A global Malmquist-Luenberger productivity index

Abstract

This paper introduces an alternative environmentally sensitive productivity growth index, which is circular and free from the infeasibility problem. In doing so, we integrated the concept of the global production possibility set and the directional distance function. Like the conventional Malmquist-Luenberger productivity index, it can also be decomposed into sources of productivity growth. The suggested index is employed in analyzing 26 OECD countries for the period 1990–2003. We also employed the conventional Malmquist-Luenberger productivity index, the global Malmquist productivity index and the conventional Malmquist productivity index for comparative purposes in this empirical investigation.

Appendix

Proposition 1

The GML index and its decomposed components are circular.

Proof

In order to save space, we substitute input and output vector in a parenthesis of a distance function with a time index. For example, D ^t(x ^t, y ^t, b ^t) is equivalent to D ^t(t). Then,

$$ \begin{aligned} \hbox{GML}^{t, t+1} \times \hbox{GML}^{t+1, t+2} &= {\frac{1 + {\bf D}^{G}(t)} {1 +{\bf D}^{G}(t+1)}} \times {\frac{1 + {\bf D}^{G}(t+1)}{1 + {\bf D}^{G}(t+2)}}= {\frac{1 + {\bf D}^{G}(t)}{1 + {\bf D}^{G}(t+2)}} = \hbox{GML}^{t, t+2}.\\ \hbox{EC}^{t,t+1} \times \hbox{EC}^{t+1,t+2} &= {\frac{1 + {\bf D}^{t}(t)} {1 +{\bf D}^{t+1}(t+1)}} \times {\frac{1 + {\bf D}^{t+1}(t+1)}{1 +{\bf D}^{t+2}(t+2)}}= {\frac{1 + {\bf D}^{t}(t)}{1 + {\bf D}^{t+2}(t+2)}} = \hbox{EC}^{t, t+2}.\\ \hbox{BPC}^{t, t+1} \times \hbox{BPC}^{t+1, t+2} &= {\frac{\frac{1 + {\bf D}^{G}(t)}{1 + {\bf D}^{t}(t)}}{\frac{1 +{\bf D}^{G}(t+1)}{1 + {\bf D}^{t+1}(t+1)}}}\times {\frac{\frac{1 +{\bf D}^{G}(t+1)}{1 + {\bf D}^{t+1}(t+1)}}{\frac{1 +{\bf D}^{G}(t+2)}{1 + {\bf D}^{t+2}(t+2)}}}= {\frac{\frac{1 +{\bf D}^{G}(t)}{1 + {\bf D}^{t}(t)}}{\frac{1 +{\bf D}^{G}(t+2)}{1 + {\bf D}^{t+2}(t+2)}}}= \hbox{BPC}^{t, t+2}. \end{aligned} $$

 $\square$

Proposition 2

If undesirable outputs are excluded from directional distance functions andg = g _y = y, then GML ≡ GM and ML ≡ M.

Proof

PPS is redefined as follows.

$$ \Uppsi ({\bf x}) = \{ {\bf y} \vert {\bf x}\,\hbox{can}\,\hbox{produce}\,{\bf y} \} $$

(11)

Then

$$ {\bf D}({\bf x}, {\bf y}; {\bf y}) = \max \{\beta \vert ({\bf x}, {\bf y} + \beta {\bf y}) \in \Uppsi ({\bf x}) \}. $$

(12)

Then D(x, y | y) = 1/D(x, y) − 1, where $D({\bf x}, {\bf y}) =\min\{\delta \vert{\bf y}/{\delta} \in \Uppsi({\bf x}) \}$ is an output-oriented distance function. This relationship can be shown as follows:

$$ \begin{aligned} {\bf D}({\bf x}, {\bf y} \vert {\bf y}) & = \max\{\beta \vert ({\bf y} + \beta {\bf y}) \in \Uppsi ({\bf x}) \}\\ & = \max \{\beta \vert D({\bf x}, (1+ \beta){\bf y}) \leq 1 \}\\ & = \max \{\beta \vert (1+\beta)D({\bf x}, {\bf y}) \leq 1 \}\\ & = \max \left\{\beta \vert \beta \leq {\frac{1} {D({\bf x}, {\bf y})}} - 1 \right\}\\ & = {\frac{1} {D({\bf x}, {\bf y})}} - 1. \end{aligned} $$

(13)

Then GML ≡ GM since

$$ \begin{aligned} \hbox{GML}^{t, t+1} ({\bf x}^t, {\bf y}^t, {\bf x}^{t+1}, {\bf y}^{t+1})&= {\frac{1 + {\bf D}^{G}({\bf x}^{t}, {\bf y}^{t}, {\bf b}^{t})}{1 + {\bf D}^{G}({\bf x}^{t+1}, {\bf y}^{t+1}, {\bf b}^{t+1})}}\\ & = {\frac{1 + \left(-1 + {\frac{1}{D^G({\bf x}^t, {\bf y}^t)}} \right)}{1 + \left(-1 + {\frac{1}{D^G({\bf x}^{t+1}, {\bf y}^{t+1})}} \right)}}\\ & = {\frac{D^G({\bf x}^{t+1}, {\bf y}^{t+1})} {D^G({\bf x}^t, {\bf y}^t)}}\\ & = GM^{t,t+1}({\bf x}^t, {\bf y}^t, {\bf x}^{t+1}, {\bf y}^{t+1}). \end{aligned} $$

(14)

The relationship between the ML and the M indices is trivial. $\square$