Pollution-Adjusted Productivity Changes: Extending the Färe–Primont Index with an Illustration with French Suckler Cow Farms

Abstract

Several approaches have been proposed in the literature to compute technical efficiency indices that account for bad outputs. The literature is, however, less rich when it comes to incorporating bad outputs in productivity indices. In this context, this article extends the multiplicatively complete Färe–Primont productivity index to a generalized version that considers pollution. This novel pollution-adjusted total factor productivity (TFP) is the ratio of an aggregated measure of good outputs to an aggregated measure of bad outputs and non-polluting inputs, in such a way that the materials balance principle is not distorted. A decomposition of this new pollution-adjusted TFP is proposed using the by-production. The latter implies considering a technology that lies at the intersection of two sub-technologies: one for the production of good outputs and the other for the generation of pollution. The pollution-adjusted TFP is further decomposed into pollution-adjusted technical change and efficiency change. The latter is further broken down into technical, scale, mix, and residual efficiency change components. A crucial advantage of the Färe–Primont index is the verification of the transitivity property that allows multi-temporal and multilateral comparisons. The generalized (pollution-adjusted) Färe–Primont TFP proposed here is illustrated with a sample of French suckler cow farms surveyed over the period of 1990 to 2013. The bad outputs considered are greenhouse gas emissions, namely methane (CH₄), carbon dioxide (CO₂), and nitrous oxide (N₂O). The results reveal a decrease in pollution-adjusted TFP by 5.57% during the period, due to technological regress of 2.23%, and to technical efficiency decrease of almost 3.34%.

Appendices

Appendix 1: Environmental Input Aggregator

We start by defining the environmental input technical efficiency for decision-making unit n as:

$$ {\displaystyle \begin{array}{c}{D}_{\mathrm{I}}^{\mathrm{E}}{\left({x}_n^t,{y}_n^t,{b}_n^t,t\right)}^{-1}=\mathrm{EIT}{\mathrm{E}}_n^t=\underset{\theta, \lambda, \mu, {x}_{\mathrm{M}}}{\min }{\theta}_n^t,\\ {}\mathrm{s}.\mathrm{t}.\kern0.75em {Y}^t{\lambda}^t\ge {y}_n^t,{X}_{\mathrm{M}}^t{\lambda}^t\le {x}_{n\mathrm{M}}^t,{X}_{\mathrm{S}}^t{\lambda}^t\le {\theta}_n^t{x}_{n\mathrm{S}}^t,{B}^t{\mu}^t\le {\theta}_n^t{b}_n^t,{X}_{\mathrm{M}}^t{\mu}^t\ge {x}_{n\mathrm{M}}^t,\\ {}{X}_{\mathrm{S}}^t{\mu}^t\le {\theta}_n^t{x}_{n\mathrm{S}}^t,{X}_{\mathrm{M}}^t{\lambda}^t={X}_{\mathrm{M}}^t{\mu}^t,{X}_{\mathrm{S}}^t{\lambda}^t={X}_{\mathrm{S}}^t{\mu}^t,{\lambda}^{\prime t}\mathbb{1}=1,{\mu}^{\prime t}\mathbb{1}=1,\kern0.75em {\lambda}^t,{\mu}^t\ge 0.\end{array}} $$

(33)

The dual of this model at the representative observation (x₀, y₀, b₀) can be written as follows:

$$ {\displaystyle \begin{array}{c}{\mathrm{D}}_{\mathrm{I}}^{\mathrm{E}}{\left({x}_0,{y}_0,{b}_0,{t}_0\right)}^{-1}=\underset{U,{V}_{\mathrm{S}},W,{Z}_{\mathrm{S}},{D}_{\mathrm{M}},{D}_{\mathrm{S}},\delta, \sigma }{\max }{U}^{\prime }{y}_0+\delta +\sigma, \\ {}\mathrm{s}.\mathrm{t}.\kern0.5em {U}^{\prime }Y-{V}_{\mathrm{S}}^{\prime }{X}_{\mathrm{S}}+{D}_{\mathrm{M}}^{\prime }{X}_{\mathrm{M}}+{D}_{\mathrm{S}}^{\prime }{X}_{\mathrm{S}}+\delta \le 0,\\ {}\begin{array}{c}-{W}^{\prime }B-{Z}_{\mathrm{S}}^{\prime }{X}_{\mathrm{S}}-{D}_{\mathrm{M}}^{\prime }{X}_{\mathrm{M}}-{D}_{\mathrm{S}}^{\prime }{X}_{\mathrm{S}}+\sigma \le 0,\\ {}{V}_{\mathrm{S}}^{\prime }{x}_{0\mathrm{S}}+{W}^{\prime }{b}_0+{Z}_{\mathrm{S}}^{\prime }{x}_{0\mathrm{S}}=1,\\ {}U,{V}_{\mathrm{S}},W,{Z}_{\mathrm{S}}\ge 0,\kern0.5em {D}_{\mathrm{M}},{D}_{\mathrm{S}},\delta, \sigma\ \mathrm{unrestricted}.\end{array}\end{array}} $$

(34)

This dual formulation can be rearranged into

$$ {\displaystyle \begin{array}{c}{D}_{\mathrm{I}}^{\mathrm{E}}{\left({x}_0,{y}_0,{b}_0,{t}_0\right)}^{-1}=\underset{U,{V}_{\mathrm{S}},W,{Z}_{\mathrm{S}},{D}_{\mathrm{M}},{D}_{\mathrm{S}},\delta, \sigma }{\max }{U}^{\prime }{y}_0+\delta +\sigma, \\ {}\mathrm{s}.\mathrm{t}.\kern0.5em {U}^{\prime }Y+{D}_{\mathrm{M}}^{\prime }{X}_{\mathrm{M}}-\left({V}_{\mathrm{S}}^{\prime }-{D}_{\mathrm{S}}^{\prime}\right){X}_{\mathrm{S}}+\delta \le 0,\\ {}\begin{array}{c}-{W}^{\prime }B-{D}_{\mathrm{M}}^{\prime }{X}_{\mathrm{M}}-\left({Z}_{\mathrm{S}}^{\prime }+{D}_{\mathrm{S}}^{\prime}\right){X}_{\mathrm{S}}+\sigma \le 0,\\ {}\left({V}_{\mathrm{S}}^{\prime }+{Z}_{\mathrm{S}}^{\prime}\right){x}_{0\mathrm{S}}+{W}^{\prime }{b}_0=1,\\ {}U,{V}_{\mathrm{S}},W,{Z}_{\mathrm{S}}\ge 0,\kern0.5em {D}_{\mathrm{M}},{D}_{\mathrm{S}},\delta, \sigma\ \mathrm{unrestricted}.\end{array}\end{array}} $$

(35)

The previous program is the linearization of the following fractional program

$$ {\mathrm{D}}_{\mathrm{I}}^{\mathrm{E}}\left({x}_0,{y}_0,{b}_0,{t}_0\right)=\mathcal{A}\left({X}_{0\mathrm{S}},{Y}_0\right)=\frac{\left({V}_{\mathrm{S}}^{\prime }+{Z}_{\mathrm{S}}^{\prime}\right){x}_{0\mathrm{S}}+{W}^{\prime }{b}_0}{U^{\prime }{y}_0+\delta +\sigma }. $$

(36)

From (36), we can then derive the cost-deflated shadow price associated with each input s and bad output as:

$$ \frac{\partial {\mathrm{D}}_{\mathrm{I}}^{\mathrm{E}}\left({x}_0,{y}_0,{b}_0,{t}_0\right)}{\partial {x}_{0\mathrm{S}}}={\mathcal{w}}_{0\mathrm{S}}=\frac{V_{\mathrm{S}}+{Z}_{\mathrm{S}}}{U^{\prime }{y}_0+\delta +\sigma }, $$

(37)

$$ \frac{\partial {\mathrm{D}}_{\mathrm{I}}^{\mathrm{E}}\left({x}_0,{y}_0,{b}_0,{t}_0\right)}{\partial {b}_0}={\mathfrak{R}}_0=\frac{W}{U^{\prime }{y}_0+\delta +\sigma }. $$

(38)

The aggregated environmental input of each decision-making unit n can be determined as:

$$ \mathcal{A}\left({X}_{n\mathrm{S}},{B}_n\right)={\mathcal{w}}_{0S}^{\prime }{x}_{n\mathrm{S}}+{\mathfrak{R}}_0^{\prime }{b}_n. $$

(39)

As in the output case, the non-zero shadow prices can be estimated using the full-dimensional efficient facets strategy and the reduced hyperplane equation in (23). The reduced form of the hyperplane associated to the input orientation is displayed in (40):

$$ {U}^{\prime }Y-{W}^{\prime }B-\left({V}_{\mathrm{S}}^{\prime }+{Z}_{\mathrm{S}}^{\prime}\right){X}_{\mathrm{S}}+\delta +\sigma =0. $$

(40)

Appendix 2: Productivity Changes and Decomposition

Table 2 Changes of pollution-adjusted productivity and components over the period 1990–2013: averages for the 49 farms

Table 3 Changes in classic productivity and components between 1990 and 2013: averages for the 49 farms