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The Optimal Tax on Products with Pirates is Low Vesa Kanniainen CESifo GmbH Poschingerstr. 5 81679 Munich Germany Phone: Fax: E-mail: Web: +49 (0) 89 9224-1410 +49 (0) 89 9224-1409 [email protected] www.cesifo.de The Optimal Tax on Products with Pirates is Low Vesa Kanniainen University of Helsinki, CESifo and HECER¤ February 1, 2009 Abstract Pirate products reduce the monopoly power of innovators providing access to low price varieties. They, however, reduce the incentives for innovations, threaten the governments’ tax revenues and provide means for false signalling within social groups. The optimal tax on products with pirates is shown to be low for these reasons but should take account of the issue of how much to invest in detecting the illegal producers. A government who cares for people’s morality invests more in detection but collects more tax revenue even when it is willing to collect less. Keyords: pirates, tax revenue, optimal tax 1 Introduction Pirate products press consumer prices. From the point of view of low income consumers, this is welcome as it helps to control for the market power of monopoly producers. However, such a disciplinary impact weakens the pro…ts of the original innovator-producer destroying the incentives to innovate in the …rst place. Moreover, tax revenue is reduced if many people buy those products. There is also a morality issue. Many people refuse to buy pirate ¤ Department of Economics, P.O.Box 17, FI-00014 University of Helsinki, phone: +358919128725, fax: +358919128736, e-mail: vesa.kanniainen@helsinki.…. A grant from the Yrjö Jahnsson Foundation is gratefully acknowledged. 1 products for the reason that it is against their moral values. However, if the reputation of being a moral person has value in private interactions, immoral people may pretend to be moral ones creating a wrong signal of their type in the social contexts. For those reasons hence, pirate products are condemned in the society. The question is how a society should invest resources to combat against them. Second, should the tax on the product which is threatened by a pirate to have a special treatment? Though there are reasons to control for and catch the producers of pirates, the issue of imitation is more complex. After all, all current innovations are built on existing information and previous innovations. In a sense, and throughout the human history, all current economic activity is based on imitating the ideas of the chain of the previous generations. Nobody can truly innovate from scratch! All innovations are fundamentally incremental. This fact would favor a norm of free imitation as no innovator develops the full value-added without the past knowledge. However, there is a social cost in terms of reduced incentives to innovate if imitators can freely enter. In the case of literature and music, the societies have come to the conclusion that temporary copyright protection is a workable social norm. As imitation often improves the quality of products, imitation must be allowed in the end. However, copying without any improvement should - perhaps - not be accepted as a norm. The theory of optimal commodity taxation when products are the subject of potential imitation has not studied the tax e¤ects on the innovation incentive in the …rst place not to mention adjustment for the cost of detection the producers of the pirate products. As the theory of optimal commodity taxation appears to have missed this point, this paper will focus on this problem. The availability of pirates has an impact on the demand elasticity of the legal products and the optimal commodity tax theory captures this mechanism. This is not su¢cient, however. In democracies, government however often are short-sighted and do not pay attention to the policy e¤ects beyond their period they are in power. The innovation incentives are precisely of such type as innovations are typically not created in a short period. Innovations mature over long terms. Consequently, a revenue-maximizing tax rate tends to be too high. Such government may not have incentives to invest in a costly catching technology which reduces the resources available to them. A long-sighted government, in contrast, pays attention also to the economic e¤ects in the long run. In particular, it recognizes that the innovation intensity tends to have 2 a favorable impact on the tax revenue over the longer term. Therefore, the revenue-maximizing tax rate tends to be lower. Moreover, it desires to limit the imitation intensities recognizing that the imitation activities reduce the innovation e¤orts in the …rst place. Thus, such a government tends to invest resources in catching technologies. We also address the questions how morality a¤ects the optimal tax rate. Are morality and investment in detections substitutes or complements? 2 Model 2.1 Firms There is one innovator and one imitator. There are two stages, an innovation stage and the market stage. The model is solved by backward induction. Innovation e¤ort requires an investment but is risky. The success probability the choice variable of the innovator but leads to increasing cost, ¡ 1 ¢ is 2 0 Imitation is without cost requiring no investment. How2 imitator has to invest in a hiding e¤ort, at an increasing cost, ¡ever, ¢ the 1 2 This is because the imitator has to organize the sales through the 2 shadow markets. The success of the imitator is conditional on the success of the innovator. The variable is the government’t investment in a atching technology. If the innovation is successful and is not succesfully imitated, the innovator is entitled to monopoly pro…t in the market stage. The government is able to collect a turnover tax only on the legal product. If the imitation is successfully carried out, the …rms compete as a duopoly in the consumer market with duopoly pro…ts denoted by 1 and 2 The illegal …rm pays no tax.1 2.2 Innovation stage The innovator’s expected pro…t over the both stages is µ ¶ 1 ¦ = [ ¡( ¡ 1 ) ]¡ 2 1 2 In an earlier paper, Kanniainen and Stenbacka (2000) analyzed innovations and legal imitation. They abstracted from illegal pirates and the issue of optimal taxation. 3 where 1 is the pro…t of the innovation in the market stage. His …rst-order condition for the innovation e¤ort, the reaction function ¡( = ¡ 1 ) The imitator’s pro…t is conditional on the innovator’s success and is µ ¶ 1 2 ¦ = 2 ¡ 2 where 2 is the imitator’s pro…t in the market stage. His …rst-order condition for the hiding e¤ort, the reaction function 2 = Thus, the catching probability operates as a discipline for the imitator. Natural conditions introduced are ·1 ·1 Nash equilibrium in the innovation stage The innovation e¤ort is given by = = ¡( ¡ +( ¡ 1 1 ) ) 2 2 The imitator’s investment in hiding is given by 2 = 2 = ¡ +( 1 ) 2 Taking the derivative, = = [ 2 ¡ +( ¡ 1 2 4 ) [ ]2 0 1 2 ]¡ 2 The positive sign is guarenteed when = is su¢ciently high. Moreover, 2 0 1 The innovator is hence sensitive to the imitator’s success. The greater is the monopoly pro…t, and the higher is the duopoly pro…t of the competitor, 1 the more is he willing to invest. The government policy ( ) a¤ects both producers. Interestingly, there are two opposite e¤ect from on the innovator. Why? With perfect copyright protection, the innovator has access to monopoly pro…t choosing the innovation e¤ort as µ ¶ 1 2 ¦ = ¡ 2 or = ¡( ¡ 1 ) A guaranteed monopoly thus provides stronger innovation incentive. The comparative statics on is straightforward. 2.3 Consumers2 People di¤er in their attitudes towards illegal products. Some may …nd it valuable to misrepresent their hidden preferences for a confused signal to their co-citizens. There are two types of consumers. An -type has moral preferences; a -type does not.3 The mass of -type consumers is in the 2 The idea of consumer signalling introduced here is discussed in another paper by Glazer, Kanniainen and Poutvaara (2008) but in another context. 3 The origin of ethical preference lies beyond our scope. A natural source is that the preferences are created by evolutionary mechanisms among human beings becoming integrated into a social contract, cf. Binmore (1998). It is appropriate to think that the ability to commit to a social norm and the option to participate in a boycott develop like a social meme introduced by Dawkins (1976) and elaborated by Blackmore (1999). 5 model; the mass of -type consumers is scaled to 1.4 Each consumer buys at most one unit of the good. In the two groups, consumers are indexed in decreasing order on [0 ] and [0 1] with respect to their basic willingness to pay for the product. Consumers = 0 and = 0 have the highest basic willingness to pay for the product, say in each group; consumers = and = 1 have zero willingness to pay for it. The willingness to pay by the remaining consumers is uniformly distributed on (0 1) in both groups. To illustrate, and ignoring moral and reputational e¤ects for the moment, the utility from consumption by consumers, say and are given by indirect utility functions = ( ¡ ) ¡ and = (1 ¡ ) ¡ where is the market price. The products become di¤erentiated if one product is a pirate while the other is a legal one. In the social context, inviduals may view it important to be considered moral so as to avoid exclusion from particular social groups, loss of friendship, and even barriers in the marriage market. We let 0 denote the cost imposed on a consumer who buys a pirate product. This can be thought of as a social pressure. 2.4 Equilibrium We consider now the case where one of the …rms is succesful in innovating, thereby incurring an innovation cost, while the other imitates. The products of the …rms, though perfect substitutes in consumption, di¤er with di¤erent images. Some consumers will refuse to buy the pirate product, of the imitating …rm say , and consider buying only from the …rm which innovates, say . More speci…cally, the high-moral consumers abstain from buying at …rm and buy only at …rm . Since in equilibrium not all may buy, we denote the number of active high-moral buyers by . The number of lowmoral consumers who stay at …rm is denoted by Some of the low types, however, buy the legal product at the …rm in order to (falsely) signal high morality. Their number is denoted by . A tax is collected on the legal product but not on the illegal one. We denote the resulting producer price at the legal …rm by and the producer price at the non-ethical pirate producer by . The resulting market equilibrium has the following structure. With the tax rate on legal products 4 The population hence consists of a mixture of individuals of homo moralis and homo oeconomicus types. 6 and from the de…nition of the marginal moral consumer the equilibrium price at …rm satis…es (1 ¡ )= 5 + , we know that (1) where (1 ¡ ) is the willingness to pay by the marginal moral consumer and + is the consumer price of the legal product. Thus, the producer price is = (1 ¡ ) ¡ We can of course have a market equilibrium where no low-moral consumer buys at …rm . However, to make the analysis interesting, we assume that the bene…t from signalling is su¢cient so that some, i.e. 0 do. The marginal low-moral consumer must be indi¤erent between the two markets. Thus, consumer prices must satisfy (1 ¡ where, to recall, Therefore, )¡( + ) = (1 ¡ )¡ ¡ (2) is the social pressure when a consumer buys at …rm L. Lemma 1. The di¤erence in the consumer prises arises from the cost of social pressure adjusted for the commodity tax, ¡ = ¡ 6 (3) There is another marginal low-moral consumer (with an index = + ) who is indi¤erent between buying at …rm or buying nothing. Thus, his net utility is (1 ¡ ¡ )¡ = (4) 5 Therefore, not all moral consumers buy. They all buy only if = , making + = 0 Otherwise, 6 We notice that all low-moral types are indi¤erent between the two markets as the social cost of pressure just matches the price di¤erence adjusted fo the tax rate. Each …rm chooses its output knowing the consumers’ behavior. 7 2.5 Equilibrium in the Market Stage In the Cournot model, …rms decide on their outputs allowing the prices to adjust.7 We denote the outputs of the innovator and the imitator by and Then, = + = (5) Thus, the number of active moral buyers is = ¡ To solve for the prices, we …rst determine the number of signalling consumers. Using ¡ = ¡ (1 ¡ ) ¡ ¡ (1 ¡ ¡ )+ = ¡ or (1 ¡ + ) ¡ (1 ¡ ¡ ) = 0 we have that ¡ + (1 + ) + =0 This gives for the number of signalling consumers (noticing = ¡ 1+ = ) (6) With the producer price = (1 ¡ )¡ with ¡ + 1+ = = 7 The behavior of …rms in duopolistic markets has been subject to some debate, cf. Kreps and Sheinkman (1983). Güth (1993) shows how quantity competition can be justi…ed without the complexities discussed by the earlier literature. 8 we obtain the producer prices = (1 ¡ = (1 ¡ )¡ + 1+ )¡ and ¡ + = = + 1+ (1 ¡ )¡ Production cost is taken to be zero. Then, the pro…ts after a successful innovation and imitation are = · = = · = + 1+ ¡ ¡ + 1+ ¡ ¡ (7) ¸ (8) (9) ¸ Reaction functions are, + 1+ = ¡ = ¡ ¡2 µ ¡ ¡ + 1+ ¡ + 1+ = ¡ = ¡ ¡2 1+ 1+ ¶ =0 1 1+ ¡ ¡ ¡ 1 1+ 1+ =0 To solve for the Nash-Cournot equilibrium, the …rst of those conditions gives: µ ¶ 2 = 1¡ ¡ 1+ 1+ 9 or µ 1+ = 2 1+ = 2 1¡ ¡ 2 1+ ¡ ¶ ¡ (1 + ) 2 1+ 2 The second gives 2 = 1+ ¡ ¡ 1+ (1 + )( ¡ ) = ¡ 2 2 Inserting, 1+ (1 + )( ¡ ) 1+ ¡ + ¡ 2 4 4 2 (1 + ) ( + ¡ 2 ) = 3 = gives = (1 + ) ( + ¡ 2 ) 3 = (1 + )( ¡ ) ¡ 2 2 Similarly, (1 + )( ¡ ) ¡ 2 2 · ¸ ¡2 + = (1 + ) 3 = We then obtain the solutions for the outputs: Proposition 1. The Cournot-Nash equilibrium is given by the market shares (1 + )( ¡ 2 + ) = 3 (1 + )( + ¡ 2 ) = 3 10 Prices Then, the total output in the market will be + = (1 + )( ¡ 2 + ) (1 + )( + ¡ 2 ) + 3 3 (1 + ) [2 ¡ 2 ¡ ] 3 Then prices can be solved, = = = = Price e¤ects of the tax + 1+ +2¡2 3 (1 ¡ )¡ + )¡ 1+ +2+ ¡3 3 (1 ¡ Taking derivatives we …nd = ¡2 3 = 1 3 Thus, we …nd that a tax policy has a greater impact on the price of the legal producer than on the illegal producer. For some reason thus, Lemma 1. Pirates are more isolated from the tax e¤ect than are the legal products. Market pro…ts As = = we obtain = = (1 + )( + ¡ 2 ) ( + 2 ¡ 2 ) 9 11 = = (1 + )( ¡ 2 + ) ( + 2 + ¡ 3 ) 9 Morality and signalling The amount of false signalling is given by ¡ 1+ = ¡ (1 ¡ 2 ¡ ) = (10) (11) Propositio 1 Societies with a large moral population have less those immoral who send a false signal of being moral. The intuition for this unexpected result has to be worked out. Moreover, Proposition 2 Societies with a high tax rate have less those immoral who send a false signal of being moral. Proof. Solving µ ¶ 1 = [(1 ¡ ) + (1 + 2 ) ¡ (2 ¡ )] 3 Thus, = ¡(2 ¡ ) 0 Proposition 3 Societies with a strong moral pressure have more those immoral who send a false signal of being moral. Proof. From above, = (1 + 2 ) 0 Monopoly pro…t Suppose that the imitator is caught for producing illegal private products (which are destroyed). The innovator then is the sole producer in the market. Both types of consumers must visit him. In both consumer classes, there is a marginal one who is indi¤erent between buying at the market price and not bying.8 We denote the marginal moral consumer by and the marginal immoral consumer by Then, the willingness to pay by the marginal moral consumer must satisfy (1 ¡ 8 )= The demand curve becomes now ‡atter. 12 + (12) This gives 1¡ + = µ 1¡ = ¶ + The willingness to pay by the marginal non-moral consumer is (1 ¡ )= + Total output then is = + = ¡ or Therefore, it holds (1 ¡ ( or ¡ )) = = (1 ¡ ( Inserting (1 ¡ à = µ 1¡ = ¡ + )) ¡ µ ¡ 1¡ 1+ ¡ + (1 + ) 1+ ! ¶¶ )¡ The monopoly pro…t is = = à 1¡ 1+ 13 ¡ (1 + ) 1+ ! Taking the derivative, we obtain the pro…t-maximizing monopoly output à ! (1 + ) = 1¡ ¡ ¡ =0 1+ 1+ 1+ 1¡ 1+ ¡ (1 + ) 1+ (1 + ) ¡ 2 ¡ = 0 1+ ¡ (1 + ) = 0 (1 + ) ¡ (1 + ) = Recalling the monopoly price = = à µ 2 1¡ 1+ ¡ 1 1 1+ ¡ 2 2 1+ The monopoly pro…t thus is (1 + ) ¶ 1+ ! = h i (1 + ) ¡ (1 + ) µ 1 1 1 + ¶ = ¡ 2 2 2 1+ µ ¶2 1+ = (1 + ) 1¡ 4 1+ It is expexted that However, if the moral e¤ect proaces the monopoly pro…t. 3 is strong enough, the duopoly pro…t ap- Short-Sighted Government, Nash Equilibrium Governments are often short-sighted in democracies. To examine the e¤ects of tax and detection policies on pirates, we consider …rst the case of a shortsighted government who does not care of the morality in the society. We 14 view the government hence as maximizing its expected tax revenue net of the detection cost, µ ¶ 1 2 [ ]= (1 ¡ ) ¡ 1 + 2 A short-sighted government maximizes T but does not take into account the impact of its tax policy on the innovation incentives. The resulting equilibrium is a Nash equilibrium. Recall the pro…ts in the market stage, 1 = (1 + )( + ¡ 2 ) ( + 2 ¡ 2 ) 9 µ ¶2 1+ = (1 + ) 1¡ 4 1+ The tax thus reduces the pro…ts of the innovator in both cases, 0 1 0 However, the investment in detection only operates as as pre-emptive measure on the imitator. A short-sighted government hence abstracts from it and does not care of long-term innovations either. Taking derivatives the optimal policy satis…es [ ] = = 1 ( 1 + (1 ¡ ) + + (1 ¡ ) )+ 1 µ + 1 (1 ¡ ) + (1 ¡ ) =0 ¶ and [ ] We hasten to state, =¡ 0 Lemma 2. The short-sighted government does not invest in detection. We plan to prove, Lemma 3. There is a positive optimal tax rate. 15 Proof. Two conditions are needed, (i) [ ] 0 (ii) [ ] is convace 2 [ ] 0 Note that (i) is satis…ed because the second term is zero at the origin. (ii) Now 2 = 1 (1 + )( + ¡ 2 ) ( + 2 ¡ 2 ) 9 = (1 + ) µ 4 Their derivatives are 1 ¶2 1+ 1¡ 1+ (1 + ) [¡2 ( + 2 ¡ 2 ) ¡ 2( + ¡ 2 )] 9 2(1 + ) = ¡ [2 + 2 + ¡ 4 ] 0 9 = = ¡ (1 + ) 2 The second derivatives are 2 µ 1 2 1¡ 2 = (1 + ) µ 2 1+ 1+ 1+ ¶ 0 8(1 + ) 9 = 2 ¶µ 1+ 1+ 1+ Su¢cient conditions for concavity 2 [ ] 1 2 0 2 ¶µ 1+ 1+ ¶ 0 are 2 1 + 2 + 2 0 To elaborate, () 1 2 + 1 2 8(1 + ) 2(1 + ) [2 + 2 + ¡ 4 ] + 9 9 = ¡4 ¡ 4 ¡ 2 + 16 = ¡ 16 Take an example, =1 = 0 5; then this expression is ¡8 ¡ 2 + 8 = ¡2 0 Similarly, 2 ( ) + 2 = ¡ (1 + ) 2 µ + (1 + ) = ¡1 + 2 Now, ¡1 + 2 3.1 0 if 2 1+ 1+ ¶µ 1+ 1¡ 1+ 1+ µ ¶µ ¶ 1+ 1+ 1+ ¶ 1+ 1+ 0 5 This complets the concavity analysis. Expected Tax Revenue With a short-sighted government, nothing is invested in catching, = 0 Then, it is optimal for the imitator to choose = 1 and the imitators pro…t becomes certain, ¦ = 0 With such a high imitation intensity, the 2 innovator’s e¤ort is given by ¡( = = ¡ 1 ) 1 which is the bene…t-cost ratio of a successful innovation. The expected revenue of a short-sighted government is [ ]= 3.2 2 1 Long-Sighted Government: Stackelberg We now consider a long-sighted government who chooses the tax rate and the catching probability of pirate producers to maximize its tax revenue but who understands that innovation incentives are important also for its tax revenue. We also assume that the government actually cares about the morality in the 17 society. This will be modelled by the idea that the government overprices every tax penny but underprices every penny which is invested in detection. Below, these e¤ects are introduced by parameters 1 1 2 1 The desire to have large tax revenue may arise from the desire to distribute income transfers or from the Leviathan motive. The government, however has to take into account the threat of the pirate products and a loss in tax revenue. It pays to invest part of tax revenue to catch the illegal production and so better control the tax revenue. The government operates as a Stackelberg leader, choosing simultaneusly the tax rate, and the investment in the catching technology, As the latter is subject of inceasing 2 the tax should be chosen optimally jointly with the investment cost 12 in the catching technology. The government’s objective now is µ ¶ 1 2 [ ] = 1[ (1 ¡ ) ]¡ 2 1 + 2 In the …rst-order conditions of the long-sighted government, we have some extra terms. [ ] µ = 1[ µ + + 1 (1 ¡ ) ¶ 1 ¡ = 0 [ ] = = 0 1 ·µ 1 + 1 + ¶ ¶ + 1 + + (1 ¡ ) µ (1 ¡ ) as the pro…ts are 1 = (1 + )( + ¡ 2 ) ( + 2 ¡ 2 ) 9 = (1 + ) 4 µ 18 1¡ 1+ 1+ ¶2 1 + + (1 ¡ ) ¡ ¶ ¸ ] ¡ 2 Rewriting the terms in the …rst-order conditions of the long-sighted government, the marginal increase tax in the expected tax revenue when the adjustment of the innovation and imitation incentives are appreciated, is µµ ¶ ¶ µ ¶ + (1 ¡ ) ¡ 1 + 1 µ ¶ µ ¶ = ( 1 + (1 ¡ ) ) ¡ ( ¡ 1 ) 0 ¡ ¢ ¡ ¢ This is negative as 0 0 and ( ¡ 1 ) 0 We have proved Proposition 5 A long-sighted government always chooses a lower tax rate than a short-sighted government. The intuition is that the government pays attention to the expact of tax on the innovation incentive in the …rst place. But how much to invest in detection? Reorganizing the terms in the …rst-order condition with respect to we obtain 1 = 1 ·µ ·µ 1 ¶ ( + 1 1 ¶ + + (1 ¡ ) µ ) ¡ (1 ¡ ) µ ¶ ¶ ¸ ¡ ( ¡ 1 ) ¸ 0 ³ ´ 0 That a long-sighted government chooses a posThis follows from itive detection rate is equivalent to the condition ³ ´ ³ ´ µ ¶ ( 1 + (1 ¡ ) ) ¡ ( ¡ 1 ) 1 = 2 In this case, the optimal size of depends negatively on . One implication is that with 0, the innovation intensity is increased. A long-sighted government invests in detection if µ ¶ µ ¶ ( 1 + (1 ¡ ) ) ¡ ( ¡ 1 ) 0 We can also prove, Proposition 6 A government who cares about morality of people invests more in detection. It also manages to collect more tax revenue. ³ ´ 1 Proof. The result follows from 12 19 4 Concluding remarks Pirates should be condemned as they reduce the innovation incentive in the …rst place. However, it is not possible to have a perfect shelter as increasing the catching probability means increasing costs. 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