We characterize the trade-offs among firms compliance strategies in a market-based program where

Transcription

1 PRODUCTION AND OPERATIONS MANAGEMENT Vol. 16, No. 6, November December 2007, pp iss $1.25 POMS doi /poms Productio ad Operatios Maagemet Society Compliace Strategies uder Permits for Emissios Ravi Subramaia Sudheer Gupta Bria Talbot College of Maagemet, Georgia Istitute of Techology, 800 West Peachtree Street NW, Atlata, Georgia 30332, USA Faculty of Busiess Admiistratio, Simo Fraser Uiversity, 8888 Uiversity Drive, Buraby, British Columbia V5A 1S6, Caada Stephe M. Ross School of Busiess, Uiversity of Michiga, 701 Tappa Street, A Arbor, Michiga 48109, USA ravi.subramaia@mgt.gatech.edu sudheerg@sfu.ca btalbot@umich.edu We characterize the trade-offs amog firms compliace strategies i a market-based program where a regulator iterested i cotrollig emissios from a give set of sources auctios off a fixed umber of emissios permits. We model a three-stage game i which firms ivest i emissios abatemet, participate i a share auctio for permits, ad produce output. We develop a methodology for a profit-maximizig firm to derive its margial value fuctio for permits ad traslate this value fuctio ito a optimal biddig strategy i the auctio. We aalyze two ed-product market scearios idepedet demads ad Courot competitio. I both scearios we fid that chagig the umber of available permits iflueces abatemet to a lesser extet i a dirty idustry tha i a cleaer oe. I additio, abatemet levels taper off with icreasig idustry dirtiess levels. I the presece of competitio, firms i a relatively clea idustry ca, i fact, beefit from a reductio i the umber of available permits. Our fidigs are robust to chages i certai modelig assumptios. Key words: evirometal compliace; pollutio permits; market-based mechaisms; auctios; game theory Submissios ad Acceptace: Received Jue 2005; revisios received Jue 2006 ad November; accepted Jauary 2007 by Luk va Wassehove. 1. Itroductio Market-based policy istrumets are ow beig cosidered for a rage of evirometal issues, from edagered species preservatio to the greehouse effect ad global climate chage (The Wall Street Joural 2006, 2005a,b,c; Stavis 1998). Emissios tradig programs for pollutats such as lead, sulfur dioxide (SO 2 ), itroge oxides (NO x ), carbo dioxide (CO 2 ), ad volatile orgaic compouds (VOCs), are prime examples. Notably, the EU greehouse gas Emissios Tradig Scheme (ETS), which came ito force i 2005, caps CO 2 emissios from more tha 12,000 idustrial plats ad is a precursor to similar systems that will be implemeted uder the Kyoto Protocol. Emissios tradig refers to a market-based mechaism that allows parties to buy ad sell permits to emit certai pollutats. It differs from the traditioal commad-ad-cotrol approach that relies o a agecy, usually the govermet, to issue stadards ad specific directives o the amouts by which firms must reduce their emissios, how they must do so, ad the pealties for failure (NRTEE 2002). The total umber of permits issued correspods to the overall emissios target of the covered sources. The fact that the target is less tha busiess as usual emissios creates permit scarcity, resultig i a market price for permits. The cocept of emissios tradig has evolved ito a cetral idea i evirometal regulatio, with well-developed theory [see Tieteberg (2001) for a comprehesive collectio of articles o emissios tradig], ad attetio has ow shifted to how tradig programs should be implemeted (Muller ad Mestelma 1998). Most research to date has focused o evaluatig the efficacy of emissios tradig programs vis-a-vis other approaches such as taxes, subsidies, ad stadards from a regulator s perspective. We istead take the perspective of a profit-maximizig firm ad focus o the trade-offs amog differet compliace strategies 763

2 764 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society that a firm has at its disposal. We cosider three such strategies ivestmet i abatemet, procuremet of permits, ad adjustmet of output levels i a program based o auctioed permits. The total umber of permits available to firms is exogeous to our model ad is decided by the regulator. To the regulator, kowledge about the iterrelatioships amog firm levers for compliace is crucial give the goals of pollutio cotrol ad a desired icreasig level of strigecy i pollutio limits. From a firm s perspective, decisios such as ivestmet i pollutio abatemet, permit biddig strategy, ad productio level must be made give policy stipulatios, the accompayig cost of compliace, ad the goal of profit maximizatio. I practice, permits are allocated by gradfatherig or by auctioig. Gradfatherig refers to the process of allocatig permits to firms free of charge, based o historical emissios or process iputs. Free allocatio is problematic i terms of efficiecy because it coveys scarcity rets to the pollutig firms (Fullerto ad Metcalf 2001). Gradfathered permits also give rise to etry barriers, because ew etrats must purchase permits from existig holders (Stavis 1998). Cramto ad Kerr (2002) argue that auctioig is superior to gradfatherig because it is much more efficiet, allows reduced tax distortios, provides greater icetives for iovatio, provides more flexibility i distributio of costs, ad reduces the eed for politically cotetious argumets over the allocatio of rets. Moreover, auctioig serves to icrease liquidity i permit markets ad to facilitate the permit price discovery process (Schmalesee et al. 1998). Despite the apparet ecoomic superiority of auctios over free distributio, the latter is implemeted because it offers a greater degree of political cotrol over the distributioal effects of regulatio (Stavis 1998). We focus o auctioig as the primary meas for allocatig permits but we do provide a brief discussio i Sectio 7.2 o how our approach ca be exteded to icorporate gradfatherig i additio to auctioig. Permit auctios are implemeted i various formats. The US EPA auctios for SO 2 permits are i the sealed-bid discrimiatory price format, ad the Chicago Climate Exchage (CCX) auctios for CO 2 permits are held i two formats sealed-bid average price ad sealed-bid discrimiatory price. We model a sealed-bid uiform price share auctio where the bidders submit a schedule of prices for varyig fractios of the block of permits beig auctioed ad receive permits at a price that equates demad ad supply. Although permits are sold i discrete uits, modelig a share or divisible-good auctio is appropriate because permits are homogeeous ad the total umber of auctioed permits is geerally large (125, ,000 i US EPA auctios of SO 2 permits). It is worth metioig that aalytical results o equilibria i divisible good auctios are scarce (Hortaçsu 2002). However, there has bee substatial debate o whether the discrimiatory price format or the uiform price format is superior (Back ad Zeder 1993; Bikhchadai ad Huag 1993; Daripa 2001). Our choice of a uiform price auctio is drive by reasos of aalytical tractability. Cramto ad Kerr (2002) argue that whe o bidder has sigificat market power, uiform pricig is early as efficiet as Vickrey pricig ad that, amog sealed-bid auctios, a uiform price auctio is probably the best. Despite the discrimiatory price format of auctios for SO 2 permits, prices have bee i a arrow rage. For example, i the EPA s March 2006 spot auctio for SO 2 permits, the differece betwee the average ad lowest wiig bids was 2.61%. I ay case, the modelig framework we employ is ot limited to our auctio format choice. As results become available i auctio theory, they ca be icluded withi our framework. We cosider two scearios of imperfect competitio i the ed-product market. I the first sceario, firms targeted by the regulator are local moopolies i the ed-product market. A example is the US Acid Rai Program for restrictig SO 2 emissios from fossil fuelfired electric utilities. I the secod sceario, firms compete for ed-customer demad. A example is the EU emissios tradig program for CO 2, which icludes major competitors i the steel idustry such as ThysseKrupp ad Arcelor. I both demad scearios, however, firms do compete for scarce emissios permits. We preset a methodology by which a firm ca derive its margial value fuctio for permits ad traslate this value fuctio ito a optimal biddig strategy i the auctio for permits. I additio, the modelig effort provides a meas for firms to deduce their optimal levels of abatemet ad output. We use the term abatemet to refer to specific techological or process chages that result i lower emissios per uit of output (e.g., through the istallatio of scrubbers) or that result i a reductio i emissios idepedet of output (e.g., through the sequestratio of exhaust gases). This is i cotrast to some studies that refer to abatemet as a black-box reductio i emissios from a baselie. I both demad scearios we fid that icreasig strigecy through a reductio i the total umber of permits iduces lower levels of abatemet from firms i a dirtier idustry tha from those i a cleaer oe (the level of dirtiess correspods to the emissios per uit of output durig productio). I additio, abatemet levels taper off with icreasig idustry dirtiess levels. I the presece of Courot competitio i the output market, firms i a relatively clea idustry ca, i fact, beefit from a reductio i the umber of available permits,

3 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society 765 ruig couter to the commo wisdom that a icreasig strigecy is detrimetal to firm profitability [also see Boveberg, Goulder, ad Gurey (2004) ad Smith, Ross, ad Motgomery (2002) for related discussios o the coditios uder which firm profits ca be preserved uder icreasig strigecy]. Our fidigs are robust to chages i certai modelig assumptios ad help i uderstadig the trade-offs amog abatemet, permits, ad output levels. The remaider of this paper is orgaized as follows. Sectio 2 icludes a review of the related literature. Sectio 3 describes the model. Sectios 4 ad 5, respectively, treat the two ed-product market scearios idepedet demads ad Courot competitio. Sectio 6 illustrates our results through a umerical example. Sectio 7 exteds the aalysis to: (i) a variat of the emissios fuctio, (ii) the possibility that permits ca be gradfathered i additio to beig auctioed, (iii) the possibility that ivestmet i abatemet ca affect the cost of productio, ad, (iv) firm heterogeeity. We discuss our results ad provide directios for future research i Sectio 8. Proofs ad tables of results are icluded i Appedices A ad B, respectively. 2. Literature To the best of our kowledge, ours is the first framework that eables a assessmet of compliace strategies i a permit-based program for emissios cotrol for varyig policy strigecy ad firm dirtiess levels. Lyo (1986) examies equilibrium properties of a rage of auctios ad other procedures for allocatig trasferable permits. The paper s focus is o trasfereutral (implyig that o et reveue is geerated for the seller) mechaisms that allocate permits efficietly, but productio relatioships or abatemet are ot modeled. Sice 1995, the MIT Ceter for Eergy ad Evirometal Policy Research (CEEPR) has published a series of empirical studies o the implemetatio of the Acid Rai Program i the US, icludig EPA auctios for SO 2 permits. Laffot ad Tirole (1996a) study the impact of spot ad futures markets for tradable permits o firms compliace decisios. Firms ca buy permits, ivest i abatemet, or stop productio ad source out. The authors fid that stad-aloe spot markets iduce excessive ivestmet. The itroductio of a futures market reduces this icetive to ivest. Laffot ad Tirole (1996b) exted this aalysis assumig iovatio is a public good ad fid that optios to pollute at a give strike price fare better tha permits from a social welfare poit of view. However, they treat abatemet ivestmet as well as the choice betwee ivestmet i abatemet ad productio as biary decisios. I additio, they do ot model permit auctios. Uold ad Requate (2001) propose a combiatio cosistig of free permits ad a meu of call optios whe there is imperfect iformatio about aggregate abatemet costs so that the regulator ca approximate the margial damage fuctio. The authors, however, do ot model permit auctios, or do they explicitly model abatemet ad productio relatioships. Suevåg (2003) provides isights, based o umerical scearios, ito two competig permit auctio desigs the stadard ascedig-clock auctio ad a alterative ascedig-clock implemetatio of Vickrey pricig where the allocatio of permits affects productio ad market shares. Motero (2002) compares R&D icetives uder four policy istrumets emissio stadards, performace stadards, tradeable permits, ad auctioed permits with various strategic iteractios i permit ad output markets. Equilibrium permit price for the case of auctioed permits is established via a Nash bargaiig argumet i a two-player game idepedet of the auctio format. This is i cotrast to our explicit derivatio of the equilibrium biddig strategy ad permit price based o the margial value fuctio for permits ad o the auctio format. Motero (2002) focuses o R&D icetives i cotrast to our characterizatio of the trade-offs amog compliace levers. Boveberg, Goulder, ad Gurey (2004) ad Smith, Ross, ad Motgomery (2002) aalyze the equity-efficiecy trade-off i a geeral equilibrium model ad focus o the fractio of permits that a regulator would wat to allocate freely to firms to preserve idustry profits. The regulator uses the reveue raised from permit auctios to lower distortioary taxes o factor iputs such as capital ad labor. Our focus, o the other had, is o uderstadig firm behavior uder a exogeously specified emissios cap. Fischer, Parry, ad Pizer (2003) model a three-stage process of iovatio, diffusio, ad emissios abatemet i a oligopoly where firms are price takers. They compare the welfare effects of emissios taxes, auctioed permits, ad gradfathered permits. Oe or may of the firms iovate ad the rest are adopters. Productio relatioships are ot explicitly treated ad the equilibrium permit price for the case of auctioed permits is exogeously specified. Noe of the above papers explicitly models ivestmet i abatemet, permit valuatio ad biddig, ad output together with isights ito equilibrium reactios of firms for varyig strigecy ad dirtiess levels. There is a growig stream of literature i operatios maagemet that studies the iterface betwee operatioal decisios ad the eviromet, at both the strategic ad the tactical level [see Guide ad Wassehove (2006a,b); Kleidorfer, Sighal, ad Va Wassehove (2005); Corbett ad Kleidorfer (2001a,b), ad the papers refereced therei]. A rage of topics has

4 766 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society bee studied i the cotext of closed-loop supply chais, icludig facility locatio, product ad process desig, ivetory maagemet, yield maagemet, the value of iformatio, cotracts, ad the impact of competitio [e.g., Bakal ad Akçali (2006); Ferguso ad Toktay (2006); Debo, Toktay, ad Va Wassehove (2005); Savaska, Bhattacharya, ad Va Wassehove (2003); Rajaram ad Corbett (2002); ad Stuart, Ammos, ad Turbii (1999)]. We complemet the aforesaid by aalyzig compliace strategies for firms uder explicit regulatory costraits durig the process of productio. 3. Model We model a three-stage game i a symmetric oligopoly. I the first stage, each of the 2 firms decides its abatemet level i from a ivestmet i 2 i pollutio mitigatio (e.g., ivestmets that eable the use of cleaer iputs or the scrubbig of flue gases). Quadratic abatemet costs are commoly modeled i the literature [e.g., see Keedy (2002) ad Parry ad Toma (2002)] ad reflect dimiishig returs to abatemet ivestmets. We model abatemet as the first-stage decisio sice it is typically a log-term decisio made at a early stage withi the plaig horizo. I programs such as the US EPA s SO 2 permit program, a prespecified umber of permits is auctioed off every year a rate that is usually faster tha the abatemet ivestmet cycle [also see Ellerma (1998)]. I the secod stage of our model, firms bid for permits i a sealed-bid uiform price share auctio coducted by the regulator. Note that if the abatemet ivestmet cycle is faster tha the permit supply cycle, a sequece of decisios where the allocatio of permits precedes the abatemet decisio might be more appropriate. However, aalysis of such a sequece is itractable due to the complex permit valuatio fuctio that results (permit valuatio is discussed i Sectio 4.2.1). Each firm submits a schedule of prices for varyig fractioal shares of the total available permits. The regulator the selects the market clearig price such that total demad equals the available supply. As i Wilso (1979), we assume that the umber of bidders is kow beforehad by all participats to be. The auctio results i a share allocatio vector s (s 1, s 2,...,s ) or, equivaletly, a permit allocatio vector ( 1, 2,..., ). B i 1 i is the total permissible pollutio level, where i s i B. The regulator s choice of the target B is assumed to be exogeous to the model. Each permit allows a firm to emit oe uit of the pollutat; e is the permit price. I the third stage, each firm i produces output y i, which results i a pollutio level [ /(1 i )]y i, ad firms redeem their permits agaist their pollutio levels. is the emissios per uit of output ad defies the clealiess of firms or, equivaletly, of techologies employed. The aforemetioed fuctioal form of the pollutio level is represetative of that i idustries such as electric utilities [see Burtraw (2000) ad Motero ad Ellerma (1998)]. Sectio 7.1 treats the case where pollutio reductio is idepedet of the output level. We do ot model the possibility that firms ca trade permits i a secodary market. Nevertheless, the allocatio of permits is efficiet because of the assumptio of symmetry, ad allowig for trade will ot chage our results. The uit cost of productio is c, which we assume to be costat i the base model. I Sectio 7.3, we exted our results to the case where the cost of productio is impacted by abatemet efforts. We cosider two scearios of imperfect competitio i the ed-product market. I the first case, each firm is a local moopoly ad faces a idepedet, iverse demad fuctio p i a by i. I the secod case, the firms compete i a Courot fashio ad face a iverse total demad fuctio p i p â bˆy, where Y i 1 y i. I practice, uused permits ca be baked for future use or trade. We therefore assume a termial value of u per uused permit. This value could represet either the value of a uused permit i a secodary market or the et preset value of beefits accruig from future use of the permit. We assume that the pealty for ot havig the requisite umber of permits to accout for emissios is large eough so that ocompliace is deterred. For example, oly 4 of the more tha 2000 uits affected by the US Acid Rai Program were short of permits to cover their emissios for the 2004 compliace year (EPA 2005). The optimizatio problem of firm i 1,...,, is Maximize i, i e,y i i p i y i i (1 i ) i y u 2 i e i cy i Subject to: 1 i y i i, y i, i 0. The costrait [ /(1 i )]y i i implies that each firm must have sufficiet permits to accout for emissios. Deote i : /(1 i ). We drop the subscript i where uambiguous, for otatioal ad typographic coveiece. We focus o symmetric subgame perfect Nash equilibria. 4. Idepedet Demads We begi with the idepedet demads case i which firms compete for scarce permits but do ot compete i the ed-product market. Aalysis of each of the subgames follows.

5 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society Productio Subgame Give abatemet levels ad a permit allocatio vector, a represetative firm s optimizatio problem i the third stage of the game is Maximize y a by y y u 2 e cy Subject to: y, y 0. is cocave i y; y* : mi {(a c u)/2b, / } is the profit-maximizig quatity, assumig u (a c)/. Sice each uit produced results i uits of emissios, / is the maximum productio quatity that keeps emissios withi the allowable limit. (c u) ca be iterpreted as the effective margial cost of productio, which is the sum of the variable cost c ad the opportuity cost u sice a termial value equal to u is lost as a result of producig oe uit of output ad emittig uits of pollutat. A upper boud o the icremetal cotributio from a permit is (a c)/.ifuexceeds this boud the there is o icetive to egage i productio i the third stage ad firms would trivially salvage permits. Note that the emissios costrait bids whe / (a c u)/2b, i.e., whe : (a c u)/2b. Firm profit ca the be writte as follows: Case (i). Whe, y* /, ad equilibrium firm profit is (a c) * C e b (1) Case (ii). Whe, y* (a c u)/2b, ad equilibrium firm profit is * U u e, (2) where (1/b)[a (c u)/2] 2 2 is the ucostraied profit from productio, give the abatemet level chose i the first stage Auctio Subgame Firms bid for permits i a uiform price sealed-bid auctio orgaized by the regulator. I order to determie its biddig strategy, a firm must aticipate the outcome of the productio subgame where oe permit is redeemed for each uit of emissios geerated i the process of productio. Thus, the value placed by a firm o a permit i the auctio is determied edogeously by the margial reveue associated with that permit i the subsequet productio subgame. This is i cotrast to most of the auctios literature where the value placed by a bidder o the item(s) beig auctioed is specified exogeously or is assumed to be draw from a exogeous distributio. We elaborate o the costructio of the margial value fuctio ad discuss its properties Margial Value Fuctio. Let v( ) deote the firm s margial reveue from a additioal permit whe it ows permits, i.e., * C / v( ) e. From (1) ad (2), we have whe, v ad whe, a c 2b 2 :, (3) v u, (4) where (a c)/ ad 2b/ 2 are costats, give the abatemet level chose i the first stage. Note that u ad whe, v( ) u. The case whe is trivial because the emissios costrait does ot bid ad firms productio decisios are uaffected by the availability of permits. The value the placed o a permit is simply its termial value. We therefore assume for the remaider of the paper that firms operate i the regio defied by i ad that their beliefs are also restricted to this regio. This is a key assumptio ad essetially implies that each firm faces a capacity costrait because of scarce emissios permits i the productio subgame. Uder ay effective cap ad trade program, the availability of permits would costrai profit-maximizatio or busiess as usual ad would result i a market price for permits. Thus, the value fuctio is restricted to be v( ). Sice the emissios costrait bids, the margial value of a permit is the shadow price correspodig to the emissios costrait. Note that (a c)/ is the margial value of a permit whe the abatemet level is ad o permits are held. Give the abatemet level, as the umber of permits secured ( ) icreases, we get closer to the ucostraied profitmaximizig quatity. The margial value of a permit thus decreases at the rate 2b/ 2. The fuctio v( ) is crucial i establishig a equilibrium of the auctio ad i uderstadig the trade-offs iheret i firm strategies for compliace. We therefore elaborate o its behavior below. Properties of v( ). i. [v( )]/ 0 for [ (a c) 4b ]/4b. Sice firms are capacity costraied by the availability of permits, the attractiveess of expadig capacity through permits icreases i iitial abatemet levels. However, for [ (a c) 4b ]/4b, capacities are relatively large ad capacity icreases through permits yield decreasig value. ii. [v( )]/ 0 for [4b (1 )]/(a c). As the capacity costrait gets tighter with icreasig, a firm would be willig to pay more for a

6 768 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society additioal permit. Beyod a threshold, however, emissios per uit of output are relatively high ad additioal permits do ot sigificatly expad capacity, leadig to decliig value of permits (see Figure 1). Also, [v( )]/ [2b(1 ) 2 ]/ 2 0 ad 2 [v( )]/ } [4b(1 ) 2 ]/ 3 0, implyig that as icreases the permit allocatio has a decreasig ifluece o the valuatio of permits. iii. v( ) decreases i the uit cost of productio c; i.e., [v( )]/ c (1/ ) 0. For higher cost of productio, cotributio per icremet of output is lower ad hece a lower value is placed o permits. iv. v( ) decreases i the price sesitivity of demad b; i.e., [v( )]/ b (2 / 2) 0. For higher price sesitivity, the larger drop i price from a additioal uit of productio results i a lower margial value of permits Equilibrium Biddig Strategy. We are ow ready to establish a equilibrium of the secod-stage game i which firms participate i a sealed-bid uiform price share auctio for emissios permits. Each firm derives its margial value fuctio as discussed above. A strategy of a firm specifies a schedule of prices for varyig fractios of the total available permits B. The regulator sets the price such that demad equals supply. As i Wilso (1979), we assume that o firm has proprietary iformatio about demad, emissios, ad cost fuctios ad, therefore, about the derived margial value fuctios. Symmetry implies that firms have a commo margial value fuctio for permits. Lemma 1 summarizes the equilibrium results. Lemma 1. For the share auctio i the secod stage, it is a optimal strategy for a firm to submit a schedule such that at each price e, the requested umber of permits is (e) B[1 2e/( B)]/( 1). The equilibrium price is e* Figure 1 Margial Value of a Permit, v( ), versus (a 2000, b 5, c 200, 0.5). 1 2 [ (B/)], ad the equilibrium umber of permits received by each firm is * B/. Proof. The proposed equilibrium satisfies (e*) B. We show that if the ( 1) firms other tha firm i submit the schedule (e), the it is optimal for firm i to also submit the same schedule (e). Assume that firms j i submit the schedule (e) ad i submits some schedule (e). The clearig price e* satisfies ad firm i s profit is 0 e* e* B 1 e* ( z)dz 2e*B/ B e* e* z2 z 2 0 e* 2e*B B 2e*B 2 e*b B 2 B 2e*2 B B. (5) The fuctio o the right-had side of (5) is cocave i e* ad is maximized with respect to e* whe 2 B B 4 B 2 e* 4B B e* 0, i.e., whe e* 1 [ (B/)]. This is exactly the price 2 that will result if firm i submits the schedule (e). The result follows. We have thus derived a valuable piece of iformatio from this subgame, amely, equilibrium permit price. A methodological cotributio of our paper lies i the geeralizatio of Wilso s (1979) aalysis of share auctios. I Wilso (1979), the value of a share is proportioal to the share fractio. But, as is the case with auctios for emissios permits, it is more likely that the margial value a bidder places o the items beig auctioed decreases with the share fractio. Although we do ot establish uiqueess of our equilibrium solutio, it is the oly plausible oe that ca be deduced [also see Wilso (1979)]. If asymmetric equilibria do exist i this symmetric game, they are ot likely to be ituitively reasoable Abatemet Subgame I the first stage of the game, each firm decides the level of abatemet, aticipatig the equilibrium umber of permits it will secure i the auctio ad the (costraied) output it will be able to produce. Substitutig the equilibrium permit price ad permit share from Lemma 1 ito (1), a firm s optimizatio problem ca be writte as Maximize * C B a c 1 2 2

7 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society 769 Subject to: 0. Propositio 1 provides the equilibrium abatemet level. Propositio 1. * C is cocave i. * B(a c)/ 4 uiquely maximizes * C. Proof. See Appedix A. Table 4 i Appedix B summarizes equilibrium results. Propositio 2 describes the effects of two importat parameters i our model the umber of available permits B ad the state of curret techology maifested i the emissios per productio uit o equilibrium abatemet *, permit price e*, output y*, ad profit * C. Propositio 2. Assume that the emissios costrait bids for all firms. We have a. y*/ B 0; * C / B 0; */ B 0; b. y*/ 0; * C / 0; */ 0;? e1 such that e*/ 0 for e1 ; c. 2 y*/ B 0; 2 * C / B 0; 2 */ B 0. Proof. See Appedix A. The behavior of equilibrium output ad profit with respect to the total umber of permits B is ituitive. As B icreases, each firm secures a larger umber of permits i the auctio, leadig to higher effective capacity, higher output, ad profit. However, the behavior of equilibrium abatemet with respect to the umber of available permits is ot so straightforward. It might be expected that abatemet ad permits to pollute are substitute strategies for compliace; i.e., whe more permits become available, firms would egage i lower levels of abatemet (Buoao et al. 2001). However, sice firms are assumed to be always costraied by permits, a icrease i the umber of available permits expads capacity, ad the larger capacity makes ivestmet i abatemet more attractive sice emissios reductios apply to a larger umber of productio uits. Ceteris paribus, firms i a dirtier idustry produce lower output ad make smaller profits tha firms i a cleaer idustry, ad alterig the umber of available permits iflueces output ad profit to a lesser extet i a dirtier idustry tha i a cleaer idustry. Moreover, firms i dirtier idustries ivest less i abatemet (i absolute terms) tha firms i cleaer idustries. This behavior may appear surprisig, but ca be uderstood whe we recall that the emissios costrait effectively acts as a capacity costrait o firms output decisios. Dirty firms (with large ) face a relatively tighter costrait tha cleaer firms ad thus produce lower output, makig a lower level of ivestmet optimal. We also ote that as the regulator varies the total umber of permits (thus varyig the strigecy of the policy), cleaer firms adjust their abatemet ivestmets to a larger extet tha dirty firms. This is related to the valuatio of permits (see Figure 1). Capacity expasio through icremetal permits or abatemet does ot brig as much beefit at higher levels of. 5. Courot Competitio Strategic iteractios i oligopoly markets ca sigificatly affect ivestmets i capacity ad R&D (Motero 2002). I this sectio, we aalyze situatios where firms ot oly compete for permits i the auctio but also compete for customer demad i a Courot fashio Productio Subgame The optimizatio problem faced by a represetative firm i the third stage of the game is Maximize y â bˆy y y u 2 e cy, Subject to: y, y 0 where Y i 1 y i. The ucostraied equilibrium output is ỹ (â c u)/bˆ( 1), assumig u (â c)/. We agai focus o situatios where the emissios costrait bids for all firms, i.e., whe / (â c u)/bˆ( 1) {or equivaletly, where [(â c u)/bˆ( 1)]}, i which case Propositio 3 applies. Propositio 3. Assume that the emissios costrait bids for all firms. If the abatemet level chose by each firm i the ivestmet stage is symmetrically, ad the umber of permits secured by each firm i the auctio stage is symmetrically, the equilibrium output of each firm i the third stage of the game where firms compete i a Courot fashio is y* /. Proof. The term / correspods to the capacity costrait o a represetative firm s productio. I the absece of this costrait, the equilibrium output would be ỹ. Sice firm profit icreases with output i the rage [0, ỹ], each firm produces up to capacity i equilibrium; the resultig equilibrium output i the third stage is y* /. Let ˆ * C deote the equilibrium profit whe the emissios costrait bids ad ˆ * U deote the ucostraied equilibrium profit. Let Z deote the total output of all other firms. We the have: Case (i). Whe, y* /, ad ˆ * C bˆ 2 2 (â c e) bˆz 2. (6)

8 770 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society Case (ii). Whe, y* (â c u)/bˆ( 1), ad â c u â c u ˆ * U â bˆ bˆ( 1) bˆ( 1) â c u bˆ( 1) u 2 e â c u c bˆ( 1) ˆ u e, (7) where ˆ (1/bˆ)[(â c u)/( 1)] 2 2 is the ucostraied profit from productio, give the abatemet level chose i the first stage Auctio Subgame As i Sectio 4.2, a firm s biddig strategy i the permit auctio is deduced from its margial value fuctio v( ). We have: Whe, v (â c) bˆz 2bˆ 2. Substitutig Z ( 1)( / ) from Propositio 3, we have v whe, â c bˆ 1 2 : ˆ ˆ, (8) v u, (9) where ˆ (â c)/ ad ˆ bˆ( 1)/ 2 are costats, give the abatemet levels chose i the first stage. ˆ ˆ u ad whe, v( ) u. The behavior of the value fuctio i the case of Courot competitio is similar to that i the case of idepedet demads. I additio, the margial value of a permit decreases i the umber of competig firms. As the market becomes more competitive, the edproduct price approaches the margial cost of productio; icremetal uits of output are therefore less valuable. We are ow i a positio to deduce a firm s optimal biddig strategy i the auctio. Lemma 2. For the share auctio i the secod stage, it is a optimal strategy for a firm to submit a schedule such that at each price e, the requested umber of permits is (e) B[1 2e/( ˆ ˆB)]/( 1). The equilibrium price is ê* 1 [ ˆ ˆ(B/)], ad the equilibrium umber of permits 2 received by each firm is ˆ * B/. Proof. See Appedix A. Substitutig the equilibrium permit price ad permit share from Lemma 2 ito the profit fuctio i (6), we have ˆ * C B â c 1 bˆb (10) 5.3. Abatemet Subgame I the abatemet subgame, a firm maximizes the profit fuctio i (10) with respect to abatemet. Propositio 4 provides the equilibrium abatemet level. Propositio 4. ˆ * C i (10) is cocave i. ˆ * {[B (â c) 2bˆB 2 ( 1)]/[2bˆB 2 ( 1) ]} uiquely maximizes ˆ * C. Proof. See Appedix A. To esure a oegative equilibrium value of abatemet, we write ˆ * max {[B (â c) 2bˆB 2 ( 1)]/[2bˆB 2 ( 1) ], 0}. Table 4 i Appedix B summarizes equilibrium results. Propositio 5 formally describes the behavior of equilibrium abatemet ˆ *, permit price ê*, output ŷ*, ad profit ˆ * C with respect to B ad. Propositio 5. Assume that the emissios costrait bids for all firms. For 2bˆB( 1)/(â c), we have ˆ * 0 ad a. ŷ*/ B 0; ˆ * C / B 0; b. ŷ*/ 0; ˆ * C / 0;? ê1 such that ê*/ 0 for ê1 ; c. 2 ŷ*/ B 0; 2 ˆ * C / B 0. For 2bˆB( 1)/(â c), we have ˆ * [B (â c) 2bˆB 2 ( 1)]/[2bˆB 2 ( 1) ] ad d. ŷ*/ B 0; ˆ * C / B 0;? ˆ 1 such that ˆ */ B 0 for ˆ 1; e. ŷ*/ 0; ˆ * C / 0;? ˆ 1 such that ˆ */ 0 for ˆ 1;? ê2 such that ê*/ 0 for ê2 ; f.? ŷ1 such that 2 ŷ*/ B 0 for ŷ1 ;? ˆ C1 such that 2 ˆ * C / B 0 for ˆ C1 ;? ˆ 2 such that 2 ˆ */ B ) 0 for ˆ 2. Proof. See Appedix A. The behavior of output with respect to B ad is ituitive ad the explaatio is similar to that i Sectio 4. However, the behavior of abatemet ad firm profit with respect to B ad is ot so obvious. Note that a firm would ivest i abatemet oly if the margial beefit it obtais from that ivestmet exceeds the margial cost. From (10) we have ˆ * C â c 2 B bˆ 1 1 B 2 2. (11) That is, the maximum margial beefit a firm ca obtai from icremetal abatemet is [(â c)/2](b/ ), at 0, where B/ is the equilibrium output level ad correspods to the capacity costrait faced

9 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society 771 Figure 2 Firm Profit, ˆ * C, versus (â 50000, bˆ 125, c 200, 50000, 25). by the firm. This maximum margial beefit is the same as that i the idepedet demads case (see proof of Propositio 1). However, whe firms are Courot competitors, expaded productio capacity from abatemet has a egative exterality o firm profit; as capacity expads, the price of output drops ad has a adverse impact o profit. A margial icrease i abatemet thus leads to a declie i margial beefit, proportioal to bˆ( 1)(B/ ) 2. Whe [(â c)/2](b/ ) bˆ( 1)(B/ ) 2 or, equivaletly, whe 2bˆB( 1)/(â c), the firm s et margial beefit from a icrease i abatemet is egative. Thus, relatively clea firms ivest zero i abatemet. I fact, if the oegativity costrait o abatemet is relaxed, firms would have a icetive to icrease their emissios ad thus limit productio capacity whe 2bˆB( 1)/(â c). Figure 2 depicts the behavior of equilibrium firm profit (at the equilibrium levels of output, permit price, ad abatemet) with respect to for differet values of B. Recall that our aalysis applies whe the emissios costrait is bidig for all firms. For very low values of (ot depicted i Figure 2), the emissios costrait does ot bid ad firm profit is ivariat with respect to. For relatively clea firms [i.e., 2bˆB( 1)/(â c)] that have a bidig emissios costrait, a decrease i the umber of available permits itriguigly results i a icrease i equilibrium profit. As the emissios costrait gets tighter, firms restrict their output. Lower idustry output drives product prices up ad, i a Courot eviromet, results i a icrease i firm profit. A similar explaatio applies to the icrease i firm profit i this regio as icreases. However, beyod a certai threshold [i.e., 2bˆB( 1)/(â c)], firms emissios per uit of output are large relative to the emissios cap; the capacity costrait is therefore stiffer ad firms beefit from icreases i B. Positive ivestmets i abatemet result ad firm profits icrease i B. Beyod a poit, abatemet tapers off with respect to ad is iflueced less by chages i B; the explaatio for this behavior is similar to that detailed for the idepedet demads case i Sectio Numerical Example Table 1 icludes a umerical example that highlights the results obtaied i Sectios 4 ad 5. Note here that the total umber of permits B costrais profit-maximizatio both whe firms are clea ( 4) ad whe firms are dirty ( 10). I the case of idepedet demads, as B icreases, each firm secures more permits, ad equilibrium output ad profit icrease. Equilibrium abatemet icreases with the supply of permits, which is drive by the assumptio that firms are costraied i profit-maximizatio by the availability of permits. For the same supply of permits, we observe higher output ad profit whe firms are clea ( 4) tha whe firms are dirty ( 10). It ca be verified that equilibrium output ad profit are elastic with respect to, that is, ( y*/ ) ( /y*) 1 ad ( * C / ) ( / * C ) 1. This implies that output ad profit ca icrease sigificatly as firms get cleaer. Ideed, whe B 150,000, we fid a 281% icrease i output ad a 230% icrease i profit whe decreases by 60% from 10 to 4. Abatemet levels are higher whe firms are cleaer. As see i Table 1, for ay level of permit supply, the abatemet levels for 4 are greater tha those for 10. It ca also be verified that firm profit is elastic to the supply of permits, i.e., ( * C / B) (B/ * C ) 1. A 33% decrease i B from 150,000 to 100,000 results i a 42% decrease i profit whe 4 ad a 38% decrease i profit whe 10. A related result is that output is elastic to the supply of permits, i.e., ( y*/ B) (B/y*) 1. This is a useful result from a regulatory stadpoit because a icreasig strigecy has sigificat impact o firm output ad, hece, o the level of emissios. I the case of Courot competitio, whe 4, a icrease i the supply of permits results i a decrease i firm profit. This is due to the icrease i idustry capacity. The uit price of output drops as a result, as does firm profit, ad firms respod by ot egagig i abatemet. I cotrast to the idepedet demads case, we observe that abatemet drops to zero whe the supply of permits goes up from 100,000 to 150,000 for 4. Uder Courot competitio, icreasig strigecy ca be beeficial to firms whe they are relatively clea. From Table 1, we observe that whe B is reduced from 200,000 to 150,000, firm profit icreases by 10% whe 4 but decreases by 13% whe 10. Productio levels are higher ad abatemet levels are lower whe 4 tha whe 10. However, whe

10 772 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society Table 1 Numerical Example Idepedet demads Competitio ( 250, a 2000, b 0.3, c 200, 50000) ( 250, â 2000, bˆ 0.03, c 200, 50000) 4 4 B * y* e* * * C ˆ* ŷ* ê* ˆ * ˆ * C 100, , , , , , , , , B * y* e* * * C ˆ* ŷ* ê* ˆ * ˆ * C 100, , , , , , , , , the supply of permits is large, it is possible that firm profit ca be greater i a dirty idustry tha i a cleaer oe, due to the idustry capacity argumet made above. Whe B 200,000, firm profit is 48, for 4 ad 50, for 10. For relatively large, it ca be show that the elasticities for the case of Courot competitio are structurally similar to those described above for the case of idepedet demads. 7. Extesios I order to test the sesitivity of our results to assumptios i the model, we explore four extesios a emissios fuctio i which the abatemet level is idepedet of the volume of productio, gradfatherig of permits i additio to auctioig, abatemet efforts impactig the cost of productio, ad firm heterogeeity. Our model is quite robust to these variatios, with our structural results largely cotiuig to hold. Proofs are omitted for brevity Emissios Fuctio We cosider a emissios fuctio of the form y S, where the abatemet level S is chose i the first stage of the game at a cost S 2. Such a emissios fuctio represets ivestmets made toward fixed-capacity abatemet, as opposed to ivestmets that result i per-uit reductio i emissios cosidered i the base model. Examples iclude efforts to capture ad sequester exhaust gases i fixed capacity geologic formatios (Brea ad Burruss 2003) a aveue that is beig actively cosidered for sequesterig CO 2. Equilibrium results are preseted i Table 5 i Appedix B. Propositio 6 summarizes the mai results for the case of idepedet demads. Propositio 6. a. y*/ B 0; * C / B 0; S*/ B 0. b. y*/ 0; * C / 0;? S1 such that S*/ 0 for S1 ;? e2 such that e*/ 0 for e2. c. 2 y*/ B 0; 2 S*/ B 0;? C1 such that 2 * C / B 0 for C1. A compariso with Propositio 2 shows that all structural attributes of our results are preserved except for the behavior of equilibrium abatemet. Recall that, i the base model, abatemet impacted emissios per uit of output; the effective productio capacity for a firm was B(1 )/ (i.e., multiplicative i ). Therefore, abatemet ad permits were complemetary strategies ad abatemet icreased with permit availability. Here, however, abatemet reduces emissios by a fixed amout, idepedet of the output level; the effective capacity is give by (B S)/ (i.e., additive i S). Therefore, abatemet ad permits are substitute strategies ad greater permit availability decreases abatemet. Thus, the effect of strigecy o abatemet ivestmet by firms depeds upo the type of abatemet firms have recourse to. Propositio 7 summarizes the results for the case of Courot competitio. Propositio 7. a. ŷ*/ B 0; Ŝ*/ B 0;? ˆ C2, ˆ C3 such that ˆ * C / B 0 for ˆ C2, 3, ad ˆ * C / B 0 for ˆ C3. b. ŷ*/ 0;? Ŝ1 such that Ŝ*/ 0 for Ŝ1 ;? ˆ C4 such that ˆ * C / 0 for ˆ C4 ;? ê3 such that ê*/ 0 for ê3. c. 2 Ŝ*/ B 0;? ŷ2 such that 2 ŷ*/ B 0 for ŷ2 ;? ˆ C5 such that 2 ˆ * C / B 0 for ˆ C5. I particular, we ote that with competitio i the ed-product market, firms i a relatively clea idustry beefit from a reductio i the umber of available permits. As discussed earlier, a icrease i the umber of permits icreases idustry capacity ad exerts dowward pressure o the price of output ad, hece, o the profitability of clea firms.

11 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society Gradfatherig i Additio to Auctioig As discussed earlier, the regulator could possibly gradfather permits based o historical emissios or process iputs. Gradfatherig ca be icorporated ito our model by allowig a fractio of the total umber of permits B to be gradfathered per firm, with the remaiig fractio 1 offered via the auctio. Workig agai by backward iductio with the assumptio that scarcity of permits costrais profit maximizatio, the equilibrium output has a similar fuctioal form (give abatemet levels) as before. Although the margial value of permits decreases with gradfatherig, the smaller umber of permits ow available i the auctio offsets the decrease i the value of permits, ad the equilibrium permit price too has a similar fuctioal form (give abatemet levels) as before. However, as expected, equilibrium abatemet is lower due to the availability of gradfathered, free permits. Equilibrium results (see Table 6 i Appedix B) chage accordigly, but most of our structural results are preserved. Propositio 8 summarizes results for the case of idepedet demads. Propositio 8. a. y*/ B 0; * C / B 0;? G 1 such that */ B 0 for G 1. b. y*/ 0; * C / 0;? G 1 such that */ 0 for G 1 ;? G e1 such that e*/ 0 for G e1. A compariso of Propositios 8 ad 2 shows that the behavior of equilibrium output, permit price, ad profit remais uchaged with the itroductio of gradfatherig. However, the behavior of equilibrium abatemet chages. Abatemet always icreases with permit availability i the idepedet demads case of the base model. However, the availability of free, gradfathered permits icreases effective productio capacity sigificatly whe is low ad makes further capacity expasio through abatemet less valuable. For larger, the capacity costrait is stiffer ad abatemet icreases with permit availability as i the base model. The behavior of abatemet with respect to is similar. Propositio 9 summarizes results for the case of Courot competitio. Propositio 9. For [2bˆB( 1) 2bˆB ( 1)]/[(â c)(1 )], we have ˆ * 0 ad a. ŷ*/ B 0; ˆ C */ B 0; b. ŷ*/ 0; ˆ C */ 0;? G ê1 such that ê*/ 0 for G ê1. For [2bˆB( 1) 2bˆB ( 1)]/[(â c)(1 )], we have ˆ * [B (â c)(1 ) 2bˆB 2 ( 1) 2bˆB 2 ( 1)]/[2bˆB 2 ( 1) 2bˆB 2 ( 1) ] ad c. ŷ*/ B 0;? G ˆ C1 ; such that ˆ C */ B 0 for G ˆ C1 ;? G ˆ 1 such that ˆ */ B 0 for G ˆ 1 ; d. ŷ*/ 0;? G ˆ C1 such that ˆ C */ 0 for G ˆ C1 ;? G ˆ 1 such that ˆ */ 0 for G ˆ 1 ;? G ê2 such that ê*/ 0 for G ê2. A compariso of Propositios 9 (parts c ad d) ad 5 (parts d ad e) idicates that the availability of free permits accetuates the distictive effects of B ad o the profits of firms competig i a Courot fashio Abatemet ad the Cost of Productio Abatemet efforts might ivolve the implemetatio of resource-efficiet processig of materials, traslatig ito a decrease i uit productio cost. O the other had, stipulatios o emissios levels could ecessitate more sophisticated processig of materials or the use of more expesive, cleaer materials to mitigate emissios, leadig to a icrease i uit productio cost. We ca icorporate such effects ito the base model by replacig the costat uit cost of productio c with the fuctio c where ( c/, ) is a costat ad is the abatemet level. Equilibrium results are preseted i Table 7 i Appedix B. Our structural results remai uchaged Firm Heterogeeity I this sectio, we relax the assumptio of symmetry. Sice the game becomes ecessarily complicated, we restrict our aalysis to a two-firm game ad explore the effects of heterogeeity i dirtiess levels i, costs of abatemet i, ad costs of productio c i, i 1, Idepedet Demads. We maitai the assumptio that firms are costraied i profit maximizatio by the availability of permits. The equilibrium output of firm i i the third stage is y* i : i (1 i )/ i. Deote i (a c i )(1 i )/ i ad i 2b(1 i ) 2 / 2 i. The margial value fuctio for permits is v( i ) i i i. I the auctio stage, firm i s equilibrium biddig strategy is i (e) B [2( j i i B)e]/[ i j j i i j B], j i. Equilibrium permit price is e* [ i j j i i j B]/[2( i j )] ad the equilibrium umber of permits received by firm i is * i : i (e*) [ i j j B]/( i j ). Note that * 1 * 2 B. The profit fuctio i, after icorporatig equilibrium output, permit price, ad permit share, is a fuctio of both i ad j. Equilibrium values of abatemet correspod to the itersectio of the two reactio fuctios * 1 ( 2 ) ad * 2 ( 1 ). Aalytical derivatio ad treatmet of these reactio fuctios is itractable ad we therefore preset a umerical example i Table Courot Competitio. Equilibrium output of firm i is ŷ* i : [ i (1 i )]/ i. Deote ˆ i [(â c i )(1 i )]/ i, ˆ i [2bˆ(1 i ) 2 j bˆ(1 i )(1 j ) i ]/ 2 i j, ad [bˆb(1 i )(1 j )]/ i j. The margial

12 774 Productio ad Operatios Maagemet 16(6), pp , 2007 Productio ad Operatios Maagemet Society Table 2 Two-Firm Asymmetric Game: Numerical Example (Idepedet Demads) Sceario * 1 * 2 y * 1 y * 2 e* * 1 * 2 * 1 * 2 Asymmetry i dirtiess a 1 4, , , , , , Asymmetry i abatemet costs b , , , , , , Asymmetry i productio costs c c 1 100, c , , c 1 100, c , , Asymmetry i dirtiess ad productio costs d 1 4, 2 6; c 1 300, c , , , 2 6; c 1 275, c , , Asymmetry i dirtiess, varyig strigecy e B 400; 1 4, , , B 350; 1 4, , a ( , c 1 c 2 200, a 2000, b 5, B 400); b ( 1 2 6, c 1 c 2 200, a 2000, b 5, B 400); c ( 1 2 6, , a 2000, b 5, B 400); d ( , a 2000, b 5, B 400); e ( , c 1 c 2 200, a 2000, b 5). value fuctio for permits is v( i ) ( ˆ i ) ˆ i i.a equilibrium biddig strategy is i (e) B [2( ˆ j ˆ i ˆ ib)e]/[ ˆ i ˆ j ˆ j ˆ i ˆ i ˆ jb ( ˆ i ˆ j)]. Equilibrium permit price is ê* [ ˆ i ˆ j ˆ j ˆ i ˆ i ˆ jb ( ˆ i ˆ j)]/ [2( ˆ i ˆ j)] ad the equilibrium umber of permits received by firm i is ˆ * i : i (ê*) ( ˆ i ˆ j ˆ jb)/ ( ˆ i ˆ j). The profit fuctio ˆ i, after icorporatig equilibrium output, permit price, ad permit share, is a fuctio of both i ad j. Equilibrium values of abatemet correspod to the itersectio of the reactio fuctios ˆ * 1 ( 2 ) ad ˆ * 2 ( 1 ). Iterestigly, we observe behavior aalogous to that described i Sectios 4 ad 5. I both demad scearios, we observe that as a firm gets dirtier, it abates less, produces less, ad obtais a smaller profit. As expected, a firm abates less whe its cost of abatemet or cost of productio goes up. I additio, there is evidece of a trade-off amog firm clealiess, the correspodig cost of productio, ad firm profit. I the example preseted i Table 2, the cleaer firm has a higher cost of productio. I the combiatio 1 4, c 1 300, 2 6, c 2 100, the cleaer firm, i.e., firm 1, secures a larger profit, thus demostratig that it ca pay to be clea eve if the uit cost of productio is higher. But i the combiatio 1 5, c 1 275, 2 6, c 2 100, the dirtier firm, i.e., firm 2, secures a larger profit. A similar trade-off is observed i Table 3. I the case of idepedet demads, firm profits icrease i the umber of available permits. However, the profit of the dirtier firm is more sesitive, i percetage terms, to chages i the umber of permits. I Table 2, whe B is reduced from 400 to 350, the cleaer firm s Table 3 Two-Firm Asymmetric Game: Numerical Example (Courot Competitio) Sceario ˆ * 1 ˆ * 2 ŷ * 1 ŷ * 2 ê* ˆ * 1 ˆ * 2 ˆ * 1 ˆ * 2 Asymmetry i dirtiess a 1 2, , , , , , Asymmetry i abatemet costs b , , , , , , Asymmetry i productio costs c c 1 100, c , , c 1 100, c , , Asymmetry i dirtiess ad productio costs d 1 2, 2 6; c 1 300, c , , , 2 6; c 1 275, c , , Asymmetry i dirtiess, varyig strigecy e B 400; 1 2, , , B 350; 1 2, , a ( , c 1 c 2 200, â 2000, bˆ 5, B 400); b ( 1 2 4, c 1 c 2 200, â 2000, bˆ 5, B 400); c ( 1 2 4, , â 2000, bˆ 5, B 400); d ( , â 2000, bˆ 5, B 400); e ( , c 1 c 2 200, â 2000, bˆ 5).