(10) |
dPr*L |
= ∂ Pr L ∂ x * L = ∂ x * L d I |
∂Pr L ∂ x*H _l___ | |
dI |
∂x*H |
∂I | ||
Note that -dPL- = - |
∂PrH *, |
∂ 2Prl _ * * ʌ * ∂x ∂x HL |
- ∂2PrH ∂ x*H ∂ x |
∂ Pri 1 — and ----= —rτ . Thus L ∂ xl nl(I ) |
may rewrite (10) as:
(11) dPr*L
dI
1 |
[ ∂2 Prh H |
~d------ Bn n |
nn Ч L H ∂ X, ∂ Xh |
(ηH +ηL )
pPrH
Ч ∂x2H
η Ln2 H
∂2 PrL 2
2 L ηHn2L
∂x l
we
Λ
JJJ
J
This gives
Proposition 5:
d Pr* L
dI
> 0 if -∂LPTl.
< ∂ xL∂ xH
(ηH +ηL )>
<
p⅛
Ч ∂x2H
nH
η L —
∂2 PrL
l ∂x2L
nL
ηH
nh
By this proposition we get:
Corollary 5.1:
dPr*L
(a) Under a type-(ι) reform with i=H, ------<
dI
if
∂2 Pr
------L- ≤ 0 ;
∂xL ∂xH
d Pr*L
(b) Under a type-(ιv) reform with i=L, ------
dI
> 0 if
∂2 PrL
------L- ≥ 0 ;
∂xL ∂xH
© Under a type-(iv) reform,
dPr*L > 0 if ∂2Prh
d I < ∂x 2 H
> ηH nL
2
< ηLnH
∂2PrL
and-----L
-0 ;
∂xL ∂xH ≤
(d) Under a type-(v) reform,
dP > 0 if
dI <
∂2PrH
∂x 2 h
(d2PrL VηH
dx 2
Ч xLL J
2
nL
> ηL nH2
∂2PrL ≥
and--0.
∂xH ∂xL ≤
19
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