Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

the materials-balance principle is the existence of at least one more output, say a production
residual r_y , to fill the materials-balance gap:

ry ⁺ F ( ^my^{, £} y ) = my      (^and ry ≥ ⁰⁾.

In other words, bringing the conventional non-linear production function y = F ( £ _y,m_y ) in
line with the materials-balance principle requires to look at production as transforming the
material input, m_y , into two distinct outputs: the consumer good, y, whose generation is the
purpose of the activity, and some production residuals, considered unwanted and environmen-
tally harmful. From r_y = m_y - F (m_y, £ _y ) follows (dry / dm_y ) = 1 - F_m which is positive due to
the assumption F_m ∈ ] 0,1[ .

At this point, production and abatement need to be linked up. Without any abatement, the
total amount of production residuals generated, r_y = m_y - F (m_y, £_y ), would be discharged
into the environment. But one can also hold back part of the production residuals from dis-
charging, say the amount a, for abatement such that only the amount e= r_y -a≥0 .

To sum up, the combined technologies of production and abatement are given by

As discussed in the introduction, environmental economists have always been serious about
joint production of wanted and unwanted outputs, about environmental damage caused by the
emission of the latter and about residuals abatement to reduce emissions. But rather than fo-
cusing on comprehensive production-cum-abatement technologies such as (2), many of them
used to employ production functions of type (1), i. e. the functional form y = Y (e, £,m),
where y, e, £ and m are defined in the same way as in (2).

The comparison between (1) and (2) readily confirms that if the production function of type
(1) is at all compatible with the production-cum-abatement technology (2) it is an incomplete

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