Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered



11

Next we insert d£y from (7) into (3):

. ∕1 ( l l F λa0 λγ    (At - F£ ) γmy ,     (At Ft )γt,( (At - Ft )Ym ,

dm (1 + Am - Fm ) = de + At dt + Amdm--ddmγ -  ------j-t- d t + ʌ-------— dm .

Yt y                    Y t y                  Yt y

Rearranging this equation yields

dmy = ɪ de + — dt + πmmdm,                                  (8)

πm1πm1πm1

where πm 1 =(1 + Am - Fm ) Yt y +( At - Ft ) Ymy > 0, πmm = AmYt y +(At - Ft ) Ym > 0

and πm t = AtY t y -( At - Ft ) Y t = ( At + Ft ) Y t + [ At Fm t -(1 + Am ) Ftt ] At > 0.

Obviously, equation (8) determines the first derivatives of a function M : (e, t, m)→ my .

Hence (6) is established.

Step 2: If abatement is efficient, the set of equations (2a) - (2f) implies a function
L :
(e, t, m) t y such that

t y = L ( e, t ,m ) .                                                                       (9)

++?

dmy from (3) is now plugged into (7) yielding after some calculations

d t y = ^de + p- d t + pm-dm,                                    (10)

πm1     πm1     πm1

where Pt: = (1 + Am - Fm ) Yt + AtYm > 0 and Pm : = (1 - Fm ) Ym - [(1 + Am )Ftm - AtFmm ] Am .

In view of (10) there is a function L : (e, t, m) ty, satisfying ty = L (e, t, m) .
++?

Step 3: The preceding steps 1 and 2 imply a production function Y : (e, t,m) y satisfying

Ye > 0 ,Yt > 0 and Ym ] 0 ,Fm [.

Invoking (6) and (9) the production function (2a) is turned into

y = F [L (e, t,m) ,M (e, t,m)] = :Y (e, t,m).                                       (11)

++?    +++

The function Y defined above obviously exhibits Ye0 and Yt0 but the sign of

Ym = Ft Lm + FmMm                                                    (12)



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