10
where λy, λa, λe, λt and λm 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
(5)
— ∙(1 + Am ) = At as well as At > Ft .
F
m
Note that (5) also follows from setting dy / da = 0 in (5c). At > Ft 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 Ym∈ ] 0,Fm[ [where Fm ∈ ] 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) → my such that10
my = M (e, t,m). (6)
+++
Rewrite (5a) as Ft (1 + Am ) = AtFm, totally differentiate this equation and combine the result
with dt = dta + dty and dm = dma + dmy from (2e) and (2f). After some rearrangements,
these operations result in
d t = γmy-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 ) Fll - FmAtl > 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.