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Supply Function Equilibria of Pay-as-Bid Auctions

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Abstract

This paper characterizes the Nash equilibrium in a pay-as-bid (discriminatory), divisible-good, procurement auction. Demand by the auctioneer is uncertain as in the supply function equilibrium model. A closed form expression is derived. Existence of an equilibrium is ensured if the hazard rate of the perfectly inelastic demand is monotonically decreasing and sellers have non-decreasing marginal costs. Multiple equilibria can be ruled out for markets, for which the auctioneer’s demand exceeds suppliers’ capacity with a positive probability. The derived equilibrium can be used to model strategic bidding behaviour in pay-as-bid electricity auctions, such as the balancing mechanism of United Kingdom. Offer curves and mark-ups of the derived equilibrium are compared to results for the SFE of a uniform-price auction.

Suggested Citation

  • Holmberg, Pär, 2009. "Supply Function Equilibria of Pay-as-Bid Auctions," Working Paper Series 787, Research Institute of Industrial Economics.
  • Handle: RePEc:hhs:iuiwop:0787
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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. Rassenti, Stephen J & Smith, Vernon L & Wilson, Bart J, 2003. "Discriminatory Price Auctions in Electricity Markets: Low Volatility at the Expense of High Price Levels," Journal of Regulatory Economics, Springer, vol. 23(2), pages 109-123, March.
    3. Bulow, Jeremy & Klemperer, Paul, 1996. "Auctions versus Negotiations," American Economic Review, American Economic Association, vol. 86(1), pages 180-194, March.
    4. Kahn, Alfred E. & Cramton, Peter C. & Porter, Robert H. & Tabors, Richard D., 2001. "Uniform Pricing or Pay-as-Bid Pricing: A Dilemma for California and Beyond," The Electricity Journal, Elsevier, vol. 14(6), pages 70-79, July.
    5. Holmberg, Pär, 2005. "Comparing Supply Function Equilibria of Pay-as-Bid and Uniform-Price Auctions," Working Paper Series 2005:17, Uppsala University, Department of Economics.
    6. Robert Wilson, 2008. "Supply Function Equilibrium in a Constrained Transmission System," Operations Research, INFORMS, vol. 56(2), pages 369-382, April.
    7. Paul Klemperer, 2002. "What Really Matters in Auction Design," Journal of Economic Perspectives, American Economic Association, vol. 16(1), pages 169-189, Winter.
    8. Federico, Giulio & Rahman, David, 2003. "Bidding in an Electricity Pay-as-Bid Auction," Journal of Regulatory Economics, Springer, vol. 24(2), pages 175-211, September.
    9. Krishna, Vijay, 2009. "Auction Theory," Elsevier Monographs, Elsevier, edition 2, number 9780123745071.
    10. Roques Fabien A. & Newbery David M. & Nuttall William J., 2005. "Investment Incentives and Electricity Market Design: the British Experience," Review of Network Economics, De Gruyter, vol. 4(2), pages 1-36, June.
    11. 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.
    12. Bower, John & Bunn, Derek, 2001. "Experimental analysis of the efficiency of uniform-price versus discriminatory auctions in the England and Wales electricity market," Journal of Economic Dynamics and Control, Elsevier, vol. 25(3-4), pages 561-592, March.
    13. Bolle, Friedel, 1992. "Supply function equilibria and the danger of tacit collusion : The case of spot markets for electricity," Energy Economics, Elsevier, vol. 14(2), pages 94-102, April.
    14. 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.
    15. Holmberg, Par, 2008. "Unique supply function equilibrium with capacity constraints," Energy Economics, Elsevier, vol. 30(1), pages 148-172, January.
    16. Natalia Fabra, 2003. "Tacit Collusion in Repeated Auctions: Uniform Versus Discriminatory," Journal of Industrial Economics, Wiley Blackwell, vol. 51(3), pages 271-293, September.
    17. repec:bpj:rneart:v:4:y:2005:i:2:p:93-128 is not listed on IDEAS
    18. 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.
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    21. Wilson, Robert, 1997. "Nonlinear Pricing," OUP Catalogue, Oxford University Press, number 9780195115826.
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    Citations

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    Cited by:

    1. Heim, Sven & Götz, Georg, 2013. "Do pay-as-bid auctions favor collusion? Evidence from Germany's market for reserve power," ZEW Discussion Papers 13-035, ZEW - Zentrum für Europäische Wirtschaftsforschung / Center for European Economic Research.
    2. Pär Holmberg, 2014. "Pro-competitive rationing in multi-unit auctions," Cambridge Working Papers in Economics 1435, Faculty of Economics, University of Cambridge.
    3. 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.
    4. Bunn, Derek & Koc, Veli & Sapio, Alessandro, 2015. "Resource externalities and the persistence of heterogeneous pricing behavior in an energy commodity market," Energy Economics, Elsevier, vol. 48(C), pages 265-275.
    5. Ollikka, Kimmo & Tukiainen, Janne, 2013. "Central bank liquidity auction mechanism design and the interbank market," Research Discussion Papers 21/2013, Bank of Finland.
    6. 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.
    7. Majid Motamedi & Shahram Moeeni & Salman Gharakhani & Iman Keyfarokhi, 2014. "The Behavior of Iranian Restructured Electricity Market in Supply Function Equilibrium Framework," International Journal of Academic Research in Business and Social Sciences, Human Resource Management Academic Research Society, International Journal of Academic Research in Business and Social Sciences, vol. 4(1), pages 178-191, January.

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    Keywords

    Supply Function Equilibrium; Pay-as-bid Auction; Discriminatory Auction; Divisible;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • L11 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Production, Pricing, and Market Structure; Size Distribution of Firms
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets
    • L94 - Industrial Organization - - Industry Studies: Transportation and Utilities - - - Electric Utilities

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