Asymmetric Innovation Agreements under Environmental

Transcription

1 December 07 Asymmetrc Innovaton Agreements under Envronmental Regulaton Naoto Aoyama, Emlson C. D. Slva

2 Impressum: CESfo Workng Papers ISSN (electronc verson) Publsher and dstrbutor: Munch Socety for the Promoton of Economc Research CESfo GmbH The nternatonal platform of Ludwgs Maxmlans Unversty s Center for Economc Studes and the fo Insttute Poschngerstr. 5, 8679 Munch, Germany Telephone +49 (0) , Telefax +49 (0) , emal offce@cesfo.de Edtors: Clemens Fuest, Olver Falck, Jasmn Gröschl group.org/wp An electronc verson of the paper may be downloaded from the SSRN webste: from the RePEc webste: from the CESfo webste: group.org/wp

3 CESfo Workng Paper No. 678 Category 9: Resource and Envronment Economcs Asymmetrc Innovaton Agreements under Envronmental Regulaton Abstract In a domestc market, a duopoly produces a homogeneous fnal good, polluton, polluton abatement and R&D. One of the frms (foregn) has superor technology. The government regulates the duopoly by levyng a polluton tax to maxmze domestc welfare. We consder the potental mplementaton of three nnovaton agreements: cooperatve research jont venture (RJV), non-cooperatve RJV and lcensng. In the cooperatve (non-cooperatve) RJV, the frms (do not) nternalze R&D spllovers. We show that, for the domestc frm, the cooperatve RJV domnates and lcensng s the least desrable alternatve. Although lcensng s domnant for the foregn frm, t s not mplementable. Both RJVs are mplementable. Whle the non-cooperatve RJV s more lkely the greater the degrees of asymmetry (n terms of effcency and R&D spllover rates) between the frms, the cooperatve RJV s more lkely the lower the degrees of asymmetry. Implementaton of both types of RJVs mprove the compettveness of the domestc frm and welfare. A subsdy polcy that nduces the foregn frm to accept a feasble cooperatve RJV when t strctly prefers a feasble non-cooperatve RJV s always welfare mprovng. JEL-Codes: D430, D60, F30, L30, L40, L50, Q550, Q580. Keywords: envronmental regulaton, nnovaton, research jont ventures, lcensng. Naoto Aoyama Aomor Publc Unversty 53-4 Yamazak, Goshzawa Japan Aomor aoyama@bb.nebuta.ac.jp Emlson C. D. Slva Department of Marketng, Busness Economcs and Law Unversty of Alberta Edmonton / Alberta / Canada emlson@ualberta.ca November 0, 07 Aoyama greatly apprecates fnancal support from the Japan Socety for the Promoton of Scence (JSPS) KAKENHI Grant Number

4 . Introducton Innovaton agreements abound and take varous forms. Two mportant examples are research jont ventures (RJVs) and lcensng. Many nnovaton agreements are nternatonal (mostly, North-North and North-South) and some nternatonal agreements occur as a natural consequence of foregn drect nvestment. 3 Whle nnovaton agreements n clean and perfectly compettve markets are socally desrable because they ncrease effcency, ther socal desrablty n drty olgopolstc ndustres depends on ther mpacts on market power and envronmental degradaton. 4 As dfferent types of nnovaton agreements typcally generate dvergent welfare mpacts, one should consder each type of nnovaton agreement very carefully n order to prevent unfortunate tradeoffs. Improvement n envronmental performance appears to be one of the man reasons why frms form RJVs. Accordng to Scott (996), one thrd of the frst RJVs fled after the enactment of the Natonal Cooperaton Research Act (NCRA) of 984 n the Hagedoorn (990) presents a detaled overvew of sx forms of nterfrm cooperaton: () jont ventures and research corporatons; () jont R&D; () technology exchange agreements; (v) customer-suppler relatonshps; (v) drect nvestment; and (v) one-drectonal technology flows. RJVs and technology exchange agreements represent nearly 34.6% of the total. For further evdence of RJVs, see, e.g., Greenlee (005) and Hagedoorn (00). Accordng to a survey for the Intellectual Property Owners Assocaton n the U.S., 7.6% of respondents lcensed out ther patents (Cockburn and Henderson (003)). A lcense s an agreement whereby the owner of ntellectual property authorzes another party to use t. Scotchmer (004), p See, e.g., Caloghrou et al. (003), Song (0) and Xu (000) for evdence of North-North nnovaton agreements. See, e.g., Asano and Matsushma (04), Borenszten et al (998), Kokko (994), Müller and Schntzer (006), Vshwasrao (994) and Yang and Maskus (009) for evdence of North-South nnovaton agreements. Accordng to Tan et al (00), Chna acqures technologes through jont ventures or by purchasng technologcal lcenses. For example, the Shangha Electrc Group acqured the desgns for turbne technology by purchasng a lcense from Alstom and obtaned access to boler and generator technologes through a jont venture wth Semens. Watson et al (0) shows that RJV and lcensng are mportant sources of nternatonal knowledge transfers n low carbon technology between the Unted Kngdom and Chna. 4 Myagwa (009) shows that a cooperatve RJV facltates colluson. Duso et al. (04) fnds strong evdence that cooperatve RJVs among compettors n the same ndustry leads to colluson, whle cooperatve RJVs among non-compettors enhance effcency. We rule out colluson. It s an nterestng topc for future research.

5 U.S. and one thrd of RJVs wthn a perod of months after the enactment of the Clean Ar Act Amendments (CAAA) of 990 relate to envronmental ssues. The objectves of Japan s Research Assocaton of Refnery Integraton for Group-Operaton (RING) are to mprove ts partcpants compettveness and envronmental performance. The goals of Canada s Ol Sands Innovaton Allance (COSIA) are to mprove ts members envronmental performances n four envronmental mpact areas: toxc talngs ponds, greenhouse gas emssons, water polluton and ground and bodversty dsturbance. 5 In great part, COSIA emerged n response to an ncrease n the strngency of an envronmental standard n 009, whose objectve was to reduce envronmental damages caused by toxc talngs ponds. 6 However, as COSIA s jont R&D actvtes encompass three addtonal areas of envronmental mpacts, ts formaton ratonale, just lke the phenomenon that Scott (996) descrbes (.e., formaton of envronmental RJVs pror to CAAA), suggests that some pollutng ndustres may form RJVs n antcpaton of new (or more strngent) envronmental regulatons. In ths paper, we consder the endogenous formaton of nnovaton agreements n a duopoly where there s a technologcal gap between the frms. Technologcal gaps are common n dynamc ndustres that experence frequent technologcal mprovement. They are also common n nternatonal markets where entrants (foregn frms) have superor technology (see, e.g., Asano and Matsushma (04) and Müller and Schntzer 5 RING and COSIA are examples of ndustrywde RJVs. RING and COSIA started ther operatons n 000 and 0, respectvely. For more detals, see and Recent papers nvestgate the determnants of partcpaton n RJVs. Accordng to Hernan et al. (003), sectoral R&D ntensty, ndustry concentraton, frm sze, technologcal spllovers, past partcpaton n R&D jon venture (RJV) and the effectveness of patents nfluence the probablty of formng RJVs. Accordng to Roller et al. (007), the determnants of partcpaton n RJVs are R&D cost sharng, frm sze dfferences, the number of frms n the R&D project, the ndustres and the mpact of R&D nvestments. 6 Alberta s former envronmental regulator, called Energy Resources Conservaton Board (ERCB) mplemented Drectve 074 ( Talngs Performance Crtera and Requrements for Ol Sands Mnng Schemes ) on February 3,

6 (006)). In keepng wth the latter, we consder a duopoly consstng of domestc and foregn frms n whch the foregn frm has superor technology. We examne four settngs: () status quo (no nnovaton); () cooperatve RJV; () non-cooperatve RJV; and (v) lcensng. Under the cooperatve RJV, the frms coordnate ther R&D efforts and nternalze R&D spllovers. Under the non-cooperatve RJV, the frms enjoy R&D spllovers but do not nternalze them. In the settng wth lcensng, the foregn frm charges a royalty fee to allow the domestc frm to have access to ts superor technology knowng how t affects competton n the output s market. The frms select a feasble agreement (RJV or lcensng), f any s avalable, n the frst stage of a multstage game of complete but mperfect nformaton that they play wth a domestc government. An nnovaton agreement s feasble f t represents a (weak) Pareto mprovement relatve to the status quo. If there s a sngle feasble agreement, ths s the frms choce. If there are multple feasble nnovaton agreements, the frms announce ther preference rankngs to each other. If the frst choces concde, the frms choose the domnant one. If the frst choces do not concde, the frms utlze a random devce (e.g., flp a far con f there are two alternatves) n ther selecton procedure. The government observes the frst stage s outcome and then sets the polluton tax n the second stage before the frms make choces of abatement, output and R&D efforts. We show that the cooperatve RJV domnates the non-cooperatve RJV and lcensng s the least preferable alternatve (ncludng the status quo) for the domestc frm and the government. For the foregn frm, however, lcensng domnates and the noncooperatve RJV s generally second best. Referrng to a feasble nnovaton scheme as mplementable f t s adoptable, we can say that lcensng s not mplementable and both RJVs are mplementable under dfferent technologcal crcumstances. 4

7 To our knowledge, ths s the frst paper that examnes the endogenous formaton of dfferent types of nnovaton agreements n the presence of a technologcal gap and envronmental regulaton. We beleve ths s qute mportant because frms are typcally asymmetrc, RJVs and lcensng are common, one of the man reasons for the formaton of nnovaton agreements s to mprove the partcpants envronmental performances and some pollutng frms may have been forward lookng when they formed RJVs. Our paper contrbutes to several branches of the lterature. There s a large lterature that nvestgates varous crcumstances under whch envronmental regulaton promotes ncentves for nnovaton n polluton abatement (see, e.g., Carrón-Flores and Innes (00), Dencolo (999), Fscher et al. (003), Laffont and Trole (996), Malueg (989), Monner-Colonques and Rubo (06), Montero (00a), and Requate (995, 997, 005b)). Requate (005a) provdes an excellent revew of the early theoretcal lterature. Partcularly relevant for ths paper are the contrbutons that study nnovaton efforts suppled by frms that are mperfectly compettve n the output market (see, e.g., Chou and Hu (00), Innes and Bal (00), Katsoulacos and Xepapadeas (996), Montero (00b), Ouchda and Goto (04, 06a, 06b), Poyago-Theotoky (007)). Unlke these papers, we consder settngs n whch nnovaton agreements are endogenous, there s a technologcal gap n the ndustry and (domestc) welfare does not take the producer surplus of the (foregn) frm that has superor technology nto account. As Innes and Bal (00), we assume the frms R&D effort levels are unobservable by each other and by the government. 7 The frms observe each other s R&D effort f they cooperate. 7 See Sappngton (98) for a dscusson of the dffculty faced by regulators to observe the regulated frms R&D efforts. Sappngton (98) consders a game of ncomplete and mperfect nformaton. We analyze games of complete but mperfect nformaton. Extendng our framework to examne the mpacts of nformatonal asymmetry s an nterestng avenue for future research. 5

8 As we analyze the effects of (antcpated) envronmental regulaton on potental nnovaton adopton and mprovement of the domestc frm s compettveness, we contrbute to the vast lterature on the Porter hypothess (see, e.g., Greaker (006), Greaker and Rosendhal (008), Jaffe and Palmer (997), Lanoe et. al (0), Mohr (00), Porter (99), Xexapadeas and De Zeeum (999)). We depart from most works n ths lterature n that nnovaton adopton, f t occurs, takes place pror to envronmental regulaton. However, the antcpated costs of envronmental regulaton play an mportant role. Innovaton occurs when the frms jon a feasble RJV. The costs of envronmental regulaton that the frms face n a RJV are lower than n the status quo, and the antcpated savngs n envronmental regulatory costs are one of the key drvers of nnovaton. The other key drver s mprovement n the domestc frm s compettveness. The frms perfect foresght results n an outcome that supports the Porter hypothess. The credble threat of hgher envronmental regulatory costs produce adopton of Pareto-mprovng nnovaton and mprovement n domestc compettveness. Asano and Matsushma (04) examne the ncentves that a foregn frm faces to transfer ts superor and clean technology to a domestc frm, whch utlzes a drty technology, n the presence of an emsson tax. Our framework dffers from thers n many respects, ncludng allowng for cost-reducng R&D n the lcensng agreement, consderng RJVs and examnng the endogenous formaton of nnovaton agreements. The paper s as follows. Secton descrbes the basc model. Secton 3 examnes the subgame perfect equlbra for the four settngs. Secton 4 evaluates the mpacts of the mplementable nnovaton agreements on domestc compettveness and welfare. It also shows that a subsdy polcy that nduces the foregn frm to accept a feasble cooperatve RJV s always welfare mprovng. Secton 5 offers concludng remarks. 6

9 . Basc Model Suppose that a duopoly, contanng domestc and foregn frms, produce a homogeneous fnal good n an economy where the market for the fnal good s closed. The producton process s drty. The foregn frm possesses an advanced technology. In the absence of a lcensng agreement, the domestc frm utlzes a basc technology. Frm, =,, produces q unts of output, a unts of abatement and r unts of cost-reducng R&D effort. 8 Let y a + q denote frm s jont producton level of abatement and output. We assume that abatement and output are perfect substtutes n producton; that s, these two producton actvtes compete equally n terms of usage of costly nputs. R&D effort ncreases the effectveness of nputs. Followng Kamen et al. (99), we postulate that, due to nformaton sharng, R&D efforts produce hgher spllovers n RJVs than n the alternatve settngs where R&D efforts are undertaken ndependently. To smplfy exposton, we assume that ndependent R&D actvtes do not generate spllovers. In general, the foregn frm may produce abatement, output and R&D at a lower cost than the domestc frm. Let frm be the foregn frm and let t and e = q a denote the polluton tax and frm s polluton emsson, respectvely. We ntally consder the case where the frms exert R&D efforts ndependently from each other. We assume that frm s total cost of producng abatement, output and R&D (ncludng the regulatory cost) s C( a, q, r; θ ) te θc( x, r) t( q a ) + = +, where 8 See, e.g., d Aspremont and Jacquemn (988), Kamen et al. (99) and Zss (994) for early models of cost-reducng R&D. The ssues we consder here also overlap wth those studed n the lterature on R&D sharng (e.g., Morasch (995), Pastor and Sandons (00), Fabrz and Lppert (007, 0)). None of these papers, however, compares dfferent forms of RJV n the presence of envronmental regulaton. 7

10 x y r = a + q r, θ θθ, s frm s effcency parameter and θ θ θ > 0 s the maxmum technologcal gap between frms. Snce the foregn frm has superor technology, θ θ. For smplcty, we let θ = θ = and θ θ n what follows. 9 We assume that c (.) s ncreasng at an ncreasng rate n x and r and separable n x and r ; namely, c c x r = 0, =,. xr Let Q= q = be the total output supply and let P P ( Q ) = be the nverse market demand functon, where P ( Q) < 0 and P ( Q) 0. Let A= a and = be the ndustry s total abatement and polluton level, respectvely. = E = e = Q A Polluton causes a monetary damage, D( E ), where D ( E) > 0 and D ( E) 0 >. The government regulates the ndustry by choosng the polluton tax level that maxmzes domestc welfare, W U ( Q) te D( E) = +Π +, whch s the sum of consumer surplus, U( Q ), frm s proft, Π, and the socal surplus from envronmental taxaton: that s, polluton tax revenue, te, mnus the monetary cost of envronmental damage, D( E ). We assume that U ( Q) > 0 and U ( Q) 0 >. 0 We also assume that the government dstrbutes the tax revenue to consumers n a lump-sum 9 In Sectons 3.4 and 4, we assume that the cost and damage functons are quadratc and the demand functon for output s lnear to facltate comparsons. Lettng θ = θ = and θ = θ, we fnd that an nteror Nash equlbrum n the status quo requres θ > Hence, n Sectons 3.4 and 4, we assume that θ = 0.6. We carry out the analyss wth general demand, cost and damage functons n Sectons to demonstrate that our most mportant results do not depend on the functonal form assumptons that we make for comparson purposes. 0 If, for example, P( Q) = Q, we have U( Q) Q =. 8

11 fashon. We neglect regulatons to deter collusve behavor n the choces of output. We assume that the government can ensure full complance wth such regulatons. The frms and government play a multstage game. In the frst stage, the frms select an nnovaton agreement, f any. In the second stage, the government sets the polluton tax after observng the frst stage s outcome. The subsequent actons and tmng depend on the frst stage s outcome. If the frst stage outcome s ether the status quo or the non-cooperatve RJV, the frms n the thrd stage choose abatement, output and R&D efforts. If the frst stage outcome s the cooperatve RJV, n the thrd stage, the frms choose R&D efforts to maxmze jont proft. In the fourth stage, they choose abatement and output. If the frst stage outcome s the lcensng agreement, the foregn frm chooses the royalty fee n the thrd stage. In the fourth stage, the frms choose abatement, output and R&D efforts. The equlbrum concept s subgame perfecton. 3. Subgame Perfect Nash Equlbra 3.. Status quo As a benchmark, we frst consder a settng n whch there s no nnovaton agreement. The players payoff functons are as follows: ( ) θ (, ) ( ) Π = P q + q q c a + q r r t q a, =,, () ( ) ( ) W = U Q +Π + te D E, () where q = q f = and vce versa, θ = θ and θ =. Consder the thrd stage. See, e.g., Slva et al. (007) for regulatory regmes wth costly enforcement that yeld such outcomes. We devate from papers n the R&D lterature, ncludng Kamen et al. (99), whch consder settngs n whch there s strategc commtment wth R&D, followng Brander and Spencer (983). The crtcal assumpton underlyng strategc commtment wth R&D s that each frm can observe the other frms R&D efforts pror to makng output or prce choces. As we ponted out before, we follow Innes and Bal (00) n assumng that a frm s R&D effort s not observable by the other frm or the government. Frms observe each other s R&D effort when they coordnate ther R&D efforts. Ths occurs n the cooperatve RJV only. 9

12 An nteror Nash equlbrum satsfes the followng condtons, for =, : t = θc = c, (3) x x ( ) ( ) θ P Q + P Q q = c + t = c + t, (4) x x c x = c, (5) r where c c( x, r) x and (, ) x c c x r r, =,. For each, equaton (3) r nforms us that frm chooses abatement at the level that equates ts margnal revenue from abatement provson to ts margnal cost of abatement provson. The margnal revenue from abatement provson s the polluton tax rate, snce ths represents the amount of regulatory cost that the frm saves per unt of abatement. From equatons (3), we know that margnal costs of abatement provson are equalzed. For each, equaton (4) shows that the optmal level of output produced by frm s determned by the equalty of ths frm s margnal revenue and ts total margnal cost from producton of output. The total margnal cost assocated wth producton of output s the sum of the margnal technologcal cost and the margnal regulatory cost. For each, equaton (5) states that frm chooses R&D effort at the level that equates ts margnal beneft from R&D effort to ts margnal R&D effort s cost. The margnal beneft from R&D effort s the margnal producton cost. Gven the modellng assumptons, the suffcent second order condtons are satsfed n the maxmzaton problems n the thrd stage. It s mportant to note that the followng condtons hold n equlbrum: ( ) ( ) P Q + P Q q = t, =,. (6) Equatons (6) follow from equatons (3) and (4). The rght sdes of equatons (6) nform us that both companes face the same effectve margnal cost (measured n terms of a 0

13 multple of the polluton tax) n the producton of output. Interestngly, ths mples that the frms supply the same output quantty n equlbrum despte the technologcal gap. Proposton captures ths result as well as other mplcatons of the condtons that characterze the equlbrum n the thrd stage. Proposton. In the status quo, the nteror equlbrum n the thrd stage yelds a a ; q = q; r r; x x. (7) Proof. See Appendx A. The last three results n (7) nform us that the foregn frm produces quanttes of jont output and R&D that are at least as large as the quanttes that the domestc frm produces n equlbrum: y y and r r. Snce q = q, we fnd that the foregn frm provdes at least as much abatement n equlbrum ( a a ). Snce the suffcent second order condtons for maxmzaton are satsfed n the thrd stage, equatons (3) (5) allow us to mplctly defne the response functons, a ( t ), q ( t ) and ( ) r q t r t, =,. For =,, the margnal responses are as follows: = 0 3P Q QP Q <, (8) ( ) + ( ) c + c c + c a = q > 0, a = q > 0, (9) xx rr xx rr t t t t θcxxcrr cxxcrr = > 0, r = > 0, (0) t t θcrr crr where a t da dt, q t dq dt and r t dr dt, =,. Equatons (8) (0) are ntutve. An ncrease n the polluton tax reduces output and ncreases both abatement and R&D efforts.

14 Consder the second stage. The government chooses t [ 0,] to maxmze () subject to the polcy responses, equatons (8) (0), =, soluton, the frst order condton s = ( ) ( ). Assumng an nteror U ( Q) qt +Π t + E+ t D ( E) E t = 0, () where Πt dπ dt. Accordng to equaton (), the optmal tax accounts for the margnal effects on consumer surplus (decreases), domestc producer surplus (ambguous sgn) and the socal surplus from taxaton (ambguous sgn). Combnng equatons (8) (), equaton () reduces to a more ntutve expresson: ( Q) + ( ) ( ) ( ) ( ) U P Q q 8 cxx + crr cxx + c rr + e + t D E = 3P Q + Q t P Q P Q + QP ( ) 0. 3 ( ) ( Q) θcxxcrr cxxcrr The numerator of the frst term on the left hand sde has an ambguous sgn because U ( Q) > 0 and P ( Q) < 0. If, for example, U( Q) = Q and P( Q) () = Q, the numerator s postve and the denomnator s negatve. The bracketed term that multples ( ) t D E has a negatve sgn. Hence, the optmal polluton tax equals the margnal polluton damage f and only f the frst two terms on the left hand sde add up to zero. If the sum of these two terms s negatve (postve), then the optmal polluton tax s lower (hgher) than the margnal polluton damage. Ths s ntutve, snce the frst term represents the margnal effect on consumer surplus and the second term s the net margnal revenue from taxaton. If the margnal effect on consumer surplus s hgher (lower) n absolute value than the net margnal revenue from taxaton, the optmal polluton tax s lower (hgher) than the margnal polluton damage.

15 3.. RJVs We now turn our attenton to RJVs. We assume that the formaton of RJVs nvolve no setup or coordnaton costs n order to focus on the key compettveness ncentves that may lead a frm to prefer one type of RJV to the other Non-cooperatve RJV If the RJV s not cooperatve, the tmng of the game played by frms and the government s dentcal to the game played n the status quo. The frms payoff functons are as follows: ( ) θ ( ( β ), ) ( ) Π = P Q q c q + a r + r r t q a, (3) ( ) ( ( β ), ) ( ) Π = P Q q c q + a r + r r t q a, (4) where 0 β β. The parameter β s the R&D spllover rate that frm enjoys, =,. 4 Snce the foregn frm has superor technology, t seems logcal to postulate that t has also superor R&D capablty. For smplcty, we assume that β = and let β β. Equaton (), agan, provdes us wth the government s payoff functon. Plug β β and β = nto payoffs (3) and (4), respectvely. Assumng nteror solutons for the frms maxmzaton problems, the equlbrum n the thrd stage satsfes condtons (3) (5). The margnal responses are equatons (8), (0) and cxx + crr β cxx + crr at = + 0, 0 qt > at = + q t >. (5) θcxxcrr crr cxxcrr θcrr 3 See, e.g., Falvey et al. (03) for a settng wth market mperfecton, RJVs and coordnaton costs. 4 As n Kamen et al. (99), RJVs produce hgher R&D spllover rates than ndependent R&D efforts because there s nformaton sharng wthn RJVs. In the settngs wth ndependent R&D efforts, β = β = 0. Unlke Kamen et al. (99), we consder asymmetrc frms. Hence, t seems reasonable to assume that the spllover rates are also asymmetrc. 3

16 Equatons (5) nform us that the R&D spllover rates nfluence the frms abatement response rates under the non-cooperatve RJV. In the frst stage, the government chooses t [ 0,] to maxmze () subject to the frms optmal responses. Assumng an nteror soluton, we obtan the followng frst order condton, whch determnes the optmal tax: ( ) ( ) ( ) ( ) ( + ) + U Q P Q q β c c = 0. + c ( ) x 8 cxx + crr xx rr e t D E 3P Q + QP Q θcrr 3P ( Q) + QP ( Q) θcxxcrr cxxcrr (6) Equaton (6) dffers from equaton () n two sgnfcant ways: () there s an extra postve term on the left hand sde ( cx crr ) θ and () the last two ratos nsde of the bracketed term that multples t D ( E) dffer from ther counterparts n the prevous equaton. These changes correspond to the strategc nteractons between R&D spllovers Cooperatve RJV If the RJV s cooperatve, each frm observes the government s polluton tax and, n the thrd stage, chooses ts R&D effort to maxmze jont proft. In the fourth stage, the frms choose abatement and output levels smultaneously. The payoff functons for the government, foregn frm and domestc frm are (), (3) and (4), respectvely. The jont proft s ( ) ( ( β ) ) ( ) Π ˆ Π +Π = PQ te θc q + a r + r, r c q + a r + r, r. (7) In the fourth stage, an nteror equlbrum satsfes condtons (3) and (4). Note that these equatons agan mply q = q. The frst and second order condtons allow us to defne the mplct response functons, a( tr,, r ) and (,, ) q tr r, =,. For =,, the set of the margnal responses are gven by (8) and the followng equatons: 4

17 a = q > 0, a = q > 0, (8) q t t t t θcxx cxx = q = 0, (9) r r a = > 0, (0) r a = β > 0, a = > 0, () r r where q, ar a r and ar a r, =, and j. Equatons r q r (8) dffer from equatons (5) because n ths case the frms take the R&D efforts as gven n the last stage. Equatons (9) show that outputs do not change wth the frms R&D efforts. These results follow from our assumpton that c (.) s separable n x and r : c c x r = 0, =,. For each, equaton (0) shows that frm ncreases xr ts abatement supply at a one-to-one rate wth ts R&D effort. Equatons () capture the margnal spllover effects that R&D efforts create n the ndustry. The abatement supply of frm () rses at one-to-one ( β ) rate wth an ncrease n frm s ( s) R&D effort. In the thrd stage, frm chooses r to maxmze (7) subject to the frms responses. Assumng an nteror soluton, the frst order condtons yeld c x c r = and ( β ) c c x r + =. () Equatons () reveal that the frms choose R&D effort levels that equate the sum of margnal benefts to margnal costs. The R&D efforts nternalze R&D spllovers. Assumng that the suffcent second order condtons hold n the maxmzaton of jont proft, equatons () allow us to mplctly defne the response functons, r ( t ), =,. The margnal responses are as follows: 5

18 r t = > 0 and θc rr r t ( + β ) = > 0. (3) c rr Pluggng the R&D response functons n (3) nto the response functons for abatement ( ) ( ) and output, we have a tr, ( t), r ( t ) and, ( ), ( ) q tr t r t, =,. In the second stage, the government chooses t [ 0,] to maxmze () subject to the frms response functons. Assumng an nteror soluton, we obtan the frst order condton that determnes the optmal tax: ( Q) + ( ) ( ) ( ) ( + ) + ( + ) U P Q q θ β crr crr β c r + cx + e 3P Q + QP Q θcrrcrr cxcrr (4) 8 4c ( xx + c + β rr ) cxx + c rr + t D ( E) 0. = 3P ( Q) QP + ( Q) θcxxcrr cxxcrr Equaton (4) s more complex than equaton (6). The ncrease n the degree of complexty comes from the fact that the ndustry nternalzes R&D spllovers Lcensng Havng examned RJVs, we now turn our attenton to lcensng. Suppose that lcensng s the outcome n the frst stage. As we dscussed before, we examne a settng n whch the foregn frm sets the lcensng royalty fee after t observes the polluton tax. After observng the polluton tax and the royalty fee, the frms choose abatement, output and R&D levels n the last stage, takng each other s choces as gven. Let φ 0 denote the foregn frm s royalty fee. The frms payoffs are ( ) φ( ) θ (, ) ( ) Π = P Q q + q + a c q + a r r t q a, (5) ( ) θ ( ) ( ) φ( ) Π = P Q q c q + a r, r t q a q + a. (6) Observe that we wrte payoffs (5) and (6) wth both frms havng the same technology. 6

19 The foregn frm earns profts from supplyng output and sellng the lcense. The domestc frm faces the addtonal cost from purchasng the lcense. The equlbrum n the fourth stage satsfes the followng condtons: ( θ x) a t c = 0, a 0, t θc 0, (7) x ( x) a t φ θc = 0, a 0, t φ θc 0, (8) x ( ( ) ( ) θ x ) ( ) ( ) q P Q + P Q q c t = 0, q 0, P Q + P Q q θc t 0, (9) x ( ( ) ( ) ) ( ) x ( ) q P Q + P Q q θc t φ = 0, q 0, P Q + P Q q θc t φ 0, (30) x ( ) r c c = 0, r 0, c c 0, (3) x r x r ( ) r c c = 0, r 0, c c 0. (3) x r x r Snce we cannot guarantee that the Nash equlbrum s nteror, we wrte the Kuhn- Tucker condtons (7) (3). As we demonstrate n Appendx B, f we assume that (, ) = +, P( Q) = Q, U( Q) = Q and D( E) E c x r x r = (as we do n Sectons 3.4 and 4), an nteror equlbrum requres θ < 0.375, whch s nconsstent wth our assumpton that θ [ 0.6,]. Mantanng ths assumpton mples that a = 0 n the equlbrum wth lcensng. The other quanttes are postve. Thus, n what follows we consder the case n whch a > 0, a = 0, q > 0, r > 0, =, : t c x = θ, (33) t < φ+ θc a =, (34) x 0 ( ) ( ) θ P Q + P Q q = c + t, (35) x ( ) ( ) θ P Q + P Q q = c + t+ φ, (36) x 7

20 c = c, =,. (37) x r Condtons (33) (37) enable us to state the followng mportant results: Proposton. Suppose that θ [ 0.6,], c( x, r) = x + r, P( Q) ( ) = Q and D( E) E U Q =. Then, n the equlbrum wth lcensng, we have = Q, a > = ; q > q; r > r; x > x. (38) a 0 Proof. See Appendx B. Unlke the prevous scenaros, the foregn frm s share n the output market exceeds the domestc frm s share. By chargng a lcense fee, the foregn frm has market advantage n the supply of output. The lcense fee also deters the domestc frm from supplyng a postve amount of abatement. Assumng that the suffcent second order condtons hold n the maxmzaton problems n the fourth stage, condtons (33) (37) enable us to mplctly defne the response functons, a ( t, φ ), q ( t, φ ) and (, ) margnal responses wth respect to φ are as follows: a φ φ r t φ, =,. For =,, the ( crr + cxx ) P ( Q) + P ( Q) q ( )( + ) 3 ( ) + ( ) θ ( ) + ( ) = q = < 0, (39) P Q c c P Q P Q Q c c P Q P Q q q φ rr xx rr xx ( crr + cxx ) P ( Q) + P ( Q) q ( )( + ) 3 ( ) + ( ) θ ( ) + ( ) = < 0, (40) P Q c c P Q P Q Q c c P Q P Q q rr xx rr xx r φ = 0, (4) r φ = ( ) + ( ) crr P Q P Q q < 0. (4) P ( Q)( crr + cxx ) 3P ( Q) + P ( Q) Q θcrrcxx P ( Q) + P ( Q) q Equatons (39) reveal that the foregn frm ncreases output and reduce abatement n 8

21 response to an ncrease n the lcense fee. Equaton (40) nforms us that the domestc frm reduces output n response to an ncrease n the lcense fee. Equaton (4) shows that foregn frm s R&D effort s unaffected by changes n the lcense fee. The same, however, s not true for the R&D effort suppled by the domestc frm. Equaton (4) shows that ths frm s R&D effort decreases wth the lcense. Before we proceed wth the analyss of the thrd stage, t s useful to present the margnal responses wth respect to the polluton tax. They are as follows: q q t t ( crr + cxx ) 3P ( Q) P ( Q)( q q) θ crrcxx ( )( + ) 3 ( ) + ( ) θ ( ) + ( ) =, P Q c c P Q P Q Q c c P Q P Q q rr xx rr xx ( crr + cxx ) P ( Q)( q q) ( )( + ) 3 ( ) + ( ) θ ( ) + ( ) =, P Q c c P Q P Q Q c c P Q P Q q rr xx rr xx (43) (44) a r c + c =, (45) rr xx t q t θcrrcxx t = > 0, (46) θc rr ( )( ) c P Q q q xx rt = P ( Q)( crr + cxx ) 3P ( Q) + P ( Q) Q θcrrcxx P ( Q) + P ( Q) q. (47) As equatons (43) (45) and (47) make t clear, we are unable to sgn q, q, a and t t t r t because we do not know, a pror, the sgn of q q. As before, equaton (46) nforms us that the foregn frm ncreases ts R&D effort n response to an ncrease n the polluton =, tax. Note that f P ( Q) 0 q < 0, t a > 0 and t q = r = 0. t t In the second stage, the foregn frm chooses φ 0 to maxmze (5) subject to the optmal responses from the thrd stage. Assumng that the soluton s nteror, the frst order condton s 9

22 ( ) q + qφ P Q q+ φ = 0. (48) Snce q > 0 and + φ >. 5 q φ < 0 (see (40)), equaton (48) requres that P ( Q) q 0 Equaton (48) reveals that the optmal lcense satsfes the equalzaton of the slope of the domestc frm s reacton functon, q φ, to the slope of the foregn frm s so-proft curve, ( ) q P Q q+ φ. Assumng that the suffcent second order condton (.e., Π < 0 ) s satsfed, φφ equaton (48) defnes the mplct functon φ ( t), the foregn frm s best response n terms of ts lcense choce wth respect to the polluton tax. We obtan: ( ) φ ( ) ( ) qt + qφt P Q q+ + q P φ Q qq t + P Q q t φ t =, (49) Π where ( ) ( ) ( ) φφ Π φφ = qφ + P Q qq φ + P Q qφ + qφφ P Q q+ φ. 6 In the frst stage, the government chooses the polluton tax accountng for all response functons. Assumng an nteror soluton, the frst order condton yelds ( ) ( ) ( ) ( ( )) φ ( ) ( t ( ) ) ( ( )) U Q Q q P Q q E t D E E t t t + U Q Qφ q P Q qφ t D E E + + φ = 0. (50) Combnng equatons (39) (47) and (50), we obtan: 5 As we demonstrate n Appendx B, ths requrement s satsfed f ( ) ( ) ( ) 0 P Q = Q and c x, r = x + r. Gven these assumptons, the suffcent second order condton s also satsfed. 6 The sgn of the rato on the rght hand sde of (49) s ambguous n general. However, t s straghtforward =, c( x, r) = ( x + r ), D( E) = E to show that the sgn s negatve f P( Q) = Q, U( Q) Q { } and θ [ 0.6,]. Gven these assumptons, we obtan the followng results: φ = t + ( + θ) and φ ( θ) t = <

23 ( Q) qp ( Q) 3( crr + cxx ) P ( Q) θ crrc xx qp ( Q) P ( Q)( q q)( crr + cxx ) U + ( ) ( )( ) ( ) 6P Q P Q q q c 4 rr + cxx θcrrcxx crr + c xx + e + t D ( E) Ψ θcrrc xx U ( Q) q 4P ( Q) + P ( Q)( q+ q) P ( Q)( crr cxx ) + + φ t Ψ Ψ ( ) + ( ) θ ( )( + ) ( ) q P Q qp Q crrcxx qp Q crr cxx t D E + = 0, Ψ (5) where ( )( ) ( ) ( ) θ ( ) ( ) Ψ P Q c + c P Q + P Q Q c c P Q + P Q q > rr xx 3 rr xx 0. To get some ntuton for equaton (5), one should contrast t to equaton (), the equaton that determnes the optmal tax n the status quo. Even though the status quo nvolves frms wth dfferent technologes but dentcal output supples and the current settng nvolves frms wth dentcal technologes but dfferent output supples, the crucal dfference between equatons () and (5) s the extra component n equaton (5), whch relates to the effect of the polluton tax on the lcense fee. The terms n equaton () capture the effects of the polluton tax on consumer surplus, tax revenue and producer surplus for the domestc frm. Lkewse, the frst term on the left sde of equaton (5) captures the effects of the polluton tax on consumer surplus, tax revenue and producer surplus for the domestc frm. The second term on the left sde of equaton (5) s the extra net margnal socal beneft of the polluton tax through ts mpact on the lcense fee. Snce the lcense fee hurts the domestc frm and reduces ts output and abatement supples (relatve to the status quo), the government has an ncentve to set the polluton tax at a level that surpasses the optmal polluton tax n the status quo. In fact, f one consders the partcular quadratc functonal forms descrbed n Proposton, the optmal polluton tax n the lcense agreement s hgher than n the status quo (see secton 4).

24 3.4. Agree to nnovate? We now compare the equlbra and examne whch, f any, nnovaton agreement s mplementable. For comparson purposes, we need to make functonal form assumptons. Let P( Q) = Q, U( Q) Q =, c( x r) ( x r ), = + and D( E) E =. Let the superscrpts S, N, C and L denote equlbrum quanttes n the status-quo, noncooperatve RJV, cooperatve RJV and lcensng settngs, respectvely. Remember that an nnovaton agreement s feasble f and only f t satsfes both frms partcpaton constrants: the agreement must represent a Pareto mprovement relatve to the status quo. To derve some ntuton for the results, let us consder the frms payoffs for two effcency rates, θ = 0.6 and θ = 0.8, as functons of β. Fgures and show payoffs under the four possble scenaros. Fgure. Foregn frms payoffs, θ { 0.6, 0.8} and [ 0,] β. As Fgure reveals, for the foregn frm, the lcensng agreement domnates all alternatves, the non-cooperatve RJV s generally second best and the cooperatve RJV domnates the non-cooperatve RJV for suffcently hgh spllover values f θ = 0.8. In addton, the status quo domnates the non-cooperatve (and the cooperatve) RJV for small spllover values ( β < 0.) f θ = 0.6. The status quo domnance relatve to the cooperatve RJV ncreases as theta ncreases. The payoffs under the RJVs ncrease wth the spllover rate that ths frm enjoys.

25 Fgure. Domestc frms payoffs, θ { 0.6, 0.8} and [ 0,] β. Fgure nforms us that, for the domestc frm, cooperatve RJV domnates noncooperatve RJV, both RJVs domnate the status quo and the status quo domnates the lcensng agreement. The payoffs under the RJVs decrease wth the spllover rate enjoyed by the foregn frm, but they fall at a faster rate under the cooperatve RJV. We now show that the lcensng agreement s not mplementable because t volates the partcpaton constrant for the domestc frm. Fgure 3 reveals that ths frm s proft n the status quo s hgher than n the lcensng agreement for θ [ 0.6,]. Fgure 3. Domestc frm prefers the status quo to the lcensng agreement. As t s apparent n Fgure, the domestc frm always prefer ether type of RJV to the status quo. Fgure 4 compares the domestc frm s payoffs under the RJVs to ts payoff n the status quo when β =. These are the domestc frm s lowest payoffs as functons of the parameters. Thus, the domestc frm prefers any RJV to the status quo. 3

26 Fgure 5 shows that, for the domestc frm, the cooperatve RJV domnates the noncooperatve RJV for θ [ 0.6,] and [ 0,] β. Fgure 4. Domestc frm s nnovaton prema, β =. Fgure 5. The cooperatve RJV domnates the non-cooperatve RJV for the domestc frm Let us now examne the foregn frm s ncentves to enter nto a RJV wth the domestc frm. As we ponted out above, the foregn frm prefers the status quo to ether form of RJV f the effcency and spllover parameters are suffcently low. From Fgure, t s apparent that the nnovaton premum that ths frm obtans under each type of RJV (.e., the dfference between ts payoff under each type of RJV and ts payoff n the status quo) ncreases wth both parameters values. Indeed, as Fgure 6 reveals, f we restrct our analyss to parameter combnatons that satsfy θ [ 0.6,] and β [ 0.,], the 4

27 nnovaton premum under the non-cooperatve RJV s always postve. Fgure 6. Foregn frm s nnovaton premum (non-cooperatve RJV). Gven the results that Fgures and 6 present, we now expect that, whenever both RJVs are feasble, the foregn frm prefers the non-cooperatve RJV for most combnatons of effcency and spllover parameter values. In the relevant range, where both frms partcpaton constrants are satsfed for at least one type of RJV, Fgure 7 shows that the foregn frm rejects the cooperatve RJV n an area wth low values for the parameters. In addton, the foregn frm prefers the non-cooperatve (cooperatve) RJV n an area of ntermedary (hgh) values for the parameters. Fgure 7. Foregn frm s nnovaton acceptance areas. 5

28 We summarze the outcomes n the frst stage n the followng proposton. Proposton 3. Suppose that θ [ 0.6,] and β [ 0.,]. Then, for parameter values n () () () area I of Fgure 7, both frms select the non-cooperatve RJV; area II of Fgure 7, the choce of RJV s random; area III of Fgure 7, both frms select the cooperatve RJV. Proof. Consder Fgure 7. For parameters values n area I, the cooperatve RJV s not feasble because t volates the foregn frm s partcpaton constrant. Hence, both frms select the feasble, non-cooperatve RJV. For parameters values n areas II and III, both RJVs are feasble. In area II, the selecton s random because the frms dsagree about the top choce. In area III, the cooperatve RJV domnates for both frms. Q.E.D. The more effcent the foregn frm s relatve to the domestc frm and the lower the spllover rate that the foregn frm enjoys the more lkely t s that the outcome n the frst stage s the non-cooperatve RJV. The lower the asymmetry between the frms (both n terms of effcency and spllover rates) the more lkely t s that the outcome n the frst stage s the cooperatve RJV. 4. Domestc Compettveness and Welfare Havng examned the crcumstances under whch each type of RJV s mplementable, we now turn our attenton to mprovements n the compettveness of the domestc frm and domestc welfare that the mplementable RJVs may promote. We frst consder the effects on domestc compettveness. The compettveness of the domestc frm mproves whenever nnovaton occurs. We are not able to capture such an mprovement f we compare the frms n terms of ther shares of the output market n the settngs wth nnovaton and wthout nnovaton. Snce 6

29 the frms produce the same output quantty n the subgame perfect equlbra for the status quo and the RJVs, they have the same share of the output market. As both frms produce multple products (abatement and output) and exert R&D efforts, one should compare ther performances n terms of ther shares of the ndustry s proft. Snce the frms face the same output prce and the same prce ncentve to produce abatement (.e., the polluton tax), a comparson of proft shares yelds a precse compettveness measure. Let σ( βθ, ) ( βθ, ) ˆ ( βθ, ) Π Π denote the domestc frm s proft share as a functon of the foregn frm s spllover and effcency rates, respectvely. Fgure 8 shows that the domestc frm s proft share n the status quo ncreases as the technologcal gap decreases and t equals half n the absence of a technologcal gap. Interestngly, we also see that the domestc frm s proft share equals half despte the exstence or not a technologcal gap n the non-cooperatve RJV f the frms are dentcal n terms of the R&D spllover rates (.e., β = ). For a lower R&D spllover rate enjoyed by the foregn frm ( β = 0.4 ), the domestc frm s proft share n the non-cooperatve RJV s always hgher than the foregn frm s share. In the cooperatve RJV, the domestc frm s proft share equals half only f there s no technologcal gap and both frms are dentcal n terms of R&D spllover rates. As the R&D spllover rate enjoyed by the foregn frm decreases from β = to β = 0.4, the domestc frm s proft share n the cooperatve RJV ncreases gven the foregn frm s effcency rate, θ. 7

30 Fgure 8. Domestc frm s proft shares (status quo and RJVs) As Fgures, 4 and 8 reveal for some partcular effcency and spllover rates, t appears that the greater the degrees of asymmetry between the frms (n terms of effcency and spllover rates), the greater the domestc frm s relatve beneft from engagng n a RJV. Fgures 9 and 0 confrm ths ntuton for θ [ 0.6,] and [ 0.,] β. Fgure 9: Domestc frm s proft share n the non-cooperatve RJV. Fgure 0: Domestc frm s proft share n the cooperatve RJV. 8

31 Relatve to the status quo, a feasble RJV yelds an ncrease n the domestc frm s proft share because the R&D spllovers enable the domestc frm to reduce ts envronmental regulatory cost. The cost savngs produced by the nnovaton would occur even f the frm faced the same polluton tax n both scenaros. However, the costs savngs are more substantal because the government s optmal response to the mplementaton of an nnovaton agreement s to reduce the polluton tax, as Fgure reveals. It s also evdent that the polluton tax n the non-mplementable lcensng agreement s hgher than n the status quo n each case. Fgure. Polluton taxes, for θ { 0.6, 0.8}. One may worry that the lower polluton tax s assocated wth a hgher amount of polluton. However, consstent wth the prevous argument that an mplementable RJV mproves envronmental performance, we see n Fgure that polluton levels under the RJVs are lower than n the status quo and n the lcensng agreement. Fgure. Polluton levels, for θ { 0.6, 0.8}. 9

32 Let us now examne the welfare effects. It s straghtforward to show that the settngs wth RJVs are domnant n terms of welfare. As before, for llustraton purposes, we plot welfare as a functon of β for two effcency rates, θ { 0.6, 0.8}. Fgure 3 shows that, n each case, welfare s hghest under the cooperatve RJV, second best under the non-cooperatve RJV and lowest under lcensng. Fortunately, the lcensng agreement s not mplementable. Fgure 4 shows that the cooperatve RJV domnates the noncooperatve RJV not only n the partcular cases examned n Fgure 3, but n general. Fgure 3. Domestc welfare, θ { 0.6, 0.8}. Fgure 4. The cooperatve RJV s domnant n terms of welfare. Snce the cooperatve RJV domnates the non-cooperatve RJV n terms of welfare, a potental welfare mprovng polcy that the government may undertake s to brbe the foregn frm to select a feasble cooperatve RJV whenever t strctly prefers 30

33 a feasble non-cooperatve RJV. Remember that ths occurs n areas I and II of Fgure 7. N C Let τ( βθ, ) Π ( βθ, ) Π ( βθ, ) for all β [ 0.,] and [ 0.6,] C ( βθ, ) ( βθ, ) N θ such that Π Π denote the lump-sum transfer payment (subsdy) that the government makes to the foregn frm to nduce t to accept the cooperatve RJV when t C N prefers the non-cooperatve RJV. Assume that τ( βθ, ) = 0 f ( βθ, ) ( βθ, ) N T C Let W ( βθ, ) and W ( βθ, ) W ( βθ, ) τ( βθ, ) Π >Π. denote the welfare levels under the non-cooperatve RJV and under the cooperatve RJV net of the subsdy τ( βθ, ), respectvely. Fgure 5 shows that such a subsdy polcy s, ndeed, welfare mprovng T N because W ( βθ, ) > W ( βθ, ) for θ [ 0.6,] and [ 0.,] β. Fgure 5. Welfare mprovng subsdy polcy. 5. Concluson RJVs and lcensng are common nnovaton agreements. These agreements occur n many natons, developed and developng, and, n several nstances, they feature partcpaton of nternatonal frms. A substantal share of such agreements derves ts motvaton from attempts of ts partcpants to mprove ther envronmental performances. There s 3

34 evdence that n some cases the frms that form RJVs dsplay forward-lookng behavor wth respect to the occurrence of new (or more strngent) envronmental regulatons. Ths paper examnes the potental formaton of RJVs when, mperfectly compettve, asymmetrc frms antcpate the effects that the nteracton between each of three dfferent forms of nnovaton agreements and envronmental regulaton produce. The duopoly contans domestc and foregn frms. The regulator (domestc government) cares about domestc welfare, whch gnores the producer surplus that the foregn frm enjoys. We obtan several mportant results. Frst, we show that lcensng s not mplementable because t makes the domestc frm worse off relatve to the status quo. Second, we demonstrate that both forms of RJVs are mplementable: the non-cooperatve RJV s more lkely the greater the degrees of asymmetres (n terms of effcency and R&D spllover rates) between the frms whle the cooperatve RJV s more lkely the lower the degrees of asymmetres. Thrd, we fnd that mplementaton of both RJVs mprove the domestc frm s compettveness and domestc welfare. Welfare mprovements are larger under the cooperatve RJV. Fnally, we show that a subsdy polcy that nduces the foregn frm to accept the cooperatve RJV when t strctly prefers the non-cooperatve RJV s welfare mprovng. 3

35 Appendx We provde the proofs for Propostons and n Appendx A and B, respectvely. In Appendx C, we show the condtons that characterze the equlbra n the status quo and jont R&D arrangements under the partcular quadratc functonal forms descrbed n Proposton. In all equlbra, a > 0 for θ 0.6. Appendx A. Proof of Proposton. Combnng equatons (3) and (4), we obtan equatons (6), whch =. Then, q q mply P ( Q)( q q ) 0 = because P ( Q) 0 <. Equatons (3) mply c c because θ and c s ncreasng at an ncreasng rate n x and separable x x n x and r. Hence, x x. Combnng c c wth equatons (5) yelds c c, x x r r whch mples r r because c s ncreasng at an ncreasng rate n r and separable n x and r. Then, x x, r r and q = q mply a a. Q.E.D Appendx B. Proof of Proposton. In ths proof, we frst show that 0 a = for θ [ 0.6,]. Suppose that a > 0. In ths case, the equlbrum condtons are as follows, =, : = 8 ( 9+ 6 ) Ε ; φ = 4θ( 9+ 6θ) Ε ; q q + θ( + θ) t θ θ = = Ε ; = 7 + ( 8+ 3 ) Ε ; P ( Q ) = 9 3+ θ ( 5+ 3θ ) Ε ; a θ ( 4 θ ) Q θ θ ( ) r = ( θ ) Ε ; r ( 3 θ ) a = 9 θ θ Ε; = Ε; = 8 + Ε; ( ) e = 9 + θ( θ) Ε; E θ( θ) e = θ 7 + 3θ 9 Ε; { θ θ( θ( θ) ) } Π = Ε ; = Ε; { θ θ( θ( θ) ) } ( θ) W θ( 4+ 3θ) Π = Ε; = 7 + Ε. 33

36 Ε > 0. Hence, a > 0 f and only f θ < where ( θ)( θ) Snce a > 0 s nfeasble, the equlbrum condtons are as follows, =, : = ( ) 5 + ( ) Γ ; φ + θ ( + θ ) + θ ( + θ ) t θ θ θ θ = Γ; = + ( + ( + ( + 8 ))) Γ ; q θ( + θ ( + 0θ ) ) q 5 θ 65 θ 47 θ 3 θ = 5 7 Γ; = 5 + ( 75+ ( 64+ ( 4+ 8 ))) Γ ; P ( Q ) + θ + θ + θ ( + θ ) Q θ θ θ θ ( ( ( 8 ))) ( ( )) = ; Γ a = 5 + 4θ 0 + θ θ + θ ; Γ 0; r = 5 + 4θ 5 + θ 7 + 0θ Γ ; a = ( ) ( ) = + ( ) Γ e θ + θ + θ ( + θ ) r θ 5 θ θ ; ( ) = Γ ; = + ( ) Γ E θ ( + θ ) 5+ 4θ ( + θ ) e θ 5 θ θ ; = Γ ; { θ θ( θ( θ( θ( θ( θ θ ))))) } ; Π = Γ ( ) ( 0 ) W θ θ θ( θ) Π = Γ θ θ 5 θ 7 θ ; ( ) where Γ + θ + θ 08 + θ( θ) 5 95 > 0. We see that the result a a 0 Combnng condtons (33) (36), we obtan P ( Q)( q q ) { ( ) } = Γ; > = holds n the relevant range, [ 0.6,] 0 θ. >, whch mples q > q. The nequaltes a > a and q > q mply that y > y. Now, suppose that x x. Ths mples that c c. Combnng ths nequalty wth condtons (37), we obtan x x c c, whch yelds r r. Now, note that x x yelds y r y r, whch r r mples that y y r r 0. But, ths contradcts y > y. Hence, we must have x > x, whch mples r > r. QED 34