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Supply function equilibria: Step functions and continuous representations

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  • Holmberg, Pär
  • Newbery, David
  • Ralph, Daniel

Abstract

In most electricity markets generators must submit step-function offers to a uniform price auction. These markets are often modelled as simpler pure-strategy Supply Function Equilibria (SFE) with continuous supply functions. Critics argue that the discreteness and discontinuity of the required steps drastically change Nash equilibria, invalidating predictions of the SFE model. We prove that there are sufficient conditions, offered quantities can be continuously varied, offered prices are selected from a finite set, and the density of the additive demand shock is not too steep, where the resulting stepped SFE converges to the continuous SFE as the number of steps increases, reconciling the apparently very disparate approaches to modelling electricity markets.

Suggested Citation

  • Holmberg, Pär & Newbery, David & Ralph, Daniel, 2013. "Supply function equilibria: Step functions and continuous representations," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1509-1551.
  • Handle: RePEc:eee:jetheo:v:148:y:2013:i:4:p:1509-1551 DOI: 10.1016/j.jet.2013.04.020
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    1. Natalia Fabra & Nils‐Henrik Fehr & David Harbord, 2006. "Designing electricity auctions," RAND Journal of Economics, RAND Corporation, vol. 37(1), pages 23-46, March.
    2. Jakub Kastl, 2011. "Discrete Bids and Empirical Inference in Divisible Good Auctions," Review of Economic Studies, Oxford University Press, vol. 78(3), pages 974-1014.
    3. repec:rje:randje:v:37:y:2006:1:p:23-46 is not listed on IDEAS
    4. Frank A. Wolak, 2007. "Quantifying the supply-side benefits from forward contracting in wholesale electricity markets," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(7), pages 1179-1209.
    5. David McAdams, 2003. "Isotone Equilibrium in Games of Incomplete Information," Econometrica, Econometric Society, vol. 71(4), pages 1191-1214, July.
    6. Robert Wilson, 1979. "Auctions of Shares," The Quarterly Journal of Economics, Oxford University Press, vol. 93(4), pages 675-689.
    7. Robert Wilson, 2008. "Supply Function Equilibrium in a Constrained Transmission System," Operations Research, INFORMS, pages 369-382.
    8. repec:spr:compst:v:60:y:2004:i:2:p:215-238 is not listed on IDEAS
    9. Frank Wolak, 2000. "An Empirical Analysis of the Impact of Hedge Contracts on Bidding Behavior in a Competitive Electricity Market," International Economic Journal, Taylor & Francis Journals, vol. 14(2), pages 1-39.
    10. Green, Richard J & Newbery, David M, 1992. "Competition in the British Electricity Spot Market," Journal of Political Economy, University of Chicago Press, vol. 100(5), pages 929-953, October.
    11. Holmberg, Pär, 2009. "Numerical calculation of an asymmetric supply function equilibrium with capacity constraints," European Journal of Operational Research, Elsevier, vol. 199(1), pages 285-295, November.
    12. Supatgiat, Chonawee & Zhang, Rachel Q & Birge, John R, 2001. "Equilibrium Values in a Competitive Power Exchange Market," Computational Economics, Springer;Society for Computational Economics, vol. 17(1), pages 93-121, February.
    13. Parisio, Lucia & Bosco, Bruno, 2003. "Market Power and the Power Market: Multi-unit Bidding and (In)Efficiency in Electricity Auctions," International Tax and Public Finance, Springer;International Institute of Public Finance, vol. 10(4), pages 377-401, August.
    14. Holmberg, Pär & Newbery, David & Ralph, Daniel, 2013. "Supply function equilibria: Step functions and continuous representations," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1509-1551.
    15. David M. Newbery, 1998. "Competition, Contracts, and Entry in the Electricity Spot Market," RAND Journal of Economics, The RAND Corporation, pages 726-749.
    16. Ilan Kremer & Kjell G. Nyborg, 2004. "Divisible-Good Auctions: The Role of Allocation Rules," RAND Journal of Economics, The RAND Corporation, vol. 35(1), pages 147-159, Spring.
    17. Green, Richard J, 1996. "Increasing Competition in the British Electricity Spot Market," Journal of Industrial Economics, Wiley Blackwell, vol. 44(2), pages 205-216, June.
    18. Andrew Sweeting, 2007. "Market Power In The England And Wales Wholesale Electricity Market 1995-2000," Economic Journal, Royal Economic Society, vol. 117(520), pages 654-685, April.
    19. Genc, Talat S. & Reynolds, Stanley S., 2011. "Supply function equilibria with capacity constraints and pivotal suppliers," International Journal of Industrial Organization, Elsevier, vol. 29(4), pages 432-442, July.
    20. Anderson, Edward, 2011. "Supply Function Equilibria Always Exist," Working Papers 04/2011, University of Sydney Business School, Discipline of Business Analytics.
    21. Par Holmberg, 2007. "Supply Function Equilibrium with Asymmetric Capacities and Constant Marginal Costs," The Energy Journal, International Association for Energy Economics, vol. 0(Number 2), pages 55-82.
    22. Ramteen Sioshansi & Shmuel Oren, 2007. "How good are supply function equilibrium models: an empirical analysis of the ERCOT balancing market," Journal of Regulatory Economics, Springer, vol. 31(1), pages 1-35, February.
    23. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Oxford University Press, vol. 53(1), pages 1-26.
    24. E. J. Anderson & H. Xu, 2004. "Nash equilibria in electricity markets with discrete prices," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(2), pages 215-238, October.
    25. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
    26. Klemperer, Paul D & Meyer, Margaret A, 1989. "Supply Function Equilibria in Oligopoly under Uncertainty," Econometrica, Econometric Society, vol. 57(6), pages 1243-1277, November.
    27. Ilan Kremer, 2004. "Underpricing and Market Power in Uniform Price Auctions," Review of Financial Studies, Society for Financial Studies, vol. 17(3), pages 849-877.
    28. David M. Newbery, 1995. "Power Markets and Market Power," The Energy Journal, International Association for Energy Economics, vol. 0(Number 3), pages 39-66.
    29. Ali Hortaçsu & Steven L. Puller, 2008. "Understanding strategic bidding in multi-unit auctions: a case study of the Texas electricity spot market," RAND Journal of Economics, RAND Corporation, vol. 39(1), pages 86-114.
    30. Bagh, Adib, 2010. "Variational convergence: Approximation and existence of equilibria in discontinuous games," Journal of Economic Theory, Elsevier, vol. 145(3), pages 1244-1268, May.
    31. Ross Baldick & Ryan Grant & Edward Kahn, 2004. "Theory and Application of Linear Supply Function Equilibrium in Electricity Markets," Journal of Regulatory Economics, Springer, vol. 25(2), pages 143-167, March.
    32. von der Fehr, Nils-Henrik Morch & Harbord, David, 1993. "Spot Market Competition in the UK Electricity Industry," Economic Journal, Royal Economic Society, vol. 103(418), pages 531-546, May.
    33. Aleksandr Rudkevich & Max Duckworth & Richard Rosen, 1998. "Modeling Electricity Pricing in a Deregulated Generation Industry: The Potential for Oligopoly Pricing in a Poolco," The Energy Journal, International Association for Energy Economics, vol. 0(Number 3), pages 19-48.
    34. Leo K. Simon, 1987. "Games with Discontinuous Payoffs," Review of Economic Studies, Oxford University Press, vol. 54(4), pages 569-597.
    35. Kastl, Jakub, 2012. "On the properties of equilibria in private value divisible good auctions with constrained bidding," Journal of Mathematical Economics, Elsevier, vol. 48(6), pages 339-352.
    36. Frank A. Wolak, 2001. "Identification and Estimation of Cost Functions Using Observed Bid Data: An Application to Electricity Markets," NBER Working Papers 8191, National Bureau of Economic Research, Inc.
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    Cited by:

    1. Xavier Vives, 2011. "Strategic Supply Function Competition With Private Information," Econometrica, Econometric Society, pages 1919-1966.
    2. Pär Holmberg, 2014. "Pro-competitive rationing in multi-unit auctions," Cambridge Working Papers in Economics 1435, Faculty of Economics, University of Cambridge.
    3. David Hudgins, 2012. "Optimal Willingness to Supply Wholesale Electricity under Asymmetric Linearized Marginal Costs," International Journal of Energy Economics and Policy, Econjournals, vol. 2(4), pages 307-317.
    4. Haraguchi, Junichi & Yasui, Yuta, 2017. "Supply Function Equilibria and Nonprofit-Maximizing Objectives," MPRA Paper 82394, University Library of Munich, Germany.
    5. F. Delbono & L. Lambertini, 2015. "Investigating the Strategic Nature of Supply Functions in Oligopoly," Working Papers wp1011, Dipartimento Scienze Economiche, Universita' di Bologna.
    6. repec:eee:eneeco:v:68:y:2017:i:c:p:124-132 is not listed on IDEAS
    7. Holmberg, Pär & Willems, Bert, 2015. "Relaxing competition through speculation: Committing to a negative supply slope," Journal of Economic Theory, Elsevier, vol. 159(PA), pages 236-266.
    8. Manzano, Carolina & Vives, Xavier, 2016. "Market Power and Welfare in Asymmetric Divisible Good Auctions," CEPR Discussion Papers 11731, C.E.P.R. Discussion Papers.
    9. Holmberg, Pär & Newbery, David & Ralph, Daniel, 2013. "Supply function equilibria: Step functions and continuous representations," Journal of Economic Theory, Elsevier, pages 1509-1551.
    10. Holmberg, Pär & Newbery, David, 2010. "The supply function equilibrium and its policy implications for wholesale electricity auctions," Utilities Policy, Elsevier, vol. 18(4), pages 209-226, December.
    11. Delbono, Flavio & Lambertini, Luca, 2016. "Ranking Bertrand, Cournot and supply function equilibria in oligopoly," Energy Economics, Elsevier, vol. 60(C), pages 73-78.
    12. Eiji Sawada, 2016. "Effect of electricity system reform on retail electricity price increases in Japan," Journal of Economic Structures, Springer;Pan-Pacific Association of Input-Output Studies (PAPAIOS), vol. 5(1), pages 1-14, December.
    13. Pepermans, Guido & Willems, Bert, 2010. "Cost Recovery in Congested Electricity Networks," Working Papers 2010/22, Hogeschool-Universiteit Brussel, Faculteit Economie en Management.
    14. Bolle, Friedel & Grimm, Veronika & Ockenfels, Axel & del Pozo, Xavier, 2013. "An experiment on supply function competition," European Economic Review, Elsevier, vol. 63(C), pages 170-185.
    15. Kimbrough, Steven O. & Murphy, Frederic H., 2013. "Strategic bidding of offer curves: An agent-based approach to exploring supply curve equilibria," European Journal of Operational Research, Elsevier, vol. 229(1), pages 165-178.
    16. Vasin, A. & Daylova, E., 2013. "Analysis of the Short-Term Efficiency of Mechanisms of the Wholesale Electricity Market," Journal of the New Economic Association, New Economic Association, vol. 18(2), pages 35-60.
    17. Weron, Rafał, 2014. "Electricity price forecasting: A review of the state-of-the-art with a look into the future," International Journal of Forecasting, Elsevier, vol. 30(4), pages 1030-1081.
    18. Pär Holmberg & Frank Wolak, 2015. "Electricity markets: Designing auctions where suppliers have uncertain costs," Cambridge Working Papers in Economics 1541, Faculty of Economics, University of Cambridge.
    19. F. Delbono & L. Lambertini, 2015. "Cournot Retrouvé under Price or Supply Function Competition," Working Papers wp1003, Dipartimento Scienze Economiche, Universita' di Bologna.
    20. Edward Anderson & Pär Holmberg, 2015. "Price Instability in Multi-Unit Auctions," Cambridge Working Papers in Economics 1538, Faculty of Economics, University of Cambridge.
    21. G P Girish & Aviral Kumar Tiwari, 2016. "A comparison of different univariate forecasting models forSpot Electricity Price in India," Economics Bulletin, AccessEcon, vol. 36(2), pages 1039-1057.

    More about this item

    Keywords

    Multi-unit auctions; Supply function equilibria; Convergence of Nash equilibria; Wholesale electricity markets;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • L94 - Industrial Organization - - Industry Studies: Transportation and Utilities - - - Electric Utilities

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