Incentives for Price Manipulation in Emission Permit Markets with Stackelberg Competition

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3 Incentves for Prce Manpulaton n Emsson Permt Markets wth Stackelberg Competton Francsco J. André and Lus M. de Castro. Dept. of Economc Analyss. Unversdad Complutense de Madrd. Campus de Somosaguas, 83 Madrd, SPAIN. October 04 ABSTRACT It has been shown n pror research that cost effectveness n the compettve emssons permt market could be affected by tact colluson or prce manpulaton when the correspondng pollutng product market s olgopolstc. We analyze these cross market lnks usng a Stackelberg model to show that under reasonable assumptons, there are no ncentves to collude for lobbyng prces up. However, ncentves for manpulatng the prce of permts up appear f there s an ntal free allocaton of permts, whch s a polcy argument aganst grandfatherng and n favor of auctonng. Ths effect s ncreasng wth the amount of permts allocated to the leader. Moreover, the changes for prce manpulaton ncrease wth those changes that tend to undermne the leader's advantage n output producton or to reduce the leader s abatement cost. JEL code: D43, L3, Q58 Key words: Emssons permts, Colluson, Market power, Duopoly, Stackelberg model.

4 . Introducton In ths paper we check the exstence of ncentves for olgopolstc frms to collude n order to nflate the prce of emsson permts when there s a leader-follower relatonshp n the output market. The use of cap-and trade (CAT) systems has become ncreasngly popular as a polcy approach to ncentvze frms to curb pollutng emssons. Important examples nclude the US SO tradng system under the framework of the Acd Ran Program of the 990 Clean Ar Act as an early applcaton or, more recently, the European Unon Emsson Tradng System (EU ETS). The man reason why CAT programs are so attractve and popular among economsts s that, theoretcally, they allow reducng emssons n a cost-effectve way by means of a prce system. As long as margnal abatement costs dffer across frms, ncentves for trade exst and the market can play a postve role n achevng a pre-specfed envronmental target at a mnmum cost. Regardless of the ntal allocaton rule chosen for the permts, the costeffectveness property s well documented n the lterature under the assumpton of perfect competton (see Montgomery (97) n a statc settng and Rubn (996) n a dynamc framework). Unfortunately, the perfect-market assumpton rarely holds n practce and t s the case that the cost-effectveness property s challenged f there s market power n ether the permt market, n the assocated product market or n both. The lterature analyzng the relatonshp between mperfect competton and emsson permts can be dvded n three dfferent branches, whether market power s ntroduced n the permts market, n the good market or n both smultaneously. Regardng market power n the permt market, the poneerng work s Hahn (984). Based on a statc model a la Stackelberg, he stated that the effcency loss due to market power depends on the ntal allocaton of permts, and the permt prce s an ncreasng functon of the leader s allocaton. The domnant frm wll manpulate the prce (upwards f t s a seller and downwards f t s a buyer), unless the ntal allocaton equals the cost-effectve one, whch requres a perfectly nformed regulator. Hagen & Westskog (998) extended the Hahn settng n a dynamc two-perod model and found a non-optmal dstrbuton of abatement n an mperfect compettve market wth bankng and borrowng. A second lne of research addresses the concurrent exstence of market power n both permt and output markets. Ths topc has receved attenton, among other authors,

5 by Msolek & Elder (989), who extended Hahn s settng to the product market and concluded that a sngle domnant frm can manpulate the permt market to drve up the frnge frm s cost n the product market. Hnterman (0) found that the threshold of free allocaton above whch a domnant frm wll set the permt prce above ts margnal abatement costs s below ts optmal emssons n a compettve market, and that overall effcency cannot be acheved by means of permt allocaton alone Ths paper fts wthn a thrd branch that consders mperfect competton n the product market but not n the permt market. The reason to choose ths lne s twofold. Frst, as noted by Montero (009) and Muller et al. (00), whle market power among frms s very common n output markets, the exstence of market power n emsson permts s more lkely to appear when the relevant players are countres rather than frms or facltes. In the latter case, typcally there are a very large number of them, whch makes t very dffcult that market power arses. It can be argued that ths s the case n the EU ETS, wth around,000 facltes nvolved and the latest steps of the European Commsson seem to be n the drecton of ncreasng even more the degree of competton (for example, by enlargng the number or nvolved sectors, centralzng the allocaton of permts or movng from grandfatherng to auctonng). As a second reason, The EU-ETS prce shock n 005 generated a great deal of nterest n ssues related to market power. Intally, the prce of allowances was far n excess of expectatons, but n Aprl 006 the prce suddenly fell and reached zero n md-007. Emprcal studes have not been able to perfectly explan those too hgh prce levels when the number of permts exceeded emssons n every year of the frst phase. Then, t s natural to wonder f the reason for those prce oscllatons mght be connected to the output market rather than the permt market n the sense that permts could be used somehow to obtan wndfall profts ether n the output market or the market of some mportant nput such as electrcty. In ths thrd lne, some artcles have shown that perfect competton n the permt market mght not be enough to render a cost-effectve outcome f the product market s not perfectly compettve. In the framework of a Cournot duopoly, Sartzetaks (997) compares the effcency of a compettve emssons market to a command-and-control As an example regardng Annex countres n the Kyoto Protocol, Russa ntally receved roughly a ffth of the permts and a thrd went to USA. Countres wth market power can easly manpulate prces up (down) through tarffs on permt exports (domestc subsdes to cleaner technologes) and also mplement polces regardng the lnkage between domestc and foregner markets. See Barrett (998) for a related dscusson. 3

6 regulaton n whch the emssons of each frm are legally fxed. Emsson tradng modfes the allocaton of emssons among frms and consequently ther producton choces. Sartzetaks (004) shows that welfare can decrease when emsson tradng s allowed between asymmetrc frms endowed wth dfferent abatement and producton technologes. The permt prce that clears the market s a weghted average of the value of emssons of frms under command and control and, therefore, the cost of the more neffcent frm s reduced whle the cost of the more effcent one s ncreased when permts trade s ntroduced. Meuner (0) analyzes the effcency of emsson permt tradng between two mperfectly compettve product markets and concludes that even f the frms are prce takers n the permt market, the ntegraton of permt markets can decrease welfare because of mperfect competton n product markets. Theoretcally, f markets are perfectly compettve, a unque global permt market that covers all pollutng actvtes would be effcent to allocate an aggregate emssons level. If markets are not perfect but some frm enjoys market power nstead, several permt markets may be more effcent than an ntegrated one. The closest to our paper s the one by Ehrhart et al. (008), whch shows that colluson n the product market may occur even f the frms are prce takers n the permt market. Under some condtons, a permt prce ncrease leads to hgher profts due to a decrease n product quanttes whch ncreases the output prce. In the ndustral organzaton lterature ths strategy s generally known as rasng rvals' costs. In the partcular case of an emsson permt market, Erhart et al (008) conclude that frms mght have ncentves to collude n order to push the prce of permts upwards. Although ths movement has the drect effect of ncreasng one s cost, snce t also rases the rval s, both frms could beneft by restrctng the quantty and ncreasng the prce. They argue that n the EU ETS, even f there s no explct market power n the permt market tself, there are loopholes n the tradng law that allow collusve behavor among frms to manpulate the prce of permts. Ths paper addresses the queston f frms nterests could be algned to push the prce of permts up (and, therefore, f there are ncentves to collude) under Stackelberg competton. So, we check f the colludng ncentves reported by Erhart et al. (008) mght stll arse n a settng that s asymmetrc n nature n the sense that there s a leader and a follower, whereas Erhart et al. restrct to purely symmetrc settngs. We frst set a general model n whch we show that the effect of a hgher permt 4

7 prce on the leader s and the follower s proft s ambguous. So, the possblty that frms beneft from a prce ncrease stll exsts, but the asymmetrc role of the frms ental that such a possblty happens under dfferent condtons for the leader and the follower, whch ntroduces the possblty that one frm s nterested n rsng and the other n decreasng the permt prce. Then, we explore a partcular case wth a separable cost functon to come up wth more accurate nsghts. As a frst central fndng, under the reasonable assumpton that the soluton s nteror (both frms produce, pollute and abate to some extent), we conclude that both frms face a proft functon that s convex n the permt prce. Moreover, wthn the relevant range, when the prce s low enough, both frms wll beneft from a further prce reducton, and for hgh enough prces, the follower would beneft from a prce ncrease whle the leader would stll prefer the prce to decrease. Therefore, n the latter range there s no room for colluson. Ths s n contrast to Ehrhart et al. (008), who set a symmetrc model and therefore, both frms nterests are always algned. The mplcaton of ths fndng s that the exstence of leadershp n output markets reduces the room for collusve agreements n the permt market. Actually, n our specfc example wth a separable functon, we conclude that the collusve regon shrnks to the extent that t dsappears. As a frst extenson, we consder the possblty that some permts are dstrbuted for free (by means of grandfatherng) and we conclude that ths possblty opens up the way for collusve agreements. In fact, apart from the two regons dentfed n the smple case, there s a thrd regon n whch both frms are nterested n pushng the prce up and ths regon s wder the more permts are dstrbuted for free. Ths result ponts out an mportant argument aganst grandfatherng n the sense that t could ntroduce ncentves to foster collusve behavor. As a second extenson, we explore the effect of asymmetres and we conclude that the chances to face an envronment that s proptous for a collusve agreement are very senstve to the confguraton of the cost parameters of both frms and the allocaton of free permts receved by the leader, but not by the follower. In short, those parameter changes that tend to undermne the leader's advantage n output producton (.e., an ncrease n the leader s cost or a decrease n the follower s cost) have the effect of makng the frms more symmetrc n a certan sense and, therefore, t ncreases the chances to observe a collusve behavour. The opposte happens wth the abatement costs: the chances for a collusve behavor tend to decrease wth the leader s and to 5

8 ncrease wth the follower s abatement cost. The reason s that, snce the leader produces more output than the follower, ts cost s more senstve to the permt prce and thus t s more dffcult for hm to get benefted by such a prce ncrease, and ths s truer the hgher hs abatement cost. On the other hand, an ncrease n the follower s abatement cost reduces the possblty that t s optmal for hm to pollute zero, whch wdens the nteror soluton range and, consequently, also the scope for agreement. The remander of the paper s organzed as follows. Secton states the basc model. In Secton 3 a partcular abatement cost functon s consdered, ncludng the basc case and the two extensons. Concludng remarks are gven n Secton 4 and all the mathematcal proofs are gathered n an appendx.. The general model We set up a smple duopoly Stackelberg model of a pollutng ndustry sector wth tradable permts. Frm s a leader and frm s a follower n the output market. Followng Erhart et al. (008) we assume no explct market power n the permt market, although we mplctly consder the possblty that they mght fnd some extramarket ways to agree n lobbyng actvtes to push the prce up or down. For the sake of comparablty wth Ehrhart et al. (008), n ths secton we assume that the frms do not enjoy any ntal allocaton of permts, and so they have to buy all the permts they need n a market at a gven prce p (the possblty of an ntal endowment of permts s consdered as an extenson n Secton 3). The game has two stages: n the frst stage, frms sequentally decde on ther output levels, x and x, a la Stackelberg, facng the nverse demand functon P X, where X : x x. In the second stage, they smultaneously choose on ther cost-mnmzng emsson levels, e and e. Erhart et al. (008) argue that, n the EU ETS, there are several loopholes n the tradng law that foster colluson. These loopholes consst n the exstence of some mechansms that create the possblty for a prce manpulaton even n the absence of market power. The most mportant of these mechansms are frst, the possblty to nfluence the ntal allocaton of permts (to make t more strngent), second, the 'opt-n' rule that enables ndustres not commtted to partcpate n the permts tradng system to do so voluntarly, thrd, the possblty to mplement project-based mechansms and pay more for these credts than they would at the market and, fourth, by payng addtonal emssons dutes. Hnterman (0) clams that ths type of prce manpulaton could explan the behavor of the prce for allowances durng the frst phase of the EU ETS, although hs model s not fully comparable to ours snce he assumes explct market power n both output and permt markets. 6

9 The cost functon of frm (, ),, C x e, depends on output (x ) and emssons (e ) and s contnuous and twce dfferentable n both arguments wth the followng propertes: C C C C 0, 0, 0, 0. x e e xe Ths functon ntegrates producton and abatement costs and reflects the fact that producng clean (wth low emssons) s more costly than producng drty. Every unt of emssons must be covered by a permt that can be obtaned n the market at a gven prce p. Consderng the cost of permt purchasng, total cost of frm s gven by, :, TC x e C x e pe. () The model s solved by backward nducton. In the second stage of the game both frms decde ther emssons levels to mnmze ther total cost,, () TC x e, whle takng ther output levels and the prce of permts as gven. If the soluton s nteror, we get the standard frst-order condton (FOC), 3 C p 0, (3) e from whch we obtan each frm s (nverse) demand for permts,, e x p. Total dfferentaton of the FOC shows that optmal emssons are ncreasng n output and decreasng n the permt prce: C 0 0, (4) C C e ex de dx e e x x C e C e de dp e p C e 0 0. (5) Usng the envelope theorem, we conclude that the mnmzed total cost functon defned as TC x, p : TC x, e x, p C x, e x, p pe x, p (6) has the followng propertes: emssons. 3 The second order condton s always fulflled due to the convexty of C n 7

10 TC e p 0, (7) TC C C e C p x x e x x, (8) C TC C C e C x e x x xe x x C e, (9) C TC C e xe xp xe p C e 0. (0) Now we move on to the frst stage, where the frms choose ther output levels. We start analyzng the follower s behavor, whch faces the followng maxmzaton problem: x,,,,, Max x x e x p p P x x x TC x p. () The FOC of ths problem s whch, solvng for P TC Pxx x 0, () X x x, gves the reacton functon of the follower, x x,p. Dfferentatng the FOC and operatng we conclude that the optmal follower's output s decreasng n the leader's output and the prce of permts: dx dx dp dx dp dx 0, TC x TC dx x p dp 0. (3) dp TC dx x The leader takes the follower's reacton functon nto account when maxmzng ts own proft. The FOC of the correspondng problem s P x TC Pxx x 0 (4) x x x and, by dfferentatng the FOC, we conclude that the leader's output supply s also decreasng n the prce of permts: 8

11 dx dp TC xp P x TC x x x 0. (5) Now we are ready to address the man queston of ths paper, namely the effect of an ncrease n the prce of permts on the frms proft. The queston s: could both frms beneft smultaneously from a prce ncrease as predcted by Erhart et al (008) n a symmetrc settng? The motvaton behnd ths queston s that, f the answer happens to be postve, both frms mght have ncentves to collude or, by any means, to lobby n order to manpulate the prce of permts up. For the sake of realsm t s relevant to ask ths queston n a settng n whch the frms play dfferent roles regardng ther market power as we can observe ths stuaton n realty. By drect dfferentaton of the proft functons we conclude that the margnal effect of the prce of permts on the profts of both frms s gven by the followng expressons: P x p X p x e, (6) P x x x e p. (7) X x p There are two mportant ponts to be stressed here. Frst, the sgn of both expressons s ambguous. The reason s that an ncrease n the prce of permts has two dfferent effects. On the one hand, t drves cost up, whch tends to reduce frms' proft but, on the other hand, t also causes output to decrease and, therefore, the product prce to ncrease, whch can be benefcal for both frms. If the second effect happens to domnate the frst, then proft wll ncrease wth the prce of permts. If ths happens smultaneously for both frms, there exst ncentves to collude n order to manpulate the prce up, as noted by Ehrhart et al. (008). Second, the condtons under whch a hgher prce s proft-enhancng are dfferent for the leader and the follower. Ths opens up the possblty of a dsagreement between the frms n the sense that one of them s nterested n a prce ncrease and the other one n a prce decrease. Ths s n contrast to Ehrhart et al. (008), where both frms are symmetrc and, therefore, ether both frms are better-off or both are worse-off after a prce ncrease. Ths asymmetry seems to reduce the scope for a prce 9

12 manpulaton agreement. At ths level of generalty t s not possble to get more specfc nsghts. For that reason, we explore a specfc case n the next secton. 3. A Separable Functon To get some addtonal nsght, n ths secton we assume a partcular abatement cost functon whch s separable from producton cost. We ntally consder that there s no grandfatherng and the cost functons of both frms are exactly the same, so that the only dfference between them s due to ther roles as leader and follower. After studyng ths basc case, we explore, frst, the effect of dstrbutng free permts to the frms and, second, the consequences of consderng cost asymmetres. 3.. Basc case Assume that the producton and abatement costs are separable n the followng way. The producton cost of frm s gven by cx, so there s a constant margnal producton cost equal to c. The (nverse) demand functon for output has the lnear form P X a bx. Every unt of output generates r unts of polluton, where r 0 s a constant coeffcent of polluton ntensty, (thus, gross emssons of frm are gven by rx ). By performng abatement actvtes, frms can reduce ther flow of polluton. Denote as q 0 the amount of emssons abated by frm. Then, net emssons are gven by e = rx q. Followng Sarzetaks (997) we assume the followng quadratc abatement cost functon, whch s common for both frms: AC q q d tq, (8) where d and t are postve parameters. Addng up the costs of producton, abatement and permt purchasng, and usng the defnton of q, we can wrte total cost as a functon of output and emssons as n the general model:, TC x e cx rx e d t rx e pe. (9) To ensure nteror soluton we bound the relevant parameters by ncludng the followng techncal assumpton: Assumpton : d p p, where p p e x p : /, 0. (0) Ths assumpton rules out unnterestng solutons n whch any of the frms ether produces zero, pollutes zero or abates zero. The lower bound for p prevents abatement from beng negatve (see equaton () below). To understand ths result, 0

13 note that d s the margnal cost of abatement at q 0. If the prce of permts s even lower than the cost of the frst unt of abatement, ths means that t wll never be proftable for the frms to abate, snce buyng permts s a cheaper opton. The upper bound for p s defned as that value of the permt prce such that, n equlbrum, t s optmal for the follower to pollute zero. 4 The reason to nclude ths assumpton s that, n our settng, the follower s emssons s the frst varable to reach a zero value as p ncreases and, therefore, ths s a suffcent condton to ensure a nonnegatve soluton. 5 Proceedng as n the general model, we solve frst the second stage, where both frms choose ther emsson levels. Endowed wth our specfc analytcal expressons, we can compute the optmal amount of emssons of frm as a functon of output: d p e x, p rx, () t from whch t s straghtforward to conclude that frm 's optmal abatement s p d q p 0, () t whch, due to separablty, s ndependent of output. Usng () n (9) we get the expresson for the mnmzed cost functon, whch reveals that margnal product cost s constant n output and ncreasng n the permt prce:, d p TC x p x c pr. (3) 4t Now, we move on to solve the frst stage, n whch both frms decde on ther output levels. By standard methods, 6 we get x x a c rp, (4) b a c rp. (5) 4b 4 The specfc expresson for p can be found n the appendx. Specfcally, t s gven by (A3) n the basc model, but t takes a dfferent form n the extensons developed below. 5 If both the follower s abatement and the follower s net emssons are nonnegatve, t s straghtforward to conclude that the follower s gross emssons, rx, are nonnegatve, whch mples that the follower s output s nonnegatve. As we show below, n equlbrum the leader always produces more and pollutes more than the follower and, thus, Assumptons ensures that all the relevant varables of the model are nonnegatve n equlbrum. 6 The follower chooses x to maxmze ts proft whle takng x as gven. The leader chooses x to maxmze ts own proft takng nto account the follower s reacton functon.

14 From (4) and (5) we conclude that the leader s output s twce as much as the follower s as n the classcal Stackelberg model wth lnear demand and constant margnal cost. As always, both frms output depend postvely on the demand ntercept, a, and negatvely on the demand slope, b, and all the cost parameters c, r and p. We can now compute the equlbrum value of both frms proft as a functon of the prce of permts, defned as: p: a b x xx TC x, p and for notatonal convenence, we defne, (6) pˆ : argmn p,. (7) p Lemma and Proposton show the man results of ths part of the paper. Lemma determnes the shape of the equlbrum proft functons and, as a straghtforward mplcaton, Proposton splts the relevant range for the permt prce n two regons wth dfferent consequences on the nterests of the frms regardng the evoluton of p. LEMMA and p are strctly convex functons of p wth d pˆ ˆ p p. p PROPOSITION If d p pˆ, a prce decrease would make the proft of both frms to ncrease. If ˆp p p, a prce ncrease wll decrease the leader s proft and ncrease the follower s proft. The results n Lemma and Proposton are llustrated n Fgure. There are two mportant facts to be underlned n ths fgure. Frst, the proft of frm s strctly convex n p wth a mnmum at p ˆ (for =, ). We can get some nsght about the reasons that underlne the shape of the proft functons by dssectng the effects of an ncrease n the prce of permts. When p ncreases, both frms respond by producng less, abatng more and, thus, emttng less. These responses have effects on both revenues and costs. Revenue s a strctly concave, nverted-u shaped functon of p. When p s low, the resultng reducton n output nduces a revenue-enhancng ncrease n the prce of output,

15 whereas, for large values of p, the negatve effect on output prevals. Regardng costs, there are three effects: producton cost decreases lnearly, abatement cost ncreases quadratcally due to the shape of the abatement functon and the cost of purchasng permts decreases quadratcally due to the substtuton of permts for abatement and, n margnal terms, the latter effect turns out to domnate all the others for large values of p. The second nsght from Fgure s that the mnma of the proft functons are unambguously ordered such that ˆp < ˆp,.e., the follower reaches a mnmum for a lower prce than the leader. As a matter of fact, t can be proved that,.e., p p for any value of p, the leader s proft decreases more or ncreases less than the follower s. As a consequence, we have that, f p < ˆp, both frms are n the decreasng part of ther proft functons, whch mples that ther proft would ncrease f the permt prce decreased. If, nstead, ˆp < p < ˆp, the follower s n the ncreasng part (and so t would beneft from a prce ncrease) whereas the leader s stll n the decreasng part (and, therefore, t would stll prefer the prce to decrease). Intutvely, the reason why the proft of frm reaches a mnmum before frm s that, beng a Stackelberg follower, t s optmal for frm to produce less than frm and, therefore, to pollute less. Ths mples that the drect effect of a prce ncrease s mlder as compared to the ndrect effect (the rval s cost rse wth the resultng ncrease n the output prce). Apparently, f p > ˆp the leader enters the ncreasng part of ts proft functon and, therefore, both frms would beneft from a hgher prce, but under our specfcaton, t s the case that ˆp p,.e., the mnmum of the leader s proft functon s reached precsely at the hghest value of the prce that s compatble wth an nteror soluton. (specfcally, e < 0 for any p pˆ ) and so there s no feasble range under whch both frms would beneft from a prce ncrease. 7 7 For analytcal smplcty (an also for the sake of realsm) we do not dscuss n detal the corner soluton range, snce the case n whch any of the frms does not pollute at all seems unnterestng. Actually, f we rule out negatve emssons, what happens to the rght of p s that, for frm, t s optmal not to pollute at all ( e 0 ) and, therefore, n ths range ts cost s not drectly affected by the prce of permts, whle the leader s s. The expressons for output and profts that we have derved are vald only under the nteror soluton scenaro. If we allow for corner solutons wth e 0, then we come up wth dfferent expressons and, usng these expressons we conclude that, n the vcnty (to the rght) of p, the leader s output and proft are stll decreasng (n the prce) whle the follower s output and proft are ncreasng. 3

16 The man consequence of Lemma and Proposton s that, n our example, there s a range where both frms are nterested n decreasng the prce but, unlke the symmetrc case developed by Ehrhart el al. (008), t s never the case that both frms smultaneously proft from a prce ncrease, and therefore they never have ncentves to lobby n order to press the prce up. Moreover, there s a range of dsagreement n whch the nterests of both frms dverge, whch can never happen n the symmetrc case. p p d ˆp ˆp p p Regon I RegonII FIGURE : Equlbrum profts as a functon of p (basc case) In ths example we have llustrated how the asymmetry between the frms (n the sense of a leader-follower relatonshp) reduces the chances for collusve behavor so much that they dsappear. In the next subsectons we show two generalzatons of ths example where the result s not so extreme n the sense that the chances for collusve agreements decrease wth respect to a purely symmetrc settng but they do not fully dsappear. 3.. Grandfatherng Up to now, for comparablty wth Erhart et al. (008), we have assumed that the frms do not have any ntal allocaton of permts and, therefore, they have to buy all the permts they need n the market. In realty, t s common that the partcpants n CAP systems receve a number of permts for free by means of a grandfatherng scheme. In fact, as t s dscussed for example n Alvarez and André (04), grandfatherng has tradtonally been the most wdespread method to dstrbute permts. 4

17 We now extend our settng to consder the possblty that some permts are ntally dstrbuted wth no cost for the frms by a grandfatherng scheme. 8 So, the frms only need to buy those permts that exceed ther ntal allocaton and, moreover, they have the opton to sell permts f they pollute less than ther ntal allocaton. Consder that both frms receve an equal allocaton of free permts, S, and denote as y the amount of permts that frm buys (f y 0 ) or sells (f y 0 ) n the market, whch can be calculated as the dfference between net emssons and the allocaton of permts: y e S rx q S, (8) from whch we get e y S,.e., the net emssons of a frm must be covered by permts that ether come from her free allocaton or are bought n the market. Therefore frm s total cost functon s now gven by the expresson:, TC x y cx rx y S d t rx y S py, (9) whch can be wrtten n terms of output and net emssons as, TC x e cx rx e d t rx e p e S. (30) Solvng the second stage of the game we conclude that the optmal levels of emssons and abatement for each frm are stll gven by () and ( ) respectvely, and t s straghtforward to obtan the optmal traded permts and the correspondng mnmzed cost functon: d p y x, p rx S, (3) t, d p TC x p x c pr ps, (3) 4t where separablty entals that the mnmzed cost functon has the same structure as n the basc case, except for the fact that the value of the endowment of free permts appears as a reducton n the cost. Lemma and Proposton are the man results of ths part. We stll use the notaton ntroduced n (6) to refer to the value of the permt prce that mnmzes each proft functon. For notatonal convenence we also defne r a c dr S :. 8b 8 Actually, the fact that the permts are dstrbuted for free s not crucal for our results. The only mportant assumpton s that the frms enjoy an exogenously gven amount of permts. 5

18 LEMMA When both frms are ntally endowed wth the same free allocaton of permts, S, the equlbrum proft functons for both frms are strctly convex wth a unque mnmum each at p ˆ for,, wth the followng way: a) If S S, then d pˆ ˆ p p. b) If S S S, then pˆ d pˆ p. c) If S S, then pˆ ˆ p d p. pˆ 0. Moreover, the relevant thresholds are ordered n S PROPOSITION When both frms are ntally endowed wth a free allocaton of permts, the followng results hold: a) If S S the relevant range of values for p has three regons: In regon I, defned by d p pˆ, both frms get better off when p decreases. In regon II, defned by ˆp < p < ˆp, the leader gets better off when p decreases and the follower gets better off when p ncreases. In regon III, defned by ˆp p p, both frms get better off when p ncreases. b) If S S S, regon I dsappears and regon II s delmted by d p pˆ. c) If S S, regons I and II dsappear and regon III s defned by the whole feasble range, d, p. p p d ˆp ˆp p p Reg. I Reg. II Reg. III FIGURE : Equlbrum profts as a functon of p (basc case) 6

19 The consequences of Lemma and Proposton are the followng. The proft of both frms s stll strctly convex n the prce of permts, wth a mnmum at prce p ˆ, =,. When grandfatherng s ntroduced, the values of the permt prce at whch the mnma are reached, ˆp and ˆp, shft to the left and they shft more the hgher the value of S. Ths shft mples, for each frm, that there s wder range of the permt prce such that t gets better-off when the prce ncreases. The reason s that the exstence of free permts makes permt purchasng less costly for frms and, moreover, t opens the way from gettng postve revenues by sellng some permts. More mportantly, f we focus on case a) (wth S S ), when grandfatherng s ncluded, we have three regons nstead of two, as llustrated n Fgure. In regon III, to the rght of ˆp, both frms proft from a prce ncrease, whle the soluton s stll nteror ( e, e 0). Techncally, the reason why ths new regon arses s that, now, the drect effect of a prce ncrease on cost s softer snce the frms have to buy fewer permts. It can also be the case, f the prce s hgh enough, that t s proftable for the frms to sell part of ther free endowment nstead of buyng addtonal permts, whch provdes a new opportunty to ncrease profts. Anyway, at ˆp we have that y 0.e., at the pont where the leader starts fndng t proftable to ncrease the prce, t s stll a net buyer of permts and so the proft-enhancng effect s not due to sellng permts yet. Moreover, f the ntal allocaton of permts s large enough, t could be the case that regon I dsappears, whch mples that the follower s always nterested n manpulatng the prce of permts up (case b) n Lemma and Proposton ), or even that both regons I and II dsappear, whch mples that both the leader and the follower are always nterested n manpulatng the prce up. Ths s, of course, the most favorable case for colluson. The focus of ths paper s on regon III snce ths s the only one n whch frms can fnd t proftable to collude n order to push the prce up. One natural queston s how large ths regon s or, n other words, how lkely t s that we fall n ths regon. To answer ths queston, we focus on case a) ( S S ), whch s the rcher one and perhaps the most realstc (the dscusson of the other two cases s straghtforward). Then, regon III s delmted by two threshold values for p: Frst, ˆp, whch s the prce above whch t s proftable, not only for the follower, but also for the leader to push the prce up. The 7

20 second threshold s the upper bound, p, whch s the hghest value of the prce compatble wth an nteror soluton. By computng the dfference between these two thresholds, we get the sze of regon III, p pˆ 4btS b tr whch depends postvely on the number of free permts, as well as the slope of the demand curve, b, and the abatement cost parameter t, whereas t depends negatvely on the emssons ntensty parameter r., 3.3. Asymmetrc cost In the prevous developments we have consdered that both frms were fully symmetrc n terms of cost functons and also, n the case of grandfatherng, n terms of free permt endowment. There are two reasons to hold ths assumpton. The frst s smplcty. The second s to focus on the leader-follower relatonshp as the (only) source of asymmetry between frms. In ths subsecton we consder the possblty that frms are asymmetrc n terms of cost and/or ntal permt endowment and we explore the effect of these asymmetres on the chances to generate a proptous envronment for a collusve behavor. In other words, we explore the effect of dfferent parameters on the sze of regon III as t was defned n the prevous subsecton. To account for cost asymmetry, we denote the producton cost of frm as cx, where c s a frm-specfc unt cost parameter. Snce we have postulated that frm s a leader and frm s a follower n the output market, t s natural to conjecture that c c,.e., the poston of the leader mght well be due to the fact that t enjoys a cost advantage, but nothng prevents us from consderng the opposte case. Analogously, frm s abatement cost functon s gven by: AC q q d tq, =,. (33) Fnally, each frm mght receve an ntal free endowment of permts, S, whch s not necessarly constant across frms. Proceedng as n the basc case, we conclude that, n the second stage, the optmal amounts of emssons, abatement and purchase of permts for each frm are gven, respectvely, by 9 9 Unlke the rest of parameters, we assume that the emssons ntensty parameters, r, s common for both frms,.e., r r r. There are two pragmatc reasons for ths smplfcaton. Frst, the senstvty analyss 8

21 d p e x, p rx, (34) t q p p d, t p d y x, p rx S, t and movng on to the frst stage we can compute the equlbrum levels of output: x x a c c rp, (35) b a c 3c rp. (36) 4b To nvestgate the chances to observe a collusve behavor we proceed by analyzng the effect of dfferent parameters on the sze of regon III. In the prevous subsecton we concluded that, smply by ntroducng a constant ntal allocaton of permts, we come up wth three dfferent cases. Now, due to the larger number of varyng parameters, by choosng the rght combnaton of them we could generate almost any magnable case. Hence, we need to bound the range of possbltes n some way to avod, on the one hand, meanngless results (such as negatve output, negatve abatement or negatve emssons) and, on the other hand, a qualtatve change n the nature of the soluton. For ths reason, n ths subsecton we ntroduce the followng assumptons: Assumpton : max, Assumpton : e e. Assumpton 3: pˆ ˆ p. d d p p, where p s defned n (0). The two frst assumptons ensure nonnegatve values for all the relevant varables. The dea s that, under any plausble scenaro, the leader wll stll be the one who produces a larger amount of output and a larger amount of emssons. As a consequence, the follower wll stll be the one who fnds t proftable to pollute zero for a lower value of p and such a value determnes the upper bound for the range that s results related to these parameters are unclear and so we do not come up wth any valuable nsght by explorng them. Second, the sgn of some equlbrum values for some of the key varables are affected by the terms r r and/or 3r r and ths fact, anyway, forces us to keep the asymmetry between these parameters bounded to avod meanngless results. 9

22 compatble wth an nteror soluton, p. If ths s the case, t s natural to accept that Assumpton 3 also holds,.e., t s easer for the follower than t s for the leader to beneft from a prce ncrease. Under these assumptons, regon III s delmted by ˆp and p and hence ts sze ncreases f p ncreases and/or ˆp decreases. Proposton 3 summarzes how the sze of ths regon depends on the parameters of the model. Table presents a taxonomy of all the relevant effects. PROPOSITION 3 The sze of regon III s ncreasng n the followng cases: a) If the leader s margnal producton cost, c, ncreases or the follower s margnal producton cost, c, decreases. b) If the parameter of the lnear term n the abatement cost functon decreases for the leader ( d ) or ncreases for the follower ( d ). c) If the parameter of the quadratc term n the leader abatement cost functon, t, decreases (provded the number of free permts s moderate) or the equvalent follower s parameter, t, ncreases. d) If the number of free permts receved by the leader, S, s ncreasng regardless of the free permts receved by the follower. Effects on Changes n parameters of the model thresholds c c d d t t S S p ˆp () 0-0 ˆ p p () Table. Summary of senstvty analyss results. () For a moderate value of S. Regardng pont a) n Proposton 3, ncreasng the leader s producton cost or reducng the follower s cost tends to erode the leader s advantage wth respect to the follower, whch has the effect of makng the frms more symmetrc n terms of ther poston n the market. The more symmetrc the frms are, the more algned ther nterests are and, hence, t s more lkely that they fnd t proftable to collude. Table reveals that ncreasng c has a double effect: on the one hand, p grows because the output of the follower ncreases, whch makes t less lkely that frm decdes not to emt at all (n other words, the range of prces under whch there s an nteror soluton 0

23 wdens). On the other hand, ˆp decreases snce, due to a hgher cost, frm tends to produce less, to emt less and, therefore, ts total cost s less senstve to an ncrease n the prce of permts. Both of these effects tend to enlarge the agreement regon. Just the opposte happens when c ncreases: frm tends to produce more and pollute more and hence ts cost becomes more senstve to an ncrease n the prce of permts (whch ncreases the value of ˆp ) whereas the follower tends to produce less and to reach sooner the pont where t fnds t proftable to stop pollutng ( p decreases), whch reduces the sze of the agreement regon. As t comes to the parameters of the abatement cost functon ( d and t ) notce that, due to separablty, each frm s parameters are only relevant for the own frm, but not for ts rval. Both the lnear and the quadratc term of frm are rrelevant to determne the value of ˆp but ncreasng any of them makes the follower s abatement cost to ncrease, whch n turn makes t less lkely to reach the pont where t decdes to pollute zero or, n other words, t enlarges the relevant feasble range. The correspondng parameters for frm do not matter to determne the value of p and ther only relevant effect s on ˆp. Assumng a moderate value of the leader's ntal endowment of permts, any ncrease n d and t makes the leader's abatement cost hgher, whch makes frm becomes more senstve to ncreases n the prce of permts. Fnally, the ntal allocaton of permts s rrelevant for the upper bound of p snce t represents just a fxed term n the cost (and the proft) functon and so the optmal decsons are not affected. The value of a frm s profts s affected by ts own endowment (not the rval s), and hence only S s relevant to determne the sze of regon III. When the leader s free endowment ncreases, ts cost becomes less senstve to an ncrease n the permts prce and, therefore, t would be more receptve to the dea of pushng the prce up, whch ncreases the chances of a collusve behavor. 4. Conclusons and polcy mplcatons We have explored the possblty that two frms that compete a la Stackelberg n the output market and are subject to a CAT system could have ncentves to manpulate the prce of permts up. We do so n a framework smlar to the one by Erhart et al. (008) wth the dfference that they restrct to symmetrc stuatons whereas we explore a stuaton that s asymmetrc n nature. The man research queston s f the ncentves

24 for ths type of collusve behavor, whch have been reported by Erhart et al. (008) for symmetrc models, stll exst n a stuaton n whch some frm has a domnant poston and other or others act as followers. In a general model, we have shown that the effect of a permt prce ncrease on the frms' proft has an ambguous sgn. Ths opens the way for the frms to beneft from a prce ncrease and the possblty to make collusve agreements n order to manpulate the prce up. Nevertheless, the asymmetrc role of each frm causes that the condtons under whch a prce s proft-enhancng are dfferent for each of them. Under a separable cost functon we show, frst, that the proft functons are strctly convex n the permt prce and, second, that the mnma of the proft functons are dfferent for both frms, whch creates a regon of dsagreement where the leader prefers that the prce goes down whereas the follower prefers t to go up. Ths stuaton s ruled out n Erhart et al. (008) by constructon snce the nterests of fully symmetrc frms are always algned. The man message s that a leader-follower relatonshp between frms reduces the scope for an agreement between the frms to manpulate the prce up. Actually, n the basc case n whch there are no free permts dstrbuted among the frms and the cost functons are symmetrc, the regon where there ncentves to collude shrnks to the extent to dsappear. Then man polcy mplcaton of ths fndng s that a stuaton of market power n the product market can prevent the exstence of ncentves for colluson n the permt market. Another polcy mplcaton of our research s that dstrbutng some permts for free (e.g. by means of grandfatherng) opens up the possblty for collusve behavour and the more permts are dstrbuted by a non-market scheme, partcularly to the frms that enjoy market power, the more ncentves for colluson. The European Unon s reducng the use of grandfatherng and ncreasng the use of auctonng to dstrbute emsson permts. The 008 revsed European Emsson Tradng Drectve establshed as a fundamental change for the thrd tradng perod, startng n 03, the mandate that auctonng of allowances s to be the default method for allocatng allowances. The arguments posed by the European Commsson (EC) to support the ntroducton of auctons are that auctonng "best ensures the effcency, transparency and smplcty of the system, creates the greatest ncentves for nvestment n a low-carbon economy and

25 elmnates wndfall profts". 0 Our results suggest an addtonal argument to reduce the use of grandfatherng (and, arguably, to ncrease the use of auctonng) as t mght ntroduce ncentves for prce manpulaton. Our fnal nsght s that the chances for the frms to fnd colluson proftable are very senstve to the cost asymmetres between them. In general terms, the more asymmetrc the frms are, the more dffcult colluson becomes. References Alvarez, F. and André, F.J. (04) Auctonng vs. Grandfatherng n Cap-and-Trade Systems wth Market Power and Incomplete Informaton, Envronmental and Resource Economcs, In press. Barret, S. (998) Poltcal Economy of the Kyoto Protocol. Oxford Revew of Economc Polcy 4: Ehrhart, KM Hoppe, C Löschel, R (008) Abuse of EU Emssons Tradng for Tact Colluson Envronmental and Resource Economcs, 4: Hagen, C and Westskog, H. (998) The desgn of a Dynamc Tradeable Quota System under Market Imperfectons Journal of Envronmental Economcs and Management 36: Hahn, R. (984) Market power and transferable property rghts Quarterly Journal of Economcs 99: Hntermann, B. (0) Market Power, Permt Allocaton and Effcency n Emsson Permt Markets Envron Resource Econ Meuner, G. (0) Emsson Permt Tradng Between Imperfectly Compettve Product Markets Envronmental and Resource Economcs, 50: Msolek, W. and Elder, H. (989) Exclusonary manpulaton of markets for polluton rghts Journal of Envronmental Economcs and Management 6: Montero, J.P. (009) Market Power n Polluton Permt Markets. The Energy Journal 30 (specal ssue ): 5-4. Muller, R. A., S. Mestelman, J. Spraggon and R. Godby (00) Can Double Auctons Control Monopoly and Monopsony Power n Emssons Tradng Markets? Journal of Envronmental Economcs and Management 44: See secton "Why are allowances beng auctoned?". Alvarez and André (04) study the effcency argument. 3

26 Montgomery, WD (97) Markets n Lcenses and Effcent Polluton Control Programs Journal of Economc Theory 5 (3): Rubn, J. (996) A model of ntertemporal emsson tradng, bankng and borrowng Journal of Envronmental Economcs and Management 3: Sartzetaks, ES (997) Tradeable Emsson Permts Regulatons n the Presence of Imperfectly Compettve Product Markets: Welfare Implcatons Envronmental and Resource Economcs 9: Sartzetaks, E.S. (004) On the Effcency of Compettve Markets for Emsson Permts Envronmental and Resource Economcs, 7: -9. 4

27 Proof of Lemma APPENDIX Usng (3) and (4) n () we get the equlbrum values for emssons: e x p, db rt a c p b tr, (A) bt bd tr a c p b tr ex, p, (A) 4bt and usng the defnton gven n (0), we compute the value of p by equatng (A) to zero: bd rt a c ex, p0 p p b tr. (A3) Usng (3) and (4) n the nverse demand expresson P X a bx we get a3c pr the equlbrum prce of output: P. Usng the expressons for x, x, e, 4 e and P together wth (9), we get the expressons for the equlbrum profts of both frms: t a c pr b d p p, 8bt 4 t a c pr b d p p. 6bt Dfferentatng twce wth respect to p we conclude that the second dervatve of both functons s postve and, thus, both of them are strctly convex n p. By nspecton of the frst dervatve, and bearng (A3) n mnd, we conclude that has a mnmum at p pˆ p, whch mples that s decreasng n p for all the feasble values of p below p. In a smlar way, we conclude that has a mnmum at tr a c 4bd pˆ :, whch mples that s decreasng n p for p p and 4b tr ncreasng for p p. The last step s to check that the thresholds are ordered n the rght way. By drect comparson we conclude that d pˆ p ac dr. 5

28 To prove that the last nequalty s true, usng (') and the defnton of e abatement ( q rx e ) we conclude that, n the relevant range, x 0. Usng the r expresson for x gven n (4) we conclude that x 0 mples ac rp, and ths nequalty, together wth the frst part of Assumpton ( d p), mples acdr. QED. Proof of Proposton The result straghtforwardly follow from Lemma : the relevant range for p s delmted by d and p. Snce s strctly convex and t reaches at mnmum at ˆp, we conclude that t s strctly decreasng between d and ˆp and strctly ncreasng between ˆp and p. Snce s strctly convex and t reaches at mnmum at ˆp s strctly decreasng between d and p. Ths completes the proof. p, t QED. Proof of Lemma Snce the expressons for e (=,) are the same as n the basc case and the mnmzed cost functon (3) s the same as () except for a constant term, t mmedately follows that the expressons for x (=,) are also the same as n the basc case. Usng these values we get the equlbrum profts of both frms t ac pr b d p 8btpS p Px xx TC x, p, 8bt tac pr 4bd p 6btpS p Px xx TCx, p. 6bt The second dervatve reveals that these functons are stll strctly convex. Dfferentatng them wth respect to p we conclude that they have respectve mnma at rt a c bd 4btS Arg mn p pˆ, p b tr rt a c4bd 8btS Arg mn p pˆ, p 4b tr and t s straghtforward that both ˆp and ˆp depend negatvely on S. 6

29 Regardng the order of the thresholds, by drect comparson we conclude that pˆ pˆ brt a c dr rst 0, but n the proof of Proposton we have proved acdr 0, whch ensures that pˆ ˆ p. Moreover, usng (0 ) we also conclude that 4btS p pˆ ˆ p. So, we have pˆ pˆ p. To determne the relatve poston of b tr d, frst recall that, from Lemma, we know d s below ˆp, n the nterval conclude the followng: pˆ, p ˆ or n the nterval pˆ, p and so we only have to check f d p. By drect comparson we r acdr pˆ d S S, (A4) 4b r acrd pˆ d S 8b S. (A5) Ths completes the proof. QED. Proof of Proposton Consder frst statement a). The results n regons I and II follow from Lemma accordng to a smlar reasonng to that used n the proof of Proposton. In regon III, between ˆp and p, t s straghtforward to conclude that both and p are strctly ncreasng n p. Statements b) and c) straghtforwardly follow form (A4), (A5) p and Assumpton. QED. Proof of Proposton 3 Usng (34) and (35) n (33) we get the equlbrum values for emssons: e x, p e x, p b d p tr a c c rp bt b d p rt a c 3c rp 4bt By mposng the non-negatvty condtons to the follower s emssons we get the upper bound value for the permt prce, p : bd rt a 3c c e 0 p p: b r t (A6) 7

30 By substtuton of the relevant varables n the proft functon we obtan the expresson for the leader s proft functon n terms of the parameters of the model: ac c rp d p x, x, y, p ps 8b 4t Dfferentatng wth respect to p we obtan p b p d 4btS rt a c c rp 4bt And by equatng ths dervatve to zero we get the mnmum value of p such that the leader fnds t proftable to push the prce up, ˆp : proposton: 0 p pˆ : p r t b rt a c c bd 4bt S (A7) By drect dfferentaton of the values of p and ˆp we obtan the results n the p rt c br t 0; p 3rt c br t 0; p 0; d p b d br t 0; p p S S 0; pˆ 4bt S brt 0; pˆ rt c brt pˆ rt c brt pˆ b d brt pˆ 0; d pˆ S 0; 0; 0; 0; 8 0 S rt 4 b b pˆ br a c c d r b S r a c c d r t p br a 3c c rd 0 t b r t where, n an nteror soluton, the numerator of the last expresson must be postve for the follower s output to be postve. QED. 8