Download The Optimal Structure and Performance

The Optimal Structure and Performance
in Two Supply Chains with Different Stackelberg Leaders
Zhong Deqiang, Li Jinghong
Management Science & Engineering Research Institute,
Hunan University of Technology, Zhuzhou, P.R.China, 412008
Abstract Aiming at the two situations of two-tier supply chain when the upstream or downstream has a
Stackelberg leader firm with deterministic demand, we analyze the effects of the structures of supply chain
and the collocations of decision-making power on product price, quantity, and profits of member firms in
supply chain, the quantity of the downstream products increases with the number of the entrants at either tier,
while the price deceases with it; The supplying price of upstream suppliers decreases with the number of the
suppliers, but it is irrelevant with the number of sellers. Moreover, with these two situations, we testify they
have the same overall optimal profit of supply chain. And when the numbers of upstream and downstream
entrants are both 2, the overall profit of supply chain reaches to maximum.
Key words supply chain, decision-making priority, price, profit
1 Introduction
Product price, quantity, and profit of member firms in supply chain are affected both by the structures of
supply chain and the arrangement of decision-making power of member firms in supply chain. Corbett [1]
studied the problems of competition and optimal structures of supply chain in a serial supply chain with
deterministic demand, analyzed the effects of the structures of supply chain on product price, quantity, and
profit of member firms in supply chain. Nowadays, some literatures researched the impacts of the
collocations of decision-making power among firms on the each firm’s performance, for example, Rajan and
Zingales [2] and Munson etc [3] studied why a firm in supply chain had power and how to estimate magnitude
of power of member firms in supply chain; Some literatures related to the theory of industrial organization
discussed the effects of decision-making power on the firm’s efficiency, in which some literatures discussed
the relations between market force and industry intensity, some discussed the effects of power distribution
on the distribution of transaction income and invest enthusiasm. And a few of literatures made quantitative
analysis in the effects of power on distribution of total supply chain benefits from the view of supply chain
[6]
. This paper further analyses the problems of competition and optimal structures of supply chain when the
upstream or downstream has a Stackelberg leader firm with deterministic demand based on Corbett [1], and
analyzes the effects of the structures of supply chain and the arrangement of decision-making power on
product price, quantity, and profits of member firms in supply chain.
2 The optimal structure and performance of a supply chain with a downstream
Stackelberg leader
Consider a supply chain with two tiers as figure 1, final product assembly (Tier 1) and parts
manufacturing (Tier 2), where tier 1 (downstream) has n 1 firms (R1, …, Rn1), tier 2 (upstream) has n 2 firms
(S1, …, Sn2) and each downstream firm has to buy a single “part” (or bundle or kit of parts) from suppliers
(parts manufacturer). Without loss of generality, assume that one part is required to produce one unit of the
final product. Assume the parts produced by the supplier (parts manufacturer) and the final products
produced by the downstream are respectively homogenous products. The price of a part (the price in Tier 2)
is w, the price of final product is p, market inverse demand function is p = a 1 – b1Q1, where Q1 is the gross
supply (demand) of final product market.
159
Tier 2
Tier 1
p
R1
S1
M
M
w
Si
Market
quantity
Q1
p
Rj
p
M
M
Sn2
Rn1
p
Figure1 the structures of two tiers supply chain
Assume there is a firm (without loss of generality we assume firm S1) that has output decision-making
priority at tier 1 (downstream), the rest of firms are followers. The profits of downstream firms is
(1)
π 1, j ( n) = ( p1 − c − w) q1, j = ( a1 − b1Q1,− j − b1 q1, j − c − w)q1, j
Where n = (n1, n 2), Q1,- j = Q 1 - q1,
j
Because the downstream firm R1 has output decision-making priority in market competition, when
calculating its market maximal profit, R1 considers the rest firms will follow its market sales. Hence, its
maximum profit is given by solving the following optimal problem
Max π 1,1 (n) = ( a1 − b1Q1,−1 − b1 q1,1 − c − w) q1,1
q1,1
(2)
( j = 2, L , n1 )
s.t. max π 1, j (n) = (a1 − b1Q1, − j − b1 q1, j − c − w)q1
q1, j
Solving the above optimization problem, from ∂π ( n) ∂q = 0 , we have q 1, j = (a1 – w – c)/(2b) –
1, j
1, j
Q1,- j /2 . Because R2, …, Rn are symmetric, then q1,2 = … = q1, j = … = q1, n , so Q1,- j = q 1 + (n1 – 2)q1, j, by
1
1
the above expression, we have
[
]
q1, j = (a1 − w − c ) b − q1,1 n1
So, π 1,1 ( n) = ( a1 − b1Q1, −1 − b1q1,1 − c − w) q1,1 = ( a1 − w − c − b1 q1,1 )q1,1 n1
(3)
From ∂π 1,1 (n) ∂q1,1 = 0 , we get
(4)
q1,1 = ( a1 − w − c ) (2b1 )
Substituting expression (4) into (3), we have
(5)
q1, j = ( a1 − w − c ) (2n1b1 )
2
n
−
1
(6)
So, we can have Q1 = q1,1 + (n − 1)q1, j = 1
(a1 − w − c)
2n1b1
2n − 1
(7)
p1 = a1 − 1
( a1 − w − c)
2n1
By (6), we get w = a1 − c − 2n1b Q2 (2n1 − 1) = a 2 − b2 Q2 , where Q2 = Q1, a2 = a 1 - c,
b2 = 2n1 b1 (2n1 − 1) .
The profit of each firm at tier 2 is π2 (n) = (w – s)q2, where q 2 = Q2/n 2. From ∂π 2 ( n) ∂q 2 = 0 , we get
Q2 = n 2 q 2 = n 2 (a 2 − s ) (b2 (n 2 + 1)) , so
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n
1
(8)
a2 + 2 s
n2 + 1
n2 + 1
Substituting a 2, b2, w into expressions (4), (5), (6), (7) and after simplifying, we get the sales, total
market sales and products’ price in equilibrium respectively
n
n2
1
1
(9)
q1*,1 =
[a1 −
( a1 − c) − 2 s − c] =
( a1 − c − s)
2b1
n2 + 1
n2 + 1
2( n 2 + 1)b1
n2
(10)
q1*, j =
( a1 − c − s) ( j = 2, L, n1 )
2n1 (n 2 + 1)b1
(2n1 − 1)n 2
(11)
Q1* =
( a1 − c − s)
2n1 ( n 2 + 1)b1
( 2n1 − 1) n2
(12)
p1* = a1 −
( a1 − c − s )
2n1 ( n2 + 1)
So, the profits of downstream firms are respectively
n
1
(13)
π 1*,1 {n} =
( 2 ) 2 (a1 − c − s) 2
4n1b1 n 2 + 1
n
1
(14)
( 2 ) 2 (a1 − c − s) 2 ( j = 2, L , n1 )
π 1*, j ( n) =
2
4n1 b1 n 2 + 1
The total output, product’s wholesale price and profit of each upstream firm in equilibrium are
respectively
(2n1 − 1) n 2
(15)
Q2* = Q1* =
( a1 − c − s)
2n1 ( n2 + 1)b1
n2
1
(16)
w* =
( a1 − c) +
s
n2 + 1
n2 + 1
Q*
2n1 − 1
(17)
π 2* (n) = ( w* − s)q * = ( w* − s) 2 =
( a1 − c − s) 2
n2
2n1 ( n2 + 1) 2 b1
By (12), (15) and (16), it is easy to have
Proposition 1 In the two-tier supply chain which the downstream has a dominant firm, the quantity
produced for the final market increases with the number of entrants at either level, and the price decreases.
Moreover, the supplying price of downstream suppliers decreases with the number of suppliers, but is
irrelevant with the number of the sellers.
The total supply chain profit is
(2n1 − 1) n 2 (n 2 + 2n1 )
(18)
π t A (n1 , n2 ) = π 1*,1 ( n) + (n1 − 1)π 1*, j ( n) + n 2π 2* ( n) =
(a1 − c − s ) 2
4n1 2 ( n2 + 1) 2 b1
Proposition 2 For any a1, b1, c and s, π t A (n 1, n2) has the following properties:
1) If there is a single entrant at either tier (n 1 = 1 or n2 = 1), then πt A (n1, n 2) increases with the number of
entrants at the other tier (n 1 or n2).
2) When n 1 > n 2 / ( n2 – 1) or n 2 > n 1 / ( n1 – 1), πt A (n 1, n2)decreases with the increase of n1 or n2.
3)
lim π t A (n1 ,1) = lim π t A (1, n 2 ) = π t* .
w=
n1 → +∞
n2 → +∞
4) π t A (n 1, n2) achieves its maximum π t * = (a 1 – c – s)2/(4b1), at (n1, n 2) equal to (2, 2).
Proof: By (18), it is easy to get
n2
∂π tA (n1 , n2 ) n1 + n2 − n1n 2
(a − c − s ) 2
=
⋅
⋅ 1
=0
3
2
∂n1
b1
2( n 2 + 1)
n1
⇔ n1 + n2 − n1n 2 = 0
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(19)
And, ∂π A (n , n ) ∂n < 0 ⇔ n > n /( n − 1)
t
1 2
1
1
2
2
∂π tA ( n1 , n 2 ) n1 + n2 − n1n2 2n1 − 1 ( a1 − c − s) 2
⋅
⋅
=0
=
∂n2
b1
( n2 + 1) 3
2n12
(20)
(21)
⇔ n1 + n 2 − n1n2 = 0
And, ∂π A ( n , n ) ∂n < 0 ⇔ n > n /(n − 1)
(22)
t
1 2
2
2
1
1
By (20) and (22), we can immediately get properties 1) and2). From (18), we can immediately testify
property 3) holds.
By (19) and (21), for first-order conditions of πt A (n1, n 2), the only one integer solution is (2, 2), and
3 × 2( 2 + 2 × 2) ( a1 − c − s ) 2 ( a1 − c − s ) 2
π tA (2,2) =
⋅
=
= π t*
2
2
b
b
4
4 × 2 (2 + 1)
1
1
So, property 4) is testified.
3 The optimal structure and performance of a supply chain with a downstream
Stackelberg leader
Consider a two-tier supply chain in figure 1as shown in the above section, but the difference lies in the
assumption that at tier 2, rather than tier 1, there is a firm (without loss of generality we assume firm S1) has
priority of output decision-making, and the rest of conditions are the same with those of the above section.
The profit of downstream firm Rj is
(23)
π 1, j = ( p1 − c − w) q1, j = ( a1 − b1Q1,− j − b1q1, j − c − w) q1, j
By ∂π 1, j
∂q1, j = 0 and symmetry, we have
a−c− w
(24)
( n1 + 1)b1
(25)
So, Q1 = n1 q1 = n1 a1 − c − w
n1 + 1
b1
n
1
(26)
p1 =
a1 + 1 (c + w)
n1 + 1
n1 + 1
The profit of either firm at tier 1 is
1 a−c−w 2
(27)
π 1 ( n) = ( p1 − c − w) q1 = (
)
b1 n1 + 1
By (25), we have
(28)
w = ( a1 − c) − b1 ((n1 + 1) n1 )Q2 = a 2 − b2 Q2
Where b = b (n + 1) n .
2
1 1
1
The profit of firm Si at tier 2 is
(29)
π 2,i ( n) = ( w − s ) q 2,i = ( a 2 − b2 Q2,− i − b2 q 2,i − s )q 2,i
Where Q2,- i = Q2 – q2, i .
Because the upstream (Tier 2) firm S1 has priority of output decision-making (Stackelberg dominance)
in supply market, S1 considers the rest firms will follow its market supply when calculating its maximum
profit of supply market, so its maximum profit is given by solving the following optimization problem.
max π 2,1 (n) = ( a 2 − b2 Q2, −1 − b2 q 2,1 − s)q 2,1
(30)
s.t. max π (n) = ( a − b Q
− b q − s) q (i = 2,L , n )
q1, j = q1 =
q2 ,i
2,i
2
2
2,− i
2 2, i
2,i
2
Similar to the solution of (2), we have the gross output and products’ wholesale price of each upstream
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firm in equilibrium are respectively
1
1
q2*,1 =
q 2*,i =
(a 2 − s )
(a 2 − s ),
2n2 b2
2b2
n (2n − 1)
2n − 1
Q2* = 2
( a1 − c − s )
(a 2 − s ) = 1 2
( n1 + 1) 2n2 b1
2n2 b2
(31)
2n 2 − 1
2n − 1
1
(a1 − c) + 2
(a 2 − s ) =
s
2n2
2n2
2n2
So, we get the sales and sale price of each downstream firm products in equilibrium are respectively
Q * Q*
n1 (2n 2 − 1)
( a1 − c − s)( j = 1, L , n1 )
q1*, j = 1 = 2 =
n1
n1
n1 ( n1 + 1)2n 2 b1
(32)
w * = a2 −
n1 ( 2n2 − 1)
( a1 − c − s)
(n1 + 1) 2n2
The profits of each upstream firm in equilibrium are respectively
n1
1
π 2*,1 = ( w* − s )q 2*,1 =
(a1 − c − s ) 2 =
(a1 − c − s ) 2
4n2 b2
4n2 ( n1 + 1)b1
p1* = a1 − b1Q1* = a1 − b1Q2* = a1 −
π 2*,i
*
= (w −
s ) q 2*,i
=
n1
4n2 2 ( n1 + 1)b1
(33)
2
(a1 − c − s ) (i = 2, L , n2 )
By (31), we easily get
Proposition 3 In the two-tier supply chain which upstream has a dominant firm, the quantity produced for
the final market increases with the number of entrants at either tier, while the price decreases with it.
Moreover, the supplying price of downstream suppliers decreases with the number of suppliers, but is
irrelevant with the number of the sellers.
The total supply chain profit is
π t B (n) = π 2*,1 + (n 2 − 1)π 2*,i + n1π 1*, j
(34)
n n ( n + 1) + n1 ( n1 + 1)(n2 − 1) + n1 ( 2n 2 − 1) 2 ( a1 − c − s ) 2
= 1 2 1
b1
4n2 2 ( n1 + 1) 2
B
Proposition 4 For any a1, b1, c and s, π t (n 1, n2) has the following properties
1) If there is a single entrant at either tier (n 1 = 1 or n2 = 1), then πt B (n1, n 2) increases with the number of
entrants at the other tier (n 1 or n2).
2) When n 1 > n 2 / ( n2 – 1) or n 2 > n 1 / ( n1 – 1), πt B (n 1, n2) decreases with the increase of n1 or n 2.
3)
lim π t B (n1 ,1) = lim π t B (1, n 2 ) = π t* .
n1 → +∞
n2 → +∞
4) π t B (n 1, n2) achieves its maximum π t * = (a 1 – c – s)2/(4b1), at (n1, n 2) equal to (2, 2).
Proof: property 3) is immediately verify hold by (34). By (34), it is easy to get
∂π tB ( n) ( 2n 2 − 1)(n1 + n 2 − n1n 2 ) (a1 − c − s) 2
=
=0
b1
∂n1
2n 2 2 ( n1 + 1) 3
(35)
⇔ n1 + n 2 − n1 n 2 = 0, or 2n 2 − 1 = 0
n ( n + n − n1 n2 ) ( a1 − c − s) 2
(36)
= 0 ⇔ n1 + n 2 − n1n 2 = 0
= 1 13 2
∂n2
b1
2n 2 (n1 + 1) 2
The above first-order condition has the only integer solution n = (2, 2). And πt B(2, 2) = (a1 – c –
2
s) /(4b1) = π t *, so, property 4) is verified. And by (35) and (36), it is easy to have
(37)
∂π tB (n1 , n2 ) ∂n1 < 0 ⇔ n1 > n 2 /( n 2 − 1)
∂π tB ( n)
163
∂π tB (n1 , n2 ) ∂n2 < 0 ⇔ n2 > n1 /( n1 − 1)
By (37) and (38), we can immediately have properties 1) and 2).
(38)
4 Conclusion
In this paper, aiming at the two situations of two-tier supply chain when the upstream or downstream
has a Stackelberg leader firm (i.e. the firm has decision-making priority) with deterministic demand, we
analyze the effects of the structures of supply chain and the arrangement of decision-making power on
product price, quantity, and profits of member firms in supply chain, the quantity of the downstream
products increases with the number of the entrants at either tier, while the price deceases with it; The
supplying price of upstream suppliers decreases with the number of the suppliers, but it is irrelevant with the
number of sellers. Moreover, with these two situations, we testify they have the same overall optimal profit
of supply chain. And when the numbers of upstream and downstream entrants are both 2, the overall profit
of supply chain reaches to maximum.
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