Performance Evaluation Techniques: An Application to Indian Garments Industry

Abstract

One plausible way to assess empirically the impact of reforms in globalization on an industry is by evaluation of its performance under these reforms in the input–output framework.

Appendices

Appendix 1

A Note on Normal–Half-Normal Model

The basic distributional assumption in this case is:

(i)$v{}_{i}\sim iid - N(0,\sigma_{v}^{2} )$

(ii)$u{}_{i}\sim iid - N^{ + } (0,\sigma_{u}^{2} )$

(iii)$u_{i}\; \& \;v_{i} - Independent$

So the joint density of $u\;\&\; v$ is:

$$ f\left( {u,v} \right) = \frac{2}{{2\pi \sigma_{u} \sigma_{v} }}\exp \left\{ { - \frac{{u^{2} }}{{2\sigma_{u}^{2} }} - \frac{{v^{2} }}{{2\sigma_{v}^{2} }}} \right\} $$

(10.17)

Now assume $\varepsilon = v - u$. Therefore the expression of joint density of $u\;\& \;\varepsilon$ is:

$$ f\left( {u,\varepsilon } \right) = \frac{2}{{2\pi \sigma_{u} \sigma_{v} }}\exp \left\{ { - \frac{{u^{2} }}{{2\sigma_{u}^{2} }} - \frac{{\left( {\varepsilon + u} \right)^{2} }}{{2\sigma_{v}^{2} }}} \right\} $$

(10.18)

The marginal density of $\varepsilon$ is:

$$ \begin{aligned} f\left( \varepsilon \right) &= \int\limits_{0}^{\infty } {f\left( {u,\varepsilon } \right)} du \hfill \\&= \frac{2}{{\sqrt {2\pi } \sigma }}\left[ {1 - \Upphi \left( {\frac{\varepsilon \lambda }{\sigma }} \right)} \right]\exp \left\{ { - \frac{{\varepsilon^{2} }}{{2\sigma^{2} }}} \right\} \hfill \\&= \frac{2}{\sigma }\varphi \left( {\frac{\varepsilon }{\sigma }} \right)\Upphi \left( { - \frac{\varepsilon \lambda }{\sigma }} \right) \hfill \\ \end{aligned} $$

(10.19)

where; $\sigma = \left( {\sigma_{u}^{2} + \sigma_{v}^{2} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ and $\lambda = \frac{{\sigma_{u} }}{{\sigma_{v} }}$

The log likelihood function of the sample of $N$ producer is:

$$ \ln L = {\text{CONS}} - N\ln \sigma + \sum\limits_{i = 1}^{N} {\ln \Upphi \left( { - \frac{{\varepsilon_{i} \lambda }}{\sigma }} \right)} - \frac{1}{{2\sigma^{2} }}\sum\limits_{i = 1}^{N} {\varepsilon_{i} } $$

(10.20)

The conditional distribution of $u|\varepsilon$ is:

$$ \begin{aligned} f\left( {u|\varepsilon } \right) &= \frac{{f\left( {u,\varepsilon } \right)}}{f\left( \varepsilon \right)} \hfill \\ \hfill \\& = \frac{1}{{\sqrt {2\pi } \sigma_{*} }}{\raise0.7ex\hbox{${\exp \left\{ { - \frac{{\left( {u - \mu_{*} } \right)^{2} }}{{2\sigma_{*}^{2} }}} \right\}}$} \!\mathord{\left/ {\vphantom {{\exp \left\{ { - \frac{{\left( {u - \mu_{*} } \right)^{2} }}{{2\sigma_{*}^{2} }}} \right\}} {\left[ {1 - \Upphi \left( { - \frac{{\mu_{*} }}{{\sigma_{*} }}} \right)} \right]}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left[ {1 - \Upphi \left( { - \frac{{\mu_{*} }}{{\sigma_{*} }}} \right)} \right]}$}} \hfill \\ \end{aligned} $$

(10.21)

where; $\mu_{*} = - \frac{{\varepsilon \sigma_{u}^{2} }}{{\sigma^{2} }}$ and $\sigma_{*}^{2} = \frac{{\sigma_{u}^{2} \sigma_{v}^{2} }}{{\sigma^{2} }}$

Estimated Technical Efficiency of ith Firm:

$$ \begin{aligned} TE_{i}& = E\left( {\exp \left\{ { - u_{i} } \right\}|\varepsilon_{i} } \right) \hfill \\ &= \left[ {\frac{{1 - \Upphi \left( {\sigma_{*} - {\raise0.7ex\hbox{${\mu_{*i} }$} \!\mathord{\left/ {\vphantom {{\mu_{*i} } {\sigma_{*} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{*} }$}}} \right)}}{{1 - \Upphi \left( { - {\raise0.7ex\hbox{${\mu_{*i} }$} \!\mathord{\left/ {\vphantom {{\mu_{*i} } {\sigma_{*} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{*} }$}}} \right)}}} \right]\exp \left\{ { - \mu_{*i} + \frac{1}{2}\sigma_{*}^{2} } \right\} \hfill \\ \end{aligned} $$

(10.22)

Appendix 2

Some Conditions for SFPF Model

Likelihood Ratio Test

Suppose, we have a three input production function $Y = f\left( {x,\,y,\,z} \right)$. In order to estimate the SFPF model, the chapter uses fixed assets, intermediate input, and man-days worked as inputs. It may be possible that some of the inputs may be unimportant. This may be judged by performing statistical significance of the parameters associated with that particular input variable using likelihood ratio test. The higher the value of the LR statistic and the closer the value of the probability to zero, the better the fit. The results of our study are summarized in Table 10.22. It is only for the year 2001–2002 for the input Fixed Asset that we find the fit is not so good. Still, we continue taking this input for the year.

Table 10.22 Likelihood ratio test for half-normal frontier model

Monotonicity Requirement

Three conditions need to be satisfied:

(i) $f_{x} = \frac{\partial Y}{\partial x} \ge 0$; (ii) $f_{y} = \frac{\partial Y}{\partial y} \ge 0$; and (iii) $f_{z} = \frac{\partial Y}{\partial z} \ge 0$

Table 10.23 reports the number of firms having positive input elasticities for each of the inputs in our study. It is to be noted that these firms that have positive input elasticity with respect to an input would satisfy the Monotonicity Requirement for that input autmatically.

Table 10.23 Firms having positive input elasticity

Quasiconcavity Requirement

For a three-input model, three conditions need to be satisfied [Notations carry their usual meanings]:

• (a) $\left| {\begin{array}{*{20}c} 0 & {f_{x} } \\ {f_{x} } & {f_{xx} } \\ \end{array} } \right| \le 0$, which is a trivial requirement and is satisfied always;

• (b) $\left| {\begin{array}{*{20}c} 0 & {f_{x} } & {f_{y} } \\ {f_{x} } & {f_{xx} } & {f_{xy} } \\ {f_{y} } & {f_{xy} } & {f_{yy} } \\ \end{array} } \right| \ge 0;$

• (c) $\left| {\begin{array}{*{20}c} 0 & {f_{x} } & {f_{y} } & {f_{z} } \\ {f_{x} } & {f_{xx} } & {f_{xy} } & {f_{xz} } \\ {f_{y} } & {f_{xy} } & {f_{yy} } & {f_{yz} } \\ {f_{z} } & {f_{xz} } & {f_{yz} } & {f_{zz} } \\ \end{array} } \right| \le 0.$

We have to check the number of firms satisfying both (b) and (c) above. Table 10.24 reports the number of firms satisfying these requirements in our study.

Table 10.24 Firms satisfying regularity property

A note of caution may apply. The estimated parameters need to be adjusted according to the form of the production function and the form in which the variables are used to derive the function for the empirical estimation. For instance,

$f_{x} = \frac{\partial Y}{\partial x} = \frac{\partial \ln Y}{\partial \ln x} \times \frac{Y}{x} = \varepsilon_{x}^{Y} \times \frac{Y}{x}$; and similarly for $f_{y} = \varepsilon_{y}^{Y} \times \frac{Y}{y}$ and $f_{z} = \varepsilon_{z}^{Y} \times \frac{Y}{z}$.

Again, for the second order derivatives $f_{xx} ,\,f_{xy} ,\, \ldots ,f_{zz}$in the matrix under (b) and (c) above need to be adjusted. For instance, $f_{xx}$ should be $\left[ {\beta_{11} + \left( {\varepsilon_{x}^{Y} } \right)^{2} \;- \;\varepsilon_{x}^{Y} } \right] \times \frac{Y}{{x^{2} }}$. Similarly, $f_{yy} = \left[ {\beta_{22} + \left( {\varepsilon_{y}^{Y} } \right)^{2} \;-\; \varepsilon_{y}^{Y} } \right] \times \frac{Y}{{y^{2} }}$ and $f_{zz} = \left[ {\beta_{33} + (\varepsilon_{z}^{Y} )^{2} - \varepsilon_{z}^{Y} } \right] \times \frac{Y}{{z^{2} }}$. Cross partials also need to be adjusted. In general, $f_{ij} = \left[ {\beta_{ij} + \varepsilon_{i}^{Y} \varepsilon_{j}^{Y} } \right] \times \frac{Y}{{x_{i} x_{j} }}\,\forall \,\,i \ne j;\,{\text{and}}\,\,\,i,j = 1,\,2,\,3$.