Price discrimination: an protection strategy against
software piracy
Marc Robert∗
ABSTRACT
In this paper, we analyze the new interest of software firms in protection policies against
piracy. These firms want to revolutionize safety in the computer industry.
However, in the mid-1980s, software firms gradually removed their copy protections. Shy and
Thisse (1999) explain that those protections have not been profitable for firms.
In contrast, in this work we show that, thanks to the development of Internet, the price
discrimination allows firms to protect their software again. Indeed, software firms can use
discrimination as a defensive strategy against the hackers. This use contrasts with this
suggested in the discrimination literature.
1. INTRODUCTION
The current objective of Microsoft Corporation is to revolutionize the security in the
computer industry with its new software “Palladium” that will be incorporated in future
version of windows 2004, “longhorn”. In order to obtain the trustworthy computer industry, in
the 1999s Microsoft sets up a consortium with microprocessor makers (Intel and AMD) and
with PC hardware makers (IBM, HP, Compaq). This alliance is the “Trusted Computing
Platform Alliance”, which promotes a new computing platform that will spot and disable both
pirated software and pirated songs or videos.
In contrast, in the mid-1980s, software firms have gradually removed
protections against copying. Shy and Thisse (1999) explain that firms had strategic incentives,
due to network effects. Indeed, the piracy protections have not been profitable for firms
because protections decreased the number of consumers using their software.
So, objective of this paper is to analyze the recently policy change on the part of
software firms. Why firms protect their software again? Which is the reason of this behaviour
change of managers’ software firms?
∗
LAMETA, Faculté Des Sciences Economiques, Espace Richter Avenue de la Mer B.P.9606 34054
Montpellier. Email: [email protected]
1
For this purpose, we study software industry in taking into account the recent
development of Internet. Since the early 2000s, the number of users of the web has increased
thus this media becomes a new canal of distribution for software firms. The Web has as main
advantage to facilitate the ability of software firms to use price discrimination. In order to
make this analysis, we develop a five stage game, inspired from Shy and Thisse (1999), where
software firms are free to adopt uniform or discriminatory pricing. Our solution concept is a
sub-game perfect equilibrium and we solve the game by backwards induction.
Our main result is that price discrimination allows software firms to protect their
software again because piracy protections become profitable again. We show that software
firms can use price discrimination as a defensive strategy against aggressive consumers, the
hackers. This reason is very different from those traditionally advanced in the study of the
discrimination. Indeed, firms do not want to earn a part of the consumer surplus or to increase
rival's cost, but they only want to protect their software against piracy. In software industry,
price discrimination is not necessarily an aggressive strategy. Our work aims to illustrate one
other of the classical explanations for which firms adopt discriminatory prices in the
oligopolistic markets. Price discrimination is not systematically harmful or beneficial strategy
for consumers or for rival firms. Discriminatory pricing is only a management strategy. What
is important, it is the way in which firms use it. The impacts of this strategy depend on firms'
reasons of its adoption.
The remainder of the article is organized in the following way: the next section
describes our model. Section 3 explains the consumers’ behavior for all protection software
degrees. The section 4, 5 and 6 determine the software equilibrium prices by all possible
pricing, in turn for unprotected industry, protected industry and partially protected industry.
Section 7 analyzes both the price policy choice of firms and their degree of software
protection choice. We derive the sub-game perfect equilibrium. Then in this section we study
the impacts of price discrimination consumers’ surplus and social welfare. We conclude the
paper with section 8.
2
2. THE MODEL
We use the Shy and Thisse (1999) framework to model price competition between two
firms A and B in the software industry. The software market is represented by a linear
segment of the length one [0, 1].
Those software firms are localized at both end of the line, firm A on the left and firm
B on the right. Each software firm produces and sells its software goods to consumers. They
also offer at each purchaser a technical assistance. They can either adopt a uniform pricing or
a discriminatory pricing. The uniform pricing with unique mill price is noted FOB (Free On
Bord) and the discriminatory pricing with delivered discriminatory prices is noted D.
If uniform pricing is adopted by firms, consumers bear transportation costs to go until
software firm. But, all consumers will pay the same mill price for the same software good.
In contrast, if discriminatory pricing is adopted, firms will deliver the product to the
consumers. Each consumer will pay a different delivered price because firms absorb a part of
transportation costs. Since they observe the consumer’s locations, they can charge locationspecific prices. Thus, firms price discriminate, setting different mill prices at the firms’ door.
Actually, users of software goods are divide up two categories. The first group is
constituted by professionals, who need the services and support provided by the software
suppliers. Moreover, they are strongly risk-averse with respect to being prosecuted for using
software illegally. In contrast, the second category is constituted by private users which are
student users or others who need not services and who are not risk-averse.
Thus, we assume that potential users of software firms A et B are professionals and
private consumers. Slive and Bernhardt (1998), Shy and Thisse (1999) make a similar
hypothesis. Professionals gain extra utility σ from services and support of software firms,
whereas private users do not derive utility. All consumers are uniformly1 distributed along the
unit interval [0, 1]. The location of a professional consumer is denoted x and the location of a
private is denoted y. The consumer localization is defined as the distance to the left endpoint
of the market. The total population in the economy has a measure of 2.
The group of professionals and private consumers have the same size. Each category
of consumers has the choice between buy software and pirate software or not uses any
software. But the professionals group will not pirate software because they are risk averse.
The number of total users (legally and illegally) of software A is noted nA and number of
users of software B is noted nB. The number of users influences consumers’ evaluation of
1
With unity density.
3
software goods. The consumer’s evaluation of software good depends both on its inherent
characteristics and on the number of users of software: the greater the number of software, the
greater the benefit of software, and so, the greater the demand. Indeed, the increase of number
of users leads to an increase in network size, thereby increasing both number of professionals
and private consumers using this software. The files generated by the same software are
completely compatible. So, the files exchange is facilitated and consumers benefit from
positive externalities who are represented by µni, where i = A, B and µ is the coefficient
measuring the importance of the network size to a software user.
In order to analyze consumers’ and firms decisions in software industry, we use multistage game. The timing of this game is the following (see details in the appendix A).
In the first stage, software firms choose between to protect their goods and not to protect,
those strategies are noted P and P respectively.
In second stage, after the protection choice, each software firms choose its pricing policy. It
is free to adopt a uniform pricing (FOB) or discriminatory pricing (D).
In third stage, firms set the price level for the selling of their software goods.
In the fourth stage, professionals and private consumers observe software prices charged by
software firms A and B, and make software choice. Consumers have different options: they
can remain in legality, buying either software good or not using any software, or they can
become hackers pirating either software good.
Our solution concept is a subgame perfect equilibrium. As with all multistage games,
we solve the game with backwards induction.
3. CONSUMERS’ DECISONS
Each consumer will buy software good, if he can find a variety i which procures at him the
positive indirect utility Ui. She will purchase the variety software that will maximize its
utility. This consumption decision is influenced both by the firms' pricing choice and the
firms' degree of protection choice. In the following, we detail as these firms’ decisions
influence consumers’ behavior purchasing.
Software firms A and B can adopt the same degree of protection against software
piracy.
If all firms possess the means of protecting their software, thereby making software piracy
prohibited, this situation is noted (P,P). Then consumers must choose between buying the
software and not using any software. But, if both software firms not want to protect software
4
or not possess the means of protecting it, situation noted ( P , P ) , the piracy become costless
for private consumers. In this case, the hackers use the software without obtain support and
services. Since some users may pirate software goods, the number of buyers can be smaller
than the number of users.
If firms use different protection policies, others cases may arise. Those situations are noted,
( P, P ) and ( P , P) , a firm protects its software whereas other firm does not. Firms adopt an
asymmetric choice because they have not the same technology level to fight against piracy, or
because they have different strategic behaviors.
Therefore, according to firms’ decisions, consumers can be placed in different industries,
unprotected industry ( P , P ) , protected industry ( P, P) , or partially protected industry ( P, P ) ,
( P , P) .
Furthermore, in each previous situations software firms can adopt the same pricing or
different pricing. In this paper, we extend previous analysis of Shy and Thisse which has only
studied the protection strategy in the uniform Mill price context. We consider a model where
firms are free to adopt uniform or discriminatory pricing policy. Four price systems may arise
in the second stage of our game: (D-D), (FOB-FOB), (FOB-D), (D-FOB). Those firms’
decisions influence price level and so consumers demand.
3.1 PROFESSIONALS’ UTILITY
When software firms use a uniform pricing, the professionals obtain the following
utility level:
U ( x, i ) ≡ σ + µ ni − CTi − piN with i=A,B
(1)
Where σ is the extra utility of professional from services and support who are provided by
software firms i; piN stands for the uniform mill price of software i and CTi stands for
transportation costs. Let those transportation costs be professionals’ costs of traveling distance
to buy the good. They can be interpreted as the costs for the consumer, in terms of utility loss,
for not consuming the favor variety of goods, but a variety at a distance d. When the transport
cost per distance unit is normalized at one, the transportation costs of professionals are
represented by x if i=A and 1-x if i= B. We assume that those potential consumers do not
5
pirate software goods even if software firms do not protect their products. Indeed, the
retaliation of software firms will be more harmful to professionals’ business activity.
Moreover, they need the support provided by software firm i. If the software’s price is less
than the extra utility generated by services, such pi < σ , professionals consider the piracy as
not profitable. Indeed, they only will obtain the following utility level U ( x, i ) ≡ µ ni − CTi .
Furthermore, if the importance of the network size to a software professional users are such
µ < 1/ 2 , all professionals will find the purchasing software profitable because U ( x, i ) > 0 ,
see Shy and Thisse (1999) lemme 1. In the remainder of this paper, we only consider cases in
which pi < σ and µ < ½.
Alternatively, when software firms use discriminatory prices, professionals obtain the
following utility level:
D
U ( x, i ) ≡ σ + µ ni − piL with i=A,B
(2)
D
Where piL stands for the delivered discriminatory prices of the software i. In this case,
Software firm i deliver the product to consumers and bears transportation costs. Each
consumer pays an individually delivered discriminatory price.
3.2 PRIVATE CONSUMERS’ UTILITY
Private consumers have different preferences from professionals’ preferences. They not derive
utility from the services and support provided by software firms to their legal customers. So,
they will have various consumption behaviors according to protections degree of software.
If software firms adopt a very weak degree of protection, private consumers will
prefer pirating software goods over buying software goods. In this case, they will obtain the
same utility level whatever the pricing choice of software firms, that is:
U ( y, i ) = µ ni − CTi ( i= A, B); CTi = y for i= A and CTi =1-y for i= B
(3)
In contrast, if software firms adopt a higher degree of protection, such private
consumers can not pirate software, some consumers will want buy software to use it. In this
case, the pricing choice of firms influences private consumers’ utility level and their demand.
6
When firms sell their software with uniform prices, a legal purchaser will obtain the following
utility level:
U ( y, i ) = µ ni − CTi − piN (i= A, B) ; CTi= y for i= A and CTi = 1-y for i= B
(4)
and when they adopt discriminatory prices a private purchaser will obtain:
D
U ( y , i ) = µ ni − piL , i=A, B
(5)
Consequently, the behavior of private consumers will be as follows:
Firstly, if software goods are unprotected, closest consumers of software firm i will choose
between to pirate software i and not to use it, while far consumers will not use any software
even if they can obtain it illegally for free. They will prefer not using software instead of
pirating software. Indeed, software goods are more distant or more different of their ideal
variety of software. The consumers’ utility of using software do not exceed its utilization cost,
such µ ni < CTi . So, in the middle of the linear market never consumers want software good.
Therefore, the group of private consumers is partitioned in three sub-groups: the pirates of
software A and pirates of software B, who are located at either end of a market, and nonusers.
The indifferent consumer between pirate software A and not using any software is yˆ A .
Formally yˆ A is the solution to equation U ( yˆ A , A) = − yˆ A + µ nA ; and the indifferent consumer
between pirate software B and not using any software is yˆ B , formally yˆ B solves
U (1 − yˆ B , B) = −(1 − yˆ B ) + µ nB .
Secondly, if software firms protect their software such as the piracy becomes impossible,
private consumers will have only two options: buy the near software or not use any software.
Thus, all private users of software goods will be legal purchasers. They are located at either
extremities of market. Instead, at the middle of market, private consumers remain not
interested by software. The last private consumers who accept to pay software good i, at the
uniform prices or discriminatory prices, are found by solving equations U ( yˆ A , A) = 0 and
U (1 − yˆ B , B) = 0 .
We now determine price levels fixed by software firms in all possible situations.
According to firm’s protection choices and firms’ pricing choices, sixteen sub-cases may
7
arise. Indeed, in each four protection situations (unprotected industry, protected industry, and
both partially protected industries) firms can adopt four price systems.
4. EQUILIBRIUM PRICES IN UNPROTECTED INDUSTRY
Without protection policy, the pricing of software firms only influence the purchasing
behaviour of professional consumers (group 1), while private consumers (group 2), pirate
software goods whatever the price type.
4.1 THE UNIFORM PRICING: THE CASE (FOB-FOB)
In this case, software firms A and B adopt the uniform price policy. Professionals pay out the
same mill price for the same software, and they have to move to buy software goods.
Consumers will bear the transportation cost of traveling distance to obtain the good. If they
do not want to move, consumers pay the price at the firm’s door plus the transportation cost
for delivery. We recall that the full price is constituted by mill price and transportation costs.
F
piL = piN + CTi . When firms do not protect their software, all private users are illegally users.
This case has been studied by Shy et Thisse (1999). We rapidly recall their results.
Among the professionals localized in x on the software market, we known that the
professional who is indifferent between buying software A and buying software B is given by
the following equality: s + µ nA − x − p AN = s + µ nB − (1 − x ) − pBN . This equality illustrates the
fact that this professional derives the same utility level from consumption of either software.
The localization of this professional will be:
xˆ =
1 + µ ( n A − nB ) + p B − p A
2
(6)
Instead, among private consumers on the software market, the indifferent consumer
between to pirate software A (B) and not to use it, is yˆ A = µ nA ( yˆ B = 1 − µnB ) solution of the
equality U ( yˆ A , A) = − yˆ A + µ nA = 0 ( U (1 − yˆ B , 2) = −(1 − yˆ B ) + µ nB = 0 ). Equilibrium prices
and firms’ profits are:
p AN * = pBN * =
1 − 2µ
1 − 2µ
and π APP − FF = π BPP − FF =
1− µ
2(1 − µ )
8
(7)
4.2 THE DISCRIMINATORY PRICING: THE CASE (D-D)
In this situation, software firms A and B sell their software at the discriminatory prices. We
analyze the purchasing behavior of professionals and of private consumers with this new
pricing.
4.2.1 The professionals’ consumption
Professional consumers benefit from delivery service offered by software firms. Each
D
professional purchaser pays out a delivered price piL . In contrary to the uniform pricing, firms
software discriminate with respect to locations purchasers absorbing part of the transportation
D
costs. Consequently, each professional x pay out an individual mill price, piN ( x) = piL − CTi ,
which is different from that of its neighbours. Software goods will be relatively cheaper for
far consumers of software firms than next consumers, because software prices decrease with
competition. Software firms sell only of consumers that accept to pay an enough higher
D
delivered price piL to cover total marginal cost CmTi which companies bear. Total marginal
cost is the sum of the marginal production cost and the marginal transportation cost. Ici, il se
résume au coût de transport CTi par logiciel.
Consumer arbitrage is considered to be prohibitively costly. Indeed, each professional
prefers buying software at the originally software makers over buying it at the other retailers.
Thus, he will obtain the support and services provided by software firms. Since resale among
professionals is not binding, a software firm’s price decision at a particular location has no
effect on actions at other locations. Delivered prices charged at different points of the market,
by the same firm, are strategically independent. Firms compete for consumers at each point on
the line. This is equivalent to Bertrand competition at each point in space. Competition in
prices will be thus represented by a succession of Bertrand’s games.
A segmentation of potential consumers appears, each software firm can sell at a subgroup of professionals. The boundary of markets is given by the intersection point of the total
marginal costs of firms net of network externalities that consumers derive from the
consumption of software goods. Indeed, the firm that is at a disadvantage due to distance will
have to set a price equal to its net total marginal cost CmTi-µni, to remain attractive for
consumers, while the other firm will set a price that just picks up all the consumers. Thus,
equilibrium
prices
will
be:
piL * = max {CmTi − µ ni ; CmT j − µ n j }
D
piL * = max { x − µ ni ; (1 − x ) − µ n j } .
D
9
i.e.
This equilibrium is different from that obtained by Lederer and Hurter (1986) or by Thisse
and Vives (1988), piL * = max {CmTi ; CmT j } , in which the networks effects has not been
D
analyzed.
In software industry, each software firms will fix on its market share discriminatory prices
equal at the net total marginal cost of its rival.
When software goods are unprotected and firms use discriminatory pricing, professional
consumers’ demand of software A and B will be:
xˆ =
1 − µ nB + µ nA
1 − µ nA + µ nB
and 1 − xˆ =
2
2
(8)
Software firm A will deliver all professionals localized between 0 and x̂ , software B will
deliver purchasers localized between x̂ and 1.
4.2.2 The private consumers’ consumption
Private consumers will not buy software goods. The lack security allows at some
private consumers y to pirate software. The others private consumers do not find software A
and B as interesting, in spite they are free. As viewed previously, the private consumers group
is divided between third sub-groups: hackers of software A at the left end of the market, the
hackers of software B at the right end of the market and at the middle of market the not users.
The last pirate y of software A is yˆ A = µ nA and the last hackers of product B is yˆ B = 1 − µ nB .
Hence, expressions of the number of total users, legally and illegally, of software A
and B are: nA = xˆ + yˆ A and nB = (1 − xˆ ) + (1 − yˆ B ) . The resolution of those equations give us
the equilibrium values n*A and nB* ,
1
2(1 − µ )
, then we obtain equilibrium prices and firms’
profits:
D
D
p AL * = (1 − x ) − µ nB* , pBL * = x − µ n*A and
. xˆ*
. xˆ*
π APP − DD = ∫ p AL * − CmTA dx = ∫ p AL * − ( x − µn*A ) dx =
D
.0
π
PP − DD
B
.1
=∫ *p
. xˆ
D
.0
LD *
B
.1
− CmTB dx = ∫ * p
. xˆ
LD *
B
1
4
1
− (1 − x − µn ) dx =
4
(9)
*
B
Software firms only earn gain by the selling software at professional consumers. Whereas,
software firms indirectly derive profits from the piracy software because that increase the
10
demand of legally costumers. In the real word, the professionals’ demand is influenced by
employments’ knowledge of software goods, which has been obtained on software pirated.
In the two previous paragraphs, we only assumed that software firms used the same
pricing, but they are free to adopt a different pricing. Consequently, asymmetric price systems
may arise (FOB, D) or (D, FOB), that we now study.
4.3 ASYMMETRIC PRICING
In this case, software firms adopt a different pricing, one sell their products at uniform prices,
whereas other sell their software at discriminatory prices. When firms adopt the asymmetric
price system (FOB-D) or (D-FOB), the firm which chooses uniform pricing will be the price
leader and the other firm will react optimally to its price. This situation can be interpreted as
the single-basing point pricing system in spatial price competition. The consumer pays the
base price, announced by the market leader, plus the transportation cost from the leader firm’s
door to the location of the consumer, whatever the firm that really supplies the consumer. The
localization of leader is reference point. It is a classical assumption in the spatial price
discrimination literature; see Philps (1983), Thisse and Vives (1988), Thisse and Vives (1992)
for a justification of this hypothesis.
If firm A adopt the FOB prices, it will charge its unique mill price p AN in the first
place. The market share of the leader is represented by the set of the locations where its price
is lower than the net total marginal cost of its competitor. Indeed, the software firm B, thanks
to its follower situation, will be able to just undercut the full price of software firm A,
F
D
p AL − µ nA . This firm B will fix lower discriminatory prices pBL than rival’s prices (such
D
F
as pBL = p AL − ε ) which will allow it to get all the demand. Thus, software B will serve all the
consumers localized up to point x. The market boundary will be this localization x who solve
F
the quality p AL − µ n A = CmTB − µ nB i.e.
p AN + x − µ n A = (1 − x ) − µ nB . Therefore, when the
price system will be FOB-D, the marginal professional consumer x will be:
xˆ =
1 − p A + µ nA − µ nB
2
(10)
Software firm A will sell their software at professional consumers localized at the left of
x̂ and software firm B will sell at professionals placed at the right.
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Moreover, all private users of software A and B are hackers, which are at the number
yˆ A = µ nA and 1 − yˆ B = µ nB respectively.
Consequently, the total number of A-software users is nA = xˆ + yˆ A and number of B-software
users is nB = (1 − xˆ ) + (1 − yˆ B ) . The solution of these equations give us the expression of nA
and nB that we substitute into equations of demands x̂ and 1 − xˆ . Software firm A will
chooses uniform price, p AN * , to maximize its profit π APP − FD = p AN . xˆ . The firm B’s best response
{
D
}
to any mill price is pBL * = max p AN * + x − µ n*A ; CmTB − µ n*B . So, it will be able to charge
D
discriminatory prices pBL
*
at the majority of professional consumers, 1 − xˆ* = 3 / 4 . The
determination of equilibrium price p AN * yields equilibrium values n*A and nB* .
When software firms adopt the asymmetric price system (FOB, D), equilibrium prices and
firms’ profits will be:
p AN * =
F
D
2µ − 1
and p AL * = pBL * = p AN * + x − µ n*A and
2( µ − 1)
π APP − FD = p AN *.xˆ* =
.1
2µ − 1
and
8( µ − 1)
.1
π BPP − FD = ∫ ˆ pBL * − CmTB dx = ∫ ˆ pBL * − (1 − x) − µ n*B dx =
D
.x
D
.x
9
16
(11)
In the case (D-FOB), software firm B becomes the price leader, it uses the uniform
pricing and charges to its consumers the mill price pBN . The rival firm A will fix its
D
F
discriminatory prices p AL once the full price, pBL , of software B is known. Consequently, we
obtain the symmetric previous results.
Whatever the pricing used in the software industry, firms only compete on the professional
consumers, whereas private consumers profit by the lack of software protection to pirate it.
Those illegally utilisations are loss of earnings to software firms. In the real world, this piracy
increase with the recent development of the Web. In order to remove software piracy,
software firms may protect their products with a security chip or a special plug that is
necessary to launch the application. These protection policies will lead some private
consumers to buy software to use it. Now we determine again equilibrium prices in the
protected industry in which software firms delete piracy software.
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5. EQUILIBRIUM PRICES IN PROTECTED INDUSTRY
The security degree adopted by software firms does not directly influence the purchasing
behavior of professional consumers. The utility expression of professionals is not modified.
So, whatever the pricing choice of software firms A and B, professionals will buy the same
quantity of protected software or unprotected software. In contrast, the protection of software
reduces the private consumers’ choice. They only are two options: buying software or not
using it. Those consumers will buy software if software’s price does not exceed utility level
that they drive of networks effects. Firms have to fix lower software’s price in order to sell at
the private consumers. Firstly, recall rapidly results of Shy and Thisse in the uniform pricing
context, then we take into account the possible firms’ adoption of discriminatory pricing.
5.1 UNIFORM PRICING: THE CASE (FOB-FOB)
As viewed in unprotected industry case, all private consumers do not want use software
goods, see Shy and Thisse (1999) lemma 2. Protection policies do not change this choice.
Thus, Shy and Thisse draw a distinction between two possible cases. They determine
equilibrium prices both for the case where some (but not all) private consumers buy software
and for the case where none of private consumers buy software.
In first, when software firms A and B charge lower prices, some private consumers
will buy software products. Indeed, the last private purchaser of software A is yˆ A whom the
utility level U ( yˆ A , A) = µnA − y A − p AN become zero. Among next private consumers of firm B
the last buyer of software B is
yˆ B , he derive from it the zero utility level
U ( yˆ B , B) = µnB − (1 − y B ) − pBN . In this case, the number of legally private users of software A
is yˆ A = µnA − p AN and that of B is 1 − yˆ B = µnB − pBN . All consumers, private and professionals
( xˆ + yˆ A and 1 − xˆ + 1 − yˆ B ), will pay out prices, p AN * and pBN * , and software firms will earn
profits:
p AN * = pBN * =
1 − 2µ
(1 − 2µ)(3 − 4 µ)
and π APP − FF = π BPP − FF =
5 − 8µ
2(1 − µ)(5 − 8µ)
(12)
Software firms A and B will charge those lower prices for values of network effects such as µ
> 0,399. Al contrary, when the importance of the network size to a software user is lower,
such as µ < 0,399, software firms can increase their profits by raising their prices. In spite the
fact that they loss a private market, they find it profitable to deviate of equilibrium lower
13
prices. The situation where none of private consumers buy and use software is the second case
studied by Shy and Thisse.
Secondly, when software A and B fix higher prices, so that private consumers y do not
buy and do not use software, they earn profits π iPP − FF − E =
(1 − µ)
. They sell their software
2
products at the professional consumers at price piN * = 1 − µ . For values µ < 0.438, never
software firms do not find profitable to deviate at this higher equilibrium prices, charging
lower prices.
In their proposition 4, Shy and Thisse conclude that the number of total software users
decrease with protection policies, and also conclude that firms’ profits decrease. Those results
are verified into cases, firstly when firms charge lower prices and values of µ are such
0.399<µ<0.438, secondly for all price levels when 0.438<µ<1/2. In their analyze Shy and
Thisse assume that software firms only adopt uniform pricing.
We now look for the impact of the discriminatory pricing on both private consumers’
behaviors consumption and profitability of software protection policies.
5.2 Discriminatory pricing: the case (D-D)
In this case, software firms A and B compete on both professionals and private markets.
In contrast to the uniform pricing context, all private consumers will buy protected
software. The software products appears more attractive with discriminatory prices because
firms deliver their software products at home of the purchasers. They bear transportation costs
and so they sell ideal variety software for each consumer. With discriminatory pricing, firms
fix higher prices at the next consumers and fix lower prices at far consumers or not very
interested consumers. Discriminatory pricing allows at software firms to decrease their prices
in order to attractive private consumers without profits loss on professionals market. Each
software firms have an influence zone on each market, which is given by the intersection
points ( x̂ and ŷ ) of the net marginal costs of two firms. Those zones overlap the market
areas where consumers’ utility levels, and so demands, are positive. The expressions of
professionals and private demands of software A and B are:
xˆ = yˆ =
1 − µ nB + µ n A
1 − µ n A + µ nB
and 1 − xˆ = 1 − yˆ =
2
2
14
(13)
The demand of professionals is the same that obtained in the unprotected industry given by
(8).
The number of total users of software A and B is different when firm protect their software
and do not protect. In substituting equilibrium values of n*A and nB* into professionals and
private’s demands we obtain xˆ * = yˆ * = 1/ 2 and 1 − xˆ* = 1 − yˆ * = 1/ 2 . For all consumers,
software firms fix their discriminatory prices equal to rival’s net marginal costs. Hence
discriminatory prices are less than uniform prices. Only consumers localized at the end of
market will pay out discriminatory prices equal at the uniform prices. Equilibrium prices and
firms’ profits are:
D
D
p AL * = CmTB − µn*B = (1 − x ) − µ and pBL * = CmTA − µn*A = x − µ
π APP − DD = π BPP − DD = 1/ 2
(14)
For equilibrium prices, we may check that consumers derive positive surplus of the
purchasing of software A and B. Indeed, consumers’ surplus of software A-users and software
PP
PP
B-users, S DD
( yˆ , A) and S DD
( yˆ , B) , are
3
3
µ − that are positive for µ>0.25.
2
8
Moreover, when firms adopt discriminatory pricing, the only stable equilibrium is the
lower prices equilibrium. This result is contrary to that is obtained with uniform pricing,
where higher prices equilibrium also may arise. Indeed, never software firm using
discriminatory prices find profitable to fix greater prices because its rival will undercut, and
the competitive intensity will leads decreasing price level.
In the protected industry, software also may choose pricing of the different nature. The
price systems D-FOB and FOB-D may take place. We study those price systems in the
following paragraph.
5.3 THE ASYMETRIC PRICE SYSTEMS
When firms choose an asymmetric price system, some consumers have to bear the
transportation cost of traveling distance to obtain software good, while others benefit to
delivered software.
In the case of FOB-D price system, software firm A becomes the price leader. A
segmentation of professional and private markets appears. The boundary of the potential
markets of each firm is delimited by x̂ and ŷ points. They are intersection points of uniform
15
F
full price p AL of the software A’s and net marginal costs of software B, CmTB − µnB on
professionals market and private market respectively. We obtain:
yˆ = xˆ =
1 − p AN + µ nA − µ nB
2
(15)
We remark that the expression of the professionals’ demand of protected software, x̂ and
1 − xˆ , is the same that of unprotected software, given by equation (10).
Software firm B (firm A) controls professional consumers from x̂ to 1 (from 0 to x̂ ), and
controls the private consumers from ŷ to 1. This software firm B is the only firm who can sell
its products to all potential consumers, professionals and privates, that thanks to
discriminatory pricing. While, software firm A sell its product to yˆ A private consumers out of
ŷ , where yˆ A = µn A − p AN . Indeed, the private consumer localized at yˆ A derives from software
D
A consumption a utility U ( y, A) = µn A − p AL null. But all purchasers of software B obtain an
positive utility U ( y, B) = µnB − pBN − (1 − y ) .
Therefore, number of legally users of software A is nA = xˆ + yˆ A , and the number of software
B-users is nB = (1 − xˆ ) + (1 − yˆ ) . Substituting x̂ , yˆ A , ŷ into those equations and then solving
simultaneously for nA and nB yields at the expressions of number users of software goods.
Those expressions are influenced by the software A mill price, they can be used to find out
the equilibrium mill price p AN * maximizing firm A’s profits. Thus, we determine equilibrium
D
discriminatory prices pBL * of firm B. When firms adopt the FOB-D prices system, equilibrium
prices and firms’ profit will be:
p AN * =
D
pBL * =
F
(2 µ − 1)
1
and p AL * = p AN * + Z − µn*A with Z = x, y and n*A = −
2(2µ − 3)
2( µ − 2)
(2 µ − 1)
µ
+Z +
and
2(2 µ − 3)
2( µ − 2)
π APP − FD = p AN * ( xˆ * + yˆ *A ) =
.1
1 − 2µ
4(µ − 2)(2 µ − 3)
.1
(16)
π BPP − FD = ∫ ˆ pBL * − (1 − x − µn*B ) dx + ∫ ˆ pBL * − (1 − y − µn*B ) dy =
. x*
D
. y*
16
D
(4µ − 7)2
2(µ − 2)2 (2 µ − 3)2
We may check that private purchasers get positive utility from software goods. Indeed, for
equilibrium value of prices, consumers’ surplus of software-users A and B, noted
PP
S FD
( yˆ A , A)
PP
S FD
( yˆ , B) ,
and
PP
S FD
( yˆ A , A) = µn*A − ∫
. yˆ *A
.0
p AN * + y dy =
are
positive.
The
1 + 13µ − 27 µ 2 + 18µ3 − 4µ4
(µ − 2)2 (2 µ − 3) 2
PP
( yˆ , B) =
values, and the second is S FD
first
expression
is
it is positive whatever µ
(7 − 4 µ)(16µ3 − 72µ2 + 88µ − 21)
it is positive for µ >
8( µ − 2) 2 (2 µ − 3)2
0,313.
When firms choose the D-FOB price system, they are placed in symmetric situation of
previous case and results are inversed. The software firm B becomes the price leader on
software markets.
Contrary to uniform pricing, lower prices equilibrium are the only stable equilibrium,
this equilibrium will appear in spite of the leader’s incitation to charge higher equilibrium
prices. This higher prices equilibrium is not sustainable because discriminatory firm
(follower) finds profitable to fix smaller prices. It will undercut leader’s prices, so its rival
firm will have to decrease its prices to remain attractive to consumers and protect its market
shares.
Effectively, with higher prices on professionals market the firm leader increase its profits
that become equal to (1 − µ) / 8 . It earn more profits than that it obtain in selling on both
consumer markets, professionals and private. A rapidly calculus give us π APP − FD −
and π BPP − DF −
1− µ
<0
8
1− µ
< 0 . The loss profits, due to loss private consumers, are overlapped.
8
However, discriminatory firm (follower) does not find profitable to increase prices as the
leader. It will loss its all private purchasers and so will earn lower profits of 9 /16 . A rapidly
calculus show that π APP − DF − (9 /16) > 0 and π BPP − FD − (9 /16) > 0 .
Software firms A and B are different objectives because the firm A sells more software on
private market than its rival B. Private consumers are more important for firm A than firm B.
In two previous sections, 4 and 5, we assumed that software firms choose the same
protection policy for their software. But others situations may arise, in which, for example, a
software firm protect its products while other do not protect it. In this case, the software
industry becomes partially protected. In the following section, we study this possibility.
17
6. EQUILIBRIUM PRICES IN PARTIALLY PROTECTED INDUSTRY
As viewed previously, the purchasing behaviors of professionals are not directly influenced
by protection policies of software firms, whereas those modify the choice of private
consumers. In the partially protected industry only firm protect its products. Consequently a
part of private consumers must buy software to use it. Thus, some private consumers pirate
software good while others buy it.
But, the price levels fixed by firms influence the choice of all potential consumers. We now
determine equilibrium prices for the four possible prices systems. At first, we rapidly recall
results of Shy and Thisse those are obtained in uniform price context, and then we analyze the
discriminatory pricing.
In the following, we assume that software firm A is more concerned about security problems
than its rival. This firm is the single firm who protect its products. All results will be
symmetric when in his turn firm B protect its products.
6.1. UNIFORM PRICING
When software firms A and B adopt uniform pricing, Shy and Thisse draw a distinction
between two possible (prices) equilibriums. Similarly to the analysis of protected industry
(sub-sections 5.1), in the first equilibrium some private consumers purchase the protected
software A because prices are small, whereas in the second equilibrium where the price of
software A is high neither private consumer buy software product. For those two prices
equilibriums, software firm B suffer from software piracy while software firm A do not suffer
from piracy thanks to its protection policy.
In the lower prices equilibrium, software firm A may sell at both professional and
private consumers. Next private consumers have to buy its products to use it, due to piracy
protections. Private buyers represent yˆ A = µnA − p AN customers out of all its customers.
Software B who is not interested in security problem, only sell at professional consumers. All
private users of its software are illegally users who are at the number of 1 − yˆ B = µnB . Those
illegally utilizations are loss of potential profits for software firm B.
The professionals’ demands of software A and B are x̂ and 1 − xˆ , given by the
equation (6). Therefore, the number of total users of software A is nA = xˆ + yˆ A and that of
software B’ users is nB = (1 − xˆ ) + (1 − yˆ B ) .
In order to maximize their profits, software firms will fix the equilibrium uniform
prices p AN * and pBN * and they will earn the following profits:
18
p
N*
A
3(2µ − 1)
16µ2 − 22 µ + 7
N*
=
et pB =
,
16 − 11
( µ − 1)(16µ − 11)
π APP − FF = p AN * ( xˆ * + yˆ *A ) =
π BPP − FF = pBN * (1 − xˆ * ) =
9(2µ − 1)(4µ − 3)
2(1 − µ)(16 µ − 11) 2
(8µ − 7)(16 µ2 − 22µ + 7)
2(1 − µ)(16 µ − 11) 2
(17)
Software firms will adopt those price levels for µ > 0.399. They do not find profitable to
deviate from this equilibrium charging higher prices.
In the high-price equilibrium, software firms only compete on the professional market
where professionals’ demand always are x̂ and 1 − xˆ . The pirate of software B always are at
the number of yˆ B . In this situation, where the numbers of users of software A and B are
nA = xˆ and nB = (1 − xˆ ) + (1 − yˆ B ) respectively, firms A and B will impose the prices p AN * and
pBN * which will maximize their profits:
p AN * =
µ2 − 6µ + 3
2µ2 − 6 µ + 3
et pBN * =
,
3(1 − µ)
3(1 − µ)
π
= p xˆ =
PP − FF
A
N* *
A
( µ2 − 6 µ + 3) 2
9(1 − µ)( µ2 − 4µ + 2)
π BPP − FF = pBN * (1 − xˆ* ) =
(2 µ2 − 6µ + 3)2
9(1 − µ)( µ2 − 4µ + 2)2
(18)
Firms will charge those price levels for values µ < 0.438. They prefer not derive from that
high price equilibrium in decreasing their prices.
Those results of Shy and Thisse, have been obtained in the uniform price context. We now
determine the influence of firms’ adoption of discriminatory pricing on price levels, when
software products are partially protected.
6.2. DISCRIMINATORY PRICING
When software firm A is the single firm to protect its products, it becomes a monopoly on the
private market. But in reality, it is in almost monopoly position. A priori, on this market the
software firm B can not compete with it. However, firm A is not free to fix the price level that
19
it wishes. Firm A is indirectly competed with the possible piracy of software B. Private
consumers may choose between to buy software A (at monopoly) and to pirate software B.
So, software firm A suffer from an indirect and illegally competition, which reduces its
market share monopoly. In order to consumers consider piracy as less attractive, firm A has to
decrease its price levels. Its objective is to capture a part of private customers in decreasing
prices until hackers’ costs. Pirates use the software B that is not their ideal variety of software
good, consequently they bear a utility loss (1 − y ) which is reduced by positive network
externalities µnB . Therefore, the real market boundary of software firm A is given by ŷ that
is the intersection point of its net marginal costs CmTA − µnA with the net utility loss of
hackers; it is the solution of equation y − µnA = (1 − y ) − µnB . Demands of professional and
private consumers of software A are xˆ = yˆ =
1 + µn A − µnB
.
2
Software firm B only sell at the 1 − xˆ professional consumers, and its software attracts
1 − yˆ B = µnB pirates. Thus, software A and B are used by n*A and nB* consumers, where
n*A = xˆ * + yˆ * =
2
1
and nB* = (1 − xˆ * ) + (1 − yˆ *B ) =
. The discriminatory pricing leads at
2−µ
2−µ
the following equilibrium prices and firms’ profits:
D
D
p AL * = 1 − z − µn*B et pBL * = z − µn*A , z = x, y ;
. xˆ*
π APP − DD = ∫ p AL * − ( x − µn*A ) dx + ∫
D
.0
.1
. yˆ *
.0
π BPP − DD = ∫ ˆ pBL * − (1 − x − µnB* ) dx =
D
. x*
D
p AL * − ( y − µn*A ) dy =
( µ − 1) 2
( µ − 2)2
2
( µ − 2) 2
(19)
For those equilibrium values, we check that all private consumers localized from 0 to
ŷ effectively buy the software A. Indeed, for values of µ such as µ > 0,275 they obtain the
PP
positive surplus S DD
( y, A) = µn*A − ∫
. yˆ *
.0
D
p AL * dy =
−4µ2 + 12µ − 3
. We also confirm that at these
2( µ − 2) 2
prices all private consumers localized from yˆ B to 1 will pirate software B. They derive from
PP
illegally utilisation of this software the positive S DD
( y, B ) .
Moreover, contrary to uniform price context in this discriminatory case the only stable
equilibrium is low-price equilibrium. Software A has never interest to sustain high-price
20
equilibrium. It will loss private customers and will make lower profits equal to
(2 µ − 1)2
( µ − 1)4
, whereas firm B’s profit will increase until
. These higher
(µ 2 − 4 µ + 2) 2
(2 − 4 µ + µ2 ) 2
prices increase profit margin of its rival B who exclusively sell at the professionals. So, in this
case software firm A will be able to increase its profit by sharply reducing its price and thus
attracting some private consumers. The war of prices will happen.
However, discriminatory prices may appear in others price systems as the asymmetric
price systems FOB-D and D-FOB. In these cases, software firms adopt different price
policies. We look for price equilibrium for various situations.
6.3. ASYMMETRIC PRICE SYSTEMS
When software goods are partially protected, the price policy chosen by firm A is the only one
price policy that influences the purchasing behavior of private consumers. Software firm B’s
decisions only influence behaviors of professional purchasers.
If the asymmetric price system FOB-D appears, the firm A impose uniform price at its
professional customers and private customers, whereas the rival firm B charge discriminatory
prices. In this case, the firm A becomes the price leader and firm B the follower. When firm B
chooses to not protect its software, this decision can be interpreted as the mean for it to fix
null prices for some consumers. This strategy has many consequences. The firm B’s laxness
modify indirectly the competition between firms on two consumers market, professional and
private. Thanks to network effects, this firm B artificially increases the total number of users
of its software, and so, artificially increases professionals’ demand of software B.
Moreover, software firm A can not earn the monopoly profit on private market in spite of the
fact that it is the single firm on it. Firm A suffer from indirect and illegally competition,
because when private consumers want to use software they have the choice between to buy
software A and pirate software B. Software firm A will sell at the private consumers if its net
F
full price, p AL * − µnA , is less than consumers’ utility loss that consumers bear when pirate
software B. So the potential private market share of software firm A is delimited by ŷ , that is
F
the solution of following equality: p AL * − µnA = (1 − y ) − µnB ⇔ p AN + y − µn A = (1 − y ) − µnB .
However, firm A will solely sell at the private consumers localized from 0 to yˆ A ( yˆ A p yˆ ),
who derive a positive utility, U ( y, A) = µnA − y − p AN , from consumption of software A. In
private market software firm B merely attract hackers that derive the positive utility
U ( y, B) from using of its software. Those pirates are localized from yˆ B to 1.
21
Due to the fact that professionals’ demands of unprotected software and protected software
are identical, the expressions x̂ and 1 − xˆ are given once again by equation (10). The total
number of users of software A and B are nA = xˆ + yˆ A and nB = (1 − xˆ ) + (1 − yˆ B ) .
When the asymmetric price system FOB-D is adopted, software firms fix equilibrium
prices, p AN * and pBL* , and earn following profits:
p AN * =
n*A =
F
2µ − 1
and p AL * = p AN * + Z − µn*A , Z = x, y and
2(4µ − 3)
D
1
, pBL * = p AN * + x − µn*A ;
4(1 − µ)
π APP − FD = p AN * ( xˆ * + yˆ *A ) =
(2 µ − 1)
8(1 − µ)(4 µ − 3)
.1
π BPP − FD = ∫ ˆ pBL* − CmTB − µn*B dx =
. x*
(8µ − 7) 2
16(4µ − 3)2
(20)
For low-price equilibrium, it can now be easily verified that all private consumers localized
from 0 to yˆ A effectively buy software A. Indeed, for all values of µ they obtain form their
purchase the positive surplus
PP
( y, A) = µ n*A − ∫
S FD
. yˆ *A
.0
that
pirate utility
.1
*
p AN + y dy =
U ( y, B )
PP
S FD
( y, B ) = µnB* − ∫ * 1 − y dy =
. yˆ B
−128µ 4 + 296µ 3 − 221µ 2 + 48µ + 4
. We also may check
32( µ − 1) 2 (4µ − 3)2
becomes null at
the
yˆ *B
and that
pirate surplus
(7 − 8µ)(40 µ2 − 63µ + 24)
is positive from yˆ *B to 1.
32( µ − 1)2 (4 µ − 3) 2
When the asymmetric price system D-FOB is adopted, software A sells again at both
professional consumers and private consumers, but in this case it choose discriminatory
pricing, while software B charge uniform prices. Software firm B sells again uniquely at
professional consumers and becomes the price leader. Once more this firm modifies the
nature of the competition between firms; because it lets private consumers pirate its software.
All private consumers choose the cheaper software (legally or illegally) in term of loss utility.
So, firm A have to decrease its discriminatory prices until level of the privates’ costs that they
bear in pirating software B. This once, firm A will sell at all potential private consumers, and
its market share are demarcated from 0 to ŷ , solution of equality: y − µnA = 0 + (1 − y ) − µnB .
The
number
of
software-A
and
software-B
22
users
becomes
nA = xˆ + yˆ
and
nB = (1 − xˆ ) + (1 − yˆ B ) . Contrary to protected industry, software firms are not placed in
symmetric situation that of FOB-D. Indeed, firm B impose following prices:
pBN * =
(2 µ − 1)
on professional market ( pBN * = 0 for private users thanks to piracy), and so full
µ−2
F
prices are pBL * = pBN * + (1 − Z ) − µnB* , where Z = x, y and nB* =
D
1
. Firm
2(2 − µ)
A charge
F
p AL * = pBL * on both private markets and professional markets. Firms’ profits are:
π BPP − DF = pBN * (1 − xˆ ) =
. xˆ
( µ − 1)(2 µ − 1)
2( µ − 2) 2
. yˆ
(21)
π APP − DF = ∫ p AL * − ( x − µn*A ) dx + ∫ p AL * − ( y − µn*A ) dy =
.0
D
D
.0
(2 µ2 − 2µ + 13)
4(µ − 2) 2
The possible hacking of software B is the means of private consumers (y) to buy it at the null
prices.
We may confirm that private consumers localized from 0 to ŷ will buy effectively software A
PP
because for µ > 0,269, they derive from their purchase the positive surplus S DF
( y, A) . We as
well can check that piracy is profitable for private consumers localized at the right of yˆ *B
PP
( y, B ) .
because they obtain from illegally utilization the positive surplus S DF
In contrast to the uniform pricing FOB-FOB, the asymmetric price systems FOB-D
and D-FOB lead at the single price equilibrium, which is enough low to allow at the private
consumers to buy software A. The high price equilibrium is unstable equilibrium. Indeed, it is
in the interest of software firm A to sustain low-price equilibrium because private customers
are very important for it. If prices level is too much higher, firm A will loss these private
consumers and will earn smaller profit that become equal to
(2 µ − 1)2
for µ > 0,4,
4(1 − µ)( µ2 − 4µ + 2)
(3 − 6 µ + µ2 ) 2
, when it is follower. Firm A finds more
when it is leader, or equal to
4( µ2 − 4 µ + 2) 2
profitable to sell at the two categories of consumers. Therefore, it will undercut rival’s high
prices. But software firm B will make more profit if it can sustain higher price equilibrium.
( µ − 1)3
Effectively, with the increase of prices it earns
when it is leader, and earns
4(4 µ − µ2 − 2)
23
(2µ2 − 6 µ + 3) 2
when it is follower. This increase of its profits is due to the fact that it does
4( µ2 − 4 µ + 2) 2
not suffer from large loss of customers because it only sells on professional consumers.
After the determination of price levels for each industry type (unprotected, protected
and partially protected) we now look for the strategic choices of software firms. Which is the
optimal price policy for software firms? Which is the optimal degree of protection of
software?
7. FIRMS’ STRATEGIC CHOICES AND THEIR IMPACTS
Shy and Thisse studied the protection policy of software firms with the assumption that firms
use uniquely uniform pricing. In the actual computer industry where the Internet is more and
more used by consumers, will the uniform pricing be chosen by software firms? If it is not
verified, will the policy of protection be again unprofitable to software firms?
7.1 THE PRICING CHOICE
In this paper, we assumed that software firms have the choice between uniform pricing
(FOB) and discriminatory pricing (D) at the second stage of the game. So, four price systems
may arise: the uniform FOB-FOB, the discriminatory D-D and the both asymmetric FOB-D
and D-FOB. Software firms will choose the pricing that will generate greater profits
according to behaviors of their rival. We determine their decisions with simple comparisons
of potential payoffs gained by these firms.
Whatever the protection policy adopted by firms at the first stage of the game, the
analysis of the payoff matrix of the second stage shows that price discrimination is the
equilibrium strategy; details of this analysis are available in the appendix B.
Indeed, in each sub-game, software firms will systematically adopt discriminatory prices. It is
a dominant strategy when firms implement no protection, protection and partially protection.
Consequently, the discriminatory price system D-D appears as the Nash equilibrium for these
third sub-games. In unprotected industry, software firms A and B will earn profits π APP − DD and
π BPP − DD that equal to 1/4, given by (9). In protected industry, they will obtain higher profits,
π APP − DD and π BPP − DD , that equal to ½, given by (14). In partially protected industry, software A
gains intermediate profit level π APP − DD , whereas firm B, who do not protect its software, earns
the most lower profits π BPP − DD , given in the equation (19).
24
We now look for the protection choice of software firms who use discriminatory
prices.
7.2 THE PROTECTION CHOICE OF SOFTWARE FIRMS AND ITS IMPACTS
In the uniform price context, Shy and Thisse show that software firms do not find profitable to
implement piracy protections due to the network effects. But the recent development on the
web makes discriminatory policy easier for software firms.
Is the discriminatory pricing choice a means to better protect software product? Is the piracy
protection policy an optimal strategy for software firms?
As Shy and Thisse we assumed that software firms can choose between two protection
policies, either it protect their software against piracy (strategy noted P ) or, al contrary, it do
not protect their products ( P ). So, third industry types may arise: the unprotected
industry ( P , P ) ,
the
protected
industry
( P, P )
and
partially
protected
industries
( P, P ) ; ( P , P) . For various situations, the analysis of the payoff matrix of software firms A
and B yields at the following results, this analysis is detailed in the appendix C.
Whatever the importance of the network size to a software user, either for law values
of µ (µ < 0.399) or for large values of µ (µ > 0.43), each software firms will adopt
systematically the protection strategy. The single sub-game perfect equilibrium is ( P, P )
where all software firms protect their software. Contrary to uniform pricing, the
discriminatory pricing make protections again profitable to software firms. Therefore, when
we take into account the discrimination strategy we obtain results that are different at the one
of Shy and Thisse.
Indeed, for values µ > 0,2765 Shy and Thisse obtain indeterminate result, because two
equilibriums may exist. When network effects are moderate (0,2765 < µ < 0,399), Shy and
Thisse show that asymmetric protection policies ( P, P ) and ( P , P ) are the equilibriums, in
which one firm protects its software and the other does not.
For stronger network effects µ>0.43, Shy and Thisse show that two other equilibriums may
arise: ( P , P ) or ( P, P) , where both firms protect or both refrain from protecting their
software. For this authors, it is reasonable to assume that unprotected equilibrium ( P , P ) will
prevail, because this equilibrium will yield strictly higher profits to both firms than ( P, P) .
The work of Shy and Thisse (1999) provides a strategic reason why the use of
software protection has declined since the mid-1980s. The reason is that a larger number of
users increases the utility of software. Thus, each firm can increase the competitive value of
25
its software by not protecting it. Alternatively, each firm can protect its software by reducing
the number of users to the number of buyers, and so making its software less attractive. Their
analysis show that an industry totally protected is not the Nash equilibrium of the game where
firms not cooperatively choose their protection policy.
Our results are inversed when firms are the possibility of discriminate between
consumers. The discrimination makes software protections again profitable. This result may
explain the recent interest of software firms for policies against piracy.
So, the use of discriminatory prices appears more profitable for software firms. But
before concluding this paper, we look for if discrimination is beneficial to consumers. Is the
selling of protected software (at the discriminatory prices) socially beneficial?
7.3 THE IMPACTS OF THE SELLING PROTECTED SOFTWARE ON WELFARE
Thanks to discriminatory pricing software firms can better protect their software
product against the piracy, while consumers benefit from an increase of their surplus.
Consumers more prefer to buy protected software at the discriminatory prices than to buy
unprotected software at the uniform prices. Moreover the discriminatory pricing also increase
the social welfare.
Indeed, when firms adopt uniform pricing and the net work effects are moderate,
0,2765 < µ < 0,399, Shy and Thisse show that firms will adopt opposed protection polices
( P, P ) or ( P , P) . When firms take these decisions, consumers obtain the surplus level2
µ3 (12 µ4 + aµ3 + bµ2 + cµ + d ) + eµ2 + fµ + 72σ − 90
,
18(µ − 1)2 ( µ2 − 4µ + 2)2
whereas
when
both
firms
use
discriminatory prices and choose to protect their software, consumers obtain the higher
surplus 2σ + 6µ − 5 / 4 .
Furthermore, when network effects are stronger such as µ>0.43, Shy and Thisse show
that two situations may arise: either firms protect their software ( P, P) or neither firms protect
software ( P , P ) .
In
this
two
situations,
the
total
consumer
surplus3
is
4σ − 8σ µ + 4σ µ2 + 24µ − 18µ2 − 7
gµ4 + hµ3 + iµ2 + jµ + 100σ − 139
and
respectively. When
4( µ − 1) 2
4( µ − 1) 2 (8µ − 5) 2
firms discriminate between consumers, the consumers’ surplus increases and becomes again
higher 2σ + 6µ − 5 / 4 .
2
3
Où a = (18σ − 159) , b = (812 − 180σ ) , c = (666σ − 2023) , d = (2646 − 1152σ ) , e = (1008σ − 1845) , f = (648 − 432σ ) .
Où g = 256σ − 640 , h = 18246 − 832σ , i = 996σ − 1898 , j = 852 − 520σ .
26
The discriminatory pricing increases firms’ profit and consumers’ surplus, so this pricing
increase social welfare. This price policy is socially beneficial because software firms and
consumers benefit from its utilization.
8. CONCLUSION
The analysis of Shy and Thisse (1999) allows explaining the fact that since the mid1980s software firms have gradually removed protection against copying. They had shown
that the piracy protections have not been profitable for software firms. When the market
expands and competition intensifies software firms have strategic incentives to remove
protection. Indeed, with this decision they increase the total number of software users because
not only legally purchasers can use software goods, and due to the strong network effects the
legal demand increase.
However, since the early 2000s, the recent development of the Web allows at software
firms more easily to use the discriminatory pricing. Thanks to this new media of distribution
software firms can adopt discriminatory prices with lower costs.
In this paper, we show that price discrimination allows software firms to protect their
software again because piracy protections become again profitable. Our work may explain the
new software firms’ interest for the protection practices. Indeed, in order to obtain the
trustworthy computer industry, in the 1999s Microsoft and important software firms set up a
consortium with microprocessor makers (Intel and AMD) and with PC hardware makers
(IBM, HP, Compaq). This alliance is the “Trusted Computing Platform Alliance”, TCPA,
which promotes a new computing platform that will spot and disable both pirated software
and pirated songs or videos.
Software firms can use price discrimination as a defensive strategy against aggressive
consumers, the hackers. This reason of its use is very different from this traditionally
suggested in the study of the discrimination. Indeed, firms do not want to earn a part of the
consumer surplus or to increase rival's cost, but they only want to protect their software
against piracy. In software industry, price discrimination is not necessarily an aggressive
strategy. Our work aims to illustrate one other of the classical explanations for which firms
adopt discriminatory prices in the oligopolistic markets. Price discrimination is not
systematically harmful or beneficial strategy for consumers or for rival firms. Discriminatory
27
pricing is only a management strategy. What is important, it is the way in which firms use it.
The impacts of this strategy depend on firms' reasons of its adoption.
28
Appendix
Appendix A: The timing of the game
1st stage: The protection policy choice of software firms A and B
Decision of software firm A
P
NP
Decision of software firm B
P
NP
P
NP
2nd Stage: the pricing choice
Decision of firm A
D
FOB
D
FOB
D
FOB
D FOB
Decision of firm B
D
FOB D
FOB
D
FOB
D
FOB D FOB D FOB D FOB
3rd stage: Determination of equilibrium software prices type A and B
4th stage: The purchasing choice of consumers
29
Appendix B: the pricing choice of software firms
Price discrimination is an dominate strategy for software firms, the system price (D,D) is the
Nash equilibrium of the sub-game, whatever the protection policy that is adopted by firms.
Indeed, in the unprotected industry ( P , P ) with values of µ such as 0.5>µ>0 the payoff matrix
is:
AB
FOB
1 − 2µ
2(1 − µ)
FOB
D
2µ − 1
8( µ − 1)
1 − 2µ
2(1 − µ)
9
16
9
16
D
1
4
2µ − 1
8( µ − 1)
As the inequalities are verified
9
> 1 − 2µ
16 2(1 − µ)
et 1 >
4
1
4
2µ − 1
,
8( µ − 1)
so the nash equilibrium is (D,D), where
software firms sell their products at the discriminatory prices.
In the case of the protected industry ( P, P) , the payoff matrix of software firms
becomes with strong network effects and with weak network effects respectively:
When 0.5>µ>0.438
A
B
When 0.399>µ>0
FOB
D
(2 µ − 1)(4 µ − 3)
2µ − 1
4(2 − µ)(2 µ − 3)
2(1 − µ)(8 µ − 5)2
F
O
B
(2 µ − 1)(4 µ − 3)
2(1 − µ)(8 µ − 5)
2( µ − 2) 2 (2 µ − 3) 2
1
2
2( µ − 2) (2 µ − 3)
2
>
1− µ
2
, as well as
2µ − 1
4(2 − µ)(2 µ − 3)
1− µ
2
(4 µ − 7)2
2( µ − 2) 2 (2 µ − 3) 2
D
1
2
As the following inequalities are verified:
2
1
2
D
1− µ
2
2( µ − 2) 2 (2 µ − 3) 2
2µ − 1
4(2 − µ)(2 µ − 3)
(4 µ − 7)2
FOB
F
O
B
(4 µ − 7)2
2
(4 µ − 7)2
D
A
B
>
2µ − 1
4(2 − µ)(2 µ − 3)
(4 µ − 7)2
2
2( µ − 2) (2 µ − 3)
2µ − 1
4(2 − µ)(2 µ − 3)
2
>
(2 µ − 1)(4 µ − 3)
2(1 − µ)(8 µ − 5)2
(4 µ − 7)2
2( µ − 2) 2 (2 µ − 3) 2
1
2
1
2
and
, consequently (D,D) will be the single Nash
equilibrium, where both software firms choose to sell their software with discriminatory
prices.
30
In the case of partially protected industry ( P, P ) , the payoffs matrix of firms becomes
with strong network effects and with weak network effects respectively:
when 0.5>µ>0.438
A
B
FOB
D
2(1 − µ)(16µ − 11)
2
16(4 µ − 3)
2(1 − µ)(16µ − 11)2
F
O
B
( µ − 2)
2
9(1 − µ)( µ − 4 µ + 2)
2µ − 1
8(1 − µ)(4µ − 3)
(8µ − 7) 2
16(4 µ − 3) 2
9(1 − µ)( µ 2 − 4 µ + 2) 2
2
2 µ2 − 2 µ + 13
4( µ − 2)
D
( µ − 1)2
( µ − 2)
2( µ − 2)2
2
( µ 2 − 6 µ + 3) 2
2
( µ − 1)(2 µ − 1)
D
2
2
2
2 µ2 − 2 µ + 13
4( µ − 2)
FOB
( µ 2 − 6 µ + 3) 2
(8µ − 7) 2
(2 µ − 1)(8 µ − 7) 2
D
A
B
2µ − 1
8(1 − µ)(4µ − 3)
9(8µ2 − 10 µ + 3)
F
O
B
when 0.399>µ>0
( µ − 2) 2
2
( µ − 1)2
( µ − 1)(2 µ − 1)
2
( µ − 2) 2
2( µ − 2)2
Direct calculations from previous matrix give us the following inequalities
2 µ2 − 2 µ + 13
4( µ − 2)2
>
9(8µ2 − 10 µ + 3)
2(1 − µ)(16µ − 11)2
( µ − 1)2
( µ − 2)
> ( µ − 1)(2µ 2− 1) et
2
2( µ − 2)
;
2
( µ − 2)
2 µ2 − 2 µ + 13
4( µ − 2)2
>
2
>
2µ − 1
8(1 − µ)(4µ − 3)
;
(8µ − 7) 2
16(4 µ − 3) 2
( µ 2 − 6 µ + 3) 2
9(1 − µ)( µ 2 − 4 µ + 2) 2
;
>
(2 µ − 1)(8 µ − 7) 2
2(1 − µ)(16µ − 11)2
(8µ − 7) 2
>
16(4 µ − 3) 2
;
( µ 2 − 6 µ + 3) 2
9(1 − µ)( µ 2 − 4 µ + 2) 2
So we may claim that the only equilibrium of this sub-game is the discriminatory price system
(D,D)
Appendix C: The protection policy choice of the software firms
In the first stage, where firms choose their own protection policy, the perfect sub-game
equilibrium is ( P, P) where both firms protect their software. For all values of µ, the payoff
matrix of firms is:
AB
NP
NP
P
1
4
( µ − 1)
2
( µ − 2) 2
2
1
4
( µ − 2) 2
2
P
1
2
( µ − 2) 2
1
2
( µ − 1)2
( µ − 2)
2
We obtain from the previous matrix the following inequalities:
2
( µ − 2)
2
> 1 and
4
1
2
2
> ( µ − 1) 2 . So,
( µ − 2)
we can confirm that protection strategy is a dominate strategy, and (P,P) is single equilibrium
of our multi-stages game.
31
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