180
J. Wang and N. Mishima
with dated technology and small scale. The higher quality of recycled materials and
the sufficient cash flow of the formal recycler make the formal recycler a higher
bargaining power on selling recycled materials.
Set the sell price of the formal recycler as P, the sell price of the informal recycler
will be βP given the bargaining power of the informal recycler as β. Therefore, the
average benefit of the informal recycler will be B i = β P − P i − c 1 . The average
benefit of the formal recycler will be B f = P − P f − c 1 − c 2 , where c is the
additional cost of the formal recycler associated with environmental treatment and
waste management in formal recycling activities.
Combining with the quantity, the profit function of the formal recycler is
π f =
⎧
⎪ ⎨
⎪ ⎩
P − P f − c 1 − c 2
(2−v)P f −P i
1−v
P f >
P i
2−v
0
v P i < P f ≤
P i
2−v
0
P f ≤ v P i
Likewise, the profit function of the informal recycler is
π i =
⎧
⎪ ⎨
⎪ ⎩
(β P − P i − c 1 ) (Pi−Pf )
1−v
P f >
P i
2−v
(β P − P i − c 1 ) (Pi−Pf )
1−v
v P i < P f ≤
P i
2−v
(β P − P i − c 1 )P i
P f ≤ v P i
As analyzed previously, no WEEE will be sent to the formal recycler from the
informal collector when v P i < P f ≤
P i
2−v
, and P f ≤ v P i . This means the formalization extent is zero when P f ≤
P i
2−v
. Since our mainly target is analyzing the
formalization extent of informal collection activities, in other words, the amount of
WEEE sent to the formal recycler from the informal collector, we will focus on the
condition of P f >
P i
2−v
in the following sections.
13.3 Game Theoretic Model
In this section, the quantity change will be analyzed under the optimal price solved
through the game theoretic model. The formal recycler Stackelberg game model is
considered in following study.
In the formal recycler Stackelberg game model, the formal recycler decides its
optimal price P f first. Then, the informal recycler makes its pricing decision P i based
on the optimal price P f . Therefore, the model is formulated as follows:
⎧
⎨
⎩
First : maxπ f
P f
T hen : maxπ i (P i )
s.t. P f >
P i
2−v
J. Wang and N. Mishima
with dated technology and small scale. The higher quality of recycled materials and
the sufficient cash flow of the formal recycler make the formal recycler a higher
bargaining power on selling recycled materials.
Set the sell price of the formal recycler as P, the sell price of the informal recycler
will be βP given the bargaining power of the informal recycler as β. Therefore, the
average benefit of the informal recycler will be B i = β P − P i − c 1 . The average
benefit of the formal recycler will be B f = P − P f − c 1 − c 2 , where c is the
additional cost of the formal recycler associated with environmental treatment and
waste management in formal recycling activities.
Combining with the quantity, the profit function of the formal recycler is
π f =
⎧
⎪ ⎨
⎪ ⎩
P − P f − c 1 − c 2
(2−v)P f −P i
1−v
P f >
P i
2−v
0
v P i < P f ≤
P i
2−v
0
P f ≤ v P i
Likewise, the profit function of the informal recycler is
π i =
⎧
⎪ ⎨
⎪ ⎩
(β P − P i − c 1 ) (Pi−Pf )
1−v
P f >
P i
2−v
(β P − P i − c 1 ) (Pi−Pf )
1−v
v P i < P f ≤
P i
2−v
(β P − P i − c 1 )P i
P f ≤ v P i
As analyzed previously, no WEEE will be sent to the formal recycler from the
informal collector when v P i < P f ≤
P i
2−v
, and P f ≤ v P i . This means the formalization extent is zero when P f ≤
P i
2−v
. Since our mainly target is analyzing the
formalization extent of informal collection activities, in other words, the amount of
WEEE sent to the formal recycler from the informal collector, we will focus on the
condition of P f >
P i
2−v
in the following sections.
13.3 Game Theoretic Model
In this section, the quantity change will be analyzed under the optimal price solved
through the game theoretic model. The formal recycler Stackelberg game model is
considered in following study.
In the formal recycler Stackelberg game model, the formal recycler decides its
optimal price P f first. Then, the informal recycler makes its pricing decision P i based
on the optimal price P f . Therefore, the model is formulated as follows:
⎧
⎨
⎩
First : maxπ f
P f
T hen : maxπ i (P i )
s.t. P f >
P i
2−v