Theory and Methodology
Determination of the optimal externality: Efficiency versus equity

https://doi.org/10.1016/S0377-2217(97)00422-0Get rights and content

Abstract

The environmental economic problem of determining the optimal level of externality is analysed in this paper within a non-traditional framework. Thus, this classic problem is formulated as a bi-criteria model defined in the private benefits–external costs space. This change of scope, together with some results in multicriteria analysis, lead to new interpretations and non-conventional predictions with respect to the conclusions derived from the Pigovian and Coasian traditions.

Introduction

A basic problem in environmental economics is the following: to determine the optimal level of pollution (externality) produced by an externality generator (e.g. a polluter firm such as a pulp mill operating plant) in an externality sufferer (e.g. a sufferer firm such as a fish factory). The traditional solution to this problem, since the pioneer work by Pigou (1932)consists in defining the optimal externality as the amount of pollution corresponding to the level of economic activity by which the marginal benefit of the polluter firm equalises the marginal external cost of the sufferer firm. The diagrammatic presentation of this problem is shown in Fig. 1 following the classical devise proposed by Turvey (1963).

The optimal social level of economic activity is given by x* while the crosshatched area of Fig. 1 represents the optimal externality or socially optimal level of damage caused by the polluter firm (e.g. the pulp mill plant) to the sufferer firm (e.g. the fish factory). The reduction in the level of economic activity from the polluter's (private) optimum Xmax to the social optimum x* (mix of interests of polluter and sufferer) can be achieved roughly speaking by governmental intervention (i.e. “à la Pigou”) or by repairing the externality through a negotiation process as Coase (1960)has proposed. A good review of different methods within the Pigovian and Coasian approaches can be seen in Pearce and Turner (1990)chs. 4–10).

Despite its logical soundness, the traditional determination of the optimal level of externality is not exempt of difficulties. In fact, the implementation of the above approach requires a reliable representation of the utility function U(B, C) defined in the private benefits–external costs space. However as this function is virtually unknown, it is usually assumed to be a simple linear and additive structure such as U(B, C)=B − C (e.g. Varian, 1987, ch. 32) what represents a too restrictive treatment.

In this paper, it will be shown how the optimal externality can be defined with a broader perspective in scenario where the utility function U(B, C) is virtually unknown. To undertake this task, the framework where the optimal externality is determined needs to be formulated in a different way. Thus, our problem is not researched in the traditional space: marginal benefit/marginal external cost–level of economic activity but in the bi-criteria space: private benefits–external costs. This change of reference framework together with some recent results in the field of multiple criteria decision making (MCDM) will lead to interesting new results and interpretations of the old problem of determining the optimal externality.

The paper is organised as follows. First, the frontier or transformation curve in the private benefits–external costs space is established. Second – a compromise programming (CP) – model adapted to our environmental context is presented. Third a connection between CP and the optimum of the utility function is thoroughly analysed. Thus the maximum of the utility function defined in the private benefits–external costs space on the corresponding frontier or transformation curve is researched within a CP context. This connection provides a good analytical tool which allows us to approximate the optimum externality, even when the utility function is practically unknown – which is a fact in most scenarios. Moreover, it is shown how the solution obtained is much general than the optimal level of externality which is traditionally obtained.

It should be noticed that the proposed approach, as happens with the classic one, works well only for the case where sufferers are firms. However, when sufferers are populations, protected areas or endangered species, the straightforward application of the approach can be problematic.

Section snippets

Transformation curve and compromise sets

As a first step in our analysis let us determine the transformation curve or production possibility frontier in the private benefits–external costs space. We will undertake our task by referring the analysis to a perfectly competitive polluter firm although the basic analytical presented is extensible to other market structures. The benefit function B for the polluter or externality generator is given byB=Px−f(x),where P is the market price, x the amount of output produced and f(x) the cost

Economic interpretations of the compromise sets

In this section we are going to research conditons under which the optimal level of externality (social optimum) lies within the bounds of the compromise set. For that purpose we need to resort to some results obtained in the MCDM field. These results are basically two lemmas and one theorem proved elsewhere, which suitably adapted to our context are the following.

Lemma 1. The L1 bound (i.e. π=1) of the compromise set is the point where the straight line W2TB − W1TC=0 intercepts the

Analytical procedure

In this section the ideas presented previously will be operationalised. With that purpose let us consider – without loss of generality and only to simplify the presentation – linear functions for the marginal cost function as well as for the marginal external costs (i.e. functions f(x) and g(x) are quadratic). Thus, expressions (1) and (2) turn into:B=Px−μx2,C=βx2.

The transformation curve (4) or (5) now becomesB=β−0.5PC0.5−μβ−1C.

As it was explained in Section 2, the anchor and nadir values are

Some illustrative examples

Let us illustrate the logic and analytical study presented in the previous sections by resorting to some simple numerical examples. Let us assume that the polluter firm can sell its output in a perfectly competitive market at a price of 10 monetary units facing to a cost function such as: f(x)=0.5x2. The production process of the polluter firm generates a external cost in the sufferer firm ruled by the function C=0.25x2.

The traditional analysis can be summarised in the following way. The

Concluding remarks

When the traditional Pigovian problem of determining the optimal externality is approached like the search of a compromise between two conflicting interests: benefits of the polluter and of the sufferer some conclusions can be established:

(a) The usual optimal level of externality implies the optimisation of a linear and additive utility function U(B, C) with two arguments: polluter's benefits and external costs of the sufferer.

(b) This solution is a point of maximum efficiency providing the

Acknowledgements

Thanks are given to Ms. Christine Mendez for her English editing. Comments raised by Enrique Ballestero and Luis Dı́az-Balteiro (Technical University of Madrid) are appreciated. Thanks are also given to one referee for his helpful suggestions which have improved the presentation and accuracy of the paper. This research was supported by the Spanish “Comisión Interministerial de Ciencia y Tecnologı́a (CICYT)” under project AGF95-0014 and “Junta de Andalucı́a” (Research Group 2081).

References (20)

There are more references available in the full text version of this article.

Cited by (6)

  • Modeling fair air traffic assignment in the vicinity of airports

    2018, Transportation Research Part D: Transport and Environment
    Citation Excerpt :

    The problems considered are of different types (allocation, location, vehicle routing, scheduling, transportation network design, etc.) and there exist a multitude of approaches to handle equity concerns: Rawlsian (minimax) and its lexicographic extension, inequality index (range, mean deviation, variance, gini coefficient,…) and aggregation function (social welfare function or equitable efficiency). Several authors propose multiple criteria analysis where efficiency and equity are combined in the objective function (Romero, 1999; Kostreva et al., 2004; Kostreva and Ogryczak, 1999; Medernach and Sanlaville, 2012). For example, Hooker and Williams (2012) propose a social welfare function that combines efficiency and equity.

  • Supporting sustainable electricity technologies in Greece using MCDM

    2006, Resources Policy

  • Design of coordinated energy and environmental policies: Use of multi-criteria decision-making

    2004, Energy Policy

  • Multiple objective and subjective criteria evaluation technique (MOSCET): An application to material handling system selection

    2023, Optimization Techniques in Engineering: Advances and Applications

  • An Integrated Approach for Robot Selection Under Utopia Environment

    2023, Lecture Notes in Electrical Engineering

  • Towards a Circular Economy for Products: An analysis of EU’s policy and regulatory framework in an ecological perspective

    2022, SSRN
View full text
Copyright © 1999 Elsevier Science B.V. All rights reserved.