Commercial Piracy and Intellectual Property Policy

Transcription

1 Commercal Pracy and Intellectual Proerty Polcy Ilkka Kema To cte ths verson: Ilkka Kema. Commercal Pracy and Intellectual Proerty Polcy. Journal of Economc Behavor and Organzaton, Elsever, 2008, 68 (1),.304. < /j.jebo >. <hal > HAL Id: hal htts://hal.archves-ouvertes.fr/hal Submtted on 3 Se 2011 HAL s a mult-dsclnary oen access archve for the deost and dsseaton of scentfc research documents, whether they are ublshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from ublc or rvate research centers. L archve ouverte lurdsclnare HAL, est destnée au déôt et à la dffuson de documents scentfques de nveau recherche, ublés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores ublcs ou rvés.

2 Acceted anuscrt Ttle: Commercal Pracy and Intellectual Proerty Polcy Author: Ilkka Kema PII: S (08) DOI: do: /j.jebo Reference: JEBO 2186 To aear n: Journal of Economc Behavor & Organzaton Receved date: Revsed date: Acceted date: Please cte ths artcle as: Kema, I., Commercal Pracy and Intellectual Proerty Polcy, Journal of Economc Behavor and Organzaton (2007), do: /j.jebo Ths s a PDF fle of an unedted manuscrt that has been acceted for ublcaton. As a servce to our customers we are rovdng ths early verson of the manuscrt. The manuscrt wll undergo coyedtng, tyesettng, and revew of the resultng roof before t s ublshed n ts fnal form. Please note that durng the roducton rocess errors may be dscovered whch could affect the content, and all legal dsclamers that aly to the journal ertan.

3 * Ttle Page (wth Full Author Detals) Commercal Pracy and Intellectual Proerty Polcy JEL codes and keywords: O34 (Intellectual Proerty rghts), L84 (Informaton and Internet Servces; Comuter Software), Pracy Corresondng author: Ilkka Kema Researcher Deartment of Economcs / RUESG P. O. Box 17 (Arkadankatu 7) Unversty of Helsnk FINLAND Tel. (offce): Tel. (moble): Fax: e-mal: lkka.kema@helsnk.f Acceted anuscrt I would lke to exress my grattude to Rune Stenbacka, Erkk Koskela, kko ustonen, and Juuso Välmäk for ther helful comments on the earler versons of ths aer. Page 1 of 48

4 Blnded anuscrt (NO Author Detals) Commercal Pracy and Intellectual Proerty Polcy Abstract I dscuss the cometton between a coyrght owner and several commercal rates who sell coes of the same nformaton good to consumers. I vew the ncreased rsk of a unshment that offerng a rate coy to a consumer causes as an advertsng cost whose value s chosen by the government. The structure of the market for rate coes s affected also by fxed costs that are caused by unshments or DR systems. I resent a systematc analyss of the effects of these olcy varables and the qualty of rate coes on the market for the consdered nformaton good. 1. Introducton The dstrbuton of llegal rate coes of nformaton goods mght have a varety of motves. Such coes are dstrbuted on the one hand by the members of eer-to-eer networks, who delver dgtal goods on the Internet wthout monetary comensaton Acceted anuscrt and who are motvated by, for examle, a feelng of dentfcaton wth the other network members, and on the other hand by commercal rates who are, more conventonally from the ersectve of the economst, motvated by the revenue that results from ther actvtes. Somewhat less obvously, the consumers of an Page 2 of 48

5 2 nformaton good mght also form a club, whch buys a sngle coy of an nformaton good, roduces coes of t, and dstrbutes one of them to each club member. 1 Gven that both commercal and non-commercal forms of racy are llegal, there s no obvous way of estmatng the extent to whch rate coes of nformaton goods are sold rather than dstrbuted for free or the effects of commercal racy on the rofts of coyrght owners. Nevertheless, the Internatonal Federaton of the Phonograhc Industry (IFPI) has estmated that aroxmately 37% of all the [musc] CDs that were urchased n 2005 globally were rate coes. 2 However, n the case of the software ndustry, t s more dffcult to fnd estmates of the revalence of commercal racy. The Busness Software Allance (BSA) ublshes yearly a racy study that contans estmates for the racy rate (.e. the rato of the number of rated software unts to the total number of nstalled software unts) for dfferent countres of the world, as well as for dfferent regons of the world. For examle, accordng to the BSA the worldwde racy rate was 35% n However, such estmates do not make a 1 For the roducer of the frst coy of the nformaton good, ths ractce resembles a stuaton n whch the club members buy a sngle coy of an nformaton good and use t successvely. The latter ractce s legal when the consdered roduct s, for examle, a book, a journal, or a vdeo tae, and n both cases, the roducer of the frst Acceted anuscrt coy mght be able to arorate ndrectly a art of the value that the good has for ts consumers. See Varan (2000). 2 Internatonal Federaton of the Phonograhc Industry (IFPI), The Recordng Industry 2006 Pracy reort,. 4. Avalable at htt:// accessed on October 29, Thrd Annual BSA and IDC Global Software Pracy Study,. 13. Avalable at Page 3 of 48

6 3 dstncton between commercal and non-commercal forms of racy. 4 Nevertheless, the other surveys of the BSA suggest that both the commercal and non-commercal forms of software racy are of consderable economc sgnfcance. 5 htt:// %20Offcal%20Verson.df, accessed on Arl 13, Ths s because the estmates of the BSA have been calculated from an estmate of the total number of nstalled software unts, whch s based on the number of the sold hardware unts and surveys concernng ther average software load, and an estmate of the number of the sold software unts, whch s based on nformaton concernng the market revenues of software vendors and software rcng. See Thrd Annual BSA and IDC Global Software Pracy Study, In one of such surveys, the BSA has nvestgated the atttudes of the onlne consumers from sx dfferent countres towards sam, commercal emals that they have receved wthout requestng or sgnng u for them (BSA, Consumer Atttudes Toward Sam n Sx Countres, 2004, avalable at htt:// ed=20654, accessed on Arl 13, 2007). In each country, more than 80% of the resondents stated that they had receved sam that was concerned wth comuter Acceted anuscrt software (bd.,. 6), and 27% reorted that, n the roduct category comuter software, they had urchased an tem or taken advantage of an offer that was suggested to them n sam (bd.,. 12). Only 31% of the resondents stated that they agreed wth the statement that they would never buy commercal software usng ths method because t s most lkely unlcensed and llegal (bd.,. 16). Page 4 of 48

7 4 There s a relatvely large economc lterature on end-user coyng. 6 Dyut S. Banerjee has recently ut forward several closely related models of the cometton between a monoolst (.e., the coyrght owner) and a sngle commercal rate (2003, 2006a, 2006b), but t nevertheless seems that untl now economsts have gven much less attenton to commercal racy than to end-user coyng. 7 Below I shall ut forward a model of the cometton between the coyrght owner and several commercal rates, to whom I shall refer as bootleggers. The roducton costs of rate coes are low, and n the case of the rate coes that are dstrbuted n an electrc form va the Internet they are almost zero. Accordngly, f ntellectual roerty rghts are not enforced, the rces of rate coes can be exected to fall to zero va Bertrand cometton. However, when bootleggers are n danger of beng unshed for ther actvtes, t may be costly for them to nform otental consumers of ther roducts, snce ths may ncrease the rsk of beng caught and recevng a unshment. For examle, f an llegally oeratng Internet ste that offers rate coes of software roducts for sale nforms ts otental customers by sendng e-mal messages to randomly chosen addresses, the rsk of a unshment s ncreased by each message. In ths case the exected cost from a unshment s analogous to an advertsng cost, whch exlans the ostve rce of rated nformaton goods. Acceted anuscrt Informaton goods can be rotected not only by coyrght and other ntellectual roerty rghts but also by dgtal rghts management (DR) systems. Dgtal rghts management tools can, broadly seakng, be dvded nto crytograhy (.e. the 6 7 For a survey, see Petz and Waelbroeck (2006a). See, however, Yao (2005), whch dscusses counterfetng on a more general level, wthout restrctng attenton to counterfeted nformaton goods. Page 5 of 48

8 5 dstrbuton of nformaton goods n an enchered format) and watermarkng (.e. embeddng nformaton nto a dgtal roduct n such a way that each coy of the good becomes dfferent). 8 Watermarks can be used to track down the erson who has orgnally bought the legtmate coy of an nformaton good from whch the rate coes on the market have been roduced, whch makes t easer enalze commercal rates. 9 Clearly, a crytograhc devce causes a fxed cost for a commercal rate, but the costs of watermarkng can be ether fxed or varable: f a bootlegger removes the watermark, ts removal causes a fxed cost, but f she sells watermarked nformaton goods, the rsk caused by the watermark ncreases wth the number of the coes that she sells. Below I shall analyze the effects of DR systems and the olcy nstruments of the government on the rofts of the bootleggers and the coyrght owner. In the model the advertsng costs that are caused by an ncreased rsk of legal sanctons also kee the rce of rate coes ostve when there are several bootleggers on the market. Accordngly, the model dffers from revous work n the same feld n so far 8 For a survey, see Eskcoglu and Del (2001). Acceted anuscrt 9 Snce a large-scale commercal rate can be exected to be able to break down a crytograhc system, watermarkng seems to be the more relevant DR tool n the context of large-scale commercal racy (cf. Eskcoglu and Del (2001), ). See also Park and Scotchmer (2005), whch contans an analyss of DR and of the dfferent effects of the use of shared and ndeendent DR systems on end-user racy. Page 6 of 48

9 6 that t rovdes tools for analyzng also the market structure of the market for rate coes as well as ts effects on the market for legtmate coes The an Features of the odel The agents of the model that s consdered below are 1) a coyrght owner who sells coes of an nformaton roduct legally, 2) K otental bootleggers who wsh to sell llegtmate rate coes of the same good, and 3) a unt mass of consumers, whch s ndexed by 0,1. The bootleggers can nform the consumers of the avalablty of ther roducts by sendng them advertsements at random. As a aradgmatc examle of a stuaton of ths tye, one mght thnk of an llegally oeratng Internet ste that sends advertsements of rated software roducts to randomly chosen e-mal addresses. The sendng of an advertsement s assocated wth a cost b, whch should be nterreted as the ncrease n exected cost of unshment that sendng a sngle advertsement causes. 10 See Banerjee (2003, 2006a, 2006b), who dscusses several closely related models Acceted anuscrt of the cometton between the coyrght owner and a sngle rate. In these models the rate coes are of a lower qualty than legtmately bought coes, and ths stos the rce of the good from snkng to zero through Bertrand cometton. However, n these models the number of the rates s by assumton 1, and they do not address the queston why the rce of the rate coes does not snk to zero through a Bertrand cometton between the rates (rather than between them and the coyrght owner). Page 7 of 48

10 7 ore recsely, I shall assume that sendng an advertsement causes an ncrease n the rsk of beng caught, that the bootlegger receves a unshment G f she s caught, and that the bootleggers are rsk neutral. In ths case the advertsng cost b s gven by 11 (1) b G. Any combnaton of and G that corresonds to the same value of b has the same effects on the markets for both legtmate and rate coes, and accordngly I shall analyze below the effects of the choce of b (rather than of and G) for these markets. 12 I shall assume that the bootleggers cannot kee track of the consumers to whom they have already sent an advertsement. Rather, each of the bootleggers sends each of the advertsements wth the same robablty to each consumer. 13 Ths mles that a 11 For a dscusson of the econometrc roblem of actually constructng an ndex that measures the strength of legal software rotecton n a gven country, see Andrés (2006, ). 12 It s easy to see that the roblem of choosng and G so that the montorng costs of the government are mzed, gven the constrant (1), does not have a welldefned soluton: f an ncrease n montorng causes costs for the montorng authortes, but an ncrease n unshments (such as fnes) does not, any ncrease of G Acceted anuscrt and decrease n that kees (1) vald seems always benefcal from the ersectve of the government. Hence, t seems that a meanngful dscusson of the otmal choce of and G would requre a more general model. 13 In a more general model t could be assumed that the bootleggers would be able to target ther advertsements at the consumers who are lkely to buy a rate coy. Ths would change the demand functon (8), but t s easy to see that the analyss of the Page 8 of 48

11 8 bootlegger mght send to the same consumer several advertsements n whch the roduct s offered for sale at dfferent rces. Ths assumton s artcularly lausble n the context of trade on the Internet snce the otental customers of bootleggers mght have several e-mal addresses. In addton, f the consumers are dvded nto grous whose members nform each other of the advertsements that they have receved, a sngle advertsement mght reach ndvduals wth dfferent e-mal addresses, and n ths case a bootlegger cannot elate the ossblty that the same grou of consumers receves many advertsements from her. In ths case one should nterret b as the average cost of reachng a sngle consumer wth a sngle advertsement. Ths mles that the descrton of the cometton between the bootleggers resembles the classcal model of advertsng by Gerard R. Butters (1977). 14 Accordngly, n the current model there wll be rce dserson n the market for the market for rate coes for a fxed value of (.e. the results whch are summarzed n Prooston 1 below) would stll reman vald. However, the analyss of the otmzaton roblem of the coyrght owner would become essentally more comlcated. Acceted anuscrt 14 See also Trole (1989, ). The current model and Butters s model of advertsng can both be contrasted wth Varan s model of sales (Varan 1980), n whch the otental customers are dvded nto nformed and unnformed consumers, and the nformed consumers know the rces chosen by all the sellers, whereas the unnformed consumers know only the rce set by the seller from whom they make ther urchase. Page 9 of 48

12 9 advertsed roduct. 15 Clearly, ths result s qute lausble n the context of the llegal busness model of the bootlegger: snce the bootleggers do not necessarly know the rces set by ther comettors or the demand that they face, they have an ncentve to fnd the otmal rce level exermentally by secfyng dfferent rces n dfferent advertsements. Further, when eer-to-eer networks are vewed as a lmtng case of commercal racy, also the fact that dentcal nformaton goods are sold by rates and dstrbuted for free on eer-to-eer networks s seen to exemlfy rce dserson n the market for rate coes. However, the current model dffers from Butters s model n several essental resects. In the currently consdered model there s an olgooly n the market for rate coes, 16 and snce the advertsng costs have been meant to reresent the exected costs from a unshment for coyrght volaton, t wll be for smlcty assumed that the coyrght owner may advertse for free and that all consumers have the oton of buyng the roduct from her. In addton, snce I wsh to model a stuaton n whch only a art of the consumer ool refers buyng a rate coy to buyng a legtmate coy, unlke Butters I shall assume that the reservaton rces of the consumers dffer. I shall assume that the reservaton rce of the consumer (where 0 1) for a legtmate coy of the good s and that for a rate coy her reservaton rce s q, 15 ore recsely, there s rce dserson whenever there are more than one actve Acceted anuscrt bootleggers on the market. See Prooston 1 below. 16 Butters s for the most art concerned wth a model wth a large number of sellers wth no market ower. See, however, Aendx A (bd., ) for a short dscusson of an olgooly of N sellers. Page 10 of 48

13 10 where q 0,1 s a constant. 17 In Banerjee (2003,. 100), from whch I have borrowed ths notaton, t was assumed that q corresonds to the robablty wth whch a rate coy s oeratonal. However, here the arameter q reresents not only the fact that a rated roduct mght be techncally of a worse qualty than a legal one or not oeratonal at all, but also a consumer reference for legally bought coes for ethcal reasons because there mght be legal sanctons aganst usng (and not just aganst sellng) rate coes or because buyng a rate coy requres gvng credt card nformaton to crals. Dfferent values of q seem lausble n the dfferent alcatons of the model. One mght exect that, for examle, musc fles downloaded from a eer-to-eer network are exerenced by ther users to be of an almost dentcal qualty wth legally bought ones, and n ths case t seems lausble to assume that q 1. However, consderatons other than techncal qualty seem more relevant n the case of llegally bought software roducts, and ths motvates the assumton that such software roducts corresond to essentally lower values of q. Fnally, f one aled the current model to rate coes of other branded artcles rather than nformaton goods, they could be assocated wth even lower values of q. 17 Clearly, one could also assume that consumers dffer also wth resect to q and not Acceted anuscrt just wth resect to. For examle, one could follow Chen and Png (2003,. 110) by assug that some consumers reject rate coes on ethcal grounds whle others do not. The remarks made n footnote 13 above aly also to models of ths tye; our dscusson of the market for rate coes for each fxed value of can be generalzed n an obvous way to such models, but choosng the otmal would be essentally more comlcated n ther context. Page 11 of 48

14 11 In the model a coyrght owner and K otental bootleggers lay the followng fourstage leader-follower game. 18 1) The coyrght owner sets the rce of legtmate coes. 2) Each otental bootlegger decdes whether to enter the market and to ay a fxed cost F. 3) The bootleggers (f any) who have entered decde to send a certan number of the advertsements to randomly chosen consumers. The bootleggers are not constraned to offer the roduct at the same rce n dfferent advertsements. 4) The consumers choose whether to buy the roduct. If a consumer has not receved any advertsements, she wll buy the roduct from the coyrght owner f 0, and she wll buy nothng otherwse. If a consumer has receved at least one advertsement, and f the lowest rce suggested n the advertsements that she has receved s, she wll buy the roduct from the coyrght owner f 0, q. If ths s not the case, she wll buy the roduct from a bootlegger at rce f q 0. If nether of these condtons s vald, she wll not buy anythng. If the consumer buys the roduct from a bootlegger at rce and f there are several bootleggers who have offered the roduct at rce to her, she wll choose one of them at Acceted anuscrt random. 18 See Banerjee (2003), whch contans a dscusson of both a leader-follower game and a Bertrand game n the context of a model wth a sngle rate. Page 12 of 48

15 12 In ths game the am of the coyrght owner s to mze her roft, whch s smly equal to her revenue (2) R D where D s the demand functon of the coyrght owner, and each bootlegger 1,2,..., k ams at mzng the roft P R ba, where R s the revenue of the th bootlegger and A s the number of the advertsements that she sends. Obvously, n equlbrum the number of the bootleggers who actually enter the market wll be (3) k k R ba F when the number of the bootleggers s k Part 4) of the defnton of the consdered game mles that f the cheaest rce at whch a consumer can buy the consdered good from a bootlegger s, she wll buy t from the coyrght owner f and only f (4) and 1 q and from the bootlegger f and only f (5) and q. 1 q Clearly, (4) mles that there are consumers who are wllng to buy the roduct from a bootlegger for the rce only f 1 q q. Ths condton, whch Acceted anuscrt must be vald for all advertsements n equlbrum, s equvalent to (6) q. If the condton (6) s vald for the cheaest rce that s suggested n the advertsements that the consumer has receved, then t wll be the case that Page 13 of 48

16 13 f q, the consumer does not buy anythng, (7) f q 1 q, the consumer buys a rate coy for rce, f 1 q, the consumer buys a legtmate coy. I shall denote the demand functon of the rate coes by shall let x. ore recsely, I x denote the roorton of the consumers who would buy a rate coy at the rce f s the cheaest rce suggested to them n the advertsements that they have receved. Clearly, (7) mles that (8) x x, 1, 1 A q q q q xb, q 1 q q 1 q, q 1 Here the case n whch x x corresonds to the stuaton n whch there are A no consumers who would buy a legtmate coy f they are offered a rate coy at rce. In Secton 4 below t wll be seen that n equlbrum the choce of by the coyrght owner and the choces of the values of by the bootleggers wll be such that x x for all the advertsements that the bootleggers send. B In the next secton I deduce the equlbrum dstrbuton of the rces of rate coes for a gven value of and a corresondng number of bootleggers Acceted anuscrt k k wll be utlzed n the subsequent dscusson of the otmzaton roblem of the coyrght owner who chooses. Ths. However, as was exlaned n the ntroducton, relable estmates of the sze of the market for rate coes and of the rsk that bootleggers face of beng caught are not avalable, and accordngly t s also nterestng to study the stuaton n whch s exogenously gven. Ths stuaton can be thought of as a model of a case n whch the coyrght owner has no nformaton Page 14 of 48

17 14 concernng the bootleggers or bases her decsons on ncorrect estmates of the arameters that characterze the market for rate coes. 3. The arket for Prate Coes Below I shall assume that there are k bootleggers, labeled 1,..., k, who have entered the market. Analogous to the notaton used n Butters (. 469), I defne A Acceted anuscrt to be the measure of the advertsements sent by bootlegger at a rce smaller than or equal to, a to be the dervatve of A whenever t exsts, and A to be the total number of the advertsements. As t was exlaned n Secton 2 above, I assume that the bootleggers cannot kee track of the consumers to whom they have already sent an advertsement and that a bootlegger mght send several advertsements to the same consumer. If there are N consumers to whom the bootleggers send altogether advertsements at random, the robablty wth whch a consumer does not receve any advertsements s N N N where N s the number of the advertsements er consumer. Clearly, n the lmt n whch the number of the consumers aroaches nfnty, ths becomes 1 lmn 1 N N e. In the currently consdered case, the mass of the consumers has been normalzed to one, and the number of the advertsements that contan a rce not larger than s Page 15 of 48

18 15 denoted by A k 1. Hence, the robablty wth whch a consumer receves at least one of these advertsements s 1 ex k A. 1 It s easy to see that f k 1 (.e. f there s just one actve bootlegger on the market), she cannot have an ncentve to secfy dfferent rces n dfferent advertsements. If the sngle bootlegger sends A advertsements that all contan the rce, her roft wll be A (9) 1 1 k P x e ba. Hence, n ths case t s otmal for the bootlegger to advertse at the rce, whch mzes rce by x. For reasons whch wll become soon obvous, I shall denote ths, and I shall ut (10) r x. Further, one can also conclude from (9) that when k 1, the otmal number of the advertsements s (11) A ln x b ln r b. However, t s clear that when there are more than one actve bootleggers on the market, there must be rce dserson n the market for rate coes: f two Acceted anuscrt bootleggers secfed the same rce n all ther advertsements, each of them could ncrease her rofts by makng a suffcently small reducton n the rce. 19 By the 19 As was exlaned n Secton 2, the concluson that the rces of rate coes vary s qute lausble. Snce the bootleggers oerate on an llegal market concernng Page 16 of 48

19 16 same argument, t can be seen that there cannot be any set of advertsements wth a ostve measure that would all contan the same rce when there are at least two bootleggers. It s also clear that t can never be otmal for a bootlegger to secfy a rce that s larger than (.e. the rce whch s otmal n the absence of cometton). Hence, when k 2, the rces secfed by the bootleggers corresond to a contnuous dstrbuton functon A k R x Aj a d 1 (12) ex for some, so that the revenue of each bootlegger s. The roblem of mzng ths quantty can be solved by the standard tools of the analyss of varatons, and ts soluton s gven by the followng rooston. PROPOSITION 1. Suose that the rce of legtmate coes s, that there are k actve bootleggers on the market, and that ther strateges yeld a symmetrc Nash equlbrum relatve to. The largest rce suggested n the advertsements has the value for whch x s mal, and the smallest rce suggested s then detered by the condton (13) r b r 11 k 1 k Acceted anuscrt where r x and r x. When k 2, the number of the advertsements wth a rce not larger than, sent by a bootlegger s gven by whch t s dffcult to fnd detaled nformaton, they can be exected to try to receve such nformaton by secfyng dfferent rces n dfferent advertsements. Page 17 of 48

20 17 (14) A 1 x 1 x ln ln k 1 x k 1 r. Prooston 1 mles that when k 2 the total number of advertsements sent by each bootlegger s gven by (15) k 1 k r r ln ln ln A r b. k 1 r k 1 b k A comarson of ths result wth (11) shows that the total number of the advertsements that the bootleggers send s ndeendent of ther number k. However, accordng to Prooston 1 an ncrease n k shfts ther rce dstrbuton downwards. The condton (13) that deteres the value of smallest rce that occurs n the advertsements can be made more ntutve by observng that (12) and (14) together mly that the revenue of each bootlegger s (16) k R x ex Aj a d 1 1 k 1 Acceted anuscrt A r da da r d r e d r 1 e x d d A Ths result states that the revenue of each bootlegger s dentcal to the revenue that she would receve f she were the only bootlegger n the market and she secfed the rce n all her advertsements. As k ncreases, the ncreased cometton n the market for rate coes s shown by the fact that, n accordance wth (13), ths rce snks from the rce that would be otmal n the absence of the other bootleggers,., towards the cost of sendng a sngle advertsement, b. Combnng (13), (15), and (16), the revenue of the bootlegger can be also wrtten A (17) 1 1 k b R r e r r r b, r Page 18 of 48

21 18 so the roft of each bootlegger s gven by r (18) P R ba r b b ln. b These results have been roved wthout assug that the demand functon x for rate coes (.e., the robablty that a consumer buys a rate coy f the lowest rce suggested to her s ) s of the artcular form (8). However, a more detaled analyss of the comaratve statcs of the market for rate coes wll deend on the x. form of The otmal rce of a rate coy n the absence of other bootleggers, Acceted anuscrt, can be deduced from (8) by elementary means. Clearly, some of the consumers who receve an advertsement wth rce wll refer a legtmate coy to a rate coy f q 1 and x x, but there wll no such consumers when q 1 and x x value when q 2 A B. It s also clear that x A, and that x B A obtans ts largest obtan ts largest value when B q 2. Snce A B, t s ossble to dstngush between three cases: f A B q 1, the otmal rce s B ; f q 1 A B, the otmal rce s ; and f q 1, the otmal rce must be the corner soluton A A B q 1. Solvng for, these results receve the form (19) q q q 2, 2 1 2, q 1, 2 1 q 2 q 1 q 2, q 2, 1 q 2. Hence, Page 19 of 48

22 19 (20) q q q 2 1, 2 1 2, x 1 q, 2 1 q 2 q 1 q 2, 1 2, 1 q 2. Now t can be concluded from (15) that the total number of the advertsements s (21) 2 q b q q q q b q ln 4 1, 2 1 2, A ln q 1 1 qb, 2 1 q 2 q 1 q 2, ln 4, 1 2. The followng rooston, whch follows easly from these results together wth Prooston 1, summarzes the comaratve statcs of the market for rate coes for a fxed value of assumton and a fxed number of actve bootleggers k. The addtonal 1 q that s made n ths rooston means that some consumers refer buyng a legtmate coy to buyng a rate coy at (.e. the cheaest rce that rate coes have on the market). In the next secton t wll be seen that ths addtonal assumton s always vald when Acceted anuscrt s chosen otmally. PROPOSITION 2. Suose that there are k actve bootleggers, that ther strateges form a Nash equlbrum relatve to the rce of legtmate coes 1 q. (a) An ncrease n the rce, and that of legtmate coes ncreases the revenue of each bootlegger, the number of the advertsements, and the largest rce that rate coes have on the market. If k 2, an ncrease n of rate coes on the market. decreases the smallest rce Page 20 of 48

23 20 (b) An ncrease n the number of the bootleggers k does not affect the total number of the advertsements or the largest rce of rate coes, but t decreases, the revenue of each sngle bootlegger and her total revenue. (c) An ncrease of the advertsng cost b does not affect Acceted anuscrt, but t decreases the number of the advertsements, and when k 2, t shfts uwards. In ths case the roft P of each bootlegger receves ts mum value for a sngle value b E of b wthn the nterval b 0, r b b E, a decreasng functon of b when b be, and lm P lm P 0. b0 br. The roft P s an ncreasng functon of b when (d) An ncrease n the qualty arameter q ncreases the mum rce of rate coes, the number of the advertsements, and the rofts of the bootleggers. The effect of an ncrease of q on can be ether ostve or negatve, but t s always the case that q 1 2. The ntutve exlanaton of the last of these results s that an mrovement n the qualty of rate coes on the one hand allows the bootleggers to charge hgher rces, but on the other hand t also ntensfes cometton by ncreasng the number of advertsements and tendng to lower the rces of rate coes. Ths leads to the redcton that when the rates have a consderable rsk of recevng a unshment, an mrovement n the qualty of rate coes can be exected to ncrease ther rce dserson. The fact that 0 when k 2 has a smlar ntutve nterretaton. Part (c) of ths result, whch s llustrated by Fgure 1 below, s concerned wth the queston how the rofts of each bootlegger deend on the arameter b. Ths queston Page 21 of 48

24 21 s mortant for obvous reasons: when the number of the bootleggers on the market deends on ther rofts n accordance wth (3), t s n the nterest of the coyrght owner that ther rofts are low. Part (c) of ths result mles that when k 2, the rofts earned by the bootleggers are, for suffcently low values of b, ncreased by an ncrease of b, whch mght ncrease the ncentves to enter the market for rate coes. I formulate ths observaton as a searate corollary. [Place Fgure 1 aroxmately here] COROLLARY 1. (a) For each fxed value of the number of the bootleggers. (b) For each fxed value of an ncrease n F always has a non-ostve effect on an ncrease n b sometmes has a ostve and sometmes a negatve effect on the number of the bootleggers. 4. The market for legtmate coes In the current model the roft of the coyrght owner s equal wth her revenue. The characterzaton of the demand R D coyrght owner s easy when 21 q 2 q D of the. In ths case s accordng to (19) such that none of the consumers who have the oton to buy a rate coy wll choose a legtmate coy, and the demand of the coyrght owner s smly Acceted anuscrt A (22) D e 1. Here the total number of the advertsements A s gven by (21), so that (23) R 4b 1 q, 1 q 2. qb q 1, 2 1 q 2 q 1 q 2, Page 22 of 48

25 22 It s now mmedately observed that when R s gven by ether of these formulas, the coyrght owner can ncrease her rofts by lowerng so that these choces of cannot be otmal for her. It s essentally more comlcated to calculate n whch 21 q 2 q D n the more nterestng case. Clearly, n ths case the consumers for whom q wll buy a legtmate coy f and only f they do not 1 receve any advertsements. The demand from such consumers s D e 2b L 1 q ka1 (24) On the other hand, a consumer would buy a legtmate coy even f she could buy a rate coy for f q (25), q 1. Puttng, 1 the demand from such consumers (f any) s seen to be (26) DH 0,1 1 q 1 1 q. Fnally, the consumers for whom 1 q 1 q. Acceted anuscrt wll buy a legtmate coy f they do not receve an advertsement that contans a rce lower than 1 (27) ed q. The demand from such consumers s 1q ka1 1q 1q D e d. Here the ndexes L, ed, and H refer to low, medum, and hgh valuaton, resectvely. The total demand of the coyrght owner s the sum of the three demand comonents: (28) D D D D. H ed L Page 23 of 48

26 23 Unfortunately, t seems that one cannot exress the value of the ntegral that occurs n (27) n terms of elementary functons. The results n the rest of ths secton are based on a change of varables n ths ntegral. If one uts (29) z 1 q q, the demand comonent (30) D s seen to equal ed q Ded e dz 1 q 1 2 ka1 q z. q Usng (8), (13), and (14), the ntegrated functon can be exressed n the form ka1 q z x e q zxq z (31) b1 q 1 1 q 1 k 1 2 k k 1 2 f z 4 q z 1 z q where f z (32) 1 k 1 k 4 1 k z z 1 k k 1 b. Puttng these results together, the demand of the coyrght owner can now be exressed n the form 2b 1 D f z dz 1 q 1 q. 1 2 (33) These results make t easy to analyze the comaratve statcs of the roft of the Acceted anuscrt coyrght owner relatve to the varables b, q, and k when the rce set by the coyrght owner s fxed. PROPOSITION 3. Assume that the rce of legtmate coes s fxed, that there are k actve bootleggers, and that ther strateges form a symmetrc Nash equlbrum relatve to. Page 24 of 48

27 24 (a) The demand D of the coyrght owner s a decreasng functon of the qualty arameter q. (b) When 21 q 2 q, the demand D s ndeendent of the market structure n the market for rate coes, but when 21 q 2 q D s a decreasng functon of k., (c) An ncrease n the fxed cost F has a non-negatve effect on n the advertsng cost b ncreases D. An ncrease D f t does not affect the number of the bootleggers. If ths s not the case, an ncrease n b can also decrease Acceted anuscrt D. Accordng to art (c) of ths rooston, an ncrease n the unshments for racy can decrease the rofts of the coyrght owner also n the absence of network effects, whch would make the market enetraton of the roduct affect the value that t has for the consumers. Ths s because an ncrease n the advertsng cost mght ncrease the rofts of the bootleggers and the number of the bootleggers who enter the market. I now turn to the otmzaton roblem of the coyrght owner, whch s the roblem of mzng the revenue R D where D s gven by (33). It turns out that although the ntegral that occurs n (33) cannot be evaluated exlctly, the dervatve of the revenue exresson. R has a surrsngly smle PROPOSITION 4. The otmal rce of legtmate coes s never such that some rate coes would be so chea that there would be no consumers who refer buyng Page 25 of 48

28 25 a legtmate coy to buyng one of them. If there are rate coes on the market, s such that ths s not the case, and f (34) 2 2 dr 1 d 1 q 1 q. dr d Ths rooston mles that for each fxed value of k the dervatve has the value zero when (35), k where 1 q 2,k denotes the mum rce of rate coes for the gven and k. Further, together wth the fact that, 0 (see Prooston 2(a) above), ths mles that for each k there can be at most one value of However, the actual number of the rates accordance wth (3), and t s a non-decreasng functon of Prooston 2(b), of,k k for whch (35) s vald. Acceted anuscrt k s detered by n. Accordng to s a decreasng functon of k, and hence, the equlbrum value detered by (35) s a decreasng functon of k. These observatons yeld the followng characterzaton of the equlbrum rce of legtmate coes. COROLLARY 2. (a) Assume that there are K k otental bootleggers and that they do not have fxed costs. If there are rate coes on the market n equlbrum, the equlbrum value of s detered by the condton (35). (b) ore generally, when there are K otental bootleggers wth fxed costs F 0, there can be at most one value of for whch (35) s vald wth k k. The Page 26 of 48

29 26 otmal choce of s ether ths value (f t exsts) or the value that mzes R n k k 0,1,..., K 1, where, k denotes the largest rce for whch the number of the actve bootleggers stays below k:, k P ; k F. Fgure 2 llustrates the frst art of ths result. It shows the revenue curve R of the coyrght owner when the number k of the bootleggers s 0, 1, and 2, and the R when k. In the stuaton of the fgure the coyrght owner lmt of would have to lower to the value n order to revent the emergence of a market for rate coes, and n equlbrum she wll let all otental bootleggers enter. In the lmt n whch the number of the actve bootleggers aroaches nfnty the otmal rce of legtmate coes aroaches the value, ot shown n Fgure 2. [Place Fgure 2 aroxmately here] Fgure 3 reresents a stuaton n whch the otental bootleggers have fxed costs, but that s smlar n other resects. The dotted lnes show the revenue of the coyrght owner n the resence of 0, 1, and 2 bootleggers. The revenue curve of the coyrght owner (whch s shown as a sold lne) has a dscontnuty at the onts,1 and,2 Acceted anuscrt at whch the number of the bootleggers changes. Accordng to the latter art of the above corollary, the equlbrum value of s ether one of the onts of dscontnuty or the mum of the curve that corresonds to one of the values k 0, k 1,. Fgure 3 llustrates the former ossblty snce n t the otmal rce Page 27 of 48

30 27 s the largest rce that suffces to block the entry of the second bootlegger (.e.,2 ). [Place Fgure 3 aroxmately here] When the number of the bootleggers k s fxed, accordng to Prooston 3(c) an ncrease of b shfts the revenue curve R uwards, and hence, t must corresond to an ncrease of the revenue of the coyrght owner. However, f b s suffcently small and k 2, an ncrease of b wll ncrease the rofts of the bootleggers. Ths decreases the values,2,,3, at whch the bootleggers enter, and f one of these values s the equlbrum value of, ths decreases of the revenue of the coyrght owner n equlbrum. The followng corollary follows mmedately from these observatons and Prooston 3. COROLLARY 3. Suose that the strateges of the coyrght owner and the bootleggers form a Nash equlbrum of the game 1) - 4). a) The revenue of the coyrght owner s a non-decreasng functon of the fxed cost F and a decreasng functon of q. Acceted anuscrt b) The revenue of the coyrght owner s sometmes ncreased and sometmes decreased by an ncrease n the advertsng cost b. Hence, our earler concluson that an ncrease n the unshments for racy mght decrease the revenue of the coyrght owner generalzes to the equlbrums of the Page 28 of 48

31 28 model as a whole. I shall conclude ths secton by resentng some other comaratve statc results that are concerned wth these equlbrums. Clearly, the comaratve statc roertes of the model wll be dfferent n the equlbrums n whch s otmal relatve to some fxed number of bootleggers, and the corner solutons n whch has the largest value suffcent to block the entry of one more bootlegger. In the followng rooston, attenton has been restrcted to the equlbrums of the former knd. These equlbrums are the ones n whch the revenue of the coyrght owner s mal also when the number of the actve bootleggers s exogenously gven, whch corresonds to settng F 0. PROPOSITION 5. Suose that the strateges of the coyrght owner and the bootleggers form a Nash equlbrum of a game of the form 1) - 4) n whch F 0. Suose further that there are K k actve bootleggers who send advertsements. (a) The rce of legtmate coes and the mum and mum rces of rate coes, and, are decreased by an ncrease n k. (b) When k 1, the rces and are ndeendent of b. When k 2, the rces, d db, and are ncreasng functons of the advertsng cost b, and d db. Acceted anuscrt (c) A rase n the qualty arameter q shfts downwards, but the dervatves d dq and d dq can be ether ostve or negatve. 5. Concludng Remarks Page 29 of 48

32 29 The role of government olcy and DR systems n reventng commercal racy was analyzed n a settng n whch the roftablty of racy was restrcted not just by these factors and the cometton wth the coyrght owner, but also by the cometton between the commercal rates. The aarently uzzlng fact that the rces of rate coes do not always snk to zero va Bertrand cometton between the rates was exlaned by drawng a dstncton between the dfferent effects that government montorng and DR systems have on the llegal busness model of the commercal rate. The costs of breakng DR systems and the exected costs of a unshment were vewed as consstng of a fxed cost of roducton and an advertsng cost. The advertsng cost deends on the number of the consumers to whom a bootlegger (.e., a commercal rate) offers her roducts and t kees the rces of rate coes above ther roducton costs. The markets for legtmate and llegtmate coes of nformaton goods were studed both when the coyrght owner chooses the rce of legtmate coes otmally and when ths s not the case, for examle, because of lack of nformaton concernng the rate coy market. The comaratve statcs of these markets was studed wth resect to the qualty of the rate coes and the olcy varables, whch were the fxed and the varable costs of the bootleggers. Some of the results of ths analyss were to be Acceted anuscrt exected, but others were more surrsng. For examle, t turned out that the revenue of the coyrght owner s decreased by an ncrease n the qualty of rate coes both when the rce of legtmate coes s otmal and when t s exogenously gven, and that, as long as ths does not affect the number of the bootleggers on the market, an ncrease n ther advertsng costs ncreases the revenue of the coyrght owner and the rces of rate coes. ore Page 30 of 48

33 30 nterestngly, t also turned out that when there are several bootleggers on the market, an ncrease n the rce of legtmate coes ncreases rce dserson n the market for rate coes and decreases ther mum rce. Snce the rate coes that have a very low rce mght be vewed as the counterart of non-commercal forms of racy n the current model, ths result can be taken to mean that f the coyrght owner chooses a suffcently hgh rce for an nformaton good, t wll smultaneously be subject to both commercal and non-commercal forms of racy. Our analyss also revealed that the effects of the two olcy varables on the revenue of the coyrght owner were dfferent: whereas t was always n the nterest of the coyrght owner that the fxed costs of the bootleggers were ncreased, ths was not true of the advertsng costs because an ncrease n the advertsng costs mght ncrease the roftablty of commercal racy and the ncentve to enter the market for rate coes. As the current model contans just a sngle nformaton good wthout any network effects, ths result s dstnct from the famlar results that the roducer of an nformaton good mght roft from racy f ths makes t easer to sell comlementary goods or servces to consumers 20 or leads network effects that ncrease the oularty of the roduct For examle, the demand for concerts by an artst mght be ncreased by the rate Acceted anuscrt coes of her recordngs (Gayer and Shy 2006). Somewhat less obvously, when the consumers do not know n advance whch nformaton roducts they refer (e.g. to whch muscal recordngs they would enjoy lstenng), the ossblty to samle rate coes mght make them wllng to ay more for ther referred roduct. See Petz and Waelbroeck (2006b). 21 See Conner and Rumelt (1991, Prooston 4 on. 133). Page 31 of 48

34 31 Snce the fxed costs of the bootleggers can be caused by techncal rotecton devces whereas, at least n the case of commercal racy on the Internet, the advertsng costs result almost comletely from the ncreased rsk of a unshment, ths result can be nterreted to mean that an mrovement n the DR systems that make an nformaton good techncally dffcult to coy s always n the nterest of the coyrght owner, but ths s not necessarly true of an ncrease n the legal rotecton of nformaton goods. I have not resented a detaled welfare analyss from whch one could have deduced the otmal values of the olcy varables of the model. Such an analyss s made roblematc by the fact that t s not clear whether a socal lanner should am at mzng welfare that s obtaned by llegal means, as through commercal racy. I also have not analyzed the effects of racy on the market enetraton of the roduct, whch would be qute essental f one wanted to nclude network effects n the current model. Incluson of network effects would lead to several nterestng questons that we have not consdered. For examle, above t was seen that an mrovement n the qualty of rate coes tends to lower the revenue of the coyrght owner, and t would be nterestng to fnd out whether ths effect could, n the resence of network effects, be balanced by the ostve effects of the ncrease n market enetraton that a qualty mrovement would cause. Acceted anuscrt Page 32 of 48

35 32 APPENDIX. PROOFS OF PROPOSITIONS. PROOF OF PROPOSITION 1. When one uts f A,, ex A a x Aj e a, j the Euler equaton, whch must be vald n the nterval, for a functon A to be for whch the functon R gven by (12) obtans ts mum value, turns out f d f (A1) 0. A d a Ths Euler equaton s equvalent to A d A x ex Aj e a x ex Aj e 0 j d j and further to (A2) x x x a. j j Snce we are consderng a symmetrc equlbrum, t can be concluded that (A3) a and further that 1 1 x k 1 x Acceted anuscrt 1 1 x u 1 x A du ln k 1 u x u k 1 x. Ths roves the valdty of (14). The result (13) can be deduced from (14) by observng that (14) and (12) mly that Page 33 of 48

36 33 (A4) A R x e a d A x e 1 x x and that the roft of each bootlegger s gven by (A5) 1 A P R ba x e ba. Snce each bootlegger could choose to send arbtrarly many advertsements wth the rce wthout havng to comete wth the other bootleggers, value that mzes rofts for the gven value of equlbrum t must be the case that so that (A6) P A A x e b A 0 x r b b ln ln., A must have the Acceted anuscrt. In other words, n On the other hand, the values of A and are connected also by the fact that the total number of the advertsements that the advertser sends s (14) mles that 1 x 1 r ln ln k 1 x k 1 r (A7) A A Together the results (A6) and (A7) mly (13).. A. The result PROOF OF PROPOSITION 2. Frst t s observed that snce by assumton, (19) mles that 21 q 2 q q 1. Keeng ths n d, t s observed that art (a) of ths rooston follows trvally from (13), (16), (19), (20), and (21), excet for the statement that 0. It can be roved by Page 34 of 48

37 34 observng that the when k 2 and q 1, the condton (13) (.e. r b r 11 k 1 k ) can be exressed n the form Hence, and b 1 41 q k q q q q q 2 2 q 1 q 1 q k r 1 k q 2 2x 1 q1 q k 1 q 2 q q qk Snce q q 2 2 0, t now follows that 0. The arts (b) and (c) of ths rooston follows trvally from (13), (15), (16), (19), and (21), excet for the clam concernng the roft P of the bootleggers. In order to demonstrate t, we assume that k 2 and observe that (18) and (13) mly that k P r 1 r 1 11 k 1 ln 1 k b b k b k (A8) 1 and that (A9) 2 1 k P k 1 r k b k b kb. Acceted anuscrt Ths mmedately mles that there s recsely one value of b 0, r for whch P b 0, and ths further mles that there are at most two values of b 0, r 2 2 for whch P b 0. It s now observed that dp db (A10) k br k Page 35 of 48

38 35 so that wthn the nterval 0,r, there can be at most one value of b for whch P b 0. On the other hand, snce the functon P s by constructon ostve n the nterval 0,r, and snce t has the lmt 0 when b 0 and when b r, t must be the case that P b 0 for at least one value b 0, r E Acceted anuscrt. Hence, P s an ncreasng functon of b when b be and that P s a decreasng functon of b when b b E. Fnally, turnng to the art (d) of ths rooston, all the statements n t also follow trvally from (13), (18), (19), (20) and (21), excet for the statement concernng q. It can be demonstrated by usng (8) for exressng (13) n the form q q b 11 k 2 k k k 1 k 4 1 q. Snce the rght-hand sde of ths equaton s constant for each constant value of q, t must be the case that (A11) q 2 11 k 11 k q. q q q 1 q Clearly, the multler of q n (A11) s always ostve, but the term on the rght-hand sde s sometmes ostve and sometmes negatve: the rght-hand sde s negatve, for examle, when k 2 and q 3 4, but t has a ostve value when q 0 and q s very small, snce n ths case 11 k 11 k 1 11 k 1. q q 1 q q q kq Ths roves the statement that q s sometmes ostve and sometmes negatve. In order to rove the clam concernng ts uer lmt, we now ntroduce the Page 36 of 48

39 36 notaton q only f and observe that the rght-hand sde of (A11) can be ostve 1 11 k 11 k 0. q 1 q 1 q Ths s equvalent to 1 k q (A12). 1 2q 1 k The artal dervatve can be ostve only when ths condton s vald, and f ths s the case, (A13) 1 2q 1 k 21 k q q k 1 11 k. q 1 1 2q 1 k 1 k q 1 q 2 Hence, now one can conclude from (A11) that f s ostve, 11 k 2 2 q q kq Gven that q 2 and that k 2, t now follows that 1 k. 2 PROOF OF PROPOSITION 3. When 21 q 2 q Acceted anuscrt. the statements of ths rooston follow trvally from (23) and the contnuty of that 21 q 2 q. Accordng to (33) n ths case D. Suose then 1 2b D f z dz q 1 2 (A14) 1 where 1 q 1 k 1 k k Below f f z b z z z wll be vewed as gven by ths formula for arbtrary real values of k although, of course, (A14) has a meanngful Page 37 of 48

40 37 economc nterretaton only when k s an nteger. Clearly, (A14) s formally vald also when 1 1 2, so that the k, snce n ths case q q ntegral on the rght-hand sde of (A14) vanshes. When q 1, (A14) mles that b D f z dz, 1 1 q and t mmedately follows that D b 0 and D q 0 observed that z z z when z 1 2, t follows that f 1 1 k z z k z z so also D k k 1 0,. When t s. Ths roves (a) and (b) when q 1., the analyss of the consdered artal dervatves s made more When comlcated by the mlct deendence of that (A15) f q 2 q, 1 q so one can conclude from (A14) that D 1 1 f 1 q q q Acceted anuscrt on b, q, and k. However, (31) mles ths makes t legtmate to leave ths deendence out of consderaton below. 0 ; Now t follows that when, (A16) 1 2 D 1 f z 2 dz 0 b q b Page 38 of 48