APPENDICES TO
“EQUILIBRIUM MARKUP PRICING STRATEGIES FOR THE DOMINANT
RETAILERS UNDER SUPPLY CHAIN TO CHAIN COMPETITION”
NOTE: The paper can be understood without reading the following Appendices. Thus,
if the paper is deemed publishable, these Appendices need not be published but can be
supplied to interested readers by the authors separately.
Appendix A: Proof of Proposition 1
The support interval for the decision variable wi is [ci, +∞), since the supplier i’s
profit would be less than 0 if wi<ci. At this situation, the supplier will quit the market. If
supplier i sets his wholesale price at ci, we have ΠSi=0; i.e., the supplier earns nothing.
However, if supplier i sets his wholesale price at +∞, all other things being kept
constant, the market demand will approach 0, which also leads to a zero profit for the
supplier. Therefore, the maximum value of ΠSi cannot be obtained at the boundaries of
wi.
Linear Demand Case
The first order conditions for Eq. (2) with respect to wi are given as follows:
  Sl1
 w  2bwl1   wl 2  a  b(U Rl1  c1 )  U Rl 2  0
 l1
.


Sl
2

  wl1  2bwl 2  a  b(U Rl 2  c2 )  U Rl1  0
 wl 2
(A1)
As we all know, if the rank of a general system of n equations with n variables
does not equal to 0, there is one and only one set of solutions. It is easily shown that the
rank of (A1) satisfies
2b, 
 4b 2   2  0 . Therefore, the linear system of equations
 , - 2b
in (A1) has one and only one set of solutions. Solving the first order conditions in (A1)
gives the optimal response wholesale price in Eq. (3a). Manipulating with the second
order condition with respect to wli leads to:
 2  Sli
 2b  0 .
 2 wli
Therefore, the second order condition in (A2) indicates that the supplier i’s profit
can be maximized by setting the wholesale price at the level of wlirf in Eq. (3a).
Iso-elastic Demand Case
The first order conditions with respect to wci are given as:
(A2)
  Sc1 K ( wc 2  U Rc 2 ) 

  (  1) wc1  U Rc1   c1   0

( wc1  U Rc1 ) 1
 wc1
.


  Sc 2  K ( wc1  U Rc1 )   (  1) w  U   c   0
c2
Rc 2
2
 wc 2
( wc 2  U Rc 2 ) 1
(A3)
Obviously, the linear system of equations in (A3) has one and only one set of
solutions which are given in Eq. (3b).
Manipulating with the second order condition with respect to wci leads to
 2  Sci
 2 wci

K    ( wcj  U Rcj ) 
wci  wci rf
( wci  U Rci )  2
 (U Rci  ci )  0 .
(A4)
Therefore, the second order condition in (A4) indicates that the stationary point of wcirf
is the local maximum of ΠSi. Since wcirf is the only stationary point of ΠSi and the
maximum value of ΠSi cannot be obtained at ci (the lower bound of wci), we can
conclude that the supplier i’s profit can be maximized by setting the wholesale price at
the level of wcirf in Eq. (3b) for the iso-elastic demand case.
Appendix B: Proof of Proposition 3
It is intuitive that the markup URi satisfies 0≤URi≤UT(=[a+θ(URj+cj)]/b-ci). If retailer i
sets her markup at the lower bound (i.e., URi=0), we have ΠRi=0; i.e., the retailer will
earn nothing; however, if the retailer i sets her markup at the upper bound (i.e., URi=UT),
the market demand for supply chain i will equal to 0, which will in turn yield zero profit
for retailer i. Therefore, the maximum value for ΠRi cannot be achieved at the
boundaries of URi.
Linear Demand Case
The first order conditions for Eq. (4) with respect to URli are given as follows:
b
  Rl1
2
2
2
2
 U  4b 2   2   2(2b   )U Rl1  bU Rl 2  a(2b   )  (2b   )c1  b c2   0
 Rl1
;


b
2
2
2
2
Rl
2

 2
 bU Rl1  2(2b   )U Rl 2  a(2b   )  (2b   )c2  b c1   0
2 

 U Rl 2 4b  
It is easily shown that
2(2b 2   2 ), b
b ,  2(2b   )
2
2
 (4b 2  b  2 2 )(4b 2  b  2 2 )  0 .
Therefore, the linear system of equations stated above has one and only one set of
solutions, which are given in Eq. (5a). Also, we can show that
 2 Rli
2b(2b2   2 )


0.
 2U Rli
4b2   2
Therefore, URli[FF] in Eq. (5a) maximizes retailer i's profit stated in Eq. (4).
Iso-elastic Demand Case
The first order conditions for Eq. (4) with respect to URci are given as follows:


  Rc1
K
  (U Rc1  c1 )    (U Rc 2  c2 ) 






   (  1)U Rc1  c1   0
U Rc1  c1 
  1  
 1

 U Rc1
.



K
  (U Rc 2  c2 )    (U Rc1  c1 ) 
  Rc 2
  (  1)U Rc 2  c2   0
 
 U   U  c  
 1
  1 
 
Rc 2
2
 Rc 2
Obviously, the linear system of equations above has one and only one set of solutions
which are given in Eq. (5b). It is easily shown that the second order conditions satisfy
 2 Rci
 2U Rci

U Rci U Rci[ FF ]

K  (  1) 2   2ci    (U Rcj  c j ) 



 0.
2 
 ci
 1 
 (  1)  
Therefore, URci[FF] in Eq. (5b) maximizes retailer i's profit.
Appendix C: Proof of the Non-negativity of Eq. (5a) and Eq. (11)
Proof of the non-negativity of Eq. (5a)
Since b>θ is assumed in Eq. (1a) to ensure that the effect of the competitor’s price
cannot exceed the effect of its own price, we know that the denominator of Eq. (5a)
satisfies
16b4  17b2 2  4 2 = (4b2  2 2  b )(4b2  2 2  b ) >0.
It is also known that, the market demand must be larger than 0 when the retail prices are
at their lower bounds (i.e., pi=ci) to ensure a positive profit for the supply chain; i.e., we
have a-bci+θcj>0 under the linear demand case. Therefore, we have
U Rli[ FF ] 
>
a(6b 2  3b 2  2 3  8b3 ) b 2 2  2(2b 2   2 ) 2
b (2b 2   2 )

c

cj
i
16b 4  17b 2 2  4 2
16b 4  17b 2 2  4 2
16b 4  17b 2 2  4 2
a (6b2  3b 2  2 3  8b3 ) b2 2  2(2b2   2 ) 2
b(2b2   2 )(bci  a )

c

i
16b4  17b2 2  4 2
16b4  17b2 2  4 2
16b4  17b2 2  4 2
2(b   )(3b2   2 )(a  bci   ci )
=
 0.
16b4  17b2 2  4 2
Proof of the non-negativity of Eq. (11)
It is obviously that Eq. (11b) is nonnegative due to the assumption “b>θ”. In the
following we prove Eq. (11a) is also nonnegative.
As we have stated above, the market demand must be larger than 0 when the retail
prices are at their lower bounds. Therefore, when the wholesale price for each supplier
is set to equal to the production cost, we have Di=a-b(URi+ci)+θ(1+ηj)cj>0 for the
hybrid scenario and Di=a-b(URi+ci)+θ(URi+ci)>0 for the [FF]-scenario. Therefore, Eq.
(11a) can be transformed as
wlirf=
a
2b2
(2b2   2 )
b
 2
c

U Rli  2
(1  lj )c j
2 i
2
2
2b   4b  
4b  
4b   2
>
a (2b   )  2b2ci  (2b2   2 )U Rli  b[b(U Rli  ci )  a ]
4b2   2
=
a (b   )  (b2   2 )U Rli  3b2ci
4b2   2
>
(b   )(b   )(U Rli  ci )  (b2   2 )U Rli  3b2ci
4b2   2
= ci .
Obviously, we have wlirf>0.
Appendix D: Proof of Proposition 4
(a) under this situation, we have
USli[FF]-URli[FF]= 
=
2b2 (2b   )(a  bc   c)(4b2  b  2 2 ) 2 (4b2  b  2 2 )
(4b2   2 )(16b4  17b2 2  4 4 )2
2b2 (2b   )(a  bc   c)(4b2  b  2 2 )2 (4b  3 )(b   )   2 
(4b2   2 )(16b4  17b2 2  4 4 )2
;
As stated in Appendix C, we have a-bc+θc>0. Together with the assumption “b>θ”, we
have “USli[FF]/URli[FF]<1” holds true no matter how the environment parameters change;
(b) under this situation, we have
USci[FF]-URci[FF]=

(  1)
2
ci -
ci
1
=
c >0.
  1 (  1) 2 i
Thus, we have “USci[FF]/URci[FF]>1” no matter how the environment parameters
change;
Appendix E: Proof of Proposition 5
According to the discussions in Appendix A, we know that the maximum value of
ΠSi cannot be obtained at the boundaries of wi.
Linear demand case
The first order conditions for Eq. (6) with respect to wli are given as follows:
  Sl1
 w  2b(1  l1 ) wl1   (1  l 2 ) wl 2  a  bc1 (1  l1 )  0
 l1
.

  Sl 2   (1   ) w  2b(1   ) w  a  bc (1   )  0
l1
l1
l2
l2
2
l2

 wl 2
It is easily shown that
2b(1  l1 ),  (1  l 2 )
 (1  l1 ), - 2b(1  l 2 )
(A5)
 (4b2   2 )(1  l1 )(1  l 2 )  0 .
Therefore, the linear system of equations in (A5) has one and only one set of solutions
which are given in Eq. (7a). Manipulating with the second order condition with respect
to wli leads to:
 2 Sli
 2b(1  li )  0 .
 2 wli
Therefore, the second order conditions are satisfied.
Iso-elastic Demand Case
The first order conditions for Eq. (6) with respect to wci are given as follows:

 
K  (1  c 2 ) wc 2 
Sc1

 (  1) wc1   c1   0

 
wc1  (1  c1 ) wc1 
 wc1
.


K  (1  c1 ) wc1 
  Sc 2
 (  1) wc 2   c2   0
 
 w 
w
(1


)
w


c
2
c2
c2
c2

The unique set of solutions for (A6) is given in Eq. (7b). And the second order
condition is also satisfied; i.e., we have
 2 Sci
 2 wci
 K 
wci  wci rf
(  1)  2  (1  cj ) wcj 
( ci ) 1  (1  ci )

0.
Appendix F: The Existence and Uniqueness of Equilibrium Makeup for [PP]
(A6)
For the iso-elastic demand case, the first order conditions for Eq. (8) with respect to ηci
are given as:


  Rc1
K c1
 (1  c1 ) c1   (1  c 2 ) c2 



  (  1)c1  1  0

(  1)(1  c1 )    1     1 
 c1
. (A7)




(1


)

c
(1


)

c
K

c


 Rc 2
c2
2
c1
1
2
    (  1)(1   )     1      1    (  1)c 2  1  0
c2
 c2
Obviously, the set of optimal solutions for (A7) is unique, and is given in Eq. (7b). And
the second order condition is also satisfied; i.e., we have
 2 Rci
 2ci

ci ci[PP]
K  (1  cj )c j 

 2    ci 1  (  1)   1
 0.
Although we can analytically prove the existence and uniqueness of equilibrium
markup for the iso-elastic demand curve, we fail to do that for linear demand curve.
Nevertheless, we can graphically show the relationship between ηl1 and ηl2 which is
depicted in Figure A1. It is obviously that, the equilibrium solution is unique.
Fig.A1. The relationship between ηl1 and ηl2, a=10, b=1, c1=c2=1, θ=0.5
Appendix G: Proof of Proposition 8
Linear Demand Case
The first order conditions for Eq. (10) with respect to wli are given as follows:
  Sli
 w  2bwli   (1  lj ) wlj  a  b(U Rli  ci )  0
 li
, i  j.
 
Slj

  wli  2b(1  lj ) wlj  a  U Rli  b(1  lj )c j  0

 wlj
It is easily shown that
2b,  (1  lj )
 , - 2b(1  lj )
(A8)
 (4b 2   2 )(1  lj )  0 . Therefore, the linear
system of equations in (A8) has one and only one set of solutions which are given in Eq.
(11a) and (11b). Manipulating with the second order condition with respect to wli leads
to:
  2  Sli
 2b  0
 2
  wli
, i j.
 2


Slj

 2b(1  lj )  0
  2 wlj

Therefore, the second order conditions are satisfied.
Iso-elastic Demand Case
The first order conditions for Eq. (10) with respect to wci are given as follows:


K (1  cj ) wcj 

Sci


  (  1) wci  U Rci   ci   0
 wci
( wci  U Rci ) 1
, i j.



K
(
w

U
)
Scj

ci
Rci

 (  1) wcj   c j   0
 
 wcj


w
(1


)
w
cj 
cj
cj 

(A9)
The unique set of solutions for (A9) is given in Eq. (12). And the second order
conditions are also satisfied; i.e., we have

 2
K  (  1)  2  (1  cj ) wcj 


Sci


0
 1
  2 wci
 (U Rci  ci )
, i  j.
 2
 2

   Slj
K  (  1)  ( wci  U Rci )

0
 2
( c j ) 1  (1  cj )
  wlj
Appendix H: Proof of Proposition 9
Anticipating how the suppliers respond to the declared markups through Eq. (11) and
Eq. (12) for the linear or iso-elastic demand case respectively, the dominant retailers
decide the optimal markups to maximize their own profits as:
ΠRi(URi|ηj)=URi∙Di(wi+URi,(1+ηj)wj),
(A10a)
ΠRj(ηj|URi)=ηjwj∙Dj(wi+URi,(1+ηj)wj),
(A10b)
According to the discussions in Appendix B, the maximum value for ΠRi cannot be
achieved at the boundaries of URi and ηj.
Linear Demand Case
The optimal values of (URli, ηlj) which maximize the retailers’ profits can be obtained
by solving the following equations:
  Rli
 w  0
 li
, i  j.
 
 Rlj  0
 wlj
(A11)
Unfortunately, we cannot prove the existence and uniqueness of the equilibrium
outcomes analytically. Nevertheless, we can graphically show that there is one and only
one set of solutions that satisfied (A11). The relationship between URli and ηlj is given in
Figure A2.
Fig. A2. The relationship between URli and ηlj, a=10, b=1, c1=c2=1, θ=0.5
Straightforward manipulation with the first order condition of Eq. (A10a) with
respect to URli gives the optimal markup for the dominant retailer i as:
URlirf=
a(2b   ) ci
b
 
(1  lj )c j .
2
2
2(2b   ) 2 2(2b2   2 )
(A12)
Substituting URlirf into the equation “∂ΠRlj(ηlj|URli)/∂ηlj=0” and solving it gives
ηlj=
 2 

3
Y1  3 Y2
31
.
(A13)
where
1  2b2c j 2 (64b6  7 6  46b2 4  96b4 2 ) ,
 2  2c j (160c j b8  240c j 2b6  4 2 (ci  4a)b5  ( (227 / 2) 4c j  12 3a)b 4
 (14 4 a  4 5ci )b3  (10 5 a  17 6c j )b 2   6 (ci  3a)b  2 7 a)
 3  2c j (192c j b6 2  128c j b8  9 5b3ci  30 3b4a  31 4b3a  10 3b5ci
 13b2 6c j  40 2b5a  89c j b4 4  4 7a  6 6ba  22 5b2a  2 7bci )
,
,
 4  (2 4c j  (bci  2a ) 3  (9c j b  3a )b 2  2b2 (bci  3a )  b3 ( c j b  a ))
(( bci  2a ) 3  3b( c j b  a ) 2  2b2 (bci  3a )  8b3 ( c jb  a ))
,
A   2 2  313 , B  2 3  91 4 , C   32  3 2  4 ,
  B  B 2  4 AC
Y1  A   2  31 

2


,


  B  B 2  4 AC
Y2  A   2  31 

2


.


Other optimal solutions (such as the optimal wholesale price, retail price, market
demand, profits) can then be determined by substituting the expressions in (A13) into
the corresponding functions.
Appendix I: Proof of Proposition 10
According to the entries in Table 2, we have
2
2 
 
     1 
wi[PP]/wi[FF]= 
=
<1;
ci  
c
i
2
2
   1   (  1)
   1
c    1 2(   )1 



   1   


Kc
ΠRi[PP]/ΠRi[FF]=  i
ΠSi
[PP]
 1 
j
c    1 2(   ) 



   1   


Kc
/ΠSi[FF]=  i
 1 
j
 Kci  1c j

   1
  1


  
2(    )


>1;

   1
 Kci  1c j    1 2(   ) 1    1
<1.





   1   

Appendix J: The Elasticity of Demand
Let ES be the elasticity of demand the supplier faces, and it can be stated as:
ESi=
dDi wi
 .
dwi Di
(A14)
When the retailer offers a fixed-dollar markup; i.e., pi=wi+URi, the price elasticity of
demand the supplier faces under an iso-elastic demand curve can be expressed as:
ESi[ FF ]  K    (wi  Ui ) 1  p j  
wi
wi
.
  


K (wi  Ui ) p j
wi  Ui
(A15)
When the retailer offers a percentage markup, i.e., pi=(1+ηi)w, the price elasticity of
demand at the wholesale level can be stated as:
ESi[ PP ]  K    (1  i )  wi  1  p j  
wi
  . (A16)
K  (1  i )  wi   p j 

From (A15) and (A16), we have
E Si[ FF ]
wi

 1.
[ PP ]
E Si
wi  U Ri
(A17)
Eq. (A17) implies that the supplier faces a more elastic demand when the retailer
offers a percentage markup pricing strategy.