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
ra2 = ma due to (2h), we obtain ra = 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 = -( AlLe + AmMe ) and R2 = -[TaAmMe +( TaAl + T ) Le ] = - Me.

While Re1 0 is obvious, Re2 =-Me <0 is explained by observing that total differentiation of
1
A

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

AmAm

above we clearly have Ta = (1/ Am ) and T£ =-( A£/Am ). The signs of the partial derivatives
Re1 and Re2 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 ra1 and ra2 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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