Long-Run Equilibrium Modeling of Alternative Emissions Allowance Allocation Systems in Electric Power Markets

Transcription

1 Long-Run Equilibrium Modeling o Alternative Emissions Allowance Allocation Systems in Electric Power Markets Jinye Z. Schulkin, Benjamin F.Hobbs and Jong-Shi Pang September 2007 CWPE 0748 & EPRG 0719

2 Long-Run Equilibrium Modeling o Alternative Emissions Allowance Allocation Systems in Electric Power Markets Jinye Z. Schulkin, Benjamin F. Hobbs, and Jong-Shi Pang September 13, 2007 Abstract A question in the design o carbon dioxide trading systems is how allowances are to be initially allocated: by auction, by giving away ixed amounts, or by allocating based on output, uel, or other decisions. The latter system can bias investment, operations, and pricing decisions, and increase costs relative to other systems. A nonlinear complementarity model is used to investigate long-run equilibria that would result under alternative systems or power markets characterized by time varying demand and multiple generation technologies. Existence o equilibria is shown under mild conditions. Solutions show that allocating allowances to new capacity based on uel use or generator type can distort generation mixes, invert the operating order o power plants, and inlate consumer costs. The distortions can be smaller or tighter CO2 restrictions, and are somewhat mitigated i there are also electricity capacity markets or minimum-run restrictions on coal plants. Subject classiication: emissions trading, allowance allocations, electricity, air pollution, auction, grandathering, cost-eectiveness, greenhouse gases, climate change, global warming, carbon dioxide, generation investment JEL Classiication Numbers: C61; L94; Q4; Q53 1 Introduction Pollution cap-and-trade policies operate by allocating or selling permits to emit (or allowances ) to eligible pollution sources, who are then allowed to trade permits among themselves so that every polluter holds a number o allowances at least equal to their emissions. I the cap allows ewer emissions than would otherwise occur, the allowances have a positive market price. Under certain assumptions, such systems result in least-cost control o the emissions [24]. The work o the irst and third author was based on research supported by the National Science Foundation under grant CMMI The work o the second author was supported by the National Science Foundation under grant ECS and by the Energy research Centre o the Netherlands (ECN). The authors thank J. Sijm o ECN, C. Norman o JHU, and K. Neuho o Cambridge University or valuable suggestions. Department o Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York , U.S.A. zhaoj2@rpi.edu. Department o Geography & Environmental Engineering and Department o Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, Maryland 21218, U.S.A. and Scientiic Advisor, ECN, Policy Studies Unit, Amsterdam, The Netherlands. bhobbs@jhu.edu. Department o Industrial and Enterprise Systems Engineering, University o Illinois at Urbana-Champaign, IL, USA. jspang@uiuc.edu. This paper was written when the author was at the Department o Mathematical Sciences and Department o Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute, Troy, New York , U.S.A. 1

3 The irst large-scale emissions cap-and-trade system was instituted in the US or electric sector SO 2 emissions by the 1990 Clean Air Act Amendments. Since then, cap-and-trade systems have been adopted in the US or NO x and proposed by the US Environmental Protection Agency or mercury. Meanwhile, the EU has leaprogged the US by adopting a cap-and-trade policy or the greenhouse gas CO 2. This system, called the Emissions Trading System (ETS), came into eect in Meanwhile, although there is no US ederal CO 2 reduction requirement as o 2007, several states have initiated their own emissions reduction eorts, the most noteworthy being the Regional Greenhouse Gas Initiative (RGGI). CO 2 capand-trade programs are anticipated to have much larger economic impacts than previous emissions trading programs. The cost involved with signiicant CO 2 reductions and the economic value o trading such allowances are likely to be about an order o magnitude greater than or NO x and SO 2 [9]. Wholesale electricity price increases in Germany and the Netherlands o 40% or more in 2005 have been blamed upon the introduction o the ETS [21]. There are many aspects o the design o a cap-and-trade system that can aect its economic eiciency and impacts upon consumers. One o the most important and most debated eatures is the initial allocation o allowances. The potentially high value o CO 2 allowances means, or example, that tens o billions o dollars o economic rents are created by the ETS system. Understandably, the power industry would preer that this rent be given to them through a ree initial assignment o allowances to existing and perhaps new power sources, while others argue that the government should auction the allowances and use the resulting revenues or tax relie or public programs [12]. However, not only income distribution is aected by who gets the economic rents associated with CO 2 allowances. Rules or initial distribution o allowances can also aect economic eiciency, potentially distorting investment, operation, and output pricing decisions and raising the social cost o reducing emissions [20]. Social cost is deined here in the manner usually used by economists as the sum o producer and consumer surpluses. We do not consider external pollution costs. As an example o such a distortion, i a contingent allocation system (also called output-based or input-based allocation) gives allowances in a way that depends on present or uture generator decisions, incentives to deviate rom least-cost investment mixes and operation are introduced. For instance, in the EU ETS, allowance allocations ater 2007 depend upon emissions in , arguably providing an incentive to expand pollution in these years. The potential or distortion is clear rom the analysis in [1] showing that the value o ETS allowances can be % o ixed plant costs. As another example, i a present polluting acility would lose its allowance allocation i it shuts down, then this provides an incentive to keep non-economic capacity in operation. On the other hand, i new investment is allocated ree allowances, this can instead create a bias towards new investment. Further i dirtier new sources are allocated more allowances per unit capacity or unit output, as is done in at least eight EU countries [15], then technology choices can be skewed. Also, i dierent jurisdictions in the same power market have dierent allocation rules, as in the EU ETS, the location o investment can also be distorted. Economic ineiciencies can result not only rom distortions in production, but also rom distortions in product pricing and consumption. As an example, ree allocation to new entry can distort overall market prices, depressing prices below socially optimal levels i entry is made artiicially cheap by the ree provision o allowances. Also, existing distortions in retail power prices arising rom average-cost based price regulation, which is still the rule in many US states and some EU countries, can be worsened by ree allowance allocation [3]. There are strong political orces that support contingent allocation schemes, and the result is that most o the national allocation plans in the EU ETS are based upon such schemes [17, 23]. This support is in spite o numerous modeling studies that have compared the relative eiciency o such schemes compared to grandathering and auctions, and that have ound the latter to be superior. In general, theoretical analyses show that allocating allowances in proportion to output tends to result in greater sectoral output (since there is an implicit subsidy o output), greater reductions in emissions rates per unit output, and higher control costs than grandathering or auctioning, i there are not other 2

4 distortions in the economy [7]. However, in the presence o other market ailures, such as ineicient tax policies or emissions policies that dier among sectors or countries, output-based policies can actually be welare improving relative to grandathering and auctions [5, 8]. As an example, i the cement industry is subject to CO 2 limits in some countries but not others, and cement is internationally traded, an output-based allocation in the regulated countries can lead to less distortion and lower social cost [4]. In this paper, however, we ocus on the eects o allocation policies on a single market sector (power) that we assume is subject to the same rules throughout the market. We propose models or evaluating the long-run implications o dierent emissions allocation schemes or economic eiciency and consumer costs o the electric power sector. We compare investment, operating, and pricing outcomes o two general allocation approaches: contingent allocation schemes that allocate allowances ree to new investment, and systems in which the initial allocation o allowances does not depend on present or uture capital or operating decisions (either grandathering or auction). The models can be used to investigate whether statements such as the ollowing are likely to be true: I the expansion o the generation park (by incumbents or newcomers) is associated with a ree allocation o emission allowances, then players will base their long-term investment decisions on the long-term marginal costs, including the costs o the CO 2 allowances, but by subtracting the subsidy that lowers the required mark-up or the ixed costs.... On balance, the power price will not be increased (ceteris paribus) [14]. We do not address other important issues concerning the design o emissions allocation systems. Some o these include [20]: transparency and transactions costs; international competitiveness o aected industries; which economic sectors are covered; possibilities or obtaining allowances by unding emissions reductions in developing countries; the eect o mechanisms, such as price ceilings, designed to stabilize prices; the value o banking schemes to buer interannual variations in emissions; or the eiciency implications o dierent ways to dispose o auction revenues. Previous modeling studies o the power sector can be divided into two groups. The irst includes detailed simulation analyses o near term (e.g., ) market developments using large-scale linear programming or other optimization-based models or calculating market equilibria. The second consists o theoretical analyses designed to show general results, oten or the long-term. Short-run analyses have considered the present mix o generation capacity in particular markets, and simulated competitive entry o new generation over the next decade or two under alternative allowance schemes. For instance, Neuho, Grubb, and Keats [16] use the IPM linear programming model to simulate eects on coal and natural gas investments, prices, and generator revenues under an exogenous (ixed) CO 2 price and no demand elasticity. As another example, Bartels and Musgens [2] apply a linear programming model ormulated or 11 European power markets, and ind that giving all new capacity the same number o allowances irrespective o emission rates ( sector benchmarking ) resulted in less distortion than uelspeciic ormulas that gave more allowances to technologies with more emissions. More coal plants were added in the latter case than under either sector benchmarking or allowance auctions. An earlier study [3] applies Haiku, a multidecadal equilibrium model or the US. The latter model, unlike the above linear programs, considers price elasticity and average cost-based regulation o retail electricity prices. As a result o ineicient retail pricing, grandathering is ound to have much higher social costs than auctioning, unlike in other studies. Finally, Palmer, Burtraw, and Kahn [18] use Haiku to determine the minimum raction o allowances that should be given away to generators in order to ensure that they would not be worse o ater the implementation o allowance trading; this number (approximately 20% or the RGGI program) was surprisingly small. In contrast to these studies, theoretical analyses tend to involve simpler models and more general conditions. Some theoretical analyses use two-period models to address the distortion that arises i decisions in one period aect emissions allocations in the next period, as in the irst two phases o the EU ETS. These demonstrate the existence o a bias towards over-investment in the irst period [2, 16, 23] to gain more allowances later. But Neuho, Martinez, and Sato [17] point out that incentives to new entry can help mitigate market power in existing concentrated markets, and can also oset a 3

5 bias towards keeping old plants running i shutting them down would cause allowances to be oreited. Other theoretical results include the ollowing. I allowances are given ree to new investment, this increases the eective demand or allowances by generators, inlating the price o emissions allowances and the cost o compliance i power demands are ixed [23]. The papers [16, 17] explore how longrun choices between two new generation technologies could be distorted by uel-speciic allocation rules compared to auctions assuming a zero-proit, ree-entry equilibrium. They show that distortions are worse i the price o allowances is ixed, as dirty technologies will signiicantly expand i given more allowances than clean technologies. However, the investment distortions are less (but power prices are higher) i instead emissions are capped, because emissions prices rise. An innovative long run analysis by Smeers and Ehrenmann [22] looks at the complications introduced by particular market ailure in the power market: the existence o market power mitigation rules in the power market that can depress returns on investment and yield suboptimal capacity additions. In a second-best analysis, they show that it is possible to design a ree allocation o allowances to new generation capacity that can largely oset those investment disincentives, and actually improve market eiciency. The models o this paper are more elaborate than other theoretical analysis o allowance allocation in power markets, with the possible exception o [22] because o its consideration o capacity market ailures. The most comparable models are those o [16, 17]. Like those models, we consider a long run, ree-entry equilibrium among more than one technology. We also share their implicit assumptions that long-run contract markets and short-run spot markets are arbitrage, that generators are price takers (although irms exhibiting market power can be handled easily), and that there are constant returns to scale in generation. The models o this paper are, however, more general than previous theoretical models in the ollowing respects. Neuho, Grubb and Keats [16] consider up to two supply technologies, and assume a ixed operating order; in particular, when they consider two technologies, they assume that coal plants are always operated in preerence to gas plants. Our model is more general in that any number o plant types can be considered, and the operating ( dispatch ) order is endogenous; this is important, because our solutions show that dispatch orders can change i allowance prices are high enough. Our model also automatically considers corner solutions, in which some plant variables are zero. Also, minimum output constraints can be imposed, relecting the reality that some types o capacity (modern coal plants) cannot be cycled on and o on a daily basis; this results in a more realistic characterization o the ability o power supply systems to adapt to changing emissions prices. Capacity markets are included in addition to energy markets, unlike other models. On the demand side, our model can consider arbitrary temporal distributions o demand, which can be price-responsive, unlike the models o Neuho and his colleagues. Finally, our models allow the number o allowances allocated per MW o new capacity (capacity-based allocation) or per MWh o energy output (sale-based allocation) to be endogenous. In particular, the allocation rule speciies the total number o allowances that are to be available to new capacity, and the amounts per MW or per MWh are automatically adjusted to achieve that target. Since several EU national allocation plans place a ceiling on the number o allowances to be allocated to new investment, some type o rationing similar to this may need to be instituted when enough entry has occurred so that ceiling is reached [20]. However, a price is paid or this added complexity; our general models are ormulated as nonlinear complementarity problems or which analytical solutions cannot be derived. This means that it is not possible to obtain general analytical results showing the equilibrium as an explicit unction o parameters, unlike [2, 16, 23]. Instead, our models need to be solved repeatedly or dierent parameter sets. Further, the inclusion o endogenous per MW or per MWh allowance allocations results in bilinear equilibrium conditions that make it more diicult to compute or show the existence o equilibria. The paper is organized as ollows. Following a statement o the model, along with several variants, we demonstrate that a solution exists under general conditions. By using path solver, the model is solved under several sets o input assumptions in order to explore the ineiciencies that result rom dierent emissions allocation systems. In particular, we calculate the ineiciency that results rom two dierent rules or allocating allowances to new investment, both o which discriminate between dierent plant 4

6 technologies: one that allocates allowances based on a per MW o capacity rule (potential emission rule), and another that allocates allowances in proportion to emissions (actual emission rule). The results show that as the percentage o allowances granted ree to new construction increases, the ineiciency (quantiied as the loss o producer and consumer surplus) also increases. We also consider how the results are aected by the simultaneous imposition o a capacity market, as well as by the presence o minimum-run constraints that more realistically simulate the operation o coal plants. 2 Model Deinition We summarize the notation used in the model ormulation; irst the parameters, which are all nonnegative, next the input unctions, and inally the models variables, which include the irms variables and the market prices o capacity and emission allowances. The physical units are noted within parentheses. Figure 1 depicts the various components in the market structure and their interrelations. Parameters: all positive except possibly CAP and CAP which can be zero, F Set o irms T Set o time periods {1,, T } CAP Minimal amount o energy that irm has to generate (MW) MC Marginal cost or irm, excluding cost o emission allowances (EURO/MWh) E Emission rate or irm (tons/mwh) F Annualized investment cost o irm s capacity (EURO/MWyr) R Ratio o allowances allocated to irm per unit o capacity relative to irm 1 R Ratio o allowances allocated to irm per weighted unit o sales relative to irm 1 E Total emission allowances supply (tons/yr): E > E GF E GF Amount o emission allowances that are grandathered or auctioned (tons/yr) H t Hours in period t (hr/yr), here assumed to be 8760/T χ Unit converter = 1 MW 2 yr/euro CAP Total capacity requirement (MW) E > E GF means that a certain volume o ree allowances is guaranteed to new entrants. For example, In EU ETS phase I, a raction o the allowances have been reserved or eligible new entrants. Functions: d t ( ) π t ( ) e NP ( ) Variables: Demand unction or energy, strictly decreasing (MW) The inverse o d t ( ); (EURO/MWh) Nonpower emission, nonincreasing (tons/yr) p t = π t s gt : Energy price during period t (EURO/MWh) p e Emission allowance price (EURO/ton) p c Capacity price(euro/mwyr) α Emission allowance or irm (tons/mwyr) s t Energy sold by irm in period t (MW) s t = s t CAP (MW) cap Capacity or irm (MW) µ t Dual variable associated with irm s capacity constraint in period t (EURO/MWyr) There are three main components in the basic model: (a) irms proit maximization problems, (b) 5

7 Firm 1 Fuel Market Customer Energy Market p t s t Firm cap Capacity Market p c CAP p t E GF (Auctioned/ grandatherd allowances) α cap (Allowances given to new capacity) d t Emission Market p e e NP (Nonpower Emissions) E = α cap + GF Total AllowanceSupply Figure 1: Market structure E market clearing conditions or the the emission, capacity, and energy markets, and (c) allowances allocation rules. Each o these components is described in detail below. 2.1 Firms optimization problems For simplicity, each irm is assumed to own only one type o generating capacity. Taking as exogenous the emission allocation α, as well as the the prices (or energy p t or t T, emission allowances p e, capacity p c ), the irm solves the ollowing proit maximization problem, whose objective unction is revenue less cost, to determine its capacity cap and sales s t : maximize cap, (s t ) H t ( p t MC t p e E ) s t + ( p c + p e α F ) cap subject to CAP s t cap, t T Although the amount o allowances granted per MW o new investment depends on the emissions per MW-year o that capacity type (see the emission rules later), each generator believes (naively) that it cannot aect that amount, and treats it as exogenous. The problem (1) is a linear program whose optimality conditions are straightorward to write down: (1) 0 s t H t ( p t + MC t + p e E ) + µ t 0, t T 0 µ t cap s t CAP 0, t T 0 cap p c p e α + F µ t 0, (2) where is the perpendicularity notation between two vectors, which in this case simply expresses the complementary slackness condition in linear programming. 6

8 When irms instead exert market power, their revenues rom energy sales change rom linear to nonlinear unctions o the sales variables: H t s t p t s gt, H t s t p t and an additional term corresponding to the derivative o the price unction p t ( ) with respect to s t will appear in the irst complementarity condition in (2). The rest o the paper ocuses on the case where all irms are price-takers. Reinements o the irms problems are possible; or instance, the model could accommodate spatially distributed generation and sales variables as well as bounded transmission; see, e.g., the previous models [13, 19], as well as linear constraints, such as a min-run capacity constraint that is o the orm s γ cap or a irm-dependent constant γ > 0. Nevertheless, the main ocus here is on the emission allocation rules to be introduced momentarily, which introduce a new dimension to electric power equilibrium problems that has not been analyzed beore. Thereore, we will work with (1) and its equivalent optimality conditions (2) rom now on. 2.2 Market clearing conditions The price p e o emission allowance is determined by the complementarity condition: 0 p e E e NP (p e ) H t E g s gt 0, (3) which stipulates that allowance price is positive only when demand or allowances equals the available supply. Notice that this ormulation assumes that allowances can be purchased rom or sold to sectors o the economy other than electric power; this is consistent with the EU ETS. The unction e NP (p e ) represents the eective demand or allowances rom other sectors [11]. Similarly, the capacity price p c is determined by the complementarity condition: 0 p c cap g CAP 0, (4) which stipulates that capacity price is positive only when demand or capacity equals the supply. The inal market clearing condition stipulates that energy supplies equal the quantity demanded: s t = d t (p t ), or all t T, or equivalently, p t = π t. F F Because this condition is an equality, the associated price p t is unrestricted in sign. 2.3 Emission allocation rules All emissions rules considered satisy the condition that the amount o allowances available or allocation equals the amount allocated to capacity: s t E E GF = T α cap. (5) We distinguish two types o emission allocation rules or determining α, both being based on certain (weighted) averages o CO 2 emission: 7

9 (I) The potential emission (or input) rule (in terms o capacity), where α cap = R cap ( E E GF ), F, (6) R g cap g provided that the denominator is positive; or equivalently, α = αr or a common endogenous variable α, due to the allowance allocation balance (5). Under this rule, the emission allowance allocated to new capacity o a particular type is ixed ahead o actual operations and is proportional to the ratio o the irm s capacity to a weighted sum o the capacity owned by all irms. (II) The actual emission (or output) rule (in terms o sales), where R H t s t α cap = H g Rg s gt ( E E GF ), F; (7) H t s t provided that the denominator is positive; or equivalently, α = α R (i cap cap > 0) or a common variable α, due to the same emission balancing constraint (5). I the R are equal or all, then this rule allocates allowances in proportion to sales. However, i R instead is the emissions rate per MWh, then allowances are allocated in proportion to emissions. In that case, in contrast to rule (I), this rule ensures that i, say, a plant emits 75% o the CO 2, it receives 75% o the allowances E E GF that are allocated to new capacity. There are simple conditions ensuring that the denominators in (6) and (7) are positive, such as when CAP > 0 or some F. Subsequently, in order or the above rules to be well-deined irrespective o whether the denominator is zero, we write α R cap or (6) α cap = α R H t s t or (7) (8) or some nonnegative variables α and α to be determined. Needless to say, other allocation rules are possible, such as some combination o the two expressions in (8), or output-based rules in which allowances are allocated to production rather than to capacity. In the rest o the paper, we ocus on rules (I) and (II). 2.4 Model solution The model seeks a set o electricity sales (s t ) (,t) F T, capacities ( cap ) F, dual variables (µ t) (,t) F T, emission allowances (α ) F, emission allowance price p e, and capacity price p c, satisying, or some nonnegative scalars α and α corresponding the emission allocation rules (I) and (II), respectively, the ollowing 8

10 conditions: 0 s t H t π t ( s gt + CAP g ) + MC t + p e E + µ t 0, (, t) F T 0 µ t cap s t CAP 0, (, t) F T 0 cap p c p e α + F 0 p e E e NP (p e ) 0 p c cap g CAP 0 (8) and α g cap g ( E E GF ) = 0. µ t 0, F H t E g ( s gt + CAP g ) 0 (9) 3 Nonlinear Complementarity Formulations To establish the existence o a solution to the model with the emission rules (I) and (II), we derive equivalent ormulations o (9) under these rules as standard nonlinear complementarity problems (NCPs). 3.1 The rule (I) For the emission rule (I): α cap = αr cap or all F, we introduce a reormulation o (9) that replaces this rule by a complementarity condition. Speciically, we multiply the emission allowance balancing constraint F α cap = E E GF by the variable p e, turn the resulting equation into an inequality, introduce the nonnegative variable σ αp e, and impose complementarity between σ and the modiied emission inequality. The resulting NCP is as ollows: 0 s t H t π t ( s gt + CAP g ) + MC t + p e E + µ t 0, (, t) F T 0 µ t cap s t CAP 0, (, t) F T 0 cap p c σ R + F µ t 0, F 0 p e E e NP (p e ) H t E g ( s gt + CAP g ) 0 0 p c cap g CAP 0 (10) 0 σ σ R g cap g ( E E GF ) p e 0. Note the dierence between (9) and (10): the ormer is a mixed NCP in the variable (x, α), where { x (s t ) (,t) F T, (µ t ) (,t) F T, ( } cap ) F, (α ) F, p e, p c ; the latter is a standard NCP in the variable { x I (s t ) (,t) F T, (µ t ) (,t) F T, ( ) cap F, p e, p c, σ 9 }.

11 The ollowing result summarizes the connection between (10) and (9) with the emission rule (I) under the mild condition (11), which postulates that, in the event where CAP + F CAP = 0, at least one irm s investment cost or capacity is not too high. Proposition 1. Assume that max CAP + F CAP, χ max F I (x, α) is a solution o (9) under the emission rule (I), then { } H t [ π t (0) MC t ] F > 0. (11) σ ( E E GF ) p e (12) R g cap g is well deined and x I is a solution o (10). Conversely, i x I is a solution o (10), then α E E GF (13) R g cap g is well deined, and with α αr or all F, (x, α) is a solution o (9) under the emission rule (I). Proo. To prove the irst assertion, let (x, α) be as given. We irst show that cap > 0 or some F. Suppose not, then cap = s t = 0 or all (, t) F T, which implies CAP + F CAP = 0. From the irst and third line in (9), we deduce, or all F, H t [ π t (0) + MC t ] + F 0; or equivalently, max F { } H t [ π t (0) MC t ] F 0, which contradicts (11). Thereore, the scalar σ in (12) is well deined; moreover, σ = p e α, yielding σr = p e α. Consequently, (10) ollows rom (9). Conversely, let x I be a solution o (10). By the same argument as beore, we deduce that cap > 0 or some F. Thereore, the scalar α in (13) is well deined; let α αr. We then have F α cap = E E GF. Consequently, (x, α) is a solution (9) under the emission rule (I). Condition (11) implies that i CAP + FCAP = 0, then max F { } H t [ π t (0) MC t ] F > 0, which allows each irm to produce zero power. I no irm sells any power, then the power price will be expected to very high at each time interval. Thereore, it is reasonable to assume that there is at least one irm which will ind it proitable to invest; i.e., whose total short-run and investment cost is less than their revenue, on a per MW o investment basis. 10

12 It turns out that i CAP + FCAP > 0, then the NCP (10) is equivalent to the set o Karush- Kuhn-Tucker (KKT) conditions o the variational inequality (VI) deined by the pair (K I, Φ I ), where K I K R + with K ( s, cap ) 0 : cap g CAP 0 and cap s t CAP 0, (, t) F T being an unbounded polyhedron in the variables s ( s t ) (,t) F T and cap ( cap ) F, and Φ I ( s, cap, p e ) H t π t E ( s gt + CAP g ) + MC + p e E F ( E E GF ) p e R g cap g R F H t E g ( s gt + CAP g ) e NP (p e ) (,t) F T is a non-monotone map. Indeed, note that Φ I is well deined on the set K I because or every element ( s, cap) K, we must have cap > 0 or some F. Letting p c and µ t be the multipliers o the unctional constraints in K, we can readily write down the KKT conditions o the VI (K I, Φ I ) and conclude that they are equivalent to the NCP (10) under the identiication (12) or σ. When CAP = 0 < FCAP, the set { ( ( st ), cap ) 0 : cap s t CAP 0, t T } K = F is the Cartesian product o separable sets. While the VI ormulation is quite compact, one obvious advantage o the NCP (10) is that it applies to the case where CAP = CAP = 0 or all F; in the latter case, the set K I contains the origin where the unction Φ I ails to be well deined. 3.2 The emission rule (II) The NCP ormulation or (9) under the emission rule (II) is somewhat dierent. For one thing, there is no change o variables in the ormulation; in particular, the variables α are kept in the ormulation. 11

13 The derivation in this subsection permits CAP + FCAP = 0. Speciically, consider the NCP: 0 s t H t π t ( s gt + CAP g ) + MC t + p e E + µ t 0, (, t) F T 0 µ t cap s t CAP 0, (, t) F T 0 cap p c α p e + F µ t 0, F 0 α α cap α R H t ( s t + CAP ) 0, 0 p e E e NP (p e ) H t E g ( s gt + CAP g ) 0 0 p c cap g CAP 0 F (14) 0 α α g cap g ( E E GF ) 0 in the variable (x, α). The ollowing result establishes the equivalence o the above NCP with (9) under the emission rule (II). Proposition 2. A pair (x, α) is a solution o (9) under the emission rule (II) i and only i it is a solution o (14). Proo. The only i statement is obvious. Conversely, i (x, α) is a solution o (14), it suices to show that the ollowing equalities hold: α g cap g ( E E GF ) = 0 and α cap α R H t E ( s t + CAP ) = 0, F. Indeed i the irst equality does not hold, then α = 0 by complementarity, yielding 0 α α cap 0. In turn, this implies α cap = 0 or all F; thus E E GF = 0, which contradicts the assumption that E > E GF. Similarly, i α cap α R H t E ( s t + CAP ) > 0 or some F, then α = 0 by complementarity, which contradicts the inequality itsel. Similar to the VI (K I, Φ I ), i CAP > 0 or all F, the NCP (14) is equivalent to the KKT conditions o the VI (K II, Φ II ), where K II K I = K R + and H t π t ( s gt + CAP g ) + MC + p e E (,t) F T ( E E Φ II GF ) p e R H t ( s t + CAP ) ( s, cap, p e ) F. cap R g H t ( s gt + CAP g ) F E H t E g ( s gt + CAP g ) e NP (p e ) 12

14 4 Existence o Solutions For the analysis in this section, we introduce the ollowing mild condition on the supply o allowances: E > e NP (0) + H t E g CAP g. (15) This condition merely says that the total number o allowances is suicient to cover the emissions resulting rom the sum across irms o the lower bounds upon generation, net o the supply o allowances rom other sectors at an allowance price o zero. This is not restrictive, because at a price o zero, it is likely that the supply rom other sectors is nonpositive (as other sectors would likely be willing to buy allowances at such a low price), and because the minimum required generation is likely to be a small raction o total generation, thereby requiring ew allowances. We also impose the ollowing condition that is a slight strengthening o (11): max H t π t g MC F t F > 0. (16) CAP This condition states that at least one irm would ind investment proitable (ixed and variable cost less than revenue) i every generator is producing just its individual lower bound. This is a mild restriction, as the power price would likely be very high when everyone is producing at their lowest possible level. The ollowing proposition shows that under these two conditions, any solution to the model (9) is nontrivial. Proposition 3. Under (15) and (16), any solution o the NCP (10) and (14) must have s t > 0 or some (, t) F T. Proo. We prove the proposition only or (14). Assume or the sake o contradiction that some solution o this NCP has s t = 0 or all (, t) F T. We claim that p e = 0. Indeed, i p e > 0, then 0 = E e NP (p e ) H t E g CAP g E e NP (0) which is a contradiction. Hence, we have, or each T, 0 H t π t g + MC t + µ t CAP H t π t g + MC t + F, CAP H t E g CAP g > 0, which contradicts (16). We recall that the temporal price unctions π t ( ) are strictly decreasing and the nonpower emission unction e NP ( ) is nonincreasing. The ollowing is the main existence theorem or the model (9) with the emission rules (I) and (II). Theorem 4. Under conditions (15) and (16), the model (9) has a solution. 13

15 We prove the above theorem via the two equivalent NCPs: (10) or emission rule (I) and (14) or emission rule (II). In turn, the proos or these two NCPs are quite similar. Both are based on the application o a undamental existence result or a general NCP summarized below. A proo o this lemma can be ound in [6, Theorem 2.6.1]. Lemma 5. Let Φ : R n R n be a continuous unction. I there exists a constant c > 0 such that all solutions o the NCP: 0 x Φ(x) + τx 0 or τ > 0 satisy x c, then the NCP: 0 x Φ(x) 0 has a solution. To avoid repetition, we present the proo or the NCP (14) only; see Subsection 4.1. We choose this NCP because there is an extra perturbation step that is needed in applying the lemma, whereas one can ollow the same argument and directly apply the lemma to the NCP (10). 4.1 Proo or the NCP (14) Toward the proo o solution existence to the NCP (14), we consider a perturbation o the unction Φ II in order to deal with the general case where some CAP = 0. Speciically, or each ε > 0, let H t π t ( s gt + CAP g ) + MC + p e E (,t) F T Φ II ( E E GF ) p e R H t ( s t + CAP ) ε ( s, cap, p e ) F, ( cap + ε ) ε + R g H t ( s gt + CAP g ) F E H t E g ( s gt + CAP g ) e NP (p e ) which is well deined on the set K II. We irst show that the VI (K II, Φ II ε ) has a solution or each ixed but arbitrary ε > 0. For this purpose, we take an arbitrary sequence o positive scalars {τ k }; or each k, let ( s ε,k, cap ε,k, µ ε,k, p ε,k e, p ε,k c ) be a tuple satisying 0 s ε,k t H t π t ( s ε,k gt + CAP g ) + MC t + p ε,k e E + µ ε,k t + τ k s ε,k t 0, (, t) F T 0 µ ε,k t 0 cap ε,k 0 p ε,k e 0 p ε,k c cap ε,k p ε,k c s ε,k t CAP + τ k µ ε,k t 0, (, t) F T (E E GF )p ε,k e R H t ( s ε,k t + CAP ) E e NP (p ε,k e ) + F (cap ε,k + ε) ε + R g H t ( s ε,k gt + CAP g ) F ( ) H t E g s ε,k gt + CAP g + τ k p ε,k e 0 cap ε,k g CAP + τ k p ε,k c 0. µ ε,k t + τ kcap ε,k 0, 14

16 We claim that under condition (15), the sequence {( s ε,k, cap ε,k, µ ε,k, p ε,k e this in several steps: irst the sequence {p ε,k e sequence {cap ε,k } or all F. }; next the sequence { s ε,k t, p ε,k c )} is bounded. We show } or all (, t) F T ; then the Boundedness o {p ε,k e }. Assume or the sake o contradiction that {p ε,k e } is unbounded. Then or an ininite index set κ {1, 2,, }, we have Without loss o generality, we may assume that p ε,k e E e NP (p ε,k e ) For any k κ such that s ε,k 0 t 0 > 0 or some pair ( 0, t 0 ), we have H t0 π t0 ( s ε,k gt 0 + CAP g ) + MC 0 t 0 + p ε,k e lim k( κ) pε,k e =. (17) > 0 or all k κ. It ollows by complementarity that H t E g ( s ε,k gt + CAP g ) + τ k p ε,k e = 0, k κ. E 0 + µ ε,k 0 t 0 + τ k s ε,k 0 t 0 = 0, which yields p ε,k e E 1 0 π t0 CAP g MC 0 t 0. (18) On the other hand, i k κ is such that s ε,k t 0 = E e NP (p ε,k e ) E e NP (0) = 0 or all (, t), then we have H t E g CAP g + τ k p ε,k e H t E g CAP g > 0, which contradicts (15). Consequently, the bound (18) holds or all k κ; contradicting the limit (17). Boundedness o { s ε,k t } or every (, t) F T. Assume or the sake o contradiction that or some pair ( 0, t 0 ) and an ininite set κ {1, 2,, }, lim k( κ) sε,k 0 t 0 =. (19) Without loss o generality, we may assume that s ε,k 0 t 0 > 0 or all k κ. It ollows by complementarity that 0 = H t0 π t0 ( s ε,k gt 0 + CAP g ) + MC 0 t 0 + p ε,k e E 0 + µ ε,k 0 t 0 + τ k s ε,k 0 t 0 H t0 π t0 CAP g + MC t + p ε,k e E 0 + max( µ ε,k 0 t 0, τ k s ε,k 0 t 0 ) which implies max( µ ε,k 0 t 0, τ k s ε,k 0 t 0 ) H t0 π t0 CAP g MC t p ε,k e E 0. 15

17 Since the right-hand side is bounded, by (19), it ollows that But this contradicts the inequality: E e NP (p ε,k e ) { s ε,k t } is bounded or all (, t) F T. lim τ k = 0. (20) k( κ) ( ) H t E g s ε,k gt + CAP g +τ k p ε,k e 0. Thereore, Boundedness o {cap ε,k } or every F. Assume or the sake o contradiction that or some 0 and an ininite set κ {1, 2,, }, lim k( κ) capε,k 0 =. (21) Thus, by complementarity, we must have p ε,k c we deduce µ ε,k 0 t assume that cap ε,k 0 which implies that = 0 or all k κ suiciently large. Since { s ε,k 0 t } is bounded, = 0 or all t T and all k κ suiciently large. Without loss o generality, we may > 0 or all k κ. It ollows by complementarity that or all k κ suiciently large, (E E GF ) p ε,k e R0 H t ( s ε,k 0 t + CAP 0 ) F 0 ( cap ε,k 0 + ε ) ε + R g H t ( s ε,k gt + CAP g ) F 0 (E E GF ) p ε,k e R0 H t ( s ε,k 0 t + CAP 0 ) + τ k cap ε,k 0 = 0, ( cap ε,k 0 + ε ) ε +. R g H t ( s ε,k gt + CAP g ) The limit (21) implies that the right-hand side tends to zero as k( κ), which is a contradiction. Hence {cap ε,k } is bounded or all F. Boundedness o {µ ε,k t } or all (, t) F T. Assume or the sake o contradiction that or some ( 0, t 0 ) F T and an ininite set κ {1, 2,, }, lim k( κ) µε,k 0 t 0 =. (22) Without loss o generality we may assume that µ ε,k 0 t 0 > 0 or all k κ. By complementarity, we deduce cap ε,k 0 s ε,k 0 t 0 + CAP 0 + τ k µ ε,k 0 t 0 = 0. Since {(cap ε,k 0, s ε,k 0 t 0 )} is bounded, (22) implies that lim τ k = 0. k( κ) Since µ ε,k 0 t 0 (E E GF )p ε,k e R0 H t ( s ε,k 0 t + CAP 0 ) µ ε,k 0 t p ε,k c + F 0 (cap ε,k 0 + ε) ε + R g H t ( s ε,k gt + CAP g ) F 0 + τ k cap ε,k 0 + τ k cap ε,k 0 16

18 and τ k cap ε,k 0 0 as k( κ), we obtain a contradiction to (22). Boundedness o {p ε,k c }. This is similar to the above proo o the µ-sequence. We have thereore completed the proo o the boundedness o the sequence {( s ε,k, cap ε,k, µ ε,k, p ε,k e, p ε,k c )} under the condition (15). This is enough to apply Lemma 5 to deduce the existence o a solution to the VI (K II, Φ II ε ) or all ε > 0 via its KKT ormulation. Let ( s ε, cap ε, p ε e) be one such solution. For each ε > 0, there exists (µ ε t, pε c) such that 0 s ε t H t π t ( s ε gt + CAP g ) + MC t + p ε ee + µ ε t 0, (, t) F T 0 µ ε t cap ε sε t CAP 0, (, t) F T (E E GF ) p ε R e H t ( s ε t + CAP ) 0 cap ε pε c + F ( cap ε + ε ) ε + µ ε t 0, F R g H t ( s ε gt + CAP g ) 0 p ε e E e NP (p ε e) ( ) H t E g s ε gt + CAP g 0 0 p ε c cap ε g CAP 0. By the same proo sequence as beore, we can show that lim sup ( s ε, cap ε, µ ε, p ε e, p ε c ) <. Let ε 0 ( s, ĉap, µ, p e, p c ) be the limit o a convergence sequence {( s ε k, cap ε k, µ ε k, p ε k e, p ε k c )} corresponding to a sequence o positive scalars {ε k } 0. It ollows ready that ( s, ĉap, µ, p e, p c ) satisies 0 s t H t π t ( s gt + CAP g ) + MC t + p e E + µ t 0, (, t) F T 0 µ t ĉap s t CAP 0, (, t) F T 0 p e E e NP ( p e ) ) H t E g ( s gt + CAP g 0 0 p c ĉap g CAP 0. By the same proo as that o Proposition 3, we can show that s 0. Since H t ( s ε k t + CAP ) H t cap ε k ( cap ε k + ε k ) ( cap ε k + ε k ) < H t H t ( s ε k t + CAP ) it ollows that the sequence ( cap ε must have at least one accumulation point. Withk + ε k ) 17

19 out loss o generality, we may assume that this sequence converges to a limit, say γ 0. Note that H t ( s t + CAP ) γ = Deine, or each F, α ĉap, i the denominator is positive. (E E GF ) R γ R g H t ( s gt + CAP g ) p e. It is easy to show that the ollowing complementarity holds: 0 ĉap p c + F α p e + µ t 0, F. With α (E E GF ) p e R g H t ( s gt + CAP g ), it is easy to see that all conditions in (14) are satisied. 5 An Application To illustrate the results that can be obtained rom our proposed models (as mentioned beore, the NCPs (10) and (14) are the workhorses in the experiments), we consider a competitive power market at a single node with the ollowing characteristics: Time periods: T = 20 periods per year, each H t = 438 hours in length Demands: d t (p t ) = a t b t p t, with a t = 500t and b t = t/2 Nonpower emission: e NP (p e ) = 0 Generator types: i = 1 (coal steam), 2 (natural gas-ired combined cycle), and 3 (natural gas-ired combustion turbine) Minimal generation: CAP 1 = 0 MW, CAP 2 = 0 MW, and CAP 3 = 0 MW Marginal costs: MC 1 = 20 $/MWh, MC 2 = 40 $/MWh, and MC 3 = 80 $/MWh Investment costs: F 1 = 120, 000 $/MW/yr, F 2 = 75, 000 $/MW/yr, and F 3 = 50, 000 $/MW/yr Firms emission rates: E 1 = 1 ton/mwh, E 2 = 0.35 ton/mwh, and E 3 = 0.6 ton/mwh Total capacity requirement: CAP = 11,000 MW, i CAP > 0. Thus, the demand unction in each period is deined so that the peak load occurs during period 20, and load is proportional to t, i the same price is aced in each period. The demand curve parameters imply that the price intercept o the inverse demand curve is $1000/MWh in each period; since equilibrium prices are usually under $100/MWh, this means that demand is relatively inelastic at the equilibrium price. A capacity market is assumed that requires 10% more capacity than the peak demand o 10,000 MW that occurs i the price in the peak period was 0$/MWh. Consumers are assumed to pay or capacity through a non-distorting customer charge; other assumptions are possible, such as allocation o capacity charges to peak energy prices, but are not explored here. We introduce the ollowing system perormance measures: Generation cost (M$/yr), total generation investment and uel costs: ( F cap + ) H t MC s ; F 18

20 Social cost (M$/yr), the cost o generation plus the cost o price-induced changes in energy consumption: PS + CS + GS, where PS is the equilibrium producer surplus, equal to the sum over all irms o the objectives in (1); CS is the equilibrium consumer surplus: ( dt(pt) H t π t (x)dx d t (p t )p t ); and GS 0 is the auction or grandathering surplus, equal to the economic rent accruing to the original owners o grandathered allowances (or, equivalently, the revenue received by the government i it instead auctioned those allowances). However, we exclude environmental costs rom this perormance measure; Consumer payments (M$/yr): p c cap + d t (p t )p t ; and F Capacity actor, the ratio o annual generation to potential generation: H ts t H. tcap In the computations, the NCPs (10) and (14) and the linear complementarity problem (23) below are solved by the path solver available on the neos server ( 5.1 The base run To provide a basis or comparison o the two emission rules, we derive an equilibrium solution in the absence o a CO 2 limit; i.e., with E = ; thus the constraint (3) is absent and p e = 0. Such an equilibrium, which we call the (emission) unconstrained solution, is the solution o the ollowing linear complementarity problem: 0 s t H t π t ( s gt + CAP g ) + MC t + µ t 0, (, t) F T 0 µ t cap s t CAP 0, (, t) F T 0 cap p c + F µ t 0, F (23) 0 p c cap g CAP 0. When the total capacity requirement CAP is 11,000 MW, capacity cap o coal, combined cycle, and combustion turbines equal 7329 MW, 1628 MW, and 2042 MW, respectively. The bulk (94%) o the energy is obtained rom coal plants, with combined cycle acilities providing nearly all o the remainder. Combustion turbines are built primarily to meet the capacity market requirement. Emissions amount to 47.4 Mtons/yr. The total cost o generation is 2049 M$/yr, which also equals consumer payments or energy, under the zero proit ree-entry assumption. Energy prices p t equal the marginal cost o generation in each period, varying rom $20 to $80/MWh, depending on the marginal source o energy; meanwhile, the capacity market price p c equals the annual cost o combustion turbine capacity, 50,000 $/MW/yr. The quantity-weighted power price is 46.1 $/MWh, o which capacity payments make up 27%. The sum o consumer surplus and producer surplus is M$/yr. 5.2 Comparative results or the allowance allocation rules Table 1 reports the main system perormance measures or a series o experiments representing alternative assumptions regarding: the type o the contingent emission allocation, i.e., rules (I) and (II) as well as the base run where E = ; the presence or absence o a capacity market; and the presence or absence o minimum output levels (in the orm o a min-run capacity constraint) or coal plants only. For each set o 19

21 assumptions, results are presented or CO 2 limits o 20 (a severe restriction) and 40 (a mild restriction) Mtons/yr, and or three cases o 0%, 50% or 100% grandathering o allowances (corresponding to 100%, 50%, and 0% o allowances granted to new generating plants; i.e., E GF /E = 0,.5, 1, respectively). We take R in (7) and R in (6) to be both equal to E /E 1. With the presence o a capacity market, runs 1 6 show the results or the potential emission rule and runs 7 12 show the results or the actual emission rule. For the purpose o the comparison, we ocus on the increase in generation cost, social cost (equal to the loss o social surplus), and consumer payments relative to the (emission) unconstrained solution obtained in Subsection 5.1. We do not report producer surplus, because it is by assumption zero in the ree entry solutions. For ease o comparison, the increases are expressed as a percentage o the production cost o the unconstrained solution (2049 $M/yr), which also equals the consumer payment in that solution. The three percentage increases are calculated, respectively, as: relative generation cost increase: 100% (generation cost M$/yr)/2049 M$/yr; relative social cost increase: 100% [20911 M$/yr - (PS + CS + GS)]/2049 M$/yr; and relative consumer payments increase: 100% ( consumer payments M$/yr )/2049 M$/yr. The imposition o the CO 2 constraint causes social surplus to decrease relative to the unconstrained value o M$/yr. This decrease is not identical to the change in generation cost because o changes in energy consumption that are caused by shits in energy prices. I demand elasticity was instead zero, then the change in social cost would be the same as the change generation cost. The 100% grandathered cases are the same or both contingent allocation rules because, o course, that level o grandathering means no allowances are allocated to new investment. In the case where all allowances are instead allocated to new plants by the potential emission rule (Runs 1,4), we see that, like the actual emission rule, the investment is greatly distorted and costs are much higher than i allowances are completely grandathered (or auctioned). In act, the distortion is worse, because it is possible or new generators to receive ree allowances even i they do not generate power. The allowances given to new investment are suiciently valuable so that it is worthwhile to build combustion turbines, even though they don t operate. In the 20 Mton/yr limit case, 95% o the combustion turbine capacity is never used, and is built just to collect ree allowances. Unexpectedly, the distortion is much worse in the mild (40 Mton/yr) limit case, making the total cost o compliance almost as high as or the 20 Mton/yr case. This conirms that it should not be assumed that the risk o distortion is less i CO 2 limits are less severe. The next set o alternative assumptions addresses the eect o assuming an energy-only market in which there is no separate market or capacity. In this case (Runs 13 24), energy prices rise much higher during peak periods to ensure that consumer demand does not exceed available capacity. There are extensive debates regarding the pros and cons o capacity markets (e.g., see [10]), which we do not consider here. Instead we merely consider the interaction o capacity and emissions markets. The main eect observable in comparing Runs with Runs 1-12 is in the case o the actual emission rule. Cost increases due to investment distortion are somewhat greater without the capacity market, especially in the 40 Mton/yr limit, and there is less investment. The decreased investment is most dramatic or combustion turbines, which are not built at all under the actual emission rule. In contrast, under the potential emission rule, the costs and generation mixes under 0% grandathering (all allowances given to new investment) are completely unaected by the presence o a capacity market. This is because the potential emission orm o the contingent rule results in overinvestment such that the capacity constraint is not binding. The inal set o alternative assumptions we consider is the imposition o a min-run constraint on coal plants, since in reality they cannot really be cycled in the way they are in the solutions that have low coal capacity actors. We assume that the output o coal plants cannot be reduced below 35% o their capacity; as a result, in periods where that constraint is binding, energy prices can actually be negative. 20