Gregor Zöttl: Emission Trading Systems and the Optimal Technology Mix

Transcription

1 Gregor Zöttl: missio Tradig Systems ad the Optimal Techology Mix Muich Discussio Paper No Departmet of coomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at

2 missio Tradig Systems ad the Optimal Techology Mix Gregor Zoettl Uiversity of Muich February 16, 011 Abstract Cap ad trade mechaisms ejoy icreasig importace i evirometal legislatio worldwide. The most promiet example is probably give by the uropea Uio missio Tradig System U TS desiged to limit emissios of greehouse gases, several other coutries already have or are plaig the itroductio of such systems. Oe of the importat aspects of desigig cap ad trade mechaisms is the possibility of competitio authorities to grat emissio permits for free. Free allocatio of permits which is based o past output or past emissios ca lead to iefficiet productio decisios of firms compare for example Böhriger ad Lage 005, Rosedahl 007, Mackezie et al. 008, Harstad ad skelad 010. Curret cap ad trade systems grat free allocatios based o istalled productio facilities, which lead to a distortio of firms ivestmet icetives, however. 1 It is the purpose of the preset article to study the impact of a cap ad trade mechaism o firms ivestmet ad productio decisios ad to aalyze the optimal desig of emissio tradig systems i such a eviromet. Keywords: missios Tradig, Free Allocatio, Ivestmet Icetives, Techology Mix. JL classificatio: H1, H3, Q55, Q58, 033 We wat to thak Veroika Grimm, Karste Neuhoff ad Yves Smeers for may valuable commets. Uiversity of Muich, Dep. of coomics, Ludwigstrasse 8, Muich, Germay, zoettl@lmu.de. 1 See the preset phase II of the U TS, compare for example Germa Parliamet 007 for the case of Germay. 1

3 1 Itroductio I the preset article we aalyze the impact of a cap ad trade mechaism o the techology mix of productio facilities ad o firms fial output decisios. We determie the optimal desig of such a mechaism for ideal market coditios but also for o-ideal situatios where competitio authorities decisios are partially costraied by requiremets of the political or legislative process. Cap ad trade mechaisms desiged to iteralize social cost of pollutio ejoy icreasig importace i evirometal legislatio worldwide. A promiet example is probably give by the uropea Uio missio Tradig System U TS, several other coutries already have or are plaig the itroductio of such systems. A importat aspect whe itroducig cap ad trade mechaisms is the possibility of competitio authorities to grat emissio permits for free. This apparetly allowed to crucially facilitate the political processes which fially lead to the itroductio of curretly adopted cap ad trade systems. As Covery009 i a recet survey o the origis ad the developmet of the U TS observes: The key quid pro quos to secure idustry support i Germay ad across the U were agreemets that allocatio would take place at Member State level [..., ad that the allowaces would be free. Very similar observatios ca also be foud i may other cotributios to that issue. 3 Clearly, a oe ad for all lump sum allocatio of permits, which is etirely idepedet of firms actios has a purely distributive impact as the semial cotributios to the desig of cap ad trade mechaisms have already illustrated See Coase 1960, Dales 1968 ad Motgomery However, the desig of free allocatios i curretly active cap ad trade systems is ot very likely to have such lump sum property but will iclude explicit or implicit features of updatig, as several authors argue. Compare for example Neuhoff et al. 006 who observe for the case of the U TS: For the phase I tradig period, icumbet firms received allowaces based o their historic emissios.[... For future tradig periods, the Member States have to agai defie NAPs for the TS. [... It is likely that the base Those iclude New Zealad, Australia, Caada, Uited States, for a overview See IA As for example Tieteberg 006 observes: free distributio of permits as opposed to auctioig them off seems to be a key igrediet i the successful implemetatio of emissios tradig programs. Boveberg et al. 008 state: The compesatio issue has come to the fore i recet policy discussios. For example, several climate chage policy bills recetly itroduced i the U.S. Cogress for example, oe by Seator Jeff Bigama of New Mexico ad aother by Seator Diae Feistei of Califoria cotai very specific laguage statig that affected eergy compaies should receive just eough compesatio to prevet their equity values from fallig. 4 For a recet discussio of the coditios uder which the idepedece property is likely to hold both i theory ad i practice, see Hah ad Stavis 010.

4 period will be adjusted over time to reflect chages i the distributio of plats over time. It is, for example, difficult to evisage that i phase II a govermet will decide to allocate allowaces to a power plat that closed dow i phase I. This suggests that some elemet of updatig of allocatio plas caot be avoided if such plas are made sequetially. Updatig of free allocatio schemes desiged to cosistetly adapt to a idustry s dyamic developmet has a impact o firms behavior, however. First, it leads to a distortio of the operatio of existig productio facilities if firms believe that curret output or emissios do have a impact o allocatios grated to those facilities i the future. 5 Secod, it has a impact o firms icetives to modify their productio facilities through upgradig, retirig ad buildig of ew facilities if free techology specific allocatios are grated for all istalled facilities. 6 Most cotributios to the literature which aalyze the impact of free allocatios ad the optimal desig of emissio tradig systems have focused o the first effect ad abstract from the latter. That is, they provide very rich isight o the impact of updatig o firms productio ad emissio decisios, but abstract from a explicit aalysis of firms icetives to modify their productio facilities. The most promiet cotributios iclude Moledia et al. 003, Böhriger ad Lage 005, Rosedahl007, Mackezie et al. 008, or Harstadt ad skelad I the preset article we wat to explicitly aalyze the impact of updatig o firms ivestmet icetives which determie their techology mix i the log ru. Sice the log ru implicatios to a large extet are resposible for the fial success of a evirometal legislatio this seems to provide a importat aspect for the ogoig debate o the optimal desig of emissio tradig systems. I order to do so we provide a aalytical framework with a edogeous emissio permit market where strategic firms chose to ivest i two differet productio techologies with differet emissio itesities which allow for productio durig a loger horizo of time. We the aalyze the impact of a cap ad trade 5 Compare for example Böhriger ad Lage 005: As a case i poit, oe major policy cocer is that [... the allocatio should accout for major chages i the activity level of firms. Free allocatio schemes must the abstai from lump-sum trasfers ad revert to output- or emissio-based allocatio. 6 A recet review of curret emissio tradig schemes by the Iteratioal ergy Agecy IA 010 reveals that most legislatios which provide free emissio permits do update their allocatio schemes: A importat detail of systems usig gradfathered allocatio is the treatmet of compaies that establish ew facilities or close dow. Curret or proposed schemes geerally provide ew etrats with the same support as existig facilities. The ratioale for this is to avoid ivestmet movig to jurisdictios without carbo pricig. A promiet example i this respect is give by the legislatio curretly observed i phase II of the U TS, i the case of Germay for example ew productio facilities receive techology specific free allocatios whe they start operatios, retirig facilities loose their allocatios, compare Germa Parliamet I a recet empirical study o phase I of the U TS Aderso ad Di Maria 011 ideed fid evidece that firms output decisios have bee iflated, possibly due to future policy desig features. 3

5 mechaism o firms techology choices ad their productio decisios. As a bechmark we determie the first best solutio. Aalogous to the previous literature, if distributioal cocers do ot matter, i a ideal market with perfectly competitive firms it is optimal to grat o free allocatio to ay techology ad to set the total emissio cap such that the permit price equals to margial social cost of pollutio. I the mai part of the paper we the aalyze the optimal desig of a cap ad trade system if the market is ot ideal. First, we cosider the case that firms behave imperfectly competitive whe makig their ivestmet ad their productio decisios. It is the optimal to grat free allocatios i order to stimulate iefficietly low ivestmet icetives. As we show, however, i a closed system with edogeous permit market it is ot optimal to implemet total ivestmet at first best levels sice this would imply a iefficietly high permit price. It ca be optimal, furthermore, to set free allocatios such as to iduce firms to choose a techology mix which is eve cleaer tha i the first best sceario i order to depress the edogeous permit price. 8 Secod, we aalyze the case where the desig of the cap ad trade mechaism is subject to political costraits as extesively discussed above ad the competitio authority has to determie the optimal market desig give those costraits. 9 We first aalyze how the optimal target o total emissios should be set i case free allocatios i all techologies are exogeously fixed. As we fid, for moderate levels of free allocatios the target o total emissios should be set such that the equilibrium permit price is above margial social cost of pollutio. For high levels of free allocatio as for example for the case of full allocatio where all permits used by a certai techology durig a compliace period are freely allocated, compare for example Germa Parliamet the total cap o emissios should be set such that the equilibrium permit price should be below margial social cost of pollutio. We the aalyze the case that free allocatio oly for a specific techology is exogeously fixed ad determie the optimal level of free allocatio for the remaiig techology. I order to avoid excessive distortios of the resultig techology mix it is typically optimal to grat free allocatio for the remaiig techology. That is, the isights obtaied from 8 Those results have a direct implicatio also for other measures desiged to stimulate ivestmet icetives of firms, as for example capacity mechaisms itroduced i electricity markets, compare for example Cramto ad Stoft Observe that to some exted this parallels the fudametal approach foud i the previous literature: Böhriger ad Lage005 provide secod best rules if for political reasos updatig has to be based o past output, Harstadt ad skelad 010 aalyze market desig i case govermets caot commit to full auctioig of permits ad Boveberg et al. 005, 008 cosider the costrait that firms have to be fully compesated for the regulatory burde. 4

6 the first best bechmark that free allocatios are ever optimal are o loger true i case allocatio to oe of the techologies is exogeously fixed. Moreover, if this techology is relatively dirty as compared to the techology with exogeously fixed allocatio the level of free allocatio should remai below the exogeously fixed allocatio. If o the cotrary the remaiig techology is relatively clea, the level of free allocatio should eve be above the exogeously fixed allocatio. Observe that the curret practice of full allocatio as curretly grated i phase II of the U TS, compare Germa Parliamet 007 for the case of Germay iduces a patter of free allocatio which is completely opposed to those fidigs. Let us fially metio that from a modelig perspective the preset paper also cotributes to the literature of peak load pricig which aalyzes optimal ivestmet decisios i several techologies. For a survey o this literature see Crew ad Kleidorfer More recet cotributios iclude hrema ad Smeers 011, Zöttl 010, or Zöttl Our framework itroduces a edogeous emissio permit market with the purpose to iteralizes social cost of emissios. This setup allows us to aalyze the optimal desig of a cap ad trade mechaism by takig ito accout firms ivestmet ad productio decisios. The remaider of the article is structured as follows: Sectio states the model aalyzed throughout this article, sectio 3 derives the market equilibrium for a give cap ad trade mechaism. I sectio 4 we determie the optimal markets desig, sectio 5 cocludes. The Model We cosider firms which first have to choose productio facilities from two differet techologies prior to competig o may cosecutive spot markets with fluctuatig demad. Iverse Demad is give by the fuctio P Q, θ, which depeds o Q R +, ad the variable θ R that represets the demad sceario. The parameter θ takes o values i the iterval [θ, θ with frequecies fθ. The correspodig distributio is deoted F θ = θ θ fθdθ.11 We deote by qθ = q 1 θ,..., q θ the vector of spot market outputs of the firms i demad sceario θ, ad by Qθ = i=1 q i total quatity produced i sceario 10 Based o those aalytical frameworks a umber of umerical studies tries to quatify the impact of a cap ad trade mechaism o firms ivestmet decisios for differet levels of a exogeously fixed permit price compare for example Neuhoff et al. 006, Matthes 006 or recetly Pahle, Fa ad Schill Mathematically we treat the frequecies associated to the realizatios of θ by makig use of a desity ad a distributio-fuctio. Notice, however, that there is o ucertaity i the framework preseted all realizatios of θ [θ, θ ideed realize, with the correspodig frequecy fθ. 5

7 θ. Demad i each sceario satisfies stadard regularity assumptios, i.e. 1 Assumptio 1 Demad Iverse demad satisfies P q Q, θ < 0, P θ Q, θ 0, P qθ Q, θ 0 ad P q Q, θ + P qq Q, θ Q < 0 for all Q, θ R. Techologies differ with respect to ivestmet ad productio cost ad emissio factors. Assumptio Techologies Firms ca choose betwee two differet techologies, t=1,. ach techology t has costat margial cost of ivestmet k t, costat margial cost of productio c t, ad a emissio factor w t which measures the amout of the pollutat emitted per uit of output. We deote total ivestmet of firm i i both techologies by x 1i ad ivestmet of firm i i techology by x i, aggregate total ivestmet is deoted by X 1 ad aggregate ivestmet i techology by X. 13 We deote aggregate output produced i sceario θ by Qθ. ach uit of output produced with techology t = 1, causes emissios w t. We deote total emissios for example of a greehouse gas produced at all markets θ [θ, θ by T. The social cost associated to emissios is deoted by DT. The competitio authority desigs a cap ad trade mechaism to iteralize this social cost. Assumptio 3 Cap ad Trade Mechaism ad Social Cost of Pollutio Total Pollutio T causes a social damage DT, which satisfies D T T 0 ad D T T T 0. A cap ad trade mechaism limits total emissios such that T T. ach uit ivested i techology t = 1, is assiged the amout A t of permits for free. Permits are tradeable, we make the followig assumptios regardig the permit market. Assumptio 4 Permit Markets storage of permits is costless. i missio permit tradig is arbitrage free ad ii Firms are price takers at the permit market. 14 We deote the market price for emissio permits by e. For give ivestmet decisios of a firm x 1i, x i we ca ow write dow margial productio cost of firm i as follows: c + w e for 0 < q i x i, Cq i, x 1i, x i = c 1 + w 1 e for x i < q i x 1i, for x 1i < q i. 1 We deote the derivative of a fuctio gx, y with respect to the argumet x, by g x x, y, the secod derivative with respect to that argumet by g xx x, y, ad the cross derivative by g xy x, y. 13 Thus, aggregate ivestmet i techology 1 is give by X 1 X. 14 Sice emissio tradig systems typically ecompass large regios several coutries i the case of the U TS this seems to be a quite atural assumptio. 6

8 To sum up, at the first stage, firms simultaeously ivest i the two differet techologies at margial cost of ivestmet k 1, k. Ivestmet choices are observed by all firms. The, give their ivestmet choices, firms compete at a sequece of spot markets with fluctuatig demad i the presece of a cap ad trade mechaism. At each spot market θ, firms simultaeously choose output q i θ which causes emissios. ach firm i has to cover its total emissios by permits. Depedig o the allocatio rule A 1, A firms obtai permits for free, cotiget o their ivestmet decisio. Firms have to purchase permits eeded i excess of the free allocatio at the permit market at price e, which is the price at which the permit market clears give the target T. 3 The Market quilibrium I this sectio we derive the market equilibrium with cap ad trade mechaism, for the case of perfect ad imperfect competitio. Observe that i the framework aalyzed, where demad fluctuates over time it is optimal for firms to ivest ito a mix of both techologies. We will cosider the case that techology allows cheaper productio but exhibits higher ivestmet cost. Those uits have to ru most of the time i order to recover their high ivestmet costthis is typically deoted baseload techology. Techology 1 has relatively low ivestmet cost but produces at high margial cost. Those uits are built i order to serve durig periods of high demad this is typically deoted peakload techology but ru idle if demad is low. I order to be able to characterize the market equilibrium for a give cap ad trade mechaism T, A 1, A, we first determie firms profits, give ivestmets x 1, x ad give spot market output qθ. π i x 1i, x i = θ B θ P Q, θ c w e q i θ, xdf θ + P X, θ c w e x idf θ 1 θ B + P Q, θ c 1 w 1 e q i θ, xdf θ + θ θ P X 1, θ c 1 w 1 e x 1idF θ c 1 + w 1 e c + w e x idf θ k A ex i k 1 A 1 ex 1i x i. Note that the permit market affects both, the firms margial productio cost as well as their ivestmet cost. No matter whether permits have bee allocated for free or have to be bought at the permit market, firms face opportuity cost of w t e whe decidig to produce oe uit of output with techology t = 1,. This opportuity cost icreases their 7

9 margial productio cost to c t + w t e, t = 1,. Ivestmet cost is affected by the firms aticipatio of a free allocatio of permits. A free allocatio is equivalet to a subsidy paid upo ivestmet: If each uit of capacity ivested is assiged A t permits, ivestmet cost k t is reduced by their value, that is by A t e for t = 1,. The critical spot market scearios 15 θ B,, idicate wether firms produce either at the capacity bouds x, x 1 that is, at the vertical pieces of their margial cost curves, or o the flat i.e. ucostraied parts of their margial cost curves. They deped o the itesity of competitio at the spot market ad are illustrated i Figure 1 both for the case of perfect ad imperfect competitio. For θ [θ, θ B firms produce the output at Figure 1: Illustratio of the critical spot market scearios. Left: The case of a perfectly competitive market, right: the geeral case with imperfect competitio. I the figure we deote margial reveue by MRQ, θ := P Q, θ + P q Q, θ Q. margial cost c. For θ [θ B, firms are costraied by their ivestmet i the base load techology ad produce X, still at margial cost c, ad prices are drive by the demad fuctio. At those demad levels, usig the peak load techology 1 is ot yet profitable. Observe that F F θ B measures the fractio of time where ivestmet i the base load techology is bidig, which we will refer to as costraied base duratio. For θ [, firms produce output at margial cost c 1, we deote 1 F as peak duratio. 16 Fially, for all realizatios above, firms are costraied by their total capacity choice X 1, ad 15 For the precise defiitio of those critical spot market scearios, see appedix A. 16 The equivalet base duratio would be give by 1 F θ = 1, it is ot explicitly itroduced, however. 8

10 prices are drive exclusively by the demad fuctio, we deote 1 F θ P as costraied peak duratio. I the subsequet lemma we characterize the market equilibrium whe firms ivest i the base load ad i the peak load techology. Lemma 1 For a give cap ad trade mechaism T, A 1, A, defie the toal ivestmet coditio Ψ I, the base ivestmet coditio Ψ II ad the permit pricig coditio 17 Ψ as follows: Ψ I := Ψ II := Ψ := θ θp θ B θ [ P X 1, θ + P q X 1, θ X 1 c 1 + w 1 e [ P X, θ + P q X, θ X c + w e df θ k 1 A 1 e df θ + 3 c 1 c + w 1 w e df θ k k 1 + A A 1 e θb + θ θ w Qe, θdf θ + w 1 X 1dF θ θ θp θ B w X df θ + θp w 1 w X df θ T w 1 Qe, θdf θ 4 quilibrium ivestmet X 1, X ad the equilibrium permit price e simultaeously solve Ψ I = Ψ II = Ψ = 0. Proof See appedix A. I the lemma, is the first order coditio that determies total ivestmet. Firms choose their total ivestmet X 1 as to equal margial profits geerated by their last ruig uit ruig at total margial cost c 1 + w 1 e to the ivestmet cost of that uit give by k 1 A 1 e. As already metioed above, uder a cap ad trade mechaism, the value of the permits reuired for productio at the spot market is part of the firms margial productio cost, the value of free allocatios is of firms margial cost of ivestmet. Now let us provide some ituitio o the determiats of the optimal base load ivestmet. Sice total ivestmet X 1 has already bee fixed it is determied by, the firms decisio whe choosig X has to be iterpreted as a decisio of virtually replacig uits of techology 1 by techology. The cost of such virtual replacemet of the margial uit give by k k 1 A A 1 e has to equal the extra profits geerated by that uit 17 For a positive permit price, we might also obtai the situatio, where productio for very low demad realizatios is suppressed ad positive output is produced oly for demad realizatios which satisfy θ : P 0, θ C0, θ e > 0. For ease of otatio we disregard this corer solutio, which could be easily icluded i the etire aalysis. 9

11 due to lower margial productio cost. Lower productio cost of oe additioal uit has two effects: First, for all demad realizatios θ [θ B, oe more uit is produced that would ot have bee produced without the replacemet; for θ [, θ oe more uit ca be produced at lower margial cost c + w e istead of c 1 + w 1 e due to the replacemet. Compare also figure 1. The market price for permits, e, depeds o the emissio target T set by the market desiger as well as the techology mix istalled by the firms. At the equilibrium permit price the market exactly clears, allowig for total emissio of T uits of the pollutat. Notice that the left had side of expressio 4 is just total productio at all spot markets multiplied by the emissio factors of the respective Techologies w 1, w, total emissios are obtaied by itegratig over emissios at all spot markets θ [θ, θ. Fially observe that lemma 1 characterizes the market solutio whe firms decide to ivest i both techologies, that is, whe ideed 0 < X < X1 obtais. First, wheever the base load techology k, c is very uattractive, 18 the oly the peak load techology k 1, c 1 is active. Secod, if the base load techology k, c is always more attractive 19 tha the peak load techology k 1, c 1, the oly techology k, c is active i the market equilibrium. Notice that i priciple the case of ivestmet i a sigle techology is covered by out framework, it obtais by elimiatig the possibility to ivest i techology, expressio the determies ivestmet i the sigle techology. To keep the otatioal burde limited, however, we do ot explicitly iclude those corer solutio i the expositio of the paper, but opted to focus o all those cases whe firms ideed choose to ivestmet i both techologies. To coclude the discussio of lemma 1 let us already at this poit metio the relevace of edogeously modelig the emissio permit market as compared to the case which assumes a exogeously fixed price for pollutio. Observe that, for a costat emissio price equilibrium ivestmet uder imperfect competitio differs from that obtaied uder ad P θ q X, θ X df θ respectively, B which correspods to the differece betwee scarcity prices ad margial scarcity profits. Sice those terms are egative ad profits cocave give our assumptios ivestmet icetives uder imperfect competitio are lower tha uder perfect competitio. That is, i the absece of a explicit market for emissio permits whe pollutio is for example taxed at some fixed level e 0 subsidies for ivestmet for example by gratig free tax vouchers A 1 > 0 ad A > A 1 respectively which exactly compesate for those differeces would iduce optimal ivestmet icetives. Sice the emissio price is edogeous i our perfect competitio by the terms θ P θ q X 1, θ X 1 P 18 That is expressio 3 yields X That is expressios 3 ad yield X X 1. 10

12 framework, however, we will obtai a differet result compare theorem. Before we ow discuss existece of the market equilibrium we itroduce the followig defiitios which will simplify the subsequet aalysis ad allow for a more ituitive discussio of our results: Defiitio 1 i We deote the impact of icreased total ivestmet o total emissios for fixed e by A 1 := Ψ = 1 F θ X1 P w 1, observe A 1 > 0. This allows to state the impact of chaged emissio price e o the equilibrium coditio Ψ I as follows Ψ I = A e 1 A 1. ii We deote the impact of icreased base load ivestmet o total emissios for fixed e by A := Ψ = 1 F θ X B w 1 F w 1. This allows to state the impact of chaged emissio price e o the equilibrium coditio Ψ II as follows Ψ II = e A A 1 A. We furthermore deote w := 1 F w 1 F θ 1 which implies A B > 0 if ad oly if w > w ad w L := F F w 1 F θ 1 which implies A B 1 + A > 0 if ad oly if w > w L. iii We deote the impact of chaged X 1 o the equilibrium coditio Ψ I by Ψ I1 := Ψ I, X1 the impact of chaged X o the equilibrium coditio Ψ II by Ψ II := Ψ II ad the X impact of chaged e o the equilibrium coditio Ψ by Ψ e := Ψ. Observe that e those three expressios are egative. Observe 0 that A 1 = 1 F θ P w 1 determies the total amout of additioally ecessary permits resultig from a additioally ivested uit of total capacity formally give Ψ by the partial derivative of total emissio with respect to X 1, i.e. X 1. A icrease of the permit price e ow has two opposig effects o total ivestmet icetives: o the oe had ivestmet icetives are reduced by the amout A 1, o the other had they icrease by A 1 due to the icreased value of free allocatios. A similar reasoig obtais for ivestmet icetives i the base load techology. A determies the total amout of additioally ecessary permits resultig from the replacemet of oe uit of the peak techology with oe uit of the base techology formally give by the partial derivative of total emissio with respect to X, i.e. Ψ X. A icrease of the permit price e has two opposig effects o total ivestmet icetives: o the oe had they are reduced by the amout A, o the other had they icrease by A A 1 due to the icreased value of free allocatios. 0 Notice that the statemets of defiitio 1 ad the subsequet discussio exclusively refer to partial derivatives. I equilibrium total emissios do ot chage sice they are capped at T. 11

13 Notice that A 1 0 whereas A ca also become egative. That is, a icreased level of total ivestmet X1 always implies additioally ecessary emissio permits. A icreased level of base ivestmet X does oly imply additioally ecessary emissio permits if the base techology is dirtier tha the peak techology. Iterestigly the cut off poit obtais for w = w < w 1, sice a icreased level of X leads to icreased emissios for θ [, θ if w > w but also leads to oe uit of additioal output for the demad levels θ [θ B,. As already argued, lemma 1 oly characterizes the market equilibrium by establishig ecessary coditios. I the subsequet lemma e ow wat to establish coditios secod order coditios for the existece of the market equilibrium. i Lemma 1 characterizes the market equi- Lemma Secod Order Coditios librium if a A 1 A 1 A 1 Ψ I1 Ψ e < 0, b A A 1 A A Ψ II Ψ e < 0 c A 1 A 1 A 1 Ψ I1 Ψ e A A 1 A A Ψ II Ψ e > A1 A 1 A 1 A A 1 A A ii If the levels of free allocatio satisfy A 1 A 1 A 1 0 ad A A 1 A A 0, the coditio i is satisfied. iii Defie by A lim 1 the highest A 1 yieldig A 1 A 1 A 1 Ψ I1 Ψ e 0, defie by A lim the highest A yieldig A A 1 A A 1 + A ΨI1 + Ψ II Ψ e 0. The secod order coditios i caot be satisfied if either A 1 A lim 1, or A A lim 1. Proof See appedix B. Part i of the lemma establishes the stadard secod order coditios which establishes egative semi defiiteess of the Hessia matrix of firms optimizatio problem. It allows the usual applicatio of the implicit fuctio theorem i order to coduct a aalysis of comparative statics for the equilibrium characterized i lemma 1. I part ii we establish coditios whe those secod order coditios are satisfied ad part iii provides a upper boud o the levels of free allocatio such that higher allocatios always violate those secod order coditios. Let us explicitly metio at this poit that our aalysis throughout this article focuses o symmetric ivestmet decisios, the secod order coditios established i lemma i guaratee that lemma 1 characterizes a uique symmetric solutio. Sice for the case of a moopolistic or a perfectly competitive market asymmetric ivestmet levels are irrelevat 1 lemma i guaratees a existece ad uiqueess of the market equilibrium i those cases. 1 For perfect competitio observe that both margial cost of ivestmet ad margial cost of productio are costat, for moopoly observe that asymmetries caot arise by defiitio. 1

14 For the case of oligopoly, whe firms behave strategically, asymmetric ivestmet levels might be relevat, however. Ideed, as we show i a compaio paper Zoettl010 for ivestmet decisios i a discrete umber of techologies symmetric equilibria ca oly exist if techologies are sufficietly differet, for sufficietly similar techologies a symmetric equilibrium of the ivestmet game always fails to exist ad asymmetric equilibria might arise. After havig established the market equilibrium, we ow determie the impact of chagig the parameters of the cap ad trade mechaism A 1, A, T i a aalysis of comparative statics. If the secod order coditios specified i lemma i are satisfied we obtai the followig results: Lemma 3 Comparative Statics of the market quilibrium i Higher free allocatio for the base load techology A always yields higher ivestmet dx i the base load techology i.e. da > 0. We furthermore obtai dx 1 da < 0 if ad oly if A1 A 1 A < 0. Defie A cross 1 as the highest A 1 yieldig A1 A 1 A Ψ e Ψ I1 A 1 A 1 A 1, we obtai dx da < dx 1 da if ad oly if w > w ad A1 A cross 1, A lim 1. ii Higher free allocatio for the peak load techology A 1 always yields higher ivestmet dx i the peak load techology i.e. 1 > dx. Defie by A total the highest A which yields A A 1 A A Ψ II Ψ e 0 ad by A cross the highest A which yields A A 1 A A 1 Ψ I1 Ψ e 0. There exists a uique w S with w < w S w 1 such that dx 1 < 0 if ad oly if w > w S ad A A total, A lim. Furthermore, we obtai dx > 0 if ad oly if w < w S ad A A cross, A lim. iii For a chage of the total emissio cap T we obtai dx 1 > 0 if ad oly if A dt 1 < A 1 we furthermore obtai dx > 0 if ad oly if A dt A 1 < A. Proof See appedix C. As we establish i the theorem, a icrease of the free allocatio A i the base load dx techology always leads to icreased base load ivestmet i.e. da > 0, see poit i, a icrease of the free allocatio A 1 i the peak load techology always leads to icreased dx1 ivestmet i the peak load techology i.e. > dx, see poit ii. The impact of As aalyzed i Zoettl010 those problems ca be overcome whe firms are allowed to choose from a cotiuum of techologies, whe existece ad uiqueess of the symmetric equilibrium ca be reestablished. Those fidigs to some exted seem to be parallel the discussio o supply fuctio equilibria, where Klemperer ad Meyer 1989 show existece ad uiqueess of the market equilibrium whe firms ca bid smooth supply fuctios, whereas vo der Fehr ad Harbord 1993 show tha a symmetric equilibrium i discrete step fuctios fails to exist., 13

15 Figure : Results of comparative statics i the degree of free allocatio. Left: For the degree of free allocatio to the base load techology A, Right: For the degree of free allocatio to the peak load techology A 1. For the case of liear demad we obtai, left: A cross 1 w = 1 F 1 w = w, A lim right: A total w = F θ 1 F θ B B w, A cross 1 0 = 1 F w 1 ad 1 1 F F θ 1 F θ B B w, A cross w S = w. such chages o the remaiig ivestmet decisios is more ambiguous. I the subsequet paragraphs we briefly sketch the cetral trade-offs, a complete proof is oly provided i the appedix, however. First cosider a variatio of the free allocatio A ad determie its impact o the system of equilibrium coditios established i lemma 1. The total differetial yields: 3 dψ I dx1 de = Ψ I1 + Ψ Ie = 0 5 da da da dψ II dx de = Ψ II + Ψ IIe + Ψ II = 0 6 da da da A dψ dx1 dx de = Ψ 1 + Ψ + Ψ e = 0 7 da da da da I order to directly evaluate the impact of the chaged emissio price de da o the equilibrium coditios for total ivestmet ad ivestmet i the base load techology, we solve 3 For a better traceability of our computatios we deote the partial derivatives Ψ I e = Ψ Ie, Ψ II e = Ψ IIe, = Ψ 1 ad Ψ X = Ψ, i a secod step we make use of A 1 ad A itroduced i defiitio 1. Ψ X 1 14

16 expressio 7 for de da dψ I da = = Ψ 1 dx1 da Ψ I1 + Ψ Ie + Ψ dx da Ψ 1 dx 1 da + ad plug ito expressio 5, which yields: Ψ Ψ Ie dx = 0 8 da Ψe Ψ I1 + A 1 A 1 A 1 dx 1 da + A 1 A 1 A dx da = 0 Observe that the coefficiet o the expressio dx 1 determies the total impact of chaged X1 o the equilibrium coditio Ψ I. This is give by the direct impact i.e. Ψ I1 ad the idirect impact which takes ito accout the impact of chaged X1 o the emissio price Ψ ad its feed back o the equilibrium coditio Ψ I i.e. Ψ 1 Ie = 1 A1 A 1 A 1. Observe that the total impact of chaged X1 o the equilibrium coditio Ψ I is egative if the secod order coditios established i lemma i are to be satisfied. This directly illustrates why dx da caot drop to zero. Furthermore, observe that the total impact of chaged X o the equilibrium coditio Ψ I is oly idirect, sice Ψ I does ot directly deped o X. That is, we oly have to take ito accout the impact of icreased X o the emissio price e ad its feedback o the equilibrium coditio Ψ I. Accordig to defiitio 1 a icrease of X leads to a icreased equilibrium emissio price if A > 0 i.e. for w > w, we obtai a decreased equilibrium emissio price if A < 0, i.e. for w < w. The impact of a icreased emissio price o the equilibrium coditio Ψ I depeds o the the degree of free allocatio Ψ I e A 1. Wheever A 1 < A 1 i.e. < 0, compare defiitio 1 a icreased emissio price leads to a decrease of firms total ivestmet activity X1. I this case the reductio of scarcity rets obtaied whe total capacity is bidig caused by the icreased emissio price domiates the icreased value associated to the permits grated for free. The reverse holds true for a high level of free allocatio, i.e. A 1 > A 1 where a icreased emissio price leads to icreased total ivestmet X1. Wheever the impact of icreased ivestmet X yields a decreased emissio price, which obtais for cleaer base load techologies for A < 0, i.e. w < w, we obtai the opposite results. I sum, dx 1 da > 0 if ad oly if A1 A 1 A > 0, as stated i the theorem. Fially expressio 8 also provides the ituitio uder which coditios we obtai i.e. also ivestmet i the peak load techology icreases. To this ed observe dx 1 da dx da that dx 1 da = dx da if ad oly if i expressio 8 the total impact of chaged X 1 is precisely of the same size as the total impact of chaged X, but of opposite sig. As show i the theorem this oly obtais i case the icrease of ivestmet i the base techology leads to a icrease of the emissio price for A > 0, i.e. w > w ad if this icrease has a sufficietly positive impact o the equilibrium coditio Ψ I, i.e. for allocatio A 1 sufficietly big for A 1 > A cross 1 > A 1. All those results are illustrated i the left graph of figure. 15

17 Likewise we ca aalyze the impact of chagig A 1, as established i theorem 3ii. Aalogous to expressios 19 0 ad 1 we ca determie the total derivative ad solve for de. After pluggig i, we obtai for dψ I dψ I + dψ II Ψ 1 = Ψ I1 + Ψ Ie + Ψ IIe Ψ I1 + A A 1 A Ψ 1 dx dψ II observe Ψ I A 1 Ψ Ψ II + Ψ IIe + Ψ Ie A 1 dx 1 + Ψ II + A A 1 A = Ψ II A 1 Ψ = e: dx = 0 9 A dx = 0 Aalogous to above the coefficiets o the expressios dx ad dx 1 determie the impact of chaged ivestmet X1 or X o both equilibrium coditios. The sum of both coefficiets is strictly egative if secod order coditios are ot to be violated compare lemma iii. This directly illustrates why dx caot reach the level of dx 1 i other words, icreased free allocatio A 1 caot leave ivestmet i the peak load techology uchaged. Furthermore, as we show, for small A both coefficiets are egative thus dx 1 ad dx have opposite sigs, sice dx 1 > dx this implies dx < 0. Observe that the coefficiet of the expressio dx 1 is icreasig i A, the coefficiet of expressio dx 1 A is icreasig i A if > 0 i.e. w > w. That is, for A high eough the coefficiets become o egative, leadig to altered mootoicity behavior. As we show i the theorem we ca establish a relative level of dirtiess w S with w S w ad w S = w i the case of liear demad, which separates the cases whe either of the coefficiets becomes zero for higher levels of A Remember the sum of both coefficiets has to be egative i order to satisfy the secod order coditios, see above. Wheever the coefficiet of dx 1 equals to zero, expressio 9 directly implies dx = 0 ad vice versa, as stated i the theorem. Fially, i theorem 3iii we provide the results of comparative statics with respect to the parameter T. For a ituitio of those results observe first of all that a icrease of the total emissio cap T leads to a reductio of the equilibrium permit price. This i tur iduces icreased total ivestmet X1 if similar to the ituitio for part i the icrease of scarcity rets which obtais due to lower emissio price domiates the decreased value of the emissio permits grated for free, i.e. A 1 < A 1. The opposite result obtais for A 1 > A 1. Similarly, the reduced emissio price iduces icreased ivestmet i the base load techology X if the total impact of reduced emissio price o the base load ivestmet coditio is egative, i.e. if ad oly if A < A 1 + A i.e. Ψ IIe < 0. If we deote total emissios which obtai i the absece of ay evirometal policy by T. Lowerig the cap o total emissios T below T correspods to the itroductio of a cap ad trade mechaism. To provide the direct coectio of our framework with curret practice i competitio policy, let us coclude this sectio by briefly discussig the impact of itroducig a cap ad trade mechaism as observed for example durig the curret phase of the U TS. I this 16

18 phase, the free allocatios grated for free to a uit of each class of techologies were such as to cover the total eeds ecessary o average to operate that uit. 4 I our framework that would correspod to levels of free allocatio A full 1 [1 F w 1, 1 F w 1 ad A full [1 F θ B w, w. 5 With a allocatio scheme A full 1, A full which aims at coverig the average total eeds of a uit of specific techology we obtai icreased total ivestmet ad icreased base load ivestmet whe itroducig the tradig system i.e. the emissio cap is lowered below T i our framework. To see this, first observe that A full 1 > A 1 which accordig to lemma 3 iii leads to a icrease of X1. Secod observe that A full A full 1 > A sice A full > 1 F θ B w ad A full 1 < 1 F w 1, which accordig to lemma 3 iii leads to icreased ivestmet i the base load techology. I order to apply our fidigs of lemma 3i cosider agai our example of a electricity markets with ligite or coal fired plats as a represetative base load techology ad ope cycle gas turbies as a represetative peak load techology. Sice ope cycle gas turbies have lower emissio factors, we obtai w > w 1, which directly implies w > w compare defiitio 1. Sice furthermore A full 1 > A 1 as established above, we ca directly coclude that a icrease decrease of the free allocatio A ot oly would yield icreased base load ivestmet but also a icreased decreased emissio price ad icreased decreased total ivestmet. After havig aalyzed the market equilibrium which obtais i the presece of a emissio tradig system ad derived its properties of comparative statics we ow proceed to the mai part of this article ad aalyze the optimal desig of a cap ad trade mechaism. 4 The Optimal Cap ad Trade Mechaism I this sectio we determie the optimal cap ad trade mechaism. We first determie the first best solutio as a bechmark, which obtais for the case of a perfectly competitive market whe a regulator ca freely choose all parameters A 1, A, T of the cap ad trade mechaism see theorem 1. We the aalyze several market imperfectios ad solve for 4 To give a specific example: I the Germa electricity market free allocatio is determied by a techology specific emissio factor which measures average emissios per uit of electricity produced tco/mwh for gaseous fuels ad tco/mwh for solid ad liquid fuels multiplied by a preestablished techology specific average usage. For ope cycle gas turbies i Germay the average usage is established at 0.11 i.e hours per year, for coal ad combied cycle gas turbies it is give by 0.86 i.e hours per year ad for ligite plats it is give by 0.94 i.e. 850 hours per year, See appedices 3 ad 4 of Germa Parliamet To be precise, i our framework average usage of the base techology is 1 X θb θ Q θdf θ F θ B, the average usage of the peak techology is X1 X Q θ X df θ + 1 F. The correspodig emissio factors are give by w 1 ad w respectively. 17 θp

19 the correspodig secod best solutios. We first determie the optimal cap ad trade mechaism which should be chose for a imperfectly competitive market see sectio 4.1. We the aalyze the case whe competitio authorities caot freely choose all parameters A 1, A, T of the cap ad trade mechaism but oly a subset of them see sectio 4.. I order to aswer all those questios we first determie total welfare geerated i a market with some cap ad trade mechaism A 1, A, T : W A 1, A, T = θ B θ [ Q P Y, θ c Y dy df θ + 0 [ Q 0 P Y, θ c 1 Y dy θ B θ df θ + [ X 0 [ X 1 0 P X, θ c Y dy P X 1, θ c 1 Y dy df θ df θ θ c 1 c X df θ k X k 1 X 1 X DT. 10 Observe, that welfare does ot directly deped o the parameters A 1, A, T chose for the cap ad trade mechaism but oly idirectly through the implied ivestmet ad productio decisios X 1, X ad Q. I order to maitai presetability of the results, we relegated all computatios to the appedix ad directly characterize the optimal cap ad trade mechaism i the subsequet lemma. Lemma 4 The optimal cap ad trade mechaism solves the followig coditios: i W A1 := dx 1 Ω I + dx Ω II = 0 ii W A := dx 1 da Ω I + dx da Ω II = 0 iii W T := dx 1 dt Ω I + dx dt Ω II D T T + e + = 0. The expressios Ω I ad Ω II determie the total impact of chaged X 1 ad chaged X respectively o total Welfare. They are defied as follows: Ω I := θ P q X1 The term df θ A 1 e A 1 Ω II := θ B dq := θ de Q P q θ B θ df θ+ dq de θp P q X θ B Q P q df θ df θ A A 1 e A. > 0 determies the impact of dq de w df θ+ dq de w 1dF θ chaged emissios o welfare for those spot markets where ivestmet is ot bidig. Proof See appedix D. We ow provide some ituitio for the coditios which characterize a optimal cap ad trade mechaism. We first cosider the optimal choice of the free allocatio to the peak 18

20 load techology give by A 1. Observe that the optimality coditios i ad ii express the impact of chaged free allocatio o total welfare exclusively through the chael of chaged ivestmet i the base load techology X ad chaged total ivestmet X1. The total impact of chaged ivestmet o total welfare is deoted by Ω I ad Ω II, this total impact ca be broke dow ito three compoets correspodig to the three summads of Ω I ad Ω II respectively. First, observe that at all those spot markets where total ivestmet is bidig i.e. for θ [θ B, ad θ [, θ respectively imperfectly competitive ivestmet behavior iduces too low ivestmet icetives, a icrease of ivestmet X or X1 leads to icreased X welfare give by the markup P q. Secod, free allocatio A 1 > 0 or A A 1 > 0 iduces too high ivestmet icetives, thus a icrease of ivestmet would lead to a reductio of welfare give by the moetary value of the free allocatio i.e. A 1 e ad A A 1 e. Notice that i a world with exogeously fixed emissio price e the optimal level of free allocatio should be chose such as to balace those two effects. 6 Sice the emissio price is edogeous i our aalysis, a additioal term obtais. A icrease of ivestmet dx1 or dx leads to icreased emissios of dx1a 1 ad dxa at those spot markets where ivestmet is bidig. Sice total emissios are capped by T, however, this ecessarily has to imply a equivalet reductio of emissios at those spot markets where ivestmet is ot bidig i.e. for θ [θ, θ B or θ [,. Sice productio decisios are also imperfectly competitive, a reductio of output leads to reduced welfare geerated at those spot markets. This impact is quatified by the term defied i the lemma. That is, takig ito accout the edogeous ature of the emissio price leads to a lower degree of optimal free allocatio A 1 tha suggested by a aalysis with exogeously fixed emissio price. The impact of a chaged emissio cap T o total welfare has a similar structure tha the impact of chaged free allocatios. Aalogous to above, a chaged emissio cap leads to chaged ivestmet icetives, the impact of chaged ivestmet icetives o welfare is give by the terms Ω I ad Ω II, which have already bee discussed above. As we will see later o i theorems 1 ad, if the levels of free allocatio are chose optimally such as to obtai Ω I = Ω II = 0 those terms will ot be relevat for the optimal choice of the emissio cap. If the levels of free allocatio are ot chose optimally, however, they have to be cosidered whe determiig the optimal level of the emissio cap T compare theorems 3, 4, 5 ad 6. 6 That is, the moetary subsidy A 1 e for example should the equate to the itegral of the markups over all relevat spot markets. The ituitio for this result i some sese parallels the quite well kow isight obtaied for a simple static model where a moopolist ca be iduced to produce first best output if he obtais a subsidy correspodig to his markup. 19

21 Apart from havig a impact o ivestmet icetives, a chaged emissio cap T leads to chaged welfare also through several other chaels. First, most apparetly a icreased emissio cap leads to icreased emissios which reduce welfare by the margial social cost of pollutio D T. Secod, observe that o the other had a icreased emissio cap leads to a welfare icrease sice it implies a reduced emissio price which allows for icreased output. The welfare icrease at each spot market is give by the chaged output multiplied by the differece betwee margial cost as perceived by the firms ad true margial cost, i.e. dqw i e, for i = 1,. Put differetly however, this correspods to the chaged pollutio at each spot market multiplied by the emissio price e, the chage i welfare at all spot markets the is simply give by the total chage of emissios multiplied by the emissio price i.e. dt e. As we will see i the subsequet theorem 1, for a perfectly competitive market the optimal cap ad trade mechaism oly balaces those two effects ad equates the margial social cost of pollutio to the emissio price i.e. e = D T. 7 Third, observe that a icreased emissio cap T leads to a reduced emissio price. This allows to reduce the welfare loss obtaied due to imperfect competitio at those spot markets where ivestmet is ot bidig ad output too low. Notice that the impact of chaged emissios o welfare at those spot markets where ivestmet is ot bidig has already bee discussed above, it is give by. Based o the fidigs of lemma 4 as the first best bechmark we ca ow directly establish the optimal cap ad trade mechaism which obtais for a perfectly competitive market Theorem 1 Optimal Market Desig, First Best Bechmark Uder competitio the optimal market desig satisfies perfect i A 1 = 0 ii A = 0 iii T : e = D T T. Proof See appedix. The theorem demostrates that i a competitive market i.e., full auctioig is uambiguously optimal i.e. o free allocatios should be grated. A brief glace to lemma 4 ad the ituitio provided reveals that ivestmet icetives of firms uder perfect competitio are optimal, positive free allocatio would lead to reduced welfare. Moreover, as coditio iii shows, the emissio target T should be set such that the equilibrium permit price equals margial social cost of evirometal damage. That is, as already discussed above, the optimal cap ad trade mechaism balaces welfare losses due to foregoe 7 This parallels the fudametal tradeoff obtaied i a simple static model where a Pigou tax should just equal to the margial social damage of pollutio. 0

22 productio at all spot markets give by e with the margial social cost of pollutio give by D T. i the subsequet two sectios we ow cosider market imperfectios which make a attaimet of the first best outcome impossible. First, we determie the desig of a optimal cap ad trade mechaism for a imperfectly competitive market see sectio 4.1. Apart from imperfect competitio, aother source of market imperfectio arises whe the competitio authorities caot freely choose all parameters A 1, A, T of the cap ad trade mechaism, but oly a subset. Such situatios arise for example whe the level of free allocatio for some of the differet techologies or the total emissio cap is exogeously fixed due to political arragemets or lobbig of firms ad the competitio authority ca oly determie the remaiig parameters see sectio Optimal Market Desig uder Imperfect Competitio After havig determied the first best bechmark theorem 1 we ow determie the optimal cap ad trade mechaism for a imperfectly competitive market. Theorem Optimal Market Desig uder Imperfect Competitio Uder imperfect competitio the optimal market desig satisfies i A 1 = 1 θ Pq X 1 e df θ A 1 ii A = 1 θp Pq X θ Pq X 1 e df θ + df θ A θ B 1 + A iii T : e = D T T. Now assume that P qθ = 0. We the obtai A 1 > 0. For w w w > w we ca obtai A = 0. we obtai A > A 1, for Proof See appedix. The optimal levels of free allocatios A 1, A uder imperfect competitio are thus typically differet from zero, a strikig differece to the result obtaied uder perfect competitio see theorem 1. The fudametal reaso why this is the case follows directly from the isights provided by lemma 1 ad the subsequet discussio of the results: Imperfectly competitive firms ot oly exercise market power at the spot markets, but also choose their capacity such that they optimally beefit from scarcity prices, implyig reduced productio ad ivestmet icetives. As already discussed i the text followig lemma 1 compare the last paragraph which discusses lemma 1, for a exogeously fixed price for pollutio e.g. a pigouvia tax at 1

23 some fixed level e optimal ivestmet icetives are obtaied by subsidizig ivestmet such as to precisely compesate for the differece betwee scarcity rets ad margial scarcity profits. To stick as close as possible to our otatio such subsidy could be made by assigig the amouts A 1 ad A of free tax vouchers to each uit ivested i either of the techologies. The optimal level of tax vouchers is the give by expressios i ad ii of theorem otice that for exogeously fixed permit price we have = 0. Remember that i our framework the expressio allowed to quatify the impact of chaged emissios at those spot markets where ivestmet is ot bidig. Positive free allocatio leads to icreased ivestmet icetives, which through a icreased emissio price ca lead to reduced output ad thus pollutio at those spot market where ivestmet is ot bidig. The terms icludig the expressio take this welfare loss ito accout. This leads to a reduced level of the optimal degree of free allocatio. As we show i the theorem, uder imperfect competitio the degree of free allocatio for the peak load techology is always positive. For the optimal allocatio for the base load techology ambiguous results obtai. If the base load techology is less emissio itesive tha the peak load techology i.e. w w 1 icreased ivestmet i the base load techology leads to reduced emissios ad thus allows for more output at spot markets where ivestmet is ot bidig. As we show this always implies A > A 1. O the other had, if the base load techology is more emissio itesive tha the peak load techology w > w 1, i.e. a icrease of base load ivestmet leads to icreased emissio price, the it might be optimal to set A < A 1 or eve A < 0 as we show. Fially cosider the optimal choice of the total emissio cap T for the case of imperfect competitio. A brief look at the optimality coditio iii established i lemma 4 reveals, that the impact of a chaged emissio cap o ivestmet decisios ca be eglected sice the levels of free allocatio are determied optimally such as to obtai Ω I = Ω II = 0. What matters, however, is the fact that a icreased emissio cap leads to a reduced emissio price which i tur allows to reduce the welfare loss iduced by imperfectly competitive productio decisios at those spot markets where ivestmet is ot bidig give by. As a result the optimal cap o total emissios is chose such as to yield a emissio price below the margial social cost of pollutio. I sum, the mai ituitio why a optimal cap ad trade mechaism with edogeous emissio price implies levels of free allocatio which are differet from zero is similar to the ituitio obtaied for the case of a exogeously fixed price for pollutio e.g. a pigouvia tax: Uder imperfect competitio firms ivestmet ad productio icetives are too low, leadig to decreased welfare. Free allocatios ca provide adequate icetives which lead to a icrease of welfare. However, for the case of a edogeous emissio price, as modeled i the preset paper, icreased ivestmet icetives also lead to a icreased emissio

24 price which i tur aggravates welfare losses at those spot market where ivestmet is ot bidig. Let us fially discuss those results i the light of recetly proposed measures thought to icrease firms ivestmet icetives, as for example observed i liberalized electricity markets. I the perceptio of may ecoomists ad policy makers ivestmet icetives i those markets are too low, oe of the reasos potetially beig market power as modeled i the preset paper. To resolve those problems of too low ivestmet icetives, several measures have bee proposed, amog them capacity mechaisms. 8 For the preset discussio we clearly have to abstract form the specific problems ecoutered whe desigig real capacity markets, ad just cosider some subsidy s t paid to the firms per uit of ivestmet made i techology t = 1,. Notice that i the preset framework it is equivalet if a moetary paymet s t or free allocatios with value A t e for t = 1, are grated to a firm per uit of ivestmet i a techology t. What exclusively matters for firms ivestmet icetives is the total value s t + A t e grated to firms per uit of ivestmet, this total value should be set at a optimal level. 9 This i tur implies, however, that, oce a cap ad trade mechaism is put i place i specific give market, the implicatios of this cap ad trade have to be take ito accout whe desigig the capacity market. More specifically the desig of a capacity market which disregards the edogeous ature of emissio prices, will lead to too high ivestmet icetives. However, we establish furthermore that also whe takig ito accout the edogeous ature of the emissio price, the subsidy grated to the peak load techology should be positive, if the base load techology is less emissio itesive it should receive a subsidy which exceeds that of the peak load techology. However, a relatively dirty base load techology should ot receive ay free allocatios. 4. Optimal Desig of a Partially Costraied Cap ad Trade Mechaism I theorems 1 ad we determied the optimal desig of a cap ad trade mechaism whe all its parameters A 1, A, T ca be freely chose by the competitio authority. We first 8 I most restructured electricity markets i the Uited States so called capacity markets are istalled i order to icrease iefficietly low ivestmet icetives, also i urope policy makers cosider their itroductio see e.g. Cramto ad Stoft Cosequetly, optimality just requires that the sum of both parameters satisfies the above optimality coditios. A immediate ad iterestig implicatio is that possible iefficiecies due to gradfatherig could be healed by capacity paymets that compesate for the distortig effect without ay efficiecy losses as log as the subsidies resultig from free allocatios are ot higher tha the sum of both parameters should be. 3

25 aalyzed the case of a perfectly competitive market, which yields the first best bechmark theorem 1 ad the the case of imperfect competitio theorem. Aother source of market imperfectio, apart from imperfect competitio, arises whe the competitio authorities caot freely choose all parameters A 1, A, T of the cap ad trade mechaism. Such rigidities might be due to political costraits ad arragemets or due to lobbig of firms. As already discussed extesively i the itroductio of this article free allocatios have bee key to guaratee the political support ecessary to itroduce cap ad trade systems, compare Covery009, Tieteberg006, Boveberg 008, or for example Grubb ad Neuhoff It is the purpose of the preset sectio to aalyze how a competitio authority should optimally desig a cap ad trade mechaism if it ca determie oly a subset of the parameters of the cap ad trade mechaism, whereas the remaiig parameters are exogeously fixed due to the above discussed problems. Theorem 3 determies the optimal degree of free allocatios for the case of exogeously fixed level of the total emissio cap T. I theorems 4, 5 ad 6 we determie the optimal degree of free allocatio to the remaiig techologies ad the correspodig level of the optimal total emissio cap T. Observe that our results obtaied i lemma 4 i priciple would allow for a detailed aalysis of those questios both for the cases of perfect ad imperfect competitio. I order to limit the otatioal burde i the preset paper we restrict ourselves to the case of perfect competitio, however. I this case the optimality coditios determied i lemma 4 read as follows W A1 := dx 1 A 1 e + dx A 1 A e = 0 11 W A := dx 1 da A 1 e + dx da A 1 A e = 0 1 W T := dx 1 dt A 1 e + dx dt A 1 A e D T T + e = We first aalyze the case of a exogeously fixed level of the cap o total emissios T, a example might be a situatio where politicias are willig to itroduce a cap ad trade mechaism but are reluctat to iduce too severe eve though optimal from a overall welfare poit of view distortios o the ecoomy. The above optimality coditios directly reveal that i a perfectly competitive market o free allocatios should be grated to firms, idepedetly of the level of the emissio cap. 31 This is summarized i theorem Due i part to the sheer scale of the U TS, govermets are subject to itese lobbyig relatig to the distributioal impact of the scheme, ad are costraied by this ad by cocers about the impact of the system o idustrial competitiveess. Few academics uderstad the real difficulties that policy-makers face whe cofroted with ecoomically importat idustries claimig that govermet policy risks puttig them at a disadvatage relative to competitors. 31 The results of theorem 3 for the case of imperfect competitio obtai aalogously, the optimal levels of free allocatio are give by coditios i ad ii established i theorem. 4

26 Theorem 3 Optimal Desig for fixed emissio cap T For ay exogeously fixed total emissio cap T it is optimal to choose the levels of free allocatio A 1 = A = 0. That is, the result obtaied i the first best bechmark theorem 1, where o free allocatio has bee foud to be optimal also obtais if the total emissio cap is ot set at a optimal level. Observe that the reverse does ot hold as we show i the subsequet theorem, however. Theorem 4 Optimal Desig for fixed allocatios A 1 ad A Suppose the iitial allocatios A 1 ad A are fixed exogeously. Defie Γ 0 A 1, A := A 1 A 1 A 1 Ψ I1 + A A 1 A A A 1 Ψ II. 14 The optimal emissio cap T has to be set such as to satisfy e = D T T for Γ 0 A 1, A = 0, e > D T T for Γ 0 A 1, A > 0, ad e < D T T for Γ 0 A 1, A < 0. Proof See appedix F. Figure 3: Choosig the optimal T for exogeously fixed iitial allocatios A 1 ad A. Left: for relatively dirty base techology, i.e. w > w, Right: for relatively clea base techology, i.e. w < w. That is, for levels of free allocatio A 1, A which are ot set optimally the optimal cap o emissios T typically does ot implemet a emissio price e equal to the social cost of 5