Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

Maintaining the standard technological assumptions of concavity, non-linearity,⁵ and smooth
factor substitution, the proper regard of materials-balance requirements will make it necessary
to also account for residuals other than the production residuals represented by the variable e
in (1). We will also demonstrate that these additional interdependent residuals do not render
incorrect the conventional analysis of pollution control based on (1) if and only if their emis-
sion doesn't contribute to environmental degradation. Insofar we provide a rigorous rationale
and justification for conventional model building. However, if the emitted production residu-
als, e, are not the only pollutants, the conditions determining allocative efficiency will be
shown to differ markedly from those derived in conventional analysis. In that case, the con-
ventional marginal cost of abating production residuals, Y_e , will turn out to deviate from the
social marginal cost of abating these residuals because we deal with a pollution problem in-
volving multiple and interdependent pollutants. This finding will be shown to have non-trivial
implications for efficiency-restoring tax schemes.

Section 2 introduces a comprehensive technology of production and residuals abatement
based on the materials-balance principle, and we will rigorously derive the production func-
tion (1) as a proper though incomplete technological subsystem of the comprehensive produc-
tion-cum-abatement technology. Moreover, the entire comprehensive production-cum-
abatement technology will be shown to be completely represented by (1) and two further pro-
duction functions mapping the domain of (1) into the abatement residuals. In Section 3 we
will incorporate the comprehensive production-cum-abatement technology developed in Sec-
tion 2 into a simple economy subject to pollution, and we will derive the pertaining rules for
an efficient allocation. If residuals other than (unabated) production residuals also cause pol-
lution, the optimality rules become complex, since all these pollutants are generated in strict
technological interdependence. Section 4 explores the consequences of that interdependence
for the design of efficiency-restoring tax schemes. Taking the conventional Pigouvian tax rule
as a benchmark we show that if residuals other than production residuals contribute to pollu-
tion in addition to the latter, it is not efficient, in general, to set the tax on production residuals
equal to the conventional marginal abatement cost, Y_e .

⁵ Ayres and Kneese (1969) developed their 'materials balance approach' in a model with strictly linear produc-
tion processes. That makes it quite easy to keep track of material balance but fails to account for realistic substi-
tution and transformation possibilities. See also Pethig (2003).

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