Setting handicaps to industrial sectors in DEA illustrated by Ethiopian industry

Abstract

In the ordinary macro-economic input–output tables, the industrial sector consists of several dozen industries and each industry in a certain sector is an aggregate of many companies in the sector. The sectoral statistics are the sum of statistics of companies in the respective sector. Usually, all sectors have the same set of inputs for producing outputs. For example, they have labour, capital and intermediate input as input and amount of production as output. We can apply traditional DEA models for evaluation of efficiency regarding all sectors by means of these common input and output factors. However, there remain concerns about comparing all sectors as a scratch race. Some sectors are in fields with matured technologies, while others are in emerging fields. Some are labour intensive, while others are capital intensive. These situations lead us to compare sectors under a handicap race. In this paper, we propose a new DEA model based on the non-convex frontiers that all associated sectors may exhibit and from which handicaps are derived. We apply this model to Ethiopian industry.

Appendix

We summarise the steps for the SAS computation as follows:

• Step 1  Input data (X, Y, Cluster). Cluster can be supplied exogenously by some clustering methods, including experts’ knowledge, or determined internally depending on the degree of scale-efficiency;

• Step 2  For $k=1,\ldots ,n,$ solve the CRS and VRS models to obtain the CRS and VRS scores, $\theta _k^{CRS}$ and $\theta _k^{VRS}$, respectively. Define the scale-efficiency $\sigma _k :={\theta _k^{CRS} }/{\theta _k^{VRS} }$;

• Step 3  Using the optimal slacks $(\mathbf{s}_k^{-*} ,\mathbf{s}_k^{+*} )$ for the CRS model, define the scale-dependent slacks as $((1-\sigma _k )\mathbf{s}_k^{-*} ,(1-\sigma _k )\mathbf{s}_k^{+*} );$

• Step 4  Define the scale-independent data set $(\overline{\mathbf{X}} ,\overline{\mathbf{Y}} ):=\left\{ {({\bar{\mathbf{x}}}_k , {\bar{\mathbf{y}}}_k )\left| {k=1,\ldots ,n} \right. } \right\}$ by:

  $$\begin{aligned}&\hbox {Scale-independent input }{ \bar{\mathbf{x}}}_k :=\mathbf{x}_k -(1-\sigma _k )\mathbf{s}_k^{-*} ,\;\hbox {Scale-independent output }\\&{ \bar{\mathbf{y}}}_k :=\mathbf{y}_k +(1-\sigma _k )\mathbf{s}_k^{+*} ; \end{aligned}$$

• Step 5  Solve the CRS model (15) for each $({ \bar{\mathbf{x}}}_k ,{ \bar{\mathbf{y}}}_k )$referring to the $(\overline{\mathbf{X}} ,\overline{\mathbf{Y}} )$ in the same cluster (k), and obtain the optimal in-cluster slacks $(\mathbf{s}_k^{cl-*} ,\mathbf{s}_k^{cl+*} )$;

• Step 6  Define the SAS by:

  $$\begin{aligned} \theta _k^{SAS} :=1-\frac{1}{m}\mathop {\sum }\limits _{i=1}^m {\frac{s_{ik}^{cl-*} +(1-\sigma _k )s_{ik}^{-*} }{x_{ik} }} ; \end{aligned}$$

• Step 7  Obtain the scale and cluster adjusted input and output (projection) $({ \bar{{\bar{\mathbf{x}}}}}_k ,{ \bar{{\bar{\mathbf{y}}}}}_k )$by:

  $$\begin{aligned} { \bar{{\bar{\mathbf{x}}}}}_k :=\mathbf{x}_k -(1-\sigma _k )\mathbf{s}_k^{-*} -\mathbf{s}_k^{cl-*} \hbox {,}\; {\bar{{\bar{\mathbf{y}}}}}_k :=\mathbf{y}_k +(1-\sigma _k )\mathbf{s}_k^{+*} +\mathbf{s}_k^{cl+*} ; \end{aligned}$$

• Step 8  We define the scale and cluster adjusted dataset by $({\mathop {\mathbf{X}}\limits ^{=}}, {\mathop {\mathbf{Y}}\limits ^{=}} ):=\left\{ {({\bar{\bar{\mathbf{x}}}}_j ,{\bar{\bar{\mathbf{y}}}}_j )\left| {j=1,\ldots ,n} \right. } \right\} .$