Coordination by Option Contracts in a Retailer-Led Supply Chain with

WANG Xiaolong (王小龙) et al:Coordination by Option Contracts in a Retailer-Led …
N * c
with ε ( q1)
because d d < q . A final interesting
N
issue is that d is not continuous in ε for a given
contract and q 1 is a kink point of the retailer’s expected
profit function. No matter what the realized
market signal is, the optimal total order never equals
1 . q This may occur with the mechanism which sets the
second-period compensation income to be max{ d− q1,
0}.Thus, the derivative of π R ( ε,
d)
will never be equal
to zero at q 1 .
Now, consider the supplier’s problem. Knowing the
retailer’s ordering behavior, the supplier will set his
first-period production quantity accordingly. When the
retailer knows the supplier’s first-period production
decision (which is related to the retailer’s ordering be-
c
havior), the retailer coordinates it to be equal to q 1 by
modifying o 1 . This is the standard Stackelberg game.
Combining Eq. (5), the overall findings for coordination
are summarized in Proposition 3.
e−w2 Proposition 3 Denote α = and β =
e−v e− w2 + o2
. To coordinate the channel, ensure that
e−v o2 < c2 − v and maintain some specific functional relationships
between the contract parameters of 1 o , o 2 ,
and e .
p− v+ s
e= p+ s− o2and
c2−v ⎡ c2 − c1⎤
o1 =
⎢
G( εT
( o2)) − o2
c2−v ⎥ ,
⎣ ⎦
where T ( 2)
o ε is determined by
−1
πR( εT( o2), F (max{ α,0}| εT(
o2)))
=
−1
πR( εT( o2), F ( β | εT(
o2)))
when β > 0 and
−1
{ εT( o2) : πR( εT( o2), F (max{ α,0}| εT( o2))) = πR( εT(
o2),
−1
F ( β | εT(
o2)))}
≠∅. Otherwise, define T ( 2) max o ε = ε .
Proof The necessity of equation e= p+ s−
p− v+ s
o2
was proved in Proposition 1. Next, show
c2−v that coordination still needs another condition,
⎡ c2 − c1⎤
o1 =
⎢
G( εT
( o2)) − o2
c2−v ⎥ .
⎣ ⎦
Recall that in this model coordination means
N c
N c
q = q and q1 = q1
. Based on the corollary in
Proposition 2, write the first-order condition for the
optimal
N
q 1 as
2 1 2 1
ε ( q1
)
0
2 2 1
∫
577
c − c − o + o + [ e− c + o −( e− v) F( q | ε )]d G(
ε ) +
∫
∞
o
(
2d
G(
ε ) = 0.
εT
o2 )
p− v+ s
Substitute e= p+ s− o2into
this equation
c2−v and rearrange
⎛ o ( 1)
2 ⎞ ε q
c2 − c1− o2 + o1+ ⎜1 − [ p− c
0
2 + s−
c2−v ⎟
⎝ ⎠ ∫
∞
( p− v+ s) F( q | ε)]d G( ε) + o d G(
ε)
= 0.
Comparing with
1
T ( 2 )
2
o ε
c
ε ( q1
)
2 1
0
2
∫
∫
c − c + [ p− c + s−( p− v+
c
N c
sFq ) ( 1 | ε)]d G(
ε ) = 0,
to let q1 = q1
, set o 1 so that
the following equality holds:
c o ε ( q1
)
2
− o2+ o1− [ p− c
0
2+ s−( p− v+ s) F( q1| ε )]d G(
ε ) +
c −v∫ Note that
2
∫
∞
o d G(
ε ) = 0.
o
ε (
2
T 2 )
c
ε ( q1
)
c
2− 1+ − 2+ − − + 1
0
c c ∫ [ p c s ( p v s) F( q | ε )]d G(
ε ) = 0
and rearranging gives
⎡ c2 − c1⎤
o1 =
⎢
G( εT
( o2)) − o2
c2−v ⎥ . □
⎣ ⎦
There are the pricing equations which assure channel
profit maximization. However, an important issue
to be considered before implementing any new pricing
scheme is whether the scheme is Pareto improving
with respect to an existing policy. In other words, will
both the retailer and the supplier be better off with the
proposed policy? To shed light on this question, examine
how the retailer’s and supplier’s expected profits
change as the contract parameters vary.
Unfortunately, due to the model complexity, closed
form solutions are not possible for some of the key
variables; thus, quantitative comparisons are almost
impossible. We cannot even prove that the monotonicity
of each side’s expected profit function as o 2 varies
because it is related to the specific forms of the distribution
functions.
3 Example
Suppose that F and G are both uniform distribution
functions with F~ U( a, b ) and G~ U(0, t ) . The critical