# Bayesian Approach to Disentangling Technical and Environmental Productivity

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}and NO

_{x}gases.

_{2}than in the emission of NO

_{x}gases. We also document a significant divergence between the electric-power-oriented technical productivity and the emission-oriented environmental productivity. Specifically, we find that, while the pooled posterior mean estimate of (annual) productivity growth is negative for electric power generation (–0.13%), it is non-negligibly positive for the SO

_{2}and NO

_{x}emissions:2 2.25% and 3.31% per annum, respectively. The cumulative eleven-year growth is 23.26% for the SO

_{2}-oriented environmental productivity, 37.98% for the NO

_{x}-oriented environmental producitivity and a mere 5.33% for the electric-power-oriented technical productivity.

## 2. The By-Production Model

**X**), linear homogeneity in Y and concavity in

**X**and Y. Similarly, the residual generation function ${H}_{p}(\xb7)$ is continuous, positively (negatively) monotonic in ${B}_{p}$ (Y), linearly homogeneous in ${B}_{p}$ and convex in Y and ${B}_{p}$.

#### Technical and Environmental Productivity

## 3. Data

_{2}(sulfur dioxide) gas emissions ${B}_{1}$ and (ii) the NO

_{x}(nitrogen oxides) gas emissions ${B}_{2}$, both measured in short-tons. The three inputs to the production are (i) the real stock of physical capital ${X}_{1}$, constructed from historical cost of plant data and deflated to constant dollars using the Handy-Whitman Index; (ii) labor ${X}_{2}$, measured in the number of employees; and (iii) energy ${X}_{3}$, i.e., the heat content of coal, oil and natural gas consumption, measured in Btu.

## 4. Econometric Strategy

#### 4.1. Priors

#### 4.2. Posterior Distribution

#### 4.3. Imposition of Restrictions

(1) | (2) | |
---|---|---|

median RNE | 0.113 | 0.615 |

median NSE | 0.0010 | 0.0012 |

draws to convergence | 150,000 | 70,000 |

median ACF at lag 50 | 0.977 | 0.312 |

#### 4.4. Improving Performance of MCMC

**V**, i.e.,

#### 4.5. Random Effects

${\gamma}_{0,i}$ | ${\delta}_{0,i}$ | |
---|---|---|

${\alpha}_{0,i}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}$ | 0.831 (0.0011) | 0.630 (0.0130) |

${\gamma}_{0,i}$ | 0.601 (0.0102) |

## 5. Results

_{2}and NO

_{x}emissions (undesirable outputs) with respect to the net generated electric power (desirable output).8 In particular, the reported input elasticity estimates imply a posterior mean estimate of the returns to scale, defined as the sum of input elasticities, of 0.90, which suggests that, on average, electric utilities operated at decreasing returns to scale during our sample period.

_{2}and NO

_{x}emissions. It is intuitive to interpret these estimates as “shadow prices” (in the elasticity form) of the power generation. The posterior mean estimates of the two shadow prices are 1.09 and 1.13. The latter implies that, on average, an increase in the net power generation by 1% requires a simultaneous increase in the SO

_{2}and NO

_{x}emissions by at least 1.09% and 1.13%, respectively. Note that emissions may increase by even more if the firm is not on the residual generating frontier, i.e., environmentally inefficient.

_{x}-oriented environmental inefficiency is noticeably skewed to the right and the distribution of the SO

_{2}-oriented environmental inefficiency exhibits apparent bi-modality. There may be many reasons for such a stark difference between the levels of technical and environmental inefficiencies across utilities. One plausible explanation is that technical inefficiency may also be capturing declines in the desirable output due to unforeseen fluctuations in the demand for electric power. Since inputs often cannot be immediately adjusted/reallocated and electric power is not easily storable, electric plants may be forced to under-utilize their facilities and labor, which our model would detect and classify as technical underperformance (inefficiency) relative to the frontier. However, such a demand uncertainty would not apply to the by-production of undesirable SO

_{2}and NO

_{x}gases given the exact physical relationship between the power generation and the associated emission of pollutant gases. The latter is also capable of at least partly explaining why environmental inefficiency (in the emission of both SO

_{2}and NO

_{x}gases) appears to be relatively more stable over time unlike the electric-power-oriented technical inefficiency, which we discuss in more detail later in the paper.

Mean | Median | St.Dev. | 95% Credible Interval | |
---|---|---|---|---|

Elasticity | ||||

Capital Elasticity | 0.2985 | 0.2984 | 0.0505 | (0.1992; 0.3959) |

Labor Elasticity | 0.4032 | 0.4043 | 0.0482 | (0.3076; 0.4935) |

Energy Elasticity | 0.2002 | 0.1998 | 0.0103 | (0.1801; 0.2205) |

RTS | 0.9018 | 0.9032 | 0.0726 | (0.7608; 1.0370) |

SO_{2} Shadow Price | 1.0873 | 1.0664 | 0.1437 | (0.8524; 1.4334) |

NO_{x} Shadow Price | 1.1275 | 1.1163 | 0.1161 | (0.9366; 1.3988) |

Inefficiency | ||||

Tech. Ineff. | 0.0905 | 0.0915 | 0.0257 | (0.0361; 0.1390) |

SO_{2} Env. Ineff. | 0.0870 | 0.0875 | 0.0351 | (0.0254; 0.1504) |

NO_{x} Env. Ineff. | 0.0458 | 0.0438 | 0.0156 | (0.0186; 0.0798) |

Efficiency Change | ||||

TEC | 0.0029 | 0.0022 | 0.0308 | (–0.0575; 0.0647) |

SO_{2} EEC | –0.0000 | –0.0004 | 0.0099 | (–0.0133; 0.0207) |

NO_{x} EEC | 0.0000 | –0.0004 | 0.0038 | (–0.0050; 0.0107) |

Technological Change | ||||

TTC | –0.0042 | –0.0027 | 0.0104 | (–0.0272; 0.0117) |

SO_{2} ETC | 0.0225 | 0.0224 | 0.0110 | (0.0024; 0.0446) |

NO_{x} ETC | 0.0331 | 0.0332 | 0.0052 | (0.0229; 0.0431) |

Productivity Growth | ||||

TPG | –0.0013 | 0.0009 | 0.0323 | (–0.0663; 0.0626) |

SO_{2} EPG | 0.0225 | 0.0220 | 0.0149 | (–0.0021; 0.0495) |

NO_{x} EPG | 0.0331 | 0.0330 | 0.0064 | (0.0211; 0.0453) |

_{2}than in the emission of NO

_{x}gases. These differences may be reflective of varying degree of strictness of environmental regulations (or the degree of their enforceability) for different pollutants across states. For instance, loose regulations for the SO

_{2}emissions could potentially explain why the SO

_{2}-oriented environmental inefficiency is considerably higher on average and is more dispersedly distributed than the NO

_{x}-oriented inefficiency. Lastly, we document little correlation between the two environmental inefficiencies as well as between the environmental inefficiencies and the technical inefficiency. Table 4 reports such Spearman rank correlation coefficients for the posterior inefficiency estimates. For instance, there appears to be virtually no relationship between the electric-power-oriented technical inefficiency and the SO

_{2}-oriented environmental inefficiency exhibited by utilities.

Inefficiency | Efficiency Change | ||||||

Tech. Ineff | 1.000 | TEC | 1.000 | ||||

SO_{2} Env. Ineff | 0.080 | 1.000 | SO_{2} EEC | –0.059 | 1.000 | ||

NO_{x} Env. Ineff | 0.112 | 0.227 | 1.000 | NO_{x} EEC | 0.080 | 0.139 | 1.000 |

Technological Change | Productivity Growth | ||||||

TTC | 1.000 | TPG | 1.000 | ||||

SO_{2} ETC | –0.001 | 1.000 | SO_{2} EPG | –0.024 | 1.000 | ||

NO_{x} ETC | –0.020 | 0.045 | 1.000 | NO_{x} EPG | –0.049 | 0.065 | 1.000 |

_{2}$EEC$ and the NO

_{x}$EEC$ are virtually zero. The left panel of Figure 2, which depicts box-and-whiskers plots of the distributions of technical and environmental efficiency change estimates for each year in our sample, confirms the relative stability of environmental efficiency levels (see the left panel of the figure). The electric-power-oriented (i.e., desirable-output-oriented) technical efficiency is however less stable over the course of the years. The mean estimates of $TEC$ are predominantly positive across utilities in 1986, 1988 and 1995, whereas a significant decrease in efficiency is documented for 1987 and 1994. However, the mean posterior estimate of $TEC$ pooled over the entire sample is a mere 0.29% (also see Table 3).

_{2}and NO

_{x}emissions (undesirable outputs) are staggering 2.25% and 3.31% per annum, respectively. Second, we find that technological change is fairly stable across the years in all dimensions, be it the intended production of electric power or undesirable by-production of emission gases. The right panel of Figure 2 confirms this observation: the distributions of $TTC$ and the two $ETC$ do not change much over the course of the years.

_{2}and NO

_{x}emissions: annual 2.25% and 3.31%, respectively. In other words, keeping input quantities constant, the net electric power generation, on average, fell by 0.13% per year during our sample period. Utilities however did a significantly better job in terms of cutting the emission of SO

_{2}and NO

_{x}gases for any fixed quantity of the net electric power generated: on average, emissions fell by respective 2.25% and 3.31% per year ceteris paribus. This disconnect between technical and environmental productivities of electric plants in our sample is also confirmed by virtually zero rank correlation coefficients between $TPG$ and $EPG$ for the SO

_{2}and NO

_{x}emissions (see Table 4).

_{2}-oriented $EPG$, 37.98% for the NO

_{x}-oriented $EPG$ and a mere 5.33% for the electric-power-oriented $TPG$.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- S.E. Atkinson, and J.H. Dorfman. “Bayesian measurement of productivity and efficiency in the presence of undesirable outputs: Crediting electric utilities for reducing air polution.” J. Econ. 126 (2005): 445–468. [Google Scholar] [CrossRef]
- S.E. Atkinson, and E.G. Tsionas. “Directional distance functions: Optimal endogenous directions.” J. Econ., 2015, in press. [Google Scholar] [CrossRef]
- S. Reinhard, C.A.K. Lovell, and G. Thijssen. “Econometric estimation of technical and environmental efficiency: An application to Dutch dairy farms.” Am. J. Agric. Econ. 81 (1999): 44–60. [Google Scholar] [CrossRef]
- S. Reinhard, C.A.K. Lovell, and G. Thijssen. “Environmental efficiency with multiple environmentally detrimental variables; estimated with SFA and DEA.” Eur. J. Oper. Res. 121 (2000): 287–303. [Google Scholar] [CrossRef]
- A. Hailu, and T.S. Veeman. “Non-parametric productivity analysis with undesirable outputs: An application to the Canadian pulp and paper industry.” Am. J. Agric. Econ. 83 (2001): 605–616. [Google Scholar] [CrossRef]
- R. Färe, S. Grosskopf, D.W. Noh, and W. Weber. “Characteristics of a polluting technology: Theory and practice.” J. Econ. 126 (2005): 469–492. [Google Scholar] [CrossRef]
- S. Murty, R.R. Russell, and S.B. Levkoff. “On modeling pollution-generating technologies.” J. Environ. Econ. Manag. 64 (2012): 117–135. [Google Scholar] [CrossRef]
- Y. Chung, R. Färe, and S. Grosskopf. “Productivity and undesirable outputs: A directional distance function approach.” J. Environ. Manag. 51 (1997): 229–240. [Google Scholar] [CrossRef]
- G. Feng, and A. Serletis. “Undesirable outputs and a primal Divisia productivity index based on the directional output distance function.” J. Econ. 183 (2014): 135–146. [Google Scholar] [CrossRef]
- R. Färe, and S. Grosskopf. “Directional distance functions and slacks-based measures of efficiency.” Eur. J. Oper. Res. 200 (2010): 320–322. [Google Scholar] [CrossRef]
- C. Fernández, G. Koop, and M.F.J. Steel. “Multiple-output production with undesirable outputs: An application to Nitrogen Surplus in Agriculture.” J. Am. Stat. Assoc. 97 (2002): 432–442. [Google Scholar] [CrossRef][Green Version]
- C. Fernández, G. Koop, and M.F.J. Steel. “Alternative efficiency measures for multiple-output production.” J. Econ. 126 (2005): 411–444. [Google Scholar] [CrossRef][Green Version]
- F. Forsund. “Good modelling of bad outputs: Pollution and multiple-output production.” Int. Rev. Environ. Resour. Econ. 3 (2009): 1–38. [Google Scholar] [CrossRef]
- R.M. Solow. “Technical change and the aggregate production function.” Rev. Econ. Stat. 39 (1957): 312–320. [Google Scholar] [CrossRef]
- W.A. Barnett, J. Geweke, and M. Wolfe. “Seminonparametric Bayesian estimation of the Asymptotically Ideal Production Model.” J. Econ. 49 (1991): 5–50. [Google Scholar] [CrossRef]
- W.A. Barnett. “Tastes and technology: Curvature is not sufficient for regularity.” J. Econ. 108 (2002): 199–202. [Google Scholar] [CrossRef]
- C.A. Pasurka. “Decomposing electric power plan emissions within a joint production framework.” Energy Econ. 28 (2006): 26–43. [Google Scholar] [CrossRef]
- C.J. O’Donnell, and T.J. Coelli. “A Bayesian approach to imposing curvature on distance functions.” J. Econ. 126 (2005): 493–523. [Google Scholar] [CrossRef]
- G. Koop, and M. Steel. “Bayesian analysis of stochastic frontier models.” In A Companion to Theoretical Econometrics. Edited by B. Baltagi. Malden, MA, USA: Blackwell Publishing Ltd., 2001. [Google Scholar]
- J.O. Berger, and J. Mortera. “Default Bayes factors for nonnested hypothesis testing.” J. Am. Stat. Assoc. 94 (1999): 542–554. [Google Scholar] [CrossRef]
- J. Mulder, H. Hoijtink, and I. Klugkist. “Equality and inequality constrained multivariate linear models: Objective model selection using constrained posterior priors.” J. Stat. Plan. Inference 140 (2010): 887–906. [Google Scholar] [CrossRef]
- J.O. Berger, and L.R. Pericchi. “The intrinsic Bayes factor for model selection and prediction.” J. Am. Stat. Assoc. 91 (1996): 109–122. [Google Scholar] [CrossRef]
- J. Geweke. “Evaluating the accuracy of sampling-based approaches to calculating posterior moments.” In Bayesian Statistics 4. Edited by J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith. Oxford, UK: Clarendon Press, 1992. [Google Scholar]
- E.G. Tsionas, and S.C. Kumbhakar. “Firm heterogeneity, persistent and transient technical inefficiency: A generalized true random-effects model.” J. Appl. Econ. 29 (2014): 110–132. [Google Scholar] [CrossRef][Green Version]
- I. Verdinelli, and L. Wasserman. “Computing Bayes factors by using a generalization of the Savage-Dickey density ratio.” J. Am. Stat. Assoc. 90 (1995): 614–618. [Google Scholar] [CrossRef]

^{1}Färe and Grosskopf [10] have recently proposed the slacks-based directional distance function which allows inefficiency to be input- and output-specific. The estimation of such slacks-based inefficiencies however is feasible under the deterministic treatment of the production technology only. In this paper, we focus on the econometric estimation of stochastic production technologies that accommodate random disturbances.^{2}Implying a ceteris paribus contraction in these emissions.^{4}Recall that $f(\xb7)={\left[F(\xb7,1)\right]}^{-1}$ and ${h}_{p}(\xb7)={\left[{H}_{p}(\xb7,1)\right]}^{-1}$. Hence, negative monotonicity of $F(\xb7)$ and ${H}_{p}(\xb7)$ imply positive monotonicity of $f(\xb7)$ and ${h}_{p}(\xb7)$.^{5}Recall that the quantity of undesirable outputs does down as desirable outputs decrease due to the complementarity of the two types of outputs.^{6}For a similar stochastic formulation, e.g., see Koop and Steel [19].^{7}For alternative ways to reduce this correlation in multiple random-effect models, see Tsionas and Kumbhkar [24].^{8}We also reestimate our model with no theoretical regularity constraints imposed. Consistent with one’s expectations, the unconstrained metrics generally have larger credible intervals. However, since the unconstrained estimates violate regularity conditions dictated by economic theory and thus have no meaningful economic interpretation, we do not report them here.

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Malikov, E.; Kumbhakar, S.C.; Tsionas, E.G.
Bayesian Approach to Disentangling Technical and Environmental Productivity. *Econometrics* **2015**, *3*, 443-465.
https://doi.org/10.3390/econometrics3020443

**AMA Style**

Malikov E, Kumbhakar SC, Tsionas EG.
Bayesian Approach to Disentangling Technical and Environmental Productivity. *Econometrics*. 2015; 3(2):443-465.
https://doi.org/10.3390/econometrics3020443

**Chicago/Turabian Style**

Malikov, Emir, Subal C. Kumbhakar, and Efthymios G. Tsionas.
2015. "Bayesian Approach to Disentangling Technical and Environmental Productivity" *Econometrics* 3, no. 2: 443-465.
https://doi.org/10.3390/econometrics3020443