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Electricity market mergers with endogenous forward contracting

Abstract

We analyze the effects of electricity market mergers in an environment where firms endogenously choose their level of forward contracts prior to competing in the wholesale market. We apply our model to Alberta’s wholesale electricity market. Firms have an incentive to reduce their forward contract coverage in the more concentrated post-merger equilibrium. We demonstrate that endogenous forward contracting magnifies the price increasing impacts of mergers, resulting in larger reductions in consumer surplus. Current market screening procedures used to analyze electricity mergers consider firms’ pre-existing forward commitments. We illustrate that ignoring the endogenous nature of firms’ forward commitments can yield biased conclusions regarding the impacts of market structure changes such as mergers. In particular, we show that the price effects of mergers can be largely underestimated when forward contract quantities are held at pre-merger levels. Whether the profits of the merged firm are greater with fixed or endogenous forward quantities is ambiguous.

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Fig. 1
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Notes

  1. 1.

    For example, in 2012 in the United States, 107 Gigawatts (GW) of generation capacity was exchanged via mergers and acquisitions, representing 10% of aggregate capacity nationwide (EIA 2013). See Moss (2008) and Federico (2011) for a survey of recent electricity market merger cases in United States and Europe, respectively.

  2. 2.

    Similarly, in the proposed merger of Exelon and PSEG (FERC 2005), the importance of forward contracts in the post-merger environment was emphasized in several equilibrium simulation studies (Morris and Oska 2008; McRae and Wolak 2009). The estimated merger price effects depended critically on the merged firm’s post-merger forward commitments.

  3. 3.

    A supplier passes the pivotal supplier test if its uncommitted capacity is less than the difference between the annual peak wholesale demand and uncommitted installed capacity of all other suppliers. A firm’s uncommitted capacity reflects its installed nameplate generation capacity, net of its native load obligations and long-term supply commitments (FERC 2015).

  4. 4.

    The competition-enhancing role of forward markets has been found to be sensitive to certain assumptions in the model. Hughes and Kao (1997) emphasize the importance of public knowledge of forward contracts. Mahenc and Salanie (2004) demonstrate that under price competition forward markets may allow firms to soften competition, while Liski and Montero (2006) and Green and Coq (2006) illustrate that forward contracts can facilitate collusion in an infinitely repeated context. de Braganca and Daglish (2016) demonstrate that firms may potentially exercise market power in the spot market to increase subsequent forward market prices.

  5. 5.

    Newbery (2009) extends Bushnell’s analysis to consider firms with asymmetric constant marginal cost and develops a residual supplier index to assess market power in electricity markets.

  6. 6.

    A discussion of Cournot models employing such cost functions and their usefulness in merger simulation can be found in Werden and Froeb (2008). A recent example employing this framework in a merger simulation is Greenfield et al. (2015).

  7. 7.

    Exogenous long-run forward commitments can reflect regulatory mandated fixed-price contracts or requirements linked with the divestment of generation assets from large incumbent suppliers (Bushnell 2007; Frutos and Fabra 2012).

  8. 8.

    Bushnell et al. (2008) illustrate that Cournot Nash equilibria approximate observed behavior well in electricity markets once firms’ forward commitments are taken into account.

  9. 9.

    For example, the relative transparency of forward contracts in electricity markets can arise because of regulatory oversight (Bushnell 2007).

  10. 10.

    Throughout the theoretical analysis we focus on interior solutions. In equilibrium, the interior solutions are verified. In the empirical analysis in Sect. 4, we have verified that the addition of non-negative forward contracting and spot market output constraints does not impact the key results of the analysis.

  11. 11.

    A firm’s profit function is strictly concave in its own strategy. This ensures that there exists a unique equilibrium level of forward contracts and spot market production for each firm. See the Proofs of Lemmas 1 and 3 for a formal derivation.

  12. 12.

    In addition to the setting where all firms endogenously choose forward contracts, Allaz and Vila (1993) consider the setting where a single firm endogenously chooses its forward position. While we focus on the setting where all firms endogenously choose their forward positions in the pre- and post-merger equilibrium, the reduced strategic incentive for forward contracting due to a merger identified in Proposition 1 is likely to hold in the setting where a single firm forward contracts for strategic reasons. As noted in Sect. 5, a detailed analysis of this case is left for future research.

  13. 13.

    More formally, suppose firms 1 and 2 merge, resulting in capital stock \(k_M = k_1+k_2\) and \(\beta _M = \frac{bk_M}{bk_M +1}\). It is straightforward to show that \(\beta _1 + \beta _2 - \beta _M > 0\). Using (11), this implies that a non-merging firm’s forward contracted ratio decreases.

  14. 14.

    In a symmetric model, Bushnell (2007) illustrates that the percentage of output that is forward contracted increases quickly as N expands. Unlike the current study, Bushnell does not consider the impacts of forward contracts on mergers.

  15. 15.

    In our symmetric setting, the merger can only decrease total surplus because it elevates prices, reduces output, and results in an asymmetric and therefore inefficient allocation of output across plants. The potential remains that a merger could increase total surplus in the asymmetric environment. See McAfee and Williams (1992) for a related discussion. This is beyond the scope of our analysis and is the subject of future research.

  16. 16.

    For detailed reviews of Alberta’s electricity market see Olmstead and Ayres (2014) and Brown and Olmstead (2017, forthcoming).

  17. 17.

    In Alberta, the distribution of generation capacity by technology is: 47.2% coal, 36.9% natural gas based, 7.2% Wind, 6.2% Hydro, and 2.4% Other (Brown and Olmstead 2017, forthcoming).

  18. 18.

    Section 4.3 discusses the estimation of unit-specific marginal cost. Wind units are excluded from Table 2 because during the sample period wind units did not submit offer prices into the pool, but were required to be price takers.

  19. 19.

    The zero bid markups are arising because firms in Alberta own a sizable amount of cogeneration facilities that systematically submit bids of zero. Our marginal cost methodology detailed in Sect. 4.3 assigns these facilities a marginal cost of zero.

  20. 20.

    Hobbs et al. (2000), Xian et al. (2004), Hu and Ralph (2007), and Yao et al. (2008) use similar empirical methods to model multi-stage equilibrium problems in electricity markets.

  21. 21.

    We restrict forward contracts to be non-negative. Further, to reduce the computation time our numerical program, we restrict forward positions to be below a firm’s maximum observed generation (in MWs) in any given month of our sample. The key qualitative results of our analysis are robust to removing this restriction.

  22. 22.

    See Brown and Eckert (2016b) for a detailed derivation of the full mixed nonlinear complementarity program.

  23. 23.

    Because EPECs reflect a set of MPECs with non-convex constraints, there could potentially be multiple local Nash equilibria resulting in different starting values converging to different equilibria (Xian et al. 2004; Hu and Ralph 2007). To alleviate concerns of multiple Nash equilibria, we use KNITRO’s multistart algorithm which randomly selects starting values for all endogenous variables and then launches the Levenberg–Marquardt algorithm to search for the solution to the EPEC from this starting point. We use the multistart algorithm to solve the EPEC 50 times for all 8760 h in our sample. For each hour, we find systematic convergence to the same forward-spot equilibrium regardless of the starting values.

  24. 24.

    See Appendix 2 for additional details on the data sources used to estimate unit-level marginal costs.

  25. 25.

    A similar approach was undertaken by Bushnell et al. (2008). Borenstein et al. (2002) use a Monte-Carlo approach to simulate the random process associated with unit availability. This approach is numerically intractable in our analysis.

  26. 26.

    Under Alberta’s ISO rules, units are required to offer MSG each hour at the lowest price offered by the unit, which is effectively always $0. Therefore, the number of MW offered at $0 acts as a useful upper bound on MSG.

  27. 27.

    For example, Neuhoff et al. (2013) assume that MSG equals 38% of capacity in an analysis of Europe.

  28. 28.

    We apply the observed output from hydro generation for each hour and analyze the amount of non-hydro production necessary to meet demand net of hydro generation. The biases associated with this approach are limited in the Alberta context because hydro represents approximately 2% of annual production.

  29. 29.

    It is numerically intractable to model each firms’ stepwise marginal cost functions in our current framework. This would require explicitly modeling the production decision of each firms’ assets for every hour in our sample, increasing the scale of the spot market complementarity conditions defined in (19) and (20) substantially.

  30. 30.

    Using the established marginal cost functions, we choose the fourth-order polynomial parameters to minimize the squared differences between the estimated marginal cost and the observed marginal cost, subject to a restriction that this function is non-decreasing. We solve this nonlinear program using the primal-dual interior point optimization method (IPOPT).

  31. 31.

    Similar to other analyses that analyze large-scale EPECs in electricity markets (e.g., Yao et al. 2008), we are restricted to estimating a linear residual market demand function due to computational constraints.

  32. 32.

    The temperature variables used in the analysis for BC and SK are modeled as quadratics for hourly cooling degrees [hourly mean degrees above \(18.33\,^\circ \hbox {C}\) (\(65\,^\circ \hbox {F}\))] and hourly heating degrees [hourly mean degrees below \(18.33\,^\circ \hbox {C}\) (\(65\,^\circ \hbox {F}\))]. The cities considered in BC and SK are Vancouver and Saskatoon, respectively. The results of the analysis are robust to the consideration of higher degree polynomials on the temperature variables and alternative large cities in each province.

  33. 33.

    Alternatively, one could treat the offers of the fringe as its marginal cost curve and the fringe’s competitive supply curve. This would yield a fringe supply curve that is a discontinuous step-function, and would require incorporating the fringe’s capacity constraint. Both of these yield computational difficulties that are beyond the scope of this paper.

  34. 34.

    Natural gas prices reflect the hourly natural gas price from Alberta’s Natural Gas Exchange (NGX). Because regional gas prices may be endogenous to local supply and demand conditions, we also used monthly natural gas prices from Henry Hub (converted to Canadian Dollars using Bank of Canada exchange rates). The Henry Hub and NGX natural gas prices are strongly correlated (\(\rho = 0.94\)). The residual demand estimation results are unaffected by the use of Henry Hub prices.

  35. 35.

    Price responsive load that is subject to wholesale price represents less than 2% of average market demand.

  36. 36.

    For each model, we estimate the model using an IV approach with Newey–West heteroskedasatic and autocorrelation robust standard errors with 24 lags.

  37. 37.

    MSA (2010a) provides evidence that forward trading in Alberta is lower than other electricity markets worldwide.

  38. 38.

    While precise data on the forward commitments of individual firms are not publicly available, we expect that firm’s are able to draw inferences about rivals’ positions. Such inferences may be drawn from participation in procurement processes (such as auctions) for fixed price contracts for retail supply, or by monitoring the Natural Gas exchange, through which much of the forward contracting occurs. Finally, ENMAX is vertically integrated and supplies retail customers both through regulated and competitive retail plans. Data on its supply through the regulated default product are publicly available.

  39. 39.

    Using Hortacsu and Puller’s approach, the highest estimated implied forward contract coverage arises from TA and TC, the firms whose generation portfolios primarily consists of base-load coal generators with a large amount of MSG.

  40. 40.

    Brown and Eckert (2016b) detail the mixed nonlinear complementarity program used to solve the post-merger EPEC.

  41. 41.

    Regulators have imposed or advocated for the imposition of requirements on firms’ forward contracting quantities to alleviate concerns over market power (Harvey and Hogan 2000; Frutos and Fabra 2012).

  42. 42.

    The results are presented as unweighted averages. Weighting hours by market demand has little impact on estimated price effects, estimated weighted and unweighted average prices differ by less than $2/MWh for all mergers.

  43. 43.

    In the recent Exelon and Constellation merger analysis, numerical simulations of wholesale market competition found that under certain circumstances wholesale prices could rise over 25% due to the merger (MPSC 2012, p. 46).

  44. 44.

    Section 4.5.3 demonstrates that the key qualitative conclusions of our analysis are robust to a more price-elastic residual demand function.

  45. 45.

    Under our assumption of perfectly inelastic demand, changes in total surplus equate to changes in domestic and imported production costs. In each of the merger cases considered in our numerical analysis, total production costs increase, so that total surplus declines.

  46. 46.

    Section 4.5.3 considers a more price-elastic residual demand and demonstrates that the merged firm’s profits can be higher when forward contracted output is held at pre-merger levels in MWs than those with endogenous forward contracts.

  47. 47.

    See MSA (2012) for an extended discussion of geographic market definition issues in Alberta’s wholesale electricity market, and issues arising regarding concentration statistics and market shares.

  48. 48.

    The majority of the mergers do not violate the provisions in Alberta’s Fair, Efficient and Open Competition Regulation (Alta Reg 159/2009) that prevents a firm from holding offer control greater than 30% of Alberta Maximum Capability.

  49. 49.

    van Eijkel et al. (2016) find that strategic and risk-hedging reasons for forward contracting create opposing predictions regarding the effect of an increase in the number of firms on firms’ incentives to sign forward contracts. The authors find that risk-hedging can increase firms’ incentives to sign forward contracts as the number of firms decrease.

  50. 50.

    For example, in Alberta, certain generating units are under long-term contracts called Power Purchase Arrangements which give the buyer of the contract offer control over the asset. In 2020, the expiry of these contracts will return offer control of the generating asset to the owner, leading to a sizable change in the market structure (MSA 2011).

  51. 51.

    Virtual divestitures of generation assets have been used in European electricity markets to reduce market concentration (Frutos and Fabra 2012).

  52. 52.

    Brown and Eckert (2016a) examine the robustness of the estimated effects of the expiry of Power Purchase Arrangements in Alberta to assumptions on forward contracts.

  53. 53.

    Using 2008–2014 data from Alberta’s wholesale electricity market, Brown and Olmstead (2017, forthcoming) use a Monte-Carlo approach that selects unit-specific non-zero coal bids into the wholesale market in low demand hours to represent each coal unit’s marginal costs. The authors demonstrate that their Monte-Carlo methodology results in marginal cost estimates that closely reflect estimates obtained using Alberta coal unit heat rates and PRB coal prices.

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Correspondence to David P. Brown.

Additional information

The authors would like to thank Michael Crew, two anonymous referees, Shourjo Chakravorty, Michelle Phillips, and seminar participants at the University of Alberta and the Canadian Law and Economics Association Conference for their helpful comments and suggestions.

Appendices

Appendix 1: A proofs of formal conclusions

Proof of Lemma 1

(2) can be rewritten as:

$$\begin{aligned} a-bQ+bq_{i}^{f}=\frac{bq_{i}}{\beta _{i}}. \end{aligned}$$
(24)

Multiplying both sides of (24) by \(\beta _{i}\) and summing across all firms yields:

$$\begin{aligned} B(a-bQ) + b\sum \beta _{i}q_{i}^{f} = bQ. \end{aligned}$$
(25)

(3) and (4) follows from (25) given \(P(Q) = a - bQ\). (5) follows from (3) and (24).

Using (2), because b and \(k_i\) are positive constants, each firm’s profit function is strictly concave in \(q_i\):

$$\begin{aligned} \frac{\partial ^2 \pi _{i}(q_{i},q_{-i})}{\partial q_i^2} = -2b - \frac{1}{k_{i}} <0. \end{aligned}$$

\(\square \)

Proof of Lemma 2

Using (3)–(5) and that \(\beta _i > 0\,\forall \,i = 1, 2, \ldots , N\), the following inequalities hold:

$$\begin{aligned}&\frac{\partial q_i}{\partial q_i^f} = \beta _i - \frac{\beta _i^2}{1+B} \mathop {=}\limits ^{s} 1 + B - \beta _i = 1 + \sum _{j\ne i} \beta _j>0; \\&\frac{\partial q_j}{\partial q_i^f} = \frac{-\beta _i \beta _j}{1+B}<0; \quad \frac{\partial Q}{\partial q_i^f} = \frac{\beta _i}{1+B} >0; \quad \text {and} \quad \frac{\partial P}{\partial q_i^f} = \frac{-b \beta _i}{1+B} <0. \end{aligned}$$

\(\square \)

Proof of Lemma 3

Using the marginal effects of forward contracting on spot market quantities identified in Lemma 2, (10) follows from (9).

Using (7) and from Lemma 2 \(\frac{\partial q_i}{\partial q_i^f}\) and \(\frac{\partial q_j}{\partial q_i^f}\) are independent of \(q_i^f\) for all \(i,j = 1, 2, \ldots , N\) with \(i\ne j\) :

$$\begin{aligned} \frac{\partial ^2 \pi _{i}\left( q_{i}^{f}, q_{-i}^{f}\right) }{\partial {q_i^f}^2}= & {} P^\prime (Q)\frac{\partial q_{i}}{\partial q_{i}^{f}} + q_{i}P''(Q)\sum _{j=1}^{N}\frac{\partial q_{j}}{\partial q_{i}^{f}}\nonumber \\&+\,P'(Q)\, \frac{\partial q_{i}}{\partial q_{i}^{f}}\, \sum _{j=1}^{N}\frac{\partial q_{j}}{\partial q_{i}^{f}}-C_{i}''(\cdot )\left( \frac{\partial q_{i}}{\partial q_{i}^{f}}\right) ^2. \, \end{aligned}$$
(26)

Since \(P'(Q) = - b<0\), \(P''(Q) = 0\), \(C_{i}''(\cdot ) = \frac{1}{k_i}>0\), and from Lemma 2 \(\frac{\partial q_{i}}{\partial q_{i}^{f}}>0\) and \(\sum _{j=1}^{N}\frac{\partial q_{j}}{\partial q_{i}^{f}} = \frac{\partial Q}{\partial q_{i}^{f}} >0 \), (26) simplifies and satisfies the following inequality:

$$\begin{aligned} \frac{\partial q_{i}}{\partial q_{i}^{f}}\left[ P^\prime (Q) + P'(Q)\, \frac{\partial Q}{\partial q_{i}^{f}}-C_{i}''(\cdot )\, \frac{\partial q_{i}}{\partial q_{i}^{f}} \right] <0. \end{aligned}$$
(27)

\(\square \)

Proof of Proposition 1

It is without loss of generality to assume that firms 1 and 2 merge resulting in capital stock \(k_M = k_1 + k_2\). Define \(\beta _M = \frac{bk_M}{bk_M+1}\) and \(\beta _j = \frac{b k _j}{b k_j +1}\) \(\forall \) \(j \ge 3\). Using (11), the difference between firm 1 and the merged firm’s proportion of forward contracted output satisfies:

$$\begin{aligned}&\frac{q_1^f}{q_1} - \frac{q_M^f}{q_M} = \frac{\sum _{j=2}^N \beta _j }{1+\sum _{j=2}^N \beta _j} - \frac{\sum _{j=3}^N \beta _j }{1+\sum _{j=3}^N \beta _j}> 0 \\&\quad \Rightarrow \left( \sum _{j=2}^N \beta _j \right) \left( 1+ \sum _{j=3}^N \beta _j \right) - \left( \sum _{j=3}^N \beta _j \right) \left( \sum _{j=2}^N \beta _j \right) = \left( \sum _{j=2}^N \beta _j \right) > 0. \end{aligned}$$

Similar reasoning applies for firm 2. Next, we illustrate that the non-merging firms reduce the proportion of output that is forward contracted. It is without loss of generality to focus on the non-merging firm 3. Denote \({q_3^f}^\prime \) and \(q_3^\prime \) to be firm 3’s post-merger forward and spot market output, respectively. Using (11) and that \(\beta _1+ \beta _2 - \beta _M >0\):

$$\begin{aligned} \frac{q_3^f}{q_3} - \frac{{q_3^f}^\prime }{q_3^\prime }= & {} \frac{\sum _{j\ne 3} \beta _j }{1+\sum _{j\ne 3} \beta _j} - \frac{\beta _M + \sum _{j=4}^N \beta _j }{1+\beta _M + \sum _{j=4}^N \beta _j }> 0\\\Rightarrow & {} \left( \sum _{j\ne 3} \beta _j \right) \left( 1+ \beta _M + \sum _{j = 4}^N \beta _j\right) - \left( \beta _M + \sum _{j=4}^N \beta _j \right) \left( 1+\sum _{j\ne 3} \beta _j \right)> 0\\\Rightarrow & {} \left( \sum _{j\ne 3} \beta _j \right) - \beta _M - \sum _{j=4}^N \beta _j = \beta _1 + \beta _2 - \beta _M > 0. \end{aligned}$$

\(\square \)

Proof of Lemma 4

Using (10) and that \(\beta _i = \beta \) for all \(i=1, 2, \ldots , N\) and \(B = N \beta \):

$$\begin{aligned}&\left[ \frac{a}{b}\frac{\beta }{1+B}-\frac{\beta ^2 N\, q_{f}}{1+B}+\beta q^{f}\right] \, \left( \frac{-\beta ^2}{1+B}\right) + q^{f}\left( \beta - \frac{\beta ^{2}}{1+B}\right) =0 \nonumber \\&\quad \Rightarrow q^f \left( \frac{\beta }{1+B}\right) \left[ \frac{\beta ^3}{1+B}N(N-1)-\beta ^2(N-1)+1-B - \beta \right] \,= \beta ^3 \frac{a}{b} \left( \frac{N-1}{(1+B)^2}\right) .\nonumber \\&\quad \Rightarrow q^{f} = \frac{\frac{(N-1)}{1+B}\beta ^{2}\frac{a}{b}}{(1+B) - \beta + (N-1)\beta ^{2}(\frac{\beta N}{1+B} - 1)} = \frac{\frac{(N-1)}{1+NB}\beta ^{2}\frac{a}{b}}{1 +(N-1)\beta - \frac{(N-1)\beta ^{2}}{1+N\beta }} \nonumber \\&\quad \quad = \frac{(N-1) \frac{a}{b}}{(1+(N-1)\beta )(1+N\beta )-(N-1)} = \frac{(N-1) \frac{a}{b}}{ \frac{\left( \frac{1+Nbk}{1+bk}\right) \left( \frac{1+(N+1)bk}{1+bk}\right) }{\left( \frac{bk}{1+bk}\right) ^2} -(N-1)} \end{aligned}$$
(28)
$$\begin{aligned}&\quad \quad = \frac{(N-1) \frac{a}{b}}{ \left( N+ \frac{1}{bk}\right) \left( N+1+\frac{1}{bk} \right) -(N-1)} = \frac{(N-1) \frac{a}{b}}{ \left( N+ \frac{1}{bk}\right) ^2 + N + \frac{1}{bk} -(N-1)}. \end{aligned}$$
(29)

(14) follows from (29).

Using (14) and recognizing that \(B = N \beta \) in this symmetric setting, (5) is simplified to:

$$\begin{aligned} q_{i}= & {} \frac{a}{b}\frac{\beta }{1+B}+q^f \beta \left[ 1 - \frac{\beta N}{1+B}\right] = \frac{a}{b}\frac{\beta }{1+B}+q^f \frac{\beta }{1+B}\nonumber \\= & {} \frac{a}{b}\frac{\beta }{1+B}+q^f \frac{\beta }{1+B} = \frac{a}{b} \frac{\beta }{1+N\beta }\left[ \frac{ \left( N+ \frac{1}{bk}\right) ^2 + N+\frac{1}{bk} }{\left( N+ \frac{1}{bk}\right) ^2 + (1+ \frac{1}{bk})}\right] \nonumber \\= & {} \frac{a}{b} \frac{1}{\frac{1}{\beta }+N}\left[ \frac{ \left( N+ \frac{1}{bk}\right) ^2 + N+\frac{1}{bk} }{\left( N+ \frac{1}{bk}\right) ^2 + (1+ \frac{1}{bk})}\right] \nonumber \\= & {} \frac{a}{b} \frac{1}{\frac{1+bk}{bk}+N}\left[ \frac{ \left( N+ \frac{1}{bk}\right) \left( N + \frac{1}{bk} +1 \right) }{\left( N+ \frac{1}{bk}\right) ^2 + (1+ \frac{1}{bk})}\right] . \end{aligned}$$
(30)

(12) follows from (30). Recognizing that \(P(Q) = a - b N q\), (13) follows from (12). \(\square \)

Proof of Proposition 2

First, we illustrate that the spot market prices increase post-merger, holding forward contracting quantities at pre-merger levels. Define \(q_{Pre}^f\) to be the symmetric pre-merger forward contracting level. Using (4) and that \(B= N \beta _{NM}\) and \(B_M = \beta _M + (N-2) \beta _{NM}\), the spot market prices pre- and post-merger, holding forward contracting quantities at pre-merger levels, equals:

$$\begin{aligned} P^{Pre}= & {} \left( \frac{1}{1+B} \right) \left( a - b q_{Pre}^f B\right) \quad \text {and}\nonumber \\ P^{Post}= & {} \left( \frac{1}{1+B_M} \right) \left( a - b q_{Pre}^f (\beta _M + B_M) \right) . \end{aligned}$$
(31)

Using (31) and that \(B= N \beta _{NM}\), \(B_M = \beta _M + (N-2) \beta _{NM}\), \(\beta _{NM} = \frac{bk}{bk+1}\), and \(\beta _M = \frac{2bk}{2bk+1}\):

$$\begin{aligned} P^{Post} - P^{Pre}= & {} \left( \frac{1}{1+B_M} \right) \left( a - b q_{Pre}^f (\beta _M + B_M) \right) - \left( \frac{1}{1+B} \right) \left( a - b q_{Pre}^f B\right)> 0\nonumber \\\Rightarrow & {} a(B-B_M) - b q_{Pre}^f \left[ (1+B) (\beta _M + B_M) - B(1+B_M) \right]> 0\nonumber \\\Rightarrow & {} a(2\beta _{NM} - \beta _M) - b q_{Pre}^f \left[ 2(\beta _M - \beta _{NM}) + N \beta _{NM} \beta _M \right]> 0\nonumber \\\Rightarrow & {} a \left( \frac{2(bk)^2}{(bk+1)(2bk+1)} \right) - b q_{Pre}^f \left[ \frac{2bk}{(2bk+1)(bk+1)} + \frac{2N (bk)^2}{(bk + 1) (2bk+1)}\right] > 0.\nonumber \\ \end{aligned}$$
(32)

Using (14), (32) simplifies:

$$\begin{aligned}&a\, bk - b \left( \frac{(N-1) \frac{a}{b}}{\left( N+ \frac{1}{bk}\right) ^2 + \left( 1 + \frac{1}{bk} \right) } \right) \left( 1+N bK \right)> 0\nonumber \\&\quad \Rightarrow N + 2 + N bk + \frac{1}{bk} + bk > 0. \end{aligned}$$
(33)

Because spot market prices are decreasing in total output Q, \(P^{Post} - P^{Pre}>0\) implies that total output decreases post-merger, holding forward contracting quantities at pre-merger levels.

Second, we illustrate that total output post-merger is lower with endogenous forward contracts (\(Q^{Post}_{Endog}\)) compared to total output holding forward contracting quantities at pre-merger levels (\(Q^{Post}_{Fixed}\)). Using (3) and \(B_M = \beta _M + (N-2) \beta _{NM}\), \(Q^{Post}_{Endog} < Q^{Post}_{Fixed}\) if:

$$\begin{aligned}&\frac{a}{b} \frac{1}{1+B_M} + \frac{\beta _M q_M^f + (N-2) \beta _{NM} q_{NM}^f}{1+B_M}< \frac{a}{b} \frac{1}{1+B_M} + \frac{\left[ \beta _M + (N-2) \beta _{NM} \right] q_{Pre}^f}{1+B_M}\nonumber \\&\quad \Rightarrow \beta _M q_M^f + (N-2) \beta _{NM} q_{NM}^f < \left[ \beta _M + (N-2) \beta _{NM} \right] q_{Pre}^f. \text { } \end{aligned}$$
(34)

Using (10) and \(B_M = \beta _M + (N-2) \beta _{NM}\), in the post-merger equilibrium the merged firms’ forward contract quantity satisfies:

$$\begin{aligned}&q_M^f \beta _M \left( 1 - \frac{\beta _M}{1+B_M} \right) = q_M \left( \frac{\beta _M \beta _{NM} (N-2) }{1+B_M} \right) \nonumber \\&\quad \Rightarrow \quad q_M^f (1+B_M - \beta _M) = q_M \beta _M (N-2). \end{aligned}$$
(35)

Using (5) and that \(B_M = \beta _M + (N-2) \beta _{NM}\) and \(\sum _{j \ne M} q_j^f = (N-2) q_{NM}^f\), the merged firms’ spot market quantity satisfies:

$$\begin{aligned} q_M= & {} \left( \frac{\beta _M}{1+B_M} \right) \left[ \frac{a}{b} - \beta _{NM} \sum _{j \ne M} q_j^f - \beta _M q_M^f + (1+B_M) q_M^f \right] \nonumber \\= & {} \left( \frac{\beta _M}{1+B_M} \right) \left[ \frac{a}{b} - \beta _{NM} (N-2) q_{NM}^f + q_{M}^f(1+(N-2)\beta _{NM}) \right] . \end{aligned}$$
(36)

Using (36), (35) simplifies:

$$\begin{aligned} q_M^f = \frac{ \beta _{NM} \beta _M (N-2) \left( \frac{a}{b} - \beta _{NM} (N-2) q_{NM}^f \right) }{\left( 1 +(N-2) \beta _{NM} \right) \left( 1+ \beta _M + (N-2) \beta _{NM} - (N-2) \beta _{NM} \beta _M \right) }. \end{aligned}$$
(37)

Using (10) and \(B_M = \beta _M + (N-2) \beta _{NM}\), in the post-merger equilibrium a non-merging firm’s forward contract quantity satisfies:

$$\begin{aligned}&q_{NM}^f \left( 1 - \frac{\beta _{NM}}{1+B_M} \right) = q_{NM} \left( \frac{ \beta _{NM} (N-3) + \beta _M }{1+B_M} \right) \nonumber \\&\quad \Rightarrow q_{NM}^f (1+B_M - \beta _{NM}) = q_{NM} \left( \beta _{NM}(N-3) + \beta _M \right) . \end{aligned}$$
(38)

Using (5) and that \(B_M = \beta _M + (N-2) \beta _{NM}\) and \(q_j^f = q_{NM}^f\) for all \(j \ne M\), a non-merging firm’s spot market quantity satisfies:

$$\begin{aligned} q_{NM}= & {} \left( \frac{\beta _{NM}}{1+B_M} \right) \left[ \frac{a}{b} - \beta _{NM} \sum _{j \ne M} q_j^f - \beta _M q_M^f + (1+B_M) q_{NM}^f \right] \nonumber \\= & {} \left( \frac{\beta _{NM}}{1+B_M} \right) \left[ \frac{a}{b} - \beta _M q_M^f + q_{NM}^f(1+\beta _M) \right] . \end{aligned}$$
(39)

Using (39), (38) simplifies:

$$\begin{aligned} q_{NM}^f = \frac{ \beta _{NM} \left( (N-3) \beta _{NM}+\beta _M \right) \left( \frac{a}{b} - \beta _{M} q_{M}^f \right) }{\left( 1 +(N-3) \beta _{NM} + \beta _M \right) \left( 1+ \beta _M + (N-2) \beta _{NM} \right) - \left( (N-3) \beta _{NM} + \beta _M \right) \beta _{NM} (1+\beta _M)}. \nonumber \\ \end{aligned}$$
(40)

Using (37) and (40), the post-merger forward contract quantities satisfy:

$$\begin{aligned} q_{NM}^f= & {} \frac{\frac{a}{b} D (H- \beta _M G)}{H E - (N-2)\beta _M \beta _{NM} G D} \quad \text {and} \quad q_{M}^f = \frac{\frac{a}{b} G(E - \beta _{NM}(N-2)D)}{HE - (N-2)\beta _M \beta _{NM} G D }, \quad \text {where} \\ D\equiv & {} \beta _{NM} \left( (N-3) \beta _{NM} + \beta _M \right) ; \text { }\nonumber \\ E\equiv & {} \left( 1+ (N-3) \beta _{NM} + \beta _M \right) \left( 1+(N-2) \beta _{NM} + \beta _M \right) - \left( (N-3) \beta _{NM} + \beta _M \right) \beta _{NM} (1+\beta _M);\nonumber \\ G\equiv & {} \beta _{NM} \beta _M (N-2); \quad \text {and}\nonumber \end{aligned}$$
(41)
$$\begin{aligned} H\equiv & {} \left( 1+ (N-2) \beta _{NM}\right) \left( 1+ \beta _M + (N-2) \beta _{NM} - (N-2) \beta _{NM} \beta _M \right) . \text { } \end{aligned}$$
(42)

Using (28) and (42), (34) can be rewritten as:

$$\begin{aligned}&\frac{(N-2)\beta _{NM} D H + \beta _M G E - 2(N-2) \beta _M \beta _{NM} D G }{HE - (N-2)\beta _M \beta _{NM} G D} \nonumber \\&\quad < \frac{(N-1)\left( \beta _M + (N-2) \beta _{NM}\right) }{\left( 1+ (N-1) \beta _{NM} \right) \left( 1 + N \beta _{NM}\right) - (N-1) }. \end{aligned}$$
(43)

Using (42), that \(\beta _{NM} = \frac{bk}{1+bk}\) and \(\beta _M = \frac{2bk}{1+2bk}\), and using Mathematica to simplify the algebraic expression, (43) can be rewritten as:

$$\begin{aligned}&\frac{2bk(1+bk(3+N +2Nbk))}{(1+3bk+2(bk)^2)^3} \bigg \{ 2+ bk \bigg [ 6 + 7N + bk \bigg ( 2 +19N + 7N^2 + 4(bk)^3 \left[ N^2 + N -2 \right] \nonumber \\&\quad + 2(bk)^2 \left[ 2N^3 + N^2 + 9N -8\right] + 2(bk) \left[ N^3 + 8N^2 + 4N -1\right] \bigg ) \bigg ] \bigg \} > 0. \end{aligned}$$
(44)

(44) holds for all \(N\ge 3\). Hence, \(Q^{Post}_{Endog} < Q^{Post}_{Fixed}\). This implies that the price increasing impact of mergers is magnified in the endogenous forward contracting environment.

Third, price increases strictly reduce consumer surplus. The price effects illustrated above imply that consumer surplus is lower post-merger and the reduction in consumer surplus is larger when forward quantities are endogenous. \(\square \)

Proof of Proposition 3

Using (4)–(6), the merged firm’s profit with endogenous forward contracts:

$$\begin{aligned} \pi _M(q_M^f, q_{NM}^f) = \left[ \frac{a - b \left( \beta _M q_M^f + (N-2) \beta _{NM} q_{NM}^f \right) }{1 + B_M} \right] q_M - \frac{(q_M)^2}{2 k_M} \end{aligned}$$
(45)

where \(k_M = 2k\), \(B_M = (N-2) \beta _{NM} + \beta _M\), and

$$\begin{aligned} q_M = \frac{a}{b} \left( \frac{\beta _M}{1 + B_M} \right) - \beta _M \left[ \frac{(N-2) \beta _{NM} q_{NM}^f + \beta _M q_M^f}{1 + B_M} \right] + \beta _M q_M^f. \end{aligned}$$

Using (4)–(6), the merged firm’s profit with fixed pre-merger forward contracts:

$$\begin{aligned} \pi _M(\overline{q}^f) = \left[ \frac{a - b \left( 2\beta _M \overline{q}^f + (N-2) \beta _{NM} \overline{q}^f \right) }{1 + B_M} \right] q_M - \frac{(q_M)^2}{2 k_M} \end{aligned}$$
(46)

where \(q_M = \frac{a}{b} \left( \frac{\beta _M}{1 + B_M} \right) - \beta _M \left[ \frac{(N-2) \beta _{NM} \overline{q}^f + 2 \beta _M \overline{q}^f}{1 + B_M} \right] + 2 \beta _M \overline{q}^f\).

Using \(\overline{q}^f\), \(q_{NM}^f\), and \(q_{M}^f\) defined in (14) and (41) and (45), (46), and Mathematica to simplify the following expression:

$$\begin{aligned}&\pi _M(\overline{q}^f) - \pi _M(q_M^f, q_{NM}^f)> 0 \nonumber \\&\quad \Leftrightarrow 1 - N + bk (17 - 9 N - 6 N^2) + (bk)^2 (112 + 4 N - 68 N^2 - 15 N^3)\nonumber \\&\quad \quad +\, (bk)^3 (398 + 268 N - 237 N^2- 170 N^3 - 20 N^4) - (bk)^4 (-870 - 1206 N\nonumber \\&\qquad + 171 N^2 + 674 N^3 + 214 N^4 + 15 N^5) \nonumber \\&\quad \quad +\, (bk)^5 (1244 + 2738 N + 928 N^2 - 1227 N^3 - 830 N^4 - 145 N^5 - 6 N^6)\nonumber \\&\quad \quad +\, (bk)^7 (1125 + 2923 N+4281 N^2 + 1088 N^3 - 2005 N^4 - 1011 N^5 {-} 133 N^6 {-} 4 N^7) \nonumber \\&\quad \quad +\, 16 (bk)^{12} (4 - 8 N + 13 N^2 - 5 N^3 - 5 N^4 + 10 N^5 - 6 N^6 + N^7)\nonumber \\&\,\,\quad - (bk)^6 (-1269 - 3621 N - 3125 N^2\nonumber \\&\quad \quad +\, 854 N^3 + 1668 N^4 + 512 N^5+ 48 N^6 + N^7) + (bk)^8 (932 + 1846 N \nonumber \\&\,\,\quad + 2741 N^2 + 2576 N^3 - 827 N^4\nonumber \\&\quad \quad -\, 1416 N^5 - 231 N^6 + 6 N^7 + N^8) + 4 (bk)^{11} (44 - 20 N + 151 N^2 - 166 N^3\nonumber \\&\,\,\quad + 182 N^4 - 66 N^5 - 2 N^6 - 10 N^7 + 3 N^8)\nonumber \\&\quad \quad +\, 4 (bk)^{10} (76 + 188 N - 48 N^2 + 250 N^3 + 25 N^4 - 89 N^6 - 2 N^7 + 4 N^8)\nonumber \\&\quad \quad +\, (bk)^9 (688 + 924 N + 1333 N^2 + 1296 N^3 + 709 N^4 - 976 N^5 \nonumber \\&\quad \quad -\, 405 N^6 + 28 N^7 + 7 N^8) > 0. \end{aligned}$$
(47)

For a given \(bk>0\), inequality (47) holds for a sufficiently large value on N. \(\square \)

Appendix 2: Marginal cost estimation

Data on natural gas unit heat rates were obtained from Alberta’s Market Surveillance Administrator (MSA), the Alberta Utility Commission, and the AESO. Coal unit heat rates were obtained from CASA (2004). Data on technology-specific variable O&M costs were obtained from the Energy Information Administration (EIA 2016). Hourly natural gas prices were obtained from Alberta’s Natural Gas Exchange. We use weekly coal prices from Wyoming’s Powder River Basin (PRB) from the Energy Information Administration to estimate the marginal cost of coal units in Alberta.Footnote 53 PRB coal is sub-bituminuous coal which is the primary coal used for electricity generation in Alberta. Further, PRB coal and Alberta’s sub-bituminuous coal have similar heat-rate contents (Alberta Energy, 2014). We adjusted the PRB coal prices and USD based variable O&M costs from USD to CAD using Bank of Canada exchange rates. The environmental compliance costs are analogous to those detailed in Brown and Olmstead (2017, forthcoming). We obtained asset-level generation capacity from MSA. Derated forced outage statistics were obtained from the North American Electric Reliability Corporation’s Generation Availability Data System Reports.

Appendix 3: 100% increase in price-elasticity of residual demand

Tables 13, 14 and 15 demonstrate that the key qualitative conclusions of our analysis are robust to a 100% increase in the average price-elasticity of residual demand.

Table 13 Actual versus estimated market outcomes—100% increase
Table 14 Merger effects: changes in average market prices—100% increase
Table 15 Percentage changes in combined profit of merging firms—100% increase

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Brown, D.P., Eckert, A. Electricity market mergers with endogenous forward contracting. J Regul Econ 51, 269–310 (2017). https://doi.org/10.1007/s11149-017-9323-0

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Keywords

  • Electricity
  • Mergers
  • Forward contracts
  • Market power
  • Regulation

JEL Classification

  • D43
  • L40
  • L51
  • L94
  • Q40