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,
dmy
----1----de +---A----d £--A F£ d £ y +---Am---dm.
(3)
1+Am-Fm 1+Am-Fm 1+Am-Fm 1+Am-Fm
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 + Fmdmy = ---ɪ---- d£ y > 0 ,
(4a)
(4b)
(4c)
1 +Am-Fm <
z√ A,F Λ,l AmF£ + A£ (1 Fm ) ι∕j ∩ 11
da = - A, d £,, - Amdm, =--d-----d d £,, < 0 and hence
£ y m y 1∣∕fZ7 y
1 +Am-Fm
dy_ = FmA£ - F1 (1 + Am ) > 0
da AmF£ + A£ (1 - Fm )<
Since Fm ∈ ] 0,1[ and Am> 0 by assumption we have 1 + Am-Fm>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,ma ), 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, ma, my and ry the wanted output F ( £ y, my ) subject to (2b) - (2f). The associated La-
grangean reads
L = y + λy [ F ( £ y,my )-y ] + λa [ A ( £ a,ma )-a ] + λe [ e + a + y - my ]
+ λ(. [£ - £ a - £ y ] + λm [m - ma - my ] ,
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