Uncertainty in optimal pollution levels: modelling and evaluating the benefit area

Abstract

This paper identifies the optimal pollution level under the assumptions of linear, quadratic and exponential damage and abatement cost functions and investigates analytically the certain restrictions that the existence of this optimal level requires. The evaluation of the benefit area is discussed and the mathematical formulation provides the appropriate methods for that to be calculated. The positive, at least from a theoretical point of view, is that both the quadratic and the exponential case obey the same form for evaluating the benefit area. These benefit area estimations can be used as indexes between different rival policies, and depending on the environmental problem, the policy that produces the maximum area will be the beneficial policy.

Keywords:

• benefit area
• damage cost
• abatement cost
• pollution

Acknowledgments

This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program ‘Education and Lifelong Learning’ of the National Strategic Reference Framework (NSRF) ‒ Research Funding Program: Heracleitus II Investing in knowledge society through the European Social Fund. We confirm that the above-mentioned funding sources did not have any involvement in the writing of the paper and the decision to submit the paper for publication. The authors would like to thank Professor Ken Willis and two anonymous reviewers for their helpful and constructive comments on an earlier draft of this paper. Any remaining errors are solely the authors’ responsibility.

Notes

1. Rabl and Holland (2008) illustrated an impact pathway framework for analysing external costs of environmental burdens, together with the inventory stage of life cycle assessment.

2. The cost functions may not behave well or may not satisfy the conditions of convexity or concavity. In the case of the damage cost function this may take place by threshold effects as well as by any irreversibility where pollution reaches a critical point at which the receptor (rivers, lakes, etc.) is damaged completely and cannot sustain any life. If one or both of the cost functions are not well behaved then our results will be different. At the same time, the distinction between flow and stock pollutants is important because for stock pollutants the persistence has to be taken into consideration due to the accumulation (and decay) of pollutant(s) in time (Perman et al. 2011). As an example, we may consider the case of F-Gases with the very high global warming potentials (Halkos 2013).

3. This is clear as if it is assumed that α<β₀ there is no intersection (no benefit area) and if we let α = β₀ the benefit area coincides with the point, namely A = B = I, that a one point area is created.

4. The numerically evaluated root is under an error ϵ, say. Therefore the values of andare approximated values.

5. Practically that results in the value of the difference MAC(z₀)-MD(z₀) is not zero, but close to zero, with a certain accuracy ±ζ, say ζ = 10⁻³ or 10⁻⁶.

6. The graphical presentation of this case is similar to Figure 6.

7. In such a case there are ‘fast’ increases/decreases of both marginal abatement and marginal damage costs, indicating possibly for the former the existence of limited and expensive control methods, and for the latter very high instant damages.

8. Equations were fitted across the range 5-55% of maximum feasible abatement. The estimated coefficients of both specifications were statistically significant in all cases with only exception in the estimate of β₁ in the quadratic specification of Spain.

9. Rabl, Spadaro, and van der Zwaan (2005) approximated the damage function by a linear function of the pollution emissions and they claimed that linearity is found to be appropriate approximation in the case of PM, SO₂ and NO_X emissions, while for CO₂ linearity is probably acceptable for emissions reductions in the ‘foreseeable’ future period.

10. Damage is a function of deposits, which depend on the transfer coefficients [d_ij] matrix as explained before. In the more complex case of both quadratic abatement and damage cost functions the total cost function may be expressed as:

TC_i = [β_0i + β_1i SR_i + β_2i SR_i²] + [γ_0i + γ_1i D_i + γ_2i D_i²] I = 1,2,…,n

To ‘calibrate’ the damage function, we assume that national authorities act independently (as Nash partners in a non-cooperative game with the rest of the world), taking as given deposits originating in the rest of the world. Specifically, we minimise TC_i with respect to SR_i and we calibrate the damage function by taking the first order conditions (for more details see Hutton and Halkos 1995, 265).

11. Estimates of c₀ were derived by assuming countries act in a Nash behaviour.

12. The empirical results presented are indicative and very sensitive to the assumptions of calibration.

13. Following Halkos and Kitsos (2005) the efficiency (Eff) of the benefit area, in comparison with the maximum evaluated from the sample of countries under investigation, can be estimated using as measure of efficiency the expression: Eff = *100

Table 2. Coefficient estimates in the case of quadratic MD and MAC functions.

Table 3. Calculated ‘calibrated’ Benefit Areas (BA^c)

This efficiency is evaluated for the same class of model, referring to different data sets in each case.

14. Germany dominates the picture in Europe as it has a very high initial abatement level (≈42%) and its calibrated damage function ensures high abatement levels (Hutton and Halkos 1995). For this reason the efficiency index was constructed on the second highest benefit area (former Czechoslovakia).