Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

15

Yet the characterization of the relationship between (1) and (2) is still incomplete from a theo-
retical point of view. One may want to know which the comprehensive set of conditions is
that must be satisfied by the functions Y, R¹ and R² from (15) to secure that these functions
are equivalent to some technology (2). Unfortunately there is little hope to make progress to-
ward this end because the properties of Y, R¹ and R² from (15) are made up of a complex
mix of first and higher-order derivatives of the functions A and F from (2). The basic produc-
tion functions A and F determine the curvature of Y, R¹ and R² in a very complex and inter-
dependent way involving a constrained maximization procedure (efficient abatement).

It is true that without a complete set of conditions we cannot decide whether a given conven-
tional production function (1) is compatible with the comprehensive technology (2) or not.
Yet the value of such a result is not so clear. The information provided in propositions 2 and 3
appears to be sufficient for most theoretical modeling exercises, since in those studies first
and higher-order derivatives are usually not quantitatively (let alone numerically) specified.
For applied research, it is hardly appropriate to start out with some ‘arbitrary’ function Y satis-
fying the properties (Y*). One would rather have to start with the empirically valid specifica-
tion of the true technology (2) anyway, since (11), (13) and (14) are and always will be de-
rived from the true empirically observable technology (2).

3. Allocative efficiency and materials balance in an economy with production, abate-
ment and pollution

We now envisage a simple economy where an (aggregate) firm applies technology (2). The
consumption of the only wanted output y is modeled as a process of material transformation
like production and abatement: it consists in turning the amount y of the consumer good into
the amount r_c of a post-consumption residual. Both are equal in weight,

y=r_c .                                                                                      (16)

Thus we now deal with four different types of residuals: e, r_a1, r_a2 and r_c , each of which has
the potential to degrade the environment after having been discharged. Denote by x an index
of the ambient concentration of pollutants, called pollution for short. We define

x=X(e,r_a1,r_a2,r_c) with X (0, 0, 0, 0) = 0                                         (17)

+≥0≥0≥0

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