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Unformatted text preview: One can maximize net beneﬁts of pollution
or net benefits of abatement, for same result If maximizigg net benefits of mllution M, B(M) is the surplus gain from being allowed to polute more. E.g.
increased PS from reduced production costs and possibly increased
PS and CS from increased goods output.
. For clarity, we often assume goods output is constant, so B(M) is
just cost savings from reduced emission control costs. In this case
we call it S(M). C(M) (or “D(M)”) is the damages from pollution
A If maximizing net benefits of abatement Z = ﬂu - M, B(Z) is the beneﬁts of abatement in the form of reduced damages from
pollution. C(Z) is the loss in PS and possibly CS from having to change
production to curtail pollution. 4' ,
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MS: MAC ‘“ (13
K Lagrange multiplier method
for constrained optimization "
0 with equality constraint ‘ , .
Ex: Maximi e area of rectan
Wig} F(X,Y) H” a. H
Subject to G(X,Y) = K 5.1. 8H! s to K-Gllsﬂ:0 Step 1: Form Lagrangian
L=|i§XY)+A K—czgxy B-HI’ADO-vaj
o J I )Y) . u + 1' K"
Step 2: Take derivative with ”J. 3.. H ,, g) g Q
respect to each control H: & , )3 g 0
variable and A (lambda). This A: 26- 5- H - 0
gives you system of '
simultaneous equations. H g A §
g , A /
Step 3: Solve that system for H: 3
each control variables: 20., 3- (B) 3 0
’ f 20 = 25 Step 4 (if desired): Solve for A .: H : 5 9- l0 lambda. It is rate at which 1,"? Imme $31., 2
maximized value of F(X,Y) M“ 3+ H = 22 increases, ifKincreases. A". s M‘” = ’2’ V3. I60 Kahuna-ell! 2
r a. ’ -
9x1 6% IN ...
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- Spring '08