Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

10

where λy, λ_a, λ_e, λ_t and λ_m are Lagrangean multipliers. Since the objective function is
linear and all terms in the cornered brackets are concave functions the Kuhn-Tucker condi-
tions are necessary and sufficient for a maximum. An interior solution implies

F

— ∙(^{1 +} Am ) = At as well as At > Ft .

F

m

Note that (5) also follows from setting dy / da = 0 in (5c). A_t > F_t follows from rearranging
^the equation ^{in (5)}: Ft (1 + Am ^- Fm ) = Fm (At ^- Ft ) > ⁰ .

With the concept of efficient abatement we now continue our inquiry into the relationship
between the production system (2) and the production function (1).

Proposition 2: If abatement is efficient, the production-cum-abatement technology (2) implies
a production function Y : D → R ₊ that exhibits the properties (Y*) defined as

(a) Y exhibits the properties (Y),

(b)   Y satisfies Y_m∈ ] 0,F_m[ [where F_m ∈ ] 0,1[ due to properties (F)],

(c)   The domain of Y is D := {(e, t, m) ∣ e ≤ m - F ( t, m)} ⊂ R+ ;

Proposition 2 will now be proved in six steps.

Step 1: If abatement is efficient, the set of equations (2a) - (2f) implies a function
M : (e, t, m) → m_y such that¹⁰

m_y = M (e, t,m).                                                                (6)

+++

Rewrite (5a) as F_t (1 + A_m ) = A_tF_m, totally differentiate this equation and combine the result
with dt = dt_a + dt_y and dm = dm_a + dm_y from (2e) and (2f). After some rearrangements,
these operations result in

d t = ^γm^y-dm + ^γ- d t - ^γ÷dιιι,                                        (7)

y   Yt     ^y Y t       Yt

yyy

where γ t y : = A t Fm, + F₁ Am t -(1 + Am ) F_ll - FmAt_l > 0, Yt : = Ft Am t - FmA tt > 0,

Ymy : = FmAt,m +(1 + Am ) Ft m - F₁ Amm - AFmm > 0 and Ym : = FmAtm - Ft Amm > 0 .

¹⁰ A plus or minus sign underneath an argument of a function indicates the (assumed) sign of the corresponding
first partial derivative.

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