Estimating and decomposing overall inefficiency by determining the least distance to the strongly efficient frontier in data envelopment analysis

Abstract

This paper proposes a new method to measure economic inefficiency of decision making units based on the calculation of the least distance to the Pareto-efficient frontier in data envelopment analysis. While all previously published approaches that have dealt with the problem of determining least distances to the efficient frontier are focus on exclusively technical inefficiency, the new methodology opens the door to applications of this approach when market prices, together with inputs and outputs, are available. Finally, the paper empirically illustrates the new method using recent data on the mandarins’ production in a Spanish eastern province.

Appendix

Proof of Proposition 2

Let $\left( {\varvec{c},\varvec{r}} \right) \in R_{ + + }^{m + s}$. Then, $\left( {\varvec{c^{\prime}},\varvec{r^{\prime}}} \right) = \frac{{\left( {\varvec{c},\varvec{r}} \right)}}{{\left\| {\left( {\varvec{c},\varvec{r}} \right)} \right\|_{q} }} \in R_{ + + }^{m + s}$ and satisfies $\left\| {\frac{{\left( {\varvec{c},\varvec{r}} \right)}}{{\left\| {\left( {\varvec{c},\varvec{r}} \right)} \right\|_{q} }}} \right\|_{q} = 1$. In this way, $\left( {\varvec{c^{\prime}},\varvec{r^{\prime}}} \right)$ is a feasible solution of the program that appears in Proposition 1. Therefore, $\varPi \left( {\varvec{c^{\prime}},\varvec{r^{\prime}}} \right) - \left( {\sum\nolimits_{k = 1}^{s} {r^{\prime}_{k} y_{k0} } - \sum\nolimits_{i = 1}^{m} {c^{\prime}_{i} x_{i0} } } \right) \ge D_{{\partial^{w} \left( T \right)}}^{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right)$. Finally, we have that $\frac{{\varPi \left( {\varvec{c},\varvec{r}} \right) - \left( {\sum\nolimits_{k = 1}^{s} {r_{k} y_{k0} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i0} } } \right)}}{{\left\| {\left( {\varvec{c},\varvec{r}} \right)} \right\|_{q} }} \ge D_{{\partial^{w} \left( T \right)}}^{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right)$ since the profit function is homogeneous of degree +1 in prices (see Färe and Primont 1995). □

Proof of Proposition 3

T in (1) can be rewritten as $\left\{ {\left( {\varvec{x},\varvec{y}} \right) \in R_{ + }^{m + s} :\sum\nolimits_{k = 1}^{s} {a_{k}^{f} y_{k} } - \sum\nolimits_{i = 1}^{m} {b_{i}^{f} x_{i} } \le \varPi \left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right),f = 1, \ldots ,F} \right\}$, where $\left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right) \in R_{ + }^{m + s}$ for all $f = 1, \ldots ,F$ since T is a polyhedral set (Proposition 1 in Briec and Leleu 2003). Trivially, we see that if $\left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right) = \left( {{\mathbf{0}}_{m} ,{\mathbf{0}}_{s} } \right)$ for some $f = 1, \ldots ,F$, where ${\mathbf{0}}_{g} = \left( {0, \ldots ,0} \right)$, then $\varPi \left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right) = 0$ and the corresponding constraint is redundant and can be deleted. Now, by Lemma 4.1 in Briec (1997), $WD_{{\partial^{w} \left( T \right)}}^{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) = \mathop {\hbox{min} }\nolimits_{f = 1, \ldots ,F} \left\{ {\mathop {\inf }\nolimits_{{\varvec{u},\varvec{v}}} \left\{ {\left\| {\left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) - \left( {\varvec{u},\varvec{v}} \right)} \right\|_{p}^{\alpha } :\left( {\varvec{u},\varvec{v}} \right) \in H_{f} } \right\}} \right\},$ where $\varvec{\alpha}= \left( {\tfrac{1}{{x_{10} }}, \ldots ,\tfrac{1}{{x_{m0} }},\tfrac{1}{{y_{10} }}, \ldots ,\tfrac{1}{{y_{s0} }}} \right)$ and $H_{f} = \left\{ {\left( {\varvec{x},\varvec{y}} \right) \in R_{{}}^{m + s} :\sum\nolimits_{k = 1}^{s} {a_{k}^{f} y_{k} } - \sum\nolimits_{i = 1}^{m} {b_{i}^{f} x_{i} } = \varPi \left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right)} \right\}$, $f = 1, \ldots ,F$. On the other hand, $\mathop {\inf }\nolimits_{{\varvec{u},\varvec{v}}} \left\{ {\left\| {\left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) - \left( {\varvec{u},\varvec{v}} \right)} \right\|_{p}^{\alpha } :\left( {\varvec{u},\varvec{v}} \right) \in H_{f} } \right\} = \mathop {\inf }\nolimits_{{\tilde{\varvec{u}},\tilde{\varvec{v}}}} \left\{ {\left\| {\left( {{\mathbf{1}}_{m} ,{\mathbf{1}}_{s} } \right) - \left( {\tilde{\varvec{u}},\tilde{\varvec{v}}} \right)} \right\|_{p}^{{}} :\left( {\tilde{\varvec{u}},\tilde{\varvec{v}}} \right) \in \tilde{H}_{f} } \right\}$, where ${\mathbf{1}}_{g} = \left( {1, \ldots ,1} \right)$ and $\tilde{H}_{f} = \left\{ {\left( {\varvec{x},\varvec{y}} \right) \in R_{{}}^{m + s} :\sum\nolimits_{k = 1}^{s} {\left( {a_{k}^{f} y_{k0} } \right)y_{k} } - \sum\nolimits_{i = 1}^{m} {\left( {b_{i}^{f} x_{i0} } \right)x_{i} } = \varPi \left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right)} \right\}$, thanks to (8) and the change of variables $\tilde{u}_{i} = {{u_{i} } \mathord{\left/ {\vphantom {{u_{i} } {x_{i0} }}} \right. \kern-0pt} {x_{i0} }},\;i = 1, \ldots ,m$, and $\tilde{v}_{k} = {{v_{k} } \mathord{\left/ {\vphantom {{v_{k} } {y_{k0} }}} \right. \kern-0pt} {y_{k0} }},\;k = 1, \ldots ,s$. Now, applying the Ascoli’s formula, we have that $\mathop {\inf }\nolimits_{{\tilde{\varvec{u}},\tilde{\varvec{v}}}} \left\{ {\left\| {\left( {{\mathbf{1}}_{m} ,{\mathbf{1}}_{s} } \right) - \left( {\tilde{\varvec{u}},\tilde{\varvec{v}}} \right)} \right\|_{p}^{{}} :\left( {\tilde{\varvec{u}},\tilde{\varvec{v}}} \right) \in \tilde{H}_{f} } \right\} = \frac{{\varPi \left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right) - \left( {\sum\nolimits_{k = 1}^{s} {a_{k}^{f} y_{k0} } - \sum\nolimits_{i = 1}^{m} {b_{i}^{f} x_{i0} } } \right)}}{{\left\| {\left( {b_{1} x_{10} , \ldots ,b_{m} x_{m0} ,a_{1} y_{10} , \ldots ,a_{s} y_{s0} } \right)} \right\|_{q} }}$. In this way, we have that $WD_{{\partial^{w} \left( T \right)}}^{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) = \mathop {\hbox{min} }\nolimits_{f = 1, \ldots ,F} \left\{ {\frac{{\varPi \left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right) - \left( {\sum\nolimits_{k = 1}^{s} {a_{k}^{f} y_{k0} } - \sum\nolimits_{i = 1}^{m} {b_{i}^{f} x_{i0} } } \right)}}{{\left\| {\left( {b_{1} x_{10} , \ldots ,b_{m} x_{m0} ,a_{1} y_{10} , \ldots ,a_{s} y_{s0} } \right)} \right\|_{q} }}} \right\}$. Finally, let us consider $\left( {\varvec{c},\varvec{r}} \right) \in R_{ + + }^{m + s}$. By the definition of the profit function, $T = \left\{ {\left( {\varvec{x},\varvec{y}} \right) \in R_{ + }^{m + s} :\sum\nolimits_{k = 1}^{s} {a_{k}^{f} y_{k} } - \sum\nolimits_{i = 1}^{m} {b_{i}^{f} x_{i} } \le \varPi \left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right),f = 1, \ldots ,F,\sum\nolimits_{k = 1}^{s} {r_{k} y_{k} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i} } \le \varPi \left( {\varvec{c},\varvec{r}} \right)} \right\}.$ In this way, by the same reasoning than above, $WD_{{\partial^{w} \left( T \right)}}^{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) = \hbox{min} \left\{ {\mathop {\hbox{min} }\nolimits_{f = 1, \ldots ,F} \left\{ {\frac{{\varPi \left( {\varvec{b}^{f} ,\varvec{a}^{f} } \right) - \left( {\sum\nolimits_{k = 1}^{s} {a_{k}^{f} y_{k0} } - \sum\nolimits_{i = 1}^{m} {b_{i}^{f} x_{i0} } } \right)}}{{\left\| {\left( {b_{1} x_{10} , \ldots ,b_{m} x_{m0} ,a_{1} y_{10} , \ldots ,a_{s} y_{s0} } \right)} \right\|_{q} }}} \right\},\frac{{\varPi \left( {\varvec{c},\varvec{r}} \right) - \left( {\sum\nolimits_{k = 1}^{s} {r_{k} y_{k0} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i0} } } \right)}}{{\left\| {\left( {c_{1} x_{10} , \ldots ,c_{m} x_{m0} ,r_{1} y_{10} , \ldots ,r_{s} y_{s0} } \right)} \right\|_{q} }}} \right\}$. Finally, $\frac{{\varPi \left( {\varvec{c},\varvec{r}} \right) - \left( {\sum\nolimits_{k = 1}^{s} {r_{k} y_{k0} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i0} } } \right)}}{{\left\| {\left( {c_{1} x_{10} , \ldots ,c_{m} x_{m0} ,r_{1} y_{10} , \ldots ,r_{s} y_{s0} } \right)} \right\|_{q} }} \ge WD_{{\partial^{w} \left( T \right)}}^{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right)$. □

Proof of Proposition 4

(1) It is evident by the definition of $PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right)$. (2) If $PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right) = 0$, by (12), $y_{k}^{M*} - y_{k0} = 0$ for all $k = 1, \ldots ,s$ and $x_{i0} - x_{i}^{M*} = 0$ for all $i = 1, \ldots ,m$. Hence, $\left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) = \left( {\varvec{x}^{M*} ,\varvec{y}^{M*} } \right)$ and, consequently, $\left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) = \arg \hbox{max} \left\{ {\sum\nolimits_{k = 1}^{s} {r_{k} y_{k} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i} } :\left( {\varvec{x},\varvec{y}} \right) \in T} \right\}$. On the other hand, if $\left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) = \arg \hbox{max} \left\{ {\sum\nolimits_{k = 1}^{s} {r_{k} y_{k} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i} } :\left( {\varvec{x},\varvec{y}} \right) \in T} \right\}$, then, under our assumptions, $\left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) = \left( {\varvec{x}^{M*} ,\varvec{y}^{M*} } \right)$ and, consequently, $PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right) = 0$ by (12). (3) It is evident by the definition of $PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right)$ since the sign of the profit function does not affect directly to the value of $PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right)$. (iv) $PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\delta \varvec{c},\delta \varvec{r},T} \right) = \frac{{\left( {\sum\nolimits_{k = 1}^{s} {\delta r_{k} \left| {y_{k}^{M*} - y_{k0} } \right|^{p} } + \sum\nolimits_{i = 1}^{m} {\delta c_{i} \left| {x_{i0} - x_{i}^{M*} } \right|^{p} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0pt} p}}} }}{{\left\| {\left( {\delta c_{1} x_{10} , \ldots ,\delta c_{m} x_{m0} ,\delta r_{1} y_{10} , \ldots ,\delta r_{s} y_{s0} } \right)} \right\|_{q} }} = PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right)$ for all $\delta > 0$ by (4). (v) Let us assume that each input $i$, $i = 1, \ldots ,m$, is transformed as follows $x_{i} \to \delta_{i} x_{i}$ with $\delta_{i} > 0$ and each output $k$, $k = 1, \ldots ,s$, is transformed as follows $y_{k} \to \mu_{k} y_{k}$ with $\mu_{k} > 0$. In this way, the original technology is transformed and denoted as $\left( {\delta ,\mu } \right)T$. This data transformation affects also to the market prices in the following way: $\left( {\varvec{c},\varvec{r}} \right) \to \left( {\frac{\varvec{c}}{\delta },\frac{\varvec{r}}{\mu }} \right) = \left( {\frac{{c_{1} }}{{\delta_{1} }}, \ldots ,\frac{{c_{m} }}{{\delta_{m} }},\frac{{r_{1} }}{{\mu_{1} }}, \ldots ,\frac{{r_{s} }}{{\mu_{s} }}} \right)$. Then, by (1) and the definition of the profit function, $\left( {\varvec{x}^{M*} ,\varvec{y}^{M*} } \right) \to \left( {\delta_{1} x_{1}^{M*} , \ldots ,\delta_{m} x_{m}^{M*} ,\mu_{1} y_{1}^{M*} , \ldots ,\mu_{s} y_{s}^{M*} } \right)$. Also, $\left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right) \to \left( {\delta \varvec{x}_{0} ,\mu \varvec{y}_{0} } \right) = \left( {\delta_{1} x_{10}^{{}} , \ldots ,\delta_{m} x_{m0}^{{}} ,\mu_{1} y_{10}^{{}} , \ldots ,\mu_{s} y_{s0}^{{}} } \right)$. Then, $PI_{p} \left( {\delta \varvec{x}_{0} ,\delta \varvec{y}_{0} ,\frac{\varvec{c}}{\delta },\frac{\varvec{r}}{\delta },T} \right) = \frac{{\left( {\sum\nolimits_{k = 1}^{s} {\left| {\frac{{r_{k} }}{{\mu_{k} }}\left( {\mu_{k} y_{k}^{M*} - \mu_{k} y_{k0} } \right)} \right|^{p} } + \sum\nolimits_{i = 1}^{m} {\left| {\frac{{c_{i} }}{{\delta_{i} }}\left( {\delta_{i} x_{i0} - \delta_{i} x_{i}^{M*} } \right)} \right|^{p} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0pt} p}}} }}{{\left\| {\left( {\frac{{c_{1} }}{{\delta_{1} }}\delta_{1} x_{10} , \ldots ,\frac{{c_{m} }}{{\delta_{m} }}\delta_{m} x_{m0} ,\frac{{r_{1} }}{{\mu_{1} }}\mu_{1} y_{10} , \ldots ,\frac{{r_{s} }}{{\mu_{s} }}\mu_{s} y_{s0} } \right)} \right\|_{q} }} = PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right)$. □

Proof of Proposition 5

From $p = 1$ and $\left( {\varvec{x}^{M*} , - \varvec{y}^{M*} } \right) \le \left( {\varvec{x}_{0} , - \varvec{y}_{0} } \right)$, we may rewritten $PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right)$ as $\frac{{\left( {\sum\nolimits_{k = 1}^{s} {r_{k} y_{k}^{M*} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i}^{M*} } } \right) - \left( {\sum\nolimits_{k = 1}^{s} {r_{k} y_{k0}^{{}} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i0}^{{}} } } \right)}}{{\left\| {\left( {c_{1} x_{10} , \ldots ,c_{m} x_{m0} ,r_{1} y_{10} , \ldots ,r_{s} y_{s0} } \right)} \right\|_{\infty } }}$. Finally, this last expression is equivalent to $\frac{{\varPi \left( {\varvec{c},\varvec{r}} \right) - \left( {\sum\nolimits_{k = 1}^{s} {r_{k} y_{k0} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i0} } } \right)}}{{\left\| {\left( {c_{1} x_{10} , \ldots ,c_{m} x_{m0} ,r_{1} y_{10} , \ldots ,r_{s} y_{s0} } \right)} \right\|_{\infty } }}$ since $\sum\nolimits_{k = 1}^{s} {r_{k} y_{k}^{M*} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{i}^{M*} } = \varPi \left( {\varvec{c},\varvec{r}} \right)$. □

Proof of Proposition 6

(1) Trivial by the definition of (16). (2) It is evident from the fact that we are calculating a mathematical distance from $\left( {\varvec{x}_{0} ,\varvec{y}_{0} } \right)$ to the set $\partial^{s} \left( T \right)$. (3) Let us assume that each input $i$, $i = 1, \ldots ,m$, is transformed as follows $x_{i} \to \delta_{i} x_{i}$ with $\delta_{i} > 0$ and each output $k$, $k = 1, \ldots ,s$, is transformed as follows $y_{k} \to \mu_{k} y_{k}$ with $\mu_{k} > 0$. In this way, the original technology is transformed and denoted as $\left( {\delta ,\mu } \right)T$. It is not difficult to prove that $\left( {\varvec{x},\varvec{y}} \right) \in \partial^{s} \left( T \right)$ if and only if $\left( {\delta \varvec{x},\mu \varvec{y}} \right) \in \partial^{s} \left( {\left( {\delta ,\mu } \right)T} \right)$. This data transformation affects also to the market prices in the following way: $\left( {\varvec{c},\varvec{r}} \right) \to \left( {\frac{\varvec{c}}{\delta },\frac{\varvec{r}}{\mu }} \right) = \left( {\frac{{c_{1} }}{{\delta_{1} }}, \ldots ,\frac{{c_{m} }}{{\delta_{m} }},\frac{{r_{1} }}{{\mu_{1} }}, \ldots ,\frac{{r_{s} }}{{\mu_{s} }}} \right)$. Then, by the definition of (16), $WD_{{\partial^{s} \left( {\left( {\delta ,\mu } \right)T} \right)}}^{p} \left( {\delta \varvec{x}_{0} ,\mu \varvec{y}_{0} ;\left( {\delta ,\mu } \right)\varvec{\alpha}_{0} } \right) = WD_{{\partial^{S} \left( T \right)}}^{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ;\varvec{\alpha}_{0} } \right)$, where $\left( {\delta ,\mu } \right)\varvec{\alpha}_{0} = \frac{{\left( {\frac{\varvec{c}}{\delta },\frac{\varvec{r}}{\mu }} \right)}}{{\left\| {\left( {\frac{{c_{1} }}{{\delta_{i} }}\delta_{i} x_{10} , \ldots ,\frac{{c_{m} }}{{\delta_{m} }}\delta_{m} x_{m0} ,\frac{{r_{1} }}{{\mu_{1} }}\mu_{1} y_{10} , \ldots ,\frac{{r_{s} }}{{\mu_{s} }}\mu_{s} y_{s0} } \right)} \right\|_{q} }}$. □

Proof of Proposition 7

Given $\left( {\varvec{c},\varvec{r}} \right) \in R_{ + + }^{m + s}$, $\left( {\varvec{x}^{M*} ,\varvec{y}^{M*} } \right) \in \partial^{s} \left( T \right)$, which implies that $\left( {\varvec{x}^{M*} ,\varvec{y}^{M*} } \right)$ is a feasible solution of model (16) and, therefore, $PI_{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ,\varvec{c},\varvec{r},T} \right) \ge WD_{{\partial^{S} \left( T \right)}}^{p} \left( {\varvec{x}_{0} ,\varvec{y}_{0} ;\varvec{\alpha}_{0} } \right)$ by (12). □