}}^{-},\dots ,s_{\mathit{\text{oMSR}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill \end{array}$$(18) yields $w\cdot s_{\mathit{\text{oSR}}}^{-*}=0_{\$},$ $(x_o^{},y_o^{})$ is an efficient firm of $L(y_o),$ and, conversely, if $w\cdot s_{\mathit{\text{oSR}}}^{-*}>0_{\$}$ we are facing an input inefficient firm.

To obtain the normalized cost inefficiency decomposition, we express $w\cdot s_{\mathit{\text{oSR}}}^{-*}$ in (17)(17) $$\begin{array}{c}\mathit{\text{CI}}\left(x_o,y_o,w\right)=w\cdot x_o^{}-C\left(y_o,w\right)=w\cdot s_{\mathit{\text{oSR}}}^{-}+w\cdot {\widehat{x}}_{\mathit{\text{oSR}}}-C\left(y_o,w\right)\\ =w\cdot s_{\mathit{\text{oSR}}}^{-}+\mathit{\text{CI}}\left({\widehat{x}}_{\mathit{\text{oSR}}},y_o,w\right)={\mathit{\text{TCI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right)+{\mathit{\text{ACI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right).\end{array}$$(17) as the product of its input technical inefficiency score, that needs to be defined, times the resulting normalizing factor. The approach we propose is simple, like in the profit case, and consists in identifying first the maximum value across the optimal objective function values associated with all J firms in our sample: (19) $$U_{\mathit{\text{SR}}}^I=\mathrm{\text{max}}\left\{w\cdot s_{\mathit{\text{jSR}}}^{-*},\forall j\in J\right\}>0_{\$}.$$(19)

Again, the existence of at least one inefficient firm guarantees that $U_{\mathit{\text{SR}}}^I$ is positive, as expressed in (19)(19) $$U_{\mathit{\text{SR}}}^I=\mathrm{\text{max}}\left\{w\cdot s_{\mathit{\text{jSR}}}^{-*},\forall j\in J\right\}>0_{\$}.$$(19) . Dividing each objective function value by $U_{\mathit{\text{SR}}}^I$ we obtain a pure number that corresponds to the input-oriented technical inefficiency measure, taking values in the interval [0,1]: (20) $$T{I}_{\mathit{\text{SR}}}^I\left(x_j,y_o\right):=\frac{w\cdot s_{\mathit{\text{oSR}}}^{-*}}{U_{\mathit{\text{SR}}}^I},\hspace{0.17em}\hspace{0.17em}j\in J.$$(20)

As in the profit case, it can be proven that $T{I}_{\mathit{\text{SR}}}^I(x_j,y_o)$ satisfies the four properties that qualifies it as a true technical inefficiency measure. Coming back to equality (17)(17) $$\begin{array}{c}\mathit{\text{CI}}\left(x_o,y_o,w\right)=w\cdot x_o^{}-C\left(y_o,w\right)=w\cdot s_{\mathit{\text{oSR}}}^{-}+w\cdot {\widehat{x}}_{\mathit{\text{oSR}}}-C\left(y_o,w\right)\\ =w\cdot s_{\mathit{\text{oSR}}}^{-}+\mathit{\text{CI}}\left({\widehat{x}}_{\mathit{\text{oSR}}},y_o,w\right)={\mathit{\text{TCI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right)+{\mathit{\text{ACI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,w\right).\end{array}$$(17) and introducing the just defined inefficiency measure, we get the next standard reverse cost inefficiency decomposition, (21) $$\mathit{\text{CI}}\left(x_o,y_o,w\right)=T{I}_{\mathit{\text{SR}}}^I\left(x_o,y_o\right)\times U_{\mathit{\text{SR}}}^I+\mathit{\text{CI}}\left({\widehat{x}}_{\mathit{\text{oSR}}},y_o,w\right).$$(21)

It is interesting to realize that $U_{\mathit{\text{SR}}}^I$ constitutes a normalizing factor due to its units of measurement, and which that, once again, is common to all firms. Therefore, we get the next reverse normalized cost inefficiency decomposition: (22) $$\begin{array}{c}{\mathit{\text{NCI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,\tilde{w}\right)=\frac{\mathit{\text{CI}}\left(x_o,y_o,w\right)}{U_{\mathit{\text{SR}}}^I}=T{I}_{\mathit{\text{SR}}}^I\left(x_o,y_o\right)+\frac{\mathit{\text{CI}}\left({\widehat{x}}_{\mathit{\text{oSR}}},y_o,w\right)}{U_{\mathit{\text{SR}}}^I}\\ =T{I}_{\mathit{\text{SR}}}^I\left(x_o,y_o\right)+{\mathit{\text{NAI}}}_{\mathit{\text{SR}}}^I\left(x_o,y_o,\tilde{w}\right).\end{array}$$(22)

6. The standard reverse revenue inefficiency decompositions

The standard reverse revenue inefficiency decomposition is analogous to its cost inefficiency counterpart but considering an output-orientation. Hence, in what follows, we present the main concepts and results related to the monetary and normalized decompositions.

In this case we start considering a VRS production possibility set generated by a finite sample of firms belonging to the output set $P(x_o):=\left\{y:(x_o,y)\in T\right\}.$ Last set is based on the observed input quantities, $x_o,$ which is the reference non-zero M-dimensional input vector of the unit under evaluation. A common vector of market output prices, $p\in {\mathbb{R}}_{++}^N,$ is known and applies to all firms of $P(x_o),$ delivering the output-revenue $p\cdot y_j$ generated by each firm $j\in J.$ The maximum revenue is achieved in at least one firm of our finite sample and is denoted as $R(x_o,p)=\mathrm{\text{max}}\left\{p\cdot y:y\in P(x_o)\right\}.$ Again, as in the two preceding cases, we assume that at least one firm of our sample is inefficient. The revenue inefficiency of firm $(x_o^{},y_o^{})$ is defined as $\mathit{\text{RI}}(x_o,y_o,p)=R(x_o,p)-p\cdot y_o.$ To decompose revenue inefficiency one needs to find, for each firm $(x_o,y_o)\in P(x_o),$ an efficient projection belonging to the frontier of $P(x_o),$ $(x_o,{\widehat{y}}_{\mathit{\text{oSR}}}),$ that satisfies $p\cdot {\widehat{y}}_{\mathit{\text{oSR}}}=p\cdot y_o^{}+p\cdot s_{\mathit{\text{oSR}}}^{+},$ with $p\cdot s_{\mathit{\text{oSR}}}^{+}\ge 0_{\$}.$ This results in the following reverse revenue inefficiency decomposition defined as an equality, relating the revenue inefficiency of firm $(x_o^{},y_o^{})$ and that of its projection $(x_o,{\widehat{y}}_{\mathit{\text{oSR}}}):$ (23) $$\begin{array}{c}\mathit{\text{RI}}\left(x_o,y_o,p\right)=R\left(x_o,p\right)-p\cdot y_o^{}=R\left(x_o,p\right)-p\cdot s_{\mathit{\text{oSR}}}^{+}+p\cdot {\widehat{y}}_{\mathit{\text{oSR}}}\\ =p\cdot s_{\mathit{\text{oSR}}}^{+}+\mathit{\text{RI}}\left(x_o,{\widehat{y}}_{\mathit{\text{oSR}}},p\right)={\mathit{\text{TRI}}}_{\mathit{\text{SR}}}\left(x_o,y_o;p\right)+{\mathit{\text{ARI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,p\right).\end{array}$$(23)

As before, equality (23)(23) $$\begin{array}{c}\mathit{\text{RI}}\left(x_o,y_o,p\right)=R\left(x_o,p\right)-p\cdot y_o^{}=R\left(x_o,p\right)-p\cdot s_{\mathit{\text{oSR}}}^{+}+p\cdot {\widehat{y}}_{\mathit{\text{oSR}}}\\ =p\cdot s_{\mathit{\text{oSR}}}^{+}+\mathit{\text{RI}}\left(x_o,{\widehat{y}}_{\mathit{\text{oSR}}},p\right)={\mathit{\text{TRI}}}_{\mathit{\text{SR}}}\left(x_o,y_o;p\right)+{\mathit{\text{ARI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,p\right).\end{array}$$(23) decomposes revenue inefficiency in the sum of technical revenue inefficiency ${\mathit{\text{TRI}}}_{\mathit{\text{SR}}}(x_o,y_o,p)$ and allocative revenue inefficiency ${\mathit{\text{ARI}}}_{\mathit{\text{SR}}}(x_o,y_{o}p).$ Moreover, the projection $(x_o,{\widehat{y}}_{\mathit{\text{oSR}}})$ should bring the greatest possible revenue gains—or, equivalently, the smallest possible revenue inefficiency with respect to maximum profit $R(x_o,p),$ meaning that we seek to maximize the market value of the output-slacks, $p\cdot s_{\mathit{\text{oSR}}}^{+}.$ The following model calculates ${\mathit{\text{TRI}}}_{\mathit{\text{SR}}}(x_o,y_o,p)$ under this criterion: (24) $$\begin{array}{cccc}{\mathit{\text{TRI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{onSR}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{o1\mathit{\text{SR}}}^{+},\dots ,s_{\mathit{\text{oNSR}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(24)

If the optimal solution of (24)(24) $$\begin{array}{cccc}{\mathit{\text{TRI}}}_{\mathit{\text{SR}}}\left(x_o,y_o,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{onSR}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{o1\mathit{\text{SR}}}^{+},\dots ,s_{\mathit{\text{oNSR}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(24) equals $p\cdot s_{\mathit{\text{oSR}}}^{+*}=0_{\$},$ $(x_o,y_o)$ is an efficient firm of $P(x_o),$ while $p\cdot s_{\mathit{\text{oSR}}}^{+*}>0_{\$}$ signals that it is inefficient. Since we are dealing with a finite sample of firms, and knowing that at least one of them is inefficient, we can calculate the maximum of the objective function value over $P(x_o),$ and call it $U_{\mathit{\text{SR}}}^O,$ (25) $$U_{\mathit{\text{SR}}}^O=\mathrm{\text{max}}\left\{p\cdot s_{\mathit{\text{jSR}}}^{+*},\forall j\in J\right\}>0_{\$}.$$(25)

Consequently, we propose the following output-oriented technical inefficiency measure, taking values in the interval [0,1]: (26) $$T{I}_{\mathit{\text{SR}}}^O\left(x_o,y_o\right):=\frac{p\cdot s_{\mathit{\text{oSR}}}^{+*}}{U_{\mathit{\text{SR}}}^O}\ge 0,j\in J.$$(26)

By reproducing the arguments shown before, it can be proved that $T{I}_{\mathit{\text{SR}}}^O(x_o,y_o)$ is a true technical inefficiency measure over the entire VRS production possibility set, since it satisfies the four conditions enounced before. Using this measure, we decompose revenue inefficiency as follows: (27) $$\mathit{\text{RI}}\left(x_o,y_o,p\right)=T{I}_{\mathit{\text{SR}}}^O\left(x_o,y_o\right)\times U_{\mathit{\text{SR}}}^O+\mathit{\text{RI}}\left(x_o,{\widehat{y}}_{\mathit{\text{oSR}}},p\right),$$(27) which constitutes the reverse revenue inefficiency decomposition for firm $(x_o,y_j)$ in two components: the technical revenue inefficiency and the allocative revenue inefficiency. Again, let us divide each term of (27)(27) $$\mathit{\text{RI}}\left(x_o,y_o,p\right)=T{I}_{\mathit{\text{SR}}}^O\left(x_o,y_o\right)\times U_{\mathit{\text{SR}}}^O+\mathit{\text{RI}}\left(x_o,{\widehat{y}}_{\mathit{\text{oSR}}},p\right),$$(27) by $U_{\mathit{\text{SR}}}^O>0_{\$},$ obtaining the next reverse normalized revenue inefficiency decomposition: (28) $$\begin{array}{c}{\mathit{\text{NRI}}}_{\mathit{\text{SR}}}\left(x_o,y_j,\tilde{p}\right)=\frac{\mathit{\text{RI}}\left(x_o,y_j,p\right)}{U_{\mathit{\text{SR}}}^O}=T{I}_{\mathit{\text{SR}}}^O\left(x_o,y_o\right)+\frac{\mathit{\text{RI}}\left(x_o,{\widehat{y}}_{\mathit{\text{oSR}}},p\right)}{U_{\mathit{\text{SR}}}^O}\\ =T{I}_{\mathit{\text{SR}}}^O\left(x_j,y_o\right)+{\mathit{\text{NAI}}}_{\mathit{\text{SR}}}^O\left(x_o,y_o,\tilde{p}\right).\end{array}$$(28)

7. Conclusions

We develop a new methodology to decompose economic inefficiency that improves the traditional approach, which resorts to duality theory and Fenchel-Mahler inequalities. Our approach is based on the so-called reverse approach suggested by Bogetoft and Hougaard (2003) and Bogetoft et al. (2006), which favours economic gains over technical improvements associated with a specific inefficiency measure. This is the first time that this paradigm is developed for each of the three possible decompositions of economic inefficiency: profit, cost and revenue. Our new approach presents some relevant novelties that deserve to be highlighted. Our first aim is to identify the best possible economic projection for each firm, i.e., the projection that brings the greatest economic gains given market prices, or, equivalently, that results in the smallest possible allocative inefficiency. Interestingly, our approach can be rendered operational through the weighted additive approach introduced by Lovell and Pastor (1995), where the weights are the input and output prices. Based on this model, we can deduce a new graph technical inefficiency measure that captures the essence of the standard reverse approach. Then, inspired by the general direct approach of Pastor et al. (2022), economic inefficiency can be decomposed relying on an equality. The two components of the decomposition correspond to the new graph technical inefficiency measure plus the allocative efficiency of the projected benchmark. The decomposition can be expressed in monetary and normalized terms by defining a specific normalization factor—yielding a normalized technical inefficiency measure that satisfies a customary set of desirable properties. The normalized profit inefficiency decomposition is unit’s invariant, as its traditional counterparts, and easier to interpret numerically.

Disclosure statement

The authors report there are no competing interests to declare.

Additional information

Funding

The authors thank the grant PID2019-105952GB-I00 funded by Ministerio de Ciencia e Innovación/ Agencia Estatal de Investigación /10.13039/501100011033. Additionally, José L. Zofío thanks the grant EIN2020-112260 funded by the Ministerio de Ciencia e Innovación / Agencia Estatal de Investigación /10.13039/501100011033, and grant H2019/HUM-5761 (INNJOBMAD-CM) funded by the Comunidad de Madrid Government.

Notes

1 See Asmild et al. (2013) for an early application to the Canadian banking industry of what Bogetoft and Hougard (2003) called ‘rational inefficiencies’.

2 The restriction that guarantees that T is a VRS technology, ${\sum }_{j=1}^J{\lambda }_j^{}=1,$ is called the convexity constraint, see Banker et al. (1984).

3 If we fix ${\rho }^{-}=1_M$ and ${\rho }^{+}=1_N,$ we then get the original additive model (Charnes et al., 1985). Depending on the weights, we obtain other variants such as the Range-Adjusted Measure (RAM), the Measure of Inefficiency Proportions (MIP), or the Bounded-Adjusted measure (BAM), as discussed in the Introduction. See chapter 6 in Pastor et al. (2022).

4 Model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) measuring technical profit inefficiency can be related to the model proposed by Ruiz and Sirvent (2011) searching for the maximum profit that a firm can attain by operating in a technically efficient manner.

5 Directly applying model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) to the traditional decomposition of profit inefficiency presented in (7)(7) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=\underset{\mathrm{(}\text{Normalized}\mathrm{)}\mathrm{ }\text{Profit}\mathrm{ }\text{Inefficiency}\mathrm{(}\Pi I_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{\frac{\Pi I\left(x_o,y_o,w,p\right)}{\mathrm{\text{min}}\left\{\frac{w_1}{{\rho }_1^{-}},\dots ,\frac{w_M}{{\rho }_M^{-}},\frac{p_1}{{\rho }_1^{+}},\dots ,\frac{p_N}{{\rho }_N^{+}}\right\}}}}\ge \underset{\text{Technical}\mathrm{ }\text{Inefficiency}\mathrm{(}T{I}_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{T{I}_{\mathit{\text{WA}}}\left(x_o,y_o,{\rho }^{-},{\rho }^{+}\right)}},$$(7) implies that the normalization factor is equal to one. This results in the monetary decomposition of profit inefficiency shown in (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10) .

6 As shown above, any firm with $(p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*})=0_{\$}$ is an efficient firm.

7 The notation $N\Pi I(x_o,y_o,\tilde{w},\tilde{p})$ means that although the value of the normalized profit inefficiency corresponds to a pure number, its value is influenced by the numerical values of the market prices. Particularly, if we would consider different market prices $(w^{\prime },p^{\prime })$ that are not proportional to the initial ones $(w,p),$ the resulting normalized profit inefficiency and, consequently, the allocative inefficiency, would yield different values.

8 See Prieto and Zofío (2001) for an early application of the RAM measure to study the effectiveness in the provision of public infrastructure and equipment in Spanish municipalities.

9 We identify with an asterisk the only projection point that does not belong to the sample, (8,11), but conforms the VRS-DEA technology (1)(1) $$T=\{\left(x,y\right):\sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}\le x_m,m=1,\dots ,M;\sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}\ge y_n,n=1,\dots ,N;\sum _{j=1}^J{\lambda }_j=1,{\lambda }_j\ge 0,j=1,\dots ,J\}$$(1) and, more precisely, the strongly efficient frontier, being a convex combination of firms A and B.

10 The unboundedness and lack of interpretability of the traditional profit inefficiency values is a common drawback of all the decompositions surveyed in the Introduction.