TARIFF AGREEMENTS AND NON-RENEWABLE RESOURCE ... - Ivie

Next, we calculate the difference
nγ
WA(x) S − WA N 2
(x) =
6r − nδ2
8r
x 2 −
3anγ
9r +5nγ −
anδ
x
4r +3nδ
+ 27a2 nr
2(9r +5nγ) 2 − 2a2 nr
(4r +3nδ) 2 . (61)
The first coefficientofthislinear-quadraticfunctionispositiveif4γ 2 − 3δ 2
is positive. By substitution of γ and δ, see (40) and (57), we obtain
1 9
n 2 25 ((20cnr +9r2 ) 0.5 − 3r) 2 − 4
3 ((3cnr + r2 ) 0.5 − r) 2 .
Let us suppose that 4γ 2 − 3δ 2 ≤ 0.Then it must be satisfied that
9
25 ((20cnr +9r2 ) 0.5 − 3r) 2 ≤ 4 3 ((3cnr + r2 ) 0.5 − r) 2 .
Takingthepositivesquarerootandreorderingweobtain
3 √ 3(20cnr +9r 2 ) 0.5 ≤ 10(3cnr + r 2 ) 0.5 +5.59r.
Squaring, reordering terms and squaring again we get the contradiction:
57600c 2 n 2 r 2 + 16172cnr 3 ≤ 0 which establishes that 4γ 2 − 3δ 2 is positive.
Now, we look at the second coefficient of the difference WA S(x) − W A N(x).
This second coefficient is positive if 4γ − 3δ is positive. By substitution we
get now
2 6
n 10 ((20cnr +9r2 ) 0.5 − 3r) − (3cnr + r 2 ) 0.5 + r
.
Let us suppose that 4γ − 3δ ≤ 0. Then it must be satisfied that
3(20cnr +9r 2 ) 0.5 ≤ 5(3cnr + r 2 ) 0.5 +4r.
Squaring, reordering terms and squaring again we get the contradiction:
11025c 2 n 2 + 3600cnr ≤ 0 which establishes that 4γ − 3δ is positive. Finally,
we compare the independent coefficients. Let us suppose that
Developing this inequality we obtain
27a 2 nr
2(9r +5nγ) − 2a2 nr
2 (4r +3nδ) ≤ 0. 2
432r 2 +648rnδ +243n 2 δ 2 ≤ 324r 2 +360rnγ +100n 2 γ 2 ,
24