ORIGINAL ARTICLE: Michael Kúnin & Krešimir Žigić (2023), On Jain’s digital piracy model: horizontal vs vertical product differentiation, Economic Research-Ekonomska Istraživanja, 36:1, 1299-1316, DOI: https://doi.org/10.1080/1331677X.2022.2086149
Article Abstract: We study how private intellectual property rights protection affects equilibrium prices in a duopoly competition between firms that offer a product variety of distinct qualities (vertical product differentiation) in a setup that is closely related to that put forward by Jain where firms offer the same qualities in equilibrium (horizontal product differentiation). Consumers may make a choice to buy a legal version, use an illegal copy (if they want to and can), or not use a product at all. Using an illegal version violates intellectual property rights protection and is thus punishable when discovered. Thus, both private and public (copyright) intellectual property rights protection are available on scene.

Introduction

Here we pre-evaluate the quality sensitivities of users who are indifferent between two of the available products, or between buying a product and not buying at all.

Indifferent users

P stands for illegal A. Its (de facto, expected) price is X.
In[]:=
θ0A=pA
Out[]=
pA
In[]:=
θ0B=pB/q
Out[]=
pB
q
In[]:=
θBA=
pA-pB
1-q
Out[]=
pA-pB
1-q
In[]:=
θBP=
X-pB
1-q
Out[]=
-pB+X
1-q

No full dependence equilibria with adjustments à la Jain (2008)

Here we explore the most interesting equilibrium structure, where the outcome differs from Jain (2008).

Profit functions (Section 3)

These are the profit functions for this equilibrium structure.
In[]:=
CΠADDD=βpAc(1-θBA/δ)+pA(1-θBA)//FullSimplify
Out[]=
pA((-1+pA-pB+q)δ+cβ(pA-pB+(-1+q)δ))
(-1+q)δ
In[]:=
CΠBDDD=βpB(c(θBA-θ0B)/δ+(1-c)(θBP-θ0B)/δ)+pB(θBA-θ0B)//FullSimplify
Out[]=
pB(pB(β+δ)-q(c(pA-X)β+Xβ+pAδ))
(-1+q)qδ

Not serving the copier segment (Proposition 1, Subsection 4.2)

Here we derive the conditions for the developers not to serve the copier segment of the market. For developer
A
, this applies to all equilibrium structures where private protection is used. For developer
B
, the analysis here is for the no full dependence equilibria; for interior full dependence equilibria, the proof is trivial,whereas for corner full dependence equilibria, it is obvious that if
X
is too low then there will be deviation, see the paper.

Not serving the copier segment by developer
A

In[]:=
ccΠA=βpAc(1-θBA/δ)+pA(1-θBA)//FullSimplify
Out[]=
pA((-1+pA-pB+q)δ+cβ(pA-pB+(-1+q)δ))
(-1+q)δ
In[]:=
ccΠAA=pA(1-θBA)//FullSimplify
Out[]=
pA(-1+pA-pB+q)
-1+q
In[]:=
ccRA=pA/.Solve[
∂
pA
ccΠA0,pA]1//FullSimplify
Out[]=
(1+pB-q)δ+cβ(pB+δ-qδ)
2(cβ+δ)
In[]:=
ccRAA=pA/.Solve[
∂
pA
ccΠAA0,pA]1//FullSimplify
Out[]=
1
2
(1+pB-q)
In[]:=
ccπA=ccΠA/.pAccRA//FullSimplify
Out[]=
-
2
((1+pB-q)δ+cβ(pB+δ-qδ))
4(-1+q)δ(cβ+δ)
In[]:=
ccπAA=ccΠAA/.pAccRAA//FullSimplify
Out[]=
-
2
(1+pB-q)
4(-1+q)
In[]:=
CoefficientList[(ccπA-ccπAA),pB]//FullSimplify
Out[]=
-
c(-1+q)β(-1+(2+cβ)δ)
4(cβ+δ)
,
cβ
2
,
cβ
4δ-4qδ

Therefore, the threshold
δ≥
1
2+cβ
applies to all equilibria discussed in the paper.

Not serving the copier segment by developer
B
(Appendix F)

In[]:=
CΠΠBDDD=pB(θBA-θ0B)//FullSimplify
Out[]=
pB(pB-pAq)
(-1+q)q
In[]:=
CRBDDD=pB/.Solve[
∂
pB
CΠBDDD0,pB]1//FullSimplify
Out[]=
q(c(pA-X)β+Xβ+pAδ)
2(β+δ)
In[]:=
-CRBDDD+Xq//FullSimplify
Out[]=
qX-
q(c(pA-X)β+Xβ+pAδ)
2(β+δ)
The following is an upper bound on
p
A
.
In[]:=
Solve[(c(-pA+X)β-pAδ+X(β+2δ))0,pA]//FullSimplify
Out[]=
pA
X(β+cβ+2δ)
cβ+δ

In[]:=
CRRBDDD=pB/.Solve[
∂
pB
CΠΠBDDD0,pB]1//FullSimplify
Out[]=
pAq
2
In[]:=
CρBDDD=CΠBDDD/.pBCRBDDD//FullSimplify
Out[]=
-
q
2
(c(pA-X)β+Xβ+pAδ)
4(-1+q)δ(β+δ)
In[]:=
CρρBDDD=CΠΠBDDD/.pBCRRBDDD//FullSimplify
Out[]=
2
pA
q
4-4q
In[]:=
CΔBDDD=(CρBDDD-CρρBDDD)4(1-q)δ(β+δ)/(βq)//Expand//FullSimplify
Out[]=
2
(c(pA-X)+X)
β+pA((-1+2c)pA-2(-1+c)X)δ
This is nonnegative and increasing in
p
A
at
p
A
=0
, so it is sufficient to check the upper bound.
In[]:=
CoefficientList[CΔBDDD,pA]//FullSimplify
Out[]=
{
2
(-1+c)
2
X
β,-2(-1+c)X(cβ+δ),
2
c
β-δ+2cδ}
In[]:=
CΔBDDD/.pA
X(β+cβ+2δ)
cβ+δ
//FullSimplify
Out[]=
2
X
(β+δ)(4
2
c
2
β
+(-1+3c(2+c))βδ+4c
2
δ
)
2
(cβ+δ)
In[]:=
CΓBDDD=(4
2
c
2
β
+(-1+3c(2+c))βδ+4c
2
δ
)
2
β
/.δβγ//FullSimplify
Out[]=
-γ+c4c+3(2+c)γ+4
2
γ

In[]:=
Solve
∂
γ
CΓBDDD0,γ
Out[]=
γ
1-6c-3
2
c
8c

In[]:=
CΓBDDD/.γ
1-6c-3
2
c
8c
//FullSimplify
Out[]=
-
3
(-1+c)
(-1+9c)
16c
Whence, the condition for firm
B
is
c≥1/9
.

Prices and profits (Subsection 4.3, Appendix D.1)

Here we obtain the no full dependence equilibrium outcome.
In[]:=
CEDDD=Solve[{
∂
pA
CΠADDD0,
∂
pB
CΠBDDD0},{pA,pB}]1//FullSimplify
Out[]=
pA
(-1+c)cqX
2
β
+β(-2+q(2+(-1+c)X)+2c(-1+q)β)δ+2(-1+q)(1+cβ)
2
δ
(cβ+δ)((-4+cq)β+(-4+q)δ)
,pB
q(2(-1+c)Xβ+(-1+q)(1+cβ)δ)
(-4+cq)β+(-4+q)δ

In[]:=
CpADDD=pA/.CEDDD
Out[]=
(-1+c)cqX
2
β
+β(-2+q(2+(-1+c)X)+2c(-1+q)β)δ+2(-1+q)(1+cβ)
2
δ
(cβ+δ)((-4+cq)β+(-4+q)δ)
In[]:=
CpBDDD=pB/.CEDDD
Out[]=
q(2(-1+c)Xβ+(-1+q)(1+cβ)δ)
(-4+cq)β+(-4+q)δ
In[]:=
CπADDD=CΠADDD/.CEDDD//FullSimplify
Out[]=
-
2
((-1+c)cqX
2
β
+β(-2+q(2+(-1+c)X)+2c(-1+q)β)δ+2(-1+q)(1+cβ)
2
δ
)
(-1+q)δ(cβ+δ)
2
((-4+cq)β+(-4+q)δ)

Other relevant equilibria (Subsection 4.4, Appendices D.2 and D.3)

Interior full-dependence equilibria

Prices and profits

Corner full-dependence equilibria

Prices and profits

Vertical versus Horizontal Product Differentiation (Section 5)

This applies to No Full Dependence equilibria. Many expressions are untractable, but they are used in the numerical example.
We need these bounds to prove the results below.

Numerical example for no full dependence

Set-up

Equilibrium

If the parameters are modified, it is necessary to check that the right solution is used, which can be either 〚1〛 or 〚2〛. If the graphs do not look right after re-evaluation, then change the index in the double square brackets. The same applies to the other bound.

CITE THIS NOTEBOOK

On Jain's Digital Piracy Model: Horizontal vs Vertical Product Differentiation​
by Michael Kúnin and Krešimir Žigić​
Wolfram Community, STAFF PICKS, November 30, 2023
​https://community.wolfram.com/groups/-/m/t/3074816