<< . .

. 26
( : 53)



. . >>

software, hence quality of the pirated software q can also be interpreted like this. For
simplicity, we also assume that the marginal costs of software production (i.e., making
copies) are nil for both firms.6
There is a continuum of consumers indexed by X, X∈[0,1]. A consumer™s willingness to
pay for the software depends on how much he/she values it “ measured by X. A high
value of X means higher valuation for the software and low value of X means lower
valuation for the software. Therefore, one consumer differs from another on the basis
of his valuation for the particular software. Valuations are uniformly distributed over the
interval [0,1] and the size of the market is normalized to 1.
A consumer™s utility function is given as:


X “ PO if buys original software

7
U= q X“ PP if buys pirated software

0 if buys none




There is no way a consumer can get pirated software which has a defect replaced, since
there is no warranty for it.8 Hence, the consumer enjoys the benefit of the pirated software
only with probability q. In the event that the pirated software purchased does not work
at all, the loss to the consumer is the price paid for it. The original software is fully
guaranteed to work. PO and P P are the prices of the original and pirated software
respectively. It must be true that PO > PP . PO - PP can be viewed as the premium a consumer
pays for buying “guaranteed-to-work” software.
In our model, just as before, network externality implies that the value of a particular piece
of software for a consumer increases as more and more consumers use it. With the
presence of a pirate, and due to lower price of the pirated software, more people tend to
buy the software, which in turn increases the number of software users in the society.




Copyright © 2005, Idea Group Inc. Copying or distributing in print or electronic forms without written
permission of Idea Group Inc. is prohibited.
On Software Piracy 181


This intensifies the network effect, and as a result, this increases the value of that
software for any potential buyers (both legal and illegal). Under this, we will consider
two cases in turn. First, where the original developer protects its software, and secondly,
where the original developer does not protect.9


Software Protection (No Piracy)

Without piracy, consumers would choose only between either buying the original one
or not buying, depending on their valuation of the software.
Thus a consumer™s utility in the presence of network externality is given by:


X + θ D NP “ P NP
U= if buys original software

0 if buys none




DNP denotes the total demand of the software under protection (i.e., no piracy) 10 and PNP
denotes the price. Now θ ∈[0, ½]11 is a coefficient which measures the importance of
network size to the software user. It can be viewed as the degree of network externalities.
For example, if θ is close to ½, it implies the stronger effect of network externality and
when θ is close to zero, there is almost no effect of any network externality at all.


Figure 1. Distribution of buyers (case of protection)

None Original


0 X 1




X is the marginal consumer who is indifferent between buying the original software and
not buying any software at all:


X+ θ D NP - P NP = 0

X = P NP - θ D NP




Copyright © 2005, Idea Group Inc. Copying or distributing in print or electronic forms without written
permission of Idea Group Inc. is prohibited.
182 Poddar


Demand for the original software is:



1 ’ X = 1- P NP + θ D NP
D NP =

1 ’ PNP
’ D NP =
1’θ


The monopolist™s profit is:


πNP = PNP . DNP

1 ’ PNP
= PNP .
1’θ


Solving for the profit-maximizing monopolist price, we get:


P*NP = ½ (1)


And demand is:


1
D*NP = 2(1 ’ θ ) (2)


Note that when θ = ½, D *NP = 1, i.e., the full market is served.
Hence, the profit of the monopolist software firm in the case is:


1
π*NP = (3)
4(1 ’ θ )



No Software Protection (Piracy)

This time, a consumer™s utility is given by:


X + θ D O + qθ D P “ P O 12 if buys original software

q X +qθ D O + q 2 θ D P “ P P 13
U= if buys pirated software

0 if buys none


Copyright © 2005, Idea Group Inc. Copying or distributing in print or electronic forms without written
permission of Idea Group Inc. is prohibited.
On Software Piracy 183


DO, PO and D P, PP are the demand and prices for the original and pirated software
respectively.
As mentioned earlier, q is the probability that the pirated software works. This time,
though, the loss to the consumer if the pirated software does not work is comprised of
the price paid for the illegal software and the intangible cost which arises from not being
able to enjoy the positive network externality.



Figure 2. Distribution of buyers (case of non-protection)


None Pirate Original



ˆ ˆ
0 Y X 1




ˆ
Like before, the marginal consumer, X , who is indifferent between buying the original
software and the pirated version is given by:


X + θDO + qθ DP “ PO = q X + qθ DO + q2θ DP “ PP
ˆ ˆ

PO ’ PP
’ θ (DO + qDP )
ˆ
X= 1’ q


ˆ
The marginal consumer, Y , who is indifferent between buying the pirated software and
not buying any software at all is given by:


q Y + qθ DO + q 2θ DP “ P P = 0
ˆ

’θ (DO + qDP )
PP
ˆ
Y=
q


The demand for original software is given by:


ˆ
DO = 1- X




Copyright © 2005, Idea Group Inc. Copying or distributing in print or electronic forms without written
permission of Idea Group Inc. is prohibited.
184 Poddar


(1 ’ q ) + θ (qPO ’ PP )’ (PO ’ PP )
’ DO =
(1 ’ q )(1 ’ θ )

and the demand for pirated software is given by:


ˆˆ
DP = X - Y

qPO ’ PP
’ DP =
q (1 ’ q )


The original firm and the pirate compete by choosing price strategically. The respective
reaction functions are given by:


1 ’ q + PP (1 ’ θ )
2(1 ’ θq )
PO(PP) =

qPO
PP( P O ) =
2


Hence, Nash Equilibrium prices are:


2(1 ’ q )
P*O = 4 ’ q ’ 3θq (4)

q (1 ’ q )
P*P = 4 ’ q ’ 3θq (5)


Equilibrium demands are:


1 ® 2 ’ 2θq 
DO =
*

1 ’ θ  4 ’ q ’ 3θq 
(6)
° »

1
DP =
*
(7)
4 ’ q ’ 3θq




Copyright © 2005, Idea Group Inc. Copying or distributing in print or electronic forms without written
permission of Idea Group Inc. is prohibited.
On Software Piracy 185


The profit of the original software firm is:


4(1 ’ q )(1 ’ θq )
π **
= (1 ’ θ )(4 ’ q ’ 3θq) 2 (8)
O




and that of the pirate is:


q(1 ’ q )
π P* =
*

(4 ’ q ’ 3θq )2 (9)



The following result summarizes the impact of the presence of the pirate in the market
under network externality.


Proposition 1
In the presence of network externality, when the pirate is present in the market, the
demand for the original firm is higher than its demand under protection, while price
under piracy is lower than under protection. Formally:


D*O > D*NP and P*O < P*NP.


Proof: Follows after comparing (and simplifying) (2) with (6) and (1) with (4) respec-
tively. Q.E.D.


So under network externality, the presence of the pirate has a positive effect on the
original firm™s demand as expected, but a dampening effect on the price due to compe-
tition. Under this, we are interested to see how these two opposing effects combine and
what would be a more profitable situation for the original firm between piracy and
protection.


Protection versus Non-Protection

We compare between the profits of the original software firm under protection and non-
protection.


Proposition 2
In the presence of network externality given a choice between employing protection
and non-protection, it is always profitable for the original software developer to
protect its software.


Copyright © 2005, Idea Group Inc. Copying or distributing in print or electronic forms without written
permission of Idea Group Inc. is prohibited.
186 Poddar


Proof: To show that π*NP “ π**O ≥ 0


Observe that:


4(1 ’ q )(1 ’ θq )
®1
1
π*NP “ π**O = (1 ’ θ )  4 ’ ( ’ q ’ θq )2 
° »
4 3

q 2 ’ 8qθ ’ 10θq 2 + 8q + 9θ 2 q 2
=
4(1 ’ θ )(4 ’ q ’ 3θq )
2




The denominator of the above expression is non-negative. We have to show that the
numerator is non-negative for all θ and q.
Simplifying the numerator, we get (1 “ θ)[8q “ q2(9θ “ 1)], to make it positive we must
8+q 8+ q
which is always true for all q ∈ (0.1) and θ ∈ ®0,  . Note that 9q is
have θ ¤
1
 2
9q ° »
decreasing in q. Q.E.D.
This result is interesting since under network externality, when the pirate is present in
the market, even if there is a positive effect on the demand of the original firm, yet the
more profitable situation for the firm, is to protect.


Proposition 3
The original software developer has got higher incentive to protect its product in the
presence of network externality as oppose to the case of without any network external-
ity.


Proof: It is easy to see that the incentive to protect increases with the degree of network
q(8 + q ’ 9qθ )
externality. Gain from protection under network externality is 4(4 ’ q ’ 3θq )2 = G (say).
Observe that G is an increasing function of θ. Q.E.D.


Discussion

In this part, we tried to argue that the prevalence of network externality in the software
user market cannot generally be held as a reason for software piracy. We showed that
in some situations, even with very strong network effect, protection instead of allowing
piracy, is the optimal measure for the original software developer. To this end one might
argue that in our model since deterring the pirate (or protection) is costless to the original




Copyright © 2005, Idea Group Inc. Copying or distributing in print or electronic forms without written

<< . .

. 26
( : 53)



. . >>