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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: [email protected]
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