Proof of Proposition 5 (Sharing Demand Information with Retailer under Upstream Competition)
Author
Anonymous Author
Title
Proof of Proposition 5 (Sharing Demand Information with Retailer under Upstream Competition)
Description
Technical proof of one of the main results
Category
Working Material
Keywords
URL
http://www.notebookarchive.org/2020-07-52zj3rg/
DOI
https://notebookarchive.org/2020-07-52zj3rg
Date Added
2020-07-11
Date Last Modified
2020-07-11
File Size
140.32 kilobytes
Supplements
Rights
Redistribution rights reserved
Download
Open in Wolfram Cloud
Proof of Proposition 5
Proof of Proposition 5
Define a very small number
Define a very small number
In[]:=
Eps:=
-10
10
The expressions for expected profits are when s1=s and s2=s.
When only manufacturer 1 shares information
The expressions for expected profits are when =s and =s.
When only manufacturer 1 shares information
s
1
s
2
When only manufacturer 1 shares information
In[]:=
Pi1PS1[s_,ϕ_,a_,σ_]:=(1+ϕ)+Pi2PS1[s_,ϕ_,a_,σ_]:=s(2(-2+s+)+(-4+s+(2+s))ϕ)(2(-1+s)+ϕ(-5-3ϕ+s(4+s+ϕ+2sϕ)))+σ(2(-2+s+)+(-4+s+2+)ϕ)(2(-1+s)+(-5+s(4+s))ϕ+(-3+s+2))
2
a
2
(4+ϕ(12+(8+s)ϕ)-(4+2(5+s)ϕ+(5+4s)))
2
s
2
ϕ
2
2
(-1+)
2
s
2
(8+4ϕ(2+ϕ)(3+ϕ)+sϕ(4+5ϕ(2+ϕ)))
sσ(1+ϕ)
2
(-2+2+(-2++s(-2+3s))ϕ)
2
s
3
s
2
2
(-1+)
2
s
2
(4+8ϕ+(4-))
2
s
2
ϕ
2
a
2
(2+3ϕ)
2
s
2
s
2
2
(-1+)
2
s
2
(8+4ϕ(2+ϕ)(3+ϕ)+sϕ(4+5ϕ(2+ϕ)))
2
s
2
s
2
s
3
s
2
s
2
ϕ
2
2
(-1+)
2
s
2
(4+8ϕ+(4-))
2
s
2
ϕ
When both manufacturers share information
When both manufacturers share information
In[]:=
PiFS[s_,ϕ_,a_,σ_]:=(1+ϕ)+
2
a
2
2
(2+ϕ)
sσ(1+ϕ)
2
(2+(2+s)ϕ)
2
2
(4+8ϕ+(4-))
2
s
2
ϕ
When both manufacturers do not share information
When both manufacturers do not share information
In[]:=
PiNS[s_,ϕ_,a_,σ_]:=s(1+2ϕ)(2(-1+s)+(-2+s+)ϕ)(2(-2+s+)+(-4+(3+s))ϕ)+σ(1+2ϕ)(2(-1+s)+(-2+s+)ϕ)(2(-2+s+)+(-4+(3+s))ϕ)
2
a
2
(2+(2+s)ϕ)
2
s
2
s
2
s
2
2
(-8-4ϕ(2+ϕ)(2+s+sϕ)+ϕ(4+ϕ(8+3ϕ+2s(1+ϕ)))+s(-4ϕ(1+ϕ)(2+ϕ)+ϕ(2+ϕ)(2+3ϕ)+2s(4+ϕ(12+7ϕ))))
3
s
2
s
2
s
2
s
2
s
2
s
2
2
(-1+)
2
s
2
(4+8ϕ+(4-))
2
s
2
ϕ
Part (a) of the proposition: Incentive to share information when other manufacturer does not share.
Part (a) of the proposition: Incentive to share information when other manufacturer does not share.
In[]:=
Pi1PS1[s,ϕ,a,σ]-PiNS[s,ϕ,a,σ]/.{σc}//Factor;%(8)//Factor
2
a
2
(1+s)
2
(1+s+sϕ)
2
(2+2ϕ-sϕ)
2
(2+2ϕ+sϕ)
2
(8+24ϕ+4sϕ+20+10s+4+5s)
2
ϕ
2
ϕ
3
ϕ
3
ϕ
2
a
Out[]=
Since (82(1+s)2(1+s+sϕ)2(2+2ϕ-sϕ)2(2+2ϕ+sϕ)2(8+24ϕ+4sϕ+202ϕ+10s2ϕ+43ϕ+5s3ϕ))/2a>0 , the sign of Pi1PS1[s,ϕ,a,σ]-PiNS[s,ϕ,a,σ] is the same as the sign of the following polynomial:
Since >0 , the sign of Pi1PS1[s,ϕ,a,σ]-PiNS[s,ϕ,a,σ] is the same as the sign of the following polynomial:
(8)/
2
(1+s)
2
(1+s+sϕ)
2
(2+2ϕ-sϕ)
2
(2+2ϕ+sϕ)
2
(8+24ϕ+4sϕ+20+10s+4+5s)
2
ϕ
2
ϕ
3
ϕ
3
ϕ
2
a
In[]:=
Let phi1hat[s,c] denote the real and positive root of Poly1[s,ϕ,c]=0 for given s and c:
Let phi1hat[s,c] denote the real and positive root of Poly1[s,ϕ,c]=0 for given s and c:
In[]:=
phi1hat[s_,c_]:=ϕ/.Solve[Poly1[s,ϕ,c]0&&ϕ≥0,ϕ,Reals][[1]];
The following plot shows that phi1hat[s,c] is unique for 0<s<1 and 0 < c < 1/4 ( c= 1/4 implies very high coefficient of variation of 1/2).
The following plot shows that phi1hat[s,c] is unique for 0<s<1 and 0 < c < 1/4 ( c= 1/4 implies very high coefficient of variation of 1/2).
In[]:=
Manipulate[Plot[phi1hat[s,c],{s,Eps,1-Eps},AxesOrigin{0,0}],{c,Eps,1/4}]
Out[]=
| |||||||
| |||||||
Part (b) of the proposition: Incentive to share information when other manufacturer shares.
Part (b) of the proposition: Incentive to share information when other manufacturer shares.
In[]:=
PiFS[s,ϕ,a,σ]-Pi2PS1[s,ϕ,a,σ]/.{σc}//Factor;%(2)//Factor
2
a
2
(1+s)
2
(2+ϕ)
2
(2+2ϕ-sϕ)
2
(2+2ϕ+sϕ)
2
(8+24ϕ+4sϕ+20+10s+4+5s)
2
ϕ
2
ϕ
3
ϕ
3
ϕ
2
a
Out[]=
1024+1024cs+11264ϕ+10240csϕ+1024cϕ+55296-512s+45312cs-1024+9728c-768c-512c+159744-4864s+116992cs-9472+40448c-5888c-5376c-768c+301312-20352s+195584cs-38144+96896c+384-18944c+384-23424c-6144c-384c+388864-49216s+221952cs-87744+148032c+2880-33216c+2880-55744c-20672c-2624c-64c+349440-75776s+174144cs-127040+150592c+9120-34368c+9024-80000c-96-38080c-64-7296c-384c+217856-76992s+94272cs-120064+103200c+15792-21024c+15312-71936c-528-41920c-368-10688c-896c+91904-51584s+34432cs-74112+46976c+16128-6972c+15216-40608c-1128-28240c-816-8928c+8-1040c+4+24832-21952s+8064cs-28864+13568c+9696-820c+8880-13904c-1164-11392c-868-4272c+28-640c+16+3840-5376s+1088cs-6464+2240c+3168+140c+2832-2624c-576-2524c-444-1090c+30-200c+20+256-576s+64cs-640+160c+432+36c+384-208c-108-236c-88-115c+9-25c+7
2
s
2
ϕ
2
ϕ
2
ϕ
2
s
2
ϕ
2
s
2
ϕ
3
s
2
ϕ
4
s
2
ϕ
3
ϕ
3
ϕ
3
ϕ
2
s
3
ϕ
2
s
3
ϕ
3
s
3
ϕ
4
s
3
ϕ
5
s
3
ϕ
4
ϕ
4
ϕ
4
ϕ
2
s
4
ϕ
2
s
4
ϕ
3
s
4
ϕ
3
s
4
ϕ
4
s
4
ϕ
4
s
4
ϕ
5
s
4
ϕ
6
s
4
ϕ
5
ϕ
5
ϕ
5
ϕ
2
s
5
ϕ
2
s
5
ϕ
3
s
5
ϕ
3
s
5
ϕ
4
s
5
ϕ
4
s
5
ϕ
5
s
5
ϕ
6
s
5
ϕ
7
s
5
ϕ
6
ϕ
6
ϕ
6
ϕ
2
s
6
ϕ
2
s
6
ϕ
3
s
6
ϕ
3
s
6
ϕ
4
s
6
ϕ
4
s
6
ϕ
5
s
6
ϕ
5
s
6
ϕ
6
s
6
ϕ
6
s
6
ϕ
7
s
6
ϕ
7
ϕ
7
ϕ
7
ϕ
2
s
7
ϕ
2
s
7
ϕ
3
s
7
ϕ
3
s
7
ϕ
4
s
7
ϕ
4
s
7
ϕ
5
s
7
ϕ
5
s
7
ϕ
6
s
7
ϕ
6
s
7
ϕ
7
s
7
ϕ
8
ϕ
8
ϕ
8
ϕ
2
s
8
ϕ
2
s
8
ϕ
3
s
8
ϕ
3
s
8
ϕ
4
s
8
ϕ
4
s
8
ϕ
5
s
8
ϕ
5
s
8
ϕ
6
s
8
ϕ
6
s
8
ϕ
7
s
8
ϕ
7
s
8
ϕ
8
s
8
ϕ
9
ϕ
9
ϕ
9
ϕ
2
s
9
ϕ
2
s
9
ϕ
3
s
9
ϕ
3
s
9
ϕ
4
s
9
ϕ
4
s
9
ϕ
5
s
9
ϕ
5
s
9
ϕ
6
s
9
ϕ
6
s
9
ϕ
7
s
9
ϕ
7
s
9
ϕ
8
s
9
ϕ
10
ϕ
10
ϕ
10
ϕ
2
s
10
ϕ
2
s
10
ϕ
3
s
10
ϕ
3
s
10
ϕ
4
s
10
ϕ
4
s
10
ϕ
5
s
10
ϕ
5
s
10
ϕ
6
s
10
ϕ
6
s
10
ϕ
7
s
10
ϕ
7
s
10
ϕ
8
s
10
ϕ
11
ϕ
11
ϕ
11
ϕ
2
s
11
ϕ
2
s
11
ϕ
3
s
11
ϕ
3
s
11
ϕ
4
s
11
ϕ
4
s
11
ϕ
5
s
11
ϕ
5
s
11
ϕ
6
s
11
ϕ
6
s
11
ϕ
7
s
11
ϕ
7
s
11
ϕ
8
s
11
ϕ
Since (22(1+s)2(2+ϕ)2(2+2ϕ-sϕ)2(2+2ϕ+sϕ)2(8+24ϕ+4sϕ+202ϕ+10s2ϕ+43ϕ+5s3ϕ))/2a>0 , the sign of PiFS[s,ϕ,a,σ]-Pi2PS1[s,ϕ,a,σ] is the same as the sign of the following polynomial:
Since >0 , the sign of PiFS[s,ϕ,a,σ]-Pi2PS1[s,ϕ,a,σ] is the same as the sign of the following polynomial:
(2)/
2
(1+s)
2
(2+ϕ)
2
(2+2ϕ-sϕ)
2
(2+2ϕ+sϕ)
2
(8+24ϕ+4sϕ+20+10s+4+5s)
2
ϕ
2
ϕ
3
ϕ
3
ϕ
2
a
In[]:=
Poly2[s_,ϕ_,c_]:=1024+1024cs+11264ϕ+10240csϕ+1024cϕ+55296-512s+45312cs-1024+9728c-768c-512c+159744-4864s+116992cs-9472+40448c-5888c-5376c-768c+301312-20352s+195584cs-38144+96896c+384-18944c+384-23424c-6144c-384c+388864-49216s+221952cs-87744+148032c+2880-33216c+2880-55744c-20672c-2624c-64c+349440-75776s+174144cs-127040+150592c+9120-34368c+9024-80000c-96-38080c-64-7296c-384c+217856-76992s+94272cs-120064+103200c+15792-21024c+15312-71936c-528-41920c-368-10688c-896c+91904-51584s+34432cs-74112+46976c+16128-6972c+15216-40608c-1128-28240c-816-8928c+8-1040c+4+24832-21952s+8064cs-28864+13568c+9696-820c+8880-13904c-1164-11392c-868-4272c+28-640c+16+3840-5376s+1088cs-6464+2240c+3168+140c+2832-2624c-576-2524c-444-1090c+30-200c+20+256-576s+64cs-640+160c+432+36c+384-208c-108-236c-88-115c+9-25c+7
2
s
2
ϕ
2
ϕ
2
ϕ
2
s
2
ϕ
2
s
2
ϕ
3
s
2
ϕ
4
s
2
ϕ
3
ϕ
3
ϕ
3
ϕ
2
s
3
ϕ
2
s
3
ϕ
3
s
3
ϕ
4
s
3
ϕ
5
s
3
ϕ
4
ϕ
4
ϕ
4
ϕ
2
s
4
ϕ
2
s
4
ϕ
3
s
4
ϕ
3
s
4
ϕ
4
s
4
ϕ
4
s
4
ϕ
5
s
4
ϕ
6
s
4
ϕ
5
ϕ
5
ϕ
5
ϕ
2
s
5
ϕ
2
s
5
ϕ
3
s
5
ϕ
3
s
5
ϕ
4
s
5
ϕ
4
s
5
ϕ
5
s
5
ϕ
6
s
5
ϕ
7
s
5
ϕ
6
ϕ
6
ϕ
6
ϕ
2
s
6
ϕ
2
s
6
ϕ
3
s
6
ϕ
3
s
6
ϕ
4
s
6
ϕ
4
s
6
ϕ
5
s
6
ϕ
5
s
6
ϕ
6
s
6
ϕ
6
s
6
ϕ
7
s
6
ϕ
7
ϕ
7
ϕ
7
ϕ
2
s
7
ϕ
2
s
7
ϕ
3
s
7
ϕ
3
s
7
ϕ
4
s
7
ϕ
4
s
7
ϕ
5
s
7
ϕ
5
s
7
ϕ
6
s
7
ϕ
6
s
7
ϕ
7
s
7
ϕ
8
ϕ
8
ϕ
8
ϕ
2
s
8
ϕ
2
s
8
ϕ
3
s
8
ϕ
3
s
8
ϕ
4
s
8
ϕ
4
s
8
ϕ
5
s
8
ϕ
5
s
8
ϕ
6
s
8
ϕ
6
s
8
ϕ
7
s
8
ϕ
7
s
8
ϕ
8
s
8
ϕ
9
ϕ
9
ϕ
9
ϕ
2
s
9
ϕ
2
s
9
ϕ
3
s
9
ϕ
3
s
9
ϕ
4
s
9
ϕ
4
s
9
ϕ
5
s
9
ϕ
5
s
9
ϕ
6
s
9
ϕ
6
s
9
ϕ
7
s
9
ϕ
7
s
9
ϕ
8
s
9
ϕ
10
ϕ
10
ϕ
10
ϕ
2
s
10
ϕ
2
s
10
ϕ
3
s
10
ϕ
3
s
10
ϕ
4
s
10
ϕ
4
s
10
ϕ
5
s
10
ϕ
5
s
10
ϕ
6
s
10
ϕ
6
s
10
ϕ
7
s
10
ϕ
7
s
10
ϕ
8
s
10
ϕ
11
ϕ
11
ϕ
11
ϕ
2
s
11
ϕ
2
s
11
ϕ
3
s
11
ϕ
3
s
11
ϕ
4
s
11
ϕ
4
s
11
ϕ
5
s
11
ϕ
5
s
11
ϕ
6
s
11
ϕ
6
s
11
ϕ
7
s
11
ϕ
7
s
11
ϕ
8
s
11
ϕ
Let phi2hat[s,c] denote the real and positive root of Poly2[s,ϕ,c]=0 for given s and c:
Let phi2hat[s,c] denote the real and positive root of Poly2[s,ϕ,c]=0 for given s and c:
In[]:=
phi2hat[s_,c_]:=ϕ/.Solve[Poly2[s,ϕ,c]0&&ϕ≥0,ϕ,Reals][[1]];
The following plot shows that phi2hat[s,c] is unique for 0<s<1 and 0 < c < 1/4 ( c= 1/4 implies very high coefficient of variation of 1/2).
The following plot shows that phi2hat[s,c] is unique for 0<s<1 and 0 < c < 1/4 ( c= 1/4 implies very high coefficient of variation of 1/2).
In[]:=
Manipulate[Plot[phi2hat[s,c],{s,Eps,1-Eps},AxesOrigin{0,0}],{c,Eps,1/4}]
Out[]=
| |||||||
| |||||||
The following shows that Poly2[s,ϕ,c] >0 for s < (193-9)14:
The following shows that Poly2[s,ϕ,c] >0 for s < :
(
193
-9)14In[]:=
Reduce1024+1024cs+11264ϕ+10240csϕ+1024cϕ+55296-512s+45312cs-1024+9728c-768c-512c+159744-4864s+116992cs-9472+40448c-5888c-5376c-768c+301312-20352s+195584cs-38144+96896c+384-18944c+384-23424c-6144c-384c+388864-49216s+221952cs-87744+148032c+2880-33216c+2880-55744c-20672c-2624c-64c+349440-75776s+174144cs-127040+150592c+9120-34368c+9024-80000c-96-38080c-64-7296c-384c+217856-76992s+94272cs-120064+103200c+15792-21024c+15312-71936c-528-41920c-368-10688c-896c+91904-51584s+34432cs-74112+46976c+16128-6972c+15216-40608c-1128-28240c-816-8928c+8-1040c+4+24832-21952s+8064cs-28864+13568c+9696-820c+8880-13904c-1164-11392c-868-4272c+28-640c+16+3840-5376s+1088cs-6464+2240c+3168+140c+2832-2624c-576-2524c-444-1090c+30-200c+20+256-576s+64cs-640+160c+432+36c+384-208c-108-236c-88-115c+9-25c+7<0&&0<s<(
2
s
2
ϕ
2
ϕ
2
ϕ
2
s
2
ϕ
2
s
2
ϕ
3
s
2
ϕ
4
s
2
ϕ
3
ϕ
3
ϕ
3
ϕ
2
s
3
ϕ
2
s
3
ϕ
3
s
3
ϕ
4
s
3
ϕ
5
s
3
ϕ
4
ϕ
4
ϕ
4
ϕ
2
s
4
ϕ
2
s
4
ϕ
3
s
4
ϕ
3
s
4
ϕ
4
s
4
ϕ
4
s
4
ϕ
5
s
4
ϕ
6
s
4
ϕ
5
ϕ
5
ϕ
5
ϕ
2
s
5
ϕ
2
s
5
ϕ
3
s
5
ϕ
3
s
5
ϕ
4
s
5
ϕ
4
s
5
ϕ
5
s
5
ϕ
6
s
5
ϕ
7
s
5
ϕ
6
ϕ
6
ϕ
6
ϕ
2
s
6
ϕ
2
s
6
ϕ
3
s
6
ϕ
3
s
6
ϕ
4
s
6
ϕ
4
s
6
ϕ
5
s
6
ϕ
5
s
6
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6
s
6
ϕ
6
s
6
ϕ
7
s
6
ϕ
7
ϕ
7
ϕ
7
ϕ
2
s
7
ϕ
2
s
7
ϕ
3
s
7
ϕ
3
s
7
ϕ
4
s
7
ϕ
4
s
7
ϕ
5
s
7
ϕ
5
s
7
ϕ
6
s
7
ϕ
6
s
7
ϕ
7
s
7
ϕ
8
ϕ
8
ϕ
8
ϕ
2
s
8
ϕ
2
s
8
ϕ
3
s
8
ϕ
3
s
8
ϕ
4
s
8
ϕ
4
s
8
ϕ
5
s
8
ϕ
5
s
8
ϕ
6
s
8
ϕ
6
s
8
ϕ
7
s
8
ϕ
7
s
8
ϕ
8
s
8
ϕ
9
ϕ
9
ϕ
9
ϕ
2
s
9
ϕ
2
s
9
ϕ
3
s
9
ϕ
3
s
9
ϕ
4
s
9
ϕ
4
s
9
ϕ
5
s
9
ϕ
5
s
9
ϕ
6
s
9
ϕ
6
s
9
ϕ
7
s
9
ϕ
7
s
9
ϕ
8
s
9
ϕ
10
ϕ
10
ϕ
10
ϕ
2
s
10
ϕ
2
s
10
ϕ
3
s
10
ϕ
3
s
10
ϕ
4
s
10
ϕ
4
s
10
ϕ
5
s
10
ϕ
5
s
10
ϕ
6
s
10
ϕ
6
s
10
ϕ
7
s
10
ϕ
7
s
10
ϕ
8
s
10
ϕ
11
ϕ
11
ϕ
11
ϕ
2
s
11
ϕ
2
s
11
ϕ
3
s
11
ϕ
3
s
11
ϕ
4
s
11
ϕ
4
s
11
ϕ
5
s
11
ϕ
5
s
11
ϕ
6
s
11
ϕ
6
s
11
ϕ
7
s
11
ϕ
7
s
11
ϕ
8
s
11
ϕ
193
-9)14&&ϕ>0&&0<c<12Out[]=
False
Part (c) of the proposition: Comparison of profits under full and no sharing
Part (c) of the proposition: Comparison of profits under full and no sharing
In[]:=
PiFS[s,ϕ,a,σ]-PiNS[s,ϕ,a,σ]/.{σc}//Factor;%8//Factor
2
a
2
(1+s)
2
(2+ϕ)
2
(1+s+sϕ)
2
(2+2ϕ-sϕ)
2
a
Out[]=
16+32s+16cs+16+32c+16c+48ϕ+48sϕ+32csϕ-48ϕ+64cϕ-48ϕ-16cϕ-64cϕ-16cϕ+48-40s+20cs-176+72c-40-44c+48-160c-48c+16-112s+4cs-164+40c+52-8c+68-144c-20-52c-72s-60+8c+66+12c+27-56c-24-24c+3-16s-8+20+4c+2-8c-8-4c+2
2
s
2
s
3
s
2
s
2
s
3
s
3
s
4
s
5
s
2
ϕ
2
ϕ
2
ϕ
2
s
2
ϕ
2
s
2
ϕ
3
s
2
ϕ
3
s
2
ϕ
4
s
2
ϕ
4
s
2
ϕ
5
s
2
ϕ
3
ϕ
3
ϕ
3
ϕ
2
s
3
ϕ
2
s
3
ϕ
3
s
3
ϕ
3
s
3
ϕ
4
s
3
ϕ
4
s
3
ϕ
5
s
3
ϕ
5
s
3
ϕ
4
ϕ
2
s
4
ϕ
2
s
4
ϕ
3
s
4
ϕ
3
s
4
ϕ
4
s
4
ϕ
4
s
4
ϕ
5
s
4
ϕ
5
s
4
ϕ
6
s
4
ϕ
5
ϕ
2
s
5
ϕ
3
s
5
ϕ
3
s
5
ϕ
4
s
5
ϕ
4
s
5
ϕ
5
s
5
ϕ
5
s
5
ϕ
6
s
5
ϕ
Since 82(1+s)2(2+ϕ)2(1+s+sϕ)2(2+2ϕ-sϕ)/2a>0 , the sign of PiFS[s,ϕ,a,σ]-PiNS[s,ϕ,a,σ] is the same as the sign of the following polynomial:
Since >0 , the sign of PiFS[s,ϕ,a,σ]-PiNS[s,ϕ,a,σ] is the same as the sign of the following polynomial:
8/
2
(1+s)
2
(2+ϕ)
2
(1+s+sϕ)
2
(2+2ϕ-sϕ)
2
a
In[]:=
Poly3[s_,ϕ_,c_]:=16+32s+16cs+16+32c+16c+48ϕ+48sϕ+32csϕ-48ϕ+64cϕ-48ϕ-16cϕ-64cϕ-16cϕ+48-40s+20cs-176+72c-40-44c+48-160c-48c+16-112s+4cs-164+40c+52-8c+68-144c-20-52c-72s-60+8c+66+12c+27-56c-24-24c+3-16s-8+20+4c+2-8c-8-4c+2
2
s
2
s
3
s
2
s
2
s
3
s
3
s
4
s
5
s
2
ϕ
2
ϕ
2
ϕ
2
s
2
ϕ
2
s
2
ϕ
3
s
2
ϕ
3
s
2
ϕ
4
s
2
ϕ
4
s
2
ϕ
5
s
2
ϕ
3
ϕ
3
ϕ
3
ϕ
2
s
3
ϕ
2
s
3
ϕ
3
s
3
ϕ
3
s
3
ϕ
4
s
3
ϕ
4
s
3
ϕ
5
s
3
ϕ
5
s
3
ϕ
4
ϕ
2
s
4
ϕ
2
s
4
ϕ
3
s
4
ϕ
3
s
4
ϕ
4
s
4
ϕ
4
s
4
ϕ
5
s
4
ϕ
5
s
4
ϕ
6
s
4
ϕ
5
ϕ
2
s
5
ϕ
3
s
5
ϕ
3
s
5
ϕ
4
s
5
ϕ
4
s
5
ϕ
5
s
5
ϕ
5
s
5
ϕ
6
s
5
ϕ
Let phi3hat[s,c] denote the real and positive root of Poly3[s,ϕ,c]=0 for given s and c:
Let phi3hat[s,c] denote the real and positive root of Poly3[s,ϕ,c]=0 for given s and c:
In[]:=
phi3hat[s_,c_]:=ϕ/.Solve[Poly3[s,ϕ,c]0&&ϕ≥0,ϕ,Reals][[1]];
The following plot shows that phi3hat[s,c] is unique for 0<s<1 and 0 < c < 1/4 ( c= 1/4 implies very high coefficient of variation of 1/2).
The following plot shows that phi3hat[s,c] is unique for 0<s<1 and 0 < c < 1/4 ( c= 1/4 implies very high coefficient of variation of 1/2).
In[]:=
Manipulate[Plot[phi3hat[s,c],{s,Eps,1-Eps},AxesOrigin{0,0}],{c,Eps,1/4}]
Out[]=
| |||||||
| |||||||
Part (d) of the proposition : Comparison three threshold values
Part (d) of the proposition : Comparison three threshold values
The following plot shows that for all values of s, phi3hat[s,c] < phi1hat[s,c] < phi2hat[s,c]
The following plot shows that for all values of s, phi3hat[s,c] < phi1hat[s,c] < phi2hat[s,c]
In[]:=
Manipulate[Plot[{phi1hat[s,c],phi2hat[s,c],phi3hat[s,c]},{s,Eps,1-Eps},AxesOrigin{0,0},PlotStyle{Red,Blue,Black}],{c,Eps,1/4}]
Out[]=
| |||||||
| |||||||
Out[]=
$Aborted
Cite this as: Anonymous Author, "Proof of Proposition 5 (Sharing Demand Information with Retailer under Upstream Competition)" from the Notebook Archive (2020), https://notebookarchive.org/2020-07-52zj3rg
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