Mathematics proof for equilibrium locations if the upstream
firm determines the globally optimal wholesale price
In the following contents, we will show that the main results in the previous
version hold even if we consider that the upstream firm maximizes its profit to
determine the “globally” optimal wholesale price.
In the third stage, the profit functions of two firms under duopoly and monopoly
can be expressed respectively as (3). Differentiating (3), we have the equilibrium
prices as (4) and the equilibrium outputs as (5). Substituting (4) into (3), we have the
equilibrium profits of two firms in the third stage as (6). Using (5.1) and (5.3), we
have the profits of the upstream firm under duopoly and monopoly market structures,
which can be expressed respectively as (7). Then, we can have the equilibrium input
prices as (8.2), (8.3) and (8.4) under duopoly and monopoly, respectively. Until now,
the results in the third and second stages are the same as the previous version.
Now, we are going to consider the “globally” optimal wholesale price under
which maximizes the upstream firm’s profit. The downstream market structure will be
duopoly (monopoly) if the upstream firm’s profit under duopoly case is higher (lower)
than that under monopoly. We substitute the input price (i.e., (8.2)) into the upstream
firm’s profit (i.e., (7.1)). The upstream firm’s profit is as following when the market
structure is duopoly.
 uD 
{ 2 (cu  cv )  t ( xd  x v )[6   (2  xd  xv )]}2
,
24 2t ( xd  xv )
if cu  cv  (2 /  ) 2  ( /  2 ).
(R.1)
Considering all the consumers are served in the market, there are two
equilibrium input prices under monopoly. Substituting (8.3) and (8.4) into (7.2),
respectively, we can have the upstream firm’s profit when the downstream market is a
monopoly under two input prices cases.
1
 uM
[k  t (1  xd ) 2  cu ](1   ), if 0  xv  1 / 2 and

 (k  cv )   2  t (1  xd ) 2 ,


2
[k  t (2 xd  2 xv  xd )  cu ](1   ), if 1/2  xv  1 and

 (k  cv )   2  t (2 xd  2 xv  xd2 ).

(R.2)
(R.3)
1. The upstream firm determines the globally optimal wholesale price
Now, we are going to compare the upstream firm’s profit under monopoly and
duopoly cases. Due to the two cases of the equilibrium input prices under monopoly,
we divide our comparison into two cases, which are 0  xv  (1/2) and (1/2) < xv  1,
respectively.
Case 1: 0  xv  (1/2)
When 0  xv  (1/2), the profit of the upstream firm under monopoly will be (R.2)
if  (k  cv )   2  t (1  xd ) 2 (corresponding to the condition in (8.3) in manuscript).
Under the above conditions, we compare (R.1) with (R.2) and find that the upstream
firm’s profit under monopoly is higher than that under duopoly if and only if k >
{ 2 (cu  cv )  t ( xd  x v )[6   (2  xd  xv )]}2
~
k c u t (1  xd ) 2 
.
24(1   ) 2t ( xd  x v )
~
Therefore, the equilibrium market structure is monopoly when k  k . Otherwise,
the upstream firm’s profit under duopoly is higher than that under monopoly.
Case 2: (1/2) < xv  1
When (1/2) < xv  1, the profit of the upstream firm under monopoly will be (R.3)
if  (k  cv )   2  t (2 xd  2 xv  xd2 ) (corresponding to the condition in (8.4) in
manuscript). Under the above conditions, we compare (R.1) and (R.3) and find that
the upstream firm’s profit under monopoly is higher than that under duopoly if and
2

{ 2 (cu  cv )  t ( xd  x v )[6   (2  xd  xv )}2
only if k > k  cu  t(2xd2xvxd2) 
.
24(1   ) 2t ( xd  x v )

Therefore, the equilibrium market structure is monopoly when k  k . Otherwise,
the upstream firm’s profit under duopoly is higher than that under monopoly.
According to the above analysis, the equilibrium downstream market structure is
duopoly if and only if the following conditions are hold, which are as following:
~
(1) 0  xv  (1/2),  (k  cv )   2  t (1  xd ) 2 and k < k ,

(2) (1/2) < xv  1,  (k  cv )   2  t (1  xd ) 2 and k < k .
Similarly, the optimal market structure is monopoly if and only if the following
conditions are hold, which are as following:
~
(3) 0  xv  (1/2),  (k  cv )   2  t (2 xd  2 xv  xd2 ) and k  k ,

(4) (1/2) < xv  1,  (k  cv )   2  t (2 xd  2 xv  xd2 ) and k  k .
2. Optimal location choice for the downstream firms
2.1 The optimal location under the duopoly market structure
In this section, we investigate the optimal locations for the two firms when we
consider the upstream firm’s decision. When the market structure is duopoly, the
profits of the two firms are (corresponding to (9.1) and (9.3) in manuscript):
 vD   ( pvD  cv )qvD  (1   )( pvD  w D )qvD ,
~
 0  (cu  cv )   3 ,0  xv  1 / 2, and k  k ,
if 

0

(
c

c
)


,
1
/
2

x

1
,
and
k

k
,
u
v
3
v

~
 0  (cu  cv )   3 ,0  xv  1 / 2, and k  k ,

0  (cu  cv )   3 ,1 / 2  xv  1, and k  k .
 dD  ( pdD  w D )qdD , if 
(R.4)
(R.5)
Differentiating (R.4) and (R.5) with respect to xv and xd, respectively, we find that the
equilibrium locations of two firms under duopoly are (corresponding to (13.1) and
3
(13.2) in manuscript):
x
D*
v
 0,
~
 0  (cu  cv )   d , and k  k D ,
if 
D
 d  (cu  cv ), and k  k ,
[t (5  3)  A ] / 3t ,
xdD*  
 1,
~
if 0  (cu  cv )   d , and k  k D
D
if  d  (cu  cv ), and k  k .
(R.6)
(R.7)


~
~
where k D  k ( xvD* = 0, xdD*  [t (5  3)  A] / 3t ) and k D  k ( xvD* = 0, xdD*  1 ).
According to the above results, we find that the optimal downstream firm’s
locations under duopoly are the same as the previous version even if the market
structure is endogenous determined by the upstream firm.
2.2 The optimal location under the monopoly market structure
Under monopoly, the firms’ profits are (corresponding to (9.2) and (9.4) in
manuscript):
 vM   ( pvM  cv )  (1   )( pvM  w M ),
~
 (cu  cv )   3 ,0  xv  1 / 2, and k  k ,
if 

(cu  cv )   3 ,1 / 2  xv  1, and k  k ,

M
d
0
~
 (cu  cv )   3 ,0  xv  1 / 2, and k  k ,
if 

(cu  cv )   3 ,1 / 2  xv  1, and k  k .
(R.8)
(R.9)
Substituting (8.3) and (8.4) into (R.8), we have the profit of the monopolist firm v
as (corresponding to (15.1) and (15.2) in manuscript):
 vM
 (k  cv )  t (1  xd ) 2  t ( xd  xv )( 2  xd  xv ),

~

if 0  xv  1 / 2, (cu  cv )   3 and k  k ,

2
2
 (k  cv )  t[(1   )( 2 xd  2 xv  xd )  xv ],


if
1
/
2

x

1
,
(
c

c
)


and
k

k
.
v
u
v
3

(R.10)
(R.11)
Differentiating (R.10) and (R.11) with respect to xv respectively, the optimal
location of the monopolist firm v is (corresponding to (16.1) and (16.2) in
4
manuscript):
~
~


where k M ≡ k ( xvM *  1/ 2) and k M ≡ k ( xvM *  1   ) .
From (R.12) and (R13), the equilibrium locations are still the same as the previous
version.
Based on the above analysis, we find that the main results are the same as the
previous version when the upstream firm considers the global equilibrium. The
conditions for the results are more complicated and it does not have any new insights.
Therefore, we request that we only show the main results as market structure
exogenous determined in the paper and have the description for the global equilibrium
on footnote 9 on page 9.
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