Transboundary Pollution, R&D Spillovers, Absorptive Capacity and International Trade

Transcription

1 Dscusson Paper No March 13, Transboundary Polluton, R&D Spllovers, Absorptve Capacty and Internatonal Trade Zeneb Dnar Abstract In ths paper, we consder a non-cooperatve and symmetrc three-stage game model composed by two regulator-frm herarches. By means of adequate emsson taxes, orgnal and absorptve research and development (R&D) subsdes we prove that regulators can reach the non-cooperatve socal optmum. In the presence of free R&D spllovers between countres, as well as the nvestment n absorptve research, the competton of frms on a common market helps non-cooperatng countres to better nternalze transboundary polluton. We fnd that n autarky and common market cases the nvestment n absorptve R&D leads to multple non-cooperatve equlbra, whch may necesstate competng regulators to coordnate an equlbrum. Interestngly, openng markets to nternatonal trade ncreases the per-unt emsson-tax and the per-unt orgnal research subsdy. It causes a hgher nvestment n orgnal research and producton, and a lower emsson rato. JEL D62 C72 H21 O32 Keywords Transboundary polluton; R&D spllovers; absorptve capacty; nternatonal trade Authors Zeneb Dnar, LAREQUAD and Faculté de Scences Economques et de Geston de Tuns, N 11 RUE GBELLI, Cté Ettadhamen 2041 Arana Tuns, dnarzeneb@yahoo.fr The author would lke to thank Slm Ben Youssef for hs rch comments. Ctaton Zeneb Dnar (2013). Transboundary Polluton, R&D Spllovers, Absorptve Capacty and Internatonal Trade. Economcs Dscusson Papers, No , Kel Insttute for the World Economy. Author(s) Lcensed under the Creatve Commons Lcense - Attrbuton 3.0

2 1 Introducton The am of ths paper s to characterze the socally optmal producton, nvestment n nventve research and development (R&D) and absorptve capacty. We used tax and subsdes rates n a game played by two regulator- rm herarches wth the negatve externalty and transboundry polluton whch usually does not lead non-cooperatng countres to the Pareto-optmalty. However, some authors showed that non-cooperatng governments can reach the rst best under some condtons (Hoel 1997, Zagonar 1998). By developng a statc two country, two-good general equlbrum model, Takarada (2005) examnated the welfare e ects of the transfer of polluton abatement technology when cross-border polluton exsts. He derved and nterpreted the condtons under whch technology transfer enrches the donor and the recpent. By d erent envronmental polcy nstruments for promotng technologcal change n polluton control such as drect controls, emssons subsdes, emssons taxes, free marketable permts and auctoned marketables permts, Mllman and Prnce (1989) showed that emssons taxes and auctoned permts proved the hghest rm ncentves to promote technologcal change. Jung et al. (1996) extended ths approach to a heterogenous ndustry. Stranlund (1997) consdered publc ad to encourage the adopton of superor emsson control technologes combned wth montorng. Ths strategy s attractve when montorng s d cult because the sources of polluton are wdely dspersed or the emssons are not easly measured as n non pont polluton problems. Technologcal ad reduces the drect enforcement e ort necessary for rms to reach the complance goal. Farzn and Kort (2000) examneted the e ect of a hgh polluton tax rate on abatement nvestment both under full certanty and when the tmng or the 2

3 sze of tax ncrease s uncertan. We showed that a hgher polluton tax encourages abatment nvestment f t does not exceed a certan threshold rate. Lao (2007) nvestgated the non-cooperatve and jontly optmal R&D subsdy polces of two exportng countres n the presence of nternatonal technology spllovers. He showed that when spllovers are low (hgh), the R&D game exhbts a negatve (postve) externalty, so the jontly optmal polcy s to tax (subsdze) R&D. Usng a non-cooperatve and symetrc three-stage game played by two regulator- rm herarches, Ben Youssef (2009) showed that free R&D spllovers and the competton of rms on the common market help non-cooperatng countres to better nternalze transfronter polluton. Surprsngly, nternatonal competton ncrease the per-unt emssons-tax and decreases the per-unt R&D subsdy. Conrad (1993) constructed a model of nternatonal duopoly wth negatve externaltes n producton n whch optmal envronmental polcy responses to foregn emssons tax and subsdy programs can be calculated. However, he dd not consder R&D possbltes and took the context of nternatonal market. Also, wth a model of mperfectly compettve nternatonal markets and wthout polluton, Spencer and Brander (1983) showed that there are natonal ncentves to subsdze R&D f export subsdes are not avalable. R&D actvtes generate nnovatons and develop the rm s ablty to dentfy, assmlate, and explot knowladge from the envronment, for ths Cohen and Levnthal (1989) were the rst to ntroduce the dea of absorptve capacty n the cost reducton R&D lterature. Contrary to D Aspremont and Jacquemn (1988, 1990) and Kamen et al. (1992), where R&D spllovers are assumed exogenous and cost free, Poyago-Theotoky (1999) showed that, when spllovers of nformaton are endogenzed, non-cooperatve rms never dsclose any of 3

4 ther nformaton, whereas they wll always fully share ther nformaton when they cooperate n R&D. Kamen and Zang (2000) modeled a rm s e ectve R&D level that re ects how both ts R&D approach ( rm spec c or general) and R&D level n uence ts absorptve capacty. Leahy and Neary (2007) spec ed a general model of the absorptve capacty process and showed that costly absorpton both rases the e ectveness of own R&D and lowers the e ectve spllover coe cent. Ths weakens the case for encouragng research jont ventures, even f there s complete nformaton sharng between rms. Mllou (2009) showed that the lack of full approprablty can lead to an ncrease n R&D nvestments. Hammerschmdt (2009) dstngushed between two types of R&D: nventve (or orgnal) R&D that generates new knowledge and absorptve R&D that allows a rm to bene t from the nventve R&D conducted by others. She found that rms wll nvest more n R&D to strengthen absorptve capacty when the spllover parameter s hgher. Ben Youssef and Zaccour (2009) were the rst to ntegrate nto the same model absorptve R&D and polluton control. They have compared the socally optmal levels of orgnal and absorptve research, and the socally optmal subsdes for both types of R&D. There are no free R&D spllovers between rms, Ben Youssef (2010) showed that the nvestment n absorptve research enables non-cooperatng regulators to better nternalze transboundary polluton. In contrast, Ben Youssef (2009) showed that free R&D spllovers and the competton of rms on a common market help non-cooperatng countres to better nternalze transboundary polluton. The d erence n our model and the model of Ben Youssef (2009) s to ntegrate the nvestment n absorptve research. We wsh to show how R&D spllovers, the nvestment n absorptve research and the competton of rms on the 4

5 common market help non-cooperatng countres to nternalze transboundary polluton. We consder a non-cooperatve and symmetrc three-stage game consstng of two dentcal regulator- rm herarches. Each rm produces one good sold on the domestc market n the thrd stage and they can nvest n orgnal and absorptve research whch drectly reduces ts emsson/output rato, n the second stage. In the rst stage, regulators announce non-cooperatvely ther per-unt emsson tax and R&D subsdes and them ams are maxmzng hs socal welfare functon to reach the non-cooperatve socal optmum. Ths game s solved backward to get a sub-game perfect Nash equlbrum. We show that regulators can nduce ther rms to mplement the non-cooperatve socally optmal levels of producton and R&D by usng three regulatory nstruments, whch are a per-unt emsson tax, a per-unt orgnal research subsdy and a per-unt absorptve research subsdy. Moreover, n autarky and common markets cases, the nvestment n absorptve R&D may leads to a multplcty of subgame perfect Nash equlbra necesstatng the coordnaton on an equlbrum, whch consttutes an ncentve for non-cooperatng countres to cooperate. Interestngly, we show that wthout R&D spllovers and the ablty to absorb (l = 0; = 0), transboundary polluton s completely not nternalzed n the autarky regme. The hgher are the ablty to absorb and the R&D spllovers, the greater s the proporton of transboundary polluton nternalzed by non cooperatng countres. Moreover, openng markets to nternatonal trade help countres to better nternalze transboundary polluton through rms competton on the common market, whch are realzed by an ncrease n the level of 5

6 the orgnal and absorptve research. Consequently, the emsson rato s lower, enablng states to produce more n common market. The paper has the followng structure. In Secton 1, we ntroduce the model. Secton 2 presents the basc model n autarky, resolves t and exhbts the role of the R&D spllovers and the absorptve capacty n the nternalzaton of transboundary polluton. Secton 3 deals wth the case where markets are opened to nternatonal trade and shows how ths contrbutes to nternalze transbounder polluton, n Secton 4 we compare the non-cooperatve socally optmal values n autarky and common markets, and n secton 5 we conclude. An appendx contans some proofs. 2 Autarky We consder a symmetrc model consstng of two countres and two rms. Frm, located n country, s a regonal monopoly and produces good n quantty q sold on the domestc market havng the followng nverse demand functon p = a bq where a; b > 0. One reason for the market structure used s that the markets of the ndustres engaged n large nvestments n R&D are usually olgopolstc. The producton process generates polluton and rms can nvest n R&D n order to lower ther xed emsson/output rato. We dstngush between nventve or orgnal research, denoted by x o, whch drectly reduces the emsson rato and costs k o (x o ) 2, where k o > 0, and absorptve research, denoted by x a, whch enables a rm to capture part of the orgnal research made by the other one, and costs k a (x a ) 2, where k a > 0. 6

7 The nnovaton actvty carred out by rms s caracterzed by postve externaltes whch mply that a proporton of each rm s R&D level gratutously spllovers to the other rm and by absorptve capacty s mplctly assumed that the ablty to absorb spllovers from other rm. The e ectve R&D level of rm s x = x o + ( + lx a )x o j,where 0 < 1 and l > 0: By normalzng the emsson per unt of producton to one wthout nnovaton, the emsson/output rato of rm s e = 1 x o ( +lx a )x o j and ts emsson of polluton s E = h 1 x o ( + lx a )x o j q. Snce rm s a regonal monopoly that pollutes the domestc envronment, t s regulated. Each regulator behaves non-cooperatvely and maxmzes hs own socal welfare functon by usng three regulatory nstruments: an emsson tax per unt of polluton t f to nduce the noncooperatve socally optmal levels of producton and polluton, a subsdy per unt of orgnal R&D level r of and a subsdy per unt of absorptve R&D level r af to nduce the non-cooperatve socally optmal levels of e ectve R&D and emsson/output rato. Therefore, each regulator chooses the non-cooperatve socally optmal per-unt emsson tax and per-unt R&D subsdes n the rst stage gven that the reacton of hs rm whch s t chooses ts optmal levels of R&D and producton n the second and thrd stages, respectvely. Ths three-stage game s solved backward to get a the subgame perfect Nash equlbrum. denotng the margnal cost of producton by > 0, the pro t of rm s f = p (q ) q q k o (x o ) 2 k a (x a ) 2, and ts pro t net of taxes and subsdes s V f = f t f E + r of x o + r af x a. Conjecture 1 We conjoncture that lm k o ;k a!+1 xof = lm k o ;k a!+1 xaf = 0 7

8 Ths conjecture s logcal because when the nvestment cost parameters are relatvely very hgh, t s socally optmal to not nvest n R&D. Transboundary polluton s also a negatve externalty among countres. Thus, the damages caused to country are D = E + E j where > 0 s the margnal damage of the domestc polluton, and > 0 s the margnal damage of the foregn polluton. 1 The consumer surplus n country engendered by the consumpton of q f CS f = Z q 0 p (u) du p (q ) q = b 2 q2. s The socal welfare of a country s de ned as the consumer surplus, mnus damages and subsdes, plus taxes and the net pro t of the domestc rm, and s equal, after smpl catons, to: S f q ; q j ; x o ; x a ; x o j; x a j = CS f D + f (1) Notce that taxes and subsdes do not appear n the socal welfare functon because the tax dmnshed from the rm s pro t s added to the consumer welfare, and the subsdes added to the rm s pro t are dmnshed from the 1 Notce that, even when and are d erent, the model stll remans symmetrc because these parameters are the same for the two countres. Ths damage functon can explan a pure transfronter polluton problem when = d(1 c) and = dc, where 0 < c < 1 s the proporton of polluton of rm j exported to country. It can also explan an nternatonal envronmental problem, when =, because damages n one country become a functon of the whole polluton. To explan how transfronter polluton can be nternalzed, we separate the negatve e ect of the foregn polluton from the one of at home polluton by separatng and. 8

9 consumer welfare. 2.1 The reacton of rms The regulator announced n the rst stage the per-unt emsson tax and the per-unt R&D subsdes, the rm reacts by choosng ts optmal research and producton levels n the second and thrd stages, respectvely. By backward nducton, the rm maxmzes n the thrd stage ts net pro t wth respect to ts producton level, and n the second stage, t maxmzes ts net pro t wth respect to ts R&D levels. The rst order condton of rm thrd stage = 0 (2) The resoluton of (2) gves: q f = a t [1 x o + ( + lx a ) x o ] 2b (3) We deduce the o o = t a = t ( + lx a ) 2b = t lx o j a j = 0 Consder the case of a postve emsson tax. When a rm ncrease ts level of orgnal or a absorptve research, ts emssons/output rato decrease enablng 9

10 t to expand ts producton. When the competng rm ncrease ts orgnal research, ths has a postf e ect on the producton of the rm: because of R&D spllovers and absorptve capacty, the emsson rato of rm decrease enablng t to expend ts producton. The symmetrc optmal producton level for each rm s obtaned from expresson (3): q f = a t [1 (1 + + lx a ) x o ] 2b (4) The rst-order condtons of rm second stage are 2 : dv f dx o = 0 (5) and dv f dx a = 0 (6) At the equlbrum, by usng (2), equatons (5) and (6) are smpl ed, and the symmetrc 3 solutons are gven by the followng equatons system : 2 The second-order condtons are ver ed n the appendx when k o and k a are hgh enough. 3 The model s symmetrc for ths we look for the symmetrc equlbrum. Further, as wll be made clear n the followng secton, the backward resoluton of the game s stopped at the second stage, whch explans why t s approprate to look for symmetrc equlbra at ths second stage. 10

11 t f q f + r of 2k o x o = 0 (7) and t f lx o q f + r af 2k a x a = 0 (8) where q f s gven by (4). 2.2 The Non-Cooperatve Socally Optmal Emsson Tax and R&D Subsdes At the rst stage, each regulator maxmzes hs socal welfare, gven by (1), wth respect to t f, r of and r af whch are the choce varables. However, ths drect method s not easy to do f the regulator looks drectly for the optmal per-unt emsson tax and per-unt R&D subsdes. Therefore, we wll use a smpler method. Indeed, the regulator maxmzes, respectvely n the thrd and second stages, hs socal welfare wth respect to the producton quantty and the R&D levels whch become the new choce varables. Then, by equalzng the socally optmal quanttes obtaned to those chosen by hs rm, he determnes the socally optmal per-unt emsson tax and per-unt R&D subsdes. The model s resolved as f t was a two-stage game. Expresson (1) can be wrtten : S f = b 2 q2 h 1 x o + ( + lx a ) x o j q h 1 x o j ( + lx a j )x o qj (9) + (a bq ) q q k o x o2 k a x a2 11

12 In the thrd stage, when regulator chooses hs socally optmal producton level, the parameter s elmnated by the dervaton of S f wth respect to q. Thus, the polluton comng from country j s completely not nternalzed. Ths s general for statc models wth a damage functon lnear wth respect to the whole polluton, or a separable one wth respect to the polluton remanng at home and the one receved from other countres. However, when he chooses hs optmal level of nventve research n the second stage, the negatve transboundary externalty s partally nternalzed f the learnng parameter and the R&D spllovers are non-nl. The hgher the absorptve parameter and R&D spllovers are, the greater proporton of transboundary polluton s nternalzed. The rst-order condton of regulator thrd stage = 0 (10) The resoluton of (10) gves ^q f = a h 1 x o ( + lx a )x o j b (11) The symmetrc expresson of (11) s ^q f = a [1 (1 + + lxa )x o ] b (12) A su cent condton for the symmetrc producton quanttes to be postve s 12

13 a > + () a > (13) Thus, the margnal domestc damage cost of polluton s lower than the maxmum wllngness to pay for the good mnus ts margnal cost of producton. The rst-order condtons of regulator second stage are 4 ds f dx o ^q o = 0 (14) and ds f dx a ^q a = 0 (15) At the equlbrum, by usng (10) ; equatons (14)-(15) s smpl ed, and the symmetrc solutons are gven by the followng equatons system b h + + lx af ^q h lx af x of + lx af 2bk o x of = 0 (16) and lx of ^q f 2k a x af = 0 (17) 4 The second-order condtons are ver ed n the appendx when k o and k a are hgh enough. 13

14 When we replace ^q f by ts symmetrc expresson, the equatons (16)- (17) become: (a ) h + + lx af h lx af h lx af 2bk o x of = 0 x of (18) and lx of h a + (1 + + lx af )x of 2bk a x af = 0 (19) The non-lnear equatons system (18) and (19), con rms the fact that when the learnng parameter and the R&D spllovers are nl (l = 0; = 0), transboundary polluton s completely not nternalzed snce dsappears from (18). Consequently, we can obtan the expressons of ^x of and ^x af explctly. The hgher l and/or s, the greater proporton of transboundary polluton s nternalzed. Thus, we can deduce the followng proposton. Proposton 2 The R&D spllovers and the nvestment n absorptve research enable non-cooperatng countres to better nternalze transboundary polluton. The hgher and/or the ablty to absorb are, the greater s the proporton of transboundary polluton nternalzed. Solvng of the non-lnear equatons system (18)-(19) gves the symmetrc socally optmal R&D levels denoted by ^x of and ^x af. Accordngly, we can not have the explct solutons. For ths reason we get the follownd proposton. Proposton 3 When k o and k a are su cently hgh, there are at least one 14

15 and at most ve couples of real solutons ^x of non-lnear equatons system gven by (18) and (19). > 0 and ^x af > 0 that solve the PROOF. See the appendx. The above proposton shows the possblty of multple symmetrc equlbra maxmzng the socal walfare. In ths case, non-cooperatng regulators have to coordnate on an equlbrum, whch consttutes an ncentve for them to fully cooperate. Therefore, the possblty to nvest n absorptve research may gve ncentves to cooperate. When k o and k a are hgh enough, the condton (13) and the conjecture guarantee that the socally optmal levels of research, producton, and polluton are postve, and that 0 + lx a 1. Snce the emsson tax and the R&D subsdes are set to ncte rms to reach the socally optmal producton and research levels whch are ^q f ; ^x of and ^x af, then from equatons (4), (7), and (8) we have the optmal emsson tax and R&D subsdes : t f = a 2b^q 1 (1 + + l^x af )^x of (20) and r of = 2k o^x o t f ^q f (21) and 15

16 r af = 2k a^x a t f l^x o ^q f (22) Proposton 4 In the autarky regme, the regulator can nduce ther rms to reach the noncooperatve socally optmal levels of producton and R&D by usng the three regulatory nstruments, whch are a per-unt emsson tax, a per-unt orgnal research subsdy, and a per-unt absorptve research subsdy. Ths proposton shows that necessty of the three regulatory nstruments to ncentve for rms to mplement the socally optmals levels of producton and R&D. By usng the conjecture, the expressons (12) and (20), we obtan: lm k o ;k a!+1 tf = 2 (a ) () < a 2 (23) Consder the case when k o and k a are hgh enough. Thus, when the margnal damage of polluton s su cently low, the tax s negatve meanng that each regulator actually subsdzes polluton (or producton because they are proportonal) to deal wth the monopoly dstorton. From (18) and (19), we have: lm ko^x o k o ;k a =!+1 (a ) + (a 2) 2b (24) and lm ka^x a k o ;k a = 0 (25)!+1 16

17 By usng (24) and (25) n (21) and (22), we get: lm k o ;k a!+1 rof = (a )2 + (a 2) b > lm k o ;k a!+1 raf = 0 (26) For ths we state the followng proposton, Proposton 5 If a > + 2, then when the nvestment-cost parameters are su cently hgh, the per-unt R&D subsdy for nventve research s hgher than the one for absorptve research. Notce that condton (13) s sats ed and k o and k a are hgh enough, then when a > + 2 the subsdy for orgnal research s strctly postve. The nvestment n absorptve research s socally desred. Ths result s smlar to a ndng of Ben Youssef (2010). 3 Internatonal trade In ths secton, t s assumed that when markets are opened to nternatonal b trade, the nverse demand foncton s p = a 2 (q + q j ). The rms pro ts are c = p (q ; q j ) q q k o (x o ) 2 k a (x a ) 2 and ther net pro ts arev c = c t c E + r oc x o + r ac x a, wth t c s the emsson tax per-unt of polluton, r oc s the subsdy per-unt of orgnal R&D level and r ac subsdy per-unt of absorptve R&D level. s the As n autarky, we make the followng conjecture Conjecture 6 lm k o ;k a!+1 xoc = lm k o ;k a!+1 xac = 0 17

18 The total consumer surplus s equally dvded between the two symmetrc countres : CS f = qz +q j 0 p (u) du p (q ; q j ) (q ; q j ) = b 8 (q + q j ) 2 And the socal welfare of country s S c q ; q j ; x o ; x a ; x o j; x a j = CS c D + c (27) 3.1 The reacton of rms By backward nducton, at the thrd stage each rm maxmzes ts net pro t wth respct to ts producton level and at second stage, t maxmzes ts net pro t wth respect to ts R&D levels. The rst-order condtons of the rms thrd stage j c = 0 j The resoluton of system (28) gves: q c = 2 h h a + tj 1 x o j + lx a j x o 3b 2t 1 x o [ + lx a ] x o j (29) The partel dervatons set for the symmetrc case are: 18

19 @q o j = 2t 3b (2 = 2t 3b (2 [ + lxa ] [ + lxa a a j = 4lxo 3b t = 2lxo 3b t Consder the case of a postve emsson tax. When a rm ncrease ts level of orgnal or a absorptve research, ts emssons/output rato decrease enablng t to expand ts producton. When the competng rm ncrease ts orgnal research, ths has tow opposte e ects on the producton of the rm: because of R&D spllovers and absorptve capacty, the emsson rato of rm decrease enablng t to expend ts producton; the second e ect s a negatve one and oblges the rm to decrease ts producton because the competng one can ncrease ts producton due to the decrease of ts emsson/output rato.when and/or l are hgh enough, the rst postve e ect domnates. When the competng rm ncrease ts absorptve capacty, ts emssons rato decrease enablng t to expend ts producton whch forces the rm to reduce ts producton. The symmetrc expresson of (29) s: q c = 2 [a t (1 (1 + + lx a ) x o )] 3b (30) The rst-order condtons of the rm s second stage are: dv c dx j o = 0 (31) and 19

20 dv c dx j a = 0 (32) At the equlbrum, by usng (28), (31)-(32) are smpl ed, and the followng equatons are sats ed for symmetrc solutons: 2 3 (2 lxa ) t c q c + r oc 2k o x oc = 0 (33) and 4 3 tc lx o q c + r ac 2k a x ac = 0 (34) where q c s gven by (30) :Ths system contans two equatons and two unknown varables whch are x oc and x ac. 3.2 The Optmal per-unt Emsson Tax and R&D Subsdes Gven that the socally optmal per-unt emsson-tax and per unt R&D subsdes are reached n the rst stage, regulators determne the socally optmal producton and R&D levels n the thrd and second stages, respectvely. Then, by equalzng the socally optmal quanttes obtaned to those chosen by the taxed and subsdzed rm, they determne the socally optmal per-unt emsson tax and per-unt R&D subsdes. The rst-order condtons of the regulators thrd stage j = 0 (35) 20

21 The resoluton of ths system gves: h 2 (a ) + x o 3 ( + lx a j ) + x o j (3( + lx a ) 1) ^q c = 2b (36) The transboundary polluton s completely not nternalzed because the above quantty does not depend on the margnal damage of the foregn polluton (). The symmetrc producton quanttes are gven by (36) s: ^q c = [a + (1 + + lxa )x o ] b (37) In order that the socally optmal producton quanttes must be postve we found the same condton n the autarky regme. Ths condton s as follows. a > + () a > (38) The rst-order condtons of regulator s second stage are ds c dx o ^qc o = 0 (39) and ds c dx a a = 0 (40) At the equlbrum, equatons (39)-(40) are smpl ed by usng (35), and the symmetrc solutons verfy the followng equatons system: 21

22 2b [ + ( + lx ac )] ^q c [3 ( + lx ac ) 1] (41) [1 (1 + + lx ac ) x oc ] 4bk o x oc = 0 and 2blx oc ^q c + l [1 (1 + + lx ac ) x oc ] x oc 4bk a x ac = 0 (42) where ^q c s gven by (37) ; the system (41)-(42) are equvalent to: 2 (a ) [ + ( + lx ac )] [1 (1 + + lx ac ) x oc ] (43) [2 + 5 ( + lx ac ) ] 4bk o x oc = 0 and lx oc [2 (a ) + (1 (1 + + lx ac )x oc ) ( 2)] 4bk a x ac = 0 (44) From ths system, we can state the followng proposton. Proposton 7 In addton, the competton of rms on the commun market enable non-cooperatng countres to better nternalze transboundary polluton. The hgher and/or the ablty to absorb s, the greater s the proporton of transboundary polluton nternalzed. As n autarky, the resoluton of the non-lnear equatons system (43)-(44) gves two equatons wth two unknown varables whch are the symmetrc socally 22

23 optmal R&D levels denoted by ^x oc and ^x ac. Moreover, we are not able to nd the explct solutons. Indeed, we have the follownd proposton Proposton 8 When k o and k a are su cently hgh, there are at least one and at most ve couples of real solutons ^x oc non-lnear equatons system gven by (43) and (44). > 0 and ^x ac > 0 that solve the PROOF. See the appendx. By equalzng the producton level chosen by rms, we determne the socally optmal emsson tax: t c = 2 (a ) 3b^q c 2 [1 (1 + + lx ac )x oc ] (45) And Equatons (33), and (34) gve the socally optmal R&D subsdes: r oc = 2k o x o 2 3 (2 lxa ) t c ^q c (46) and r ac = 2k a x a 4 3 tc lx o ^q c (47) Thus, we can establsh the followng proposton. Proposton 9 When there s a common market, by usng the three regulatory nstruments, whch are a per-unt emsson tax, a per-unt orgnal research subsdy, and a per-unt absorptve research subsdy, regulators can push ther 23

24 rms to mplement the noncooperatve socally optmal levels of producton and R&D. By usng the conjecture, (45) and (46), we obtan: lm k o ;k a!+1 tc = 3 (a ) 2 < 0 () < a 3 (48) Therfore, when the margnal damage of polluton s hgh enough, the regulator taxes polluton (or producton) and when ts low enough, he subsdzes producton to deal wth the monopoly dstorton. Further, from (43) and (44), we have: lm k o ;k a!+1 ko x oc = 2 (a ) + (2a 2 5) + 4b (49) and lm k o ;k a!+1 ka x ac = 0 (50) By usng (50) and (49) n (47) and (46), we get: lm k o ;k a!+1 roc 2(a )[3( 1)+(2 )(a )]+3[2(a )+(1 3)] = (51) 2b lm k o ;k a!+1 rac = 0 (52) The followng proposton compares the subsdy rates of e orts n orgnal and absorptve R&D, when markets are opened to nternatonal trade. 24

25 Proposton 10 When markets are opened to nternatonal trade, and when the nvestment-cost parameters are su cently hgh, the per-unt R&D subsdy for nventve research s hgher than the one for absorptve research, when < 1 3. Ths proposton mply that when k o and k a are hgh enough. Thus, when the margnal damage of polluton s su cently low, the subsdy for orgnal research s always postve. 4 Common market versus autarky In ether case we have studed each regulator chooses the socally optmal producton and R&D levels and, by means of the per-unt emsson tax, a per-unt orgnal research subsdy, and a per-unt absorptve research subsdy, puches hs rm to mplement them for the two market regmes. For ths, we compare the taxes and subsdes also that the orgnal research of the both cases. Subsequently, to smplfy our computatons, we wll replace the margnal damage of the domestc polluton by the margnal damage of the foregn polluton, e =. Proposton 11 When the markets are opened to nternatonal trade, and when the nvestment-cost parameters are su cently hgh, the orgnal research ncreases. PROOF. See the appendx. 25

26 The better nternalzaton of transboundary polluton generated by competton n the common market s realsed by an ncrease of the level of the orgnal research when the nvestment-cost parameters are su cently hgh. Consequently, the absorptve research ncrease and the emsson rato s lower, whch encourages rms to produce more n common market. Proposton 12 The per-unt emsson tax and the per-unt R&D subsdy for nventve research are greater n common market than n autarky, when the nvestment-cost parameters are su cently hgh. PROOF. See the appendx. Openng markets to nternatonal trade and nvestment n absorptve research better nternalze transboundary polluton and ths procure by an ncrease of the emsson tax and of the R&D subsdy for nventve research. Ths result d ers from that of Ben Youssef (2009) where he has shown that the emsson tax s hgher but the R&D subsdy s lower when s low enough. We note that f there s no negatve externalty between countres, so many optmal values n autarky and commun market become equal. In fact, f = 0, equatons system (18)-(19) and (43)-(44) and show that the R&D levels are equal whch mples that producton, polluton, and socal welfare are equal n the two market regmes. 26

27 5 Concluson We have developed a non-cooperatve three-stage game model composed by two regulator- rm herarches n presence of transborder polluton, the R&D spllovers and the absorptve capacty. We study the e ects of the postve R&D externalty, the ablty to capture part of orgnal research developed from other rms and nternatonal trade on the nternalszaton of the transboundary pollluton. Frms have the possblty to nvest n orgnal and n absorptve research to reduce ther emsson/output rato. Indeed, we evaluate the mpact of nternatonal competton on the orgnal research, by means the emsson-tax and the R&D subsdes. We show that free R&D spllovers, the nvestment n absorptve research and the common markets enable non-cooperatng countres to better nternalze transboundary polluton. The hgher the learnng parameter of absorptve capacty and the R&D spllovers are, the hgher the proporton of transboundary polluton nternalzed s. Interestngly, for autarky and nternatonal trade cases the learnng ablty of rms may lead to multple subgame perfect Nash equlbra necesstatng noncooperatng countres to coordnate on an equlbrum, whch consttutes an ncentve for them to cooperate. Accordngly, transboundary polluton wll be completely nternalzed and the rst best outcome may be reached. Openng markets to nternatonal trade helps competng countres to better nternalze transboundary polluton through the competton rms on the common market. Consequently, the per-unt emsson tax and the per-unt subsdy for nventve research ncrease wth market opened for nternatonal trade. 27

28 6 Appendx A) In autarky case, the second-order condtons of the rms second stage consder the Hessan matrx: 0 H V = d 2 V f dx o2 d 2 V f dx o xa d 2 V f dx o xa d 2 V f dx a2 1 C A By usng the rst-order condtons gven by (5) and (6), we can determne the second dervatves constng matrx H V f whch can be wrtten as: 0 1 g 1 2k o g 2 H V = B g 2 g 3 2k a A where g ;=1;2;3 are polynomal functons n t f and x of (symmetrc case). Snce lm k o ;k a!+1 xof and lm k o ;k a!+1 tf are nte numbers, then g take nte values when k o and k a tend to +1: Therefore, when k o and k a are su cently hgh: a. d2 V f dx o2 < 0 and d2 V f dx a2 < 0 b.det H V = (g 1 2k o ) (g 3 2k a ) g 2 2 > 0: Thus, we have a maxmum when k o and k a are hgh enough. B) Second-order condtons of the regulators seconds stage, n autarky case, consder the Hessan matrx: 28

29 0 H S = d 2 S f dx o2 d 2 S f dx o xa d 2 S f dx o xa d 2 S f dx a2 1 C A By usng the rst-order condtons gven by (14) and (15), we can determne the second dervatves constng matrx H S f 0 1 f 1 2k o f 2 H S = B f 2 f 3 2k a A whch can be wrtten as: where f ;=1;2;3 are polynomal functons n t f and x of (symmetrc case). Snce lm k o ;k a!+1 xof = lm k o ;k a!+1 xaf k a tend to +1: = 0, then f take nte values when k o and Therefore, when k o and k a are su cently hgh: a. d2 S f dx o2 < 0 and d2 S f dx a2 < 0 b.det H S = (f 1 2k o ) (f 3 2k a ) f 2 2 > 0: Thus, we have a maxmum when k o and k a are hgh enough. C) Proof of proposton 3 From, we deduce: x af = l h a + (1 + ) x of x of 2bk a 2 l 2 x of2 (53) Expresson (18) s equvalent to: 29

30 (a ) + (a 2) + l (a 2) x af (54) + [(1 + ) ( + 2) 2bk o ] x of + [ + 2 (2 + 1)] lx af x of +2l 2 x af2 x of = 0 by usng (53) n (54), and then multplyng by 2bk a 2 2 l 2 x of2, we get a polynomal functon of degree 5 n x of ; A x of = 0. The constant term of A s 4b 2 [ (a ) + (a 2)] k a2 > 0 and the coe cent of s 2bk o 4 l 4 < 0. 5 x of We have A (0) > 0 and lm A x of k o ;k a =!+1 one and at most ve real and postve roots ^x of. Snce ^x of (54) and condton (13), we have ^x af 1, then A x of admts at least > 0, from expresson > 0 when k o and k a are hgh enough. D) Second-order condtons of the rms second stage n trade nternatonal case consder the Hessan matrx: 0 H V = d 2 V c dx o2 d 2 V c dx o xa d 2 V c dx o xa d 2 V c dx a2 1 C A By usng the rst-order condtons gven by (31) and (32), we can determne the second dervatves constng matrx H V 0 1 g 1 c 2k o g c 2 H V = B g2 c g3 c 2k a A whch can be wrtten as: where g c ;=1;2;3 are polynomal functons n t f and x of (symmetrc case). 30

31 Snce lm k o ;k a!+1 xoc and lm k o ;k a!+1 tc are nte numbers, then g c take nte values when k o and k a tend to +1: Therefore, when k o and k a are su cently hgh: a. d2 V c dx o2 < 0 and d2 V c dx a2 < 0 b.det H V = (g c 1 2k o ) (g c 3 2k a ) g 2 2 > 0: Thus, we have a maxmum when k o and k a are hgh enough. E) Second-order condtons of the regulators seconds stage consder n trade nternatonal case the Hessan matrx: 0 H S = d 2 S c dx o2 d 2 S c dx o xa d 2 S c dx o xa d 2 S c dx a2 1 C A By usng the rst-order condtons gven by (39) and (40), we can determne the second dervatves constng matrx H S whch can be wrtten as: 0 1 f 1 2k o f 2 H S = B f 2 f 3 2k a A where f ;=1;2;3 are polynomal functons n x ac and x oc (symmetrc case). Snce lm k o ;k a!+1 xoc = lm k o ;k a!+1 xac k a tend to +1: = 0, then f take nte values when k o and Therefore, when k o and k a are su cently hgh: a. d2 S c dx o2 < 0 and d2 S c dx a2 < 0 31

32 b.det H S = (f 1 2k o ) (f 3 2k a ) f 2 2 > 0: Thus, we have a maxmum when k o and k a are hgh enough. F) Proof of the proposton 8 From (44), we deduce: x ac = l [2 (a ) + ( 2) (1 (1 + ) xoc )] x oc 4bk a + l 2 ( 2) x oc2 (55) Expresson (43) s equvalent to: (2a 2 2 ) + (2a 2 5) (56) +l (2a 2 5) x ac + [ (1 + ) (2 + 5 ) 4bk o ] x oc +2 ( + 5 2) lx ac x oc + 5l 2 x ac2 x oc = 0 by usng (55) n (56), and then multplyng by [4bk a + l 2 ( 2) x oc2 ] 2, we get a polynomal functon of degree 5 n x oc ; B (x oc ) = 0:The constant term of B s 8b 2 [ (2a 2 2 ) + (2a 2 5)] k a2 > 0 and the coe cent of (x oc ) 5 s 4 ( 2) bk o 2 l 4 < 0. We have B (0) > 0 and lm k o ;k a!+1 B (xoc ) = one and at most ve real and postve roots ^x oc expresson (56) and condton of a > + () a when k o and k a are hgh enough. 1, then B (x oc ) admts at least > 0. Snce ^x oc > 0, from >, we have ^x ac > 0 G) Proof of proposton 10 32

33 To compare the lm k o ;k a!+1 roc et lm k o ;k a!+1 rac, we assume that = and we used the condton of a > + () (2 )(a ) > (2 ) () 3 ( 1) + (2 )(a ) > (2 ) + 3( 1); thus, we have 2(a )[3( 1) + (2 )(a )] + 6(a ) (1 5) > [2(a ) (5 1) + (5 19)] And after all calculaton s obtaned 2(a )[3( 1)+(2 )(a )]+6(a )+3 2 (1 5) > 3 2 (1 3): Indeed, f 3 2 (1 3) s postf then < 1 3. H) Proof of proposton 11 From (24) and (49) ; we have because (1 lm k o ;k a!+1 ko x oc lm k o ;k a!+1 ko x of = ) > 0. Snce k o and k a are hgh enough, then x oc mply that e c < e f and we also have ^q c > ^q f : (1 ) > 0 4b > x of ; ths I) Proof of proposton 12 From (23) and (48), and the condton of a > + () a >, we have lm k o ;k a!+1 tc then t c > t f. lm k o ;k a!+1 tf = (a ) 2 < 0. Snce k o and k a are hgh enough, And from (24) and (49), and by usng the condton (38), we have lm k o ;k a!+1 roc lm k o ;k a!+1 rof = 2(a )[3 (a )(1+)] 32 ( 1) 6b > 0. Snce k o and k a are hgh enough and when the margnal damage of polluton s su cently low, then r oc > r of. 33

34 References [1] Ben Youssef, S. (2009). Transboundary polluton, R&D spllovers and nternatonal trade. The Annals of Regonal Scence, 43, [2] Ben Youssef, S., & Zaccour G. (2009). Absorptve capacty, R&D spllovers, emssons taxes and R&D subsdes, HEC Montréal: Les Cahers du GERAD No. G , MPRA Paper No [3] Ben Youssef, S. (2010). Transboundary polluton and absorptve capacty. Envronmental Modelng Assessment, 16, [4] Conrad, K. (1993). Taxes and sudsdes for polluton-ntensve ndustres as trade polcy. Journal of Envronmental Economcs and Management, 25, [5] Cohen, W. M., & Levnthal, D. A. (1989). Innovaton and learnng: the two faces of R&D. The Economc Journal, 99, [6] D Aspremont, C., & Jacquemn, A. (1988). Cooperatve and noncooperatve R&D n duopoly wth spllovers. The Amercan Economc Revew, 78, [7] D Aspremont, C., & Jacquemn, A. (1990). Cooperatve and noncooperatve R&D n duopoly wth spllovers: erratum. The Amercan Economc Revew, 80, [8] Farzn, Y.H. & P.M. Kort. (2000). Polluton abatement nvestment when envronmental regulaton s uncertan. Journal of Publc Economc Theory, 2, [9] Jung, C., K. Krutlla & R. Boyd. (1996). Incentves for advanced polluton abatement technology at the ndustry Level: an Evaluaton of Polcy Alternatves. Journal of Envronmental Economcs and Management, 30,

35 [10] Hammerschmdt, A. (2009). No pan, no gan: an R&D model wth endogenous absorptve capacty. Journal of Insttutonal and Theoretcal Economcs, 165, [11] Hoel, M. (1997). Coordnaton of envronmental polcy for transboundary envronmental problems? Journal of Publc Economcs,66, [12] Kamen, M. I., Muller, E., & Zang, I. (1992). Research jont ventures and R&D cartels. The Amercan Economc Revew, 82, [13] Kamen, M. I., & Zang, I. (2000). Meet me halfway: research jont ventures and absorptve capacty. Internatonal Journal of Industral Organzaton, 18, [14] Leahy, D., & Neary, J. P. (2007). Absorptve capacty, R&D spllovers, and publc polcy. Internatonal Journal of Industral Organzaton, 25, [15] Lao, P.-C. (2007). Internatonal R&D rvalry wth spllovers and polcy cooperaton n R&D subsdes and taxes. Internatonal Economc Journal, 21, [16] Mllman, S.R. & R. Prnce. (1989).Frm ncentves to promote technologcal change n polluton control. Journal of Envronmental Economcs and Management, 17, [17] Mllou, C. (2009). Endogenous protecton of R&D nvestments. Canadan Journal of Economcs, 42, [18] Poyago-Theotoky, J. (1999). A note on endogenous spllovers n a nontournament R&D duopoly. Revew of Industral Organzaton,15, [19] Spencer, B.J & Brander, J.A. (1983). Internatonal R&D rvalry and ndustral strategy. Revew of Economcs Studes, 50,

36 [20] Stranlund, J.K. (1997). Publc Technologcal Ad to Support Complance to Envronmental Standards. Journal of Envronmental Economcs and Management, 34, [21] Takarada, Y. (2005). Transboundary polluton and the welfare e ects of technology transfer. Journal of Economcs, 85, [22] Zagonar, F. (1998). Internatonal polluton problems: unlateral ntatves by envronmental groups n one country. Journal of Envronmental Economcs and Management, 36,

37 Please note: You are most sncerely encouraged to partcpate n the open assessment of ths dscusson paper. You can do so by ether recommendng the paper or by postng your comments. Please go to: The Edtor Author(s) Lcensed under the Creatve Commons Attrbuton 3.0.