OPTIMAL ALLOCATION OF TRADABLE EMISSION PERMITS UNDER UPSTREAM-DOWNSTREAM STRATEGIC INTERACTION

Transcription

1 OPTIMAL ALLOCATION OF TRADABLE EMISSION PERMITS UNDER UPSTREAM-DOWNSTREAM STRATEGIC INTERACTION GIUSEPPE DE FEO 1 ; JOANA RESENDE 2 ; MARIA-EUGENIA SANIN 3 Verson: In ths paper we analyze envronmental regulaton based on tradable emsson permts n the presence of strategc nteracton n an output market wth d erentated products. We characterze rms equlbrum behavor n the permts and n the output market and we show that both rms adopt "rval s cost-rsng strateges". Then, we study the problem of the regulator that ams to maxmze socal welfare, proposng an e cent crteron to allocate permts between rms. We nd that the optmal allocaton crteron requres a perfect balance between the d erence on rms prce-cost margns n the permts and the d erence on rms mark ups n the output market. In lght of the prevous result, we use a smulaton to obtan the optmal allocaton of permts between rms as a functon of output market characterstcs, n partcular as a functon of goods substtutablty. 1. INTRODUCTION The nterplay between some of the tradable emsson permts (TEP) markets mplemented so far and the correspondng output market has rased some concern regardng non-compettve emsson tradng (see Montero, 2009 for a survey on the exstng lterature on ths pont). For example, accordng to Kolstad and Wolak (2008), the olgopolstc rms partcpatng n the Calfornan electrcty market (CAISO) behaved strategcally n the Los Angeles market for NOx emssons called RECLAIM (Regonal Clean Ar Incentves Market). Such rms, shown to exert unlateral market power n the CAISO by Wolak (2003), were allocated 56% of total ntal stock of permts. In the same lne, Chen et al. (2006) compute equlbrum behavor consderng the nteracton between the NOx budget program and the Pennsylvana-New Jersey-Maryland (PJM) electrcty market. Due to the hgh concentraton of the PJM market, sx large electrcty generators alone account for 90% of emssons n the referred permts market. In ths context, Chen et al. (2006) 1 Unversty of Strathclyde and Unverstà degl Stud d Pava. E-mal: guseppe.defeostrath.ac.uk. The authors thank Rabah Amr, Paul Belle amme, Therry Brechet, So a Castro, Jean Gabszewcz, the Edtors Arel Dnar and Foravante Patrone, and two anonymous referee report for ther helpful comments. The usual dsclamer apples. 2 Unversty of Porto and Cef.up. E-mal: resendefep.up.pt. Fnancal support from FCT s deeply acknowledged (Research grant PTDC/EGE-ECO/115625/2009). 3 Unversty of Montpeller 1, UMR5474 LAMETA, and Ecole Polytechnque, Pars, France. E-mal: maraeugena.sannpolytechnque.edu 1

2 nd that a Stackelberg leader wth a long poston n the permts market could gan substantal pro ts by wthholdng permts and drvng up permts costs for rval producers. More recently, Tanaka and Chen (2012) analyze the n uence of permts allocaton on strategc rms n a forward electrcty market that could produce shfts n the spot electrcty market. In partcular they nd that the more allowances are allocated to the less pollutng rm, the more ths rm wll contract n the forward market leadng to a declne n spot electrcty prces. Supporters of envronmental regulaton based on TEP markets argue that the creaton of such market makes t possble to reach the polluton reducton target (re ected n "the cap") n a coste ectve manner (Montgomery, 1972) and wth a mnmum nformaton cost for the regulator, n partcular concernng the polluton abatement technology avalable. Ths argument has been challenged, rst, by Hahn s (1984) domnant-frnge model whch argues that the exstence of market power reduces the cost-e ectveness of TEP markets because a domnant rm manpulates the prce of permts to reduce ts own emsson abatement costs. Msolek and Elder (1989), nspred n Salop and Sche man (1987), and Eshel (2005), also relyng on a domnant-frnge settng, show that technologcal lnkages between permts and output markets would gve rase to "rval s cost-rsng strateges" by the domnant rm, whch would ntroduce an addtonal type of market dstorton. Fehr (1993) and Sartzetaks (1997) have also challenged the e ectveness of envronmental regulaton based on TEP markets, showng that, n a context of strategc permts tradng, TEP markets could lead to monopolzaton or excessve entry barrers. Although the last two papers have consdered strategc nteracton n the output market, ther obectve s to focus on the monopolzaton or no-entry corner solutons nstead of assessng the e ects of strategc nteracton on optmal permts allocaton. In fact, Fehr (1993) assumes that downstream rms buy permts for a gven supply, whereas Sartzetaks (1997) assumes a compettve permts market. To our knowledge only Tanaka and Chen (2012) and De Feo et al. (2012) consder a model of strategc nteracton both n the permts and n an output market. The former s mostly nterested on the way electrcty prces are a ected by the lnkages between a forward electrcty market and the permts and spot electrcty markets, whle the latter s only nterested on the way nteracton on the permts market mpacts the nal prce n a homogeneous good market. The man contrbuton of ths paper s to show the dependence of TEP market e ectveness on the way rms nteracton n the output market takes place. We assume that two asymmetrc (n terms of producton and abatement technology) Cournot producers of a d erentated pollutng good also meet n a TEP market where they trade permts that were freely allocated to them. Wth ths purpose we generalze De Feo et al (2012) s game to the case generc of prce competton n a d erentated output market wth general demand and abatement functons. The former nd that the leader n the permts market always marks-up the prce of permts to ncrease ts market share n the output market. D erently from the prevous, heren we pn down a full characterzaton of rms strateges as a functon of ther relatve (producton and abatement) e cency as well as the 2

3 degree of output d erentaton. In ths context the prce-makng rm n the permts market may mark permts up or down. D erently from De Feo et al. (2012) we nd a prce-makng rm n the permts market may set a dscount on the permts prce f he s a net buyer. Ths occurs when the degree of d erentaton n the output market s enough hgh. Moreover, even when he s a net seller, the strength of hs cost-rsng strateges depends on hs relatve e cency as compared to hs rval. In lght of these results, we show how the net mpact of the smultaneous adopton of rval s cost-rsng strateges on cost-e ectveness of envronmental regulaton depends on rms poston n the permts market (buyer or seller), on the nterplay of rms cost structures and on the degree of substtutablty between goods. Altogether, these factors determne the way TEP should be allocated to promote e cency. In ths lne, we propose an optmal crteron to allocate permts between rms, when the regulator ams to maxmze total welfare. We nd that the optmal allocaton crteron s the one guaranteeng that the d erence on each rm s prce-cost margns n the permts and n the output market are equalzed for the last permt dstrbuted. Accordngly, the optmal crteron we derve (accountng for strategc nteracton n the output market) departs from the so-called equmargnal prncple that has been generally adopted by the envronmental lterature (see for example Requate, 2005). The equmargnal abatement costs crteron assumes perfect competton n the market that s subect to the envronmental regulaton. Whle, under perfect competton n the output market equmargnalty ensures the mnmzaton of total abatement costs and, consequently, maxmzaton of total welfare, that s not the case n our set-up. 2. MODELLING FRAMEWORK We model rms nteracton n the permts market and sequentalty of decsons as n De Feo et al. (2012) but accountng for general demand and abatement functons as well as a d erentated output market. We consder two rms that meet both n the permts and n the output market. In the permts market, one of the rms (say rm ) s assumed to move rst, settng the prce of permts and clearng the permts market. In the output market the two rms ( rm and rm ) compete n quanttes producng mperfect substtute goods. Quantty y k represents the producton of good k = ; and p k represents ts prce. The nverse demand functon for each avalable varant k of the good s gven by p k (y k ; y k ), where y k represents the output producton of rm k 0 s rval. Frms may have asymmetrc producton technologes. The producton cost functon of rm k s gven by c k [y k ], wth c 0 k [y k] > 0 and c 00 k [y k] 0: 4 The producton of goods and generates pollutng emssons as a by-product. The parameter > 0 represents output pollutng ntensty. We consder that rms must comply wth envronmental 4 The assumpton of non-decreasng margnal producton costs has also been used n other papers studyng upstream-downstream nteractons n TEP markets (see for example, Eshel 2005). 3

4 regulaton based on a cap and trade system: () permts are freely dstrbuted up to the cap S and; () each rm k needs to hold an amount of permts equal to the non-abated pollutng emssons generated by the producton of y k. A percentage of total permts s receved by rm and a percentage 1 by rm. The parameter 2 [0; 1] and the cap S are an exogenous choce of the regulator. In ths paper we restrct our attenton to the allocaton decson (as ponted out by Eshel (2005) the regulator s choces concernng permts allocaton and the cap on total permts can be studed separately). 5 When rms ntal permts holdngs, S and (1 ) S; respectvely, do not concde wth the pollutng emssons caused by the producton of the optmal quantty of output yk, rms may ether abate some of the extra pollutng emssons or engage n permts tradng. Then, the envronmental regulaton mples the followng restrcton: y k = E k + a k ; k = ; ; where E k 0 denotes the amount of TEP that rm k must hold to comply wth the envronmental rule and a k 0 stands for the level of emssons abated. The abatement of pollutng emssons s costly. Frms may have asymmetrc abatement technologes. More precsely, to abate a k pollutng emssons, rm k ncurs a cost of h k (a k ) ; 6 wth h 0 k (a k) > 0 and h 00 k (a k) > 0: 7 Under the prevous assumptons, rms pro ts wrte as follows: (y ; y ; E ; q) = p (y ; y ) y c (y ) +q (S E ) h (y E ) ; (y ; y ; E ; q) = p (y ; y ) y c (y ) q (S E ) h (y + S E ) : where () S E = x = x corresponds to the amount of permts sold (or bought, when x < 0) by rm ; () y E = a corresponds to the amount of pollutng emssons abated by rm and () y S + E = a corresponds to the amount of pollutng emssons abated 8 by rm : Assumpton: Throughout the paper we assume that: (A1) The demand functons are lnear wth p k(y k ;y k ) y k < 0. (A2) The goods are mperfect substtutes wth p k(y k ;y k ) y k < 0. 5 The cap on polluton s generally xed by the regulatory authorty wth the help of experts n the lght of the polluton control target (e.g. the IPCC 1990 Scent c Assessment n the case of the Kyoto protocol and ts European sde agreement for the creaton of the EU-ETS). 6 Smlarly to Eshel (2005) and De Feo et al. (2012), ths paper assumes total cost ndependence between producton and abatement. Snce we are nterested n analyzng the demand-sde mechansms leadng to rval s rsng cost strateges, the study of cost dependence between output producton and abatement s beyond the scope of ths paper. 7 The assumpton of strctly ncreasng margnal abatement costs s n lne wth other papers studyng TEP markets, such as Montero (2009) and Lsk and Montero (2011). 8 From the market clearng condton, x = x : Snce x = S E and x = (1 )S E ; we obtan that E + E = S and, therefore, a = y E can be wrtten as a = y S + E : 4

5 (A3) 2 k y k y k > 2 k yk 2 ; or: p k (y k ;y k ) y k > 2 p k(y k ;y k ) y k c 00 k (y k ) 2 h 00 k (y k E k): Note that Assumpton (A2) mples 2 k y k y k = p k(y k ;y k ) y k, whch ensures strategc substtutablty,.e. downward slopng reacton functons n the output market. Addtonally, (A3) ensures that the equlbrum n the output market s unque. 9 A su cent condton when producton and abatement costs are convex s that p k(y k ;y k ) y k < p k(y k ;y k ) y k : Under the prevous assumptons, we model nteracton n the output (downstream) and the permts (upstream) market usng a three stages sequental game. The players are the two rms and the tmng of the game s the followng: n the rst stage, rm sets the prce of permts (q 2 R + ); n the second stage, rm observes the prce q and t chooses the amount of permts to use for producton (E 2 R + ); whch determnes the amount of permts to buy or sell (x 2 R). Frm clears the permts market. Fnally, n the thrd stage, gven rms permts holdngs after tradng (E 2 R + ), rms smultaneously nteract n the output market, strategcally competng on quanttes (y 2 R + and y 2 R + ) and abatng the extra pollutng emssons (a k 2 R + ; wth a k = y k E k ). The payo s of the game are the sum of rms pro ts n the permts market and n the output market. We rely on the noton of subgame perfect Nash equlbrum (SPNE) to nvestgate rms optmal behavor n the context of our sequental game. In the followng subsecton, we look for the SPNE, usng backward nducton technques Subgame Perfect Nash Equlbrum We sequentally solve the three-stages game prevously descrbed, startng from the last stage of nteracton. In the last stage, both rms smultaneously choose the output levels that solve the followng pro t maxmzaton problem: max k (y k ; y k ; E ; q) ; y k 0 where each rm takes as gven the output level of the rval rm (n the sprt of Cournot competton) as well as the outcomes n the permts market (to be determned n the rst two stages of the game). The soluton 10 to the prevous problem de nes rm k s best response functon n the output 9 More precsely t ensures that the downward slopng reacton functons are contractons;.e., ther slope s always smaller than 1 n absolute value. 10 The Cournot equlbrum s nteror as long as the margnal revenue s hgher than the margnal cost when y k = 0. When ths s not the case, equlbrum outcomes may correspond to corner solutons, n whch rms optmal behavor may d er from the rst order condtons derved n ths paper. A su cent condton to guarantee that we are n an nteror soluton s to assume that p k [0; y k ] > c 0 k [0] + h0 k [0] 8y k. 5

6 market, k = ; and s drectly obtaned from the rst order condton (FOC): p k (y k ; y k ) + p k(y k ;y k ) y k y k = c 0 k (y k) + h 0 k (a k): (1) The equlbrum output levels n the thrd stage are the quanttes for whch both rms are gvng ther best response to the rval s output choce. In other words, the equlbrum vector y (E ; q) ; y (E ; q) corresponds to the soluton of the two-equatons system that s obtaned when we set k = and k = n condton (1). Condton (1) shows that, n equlbrum each rm chooses the output level y k (E ; q) for whch there s a perfect balance between the margnal revenue and the margnal cost (ncludng abatement margnal costs). It also llustrates rms ablty to explot the upstream-downstream strategc lnkages to pass-through to consumers (at least part of) the addtonal cost of envronmental regulaton. 11 In the followng Lemma, we summarze how the outcomes n the permts market a ect output decsons va margnal abatement costs: Lemma 1. The larger the amount of permts used for producton by rm k; the larger ts equlbrum output level and the lower the equlbrum output level for ts rval k. It follows: y k (E k) E k > 0; y k (E k) E k < 0: (2) Proof. The proof reles on the study of the cross-partal dervatves 2 k y k E k and 2 k y k E k (supermodularty and complementarty, see Amr, 2005). Wthout loss of generalty, suppose k =. Snce and 2 y E = h 00 (y E ) > 0 2 y E = h 00 (y S + E ) < 0 the larger E, the larger y (E ) and the smaller y (E ). Lemma 1 shows that rms equlbrum output producton ncreases wth ther level of permts holdngs after tradng (decreasng wth the rval s level). Everythng else the same, the larger the amount of permts hold by rm k after tradng, the lower ts abatement needs. As a result, rm k 0 s margnal abatement cost s lower and rm k s able to ncrease ts output producton. Ths drect e ect s renforced by the strategc substtutablty of rms output decsons. Accordngly, the result n (2) s ndependent of whch of the two rval rms s the most e cent n terms of 11 The extent to whch rms are able to pass-through the envronmental costs to consumers has been wdely dscussed n the context of the debate about wndfall pro ts n olgopoles subect to envronmental regulaton based on TEP. 6

7 abatement. In the second stage, rm chooses the amount of permts to use n producton (E ) after observng the prce of permts (q). When decdng E ; rm antcpates the strategc nteracton that wll take place n the output market. Formally, rm solves the followng optmzaton problem: max (y ; y ; E ; q) E 0 s.t. y = y (E ) and y = y (E ) The restrctons follows from rm 0 s ablty to antcpate how ts decsons n the permts market a ect output competton n the thrd stage. The soluton to the prevous problem de nes rm 0 s equlbrum permts endowments after tradng condtonal on the prce of permts, E (q): In the nteror equlbrum, E (q) s obtaned from the FOC of the prevous problem, that after rearrangng 12 becomes q h 0 (y (E )) = p(y ;y ) y y y (E) (3) E In equlbrum, E (q) s such that rm does not have any opportunty to ncrease ts total pro ts by tradng-o output pro ts by pro ts due to permts transactons. 13 The equlbrum condton (3) shows that, when decdng E, rm takes nto account the margnal pro tablty of permts transactons (q h 0 (:)) as well as the mpact of E n the pro ts n the output market (whch s gven by the LHS of (3)). Ths behavor leads rm to adopt a rval s cost-rsng strategy". Despte beng a follower n the permts market, rm does not behave as a typcal prce-taker. D erently from the domnant-frnge settng (e.g. Eshel 2005), n the context of our model, rm has ncentves to over-use emsson permts to decrease ts abatement needs (whle ncreasng the rval s abatement needs) and become more compettve n the output market. The followng Lemma demonstrates that rm s behavor n the permts market s ndeed consstent wth a rval s cost-rsng strategy. 14 Lemma 2. For a gven prce of permts q, rm abates less than e cently. The d erence 12 The FOC s d = + y de E y (E (E ) + y ) E y (E ) (E = 0 ) E Snce =y (E ) = 0 because rm optmally chooses ts quantty n the thrd stage, the FOC reduces to E + y y (E ) (E = 0; ) E y ;y where = q + h E 0 (y (E )) and y (E ) = p y y : 13 Our analyss s vald when the second order condton d2 [E ;q] de 2 < 0 holds. See Appendx for ts explct computaton. 14 The result n Lemma 2 s n lne wth the result n the smpler setup of De Feo et al. (2012). 7

8 between permts prce and rm s margnal abatement costs s gven by: q h 0 (y (E )) = p(y ;y ) y y y (E) E > 0 (4) 1. Proof. The proof s straghtforward snce p(y ;y ) y < 0 and y (E) E < 0, as shown n Lemma Due to strategc nteracton n the output market, the prce-takng rm n the permts market ends up behavng strategcally: rm strategcally chooses to forego pro ts from permts transactons to bene t from a better poston n relaton to ts rval n the output market. D erently from a typcal prce-taker, rm uses ts decsons concernng E to reduce ts overall margnal costs, whle ncreasng ts rval margnal costs (recall that the rval s responsble for clearng the permts market). Ths strategy allows rm to ncrease ts market share n the output market, obtanng hgher output pro ts than the pro t level correspondng to the choce of E that leads to q = h 0. Fnally, n the rst stage, rm quotes the prce q that solves the followng optmzaton problem: max q0 (y ; y ; E ; q) s.t. E = E (q); y = y (E (q)) and y = y (E (q)): Snce rm antcpates nteractons n the subsequent stages of the game, we need to account for the prevous restrctons that state that rm s able to antcpate both () rm 0 s permts holdngs after tradng n the second stage,.e. E = E (q); and () how the decsons n the permts market a ect output competton n the thrd stage,.e. y k = yk (E (q)); k = ; : Concentratng exclusvely on the case of nteror solutons, the equlbrum prce of permts s obtaned from the rst order condton to the prevous problem d dq = 0, yeldng: q + E E q = y E y E q y y E E q (5) The equlbrum prce of permts q that sats es the prevous guarantees that rm s explotng all exstng pro t opportuntes (consderng the permts market as well as the output market). In equlbrum, varatons n output pro ts nduced by margnal varatons n q (gven by the RHS n condton (5)) are exactly compensated by varatons n the pro ts assocated wth permts transactons (gven by the LHS n condton (5)). The optmalty condton shows that the equlbrum prce of permts q depends on rm 0 s market power n the permts market but also on rms ablty to explot the technologcal lnkages between permts and output markets. As a result, the 8

9 equlbrum prce of permts wll, n general, d er from rm 0 s margnal abatement cost. 15 followng Lemma dent es the mechansms drvng rm away from margnal cost prcng n the permts market. Lemma 3. Everythng else the same, the equlbrum prce of permts (q ) may be ether hgher or lower than the e cent one, bq = h 0, wth: q h 0 = p(y;y) y y E y + x E The =q : (6) Proof. Equaton 6 s obtaned by substtutng the sngle elements n (5) and rearrangng. y The rst determnant of the d erence between the equlbrum prce of permts q and the margnal abatement cost h 0 corresponds to the postve term p(y;y) E y (E (q)) > 0 n condton (6). Ths term s assocated wth rm s reacton to rm 0 s cost-rsng strategy n the second stage: In stage 2, rm wshes to purchase more/ sell less permts than a compettve rm would do. Ths shfts upwards rm s demand/ supply of permts. Antcpatng such behavor, n the rst stage, rm tres to make rm 0 s rsng-cost strategy less e ectve by chargng an addtonal p (y ;y ) y y mark-up equal to E y (E (q)) > 0. The second determnant of the gap q s assocated wth the market power of rm n the h 0 permts market. Ths determnant corresponds to E (q)=q n condton (6). Focusng exclusvely on ths e ect, condton (6) mples that, everythng else the same, when rm s a net-seller of permts (x < 0), t s wllng to quote an addtonal mark-up16 over h 0. In contrast, when rm s a net-buyer of permts (x x > 0) ; there s a prce dscount equal to (q) E (q)=q. In lght of (6), the gap q may be postve or negatve. When rm s a seller of permts h 0 x (q) t wll certanly be postve. In contrast, when rm s a buyer of permts, f the e ect y x (q) E (q)=q s very strong, rm mght be nterested n markng down the prce of permts n order to ncrease ts pro ts from permts tradng. Ths possblty s ruled out n De Feo et al. (2012). Due to perfect substtutablty, n ther paper t s always pro table to mark up the prce of permts n an nteror soluton to become more compettve n the output market. Heren, nstead, we see how the degree of output competton (whch s related to the degree of product d erentaton 17 ) and abatement 15 The sngle elements of equaton (5) are the followng. On the LHS: q = E S = x (q) E = q h 0 (y S + E ) and E q < 0 because 2 E q = 1 < 0: On the RHS, we have that y p (y ;y ) y y < 0; and y E y E E q < 0 as shown n Lemma Recall that E (q) q < 0: 17 Competton s ntense when goods are closer substtutes. = 0;by the FOC of rm n the thrd stage: In addton, y = 9

10 cost asymmetres nteract to determne whch strategy s more pro table for the rm. Subtractng (6) from (4) we can compute the equlbrum value of the gap between rms margnal abatement costs: h 0 (a ) h0 (a ) = x (q) E (q)=q p y y E y p y y E y : (7) The prevous equalty shows that the equlbrum d erental between rms abatement costs can be decomposed n three e ects. The rst e ect s gven by 0 s market power n the permts market (Hahn, 1985). x (q) E (q)=q and t stems from rm Ths e ect may be postve or negatve (dependng on whether rm s a buyer or a seller of permts, respectvely). The second e ect s p gven by y y E y > 0 and t stems from rm 0 s ablty to adopt a rsng rval s cost strategy. Ths e ect s equvalent to the output market e ect emphaszed n the domnant-frnge lterature (see, e.g. Eshel, 2005). Fnally, the thrd e ect s gven by rm s ablty to adopt rsng rval s cost strateges as well. p y y E y < 0 and t s assocated wth Whether ths thrd e ect ncreases or decreases the gap between margnal abatement costs depends on rms poston n the permts market, as well as on the nterplay of rms n the permts market and n the output market 18. In the context of our general framework, rms behavor s n uenced by several determnants (demand con guraton, producton costs, abatement technologes, polluton ntenstes and regulatory decsons) that a ect equlbrum outcomes n potentally d erent drectons. In Appendx B we consder spec c functonal forms to llustrate the strategc nteractons prevously dscussed. In the remander of ths secton, we hghlght how the key parameters of our model (namely, polluton ntensty, degree of substtutablty n the output market, and rms abatement technologes) a ect equlbrum outcomes. We leave the analyss of welfare and regulatory mplcatons for next secton. Frst, t s worth notng that n ths paper we have assumed that the two rms have the same polluton ntensty, somehow 19 mplyng the use of the same producton technology by the two rms. Assumng d erent polluton ntenstes would be d erent from assumng d erences n 18 When rm s a net buyer of permts and the rst term n the RHS of (6) s hgher n absolute value than the second term, permts prces are under-prced, leadng to h 0 (a ) < h0 (a ): The fact that permts are under-prced generate a downward move along rm s supply of permts. Ths s further aggravated by the upward shft n the supply of permts that s assocated wth rm 0 s rsng rval s cost strateges. When rm s a net-seller of permts, they are always over-prced (q > h 0 a ). The hgher prce of permts nduces a reducton n the quantty of permts demanded by rm due to a move along rm 0 s demand curve for permts. However, as descrbed n Lemma 2, rm 0 s strategc behavor also shfts permts demand curve upwards. As the two e ects move n opposte drectons, t s not possble to say a pror, whch rm has the lowest margnal abatement cost. When p y y y E s small enough, we have that h 0 (a ) > h0 (a ): In contrast, when the adopton of rsng rval s cost strategy by rm more than compensates rm 0 s over-prcng strategy n the permts market, we have that rm s the one wth the lowest margnal abatement cost (wth h 0 (a ) < h0 (a )). 19 Somehow because the producton technology s gven both by the polluton ntensty ; whch re ects the use of a pollutng nput whch s the same for both rms, and the separable producton cost c k (y k ): 10

11 abatement technologes and cannot be fully captured by usng smple scale factors n the abatement functon. It s therefore worth dscussng what would be the e ect of havng a rm usng a less pollutng producton process. Everythng else the same, a lower would shft outwards the reacton functon n the output market of the less pollutng rm (and make t atter). The result would be that the less pollutng rm would produce more n equlbrum, for any gven amount of permts held, and the most pollutng rm would produce less. As for the permts market, we rst explore the case n whch rm has a lower polluton ntensty than rm. Lookng at equaton (4), when comparng the results wth rm havng a lower we can dentfy a drect e ect, for whch rm s wllng to accept a lower prce for the supply of any amount of permts (and s therefore wllng to sell more permts for any gven prce q): h 0 decreases wth a lower polluton ntensty. However there s also a strategc e ect whch goes n the opposte drecton. 20 In equaton (4) the term on the RHS ncreases wth a lower snce y s larger for any gven E. Ths term shows that the less pollutng rm s wllng to wthhold permts n order to gan a strategc advantage n the output market. The same reasonng can be done n the case n whch rm has a polluton ntensty that s lower than rm. Lookng at (6) we see that there s a drect e ect of a lower on the demand of permts by rm that would choose a lower q (and therefore a lower E ) on the supply schedule of rm because of ts lower margnal abatement cost h 0. As n the case of rm there s also a strategc e ect related to the rsng-rval-cost strategy of rm. The rm wth lower has an ncentve to wthhold polluton permts n order to gan an advantage on the output market. Everythng else the same, the term on the RHS of (6) ncreases wth a lower snce y s larger for any gven E. Regardng the degree of product d erentaton n the output market, t a ects rms optmal decsons not only n the output market but also n the permts market. As shown n (4), the degree of substtutablty between good and determnes the magntude of q-h 0 for rm. Smlarly, the degree of substtutablty between good and determnes the magntude of the rst term n (6). When products are very d erentated, a reducton n rms margnal abatement cost has a lower mpact on rms relatve compettveness n the output market than n the case of close substtutes. Accordngly, when products are sgn cantly d erentated, the rval s cost-rsng strateges tend to become less e ectve and the equlbrum prce of permts s mostly n uenced by rm 0 s poston n the permts market. Instead, when products are close substtutes 21, rms relatve compettveness n the output market depends more sgn cantly on ther margnal abatement cost and therefore rm tends to mark up the prce of permts to decrease ts abatement needs (whle ncreasng the rval s abatement needs) and become more compettve n the output market. The asymmetry between margnal abatement cost functons also a ect the relatve magntude of the two e ects n (6) and, as we wll see n the followng secton, the optmal allocaton of 20 We thank an anonmous referee for pontng out ths e ect. 21 Ths s the case of De Feo et al. (2012) who consder perfectly substtute goods. 11

12 permts between rms. As the relatve sze of the permts market ncreases n relaton to the output market, f rm s a net buyer of permts, t tends to be more nterested n markng-down the prce of permts than n ncreasng ts rval s cost n the output market. 3. REGULATION AND POLICY IMPLICATIONS From Lemma 2, we conclude that strategc nteracton n the output market results n an upward shft of rm 0 s demand curve (or supply curve) of permts, whch leads rm to under-nvest n polluton abatement, for any gven prce of permts. Dependng on the nterplay between the market power n the permts market and the ncentve to adopt a rval s cost-rsng strategy, rm mght be nterested n abatng more or less than e cently. Envronmental lterature usually presents the maxmzaton of socal welfare as the man obectve of the envronmental regulator. Wth ths purpose the regulator maxmzes welfare that depends postvely on consumer s surplus (CS) and pro ts ( + ) and negatvely on a damage functon that depends on total emssons, and that heren we assume lnear. Snce total emssons n a cap-and-trade system are xed equal to S our lnear damage functon as does not depend on the permts allocaton rule. Then, the regulator maxmzes W = CS + + as wth respect to antcpatng subsequent equlbrum outcomes, q () ; E (q ()) and yk (E (q ())) : Accordngly, we consder that the regulator s able to antcpate the mpacts of not only n the permts market but also on the output market. In partcular, the regulator s able to antcpate that along the equlbrum path, the output level of the rm recevng more permts ncreases whle the output of the rm recevng less permts decreases (see the Appendx for more detal on ths): y > 0 and y < 0 (8) The mechansm behnd ths result s the followng: rst, an ncreases of leads to a decrease n the permts prce. Ths reducton of the prce of permts entals a downward move along rm 0 s demand curve for permts (or supply of permts, f rm s a net-seller), whch, n turn, leads to an ncrease n the use of permts for producton by rm : As a consequence, rm 0 s margnal abatement costs are lower than before, yeldng an ncrease n rm 0 s output producton. opposte occurs to rm ; that looses market share after an ncrease of. The mpact on total output and consumers welfare depends on whether the ncrease of y k more than compensates the decrease of y k after the varaton of : The 12

13 Gve the regulator s maxmzaton problem, the followng FOC s vald for nteror solutons 22 : CS + + = 0 In Appendx C we show how the prevous FOC results n the followng optmal permts allocaton rule n the case of symmetrc demands n the output market: de d h0 h 0 de = d y y p + y y p : (9) E y E y Bascally such allocaton rule ensures that any asymmetry on rms equlbrum margnal abatement costs s compensated by an equal asymmetry n the strategc e ect on the output market n equlbrum. The result n (9) volates the so-called equmargnal prncple that has been generally adopted by the envronmental lterature (see for example Requate, 2005) that assumes perfect competton n the market that s subect to the envronmental regulaton (see for example Montgomery, 1972). Under perfect competton n the output market equmargnalty ensures the mnmzaton of total abatement costs and, consequently, maxmzaton of total welfare. Due to the strategc nteracton n the output market our result d ers from the equmargnal prncple. In fact (9) can be reformulated as we do n the followng proposton. Proposton 1. The optmal allocaton of permts s the one that ensures that the d erence on each rm s prce-cost margns n the permts and n the output market are equalzed for the last permt dstrbuted: q h 0 y de E d = (q h 0 ) y de E d where we note markups as = p c 0 h 0 and = (p c 0 h 0 ) respectvely. Proof. See Appendx C. From Proposton 1 t follows that the optmal allocaton of permts s the one guaranteeng that, for the last permt dstrbuted, the d erence on each rm s prce-cost margn n the permts market s balanced wth the d erence on each rm s mark-up n the output market, where the latter s weghted by y k E k From (10) we get (10) whch measures the e ect of the allocaton on rm s output producton. h 0 h 0 = y E y E : (11) 22 Our analyss s only vald when the condtons that guarantee the exstence of an nteror soluton exst. In our general setup t s not possble to derve explctly the condtons that guarantee that an nteror soluton exsts. Whenerver exstence condtons for an nteror soluton are volated, a corner soluton arses and the regulator must allocate all the avalable emsson permts to only one of the rms. 13

14 The gap n margnal abatement costs must compensate for the relatve strength of rms strategc e ects n the output market. For each rm k; the strategc e ect s captured by y k E k k, whch measures the margnal pro tablty obtaned by rm k n the output market when t uses an addtonal permt n producton. When rm 0 s strategc e ect y E > 0 s hgher than the one of rm ; gven by y E > 0; the regulator should allocate permts so that h0 > h0 : The opposte occurs when rm 0 s strategc e ect s stronger than the one of rm : Note that, for rm k; the margnal pro tablty of recevng an addtonal permt to use n producton s gven by the savngs n the margnal abatement cost h 0 k plus the margnal pro tablty of the output ncrease that such permt wll generate, that s entaled by strategc nteracton and t s gven by y k E k k n our set-up. Accordngly, an alternatve approach to nterpret the condton (11) s to state that the welfare maxmzng permts allocaton must equalze rms total margnal pro tablty of recevng an addtonal permt, as can be seen from restatng (11) as h 0 + y E = h 0 + y E : A regulator that consders strategc nteracton n the output market s neglected would allocate permts so as to ensure equmargnalty of abatement costs. For the sake of comparson let us note the equmargnal allocaton rule as EM, whch solves: 23 p y y + p y y = SEM E (q) y E y E E (q)=q : (12) Recall that the sgn of the LHS of (12) s a pror undetermned. The term assocated wth rm 0 s rsng cost-e ect, p y y E y ; s negatve, whereas the term assocated wth rm s rsng-cost p e ect, y y E y p y ; s postve. When the rst term s domnant, so that y E y < p y y E y ; the LHS of (12) s negatve. Ths takes place, when one or more of the followng occurs: p y ; s su cently smaller than p y the equlbrum output of rm s lower than the equlbrum output of rm, and/ or rm 0 s abatement technology s sgn cantly less e cent than the one of rm : In ths case, the equmargnal allocaton rule dstrbutes more permts to rm (recall that E (q)=q s negatve). By makng rm a net-seller of permts, the regulator s nducng rm to buy permts n the market and therefore, rm wll have addtonal ncentves to mark-down the prce of permts (ths corresponds to the e ect E =q, n equaton (6)). Ths partally o sets the strength of rm s rval s cost rsng e ect, restorng equmargnalty of abatement. The opposte stuaton takes place when the strength of the rsng rval s cost e ect s hgher for rm than for rm (whch occurs, for example, when rm s endowed wth a more e cent abatement 23 To obtan condton (12), let us consder the results n the rst and second stage of the game and substract equatons such that h 0 a = h 0 a : x 14

15 technology). In that case the LHS of (12) s postve and equmargnalty requres rm to be a net-seller of permts. By dong ths, the regulator ensures that rm wll set a postve mark-up on the prce of permts, whch countervals the strength of rm 0 s rval s rsng cost e ect. In addton, our model also predcts that, everythng else the same, the fracton of permts allocated to rm to ensure equmargnalty tends to ncrease f ths rm becomes more ne cent n relaton to rm : 24 Fnally, t s worth notng that the allocaton schemes W E 2 [0; 1] as well as EM 2 [0; 1] may not always exst n our set-up. In partcular, the condtons for the exstence of an nteror soluton to the regulator s problem may be volated when the degree of asymmetry between rms s very hgh n terms of abatement. Same thng wth the equmargnal allocaton: when the mpossblty to restore equmargnalty n abatement s caused by the asymmetry n abatement technologes, the regulator should allocate all the avalable emsson permts to the rm wth the hghest margnal abatement cost 25. Optmal allocaton rules may change consderably dependng on the regulator s obectve. Some examples of possble obectves to be pursued by the regulator are: equalzaton of abatement costs, maxmzaton of ont pro ts, maxmzaton of consumers surplus (even wth some preference over some category of consumers), maxmzaton of total welfare, or other. Dependng on the scope of the regulator s actvty and the spec c obectve pursued by the regulator, there may be contradctory recommendatons regardng the optmal allocaton rule : In the followng subsecton, we rely on an example to study ths pont The mportance of output demand characterstcs In ths subsecton we rely on a smulaton 26 to llustrate the polcy mplcatons of our model. In partcular, we stress the mportance of output demand characterstcs n the de nton of the optmal allocaton rules by the envronmental regulator. In lne wth the example presented n Appendx B, we consder an ndustry n whch: () the nverse demand for good k s gven by p k (y k ; y k ) = 25 + " 2y k y k ; wth 0 < < 2 and " = 0:1 f k = ; () c k [y k ] = (y k ) 2 4 ; () = 0:8; (v) S = 3; and, (v) rms abatement technologes are gven by h (a ) = 1:1(a ) 2 2 and h (a ) = (a)2 2 : We compute optmal allocaton rules for progressvely hgher degrees of substtutablty between goods, comparng outcomes for = 0:01; = 0:4; = 1:5; and = 1:99. Concernng the mpact of the obectve pursued by the regulator, we compare optmal allocaton 24 For nstance, n the context of the example presented n Appendx B, we would obtan that EM ncreases from 50% to 54:9% when ncreases from = 1 to = 1:1. 25 For nstance, n the context of the example presented n Appendx B, we obtan the followng corner solutons: () EM = 1 when rm s endowed wth the least e cent abatement technology ( = 4) ; and () EM = 0 when rm s endowed wth the most e cent technology ( = 0:4) : 26 In fact t s hard to derve general polcy mplcatons n our framework snce equlbrum outcomes are a ected by several determnants (demand con guraton, producton costs, abatement technologes, polluton ntenstes and regulatory decsons) wth potentally opposte e ects. 15

16 rules under four d erent regulatory obectves: () equmargnalty of abatement e ort (EM); () maxmzaton of rms ont pro ts (JP ); and, nally, () maxmzaton of total socal welfare (W ). To compute consumer s surplus, we use the partal equlbrum analyss from Belle amme and Petz (2010). Table 3 summarzes our results: Substtuton EM JP W = 0:01 EM = 54:1% JP = 54% W = 69% = 0:4 EM = 54:4% JP = 53% W = 61% = 1:5 EM = 54:9% JP = 51% W = 60% = 1:99 EM = 55:2% JP = 51% W = 60% Table 3: Optmal permts allocatons Readng each lne separately, we conclude that the optmal allocaton of permts vares accordng to the regulator s obectve. When the regulator wants to promote equmargnalty of polluton abatement (EM), he voluntarly chooses to gnore the mpact of permts decsons on the output market. Table 3 shows that, under a equmargnalty allocaton rule, the regulator must allocate more permts to the rm that owns the less e cent abatement technology (as descrbed n the prevous secton). In relaton to the way output market characterstcs n uence the equmargnalty allocaton rule, table 3 shows that as substtutablty decreases, rsng-rval s cost strateges are weaker (due to less competton) and therefore the lower needs to be the compensaton 27 to the least e cent rm. It s worth notng that the equmargnalty allocaton rule s not very senstve to changes n the degree of output substtutablty. In fact output demands reman perfectly symmetrc, wth p y = p y and therefore, the degree of product substtutablty only a ects the strength of rval s cost-rsng e ects ndrectly, through ts e ect on output producton (and therefore n rms abatement needs). Turnng now to the maxmzaton of ont pro ts (JP), the optmal allocaton of permts once more favours the least e cent rm : Ths s the case because, when gvng more permts to the least e cent rm, the ncrease n ts output producton (and ts pro ts) more than compensates the decrease n ts rval s producton (and pro ts). The changes n the pro t-maxmzng allocaton as substtutablty changes are stronger than n the cost-e ectve allocaton. Fnally, the welfare maxmzng (W) allocaton rule gves even more permts to the least e cent rm than the pro t maxmzng allocaton rule. Ths s the case because the latter only accounts for the fact that the ncrease n the least e cent rm s output producton after an ncrease n more than compensates the decrease n the most e cent rm s output producton. Instead, the welfare maxmzng allocaton rule also accounts for the fact that the mentoned changes n quanttes a ect prces and consequently consumer s surplus. In partcular, for the values of the parameters that we 27 Compensaton n ths context takes place through a hgher permts allocaton. 16

17 consdered, we observe that the reducton n the prce p more than compensates the ncrease n p : Ths leads to permts allocaton rules more based towards rm (around 60%) Concernng the mpact of output substtutablty on the welfare-maxmzng allocaton rule, we nd that as goods become less substtutable, rm must receve a hgher fracton of permts to compensate the fact that p s hgher than p ). (due to the hgher margnal abatement costs of rm It s worth notng that the amount of permts allocated to the least e cent rm ncreases as we move rghtwse n Table 3. Ths s because, when gvng more permts to the least e cent rm, the cost-e ectve allocaton rule takes nto account the bene t n terms of cost mnmzaton whle the pro t-maxmzng allocaton takes nto account, addtonally, the postve e ect on overall output quantty and, nally, the welfare-maxmzng allocaton consders the bene ts for consumers n terms of prces (although one group of consumers s favoured n detrment of the other) CONCLUDING REMARKS We have shown how the possblty of the smultaneous adopton of rval s cost-rsng strateges n an olgopolstc market may ether aggravate of amelorate the cost-e ectveness of envronmental regulaton based on TEP markets. In partcular, we found that t depends on each rms poston as buyer or seller of permts as well as on the nterplay of abatement and producton cost structures (and ts mpact n the output market wth d erentated products). In lght of the prevous results, we have nvestgated optmal crtera to allocate permts between rms as a functon of the regulator s obectve, addressng n more detal the characterzaton of permts allocaton that lead to cost-e ectveness of polluton abatement. We nd that the optmal allocaton of permts s the one that ensures that the d erence on each rm s prce-cost margns n the permts and n the output market are equalzed for the last permt dstrbuted. Our analyss also reveals that optmal allocaton rules may be extremely senstve to the spec c characterstcs of the output market. Then, n lne wth Sartzetaks (1997), ths paper emphaszes that the regulator s often faced wth extremely demandng nformaton needs. 28 The analyss regardng the regulatory possbltes as a functon of goods substtutablty becomes partcularly relevant n the case of envronmental polces that may a ect the redstrbuton of producton between goods and, through that redstrbuton, harm a spec c group of consumers (n the case of our example, those would be the consumers of good ). 17

18 REFERENCES [1] Belle amme, P and Petz, M. [2010] Industral Organzaton Markets and Strateges, Cambrdge Unversty Press. [2] Amr, R. [2005]. Supermodularty and Complementarty n Economcs: An Elementary Survey, Southern Economc Journal, 71(3), [3] Chen, Y., Hobbs, B.F., Ley er, S. and Munson T.S., [2006], Leader-follower equlbra for electrc power and NOx allowances markets, Computatonal Managment Scence, 3, [4] G. de Feo, J. Resende, M-E Sann, [2012], Emsson permts tradng and downstream strategc nteracton, forthcomng The Manchester School. [5] D. M. D. Eshel, [2005], Optmal allocaton of tradable polluton rghts and market structures, Journal of Regulatory Economcs, 28:2, [6] Fehr, N.-H. M. v. d. [1993], Tradable Emsson Rghts and Strategc Interacton, Envronmental and Resource Economcs, 3, [7] Hahn, R. [1985], Market power and transferable property rghts. Quarterly Journal of Economcs, 99, [8] IPCC [1990], Clmate change: The IPCC Scent c Assessment. [9] Kolstad, J. and F. Wolak, [2008], Usng emsson permts prces to rse electrcty prces: Evdence from the Calforna electrcty market, Harvard Unversty, mmeo. [10] Lsk, M and Juan-Pablo Montero, [2011], Market Power n an Exhaustble Resource Market: The Case of Storable Polluton Permts, Economc Journal, 121(551), [11] Msolek, W. S. and H. W. Elder [1989], Exclusonary manpulaton of markets of polluton rghts. Journal of Envronmental Economcs and Management, 16, [12] Montero, J.P. [2009], Market Power n Polluton Permt Markets. The Energy Journal, 30 (Specal I), [13] Montgomery, D. W. [1972], Market n Lcenses and E cent Polluton Control Programs, Journal of Economc Theory, [14] Requate, T. [2005]. Tmng and Commtment of Envronmental Polcy, Adopton of New Technology, and Repercussons on R&D, Envronmental & Resource Economcs, 31(2), ,

19 [15] Salop, S.C. and D.T. Sche man [1983], Rasng rvals costs, Amercan Economc Revew Papers and Proceedngs, 73, [16] Sartzetaks E.S. [1997], Tradable Emsson Permts Regulatons n the Presence of Imperfectly Compettve Product Markets: Welfare Implcatons, Envronmental and Resource Economcs, 9, [17] Tanaka, M. and Y. Chen [2012], Emssons Tradng n Forward and Spot Markets of Electrcty, The Energy Journal, 33(2) [18] Wolak, F.A. [2003b], Measurng Unlateral Market Power n Wholesale Electrcty Markets: The Calforna Market 1998 to 2000, Amercan Economc Revew, May,

20 Proof of Lemma 1 Appendx A By applyng the theory of supermodular games, the sgn of the e ect of E k on y k and y k depends on the sgn of the cross-partal dervatves 2 k y k E k and 2 k y k E k. 29 Wthout loss of generalty, suppose k =. Snce 2 y E = h 00 (y E ) > 0 and 2 y E = h 00 (y S + E ) < 0 the larger E, the larger y (E ) and the smaller y (E ). SOC of the second-stage optmzaton problem The second order condton of the problem can be wrtten as d 2 de 2 = (d =de ) + (d =de ) y? + (d =de ) E E E y? y? y? where the rst term s: (d =de ) y? = 1 h 00 E E (y? E ) + P y?; y? y? y? E ; the second term s: (d =de ) y? y? E = " h 00 (y? E ) + 2 p y?; y? y? c 00 (y? ) 2 h 00 (y? E ) # y? E ; and nally the thrd term s: (d =de ) y? y? E = p y?; y? y? y? : E E y? Therefore the SOC can be wrtten as follows: y? 1 h 00 E (y? E )+ p y?; y? y? y? E " 1 + y? + E 2 p y?; y? y? c 00 (y? ) # y? E < 0 Proof of (8) 29 See Amr (2005) for a survey of ths approach. 20

21 Applyng the chan rule, the dervatves y k y k = y k (q; E ) E can be decomposed as follows: E (q) q q ; where the dervatves y k (q;e) E have already been obtaned n Lemma, wth y k (E k) y k (E k) E k < 0: We have also seen that E q < 0 because 2 E q 2 q < 0 (complementarty and supermodularty, see Amr, 2005). Accordngly, along the equlbrum path < 0: Smlarly, q E k > 0;and < 0 because and y y = y (q;e) E = y (q;e) E E (q) q q > 0; E (q) q q < 0: As stated n (8) after a margnal varaton of ; the equlbrum output of rm changes n the same drecton, whle the equlbrum output of rm changes n the opposte drecton. Appendx B: an example Heren we rely on an example to llustrate how relevant parameters, such as rms relatve e cency n terms of abatement, may a ect equlbrum market outcomes. 30 We consder that () the nverse demand for good k s gven by p k (y k ; y k ) = 25 2y k 1:5y k ; () rms producton technology s smlar, wth c k [y k ] = (y k) 2 4 ; () the ntensty of polluton s equal to = 0:8; (v) the total stock of emsson permts s exogenously xed by the regulator to meet the polluton control target and we consder t to be S = 3; whch constrans rms producton plans. Fnally, (v) rms abatement technologes are gven by h (a ) = (a)2 2 and h (a ) = (a)2 2 : In what follows, we compute the equlbrum outcomes as n Secton 2.1 consderng the spec c functons detaled n ponts () to (v). Regardng rm 0 s abatement technology, we consder three possble scenaros: rm s more e cent n terms of abatement ( = 0:4); rms have symmetrc abatement technologes ( = 1); and rm s more e cent n terms of abatement ( = 4) : In the followng table we reproduce the predctons of our model regardng equlbrum prces (both for 30 Note that a smlar analyss could be performed to study the n uence of other parameters, such as output demand, producton costs or the polluton cap S: For the sake of smplcty, n the paper we present only an example that analyses the e ects of abatement technologes and, at the end of the secton, we comment on other potentally relevant parameters. 21