Economic Modelling 33 (2013) 526–532

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Economic Modelling

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Effect of piracy on innovation in the presence of network externalities☆

Dyuti BanerjeeDepartment of Economics, Monash University, Clayton, VIC 3800, Australia

☆ I would like to thank Lata Gangadharan, Birendra Raat the WEAI conference for the useful comments and sugapplies.E-mail address: dyuti.banerjee@buseco.monash.edu.au.

1 In the Piracy Study (2005), Business Software Alliansoftware industries crippled from competition with hThe International Federation of the Phonographic Industcial Piracy Reports argues that, “The illegal music tradenovation, ⋯”

0264-9993/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.econmod.2013.04.004

a b s t r a c t

a r t i c l e i n f o

Article history:Accepted 3 April 2013

JEL classification:D21D43L13L21L26O3

Keywords:InnovationMarket uncertaintyNetwork effectPiracy effectTechnological uncertainty

This paper analyzes the impact of simultaneous increases in piracy (piracy effect) and network externalities(network effect) on R&D investment. A single firm's R&D investment increases (or decreases) if the networkeffect (or piracy effect) is dominant. With R&D competition, if the firms “significantly” differ with respect totheir R&D efficiencies and if the piracy effect dominates the network effect then the less efficient firm's R&Dinvestment increases and that of the more efficient firm's decreases. In this case, the overall probability ofsuccessful innovation increases. The reverse holds if the network effect dominates the piracy effect. If thefirms are “less” asymmetric then their R&D investment either increases or decreases depending on the rela-tive strengths of the piracy and network effects.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Digital piracy has generally been perceived as having a damaginginfluence on software and media industries due to the high magni-tude of loss in retail sale since such products can be copied at a lowcost (Marshall, 1999; Straub and Nance, 1990), and also due to thepossible detrimental effects on the incentive to innovate. Such claimshave been put forth by organizations like the Business Software Alli-ance (BSA) and the International Federation of the Phonographic In-dustry (IFPI) on numerous occasions.1 Jaisingh (2009), Qiu (2006),and Novos and Waldman (1984) render support to the negative im-pact of piracy on the incentive to innovate under different contextsbut with a single innovating firm.

While industries like software is plagued by piracy, they also ben-efit from the presence of network externalities that act like a mediumof advertisement by adding to a consumer's utility due to the usage byother consumers. Consequently, in such cases, Shy and Thisse (1999)

i, Ranjan Ray, and participantsgestions. The usual disclaimer

ce (BSA) mentions that “localigh-quality pirated software”.ry (IFPI) in its, 2005 Commer-is destroying creativity and in-

rights reserved.

and Takeyama (1994) show that copyright holders can benefit fromlax copyright enforcement.2 However, this literature does not consid-er innovation and treats it as a sunk cost.

An accurate assessment of the impact of piracy on innovationmust take into consideration the presence of network externalities,which is yet to be addressed. In this paper, I bridge the gap betweenthe literature on piracy and innovation, and that on piracy and net-work externalities, by examining the effect of simultaneous increasesin piracy (hereafter referred to as the piracy effect) and network ex-ternalities (hereafter referred to as the network effect) on the incen-tive to innovate. This is studied in the context of a single innovatingfirm and also in the presence of R&D competition.

I show that the R&D investment of a single innovating firm facingtechnical uncertainty (defined later in the main text) increases if thenetwork effect is stronger than the piracy effect. This finding is con-trary to that in Banerjee and Chatterjee (2010), who do not considernetwork externalities and show that an increase in piracy unani-mously reduces the R&D investment of an innovating firm.

With R&D competition, I show that if the innovating firms are suf-ficiently asymmetric with respect to their R&D efficiencies and if thenetwork effect dominates the piracy effect, then the R&D investmentof the more efficient firm increases and that of the less efficient firmdecreases. The reverse is true if the piracy effect is dominant. So the

2 Conner and Rumelt (1991), and Nascimento et al. (1988) discusses the role of net-work externalities on the marketing of software without considering innovation.

527D. Banerjee / Economic Modelling 33 (2013) 526–532

introduction of network externalities always results in an increase inone of the firm's R&D investment when the firms are sufficientlyasymmetric. In contrast, in Banerjee and Chatterjee (2010) an in-crease in piracy may only increase the R&D investment of the less ef-ficient firm.

If the innovatingfirms are less asymmetricwith respect to their R&Defficiencies, then their R&D investment either increases or decreasesdepending on the relative strengths of the network and piracy effects.On the contrary, Banerjee and Chatterjee (2010) show that R&D invest-ment always decreases when the firms are less asymmetric.

So the significant contribution of the inclusion of network exter-nalities in the analysis is that it generates a broader spectrum of re-sults and contradicts the standard claim that piracy reduces theincentive to innovate.3 Thus the consideration of network externali-ties may be relevant for the design of appropriate anti-piracy policies,which however, is not the focus of this paper.

While this paper examines the effect of piracy on innovationthrough the demand side via network externalities, El Harbi andGrolleau (2008) and Easley et al. (2003) consider the supply side ef-fects and show that piracy can have a beneficial impact on innovationdue to a positive feedback effect that provides direction to innovatingfirms for further R&D.4 There can also be tacit reciprocity (Kolm,2006) in knowledge exchange between the innovating and piratingfirms in which case the innovating firm accepts piracy (Barnett,2005; Barnett et al., 2010; Raustiala and Sprigman, 2006).

This paper is organized as follows. Section 2 contains the modelwith a single innovating firm. In Section 3 I analyze the case withR&D competition between two innovating firms. Section 4 containsthe concluding remarks.

2. The model

Let us consider the market for a digital/information good, like soft-ware, which has positive network externalities and also faces com-mercial piracy. There is a single innovating firm, who invests in R&Dto develop a new product in order to increase its profit above a reser-vation level, π . For simplicity I assume π ¼ 0. The innovating firm anda pirating firm (hereafter referred to as the pirate), who makeunauthorized copies of the innovating firm's product (original prod-uct), play a sequential game consisting of the following stages.

Stage 1: The innovating firm chooses a level of R&D investment Rto develop a product.Stage 2: Conditional on the successful development of the productin Stage 1, the innovating firm chooses a price pm.Stage 3: The pirate chooses a price, pc.

The sequential form of the game is motivated by the fact that thepirate may have to wait for the first few units of the original productto go on sale, purchase one, copy it, and then begin production andsales. Thus the innovating firm may enjoy a period of lead-time.

3 The analysis of the simultaneous increases in network externality and piracy is ex-haustive in the sense that the result for an increase (or a decrease) in network exter-nality and a decrease (or an increase) in piracy are qualitatively similar to thedomination of network effect (piracy effect) over piracy effect (network effect).

4 When hackers used Valve Software's Half Life game engine to develop a gamecalled Counter Strike, Valve, a gaming company, took the illegal game software andmarketed it themselves, selling over 1.5 million copies (Barnes, 2005). Apple Comput-er, in a strategic reaction to P2P file sharing technologies, launched the iTunes onlinemusic library that was easy to navigate and explore, with free music previews, andallowed flexible download and copying for personal use. See Choi and Perez (2007)for anecdotal evidences on legal firms adopting technologies used by illegal P2P filesharers. In design based industries, the good in being pirated is the signal of the highquality of the legal product, and products which ‘are not faked are considered too weakto generate consumer demand and are consequently not produced’ (Whitehall, 2006).Ritson (2007) says that pirated goods are indicative of heralding a brand's renaissanceand a brand dies if no copies appear in the market.

The probability that the innovating firm is successful in develop-ing the product is kα(R) where 0 ≤ kα(R) ≤ 1. k can be viewed asthe R&D efficiency parameter and I assume that α′(R) > 0 andα″(R) ≤ 0.5 Thus, technological uncertainty in the model is captured

by kα(R).6 I further assume that − α′ ′ Rð Þα′ Rð Þ is decreasing in R meaning

that the curvature of α(R) is decreasing in R. So for example α(R)can take the form α Rð Þ ¼

ffiffiffiR

por it can take the form of a contest func-

tion α Rið Þ ¼ aRiaRiþRj

when there is R&D competition between two firms,

which is discussed in Section 3. These functional forms satisfy the as-

sumption that − α′ ′ Rð Þα′ Rð Þ is decreasing in R.

As in Banerjee (2003), the difference between the original and thepirated products is with respect to the reliability of the latter, which isdenoted by the parameter q. The pirate operates in an illegal mannerusing a makeshift arrangement. So if the pirated product turns out tobe defective it cannot be replaced. That is, the pirated product comeswith no warranty. The original product receives full warranty. So q isthe probability that the pirated software is operational which is com-mon knowledge and we assume q ∈ (0,1).7

I consider a continuum of consumers indexed by θ, which repre-sents the consumers' valuation of the product. I assume that θ followsa uniform distribution and lies in the interval θ ∈ [0,1]. Each consum-er is assumed to buy only one unit of a product, if at all, and can eitherbuy the original or the pirated product. Also there is the presence ofnetwork externalities, which means that a consumer's utility is in-creasing in the number of other consumers using either the originalor the pirated product. A consumer buying the original productenjoys θ, and the network externalities generated by the buyers ofthe original and pirated products provided the latter is operationalwhich occurs with probability q. A consumer buying the pirated prod-uct enjoys θ and the network externalities only if the pirated productis operational. I assume that if the pirated product is operational forone buyer then it will be operational for all other buyers as well.Thus the utility of a type-θ consumer as in Banerjee (2003) is as fol-lows.

U θð Þ ¼θ−pm þ βDm þ qβDc; if the consumer buys the original product;q θþ βDm þ βDcð Þ−pc; if the consumer buys the pirated product;0; if the consumer buys nothing:

8<:

ð1Þ

β is a measure of the extent of network externalities and I assumethat β ∈ (0,1).8

The marginal consumer θm who is indifferent between pur-chasing the original product and the pirated product satisfies,

θm ¼ pm−pc−βDm 1−qð Þ1−qð Þ . The marginal consumer θc who is indifferent

between purchasing the pirated product and not buying anythingsatisfies, θc ¼ pc

q −β Dm þ Dcð Þ. As in Banerjee (2006), I assume that

θm b 1, θm − θc > 0, and θc > 0. The condition θm b 1 implies that(1 − q) > (pm − pc), whichmeans that some consumers buy the orig-inal product. The condition θm − θc > 0, which implies that pc b qpm,means that some consumers buy the pirated good. The assumptionθc > 0 means that the market is not fully covered, that is, there aresome consumers who buy nothing.

5 These properties ensure that the second order condition for profit maximizationholds.

6 Kultti et al. (2006) also refers to technological uncertainty involved in the innova-tion of digital products like software.

7 q can also be interpreted as an exogenous index of the poor quality of the piratedproduct. We set this bound to ensure that the profits are not indeterminate.

8 This assumption ensures positive profit for the innovating firm.

10 dr�� and dr�� can be compared because q and β are unit free numbers in the same do-

528 D. Banerjee / Economic Modelling 33 (2013) 526–532

Using the expressions for θm and θc I get the demand functions forthe original and pirated products as given in Eq. (2).

Dm ¼ 1−θm ¼ 11−β

− pm−pc1−qð Þ 1−βð Þ ;

Dc ¼ θm−θc ¼qpm−pc

q 1−qð Þ 1−βð Þð2Þ

I define the level of piracy as the pirate's market share in the total de-mand. This is denoted by s and is given in Eq. (3).

s ¼ Dc

Dm þ Dcð3Þ

Lemma 1 summarizes the effect of an increase in q on the pirate'smarket share and the proof is given in the Appendix A.

Lemma 1. The pirate's market share s, is monotonically increasing in q.Intuitively, an increase in q increases the demand for the pirated

product and decreases that of the original product. This is evident

from the fact that dDcdq ¼ q qpm−pcð Þþpc 1−qð Þ

q2 1−qð Þ2 1−βð Þ > 0 and dDmdq ¼ − pm−pcð Þ

1−βð Þ 1−qð Þ2 b0.

So the numerator in s increases more than the denominator resultingin a net increase in s due to an increase in q. This positive monotonicrelationship between s and q thus allows me to refer to an increase inq as an increase in piracy. An increase in β within the intervalβ ∈ (0,1) will henceforth be referred to as an increase in networkexternalities.

Using the demand functions as given in Eq. (2), I get the expectedprofits of the innovating firm and the pirate as follows.

Eπ ¼ kα Rð Þ pm−xð ÞDm−R ¼ kα Rð Þ 11−β

− pm−pc1−qð Þ 1−βð Þ

� �pm−xð Þ−R;

Eπc ¼ kα Rð Þ pc−xð ÞDc−γFf g ¼ kα Rð Þ qpm−pcð Þ pc−xð Þq 1−qð Þ 1−βð Þ −γF

� �:

ð4Þ

x is the constant marginal cost of production, which is assumed tobe the same for both firms.9 I assume that the pirate can copy theproduct with certainty if the original product is successfully devel-oped. γ is the probability of detecting the pirate in which case hepays a penalty F. So γF is the expected penalty, which acts like anentry cost. The parameters γ and F are used for the purpose of com-pletion of the model and as a justification for considering a single pi-rate. They will not be used for any comparative static analysis becausethe focus of this paper is to examine the impact of increases in piracyand network externalities on innovation and not on the efficacy oflegal mechanisms for IPR protection, for which see Yang (2013),Banerjee (2011), and Teece (1991), for examples.

I begin the analysis from stage 3 of the game. The pirate's reactionfunction, using Eq. (4), is pc ¼ qpmþx

2 . Let r ¼ 11−β−

pm−pc1−qð Þ 1−βð Þ

� �pm−xð Þ

denote the realized stage 2 profit of the innovating firm if its innova-tion is successful in stage 1. Substituting pc ¼ qpmþx

2 in r and maximiz-ing it with respect to pm yields the innovating firm's stage 2equilibrium price and realized profit (denoted by “**” in the super-script) as given in Eq. (5).

p��m ¼ 2 1−qð Þ þ 3−qð Þx2 2−qð Þ

r�� q;βð Þ ¼ 1−qð Þ 2−xð Þ28 1−βð Þ 2−qð Þ

ð5Þ

The rate of change of r⁎ ⁎(q,β) due to an increase in q, that is dr��dq , is

referred to as the piracy effect and the rate of change of r⁎ ⁎(q,β) dueto an increase in β, that is dr��

dβ , is referred to as the network effect.Lemma 2 summarizes the results for the comparative static analysis

9 x is assumed to be low enough to ensure positive profits to the firms.

of r⁎ ⁎(q,β) with respect to q and β. The proof is included in themain text because it is instructive.

Lemma 2. (i) The equilibrium realized stage 2 profit of the innovatingfirm, r⁎ ⁎(q,β), is decreasing in q and increasing in β. (ii) The network

effect exceeds the piracy effect if qbq where q ¼ 3−ffiffiffiffiffiffiffiffi5−β

p2 . The reverse is

true if q > q.

Proof of Lemma 2. (i) δr�� q;βð Þδq ¼ − 2−xð Þ2

8 1−βð Þ 2−qð Þ2 b0 and δr�� q;βð Þδβ ¼

1−qð Þ 2−Cð Þ28 1−βð Þ2 2−qð Þ > 0. (ii) δr�� q;βð Þ

δβ

− δr�� q;βð Þδq

¼ 2−xð Þ2 1þβð Þ−3qþq2f g8 1þβð Þ2 2−qð Þ2 . This is pos-

itive if (1 + β) − 3q + q2 > 0, because the coefficient of this ex-pression is always positive. Solving q for (1 + β) − 3q + q2 = 0

yields q ¼ 3�ffiffiffiffiffiffiffiffi5−β

p2 . Since by assumption q ∈ (0,1) hence, q ¼

3−ffiffiffiffiffiffiffiffi5−β

p2 . So δr�� q;βð Þ

δβ

− δr�� q;βð Þδq

> 0 if qbq and δr�� q;βð Þδβ

− δr�� q;βð Þδq

b0 if

qbq, because (1 + β) − 3q + q2 is decreasing in q sinceq ∈ (0,1). Q.E.D.

Lemma 2 that specifies the conditions for which the piracy effectdominates the network effect and vice versa, can be intuitivelyexplained as follows.10 An increase in β increases the innovatingfirm's demand thereby increasing its realized stage 2 profit. This is ev-ident from the fact that Dm is increasing in β while pm⁎ ⁎ is indepen-dent of β. However, an increase in q reduces the demand and theprice for the original product thereby reducing the stage 2 realizedprofit of the innovating firm. Part (ii) of Lemma 2 shows that the net-work effect dominates the piracy effect if for any given β, q is below

the critical level q ¼ 3−ffiffiffiffiffiffiffiffi5−β

p2 . Using (1 + β) − 3q + q2 > 0 we can

also say that the network effect dominates the piracy effect if forany given q, β satisfies β > 3q − q2 − 1, that is, β exceeds the criticallevel β where β ¼ 3q−q2−1. For the rest of the paper I will use q forthe analysis.

Substituting r⁎ ⁎(q,β) in Eq. (4) and maximizing the innovatingfirm's expected profit with respect to R yields the equilibrium R&D in-vestment. Let RP and EπP denote the equilibrium R&D investment andexpected profit of the innovating firm. Using the conditions summa-rized in Lemma 2 I now determine the impact of simultaneous in-creases in piracy and network externalities on the R&D investment.These results are summarized in Proposition 1 and the proof isgiven in the Appendix A.

Proposition 1. (i) The equilibrium R&D investment of the innovating firm

satisfies α′ RP� �

¼ 1kr�� and its expected profit in equilibrium is EπP =

kα(RP)r⁎ ⁎ − RP.11 (ii) Increases in piracy and network externalitiesresults in an increase (a decrease) in the equilibrium R&D investmentand the expected profit of the innovating firm if the network effect (piracyeffect) dominates the piracy effect (network effect). The equilibrium R&Dinvestment and the expected profit of the innovating firm remainunchanged if the two effects are the same.

Proposition 1 implies that an increase in piracy may not neces-sarily result in a decrease in the R&D investment even whenthere is a single innovating firm. The intuition behind this resultis as follows. If the network effect dominates the piracy effectthen there is a net increase in the demand for the original productand its price. This results in an increase in the innovating firm'sstage 2 realized profit, which provides the incentive to increaseits R&D investment in stage 1. The reverse is true when the piracyeffect dominates the network effect. These findings contradict the stan-dard claim that piracy retards the incentive to innovate in the contextof a single innovating firm.

dq dβmain (0,1).11 k≥ RP

α RPð Þr�� ensures non-negative profit.

Rj

B

A RPjBR

C

D RPiBR

Ri

Fig. 1. The piracy effect dominates the network effect.

Rj

B C

A

529D. Banerjee / Economic Modelling 33 (2013) 526–532

3. R&D competition

In this Section 1 consider R&D competition between two innovatingfirms i and j. Kultti et al. (2006)mention, “our discussionswith industrypractitioners confirm, that the simultaneous model of innovation char-acterizes especially network industries like software”. Thus in stage 1 ofthe game I consider the two innovating firms competing simultaneous-ly in R&D investment and the winner of the copyright engages in pricecompetition with the pirate in stage 2.12 The stage 2 results are thesame as in Eq. (5).

In stage 1 of the game an innovating firm can win the copyright ifit is successful in innovation and the rival firm is not. If both firms aresuccessful then each firm receives the copyright with equal probabil-ity. So the probability that firm i receives the copyright is,

kiα Rið Þ 1−kjα Rj � �þ kikjα Rið Þα Rjð Þ

2 : Hence, the expected profit offirm i is,

Eπi ¼ kiα Rið Þ 1−kjα Rj

� �� �þkikjα Rið Þα Rj

� �2

0@

1Ar��−Ri: ð6Þ

Firm i's reaction function is, 13

dEπi

dRi¼ kiα

′ Rið Þ 1þkjα Rj

� �2

0@

1Ar��−1 ¼ 0⇒kiα

′ Rið Þ 2−kjα Rj

� �� �¼ 2

r��

0@

ð7Þ

The effect of increases in piracy and network externalities on thereaction functions of the innovating firms is summarized inLemma 3 and the proof is given in the Appendix A.

Lemma 3. Increases in piracy and network externalities shift the reac-tion functions of the two firms: (i) downward if the piracy effect domi-nates the network effect q > qð Þ; (ii) upward if the network effectdominates the piracy effect q > qð Þ.

Lemma 3 implies that if the piracy effect is dominant then eitherthe R&D investment of both firms decrease or the R&D investmentof one firm increases and that of the other decreases. If the networkeffect is dominant then either the R&D investment of both firms in-crease or the R&D investment of one firm increases and that of theother decreases. These two cases are diagrammatically representedin Figs. 1 and 2.

In Figs. 1 and 2, point A represents the initial equilibrium R&D in-vestments denoted by (RiRP,RjRP). Points B, C and D represent the pos-sible equilibria after incorporating the piracy and network effects. Ifthe firms are symmetric with respect to R&D efficiency, then in-creases in piracy and network externalities increases or decreasesthe equilibrium R&D investments depending on the relative strengthsof the network and piracy effects.

Let us consider the case where the firms are asymmetric with re-spect to the R&D efficiency, that is, ki b kj. To determine which firm'sR&D investment increases when piracy and network externalities in-crease requires the comparative static analysis of the firms' equilibri-um R&D investment with respect to the efficiency parameters. Theresult is summarized in Lemma 4 and the proof is provided in theAppendix A.

12 Since this paper is motivated by copyright examples, the competition between thetwo firms is like a copyright race. While copyright and patent are under the generalumbrella of “intellectual property”, there are significant differences between the twomainly in terms of copying and piracy.

13 The second order conditions require ∂2Eπi∂Ri

2 b0 and Dj j≡∂2Eπi

∂Ri2 − ∂2Eπi

∂RiδRj

− ∂2Eπi

∂RiδRj

∂2Eπj

∂Rj2

> 0: We

have ∂2Eπi∂Ri

2 ¼ kiα′ ′ Rið Þ 1− kjα Rjð Þ2

� �r��b0 because, by assumption, α′′(Ri) b 0.

Lemma 4. For a given level of piracy and network externalities, an in-crease in a firm's efficiency parameter increases its equilibrium R&D in-vestment. An increase in the rival firm's efficiency parameter decreasesthe firm's equilibrium R&D investment.

Intuitively, an increase in a firm's efficiency parameter shifts its reac-tion function to the right. That is, the marginal gain from an increase inR&D investment outweighs the loss from doing so. Consequently, thefirm's equilibrium R&D investment increases. The opposite is truewhen the rival firm's efficiency parameter increases.

I now turn to the analysis of whether the more or the less efficientfirm's equilibrium R&D investment increases or decreases when thenetwork effect dominates the piracy effect and vice versa. Such an

analysis requires the derivation of the expressions for dRiRP

dq , dRiRP

dβ , anddRi

RP

dβ − − dRiRP

dq

� �, which is the difference between the absolute effects

of increases in piracy and network externalities on a firm's equilibri-um R&D investment, and are provided in the Appendix A. These ex-pressions are given in Eqs. (8), (9) and (10).

dRiRP

dq¼ A

dr��

dqð8Þ

dRiRP

dβ¼ A

dr��

dβð9Þ

D

RPiBR RP

jBR

Ri

Fig. 2. The network effect dominates the piracy effect.

530 D. Banerjee / Economic Modelling 33 (2013) 526–532

dRiRP

dβ− − dRi

RP

dq

!¼ A

dr��

dβ− − dr��

dq

� �� �

A ¼ 2kjr��2

1Dj jα

′ RjRP

� �2−kiα Ri

RP� �

−α′ ′ Rj

RP� �

α′ RjRP

� � −kiα

′ RiRP

� �2−kiα Ri

RP � �

0@

1A

0@

ð10Þ

From Eqs. (8) and (9) we observe that dRiRP

dβ and dRiRP

dq have opposite

signs because dr��dq b0 and dr��

dβ > 0. The sign of the expression in Eq. (10)

depends on the signs of A and dr��dβ − − dr��

dq

� �� �. In expression A we

know that |D| > 0 from the second order condition (shown infootnote 13), and 2kj α′ Rj

RP �

2−kiα RiRP

� �> 0

. So the sign of A

depends on the sign of − α′ ′ RjRPð Þ

α′ RjRPð Þ −

kiα′ RiRPð Þ

2−kiα RiRPð Þð Þ

� �. Thus the sign of

dRiRP

dβ − − dRiRP

dq

� �depends on the sign of − α′ ′ Rj

RPð Þα′ Rj

RPð Þ −kiα′ Ri

RPð Þ2−kiα Ri

RPð Þð Þ

� �and

on that of dr��dβ − − dr��

dq

� �� �.

The result for the effect of increases in piracy and network external-ities on the firm's R&D investments are summarized in Proposition 2.The proof is given in the Appendix A.

Proposition 2. Consider increases in piracy and network externalities.(i) If the two firms are “sufficiently” asymmetric with respect to theirR&D efficiencies and if the piracy effect dominates the network effectthen the less efficient firm's equilibrium R&D investment increases andthat of the more efficient firm decreases. The reverse is true if the networkeffect dominates the piracy effect. (ii) If the of the two firms are “less”asymmetric then the equilibrium R&D investment of both firms eitherincreases or decreases depending on the relative strengths of the piracyand the network effects.

Part (i) of Proposition 2 with the piracy effect dominating the net-work effect is represented by either point B or point D in Fig. 1depending on whether firm i or j is the less efficient firm. The samepoints in Fig. 2 represent the case where the network effect dominatesthe piracy effect. Part (ii) of Proposition 2 is diagrammatically repre-sented by point C in Figs. 1 and 2. I nowprovide an intuitive explanationfor Proposition 2.

When the piracy effect dominates the network effect it reducesthe stage 2 realized profit of an innovating firm. The market uncer-tainty, which is the probability of winning the copyright if bothfirms are successful, is the same for both firms. The only differencebetween the two firms is with respect to their R&D efficiency pa-rameters, which are a measure of the probability of successful in-novation in the presence of technological uncertainty.14 Thus afirm's stage 1 expected profit is dictated by the relative strengthsof the probability of success and the cost of R&D. An increase inthe R&D investment decreases its expected profit because the costgoes up. However, an increase in the R&D investment increasesthe stage 1 expected profit because the probability of successincreases.

Starting from the symmetric situation I allow ki to take smallervalues and kj to take higher values, which capture the asymmetric sit-uation. From Lemma 4 we know that lower values of ki and highervalues of kj implies lower RiRP and higher RjRP. By assumption α(R) isconcave and that its curvature is decreasing in R. So for firm i a mar-ginal increase in its R&D investment (RiRP) results in a higher increasein its probability of success (kiα(RiRP)) compared to firm j. So thelower is the value of ki, the higher is the increase in the probabilityof success of firm i for a marginal increase in its R&D investment. Sim-ilarly, the higher is the value of kj, the lower is the increase in the

14 I use the definitions of technological and market uncertainty from Shy (2000).

probability of success of firm j for a marginal increase in its R&Dinvestment.

This means that for firm i if its efficiency parameter is low then thegain from a marginal increase in R&D investment outweighs the costresulting in a higher profit. The opposite is true for firm j. So if the pi-racy effect dominates the network effect then the less efficient firm'sequilibrium investment increases and that of the more efficient firmdecreases. The reverse argument holds for the case where the net-work effect dominates the piracy effect.

Proposition 2 implies that an increase in piracy result in an in-crease in the R&D investment of one of the firms if they are sufficient-ly asymmetric. The dominance of the piracy effect or the networkeffect dictates whether the R&D investment of the less or the more ef-ficient firm will increase. Both firms reduce their R&D investment ifthe piracy effect dominates the network effect and if the firms areless asymmetric.

I now discuss the impact of increases in piracy and network exter-nalities on the overall probability of a successful innovation. If the twofirms are less asymmetric with respect to the efficiency parametersand if the piracy effect dominates the network effect then fromProposition 2 we know that both firms will reduce their R&D invest-ment. Thus the overall probability of a successful innovation de-creases. If the network effect dominates the piracy effect then bothfirms will increase their R&D investment levels and therefore theoverall probability of a successful innovation increases. Proposition3 summarizes the overall probability of a successful innovationwhen the firms are sufficiently asymmetric. We discuss the proof inthe main text.

Proposition 3. Consider increases in piracy and network externalities. Ifthe firms are sufficiently asymmetric with respect to the R&D efficiencyparameters and the piracy effect dominates the network effect, thenthere is an overall increase in the probability of a successful innovation.If the network effect dominates the piracy effect then the overall proba-bility of successful innovation decreases.

If the firms are sufficiently asymmetric and the piracy effectdominates the network effect, then from Proposition 2 we knowthat the less efficient firm will increase its R&D investment andthe more efficient one will reduce it. The increase in the probabilityof success of the less efficient firm exceeds the decrease in theprobability of success of the more efficient firm because α(R) is in-creasing and concave in R, and that its curvature is decreasing in R.This means that the domination of piracy effect over network effectmay result in an increase in the overall probability of success. Thereverse is true when the network effect dominates the piracy effect.

4. Conclusion

This paper analyzed the impact of increases in piracy and networkexternalities on the incentive to innovate. In the case of a single inno-vating firm facing technological uncertainty, I showed that the incen-tive to innovate increases when the network effect is stronger thanthe piracy effect.

However, with R&D competition, which generates market uncer-tainty on top of technological uncertainty, I showed that increases in pi-racy and network effects result in an increase in the R&D investment ofone firm if the competing firms significantly differ with respect to theefficiency in R&D investment. Specifically, if the piracy effect (networkeffect) is stronger than the network effect (piracy effect) then the less(more) efficient firm's R&D investment increases and that of the more(less) efficient firm's investment decreases.

I also showed that in the above case if the piracy effect dominatesthe network effect then the overall probability of successful innovationof a new product increases. The reverse is true when the network effectdominates the piracy effect. This paper thus showed that increases in pi-racy do not necessarily have a negative impact on innovation

531D. Banerjee / Economic Modelling 33 (2013) 526–532

Appendix A

Proof of Lemma 1. Using Eq. (2) we get s ¼ qpm−pc1−qð Þ q−pcð Þ.

dsdq ¼

q qpm−pcð Þþpc 1−qð Þ− qpm−pcð Þð Þ1−qð Þ2 q−pcð Þ2 > 0, because (qpm − pc) > 0 and (1 − q) −

(qpm − pc) > 0 following the assumptions θm − θc > 0 andθm b 1. Q.E.D.

Proof of Proposition 1. ∂Eπ∂R ¼ kα′ Rð Þr��−1 ¼ 0⇒α′ RP

� �¼ 1

kr��

α″ RP� �

dRP

dq ¼ −1r��2

dr��dq ⇒ dRP

dq ¼ −1r��2α″ RPð Þ

dr��dq b0 and α″ RP

� �dRP

dβ ¼ −1r��2

dr��dβ ⇒

dRP

dβ ¼ −1r��2α″ RPð Þ

dr��dq > 0. This follows from Lemma 2 and the property

that α″(R) b 0. Now dRP

dβ −−dRP

dq ¼ −1r��2α″ RPð Þ

dr��dβ −−dr��

dq

� �. The first expres-

sion is positive because α″(RP) b 0 hence the sign depends on that ofthe expression in parenthesis. The sign of this expression follows

from Lemma 1. So dEπP

dq ¼ kα′ RP� �

dRP

dq þ kα RP� �

dr��dq b0 since dRP

dq b0 and

dr��dq b0. dEπP

dβ ¼ kα′ RP� �

dRP

dβ þ kα RP� �

dr��dβ > 0 since dRP

dβ > 0 and dr��dβ > 0.

So dEπP

dβ − − dEπP

dq

� �¼ kα′ RP

� �dRP

dβ −−dRP

dq

� �þ dr��

dβ −−dr��dq

� �� �¼kα′ RP

� �1− −1

r��2α″ RPð Þ� �

dr��dβ −−dr��

dq

� �. Since kα′(RP) > 0 and 1− −1

r��2α″ RPð Þ� �

> 0

hence the sign depends on the sign of dr��dβ −−dr��

dq

� �which is stated in

Lemma 2. Q.E.D.

Proof of Lemma 3. To find the effects of changes in q and β on the reac-tion function of firm iwe differentiate its reactionwith respect to q and βassuming that Rj remains unchanged. Differentiation with respect to q

yields, kiα′ ′ Rið Þ ð1þ kjα Rj �2

� �r��

dRi

dqþ kiα′ Rið Þ

�1þð kjα Rj

�2

�dr��

dq¼

0 ⇒dRi

dq¼

−kiα′ Rið Þ�ð1þ kjα Rj

�2

�dr��

dq

kiα′ ′ Rið Þ ð1þ kjα Rj �2

� �r��

b0;because the numerator is

positive sincedr��dq b0 fromLemma2 and by assumptionα″(R) b 0. Similar-

ly, dRidβ ¼

−kiα′ Rið Þ�

ð1þkjα Rjð Þ2

�dr��dβ

kiα′ ′ Rið Þ ð1þkjα Rjð Þ2

� �r��

> 0; because from Lemma 2 we know that

dr��dβ > 0 and by assumptionα″(R) b 0. Thismeans that an increase in pira-

cy shifts down firm i's reaction function and the opposite is true for an inincrease in the network effect. The overall effect of increases in piracy andnetwork externalities is obtained by taking the difference of the absolutevalue of the effects which is,

dRidβ − − dRi

dq

� �¼

−kiα′ Rið Þ�

ð1þkjα Rjð Þ2

�kiα′ ′ Rið Þ ð1þkjα Rjð Þ

2

� �r��

dr��dβ − − dr��

dq

� �� �. The sign

of this depends on the sign of dr��dβ − − dr��

dq

� �� �which we get from

Lemma 1. That is dr��dβ − − dr��

dq

� �� �> 0 if qbq and dr��

dβ − − dr��dq

� �� �b0 if

q > q. Q.E.D.

Proof of Lemma 4. Total differentiation of kiα′ RiRP

�2−kjα Rj

RP � � ¼

2ri��

and kjα′ RjRP

�2−kiα Ri

RP � � ¼ 2

rj��with respect to ki and solving for

dRiRP

dkiand dRj

RP

dkiusing Cramer's rule yields, dRi

RP

dki¼ kjα′ Rj

RPð ÞDj j −f

α′′ RjRP

�2−kiα Ri

RP � �

2−kjα RjRP

� �þ kikjkiα RiRP

�α′ Rj

RP � �2g > 0

and dRjRP

dki¼ −kikjα′ Rj

RPð Þ 2−kjα RjRPð Þð Þ

Dj j −α′ ′ RjRPð Þα Rj

RPð Þ− α′ RiRPð Þð Þ2f gb0. Q.E.D.

Derivation of dRiRP

dq , dRjRP

dq , dRiRP

dβ , dRjRP

dβ , dRiRP

dβ − − dRiRP

dq

� �and dRj

RP

dβ − − dRjRP

dq

� �.

1. To derive the expressions for dRiRP

dq and dRjRP

dq we substitute RiRP ;Rj

RP �

in the first order condition in Eq. (8) and perform total differentiation

with respect to q. This yields, kiα′′ RiRP

�2−kjα Rj

RP � � dRi

RP

dq − kikjα′

RiRP

�α′ Rj

RP � dRj

RP

dq ¼ − 2r��2

dr��dq and kjα′′ Rj

RP �

2−kiα RiRP

� � dRjRP

dq −

kikjα′ RiRP

�α′ Rj

RP � dRi

RP

dq ¼ − 2r��2

dr��dq Using Cramer's rule to solve for dRi

RP

dq

and dRjRP

dq gives us the following two equations.

dRi

RP

dq¼ −

2kjr��2

dr��

dq1Dj jα

′ RjRP

� ��2−kiα Ri

RP� � α′ ′ Rj

RP� �

α′ RjRP

� � þkiα

′ RiRP

� �2−kiα Ri

RP � �

0@

1A dRj

RP

dq

¼ − 2kir��2

dr��

dq1Dj jα

′ RiRP

� �2−kjα Rj

RP� � α′′ Ri

RP� �

α′ RjRP

� � þkjα

′ RjRP

� �2−kjα Rj

RP� �� �

0@

1A

0@

2. To derive the expressions for dRiRP

dβ and dRjRP

dβ we proceed analogouslyby performing the total differentiation with respect to β which yields,

kiα′′ RiRP

�2−kjα Rj

RP � � dRi

RP

dβ −kikjα′ RiRP

�α′ Rj

RP � dRj

RP

dβ ¼ − 2r��2

dr��dβ and

kjα′′ RjRP

�2−kiα Ri

RP � � dRj

RP

dβ −kikjα′ RiRP

�α′ Rj

RP � dRi

RP

dβ ¼ − 2r��2

dr��dβ .

Using Cramer's rule to solve for dRiRP

dβ and dRjRP

dβ gives us the following two

equations.

dRiRP

dβ¼ −

2kjr��2

dr��

dβ1Dj jα

′ RjRP

� �2−kiα Ri

RP� �� � α′ ′ Rj

RP� �

α′ RjRP

� � þkiα

′ RiRP

� �2−kiα Ri

RP � �

0@

1A

dRjRP

dβ¼ − 2ki

r��2dr��

dβ1Dj jα

′ RiRP

� �2−kjα Rj

RP� � α′ ′ Ri

RP� �

α′ RiRP

� þkjα

′ RjRP

� �2−kiα Ri

RP � �

0@

1A:

0@

dRiRP

dβ− −dRi

RP

dq

!

¼2kjα′ Rj

RP� �

2−kjα RjRP

� �� �Dj jr��2 −

α′ ′ RjRP

� �α′ Rj

RP� � −

kiα′ Ri

RP� �

2−kiα RiRP

� �0@

1A dr��

dβ− − dr��

dq

� �� �

dRjRP

dβ− −

dRjRP

dq

!

¼2kiα′ Ri

RP� �

2−kjα RjRP

� �� �Dj jr��2 −

α′ ′ RiRP

� �α′ Ri

RP � −

kjα′ Rj

RP� �

2−kjα RjRP

� �� �0@

1A dr��

dβ− − dr��

dq

� �� �

Proof of Proposition 2. We begin with the symmetric case whereki = kj. In this case Ri

RP = RjRP. Changes in piracy and network effect

will have symmetric effects on these equilibrium values, thus, the

signs of dRiRP

dβ − − dRiRP

dq

� �and dRj

RP

dβ − − dRjRP

dq

� �are the same. So either

they are positive or negative depending on whether qbq or q > qwhich determines which effect dominate as given in Lemma 3.Starting from the point of symmetry let us consider ki taking smallervalues and kj taking higher values. Lower values of ki and highervalues of kj implies lower Ri

RP and higher RjRP. Consequently,

− α′ ′ RiRPð Þ

α′ RiRPð Þ becomes higher and − α′ ′ Rj

RPð Þα′ Rj

RPð Þ becomes lower because, by

assumption, − α′ ′ Rð Þα′ Rð Þ is decreasing in R. As − α′ ′ Rj

RPð Þα′ Rj

RPð Þ becomes lower, a

lower ki can sustain the inequalitykiα′ Ri

RPð Þ2−kiα Ri

RPð Þð Þ > − α′ ′ RjRPð Þ

α′ RjRPð Þ because

from the first order condition given in Eq. (8) we know that ki must

satisfy kib 2α Ri

RPð Þ. So in this case 0 > − α′ ′ RjRPð Þ

α′ RjRPð Þ −

kiα′ RiRPð Þ

2−kiα RiRPð Þð Þ which is

an important determinant in the sign of dRiRP

dβ − − dRiRP

dq

� �. As − α′ ′ Ri

RPð Þα′ Ri

RPð Þbecomes higher due to a small ki, it needs a very high kj to sustain

the inequalitykjα′ Rj

RPð Þ2−kjα Rj

RPð Þð Þ > − α′ ′ RiRPð Þ

α′ RiRPð Þ . However, kj is restricted by

kjb 2α Rj

RPð Þ and a higher RjRP means a lower 2α Rj

RPð Þwhich further restricts

the possibility of a high kj. Thus for firm j, − α′ ′ RiRPð Þ

α′ RiRPð Þ −

kjα′ RjRPð Þ

2−kjα RjRPð Þð Þ > 0

which is an important determinant of the sign of dRjRP

dβ − − dRjRP

dq

� �. Sup-

pose the increase in the network effect dominate the effect of an in-

crease in the piracy rate, that is, dr��dβ − − dr��

dq

� �� �> 0 (that is, qbq)

which implies that the reaction functions shift up. So for firm i,

532 D. Banerjee / Economic Modelling 33 (2013) 526–532

dRiRP

dβ − − dRiRP

dq

� �b 0, and for firm j, dRj

RP

dβ − − dRjRP

dq

� �> 0. Thus if the net-

work effect dominates the piracy effect then the more efficient firm'sequilibrium R&D investment increases and that of the less efficient

firm's decreases. When q > q which implies that dr��dβ − − dr��

dq

� �� �b0,

that is the piracy effect dominates the network effect, then,dRi

RP

dβ − − dRiRP

dq

� �> 0 for firm i, and dRj

RP

dβ − − dRjRP

dq

� �b 0 for firm j. That

is, the less efficient firm's equilibrium R&D investment increasesand that of the more efficient firm's decreases. Q.E.D.

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