Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

We now turn to proving proposition 1(ii). Let (2a) and (2c) hold as equalities and consider
(2a), (2c) and (2d) - (2f) to turn (2b) into my = e + A (£ - £y,m - my ) + F (£y,my ) . Total dif-
ferentiation of this equation yields, after some rearrangement of terms,

----1----de +---A----d £--A F^£ d £ _y +---Am---dm.

1+A_m-F_m    1+A_m-F_m    1+A_m-F_m     1+A_m-F_m

Next we set de = d£ = dm = 0 in (4), totally differentiate (2a) and (2c) to obtain, under con-
sideration of (2e) and (2f),

,                      F£ ^{(1 +} Am ^)- FmA',/ > > ∩

^dy = ^F£^{d £} y ⁺ F_m^dmy =   ---ɪ---- ^d£ y > ^{0 ,}

1 +A_m-F_m       ^<

z√     A,F     Λ,l       AmF£ ⁺ A£ ⁽¹ Fm ) ι∕_j ∩     11

da = - A, d £,, - A_mdm, =--d-----d d £,, < 0 and hence

^£ y ^m y      1∣∕fZ7      y

1 +A_m-F_m

dy_ = F_mA£ - F₁ (1 + A_m ) > ₀
da  AmF£ + A£ (1 - Fm )<

Since F_m ∈ ] 0,1[ and A_m> 0 by assumption we have 1 + A_m-F_m>0 . However, the numera-
tor on the right side of (4a) may attain either sign so that y may increase or decrease by shift-
ing inputs from production, ( £ y,my ), to abatement, ( £ _a,m_a ), while leaving net emissions e
unchanged. Equation (4c) demonstrates that for given (e, £, m) shifting the inputs labor and
material between production and abatement affects both the amount of production residuals
abated and the amount of consumer goods produced. The sign of dy/da in (4c) is unclear.

The message of proposition 1(i) is that if there is a function of type (1) implied by (2) it will
be concave. Unfortunately, proposition 1(ii) informs us that there is a correspondence rather
than a function. Yet this lack of uniqueness can be overcome in a natural way since we are
interested, of course, in the level of abatement which, for any given (e, £, m), yields the
maximum possible amount of the wanted output. That particular level of abatement will be
called efficient. To characterize the efficient abatement we maximize with respect to a, £ _a,
£ y, m_a, my and ry the wanted output F ( £ y, my ) subject to (2b) - (2f). The associated La-
grangean reads

L = y ^{+ λ}y [ F ( ^£ y^,my )^-y ] ^{+ λ}a [ A ( ^£ a^,ma )^-a ] ^{+ λ}e [ e ^{+ a +} y ^- my <sup>]
+ λ</sup>(. [^{£ - £} a ^{- £} y ] ^{+ λ}m [m ^- ma ^- my ^{] ,}

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