Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

14

First we specify the functions R¹ and R² from (13) and (14), respectively. As for the abate-
ment residuals of type 1, we combine (2c), (2e) - (2g), (8) and (10) to obtain (13). Since
r_a2 = ma due to (2h), we obtain r_a = T(a, £_a ) (with T as defined in proposition 3(i)). When
combined with (9), (13), (2e) and (2g), this equation yields (14). Concavity of the functions
R¹ and R² follows immediately from the convexity of the set Ω established in the proof of
proposition 1(i).

To see how the generation of abatement residuals reacts on the emission of production residu-
als we take the partial derivatives

R1 = -( A_lLe + AmMe ) and R2 = -[TaA_mMe +( TaA_l + T ) Le ] = - M_e.

While R_e¹ < 0 is obvious, R_e² =-M_e <0 is explained by observing that total differentiation of
1A

a = A (£ ,m ) yields dm = —da---d£ and that in view of the definition of function T

AmAm

above we clearly have T_a = (1/ A_m ) and T_£ =-( A_£/A_m ). The signs of the partial derivatives
R_e¹ and R_e² are as expected: If the scale of abatement is stepped up (such that the emissions e
are reduced) more abatement residuals are generated and vice versa. Checking the remaining
partial derivatives of (13) and (14) reveals that their sign is ambiguous. This indeterminacy
demonstrates that the impact of £ and m on r_a1 and r_a2 is quite complex in spite of the simple
hypotheses (2g) and (2h).

In view of the propositions 2 and 3 the technology (2) is completely and compactly described
by

y = Y (e, £,m), r1 = R¹ (e, £,m) and ra2 = R² (e, £,m) .                             (15)

where Y, R¹ and R² are as defined in (11), (13) and (14), respectively. In proposition 2 the
function Y from (11) has been shown to satisfy properties (Y*) while the curvature of R¹ and
R² is less well known (proposition 3(ii)). With proposition (2) our main goal is clearly
achieved, namely to show that the properties (Y) are necessary conditions for functions (1) to
represent a subsystem of the comprehensive technology (2). In proposition (2) we even
proved that the properties (Y) are not sufficient since the properties (Y*) are more restrictive
than the properties (Y).

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