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Open versus closed loop capacity equilibria in electricity markets under perfect and oligopolistic competition

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Abstract

We consider two game-theoretic models of the generation capacity expansion problem in liberalized electricity markets. The first is an open loop equilibrium model, where generation companies simultaneously choose capacities and quantities to maximize their individual profit. The second is a closed loop model, in which companies first choose capacities maximizing their profit anticipating the market equilibrium outcomes in the second stage. The latter problem is an equilibrium problem with equilibrium constraints. In both models, the intensity of competition among producers in the energy market is frequently represented using conjectural variations. Considering one load period, we show that for any choice of conjectural variations ranging from perfect competition to Cournot, the closed loop equilibrium coincides with the Cournot open loop equilibrium, thereby obtaining a ‘Kreps and Scheinkman’-like result and extending it to arbitrary strategic behavior. When expanding the model framework to multiple load periods, the closed loop equilibria for different conjectural variations can diverge from each other and from open loop equilibria. We also present and analyze alternative conjectured price response models with switching conjectures. Surprisingly, the rank ordering of the closed loop equilibria in terms of consumer surplus and market efficiency (as measured by total social welfare) is ambiguous. Thus, regulatory approaches that force marginal cost-based bidding in spot markets may diminish market efficiency and consumer welfare by dampening incentives for investment. We also show that the closed loop capacity yielded by a conjectured price response second stage competition can be less or equal to the closed loop Cournot capacity, and that the former capacity cannot exceed the latter when there are symmetric agents and two load periods.

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Notes

  1. For completeness, let us consider the explicit non-negativity constraint \(0 \le q_{i}\) in the optimization problem (6) and let us define \(\mu _i\ge 0\) as the corresponding dual variable. Then, due to complementarity conditions arising from the KKT conditions, we can separate two cases, the one where \(\mu _i=0\) and the other where \(\mu _i>0\). The first case will lead us to the solution presented in the paper, and case \(\mu _i>0\) will lead us to a solution where \(\mu _i = t (\delta - P_0)\). Considering that we assumed \(P_0 > \delta \), this yields a contradiction to the non-negativity of \(\mu _i\). Hence, this cannot be the case and therefore we omit the non-negativity constraint.

  2. Taking the first derivative of the objective function in (6) with respect to \(q_i\) yields: \(t p(q_i,q_{-i}) - t \theta q_i - t \delta \). Then, the second derivative is \(- 2 t \theta \), which is smaller or equal to zero for each value of \(\theta \) in \([0,1/\alpha ]\), which yields concavity of the objective function.

  3. Let \(D_0 =1, t=1, \alpha =1, \beta =1/2\) and \(\delta =0\), then the open loop Cournot solution is \(p = 2/3\), with \(x = 1/6\) for each firm. In this case, with these cost numbers, the open loop Cournot equals the closed loop equilibrium with \(\theta =2/ \alpha \) (collusion, \(\varPhi = 1\)).

  4. Total Profit is defined as \(\sum _l t_l (p_l - \delta ) (q_{il}+q_{-il}) - \beta (x_i + x_{-i})\). CS is defined as the integral of the demand curve minus payments for energy, equal here to \(\sum _l t_l (P_{0l}-p_l) (q_{il}+q_{-il})/2\). Market efficiency (ME) is defined as CS plus total profits.

  5. This is proven by demonstrating that for smaller \(\theta \), the second stage prices will be lower and closer to marginal operating cost in load periods for those periods that capacity is not binding.

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Acknowledgments

The first author was partially supported by Endesa and also thanks EPRG for hosting her visit to the University of Cambridge in July 2010. The second author was supported by the UK EPSRC Supergen Flexnet funding and the US National Science Foundation, EFRI Grant 0835879. The authors would like to thank Frederic Murphy for providing helpful comments. We also thank two anonymous referees for their comments.

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Authors and Affiliations

  1. Instituto de Investigación Tecnológica, Escuela Técnica Superior de Ingeniería (ICAI), Universidad Pontificia Comillas, 28015 , Madrid, Spain

    S. Wogrin, E. Centeno & J. Barquín

  2. Department of Geography and Environmental Engineering, and Environment, Energy, Sustainability and Health Institute, The Johns Hopkins University, Baltimore, MD, 21218, USA

    B. F. Hobbs

  3. Cambridge Judge Business School and Electricity Policy Research Group, University of Cambridge, Cambridge, CB2 1AG, UK

    D. Ralph

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  1. S. Wogrin
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Correspondence to S. Wogrin.

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Wogrin, S., Hobbs, B.F., Ralph, D. et al. Open versus closed loop capacity equilibria in electricity markets under perfect and oligopolistic competition. Math. Program. 140, 295–322 (2013). https://doi.org/10.1007/s10107-013-0696-2

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