Endogenous timing decisions for product R&D investment competition with demand spillovers in a horizontally differentiated duopoly

Transcription

1 DISCUSSIO PAPER SERIES Dscusson paper o. Endogenous tmng decsons for product R&D nvestment competton wth demand spllovers n a horzontally dfferentated duopoly Tsuyosh Toshmtsu School of Economcs, Kwanse Gakun Unversty October, 04 SCHOOL OF ECOOMICS KWASEI GAKUI UIVERSITY -55 Uegahara Ichban-cho shnomya , Japan

2 Endogenous tmng decsons for product R&D nvestment competton wth demand spllovers n a horzontally dfferentated duopoly Tsuyosh Toshmtsu School of Economcs, Kwanse Gakun Unversty Abstract By focusng on the constructve and combatve spllover effects of the frms nvestment n research and development (R&D), we develop a horzontally dfferentated duopoly model n whch R&D nvestment used to mprove product ualty nfluences consumer preferences and the choce of consumpton goods. Applyng the framework of endogenous tmng decsons to the model, we eamne the mutually benefcal tmng of product R&D nvestment and demonstrate that, f there are asymmetrc demand spllovers between the frms, a natural Stackelberg eulbrum perssts n noncooperatve product R&D nvestment competton n whch the frm producng the product wth weaker (stronger) demand spllovers moves frst (second) to commt to the nvestment, regardless of the mode of competton. We consder the outcome of the endogenous tmng decsons, based on the vew of endogenous sunk costs (.e., The Sutton Approach). Furthermore, we address process R&D nvestment competton wth technology spllovers under endogenous tmng. JEL classfcaton: L3, L5 Keywords: endogenous tmng; natural Stackelberg eulbrum; product R&D nvestment; demand spllovers; horzontally dfferentated Cournot duopoly; endogenous sunk cost School of Economcs, Kwanse Gakun Unversty, -55, shnomya, Japan, ; emal: ttsutomu@kwanse.ac.p

3 . Introducton. Purpose We develop a horzontal product dfferentaton model that ncludes product research and development (R&D) nvestment wth demand spllovers n pre-market competton. Applyng the framework of endogenous tmng decsons.e., the etended game wth observable delay developed by Hamlton and Slutsky (990) to the developed model, the frst purpose of the paper s to demonstrate how the order of product R&D nvestment s endogenously determned n the presence of demand spllovers. Thus, n a Stackelberg (seuental) game, the order of the moves of players s eogenously gven; f we permt players to move smultaneously or seuentally, the dstrbuton of moves can be determned. We consder under what condtons the frst- or second-mover advantage perssts for players. We demonstrate the counterntutve outcome that a small frm wth small demand spllovers (and a small effect on the market) commts to the nvestment frst, whereas a large frm wth large demand spllovers commts to the nvestment second. Ths order of nvestments s benefts both frms. Thus, the small frm gets a share of the market enhancement generated by the product R&D of the large frm. In other words, the small frm employs a free-rde strategy. The other purpose of ths paper s to eplan the economc mplcatons of product R&D nvestment under endogenous tmng by employng the perspectve of endogenous sunk costs,.e., the Sutton approach (see Sutton, 989, 99, 998; Etro, 007, 03). As shown below, that approach posts that the sze of product R&D nvestment under endogenous tmng represents entry costs and the level of ualty. Furthermore, the dfference n the sze of the Lee et al. (999) address whether small and medum enterprses can free-rde on the large frm s market development efforts, takng nto account ther resource dsadvantage.

4 nvestments between the frms depends on the dfference n the degree of spllovers and product substtutablty. Accordngly, the natural Stackelberg eulbrum under asymmetrc spllovers mples that the small (large) frm enters to the market frst (second) ncurrng low (hgh) entry costs n the pre-market competton and competes on uanttes or on prces, provdng a low (hgh) ualty product.. Lterature There have been varous contrbutons to the related lterature analyzng the choce of the respectve roles for frms n a market (or the tmng decson for strategc varables such as prce, uantty, process and product R&D nvestment, advertsng, and other frm actvtes). For nstance, Bulow, et al. (985) demonstrate that a frm prefers to be a leader (follower) f the strategc relatonshps between the applcable frms ndcates that the frms are substtutes (complements) wth respect to relevant strategc varables (.e., prce and uantty), or euvalently, f the slopes of the reacton functons are negatve (postve) n the relevant space. In partcular, when goods are substtutes, f the frms compete on uanttes (prces), a strategc substtute (complement) relatonshp holds between them. Thus, both frms prefer to be a leader (follower), such that the frms would engage n a smultaneous ash game, as opposed a seuental Stackelberg game. Gal-Or (985) and Boyer and Moreau (987) have shown these results n the case of horzontal product dfferentaton. Based on a horzontally dfferentated duopoly model wth a lnear demand and asymmetrc constant margnal costs, Yang et al. (009) compare prce and uantty competton under endogenous tmng and demonstrate that endogenous tmng n the However, Albaek (990) shows that the Stackelberg eulbrum s endogenously determned gven cost uncertanty, n whch the frm wth the larger (smaller) cost varance wll be the leader (follower). 3

5 Bertrand duopoly leads to two seuental move games, n whch one frm moves frst and the other moves second. Furthermore, these authors show that endogenous tmng n the Cournot duopoly leads to a smultaneous move game, n whch both frms move frst. Recently, Tremblay et al. (0) develop a model n whch both the tmng of play and the strategc choce varables (uantty and prce) are endogenous. The authors show that the dynamc Cournot-Bertrand outcome can be a subgame perfect ash eulbrum, n whch the frm choosng uantty (prce) moves frst (second). Wth respect to the lterature analyzng non-prce market competton such as R&D nvestment and advertsng under endogenous tmng, Amr et al. (000) n an analyss that s somewhat smlar to the analyss n the present paper consder the endogenous tmng of process (.e., cost-reducng) R&D nvestment wth technology spllovers, applyng D Aspremont and Jacuemn (988) and De Bondt and Henrues (995). 3 In partcular, t s assumed n these papers that a spllover effect arses because a rval frm s R&D nvestment stmulates the avalablty of technologcal knowledge (.e., ncomng spllovers). These authors demonstrate the estence of a unue eulbrum n the assgnment of the leader and follower roles n whch the stronger frm that s better at absorbng knowledge spllovers leads and the other frm follows. By contrast, Atallah (005) assumes that a spllover effect arses n the form of leakage of technologcal nformaton from a rval frm s R&D nvestment. In ths case, the outcome s the 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 nvestment. We address process R&D nvestment competton wth technology spllover 3 De Bondt and Henrues (995) assume asymmetrc spllover between frms. See Tesorere (008) and Vandekerckhove and De Bondt (008). 4

6 under endogenous tmng and focus on two types of technology spllovers,.e., ncomng and leakage spllovers. In the analyss of process R&D nvestment competton under endogenous tmng, researchers address the technology-sde spllovers among frms or ntra-ndustry (.e., busness to busness). However, ths paper focuses on demand-sde spllovers (.e., busness to (potental) customers) and addresses demand-enhancng nvestments such as product R&D nvestment (.e., ualty-mprovng) and advertsng. As dscussed above, by ntroducng the effect of product R&D nvestment on the demand sde nto the conventonal utlty functon, we develop a horzontally dfferentated duopoly model (e.g., Boyer and Moreau, 987; Häckner, 000). 4 At ths uncture, we should note that, because product R&D nvestment affects consumer preferences and the demand sde of the market, the nvestment consdered n our model resembles persuasve advertsng. In ths case, followng the termnology that Marshall (99, pp ) uses to eplan the effects of advertsng, we consder both the combatve and constructve effects assocated wth product R&D nvestment on the demand sde. Regardng the combatve effect, an ncrease n product R&D nvestment by a frm poaches the customers of the rval frm, thereby reducng the profts of the rval. However, an ncrease n product R&D nvestment by a frm may also attract new customers from outsde markets and epand the potental sze of the market. As a result, product R&D nvestment can also ncrease the customers of the rval frm and ncrease the profts of the rval. We refer to ths result as the constructve effect of product R&D nvestment. Smlar ssues have also receved attenton n the contet of a vertcally dfferentated 4 In a dfferent contet, Gavazza (0) theoretcally and emprcally consders the role of demand for frms product varetes and demand spllovers n determnng market conduct and market structure n the mutual fund ndustry. Furthermore, Cleff et al. (009) emprcally address demand-orented nnovaton strategy n the European energy producton sector. 5

7 duopoly model wth fed conve cost of ualty. For eample, Aok and Prusa (997) and Aok (003) demonstrate that frms select dstnctve ualtes and that a frm producng a hgh-ualty product earns hgher profts than a frm producng a low-ualty product, regardless of the mode of competton. In ths case, the leader (or follower) n a seuental Stackelberg game must decde to produce a hgh- (or low-) ualty product. However, because both frms prefer to commt to the producton of a hgh-ualty 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 ualty choces, only smultaneous-move eulbra can arse. Jn (004) also eamnes ths ssue n the contet of a vertcally dfferentated Cournot duopoly and demonstrates that the outcomes of the endogenous tmng game depend on whether frms are able to choose ther relatve poston n the ualty space before decdng the tmng of ualty choces. In other words, f frms cannot select ther relatve poston, smlar to the result n Lambertn (999), only smultaneous move eulbra persst. In ths case, the frms have an ncentve to move frst because the frst mover can earn hgher profts than the second mover. Alternatvely, f both frms can choose ther relatve poston, only seuental-move eulbra emerge. In ths case, the frm choosng the producton of the low- (hgh-) ualty product decdes to be the frst (second) mover. In other words, the strategc complement (substtute) relatonshp for the frm producng a low (hgh)-ualty product holds n the ualty space. The remander of the paper s structured as follows. In Secton, we develop a horzontally dfferentated Cournot duopoly model based on the assumpton of a uaslnear utlty functon that ncorporates demand spllovers assocated wth product R&D nvestments. 6

8 In Secton 3, we frst consder the strategc relatonshps between frms and demonstrate the estence of the subgame perfect ash eulbrum (SPE) n the noncooperatve product R&D nvestment competton. Then, applyng the framework of endogenous tmng decsons to the developed model, we consder the mutually benefcal tmng of product R&D nvestments and demonstrate the estence of a natural Stackelberg eulbrum under asymmetrc demand spllovers. Furthermore, we consder the economc mplcaton of the outcomes n the endogenous tmng game, based on the vew of endogenous sunk costs,.e., the Sutton approach. In Secton 4, we eamne the endogenous tmng decson n the cases of process R&D nvestment competton n the presence of two types of technology spllovers. et we address the case of a horzontally dfferentated Bertrand duopoly. Fnally, n Secton 5, we summarze the man results and dscuss some remanng ssues.. The model. Demand functon and product R&D nvestment wth demand spllovers We assume a duopolstc Cournot competton n a market wth horzontally dfferentated products. The frms compete n a two-stage game. In Stage, each frm smultaneously chooses product R&D nvestment,, and n Stage, each frm smultaneously chooses output,,,. We confne our attenton to the SPE n the two-stage game by solvng the model usng backward nducton. We focus on the spllover effects generated by product R&D nvestment on the demand sde. In partcular, we consder whether an ncrease n product R&D nvestment ncreases the wllngness to pay of consumers for these products, thus epandng the potental sze of the 7

9 market, whch may also ncrease demand for the rval frm s products. To hghlght ths effect, we assume that the utlty functon of a representatve consumer s gven by [, ], V U and, ; 0 U α ( ) ( ) { Ω[, ]( ) ω[ ] ω[ ] }, () where 0 s the consumpton of the outsde (numerare) good and p 0, α > 0, and ( 0,) θ s a parameter representng the degree of substtutablty between the products. Wth respect to the second term n brackets n (), we assume that [ ] Ω Ω, s an Ω ncreasng functon of product R&D nvestment,.e., > 0, whch s assocated wth aggregate market demand. 5 Furthermore, ω [ ] ω s also an ncreasng functon of ω product R&D nvestment,.e., > 0, whch s assocated wth frm s ndvdual demand. For tractablty, we make the followng assumpton. Assumpton Ω where ε ( 0),, s the coeffcent of the margnal effect () [ ],, ε ε of an ncrease n product R&D nvestment of frm on the potental sze of the market. ω where β ( 0),, s the coeffcent of the margnal effect of an () [ ] β, ncrease n product R&D nvestment of frm on ts own market. The budget constrant s gven by Y p p, where Y s the gven level of 0 5 If Ω[ ] 0,, the utlty functon s smlar to that n Häckner (000). See also the append n Symeonds (003). 8

10 ncome of a representatve consumer. Thus, we obtan the followng epresson for the optmal behavor of the representatve consumer. 6 U A [, ] p, () where A, ] α ( ε β ) ε,,,,. In partcular, A, ] n () [ [ mples that the potental sze of the market depends not only on the nvestment of frm but also on that of frm. Thus, ε β s the own effect of frm s nvestment on ts potental market sze and ε s the spllover effect of frm s nvestment. We obtan the followng nverse demand functon. p A, ],,,,. (3) [. Product R&D nvestment cost functon To smplfy the analyss, we assume that the cost ncurred n product R&D nvestment s f gven by F [ ], f > 0,,. We also assume that the margnal cost of producton s constant,.e., ( α > ) c > 0,,. c 3. Endogenous tmng and product R&D nvestment wth demand spllovers 3. oncooperatve product R&D nvestment competton wth demand spllovers 6 Unlke Levn and Ress (988), who use a multplcatve margnal utlty functon to make emprcal estmaton, we employ an addtve margnal utlty functon to smplfy the analyss. By dong so, the effect of product R&D nvestment can be epressed as the vertcal shft of the nverse demand functon. 9

11 f In Stage, frm chooses ts output to mamze proft,.e., Π ( p c). The Π frst-order condton for mamzng the proft of frm s gven by p 0. Takng (3) nto account, we derve A [, ] 0. Thus, the Cournot ash eulbrum for frm n the second stage s gven by ( A c) ( A c) A Φ Γ ( )( ) ( )( ) [, ],,,,, (4) where Α ( θ )( α c) > 0, Φ ( θ ) ε β > 0, and Γ θ ) ε θβ. Gven ( (4), t follows that [ ] θ, ( < )0 Γ ( < )0 ( < ) θ E, (5) ε where E,,, s the strength of demand spllovers. If ε > 0 and β 0, ε β then E, whch mples full spllovers. If ε 0 and β > 0, then E 0, whch mples no spllovers. Thus, t follows that [ 0,],,. E 7 Euaton (5) shows that an ncrease n the product R&D nvestment of frm has two effects on the output of frm. The frst s a postve effect,.e., θ ) ε, whereby an ( ncrease n the nvestment of frm ncreases total market demand for both products, whch, n turn, ncreases frm s ndvdual demand and s known as the postve constructve effect. The second s a negatve effect,.e., θβ, whereby an ncrease n the nvestment of frm ncreases ts own demand, whch n turn decreases frm s ndvdual demand as a result of the substtutablty between the competng products, whch s known as the negatve 7 We gnore the case where both ε 0 and β 0 smultaneously hold. 0

12 combatve effect. For that reason, f the constructve effect s larger (smaller) than the combatve effect, then an ncrease n the rval frm s product R&D nvestment ncreases (decreases) the output of the frm. f Let us epress the proft functon n Stage as Π [, ]. Based on (4), the frst-order condton s gven by Π Φ [, ] f 0, Λ,,,, (6) where Λ ( θ )( θ ) > 0. Usng (4) and (6), we derve the reacton functon of frm as follows: Φ Α ΦΓ ],,,,, (7) Λ f Φ Λ f Φ [ where we assume Λ f > Φ,,, to mantan the second-order condton. Based on (7), and regardng the cross effect, whch mples a strategc relatonshp between the frms, we derve the followng relatonshp: d d ΦΓ ( < )0 Γ ( < )0 E Λ f Φ θ ( < ). (8) Furthermore, the eternal effect on proft s gven by dπ d Γ θ θ ( < )0 Γ ( < )0 E ( < ). ( θ )( θ ) (9) To proceed wth the analyss, we assume as follows. Assumpton Demand spllovers for product (frm) are at least as strong as those for product (frm) ;.e., E E.

13 Gven (8), (9), and Assumpton, we derve the followng lemma. Lemma θ () If E E >, each frm s reacton curve slopes upward. Hence, an ncrease n the product R&D nvestment of frm () ncreases the proft of frm (). θ () If > E E, each frm s reacton curve slopes downward. Hence, an ncrease n the product R&D nvestment of frm () decreases the proft of frm (). () If E > θ > E, the reacton curve for frm slopes downward, whereas that for frm slopes upward. Hence, an ncrease n the product R&D nvestment of frm () decreases (ncreases) proft of frm (). Regardng Lemma () (Lemma ()), f demand spllovers for both frms are stronger (weaker) than half of the level of product substtutablty, an ncrease n the product R&D nvestment of the frm ncreases (decreases) the demand for the product of the rval frm. Ths result n turn ncreases (decreases) the rval frm s output and nvestment, and the rval frm s proft ncreases (decreases). Smlarly, the frm s proft ncreases (decreases) wth an ncrease n the rval frm s nvestment, whch confrms that the strategc relatonshp between the frms s complementary (substtutonary). See Fgure (). <ISERT FIGURES AD HERE>

14 Regardng Lemma (), when there are asymmetrc demand spllovers between the frms such that the spllover of frm s larger and that of frm s smaller than half of the product substtutablty, an ncrease n frm s nvestment ncreases frm s output, whereas an ncrease n frm s nvestment reduces frm s output. In ths case, frm ncreases ts output and nvestment, whereas frm reduces ts output and nvestment. As a result, frm s proft ncreases, whereas frm s proft falls. Thus, a relatonshp of strategc substtutablty for frm arses, whereas one of strategc complementarty for frm arses. See Fgure 3. <ISERT FIGURE 3 HERE> Takng (7) nto account, we derve the SPE n the noncooperatve product R&D nvestment competton as follows. { Λ f Φ ( Φ Γ )}, ΑΦ,,,, (0) D Φ Λ f Φ ΦΓ ΦΓ D Λ f > To acure a postve eulbrum where { }{ } 4 0. gven by (0), we assume Λ f > Φ ( Φ Γ ) Φ ( θ ) β,,. For the analyss below, n comparng the eulbrum nvestment level of both frms, we obtan the followng relatonshp: > ( < ) ( θ )( ε ε ) Λ f { Λ f ( θ ) ΦΦ }( β β) > ( < )0. Thus, f the effect of the nvestment of frm on market demand s larger than that of frm, the nvestment level of frm s larger than that of frm. In vew of Assumpton, for eample, f the combatve effect of both frms s almost eual,.e., β β, or f the combatve effect of frm s not suffcently lower than that of frm, and the constructve effect of frm s larger than that of frm,.e., ε > ε, then t holds that >. 3

15 3. Endogenous tmng decson and a natural Stackelberg eulbrum We now proceed to analyze the mutually benefcal tmng of product R&D nvestment n the presence of demand spllovers. By applyng the defnton used by Albaek (990), we etend the game-theoretc framework n Hamlton and Slutsky (990) for the endogenous tmng of an observable delay. In other words, we show the condtons necessary to sustan a natural Stackelberg eulbrum,.e., two players are able to determne the tmng of ther actons and the actons themselves. If the players choose ther actons at dfferent tmes, the player that chooses last can observe the acton chosen by the ntatng player. Hence, there s a seuental-play subgame and the Stackelberg eulbrum. If the players nstead choose ther actons at the same tme, there s a smultaneous-play subgame and the ash eulbrum. Conseuently, based on the etended game for the endogenous tmng of an observable delay, we compare the payoffs n the smultaneous-play game wth those from the two seuental-play games. In ths case, f one frm wants to be the frst mover (superscrpt F), whle the other frm wants to be the second mover (superscrpt S), and nether frm prefers to play a smultaneous ash game (superscrpt ), then a natural Stackelberg eulbrum results. Wthout any unnecessarly complcated calculatons and takng Lemma nto account, the relatonshps between the frms profts descrbed below are easly derved (see Fgures,, and 3). Lemma θ () If E E >, t follows that S Π > Π and Π > Π,,. F 4

16 θ () If > E E, F S t follows that Π > Π > Π,,. () If E > θ > E, t follows that F > Π S Π and Π > Π, and that Π F > Π > Π S. Under Lemma (), ust as n a standard Bertrand prce competton model, both frms prefer beng ether a frst mover or a second mover to playng a smultaneous ash game. However, each frm prefers beng a second mover to beng a frst mover because t holds that S F S Π > Π,,. Thus, there are two Stackelberg eulbra,.e., F, ) and ( S F (, ), located n the Pareto-superor sets (see S and S n Fgure ). As proven n Lemma n Yang et al. (008), ths result mples that the endogenous tmng game leads to two seuental games. Smlarly, under Lemma (), as n a standard Cournot uantty competton model, both frms prefer beng a frst mover to beng a second mover and playng a smultaneous ash game (see Theorem V (A) n Hamlton and Slutsky, 990; and Lemma n Yang et al., 008). Furthermore, the two Stackelberg eulbra are not located n the Pareto-superor sets (see S and S n Fgure ). Under Lemma (), n whch there are asymmetrc spllovers between the frms, frm prefers beng ether a frst mover or a second mover to playng a smultaneous ash game, whereas frm prefers beng a frst mover to beng a second mover and playng a smultaneous ash game. In ths case, because frm epects that frm wll take the frst move, frm wll take the second move. Takng nto account the conectural process, frm wll commt to the nvestment n advance. In other words, ths commtment s credble and 5

17 S preferable for frm because t holds that Π > Π. Therefore, based on Theorem V (B) n Hamlton and Slutsky (990), we derve the followng result. 8 F Proposton The frm producng the product wth weaker (stronger) demand spllovers chooses the frst (second) move to commt to product R&D nvestment. For eample, consder the stuaton n whch a small frm that does not produce the recognzed brand whose product has only a small effect on market demand decdes n advance to nvest on a small scale strategcally. Ths nvestment makes the large frm wth the recognzed brand where the product and the frm tself have a substantal effect on market demand ncrease ts product R&D nvestment. Accordngly, market demand ncreases, whch enables both frms to rase output. As a result, both frms make hgher profts. In other words, the small frm free-rdes on the demand spllovers generated by the large frm s product R&D nvestment. Based on the defnton n Albaek (990), we defne the outcome under endogenous tmng decson gven n Proposton as a natural Stackelberg eulbrum,.e., ΑΨ { Λ f Φ ( Φ Γ )} F Α F Γ Φ S F and, D 4ΦΓ ΨΓ Λ f F Λ f F Φ( Λ f Φ ) ΦΓ Γ where Ψ > 0. Λ f Φ See Append A. We consder the natural Stackelberg eulbrum located n the Pareto-superor sets n the case of asymmetrc spllovers (see pont S n Fgure 3). Compared wth the SPE n the 8 In a rent-seekng model, Lennger (993) demonstrates that a weaker (stronger) player moves frst (second). 6

18 natural Stackelberg eulbrum, when frm () produces the product wth stronger (weaker) demand spllovers, ts product R&D nvestment ncreases (reduces),.e., S > and F <. In ths case, regardng the output of frm, based on (4), we have the followng relatonshp: S S Φ S Φ [, ] > ( ) [, ] Φ ( ) Γ ( ) > ( )0. () S Thus, t drectly follows that >, because Γ < 0. Compared wth those n the SPE, the product R&D nvestment costs ncrease. However, the etent of the ncrease n the output of frm s suffcently large. Thus, t follows that Π > Π. Conversely, we derve the output of frm as follows: Φ S Φ Φ S [, ] > ( ) [, ] Φ ( ) Γ ( ) > ( )0. () S Φ S Hence, we derve { ( ) ( )} sgn { ( )}. sgn Φ Γ Γ ΦΛ f Φ ΦΦ Γ Γ It follows that Φ Λ f Φ ( Φ Φ Γ Γ ) 0, because Ψ 0 as denoted above. Thus, the > output of frm decreases more than that n the SPE. However, although the output decreases because the etent of the decrease n product R&D nvestment costs outweghs that of the decrease n revenue, t follows that Π > Π. F Furthermore, the relatonshp between the product R&D nvestments of both frms n the S natural Stackelberg eulbrum,.e., > ), s generally ambguous because t F ( depends on the parameters for spllovers and product substtutablty,.e., ε, β,,, and θ. However, for eample, assumng ε ε > ε 0, β β β 0, and > > ( θ ) ε θβ > 0, we can derve S F ( > ) > > (see Append B). Based on (4), t S S F F S F [ follows that, ] > [, ] n the Cournot duopolstc market. Compared wth SPE, the Pareto optmalty of terms of profts perssts n the natural 7

19 8 Stackelberg eulbrum; however, the effect on consumer surplus s ambguous. That s, consumer surplus s gven by [ ] ( ),, p p U CS where [ ], ), (., We derve as follows. ( ) ( ), d dcs where 0 ) ( < Φ Λ Λ Γ Λ f f and. 0 ) ( ) ( > Φ Λ Λ Φ Γ Γ Φ Λ Φ f f The frst (second) term epresses the effect of a change of the nvestment of frm on surplus regardng consumpton of product (). Unless the magntude of the effect epressed n the frst term s suffcently large, a decrease n the nvestment of frm from the SPE to the natural Stackelberg eulbrum reduces consumer surplus. Furthermore, evaluatng the socal optmal nvestments at the SPE, we derve ( ) ( ), 3 0 W Γ Φ Λ Γ Φ Λ Π,,,, where 0 ) ( ) )( ( > Γ Φ β θ ε θ θ θ and, ) )(3 ( 3 θβ ε θ θ θ Γ Φ,. If t holds that,,, ) )(3 ( > ε θβ θ θ then we have. 0 0 > Π W Thus, we understand that the levels of product R&D nvestment n the SPE s lower than those n the socal optmalty. Consderng, F S > > > ths mples ether that the levels of the nvestments of both frms n the natural Stackelberg eulbrum are lower than the levels n the socal optmalty or that at least the level of frm, whch s a frst mover wth a small mpact on the market, s lower than that n the socal optmalty.

20 3.3 Implcatons: Product R&D nvestment as endogenous sunk costs 9 Based on the semnal works, of John Sutton (989, 99, 998),.e., the Sutton approach denoted by Etro (03), wth respect to the outcomes demonstrated above,.e., Proposton, we take the vew that product R&D nvestment cost n our model can be regarded as endogenous sunk costs. 0 In ths case, when a small frm commts to a frst move to nvest, t mples that the small frm ncurs a low nvestment costs to enter the market n advance. Ths low entry costs leads to low levels of ualty. Otherwse, f t chooses a second move, the small frm must ncur large nvestment costs. However, a large frm prefers the second move to the frst move, whch mples that the large frm makes a substantal nvestment to mprove the ualty level and ncurs large entry costs enhancng aggregate market demand. As a result, the market structure s organzed such that the small (large) frm competes by provdng a low (hgh) ualty of the products and servces. Secondly, we consder how product R&D nvestments under endogenous tmng affect potental market sze (.e., A, ] α ( ε β ) ε, [,,, ), regardng both frms n pont S and S n Fgure 3. In ths case, we derve as follows. and A F S S F [ S] A [ S] ( ε β)( ) ε( A [ S] A [ S] ( ε β)( ) ε ( S F > S F S < F where 0 and 0. For eample, n assumng ε ε > ε 0, F S ) ), β β β 0, ( θ ) ε θβ > 0, and ε > β, f a small (large) frm ncurs a low (hgh) > entry (.e., nvestment) cost, the potental market sze may ncrease more than n the opposte 9 Ths part s based on the suggestons of the edtor and anonymous referees. 0 See also Matraves (999) and Gruber (00). 9

21 case. As a result, the market sze s determned by the magntude of entry costs through the endogenous tmng decsons before Cournot competton n the market. Fnally, compared wth the outcomes based on the vertcal dfferentaton model,.e., Jn (004), we can alternatvely nterpret the outcomes under endogenous tmng as follows. Let us assume θ. Ths assumpton mples that the products are homogenous and a horzontal product dfferentaton between the products thus vanshes. Hence, we can confrm that Lemmas and and Proposton hold. In other words, our model addressng product (.e., ualty-mprovng) R&D nvestment s formally smlar to the vertcal dfferentaton model. Jn (004) demonstrates that the frm producng the low (hgh) ualty product chooses the ualty level frst (second). However, n our model, a small (large) frm wth a small (large) effect on market demand commts to a small (large) product R&D nvestment frst (second). As a result, the small (large) frm provdes the low (hgh) ualty product. 4. Dscussons 4. Endogenous tmng of process R&D nvestment competton wth technology spllovers As dscussed n the Introducton, t s worth relatng the demand spllovers eamned to the technology spllovers. Furthermore, we demonstrate that the outcomes of endogenous tmng depend on the type of technology spllovers. We assume the followng standard nverse demand functon n a horzontally dfferentated products market: Gven the eample, we can derve the same result by comparng the SPE and the natural Stackelberg eulbrum,.e., and S. See Lambertn and Tedesch (007), and Lambertn and Tamper (0). 0

22 p α,,,,. (3) Frst, we assume the mpact of technology spllover on the margnal cost of producton presented by De Bondt and Henrues (995) and Amr, et al. (000) as follows. c c ε,,,,, (4) where ε represents technology spllovers, n whch [ 0,] ε measures the etent to whch the benefts of frm s R&D nvestment are avalable to frm, whch mples that frms can reap the rewards of another frm s technology,.e., ncomng spllovers. In ths case, the Cournot ash eulbrum s gven by Α ( ε ) (ε ) Λ,,,,. (5) If the level of technology avalable to frm from the rval frm s nvestment s hgher (lower) than half of the product substtutablty, then an ncrease n the nvestment by frm ncreases (decreases) the output of frm as follows. ( < )0 ε ( < ),,,,. (6) Alternatvely, we assume the mpact of the technology spllover on the margnal cost of producton presented by Atallah (005) as follows. c c ε,,,,, (7) where ε s the technology spllover, n whch [ 0,] ε represents the leakage of technologcal knowledge from frm s R&D nvestment to frm. In Stage, the Cournot ash eulbrum s gven by Α ( ε ) (ε ) Λ,,,,. (8) If the level of technologcal leakage from frm s hgher (lower) than half of the product

23 substtutablty, then an ncrease n nvestment by frm ncreases (decreases) the output of frm as follows. ( < )0 ε ( < ),,,,. (9) Because t follows that E ε,,, n assumng ε,,, n (4), we derve β (9). Thus, the effects of technologcal leakage from the rval frm assumed by Atallah (005) are formally euvalent to the effects of the demand spllovers n our model. To demonstrate that the endogenous tmng of process R&D nvestment depends on the θ type of technology spllovers, let us assume ε > > ε. In ths case, n the models of De Bondt and Henrues (995) and Amr et al. (000), f frm () has a hgher (lower) capablty of absorbng the technology generated by frm s ( s) process R&D nvestment, then the reacton functon s upward (downward) slopng. Thus, frm () has a strategc complementary (substtutonary) relatonshp wth frm (), and we determne that the frm wth a hgher (lower) capablty of absorbng the technology chooses to move frst (second). The result s the same as Theorem 5 n Amr, et al. (000). Conversely, the model n Atallah (005) predcts that the frst mover s frm, whch suffers only a small leakage of technologcal knowledge from ts own process R&D nvestment. Thus, we summarze the result as follows. Corollary The frm wth smaller (larger) technology spllovers of process R&D nvestment chooses to nvest frst (second). Ths result s contrary to the results of De Bondt and Henrues (995) and Amr et al.

24 (000), but s formally smlar to Proposton n the case of demand spllovers. 4. The Bertrand duopoly case wth demand spllovers We confrm whether the endogenous tmng decson depends on the mode of competton. Consderng (), the drect demand functon of product s gven by α( ) {( ) ε β } {( ) ε β } Σ p p,,,,, (0) where Σ ( θ )( θ ). In ths case, we obtan θ ( θ ) ε θβ Σ ( < )0 Ε ( < ) θ. () In Stage, frm chooses a prce to mamze ts proft,.e., Π p c) F [ ]. The ( Π p c frst-order condton for mamzng the proft of frm s gven by 0. p Σ Consderng (0), we derve { ε ( ) β } { ε ( ) β } p p c 0. α ( ) Hence, the prce of product n the second stage s gven by p B B B Α Φ Γ c p[, ],,,,. () Λ B We note parameters Α ( θ )( θ )( α c) > 0, Φ ( θ )( θ ) ε ( θ ) β > 0, B and Γ θ )( θ ) ε θβ, where superscrpt B denotes Bertrand duopoly. Gven (), ( B the followng relatonshp holds: p [, ] ( < )0 Γ B ( < )0 Θ Ε [ ] ( < ) θ, () Ε 8 Ε where Θ[ Ε ] < and [ ] > 0, Θ Ε,. 3

25 4 Euaton () demonstrates that f the parameter representng the strength of demand spllovers for product,.e., [ ] Ε Θ, s larger (smaller) than a certan value of product substtutablty,.e.,, θ then an ncrease n the product R&D nvestment of frm ncreases (decreases) the prce of product. That s, frm ncreases (decreases) ts prce because the constructve effect on product s demand that s nduced by an ncrease n total market demand s larger (smaller) than the combatve effect on product s demand resultng from the substtutablty between the products. In Stage, frm chooses ts proft-mamzng product R&D nvestment. Hence, the proft functon s represented by [ ],, f R Π where [ ] [ ].,, Σ p R The frst-order condton s gven by: [ ], 0, Λ Φ Σ Π B f p Φ R.,,, (3) We derve the second-order condton, the cross effect, and the eternal effect on the proft of the rval frm, as follows. 0, < Λ Φ Σ Π f Φ R B (4) )0, ( )0 ( < Γ < Λ Γ Λ Φ Σ Π B B B (5) and [ ], )0 ( )0 (, < Γ < Λ Γ Σ Π B B p (6) where.,,, To proceed wth the analyss, t follows under Assumpton that [ ] [ ] Ε Θ Θ Θ Ε Θ

26 and that Θ Θ f and only f Ε Ε. Gven (5) and (6), we derve the followng lemma. Lemma 3 () If Θ Θ >, each frm s reacton curve slopes upward. Hence, an ncrease n the θ product R&D nvestment for product () ncreases the revenue of frm (). () If θ > Θ, each frm s reacton curve slopes downward. Hence, an ncrease n Θ the product R&D nvestment for product () decreases the revenue of frm (). () If Θ > θ >, frm s reacton curve slopes downward, whereas frm s reacton Θ curve slopes upward. Hence, an ncrease n the product R&D nvestment for product () decreases (ncreases) the revenue of frm (). Gven Lemmas and 3, we can obtan the same results n the case of a horzontally dfferentated Bertrand duopoly as those n Proposton. However, n the vertcal dfferentaton model, the endogenous tmng of the ualty decson depends on the mode of completon, e.g., Lambertn (996, 999) and Jn (004); however, the endogenous tmng decson of the product R&D nvestment competton n a horzontally dfferentated model s ndependent of the mode of competton. Furthermore, by the same method employed n the analyss of the Cournot duopoly case, assumng ε ε > ε 0 and β β β 0, we B S B B B F can derve > > >. > 5

27 5. Concludng Remarks We have consdered endogenous tmng decsons based on a model of product (ualty-mprovng) R&D nvestment wth demand spllovers n a horzontally dfferentated duopoly. Furthermore, we have addressed the same problem n the case of process (cost-reducng) R&D nvestment wth technology spllovers and n the case of a horzontally dfferentated Bertrand duopoly. We have found that f the strategc relatonshp of the rval frm regardng the frm s one of substtutablty (complementarty), and the eternal effect on the rval frm s proft s negatve (postve), the frm chooses to move frst (second) under endogenous tmng, regardless of the mode of competton. In ths case, the Stackelberg eulbrum results, whch s Pareto-superor for the frms but may also decrease consumer surplus. We have consdered the economc mplcatons of the Stackelberg eulbrum under endogenous tmng, consderng the vews of endogenous sunk costs and endogenous entry. In partcular, when there are asymmetrc frms wth varous specfc resources and propertes, the market structure s organzed such that a small frm that affects the market demand lttle or not at all commts to a low nvestment (.e., entry) cost early, whereas a large frm that eerts substantal nfluence on the market commts to a hgh nvestment (.e., entry) cost. As a result, the former (latter) competes n the market wth a lower (hgher) ualty product. In terms of future developments, the duopoly model mght be etended to an olgopoly model, and we mght consder an endogenous market structure (see Etro, 03). Furthermore, we have shown that the endogenously decded order n the natural Stackelberg eulbrum s preferable to other eulbra for both frms, whch mples that both frms non-cooperatvely collude wth respect to nvestment levels before market 6

28 competton. However, we have not addressed cooperatve R&D nvestment, R&D agreements, or ont ventures n the presence of spllovers. 3 However, snce the semnal paper of D Aspremont and Jacuemn (988), the problem has been analyzed theoretcally and emprcally n much of the related lterature. For eample, Marn and Rodano (03) and Marn et al. (04) recently analyzed the possblty of formng R&D agreements (cooperatve ventures). Furthermore, Foros et al. (00) consder roamng polcy n the market for moble telecommuncatons. In ths model, frms collude at the nvestment stage, although they compete n the retal market,.e., sem-colluson. In the future, we wll eamne how R&D agreement and cooperaton affect endogenous entry and market structure. Append A: The Stackelberg eulbrum n the case of asymmetrc spllovers θ In the case of asymmetrc spllovers,.e., Γ > 0 > Γ E > > E, we assume that frm () chooses to move frst (second) n product R&D nvestment. Takng nto account the reacton functon of frm gven by (7), frm chooses ts product R&D nvestment to f mamze the proft gven by Π [ ( ), ]. The frst-order condton s gven dπ Ψ by [ ( ), ] f 0, d Λ Φ( Λ f Φ ) ΦΓ Γ where Ψ. Λ f Φ To sustan an nteror eulbrum, we assume Ψ > 0. Furthermore, the second-order condton s Λ f Ψ > 0. Based on the frst-order condton and (4), we have 3 One of our anonymous referees comments on ths pont. Wth respect to cooperatve product R&D nvestment (.e., sem-colluson) n the case of the Bertrand duopoly, see Toshmtsu (0). 7

29 Ψ Α Ψ Γ ( Λ f Φ Ψ ) 0. (A.) Substtutng the reacton functon of frm gven by (7) nto (A.), we derve the Stackelberg eulbrum as follows. Φ { Λ f Φ ( Φ Γ )} ΑΨ D 4Φ Γ Ψ Γ (A.) and S FΑ FΓ F. (A.3) Λ f F Λ f F In the case of asymmetrc spllovers, bearng n mnd (0) and (A.), we obtan F > drectly. Furthermore, because the reacton functon of frm s a decreasng functon of the F ( nvestment of frm, as n (A.3), t follows that ) > ( ). S Append B: An eample From (A.3), we have the followng relatonshp: S F F ( > ) F Α > ( ){ H F Γ }, (B.) > where H Λ f Φ 0. Furthermore, gven the assumpton regardng the parameters, we have Φ ( θ ) ε β > 0, Φ β > 0, Γ ( θ ) ε θβ > 0, and Γ θβ < 0. Substtutng (A.) nto the rght-hand sde of (B.), through complcated and tedous calculatons, we obtan the followng relatonshp: S Φ > ( ) HZ 4Φ Γ ΓY > ( )0, (B.) Where Z Λ f Φ Φ ) Φ Φ ( Φ Γ Γ ) Φ Γ Γ > 0 and Y Φ Γ Γ 0. ( Because Γ < 0, t follows that H Z 4Φ Γ ΓY > 0. Thus, we have >. S > F 8

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34 S Π 0 Fgure The case of strong demand spllovers: E E θ > The shaded area represents the Pareto-superor sets 33

35 0 Fgure θ The case of weak demand spllovers: > E E The shaded area represents the Pareto-superor sets 34

36 0 Fgure 3 The case of asymmetrc demand spllovers: E > θ > E The shaded area represents the Pareto-superor sets 35