International Trade and Environmental Policy Under Imperfect Competition

Transcription

1 Internatonal Revew of Envronmental and Resource Economcs, 007, 1: Internatonal Trade and Envronmental Polcy Under Imperfect Competton Cees Wthagen Department of Spatal Economcs, VU Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, the Netherlands; emal: and Department of Economcs, Tlburg Unversty, P.O. Box 90153, 5000 LE Tlburg, the Netherlands ABSTRACT The objectve of the present paper s to revew the lterature on the lnk between envronmental polcy and nternatonal trade, wth a focus on mperfect competton on the world output markets. Specal attenton wll be pad to the lterature on olgopoly and strategc government polcy and ts potental consequences for the ecologcal dumpng debate. We address partal equlbrum as well as general equlbrum approaches, and emphasze the fact that opposng conclusons can be reached. Keywords: Ecologcal dumpng; mperfect competton; nternatonal trade; polluton. 1 INTRODUCTION The polluton haven hypothess s defned by Taylor (004, p. 1) as a predcton that lberalzed trade n goods wll lead to the relocaton of polluton ntensve producton from hgh ncome and strngent envronmental regulaton countres, to low ncome and lax envronmental regulaton countres. The polluton haven hypothess was frst envsaged by Copeland and Taylor (1994). Ths semnal pece of work was followed by a large number of other mportant contrbutons by the same authors and culmnated n ISSN ; DOI / Cees Wthagen

2 15 Wthagen ther recent book (Copeland and Taylor 003). A recent ssue of Advances n Economc Analyss and Polcy and Contrbuton to Economc Analyss and Polcy (004) was entrely devoted to the polluton haven hypothess (see Fullerton 006), ncludng an excellent survey by Taylor (004). Therefore, there appears to be not much room for another survey on the topc. However, most of the lterature on the polluton haven hypothess s amng at emprcal applcatons 1 and relatvely mnor attenton s pad to the economc theory behnd the hypothess. Moreover, most of the exstng theory focuses on condtons of perfect competton. It seems therefore worthwhle to pay attenton to mperfect competton, thereby relatng the polluton haven hypothess to the exstng theory of strategc trade polcy and the envronment. The latter topc was addressed earler by Ulph (1997), but the present artcle ams to ncorporate some more recent developments. In the Copeland-Taylor approach a dstncton can be made between exogenous and endogenous envronmental polcy. In the former case envronmental polcy s mplemented ndependently of preferences or endowments, whereas n the latter envronmental polcy s amng at maxmzng socal welfare. In our treatment of nternatonal trade and envronmental polcy, we wll always concentrate on the latter type of polces. We wll treat the Copeland-Taylor model n Secton of ths artcle. We am at a formal treatment, as n the rest of ths artcle, because such an approach allows for a precse statement of the results and llustrates the exact role of the assumptons made. Startng n Secton 3, we revew the exstng lterature on trade and the envronmental n a non-compettve settng. Ths revew s bref because t partly overlaps wth Ulph s (1997) earler excellent contrbuton. In Secton 3, we also present a rather general model, n a partal as well as n a general equlbrum settng, that serves as the vehcle for n the remander of the paper. We also explore the case of perfect competton and the large country case. Secton 4 goes nto olgopoly, and n partcular treats the ssue of strategc envronmental polcy. Thereby a dstncton s made between Cournot and Bertrand competton. In ths secton, the partal equlbrum outcomes are derved and are related to the exstng lterature that already ncorporates most of them. Secton 5 dscusses olgopoly and strategc envronmental polcy n general equlbrum models. It shows that the outcomes of the partal equlbrum models regardng optmal envronmental polcy may be reversed. Fnally, Secton 6, concludes. Unfortunately, space restrcts the topcs that can be addressed. Hence, we do not go nto the mportant ssue of technology transfers that mght counterbalance the polluton haven effect. As D Mara and Smulders (004) observe, f rcher countres specalze n the producton of clean commodtes they mght also drect R&D efforts to cleaner technology that mght dffuse to other countres. Also Golombek and Hoel (004) address technology transfer. Another mportant aspect of nternatonal trade and the envronment s the property rghts dfferental that mght exst between developed and less developed countres and that mght lead to overexplotaton of natural resources and consequently welfare losses. The semnal papers here are by Chchlnsky (1993 and 1994). More recent contrbutons are by Brander and Taylor (1997) and Karp et al. (001). Fnally, we wll not go nto locaton choces as a consequence of envronmental 1 A lst of references to the emprcal lterature s gven at the end of the paper.

3 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 153 polcy. It wll be assumed throughout that all polcy effects are materalzed only n trade patterns and not n a change of locaton of plants or of persons. For the lterature on locaton aspect of frms see e.g., Markusen et al. (1993 and 1995), Motta and Thsse (1994) and Hoel (1997). Elbers and Wthagen (004), Haavo (003a and 003b) and Hoel and Shapro (003) study locaton choces of workers. Fnally, before embarkng on the survey, a note on notaton. Although we wll revew numerous contrbutons we am at provdng a unform presentaton n terms of notaton. The symbols W, U, D, F, c, p, z, τ wll denote socal welfare, utlty from consumpton, envronmental damage, producton functon, consumpton, prce, polluton and emsson taxes, respectvely. Subscrpts refer to dfferent commodtes and superscrpts to frms. The dstncton between home and foregn s usually ndcated by superscrpt h and f. THE COPELAND-TAYLOR MODEL An extremely nfluental model s the Copeland-Taylor model, descrbed n a number of publcatons by these authors and recently comprehensvely treated n ther 003 book, from whch we borrow extensvely n ths secton. Our am s to sketch how trade patterns depend on envronmental polcy n a compettve world. By way of ntroducton for the reader unfamlar wth the trade envronmental polcy lnk, we frst summarze what can be called the textbook approach, followed e.g., n Perman et al. (003) or Pearson (000). The smplest settng s a two country two goods world n a partal equlbrum framework. The usual strategy s to derve the autarky equlbrum allocaton and prces and to show that, n the absence of envronmental externaltes, openng up to free trade at gven world market prces s benefcal from a welfare perspectve. However, f producton brngs along local polluton and envronmental polcy s absent, the country exportng the drty commodty may suffer a welfare loss because of the ncreased negatve externalty from polluton followng ncreased producton. But then t s shown that an optmal Pgouvan tax on polluton wll restore the result that free trade ncreases welfare n both countres engaged n trade. The case of transboundary polluton s somewhat more complcated to analyze, because that would requre coordnated polcy. In the sequel we shall not address ths ssue. Copeland and Taylor (003) construct a general equlbrum two country (North and South), two goods, two factors model, n order to dentfy whch of the two countres s gong to be the exporter of the drty commodty. Two hypotheses are nvestgated: the polluton haven hypothess and the factor endowment hypothess, both defned n the Introducton of ths artcle. The factors of producton are labeled captal and labor; they are mmoble across the two countres but moble between sectors. Ther ntal endowments are k and l, respectvely. The producton functon of the frst commodty s y 1 = z α [F 1 (k 1, l 1 )] 1 α wth 0 <α<1 and F 1 lnearly homogeneous n captal and labor nput. Emssons z can be nterpreted as arsng from the use of a producton factor such as energy. An alternatve nterpretaton ncludes abatement. Wthout abatement emssons would be proportonal to output, wth factor of proportonalty equal to unty. Abatement requres

4 154 Wthagen labor and captal, n the same proporton as n the normal producton process. Ths s modeled as y 1 = (1 ϑ)f 1 (k 1, l 1 ), z = (1 ϑ) 1/α F 1 (k 1, l 1 ), where ϑ s the abatement level, varyng between 0 and 1. Output s used for consumpton (c 1 ) and net exports (x 1 ). The second commodty s clean. Domestc consumpton s c and net exports are x. The gven world market prce s p. Producton of the drty commodty s always more captal ntensve than the clean commodty: (k 1 /l 1 ) > (k /l ). The nstantaneous utlty functon U depends on consumpton and s homothetc. Damage D s ncreasng and convex n emssons. The am of the government s to maxmze socal welfare subject to W (c 1, c, z) = U(c 1, c ) D(z), c 1 + x 1 = z α [F 1 (k 1, l 1 )] 1 α, (1) c + x = F (k, l ), () px 1 + x = 0, (3) k = k 1 + k, (4) l = l 1 + l. (5) Equaton (3) defnes the equlbrum on the current account of the balance of trade, wth the second commodty as the numerare. The Lagrangan reads L = U(c 1, c ) D(z) + µ 1 [z α [F 1 (k 1, l 1 )] 1 α c 1 x 1 ]+µ [F (k, l ) c x ] + ν[ px 1 + x ]+ r[k k 1 k ]+ w[l l 1 l ]. Necessary condtons for an nteror soluton are: U c 1 = µ 1, U c = µ, (6) D (z) = µ 1 αz α 1 [F 1 (k 1, l 1 )] 1 α, (7) µ 1 (1 α)z α F1 α F 1 = r, k 1 µ 1 (1 α)z α F1 α F 1 = w, (8) l 1 F F µ = r, µ = w, (9) k l µ 1 = νp, µ = ν. (10) Denote the optmal value of the varables and the correspondng Lagrangan multplers by hats. Defne r = ˆ r, w = ˆ w, τ = D (ẑ). ˆµ ˆµ ˆµ

5 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 155 Then, () (ĉ 1, ĉ ) maxmzes U(c 1, c ) subject to pc 1 + c rk + wl + τẑ = rk + wl +[pẑ α [F 1 (ˆk 1, ˆl 1 )] 1 α rˆk 1 wˆl 1 τẑ] +[F (ˆk, ˆl ) rˆk wˆl ]+τẑ = pẑ α [F 1 (ˆk 1, ˆl 1 )] 1 α + F (ˆk, ˆl ). () (ẑ, ˆk 1, ˆl 1 ) maxmzes pz α [F 1 (k 1, l 1 )] 1 α rk 1 wl 1 τz. () (ˆk, ˆl ) maxmzes F (k, l ) rk wl. So, consumers maxmze utlty subject to ther budget constrant, producers maxmze profts, gven the rate of return r, the wage rate w and the tax on emssons τ, whch s set equal to margnal damage, n money terms. Therefore, (ĉ 1, ĉ, ˆk 1, ˆk, ˆl 1, ˆl, ẑ, q, r, w) consttutes a general compettve equlbrum. Hence by an approprate choce of the polluton tax the frst-best optmum can be mplemented as a general equlbrum. It s mportant to realze that ths s due to the fact that the world s perfectly compettve. Due to the fact that the utlty functon U s homothetc, relatve demand RD( p) = c 1 /c s a functon of p only. It s decreasng. For a gven z, natonal ncome pz α [F 1 (k 1, l 1 )] 1 α + F (k, l ) s maxmzed n a general equlbrum, subject to (4) and (5). Hence, we can wrte y 1 = y 1 (k, l, e, z) and y = y (k, l, e, z), where e = x/y 1 s the polluton ntensty n the pollutng sector. Due to constant returns to scale these functons are lnearly homogeneous n (k, l). Therefore, relatve supply s RS( p, e, k/l) = y 1 ( p, e, k/l)/y ( p, e, k/l). It can be shown that under the assumptons made, relatve supply s ncreasng n the prce as well as n the captal labor rato (the latter because the drty sector s more captal ntensve). Now, consder frst the case where North s relatvely rch, but wth the same relatve factor endowment as South. If envronmental polces would be such that the polluton ntenstes are dentcal, the equlbrum autarky prce p would be dentcal, and there would be no reason for trade. However, wth hgher natonal ncome, socal welfare maxmzaton requres a more strngent envronmental polcy. Therefore the polluton ntensty wll be lower n the North than n the South, mplyng that the autarky prce n North wll be hgher. Ths mples that when the countres open up to trade, North s gong to mport the drty commodty and South s gong to produce more of the drty commodty, thereby causng more polluton as well. Ths s the polluton haven hypothess. However, f envronmental polcy s set optmally n both countres, both countres wll beneft from trade. Whether world polluton ncreases or decreases depends on the so-called ncome elastcty of margnal damage. As a consequence of trade ncome ncreases, whch may call for a strong or a weak polcy response, represented by the elastcty. If the requred polcy response s weak, total polluton may ncrease, otherwse t wll ncrease. A second experment s to change the relatve endowments. If the North s relatvely captal abundant, then wth dentcal emsson ntenstes the North, whose producton of the pollutng commodty s relatvely captal ntensve, wll export the drty commodty.

6 156 Wthagen These consderatons clearly demonstrate the ssues at stake. Dfferences n the relatve abundance of captal enhance the factor endowment hypothess: North wll specalze n the producton of the drty commodty. Beng rch, whle relatve endowments are dentcal, supports the polluton haven hypothess through the polcy effect. 3 IMPERFECT COMPETITION; AN INTRODUCTION 3.1 Introducton Ideally, the Copeland-Taylor model of the prevous secton s extended so as to ncorporate mperfect competton n a full-fledged general equlbrum model, to nvestgate the mpact of strategc behavor of governments. However, to the best of our knowledge no lterature on ths topc has developed yet. Therefore, our objectves n the sequel of ths survey are relatvely modest. The am of the present secton s threefold. We frst ntroduce a model that s employed n the sequel of ths artcle. It s desgned to address the man queston posed n ths lterature, namely whether strategc consderatons may lead governments to treat dfferent sectors n the economy dfferently. In partcular, a dstncton wll be made between the sheltered and the exposed sectors of the economy. The exposed sectors export ther commodtes, whereas the sheltered sectors do not. Ths allows us to analyze the queston whether from a socal welfare pont of vew t could be optmal to protect the exposed sectors, n the sense of makng them subject to less strngent envronmental polcy. The second objectve s to characterze the socal optmum for the case of perfect competton and the large country case. Thrd, we deal wth the mplementaton of the frst-best n a decentralzed economy. 3. The Model The man ngredents of the model we employ are borrowed from the model constructed by Rauscher (1994) n hs artcle on general equlbrum and envronmental polcy. We do not take nto account abatement because for most of the results ths s not essental. There are fve commodtes: three consumer commodtes, captal and a raw materal. The frst consumer commodty s produced and consumed domestcally only. Producton takes place n the so-called sheltered sector. Ths sector s composed of many frms that behave compettvely. Aggregate technology s descrbed by a producton functon (F 1 ), havng captal (k 1 ) and the raw materal (z 1 ) as nputs. Consumpton s denoted by c 1. The second class of consumer commodtes s produced domestcally n n( 1) sectors, ndexed by = 1,,..., n. Part of the output of sector s consumed domestcally (c ), part of t s exported (x ). These sectors are called exposed. Each ndvdual sector can consst of dfferently behavng frms but producng a homogeneous commodty. We allow for the case that the exposed sector conssts of small number of frms (ncludng a sngle frm) and of a large number of frms. Another dstncton made s between the country beng large or small on the market of an exported commodty. All ndvdual frms producng a varety for whch the economy s small, can be aggregated and

7 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 157 descrbed as a representatve compettve frm. Some of the large sectors each may contan a contnuum of compettve frms, whch we aggregate nto a representatve frm as well. Each of the other large sectors conssts of a fnte number of frms explotng ther market power. The dfferent market structures to be dscussed all dffer n the specfcaton and composton of the exposed sector. Hence, for each case the detals of the sector receve due attenton. For sector the technology s descrbed by F, wth as nputs captal (k ) and a raw materal (z ). The thrd consumer commodty cannot be produced domestcally; t needs to be mported. Consumpton s denoted by c 3. The thrd commodty s taken as the numérare. Captal s mmoble nternatonally but moble between domestc sectors. Some emprcal as well as theoretcal support for the assumpton of nternatonal mmoblty of captal can be found n Gordon and Bovenberg (1996). The economy s endowment s gven by k. The rate of return on captal s denoted by r. The raw materal s n prncple freely avalable n unlmted amounts. However, processng of the raw materal causes polluton, proportonal to producton, wth factor of proportonalty equal to unty. Contrary to Copeland and Taylor (003) both sectors are pollutng. Polluton s damagng. The government therefore leves taxes τ 1 and τ ( = 1,,..., n) per unt of raw materal used n the sheltered and the exposed sectors, respectvely. The taxes can be dfferentated between as well as wthn sectors. The tax revenues are recycled to the consumers n a lump sum fashon. Alternatvely, the government mposes emsson celngs, that mght or mght not be mplemented through a system of tradable permts. The ncome of the representatve consumer conssts of the value of the captal endowment rk, the tax revenues τ 1 z 1 + n τ z, and the profts of the frms, whch amount to p 1 F 1 (k 1, z 1 ) rk 1 τ 1 z 1 and n { p F (k, z ) rk τ z } for the sheltered and the exposed sectors, respectvely, assumng for the moment that the domestc prce of the second commodty equals ts world market prce. Under the assumpton of full employment of captal (n a stuaton where frms maxmze profts) total ncome bols down to (n shorthand) p 1 F 1 + n p F. The consumer maxmzes utlty, takng prces and ncome gven. Preferences consst of two parts. Frst, they depend on the consumpton of the consumer goods. Ths s represented by a utlty functon, denoted by U(c 1, c, c 3 ), where c = (c 1, c,..., cn ). The utlty functon s assumed to have all the usually desred propertes such as concavty, dfferentablty and monotoncty. Second, consumers experence damage from polluton. Wth a lttle abuse of notaton, ths part of the preferences s gven by the (convex and ncreasng) damage functon ( D(z 1, z ) = D z 1 + z ), wth z = (z 1, z,..., zn ). So, t s assumed that polluton s only local. Total socal welfare s W (c 1, c, c 3, z 1, z ) = U(c 1, c, c 3 ) D(z 1, z ).

8 158 Wthagen Fnally, we mpose equlbrum on the current account of the balance of payments. Summarzng: k 1 + c 1 = F 1 (k 1, z 1 ), (11) c = [F (k, z ) x ], = 1,,..., n, (1) c 3 = k p x, (13) = k. (14) Demand for the exported commodty s stll to be specfed. It depends on the market structure under consderaton. In the sequel we wll occasonally employ specfc functonal forms to llustrate the results and to perform numercal exercses. In those exercses utlty s logarthmcally addtve, envronmental damage s quadratc, producton functons are Cobb Douglas. U(c 1, c, c 3 ) = ln c 1 + ln c + ln c 3, (15) F 1 (k 1, z 1 ) = k1 α z1 α 1, (16) F (k, z ) = (k )β (z )1 β, (17) [ D(z 1, z ) = 1/ z 1 + z]. (18) 3.3 Frst-Best; Perfect Competton and the Large Country Case In ths secton, we characterze the socal optmum for the case of perfect competton as well as for the case of a large country. Wth perfect competton each of the producers n the exposed sector takes the world market prce as gven. The world market prce s also equal to the domestc prce. The large country case can be modeled n several ways. One can mantan the assumpton of a contnuum of prce takng producers n an exposed sector but n addton assume that the government can explot the fact that the country as a whole s a large player on the world market. Alternatvely, t can be assumed that there s a sngle exportng frm of the commodty for whch the country s large, that explots ts monopoly power tself. We go nto both alternatve settngs. The frst-best optmum s defned as the allocaton that maxmzes socal welfare,.e., utlty of the representatve agent mnus the dsutlty of polluton damage, subject to the restrctons mposed by the technology and the condton of equlbrum on the current account of the balance of payments. In mathematcal terms t s the soluton of

9 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 159 the maxmzaton of ( W (c 1, c, c 3, z 1, z ) = U(c 1, c, c 3 ) D z 1 + z ), subject to (11) (14). The Lagrangan of the problem reads: ( ) L = U(c 1, c, c 3 ) D z µ 1 [F 1 (k 1, z 1 ) c 1 ] + z µ [F (k, z ) x ( p ) c ] + µ 3 [ ] [ p x ( p ) c 3 + r k k 1 where x ( p ) denotes world demand for varety. Assumng an nteror soluton we fnd as necessary condtons k ], U c 1 = µ 1 ; U c = µ, = 1,,..., n; U c 3 = µ 3, (19) µ F k µ 1 F 1 k 1 = r; µ 1 F 1 z 1 = D, (0) = r; µ F z µ dx dp = D, = 1,,..., n, (1) + µ 3 [x ( p ) + dx p dp ] = 0, () where prmes refer to dervatves. In the sequel hats denote the soluton of ths problem: (ĉ 1, ĉ, ĉ 3, ˆp, ˆk 1, ˆk, ẑ 1, ẑ, ˆx ), where k = (k 1, k,..., kn ), z = (z 1, z,..., zn ), x = (x 1, x,..., xn ) and p = ( p 1, p,..., pn ). The nterpretaton of the necessary condtons s straghtforward. Wth µ 1 the margnal value (n utlty terms) of one unt of output of the sheltered sector, the second part of (0) says that the margnal value of a polluton nput n ths sector should equal margnal damage (dsutlty of polluton). Ths also holds for the exposed sectors, wth µ nterpreted as the margnal value of a unt of varety. Equaton () says that an ncrease of the world market prce of varety causes lower demand, so that more output can be used for domestc consumpton, whch s benefcal. But the export revenues decrease, mplyng less ncome avalable for the mport of the thrd commodty. The next step s to nvestgate how the frst-best optmum can be realzed n a decentralzed settng. Many results regardng the mplementaton of the frst-best optmum

10 160 Wthagen n a decentralzed economy are straghtforward modfcatons of earler work by, for nstance, Hoel (1996) and Rauscher (1997), and they have become standard nferences n the theory of nternatonal trade (see e.g., Dxt 1985). Nevertheless, we state them explctly here for later reference. Defne t = 1ˆε, p 1 = ˆµ 1, q ˆµ = ˆµ = p 3 ˆµ (1 t ), ( = 1,,..., n), p 3 = 1, 3 τ = τ 1 = τ = D (ẑ) ˆµ 3, ( = 1,,..., n), r = ˆ r ˆµ 3 wth ˆε s the prce elastcty of world market demand for the second commodty whch, evaluated at the optmum (smaller than mnus unty). So, q s the value n monetary terms of varety. Due to the concavty/convexty assumptons on the functons nvolved the necessary condtons correspondng wth the frst-best socal optmum are also suffcent. Observe the followng: The par (ˆk 1, ẑ 1 ) maxmzes profts p 1 F 1 (k 1, z 1 ) rk 1 τz 1 of (aggregate) frm 1. Ths s the case because n the frst-best the par maxmzes [ ˆµ 1 ˆµ 3 F 1 (k 1, z 1 ) ˆ r ] k 1 D (z) z 1. ˆµ 3 ˆµ 3 ˆµ 3 For the same reason (ˆk, ẑ ) maxmzes profts q F (k, z ) rk τz of (aggregate) frm of the exportng sector. The trple (ĉ 1, ĉ, ĉ 3 ) maxmzes U(c 1, c, c 3 ) subject to p 1 c 1 + q c + p 3c 3 = p 1 F 1 + q F + T, where T denotes recycled export tarff revenues. The world market p maxmzes ( p q )x ( p ): export revenues mnus the opportunty costs of consumpton of the exported commodty foregone. Fnally, all markets clear at the proposed prces. Hence, we can state the followng: The frst-best optmum can be mplemented n a decentralzed economy by: () Imposng a unform emsson tax. () Imposng export taxes on the frms n sectors where the economy s large but where the ndvdual frms do not explot ths. () Correctng non-envronmental domestc dstortons. Ths result about mplementaton s well known from the general theory of nternatonal trade (see also Neary 006). It mples that n the case of perfect competton on the world

11 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 161 market for the exported commodty, t s optmal not to mpose an mport tarff (t = 0 f ˆε = ) and to tax emssons accordng to ther margnal damage. Hence, wth a unform Pgouvan emsson tax the socal optmum can be reached n a tradng economy. The stuaton s dfferent when the country s large, so that world demand depends on the prce set by the country (assumng for the moment there s no perceved nteracton wth other foregn players). If, as s commonly assumed n the lterature, each ndvdual domestc frm s small on the world market, but that the sector producng the exported commodty, s large on aggregate, then t s optmal to mpose an export tax, but stll tax emssons accordng to margnal damage, whch s equal for all frms. Fnally, f frms are large and act as such on the world market, there s clearly no need for an export levy. But, f these frms act non-compettvely on the home market, a correcton of ths externalty s n order. One may also thnk of a stuaton wth an upper bound z on total polluton, mposed by e.g., an nternatonal treaty. Hence the constrant to the optmzaton problem s z 1 + n z z. If socal welfare ncludes polluton damage and f the emsson constrant would not be bndng n the optmum, addng the constrant s not nterestng. So, we assume that the celng s bndng. In order to comply wth the nternatonal norm, the government may levy emsson taxes τ 1 and τ ( = 1,,..., n) per unt of raw materal nput n all sectors of the economy. It s easy to see that the settng does not change the way the optmum can be mplemented n a decentralzed economy by means of taxes. Alternatvely, the government may nstall a system of tradable emssons permts. Such a system s called unform f trade s allowed among all domestc sectors, ncludng the sheltered sector. The system s dfferentated f ndvdual sectors have ther own system, wth trade lmted to those frms belongng to the ndvdual sectors. Dfferental envronmental polcy across sectors s not needed f the country s small on all world markets, f ts ndvdual frms explot ther monopoly power, or f the government can levy an export tax on large but compettve sectors. We wll return to emsson caps n subsecton Second-Best n the Large Country Case We now assume that, due to nternatonal regulatons, t s not feasble to use tarffs as an nstrument. It has been shown above that trade tarffs are not needed to mplement the frst-best optmum f there s perfect competton. Therefore, our attenton wll be restrcted to the large country case. In order to avod the necessty of correcton of externaltes on the domestc output markets, we wll assume that all ndvdual frms are prce takers, or that they do not supply to the home market. The queston addressed s agan whether t s optmal to dfferentate between domestc sectors wth respect to the emsson taxes and whether taxes are below the margnal damage or not. Suppose the government has set Pgouvan emsson taxes; they equal margnal damage (τ = D /( U/ c 3 )) and are unform over the sectors. We frst nvestgate the queston how socal welfare changes f the government margnally devates from ths polcy, gven the market behavor of ndvdual agents. For the case of ndvdually prce takng frms

12 16 Wthagen we have dw = U c 1 dc 1 + = U c 3 = U c 3 + = U c 3 [ ] U c dc + U dc 3 D dz 1 + dz c 3 [ [ ]] p 1 dc 1 + p dc + dc 3 τ dz 1 + [ { F1 p 1 dk 1 + F } 1 dz 1 + k 1 z 1 { x ( p )dp + dx p ( p ) x ( p )dp = U c 3 dp dp p dz { F dk + F dz dx ( p ) k } [ τ dz 1 + x ( p )dp dτ dτ. z dz For all sectors that behave as large sectors by themselves we have dc = 0 and dp x ( p ) = rdk + τdz. Hence dw = U x c ( p ) p 3 τ dτ, where the summaton s taken over the compettve sectors. It s welfare mprovng to ncrease the emsson tax for these compettve sectors snce the hgher tax wll decrease supply, thereby rasng the prce (see also Krutlla 1991). Therefore, the hgher emsson tax helps to reduce overall polluton. Ths result obvously also holds n case of an exogenous upper bound on polluton. Margnally decreasng or ncreasng the tax on emssons n the sheltered sector does not yeld hgher socal welfare. To see ths, F observe that n equlbrum p 1 1 z 1 = τ 1 from proft maxmzaton n the sheltered sector. Moreover, U U c 1 = p 1 c3. Therefore U F 1 U c 1 z 1 = τ 1 c3. Consder a margnal varaton of z 1 only. In an optmum, ths varaton should not allow for a welfare mprovement. Hence [ ( )] d U(c 1, c, c 3 ) D z 1 + z = 0. dz 1 Stated otherwse U F 1 U = τ 1 = D. c 1 z 1 c 3 In what follows we shall perform a global analyss of the problem. Ths allows for a calculaton of globally optmal envronmental taxes, rather than calculatng the effect of a margnal devaton from taxes equal to margnal damage. Global welfare optmzaton on the part of the government bols down to the maxmzaton of utlty mnus damage, ]] dp dp }

13 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 163 takng the constrants outlned above nto account. We use the functonal forms ntroduced n subsecton 3.1. World demand s soelastc: x ( p ) = p ε (ε<0). It follows from utlty maxmzaton subject to the budget constrant that p 1 c 1 = p c = c 3 = 1 3 [ p 1F 1 + p F ]. (3) Together wth the condtons for market equlbrum,.e., F 1 = c 1, F = c + x( p ), ths yelds: p 1 c 1 = p x( p ), c = x( p ), c 3 = p x( p ). (4) It follows from proft maxmzaton that equlbrum prces are on the factor prce fronters, correspondng wth zero profts, defned by: ( r ) ( ) α 1 α τ1 p 1 = = f 1 (r, τ 1 ), (5) α 1 α ( ) r β ( ) 1 β τ p = = f (r, τ ). (6) β 1 β Factor demands are: ( r ) ( ) α 1 1 α τ1 ( r ) ( ) α 1 1 α τ1 p x( p ) k 1 (r, τ 1, p 1, p ) = F 1 =, α 1 α α 1 α p 1 (7) ( r ) ( ) α α τ1 ( r ) ( ) α α τ1 p x( p ) z 1 (r, τ 1, p 1, p ) = F 1 =, α 1 α α 1 α p 1 (8) ( ) r β 1 ( ) 1 β ( ) τ r β 1 ( ) 1 β τ k (r, τ, p ) = F = x( p ), β 1 β β 1 β (9) ( ) r β ( ) β ( ) τ r β ( ) β τ z (r, τ, p ) = F = x( p ). β 1 β β 1 β (30) Straghtforward manpulatons, usng k 1 + k = k, yeld After substtuton we have k 1 = αk α + β, k 1 = βk α + β. W = ln k α 1 z1 α 1 + ln 1 kβ z1 β + ln ( 1 kβ z1 β ) 1+1/ε 1 [z 1 + z ]. Maxmzaton wth respect to z 1 and z gves z 1 = (1 α)/ (1 α) + (1 β)( + 1/ε), z = (1 β)( + 1/ε)/ (1 α) + (1 β)( + 1/ε).

14 164 Wthagen Then, from the fact that τ 1 z 1 τ z = (1 α)p 1F 1 (1 β)p 1 F 1, t follows that τ = + 1/ε τ 1 >τ 1. So, for these specfcatons t s optmal to set the emssons tax hgher for the exposed sector than for the sheltered one, not only locally, but also globally. The same result s obtaned n case there s an exogenous upper bound on total emssons. A tradable permts system should then be dual, allowng for a separate permts market n each sector, yeldng dfferent permt prces for the two sectors. In an unconstraned tradable permts system the exposed sector would demand more emsson permts and supply more to the world market than s optmal. In the case of more than one export sector, multple separate sectoral permts markets should be establshed. Ths polcy s generally more dffcult to mplement, and t wll be ntrcate to ensure perfect competton as well. A polcy desgn based on dfferentated emssons taxes s therefore lkely to be more approprate. 4 OLIGOPOLY; PARTIAL EQUILIBRIUM 4.1 Introducton In ths secton, we start wth a bref survey of the prevous lterature on olgopoly. The analyss takes place wthn a partal equlbrum framework along the lnes developed by Conrad (1993), Kennedy (1994) and Barrett (1994). Then we develop and study a smplfed partal equlbrum model that serves as a benchmark for the general equlbrum approach of the next secton. A dstncton wll be made between Cournot and Bertrand. The basc dea n all the contrbutons dscussed here goes back to the Brander and Spencer (1985) model where governments can take actons that consttute a commtment of domestc frms toward ther compettors. In a two-stage game nterpretaton the government makes a strategc choce before the domestc frm takes an acton wth regard to prce settng. In the envronmental polcy lterature both taxes and standards predomnantly act as a precommtment devce for the government. A dscusson of other precommtment devces, lke subsdes on R&D nstrument, and ther consequences can be found n amongst others, Ulph (1996). 4. The Kennedy/Conrad/Barrett models In ths secton, most attenton s pad to Kennedy s (1994) contrbuton because t sketches the most general settng. Kennedy (1994) analyses a model that ncludes many of the elements that play a crucal role n the remander of ths survey. It s a two-country model, where each country has n dentcal frms, producng a homogeneous commodty. Producton of the representatve frm n the home country s y, and t exports x. The

15 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 165 nverse domestc demand functon reads p(c) = p(ny nx + nx f ), where x f denotes the exports of the representatve foregn frm. The foregn domestc demand functon reads p(c f ) = p(ny f nx f + nx). Margnal producton costs are ϑ. Polluton s an nevtable by-product of producton. Emssons by the representatve home frm are z = y/ϑ. The parameter ϑ s chosen by each ndvdual frm. Polluton damage n the home country s D(z, z f ) = D(nz+χnz f ). Ths specfcaton allows for transboundary polluton: fracton χ of foregn emssons s deposted n the home country, and vce versa. Emssons are taxed at a rate τ. Profts of the representatve frm read π(y, x, ϑ) = p(c)[ y x]+p(c f )x ϑy τy/ϑ wth c = y x + (n 1)[ȳ x]+nx f, c f = n[ y f x f ]+x + (n 1) x, where ȳ and x denote output and net exports of each competng domestc frm, taken as gven. When determnng optmal producton, export and cost parameter, the frm also takes foregn exports as gven. The necessary condtons for an ndvdual optmum read π ( y = 0:p (c)[ y x]+p(c) = ϑ + τ ), (31) ϑ π x = 0: p (c)[ y x] p(c) + p (c f )x + p(c f ) = 0, (3) π ϑ = 0:ϑ = τ. (33) The Nash equlbrum s symmetrc. All frms, be t domestc or foregn, are subject to the same emsson tax. Hence they wll choose the same abatement technology, and all outputs are dentcal. Moreover, c = c f. It follows from (3) that x = 1 y. Then we have np + cp = 4nt. (34) The socal optmum can be formulated as the outcome of the maxmzaton of the sum of each country s consumer surplus, producer surplus and tax revenues, mnus polluton damage W + W f = U(c) p(c)c + p(c)[ny nx]+p(c f )nx ϑny τ ny ϑ + τ ny ϑ ( ) ny D ϑ + χnyf ϑ f + U(c f ) p(c f )c f + p(c f )[ny f nx f ]+p(c)nx f ϑ f ny f τ f nyf ϑ f + τ f nyf ϑ f ( ny f D ϑ f + χ ny ) ϑ [ ( = U(ny) tny D (1 + χ) ny )], t where t has been taken nto account that the emsson tax wll be unform over the countres. Moreover, (33) has already been ncorporated. A necessary condton for

16 166 Wthagen optmalty s [ p t] ny t ny = 1 + χ [ t D t ny ] ny. (35) t If markets were perfectly compettve then ny t = 0 and τ = (1 + χ)d so that the emsson tax would equal margnal damage. By usng (34) t can be shown that n the case of olgopoly the optmal tax rate s smaller than margnal damage. Fnally, Kennedy studes the stuaton where the two governments play a Nash game aganst each other. A government sets ts own tax rate, and takes the other government s tax rate as well as frms behavor as gven. The reacton functon for the home country s p + 1 p y = t. In equlbrum, we have np + p c = n(t + t f ), where f refers to the foregn country. Each country s assumed to be able to calculate the change n the equlbrum upon a change n ts own tax rate, gven that the other country s tax rate remans unchanged. It s argued that n the Nash equlbrum the emsson tax s below the effcent one gven by (31). Three elements play a role. Frst, the so-called transboundary externalty effect s responsble for a too low emsson tax, at least f polluton s transboundary. Second, a rent capture effect occurs. A unlateral decrease of the emsson tax, startng from the effcent one, n one country wll boost net exports, because t has a greater mpact on producton than on domestc consumpton. Also the rent capture effect calls for lower envronmental taxaton. The thrd effect s brought about by the polluton shftng. A decrease of the tax rate wll ncrease domestc producton and thereby polluton. Obvously, the effect s necessarly zero f polluton s purely transboundary. If t s only partly transboundary then the effect s postve. Therefore, there would be an ncentve to ncrease the local taxes. It s formally shown by Kennedy that the second effect domnates the thrd one. Hence the overall result s a tax rate below the effcent one, n both countres. Conrad (1993) studes a two-country world, wthn each country a sngle frm producng a homogeneous commodty that s sold on a thrd market only. So, domestc consumpton s not taken nto account. Polluton s purely global and unformly mxng. Each frm has the dsposal of an abatement technology. Conrad shows that the polluton tax rate n the Nash equlbrum s below the tax rate that would emerge under socal welfare maxmzaton, takng nto account the fact that polluton s transboundary. Ths s what one would expect, snce the model s less general than Kennedy s. However, Conrad also makes another contrbuton by consderng other polcy nstruments such as a subsdy on abatement actvtes. Ths subsdy, together wth a hgh emsson tax, mght be welfare mprovng, compared wth the case where subsdes are absent. In any case, abatement s enhanced. Also Barrett (1994) s one of the forerunners on olgopoly and strategc envronmental polcy. Hs model resembles Conrad s n the sense that the two olgopolsts n two countres produce for a thrd market only. But n hs model, a standard s the polcy nstrument and polluton s purely local. In the polcy games played by the two governments, emsson standards are set below standards correspondng wth margnal The mathematcs s not reproduced here.

17 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 167 damage n the case of Cournot competton, and wth standards hgher than margnal damage when competton s Bertrand. So, wth Cournot competton polcy s too lax, and wth Bertrand competton polcy s too strct. When domestc output s produced by multple frms, these results do not necessarly hold anymore, and the standards mght be too lax or too strngent. We wll return to the Barrett model n the sequel. 4.3 Partal Equlbrum Cournot Competton We wll argue n Secton 5 that the polcy recommendatons followng from a partal equlbrum approach mght drastcally dffer from those derved n a general equlbrum. Ths wll be llustrated by means of an olgopoly model closely related to the ones dscussed n the prevous subsecton. To make the pont we employ a model that n some respects s smpler than those dscussed above. We abstract from transboundary polluton, and from the presence of an abatement technology. We also restrct ourselves to a thrd market on whch all producton of the exposed sector s sold. There s no great loss n lookng at local polluton only. It should be clear from the prevous subsecton what the effect of transboundary polluton s. Neglectng abatement s more serous. It s assumed that there s a pollutng nput, so that n the case of standards ths would just mean an upper bound on the use of the producton factor, whereas taxes have bascally the same effect. In ths way, by not ncorporatng abatement as a potental actvty to whch resources can be devoted, we reach a consderable smplfcaton, wthout loosng the need for envronmental polcy. The absence of the home market s n the sprt of earler work by Conrad (1993) and Barrett (1994). In the present subsecton, we ntroduce the model wth Cournot competton. In the partal equlbrum approach the government takes as gven all that occurs n the domestc sheltered sector. The government s not nterested n total socal welfare per se: e.g., polluton damage caused by the sheltered domestc sectors s not taken nto account. In ths settng, socal welfare can be wrtten as W = { p(x, x f )x τ z } D(z) + τ z. Here p(x, x f ) = p(x 1, x,..., x n, x f ) = p( n (x + x f )) s the world market prce of the (homogeneous) exported commodty, dependng on output x of sector ( = 1,,..., n) and foregn supply x f. Emssons by sector are z, taxed at the rate τ s the emsson tax. We have x = F (z ). There s one foregn suppler of the exported commodty. Damage s D(z) = D(z 1, z,..., z n ) = D( n z ). Hence, the government seeks to maxmze export revenues mnus socal costs, the latter consstng of captal costs (whch are exogenous) and the external damage costs caused by emssons of the exportng sector. All emsson tax revenues n τ z are recycled to the consumer. In Barrett s Nash game each government takes output of domestc frms as gven. In such crcumstances the socally optmal standard arses from the equalty of margnal abatement costs and margnal damage. Implementaton of ths rule requres nformaton about the abatement cost functon. In our model, wth taxes and n the absence of abatement, an analogous approach would be to assume that the government knows the

18 168 Wthagen cost functon or the producton functon, from whch demand for the raw materal (and hence emssons) can be derved. Proft maxmzaton on the part of frm mples { p + p x } F z = τ, = 1,,..., n. (36) We frst perform a local analyss, startng from a stuaton where the government sets the emsson tax equal to margnal damage: τ = D. From dx = F dz and (36) we have z τ dz =[p + p x ]dx. Hence dw = [x dp + pdx ] D (z) = x 1 p { dz, } { } dx + dx f + pdx 1 D dz 1 + x p dx + dx f + pdx D dz + +x n p { dx + dx f } + pdx n D dz n, = x 1 p {dx + dx 3 + +dx n }+x p {dx 1 + dx 3 + +dx n } + +x n p {dx 1 + dx + +dx n 1 }+ x p dx f, ( ) ( ) ( ) = x x 1 p dx 1 + x x p dx + + x x n p dx n ( ) + x p dx f. For dentcal domestc frms we have dw = n(n 1) xp d x + n xp dx f, where x s supply of the representatve domestc producer. So, f dx f s assumed to be zero, t s optmal to ncrease the tax n order to decrease supply and enhance socal welfare. Assume there s one domestc producer. Then dw = xp dx f = xp dx dx f dτ. (37) dτ dx If the polcy maker assumes that a change n ts polcy does not affect foregn supply (dx f = 0), t s optmal to keep the tax at the so-called envronmentally optmal level. In the rght-hand sde of (37) several factors appear. Clearly x > 0 and p < 0. Moreover, dx/dτ <0snce the reacton curve of the domestc frm shfts outward as the tax rate decreases (ths s a consequence of the second order condton for the maxmzaton of profts by the domestc frm). If the Nash equlbrum on the product market s stable,

19 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 169 and f the polcy maker takes nto account that the outward shft of the domestc reacton curve leads to an equlbrum wth smaller foregn supply, the polcy maker sets the emsson tax below margnal damage. Wth multple domestc producers (n > 1) t mght be optmal to have hgher emsson taxes, because two types of mpacts can be dstngushed due to a decrease n the emsson tax rate. On the one hand, producton ncreases, thereby enhancng profts from exports on the world market due to the fact that foregn equlbrum supply decreases, as before. On the other hand, the ncrease of supply by the domestc frms may decrease ther proftablty on the world market. The total effect s therefore ambguous. Barrett (1994) and Ulph (1997), who employ a somewhat dfferent model, fnd the same result. In order to llustrate the results obtaned above we return to the example of a Cobb Douglas technology and a quadratc damage functon of subsecton 3.1 n the prevous secton, wth a lnear demand functon. There s only a sngle domestc producer. Demand for the raw materal s ( ) r β ( ) τ β z = x, β 1 β where r s to be consdered fxed. Wthout strategc behavor on the part of the government the emsson tax rate equals margnal damage: τ = D (z) = z. Combnng the two equatons, we get a relatonshp between the emsson tax and output: ( ) β ( ) β r 1+β 1 1+β 1 τ = x 1+β. β 1 β Paraphrasng Barrett, ths schedule can be called the envronmentally optmal emsson tax. For each gven tax rate the exportng frm maxmzes ts profts wth respect to the raw materal, yeldng ( ) τ 1 β ( ) r β 1 ax ax f =. 1 β β Note that the reacton curve s downward slopng for all tax rates. The next queston s what happens when the government decreases the emsson tax, startng from the envronmentally optmal one defned above. Clearly, gven the foregn country s output, the domestc frm s reacton curve shfts outward and the new equlbrum has hgher domestc producton and less foregn producton, assumng a stable Nash equlbrum on the output market. What s the mplcaton for socal welfare? Accordng to (37) we have Takng nto account that dw dτ = axdxf dx dx dτ. dx dτ < 0 and dx f dx < 0,

20 170 Wthagen we conclude that t s ndeed optmal for the government to set an emsson tax lower than the envronmentally optmal one. Ths s not only true under the assumpton that the foregn government does not react strategcally, but the concluson holds as well f the foregn government does react strategcally. 4.4 Partal Equlbrum Bertrand Competton Next we consder Bertrand olgopoly on the world markets. We mantan the assumpton that there s only one foregn compettor, chargng a prce p f. Socal welfare reads W = { p x ( p, p f ) τ z } D(z) + τ z, where p = ( p 1, p,..., p n ) s the vector of prces charged by the domestc frms, and x = F (z ). Frm maxmzes profts, p F (z ) τ z, subject to F (z ) = x ( p, p f ). Ths yelds { } p + x 1 F x / p z = τ, = 1,,..., n. (38) Takng nto account proft maxmzaton, F (z ) = x ( p, p f ) and τ = D (z) the varaton n socal welfare s dw = = {x dp + p dx } D (z) = x1 dz } {x dp + p F z dz D dz j=1 + + xn x 1 xj / p 1 x j / p j j=1 dp1 + x x n xj / p n x j / p j dpn j=1 j=1 x xj / p x j / p j dp x j xj / p f x j / p j dpf. As n the case of Cournot competton the result s ambguous f there s more than one domestc suppler. Therefore, we assume that n = 1. Then dw = x dp dp f dp f dτ dτ. The best reply functon of the domestc country s ncreasng: for a fxed emsson tax the optmal prce ncreases as the foregn prce s ncreased. Moreover, n a stable market equlbrum, an ncrease n the tax rate wll cause a downward shft n the domestc reply

21 Internatonal Trade and Envronmental Polcy Under Imperfect Competton 171 functon and hence decrease the equlbrum prces. So, an ncrease n taxes, startng from taxes equal to margnal damage, wll ncrease socal welfare. Henceforth the optmal taxaton s larger than margnal damage. So, the usual outcome that Cournot competton and Bertrand competton lead to dfferent polcy conclusons s reached here as well. We end ths secton wth an llustraton. Wth lnear demand and a Cobb Douglas technology, proft maxmzaton entals: ( ) τ 1 β ( ) r β 1 ap + ap f = a. 1 β β The domestc reacton functon s upward slopng n the prce charged by the foregn compettor. Moreover, an ncrease of the emsson tax wll cause a downward shft of the reacton curve. Hence, gven the foregn reacton curve, t s optmal to set an emsson tax hgher than margnal damage. 5 OLIGOPOLY; GENERAL EQUILIBRIUM 5.1 Introducton The prevous secton addressed the queston whether governments have an ncentve to devate from the Pgouvan envronmental tax rule n the case of an olgopoly, n the context of a partal equlbrum model. In the absence of a home market and wth a sngle suppler per country the answer s n the affrmatve. Moreover, n these crcumstances the emssons tax s too low n the case of Cournot competton on the world market, whereas the reverse holds for Bertrand competton. If there are multple domestc frms these results do not generally hold anymore. Kennedy provdes an example where the tax s stll below the Pgouvan one n a coordnated optmum and even lower n the Nash game played between the two governments. In the present secton, we revst the queston posed n a general equlbrum settng, keepng as much as possble the sprt of the models studed above. As before we abstract from domestc consumpton of the exported commodty. There s a sngle compettor on the world market. And we do not allow for abatement. The man modfcaton s that the economy now has two sectors, both usng a pollutng nput as well as captal, assumed moble across sectors, but mmoble between countres. The motvaton for analyzng olgopoly n a general equlbrum settng s that the strategy of ncreasng domestc producton by relaxng emsson standards has an effect on the allocaton of captal n the economy through the rate of return. In partcular, by makng the pollutng nput less expensve also the sheltered sector mght be stmulated. Ths effect s neglected n a partal equlbrum settng. It wll be shown below that the effect can be mportant and may mply polcy recommendatons opposte to the ones obtaned for the partal equlbrum dscussed n the prevous secton. In the next subsecton 5. we consder Cournot competton, Bertrand s dealt wth n subsecton 5.3. Fnally, n 5.4 we consder the case of an emsson celng.

22 17 Wthagen 5. General Equlbrum: Cournot Competton The am of ths secton s frst to derve and characterze the socal optmum for the home economy f the supply by the foregn frm s taken as gven. It s shown that a unform emsson tax cannot mplement ths allocaton. The underlyng dea s smple. In the economy there are two dstortons: one s the envronmental dstorton and the second s the fact that the world market s non-compettve. The latter dstorton s fully exploted f there were a sngle domestc frm actng as an olgopolst on the world market. Here the absence of a domestc market s crucal of course. The former dstorton can be solved ndependently, usng unform emsson taxes equal to margnal damage. If there are multple domestc frms the latter dstorton s not fully nternalzed. The formal proof of ths s qute smlar to the exercses performed before. It s gven for the sake of completeness. The frst-best optmum s the soluton of maxmzng socal welfare ( ) W = U(c 1, c 3 ) D z 1 + subject to c 3 = z c 1 = F 1 (k 1, z 1 ), (39) p (x, x f )x = p x j + xf x, (40) j=1 x = F (k, z ), = 1,,..., n, (41) k 1 + = k. (4) k So, the exposed sector only exports. The export revenues are used for the mport of a thrd commodty. The world market prce of the exported commodty s p and depends on supply from the home country x = (x 1, x,..., xn ) and foregn supply xf. The Lagrangan reads ( ) L = U(c 1, c 3 ) D z µ 1 [F 1 (k 1, z 1 ) c 1 ]+ µ [F (k, z ) x ] + µ 3 p z [ x j + xf x c 3 + r k k 1 j=1 The necessary condtons are U c 1 = µ 1, k ]. U c 3 = µ 3, (43) µ 1 F 1 k 1 = r; µ 1 F 1 z 1 = D, (44)