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 …
c
Proof First, prove the concavity of Π ( q1)
. This
c c
only needs to prove that Π ( ε, q1) = max π ( ε,
q)
is
qq1 concave in q 1 .
Note that
c * *
⎧⎪ π ( ε,
q ), if q c
1 q ;
Π ( ε,
q1)
= ⎨ c
⎪⎩ π ( ε,
q1),
otherwise,
*
c c
where q maximizes π ( ε , q).
Now π ( ε , q)
is obvi-
2 c
∂ π ( ε,
q)
ously concave in q because =−( p− v+ s)
⋅
2
∂q
c
f( q| ε ) 0. Therefore, Π ( q1)
is also concave in q 1 .
The optimal total production quantity follows immediately
from the fact that the second-period problem
is a simple newsvendor problem with initial inventory
q 1.
Intuitively this quantity is positively related to the
market signal. When the signal is not strong enough,
the central planner will not produce a second sum. This
idea is exhibited in Lemma 1 with a threshold value for
c ⎧ c p− c2+ s
the market signal: ε ( q1 ) = sup
⎫
⎨ε : Fq ( 1 | ε ) ⎬ .
⎩
p− v+ s ⎭
This is the strongest signal that will still result in no
second-period production. With this notation, suppose
that the central planner’s expected profit is maximized
at q , and then rewrite the problem as
c
1
c c
1 2 1
c
1
c
ε ( q1
)
∫ 0
c c
1
∞
c c
∫ π ( ε, q )d G(
ε),
c
ε ( q1
)
Π ( q ) = ( c − c ) q + π ( ε, q )d G(
ε)
+
c 1 p c2 s
where q F
p v s ε
− ⎛ − + ⎞
= ⎜ ⎟ . Because
⎝ − + ⎠
c
Π ( q1)
is
concave in q 1 , the first order condition works, i.e., an
optimal first-period production quantity is implied by
c
ε ( q1
)
c
2 1
0
2 1
∫
c − c + [ p− c + s−( p− v+ s) F( q | ε)]d G(
ε)
= 0.
□
c
Lemma 1 introduces the concept of ε ( q1
) which
partitions the market signals into two sets. If
c
ε ε(
q1
) , then the second-period production is not
necessarily positive. Otherwise, it is exactly positive.
1.2 Decentralized system performance without
coordination
Here model the decentralized system as a Stackelberg
game. The retailer sets orders in each period as mentioned
above. The supplier in turn arranges production
573
based on the orders.
Let d i denote the retailer’s order in period i. The
problem is to choose d 1 and d2 = d − d1
to maximize
the expected profit. The problem structure is analogous
to that in the centralized system.
∞
max ΠR( d1) =− wd 1 1+∫ ΠR( ε, d1, d2) d G(
ε),
d10
0
where ΠR( ε, d1, d2) = max πR( ε,
d1, d2)
and
d20
πR( ε , d1, d2) = − w2d2 + pEmin{ D, d1+ d2}
+
vd ( 1+ d2 − Emin{ Dd , 1+ d2})
−
sE[ D − min{ D, d + d }].
Rearranging,
1 2
max ΠR( d1) = ( w2 − w1) d1+∫ΠR( ε, d1) d G(
ε)
d10
0
where ΠR( ε, d1) = max πR( ε,
d)
and
R 2
d d1
π ( ε, d) = ( p− w + s) d−( p− v+ s) F( x| ε)d
x−sED. * *
Let ( d1, d ) denote the optimal solution to the retailer’s
problem. The solution structure is given in
Lemma 2.
Lemma 2 The argument of the retailer’s first period
problem, Π R( d1)
, is concave in the initial order
quantity d 1 . The optimal total order is
* *
⎧d1,
if ε ε(
d1);
* ⎪
d = ⎨ −1
⎛ p− w2+ s ⎞
⎪F
⎜ ε ⎟,
otherwise,
⎩ ⎝ p− v+ s ⎠
where
* ⎧ * p− w2+ s⎫
ε( d1) = sup ⎨ε : F( d1
| ε)
⎬ and the first-
⎩ p− v+ s ⎭
period optimal order quantity is implied by
*
ε ( d1
)
*
2 1
0
2 1
∫
w − w + [ p− w + s−( p− v+ s) F( d | ε)]d G(
ε)
= 0.
The proof is quite similar to that of Lemma 1. Here,
*
ε ( d1
) also partitions the market signals into two sets.
*
If ε ε(
d1
) , then the retailer will not set a second
order. Note the differences from the notations defined
in Section 1.1. Furthermore, for a given x, ε ( x) < ε ( x)
,
which coincides with the phenomenon that double
marginalization makes the buyer conservative. Compared
with the central planner, given the same inventory
level, the buyer in the decentralized system needs
a stronger market signal, which induces a more
optimistic demand estimate, to build up to that level.
The supplier’s problem is to choose production
∞
∫
0
d