
227A Game of Competitive Investment
where
N(k, b) = 2(k − 1)
2
∙ b
3
+ 6k(k − 1) ∙ b
2
+ k(3k − 1) ∙ b − k
2
, (8.81)
and
D(k, b) = 2 ∙ (kb − b + k)
3
. (8.82)
From Equation 8.35, we have, when treating A
n
as a function of n, γ, and α,
∂
∂
−
−
−
−⋅
A
n
n
n
=
(4 3)
=
(4 3)
2
αγ
. (8.83)
In view of Q(A
n
, B
n
) = 0, this leads to
∂
∂
−
−⋅+
−⋅
A
n
nnB
n
=
(4 3) (1 )
(2 )
3
2
. (8.84)
Combining the preceding while treating C
n
as a function of n, γ, and α, we
obtain
∂∂∂∂ +∂ ∂
×
CnGkbk Gkbb
nknb B
n
knbB
n
// /=(,) (,)
=,==,=
||
*
//dw adaAnn
J
wa B
n
n
() =( 1) (1 )
(
()=
|
*
×∂ ∂−−⋅ ⋅+
⋅ nnB nnnBBn B
nnnn
,)[(2 )( )(21)],
23
/ −⋅ −+ ⋅−
(8.85)
where
J(k, b) = 3b
3
− b(1 + 9b(1 + b)) ∙ k + (1 + b)(8 ...