Restructuring the Electricity Industry: Vertical Structure and the Risk of Rent Extraction

Boom, Anette; Buehler, Stefan Working Paper Restructuring the Electricity Industry: Vertical Structure and the Risk of Rent Extraction Working paper, No. 2-2014 Provided in Cooperation with: Department of Economics, Copenhagen Business School (CBS) Suggested Citation: Boom, Anette; Buehler, Stefan (2014) : Restructuring the Electricity Industry: Vertical Structure and the Risk of Rent Extraction, Working paper, No. 2-2014, Copenhagen Business School (CBS), Department of Economics, Frederiksberg, https://hdl.handle.net/10398/9031 This Version is available at: https://hdl.handle.net/10419/208576 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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DK-2000 Frederiksberg Restructuring the Electricity Industry: Vertical Structure and the Risk of Rent Extraction Anette Boom and Stefan Buehler  Restructuring the Electricity Industry: Vertical Structure and the Risk of Rent Extraction Anette Boom∗ Copenhagen Business School Stefan Buehler† University of St. Gallen March 14, 2014 Abstract We study the role of vertical structure in determining generating capacities and retail prices in the electricity industry. Allowing for uncertain demand, we compare three market configurations: (i) integrated monopoly, (ii) integrated duopoly with wholesale trade, and (iii) separated duopoly with wholesale trade. We find that equilibrium capacities and retail prices are such that welfare is highest (lowest) under separated (integrated) duopoly. The driving force behind this result is the risk of rent extraction faced by competing integrated generators on the wholesale market. Our analysis suggests that vertical structure plays an important role in determining generating capacities and retail prices. Keywords: Electricity, Investments, Generating Capacities, Vertical Integration, Monopoly and Competition. JEL-Classification: D42, D43, D44, L11, L12, L13 ∗Corresponding author: Copenhagen Business School, Department of Economics, Porcelænshaven 16 A, DK-2000 Frederiksberg, e-mail: [email protected] †University of St. Gallen, Department of Economics, FGN-HSG, Varnbüelstr. 19, CH-9000 St. Gallen, e-mail: [email protected]h 1 Introduction Electricity markets around the world have been restructured in an effort to improve their performance. In several countries, legislators have allowed competition into statutory integrated monopoly and implemented regulations such as vertical unbundling to safeguard entrants and consumers from potentially harmful strategic behavior by integrated generators.1Yet, no consensus seems to have emerged as to which market configuration works best. A particular concern is that allowing competition into electricity markets might undermine investments in generating capacity (see, e.g., Joskow (2006), and Joskow and Tirole (2006)). In this paper, we study the role of vertical market structure in determining investments in electricity generating capacity, retail prices, and welfare. We compare three market configurations that vary with respect to vertical market structure and the extent to which firms compete at the wholesale and the retail level: (i) integrated monopoly, (ii) integrated duopoly with wholesale trade, and (iii) separated duopoly with wholesale trade. Throughout the analysis, we allow for uncertain demand at the retail level. A key feature of our analysis is that both capacity decisions and retail prices are determined before the state of retail demand is known. After retail demand is realized, the wholesale price for electricity is determined in a uniform-price auction due to von der Fehr and Harbord (1997) and (1993), and deliveries and payments are exchanged. This setting implies that wholesale prices react to changes in retail prices, whereas retail prices cannot react to changes in wholesale prices. The timing reflects a peculiarity of the electricity industry: capacity decisions are made under uncertainty about future demand, just as retail delivery contracts are signed before the state of demand is realized. The wholesale market then attempts to balance supply and demand based on available capacity and effective retail demand. Our main analysis will assume that a blackout occurs if the balancing act in the wholesale market fails. In an extension, we will also consider the case where 1In the UK, for example, the industry was vertically separated into three generating firms, the National Grid company, and 12 regional distribution companies by the Electricity Act in 1989. However, some regional distribution companies later re-integrated vertically into generation (Newbery, 1999, 2005). The Californian restructuring bill from 1996 also forced the regulated utilities to sell lots of their generation facilities (Borenstein, 2002). The European Union ruled in its Directive 2003/54/EC concerning common rules for the internal market in electricity adopted on 26 June 2003 that electricity generating firms which are integrated into transmission and distribution have to be functionally disintegrated. 1 blackouts may be avoided by the rationing of retail demand.2 Our main results are the following. First, aggregate generating capacity is highest under integrated duopoly and lowest under integrated monopoly. The separated duopoly yields an intermediate level of generating capacity. The driving force behind this result is the rent extraction risk faced by an integrated duopoly generator: if individual capacity turns out to be too small to serve own retail demand, an integrated generator must buy electricity from its competitor in the wholesale market, thereby fully dissipating its rent. To avoid this outcome, an integrated duopoly generator will not only choose a large generating capacity, but also set a high retail price. Vertical separation eliminates this risk of rent extraction, as electricity generators are not committed to serve an uncertain demand at a pre-determined retail price. As a result, retail prices are lower and demand is higher than under vertical integration. In effect, vertical separation reduces the investment-enhancing effect of introducing competition into statutory monopoly, but does not fully eliminate it. Second, equilibrium retail prices are lowest under separated duopoly and highest under integrated duopoly. The integrated monopoly yields an intermediate level of retail prices. Intuitively, the result follows again from the risk of rent extraction, which induces an integrated generator to charge a higher retail price. This result supports the notion that allowing competition into the electricity industry (alone) does not necessarily reduce retail prices. Third, the combined effects of restructuring on investments in generating capacities and retail prices are such that social welfare is highest under separated duopoly and lowest under integrated duopoly. The integrated monopoly yields an intermediate level of social welfare. To understand the intuition for this result, note that irrespective of market configuration capacity is always large enough to satisfy retail demand at the relevant retail price (i.e., blackouts do not occur in equilibrium). This implies that low capacity investments do not have an adverse effect on welfare per se. Increasing capacity investments has two effects. (i) it raises generation costs without improving supply security; (ii) it increases the maximum level of retail demand that can be served without causing blackouts. To benefit from higher capacities, retail prices must decrease, which requires a restructuring from integrated monopoly to separated duopoly. Therefore, the welfare ranking is essentially a reversed ranking of the price levels under the various industry configurations. Allowing for rationing of the retail demand does not undermine our analy- 2See section 5.1. 2 sis. The equilibrium outcome for the integrated duopoly remains unaffected, as the risk of rent extraction induces firms to choose capacities which make rationing unnecessary. In the other market configuration, equilibrium capacities and retail prices are no longer distorted upwards to avoid blackouts. Instead, some rationing occurs for very high demand realization, leading to a lower supply security compared to the integrated duopoly. Yet, this higher supply security affects the welfare ranking of the integrated duopoly configuration only for very low capacity costs. Our paper contributes to the extensive literature on the impact of demand uncertainty on capacity choices (Drèze and Sheshinski (1976), Gabszewicz and Poddar (1997), von der Fehr and Harbord (1997), Castro-Rodriguez et al. (2009), Boom (2002) and (2009), Borenstein and Holland (2005)), Murphy and Smeers (2005), and Grimm and Zoettl (2013)). The key difference to this literature is that we focus on the role of vertical market structure in determining capacity choices. We also contribute to the much scarcer literature on the competitive effects of changing the electricity industry’s vertical structure. Previous contributions to this strand of the literature focus either on the loss of vertical economies due to the separation of generation and distribution of electricity (see e.g. Kwoka et al. (2010) and Kwoka (2002)), or on the effect of forward contracts on the wholesale prices (see e.g. Bushnell (2007), Mansur (2007), Bushnell et al. (2008), de Frutos and Fabra (2012) and Bosco et al. (2012)). None of these papers studies the role of vertical structure in determining the investments in generating capacities.3We identify a new effect, the risk of rent extraction associated with vertical integration, as an important determinant of investments in generating capacities. The remainder of the paper is structured as follows. Section 2 introduces the analytical framework. Section 3 discusses the benchmark cases of social optimum, integrated monopoly, and integrated duopoly. Section 3.4 studies the equilibrium outcome under separated duopoly. Section 4 compares the various market configurations and derives the main results. Section 5 discusses some limitations and explores rationing at the retail level as an extension. Section 6 concludes. 3Some authors interpret the integration of electricity generators into the retail market as forward contracting in the tradition of Allaz and Villa (1993), abstracting from the fact that vertically integrated firms commit on retail prices rather than retail quantities. It is worth noting that their empirical observation of lower wholesale prices with vertical integration is in line with our model if firms successfully avoid rent extraction (the wholesale price is then low despite a high retail price). 3 2 Analytical Framework In this section, we outline the analytical framework for the various market configurations considered below, building on Boom (2009) and Boom (2007). 2.1 Demand Suppose that customers’ surplus is given by V(x, ε, r) = U(x, ε)−rx =x−ε−(x−ε)2 2−rx, (1) where x≥0is the amount of electricity consumed, r≥0is the retail price per unit of electricity, and εis a demand shock, uniformly distributed on the interval [0,1].4Maximizing V(x, ε, r)with respect to xyields the linear retail demand for electricity x(r, ε) = max{1 + ε−r, 0}.(2) If there is more than one retailer, consumers subscribe to the retailer offering the lowest retail price (electricity is a homogeneous good). If retail prices are identical, consumers choose each retailer with equal probability. 2.2 Supply We will compare three market configurations that differ in the number of active firms and the vertical market structure and the social optimum:5 (i) social optimum; (ii) integrated monopoly; (iii) integrated duopoly with wholesale trade: Two integrated firms can buy and sell electricity on the wholesale market and serve retail demand; (iv) separated duopoly with wholesale trade: Two separated generators sell to the wholesale market, and two separated retailers buy from the wholesale market to serve retail demand. 4This specification implies that a positive demand shock (ε > 0) is associated with a negative effect on consumer surplus. Our key results do not depend on this specification (see section 5.2 below for a discussion). 5We abstract from the chain of (vertically separated) monopolies, which does not give rise to a sensible market configuration in our setting. 4 For simplicity, we assume that the marginal cost of generating electricity is constant and normalized to zero. The total cost of electricity generator i=A, B is then given by C(ki) = zki,(3) where zis the constant unit cost of capacity and kiis the generating capacity installed by firm i.6We assume that capacity cost zsatisfies 0≤z < 1 2(4) to ensure strictly positive capacity investments in all market configurations. For simplicity, we further assume that the marginal cost of selling electricity to consumers is constant and normalized to zero 2.3 Timing The timing reflects some key features of the electricity industry. We first consider the duopoly configurations, which presume the following five stages: (1) In the first stage, generators i=A, B decide on capacities kibefore retail demand is known. In the integrated duopoly, capacity decisions are taken simultaneously. In the separated duopoly, we consider both simultaneous and sequential capacity decisions (assuming that Amoves before B). (2) In the second stage, retailers `=C, D simultaneously set retail prices r` in the separated duopoly, whereas generators i=A, B simultaneously set retail prices riin the integrated duopoly. Consumers buy from the firm with the lower retail price, or, if prices are identical, from each firm with equal probability. (3) In the third stage, the demand shock ε∈[0,1] is realized. Since retail prices are already set, retail demand is fixed henceforth. (4) In the fourth stage, generators bid prices pAand pBfor their full capacity ki, i =A, B to an auctioneer. The auctioneer determines the market clearing wholesale price p(if such a price exists) and the amount of electricity generators may supply to the grid. 6Firm indices may be ignored if there is only one generator. 5 (5) Finally, in the fifth stage, if supply and demand are balanced, deliveries and payments are exchanged. If supply and demand cannot be balanced, a blackout occurs and market exchange is interrupted. The monopoly configuration and the social welfare benchmark reflect the timing in the duopoly scenarios as closely as possible. In particular, both the social planner and the integrated monopoly must choose their capacity and retail price before retail demand is known. 2.4 Wholesale Market As the wholesale price is determined after the retail price, the timing is reversed relative to the standard literature on vertically related markets. This reversion reflects the peculiarity that retailers must specify the terms of delivery before retail demand is known and then buy electricity on behalf of their customers on the wholesale market. That is, the retail market clears in the long run, whereas the wholesale market clears in the short run. Even though the wholesale price cannot affect retail prices, it is an important determinant of investments in generating capacity, as it affects the returns on investment for electricity generators. To fix ideas, we assume that the wholesale price is determined by a uniform-price auction due to von der Fehr and Harbord (1997) and (1993).7Uniform-price auctions were used for the Electricity Pool in England and Wales before the reform in 2001, and are still in use elsewhere (e.g., for the Nord Pool in Scandinavia, or the Spanish wholesale market).8 The uniform-price auction we employ requires each firm ito bid a price piat which it is willing to supply its total capacity.9The auctioneer then attempts to balance supply and demand on the grid, arranging the bids in ascending order and determining the marginal bid which equates supply and demand.10. The price of the marginal bid is the spot market price paid to all generators 7An alternative approach, based on Klemperer and Meyer (1989), has been suggested by Green and Newbery (1992). They assume that firms bid differentiable supply functions, whereas von der Fehr and Harbord (1997) and (1993) assume that they bid step functions. 8See Bergman et al. (1999), Crampes and Fabra (2005) and Newbery (2005). 9That is, we abstract from the problem of strategic capacity withholding (see Crampes and Creti (2005), and Le Coq (2002)). 10For simplicity, we ignore transmission constraints, although they might interact with constraints in the generating capacity. See Wilson (2002) for insights into this problem and for the analysis of isolated transmission constraints Borenstein et al. (2000), Joskow and Tirole (2000) and Léautier (2001) 6 Note that a separated duopoly generator does not face the risk of rent extraction, since it is not committed to serve any specific level of retail demand. Best-response bidding now requires each generator to either undercut its competitor, or to bid the maximum price pi=rsd at which retailers break even. The next proposition characterizes the resulting Nash equilibria in price bids. Proposition 5 (wholesale prices) Depending on capacity levels (kA, kB) and the retail price rsd, there are the following types of Nash equilibria in price bids: (i) If kA+kB< x(rsd, ε), any pair (pA, pB)forms a Nash equilibrium in price bids. No wholesale price can equate supply and demand, and a blackout occurs. (ii) If ki≥x(rsd, ε)> kj, with i, j =A, B and i6=j, the Nash equilibrium in pure strategies is characterised by pi=rsd and pj< rsd(x(rsd, ε)− kj)/ki. The resulting equilibrium wholesale price is psd =rsd, and firms sell the quantities yi=x(rsd, ε)−kjand yj=kj. (iii) If kA+kB≥x(rsd, ε)>max{kA, kB}, there are two types of Nash equilibria in pure strategies: one with pA=rsd and pB< rsd(x(rsd, ε)− kB)/kA, and another with pB=rsd and pA< rsd(x(rsd, ε)−kA)/kB. The wholesale price is the same (psd =rsd)for both types of equilibria, but the quantities sold in equilibrium differ: in the former yA= x(rsd, ε)−kBand yB=kB, whereas in the latter yA=kAand yB= x(rsd, ε)−kA. (iv) If min{kA, kB} ≥ x(rsd, ε)the Nash equilibrium pA=pB= 0 is unique. The resulting equilibrium wholesale price is psd = 0, and firms sell the quantities yA=yB=x(rsd, ε)/2. Proof: Follows from appendix A of Le Coq (2002) or the proofs of proposition 1-3 in Crampes and Creti (2005), using that marginal generating costs are constant and normalized to zero by assumption and that the maximum wholesale price with positive demand is p=rsd. Proposition 1 is illustrated in Figure 1. Area Acorresponds to case (i), where demand exceeds aggregate capacity, so that a blackout occurs. Areas Band Dare associated with case (ii): In area B, firm A(B, respectively) is the large (small) firm. In area D, these roles are reversed. In both cases, the large firm bids the maximum price rsd, whereas the small firm bids just low 13 6 - kA kB A x(r∗, ε) x(r∗, ε) B C D E  p∗=r∗ p∗=r∗ p∗=r∗ p∗= 0 Figure 1: Prices on the Wholesale Market enough to avoid undercutting by the large firm. In area C, which corresponds to case (iii), the difference in installed capacities is smaller than in either B or D, and two types of equilibria are possible: Either the large or the small firm bids the maximum price, and the other firm bids low enough to avoid undercutting. In both cases the equilibrium wholesale price is psd =rsd. Finally, area Ecorresponds to case (iv). Here, each firm’s capacity is sufficent to satisfy aggregate demand. Therefore, price bidding yields a Bertrand-type equilibrium. Note that there are multiple pure-strategy Nash equilibria for cases (i)–(iii), as any lower bid that avoids undercutting and negative profits is admissible. These equilibria are pay-off equivalent for cases (i) and (ii), but not for case (iii), where the volume of dispatched electricity yi(pA, pB)depends on the type of equilibrium played. To deal with this multiplicity problem, we impose the following assumption:18 Assumption 1 If capacities satisfy kA+kB≥x(rsd, ε)>max{kA, kB}, generators coordinate on the Nash equilibrium where the large-capacity firm bids the maximum price and the small-capacity firm bids low enough to avoid undercutting by the large firm. If generators have equal capacities, they play each type of equilibrium with equal probability. 18The assumption is equivalent to applying risk-dominance as a selection criterion. See Boom (2008) for a detailed discussion. 14 3.4.2 Retail Market We first note that for retailers to obtain non-negative profits, the demand shock must satisfy ε∈(ε, ε), where ε≡r−1is the critical value below which demand is zero, and ε≡min{kA, kB}+r−1is the maximum value for which generating capacities are large enough to avoid that generators extract rents from retailers. With this in mind, and recalling that retailers compete à la Bertrand, the expected profits of retailer `=C, D are given by π`(r`, rt) =              0if r`> rt, 1 2Rmax{0,min{¯ε,1}} max{0,ε}r`(1 + ε−r`)dε if r`=rt, Rmax{0,min{¯ε,1}} max{0,ε}r`(1 + ε−r`)dε if r`< rt, (11) with `, t =C, D, and `6=t. Equation (11) indicates that retailers undercut each other until they reach zero profits. Therefore, the following Nash equilibrium in retail prices emerges. Proposition 6 (retail prices) Depending on the capacity levels (kA, kB), there are the following Nash equilibria in retail prices. (i) If min{kA, kB} ≥ 1there is a unique pure-strategy Nash equilibrium with rC=rD= 0. (ii) If min{kA, kB}<1all pure-strategy Nash equilibria are characterised by rC≤1−min{kA, kB}and rD≤1−min{kA, kB}. Proof: Suppose that r`> rtwith `, t =C, D and `6=t. This can only be an equilibrium if rt≤1−min{kA, kB}and r`≤1−min{kA, kB}, because otherwise firm `could increase its profits by undercutting and firm tby increasing its price. Suppose, alternatively, that r`=rt. Then either r`= rt= 0 if min{kA, kB} ≥ 1, or r`=rt<1−min{kA, kB}if min{kA, kB}<1, because otherwise each retailer could double its profit by undercutting. Proposition (6) shows that, due to Bertrand competition, retailers cannot realize strictly positive profits, no matter whether the equilibrium is unique (case (i)) or not (case (ii)). In case (i) the generators’ minimum capacity is not small enough to ensure that the rents are shifted from retailers to the generators for all demand realizations at a positive retail price, meaning 15 that the wholesale market equilibrium is for some small εand all positive retail prices always located in area Eof figure 1. Therefore the retailers compete each other down to r= 0 in a Bertrand type manner. In case (ii) the minimum capacity of the generators is small enough that all rents are shifted from the retailers to the generators at a positive retail price even if the demand realization εis close to zero. Thus, for the retail price range given in (ii) the wholesale equilibrium is for all εlocated in either A,B,C or Dmeaning that the retailers realize zero profits before they compete the retail prices down to zero. To deal with the multiplicity problem in case (ii), we introduce the following assumption regarding equilibrium selection. Assumption 2 If min{kA, kB}<1, retailers choose the Nash equilibrium with rC=rD= 1 −min{kA, kB}. Assumption 2 imposes that retailers select the equilibrium in which they choose the highest possible price which generates zero profits. 3.4.3 Capacity Investments Separated duopoly generators i=A, B must anticipate the impact of their capacity choices kion the retail market and the wholesale market. Due to Bertrand competition at the retail level, potential rents are shifted to generators. Provided that aggregate capacity exceeds retail demand, the wholesale price is given by psd =rsd = max{0,1−min{kA, kB}}.(12) With a strictly positive retail price rsd, demand is characterized by x(rsd, ε) = 1+ε−1+min{kA, kB}=ε+min{kA, kB}. Therefore, generator i’s expected profits are given by Πi(ki, kj) =                  max{0,1−kj}Rmin{1,ki} 0εdε −zkiif ki> kj, max{0,1−kj} 2hRmin{1,ki} 0εdε +Rmin{1,kj} 0kidεi −zkiif ki=kj, max{0,1−ki}Rmin{1,kj} 0kidε −zkiif ki< kj, (13) 16 with i, j =A, B and i6=j.19 To understand (13), suppose min{kA, kB}<1 and x(rsd, ε)≤kA+kB, and first consider the case where ki> kj. Firm i then bids high and serves residual demand max{x(r∗, ε)−kj,0}= max{1 + ε−1 + kj−kj,0}=ε. Next, consider the case where ki< kj: Firm i now bids low and delivers its total capacity up to the level of demand (i.e., min{ki,1+ε−1+ki}=ki). Finally, if capacities are identical (ki=kj), firm i bids high or low with probability one half each. As noted above, the condition x(rsd, ε)≤kA+kBmust hold, since generators cannot sell electricity in the event of a blackout. This condition is equivalent to ε+min{kA, kB} ≤ kA+kB or ε≤max{kA, kB}, if min{kA, kB}<1, which explains the upper bound for integration in (13). As we show in Appendix A, generator i’s best response is to choose a higher capacity than its competitor (ki= 1 > kj) if the competitor’s capacity is relatively low, and to choose a lower capacity, ki= max{0,min{(kj−z)/(2kj),(1 −z)/2}}, if the rival’s capacity is relatively high.20 This is quite intuitive, since both residual demand and the wholesale price are large if the competitor’s capacity is small. It therefore pays to install a large capacity. In contrast, if the competitor’s capacity is large, it is more profitable to install a small capacity which is completely sold and supports a higher wholesale price. The next proposition summarizes the results for the case with simultaneous capacity choices. Proposition 7 (simultaneous capacity choices) With simultaneous capacity choices, the existence of a subgame-perfect Nash equilibrium (SPNE) in pure strategies is not guaranteed. (i) If 0≤z < 1/3, there are two asymmetric SPNE in pure strategies with capacities ksd i= 1 and ksd j= (1 −z)/2,i, j =A, B and i6=j. (ii) If 1/3≤z < 1/2, there is no SPNE in pure strategies. Proof: Solving the system of best-response functions (25) and (26) in Appendix A for equilibrium capacities yields results (i) and (ii). 19With min{kA, kB} ≥ 1, the wholesale price is zero and none of the generators will realize positive profits. 20See (25) and (26) in Appendix A for a detailed description of firm i’s best response function. 17 - 6 kA kB 1 - 6 kA kB 1 11 0≤z < 1/31/3≤z < 1/2 q p Figure 2: Best Responses in Capacities Figure 2 illustrates that there exists no pure-strategy Nash equilibrium with simultaneous capacity choices and high capacity costs z. The next proposition shows that this non-existence problem disappears if separated generators choose capacities sequentially. For the sake of concreteness, it assumes that firm Amoves first. Proposition 8 (sequential capacity choices) Suppose that firm Amoves first. (i) If 0≤z < 1/3, there is a unique SPNE in pure strategies where firm Achooses ksd A= (1 −z)/2and firm Bchooses ksd B= 1. (ii) If 1/3≤z < 1/2, there is a unique subgame perfect Nash equilibrium in pure strategies where firm Achooses ksd A= 1 −2zand firm Bchooses ksd B= 1. Proof: Substituting firm B’s best response function kB(kA)into ΠA(kA, kB) in Appendix A and maximizing with respect to kAyields results (i) and (ii). Note that the first mover, generator A, prefers to be the small-capacity firm: The small-capacity firm bids low and sells its total capacity, whereas the large-capacity firm, generator B, bids high, thereby determining the wholesale price, and serves only residual demand. 18 4 Ranking Market Configurations We rank the various market configuration with respect to aggregate capacity, retail prices, and levels of social welfare. In doing so, we use the following notation. The first-best optimal level of aggregative capacity is given by ks. Under integrated monopoly, capacity is denoted as km. Under integrated duopoly, aggregate capacity is kid = 2ˆ k, whereas under separated duopoly it is ksd =ksd A+ksd B. We use similar notation to distinguish retail prices and levels of social welfare in the alternative market configurations. Proposition 9 (ranking) Suppose that capacity decisions are either taken sequentially by the separated generators or that 0≤z≤1/3, and that integrated generators co-ordinate on the pareto-dominant competitive equilibrium. Then the ranking of market configurations in terms of (i) aggregate capacity levels is ks≥kid ≥ksd ≥km;(14) (ii) of retail prices is given by rid ≥rm≥rsd ≥rs;(15) (iii) and of welfare levels is given by Ws≥Wsd ≥Wm≥Wid.(16) Proof: (i) Follows from comparing Propositions 1, 2, 8 and 4. (ii) Follows from comparing Propositions 1, 2, 6 and 3. (iii) Since blackouts do not occur irrespective of market configuration, social welfare is given by W(k) = Z1 0 U(x(r, ε), ε)dε −zk, (17) where kdenotes total capacity. Substituting U(x(r, ε), ε)from (1), x(r, ε) from (2) and plugging in equilibrium values for rand kfor each market configuration, yields the associated welfare levels. Comparing these welfare levels completes the proof. Proposition 9 shows that, compared to the social optimum, capacity levels are inefficiently low and retail prices inefficiently high in all market configurations. We now want to discuss the intuition for the ranking of these configurations. 19 Let us first consider aggregate capacity levels. Result (i) shows that capacity levels are highest under integrated duopoly and lowest under integrated monopoly. The separated duopoly yields an intermediate level of aggregate capacity. To understand this result, consider the investment incentive of an integrated monopoly generator. Adding another integrated generator introduces competition both at the wholesale and the retail level. Since an integrated duopoly generator now faces the risk of rent extraction, it has an incentive to increase its investment relative to the integrated monopoly (kid > km). Next, consider the impact of vertical separation on the investment incentives of duopoly generators. After vertical separation, generators trade with separated retailers (rather than themselves) on the wholesale market. Since they are no longer committed to serve any predetermined level of retail demand, generators do not face the risk of rent dissipation by their rival, and they thus install smaller capacities than integrated duopoly generators (kid > ksd). Result (i) indicates that vertically separating the duopoly eliminates the investment-enhancing effect of rent extraction but leaves a positive investment effect due to introducing competition (ksd > km). Result (ii) shows that retail prices are highest under integrated duopoly and lowest under separated duopoly (apart from the social optimum). The integrated monopoly yields an intermediate level of retail prices. The intuition for high retail prices under integrated duopoly parallels that for high aggregate capacity: Integrated duopoly generators face the risk of rent extraction and thus have an incentive to set a high retail price to keep demand low (in addition to making high investments to serve retail demand). This risk does not exist under integrated monopoly or separated duopoly. Also note that retail prices are lowest in the separated duopoly, where retail competition disciplines retail prices. Finally, proposition 9 (iii) indicates that the combined effects of restructuring on capacity levels and prices are such that social welfare is highest under separated duopoly and lowest under integrated duopoly. The integrated monopoly yields an intermediate level of social welfare. To understand the intuition for the result, it is important to note that, irrespective of market configuration, aggregate installed capacity is always large enough to satisfy retail demand at the relevant equilibrium retail price, so that blackouts do not occur in equilibrium. This implies that raising capacity increases capacity costs rather than supply security. These increases in capacity costs must be weighed against the effects of changes in retail prices for the construction of the welfare ranking. Since both total capacity and retail prices are higher in the integrated duopoly than in the successive duopoly, welfare must be lower in the integrated duopoly. The welfare effect of changing from integrated 20 monopoly to separated duopoly is less obvious: Total capacity is higher, but retail prices are lower in the separated duopoly. Proposition 9 (iii) shows that the positive effect of lower retail prices dominates the negative effect of higher capacity costs, so that the separated duopoly performs better than the integrated monopoly. 5 Extensions and Limitations So far, we have deliberately abstracted from a number of real-world issues to highlight the role of vertical structure in determining generating capacities and retail prices. In the following subsections, we consider various extensions and limitations of our analysis.21 First, we explore the case where blackouts may be avoided by the rationing of retail demand. Next, we consider alternative specifications of demand. Finally, we discuss alternative specifications of supply. 5.1 Rationing In practice, system operators attempt to ration retail demand to avoid (or at least limit) blackouts. However, rationing is difficult to model as it is typically implemented in unsystematic ways and generally not in line with theoretical rationing models.22 By abstracting from the possibility of rationing, our main analysis has maximized the punishment of generators for providing insufficient capacity. We now consider the other extreme where demand can be rationed and generators are not punished for providing insufficient capacity.23 Specifically, we assume that if demand exceeds aggregate supply on the wholesale market (x(r, ε)> kA+kB), blackouts may be avoided by the rationing of retail demand. Specifically, we suppose that rationing leads to a so-called “brownout”, that is, generators sell their total capacity to consumers.24 21We are grateful to the referees for prompting us to study some of these extensions. 22For instance, in case of excess demand due to an unplanned outage of facilities (e.g., a power station or a transmission line), consumers in the neighborhood are temporarily cut off from service to avoid a spread of the blackout. 23In reality, unsystematic rationing is likely to provide at least some punishment, as valuable consumers might be cut off from service. 24Note that in our model rationing is always efficient because we have a representative consumer model. 21 5.1.1 Social Optimum With Rationing With rationing the social planner maximizes W(r, k) =          R1 0U(x(r, ε), ε)dε −zk if r≥2−k, Rk−1+r 0U(x(r, ε), ε)dε+ R1 k−1+rU(k, ε)dε −zk if 2−k > r ≥max{1−k, 0}, R1 0U(k, ε)dε −zk if 1−k > r ≥0 (18) with respect to rand k, respectively. The key difference to the welfare function given in (5) is that consumers now get served up to capacity if demand exceeds capacity. In this case consumers experience a brownout rather than a blackout. Unsurprisingly, for any given capacity level, the optimal retail price with rationing is smaller than without rationing and effectively implies rationing for large demand shocks. The optimal capacity is also smaller than without rationing, inducing a retail price at the level of marginal cost (r= 0).25 5.1.2 Integrated Monopoly With Rationing The monopolist maximizes π(r, k) =                      R1 max{r−1,0}r(1 + ε−r)dε −zk, if r≥2−k, Rk−1+r max{r−1,0}r(1 + ε−r)dε +R1 min{k−1+r,1}rkdε −zk, if max{0,1−k} ≤ r <2−k, R1 0rkdε −zk, if r≤max{0,1−k} (19) with respect to rand k, respectively. Again, for any given capacity level, the profit maximizing retail price is smaller than without rationing. The chosen capacity level is also smaller, as insufficient capacity is no longer punished. In effect, the profit-maximizing retail price is smaller than without rationing, and rationing kicks in for large demand shocks.26 25See proposition 10 in Appendix B.1. 26See proposition 11 in Appendix B.1 for further details. 22 There is ample scope for future research. First, it would be interesting to allow for endogenous (and possibly asymmetric) vertical integration decisions, as suggested by Buehler and Schmutzler (2005) and Buehler and Schmutzler (2008). Doing so would further enrich our understanding of the firms’ strategic investment decisions. Second, the discrimination of non-integrated competitors has rarely been considered in the context of electricity. Third, it would be useful to study models with different mechanisms determining wholesale prices and with more than two competitors to better understand the robustness of the risk of rent dissipation. Acknowledgements We are grateful to the editor, Michael Crew, and two anonymous referees for many helpful comments and suggestions. We further thank Gregor Zöttl, Nicholas Shunda, Chloé Le Coq, as well as seminar audiences at Copenhagen University, the University of Groningen, the University of Utrecht, the DIW Berlin, and numerous conferences and workshops for useful discussions. We gratefully acknowledge financial support from the Swiss National Science Foundation through grants PP0011-114754 and PP00P1-135143 and from the Social Science Section of the Danish Council of Independent Research through the Risky Power grant. Appendix A Firm i’s Best Response in Capacity in a Separated Duopoly. For kj≥1firm i’s profit function (13) translates into Πi(ki, kj) =   −zkiif ki≥1, (1 −ki)ki−zkiif 0≤ki≤1. (23) 29 If 0≤kj<1holds, firm i’s profit function becomes Πi(ki, kj) =                    1−kj 2−zkiif ki≥1, (1−kj)k2 i 2−zkiif kj< ki≤1, 1 2h(1−kj)k2 i 2+ (1 −ki)kikji−zkiif ki=kj, (1 −ki)kikj−zkiif 0≤ki< kj. (24) The best response of firm iwhich is derived from maximizing (23) or (24), respectively, with respect to kiyields ki(kj) =            1−z 2if kj≥1, kj−z 2kjif 1−z−√1−2z−2z2 3≤kj≤1, 1if 0≤kj≤1−z−√1−2z−2z2 3, (25) for 0≤z≤1/3. If 1/3< z ≤1/2holds, the maximization of (23) and (24) with respect to kiresults in ki(kj) =                  1−z 2if kj≥1, kj−z 2kjif z≤kj≤1, 0if 1−2z≤kj≤z, 1if 0≤kj≤1−2z. (26) B Rationing B.1 The Social Planner’s and the Monopolist’s Capacity Choice with Rationing Proposition 10 (social optimum with rationing) The welfare maximizing retail price and capacity are given by rsr = 0 and ksr = 2 −√2z. Proof: Taking the first derivative of (18) with respect to rsetting it equal to zero and solving for ryields r= max{0,1−k}as the optimal retail price for a given capacity for the social planner. Substituting this into (18) and solving the first order condition with respect to kyields the optimal capacity given in the proposition if 0≤z≤1/2. 30 Proposition 11 (integrated monopoly with rationing) The profit maximizing capacity is given by kmr =21 −2z 12 −(1 −ı√3)(153 + 252z+ 4z2) 24g(z)+(1 + ı√3)g(z) 24 with ıbeing the imaginary number32 and g(z)≡1269 + 5778z+ 1188z2−8z3 +12√3√−4563 −7020z+ 16137z2−7454z3+ 1272z4−72z51 3. The retail price is given by rmr =1 3[2(1 −kmr) + p1+4kmr + (kmr)2]. Proof: Taking the first derivative of (19) with respect to r, setting it equal to zero and solving for ryields r= 1/3[2(1−k)+√1+4k+k2]as the profit maximizing retail price for a given capacity. Substituting this into (19) and solving the first order condition of profit maximization with respect to k yields the profit maximizing capacity given in the proposition. B.2 The Integrated Duopoly with Rationing. When the firms choose their retail prices their profits depend on whether they undercut, match or overcharge their rival’s retail price. Undercutting yields the following expected profit net of capacity costs. πi(ri, rj) = (Rmin{1,ki+ri−1} max{0,ri−1}rix(ri, ε)dε if 1−ki≤ri< rj, 0if 0≤ri<min{1−ki, ri},(27) that does not differ from the expected profit without rationing. Rationing does not play a role because, if firm iundercuts firm jon the retail market, firm jdoes not have any retail demand, and firm isupplies all consumers. If firm ican however not serve its retail demand, firm jcan exploit this on the wholesale market and shift all the rents to firm j. This happens already for smaller demand shocks than those for which the system operator would start to ration demand. 32Note that for the relevant range of zthe profit maximizing capacity kmr ∈[5/4,0] and is a rational number that monotonously decreases in z. 31 If firm imatches its rival’s retail price, then its expected profit net of capacity costs depends on whether ki< kj,ki=kj or ki> kjholds. With ki< kj, the structure of its expected profit does not differ from the situation without rationing and is given by πiri=rj=(Rmin{1,2ki+rj−1} max{0,rj−1} rjx(rj,ε) 2dε if rj≥1−2ki, 0if 0≤rj<1−2ki.(28) Due to the same logic than before nothing changes due to rationing. However, with ki=kj, different from the no rationing case, the firm can now continue to sell up to its capacity even if it cannot satisfy its own retail demand and its expected profit net of capacity costs changes to πiri=rj=Z1 max{0,rj−1} rjmax x(rj, ε) 2, kidε. (29) This happens because the rival cannot exploit the firm on the wholesale market when the firm’s retail demand exceeds its capacity because this happens for the other firm at exactly the same demand shock. If ki> kjthen firm i, when matching firm j’s retail price, can appropriate all its rival’s rents in the wholesale market, if firm jis not able to serve its (rationed) retail demand because firm iis always able to serve its (rationed) retail demand. Its expected profit net of capacity costs is then πiri=rj=                      R1 max{0,rj−1} rjx(rj,ε) 2dε if rj≥2−2kj, R2kj+rj−1 max{0,rj−1} rjx(rj,ε) 2dε+ R1 2kj+rj−1rjmin{x(rj, ε), ki+kj}dε if 1−2kj < rj<2−2kj, R1 0rjmin{x(rj, ε), ki+kj}dε if 0< rj <1−2kj. (30) If firm icharges a higher retail price than its rival it can only earn positive revenues exactly in the case where its rival cannot serve the (rationed) retail demand of its retail customers at its price rjresulting in firm iappropriating all the rents via the wholesale auction. Thus, the expected profit net of capacity costs is: ¯πi(ri, rj) =            0if ri> rj≥2−kj, R1 max{0,rj−1+kj}rjmin{x(rj, ε), ki+kj}dε if 0≤rj <min{ri,2−kj}. (31) 32 When solving for the two firms best responses in retail prices it turns out that they are close to identical with the ones derived in Boom (2007). They are given by If firm iundercuts its rival’s retail price, its best response from below is then ri(rj) =      max 2−ki,3 4if rj>max 2−ki,3 4 rj−µif 0≤rj≤max 2−ki,3 4(32) with µ→0being the smallest unit in which retail prices can be announced. If firm isets ri> rj, then it is indifferent between all prices that satisfy this restriction, because its profit, given in (31), does not depend on the level of ri. The overall best response is determined by the comparison of πi(ri(rj), rj), derived from (27) and (32), with πi(ri, rj)ri=rjfrom (28), (29) or (30), respectively, and with ¯πi(ri, rj)defined in (31). Suppose that ki> kj. Then the overall best response of firm ifor ki> kj≥ q5−k2 jis given by ri(rj) = ri(rj)from (32). For min nki,p5−k2 io> kj≥ 0the overall best response is ri(rj)                                = max 2−ki,3 4if rj>max 2−ki,3 4, =rj−µif ˆr < rj≤max 2−ki,3 4, > rjif max{0,1−2kj}< rj≤min {ˆr, max 2−ki,3 4, ≥rjif 0< rj≤min {1−2kh, max 2−kj,3 4. (33) with ˆr=                3−√2(k2 j+k2 i)−1 2if ki≥kj>min{p1−k2 i, ki−1}, 1−kjif 0≤kj≤ki−1, 2−qk2 j+k2 iif 0≤kj≤min{p1−k2 i, ki}. (34) Firm j’s overall best response in retail prices is the equivalent to ri(rj)with firm jalways undercutting firm iif ki> kj≥q5 2. For min{ki,q5 2}> kj≥ 33 max 0,ki−1 2firm j’s best response in retail prices is given by rj(ri)                                = max 2−kj,3 4if ri>max 2−kj,3 4, =ri−µif max 2−kj,3 4≥ri> r0 i, =riif r0 i≥ri> r00 i, > riif r00 i≥ri>0, ≥0if ri= 0, (35) where the critical prices r0 iand r00 ifor the rival with the larger capacity is defined as: r0 i= max    3−q4k2 j−1 2,2−√2kj  ,(36) and r00 i= max    3−p4k2 i−1 2, 5−q12k2 j+ 6k2 i−2 3,0  (37) for ki>1, r00 i=         max 3−√4k2 i−1 2,5−√12k2 j+6k2 i−2 3if ki≥kj≥q1−k2 i 2, 2−q2k2 j+k2 iif q1−k2 i 2> kj≥0 (38) for 1 √2< ki≤1and r00 i= max n2−√2ki,2−q2k2 j+k2 io(39) for 0≤ki<1 √2. 34 For ki−1 2> kj>0firm j’s best response in retail prices is rj(ri)                                        = 2 −kjif ri>2−kj, =ri−µif 2−kj≥ri> r0 i, =riif r0 i≥ri≥1−2kj, ≥0if 1−2kj> ri≥2−ki, > riif 2−ki> ri>0, ≥0if ri= 0, (40) where r0 jis defined in (36). Now suppose that kA=kB=k, then each firm has the same best response in retail prices. If k > q5 2, then both firms’ overall best response is given by ri(rj)from equation (32). If 0≤k < q5 2, then each firm’s best response in retail prices is given by: ri(rj) =                                = max 2−k, 3 4if rj>max 2−k, 3 4, =rj−µif max 2−k, 3 4≥rj>ˆr ≥rjif rj= ˆr > rjif ˆr > rh>0, ≥0if rj= 0, (41) with ˆrfrom (34) with ki=kj=k. Then again there is always a Nash equilibrium with rA=rB= 0. Since the Nash equilibria in retail prices depend on when these best response functions change from undercutting to matching and then to overbidding, and since the retail prices at which this happens do not change, the possible Nash equilibria in retail prices are also close to the same as in proposition 3 with the exception of case (v) where now the only possible Nash equilibrium in retail prices implies ri=rj= 0. 35 B.3 The Separated Duopoly with Rationing Firm i0sbest response with i=A, B can be derived from maximizing the profit function (22). It is given by ki=(1−z 1−kjif kj≤(3 −√1+8z)/4, min kj−, 1−z 2if kj>(3 −√1+8z)/4,(42) with →0. The best response function is downward sloping, jumps downward at the given threshold and finally increases up to the given limit. Proposition 12 (separated duopoly with rationing) If the two generators Aand Bsimultaneously choose their capacities, a subgame perfect Nash equilibrium in pure strategies does not exist. 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