CHAPTEE 4 MODELING
4.1 GSCM DRIVEN BY PRICE AGENDA BASED ON TAX AND SUBSIDY
This scenario aims to evaluate whether it is sustainable to remain a price agenda negotiating process based on the criteria that government charges an environmental tax to the end-of-life product manufacturers. Under the price agenda negotiation process, government decides its optimal economic involvement approach in the first stage. After that, in the second stage, manufacture and reversed-logistic supplier simultaneously bargain over price. In the third stage, the manufacture determines its output strategy and purchasing quantity based on the price from bargaining.
By assumption 1-3, I use backward induction to determine the manufacturer’s (mi) optimal production output which maximizes its profit. Due to our model aims to examine the negotiation process in terms of recycling raw material, we simply assume the price of traditional raw material is the same to all given manufacturers. Therefore, the relation among traditional raw material can be derived as wiT =wTj =wT and The profit function can be implied as
(6) Maxπ =(a−yi−yj)yi−wiRσRyi−wTσTyi−cmyi − fyi
where wiR , wT,cm separately represent the prices of procuring raw material as well as recycling raw material by manufacturer iand production cost; while represents the environmental tax that government charges.
f
The first order condition of Eq.(6) provides the manufacturer i’s reaction function,
) 2 (
R w c f
w y y a
R m
T T R i j j
i
−
−
−
−
= − σ σ
. Then equilibrium outputs, profits and material purchasing levels are:
3 ) 2
,
*( a w w w c f
w w
y m
T T R R j R R R i
j R i i
−
−
− +
= − σ σ σ
(7)2
2
* R
*( iR, j ) i
i w w = y
π
(8)
The reversed-logistic supplier conducts unused-good collections and material revised. According to spirit of GSCM, in order to enhance the recycling of used-goods, government subsidies the collection parts by quantities. Therefore, the reversed-logistic supplier’s profit function can be expressed as
r sy r y c y c y
wiR R i R i l i i
R
i = σ − σ − ⋅ + ⋅
π (9) where r is the rate of return.
In the scenario of price negotiation type, the negation agenda only conducts selling price; therefore, we aim to prove that the reversed-logistic supplier has no incentive to switch the current agenda. Given by the assumption 4, the negotiation model can be expressed as
(10) The left side of Eq.(10) means the manufacture’s optimal profit strategy while he agrees with the current negotiation agenda and the right side means reversed-logistic
[ ]πiM* (1−B)
[
(πiRϕyi*]
B
2 By solving and =0
j j
dy dπ 0 ,
=
i i
dy dπ
we can infer Eq.(7)
suppliers’ bargaining utility, where B denotes the bargaining power. To determine the selling price in this negotiation process, we can first rewrite Eq. (10) as
*
Substituting Eq. (8) and Eq. (9) and from the foc of Eq. (11), we can obtain the reaction function of manufacturer/reversed-logistic supplier bargaining unit I,
[ ]
, the selling prices are strategic complements for manufacturer/reversed-logistic supplier bargaining unit. In addition to Eq.(12), we obtain a stable solution for the negotiated selling price,
[ ]
Substitute Eq.(7) with Eq.(13), we can get
)
Whether our negotiation result is sustainable can be discussed vie a typical Stackelberg game theory. Based on the assumption of Stackelberg game, two firms are conducting sequentially quantity competition. The leader makes production first and the follower observes his action. Therefore, we can view leader’s optimal strategy as follower’s reaction strategy. Previous literatures imply that moving first maybe more
3 Inferred from
) 0
profitable when the leader remains a incumbent monopoly in industry since it can expand his quantity and drives out any possible entrants.
Similarly, we assumed manufacturer i as the game leader whose best strategy is to purchase more material and expand his quantity that would stop manufacturer j coming in; namely, manufacturer i tends to veto current agenda, switching to guaranteed contract agenda while the reversed-logistic suppliers’ bargaining power is low. Consequently, leader can take advantage Stackelberg leader’s rent.
However, in some extreme situation when reversed-logistic suppliers set maximization of market share as its only priority, that is φ＝0, the price agenda is never sustainable. It is a normal intuition that when reversed-logistic supplier aims to grab more market share by expanding output, it is much easier for manufacturer i to bargain for a cheap contract regarding to further procurements and both participants have the incentive to veto the price agenda, switching to price/guaranteed contract agenda. Nevertheless, this situation seldom exists in real business case.
Proposition 1：In normal situation that when reversed-logistic suppliers is risk-nature, the price agenda will sustain, if and only if when the bargaining power B is sufficiently high (B＞0.5)5.
Above discussion, leader has the incentive to veto current price agenda while the reversed-logistic supplier’s bargaining power is low. Therefore, only when B＞0.5, the agenda can be sustainable. Since, the reversed-logistic suppliers have more power to acquire better price when the leader is switching to conduct guaranteed contract.
Higher price means more cost and it will eliminate leader’ Stackelberg rent. As a result, when B＞0.5, the price agenda will sustain.
When it comes to the influence of the government’s economic involvement approaches, we can use mathematical approach to discuss the relationship among
5 Further induction is expressed in Proof 1
environmental tax, subsidy and output level.
Note that,
) 0 3 2 4 ( )3 2 4
* (
+ <
− −
=− B Bϕ
B df
dyi (15a)
* 0
>
ds= R
dyi (15b)
0 ) 3 2 4 (
*
<
+
− −
= R
i
B B B
df
dw ϕ ϕ σ (15c)
ds=
dwi* 0
) 3 2 4 ( ) 2 2 4
( <
+ + −
− − R
B B B
B ϕ ϕ σ (15d)
Eq. (15a) indicates that as environmental tax increases, the manufacturer tends to reduce its output level and negatively impacts the demand of all raw materials (contains traditional and recycling raw material) since the further increment of tax burden. As a result, the decline of recycling raw material drives the selling price to go down, just like Eq. (15c) implies. On the other hand, Eq. (15d) indicates that the increment of subsidy to reversed-logistic supplier diminishes its operation cost which also drives the selling price to go down as well. Consequently, lower selling price then stimulates the demand of recycling raw material and the manufacturer proceed to expand its output level, just like Eq. (15b) implies.
The last step we are going to discuss is government’s approaches. By assumption (5), we can further rewrite government’s objective function as:
MaxSW y y D y V y r
j i
n n i j n ij
R n j
i n
n R
n M
n j
i + + + − × + × ⋅
= ∑ ∑ ∑ ∑
=, =, =, =,
)2
2(
1 π π σ (16)
where and respectively represent the incremental environment marginal cost and producing a unit product and recycling a unit end-of-life product. In addition, and respectively represent the total revenue from end-of-life product manufacturers and reversed-logistic suppliers. Based on the assumption 6, we assumed that all economic approaches have to be satisfied with the finance balance; therefore, it can be constructed a constraint function in terms of
D V
benefit which is induced by
M
πn πnR
government’ strategy. The constraint is as follow: Combining Eq. (15) and Eq. (16), we can formulate government’s objective strategy as an incentive-constrained optimization problem:
Max ∑ ∑ ∑ ∑ We use Lagrangian Function to solve this problem. Thus, Eq. (17) can be rewrote as:
∑ ∑ ∑ ∑ ∑ ∑
ws the bounded budget balance condition.
which means government’s optimizing strategy follo
Since yi* >0, it can obviously conclude that f = s>0 and = =0 ds dL df
dL .
Therefore, combing with Eq. (20) and Eq. (21) we can infer the relation between subsidy and environmental tax as follow:
)
However, based on assumption 9, government’s involvement approach has to satisfy
4.2GSCM DRIVEN BY PRICE AND GUARANTEED CONTRACT AGENDA ON TAX AND