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以綠色導向政策探討低經濟價值再生材料之回收供應鏈
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Appendix for Theorem Proofs
Proof of Lemma 1.
The virgin material based model problem and the profits can be rewritten as
To find the optimal
q
vector, take derivative of
π
m
with respect to
q
. Let : = (I+11′)
−1
.
Drawing from the first-order condition, we can readily derive the corresponding equilibrium
solution
For the intermediary leader, substitute
q*
into
π
x
, and evaluate the Nash equilibrium as
follows:
Therefore the equilibrium solutions
w
x
and
q
*
(0S)
are then obtained as
Proof of Lemma 2.
For the intermediary leader, substitute
q
* into
π
x
and
π
y
, and evaluate the Nash
equilibrium as follows:
Therefore the equilibrium solutions
w
x
,
w
x
, and
q
*
(2S–)
are obtained as