Impact of piracy on innovation at software firms and implications for piracy policy

Decision Support Systems 46 (2009) 763–773

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Decision Support Systems j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / d s s

Impact of piracy on innovation at software firms and implications for piracy policy Jeevan Jaisingh ⁎ ISOM Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

a r t i c l e

i n f o

Available online 27 November 2008 Keywords: Software piracy Policy Quality Innovation

a b s t r a c t In this paper, we look at how innovation in the presence of piracy is affected by the policy choice of alliances such as the Business Software Alliance (BSA). Surprisingly, we find that a stricter piracy policy, that increases the perceived cost to using pirated software for end-users, may in some cases lead to an increase in piracy, and a decrease in product quality. The implication for a social planner is that in a monopoly market, an increase in the policy variable, could act as a disincentive for innovation. In a competitive market an increase in the policy variable provides an incentive for innovation. © 2008 Elsevier B.V. All rights reserved.

1. Introduction A Business Software Alliance (BSA) commissioned study in 2006 [3], found that $34 billion was lost due to piracy of software in 2005. In its fight against piracy, the software industry has combined forces through alliances such as the BSA, and the software publishers alliance (SPA). These alliances educate consumers on software management and copyright protection, cyber security, trade, e-commerce, and other Internet-related issues. They also work with law enforcement agencies in several countries to take action against pirates. The main advantage of these alliances is the tremendous economies of scale in fighting piracy. The BSA study found that countries varied in their piracy rate, ranging from Vietnam at 90%, to the United States at 21% [3]. The large variance in piracy rates could possibly be attributed to cultural differences, and the actions taken by alliances such as the BSA and the SPA in the specific countries. An often cited reason by software firms for building up their case against piracy, is the detrimental impact of piracy on innovation. “Strong intellectual property protection spurs creativity, which opens new opportunities…” — CEO, BSA [11]. Innovation with reference to software companies could mean several things — producing more types of software, improving the quality of their existing software, R&D investment etc. In the context where the software firm already has existing software, we use the “quality of software” as a surrogate for the “innovation” taking place at the software firm. We look at how the quality of software is affected by the policy choice of alliances such as the BSA when piracy is prevalent. The BSA and its members invest significant resources in educating users about copyright, its value, and enforcing copyright laws [4]. Does effort spent in educating users about the harmful aspects of piracy,

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and taking action against end-users using pirated software always result in higher quality software? What is the impact of competition between software manufacturers on prices and quality choice? These are the issues we address in this paper. These issues are of interest not just to alliances such as the BSA, and software firms, but also to a social planner such as the government in deciding public policy regarding piracy. More and more papers in the area of Information Systems (IS) are looking at issues relating to public policy from a decision support perspective [5,13,18,20]. Empirical studies on the impact of piracy on the quality choice of firms, are understandably hard because of the inherent difficulty in operationalizing a measure for quality. The analytical approach will provide intuition on the long term impact of piracy policies on welfare, quality choices, and price competition. The two most common forms of piracy are end-user piracy (friends sharing copyrighted software) and counterfeiting (large scale duplication of illegal software). This paper is about the latter. Software companies estimate they lose 15% of the industry's worldwide sales to counterfeiting [14]. Counterfeit versions of popular software are available at a much lower price on the Internet and from street peddlers. For example, Microsoft's Office XP is available at auction sites for $200 (original retail price $479). According to Microsoft, there's a 90% chance the discs are counterfeit [19]. Organized piracy or counterfeiting is thus a major cause of concern to software firms. Surprisingly, counterfeiting has received little attention in the piracy literature.1 The impact of piracy policies on innovation, and pricing strategies of the firm in the presence of a counterfeit supplier, is the focus of this paper. The environment that we study consists of a firm that develops software, a pirate that creates an illegal copy of the software, and an alliance such as the BSA which implements piracy policies (legal action, educating users about the detrimental effects of piracy etc.). The market for the software is shared between the firm and the pirate. Consumers differ in their ethical cost to pirate. The ethical cost to 1

The only known work dealing specifically with counterfeiting is Banerjee [2].

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pirate is a function of the level of the policy choice variable. A stricter piracy policy increases the ethical cost for consumers to pirate, thus affecting the demand for pirated products. We develop a game theoretic model in which the BSA chooses the piracy policy first. After observing the policy choice by the BSA, the firm chooses the quality of software. In the final stage, after quality has been chosen, the firm and the pirate choose prices simultaneously, knowing how prices are going to affect demand. We use backward induction to solve the game: we first determine the optimal prices for both firm and pirate, given that they choose prices simultaneously, and that they observe the quality choice by the firm and the policy choice of the BSA. Then based on these optimal prices, we calculate the optimal quality chosen by the firm, given that the firm observes the policy choice of the BSA. Finally based on the optimal quality choice of the firm, and the optimal prices charged by both the firm and the pirate, we calculate the optimal policy choice by the BSA. We find that when the firm knows it has a superior product, and that the policy choice by the BSA is low, then it should price aggressively to make it unprofitable for the pirate to exist in the market. Also, when the firm knows it doesn't have a high quality product, but the BSA has chosen a high policy choice, it does not engage in an aggressive pricing strategy, and leaves a small segment of the market for the pirate. Surprisingly, we find that a stricter piracy policy that increases the perceived cost to using pirated software for end-users, may in some cases lead to an increase in piracy (demand for pirated products), and a decrease in product quality. Thus an active BSA, that tries to educate consumers, and takes legal action against consumers, may actually be promoting piracy and hurting innovation in some cases. An intuitive rationale for this is that, in some regions, quality choice by the firm and the policy choice by the BSA are strategic substitutes in the fight against piracy. Thus an increase in the policy variable makes the firm choose a lower quality. Regarding the optimal policy choice by the BSA, we find that, when the likelihood that the pirated software will be functional is low, then it is nonoptimal for the BSA to expend any cost to reduce piracy. When it is medium, then a high level of policy choice is optimal only for high values of technology cost and low values of policy cost. When the likelihood that the pirated software will be functional is high, then, the BSA should almost always choose a high level of policy choice. In the presence of competition, the threat of losing market share to the competitor makes it non-optimal for a firm to choose a lower quality choice. Thus as opposed to the monopoly case, we find that optimal quality choice always increases with the policy choice. The rest of the paper is organized as follows: in Section 2 we cover the relevant literature. The basic model is set up in Section 3. The impact of competition between software firms is covered in Section 4. Finally we discuss the managerial implication of the results and directions for future research in Section 5. All proofs are in the appendix. 2. Background Novos and Waldman [16] distinguish between two types of effects of piracy on social welfare: the first effect is the loss due to underproduction, and the second is the positive welfare effect due to decrease in loss due to under-utilization. The underproduction loss comes from the decreased incentive for firms to innovate in the presence of piracy, which leads to reduced variety or reduced quality in the long term. Decrease in loss due to under-utilization results from more consumers getting to use the software as a result of piracy. In this paper our focus is on the loss due to underproduction. Specifically, we look at how innovation (quality) is affected by the policy choice of alliances such as the BSA, when piracy is prevalent. Most of the software piracy literature ([7,8,17]) addresses end-user piracy as opposed to organized pirating by a firm (i.e. counterfeiting) [2]. Previous papers that have looked at end-user piracy have studied the impact of protection strategies on prices and profits in a monopoly

setting [8], and in a duopoly setting [17]. They find that when network externalities are high enough then non-protection is optimal. Chen and Png [7] argue that a social planner, has three policy choices: to tax the copying medium, to subsidize legal sales and to fine offenders. They find that, from a welfare perspective, providing subsidies to users is optimal compared to taxing the copying medium, or penalizing copiers. Banerjee [2] studies the optimal monitoring rate and fine to charge a pirate in a setting where the market is shared between the firm and the pirate. The paper finds that welfare maximization may or may not result as the socially optimal outcome. The impact of the reliability of the pirated software and the impact of network externality on the policy variable are also studied. Chen and Png [7] mention that the long term impact of policy choice on innovation has received little attention. Novos and Waldman [16] look at the impact on quality choices by a firm in a end user piracy setting. They find that quality choices are below the socially optimal level in the presence of piracy. The setting in our paper is perhaps closest to Banerjee [2]. Both papers look at counterfeiting. We explore the impact of policy on quality and price choices of firms, Banerjee examines the impact on prices. Policy affects the supply of pirated software in Banerjee [2], while it affects the demand side of the market in our paper. From a research question perspective, this paper is in the same vein as Novos and Waldman [16]. However, they look at an end user piracy setting, and the impact of policy on the quality choices of firms is not studied. A working paper [1] studies the impact of intellectual property right protection on innovation. They operationalize the intellectual property right protection through a ‘reproduction cost’ and a ‘degradation cost’. Both are exogenous parameters that allow comparative statics on optimal quality choice. Unlike this paper [1], our policy variable is endogenous and we study the impact of competition. Gopal and Sanders [9] distinguish between preventive and deterrent measures to fight software piracy and show in a setting with club formation that using deterrent measures is optimal. Another paper by the same authors [10] develops an economic model that provides the rationale for the reluctance of a number of governments to aggressively enact and enforce intellectual property rights. The model incorporates the incentive structures for governments, software publishers, and individual consumers. 3. Model There is a firm that produces a product (software) of quality, q, at price, p. Quality of the software means two things: first software that has fewer bugs is of a higher quality. Second, software with more features (e.g. graphical analysis, regression analysis etc. in mathematical analysis software) is of a higher quality. All consumers value the software equally at its quality. There exists a pirate2 who sells a pirated version of this product at price, pc.3 The quality of the pirated software is, ϕq (0 ≤ ϕ ≤ 1). ϕ is the likelihood that the pirated software is functional.4 An article in PC Magazine [19] notes that consumers can face a variety of problems with counterfeit software: it can contain a virus, or it may not be functional; after all counterfeiters are not known for their quality control. Alternately, ϕ captures the 2 Assuming more than one pirate is not going to change the results qualitatively. What is crucial to the model is that the firm knows that a pirate or pirates are going to respond to the price set by the firm by setting the price of the pirated version. Assuming one pirate simplifies the analysis significantly. 3 Counterfeiters make huge profits on the margins between the marginal cost of duplication and the price at which it is sold. Global organized crime has taken over the highly profitable business of counterfeiting software [15]. Clearly, if at any time counterfeit software were to become a commodity, then the counterfeiter would not have an incentive to pirate. 4 We assume ϕ, the choice of copying technology by the pirate, to be exogenous. A pirate would typically use the same copying technology to copy different software [12]. Thus a pirate’s choice of technology spans across several software markets and an exogenous ϕ is more appropriate.

J. Jaisingh / Decision Support Systems 46 (2009) 763–773 Table 1 Notation Notation

Description

q p ϕ pc η z df, dc πf, πc k c i, j x

Quality of software. Price charged by firm. Likelihood that the pirated version is functional. Price charged by pirate. Policy choice by BSA. Ethical propensity not to pirate — consumer's type. Firm and pirate's demand. Profit of firm and pirate. Quality cost coefficient. Policy cost coefficient. Indexes for firms in duopoly. Location based preference cost.

degradation in the quality of the product due to lack of customer support, documentation etc.5 Alliances such as the BSA implement policies, or take actions to prevent piracy. These policies/actions may be in the form of legal action against users, or educating users about the harms of piracy. The BSA benefits from tremendous economies of scale in fighting piracy. Although the firm is a monopolist in its market, there are other software firms which also rely on the BSA to prevent piracy. These policies of the BSA increase the perceived cost to the user of using pirated software. We denote this policy/action measure by η, (0 ≤ η ≤ η̅). A higher η (stricter policy), corresponds to a more active role by the BSA to prevent piracy. Consumers (denoted by z) differ in the ethical propensity to use pirated software. z is assumed to be uniformly distributed on [0,1]. Given a consumer, z, the ethical cost to a consumer of using pirated software is, zη.6 This formulation of cost ensures that the greater the policy measure, the greater the cost to pirate for a consumer of type z, and for a given policy measure, individual consumers may differ in their ethical cost to pirate. A consumer with a higher z, has a higher ethical cost to pirate. We summarize our notation in Table 1. The utility of a consumer of type z: 8 < q−p U ðq; zÞ = q−pc −zη : 0

9 if consumer buys from firm = if consumer buys from pirate ; if consumer does not buy

ð1Þ

Let zˆ be the consumer type indifferent between buying from the firm and buying from the pirate. So q − p = ϕq − pc − zˆη. zˆ =

p−pc + q−q η

ð2Þ

The firm's demand:   p−pc + q−q d = 1−ˆz = 1− η

ð3Þ

The pirate's market share is zˆ.7

5 We assume that consumers are aware that they are buying illegal software when they buy from the pirate. It is also possible, that some of the counterfeit software may be packaged, such that consumers may be duped into thinking that they are buying the original version. However, we believe that consumers who are actually duped are a minority. 6 Some other papers [5,6] have also used the concept of moral cost although how the cost has been modeled is slightly different from the previous work [5,6] in the sense that the moral cost also depends on the policy choice. The multiplicative nature of the piracy cost is to ensure that the marginal increase in piracy cost due to an increase in piracy policy (η) is more for the ethical consumers (large z) than for the non-ethical ones (small z). With an additive specification, an increase in privacy policy would increase the piracy cost for all consumers, ethical or non-ethical, uniformly, which we believe is not reasonable. 7 Note that consumers who have a lower ethical cost will pirate. We assume here that the market is fully covered.

765

The firm pays membership fees to the BSA for fighting piracy. The membership fees variable is exogenous and for simplicity we assume that it is zero. Positive membership fees will only reduce the profit function of the firm by this fixed amount and will not qualitatively affect the nature of the results. Assuming a convex cost function (12 kq2 ; k N 0) for the firm to develop a product of quality q, the corresponding profit functions for the firm and the pirate are:    p−pc + q−q 1 − kq2 πf = p 1− η 2 πc = pc

  p−pc + q−q η

ð4Þ

ð5Þ

The BSA chooses η to maximize overall surplus. The game played between the firm, the pirate, and the BSA is specified in extensive form as follows: Stage 1: The BSA chooses the policy measure η. Stage 2: Having observed η, firm chooses quality q. Stage 3: Firm and the pirate choose prices simultaneously (Bertrand Competition). We work backwards — first determining the optimal prices for both firm and pirate given that they choose prices simultaneously and that they observe the quality choice by firm (1). Then based on these optimal prices, we calculate the optimal quality chosen by the firm (2) and then the optimal choice of η by the BSA (3). 3.1. Pricing subgame The pirate and the firm can observe the quality choice made by the firm in the previous stage, and the policy choice made by the BSA in stage 1. Both choose prices to maximize profits. Proposition 1. Given a policy choice, η, by the BSA, and a quality choice, q, by the firm, the optimal price for the firm, and for the pirate, and the demand for the firm are: 8 > > > > > > <

q q q 2 ; d = 1− if < 2 2η η 2+ qð1−Þ + 2η η−qð1−Þ 2η + qð1−Þ 2 q 1 : p = ; pc = ; d = if   > 3 3 3η 2 +  η 1− > > > q 1 > > : N p = qð1−Þ; pc = 0; d = 1 if η 1− p = q; pc =

q η is defined as the ‘quality to policy’ ratio. When the ratio is small (ηq < 2 2+ ) the share of the market taken up by the pirate (q 2η) is small. Hence it is optimal for the firm to extract all the surplus from its consumers. It charges a price equal to quality while the pirate charges half its expected quality. The optimal price set by the firm and the pirate increases in quality while the demand decreases with quality. A change in policy has no effect on the prices but decreases piracy 1 (demand for pirate). When the quality to policy ratio is high (qη N 1− )a big chunk of the market can potentially be lost to the pirate. Hence it becomes optimal for the firm to lower its price to a level (q(1 − ϕ) where the pirate is forced to leave the market. In this region, an increase in η has no effect on prices or piracy. An implication from the pricing in the two regions is that when the firm knows it has a superior product and the policy choice by the BSA is low, then it should price aggressively to make it unprofitable for the pirate to exist in the market. Also, when the firm knows it doesn't have a high quality product but the BSA has chosen a high policy choice, it does not engage in an aggressive pricing strategy and leaves a small segment of 1 the market for the pirate. For the range 2 2+   ηq  1− the following observations can be made: the price set by the firm (pirate) is an increasing (decreasing) function of quality choice. Similarly the demand for the firm is an increasing function of quality, which

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Fig. 1. Plots of change in optimal quality, prices, demand and profits with η.

means that the demand for the pirated software (henceforth called piracy) is a decreasing function of quality. The impact of policy choice η on prices is less certain since it depends on how the quality choice changes with η. There is a direct effect (+ve) on the price of the firm due to an increase in η and an indirect effect because of the effect of η on q. Similarly, there is a positive effect of an increase in η on the price of the pirate. This effect is because of the increase in the price of the firm with η, and hence is smaller. The price set by the pirate is also affected by the indirect effect of η on q An increase in η has a negative effect on the demand of the firm through an increase in price and an indirect effect because of the effect of η on q. 3.2. Quality choice As found in the previous section, the optimal pricing strategy is different for the three regions specified. The quality choice problem for the firm is8: max p⁎ d⁎ − 12 kq2 , subject to the boundary conditions of each of the three regions. For a given value of η, the firm chooses quality to maximize profits in each of the three regions. 2

2

2

Þ Þ Þ Denoting η1 = 2ð1− , η2 = 0:82ðk1−Þ , η3 = ð1− , η4 = 2ð1− , η5 = 2− , 2k 3k 3k k 2 1 η6 = k and assuming that N 3, the following is true 0 < η1 < η2 < η3 < η4 < _ η5 < η6 < η

Proposition 2. (a) Given a policy choice, η, by the BSA, the optimal quality choice by the firm, optimal price set by firm and pirate, optimal profit, and demand for firm are: Range of η

q⁎⁎

p⁎⁎

0 ≤ η ≤ η2

1− k 4ð1−Þη ½9kη−2ð1−Þ2 

ð1−Þ2 k 6η2 k ½9kη−2ð1−Þ2 

2η 2+

2η 2+

η2 < η ≤ η4 η4 < η ≤ η5 _ η5 < η < η

η  + kη

η  + kη

p⁎⁎ c 0

d⁎⁎ 1

½9kη−2ð1−Þ2 

π ⁎⁎ f ð1−Þ2 2k 4η2 k ½9kη−2ð1−Þ2 

η 2+

2η½2−kη ð2 + Þ2

2 2+

η½3kη−2ð1−Þ2 

η 2ð + kηÞ

η 2ð + kηÞ

6ηk

½9kη−2ð1−Þ2   + 2kη 2ð + kηÞ

respectively. 8 Here p⁎ and d⁎ represent the optimal price and demand in stage 3 for each of the three regions specified in Proposition 1.

(b) Change in optimal values with an increase in the policy variable is summarized below: Range of η

@q @η

@p @η

@p c @η

@π f

@d @η

0 ≤ η ≤ η2 η2 < η ≤ η4 η4 < η ≤ η5 _ η5 < η < η

0 − + +

0 + + +

0 + + +

0 + + +

0 − 0 +

@η

The above result shows the optimal choices in stage 2 given a policy choice in stage 1. We find that for a range of the policy variable (η2 <η≤η4), quality choice by the firm and the demand for the product actually decrease with a stricter policy η. Also, in the range (η2 ≤η≤η2), the policy variable has no impact on quality, prices, profit or demand. Software firms often cite the detrimental impact of piracy on the incentive to innovate as a reason for increased action/policy against piracy. Indeed we find that for ηNη4, that is the case. However, we find that in the range (η2 <η≤η4), a firm chooses a higher quality and the piracy is actually lower when the piracy policy is relaxed. Thus an active BSA that tries to educate consumers and takes legal action against consumers is actually promoting piracy and hurting innovation in this range. Novos and Waldman [16] find a similar result with respect to the impact of copyright protection. Their result shows that for a certain type of consumer distribution, with increased protection, the quality of software may be lower while with another distribution, the quality of software may be higher. In our paper however, with the same distribution, in certain ranges a higher policy variable could have a negative effect on quality. Graphically the above result is represented 2in Fig. 1. When the policy variable is below the threshold η2 = 0:82ðk1−Þ , it is optimal for the firm to price low enough to force the pirate out of the market. Understandably, the policy variable cannot affect the price, profit, or demand in this range and hence flat curves in this region. Beyond this threshold level, the firm has a higher profit if it shares the market with the pirate. It raises its price which results in the pirate finding it profitable to enter the market. Note that the profit and price charged by the firm are increasing in η beyond η2. _  Lemma 1. In the regions (η2 < η ≤ η4 and (η5 < η < η, the sign of @q@η is the _ @2 d  @2 d  same as the sign of @η@q. For (η2 < η  η4 Þ @η@q < 0 and for (η5 < η < η) 2  @ d @η@q N0.

J. Jaisingh / Decision Support Systems 46 (2009) 763–773

767

Fig. 2. TSImax and TSIV max for a) ϕ = 0.60 b) ϕ = 0.90 and c) ϕ = 0.99.

2 

@ d In the region (η2 < η ≤ η4 since @η@q = − 1− < 0, quality choice and 3η2 policy are strategic substitutes in the fight against piracy. Since quality and policy are strategic substitutes, increase in the policy variable η  would lead to the firm choosing a lower quality i.e. @q@η < 0. Other results regarding the impact of η on price, demand, and profits follow intuitively from the above result and the discussion after Proposition 1. The indirect effect of increase in η on quality is negative, hence the demand decreases and the price of the pirate increases with η. The direct effect of increase in η, dominates the indirect negative effect of η on quality and so the price of the firm increases with η. The firm increases its price with a stricter piracy policy and extracts more surplus from its loyal customers. Þ For ηNη4 = 2ð1− , the price follows the quality and all consumers 3k who buy legally earn a surplus of zero. In the range η4 < η ≤ η5 the policy variable is effective to the extent that a higher η leads to a higher quality choice. However, a higher policy variable does not lead to lower piracy as seen by the flat demand plot in this range. _ For η5 < η ≤ η, an increase in η leads to higher quality and lower @ 2 d piracy. Since @η@q = 2η2 N0 (from Lemma 1), quality choice and policy are strategic complements in the fight against piracy. Since quality and policy are strategic complements, increase in the policy variable η  would lead to the firm choosing a higher quality i.e. @q@η N0. However _ even at η = η, there is still a demand for the pirated product i.e. the demand for the firm <1.

3.3. Policy choice The BSA chooses the policy variable to maximize the total surplus. Left alone, the BSA will only care about the profit of the firm. However, given that the BSA depends on the government for support in implementing its policies, the government can have enough influence on the BSA for it to consider maximizing the surplus of the legal consumers also. Depending on the bargaining power between the government and the BSA, η could be chosen to maximize the profit of just the firm, to also maximizing surplus of legal consumers. In this section we have done the analysis for the case where the BSA chooses to maximize the surplus of the firm and the consumers who buy

legally. At the end of this section, we also present the results for the case where the BSA only maximizes the surplus of the firm.9 The BSA maximizes this surplus in each of the four regions and picks the strategy where the maximized total surplus is the greatest. Consumer surplus of the consumers who buy legally from the firm10: CS = ðq −p Þ  d

ð6Þ

Assuming that the BSA bears a cost (12 cη2 where c N 0) for implementing a policy η The BSA's maximization problem is maximize 1 2 total surplus (TS) i.e. max π f + CS− 2 cη subject to the boundary conditions of each of the four regions. 8 > 1−2 1 2 > > − cη if 0  η  η2 > > > 2k 2 > > 2 > 8η k ð 1− Þ ð 2 + Þ 1 2 > > > i2 − cη if η2 < η  η4 >h < 2 9kη−2ð1−Þ2 TS = > > 2η½2−kη 1 2 > > − cη if η4 < η  η5 > > > ð2 + Þ2 2 > > > > η 1 > > − cη2 if η5 < η < η : 2ð + kηÞ 2

ð7Þ

Proposition 3. The optimal policy choice, for each of the four regions 2 4 are: i) η I = 0, ii) ηII ≈ 0:82ðk1−Þ iii) ηIII = 2− if kc < and 2k ð2−Þð + 2Þ2  2 1 2 23 y−23 c 4 4 ηIII = if < kc < 2ð1 + 2Þ 2 iv) ηIV = where 6cky ð1−Þð + 2Þ ð2−Þð + 2Þ2 4k + cð + 2Þ2 1     qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 p ffiffiffi 2 y = c2 4c2 + 27k + 3 3 8c k+ 27kk . Here ηi denotes the optimal policy choice in region i = I, II, III, IV. In region I (0 ≤ η ≤ η2), the profit of the firm and the consumer surplus do not depend on the policy choice. The BSA thus chooses the minimum

9

The results for both cases are similar. See Appendix. Here q⁎⁎, p⁎⁎ and d⁎⁎ are different for each of the four regions specified in Proposition 2. 10

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J. Jaisingh / Decision Support Systems 46 (2009) 763–773

policy choice ηI = 0. In region II (η2 < η ≤ η4), TS is a decreasing function of η. Again the BSA2 chooses the minimum level of policy choice in this range, ηII ≈ 0:82ðk1−Þ = η2 . In region III (η4 < η ≤ η5), consumers who buy from the firm get zero surplus since the firm charges a price equal to quality. The BSA maximizes the profit of the firm net its cost to implement the policy. The profit of the firm is a concave function (as seen from Fig. 1) while the cost function of the BSA is convex. Thus if a maxima exists in this range, it is ηIII =

4 . 4k + cð + 2Þ2

The constraints for

(see Fig. 2c). A policy choice η⁎⁎⁎= 0 is thus almost never optimal in this case. Any intermediate level of policy choice is non-optimal and for specific ranges of policy choice, an increase in policy may hurt innovation and promote piracy. Policy choice when BSA only maximizes profit of firm: 2

Þ Region I: 0 ≤ η ≤ η2. Profit of the firm (ð1− ) which is independent of η. 2k BSA will maximize:

ð1−Þ2 1 2 − cη 2 2k

this solution come from the restrictions that the maxima must be in the specified range of the policy variable. The other solution in region III is a corner solution, when the constraint η ≤ η5 binds. In region IV _ (η5 < η < η), consumer surplus is again zero. The profit function of the firm is concave (as seen from Fig. 2 1) while the cost function is convex. 1

The maxima is at ηIV =

BSA will choose ηI = 0 and so

2

23 y−23 c 6cky i

TSImax =

. The maximized TS in each region can

be obtained by substituting η in the corresponding TS given in Eq. (7). Denoting the maximized TS in region i by TSimax, the global maxima can be obtained by comparing all four TSimax. It is a fairly involved algebraic problem to obtain the conditions under which each one of the solutions is the global maxima. Instead we evaluate TSimax for i =I, II, III, IV, varying the exogenous parameters c in the range [0.001,10] (increments 0.50), k in the range [0.001,10] (increments 0.50), and ϕ in the range [0.34,0.99] (increments 0.05). We find that in all cases, either TSImax or i I IV TSIV max dominates the other TSmax. We plot TSmax and TSmax for three different parameter values of ϕ (0.60, 0.90 and 0.99) and different values of c and k (see Fig. 2). When the likelihood that pirated software will be functional is low (ϕ =0.60), then the expected utility of consumers from pirating is low. In this case, TSImax dominates TSIV max for all values of c and k (see Fig. 2a). The BSA will choose a policy η⁎⁎⁎ =0, i.e. it is non-optimal for the BSA to expend any cost to prevent piracy. When the likelihood that the pirated software will be functional, is medium (ϕ =0.90), then for high I values of k and low values of c, TSIV max dominates TSmax (see Fig. 2b). The BSA chooses a high level of policy choice η⁎⁎⁎ =ηIV in this region. Consumers earn zero surplus for this choice since the firm does not price aggressively knowing that the pirate cannot take a big chunk of the market because of the high η. Thus, although the quality of the product is higher consumers are left indifferent between buying and not buying. The consumers who buy from the firm are actually better off in terms of surplus when the BSA chooses a policy of η⁎⁎⁎= 0. When the likelihood that the pirated software will be functional is high (ϕ =0.99), then the BSA will choose a high level of policy η⁎⁎⁎ =ηIV for almost all values of c and k

ð57Þ

ð1−Þ2 2k

ð58Þ

Region II: η2 < η ≤ η4. TSII : = h

4η2 k 9kη−2ð1−Þ2

1 i − cη2 2

ð59Þ

Maximizing this, the optimal solution is ηII = 2ð1 + 2Þ ð1−Þð2 + Þ < kc < ð1−Þ4ð2 + Þ. TSIImax = 2½4k−c . 9k2 ð2 + Þ 4−2 ð3 + Þ

2 ð1−Þ 3k

if

. Consumer surplus is zero in this region. Region III: η4 < η  η5 = 2− 2k Hence the optimal policy is the same as in Proposition 3. _ Region IV: η5 < η < η. Consumer surplus is zero in this region. Hence the optimal policy is the same as in Proposition 3. The optimal policy choice, for each of the four regions are: i) ηI = 0, ii) ηII =

2ð1−Þ 3k

2ð1 + 2Þ if 4− < kc < ð1−Þð42 + Þ iii) ηIII = 2 ð3 + Þ

4 4 ηIII = if < kc < 2ð1 + 2Þ 2 4k + cð + 2Þ2 ð1−Þð + 2Þ ð2−Þð + 2Þ2   q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 pffiffiffi 8c 2 + 27k 2 2 y = c 4c + 27k + 3 3 k k

2− 2k

4 if kc < and ð2−Þð + 2Þ2  2 1 2 23 y−23 c iv) ηIV = where 6cky

Thus the only change in the optimal policy choice when the BSA does not maximize the surplus of legal consumers compared to the case where the BSA also maximizes the surplus of legal consumers is in region II.

Fig. 3. Demand for software i and j and their pirated versions.

J. Jaisingh / Decision Support Systems 46 (2009) 763–773

769

This assumption is made for tractability. If this assumption is not made, then the demand functions become very complex. This technical assumption means that the pirated software steal market only from their respective original versions. The game is played out as before: first, BSA chooses the policy, then the firms choose the qualities qi and qj. Finally firms and pirate choose prices to charge. The total market size is now a unit square. The demand for i, j and their pirated versions is shown in Fig. 3. The demand for pirated versions of i, j and that of firms i and j are: di =

xi ð1−zi Þ 2 

dj = Fig. 4. Response function qi(qj, η).

Running a simulation to check for optimal policy choice (as in Section 3), we again find that either TSImax or TSIV max dominates the other TSimax. The rest of the analysis is same as in Section 3. Thus for the case when the BSA maximizes the profit of the firm only, we still i find that TSImax or TSIV max dominates the other TSmax. Since these two surpluses are the same as in the case where the BSA also maximizes the profit of the legal consumers, the rest of the analysis is the same. 4. Competition

qi −x−pi if consumer buys from firm i ðqi −xÞ−pci −zη if consumer buys the pirated software i U= qj −ð1−x if consumer buys from firm j   Þ−pj > >  qj −ð1−xÞ −pcj −zη if consumer buys the pirated software > > : 0 if consumer does not buy

xi ð1−zi Þ 2    1−xj 1−zj dcj = ð1−x0 Þ− 2

dci = x0 −

ð11Þ ð12Þ ð13Þ

respectively. The profit functions of each firm are: 1 πl = pl dl − kl q2l ; 2

ð14Þ

where l = i or j. The profit function of the pirate is:

In this section, we look at quality choice by the firm in the presence of competition. Consider a software market in which there are two firms, i, and, j. The firms produce software (also indexed by i and j respectively) that is in direct competition with each other. Duopoly reflects the market situation in several software markets such as the market for mathematical packages, where Mathematica and Maple are the primary players and statistical software where SAS and STATA are in direct competition with each other. The two firms are located at the two end points of the Hotelling line (unit distance), i.e. they are horizontally differentiated. Consumers are uniformly distributed along the Hotelling line, and differ in their preference for features provided by the competing software. For example, the interface of the software could be one feature along which the products are horizontally differentiated. One firm could offer a GUI (graphical user interface) while the other could offer a command mode interface. Some consumers may value the GUI of one product while others may value the command mode interface of the other because of past experience and flexibility in analysis. A location on the Hotelling line thus represents the location of the ideal product for the consumer at that location. If the location of the firm's software is not at the consumer's ideal product location, then the consumer bears a cost — which is the fit cost or the transportation cost. The preference for features is modeled by the transportation cost, x, as in standard horizontal differentiation models. There exists a pirate who provides a pirated version of the competing software. Consumers choose whether to buy from one of the firms, or to buy the pirated version of one of the software. Let qi (qj) be the quality and pi (pj) be the price chosen by firm i (j) for its software. The pirate correspondingly sets a price pci (pcj) for the pirated versions of i (j). Consumer's utility is given as: 8 > > > > <

  1−xj 1−zj 2

ð10Þ

9 > > > > = > j> > > ; ð8Þ

Technical assumption:       1−qj ð1−Þ−η−pcj + pj qi ð1−Þ + η + pci −pi  +  qi −qj + pj −pi < < ð9Þ 2 ð1−Þ ð1−Þ

πc = pci dci + pcj dcj :

ð15Þ

Our solution procedure is the same as in the monopoly case. First, we calculate the optimal prices of the firms and the pirate and then substitute them in the profit function of the firms. The firms then calculate the optimal quality choice. The price charged by each firm is found to be increasing in its quality and decreasing in the quality of the other software. For brevity we leave out an exhaustive analysis of the duopoly case and instead concentrate on how the quality choice of one firm changes with the quality choice of the other firm and the policy choice of the BSA. The response function of how the optimal quality choice of a firm changes with the quality choice of the other firm and the policy choice of the BSA is plotted in Fig. 4. As seen in Fig. 4, the optimal quality choice qi increases with policy choice η, the higher the quality choice of the other firm qj and the lower the optimal qi Thus as opposed to the monopoly case, we find that optimal quality choice increases with the policy choice. When the firm decreases its quality choice qi, the total share of the demand for product i (both legal and pirated demand) decreases, and is picked up by firm j and its pirated version. Thus the threat of losing market share to firm j and its pirated version makes it non-optimal for firm i to choose a lower quality choice for a higher policy choice as found for some regions in the monopoly case. 5. Discussion In this paper our focus was on how innovation (quality) is affected by the policy choice of alliances such as the BSA when piracy is prevalent. We considered a market where the firm and the pirate share the demand for the product, and consumers differ in their ethical cost to pirate. 5.1. Managerial implications Implications for the software firm: • Given that the software firm can observe the policy choice by the BSA and also the quality of its own software, then, when the firm knows it has a superior product and the policy choice by the BSA is low, it should price aggressively to make it unprofitable for the

770

J. Jaisingh / Decision Support Systems 46 (2009) 763–773

pirate to exist in the market. Also, when the firm knows it doesn't have a high quality product and the BSA has chosen a high policy choice, it should not engage in an aggressive pricing strategy and leave a small segment of the market for the pirate. Implications for the BSA: • In a monopoly market, an increase in the policy variable (in the form of legal action against pirating users or educating users about the harms of piracy) could act as a disincentive for innovation. This happens because, in certain regions, the policy choice by the BSA and the quality choice by the firm become strategic substitutes. Thus an increase in the policy variable leads to a decrease in the quality choice by the firm. • If the likelihood that pirated software will be functional is low, then it is non-optimal for the BSA to expend any cost to reduce piracy. When the likelihood that the pirated software will be functional is medium, then a high level of policy choice is optimal only for high values of technology cost and low values of policy cost. When the likelihood that the pirated software will be functional is high, then, the BSA should almost always choose a high level of policy choice. • In a duopoly setting, as opposed to the monopoly case, a higher policy choice always provides an incentive for the firm to innovate. The threat of losing its market share to the other firm prevents the firm from reducing its quality choice. The following example could be used to get a better sense of the results. In countries like Vietnam where the actions taken by BSA to prevent piracy are low or don't have an impact, i.e. the policy variable is very low, we would expect that firms that develop software would price the pirate out of the market. In countries where the policy variable is at an intermediate level, increasing policy choice could have an undesirable impact in the sense that software developers have a lower incentive to develop higher quality software. This happens when the quality choice of the firm and the policy choice of the BSA are strategic substitutes. Finally, in countries where the policy variable is high, for example in the Scandinavian countries, the piracy policy will work as expected. That is, one would find that software companies in these countries will have a higher incentive to develop better quality software as the policy variable is increased. However, note that the above results will primarily apply if the software firm is a monopoly in the market. If there is competition among software firms, a higher policy choice always provides an incentive for the firm to innovate because of the threat of losing market share to the competitor. 5.2. Limitations and future research The results however, are subject to the limitations of the model. One limitation of the model is that we assume that all consumers value software equally. This, for obvious reasons is not realistic. One consequence of this assumption is that we find threshold type pricing policies. Another consequence of the above assumption is that the market is always covered. A direction for future research could be to model heterogeneity in valuations. To keep things tractable, one will have to sacrifice on heterogeneity in terms of ethical propensity to pirate, since otherwise the consumer's type would be two dimensional. In this paper we do not consider the protection strategies by the firm. Future research can look at protection as a strategic choice by the firm. We assume a functional form for the ethical cost to pirate. Our choice is motivated by keeping the model tractable and using a functional form where the cost increases with the policy choice and increases more for the consumer who has a greater propensity not to pirate. The ethical cost function can also be a function of price, i.e. the consumers can bear a higher cost for pirating more expensive software. Our framework cannot be extended to this ethical cost function because it leads to corner solutions in the final stage. Future

research can try out this alternate ethical cost function in an alternate framework. We assume that the membership fees that the firm pays to the BSA to be zero. Given that the BSA enjoys economy of scale in fighting piracy, the membership fees will be a very small percentage of the total cost to fight piracy. Hence the results will not change qualitatively by assuming a positive membership fees. We have ignored the cost to the counterfeiter — penalty for being caught and cost of copying equipment. The nature of the cost to the counterfeiters is such that it is a fixed cost. It is possible to consider all these fixed costs, but they will not change the results qualitatively. To reduce the notational complexity of the paper we chose not to include these fixed costs in the analysis. Another direction for future research will be to look at the impact of multiple pirates in the market and considering the competitive interactions between them. In the paper we have assumed that the interests of the firm are aligned with that of the incentives of the BSA. Future research can consider modeling a social planner maximizing the welfare of the entire industry where each firm may have different incentives with respect to piracy. 5.3. Conclusions This paper helps to provide a better understanding of how innovation in the presence of piracy is affected by the policy choice of alliances such as the Business Software Alliance (BSA). Counterintuitive results, such as the one that increasing the perceived cost to using pirated software for end-users may in some cases lead to a decrease in product quality, have significant policy implications. In the light of these results there is a need for longitudinal studies that look at the long term impact of piracy policies on innovation at software firms.

Appendix A Proof of Proposition 1. The consumers should get a non-negative surplus, i.e., p ≤ q. The demand for the firm should be less than one, i.e., p ≥ pc + (1−ϕ). The firm maximizes her profit subject to these two constraints. The Lagrangian for the profit maximization problem for the firm is:    p−pc + q−q 1 − kq2 −λ1 ð p−qÞ−λ2 ½pc + qð1−Þ−p Lðp; λ1 ; λ2 Þ = p 1− η 2 ð16Þ The pirate maximizes her profit (Eq. (5)) wrt pc. The first order conditions FOC are: @L 2p−pc −qð1−Þ = 1− −λ1 + λ2 = 0 @p η

ð17Þ

@L @L = q−p  0; λ1  0 and λ1 = 0 @λ1 @λ1

ð18Þ

@L @L = p−pc −qð1−Þ  0; λ2  0 and λ2 = 0 @λ2 @λ2

ð19Þ

@πc p−2pc −qð1−Þ =0 = η @pc

ð20Þ

Case 1. λ1 = 0, and λ2 = 0. From the FOC, we obtain the reaction functions for the firm and the pirate: pðpc Þ =

qð1−Þ + η + pc 2

ð21Þ

pc ðpÞ =

p−qð1−Þ 2

ð22Þ

J. Jaisingh / Decision Support Systems 46 (2009) 763–773

a) λ1 = 0 and λ2 = 0: Solving Eqs. (29) (30) and (31)

Solving Eqs. (21) and (22), p =

qð1−Þ + 2η 3

4ð1−Þη i q  = h 9kη−2ð1−Þ2

ð23Þ

η−qð1−Þ pc = 3

ð24Þ

6η2 k i p  = h 9kη−2ð1−Þ2

3

2 1 6−η η 7 6 7 4 1 25 − η η

2η + qð1−Þ 3η

ð33Þ

h i η 3kη−2ð1−Þ2 h i p c = 9kη−2ð1−Þ2

ð25Þ

This matrix is negative definite and hence the second-order condition (SOC) for a maxima is satisfied. This solution is stable. Substituting Eqs. (24) and (23) in Eq. (3) we obtain: d =

ð32Þ

2 ð1−Þ2 . The optimal SOC is satisfied if 9kη − (1 − ϕ)2 N 0 i.e. ηN 9k price for the firm and the pirate are obtained by substituting Eq. (32) in Eqs. (23) and (24):

The Hessian matrix obtained from the FOC is 2

771

ð34Þ

The demand for the firm is obtained by substituting Eqs. (32), (33) and (34) in Eq. (3): 6ηk i d  = h 9kη−2ð1−Þ2

ð26Þ

ð35Þ

This is a valid solution only if the pc⁎ ≥ 0, d⁎ ≤ 1, and q − p⁎ ≥ 0. Now η pc⁎ ≥ 0 and d⁎ ≤ 1 gives us q  1− , while q − p⁎ ≥ 0 give us q  22η + .

Similarly optimal profit for firm is:

2η−qð2 + Þ q  . Case 2. λ1 N0, and λ2 = 0. p⁎ = q, pc = q 2η 2 , d = 1− 2η and λ1 = This is a solution provided q < 22η . SOC is satisfied since the number + of variables is equal to the number of active constraints.

h π f =

Case 4. λ1 N0, and λ2 N 0. Both constraints cannot bind at the same time — no solution exists. Proof of Proposition 2. i) Case 1: q < 22η +  Substituting optimal prices from Proposition 1 in Eq. (4) and maximizing wrt q, we get the FOC: q −kq = 0 η

2− tively. For this to be a valid solution,  +η kη < 22η + , i.e., ηN 2k .

η ii) Case 2: 22η +   q  1−Substituting Eqs. (23) and (24) in Eq. (4), and writing the Lagrangian for the maximization problem:

 Lðq; λ1 ; λ2 Þ =

    qð1−Þ + 2η 2η + qð1−Þ 1 2 η − kq −λ1 q− 3 3η 2 1−   ð28Þ 2η −λ2 −q 2+

i

ð36Þ

2ð1−Þ

SOC is satisfied since the number of variables is equal to the number of active constraints. ð1−Þ c) λ1 = 0 and λ2 N 0. Then p  = η and λ2 = 2½3kη−2 3ð2 + Þ , provided 2 ηN 3k ð1−Þ. The optimal prices, demand and profit are p  = 2η , p = η , d  = 2 and π  = 2η½2−kη2. Also η  2

ð27Þ

SOC is − η −k < 0. From Eq. (27) we get q  =  +η kη. Optimal prices of firm and pirate, profit of firm, and demand are η η  = p  =  +ηkη, p and d  = 2ð++2kη respecc = 2ð + kηÞ, π 2ð + kηÞ kηÞ

9kη−2ð1−Þ2

The price charged by the pirate pc⁎⁎ should be greater ≥0 for it 2 ð1−Þ2 . Consumers surplus to be realistic. This gives us η  3k obtained by each consumer buying legally should be greater 2 than zero, i.e. q⁎⁎ − p⁎⁎ ≥ 0, which gives us η  3k ð1−Þ. Additionally demand should be less than one, which also 2 2 gives us the restriction η  3k ð1−Þ2 . Thus 3k ð1−Þ2  η  2 ð 1− Þ satisfy all the above constraints. 3k η Þ2 −3kη b) λ1 N0 and λ2 = 0. Then q  = 1− and λ1 = 2ð1− 3ð1−Þ , provided 2 ð1−Þ2 . The optimal prices, demand and profit are η < 3k η½2ð1−Þ2 −kη respectively. p⁎⁎ = η, pc⁎⁎ = 0, d⁎⁎ = 1 and π 2 f =

Þ−η Case 3. λ1 = 0, and λ2 N 0. p⁎ = q(1 − ϕ), pc⁎ = 0, d⁎ = 1 and λ2 = qð1− 2η . η This is a solution provided qN 1− . SOC is satisfied since the number of variables is equal to the number of active constraints.

1−

4η2 k

2+

c

2+

2+

f

ð2 + Þ

k

for a non-negative firm profit. SOC is satisfied since the number of variables is equal to the number of active constraints. d) λ1 N0 and λ2 N 0. Both λ1 and λ2 cannot be positive at the same time for a positive ϕ. η iii) Case 3: qN 1− Substituting optimal prices from Proposition 1 in Eq. (4), and maximizing wrt to q, we get the FOC: ð37Þ

1−−kq = 0 SOC is −k < 0. From Eq. (37) we get q  =

1− . k

@L 2qð1−Þ2 + 4ηð1−Þ = −kq−λ1 + λ2 = 0 @q 9η

ð29Þ

Optimal prices of firm 2  = ð1−Þ , pc⁎⁎ = 0, and pirate, profit of firm and demand are p k Þ2 π  = ð1− , and d⁎⁎ = 1 respectively. For this to be a valid solution 2k η ð1−Þ2 1− N i.e. η < . 1− k k Making a realistic restriction that N 13, the following relationship is true:

@L η @L −q  0; λ1  0 and = λ1 = 0 @λ1 1− @λ1

ð30Þ

0<

@L 2η @L  0; λ2  0 and = q− λ2 = 0 @λ2 2+ @λ2

ð31Þ

Þ When η < ð1− and ηN 2− , there are two potential solutions and 2k k the firm will pick the one which gives the greater profit.

FOC are:

2 ð1−Þ2 2 ð2−Þ 2 ð1−Þ2 < ð1−Þ < < < 3k 3k 2k k k 2

772

J. Jaisingh / Decision Support Systems 46 (2009) 763–773 2

2

Þ Þ Range 1: 0  η < 2ð1− . The relevant profits to compare are ð1− 3k 2k 2 η½2ð1−Þ −kη (from Case 3) and (from Case 2b). For this range 2ð1−Þ2 2 η½2ð1−Þ −kη η½2ð1−Þ2 −kη is increasing in η. The value of at of η, 2 2

η=

2ð1−Þ 2ð1−Þ2 Þ2 is 4ð1− 3k 9k

2ð1−Þ

2

Þ < ð1− . Thus the firm will choose the 2k

strategy (quality) specified in Case 3. 2ð1−Þ2 3k

Range 2:

2

Þ  η < ð1− . The relevant profits to compare are k 2

2

4η k 4η k (from Case 2a). N ½9kη−2ð1−Þ2  ½9kη−2ð1−Þ2  0:82ð1−Þ2 if ηN . Thus the firm will choose the strategy k

(from Case 3) and ð1−Þ2 2k

ð1−Þ2 2k

(quality) specified in Case 3, if η 

0:82ð1−Þ2 k

and choose

2 ηN 0:82ðk1−Þ .

quality specified in Case 2a, if ð1−Þ2 Þ  η  2ð1− . The only possible solution is the one k 3k specified in Case 2a. Þ Range 4: 2ð1− < η  2− . The only possible solution is the one 3k 2k specified in Case 2c. Range 5: 2− < η  2k. The relevant profits to compare are 2ð η+ kηÞ (from 2k Case 1) and 2η½2−kη2 (from Case 2c). In this range 2ð η+ kηÞ N ð2 + Þ 2η½2−kη . Thus the firm will choose the strategy (quality) ð2 +  Þ2 specified in Case 1. Range 6: 2k < η < η. The only possible solution is the one specified in Case 1. b) Out of the four rows of the table in Proposition 2a, the signs of the first and bottom two rows are easy to verify. We calculate the signs of the derivatives for row two. Taking the derivative of the optimal quality, prices, profit and demand wrt η: Range 3:

@q  @η







ðq Þ 4 @p @q

q  =  ðq Þ @d ðq Þ 3 k−2 @p @q @q

Using the fact that @

=h

−8ð1−Þ

9kη−2ð1−Þ2

i2 < 0

ð39Þ

p ðq Þ @q@η

= 0,

  2 2 ðq Þ @p ðq Þ 8 @ d@η@q @q @q  =

2 @η @p ðq Þ @d ðq Þ 3 k−2 @q @q

ð47Þ

 2 The denominator and @p are positive and so the sign of @q @ 2 d < 0. determined by the sign of @η@q

@q @η

is

_ ii) For η5 < η ≤η, p⁎ = q and d  = 1− q 2η. On similar lines we can show that @q @η

@ 2 d @η@q

2 @p @q @p @d 2 k−2 @q @q 2 

=

d 2@@η@q

. The sign of ð Þ which is positive (2η2 ).

@q @η

is again determined by the sign of

Proof of Proposition 3. 2

Þ ) and consumer surplus Region I: 0 ≤ η ≤ η2. 2Profit of the firm (ð1− 2k ð1−Þ ð1−Þ (1− − = ) are independent of η. Total surplus: k k k

TSI =

ð1−Þ2 ð1−Þ 1 2 − cη + k 2 2k

ð48Þ

BSA will choose η = 0 and so TSImax =

ð40Þ

Now from the condition 23 ð1−Þ2  kη, 4(1 − ϕ)2 ≤ 6kη so the term in the square bracket of the numerator 4(1 − ϕ)2 − 9kη ≤ 6kη − 9kη = −3kη. That is 4(1 − ϕ)2 − 9kη < 0 Hence Eq. (40) becomes: @p  −6kη½−ve =h i2 N0 @η 9kη−2ð1−Þ2

ð41Þ

h i 2 @π  −4kη 4ð1−Þ −9kη = h i2 N0 @η 9kη−2ð1−Þ2

ð42Þ

ð1−Þ2 ð1−Þ 1−2 = + k 2k 2k

@d  −12kð1−Þ2 =h i2 < 0 @η 9kη−2ð1−Þ2

ð43Þ

Proof of Lemma 1. Rewriting profit function for the firm as: 1 πðqÞ = p ðqÞd ðqÞ− kq2 2

ð44Þ

Taking FOC and equating to 0: @d ðq Þ   d ðq Þ + p ðq Þ−kq  = 0 @q

ð49Þ

Region II: η2 < η ≤ η4. 8η2 kð1−Þð2 + Þ 1 2 TSII : = h i2 − cη 2 9kη−2ð1−Þ2

ð50Þ

Total surplus (TS) is decreasing in η, then TS is maximized II ð1−Þ2 . TS max is obtained by substituting at ηII ≈ 0:82 k 2 II 0:82 η = k ð1−Þ in (7). Region III: η4 < η  η5 = 2− . Consumer surplus is zero in this region. 2k Hence the BSA maximizes the profit of the firm, net the cost to implement a policy η subject to the constraint that η  2− . The Lagrangian for the profit maximization is: 2k Lðη; λ1 Þ =

@q

2 

ð46Þ

3

h i 2 @p  −6kη 4ð1−Þ −9kη = h i2 @η 9kη−2ð1−Þ2

@p ðq Þ



ðq Þ i) In the region (η2 < η ≤ η4), from Eqs. (23) and (26) @p @q = 1− 3 and @d ðq Þ 1− = . Substituting in Eq. (45) and solving for q⁎⁎, and using 3η @q     ðq Þ ðq Þ the fact that η @d @q = @p @q , we get

  2η½2−kη 1 2 2− − cη −λ1 η− 2 2 2k ð2 + Þ

ð51Þ

The FOC are: @L 4½1−kη = −cη−λ1 = 0 @η ð2 + Þ2

ð52Þ

@L 2− @L  0; λ1  0 and = −η + λ1 = 0 @λ1 2k @λ1

ð53Þ

i) λ1 N0. Solving Eqs. (52) and (53) we get ηIII = ð45Þ

λ1 =

4k−cð2−Þð + 2Þ 2kð + 2Þ2

2

. This

Optimal total surplus is

2− 2k

and

4 solution exists if < . ð2−Þð + 2Þ2 ½4k + cð2 −4Þð2−Þ III TSmax = 8ð2 + Þk2 c k

J. Jaisingh / Decision Support Systems 46 (2009) 763–773

ii) λ1 = 0. Solving Eqs. (52) and (53) we get ηIII = SOC is

− 4k 2 ð + 2Þ

4 . 4k + cð + 2Þ2

−c < 0, hence it is a maxima. This

solution must satisfy η4 < ηIII < η5. From ηIII < η5 we get 4 c N k ð2−Þð + 2Þ2

and from η4 < ηIII we get

c k

Thus this solution is feasible for the range c k

<

2ð1 + 2Þ . ð1−Þð + 2Þ2

TSIII max =

2ð1 + 2Þ . ð1−Þð + 2Þ2 4 < ð2−Þð + 2Þ2

<

8 ð2 + Þ2 ð4k + cð + 2Þ2 Þ

Region IV: η5 < η < η̅. Consumer surplus is zero in this region. Hence the BSA maximizes total surplus, equal to profit of the firm, net the cost to implement a policy η TSIV =

η 1 − cη2 2ð + kηÞ 2

ð54Þ

−cη = 0

ð55Þ

FOC is:  2

2ð + kηÞ

−k ð + kηÞ2

SOC (

ηIV =

−c < 0) is satisfied. Solving Eq. (55), we get:



2 1 2 23 y−23 c

6cky "  13 pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where y = c2 4c2 + 27k + 3 3 8c k+ 27kkÞ .

ð56Þ

References [1] S.H. Bae, J.P. Choi, A Model of Piracy, Michigan State University, 2003. [2] D.S. Banerjee, Software piracy: a strategic analysis and policy instruments, International Journal of Industrial Organization 21 (2003). [3] BSA, First BSA and IDC piracy study 2006, 2006, http://www.bsa.org/hongkong/ press/newsre leases/2006-Global-Piracy-Study.cfm. [4] BSA Enforcement. http://global.bsa.org/eupolicy/enforcement/.

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