On endogenous Stackelberg leadership: The case of horizontally differentiated duopoly and asymmetric net work compatibility effects

Transcription

1

2 On endogenous Stackelberg leadershp: The case of horzontally dfferentated duopoly and asymmetrc net work compatblty effects Tsuyosh TOSHIMITSU School of Economcs,Kwanse Gakun Unversty Abstract Introducng product compatblty assocated wth network externaltes (hereafter, network compatblty effects) nto a horzontally dfferentated duopoly model, we consder how network compatblty effects and the level of product substtutablty affect endogenous tmng decsons n the cases of quantty- and prce-settng competton. In partcular, we demonstrate the followng. Frst, gven asymmetrc network compatblty effects between the products of the frms, there s Stackelberg equlbrum where the frm provdng a product wth a larger network compatblty effect than some certan level of product substtutablty emerges as a leader (follower), whereas the frm provdng a product wth a smaller network compatblty effect than some certan level of product substtutablty emerges as a follower (leader) n the case of quantty (prce)-settng competton. Second, the Stackelberg equlbrum s Pareto-superor for both frms compared wth other equlbra. However, wth alternatve formulaton determnng network sze, wth respect to the endogenous Stackelberg leader follower relatonshp, the revers holds. Keywords: Stackelberg equlbrum; Nash equlbrum; leader-follower; product compatblty; network externalty; product substtutablty; fulflled expectatons; horzontally dfferentated products JEL classfcaton: D, D43, D6, L5. Introducton In the feld of ndustral organzaton, many studes have analyzed the choce of frm

3 roles n market or tmng decsons wth respect to strategc varables ncludng prce, quantty, and other frm actvtes (e.g., R&D, advertsng). Wth the frst- and second-mover advantage, when comparng the Stackelberg and Cournot Nash equlbra, we fnd that frms prefer to be leaders (followers) f the strategc relatonshps between them ndcate that they are substtutes (complements) wth respect to the relevant strategc varables,.e., prce and quantty. Equvalently, the same s true f negatve (postve) slopes of the reacton functons are found n the relevant strategc varable space. For example, Gal-Or (985) demonstrates that frms are wllng (unwllng) to commt frst when the reacton functons are downward (upward) slopng. In ths case, the frms have a frst (second)-mover advantage. Furthermore, Dowrck (986) consders the condtons where frms agree upon the choce of role of leader and follower n the Stackelberg duopoly model and demonstrates that each frm prefers to be a leader when the slope of the reacton functons s downward. In contrast, each frm prefers that the other frm wll be the leader when the slope of the reacton functons s upward. Hamlton and Slutsky (990) construct the extended endogenous tmng game of an observable delay framework and consder the endogenous tmng of smultaneous versus sequental moves. That s, the players determne both the tmng and the acton. If players choose ther actons at dfferent tmes, then the player choosng a later tme observes the acton taken by the ntatng player, gvng rse to a sequental-play subgame. Thus, Stackelberg equlbrum holds. If the players nstead choose actons at the same tme, then a smultaneous-play subgame takes place. Thus, Nash equlbrum holds. As shown below, we use the framework n Hamlton and Slutsky (990), to consder the endogenous tmng decsons governng quanttes and prces, and

4 demonstrate that Stackelberg equlbrum holds gven asymmetrc network compatblty effects between frms. In terms of related analyss, Robson (990) analyzes a model of prce-settng duopoly ncludng the endogenous choce of strategc tmng by ntroducng tmng costs nto an extended game. Elsewhere, van Dame and Hurkens (999) consder a lnear quantty-settng duopoly ncludng asymmetrc costs between the frms and analyze a two-stage game where each frm can ether commt to a quantty n stage or wat untl stage. They show that based on the rsk domnance consderatons n Harsany and Selten (988), the low-cost frm wll emerge as the endogenous Stackelberg leader, whereas commtment s rsker for the hgh-cost frm (see also Amr and Grllo, 999). Smlarly, van Dame and Hurkens (004) address a lnear prce-settng duopoly and show that the low-cost frm wll emerge as the endogenous prce leader (see also Amr and Stepanova, 006). In other work, L (04) examnes prce leadershp n a vertcally dfferentated product market and shows that the hgh-qualty frm actng as prce leader rsks domnatng the other equlbrum when the low-qualty frm leads. Fnally, usng a horzontally dfferentated duopoly model wth lnear demand and asymmetrc constant margnal costs, Yang et al. (009) compare prce and quantty competton under endogenous tmng and demonstrate that endogenous tmng n the prce-settng duopoly leads to two sequental move games n whch one frm moves frst and the other frm moves second,.e., two Stackelberg equlbra. Furthermore, they show that endogenous tmng n the quantty-settng duopoly leads to a smultaneous-move game, n whch both frms move frst,.e., a Cournot Nash equlbrum. In the abovementoned studes, the strategc space s assumed to be exogenously 3

5 gven,.e., a quantty- and prce-settng duopoly. However, Sngh and Vves (984) consder the endogenous choce of strategc varables,.e., prces and/or quanttes, as well as the dstrbuton of roles n duopoly games. That s, provded the products are substtutes (complements), t s a domnant strategy for each frm to choose the quantty (prce) contract (see also Boyer and Moreaux, 987a, 987b). Further, Tremblay and Tremblay (0) develop a model n whch both the tmng of play and the strategc choce varables,.e., quantty and prce, are endogenous. They show that the dynamc Cournot Bertrand outcome can be a subgame perfect Nash equlbrum, n whch the frm choosng quantty (prce) moves frst (second). Smlar ssues have also receved attenton n the context of a vertcally dfferentated duopoly model wth a fxed convex cost functon for qualty. For example, Aok and Prusa (997) and Aok (003) demonstrate that frms select dstnctve qualtes and that a frm producng a hgh-qualty product earns larger profts than a frm producng a low-qualty product, regardless of the competton mode. In ths case, the leader (follower) n a sequental Stackelberg game must decde to produce a hgh (low) -qualty product. However, because both frms prefer to commt to the producton of a hgh-qualty product, they both choose to move frst. Lambertn (996, 999) consders endogenous tmng wth a vertcally dfferentated Bertrand duopoly and demonstrates that f frms endogenously decde the tmng of qualty choces, only smultaneous-move equlbra can arse. Jn (004) examnes ths same ssue n the context of a vertcally dfferentated Cournot duopoly and establshes that the outcomes of the endogenous tmng game depend on whether Tremblay et al. (03) show that the degree of product dfferentaton s suffcently large for the Cournot Bertrand equlbrum to be stable. 4

6 frms are able to choose ther relatve poston n the qualty space before they determne the tmng of ther qualty choce. In other words, f frms cannot select ther relatve poston, n lne wth the result n Lambertn (999), only smultaneous-move equlbra persst. In ths case, the frms have an ncentve to move frst because the frst mover can earn larger profts than the second mover. Alternatvely, f both frms can choose ther relatve poston, only sequental-move equlbra emerge. In ths case, the frm choosng to produce the low (hgh)-qualty product decdes to be the frst (second) mover. In a more recent study, Lambertn and Tamper (0) determne that a frm producng a low (hgh)-qualty product takes the leader s (follower s) role n a vertcally dfferentated quantty-settng duopoly model (see also Lambertn and Tedesch, 007). We have surveyed the related lterature publshed up to now that addresses the dstrbuton of roles n market competton. Furthermore, we refer to the lterature consderng the same ssue n the case of the precommtment of the strategc varables, such as process (cost reducng) and product (qualty mprovng) R&D, and advertsng. For example, Amr et al. (000) consder the endogenous tmng of process R&D wth technology spllovers by applyng the frameworks n D Aspremont and Jacquemn (988) and De Bondt and Henrques (995). In partcular, these studes assume that a spllover effect arses because a rval frm s R&D stmulates the avalablty of technologcal knowledge,.e., ncomng spllovers. They then demonstrate the exstence of a unque equlbrum n the assgnment of the leader and follower roles n whch one frm that s better at absorbng knowledge spllovers leads and the other frm follows. In contrast, Atallah (005) assumes that a spllover effect arses from the leakage of technologcal nformaton from a rval frm s R&D. In ths case, the result s the See also Tesorere (008), and Vandekerckhove and De Bondt (008). 5

7 opposte of that presented by Amr et al. (000) and others. In other words, the frst mover s a frm that suffers only a small leakage of technologcal knowledge from ts own process R&D. Lke Atallah (005), Toshmtsu (0) consders the dstrbuton of roles n product R&D (or advertsng) nvestment competton n the presence of demand spllovers and shows that the frm producng that product wth low (hgh)-demand spllovers wll emerge as the leader (follower), rrespectve of the competton mode. To sum up, the nature of the reacton functons determne the frst- or second-mover advantage, the endogenous dstrbuton of these roles, and the leader-follower relatonshp. Put dfferently, these are strategc relatonshps (.e., strategc substtutes or complements) between frms. In turn, these strategc relatonshps depend on the propertes (.e., substtutes or complements) between products or the strategc varables (.e., quantty or prce), as addressed n Boyer and Moreaux (987a, 987b). Further, for endogenous leadershp to hold n a duopolstc game, there are requrements for certan asymmetrc characterstcs between the frms themselves, the attrbutes of ther products, and ther strategc varables. For example, extendng the strategc taxonomy of Fudenberg and Trole (984), Tombak (006) examnes a strategc asymmetry two-stage game where one frm regards ts rval s second-stage strategc varable as a strategc complement, whle the other frm regards ts rval s second-stage strategc varable as a strategc substtute. In ths paper, by focusng on product compatblty and network externaltes,.e., the network compatblty effects, as the characterstcs of the products, we reconsder the dstrbuton of roles n a horzontally dfferentated duopoly. As for the asymmetrc strategy space (.e., prce vs. quantty) n Tremblay et al. (0) and the asymmetrc 6

8 product qualty (.e., low vs. hgh qualty) n Lambertn and Tamper (0), we assume that the degree of network compatblty effects s exogenously gven and asymmetrc. Under these crcumstances, we examne endogenous Stackelberg leadershp n network product and servce ndustres. That s, even though the products provded by competng frms are substtutes, at least n terms of horzontal product dfferentaton such as brand names, t s possble for the strategc relatonshps between the frms to be complements. Ths corresponds well wth the propertes of products assocated wth network externaltes, ncludng nternet servces n the ITC ndustry and applcaton software (e.g., word processng, spreadsheet, and database products). Usng ths approach, we dentfy wth whch of these characterstcs a frm (or product) wll emerge as ether a leader or a follower, based on the framework of endogenous tmng decsons,.e., the extended game wth observable delay as developed by Hamlton and Slutsky (990). Furthermore, we reconsder that the results derved depend on both the competton mode and the formulaton of network sze.. Endogenous leader follower relatonshp and network compatblty effects. Cournot Nash equlbrum and strategc relatonshps We consder quantty competton n a horzontally dfferentated product market assocated wth product compatblty and network externaltes. Based on the framework n Economdes (996), we assume a lnear nverse demand functon for product as follows: e p A ( q q ) f ( S ),,,,, () 7

9 where A s the ntrnsc market sze, q q ) s the quantty of frm (), and 0, represents the degree of product substtutablty. The network externalty e S functon s gven by f ( ), where S e s the expected network sze of frm. Based on the concept of a fulflled expectatons equlbrum presented by Katz and Shapro e (985), we assume that S S, where S s the real network sze of frm. Followng Chen and Chen (0), we assume a lnear network externalty functon; f S ) as, where a 0, denotes the network externalty parameter for network ( sze. Furthermore, usng equaton (3.5) n Shy (00, p. 6), the network sze of frm s gven by: ( S q q,,,,, () where 0,,,, s the degree of product s compatblty wth product. Equaton () mples that frm wll provde a compatble product whch the rval frm s product can operate. If ( 0),,, product operates (does not operate) perfectly wth product. Gven equaton (), q ),,,,, s the own (ncomng) effect on network sze. ( q Based on equatons () and (), the nverse demand functon for frm s gven by: p A a) q ( a ) q,,,,. (3) ( Regardng equaton (3), we assume that the own-prce effect exceeds the cross-prce effect:.e., dp dp,,,,. Thus, t follows that a a,,. dq dq Although the products are horzontally dfferentated n terms of ther ntrnsc attrbutes,.e., 0, the relatonshp between the products wll be complementary 8

10 (substtutonary) n terms of ther operatonal propertes f the degree of product compatblty wth the network externalty s larger (smaller) than that of ther product substtutablty,.e., a ( ),,. As shown below, the nature of the products determnes the strategc relatonshps between the frms and the external effects on ther profts. Hereafter, we denote product compatblty wth a network externalty as the network compatblty effect,.e., a. To smplfy the analyss, we assume that producton costs are zero. Thus, the proft functon s p q,,. In vew of equaton (3), we derve the reacton functon for frm as follows. A a q ( a q, ) ( a),,,. (4) Gven equaton (4), the strategc relatonshp between the frms depends on the degree of product substtutablty and network compatblty effect: q q ( )0 ( ) a,,,,. (5) Equaton (5) mples that a strategc substtutonary (complementarty) relatonshp between the frms arses f the degree of product substtutablty s larger (smaller) than that of the network compatblty effect. In partcular, even though the two products are substtutable, a relatonshp of strategc complementarty arses under quantty competton when a. Usng the frst-order condton to maxmze proft, the proft functon s represented by p q ( a) q,,. Thus, we derve the external effect of an ncrease n frm on the proft of frm as follows: q ( a) q ( )0 ( ) a,,,,. (6) q q 9

11 For the followng analyss, we assume asymmetrc compatblty between the frms: Assumpton 0. Gven equaton (4), we derve the followng Cournot Nash equlbrum: N q ( a) ( a ), A,, (7) D where D ( a) ( a )( a ) 0 and ( a) ( a ) 0,,. 4 Both of these condtons are satsfed because the own-prce effect exceeds the cross-prce effect. Gven equaton (7), superscrpt N denotes the Cournot Nash equlbrum. Takng equatons (5) and (6) nto account, we categorze the Cournot Nash equlbrum across three cases. In case () (()), a strong (weak) network compatblty effect arses: ) a,,, When a strong (weak) network compatblty effect ( arses, accordng to the strategc complementarty (substtutonary) relatonshp under quantty competton, the reacton curves for both frms are upward (downward) slopng. See Fgure (). In case (), where asymmetrc network compatblty effects arse,.e., a a, the reacton curve of frm () s upward (downward) slopng. 3 See Fgure 3.. Endogenous leader follower decson and a natural Stackelberg stuaton Applyng the framework of endogenous tmng decsons,.e., the extended game 3 It s characterzed by a game wth strategc heterogenety (see Monaco and Sabarwal, 05). One player exhbts a strategc complement and the other a strategc substtute. 0

12 wth observable delay developed by Hamlton and Slutsky (990), we demonstrate an endogenous leader follower decson and derve a natural Stackelberg stuaton. We assume frm () s a leader (follower). Consderng equaton (4), we derve the Stackelberg equlbrum under quantty competton as follows: q q L F ( a) ( a ), A (8) D ( a )( a ) A D ( a)( a ), (9) ( a){ D ( a )( a )} where,,,. We can smlarly obtan the outcomes n the case of the opposte roles,.e., frm () s a follower (leader). Gven equatons (8) and (9), superscrpt L (F) denotes the leader (follower) n the Stackelberg equlbrum. Usng equatons (7), (8), and (9), and comparng the quanttes n the Nash equlbrum and n the Stackelberg equlbrum, we derve the followng relatonshps: q q q L F L N ( ) q ( a )( a ) ( )0, (0) N ( ) q a ( )0, () F ( ) q ( a )( a ) ( )0, () where,,,. We now compare the profts n the Nash and Stackelberg equlbra. Gven Assumpton, and consderng equatons (6), (0), (), and (), we derve the followng results as n Lemma. Lemma () If a a, then L q N F L N q q and,,. F

13 () If a a 0, then L q F L N F q q and,,. N () If a a 0, then q q q, q q q,, F L and. N N F L F N L L N F In Lemma (), where the degree of the network compatblty effect s larger than that of product substtutablty, both frms prefer beng the follower to beng the leader and to playng a smultaneous-move game. In partcular, both frms have a second-mover advantage. Thus, consderng Theorem V (A. ) n Hamlton and Slutsky (990) and Lemma n Yang, et al. (009), there are multple equlbra n the extended game wth observable delay,.e., two Stackelberg equlbra. See S and S n Fgure. In Lemma (), f the degree of the network compatblty effect s smaller than that of product substtutablty, both frms prefer beng the leader to beng the follower and to playng a smultaneous-move game. In ths case, both frms have a frst-mover advantage. Thus, consderng Theorem V (A. ) n Hamlton and Slutsky (990) and Lemma n Yang, et al. (009), there s a unque smultaneous-move game equlbrum n the extended game wth observable delay,.e., a Cournot Nash equlbrum. See N n Fgure. In Lemma (), there s asymmetry between the frms wth respect to the degree of the network compatblty effect. That s, f the degree of the network compatblty effect for frm () s larger (smaller) than that of the product substtutablty, frm prefers beng the leader to beng the follower and to playng a smultaneous-move game, whereas frm prefers beng the follower to beng the leader and to playng a

14 smultaneous-move game. Thus, consderng Theorem V (B) n Hamlton and Slutsky (990), the sequental move game equlbrum n the extended game wth observable delay s unque,.e., a Stackelberg equlbrum. See S n Fgure 3. Here, we use the defnton of a Natural Stackelberg Stuaton (NSS) presented by Albaek (990) as follows. Defnton In a NSS, one frm prefers beng the leader to beng the follower and to playng a smultaneous-move game, and the other frm prefers beng the follower to beng the leader and to playng a smultaneous-move game. Furthermore, we assume the necessary condton for a NSS to hold as follows. Assumpton a. Otherwse,.e., a, the strategc relatonshp under quantty competton s always substtutonary, rrespectve of the product compatblty level (.e., ). Based on Lemma (), there s a unque Cournot Nash equlbrum. Therefore, we summarze the analyss above as Proposton. Proposton There s a NSS where the frm provdng the product wth a larger (smaller) network compatblty effect than a certan product substtutablty level s the leader (follower) under quantty competton. 3

15 Because the degree of the network compatblty effect s larger (smaller) than that of product substtutablty, a(n) ncrease (decrease) n the output of the rval frm ncreases the proft of the relevant frm. In ths case, the frm provdng the product where the degree of the network compatblty effect s larger than that of product substtutablty has a strong ncentve to choose beng a frst mover so as to commt a N F L small amount of output,.e., q q q. On the other hand, the frm provdng the product where the degree of the network compatblty effect s smaller than that of the product substtutablty prefers beng a second mover to a frst mover because the small output of the rval frm ncreases ts proft. As a result, the natural Stackelberg stuaton under quantty competton holds where ths equlbrum s Pareto-superor for both frms when compared wth the other equlbra. 3. Prce competton and an alternatve formulaton of network sze 3. Bertrand Nash equlbrum and strategc relatonshps Takng equaton (3) nto account, we derve the drect demand functon of frm as follows. {( a) ( a )} A ( a) p ( a ) p q,,,,. (3) ( a) ( a )( a ) Based on equaton (3), the reacton functon for frm s gven by {( a) ( a )} A a p p, ( a) ( a),,,. (4) 4

16 Thus, the strategc relatonshp between the frms depends on the degrees of product substtutablty and the network compatblty effect as follows: p p ( )0 ( ) a,,,,. (5) Equaton (5) mples that a strategc complementarty (substtutonary) relatonshp between the frms holds f the degree of product substtutablty s hgher (lower) than that of the network compatblty effect. In partcular, even though the two products are substtutable, a relatonshp of strategc substtutonary s sustaned under prce competton when a. Furthermore, we derve the external effect of an ncrease the prce of frm on the proft of frm as follows: p ( )0 ( ) a,,,,. (6) Gven equaton (4), we derve the followng Bertrand Nash equlbrum: BN p ( a) ( a)( a ) ( a)( a), A,, (7) D where ( a) ( a)( a ) ( a )( a ) 0,. Ths condton s satsfed because the own-prce effect exceeds the cross-prce effect. Gven equaton (7), superscrpt BN denotes the Bertrand Nash equlbrum. Takng equatons (5) and (6) nto account, we categorze the Bertrand Nash equlbrum nto three cases. In case () (()), where a strong (weak) network compatblty effect,.e., ) a,,, holds, accordng to the strategc ( substtutonary (complementarty) relatonshp under prce competton, the reacton curves of both frms are downward (upward) slopng. See Fgure (). The strategc 5

17 relatonshps n these two cases le opposte to those under the quantty competton case. In case (), where the asymmetrc network compatblty effect,.e., a a, arses, the reacton curve of frm () s downward (upward) slopng. See Fgure Endogenous leader follower decson and a NSS By followng the same procedure as n quantty-settng competton, we derve the leader s prce n the Stackelberg equlbrum: p BL ( a) ( a)( a ) ( a)( a). A (8) D ( a )( a ) Substtutng equaton (8) nto equaton (4), we obtan the follower s prce: BF p A ( a) H ( a ) G, (9) ( a){ D ( a )( a )} ( a)( a) where H D 0 and G ( a) ( a )( a ) 0. Gven equatons (8) and (9), superscrpt BL (BF) denotes a leader (follower) n a Stackelberg equlbrum. 4 Usng equatons (5), (7), (8), and (9), wth respect to the prces n the Nash equlbrum and n the Stackelberg equlbrum, we derve the followng: p p p BL BF BL BN ( ) p ( a )( a ) ( )0, (0) BN ( ) p a ( )0, () BF ( ) p ( a )( a ) ( )0, () where,,,. We also compare the profts n the Nash and Stackelberg equlbra. Under 4 Because H G {( a) ( a)( a)} 0 and a a hold, t BF follows that p 0. 6

18 Assumpton, and equatons (6), (0), (), and (), wth respect to prces and profts, we derve the followng outcomes as n Lemma (see Fgures,, and 4). Lemma () If a a, then () If a a 0, then BL p BL p BF BL BN BF p p and,,. BN BF BN BF BL BN p p and,,. () If a a 0, then p p p, p p p, BF BL BN BL BN, and. BF BF BN BL BN BF BL In Lemma (), where the degree of the network compatblty effect s larger than that of product substtutablty, both frms prefer beng the leader to beng the follower and to playng a smultaneous-move game. That s, both frms have a frst-mover advantage. Thus, consderng Theorem V (A. ) n Hamlton and Slutsky (990) and Lemma n Yang et al. (009), there s a unque smultaneous move equlbrum n the extended game wth observable delay,.e., a Bertrand Nash equlbrum. See N n Fgure. In Lemma (), where the degree of the network compatblty effect s smaller than that of product substtutablty, both frms prefer beng the follower to beng the leader and to playng a smultaneous-move game. That s, both frms have a second-mover advantage. Thus, consderng Theorem V (A. ) n Hamlton and Slutsky (990) and Lemma n Yang et al. (009), there are multple equlbra n the extended game wth observable delay,.e., two Stackelberg equlbra. See S and S n Fgure. In Lemma (), there s asymmetry between the frms n terms of wth the degree 7

19 of the network compatblty effect. That s, when the degree of the network compatblty effect of frm () s larger (smaller) than that of product substtutablty, frm prefers beng the follower to beng the leader and to playng a smultaneous-move game, whereas frm prefers beng the follower to beng the leader and to playng a smultaneous-move game. Thus, consderng Theorem V (B) n Hamlton and Slutsky (990), the sequental move equlbrum n the extended game wth observable delay s unque,.e., a Stackelberg equlbrum. See S n Fgure 4. Therefore, gven Assumpton, we summarze the above analyss as Proposton. Proposton There s a NSS where the frm provdng the product wth a strong (weak) network compatblty effect s the follower (leader) under prce competton. Ths result n Proposton s contrary to that n Proposton. That s, the frm havng weak (strong) network compatblty s the Stackelberg prce leader (follower). If the network compatblty effect s smaller (larger) than a certan level of product substtutablty, a(n) ncrease (decrease) n the prce of the rval frm ncreases the proft of the frm. In ths case, the frm provdng the product wth a weak network compatblty effect has an ncentve to choose to be the frst mover and to commt the BN BF BL lowest prce,.e., p p p. On the other hand, the frm provdng the product wth a strong network compatblty effect prefers beng the second mover to the frst mover because the low prce of the rval frm,.e., p BL, ncreases ts proft. As a result, the natural Stackelberg stuaton under prce competton arses. Ths equlbrum s 8

20 Pareto-superor for both frms compared wth the other equlbra. Furthermore, wthout the network compatblty effect,.e., 0,,, t follows that the follower s prce s lower than the leader s prce, smlar to the equlbrum prce n the standard Stackelberg competton. However, t follows n our a model that BF BL p p because a a. That s, the follower can set a hgher prce than the leader s prce because the follower s product s suffcently compatble wth the leader s product. 3.3 Alternatve formulaton of network sze Wth respect to the formulaton of network sze, Chen and Chen (0) assume: S q q,,,,, (3) where mples the spllover (or leakage) effect from frm on the network sze of q frm. That s, frm provdes a product that a user of product can operate and that works n product. Here, we also assume a a. Under quantty competton, we change equatons (5) and (6) as follows: q q q ( )0 ( ) a, ( )0 ( ) a, (4) (5) where,,,. Thus, the strategc relatonshps and the external effect depend on the level of product substtutablty and the rval frm s network compatblty effect. That s, f the rval frm provdes the product wth the network compatblty effect larger (smaller) than a certan level of product substtutablty, the reacton curve for the 9

21 competng frm s upward (downward) slopng and the external effect on proft s postve (negatve). Therefore, the frm provdng the product wth a strong (weak) network compatblty effect wll emerge as the follower (leader) under quantty-settng competton. Ths result les opposte to Proposton. Smlarly for prce-settng competton, we obtan the followng: p p p ( )0 ( ) a, ( )0 ( ) a, (6) (7) where,,,. If the rval frm provdes the product wth the network compatblty effects larger (smaller) than some level of product substtutablty, then the reacton curve of the frm s downward (upward) slopng and the external effect on the proft s negatve (postve). Therefore, the frm provdng the product wth a strong (weak) network compatblty effect wll emerge as the leader (follower) under prce-settng competton. Ths result les opposte to Proposton. 4. Concluson In the model n ths paper, the propertes of a frm-specfc product,.e., product substtutablty and network compatblty, determne the strategc relatonshps and the external effects. In partcular, we have demonstrated that, n a quantty (prce)-settng duopoly, f the network compatblty effect s smaller than a certan level of product substtutablty, the strategc relatonshp of the rval frm regardng the frm s 0

22 substtutonary (complementarty) and the external effect on the rval frm s proft s negatve (postve). In general, f the strategc relatonshp of the rval frm regardng the frm s one of substtutonary (complementarty) and the external effect on the rval frm s proft s negatve (postve), then the frm chooses to be the leader (follower) wthn the framework of an endogenous tmng decson game. Based on ths result, we showed that gven asymmetrc network compatblty effects between the frms products, the frm provdng the product wth a larger (smaller) network compatblty effect than a certan level of product substtutablty s the Stackelberg leader under quantty (prce) competton. In other words, we propose that a natural Stackelberg stuaton holds n a game wth strategc heterogenety. We apprecate that our model depends on specfc assumptons, e.g., lnearty of the functons. However, by focusng on the propertes of the products assocated wth network externaltes and compatblty, we have llustrated whch effects of these propertes determne the endogenous dstrbuton of roles,.e., Stackelberg leadershp. Further, although we assumed exogenously determned product compatblty, we should consder the level of product compatblty as a frm s strategc varable. Thus, we should examne Stackelberg leadershp n the context of endogenous product compatblty choce (see, e.g., Toshmtsu, 04). For smlar reasons, although we assume that the asymmetrc network compatblty effects are also exogenously gven, we should consder how asymmetrc network compatblty effects arse between frms. References Albaek, S., 990. Stackelberg leadershp as a natural soluton under cost uncertanty, Journal of Industral Economcs 38, Amr, R., Grllo, I., 999. Stackelberg versus Cournot equlbrum, Games and

23 Economc Behavor 8, Amr, M., Amr, R., Jn, J., 000. Sequencng R&D decsons n a two-perod duopoly wth spllovers, Economc Theory 5, Amr, R., Stepanova, A., 006. Second-mover advantage and prce leadershp n Bertrand duopoly, Games and Economc Behavor 55, 0 Aok, R., 003. Effect of credble qualty nvestment wth Bertrand and Cournot competton, Economc Theory, Aok, R., Prusa, T. J., 997. Sequental versus smultaneous choce wth endogenous qualty, Internatonal Journal of Industral Organzaton 5, 03 Atallah, G., 005. R&D cooperaton wth asymmetrc spllovers, Canadan Journal of Economcs 38, Boyer, M., Moreaux, M., 987a. On Stackelberg equlbra wth dfferentated products: The crtcal role of the strategc space, Journal of Industral Economcs 36, 7 30 Boyer, M., Moreaux, M., 987b. Beng a leader or a follower: Reflectons on the dstrbuton of roles n a duopoly, Internatonal Journal of Industral Organzaton 5, 75 9 Chen, H-C., Chen, C-C., 0. Compatblty under dfferentated duopoly wth network externaltes, Journal of Industry, Competton, and Trade, D Aspremont, C. and Jacquemn, A., 988, Cooperatve and noncooperatve R&D n duopoly wth Spllovers, Amercan Economc Revew 78, De Bondt, R., Henrques, I., 995. Strategc nvestment wth asymmetrc spllovers, Canadan Journal of Economcs 8, Dowrck, S., von Stackelberg and Cournot duopoly: Choosng roles, RAND Journal of Economcs 7, 5 60

24 Economdes, N., 996. Network externaltes, complementartes, and nvtatons to enter, European Journal of Poltcal Economy, 33 Fudenberg, D., Trole, J., 984. The fat-cat effect, the puppy-dog ploy, and the lean and hungry look, Amercan Economc Revew Papers and Proceedngs, 74, Gal-Or, E., 985. Frst mover and second mover advantages, Internatonal Economc Revew 6, Hamlton, J. H., Slutsky, S. M., 990. Endogenous tmng n duopoly games: Stackelberg or Cournot equlbra, Games and Economc Behavor, 9 46 Harsany, J., Selten, S. M., 988. A General Theory of Equlbrum Selecton n Games, MIT Press, Cambrdge, MA. Jn, N., 004. Endogenous tmng n a vertcally dfferentated duopoly wth quantty competton, Htotsubash Journal of Economcs 45, 9 7 Katz, M., Shapro, C., 985. Network externaltes, competton, and compatblty, Amercan Economc Revew 75, Lambertn, L., 996. Choosng roles n a duopoly for endogenously dfferentated products, Australan Economc Papers 35, 05 4 Lambertn, L., 999. Endogenous tmng and the choce of qualty n a vertcally dfferentated duopoly, Research n Economcs 53, 0 09 Lambertn, L., Tamper, A., 0. Low-qualty leadershp n a vertcally dfferentated duopoly wth Cournot competton, Economcs Letters 5, Lambertn, L., Tedesch, P., 007. Would you lke to enter frst wth a low-qualty good?, Bulletn of Economc Research 59, 69 8 L, Y. 04. Prce leadershp n an vertcally dfferentated market, Economc Modellng 38,

25 Monaco, A. J., Sabarwal, T. 05. Games wth strategc complements and substtutes, Economc Theory, forthcomng Robson, A., 990. Duopoly wth endogenous strategc tmng: Stackelberg reganed, Internatonal Economc Revew 3, Shy, O., 00. The Economcs of Network Industres, Cambrdge Unversty Press, Cambrdge, UK. Sngh, N., Vves, X., 994. Prce and quantty competton n a dfferentated duopoly, RAND Journal of Economcs 5, Tesorere, A., 008. Endogenous R&D n lner duopoly wth one-way spllovers, Journal of Economcs Behavor and Organzaton 66, 3 5 Tombak, M. M., 006. Strategc asymmetry, Journal of Economc Behavor and Organzaton, 6, Toshmtsu, T., 0. Qualty competton and a demand spllover effect: A case of product dfferentated duopoly, Dscusson Paper Seres No. 89, School of Economcs, Kwanse Gakun Unversty Toshmtsu, T., 04. Endogenous product compatblty choce under Cournot competton wth a network externalty, Dscusson Paper Seres No. 5, School of Economcs, Kwanse Gakun Unversty Tremblay, C. H., Tremblay, V. J., 0. The Cournot Bertrand model and the degree of product dfferentaton, Economcs Letters, Tremblay, V. J., Tremblay, C. H., Isaryawongse, K., 03. Endogenous tmng and strategc choce: The Cournot-Bertrand model, Bulletn of Economc Research 65, Yang, X., Luo, Y., Wu, H., 009. On the comparson of prce and quantty competton 4

26 under endogenous tmng, Research n Economcs 63, 55 6 Vandekerckhove, J., De Bondt, R., 008. Asymmetrc spllovers and nvestment n research and development of leaders and follower, Economcs of Innovaton and New Technology 7, van Dame, E., Hurkens, S., 999. Endogenous Stackelberg leadershp, Games and Economc Behavor 8, 05 9 van Dame, E., Hurkens, S., 004. Endogenous prce leadershp, Games and Economc Behavor 47,

27 F z z ( z ) N 0 Fgure Strong network compatblty effects: z or p,,. q a a The shaded area represents the Pareto-superor sets. 6

28 z N 0 Fgure Weak network compatblty effects: a a z or p,,. q The shaded area represents the Pareto-superor sets. 7

29 q q q ) ( N 0 Fgure 3 Quantty-settng duopoly wth asymmetrc network compatblty effects: a a S s a natural Stackelberg stuaton where frm () s the leader (follower). The shaded area represents the Pareto-superor sets. 8

30 p p ( p ) N 0 Fgure 4 Prce-settng duopoly wth asymmetrc network compatblty effects: a a S s a natural Stackelberg stuaton where frm () s the leader (follower). The shaded area represents the Pareto-superor sets. 9