Journal of Service Science and Management, 2011, 4, 27-34
doi:10.4236/jssm.2011.41005 Published Online March 2011 (http://www.SciRP.org/journal/jssm)
Copyright © 2011 SciRes. JSSM
27
Optimal Capacity Expansion Policy with a
Deductible Reservation Contract
Jianbin Li1, Minghui Xu2*, Ruina Yang3
1School of Management, Huazhong University of Science and Technology, Wuhan, China; 2* School of Economics and Management,
Wuhan University, Wuhan, China; 3Department of Industrial Engineering and Logistics Management, The Hong Kong University of
Science and Technology, Hong Kong, China.
Email: jimlee@amss.ac.cn, xu_mingh@yahoo.com.cn, rnyang@ust.hk
Received October 18th, 2010; revised November 27th, 2010; accepted December 2nd, 2010.
ABSTRACT
This paper investigates an optimal capacity expansion policy for innovative product in a context of one supplier and
one retailer. With a fully deductible contract, we employ the Stackelberg game model to examine the negotiation proc-
ess of capacity expansion in a single period. We first derive the retailers optimal reservation strategy and then char-
acterize the optimal capa city expansion policy for the supplier. We also investigate the impacts of reservation pr ice on
the optimal strateg y of capacity reservation and expansion as well as th e suppliers expected profits.
Keywords: Supply Chain, Capacity Expansion, Deductible Reservation Contract, Stackelberg Game
1. Introduction
This paper is concerned with capacity expansion policy
for innovative product in a setting of one supplier and
one retailer. Evidently, capacity management is an im-
portant issue for innovative product, which is often char-
acterized by volatile demand, short life cycle and long
lead time. In fact, due to the highly volatile demand, the
supplier often suffers from capacity shortage with the
adoption of exact capacity expansion policy. Therefore,
the retailer also loses revenues and his market reputation
is damaged. Despite the need for higher revenue and im-
proved service levels, the supplier may not be ready to
expand capacity proactively because of financial risks
due to higher capacity cost, long (capacity) lead time and
high demand volatility. However, if the retailer agrees to
share the financial risks by forward reservation, then the
supplier may be motivated to expand capacity more ag-
gressively. In this paper, the retailer reserves a capacity
prior to demand realization, and in exchange, the supplier
commits to have the “excess” capacity in addition to the
reservation amount. This kind of capacity expansion pol-
icy provides a win-win situation for both the supplier and
the retailer.
In the paper, we employ the fully deductible contract:
the retailer pays a fee upfront for each unit of capacity
reserved. When the retailer actually utilizes the reserved
capacity (i.e., placing a firm order), the reservation fee is
deductible from the order payment. However, if the re-
served capacity is not fully utilized within the specified
time period, the reservation fee associated with unused
capacity is not refundable. Interestingly, supplier’s an-
nouncement of excess capacity is a unique feature of the
deductible reservation (DR) contract.
We consider a two-level supply chain in which a sup-
plier offers an innovative product to one retailer facing a
stochastic demand. Throughout the paper, we assume
that the reservation price of the DR contract is exoge-
nously determined. Obviously, the negotiation process
for capacity expansion policy can be described as a
Stackelberg game in which the supplier is the leader and
the retailer is the follower. The objective of the current
paper is to design an appropriate capacity expansion pol-
icy that allows both the supplier and the retailer to opti-
mize their expected profits. Specifically, with an exoge-
nously given reservation contract, we firstly analyze the
retailer’s optimal strategy, and then study the supplier’s
optimal capacity expansion policy. Finally, we illustrate
the impacts of reservation price on the optimal capacity
expansion policy and provide with some managerial in-
sights.
The literature on capacity reservation is fairly abun-
dant. There are some earlier literature related to capacity
reservation mainly discuss the retailer’s optimal strategy.
Optimal Capacity Expansion Policy with a Deductible Reservation Contract
28
Sample references are [1-3]. Moreover, the work in this
filed can be divided into two main categories in terms of
retailer’s motivations to reserve capacity. The first cate-
gory considers the case in which the retailer reserves a
certain portion of the future capacity to achieve potential
cost reduction, for the sample references we refer readers
to [4-10].
The second category, including [11-13], studies the
problem that the retailer is motivated to offer early com-
mitment on the future capacity so as to ensure a certain
level of production availability. Moreover, Cachon and
Lariviere [14] investigate capacity contracting in the
context of supplier-buyer forecast coordination. Murat
and Wu [15] show that, by fully deductible reservation
contracts, the supplier has the incentive to expand the
capacity proactively. They conclude that as the buyer’s
revenue margin decreases, the supplier faces a sequence
of four profit scenarios with decreasing desirability. Jin
and Wu [16] propose a capacity expansion policy that the
supplier will have excess capacity in addition to reserva-
tion amount from the buyer. With a deductible reserva-
tion contract, they show that supply chain coordination
can be achieved and both players benefit from supply
chain coordination.
Evidently, our work on capacity expansion policy for
innovative product mainly differs from earlier work in
four aspects. First of all, the papers reviewed above
mainly discuss the retailer’s decision-making behavior,
with little concern on supplier’s perspective. We investi-
gate the supplier’s optimal strategy on capacity expan-
sion policy in addition to the retailer’s optimal strategy.
Secondly, most existing papers consider endogenous
wholesale price. However, in this paper we assume that
the wholesale price is determined exogenously by the
market or by earlier negotiations. Thirdly, the papers re-
viewed above assume that the supplier does not build any
capacity without retailer’s upfront commitment. How-
ever, with knowledge of market demand information, the
supplier has the incentive to build capacity even without
retailer's commitment. Finally, different from the per-
spective of supply chain coordination, we pay our atten-
tion on the players’ interactions through modeling the
process as a Stackelberg Game to derive optimal capacity
expansion policy.
The rest of this paper is organized as follows. Section
2 presents the model. Section 3 discusses the retailer’s
optimal reservation strategy with an exogenously given
reservation contract, and then describes the optimal ca-
pacity expansion policy for the supplier. Section 4 inves-
tigates the impacts of reservation price on the optimal
capacity reservation and expansion policy. Finally, Sec-
tion 5 concludes the paper.
2. Model Description
We consider a two-echelon supply chain in a single pe-
riod, in which a supplier (called her) sells an innovative
product to a retailer (called him). The retailer faces a
stochastic demand D, with the probability density func-
tion
f
and cumulative distribution function
F
.
Assume that the two parties hold symmetrical informa-
tion about the market demand and the cost structure. The
initial capacity level of the supplier is assumed to be zero.
In the model, we propose a capacity reservation contract
with fully deductible payment and assume that the reser-
vation price r is an exogenously given constant parame-
ter. To encourage the retailer to reserve capacity more
readily, we let rw
, where w is the unit purchasing
price charged by the supplier.
The sequence of the events is as follows:
1) At stage 0, the supplier announces the excess capac-
ity E, which is the amount of capacity the supplier pre-
pares to have in addition to (and regardless of) the re-
tailer’s reservation amount R.
2) Based on the excess capacity E and the demand
forecasting information, the retailer decides the reserva-
tion amount R and pays rR to the supplier.
3) After receiving R, the supplier expands her capacity
to R + E with marginal cost c.
4) At stage1, the demand D is realized, then the re-
tailer places an order
min ,DR E, with the unit pur-
chasing cost w. The selling price for each product is p
and any unmet demand will be lost.
5) The supplier deducts the amount of
min ,rDR
from the retailer’s purchasing cost, but keeps the amount
max, 0rRD.
6) The supplier salvages the residual capacity with unit
salvage value s.
Obviously, the above negotiation process for capacity
expansion policy can be modeled as a Stackelberg game,
in which the supplier is the leader and the retailer is the
follower. The supplier has complete visibility to the re-
tailer’s decision-making process. Suppose the two parties
in the supply chain are risk-neutral. The aim of this paper
is to characterize the optimal capacity expansion policy
that allows both the supplier and the retailer to maximize
their respective expected profits. In order to avoid trivial
cases, we assume that
s
cwp
. As salvaging re-
sidual capacity will incur additional logistical and proc-
essing costs, we assume that the salvage value s is strictly
less than the capacity expansion cost c.
3. The Optimal Capacity Expansion Policy
3.1. The Optimal Strategy for the Retailer
As reservation price r of the fully deductible reservation
Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 29
contract is an exogenous given constant parameter, the
retailer offers an early commitment on a certain portion
of future capacity just before the supplier expands capac-
ity. When the stochastic demand is realized, the retailer
places an order and the reservation cost can be deducted
from the purchasing cost. In this situation, at stage 0 with
the excess capacity E offered by the supplier, the retailer
determines the optimal reservation amount
ˆ
RE that
maximizes his expected profit.
 

 
0
0
ˆmin ,min ,
d
d.
M
R
RE
REp wDR ErR rDR
pwRERx Fx
pwREx Fx
 
 
 
(1)
The first term on the right-hand side of Equation (1)
denotes the retailer’s profit from selling the innovative
product, the second and the third term represents the re-
tailer’s effective reservation cost paid to the supplier so
as to ensure a certain level of availability.
Evidently, the more the effective reservation cost is,
the higher the available capacity level will be in future.
Lemma 1 Given the supplier’s excess capacity E, there
exists a unique optimal reservation amount
ˆ
RE to
maximize the retailer’s expected profit, which is deter-
mined by


 
ˆˆ
1rF REpwF REE

 

.
(2)
Proof. For any given E, taking the first and second de-
rivatives of with respect to R, we get that

ˆMR
  


2
2
ˆ
1
ˆ
0.
M
M
Rp wFRErFR
R
RpwfRE rfR
R
  


  
,
This implies that is strictly concave in R,
and hence the optimal reservation amount

ˆMR
ˆ
RE is uni-
quely determined by the first order condition given in
Equation (2).
Lemma 1 shows that, for a given E, the retailer’s op-
timal strategy is to reserve , which is uniquely
determined by Equation (2). It follows from Lemma 1
that the optimal reservation amount

ˆ
RE
ˆ
RE is mono-
tonically decreasing in E. To see this, differentiating
Equation (2) on both sides with respect to E and rear-
ranging items, we get that





ˆˆˆ
rfR EpwfR EER EEp

 

w. , which implies that

ˆ0fRE E

ˆ0
RE E. Therefore, for a certain level of future
demand, the larger the excess capacity is, the smaller the
possibility of disruptions in future supply will be. Since
the supplier prepares to set a higher level of excess ca-
pacity (i.e., increasing E), the retailer will reserve less. In
this scenario, the supplier undertakes more financial risks
in contrast with the retailer. Furthermore, we can see that

ˆ
F
RE 0
(and hence ) as ; and

ˆ0
RE0E

FRE
ˆ
pw pwr
 as . E
3.2. The Optimal Capacity Expansion Policy for
the Supplier
In anticipation of the retailer’s optimal response behavior
for any given E, we proceed to investigate the supplier’s
optimal capacity expansion policy that maximizes her
expected profit. By taking the retailer’s response function
ˆ
RE into account, the supplier’s expected profit can be
expressed as
 







ˆ
ˆmin ,
ˆˆ
ˆˆ
min,.
SEEw DREE
sREE DrRE
rDREcREE
 


(3)
where
ˆ
RE is an implicit function of E given in Equa-
tion (2). On the right hand side of Equation (3), the first
term is the supplier’s revenues from delivering the inno-
vative product to the retailer; the second term denotes the
supplier’s revenues from salvaging the residual capacity;
the third and the forth terms represent the retailer’s effec-
tive reservation cost paid to the supplier and the last term
is the cost of expanding capacity.
To derive an explicit expression of and make
future analysis easier, throughout the paper we mainly
discuss the scenario that the customer demand is uni-
formly distributed (other distributions can be analyzed
similarly).

ˆ
RE
We assume that the customer demand D is uniformly
distributed over the interval
0,
with 0
. Note
that the assumption of uniform distribution is a simplifi-
cation of reality, but it is sufficient to capture the main
features of capacity reservation policy and derive mana-
gerial insights in practice. Specifically, from Lemma 1
we get that,

 
ˆ,
pw E
RE pwr

 (4)
and the total capacity of the supplier after capacity ex-
pansion is

ˆ.
pw rE
RE Epwr

  (5)
Different values of E represent different capacity ex-
Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract
30
pansion strategies. means exact capacity expan-
sion policy; represents aggressive capacity ex-
pansion policy with potentially higher gains and
0E
0E
0E
represents overbooking. However, credibility with the
retailer is crucial for the supplier in the industry, so
overbooking is not considered as an acceptable business.
We will only consider the case where . Obviously,
the total capacity should be no more than the maximum
possible demand
0E
, i.e.,
ˆ
RE E
, which turns
out to be E
. Therefore, we will confine our analysis
on
0,E
in the rest of the paper.
By Equation (2), the supplier’s objective function can
be reformulated as:
 





22
ˆ
ˆ
ˆˆ
.
22
SEwcREE
RE ERE
ws r
 
 
(6)
Now, we characterize the properties of supplier’s ex-
pected profit function with an exogenously given reser-
vation price r (
0,rw), which are stated in the fol-
lowing lemma.
Lemma 2 Let

12pps

,

1pw ws
 , then we have the following results.
1) If 1
wp
, then is convex in E for any

ˆSE
0,rw;
2) If 1
wp
, then is convex in E for any

ˆE
S

ˆE
p

w
w
1
0,r

, and is concave in E for any
S

1,rpw



.
Proof. Since
 
ˆ
REEpwp wr  and



ˆ
REEErp wr
by Equation (4), we
get from Equation (6) that the first and the second deriva-
tives of with respect to E are

ˆSE
 


 

 

 


2
2
ˆˆˆ
ˆ
ˆ
ˆ
SRE E
ERER
wc r
EEE
RE E
RE E
ws E
rs crE
pwr pwr
pw wsr

 
 





 



E
(7)
and
  

2
2
22
ˆ
.
Srw sp ww sr
E
Epwr

 
 

Obviously, we know that if

2
rpwws 
1pw
, then we have

2
ˆ0
SEE 
2
; and if
1
rp
w
, then
22
ˆ0
SEE
.
Since 0rw
p, we know that at

1pw w

1
w
. Therefore, we get that 1) 1

pw w
 when
1
wp
, and hence
ˆSE is convex in E for any
0,rw; 2)
pw w
1
when 1
wp
, and hence
ˆSE is convex in E for any given
1
0,rp



w
and
ˆSE is concave in E for any given
1,rpw
w
.
Lemma 2 indicates that if the purchasing price is no
more than 1p
, then with any exogenously given r, the
supplier’s expected profit is decreasing in E as the excess
capacity is smaller than a critical point while increasing
in E as the excess capacity exceeds the critical point. On
the other hand, if the purchasing price is larger than 1p
,
then when r is not greater than a threshold
w
1,
the supplier’s expected profit is decreasing in E as the
excess capacity is smaller than a critical point while in-
creasing in E as excess capacity exceeds the critical point;
when the reservation price exceeds the threshold, the
supplier’s expected profit is increasing in E as the excess
capacity is smaller than a critical point while decreasing
in E as excess capacity exceeds the critical point.
p
Following from Lemma 2, we can obtain the supplier’s
optimal level of excess capacity with any given reserva-
tion price
0,rw.
Proposition 1 Let
22cs ps

and
2)pwcswc
 .
1) If pw )( 21
, then the supplier will set the
optimal excess capacity ˆ0E
;
2) If
wp


12
, then

 
 
2
2
2
1
0, if 0,,
ˆ
,if ,.
()
rpw
Ewcr pwrpww
wsr pw



 





 

Proof. From Equation (7), we have that
 


 

2
0
2
2
2
ˆ
,
S
E
Ersc r
Epwr
pwr
pw wsr
rw crpw
pwr
 




 


 
ˆ
0.
S
E
Ersc
Epwr
 

We consider the following cases by noting that 12
.
Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 31
1) 1
wp
.
For this case, we get that
1
wpw
 and hence
is convex in for any given

ˆSEE
0,rw
p
(by
Lemma 2). Moreover, 1
w
implies that

2
pw

0wsr. Then, it follows from Equation (7) and
s
c that

ˆ
 0
EE 
S in
0,E
ˆ
E
. Therefore,
the supplier obtains her maximum profit at .
0
2) .

11
p

2
For this case, we get that
pw

12
pw wpw

 

ˆEE
.
From Lemma 2, we know that is convex in
S
with a given . Similar to case (a), we
1
0,rp

w
get that the optimal excess capacity is .
ˆ0
E
ˆ
For any given , is con-

1,rpw

w

SE
cave in E (Lemma 2(2)). Since , we
2
rw pw
 
know that

0
ˆ|0
SE
EE
 , which implies that the
optimal excess capacity is ˆ0
E
.
3) .

12
wp



w


12
p
implies that

21
wpw p

.
w Following from Lemma 2, we know that
ˆES
is convex in E when
1
0,rp


,wpw

w
and concave in
E when . For both cases,
12
rp
since is decreasing over

ˆSE
0,E
by noting
that

0
ˆS
 |0
E
EE
 , we can conclude that the op-
timal excess capacity is .
ˆ0
E
When , is concave in E,

2,rpw

w

ˆSE
and

0
ˆ0
SE
EE
 . Therefore, the optimal excess
capacity can be derived from the first order condi-
tion determined by Equation (7), which is
ˆ
E
 

2
1
ˆ.
wcr pw
Ewsr pw
 



In conclusion, 1) when , the opti-
mal excess capacity is for any exogenously given
12
0wp

 
0
ˆ
E
0,rw; and 2) when
12
wp

 , we have

 

2
2
2
1
0,if 0,,
ˆ
,if ,.
rpw
Ewcrpwrpw
wsrpw


 


w


The proof is completed.
Obviously, Proposition 1 clearly implies that the opti-
mal capacity expansion policy for the supplier is to adopt
the exact capacity expansion policy if
12
w

p
.
Moreover, the intuition underlying Proposition 1 is clear.
If the purchasing price is no more than a threshold---
, which means that the retailer’s marginal
profit by selling one unit of innovative product is larger
than a critical value---
12
p

12
1p


ˆ0E
, then the retailer
has an incentive to reserve a larger amount capacity be-
cause the revenue loss due to capacity shortage is very
big, and the retailer is willing to undertake more financial
risk for capacity expansion to ensure a higher level of
capacity availability. By observing this, the supplier be-
lieves that the retailer’s reservation amount is large
enough to meet the future demand and thus take the exact
capacity expansion policy with .
On the other hand, if the purchasing price exceeds the
threshold
12
p

, which means the retailer’s mar-
ginal profit is smaller than the critical value, then the
retailer is encouraged to reserve more to ensure a higher
level of capacity availability in future with a smaller res-
ervation price
pw
2. In this situation, the sup-
plier will also adopt exact capacity expansion policy with
r
ˆ0E
. However, when the reservation price is larger
than
pw
2
, the retailer will reserve less. To avoid
future capacity shortage, the supplier will expand the
capacity aggressively with . Therefore, the sup-
plier and the retailer’s optimal strategies with any ex-
ogenous constant
ˆ0E
0, wr can be summarized as the
following proposition.
Proposition 2 1) If
12
wp

 , then ˆ0E
and
ˆˆˆ
RREpw pwr
.
 
2) If
12
wp

 , then


 
2
2
0, ,
ˆ
, .
pw rpw
pw
Rpw rpww
wsr
 





2
,if
,if
r
cs
p w


 
 

2
2
2
1
0, if
,if
0,,
ˆ
,.
rpw
Ewcrpwrpww
wsrpw










Hence, the supplier’s total capacity is
ˆˆ
RE p
 
wpwr
if
p
w
2
0,r
; and
 
22
r pw 
ˆˆ
RE wcs
 
r pww
 
 
if
2,rpw
w
.
Proof. The results follow directly from Proposition 1
and Equation (4).
4. The Impact of Reservation Price
In this subsection, we investigate the impacts of reserva-
tion price r on the optimal capacity reservation policy.
Specially, we would like to show how r affects the opti-
mal excess capacity , the retailer’s optimal reservation
amount and the supplier’s capacity level
ˆ
E
ˆ
R
E
Rˆˆ
.
Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract
32
4.1. The Case
12
wp


The following proposition presents the results of com-
parative statics of reservation price.
Proposition 3 If , then 1) is de-
12
wp


0,rp

w
ˆ
R
ˆ
R
creasing in r over ; and 2) is de-
2

ˆˆˆ
creasing in r and and
ERE
are both increasing in
r over .

2,rpw


w
Proof. For part 1), we know that and it is easy to
see that
ˆ0E
 
ˆˆˆ
RREpw pwr
 
w
is
decreasing
in r. We now consider part 2). For ,

2,rpw


we know that

2
ˆ
Rpwcswsrpw

 

 
11
cs rpw
 

(Proposition 2). Then,

1
2
1
ˆ
d0.
d
cs
R
rrpw

 



(8)
By noticing that 21
, it follows from Proposition 2
that for

2,rpw



w




21
2
1
ˆ
d0.
d
wc pw
E
rws
rpw






w
(9)
which indicates that is increasing in r over
. Combining the above two equations
and rearranging items, we get that
ˆ
E

2,rpw




2
2
2
1
ˆˆ
d
0.
d
RE pw cs
rws rpw





(10)
Hence, is creasing in r for the case
ˆˆ
RE
1
w

and .
2p

2,rpw


w
Proposition 3 shows that, under the condition that the
purchasing price w is larger than
and the
reservation price r exceeds 2, the supplier’s
optimal excess capacity increases while the retailer’s
optimal reservation amount decreases as r increases. This
is because the retailer takes more risks of over reserva-
tion and the supplier benefits more from the capacity
reservation. Even though, the supplier’s optimal capacity
level is still increasing in r since the decreasing
rate of is smaller than the increasing rate of .
12
p


pw
ˆˆ
RE
ˆ
Rˆ
E
Corollary 1 If , then for
12
wp


w
2,rpw

, the supplier’s optimal expected pro-
fit is decreasing in r.

ˆ
SE
Proof It follows from Proposition 2 and Equation (10)
that




 
 





 

 
2
2
2
2
2
1
22
2
2
11
42
3
3
1
ˆˆ ˆˆ
d
d
.
wsRE RE
wc r
wcrpw
wc wswsrpw
pw cs
ws rpw
pw cspw cs
wsrpw ws rpw
pw cs
ws rpw

 















 








From Equation (8) and Proposition 2, we have
 



 
1
22
1
22
3
2
1
ˆˆ
d
d
.
rp wc sc s
rR R
rwsrpwrpw
rp wc s
ws rpw

 
 
 


 

Therefore, we get that






 

 
2
42
3
3
1
22
3
2
1
22
2
2
1
ˆˆˆ
dd
dd
ˆˆ ˆ
d
d2
2
SEwsRE
wc
rr
rR RR
r
pw cs
ws rpw
rp wc s
ws rpw
pw cs
ws rpw




 














 
ˆˆ
RE
22
2
2
1
0.
2
pw cs
ws rpw

 


Hence, when
12
w

 p

ˆ
SE, is decreasing
in r for
,rpww
2
.
From Corollary 1 we get that the retailer’s optimal
expected profit
ˆ
MR is increasing in r by noting that
the total supply chain profit is unrelated to r. In this
situation, the high reservation price results in much low
reservation amount of capacity. Consequently, the sup-
plier needs to build excess capacity to match the demand
Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 33
in the future. However, the increasing costs of building
up the capacity have a negative effect on the supplier’s
optimal expected profit, that is, the supplier’s optimal
expected profit is decreasing in r. Although, the expected
profit for the entire supply chain can be increased since it
will alleviate the effects of double marginalization. In
view of this point, it is advised that the supplier should
not to choose a higher reservation price in a decentralized
supply chain.
If we choose the reservation price r in the interval
2
0, pw
ˆ
, then it follows from Proposition 2 that
0E. Taking derivative of with respect to r,
we get that

ˆ
SE













2
2
22
3
ˆˆ
dˆˆ
d
dd
2
2
222.
SEwsrR RR
wc
rr
wsr pw
wc pwr
pw pw
pwr pwr
pw
pwr
pw crpwpw cs




 




 
 






 

  


2
(11)
Let

322 2pw cs pw c
 . It can be ve-
rified that 23
, therefore, for this case,
0
S
is
increasing for smaller r () and decreasing
for larger r (32
). For smaller r,
the supplier can benefit from the reservation since the
retailer decreases his amount of reservation capacity and
undertakes some risk of higher demand as r increases.
However, for larger r, the supplier may lose some profit
when r is increased since the reservation capacity is de-
creased too much.
3
rp

r pw
 
w

pw

In summary, the optimal profit for the supplier is uni-
modal in r when
12
w


3
r
p
w
and the optimal res-
ervation price can be set to

p
.
4.2. The Case
12
wp

When
12
w


ˆ0Ep
, the supplier will not expand her
capacity, i.e., . For the retailer, the optimal reser-
vation amount is

ˆ
Rpwpwr
 . If the res-
ervation price approaches zero, the retailer would set the
reservation amount at the highest possible demand λ.
However, as r increases, the retailer’s optimal reservation
amount will decrease.
Proposition 4 If
12
wp

 , then (and
) is decreasing in r. And the supplier’s expected
profit
ˆ
R
ˆˆ
RE
)0(
S
is unimodal.
Proof. Since ˆ0E
, it is easy to see that
ˆˆˆ
E
RR pwrpw
 is decreasing in r.
The derivative of
0
S
with respect to r is the same
as Equation (11). By noting that 23
and
2pw
w
, it follows from Equation (11) that, if
3pw
w
, then
0
S
is increasing in r over
0, w; and if
pw
2
3
wp

w, then
0
S
is increasing in r over

pw
3
0,
and decreasing in
r over
pw
3,w
.
For the case
12

w p

0
S
, is increasing
for smaller r (
3
rpw
) and decreasing for larger r
(
3p
 
r). For smaller r, the supplier can benefit
from the reservation since the retailer decreases his
amount of reservation capacity and undertakes some risk
of higher demand as r increases. However, for larger r,
the supplier may lose some profit when r is increased
since the reservation capacity is decreased too much.
w
In the previous analysis, we implicitly assume that the
supplier always accept the retailer’s capacity reservation.
However, the deductible reservation contract can be
conducted only if the supplier could earn some profits.
Now we identify the condition under which the supplier
has an incentive to accept the retailer’s capacity reserva-
tion. For this case, if the supplier accepts the retailer’s
reservation, then the supplier’s optimal expected profit is
 





2
ˆ
ˆ
2
ˆ
2
ˆ
2 2.
R
wc
R wsr
wsr pw
Rwc pwr
R
pwr
pw crwcspw


 






 
0
2
S

Therefore, the supplier accepts the retailer’s reserva-
tion only if
 
22pw crwcspw 0

, i.e. ,
22rcwspwpw 
w
c
c
. This condition holds
if the purchasing cost is high relative to the capacity
building cost . Then the supplier has an incentive to
accept the retailer’s capacity reservation; otherwise, the
supplier will raise the reservation price r or unit purchase
price w so that she can obtain some profits.
5. Concluding Remarks
Capacity management plays a significant role on innova-
tive product. In the paper, the capacity expansion policy
not only provides with a risk-sharing mechanism for both
the supplier and the retailer, but also improves the re-
tailer’s potential revenue. Specifically, we propose a
fully deductible contract where the retailer reserves fu-
ture capacity with a fee that cab be deducted from the
Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract
Copyright © 2011 SciRes. JSSM
34
purchasing price. Additionally, the supplier’s ex ante an-
nouncement of “excess” capacity is a unique feature of
the deductible reservation contract. Given the reservation
contract, we figure out the optimal capacity expansion
policy and also study the effects of reservation price on
the optimal strategy as well as the supplier’s optimal
profit. Finally, we address the issues of how to set the
reservation price from the perspective of the supplier in
different situations.
6. Acknowledgements
This research was partially supported by the NSFC Grant
Nos. 70901029 and 70901059, and the Fundamental Re-
search Funds for the Central Universities (Grant No.
105-275171).
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