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臺大管理論叢
第
27
卷第
2
期
177
Proof of Proposition 1.
The proof can be divided into 3 parts:
(i) To find the optimal
q
vector, take derivative of
π
m
with respect to
q
. We can readily
derive the corresponding equilibrium solutions and they are given by
From backward induction, to find the optimal
q
i
for the manufacturer follower, we take
derivative of
π
m
with respect to
q
i
. Drawing from the first-order condition, we can readily
derive the corresponding equilibrium solution
Let : = (I+11′)
−1
. The solution satisfies the following fixed-point equation
(1)
The optimal solution exits and is unique since the Hessian is negatively definite, i.e.,
(ii) For the intermediary supplier leaders, plug
q
* into
π
x
and
π
y
and obtain the supplier
profits:
The optimal solutions exit and are unique because the Hessian matrix is negatively
definite. Evaluate the Nash equilibriums:
Write the equations in matrix form
We have
(2)
(3)