Download Pricing Decisions and Coordination in a Two-echelon Supply Chain

Pricing Decisions and Coordination in a Two-echelon Supply Chain
with Two Competing Retailers
XU Beibei1, ZHANG Hanjiang2
School of Economics and Trade, Hunan University, P.R. China, 410079
1. [email protected] , 2. [email protected]
Abstract: This paper studies pricing decision and coordination in a two-echelon supply chain which
consists of one manufacturer and two nonidentical retailers who use retail price to compete for the end
customers with complete information. The manufacturer takes Stackelberg leadership in dictating the
wholesale pricing terms by implementing a two-part tariff. The sub game perfect Nash equilibrium has
been get by use backward induction method. The retailer with larger market base or lower retail price
has comparative advantage in the market. And the effect of the two-part tariff is equal to the quantity
discount, which means the wholesale price is concerned with the order quantity (market demand). The
structure of wholesale pricing mechanisms that are linear in the order quantity can coordinate the system
under limiting condition.
Keywords: two-echelon supply chain; pricing decision; coordination; game theory
1 Introduction
Supply chain coordination is an important topic in the marketing science literature. Generally, a
supply chain is coordinated when the members of the supply chain achieve the system equilibrium when
the members of the supply chain acting rationally make the decisions for the objective of maximizing
their own profitability. Some coordination mechanisms, such as two-part tariff, quantity discount
schedule and revenue-sharing schedule are used to regulate the relationship among the supply chain’s
members.
Weng[1] developed a coordinated quantity discount policy, and analyzed the impact of coordination
on members’ decisions and the supply chain profit in a general newsboy model. He found that the
optimal all-unit quantity discount scheme is equivalent to the optimal incremental quantity discount
scheme in achieving the coordination of a supply chain. Tiaojun Xiao and Xiangtong Qi[2] studied the
price competition and coordination of a supply chain with one manufacturer and two competing retailers
after the demand and production cost of the manufacturer are disrupted. They develop the conditions
under which the supply chain is coordinated and discuss how the cost disruption may affect the
coordination mechanisms of an all-unit quantity discount and an incremental quantity discount (menu of
two-part tariffs). Ingene and Parry[3] studied the price competition and the coordination of a supply
chain with one manufacturer and two competing retailers with three classes of mechanisms, the linear
quantity discount scheme, the sophisticated Stackelberg two-part tariff and the menu of two-part tariffs.
They compared them from the viewpoint of maximizing the manufacturer’s profit. Chen[4] studied
two-part tariffs as well as the all-unit quantity discount scheme cannot be guaranteed to perfectly
coordinate the supply chain with multiple independent retailers when the inventory cost is considered.
Raju and Zhang[5] considered a channel with one manufacturer serving a dominant retailer and a fringe.
They showed that coordination of the channel through a two-part tariff can be beneficial to the
manufacturer.
In our model, we assume the two-echelon supply chain which consists of one manufacturer and two
nonidentical retailers who use retail price to compete for the end customers with complete information,
which is similar with [2][3][4][5]. Also we assume the market demand and the unit cost are fixed,
without any disruption, which is different with [1][2]. We assume the market demand of a retailer is
concerned with his own retail price and the rival’s retail price, which is the same with [2][3]. Our
emphasis is the pricing decisions of the retailers and manufacturer when implementing a two-part tariff.
We use backward induction method to get the sub game perfect Nash equilibrium of the
manufacturer-Stackelberg game. At last, we show that the structure of wholesale pricing mechanisms
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that are linear in the order quantity can coordinate the system under limiting condition.
2 Problem descriptions and assumption
We assume that the two-echelon supply chain which consists of one manufacturer and two
nonidentical retailers who use retail price to compete for the end customers with complete information
in a Bertrand market. And the retailers cannot resell merchandise to each other. All parties behave
independently to maximize their individual profits. Specifically, the definition of each notation in the
model is as follows:
Di
---------- market demand for retailer i
pi
---------- retail price of retailer i
ai
---------- market base of retailer i
∆pi --------- (pi – pj), retail price differentiation
b ---------- sensitivity of a retailer’s demand to its own retail price
d --------- substitutability coefficient of the two retailers’ products
W ---------- wholesale price
w ---------- unit wholesale price
φ ---------- fixed fee component of wholesale payment
cm ---------- unit cost of manufacturer
ci
---------- unit cost of retailer i
np ---------- unit profit margins
ΠRi --------- profit of retailer i
ΠM --------- profit of manufacturer
Retailer i purchases the product from the manufacturer with a unit wholesale price w and adds
some values to the product with a unit cost ci, which determine the retail price pi to sell the product,
i (1, 2), and j=3-i. The added values make the products of the two retailers differentiable so as to meet
different market demand, but they are still have extensive substitution.
We assume that the market demand for retailer i is Di (pi ,pj)=ai −bpi +dpj, where ai is the aggregate
potential market demand scale of retailer i; b is the sensitivity of a retailer’s demand to its own retail
price; d is the substitutability coefficient of the two retailers’ products, b>d>0. The manufacturer
decides the wholesale price by implementing a two-part tariff, that is Wi=w+φ/Di, where φ is a constant.
The above information is common knowledge to all members of the supply chain who play a one-shot
game.
∈
3 Pricing decision of the competing retailers
In this section, the manufacturer takes Stackelberg leadership in dictating the wholesale pricing
terms. The optimum pricing decision of the retailers is concerned with the manufacturer’s pricing
decision. The retailers also know the contingent action plan of the manufacturer. Once the manufacturer
decides the unit wholesale price at w, the retailers must be decide the retail price at p(w). So we can get
the solution to sub game perfect Nash equilibrium by backward induction.
The retailers behave independently to maximize their individual profits without knowing the rival’s
pricing strategy, i.e., they determine retail prices simultaneously. In short, they play a static game with
complete information in a Bertrand market.
For a given w, the maximum profit of retailer i is given by solving the following optimal problem
max Π Ri ( pi , p j ) = ( pi − w − ci )( ai -bpi +dp j ) − ϕ
pi , p j
Solving the first-order conditions and the second-order conditions, we have
∂Π Ri ∂pi = ai -2bpi +dp j + bw + bci , ∂ 2Π Ri ∂pi 2 = -2b < 0
.
Setting ∂ПRi/∂pi=0 we can obtain the optimal retail price pi* as:
，
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a j + bc j
ai + bci
bw
+d 2
+
2
2
2
4b − d
4b − d
2b − d .
So we can set the optimal market demand and profits of the retailer:
a j + bc j
a + bc
b( d − b)
Di∗ = ai + ( d 2 − 2b 2 ) i 2 i2 + bd 2
+w
4b − d
4b − d 2
2b − d ,
pi ∗ = 2b
a + bc j
a + bc j
 a + bc

a + bc
2( d − b)
( d − b) w 
Π i ∗ =  2b i 2 i2 + d j 2
+w
− ci  ai + (d 2 − 2b 2 ) i 2 i2 + bd j 2
+b
−ϕ
2
2
4b − d
2b − d
4b − d
4b − d
2b − d 
 4b − d

.
Proposition 1 The retailer’s maximum profit is convex in the unit wholesale price w.
Proof. The first derivative of the retailer’s maximum profit with respect to w, are
a j + bc j
d Π i ∗ ( w) 2( d − b) 
ai + bci
( d − b) w 
2
2
=
+ bd 2
+b
 ai + ( d − 2b ) 2
+
2
2
dw
2b − d 
4b − d
4b − d
2b − d 
a + bc j

b(d − b)  ai + bci
2(d − b)
+ d j2
+w
− ci 
 2b 2
2
2
2b − d  4b − d
4b − d
2b − d

and the second derivative is
d Π i ∗2 ( w) 2b(d − b) 2
=
>0
dw2
(2b − d ) 2
.
So we can conclude that the retailer’s maximum profit is convex in the unit wholesale price w.
Proposition 2 The equilibrium for the horizontal interaction between retailers displays the following
characteristics:
∗
∗
(1) ∂pi ∂w > 0, ∂Di ∂w < 0 . The optimum retail price p * and market demand D * increase with
i
i
unit wholesale price w. Because the increasing of the unit wholesale price results in higher cost of the
retailers, which is partially transferred to the customers as a higher retail price. So the demand of the
retailer decreases, which in turn influences the profit of the manufacturers.
∗
∗
∗
∂pi∗ ∂c j > 0 ∂Di ∗ ∂c j > 0
∂Π i ∗ ∂c j > 0
(2) ∂pi ∂ci > 0 , ∂Di ∂ci < 0 and ∂Π i ∂ci < 0 ;
,
and
.
The retail price increases with his unit cost and his rival’s; The market demand and the profit both
decrease with his unit cost but increase with his rival’s. So the unit cost has multiply effect on the
market, as the decreasing of the unit cost can enhance the advantage of his own and weak the advantage
of his rival.
Proposition3 The retailer with larger market base or lower retail price has comparative advantage in the
market.
Parameterize the unit profit margins (pi-w-ci ) as np. It is easy to get
a − a j b(ci − c j )
b
b(b + d )
∆Di ∗ = Di ∗ − D j ∗ =
(ai − a j ) −
(ci − c j )
∆pi∗ = pi∗ − p j ∗ = i
+
2b + d
2b + d
2b + d
2b + d ,
,
−
a
a
+
(
b
d
)
j
∆npi∗ = npi∗ − np j ∗ = i
−
(ci − c j )
2b + d 2b + d
.
∗
∗
∗
(1) ∂∆pi ∂∆ai > 0, ∂∆Di ∂∆ai > 0, ∂∆npi ∂∆ai > 0 . When the retailers are symmetrical in unit
cost, the retailer with larger market base achieves higher retail price, demand and unit profit margins, so
does higher profit.
∗
∗
∗
(2) ∂∆pi ∂∆c i > 0, ∂∆Di ∂∆ci < 0, ∂∆npi ∂∆ci < 0 . When the retailer are symmetrical in market
base, the retailer with lower unit cost achieves lower retail price, but higher demand and unit profit
margins, so does higher profit.
This emphasizes the market base and unit cost as measures of comparative advantage. In general,
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the differences between the outcomes of the retailers reflect the tension between the two countervailing
forces. For instance, a retailer decreases the unit cost may somewhat offset his disadvantage of smaller
market base.
4 The two-echelon system
Now that we have characterized the interaction between the two competing retailers, we can gain
further insight to the supply chain performance by incorporating the role of the manufacturer. Because
the manufacturer forecasts that the retailers will decide their optimum retail price p* based on w(p), the
manufacturer have to solves the following optimization problem on the first stage:
max Π M ( w) = ( w − cm )  Di ( p ∗i ( w)) + D j ( p ∗ j ( w)) 
w
b( d − b)
2b(d − b) 
 b
w + 2ϕ
=( w − cm ) 
( ai + a j ) +
(ci + c j ) +
2
b
d
2
b
d
2b − d
−
−


Solving the first-order conditions and the second-order conditions, we have
∂Π M
b
b ( d − b)
4b(d − b)
2b(d − b)
∂Π 2 M 4b( d − b)
=
( ai + a j ) +
(ci + c j ) +
w−
cm
=
<0
2
∂w
2b − d
2b − d
2b − d
2b − d
2b − d
, ∂w
.
Setting ∂ΠM/∂w=0 we can obtain the optimal retail price w* as:
a + a j cm ci + c j
+ −
w∗ = i
4(b − d ) 2
4 .
We can set the optimal profit of the manufacturer:
 ai + a j ci + c j cm   b( ai + a j )
(d − b)cm ci + c j 
ΠM ∗ = 
−
− 
+b
−
+ 2ϕ
4
2   2(2b − d )
2b − d
2 
 4(b − d )
.
It is a common sense that if the manufacturer is willing to sell the product, it is necessary that the
，
ai + a j
unit wholesale price must be higher than the unit cost, i.e. 4(b − d )
−
ci + c j
4
−
cm
>0
2
.
Proposition4 The equilibrium of the manufacturer displays the following characteristics:
∗
∂w∗ ∂ci , j < 0 ∂Π M ∂ci , j < 0
(1)
,
. When the unit cost of either retailer decreases, the wholesale
price and the profit of the manufacturer will increase.
∗
∂w∗ ∂ai , j > 0 ∂Π M ∂a i , j > 0
(2)
,
.When market base of either retailer increases, the wholesale
price and the profit of the manufacturer will increase.
∗
∗
(3) ∂w ∂cm > 0 , ∂Π M ∂cm < 0 . When the unit cost of the manufacturer increase, the wholesale
price will increase, and the profit of the manufacturer will decrease.
Substituting the expression of w* into the expressions in section3, we can get pi *(w*), Di *(w*) and
∗
∗
∗
∗
Π *(w*). Also we have Wi = w + ϕ / Di ( w ) , from the structure of this expression, we can
i
conclude that the wholesale price of retailer i is concerned with the order quantity, and decreases with it.
So we can see the two-part tariff mechanism is equal to the quantity discount mechanism.
5 System coordination
In this section, we consider the question of whether the two-part wholesale tariff mechanism is
capable of achieving the system equilibrium when the members of the supply chain make decisions for
the objective of maximizing their own profitability.
The system’s maximum profit is given by solving the following optimal problem
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max Π s ( pi , p j ) = Π Ri ( pi , p j ) + Π Rj ( pi , p j ) + Π Ri ( w)  = ( pi − cm − ci ) + ( p j − cm − c j )  ( ai -bpi +dp j )
p1 , p2
.
bai + da j
c +c
pˆ i ∗ =
+ m i
2
2
2(b − d )
2 .
From the first-order condition, we have
So we can set the optimal market demand and profits of the retailer:
dc − bci (d − b)cm   bai + da j cm + ci 
a dc j − bci (d − b)cm Π
ˆ ∗ =  ai + j
+
i
2
  2(b2 − d 2 ) − 2 
Dˆ i∗ = i +
+
2
2


.
2
2
2
,
The wholesale price can coordinates the system only if it satisfies the following condition:
a j + bc j
bai + da j cm + ci
a + bc
bw
2b i 2 i2 + d 2
+
=
+
2
4b − d
4b − d
2b − d 2(b 2 − d 2 )
2
So we have
3d 2bai + d (d 2 + 2b 2 )a j d 2ci + 2bdc j (2b − d )cm
w=
−
+
2b(b 2 − d 2 )(2b + d )
2b(2b + d )
2b
.
Proposition5 The coordinating w displays the following characteristics:
(1)If the manufacturer is willing to participate in the channel, it is necessary that their wholesale
price is higher than the unit cost of the manufacturer, i.e.
3d 2bai + d ( d 2 + 2b 2 ) a j d 2ci + 2bdc j dcm
>
+
2(b 2 − d 2 )(2b + d )
4b(2b + d )
2b
(2) The coordinating w increases with the market base of the retailers and the unit cost of the
manufacturer. So, an increase in unit wholesale price is required to spur the retailers to expand the
market base. And an decrease in unit cost of the manufacturer must need an corresponding decrease in
the unit wholesale price.
(3) The coordinating w decreases with the unit cost of the retailers. So, an increase in unit
wholesale price is required to reduce the unit cost of the retailers.
6 Conclusions
This paper studies pricing decision and coordination in a two-echelon supply chain which consists
of one manufacturer and two nonidentical retailers who use retail price to compete for the end customers
with complete information. The manufacturer takes Stackelberg leadership in dictating the wholesale
pricing terms by implementing a two-part tariff. So we use backward induction method to get the sub
game perfect Nash equilibrium. Firstly, we have investigated the equilibrium of the two retailers who
behave independently to maximize their individual profits. This has yielded a number of insights and
testable implications. We conclude that the retailer’s maximum profit is convex in the unit wholesale
price and the retailer with larger market base or lower retail price has comparative advantage in the
market. Having characterized the interaction between the retailers, we gain further insight to the supply
chain performance by incorporating the role of the manufacturer. And we have shown that the effect of
the two-part tariff is equal to the quantity discount, which means the wholesale price is concerned with
the order quantity (market demand). We also show that the structure of wholesale pricing mechanisms
that are linear in the order quantity can coordinate the system under limiting condition.
There are several possible directions for future research. The assumption of complete information
may be limiting. For instance, when one retailer’s unit cost is a private information, the parties only
know the probability distribution of his unit cost, how the performance of the supply chain will be.
Second, we have considered the market demand and the unit cost is certain. It would be an interesting
exercise to extend the analysis to the situation of cost disruption and market demand disruption. Finally,
we can expand the supply chain with multiple manufacture and multiple retailers.
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management science 2008; (36):741-753
[3] Ingene CA, Parry ME. Is channel coordination all it is cracked up to be? Journal of Retailing
2000;76(4):511–47.
[4] Chen FR, Federgruen A, Zheng YS. Coordination mechanisms for a distribution system with one
supplier and multiple retailers. Management Science 2001;47(5):693–708.
[5] Raju, R., & Zhang, Z. J. (2005). Channel coordination in the presence of a dominant retailer.
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