13
defined above. The upper bound which is placed on e in D is due to the fact that e is an output
and the law of mass conservation doesn't allow for an arbitrary expansion of a material output
in a production process with a limited (finite) amount of material input. This completes the
proof of proposition 2.
Proposition 2 constitutes an important step toward reconciling the use of the production func-
tion (1) with material-balance requirements. It shows (i) that if efficient abatement is presup-
posed, the output of the consumer good is uniquely determined by (e, £, m) and (ii) that the
implied production function satisfies the properties (Y) since it satisfies the properties (Y*).
Thus we confirmed that the properties (Y) are necessary for any production function to be
compatible with the technology (2). We also showed, however, that (2) imposes further con-
straints on the function Y from (11) concerning its domain and the derivative Ym . Both these
additional restrictions can be easily violated if production functions Y of type (1) are em-
ployed that exhibit the properties (Y) only. In particular, Ym∈ ] 0,Fm[ implies that popular
parametric production functions such as Cobb-Douglas functions don’t qualify for represent-
ing technologies of type (2).
In proposition 2 we didn't account for the residuals resulting from the abatement process.
Now we make up for this omission in
Proposition 3:
(i) Provided that the abatement activity is always kept at an efficient level, the produc-
tion-cum-abatement technology (2) is equivalent to three functions Y , R1, and R2 that
map (e, £,m)∈ D into Y(e, £,m) ,R1 (e, £,m) and R2 (e, £,m), where D and Y are
specified as in proposition 2, and where
ra 1 = R1 ( e, £ ,m ) : = A [£ - L ( e, £ ,m ) ,m - M ( e, £ ,m )]≥ 0, (13)
ra a2 = R2 ( e, £ ,m ) : = T ɛ R1 ( e, £ ,m ), £ - L ( e, £ ,m )J ≥ 0, and (14)
T : ( a, £ a ) → ma is a function such that ma = T ( a, £ a ), if and only if a = A ( £ a, ma ).
(ii) The functions R1 and R2 are concave. Its derivatives are indeterminate in sign except
for Re1 < 0 and Re2 < 0 .