Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

Next we insert d£_y from (7) into (3):

. ∕₁ ⁽ _l_l F _λa₀_λ^γ⁽At ^- F£ ) ^γmy , (At — F_t )γ_t,( (A_t - F_t )Ym ,

dm (1 + A_m - F_m ) = de + A_t dt + Amdm--ddm_γ - ------^j-^t- d t + ʌ-------— dm .

Yt y Y t y Yt y

Rearranging this equation yields

dm_y = ɪ de + — dt + ^πmmdm, (8)

πm1πm1πm1

where ^πm 1 =(¹⁺ Am ^-^Fm ) Yt _y ⁺( ^At ^-^Ft ) Ym_y > ^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)→ m_y .

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)

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

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

πm1 πm1 πm1

^wh^ere Pt^: = (¹⁺ Am ^-^Fm ) Yt ⁺ AtY_m > 0 and Pm ^: = (¹^-^Fm ) Ym ^- [(¹⁺ Am )^Ftm ^- At^Fmm ] Am .

In view of (10) there is a function L : (e, t, m) → t_y, satisfying t_y = L (e, t, m) .
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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)

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The function Y defined above obviously exhibits Y_e > 0 and Y_t > 0 but the sign of

Ym = Ft Lm + FmMm (12)