Impact of structure, market share and information asymmetry on supply

contracts for a single supplier multiple buyer network

Indranil Biswas a,*, Balram Avittathur b, Ashis K Chatterjee b

a Indian Institute of Management Lucknow, Lucknow 226013, India

b Indian Institute of Management Calcutta, Calcutta 700104, India

Supplementary Material(s)

Appendix A: Tables

In this section we present the optimal parameters i.e. per unit prices, retail prices, and retail

quantities for three contracts types, namely Quantity Discount (index: QD), Wholesale Price

(index: WP), and Linear Two-part Tariff (index: LTT) for all the four cases of supply chain

structure (Cases: DA, DF, PA, and PF) as presented in Table 1. All these relevant results are

subsequently presented through Table A1 and A2.

* Corresponding Author. Tel.: +91 983 099 7908. E-mail address: indranil@iiml.ac.in

1

Table A1: Optimal Contract Analysis for Completely Decentralised Supply Chain

Contract type

Parameter Information AvailabilityFull (Case: DF) Asymmetric (Case: DA)

Quantity Discount(index:

QD)

Per unit price wDFQD= θ bR

2 γ (a−b ) K (R+1 )+ s−Rc

R+1 wDAQD= θ bR

2 γ (a−b ) K (R+1 )+

s−RE (c )R+1

Optimal Retail Price pDFi

QD = θbγ (a+b ) (k i+

ba−b )+ (wDF

QD+c ) (δ +γ )a−b

pDAiQD = θb

γ (a+b ) (k i+b

a−b )+ (wDAQD+c ) (δ +γ )

a−bOptimal Order

Quantity qDFiQD =

γ ( δ+γ )b { pDFi

QD −(wDFiQD +c )} qDAi

QD =γ ( δ+γ )

b { pDAiQD −(wDAi

QD +c )}

Wholesale price

(index: WP)

Per unit price wDFWP= θ

4 δ+ 1

2( s−c ) wDA

WP= θ4δ

+ 12 {s−E (c ) }

Optimal Retail Price pDFi

WP = θ(2δ+3 γ ) (k i+

γ2 δ +γ )+ (wDF

WP+c) (δ+γ )2δ+γ

pDAiWP = θ

(2 δ+3 γ ) (k i+γ

2 δ +γ )+ (wDAWP+c) (δ+γ )

2 δ+γ

Optimal Order Quantity qDFi

WP =θ ( δ+γ )

(2 δ+3 γ ) (k i+γ

2 δ+γ )− (wDFWP+c) δ (δ +γ )

2δ +γqDAi

WP =θ ( δ+γ )

(2δ+3 γ ) (k i+γ

2δ+γ )− (wDAWP+c) δ (δ +γ )

2δ +γ

Linear Two-part

Tariff(index: LTT)

Per unit price wDFLTT=s+

γ {θ−2 δ (s+c ) }4 δ (δ+γ )

wDALTT= 1

δ +γ [ γθ4 δ

+(δ+ γ2 ) {s−E ( c ) }+δcmax ]

Franchise Fee LDFiLTT=(δ+γ ) [ θ

2 δ+3 γ (ki+γ

2δ+γ )− δ (wDFLTT +c )

2δ+γ ]2

− πBi LDAiLTT=(δ+γ )[ θ

2δ+3 γ (k i+γ

2δ+γ )− δ (wDFLTT +cmax )2 δ +γ ]

2

− πBi

Optimal Retail Price pDFi

LTT= θ2 δ +3 γ (k i+

γ2 δ+γ )+ (δ+γ ) (wDF

LTT +c )2δ +γ

pDFiLTT= θ

2 δ +3 γ (k i+γ

2 δ+γ )+ (δ+γ ) (wDALTT +c )

2δ +γ

Optimal Order Quantity qDFi

LTT= (δ+γ ) [ θ2 δ+3 γ (k i+

γ2 δ+γ )− δ (wDF

LTT+c )2 δ+γ ] qDFi

LTT= (δ+γ ) [ θ2 δ+3 γ (k i+

γ2 δ+γ )− δ (wDA

LTT+c )2 δ+γ ]

Note: (i) a=(δ +γ ) {2−v (δ +γ ) }

, b=γ {1−v (δ +γ ) }

, K=1− δ+γ

a−b, R=1+ γ (δ+γ ) vK

b; (ii) In linear two-part tariff contract

πBi represents

the reservation profit level of the ith buyer.

2

Table A2: Optimal Contract Analysis for Partially Decentralised Supply Chain

ContractTypes Parameter Buye

rInformation Availability

Full (Case: PF) Asymmetric (Case: PA)

Quantity Discount

Per unit price

Optimal per unit price satisfies the following relation:γ ( δ+γ )

b { pPF2QD −(wPF

QD+c ) }+ Φ (δ , γ )1−vΦ (δ , γ ) (wPF

QD−s )=0

Optimal per unit price satisfies the following relation:γ ( δ+γ )

b [ E { pPA 2QD (c ) }−{wPA

QD+E (c ) }]+ Φ (δ , γ )1−vΦ (δ , γ ) (wPA

QD−s )=0

Optimal Retail Price

1 pPF1QD = 1

m1{θ(k1+

bγ

k2)+(wPFQD+s+2 c)−m2 pPF 2

QD } pPA 1QD = 1

m1{θ (k1+

bγ

k2)+(wPAQD+s+2 c )−m2 pPA 2

QD }2 pPF2

QD = bθγD ( γk1+bk 2

m1+k2)+ δ+γ

D [wPFQD+c+ b

m1(wPF

QD+s+2 c )] pPA 2QD = bθ

γD ( γk 1+bk 2

m1+k2)+ δ +γ

D [wPAQD+c+ b

m1(wPA

QD+s+2 c) ]Optimal Order

Quantity

1 qPF 1QD =(δ+γ ) { pPF 1

QD −(s+c ) } qPA 1QD =( δ+γ ) { pPA1

QD −(s+c ) }2 qPF 2

QD =γ (δ +γ )

b { pPF 2QD −(wPF

QD+c )} qPA 2QD =

γ (δ+γ )b { pPA2

QD −(wPAQD+c ) }

Wholesale price

Per unit price

wPFWP=ξ ( δ , γ )[ θ

2 δ+3 γ (k 2+γ

2δ +γ )+ (δ+2 γ ) s2 δ+3 γ

− δc2 δ +γ ] wPA

WP=ξ ( δ , γ )[ θ2 δ+3 γ (k 2+

γ2δ +γ )+ (δ+2 γ ) s

2 δ+3 γ−

δE (c )2 δ +γ ]

Optimal Retail Price

1 pPF1WP = θ

2 δ+3 γ (k1+γ

2 δ+γ )+ δ +γ2 δ+γ

(s+c )+γ (δ +γ ) (wPF

WP−s )(2 δ +γ ) (2 δ +3 γ )

pPA 1WP = θ

2 δ+3 γ (k1+γ

2 δ+γ )+ δ+γ2 δ+γ

(s+c )+γ (δ+γ ) (wPA

WP−s)(2 δ+γ ) (2 δ+3 γ )

2 pPF2WP = θ

2 δ+3 γ (k2+γ

2 δ+γ )+ δ+γ2 δ +γ (wPF

WP+c )−γ (δ +γ ) (wPF

WP−s )(2 δ+γ ) (2 δ+3 γ )

pPA 2WP = θ

2 δ +3 γ (k2+γ

2 δ+γ )+ δ+γ2 δ+γ (wPA

WP+c)−γ (δ+γ ) (wPA

WP−s)(2 δ+γ ) (2 δ+3 γ )

Optimal Order

Quantity

1 qPF 1WP =(δ+γ ) { pPF 1

WP −(s+c ) } qPA 1WP =( δ+γ ) { pPA 1

WP −(s+c )}2 qPF 2

WP =(δ+γ ) { pPF2WP −(wPF

WP+c )} qPA 2WP =( δ+γ ) { pPA 2

WP −(wPAWP+c ) }

3

Note: b=γ {1−v (δ +γ ) }

, m1=2 ( δ+γ )−b

,

m2=δ (1+ bγ )+b

,

D= (δ+γ )(1+ bγ )+bm 2

m1,

ξ (δ , γ )= (2 δ+3 γ ) (2δ+γ )(2δ+3 γ ) (2 δ+γ )−γ 2

,

Φ (δ , γ )= γ (δ +γ )b [ (δ+γ )

D (1+ bm1)−1]

Table A2: Optimal Contract Analysis for Partially Decentralised Supply Chain (contd.)

Contract

Types

Parameter

Buyer

Information AvailabilityFull (Case: PF) Asymmetric (Case: PA)

Linear Two-part

Tariff

Per unit price

Optimal per unit price satisfies the following relation:γ2 { pPF 2

LTT (wPFLTT )−(wPF

LTT +c )}=2 ( δ+γ )2 (wPFLTT−s)

Optimal per unit price satisfies the following relation: (wPA

LTT−s)+2 [ pPA 2LTT (w , cmax)−(wPA

LTT +cmax )]=2 ξ ( δ , γ ) [ E { pPA 2LTT (w ,c ) }− {wPA

LTT +E (c ) }]Franchis

e FeeLPF

LTT=(δ+γ ) { pPF 2LTT −(wPF

LTT+c )}2−π B 2 LPALTT=(δ+γ ){ pPA 2

LTT−(wPALTT+cmax) }2− π B 2

Optimal Retail Price

1 pPF1LTT = θ

2 δ+3 γ (k1+γ

2δ+γ )+ δ +γ2δ+γ

(s+c )+γ (δ +γ ) (wPF

LTT −s )(2δ+γ ) (2δ+3 γ )

pPF1LTT = θ

2δ+3 γ (k1+γ

2δ+γ )+ δ +γ2δ+γ

(s+c )+γ (δ +γ ) (wPF

LTT −s )(2δ+γ ) (2δ+3 γ )

2 pPF2LTT = θ

2 δ+3 γ (k2+γ

2 δ+γ )+ δ+γ2 δ +γ (wPF

LTT+c )−γ (δ+γ ) (wPF

LTT−s)(2 δ+γ ) (2 δ+3 γ )

pPF2LTT (wPF

LTT , c)= θ2 δ+3 γ (k 2+

γ2 δ +γ )+ δ+γ

2 δ+γ (wPFLTT +c )−

γ (δ +γ ) (wPFLTT−s)

(2 δ+γ ) (2 δ+3 γ )Optimal Order

Quantity

1 qPF 1LTT =(δ+γ ) { pPF 1

LTT −(s+c ) } qPA 1LTT= (δ+γ ) { pPA 1

LTT−( s+c ) }2 qPF 2

LTT =(δ+γ ){ pPF2LTT −(wPF

LTT+c )} qPA 2LTT= (δ+γ ) { pPA 2

LTT−(wPALTT +c )}

Note: (i)

ξ (δ , γ )= (2 δ+3 γ ) (2δ+γ )(2δ+3 γ ) (2δ+γ )−γ 2

, (ii) In linear two-part tariff contract πB 2

represents the reservation profit level of the second buyer.

4

Appendix B: Proofs

Centralised Supply Chain:

The central planner’s problem is given by (5). Using (1) in (5), we obtain the following two

equations about optimal retail prices from the first order conditions

∂ πC

∂ p1=0⇒2 (δ+γ ) p1−2 γp2=θk1+δ (c+s)

(B1)

∂ πC

∂ p2=0⇒2 (δ+γ ) p2−2 γp1=θk2+δ (c+s)

(B2)

By solving (B1) and (B2), we obtain the optimal retail prices and using them in (1) we obtain

the optimal order quantities and they are presented by (6). The total profit of the centralised

supply chain is given by,

πC=∑i=1

2

{ pCi−( c+s ) }qCi=δ4 { θ2

δ ( δ+2 γ ) (k12+k2

2+ γδ )−2θ

δ(c+s )+2 (c+s )2}

If all other parameters are kept constant, πC becomes a function of (k1 , k2) . Using k1+k2=1 in

the above expression, the condition for πC to reach its minimum level is calculated.

Calculations pertaining to Table A1:

(I) Completely Decentralised Supply Chain and Full Information Availability (Case: DF)

(A) Quantity Discount Contract (QD): In the case of completely decentralised supply chain and

full information availability, when the supplier offers quantity discount contract the ith buyer’s

profit function takes the following form:

πBi=pi qi−(wqi−12

vqi2)−cqi

(B3)

Using (1) in (B3), we obtain the below mentioned relation about the optimal retail prices from

the first order condition.

5

∂ πBi

∂ p i=0⇒ {2−v (δi+γ )} (δi+γ ) pi−{1−v (δi+γ ) }γp j={1−v (δi+γ ) }θk i+(w+c ) (δi+γ )

(B4)

For i=1 , 2 and i≠ j , (A4) represents a set of two equations. By simultaneously solving them

we obtain pDF 1QD

and pDF 2QD

. From the second order condition we have

∂2 π Bi

∂ pi2 =−{2−v (δ i+γ )} (δi+γ )<0 (i=1 , 2 )

(B5)

From the definition of the quantity discount contract we have v∈[ 0 , v ) and

v=min ( 2δ1+γ , 2

δ2+γ ); thus (B5) is always negative and pDFi

QD represents the optimal retail

price for buyer i. Using pDF 1QD

and pDF 2QD

we calculate the optimal order quantities qDF 1QD

and qDF 2QD

. All of them are presented in Table A1.

In this case the supplier’s optimization problem is given by,

maxw

{π S ( w ) }=maxw ( (w−s ) {qDF 1

QD (w )+qDF 2QD (w ) }− v

2 [ {qDF 1QD (w ) }2+{qDF 2

QD (w ) }2]) (B6)

From the first order condition of (B6) the optimal per unit price wDFQD

is calculated for an

exogenously decided discount rate v. The result is presented in Table A1.

(B) Wholesale Price Contract (WP): By putting v=0 in all the optimal results of the quantity

discount contract we obtain: pDF 1WP

, pDF 2WP

, qDF 1WP

, qDF 2WP

, and wDFWP

. All the results are presented in

Table A1.

(C) Linear Two-part Tariff Contract (LTT): In this case the ith buyer’s profit function is given by

the following equation:

πBi= { pi− (w+c ) }q i−LDFiLTT

(B7)

6

By definition of linear two-part tariff contract, LDFiLTT

is independent of the order quantity.

Therefore by using (1) in (B7), we obtain the below mentioned relation about the optimal retail

prices from the first order condition.

∂ πBi

∂ p i=0⇒2 (δ +γ ) pi−γp j=θk i+ (w+c ) (δ +γ )

(B8)

For i=1 , 2 and i≠ j , (A8) represents a set of two equations. By simultaneously solving them

we obtain pDF 1LTT

and pDF 2LTT

. Using pDF 1LTT

and pDF 2LTT

in (1) we calculate the optimal order quantities

qDF 1LTT

and qDF 2LTT

. All of the results are presented in Table 3. In the case of full information

availability, the supplier can design the franchise fee (LDFiLTT

) such that she is able to extract all the

profit from the ith buyer apart from the buyer’s reservation profit (πBi ). From (B7) we can

observe that the optimal franchise fee (LDFiLTT

) takes the following form:

LDFiLTT={ pDFi

LTT−(w+c ) }qDFiLTT− π Bi=(δ+γ )[ θ

2 δ+3 γ (k i+γ

2 δ +γ )− δ ( w+c )2 δ+γ ]

2

−π Bi (B9)

The supplier’s profit maximization problem is given by the following equation:

maxw

πS (w )=maxw

{(w−s ) [qDF 1LTT (w )+qDF 2

LTT (w ) ]+( LDF 1LTT + LDF 2

LTT ) } (B10)

Using the expressions for optimal prices, order quantities, and (B9) in (B10), we calculated the

optimal per unit price (wDFLTT

) from the first order condition.

(D) Nonlinear Two-part Tariff Contract (NTT): In the case of full information, linear and

nonlinear two-part tariff contracts are equivalent in nature. Therefore all the optimal results for

this case are equal to those obtained for linear two-part tariff contract.

(II) Completely Decentralised Supply Chain and Asymmetric Information (Case: DA)

(A) Quantity Discount Contact (QD): In the case of asymmetric information the buyers calculate

their optimal order quantities by maximizing their individual profits. Therefore the expressions

7

for optimal retail prices and order quantities remain identical to those of full information case

(Case: DF). As the supplier does not know the exact cost of the buyers, she maximizes her

expected profit. The supplier’s optimization problem is:

maxw

E {π S (w ) }=maxw

∫c min

cmax

( (w−s ) {qDA 1QD (w )+qDA 2

QD (w ) }− v2 [ {qDA 1

QD ( w ) }2+{qDA 2QD (w ) }2])dF (c )

(B11)

By solving (B11) we obtain the optimal per unit price wDAQD

. All the optimal results are presented

in Table A1.

(B) Wholesale Price Contract (WP): By putting v=0 in all the optimal results of the quantity

discount contract we obtain: pDA 1WP

, pDA 2WP

, qDA 1WP

, qDA 2WP

, and wDAWP

. All the results are presented in

Table A1.

(C) Linear Two-part Tariff Contract (LTT): Following the argument presented for quantity

discount contract, in this case also the optimal price and quantity expressions remain same to

those obtained for the full information case. However, as the supplier does not know the buyer’s

exact cost structure, she charges the minimum possible franchise fee so that the buyer is assured

of her reservation profit level. From (B9) we can observe that the minimum franchise fee is

given by,

LDAiLTT|min=(δ +γ )[ θ

2δ+3 γ (k i+γ

2δ +γ )− δ (w+cmax )2 δ+γ ]

2

− π Bi (B12)

Therefore the supplier’s expected profit maximization problem is given by the following

equation:

maxw

E {π S (w ) }=maxw

∫c min

cmax

{(w−s ) [qDA 1LTT (w )+qDA 2

LTT (w ) ]+( LDA 1LTT |min+ LDA 2

LTT |min ) }dF (c ) (B13)

By solving (B13) we obtain the optimal per unit price wDALTT

. All the optimal results are presented

in Table A1.

8

(D) Nonlinear Two-part Tariff Contract (NTT): In the case of asymmetric information nonlinear

two-part tariff contract corresponds to a menu contract, from which the buyer chooses her

preferable contract form. The supplier offers a menu contract {w ( c ) ,Li ( c ) } to each buyer. In this

case, buyer i’s profit function is given by the following equation,

πBi ( c )=(δ +γ )[ θ2 δ +3 γ (k i+

γ2 δ+γ )− δ {w ( c )+c }

2δ+γ ]2

−LDAiNTT ( c )

(B14)

According to Revelation Principle, we have

∂ πBi ( c )∂ c

|c=c=0. Therefore from (B14) we have

d {LDAiNTT (c ) }dc

=−2 δ (δ+γ )

(2 δ+γ ) [ θ2 δ+3 γ (k i+

γ2δ +γ )− δ {w (c )+c }

2 δ+γ ] d {w (c ) }dc (B15)

The above equation is presented as (8) in Proposition 1. The supplier’s expected profit

maximization problem is given by

maxw , L1 , L2

E ( πS )= maxw, L1 , L2

E {(w−s ) (q1+q2 )+ ( L1+L2 )}

s .t .d {LDAi

NTT ( c ) }dc

=−2δ (δ+γ )(2δ+γ ) [ θ

2δ +3 γ (ki+γ

2δ+γ )− δ {w (c )+c }2δ+γ ] d {w (c ) }

dc(i=1 , 2 )

This becomes a dynamic optimization problem. Employing calculus of variation the problem is

solved by defining dw ( c )/dc=u (c ) , then the maximization problem can be rewritten as follows:

maxw , L1 , L2

E ( πS )¿maxu ( c )

∫cmin

cmax

{ δ+γ2δ+γ

{w (c )−s} [θ−2δ {w ( c )+c }]+ [ L1 (c ) +L2 (c ) ]}f (c ) dc

s .t .d {w (c ) }

dc=u (c )

d {LDAiNTT (c ) }dc

=−2 δ (δ+γ )

(2 δ+γ ) [ θ2 δ+3 γ (k i+

γ2δ +γ )− δ {w (c )+c }

2 δ+γ ] d {w (c ) }dc

(i=1 , 2 )

The corresponding Hamiltonian is given by:

9

H={ δ+γ2 δ+γ {w (c )−s }[θ−2 δ {w (c )+c }]+[ L1 (c )+L2 (c ) ]}f (c )

−∑i=1

2

λ i (c )u (c ) {2δ (δ +γ )(2 δ+γ ) [ θ

2δ+3 γ (ki+γ

2δ+γ )− δ {w (c )+c }2δ +γ ]}+λ3 (c ) u ( c )

The necessary optimality conditions are:

∂ H∂u (c )

=0 (B16)

∂H∂ L1 (c )

=−d {λ1 ( c ) }

dc (B17)

∂ H∂ L2 (c )

=−d {λ2 (c ) }

dc (B18)

∂ H∂w (c )

=−d {λ3 (c ) }

dc (B19)

By simultaneously solving (B16) – (B19), we obtain the optimal per unit price and it is presented

by (7) in Proposition 1.

Calculations pertaining to Table A2:

Partially Decentralised Supply Chain

In the case of partially decentralised supply chain, the supplier and buyer 1 forms a loosely

coupled vertical chain. The supplier transfers the order quantity to buyer 1 at the marginal cost of

s in order to exploit the advantages of vertical integration and in addition charges a fixed fee of

πS−B 1 . The fixed fee is exogenously decided and not dependent on the order quantity. The

supplier calculates the optimal contract to maximize her profit from the transaction with buyer 2.

Thus the profits of different supply chain agents are as follows:

πB 1={ p1−( s+c ) } q1−π S−B 1 (B20)

10

πB 2=( p2−c )q2−T ( ¿ ) (B21)

πS=T (¿ )−sq2+ πS−B 1 (B22)

Using (B21) – (B22) and following the argument above for calculating the optimal prices, order

quantities and contract forms for completely decentralised supply chain (case DF and DA), the

problem for partially decentralised supply chain (case PF and PA) are solved.

Proof of Proposition 7: From Proposition 6, the optimal per unit prices for LTT contract are:

wDFLTT=wPF

LTT=s , when

γδ→0

. Putting these optimal values in the expression for retailer’s

order quantity we find: qDFiLTT= qCi , pDFi

LTT= pCi , qDF 2LTT =qC 2 , and pDF 2

LTT = pC 2 .

In case DF, when

γδ→0

, for QD contract the optimal retail price, order quantity, and per unit

price charged by the supplier take the following forms respectively:pDFi

QD =(1−δv ) θk i+δ ( w+c )

δ (2−δv )

, qDFi

QD =θk i−δ (w+c )

2−δv , wDF

QD=θ−2 δc+δs (2−δv )

δ (4−δv ) . Under the condition of channel

coordination, the following relationship must be satisfied: ∑i=1

2

qDFiQD =∑

i=1

2

qCi. This relationship

yields: v¿=2

δ and wDF

QD (v¿= 2δ )= θ

2 δ−c

.

In case PF, when

γδ→0

, for QD contract the optimal retail price, and order quantity of buyer 2,

and per unit price charged by the supplier take the following forms respectively:

pPF2QD =

(1−δv )θk2+δ ( w+c )δ (2−δv ) ,

qPF 2QD =

θk2−δ (w+c )2−δv ,

wPFQD=θ−2 δc+δs (2−δv )

δ (4−δv ) . In this case,

qPF 1QD =qC 1 , therefore channel coordination would happen iff qPF 2

QD =qC 2 . This relationship

yields: v¿=2

δ and wPF

QD(v¿= 2δ )= θk2

δ−c

.

11

Calculations pertaining to Section 5.3 and Table 3:

(A) Analysis of WP contracts:

(I) Case DF: With WP contract, the supplier’s optimal profit is expressed as follows:

πS=δ +γ

8 δ (2 δ +γ ){θ−2 δ (s+c ) }2

(B23)

We can observe that the supplier’s profit is also decreasing in c, and at the cost point, c=β DFWP

,

the supplier’s profit reaches her reservation level, πS= π S . Then from (B23) we have:

βDFWP= 1

2δ [θ−√ 8 δ (2 δ+γ )(δ+γ )

π S]−s

In the Case DF, with WP contract the ith buyer’s optimal profit is given by (B24).

πBi={ pDFiWP −(wDF

WP+c)} qDFiWP =( δ+γ )[ θ

2 δ+3 γ (k i+γ

2 δ+γ )− δ (wDFWP+c )

2δ+γ ]2

(B24)

Since the buyer’s profit is decreasing in c, at the cost point, c=αDFiWP

, the buyer’s profit

researches her reservation level, πBi= π Bi . Then using these relations in (B24) we have:

wDFWP+αDFi

WP =2δ+γδ [ θ

2 δ+3 γ (k i+γ

2 δ+γ )−√ πBi

δ+γ ] (B25)

From the expression of optimal wholesale price we obtain: wDF

WP+αDFiWP = θ

4 δ+ 1

2 (s+αDFiWP )

.

Equating this to (B25) we obtain the cut-off point for the ith buyer as follows:

αDFiWP =

2 (2δ +γ )δ [ θ

2δ+3 γ (k i+2δ−γ2 δ +γ )−√ πBi

δ +γ ]−s

12

(II) Case DA: In the presence of a supplier cut-off point (βDAWP

), the supplier profit optimization

problem, for the Case DA with WP contract, is reformulated as follows:

maxw , βDA

WPE ( πS )=max

w ,β DAWP [∫cmin

β DAWP

δ+γ2δ+γ

(w−s ) {θ−2δ ( w+c ) } f (c ) dc+ ∫β DA

WP

cmax

πS f (c ) dc ]The first order conditions of the above maximization problem yields:

∂E (π S)∂ β DA

WP =f (β DAWP )[ δ+γ

2δ+γ( w−s ){θ−2δ (w+βDA

WP) }− πS ]=0 (B26)

∂ E (π S)∂w

= δ +γ2 δ+γ [ {θ−2 δ (2 w−s ) } F (β DA

WP )−2 δE (c|c<β DAWP )]=0

(B27)

From (B26) and (B27) the condition for supplier’s cut-off point is calculated. This cut-off point

also needs to be such that the buyer’s can make at least their reservation profit. This results in the

following additional condition:

(δ +γ )[ θ2 δ+3 γ (k i+

γ2δ +γ )− δ (w +βDA

WP)2 δ+γ ]

2

≥ π Bi ( i=1,2 )

(III) Case PF: In this case for WP contract, for low level of substitutability among products, that

is

γδ→0

, the supplier’s optimal profit is expressed as follows:

πS=(wPFWP−s ) qPF 2

WP + π S−B 1=δ8 [ θk2

δ−(c+s)]

2

+ πS−B 1 (B28)

As the supplier’s profit is also decreasing in c, at the cost point, c=β PFWP

, the supplier’s profit

reaches her reservation level, that is πS= π S . Then from (B28) we have:

βPFWP=

θk 2

2 δ−s−√ 8

δ ( πS− π S−B 1)

In the Case PF, with WP contract the 2nd buyer’s optimal profit is given by (B29).

13

πB 2={ pDF 2WP −(wDF

WP+c ) }qDF 2WP =δ [ θk2

2 δ−1

2 (wPFWP+c ) ]

2

(B29)

Since the buyer’s profit is decreasing in c, at the cost point, c=α PF 2WP

, the buyer’s profit

researches her reservation level, πB 2= πB 2 . Then using these relations in (B29) we have:

wPFWP+αPF 2

WP =θk2

δ −2√ π B 2

δ (B30)

From the expression of optimal wholesale price we obtain: wPF

WP+α PF 2WP =

θk2

2 δ+1

2 (s+α PF2WP )

.

Equating this to (A30) we obtain the cut-off point for the ith buyer as follows:

α PF 2WP =

θk2

δ −4 √ πB 2

δ −s

The calculation of buyer 1’s cut-off point is relatively straight forward. Her optimal profit is

given by (B31).

πB 1={ pDF 1WP −(s+c ) }qDF 1

WP − πS−B 1=δ [ θk1

2 δ−1

2( s+c )]

2

− π S−B 1 (B31)

Since the buyer’s profit is decreasing in c, at the cost point, c=α PF 1WP

, the buyer’s profit

researches her reservation level, πB 1= πB 1 . Then using these relations in (B31) we have:

αPF 1WP =

θk1

δ −2√ π B 1+ π S−B 1

δ −s

(IV) Case PA: With WP contract and in the presence of a supplier cut-off point (βPAWP

), the

supplier profit optimization problem, for the Case DA, is reformulated as follows:

maxw , βPA

WPE ( πS )=max

w ,β PAWP [∫cmin

β PAWP

δ2

(w−s ) {θk2

δ−(w +c )}f (c )dc+ ∫

β PAWP

cmax

π S f (c )dc + πS−B1]14

The aforementioned relation is derived for low level of substitutability among products, that is γδ→0

. The first order conditions of the above maximization problem yields:

∂ E (π S)∂ βPA

WP =f (β PAWP ) [ δ

2(w−s ) {θk2

δ−(w+β PA

WP )}− π S ]=0 (B32)

∂ E (π S)∂w

=δ2 [{θk2

δ−(2 w−s )}F ( βPA

WP)−E (c|c<β PAWP )]=0

(B33)

From (B32) and (B33) the condition for supplier’s cut-off point is calculated. This cut-off point

also needs to be such that buyer 2 can make at least her reservation profit. This results in the

following additional condition:

δ [ θk2

2δ−1

2 (w+ βPAWP ) ]

2

≥π B 2

(B) Analysis of LTT contracts:

(I) Case DF: With LTT contract and low level of substitutability among products, that is

γδ→0

, the supplier’s optimal profit is expressed as follows: πS=δ∑

i=1

2 [{θk i

2δ−(wDF

LTT+c)}2

− π Bi].

Under the condition of channel coordination we have: wDFLTT=s and using this expression of per

unit price in the supplier’s profit function we obtain:

πS=θ

4 δ (∑i=1

2

ki2−1

2 )+ δ8 {θ

δ−2 (c+s )}

2−∑

i=1

2

π Bi (B34)

We can observe that the supplier’s profit is decreasing in c, and at the cost point, c=β DFLTT

, the

supplier’s profit reaches her reservation level, πS= π S . Then from (B34) we have:

15

βDFLTT=

12 {θ

δ −√ 8δ ( π S+∑

i=1

2

π Bi)−2θ2

δ 2 (∑i=1

2

k i2−

12 )}−s

(II) Case PF: With LTT contract and under the condition of channel coordination, the supplier’s

profit function is given by the expression: πS=

δ4 {θk 2

δ−(c+s )}

2

− π B 2+ πS−B 1. As the

supplier’s profit is decreasing in c, and at the cost point, c=β PFLTT

, the supplier’s profit reaches

her reservation level, πS= π S . Using this relation in supplier’s profit function we obtain:

βPFLTT=

θk2

δ−s−√ 4

δ ( πS+ πB 2− πS−B 1 )

16