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

Disclosure statement

The authors report there are no competing interests to declare.

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.

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.