Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

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 Y_m . 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, Y_m∈ ] 0,F_m[ 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

r_{a 1} = R¹ ( e, £ ,m ) : = A [£ - L ( e, £ ,m ) ,m - M ( e, £ ,m )]≥ 0,                         (13)

r_a a2 = R² ( e, £ ,m ) : = T ɛ R¹ ( e, £ ,m ), £ - L ( e, £ ,m )J ≥ 0, and                         (14)

T : ( a, £ _a ) → m_a is a function such that m_a = T ( a, £ _a ), if and only if a = A ( £ _a, m_a ).

(ii) The functions R¹ and R² are concave. Its derivatives are indeterminate in sign except
for R_e¹ < 0 and R_e² < 0 .

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