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

TSINGHUA SCIENCE AND TECHNOLOGY 

ISSN 1007-0214 22/22 pp570-580 

Volume 13, Number 4, August 2008 

Coordination by Option Contracts in a Retailer-Led Supply Chain 

with Demand Update * 

WANG Xiaolong (王小龙), LIU Liwen (刘丽文) ** 

School of Economics and Management, Tsinghua University, Beijing 100084, China 

Abstract: This research examines how to use an option contract to coordinate a retailer-led supply chain 

where the market information can be updated. Based on Stackelberg game theory, we build a mode with one 

supplier and one retailer in which the retailer designs contracts to coordinate the supplier’s production in a 

two-mode production environment. This focuses on an option contract that consists of two option prices and 

one exercise price. By theoretical analysis and numerical example, we find that such a contract can coordinate 

the supplier and retailer to act in the best interest of the channel. The optimal pricing conditions are 

given as follows: First, option prices should be negatively correlated to the exercise price and should be in a 

relevant range. Second, the first-period option price should be no greater than the second-period price and 

should be linearly correlated to the second-period option price when the latter is beyond some threshold. The 

results show that such option contracts can arbitrarily allocate the extra system profit between the two parties 

so that each party is in a win-win situation. 

Key words: supply chain coordination; option contracts; retailer-led; demand update 

Introduction 

Contemporary business puts more and more emphasis 

on channels, shifting power from supplier to retailer at 

the downstream end of a supply chain, which is the 

closest to the customers. Messinger and Narasimhan [1] , 

Ailawadi [2] , and Bloom and Perry [3] offer empirical 

evidence on how power has transferred in several international 

retailing markets. Such shifting enables the 

retailers to lead the supply chain in some sense. They 

then begin to raise stringent requirements on the suppliers’ 

production, compared with the traditional case 

where the supplier dominates the supply chain and coordinates 

the retailer’s ordering behavior. Another 

Received: 2006-10-24 

* Supported by the National Natural Science Foundation of China 

(Nos. 70532004 and 70621061) 

** To whom correspondence should be addressed. 

E-mail: liulw@sem.tsinghua.edu.cn 

Tel: 86-10-62783553; Fax: 86-10-62785876 

aspect is that the market situation often changes greatly 

long after the retailer has set an order. Some retailers, 

like Wal-Mart, deal with such problems by ordering 

less but more frequently. However, this volatile market 

puts great pressure on the suppliers who must make 

wise production/capacity building decisions to cope 

with the irregular orders which mirror a volatile market. 

Unfortunately, quite often suppliers fail to do that. In 

an effort to reduce mismatches between supply and 

demand, many industries, like the fashion industry (as 

reported by Fisher et al. [4] ) and the consumer electronics 

industry, are moving to tighten coordination across 

their supply chains. One important initiative for suppliers 

is to develop faster, but typically more expensive, 

production modes that are capable of producing a second 

run of products closer to the selling season. This 

mode allows both sides to take advantage of updated 

demand forecasts. Also, the retailer can better manipulate 

the supplier’s production so that channel coordination 

is reached. This phenomenon has become more

WANG Xiaolong (王小龙) et al:Coordination by Option Contracts in a Retailer-Led … 

common with advances in computer-based manufacturing 

technologies which significantly reduce production 

run times and product changeover times. 

However, the addition of another production mode 

is not enough to reach channel coordination due to 

double marginalization which refers to the fact that a 

party’s relative cost structure becomes distorted when 

a transfer price is introduced within a supply channel. 

For example, introducing a wholesale price into a single-stage 

newsvendor model causes the ratio of the 

buyer’s underage and overage costs to fall short of the 

channel ratio. As a result, the buyer orders less than the 

channel optimal quantity. In a two-stage production 

environment, double marginalization threatens to impact 

not only the total production quantity but also 

proper allocation of production between periods. The 

double marginalization concept was first introduced in 

the economics literature by Spengler [5] , who raised the 

issue in terms of inappropriate prices rather than inappropriate 

quantities. Tirole [6] provided further details 

on the phenomenon. Many works in the operations 

management field, like those of Lariviere and 

Porteus [7] , have shown that price-only contracts generally 

fail to coordinate the supply chain due to double 

marginalization. However, other contracts have proven 

to efficiently coordinate the supply chain. These contracts 

include buyback contracts [8] , revenue sharing 

contracts [9] , sales rebate contracts [10] , quantity discount 

contracts [11] , and quantity flexibility contracts [12] . 

The goal of this paper is to provide guidance in designing 

an option contract in a retailer-led supply chain, 

in which the supplier’s production needs coordinate 

with the existence of a market demand information 

update. The researches on option contracts in supply 

chain management are mainly divided into economic 

and operational areas. Typical economic works include 

Newbery [13] and Wu et al. [14] Typical operational 

works include Donohue [15] and Barnes-Schuster et 

al. [16] Our contract pricing scheme builds most closely 

upon the work of Barnes-Schuster et al. [16] , which considers 

a combination of committed orders and options. 

Barnes-Schuster et al. [16] analyze a two-period model 

with correlated demand. The present analysis studies a 

two-period model with demand information update in 

which the market demand does not realize until the end 

of the second period. 

Specifically, consider one supplier and one retailer. 

571 

At the beginning of the first period both parties anticipate 

a demand information update that will appear at 

the beginning of the second period and they take this 

into account in their first-period decisions. The coor- 

dinating contract is designed as follows: The option 

contract has three parameters for two option prices 1 o 

and o 2 and one exercise price e. The option prices 

serve as the unit compensation for the supplier’s production 

quantities. When the realized demand is beyond 

the retailer’s total order, the retailer will purchase 

the supplier’s excess inventory, if any, to meet the 

market demand at the unit price e. With this contract, 

the retailer’s decision is to determine orders at the beginning 

of each period and the supplier’s decision is to 

determine the production allocation between the two 

periods. This project aims to find an optimal set of 

contracts to coordinate the channel and to realize a 

proper profit allocation. 

1 Two-Period Model 

The two-period model depicts the dynamics of a retailer 

who purchases from one main supplier and markets 

these products to a number of retail outlets, reselling 

at retail price p for a single selling season. 

Unlike traditional coordination models, the retailer in 

the model leads the supply chain and designs a proper 

contract to coordinate the supplier’s production quantities. 

The specific contract form will be investigated. 

The retailer utilizes two periods of demand information 

to determine the order quantities. During the first period, 

the demand prediction is relatively less accurate 

with a distribution function denoted as F(D). At the 

beginning of the second period (market demand not yet 

realized), more current market information is available 

which helps the retailer fine-tune the forecast where an 

effective market signal will be observed depending on 

specific industry situations and the lengths of the two 

periods which may not necessarily be equal. This updated 

information is the market signal ε. Let G(·) denote 

its estimated cumulative distribution function and 

let F(·|ε) denote the demand distribution function conditional 

on the market signal. Assume that ε is expressed 

in the same units as the total demand and is 

also non-negative. Market demand is stochastically 

increasing in the market signal, i.e., given x, F( x| ε h ) < 

F( x| ε l ) for all ε h > εl 

(ε h and ε l are two arbitrary

572 

market signals). Assume that all distributions are continuous, 

invertible, doubly differentiable, and independent 

of the wholesale and retail prices. Furthermore, 

they are all common knowledge to both parties from 

the beginning of period 1. 

At the beginning of period 1, the retailer orders 1 d 

units at a unit wholesale price w 1 based on a rough 

knowledge of the demand distribution. Once ε is 

observed, the retailer will adjust the order quantities by 

ordering another d2 units at a higher unit wholesale 

price w 2 . Note that the retailer’s second order is contingent 

on the specific market signal. If the signal exhibits 

a rather weak demand in the future, the retailer 

will not set another order, i.e., d 2 = 0 . 

The supplier produces over two time periods in response 

to the retailer’s orders, with orders filled in one 

shipment before the selling season begins. The supplier 

has two different production modes. One is the cheap 

mode which incurs a unit production cost c 1 , and the 

other is the expensive mode which incurs a unit production 

cost c2 > c1. 

Assume that production costs 

include the cost of holding inventory until delivery and 

the cost of delivery itself. Other cost parameters include 

a shortage penalty s paid by the retailer to the 

customers for each unit of unfilled demand, and a salvage 

value v received for each unit of inventory remaining 

at the end of the season no matter to which 

side it belongs. Suppose that all these prices and costs 

are exogenous and the analysis is limited to the most 

relevant and interesting case where p > w2 > w1 > c2 

> 

c1 > v. 

Next assume that the lead time for the cheap 

mode (and the timing of the market signal) is such that, 

to ensure completion before the start of the selling 

season, the supplier must begin production before observing 

the market signal. In contrast, the lead time for 

the expensive mode is short enough to allow production 

after the market signal is observed. Therefore, the 

supplier’s problem is to allocate production quantities 

between the two modes. During the first period, the 

supplier needs to determine how many products, denoted 

as q 1 , would be produced among the cheap 

mode. In the second period after a market signal is ob- 

served, the supplier has to determine how many products, 

denoted as q 2 , will be produced among the expensive 

mode, if necessary. Denote d = d1+ d2 

and 

Tsinghua Science and Technology, August 2008, 13(4): 570-580 

q= q1+ q2. 

1.1 Centralized system performance: 

A benchmark 

To provide a benchmark, first consider the problem 

where the supplier and the retailer are each owned by a 

risk-neutral entity. The owner aims to maximize his 

own expected profit by choosing first and second period 

production quantities ( q1, q 2) 

which solve the 

following two-period optimization problem. 

∞ 

c c 

max Π ( q1) =− cq 1 1+∫Π ( ε, q1, q2)d G( 

ε) 

(1) 

q10 

0 

c c 

where Π ( ε, q1, q2) = max π ( ε, 

q1, q2) 

and 

q20 

c 

π ( ε , q , q ) = − c q + pEmin{ D, q + q } + v( q + q − 

1 2 2 2 1 2 1 2 

Emin{ D, q1+ q2}) −sE[ D− min{ D, q1+ q2}] 

(2) 

c 

Here, Π ( ε , q1, q2) 

represents the second-stage 

problem after the market signal is observed and the 

demand forecast is updated accordingly. To express 

the second-stage problem independently of q 1 , substitute 

q for q1+ q2 

and rearrange the terms. This 

new formulation yields the same objective solution as 

the problem in Eq. (1). 

∞ 

c c 

max Π ( q1) = ( c2 − c1) q1+∫Π ( ε, q1)d G( 

ε) 

(3) 

q10 

0 

c c 

where Π ( ε, q1) = max π ( ε, 

q) 

and 

qq c 

π ε 2 

ε 

0 

1 

q 

( , q) = ( p− c + s) q−( p− v+ s) F( x| )dx−sED 

(4) 

Note that given q 1 , the owner faces a newsvendor 

problem in period 2 but with an initial inventory and 

c c 

an adjusted demand distribution. Let ( q1, q ) denote 

the optimal solution to the central problem. The solution 

structure is given by Lemma 1. 

Lemma 1 The argument of the central first period 

c 

problem, Π ( q1) 

, is concave in the initial production 

quantity q 1 . The optimal total production quantity is 

c c 

⎧q1, 

if ε ε( 

q1); 

c ⎪ 

q = ⎨ −1 

⎛ p− c2+ s ⎞ 

⎪F 

⎜ ε ⎟, 

otherwise, 

⎩ ⎝ p− v+ s ⎠ 

c ⎧ c p− c2+ s⎫ 

where ε ( q1) = sup ⎨ε : F( q1 

| ε) 

⎬ and the 

⎩ p− v+ s ⎭ 

first-period optimal 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 

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

574 

quantities ( q1, q 2) 

that maximize his own expected 

profit, subject to the retailer’s ordering behavior. His 

decision on the second-stage production is relatively 

straightforward, i.e., 

* * * 

q2 = max{ d − q1,0} 

. 

The difficulty lies in his decision on q 1 . Obviously 

he should at least produce d 1 , but should he produce 

more? The answer depends on his estimate on the likelihood 

of the retailer’s second order. For such a decision 

his problem is 

ε ( d1) 

max Π ( q ) = wd − cq + v( q − d )d G( 

ε) 

+ 

q1d1 S 1 1 1 1 1 

0 

1 1 

ε ( q1 

) 

∫ ε ( d1) 

∞ 

wd 2 2 vq1 d G 

∫ 

ε ( q1 

) 

∫ 

[ + ( − )]d ( ε ) + 

[ wd −c( d−q)]d G( 

ε ). 

2 2 2 1 

The solution to this problem is given by Lemma 3. 

Lemma 3 Given the retailer’s ordering behavior, 

the supplier will maximize his expected profit by 

setting 

* * 

q1 d1 q′ 1 

* * * 

= max{ , } and q2 = max{ d − q1,0}, 

where 1 q′ is implied by c2 − c1 

G( ε ( q′ 

1)) 

= . 

c2−v Because this problem has a newsvendor structure, 

the first order condition works. The proof is straightforward 

so the details are omitted. These problems 

have reviewed the benchmark and decentralized uncoordinated 

system performance. Although the precise 

relationships between * * 

d 1 , q 1, 

and c 

q 1 depending on 

factors like the ratio of the two production costs, the 

ratio of the two wholesale prices, and the estimate of 

the market signal cannot be determined, the channel is 

certainly not coordinated. This is implied by Lemma 2 

which states that for a given market signal the retailer’s 

total order quantity is less than the optimal total inventory 

required in a centralized system. The next section 

focuses on how an option mechanism coordinates 

the supply chain and realizes profit allocation. 

2 Channel Coordination 

To coordinate the channel, the retailer must choose the 

proper contract parameters so that the supplier’s production 

quantities in each period are equal to the opti- 

* c 

mal solutions in the centralized system, i.e., q = q 

and 

1 1 

* c 

q = q . To do so the retailer must give incentives 


to the supplier to overproduce in each period. Recall 

that q1d1 and q2 max{ d − q1,0} 

. Thus d 1 and 

max{ d − q1,0} 

are the minimum production number 

for the supplier’s production in each period. The coordinating 

contract is defined as follows: The option contract 

has three parameters of two option prices o 1 and 

o 2 and one exercise price e . The option prices are 

the unit compensation for the supplier’s production 

quantities beyond the minimums in each period. When 

the realized demand exceeds the retailer’s total order, 

he will purchase the supplier’s excess inventory, if any, 

to meet the market demand at a unit price e . 

The sequence of events is summarized as follows: 

(1) At the beginning of period 1, the retailer sets a 

firm order d 1 . The retailer also announces an option 

contract with option prices o 1 and o 2 and exercise 

price e. The supplier, in turn, decides to produce 

q1 d1 

of goods in the cheap mode which will produce 

an income of o1( q1− d1) 

. 

(2) At the beginning of period 2, after the market 

signal is observed, the retailer orders d 2 . The supplier 

selects a second-period production, q2 max{ d− q1,0} 

, 

that involves the expensive mode and produces an income 

of o2( q2 −max{ d − q1,0}) 

. 

(3) At the beginning of the selling season, after the 

demand is realized, the retailer purchases additionally 

m= min{max{ D−d,0}, q− d} 

number of goods at 

unit price e . The supplier delivers all that the retailer 

wants. The problem assumes that the supplier is obligated 

to fill the retailer’s entire order and can neither 

sell directly to the final customer (i.e., the retailer’s 

customer) nor supply inventory in excess of the retailer’s 

total order quantity. 

With the option contract, the supplier’s problem 

becomes 

max Π ( q ) = −( c − o ) q + ( w − o ) d + 

q1d1 S 1 1 1 1 1 1 1 

∫ 

0 

∞ 

Π ( ε, q , q )d G( 

ε), 

S 1 2 

where ΠS( ε, q1, q2) = max πS( ε, 

q1, q2) 

and 

q2max{ d−q1,0} π ( ε , q , q ) = − c q + o ( q −max{ d − q ,0}) + 

S 1 2 2 2 2 2 1 

d 

wd + vq ( + q − d)d F( x| 

ε ) + 

∫ 

2 2 

0 

1 2 

q1+ q2 

d 

∫ 

[( ex− d) + vq ( − x)]d Fx ( | ε ) +


∞ 

∫ eq ( 1+ q2 −d)d 

F( x| ε ). 

q1+ q2 Here, ΠS( ε , q1, q2) 

represents his second-stage problem. 

After making the supplier’s second-stage problem 

independent of q 1 and after some algebra, the sup- 

plier’s problem is 

max Π ( q ) = ( c −c − o + o ) q + ( w − o ) d + 

q1d1 S 1 2 1 2 1 1 1 1 1 

∫ 

0 

∞ 

Π ( ε, q )d G( 

ε), 

S 1 

where ΠS( ε, q1) = max πS( ε, 

q) 

and 

qmax{ d, q1} 

π S( ε, 

q) = ( e− c2+ o2) q+ w2d2−ed−o2max{ d−q1,0} − 

q 

( e−v) ∫ F( x| ε )d x . 

d 

N N 

Let ( q , q 1 ) denote the solution to this problem, and 

then we have Proposition 1. 

Proposition 1 Given an option contract specified 

by the retailer, the supplier’s optimal total production 

quantity is 

N ⎧ N −1⎛e− 

c2 + o2 

⎞⎫ 

q = max ⎨q1, F ⎜ ε ⎟⎬ 

⎩ ⎝ e−v ⎠⎭ 

. 

The proof is relatively straight forward since the supplier’s 

problem follows a newsvendor structure. 

N 

Proposition 1 shows clearly that to coordinate q = 

c 

e− c2 + o2 p− c2 + s 

q , it is necessary to set = , i.e., 

e−v p− v+ s 

p− v+ s 

e= p+ s− o2 

(5) 

c2−v Based on Eq. (5), e v e+ o > c to make 

> and 2 2 

−1⎛ e− c2+ o2 

⎞ 

F ⎜ | ε ⎟ meaningful. Economically these two 

⎝ e−v ⎠ 

conditions prevent the supplier from suffering losses. 

e+ o2is 

all the revenue that the supplier can get from 

one unit of product if it can be sold after the market 

demand is realized. If e+ o2 < c2, 

the supplier has no 

incentive to overproduce in the second period, and thus 

channel coordination is never reached. Since e= p+ s− 

p− v+ s 

o2 

, the two conditions are equivalent with 

c2−v o2 < ( c2 − v) 

. Therefore, o2 < ( c2 − v) 

from now on. 

Next, consider the retailer’s problem. In all the fol- 

p− v+ s 

lowing parts, assume that e= p+ s− o2holds 

c2−v except where specified. 

The retailer’s problem is 

575 

∞ 

max ΠR( d1) =−o1( q1−d1) − wd 1 1+∫ ΠR( ε, d1) d G( 

ε), 

d10 

0 

where ΠR( ε, d1) = max πR( ε, 

d) 

and 

dd1 π R( ε , d) = −w2d2 −o2( q2 −max{ d − q1,0}) 

+ 

pE min{ D, q} + v( d −Emin{ D, d}) 

− 

eE min{max{ D −d,0}, q −d} − 

sED ( − Emin{ Dq , }). 


max Π ( d ) = ( o − o ) q + ( w − w + o ) d + 

where 

d10 

R 1 2 1 1 2 1 1 1 

∫ 

∞ 

Π 

0 

R( ε, d1)d G( 

ε), 

ΠR( ε, d1) = max πR( ε, 

d) 

and 

dd R 2 

1 

π ( ε, d) = ( e−w ) d −( e− v) F( x| ε)dx+ 

o max{ d − q ,0} + ( p+ s−e−o ) q− 

2 1 2 

∫ 

( p+ s−e) F( x| ε )dx−sED. 

0 

q 

N N 

Let ( 1 , d d ) denote the optimal solution to the retailer’s 

problem. Proposition 2 describes the retailer’s 

ordering behavior. 

e−w2 e− w2 + o2 

Proposition 2 Denote α = , β = , 

e−v e−v N −1 

N 

and γ ( ε) = max{ d1, F (max{ α,0}| ε)}. 

Define ˆ( ε d1 

) = 

N 

N 

sup{ ε: Fd ( 1 | ε) max{ α,0}} 

and allow ˆ( ε d1 

) = +∞ 

if ε max is infinite. For a given option contract, the 

optimal ordering behavior for the retailer has the following 

structure: 

(1) After the market signal ε has appeared, the optimal 

total order quantity is determined as follows: 

(i) If Fq ( 1 | ε ) β , 

N 

d = γ ( ε ) ; 

(ii) Otherwise, 

−1 

⎧ γε ( ), if πR( ε, F (max{ α,0}| ε)) 

> 

⎪ N −1 

d = ⎨ πR( ε, F ( β | ε)); 

⎪⎩ −1 

F ( β | ε), 

otherwise. 

(2) The optimal first-period order quantity is given 

by 

N ˆ( ε d1 

) 

N 

2 1 1 

0 

2 1 

∫ 

w − w + o + [ e−w −( e− v) F( d | ε)]d G( 

ε) 

= 0 

when e+ o1 > w1. 

Otherwise, N 

d1 ≡ 0 . 

Corollary For some second-period option price(s) 

−1 

o 2 such that β > 0 and set A= { ε : πR( ε, F (max{ α, 

−1 

0}| ε )) π ( ε, F ( β | ε))} 

is not empty, there exists a 

R 

∫ 

0 

d

576 

signal T ( 2) 

o ε such that 

N −1 

only if T ( 2) 

o ε ε 

greater than 1 q and T ( 2) max o ε ε 

d = F ( β | ε) 

> q1if 

and 

 

N 

. Otherwise, d will never be 

= for such cases. 

Proof First, prove part (1). Omitting all the irrelevant 

parts, the retailer’s second-period problem can be 

rewritten as 

d 

⎧( e−w2) d −( e−v) F( x| ε )d x, 

⎪ ∫ 0 

⎪ if dq1; max πR( ε, 

d) 

= ⎨ 

dd d 

1 ⎪( e− w2 + o2) d −( e−v) F( x| ε )d x, 

0 

⎪ 

∫ 

⎪⎩ otherwise. 

Then, the unconditional optimal N 

d may not be 

unique but can be either 

−1 

F (max{ α,0}| ε ) or 

−1 

N −1 

F (max{ β,0}| ε ) or both. However, if d = F ( β | ε ) , 

−1 

there must be F ( β | ε) 

> q1, 

i.e., 0 < Fq ( 1 | ε ) < β , so 

that max{ d − q1,0} = d − q1 

makes sense. This completes 

the proof of part (i) when Fq ( 1 | ε ) β and 

N 

d = γ ( ε ) . 

Now consider 0 < Fq ( 1 | ε ) < β . The criterion is sim- 

N 

ple but clear. Whether d = γ ( ε ) or N 1 − 

d = F ( β | ε ) 

depends on which can bring more expected profit, i.e., 

−1 

⎧ γε ( ), if πR( ε, F (max{ α,0}| ε)) 

> 

⎪ N −1 

d = ⎨ πR( ε, F ( β | ε)); 

⎪⎩ −1 

F ( β | ε), 

otherwise. 

This completes the proof of part (1) in Proposition 2. 

Next, show the link between the retailer’s second-period 

expected profit and the market signal to 

estimate the optimal total order from the beginning of 

period 1 based on the market signal conditions, which 

is the core idea of the corollary. 

−1 

When β > 0, let π ( ε, F (max{ α,0}| ε)) = π ( ε, 

−1 

F ( β | ε )) . Note that 

R R 

d ≡ d when β 0 because 

N N 

1 

β α . Suppose that there exists a solution T ( 2) 

o ε , 

and then for T ( 2) 

o ε the retailer is indifferent towards 

the choice of 

1 − 

−1 

F (max{ α,0}| ε ) and F ( β | ε ) . Obviously, 

for stronger signals T ( 2), 

o ε ε > 

N −1 

d = F ( β | ε ) > 

q 1 . Otherwise, if such a T ( 2) 

o ε does not exist, or 

−1 −1 

equivalently, πR( ε, F (max{ α,0}| ε)) > πR( ε, 

F ( β | ε )) 

for any signals, define T ( 2) max o 

N 

ε = ε . Thus, d = 

−1 

F ( βε | ) > q1is 

impossible because the market signal 

cannot be greater than its maximum. 


Next consider part (2) which depicts the first-period 

ordering behavior. If o2=0, then the retailer’s second-period 

problem is given by the following: 

For any α , similar to the reasoning in proof of 

Lemma 1, 

d is implied by 

2 1 1 

N 

1 

N ˆ( ε d1 

) 

∫ 0 

2 

N 

1 

α , i.e., e w2 

N 

1 

w − w + o + [ e−w −( e− v) F( d | ε)]d G( 

ε) 

= 0. 

Particularly, when 0 , inequality 

Fd ( 

N 

| ε ) 0 holds for any signal ε , then ˆ( ε d1 

) 

must be equal to ε max , which is infinite in our model. 

Under such cases, we can rewrite the above equation 

as 

+∞ 

N 

2 1 1 

0 

2 1 

∫ 

w − w + o + [ e−w −( e− v) F( d | ε)]d G( 

ε) 

= 0. 


+∞ 

N 

1 1 

0 

1 

∫ 

e− w + o −( e− v) F( d | ε)d G( 

ε) 

= 0. 

Note that 0 

∞ 

N 

∫ Fd ( 

0 

1 | ε)d G( 

ε) 

1 

special cases when e+ o1 w1 

, then 

. There are some 

N N 

d = d1= 

0. 

Note that e− w1+ o1 e−v cannot succeed because 

it requires o1 w1−v, which would incentivize the 

supplier to produce infinitely in period 1 and, thus, is 

ridiculous. 

In summary, the optimal first-period order quantity 

d is given by 

N 

1 

N ˆ( ε d1 

) 

N 

2 1 1 

0 

2 1 

∫ 

w − w + o + [ e−w −( e− v) F( d | ε)]d G( 

ε) 

= 0 

N 

when e+ o1 > w1. 

Otherwise, d1 ≡ 0 . □ 

Proposition 2 shows that the retailer’s ordering behavior 

is closely linked to the option price o 2 and the 

market signal ε . Different option prices result in different 

ordering behavior. Under some circumstances, 

the optimal total order may not even be unique. Signals 

such as ˆε and ε T can be used to categorize the retailer’s 

orders. These marking signals are important in 

models concerned with demand information update. 

The conclusion in the corollary is meaningful for disclosing 

the relationship between the retailer’s second-period 

profit curve and the market signal that optimal 

total order more than q 1 is possible if and only 

if T ( 2) max o ε ε < exists. T ( 2) 

o ε must be always greater 

than ε ( q1) 

for a given q 1 because the conservative 

retailer has incentive to choose an optimal total order 

greater than q 1 with T ( 2) 

o ε that he will never do


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

578 

factor in models considering demand update is how the 

market signal updates the demand information, i.e., 

how to define F( x| ε ) . Make the following assumptions 

related to the market signal: 

(1) The market signal does not change the demand 

distribution pattern, i.e., F( x| ε ) is still uniform. 

(2) The market signal only affects the mean of the 

demand distribution, not the variance. As the signal 

becomes stronger the mean shifts up accordingly. For 

uniform distributions, this implies that if F( x| ε ) ~ 

U( f1( ε ), f2( 

ε )), then f2( ε) − f1( ε) 

≡b−a and f1 ( ε ) and 

f2 ( ε ) are both increasing in ε . 

(3) A market signal as mathematically expected 

maintains the original demand distribution function. 

That is, if ε = 0.5t , then F( x| ε ) = F( x) 

. Moreover, 

assume 

F( x| ε )~ U( a−4(0.5 t−ε), b−4(0.5 t− 

ε)) 

. 

To avoid uninteresting cases, define a> 2t. 

Consider the following numerical example. Suppose 

the original demand distribution is uniform with 

a = 1000 and b = 2000 . The market signal is uniformly 

distributed between zero and t = 150. 

The cost 

parameters are c1= 50, c2= 65, w1= 60, w2= 80, v= 

20, 

s = 30, 

and p = 100 . 

For any o 2 , 

p− v+ s 

e= p+ s− o . 

(1) Calculate e according to 

(2) Solve for T ( 2) 

c2−v 2 

o ε . 

If β > 0 , εT( o2) 

= min{150, ε′ T}, 

where T 

ε′ is determined 

by 

−1 −1 

π R( ε′ T, F (max{ α,0}| ε′ T)) = πR( ε′ T, F ( β | ε′ 

T)) 

; 

If β 0 , T ( 2) 

150. o ε = 

⎡ 

(3) Calculate o 1 according to o1 = ⎢G( 

εT 

( o2)) 

− 

⎣ 

c2 − c1 ⎤ 

o2 

c2−v ⎥ . 

⎦ 

N 

(4) Solve for d 1 . 

N 

If e+ o1 w1, 

d1 ≡ 0 ; 

If e+ o1 > w1, 

substitute 

N 

N d1−[ a− 2 t+ ( b−a)max{ α,0}] 

ˆ( ε d1 

) = (which is 

4 

N N 

derived from Fd ( | ˆ ε( d )) = max{ α,0} 

) into 

1 1 

N ˆ( ε d1 

) 

N 

2 1 1 ∫ 0 

2 1 

w − w + o + [ e−w −( e− v) F( d | ε)]d G( 

ε) 

= 0. 


If ˆ( ε d ) < 150, 

the results can be directly used. 

If ˆ( ε d ) 150, 

then 

N 

1 

N 

1 

N 

d 1 is implied by 

150 

N 

2 1 1 

0 

2 1 

∫ 

w − w + o + [ e−w −( e− v) F( d | ε)]d G( 

ε) 

= 0. 

(5) Calculate each party’s expected profit. 

Figure 1 shows how the retailer’s first-period order 

changes as o 2 varies. As expected, the retailer sets 

N 

d ≡ 0 when 1 1 , e o w 

o 

1 

p− w1+ s 

c2v p− c1+ s 

+ or equivalently, 2 

( − ) = 39.375. 

Such an ordering behavior 

directly influences each party’s expected profit curve 

shown in Fig. 2. 

Fig. 1 Retailer’s optimal first-period order quantity 

Figure 2 compares each party’s expected profit after 

coordination with that before coordination which is 

shown by the benchmark curve. The powerful retailer, 

who acts as a contract designer, will never be worse off, 

but the results are not so optimistic. The supplier’s expected 

profit even becomes negative for some cases! 

Notably, with these settings, the first-best expected 

c 

profit Π = 221 209. 

Before coordination, however, 

Π R = 175 960 and Π S = 16 437. This leaves an improvement 

of 28 812 which is almost twice the supplier’s 

expected profit before coordination. 

Fig. 2 Expected profit for each party before and after 

coordination


Table 1 illustrates the profit allocation issues in detail. 

For these calculations, the new contract is feasible 

only when 34.8o2 43.8. 

For low o 2 in this range, 

each party becomes strictly better off. In particular, 

consider the best profit allocation situations for which 

each side gets a double-digit share of the extra profit. 

When o 2 is around 40, the supplier will get more 

than 60% of the extra system profit which is double 

what he earned before coordination. Such a rosy profit 

improvement surely offers a strong incentive for the 

supplier to accept the new contract. Most interestingly, 

such profit allocations correspond to o2 39.6 which 

N N 

implies e+ o1 w1 

and d = d1= 

0. Thus, order 

postponement (until the demand uncertainty is resolved) 

can be beneficial to each party. This is contrast 

to the traditional thought that suppliers always welcome 

as-early-as-possible orders. Our opinion is that 

suppliers hate late orders mostly because they make 

the work of arranging production schedules more difficult. 

However, a properly designed incentive mechanism, 

which both clarifies the quantity that the supplier 

should produce and brings extra profit, eliminates 

these negative concerns. Such a result, however, needs 

further proof from theory and practice. 

Table 1 Profit allocations between the two parties 

Extra profit allocation on supplier 

Second-period option price 

(% of improvement scope) 

0 100 

34.8 1 

… … 

39.6 67 

39.9 63 

40.2 58 

40.5 53 

40.8 48 

41.1 44 

41.4 39 

41.7 35 

42.0 30 

42.3 26 

… … 

43.8 3 

Note: For simplicity, we state here only the allocation status for the supplier. 

Its counterpart for the retailer can be calculated since the sum is equal 

to 1. 

4 Conclusions 

579 

This analysis considers the coordination of option contracts 

in a retailer-led supply chain where the retailer 

designs the contract and the supplier’s production 

needs to be coordinated with the contract. Consider the 

case where the supplier operates two modes of production 

and the retailer has the ability to update the demand 

forecast as the selling season approaches. Within 

this contract class, a set of pricing conditions are defined 

that align the supplier and retailer to act in the 

best interests of the channel. The analysis leads to 

some interesting results. 

First, option prices should be negatively correlated 

to the price and should be in a relevant range. This 

gives a trade-off between the higher unit compensation 

the supplier requires and the lower repurchasing price 

he will get. To assure that the option mechanism is an 

effective incentive, the option prices should be within a 

relevant range. The second-period option price should 

be no greater than the difference between the second- 

period unit production costs and the salvage value. 

Otherwise, the supplier will earn little due to the rapidly 

decreasing exercise price and the coordination will 

fail. 

Second, the first-period option price should be no 

greater than the second-period price and should be 

linearly correlated to the second-period option price 

when the latter is beyond some threshold. This result is 

true regardless of the specific functional form of the 

market demand or market signal. 

Third, the retailer’s ordering behavior is influenced 

by the dual effect of the predictive power of the new 

market information and option prices. Some extreme 

pricing cases may result in the ordering behavior being 

independent of the new market information. The influence 

pattern is so complex that there is no closed 

form solution to the retailer’s first-period optimal order 

quantity. Fortunately, this does not prevent exploring 

the optimal pricing conditions which are stated in 

Proposition 3. A numerical example is given for the 

profit allocation issues. An interesting result is that 

order postponement may be beneficial to both parties, 

though this needs further theoretical proof.

580 

References 

[1] Messinger P, Narasimhan C. Has power shifted in the 

grocery channel. Marketing Science, 1995, 14(2): 189-223. 

[2] Ailawadi K. The retail power-performance conundrum: 

What have we learned. Journal of Retailing, 2001, 77: 

299-318. 

[3] Bloom P, Perry V. Retailer power and supplier welfare: 

The case of Wal-Mart. Journal of Retailing, 2001, 77: 

379-396. 

[4] Fisher M, Hammond J, Obermeyer W, et al. Making supply 

meet demand in an uncertain world. Harvard Business 

Review, 1994, 72(3): 83-93. 

[5] Spengler J. Vertical integration and antitrust policy. Journal 

of Political Economy, 1950: 347-352. 

[6] Tirole J. The Theory of Industrial Organization. Cambridge, 

Boston, USA: The MIT Press, 1988. 

[7] Lariviere M, Porteus E. Selling to the newsvendor: An 

analysis of price-only contracts. Manufacturing and Service 

Operations Management, 2001, 3(4): 293-305. 

[8] Pasternack B. Optimal pricing and returns policies for 

perishable commodities. Marketing Science, 1985, 4(2): 

166-176. 

[9] Cachon G, Lariviere M. Supply chain coordination with 


revenue-sharing contracts: Strengths and limitations. Management 

Science, 2005, 51(1): 30-44. 

[10] Taylor T. Supply chain coordination under channel rebates 

with sales effort effects. Management Science, 2002, 48(8): 

992-1007. 

[11] Moorthy K S. Managing channel profits: Comment. Marketing 

Science, 1987, 6: 375-379. 

[12] Tsay A. Quantity-flexibility contract and supplier-customer 

incentives. Management Science, 1999, 45(10): 

1339-1358. 

[13] Newbery G. Competition, contracts, and entry in the electricity 

spot market. RAND Journal of Economics, 1998, 29: 

726-749. 

[14] Wu D J, Kleindorfer P R, Zhang J E. Optimal bidding and 

contracting strategies for capital-intensive goods. European 

Journal of Operational Research, 2002, 137: 

657-676. 

[15] Donohue K. Efficient supply contracts for fashion goods 

with forecast updating and two production modes. Management 

Science, 2000, 46(11): 1397-1411. 

[16] Barnes-Schuster D, Bassok Y, Anupindi R. Coordination 

and flexibility in supply contracts with options. Manufacturing 

& Service Operations Management, 2002, 4(3): 

171-207.

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

Create successful ePaper yourself

Delete template?

Save as template?