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 rc of a post-consumption residual. Both are equal in weight,
y=rc . (16)
Thus we now deal with four different types of residuals: e, ra1, ra2 and rc , 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,ra1,ra2,rc) with X (0, 0, 0, 0) = 0 (17)
+≥0≥0≥0