Do Generation Firms in Restructured Electricity Markets Have Incentives to. Support Social-Welfare-Improving Transmission Investments?

Transcription

1 Do Generation Firms in Restrutured Eletriity Markets Have Inentives to Support Soial-Welfare-Improving Transmission Investments? * Enzo E. Sauma a,**, Shmuel S. Oren b a Pontifiia Universidad Católia de Chile, Industrial and Systems Engineering Department; Avenida Viuña Makenna # 4860, Raúl Deves Hall, Piso 3, Maul, Santiago, Chile; esauma@ing.pu.l b University of California Berkeley, Industrial Engineering and Operations Researh Department; 4141 Etheverry Hall, Berkeley, CA , U.S.A.; oren@ieor.berkeley.edu Abs trat This paper examines the inentives that generation firms have in restrutured eletriity markets for supporting long-term transmission investments. In partiular, we study whether generation firms, whih arguably play a dominant role in the restrutured eletriity markets, have the inentives to fund or support inremental soial-welfare-improving transmission investments. We examine this question in a two-node network and explore how suh inentives are affeted by the ownership of finanial transmission rights (FTRs) by generation firms. In the analyzed twonode network, we show both (i) that the net exporter generation firm has the orret inentives to inrease the transmission apaity inrementally up to a ertain level and (ii) that, although a poliy that alloates FTRs to the net exporter generation firm an be desirable from a soial point of view, suh a poliy would dilute the net-importer-generation-firm s inentives to support transmission expansion. Moreover, if all FTRs were alloated or autioned off to the net exporter generation firm, then it is possible to inrease both onsumer surplus and soial welfare while keeping the net exporter generation firm revenue neutral. * The work reported in this paper was partially supported by the Power System Engineering Researh Center (PSERC) and by the Center for Eletri Reliability Tehnology Solutions (CERTS) through a grant from the Department of Energy, and by the FONDECYT Grants No and No ** Corresponding author. Tel.: ; Fax: ; E-ma il: esauma@ing.pu.l.

2 2 Keywords: trans mission investment inentives, market power, finanial transmission rights, power systems eonomis. JEL Classifiations: D43, L13, L52, L Introdution Although seurity onstrained dispath is intended to ensure reliability of the power system, there is growing evidene that the U.S. transmission system is under stress (Abraham, 2002). In fat, the National Transmission Grid Study of the U.S. Department of Energy (Abraham, 2002) delares: Growth in eletriity demand and new generation apaity, lak of investment in new transmission failities, and the inomplete transition to fully effiient and ompetitive wholesale markets have allowed transmission bottleneks to emerge. These bottleneks inrease eletriity osts to onsumers and inrease the risks of blakouts. The inreased use of the system has led to transmission ongestion and less operating flexibility to respond to system problems or omponent failures. This lak of flexibility has inreased the risk of blakouts. From an eonomi perspetive, inreased ongestion redues the ability to import power from remote heap generators, thus raising the ost of energy. It also impedes trade and ompetition, whih in turn makes onsumers more vulnerable to the exerise of market power. The so-alled Standard Market Design (FERC, 2002), whih prevails (or is in the proess of being implemented) in the restrutured eletriity markets in the US, relies on loational marginal pries for energy to prie and manage ongestion and to signal the need for eonomially driven transmission investments 1. Studies addressing the insuffiieny of inentives for investment in the U.S. eletriity transmission system are sparse. Moreover, none of the inentive strutures proposed in the literature have been broadly adopted. 1 While loational marginal pries provide the right inentives for generation firms to operate effiiently, investments in transmission systems are generally driven by either reliability motives or by the searh for a satisfatory rate of return (merhant investment). Many transmission investments in the US are driven by reliability onsiderations while the eonomi analysis serves for impat assessment and ost alloation (Abraham, 2002).

3 3 Bushnell and Stoft (1996) apply the definition of finanial transmission rights (rights that entitle holders to reeive finanial benefits derived from the use of the apaity) in the ontext of nodal priing systems. They use a transmission rights alloation rule based on the onept of feasible dispath, originally proposed by Hogan (1991), and prove that suh a rule an redue or, under ideal irumstanes, eliminate the inentives for a detrimental grid expansion while rewarding effiient investments. The paper by Bushnell and Stoft (1996) is based on the idea that transmission investors are granted finanial rights (whih are tradable among market partiipants) as a reward for the transmission apaity added to the network. 2 This sheme, in ontrast with the atual rate-ofreturn-regulation regime, ould provide, in prinipal, the orret inentive for new entrants to invest in new transmission apaity. The main idea in (Bushnell and Stoft, 1996) is that a transmission investor is allowed to selet any set of transmission rights whih, when ombined with the existing set, orresponds to a dispath that is feasible under the onstraints of the newly modified grid. An investor who reates an intentionally ongested line, whih effetively redues the feasible set of dispathes, would, therefore, be required to aept a set of transmission rights and obligations that exatly anel the flows that are no longer feasible in the resulting, lower apaity network. The onept of feasibility, thereby, provides some hek on the inentive to reate ongestion. Bushnell and Stoft (1996) show that, under ertain onditions, the mentioned simultaneous feasibility test an effetively deter detrimental investments. However, these onditions are very stringent. They assume that transmission investments are haraterized by no-inreasing returns to sale, there are no sunk osts, nodal pries reflet onsumers willingness to pay for eletriity and reliability, all network externalities are internalized in nodal pries, transmission network 2 The onept of a deentralized alloation of finanial transmission rights was originally developed by Hogan (1991; 1992), under the name of ontrat network regime.

4 4 onstraints and assoiated point-to-point apaity are non-stohasti, there is no market power, markets are always leared by pries, and the system operator has no disretion to affet the effetive transmission apaity and nodal pries over time. Joskow and Tirole (2003) reexamine the model by Bushnell and Stoft (1996) after introduing assumptions that more aurately reflet the physial and eonomi attributes of real transmission networks. They show that a variety of potentially signifiant performane problems then arise. In partiular, they laim that the definition of transmission rights by Bushnell and Stoft (1996) does not adequately aount for the stohasti and dynami physial attributes of transmission networks. Thus, they argue that property rights that are ontingent on exogenous variations in transmission apaity and reflet the diversifiation attributes of new investments would be required. Unfortunately, defining and alloating these ontingent rights is also like ly to be inonsistent with the development of liquid ompetitive markets for these rights or derivatives on them. In addition, Joskow and Tirole (2003) argue that the diffiulty of orretly assigning finanial transmission rights (FTRs) is another deterrent to invest in the transmission system. In (Bushnell and Stoft, 1996), the alloation of FTRs is made by an independent system operator (ISO) who looks for feasibility of the network using a sequene of simulations of the system. However, these types of assignments may be subjetive, espeially in the ase of alloating inremental network investments (investments that involve upgrades of existing failities). In this sense, our paper gives some insights about the possibility of using the alloation of FTRs to align the inentives for transmission expansion of the soiety and of the net exporter generation firms. The diffiulty of orretly assigning FTRs is also addressed in Barmak et al. (2003). Differently from Joskow and Tirole (2003), they mention two other important reasons for the ineffiieny of FTRs with respet to inentives for transmission investment: (i) a transmission investment that eliminates ongestion results in FTRs that are worthless, and (ii) it may be

5 5 diffiult for transmission owners (TOs) to apture other benefit streams resulting from transmission investment. Joskow and Tirole (2000) analyze how the alloation of transmission rights assoiated with the use of power networks affets the operational behavior of generation firms and onsumers with market power. Their analysis, as well as the analysis in (Joskow and Tirole, 2003), fouses on an always-ongested two-node network where there is a heap generation monopolist in an exporting region that has no loal demand and an expensive generation monopolist in an importing region that ontains the entire-system demand. They onlude that if the generation firm in the importing region has market power, their holding finanial transmission rights enhanes that market power sine the FTRs give it an extra inentive to urtail its output to make the rights more valuable. In setion 3.2 of this paper, we reah the same onlusion and, in addition, we analyze the onsequenes of this finding on the inentives that generation firms have to support soialwelfare-improving transmission expansions. Joskow and Tirole (2000) also onlude that, onsidering there is no loal demand at the node where the net exporter generation firm is loated, soial welfare is likely redued by the ownership of FTRs by the net importer generation firm beause this would inentivize the net importer firm to inrease pries. In setion 3.2 of this paper, we show that alloating FTRs to a net exporter generation firm who both has loal market power and faes loal demand with some elastiity, may ompensate this soial-welfare-reduing effet due to the inentive of the net exporter firm to redue its nodal prie to make transmission rights more valuable. Several related studies try to improve the inentive strutures for transmission investment by dealing with the generator s motivation to exerise market power. In (Cardell et al., 1997), (Joskow and Tirole, 2000), (Oren, 1997), and (Stoft, 1999), the authors study the impliations of the exerise of market power in ongested two- and/or three-node networks where the entire system demand is onentrated in only one node. The main idea behind these papers is that if an

6 6 expensive generator with loal market power is required to produe power as a result of network ongestion, then the generation firm owning this generator may have a disinentive to relieve ongestion. Borenstein et al. (2000) present an analysis of the relationship between transmission apaity and generation ompetition in the ontext of a two-node network in whih there is loal demand at eah node. The authors argue that relatively small transmission investment may yield large payoffs in terms of inreased ompetition. However, they only onsider the ase in whih generation firms annot hold transmission rights. In setion 3.2 of this artile, we extend this analysis to allow both loal demand at eah node of the network and the possibility that generation firms hold finanial transmission rights. The California Independent System Operator (CAISO) has reently developed a Transmission Eonomi Assessment Methodology (TEAM) for assessing transmission expansion projets, whih is based on the gains from trade priniple (Sheffrin, 2005), (CAISO, 2004). Although TEAM onsiders alternative generation-expansion senarios with and without transmission upgrades, as far as we know, this generation-expansion analysis does not take into aount the potential strategi response to transmission investment from generation firms who may alter their investment plans in new generation apaity. This rationale underlines ommon wisdom that prevailed in a regulated environment justifying the onstrution of transmission between heap and expensive generation nodes on the grounds of reduing energy ost to onsumers. However, as shown by Sauma and Oren (2006), suh rationale may no longer hold in a ma rket-based environment where market power is present. On the other hand, FERC has reently proposed transmission priing reforms designed to promote needed investment in energy infrastruture (FERC, 2005). Basially, FERC proposes an inrease in the rate of return on equity, espeially for stand-alone transmission ompanies (Transos), in order to both attrat new investment in transmission failities and enourage formation of Transos. This FERC proposal is based on the idea that inentives may be more

7 7 effetive in fostering new transmission investment for Transos than for traditional publi utilities that are dependent upon retail regulators for some portion of their transmission ost reovery. In this paper, we fous on the inentives that generation firms at generation pokets have to support inremental soial-welfare-improving transmission expansions and how these inentives are affeted by the ownership of finanial transmission rights (FTRs). We are interested in analyzing the effet of loal market power on suh inentives when onsidering both that generation firms an hold FTRs and that generation firms annot hold FTRs. For simpliity, we will assume through this artile that transmission line apaities are stati and deterministi. The rest of the paper is organized as follows. Setion 2 studies the distributional impats of transmission investments. In setion 3, we explore how FTRs alloation may be used to align the inentives for transmission expansion of the soiety and of the net exporter generation firms, in the ontext of a two-node network. We illustrate the theoretial results obtained in setion 3 through a numerial example presented in setion 4. Setion 5 onludes the paper. 2. Distri butional Impats of Transmission Investments Before analyzing the transmission investment inentives of generation firms, it is worth to emphasize the well-known fat that transmission expansions generally have distributional impats, whih ould potentially reate onflits of interests among the affeted parties. The key issue is that, while soiety as a whole may benefit from inremental mitigation of ongestion, some parties may be adversely affeted. In general, transmission investment effets rent transfers from load poket generators and generation poket onsumers to load poket onsumers and generation poket generators. However, load poket onsumers and generation poket generators annot simply deide to build a line linking them. Their deision will be subjet to srutiny by not only an ISO, but also state and federal energy and environmental regulators. In this type of environment, the losers from

8 8 transmission investment ould be expeted to expend up to the amount of rents that they stand to lose to blok the transmission investment. This rent dissipation is wasteful. Moreover, it may blok soially benefi ia l projets from being built. Nevertheless, it is important to mention that the usual oordination problem faed by the benefiiaries of a transmission expansion also applies to the losers from the expansion. The following examples illustrate the distributional impats of transmission investments and the potential inentives that some market partiipants ould have to e xer ise politial power in order to blok a soial-welfare-improving transmission expansion projet. Consider a network omposed of two ities satisfying their eletriity demand with loal generation firms. For simpliity, assume there exists only one (monopolist) generation firm in eah ity, whih have unlimited generation apaity. We assume that the marginal ost of supply at ity 1 is lower than that at ity 2. In partiular, suppose the marginal osts of generation are onstant 3 and equal to zero at ity 1 and $20/MWh at ity 2. Assume the inverse demand funtions are linear, given by P 1 (q) = q at ity 1 and by P 2 (q) = q at ity 2, in $/MWh. Under the monopolisti (self-suffiient-ities) senario, the ity 1 firm optimally produes q (M) 1 = 500 MWh (on an hourly basis) and harges a prie P (M) 1 = $50/MWh while the ity 2 firm optimally produes q (M) 2 = 250 MWh and harges a prie P (M) 2 = $70/MWh. With these marketlearing quantities and pries, the firms profits are Π 1 (M) = $25,000/h and Π 2 ( M) = $12,500/h, respetively. The onsumer surpluses are CS 1 (M) = $12,500/h for ity 1 onsumers and CS 2 (M) = $6,250/h for ity 2 onsumers. 4 3 The assumption that marginal osts of supply are onstant is not ritial, but it simplifies alulations. 4 Under monopoly, a firm optimally hooses a quantity suh that the marginal ost of supply equals its marginal revenue. If the marginal ost of prodution is onstant and equal to and the demand is linear, given by P(q) = a b q, where a >, then the monopolist will optimally produe q (M) = (a )/(2b) and harge a prie P (M) = (a+)/2, making a profit of Π ( M) = (a ) 2 /(4b). Under these assumptions, the onsumer surplus is equal to CS ( M) = (a ) 2 / (8b).

9 9 Now, onsider the senario in whih there is unlimited transmission apaity between the two ities. This situation orresponds to a duopoly faing an aggregated demand given by (in $/MWh): Q, if Q < 100 P(Q) =, where Q = q 1 + q Q, if Q 100 We assume that generation firms behave as Cournot oligopolists in this ase. Under this senario, the firm at ity 1 optimally produes q (D) 1 = 633 MWh (on an hourly basis) while the firm at ity 2 optimally produes q (D) 2 = 333 MWh. The prie harged by both firms is equal to P (D) = $42.2/MWh. With these new market-learing quantities and prie, the firms profits are Π (D) 1 = $26,741/h and Π (D) 2 = $7,407/h, respetively. 5 Furthermore, the onsumer surpluses are CS (D) 1 = $16,691/h for the ity 1 onsumers and CS (D) 2 = $15,124/h for the ity 2 onsumers. In this example, by linking both ities with a high-apaity transmission line, we replae some expensive power produed at ity 2 by heaper power generated at ity 1, whih makes ity 2 onsumers learly better off. Unfortunately, this is not the only impliation of the onstrution of suh a transmission line. The ity 2 firm redues its profit beause its retail prie dereases as result of the ompetition between generation firms introdued by the new transmission line. Indeed, the numerial results reveals that the onstrution of the transmission line has the following onsequenes: the ity 1-onsumers surplus inreases from $12,500/h to $16,691/h, the ity 2-onsumers surplus inreases from $6,250/h to $15,124/h, the ity 1-firm s profit inreases fro m $25,000/h to $26,741/h, and the ity 2-firm s profit dereases from $12,500/h to $7,407/h. From these results, it is lear that the ity 2 firm (load poket generator) will oppose the onstrution of the line linking both ities beause this line will derease its profit, transferring its 5 Under duopoly, the Cournot firms simultaneously hoose quantities suh that their marginal ost of supply equals their marginal revenue, but assuming the quantity produed by the other firm is fixed. If the marginal osts of prodution are onstant for both firms, given by 1 and 2 respetively, and the aggregate inverse demand is linear, given by P(Q) = A B Q, where A > 1 and A > 2, then firm i will optimally produe q i (D) = (A 2 i + j ) / (3B), with j i and i {1,2}. Under these assumptions, the duopolisti prie will be P (D) = (A ) / 3 and firm i will make a profit of Π i (D) = (A 2 i + j ) 2 / (9B), with j i and i {1,2}.

10 10 rents to the other market partiipants. Consequently, depending on the relative politial power of the ity 2 firm, this network-expansion projet ould be bloked, even though it ould be soially benefiial (depending on the transmission investment osts) 6. The problem of rent transfer may arise even in the absene of market power. To illustrate this fat, assume that ity 1 (generation poket) has 1,000 MW of loal generation apaity at $10/MWh marginal ost and another 500 MW of generation apaity at $20/MWh marginal ost, with 600 MW of loal demand, while ity 2 has 800 MW of generation apaity at $30/MWh marginal ost and loal demand of 1,000 MW. Furthermore, assume that all generation power is offered at marginal ost and that a 300 MW transmission line onnets the two ities. Under this senario, the market learing pries are $10/MWh in ity 1 and $30/MWh in ity 2 and 300 MW are exported from ity 1 to ity 2. A 300 MW inrease in transmission apaity would allow replaement of 300 MW of load served at $30/MWh by imports from ity 1, of whih 100 MW an be produed at $10/MWh and another 200 MW an be produed at $20/MWh. The soial benefit fro m suh an expansion is, therefore, $4,000/h. Assuming that the amortized upgrade osts is below $4,000/h, the upgrade is soially benefiial. The market onsequenes of suh an upgrade are that the ma rket learing prie at ity 1 inreases fro m $10/MWh to $20/MWh while the market learing prie at ity 2 stays $30/MWh as before, with 600 MW being exported from ity 1 to ity 2. Thus, onsumers and generators in ity 2 are neutral to the expansion, onsumer surplus in ity 1 will drop by $6,000/h, generator s profits in ity 1 will inrease by $10,000/h, and the merhandising surplus of the system operator will re ma in unhanged (the ISO merhandising surplus on the pre-expansion imports drops $3,000/h, but it piks up $3,000/h for the inremental imports). Clearly, suh an expansion is likely to fae stiff opposition from onsumers in ity 1, 6 Note that, in general, building transmission to eliminate all ongestion is not neessarily optimal (espeially when onstrution ost is aounted for), but it an be superior to the ase of no onnetivity. In our example, we do not advoate elimination of ongestion, but use these two polar extremes for illustrative purposes.

11 11 but it would be strongly favored by the generators at ity 1, who would be more than happy to pay for it (as long as the amortized investment ost does not exeed $10,000/h). In fat, generators at node 1 would favor suh an investment even if its amortized ost exeed the $4,000/h benefits, whih would make suh an investment soially ineffiient to the detriment of ity 1 onsumers. By ontrast to the above example, a small inremental upgrade of 90 MW in the transmission apaity would be soially benefiial inreasing soial surplus by $1,800/h without affeting the market learing pries in either ity. In suh a ase, neither the generators nor the onsumers on either side will benefit (or be harmed) by the expansion and, thus, the entire gain will go to the ISO in the form of merhandising surplus. In suh a ase, a merhant transmission owner ould be indued to undertake the transmission upgrade in exhange for finanial transmission rights (FTRs) that would entitle her to the loational marginal prie differenes for the inremental apaity, thus allowing the investor to apture the entire soial surplus gain due to the expansion. In the following setion, we will further explore how FTR alloation may be used to align the inentives for transmission expansion of the soiety and of some market partiipants. 3. Transmission Investment Inentives of Generation Firms In analyzing the transmission investment inentives of generation firms, onsidering the impliations of the exerise of loal market power by generators beomes ruial. Here, we study this idea in the ontext of a radial, two-node network and explore how the investment inentives are affeted by the ownership of finanial transmission rights (FTRs) by generation firms. The analysis in this setion shows that the net exporter generation firm has the orret inentives to inrease the transmission apaity inrementally up to ertain level. We also show that, although alloating FTRs to the net exporter generation firm an inrease its inentives to support a soialwelfare-improving transmission expansion, suh a poliy would dilute the net-importergeneration-firm s inentives to support the apaity expansion. We also show that, if all FTRs

12 12 were alloated or autioned off to the net exporter generation firm, then it is possible to inrease both onsumer surplus and soial welfare while keeping the net exporter firm revenue neutral. As general framework for the analysis presented in this setion, we assume that the transmission system uses loational marginal priing, generation firms behave as Cournot oligopolists, transmission losses are negligible, all transmission rights are finanial rights (whose holders are rewarded based on ongestion rents), and network investors are rewarded based on a regulated rate of return administered by a non-profit ISO, whih manages transmission assets owned by many investors. The main two reasons for this hoie are: (i) many of the U.S. transmission systems atually use this type of sheme and (ii) this struture has been proposed by FERC as part of its Standard Market Design (FERC, 2002). Consider a network omposed of two nodes linked by a transmission line of thermal apaity K. The non-depreiated apital and operating osts of the link are assumed to be reovered separately from onsumers (for instane, in lump-sum harges net of revenues produed by selling transmission rights) and we do not onsider these osts further in our analysis. For simpliity, we assume that there is only one generation firm at eah node, having unlimited generation apaity. We assume that the prodution ost funtions of the two firms, say C 1 (q) and C 2 (q), are onvex and twie differentiable in the firms outputs (i.e., the firms marginal osts of generation are ontinuously non-dereasing in the firms outputs). We also assume that the inverse demand funtion at eah node of the network, say P 1 (q) at node 1 and P 2 (q) at node 2, is ontinuous, onave, and downward sloping. Moreover, we suppose that, if the two markets were ompletely isolated (i.e., no onneted by any transmission line), the generation firms would produe outputs q M 1 and q M 2 suh that P 1 (q M 1 ) < P 2 (q M 2 ). 7 7 This would be the ase if, for example, both generation firms faed equal demand urves (i.e., P 1 (q) = P 2 (q)) and the marginal ost of supply at node 1 were lower than that at node 2 over the relevant range (i.e., C 1 (q 1 M ) < C 2 (q 2 M ) ).

13 13 Let q i (i = 1,2) be the quantity of energy produed by the generation firm loated at node i, and let q t be the net quantity exported from node 1 to node 2. This quantity (q t ) depends on both nodal pries and, thus, depends on both q 1 and q 2. Moreover, q t must satisfy the transmission apaity onstraints (i.e., it must satisfy K q t K, where a negative q t represents a net flow from node 2 to node 1). Our analysis onsiders two senarios: first, a senario in whih generation firms annot hold transmission rights and seond, a senario in whih generation firms an hold FTRs. 3.1 Senario I: generation firms annot hold transmission rights Assume generation firms annot hold transmission rights (and, thus, their bidding strategy is independent of the ongestion rent). Aordingly, in this ase, the profit of the generation firm loated at node 1 (heapgen) is π 1 (q 1 ) = q 1 P 1 (q 1 q t ) C 1 (q 1 ) and the profit of the generation firm loated at node 2 (deargen) is π 2 (q 2 ) = q 2 P 2 (q 2 + q t ) C 2 (q 2 ). Impliit on these definitions is the assumption that eah market partiipant must trade power with an ISO, at the nodal prie of its loal node. Thus, the generation firm loated at node i will reeive a payment equal to the nodal prie at node i times the quantity produed and the onsumers at node j will pay an amount equal to the nodal prie at node j times the quantity onsumed. Consequently, the nodal prie that eah firm faes is determined by loal generation plus imports. When generation firms annot hold transmission rights, it is relat ively simp le to analy ze the inentives that generation firms with loal market power have to support soial-welfare-improving transmission investments. We ould argue that, by ongesting the system, 8 generation firms have the ability to exerise their loal market power and deliberately withhold their outputs so that they an inrease their profits. However, we must be autious in the analysis of the equilibrium 8 In this artile, the term ongestion is used in the eletrial engineering sense: a line is ongested when the flow of power is equal to the line s thermal apaity, as determined by various engineering standards.

14 14 onditions beause nodal pries, P 1 (q 1 q t ) and P 2 (q 2 + q t ) in our example, are disontinuous at the point where the transmission line beomes ongested (i.e., at q t = ± K). In (Borenstein et al., 2000), the authors use a two-node network similar to the one desribed above. They showed that, as the thermal apaity of the transmission line, K, inreases from zero, one of two possible outomes is obtained: 9 0 < K < K passive/aggressive (P/A) Nash equilibriu m e xists Case 1: K < K < K* no pure-strategy Nash equilibriu m e xists K* < K unonstrained Nash-Cournot equilibrium exists or 0 < K < K* P/A Nash equilibriu m e xists Case 2: K* < K < K both P/A and unonstrained Cournot Nash equilibria e xists K < K unonstrained Nash-Cournot equilibrium exists where K orresponds to the largest line apaity that an support a P/A Nash equilibriu m (i.e., a pure-strategy Nash equilibriu m in whih the transmission line is ongested with net flow from the lower-prie under monopoly market to the higher-prie market) and K* represents the smallest transmission line apaity that an support an unonstrained 10 Nash-Cournot duopoly equilibrium (i.e., a Nash-Cournot duopoly equilibriu m in whih K is high enough so that the line is never ongested). One an derive the best-response (in quantity) funtions of eah firm for eah one of the two previous ases. Figure 1, reprodued from (Borenstein et al., 2000), illustrates the best-response funtions in ase 2 (i.e., the overlapping equilibria ase), where firm s and n are the heapgen and the deargen, respetively, and where q m n+, q m n-, q m s+, and q m s- represent the profit-maximizing 9 See Theorem 5 in (Borenstein et al., 2000). 10 In this paper, the term unonstrained refers to the fat that the transmission onstraint is not binding.

15 15 output (PMO) for firm n when it is ongesting the line to s, the PMO for firm n when it is produing its optimal passive output, the PMO for firm s when it is ongesting the line to n, and the PMO for firm s when it is produing its optimal passive output, respetively. Figure 1. Best-response funtions in the overlapping equilibria ase. Reprodued from Figure 7 in (Borenstein et al., 2000). When firm n is produing nothing, the best response of firm s is to produe its optimal quantity given that the line will be ongested from s to n. As n s output rises, eventually it reahes the point at whih it beomes more profitable for s to swith to a muh less aggressive output response. Pratially any asymmetry (in either osts or demand) will result in a pure-strategy P/A equilibriu m for a suffiient small line. As the apaity of the line inreases, export from the lowprie market (s) inreases. This shifts rightward the demand that s faes and, thus, rises the prie at s. As exports into n inrease with the inrease in K, firm n will redue prodution, but by less than the inrease in imports to n, so the prie in n will drop. The higher K makes it less attrative for n to allow the line to be ongested into its market. For a line apaity greater than some level, firm n

16 16 is better off ating more aggressively, whih eliminates the P/A equilibrium. Moreover, as K inreases, eventually a point must be reahed at whih a pure-strategy unonstrained Cournot duopoly equilibrium an be supported, as Figure 1 suggests. Aordingly, if the transmission line apaity is high enough (i.e., K > Max{K, K*}), then an unonstrained Nash-Cournot duopoly equilibrium exists and it orresponds to the unique purestrategy Nash equilibrium. In this ase, there is no ongestion at the Nash equilibrium and q t is far enough from ± K so that both P 1 (q 1 q t ) and P 2 (q 2 + q t ) are ontinuous and differentiable over the relevant range. Thus, the unonstrained Nash-Cournot duopoly equilibrium (in whih eah firm maximizes its profit taking the output of the other firm as fixed subjet to the fat that nodal pries must be equal at both nodes) is haraterized by the following system of equations (first order optimality onditions): P 1 (q 1 q t ) + q 1 d( P(q 1 1 qt )) P 2 (q 2 + q t ) + q d 2 ( P 2(q2 qt )) = C 1 (q 1 ), (1) dq 1 + = C 2 (q 2 ), (2) dq P 1 (q 1 q t ) = P 2 (q 2 + q t ), (3) K < q t < K, (4) q 1, q 2 0 (5) These optimality onditions are only valid under the assumption that, at the equilibrium, q t is far enough from ± K. The only way to guarantee this fat is by ensuring that the transmission line apaity is high enough so that the line is never ongested. However, this is not an interesting ase to analyze from the point of view of the transmission investment inentives beause generation firms have obviously no inentives to support an inrement in the apaity of a line that has large exess apaity. 2

17 17 On the other hand, if the transmission line apaity is low enough (i.e., K < Min{K, K*}), then generation firms at aording to a Nash equilibriu m in whih the transmission line is ongested with net flow from the lower-prie (under monopoly) market to the higher-prie market (i.e., a P/A Nash equilibriu m). In this ase, q t = K (i.e., the line is ongested with net flow from node 1 to node 2) and the disontinuity of both P 1 (q 1 q t ) and P 2 (q 2 +q t ) at the point where the line is ongested beomes problemati in the sense that, as q t approahes to K, d( P(q 1 1 q )) / dq and 1 d ( P 2 (q2 + qt )) / dq are not well defined and, thus, equations (1) and (2) annot orretly 2 represent the optimality onditions. In this ase, as mentioned before, as the apaity of the line inreases, eventually a point must be reahed at whih a pure-strategy unonstrained Cournot duopoly equilibrium an be supported. Moving the line apaity from slightly below this level to slightly above this level may ause a disontinuous jump of the equilibrium from a P/A equilibrium to an unonstrained Cournot equilibrium. Consider a P/A point of operation, (q 1, q 2 ), that maximizes the firms profits given that the quantity exported from node 1 to node 2 is fixed and equal to the line apaity (i.e., subjet to the fat that the line is ongested with flow from node 1 to node 2). That is, q 1 is the profitmaximizing output of the heapgen when it faes an inverse demand urve given by P 1 (q 1 K), whih is the heapgen s native inverse demand shifted rightward by K, and q 2 is the output of the deargen when it maximizes its profit given the residual inverse demand it faes, P 2 (q 2 + K), whih is the deargen s native inverse demand shifted leftward by K. In this ase, the heapgen effetively ats as a monopolist on the rightward-shifted inverse demand urve and the deargen effetively ats as a monopolist on its residual inverse demand urve. Borenstein et al. (2000) show that, for suffiiently small transmission apaity, the quantities (q 1, q 2 ) are the unique pure-strategy Nash equilibriu m. 11 Although the proof presented in (Borenstein et al., 2000) orretly analyzes the inentives that the generation firms have not to deviate from the equilibrium, the fat that both t 11 See Theorem 4 in (Borenstein et al., 2000).

18 18 P 1 (q 1 q t ) and P 2 (q 2 +q t ) are disontinuous at the point where the line is ongested and the assoiated omplexities are not expliitly addressed in the proof. In (Sau ma, 2005), an alternative proof is provided showing that (q 1, q 2 ) is a pure-strategy Nash equilibrium, that aounts for all possible disontinuities. We omit the detailed proof due to spae limitation and summarize in Table 1 below the basi rationale. Table 1. Rationale of proof that (q 1, q 2 ) is a Nash equilibriu m. Firm Deviation possible senarios onsequene Cheapgen Derease output q 1 q 1 ε (ε > 0) (i) q t unhanged and (q 1 q t ) dereases by ε π 1 dereases. Cheapgen Deargen Deargen Inrease output q 1 q 1 + ε (ε > 0) Inrease output q 2 q 2 + ε (ε > 0) Derease output q 2 q 2 ε (ε > 0) (ii) q t dereases by ε and (q 1 q t ) unhanged (iii) both q t and (q 1 q t ) derease by less than ε. (i) q t unhanged and (q 1 q t ) inreases (ii) q t dereases and (q 1 q t ) inreases (i) q t unhanged and (q 2 +q t ) inreases by ε (ii) q t dereases by ε and (q 2 +q t ) unhanged (iii) q t dereases by less than ε and (q 2 +q t ) inreases (i) q t unhanged and (q 2 +q t ) dereases (ii) both q t and (q 2 +q t ) derease Line still ongested, P 1 (q 1 q t ) inreases, (q 1 q t ) dereases Line deongested it is optimal to ongest the line aga in. Line deongested, π 1 dereases it is optima l to ongest the line aga in. Line still ongested, P 1 (q 1 q t ) dereases, (q 1 q t ) inreases π 1 dereases. Line deongested, P 1 (q 1 q t ) dereases it is optimal to ongest the line again. Line still ongested, P 2 (q 2 +q t ) dereases, (q 2 +q t ) inreases π 2 dereases. Line deongested it is optimal to a llo w a ongested line again. Line deongested, π 2 dereases it is optima l to a llo w a ongested line again. Line still ongested, P 2 (q 2 +q t ) inreases, (q 2 +q t ) dereases π 2 dereases. Line deongested, P 2 (q 2 +q t ) inreases it is optima l to a llo w a ongested line again.

19 19 Now, we analyze the inentives/disinentives that the generation firms have to support an inrease in the apaity of the transmission line wh ile the Nash equilibrium haraterized by (q 1, q 2 ) prevails. 12 Here, we will assume that suh an inrease in the transmission apaity is desired beause it would inrease both the total onsumer surplus and the soial welfare, as it is more likely to happen in a ongested radial network aording to the gains from trade eonomi priniple (Sheffrin, 2005). Suppose the thermal apaity of the transmission line is inreased by a small positive amount, K, suh that the P/A Nash equilibriu m is still supported. Then, the heapgen will at as a monopolist on the (K+ K)-rightward-shifted inverse demand urve and, onsequently, it will reoptimize its profit by inreasing its output so that q t is augmented by K (i.e., ongest the line ( again). Aordingly, the heapgen s new optimal output, q K+ K) 1, will be larger than q 1 and the ( new optimal prie at node 1, P 1 (q K+ K) 1 (K+ K)), will be greater or equal to that before the expansion (beause the onsumption at node 1 must either derease or remain equal at the new optimum). Lemma 1 formally proves these fats. 13 Lemma 1: In the two-node network desribed in this setion, assume that a passive/aggressive Nash equilibrium is ahieved and that a passive/aggressive Nash equilibrium is 12 Hereafter in this setion, we assume that (q 1, q 2 ) is an interior passive/aggressive Nash equilibrium, where by interior we will understand that it is a passive/aggressive Nash equilibrium that prevails when the line apaity is inreased by a small amount. 13 An intuitive way to understand the results proved on lemma 1 is the following. When the thermal apaity of the transmission line inreases by K, the heapgen ould inrease its output in K and keep the same retail prie at node 1 (making node 1 onsumers indifferent and node 2 onsumers better off), obtaining an extra profit equal to K P 1 (q 1 K). However, the fat that the heapgen now faes a higher demand motivates it to exerise its loal market power, reduing its output from the theoretial q 1 + K (while, of ourse, still resulting in an output greater than q 1 ) in order to inrease the prie at node 1 and, thus, inrease its profit. That is, the heapgen will now at as a monopolist on the (K+ K)-rightward-shifted inverse demand urve and reoptimize its profit by inreasing its output in suh a way so that the line is ongested and the profit gained due to the nodal prie inrease, q 1 (K+ K) ( P 1 (q 1 (K+ K) (K+ K)) P 1 (q 1 K) ), is larger than the profit lost due to the fat that the output is inreased by less than K, ( q 1 + K q 1 ( K+ K) ) P 1 (q 1 K). Figure 2 illustrates these fats.

20 20 still supported when making an inremental transmission investment. Then, the hange in the equilibrium heapgen s output due to an inremental transmission expansion is positive, but smaller than the hange in the transmission apaity. Proof. Assume that the Nash equilibrium haraterized by (q 1, q 2 ), with q 1 > 0 and q 2 > 0, is ahieved and that a P/A Nash equilibrium is still supported when making an inremental transmission investment. Sine generation firms annot hold transmission rights, the profit of the heapgen at the equilibrium is: π 1 *(q 1,K) = q 1 P 1 (q 1 K) C 1 (q 1 ). Hene, the first order optimality ondition is: ddq π 11 *= 0, or equivalently: P 1 (q 1 K) + q 1 P 1 (q 1 K) C 1 (q 1 )= 0. Then, 2 d(qk)dkdq ( π 111 *, = 0 ), or: dqdqdqdq 1111 **** ( ) + = ( ) ( ) ( ) PqKPqKqPqKCq * 1+ * * * 1 *0 dkdkdkdk (6) or equivalently: 1 PqKqPqK * * * ( ) + ( ) ( ) ( ) ( ) dq * = (7) dk 2 * * * * + PqKqPqKCq Sine q 1 * > 0, osts funtions are onvex, and the inverse demand funtions are ontinuous, onave, and downward sloping, every term of both the numerator and the denominator of the right-hand side of (7) is negative. Thus, dqdk 1 * is positive. Furthermore, sine + PqKqPqKCq ( ) ( ) ( ) > * * * ( ) ( ) 2 * * * * PqKqPqK +, we have that dqdk 1 * < 1, whih implies that the hange in the equilibrium heapgen s output due to an inremental transmission expansion is smaller than the hange in the transmission apaity. Following lemma 1, it beomes evident that the heapgen will have positive inentives to support this transmission expansion beause it inreases the heapgen s profit. Figure 2 illustrates this situation (where MR 1 (K) represents the marginal revenue when the heapgen faes the K-

21 21 rightward-shifted inverse demand urve and MR 1 ( K+ K) orresponds to the marginal revenue when the heapgen faes the (K+ K)-rightward-shifted inverse demand urve). Proposition 1 summarizes this intuitive result. Figure 2. Transmission investment inentives of the heapgen in the two-node network $/MWh MR 1 (K+ K) MR 1 (K) P 1 (q 1 (K + K) (K+ K)) P 1 (q 1 K ) P 1 (q) P 1 (q K) C 1 (q) P 1 (q (K+ K)) q 1 K q 1 (K + K) K Cheapgen s output (MW) Proposition 1: Ass ume that generation firms annot hold transmission rights. In the two-node network desribed in this setion, 14 the net exporter generation firm (i.e., the heapgen) has positive inentives to support an inrease in the transmission apaity up to any level so that a passive/aggressive Nash equilibrium is still supported. Proof. Assume that the Nash equilibrium haraterized by (q 1, q 2 ), with q 1 > 0 and q 2 > 0, is ahieved and that a P/A Nash equilibrium is still supported when making an inremental 14 Reall that the two-node network used here assumes a single transmission line of thermal apaity K and that there is only one generation firm at eah node, having unlimited generation apaity. We also assume that the prodution ost funtions of the two firms are onvex and twie differentiable in the firms outputs. We also assume that the inverse demand funtion at eah node of the network is ontinuous and downward sloping. Moreover, we suppose that, if the two markets were ompletely isolated (i.e., no onneted by any transmission line), the generation firms would produe outputs q 1 M and q 2 M suh that P 1 (q 1 M ) < P 2 (q 2 M ).

22 22 transmission investment. Sine generation firms annot hold transmission rights, the profit of the heapgen at the equilibrium is: π 1 *(q 1,K) = q 1 P 1 (q 1 K) C 1 (q 1 ). By using the envelope theorem, we obtain: ( (q, K) ) 1 * 1 d K d π = q 1 P 1 (q 1 K) ( 1) = q 1 P 1 (q 1 K) (8) Sine q 1 > 0 and the inverse demand funtions are ontinuous and downward sloping (i.e., P 1 (q 1 K) < 0), we have fro m (8) that: d ( π (q, K) ) 1 * 1 d K > 0. This is, the equilibriu m heapgen s profit inreases as the transmission apaity inreases, as long as a P/A Nash equilibriu m is still supported. Consequently, the heapgen has positive inentives to support an inrease in the transmission apaity up to any level so that a P/A Nash equilibriu m is still supported. On the other hand, when the line apaity is inreased by the small positive amount, K, the deargen s best response is to produe its optimal passive output. That is, the deargen will at as a monopolist on its residual, (K+ K)-leftward -shifted, inverse demand urve and reoptimize its ( profit by dereasing its output. The new optimal output, q K+ K) 2, will be sma lle r than q 2 and the ( new optimal prie at node 2, P 2 (q K+ K) 2 + (K+ K)), will be smaller or equal to that before the expansion (beause the onsumption at node 2 must either inrease or remain equal at the new optimum). Lemma 2 formally proves these fats An intuitive way to understand the results proved on lemma 2 is the following. If the deargen kept its output at the q 2 level even after inreasing the thermal apaity of the line by K, the prie at node 2 would derease from P 2 (q 2 +K) to P 2 (q 2 +K+ K), produing a lost in the deargen profit (with respet to the pre-expansion situation) equal to q 2 (P 2 (q 2 +K) P 2 (q 2 +K+ K)). However, the deargen ould exerise its loal market power and redue its output in order to inrease the prie at node 2 with respet to the theoretial prie P 2 (q 2 +K+ K) and, thus, inrease its profit with respet to the situation in whih the deargen keeps the output at the q 2 level. That is, the deargen will now at as a monopolist on the (K+ K)-leftward-shifted inverse demand urve and reoptimize its profit by reduing its output in suh a way so that the line is ongested and the gain in profit, q 2 ( K+ K) (P 2 (q 2 ( K+ K) +K+ K) P 2 (q 2 +K+ K)), is larger than the lost in profit,

23 23 Lemma 2: In the two-node network desribed in this setion, assume that a passive/aggressive Nash equilibrium is ahieved and that a passive/aggressive Nash equilibrium is still supported when making an inremental transmission investment. Then, the hange in the equilibrium deargen s output due to an inremental transmission expansion is negative and smaller, in absolute value, than the hange in the transmission apaity. Proof. Assume that the Nash equilibrium haraterized by (q 1, q 2 ), with q 1 > 0 and q 2 > 0, is ahieved and that a P/A Nash equilibrium is still supported when making an inremental transmission investment. Sine generation firms annot hold transmission rights, the profit of the deargen at the equilib riu m is: π 2 *(q 2,K) = q 2 P 2 (q 2 + K) C 2 (q 2 ). Hene, the first order optimality ondition is: ddq π 22 *= 0, or equivalently: P 2 (q 2 +K) +q 2 P 2 (q 2 +K) C 2 (q 2 )= 0. Then, 2 d(qk)dkdq ( π 222 *, = 0 ), or: dqdqdqdq 2222 **** ( = ) ( ) ( ) ( ) PqKPqKqPqKCq *1+ ** *1 *0 dkdkdkdk (9) or equivalently: PqKqPqK ** * ( ) ( ) ( ) ( ) ( ) dq * = (10) dk 2 ** * * ++ + PqKqPqKCq Sine q 2 * > 0, osts funtions are onvex, and the inverse demand funtions are ontinuous, onave, and downward sloping, every term of the numerator of the right-hand side of (10) is positive and every term of the denominator of the right-hand side of (10) is negative. Thus, dqdk 2 * is negative. Furthermore, sine 2 ** * * ++ + PqKqPqKCq ( ) ( ) ( ) > ( ) ( ) + + PqKqPqK ** * 22222, we have that dqdk 2 * < 1, whih implies that the hange in (q 2 q 2 ( K+ K) ) P 2 (q 2 +K+ K), due to the redution in the output with respet to the hypothetial ase that the deargen keeps the output at q 2. Figure 3 illustrates these fats.

24 24 the equilibriu m deargen s output due to an inremental transmission expansion is smaller, in absolute value, than the hange in the transmission apaity. Following lemma 2, it beomes evident that the deargen will have disinentives to support this transmission expansion beause it dereases the deargen s profit. Figure 3 illustrates this situation (where MR (K) 2 represents the marginal revenue when the deargen faes the K-leftwardshifted inverse demand urve and MR 2 ( K+ K) orresponds to the marginal revenue when the deargen faes the (K+ K)-leftward-shifted inverse demand urve). Proposition 2 summarizes this intuitive result. Figure 3. Transmission investment inentives of the deargen in the two-node network. $/MWh MR 2 (K) MR 2 (K+ K) P 2 (q + (K+ K)) P 2 (q 2 + K ) C 2 (q) P 2 (q 2 (K + K) + (K+ K)) P 2 (q + K) P 2 (q) q 2 (K + K) q 2 K K Deargen s output (MW) Proposition 2: Ass ume that generation firms annot hold transmission rights. In the two-node network desribed in this setion, the net importer generation firm (i.e., the deargen) has disinentives to support an inrease in the transmission apaity up to any level suh that a passive/aggressive Nash equilibriu m is still supported. Proof. Assume that the Nash equilibrium haraterized by (q 1, q 2 ), with q 1 > 0 and q 2 > 0, is ahieved and that a P/A Nash equilibrium is still supported when making an inremental

25 25 transmission investment. Sine generation firms annot hold transmission rights, the profit of the deargen at the equilibriu m is: π 2 *(q 2,K) = q 2 P 2 (q 2 + K) C 2 (q 2 ). By using the envelope theorem, we obtain: ( *(q 2, K) ) d π 2 = q 2 P 2 (q 2 + K) (+1) = q 2 P 2 (q 2 + K) (11) d K Sine q 2 > 0 and the inverse demand funtions are ontinuous and downward sloping (i.e., P 2 (q 2 + K) < 0), we have fro m (11) that: d ( π *(q 2, K) ) 2 d K < 0. Th is is, the equilibriu m deargen s profit dereases as the transmission apaity inreases, as long as a P/A Nash equilibrium is still supported. Consequently, the deargen has disinentives to support an inrease in the transmission apaity up to any level suh that a P/A Nash equilibrium is still supported. Summarizing, when the equilibrium haraterized by (q 1, q 2 ) is aomplished, the heapgen has inentives to support an inrease in the apaity of the transmission line by some small positive amount (suh that the P/A Nash equilibrium is still supported) while the deargen has disinentives to support suh a transmission expansion. However, this analysis is only valid for small inremental expansions of the line. As the size of the line upgrade inreases, the P/A Nash equilibriu m may no longer be supported (i.e., the best response of the deargen ould be to inrease signifiantly its output so that it either deongests the line or ongests the line with net flow in the opposite diretion). If this ourred, then it is unlear whether the heapgen would still have inentives to support the expansion of the transmission line. In fat, if the network upgrade were large enough so that it led to an unonstrained Nash-Cournot duopoly equilibrium, then suh an investment would likely redue the profits of both generators. 16 All these results are illustrated 16 If the two markets are omparable and the two firms have similar generation osts, then we obtain the well-known result that a large enough investment that moves the pure-strategy Nash equilibrium from a P/A Nash equilibrium to an unonstrained Nash-Cournot duopoly equilibrium redues the profits of both generators beause nodal pries disontinuously jump down (although firms outputs inrease).

26 26 through a simple numeria l e xa mp le, presented in setion 4.1, where demand funtions are linear and generation firms have onstant marginal osts. A remaining question in our analysis is what happens with the generation firms inentives to support inremental soial-welfare-improving transmission expansions when the line apaity is neither too small nor too high (i.e., when K is suh that Min{K,K*} < K < Max{K,K*}). Suh analysis is omplex beause the existene of a pure-strategy Nash equilibrium is not guaranteed in this ase. Although we leave this analysis as future work, our intuition is that, even under mixedstrategy Nash-Cournot equilibria, expeted nodal pries will deline as the line apaity inreases. With a very small transmission apaity, for instane, nodal pries should be very lose to the monopoly levels. If they were not, then either firm ould improve its expeted profit by simply admitting imports of K and produing the optimal passive output as a pure strategy. With K near K*, the lower bounds on pries provided by the optimal passive output responses should be muh weaker and the mixed strategy would be more likely to result in lower expeted pries. 3.2 Senario II: generation firms an hold FTRs Assume now that generation firms an hold some FTRs. In partiular, suppose that the heapgen and the deargen hold frations α and (1 α) of the K FTRs available from node 1 to node 2 (α [0,1]), respetively. Thus, in our two-node network, the heapgen now maximizes the following profit funtion (making rational expetations of the deargen s outome): π 1 (q 1, α) = q 1 P 1 (q 1 q t ) C 1 (q 1 ) + α K [ P 2 (q 2 + q t ) P 1 (q 1 q t ) ] (12) Likewise, the deargen now maximizes the following profit funtion (making rational expetations of the heapgen s outome): π 2 (q 2, α) = q 2 P 2 (q 2 + q t ) C 2 (q 2 ) + (1 α) K [ P 2 (q 2 + q t ) P 1 (q 1 q t ) ] (13)

27 27 Generation firms must aquire their FTRs through some type of alloation sheme or aution. In this setion, we assume that FTRs are alloated free of harge diretly to the market partiipants. 17 If the transmission line apaity were high enough (i.e., K > Max{K, K*}) 18 so that an unonstrained Nash-Cournot duopoly equilibrium would exist (and it would orrespond to the unique pure-strategy Nash equilibrium), then there would be no ongestion at the equilibrium. This means that the nodal pries at both ends of the unongested line would be equal. Aordingly, all FTRs would beome worthless due to the zero nodal prie differene. Consequently, when the transmission line apaity is high enough, so that there is no ongestion at the Nash equilibrium, the fat that generation firms an hold FTRs does not make any differene in profits as ompared to the benhmark ase (without FTRs). Thus, in th is ase, the unonstrained Nash-Cournot duopoly equilibrium is haraterized by the same system of equations (first order optimality onditions) as in the benhmark ase, i.e. equations (1) to (5). As we mentioned in the ase without FTRs, this is not an interesting ase to analyze from the point of view of the transmission investment inentives beause generation firms have obviously no inentives to support an inrement in the apaity of a line that has exess apaity. On the other hand, if the transmission line apaity were low enough (i.e., K < Min{K, K*}) so that a P/A Nash equilibrium were supported, then the transmission line would be ongested with net flow from node 1 to node 2 (i.e., q t = K) at the unique pure-strategy Nash equilibriu m In some areas, FTRs are autioned off among the market partiipants and, then, the revenues olleted from the aution proess are alloated to the load on a prorate basis. In ontrast, in some other areas, FTRs are alloated diretly free of harge to the market partiipants (on the basis of laims). This last sheme is the one assumed in this artile. 18 Here, we maintain the same notation as in the ase without FTRs. That is, K orresponds to the largest line apaity that an support a P/A Nash equilibrium and K* represents the smallest line apaity that an support an unonstrained Nash-Cournot duopoly equilibrium. 19 The proof that the outome (q 1 (α), q 2 (α)), whih maximizes the generation firms profits given both that the line is ongested with flow from node 1 to node 2 and that α has a fixed value, is a Nash equilibrium is analogous to the ase without FTRs.

28 28 In this ase, we an analyze the inentives/disinentives that the generation firms have to support an inrease in the apaity of the transmission line, while a P/A Nash equilibrium is still supported, in a similar way as in the benhmark ase (without FTRs). When the P/A Nash equilibrium is supported, the heapgen maximizes its profit as if it had monopoly power over its K-rightward-shifted inverse demand funtion, but having two revenues streams now: a first stream of revenue from sales of energy and a seond stream of revenues from the ongestion rents from the FTRs. Consequently, while the P/A Nash equilibrium prevails, the heapgen effetively inreases the prie elastiity of its residual demand urve by holding FTRs. 20 Proposition 3 establishes the same result as in proposition 1 in the ase that generation firms an hold FTRs. This is, in the two-node network desribed in this setion, the heapgen has positive inentives to support an inrease in the transmission apaity up to any level so that a P/A Nash equilibrium is still supported. Proposition 3: In the two-node network desribed in this setion, the net exporter generation firm (i.e., the heapgen) has positive inentives to support an inrease in the transmission apaity up to any level so that a passive/aggressive Nash equilibrium is still supported. Proof. When assuming that generation firms annot hold transmission rights, the proof is idential to the proof of proposition 1. Now, assume generation firms an hold FTRs. Suppose that the heapgen and the deargen hold frations α and (1 α) of the K FTRs available from node 1 to node 2 (α [0,1]), respetively. 20 When holding FTRs on the ongested line, the heapgen has inentive to inrease the nodal prie differene. To do that, it would inrease its output and, thus, derease its nodal prie with respet to the benhmark(no FTRs)-ase levels. Aordingly, at the profit-maximizing output, P 1 (q 1 K) would be lower (and q 1 K would be higher) when α > 0 (holding FTRs) than when α = 0 (without FTRs). Thus, sine demand is downward sloping, we would have that P(qK 11 ) whih orresponds to the prie elastiity of the residual demand urve (qkp(qk 111 )') is less negative (i.e., less inelasti) when α > 0 (holding FTRs) than when α = 0 (without FTRs).

29 29 Assume that a Nash equilibrium haraterized by (q 1 (α),q 2 (α)), with q 1 (α)>0 and q 2 (α)>0, is ahieved and that a P/A Nash equilibrium is still supported when making an inremental transmission investment. The profit of the heapgen at the equilibrium is: π 1 *(q 1 (α),k) = q 1 (α) P 1 (q 1 (α) K) C 1 (q 1 (α)) + α K [ P 2 (q 2 (α)+k) P 1 (q 1 (α) K) ] (14) By using the envelope theorem, we obtain: d ( π (q ( α), K) ) 1 * 1 d K = q 1 (α) P 1 (q 1 (α) K)+ α [P 2 (q 2 (α)+k) P 1 (q 1 (α) K)]+α K P 1 (q 1 (α) K), or equivalently: ( (q ( α), K) ) d π = [q 1 (α ) α K] P 1 (q 1 (α) K) + α [P 2 (q 2 (α)+k) P 1 (q 1 (α) K)], (15) 1 * 1 d K Sine q 1 (α) > K > α K in the P/A Nash equilibrium (beause the heapgen is exporting power and the line is ongested) and the inverse demand funtions are ontinuous and downward sloping, the first term of the right-hand side of (15) is positive. The seond term is also positive beause the equilibrium prie at node 2 must be greater than the equilibrium prie at node 1 in order to have power flowing from node 1 to node 2 in the P/A equilibrium (otherwise, if P 2 (q 2 (α)+k) < P 1 (q 1 (α) K)), it obviously would be more profitable for the deargen to at more aggressively than just produing the passive response of the P/A equilibrium). Consequently, from (15), we get that: d ( π (q ( α), K) ) 1 * 1 d K > 0. That is, the equilibrium heapgen s profit inreases as the transmission apaity inreases, as long as a P/A Nash equilibrium is still supported. Consequently, the heapgen has positive inentives to support an inrease in the transmission apaity up to any level so that a P/A Nash equilibrium is still supported. Now, we are interested in studying the behavior of the heapgen s inentives for supporting a line expansion (as disussed in propositions 1 and 3) when the heapgen hanges its share of FTRs. In order to do this, we previously need to analyze the behavior of both the optimal

30 30 heapgen s output and the optimal heapgen s profit with respet to hanges in the heapgen s share of FTRs. The optimal heapgen s output, q 1 *(α), is inreasing ontinuously in α, from q 1 *(0) (benhmark ase) to q 1 *(1). This monotoniity is based on the rationale that, the more generation firms internalize the ongestion rents, the higher the ongestion rents are due to the firms ability to influene nodal pries. As the fration of FTRs that the heapgen holds inreases, the heapgen is more likely to sarifie some profits it would otherwise earn from supplying energy in order to inrease the profits it reeives in the form of dividends on the FTRs it holds. Aordingly, while the P/A Nash equilibrium is supported, the larger α, the stronger the heapgen s inentive to inrease its prodution (and, in this way, derease the prie at node 1, for the benefit of the onsumers loated at node 1) in order to raise its equilibrium profit. Consequently, the equilibrium heapgen s profit is inreasing in α. These results are summarized in lemma 3. Lemma 3: In the two-node network desribed in this setion, assume that a passive/aggressive Nash equilibrium is supported. Suppose also that the heapgen holds fration α of the K FTRs available from node 1 to node 2 (α [0,1]). Then, the hange in the equilibrium heapgen s output due to an inrease in the heapgen s share of FTRs is positive and smaller than the produt between the transmission apaity and the inrease in the heapgen s share of FTRs (i.e., 0 < dqd 1 * α < K). Moreover, the hange in the equilibrium heapgen s profit due to an inrease in the heapgen s share of FTRs is positive (i.e., ddπα 1 * > 0). Proof. Assume that the Nash equilibrium haraterized by (q 1, q 2 ), with q 1 > 0 and q 2 > 0, is ahieved. Sine generation firms an hold transmission rights, the profit of the heapgen at the equilibrium is given by (14). Hene, the first order optimality ondition is: ddq π 11 *= 0, or equivalently: P 1 (q 1 K) + q 1 P 1 (q 1 K) C 1 (q 1 ) α K P 1 (q 1 K) = 0. Then, 2 ( παα 111 *(), = 0 ) d(qk)ddq, or:

31 31 dqdqdqdq 1111 **** ( ) + ( ) ( ) ( ) PqKPqKqPqKCq * + * * * *... dddd αααα or equivalently: 1... * * 0 = KPqKKPqK dq1 ( ) α ( ) 1111 KPqK 11 * ( ) ( ) ( α ) ( ) ( ) * dα dq * = (16) dα 2 * * * * + PqKqKPqKCq Sine q 1 * > K > α K in the P/A Nash equilibrium, osts funtions are onvex, and the inverse demand funtions are ontinuous, onave, and downward sloping, every term of both the numerator and the denominator of the right-hand side of (16) is negative. Thus, positive. Furthermore, sine 2 * * * * PqKqKPqKCq ( ) ( α ) ( ) ( ) dqd 1 * α is + > PqK 11 ( ) *, we have that dqd 1 * α < K. Moreover, by using (14) and the envelope theorem, we get that: *(), ( πα11 ) d(qk) d α = K [P 2 (q 2 (α)+k) P 1 (q 1 (α) K)] (17) The right-hand side of (17) is positive beause the equilibrium prie at node 2 must be greater than the equilibrium prie at node 1 in order to have power flowing from node 1 to node 2 in the P/A equilibrium. Consequently, ddπα > 0. 1 * Now, we use lemma 3 to prove proposition 4, whih establishes that, while a passive/aggressive Nash equilibrium prevails, the more FTRs the heapgen holds, the more inentive it has to support an inremental transmission expansion. Proposition 4: In the two-node network desribed in this setion, assume that a passive/aggressive Nash equilibrium is ahieved and that a passive/aggressive Nash equilibriu m is still supported when making an inremental transmission investment. Moreover, assume

32 32 generation firms an hold FTRs. If the transmission apaity is suffiiently s ma ll, then the hange in the equilibrium heapgen s profit due to an inremental transmission expansion is inreasing in the fration of FTRs that the heapgen holds (i.e., d π * d is in reasing in α). 1 K Proof. Ass ume that generation firms an hold FTRs. Suppose that the heapgen and the deargen hold frations α and (1 α) of the K FTRs available from node 1 to node 2 (α [0,1]), respetively. Assume that a Nash equilibrium haraterized by (q 1 (α), q 2 (α)), with q 1 (α) > 0 and q 2 (α) > 0, is ahieved and that a P/A Nash equilibriu m is still supported when making an inremental transmission investment. Using (15) to take derivative of the funtion d π * d with respet to α, we obtain: ( πα11 ) *(), d d(qk) dqdq 11** * * * +... = KPqKqKPqK ddkdd ααα 1 K ( ) α ( ) dq1 ( + ) ( ) α ( )... +** * PqKPqKPqK * dα (18) From lemma 3, we know that dqd 1 * α is positive. Thus, onsidering that (i) q 1 * > α K in the P/A Nash equilibrium, (ii) the inverse demand funtions are ontinuous, onave, and downward sloping, and (iii) the equilibrium prie at node 2 must be greater than the equilibrium prie at node 1 in order to have power flowing from node 1 to node 2 in the P/A equilib riu m, we onlude that all terms of the right-hand side of (18) other than the first one are positive. Unfortunately, the first term of the right-hand side of (18) is negative beause dqd 1 * α < K, as we proved in lemma 3. Aordingly, the derivative of the funtion d π * d with respet to α, 1 K will be positive if the absolute value of the first term of the right-hand side of (18) is smaller than the sum of the other terms, whih is likely to happen. A suffiient ondition for this is that the

33 33 transmission apaity, K, is suffi iently small so that the right-hand side of (18) is positive, whih implies that ddkπ * is inreasing in α. 1 The previous propositions ass ume that FTRs are alloated free o f harge diretly to the generation firms. If generation firms must aquire their FTRs through some type of aution, the autioneer ould sell the FTRs reated by a transmission expansion to the heapgen up to a prie suh that the extra expenditure inurred to aquire the FTRs equals the differene in the heapgen s profit between before and after the expansion. In suh a ase, and assuming that an inrease in the transmission apaity would inrease both the total onsumer surplus and the soial welfare (Sheffrin, 2005), it would be possible to leave the heapgen revenue neutral and, at the same time, improve both onsumer surplus and soial welfare. This would mean that we ould use this type of inentive as an instrument to indue inremental transmission expansions that are soial-welfare improving. Proposition 5 summarizes this result. Proposition 5: In the two-node network desribed in this setion, assume that a passive/aggressive Nash equilibriu m is ahieved and that a passive/aggressive Nash equilibriu m is still supported when making an inremental transmission investment. Ass ume also that generation firms an hold FTRs. Moreover, assume that an inrease in the transmission apaity would inrease both onsumer surplus and soial welfare. If all FTRs were autioned off to the net exporter generation firm, then it is possible to inrease both onsumer surplus and soial welfare while keeping the net exporter generation firm revenue neutral. Proof. Assume generation firms an hold FTRs, wh ih must be aquired through some type of aution. Suppose that an inremental transmission expansion is desired in the desribed twonode network beause it inreases both onsumer surplus and soial welfare, as it is more likely to happen in a ongested radial network aording to the gains from trade eonomi priniple (Sheffrin, 2005). Then, an autioneer ould sell the FTRs reated by the transmission expansion to the heapgen for a prie suh that the extra expenditure inurred to aquire the FTRs equals the

34 34 differene in the heapgen s profit between before and after the expansion (proposition 3 ensures that the heapgen s profit inreases within this expansion). Then, proposition 5 is true by onstrution, whih implies that this type of inentives an be used as an instrument to indue desired inremental transmission expansions, leaving the net exporter generation firm revenue neutral. On the other hand, while a P/A Nash equilibriu m is still supported, the deargen maximizes its profit as if it had monopoly power over its K-leftward-shifted inverse demand funtion, but having now also two revenues streams: a first stream of revenue fro m energy sales and a seond revenue stream from the ongestion rents. As the fration of FTRs that the deargen holds inreases, the deargen is more likely to sarifie some profits it would otherwise earn from supplying energy in order to inrease the profits it reeives in the form of dividends on the FTRs it holds. Aordingly, while the P/A Nash equilibrium prevails, the smalle r α, the stronger the deargen s inentives to derease its prodution and, in this way, inrease the prie at node 2. Consequently, while the P/A Nash equilib riu m prevails, the deargen effetively redues the prie elastiity of its residual demand urve and inreases its loal market power by holding FTRs. Proposition 6 states a similar result as in proposition 2 in the ase that generation firms an hold FTRs. In this ase, the deargen s inentives to support an inrease in the transmission apaity are unertain. Proposition 6: Assume generation firms an hold FTRs. In the two-node network desribed in this setion, while a passive/aggressive Nash equilibrium prevails, the inentives that the net importer generation firm (i.e., the deargen) has to support an inrease in the transmission apaity are ambiguous. Proof. Assume generation firms an hold FTRs. Suppose that the heapgen and the deargen hold frations α and (1 α) of the K FTRs available from node 1 to node 2 (α [0,1]), respetively.

35 35 Assume that a Nash equilibrium haraterized by (q 1 (α),q 2 (α)), with q 1 (α)>0 and q 2 (α)>0, is ahieved and that a P/A Nash equilibrium is still supported when making an inremental transmission investment. The profit of the deargen at the equilibriu m is: π 2 *(q 2 (α),k) = q 2 (α) P 2 (q 2 (α)+k) C 2 (q 2 (α)) + (1 α) K [P 2 (q 2 (α)+k) P 1 (q 1 (α) K)] (19) By using the envelope theorem, we obtain: d ( π *(q 2 ( α), K) ) 2 d K = q 2 (α) P 2 (q 2 (α)+k) + (1 α) [P 2 (q 2 (α)+k) P 1 (q 1 (α) K)] + + (1 α) K P 2 (q 2 (α)+k), or equivalently: *(), ( πα22 ) d(qk) dk = [q 2 (α)+(1 α) K] P 2 (q 2 (α)+k) + (1 α) [P 2 (q 2 (α)+k) P 1 (q 1 (α) K)] (20) Sine q 2 (α) > 0 and the inverse demand funtions are ontinuous and downward sloping, the first term of the right-hand side of (20) is negative. The seond term is positive beause the equilibrium prie at node 2 must be greater than the equilibrium prie at node 1 in order to have power flowing from node 1 to node 2 in the P/A equilibriu m. Consequently, aording to (20), we annot guarantee the sign of d ( π *(q 2 ( α), K) ) 2 d K. This sign will be negative if the energysales revenue stream is stronger than the revenue stream from the ongestion rents and positive in the opposite ase. Thus, while a P/A Nash equilibrium prevails, the inentive that the deargen has to support an inrease in the transmission apaity is amb iguous. Additionally, as we did in the ase of the heapgen, we an use (20) to argue about the monotoniity of ddkπ * 2 with respet to α. This result is summarized in proposition 7. Proposition 7: In the two-node network desribed in this setion, assume that a passive/aggressive Nash equilibrium is ahieved and that a passive/aggressive Nash equilibriu m is still supported when making an inremental transmission investment. Moreover, assume generation firms an hold FTRs. If the transmission apaity is suffiiently small, then the hange

36 36 in the equilibrium deargen s profit due to an inremental transmission expansion is dereasing in the fration of FTRs that the heapgen holds (i.e., ddk π 2 * is dereasing in α). Proof. Assume generation firms an hold FTRs. Suppose that the heapgen and the deargen hold frations α and (1 α) of the K FTRs available from node 1 to node 2 (α [0,1]), respetively. Assume that a Nash equilibrium haraterized by (q 1 (α), q 2 (α)), with q 1 (α) > 0 and q 2 (α) > 0, is ahieved and that a P/A Nash equilibriu m is still supported when making an inremental transmission investment. Using (20) to take derivative of the funtion ( πα22 ) *(), d d(qk) dqdq 22** **1 *+... = KPqKqKPqK ddkdd ααα ddk π 2 * with respet to α, we obtain: ( ) ( α ) ( ) dq2 ( ) ( ) ( α ) ( )... ** 1 * PqKPqKPqK * dα (21) In the same way o f lemma 3, it is easy to prove that 0 < dqd 2 * α < K. Thus, onsidering that (i) q 2 * > 0, (ii) the inverse demand funtions are ontinuous, onave, and downward sloping, and (iii) the equilibrium prie at node 2 must be greater than the equilibrium prie at node 1 in order to have power flowing from node 1 to node 2 in the P/A equilibrium, we onlude that all terms of the right-hand side of (21) other than the first one are negative. The first term of the righthand side of (21) is positive beause dqd 2 * α < K. Aordingly, the derivative of the funtion ddk π 2 * with respet to α, will be negative if the first term of the right-hand side of (21) is smaller than the absolute value of the sum of the other terms, whih is likely to happen. A suffiient ondition for this is that the transmission apaity, K, is suffiiently sma ll so that the right-hand side of (21) is negative, wh ih implies that ddk π 2 * is dereasing in α.

37 37 Proposition 6 says that we annot guarantee that the deargen s profit inreases when an inremental soial-welfare-improving transmission expansion ours and, thus, we annot guarantee that the deargen has the orret inentives to support suh an expansion. Furthermore, proposition 7 tells us that, even if the deargen has the right inentives to support an inremental soial-welfare-improving transmission expansion, those inentives would like ly derease as more FTRs are alloated to the heapgen (i.e., as α inreases). Therefore, although alloating FTRs to the net exporter generation firm an inrease its inentives to support a soial-welfare-improving transmission expansion, suh a poliy would dilute the net-importer-generation-firm s inentives to support the apaity expansion. Consequently, a soially onerned regulator who wants to align the inentives for transmission expansion of the soiety and of the net exporter firm must be aware that alloating FTRs to the net exporter firm would also inrease the opposition of the net importer generation firm to support the expansion. Finally, we like to reiterate, that the analysis in this-setion is only valid for suffiiently small transmission upgrades suh that the transmission line apaity does not exeed K. However, the value of K inreases as α inreases. Thus, under this seond senario, both generation firms will support a passive/aggressive Nash equilibrium up to a line apaity that not only exeeds the benhmark ase threshold, but is even larger as more FTRs are alloated to the heapgen. 4. Numerial Example In this setion, we use the same numeria l e xa mp le employed in setion 2 to illustrate the previous-setion findings about the inentives that generation firms have to support inremental soial-welfare-improving transmission expansions under both senarios: with and without FTRs. This is, under both senarios, we assume that the inverse demand funtions are given by P 1 (q) = q at node 1 and P 2 (q) = q at node 2 (in $/MWh) and that the marginal

38 38 osts of generation are zero for the heapgen and $20/MWh for the deargen. We also as sume now that there is a transmission line onneting both nodes. 4.1 Senario I: generation firms annot hold transmission rights If the apaity of the line linking both nodes were very high, then the transmission apaity onstraint would not be binding and the firms would ompete as Cournot duopolists in the ombined market. In suh a ase, at the unique pure-strategy Nash equilibriu m, the heapgen would hourly produe 633 MWh while the deargen would hourly generate 333 MWh and the ma rket- learing prie would be $42.2/MWh at both nodes. The smallest transmission apaity that an support an unonstrained Nash-Cournot duopoly equilibriu m, K*, is approximately equal to 115 MW in this nume ria l e xa mp le. 21 With K = K*, the deargen is indifferent between produing its unonstrained Nash-Cournot equilibriu m hourly output (i.e., 333 MWh) and produing its optimal passive response (i.e., 193 MWh), given that the heapgen is produing 633 MWh (i.e., its unonstrained Nash-Cournot equilibrium hourly output). At any larger K, eah generation firm would stritly prefer the unonstrained Nash-Cournot duopoly equilibrium outome to its optimal passive output response when the other firm produes its unonstrained Nash-Cournot equilibrium quantity. 21 We omputed K* as follows. The deargen s profit, when a line of apaity K is ongested into its market, is given by π 2 (q 2 ) = q 2 P 2 (q 2 +K) C 2 (q 2 ) = q 2 [ (q 2 +K)] 20 q 2 = ( K) q (q 2 ) 2, and the first order optimality ondition of the deargen s profit maximization problem implies that q 2 * = 2.5 ( K), where q 2 * is the deargen s optimal passive output. Thus, the deargen s profit from produing its optimal passive output is: π 2 (q 2 *) = ( K) q 2 * 0.2 ( q 2 *) 2 = 0.05 (500 K) 2. Consequently, the line apaity that makes the deargen indifferent between produing its unonstrained Nash-Cournot duopoly equilibrium output, q 2 UCDE, and produing its optima l passive output, q 2 *, given that the heapgen is produing its unonstrained Nash-Cournot duopoly equilibrium output, must satisfy the ondition π 2 (q 2 U CDE ) = π 2 (q 2 *), or equivalently, 7,407 = 0.05 (500 K*) 2. Thus, K* = 500 (,407 / 0.05 ) MW.

39 39 For a transmission line of apaity slightly less than K*, K = 110 MW for instane, the unonstrained Nash-Cournot equilibriu m is not attainable; the deargen would (just barely) prefer to produe the optimal passive output than play its Cournot best response to the heapgen produing its Nash-Cournot equilibriu m quantity. But if the deargen produed its optimal passive output (i.e., 195 MWh), then the heapgen would revert to sell its profit-maximizing quantity that ongests the transmission line (i.e., 555 MWh). This amount is smaller than the heapgen s Nash- Cournot equilib riu m quantity (i.e., 633 MWh). As the heapgen redues its output, produing its optimal passive output beomes less attrative to the deargen. If that were the ase, then the deargen would jump to produe its Cournot best response to 555 MWh, whih is 373 MWh. With the line unongested, however, the heapgen would then respond with its Cournot best response of 614 MWh, and the proess would one again iterate toward the unonstrained Nash-Cournot equilibrium. However, beause the line apaity is just slightly below the level that an support the Nash-Cournot equilibrium, as the heapgen s output approahes its Nash-Cournot equilib riu m quantity (i.e., 633 MWh), and stritly before it equals that quantity, the deargen will one again revert to produe its optimal passive output. Consequently, no pure-strategy Nash equilib riu m exists in this ase. This situation will our for any line apaity between K and K*. The largest line apaity that an support a P/A Nash equilibriu m, K, is approximately equal to 53.6 MW in this numerial e xa mple. 22 With K = K, the deargen is indifferent between 22 To ompute K, we proeed as follows. The heapgen s profit, when a line of apaity K is ongested from its market, is given by π 1 (q 1 ) = q 1 P 1 (q 1 K) C 1 (q 1 ) = q 1 [ (q 1 K)] 0 = ( K) q (q 1 ) 2, and the first order optimality ondition of the heapgen s profit maximization problem implies that q 1 * = 5 ( K), where q 1 * is the heapgen s optimal aggressive output. Thus, the deargen s Cournot best response to q (BR 1 * is a quantity q ) 2 satisfying: (BR q ) 2 = Argmax {q 2} π 2(q 2 ), where π 2(q 2 ) = q 2 P(q 1 * + q 2 ) C 2 (q 2 ) = = q 2 [ (q 1 * + q 2 )] 20 q 2 = = q 2 [ (5 ( K) + q 2 )] 20 q 2 = = ( K) q (q 2 ) 2.

40 40 produing its Cournot best response to the heapgen s aggressive output and produing its optimal passive output. At any smaller K, eah generation firm would stritly prefer the P/A Nash equilibrium outome to its Cournot best response when the other firm produes its P/A Nash equilibrium quantity. Summarizing, for a line of apaity smaller than 53.6 MW (i.e., for K suh that 0 < K < K ), the P/A Nash equilibrium haraterized by q 1 = 5 ( K) and q 2 = 2.5 ( K) exists and is the unique pure-strategy Nash equilibriu m; for a line o f apaity between 53.6 MW and 115 MW (i.e., K < K < K*), no pure-strategy Nash equilibrium exists; and for a line of apaity higher than 115 MW (i.e., K* < K), the unonstrained Nash-Cournot equilibrium haraterized by q UCDE 1 = 633 MWh and q UCDE 2 = 333 MWh is the unique pure-strategy Nash equilibriu m. Now, suppose that the apaity of the transmission line onneting the heapgen and the deargen is urrently 50 MW. With this transmission apaity, the resulting equilibriu m will be the one shown in the first olumn of Table 2. Table 2. Equilibria in the two-node network, without onsidering FTRs a Equilibrium with K = 50 MW Equilibrium with K = 52 MW Equilibriu m with K > 115 MW q 1 = 525 MWh q 1 = 526 MWh q 1 = MWh q 2 = 225 MWh q 2 = 224 MWh q 2 = MWh P 1 = $52.5/MWh P 1 = $52.6/MWh P 1 = $42.2/MWh P 2 = $65/MWh P 2 = $64.8/MWh P 2 = $42.2/MWh (B The first-order optimality ondition implies that q R) 2 = 0.25 (1600 K). Thus, the deargen s profit from produing the Cournot best response to q 1 * is: π 2 (q (BR) (B 2 ) = ( K) q R) ( BR (q ) 2 2 = (1600 K) 2 / 240. Consequently, the line apaity that leaves the deargen indifferent between produing its (BR) Cournot best response to the heapgen s aggressive output (i.e., q 2 ) and produing its optimal passive output (i.e., q (B 2 *) must satisfy π 2(q R) 2 ) = π 2 (q 2 *). Realling that the deargen s profit when produing its optimal passive response to q 1 * is π 2 (q 2 *) = 0.05 (500 K) 2, we onlude that K must satisfy the following equality: (1600 K ) 2 / 240 = 0.05 (500 K ) 2. Thus, we have 500* , K ' = MW. 12 1

41 41 π 1 = $27,563/h π 1 = $27,668/h π 1 = $26,741/h π 2 = $10,125/h π 2 = $10,035/h π 2 = $7,407/h CS 1 = $11,281/h CS 1 = $11,234/h CS 1 = $16,691/h CS 2 = $7,563/h CS 2 = $7,618/h CS 2 = $15,123/h W = $56,531/h W = $56,554/h W = $65,963/h a CS i denotes to the onsumer surplus at node i and W denotes the total soial we lfa re (not aounting for transmission investment osts). If the apaity of the transmission line were inreased by a large-enough amount suh that it beame greater than K*, then the transmission apaity onstraint would not be binding and the firms would ompete as Cournot duopolists in the ombined market. As result of that, the heapgen would earn a profit of $26,741/h and the deargen would earn a profit of $7,407/h, wh ih would result in a redution in profits for both generation firms as ompared to the pre-e xpansion situation. Consequently, neither the heapgen nor the deargen have inentive to support suh an investment, although it may imp rove soial welfare (fro m $56,531/h to $65,963/h, without onsidering any investment ost). On the other hand, if the thermal apaity of the transmission line were slightly inreased fro m 50 MW to 52 MW (note that 52 MW < K ), then the resulting equilibrium would be the one shown in the seond olumn of Table 2. Comparing the results obtained when K = 50 MW and when K = 52 MW, we verify that, as the transmission apaity inreases from 50 MW to 52 MW: (i) the heapgen inreases its output at the equilibrium (in agreement with lemma 1), (ii) the equilibrium prie at node 1 inreases, (iii) the heapgen s profit inreases (whih onfirms the heapgen s inentives to support this transmission expansion), (iv) the deargen redues its output at the equilibriu m(in agreement with lemma 2), (v) the equilibrium prie at node 2 dereases, (vi) the deargen s profit dereases (whih onfirms the deargen s disinentives to support this transmission expansion), and (vii) soial welfare inreases. Consequently, these results verify that, while a P/A Nash equilib riu m prevails, the heapgen has inentives to support an inrease in the apaity of the transmission line while the deargen has disinentives to support suh an expansion.

42 42 As mentioned before, this onlusion is only valid for upgrades that inrease the apaity of the line up to K. 4.2 Senario II: generation firms an hold FTRs Now, we assume that all FTRs are alloated free of harge diretly to the generation firms. For illustrative purposes, suppose that the heapgen holds 80% of the available FTRs and the deargen holds the remaining 20% (i.e., α = 0.8). In this ase, Table 3 presents the resulting equilibria when the transmission apaity is 50 MW and when it is 52 MW. Table 3. Equilibria in the two-node network, when α = 0.8 a Equilibrium with K = 50 MW Equilibrium with K = 52 MW q 1 = 545 MWh q 1 = MWh q 2 = 220 MWh q 2 = MWh P 1 = $50.5/MWh P 1 = $50.5/MWh P 2 = $66/MWh P 2 = $65.8/MWh π 1 = $28,143/h π 1 = $28,262/h π 2 = $10,275/h π 2 = $10,189/h CS 1 = $12,251/h CS 1 = $12,241/h CS 2 = $7,290/h CS 2 = $7,333/h W = $57,959/h W = $58,025/h a CS i denotes the onsumer surplus at node i and W denotes the total soial welfare (not aounting for transmission investment osts). By omparing Table 2 and Table 3, we observe that, by holding some FTRs, both generation firms inrease their profits with respet to the benhmark ase. Furthermore, we notie that, when holding FTRs, the heapgen has inentives to inrease its prodution (and, in this way, to derease its nodal prie) while the deargen has inentives to derease its prodution (and, in this way, to inrease its nodal prie) in order to inrease their revenues from ongestion rents, as we predited in the previous setion.

43 43 As in the benhmark ase, by omparing the two olumns of Table 3, we observe that the heapgen has positive inentives to support an inrease from 50 MW to 52 MW in the transmission apaity while the deargen has disinentives to support suh an expansion. Moreover, by omparing Table 2 and Table 3, we note that the hange in the equilibrium heapgen s profit due to the inremental transmission expansion is greater in the ase where the heapgen an hold FTRs (and, in fat, it is inreasing in α, as shown in Figure 4). This result suggests that, while the P/A Nash equilibrium prevails, it would be more likely that the heapgen supports an inremental soial-welfare-improving transmission expansion when it holds FTRs than when it does not hold FTRs. By varying the values of α, it is straightforward to verify both that the larger α, the stronger the heapgen s inentive to inrease its prodution (and, in this way, to derease its nodal prie). Furthermore, the larger α, the weaker the deargen s inentive to redue its prodution (and, in this way, to raise its nodal prie). Aordingly, when the heapgen holds all the available FTRs, the onsumers loated at node 1 benefit the most from the nodal prie redution while the surplus of the onsumers loated at node 2 remains at the benhmark s level (beause the deargen has no extra inentive to redue its prodution and, thus, inrease its nodal prie when α = 1). Consequently, the value of α that maximizes both onsumer surplus and soial welfare is α = 1, as it is evident in Figure When α = 1 and K = 50 MW, we obtain a Nash equilibrium haraterized by: q 1 = 550 MWh, q 2 = 225 MWh, P 1 = $50/MWh, and P 2 = $65/MWh. In this ase, soial welfare is W = PS + CS = π 1 + π 2 + CS 1 + CS 2 = $28,250/h + $10,125/h + $12,500/h + $7,563/h = $58,438/h (without onsidering any investment ost). This soial welfare represents an inrease of 3.4% with respet to the ase without FTRs.

44 44 Figure 4. Evolution of equilibrium quantities as α inreases. π 1* ($/h) α CS* ($/h) 20,000 19,500 19,000 18,500 18,000 17,500 17, α π 2* ($/h) α W* ($/h) 59,000 58,500 58,000 57,500 57,000 56,500 56,000 55, α Figure 4 shows the evolution of several equilibrium quantities, as α inreases, when K = 50 MW. In Figure 4, π * 1 orresponds to the hange in the equilibrium heapgen s profit due to an inremental transmission expansion from 50MW to 52 MW; π * 2 is the hange in the equilib riu m deargen s profit due to an inremental transmission expansion from 50 MW to 52 MW; CS * is the equilibrium total onsumer surplus (K = 50 MW); and W * represents the equilibrium soial welfare (K = 50 MW). In this figure, we verify both that π * 1 is inreasing in α, as proposition 4 states, and that π * 2 is dereasing in α, as stated in proposition 7. Using a proedure similar to the one followed in the benhmark ase, we an ompute the largest line apaity that an support a P/A Nash equilibrium, K, for different values of α. This is illustrated in Figure 5. As Figure 5 suggests, K inreases as α inreases. For instane, with α = 0.8, we obtain K = 90 MW and, with α = 0.5, we obtain K = 88 MW. Consequently, as more FTRs are alloated to the heapgen, both generation firms will support a P/A Nash equilibrium up to a larger transmission line apaity.

45 45 Figure 5. Evolution of K as α inreases. K (MW) α 5. Conlusions In this paper, we analyzed how the exerise of loal market power by generation firms alters the firms inentives to support inremental soial-welfare-improving transmission investments in the ontext of a two-node network. We explored how suh inentives are affeted by the ownership struture of FTRs and how the FTRs alloation may be used to align the inentives for transmission expansion of the soiety and of the net exporter generation firms. Our analysis showed that, in the two-node network desribed, the net exporter generation firm (i.e., the heapgen) has positive inentives to support an inrease in the transmission apaity up to any level so that a passive/aggressive Nash equilibrium is still supported. We also proved that the hange in the equilibrium heapgen s profit due to an inremental transmission expansion will like ly be inreasing in the amount of FTRs that are alloated to the heapgen. Moreover, we showed that, if all FTRs were alloated or autioned off to the net exporter generation firm, then it is possible to inrease both onsumer surplus and soial welfare while keeping the net exporter generation firm revenue neutral Our onlusions are based on the stati model proposed in this artile. However, we reognize that transmission investments usually affet mult iple time periods, in whih generators have different ost strutures and fae different demands. If a firm s inentives hange from hour to