Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

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is unclear. To prove Y_m∈ ] 0,F_m[ , let (2a) and (2c) hold as equalities and combine the equa-
tions (2a) - (2f), (6) and (9) to rewrite (2b) as

e + A [t - L (e, t, m), m -M (e, t, m)] + F [L (e, t, m), M (e, t, m)] = M (e, t, m) .

We differentiate this equation with respect to m (keeping e and t constant):

- At L_m + Am (1 - Mm ) + F_t L_m + FmMm = M_m
which yields, after rearrangement of terms,

Am ^{- (1 +} Am ^- Fm )Mm
At ^- Ft

Next we substitute this term for L_m in (11) and get

AmFt  Γ ^Ft (1 ⁺ Am ^- Fm ) _ _f 1 _m = AmFt

A_t - F_t       A_t - F_t        m m   A _t - F_t ^,

since owing to (5) the term in the cornered brackets is zero. Moreover, A_t, Am and F_t are
positive due to the properties (A) and (F), and A_t - F_t is positive due to (5). Therefore

AF                           AF F(1 - F)

Ym = _l'_l > 0. By invoking (5) again we find Ym = -^mFA- = Fm - ^{t ( m} ) < Fm .

At ^- Ft                                    At ^- Ft        At ^- Ft

Step 4: The production function Y from (11) is concave.

In view of the complex terms constituting its first derivatives there is no way to further spec-
ify its curvature by determining the sign of its second-order derivatives. Yet concavity of Y is
straightforward from the convexity of the set Y that was established in proposition 1(i).

Step 5: The domain of the production function Y

We now determine the domain D of function Y by observing that (3) yields

— = ^{1 +} Am ^- Fm > ^{0 and} =-- At ^- Ft > ⁰
dm_y                     d t _y

for given t and m. That implies, in turn,

Consequently, for given inputs t and m the amount of production residuals emitted is largest
when no abatement takes place at all. As an implication, the domain of Y from (11) is D as

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