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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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    Cited by:

    1. Edward Anderson & Pär Holmberg, 2023. "Multi-unit auctions with uncertain supply and single-unit demand," Working Papers EPRG2310, Energy Policy Research Group, Cambridge Judge Business School, University of Cambridge.
    2. Heim, Sven & Götz, Georg, 2021. "Do Pay-As-Bid Auctions Favor Collusion? Evidence from Germany's market for reserve power," Energy Policy, Elsevier, vol. 155(C).
    3. Pär Holmberg, 2017. "Pro‐competitive Rationing in Multi‐unit Auctions," Economic Journal, Royal Economic Society, vol. 127(605), pages 372-395, October.
    4. 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.
    5. Baisa, Brian & Burkett, Justin, 2020. "Discriminatory price auctions with resale and optimal quantity caps," Theoretical Economics, Econometric Society, vol. 15(1), January.
    6. 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.
    7. Ollikka, Kimmo & Tukiainen, Janne, 2013. "Central bank liquidity auction mechanism design and the interbank market," Research Discussion Papers 21/2013, Bank of Finland.
    8. Willems, Bert & Pollitt, Michael & von der Fehr, Nils-Henrik & Banet, Catherine, 2022. "The European Wholesale Electricty Market: From Crisis to Net Zero," Other publications TiSEM 2f225964-853e-4d30-a46d-0, Tilburg University, School of Economics and Management.
    9. Pär Holmberg & Frank Wolak, 2015. "Electricity markets: Designing auctions where suppliers have uncertain costs," Working Papers EPRG 1523, Energy Policy Research Group, Cambridge Judge Business School, University of Cambridge.
    10. Ollikka, Kimmo & Tukiainen, Janne, 2013. "Central bank liquidity auction mechanism design and the interbank market," Bank of Finland Research Discussion Papers 21/2013, Bank of Finland.
    11. 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.
    12. Pycia, Marek & Woodward, Kyle, 2021. "Auctions of Homogeneous Goods: A Case for Pay-as-Bid," CEPR Discussion Papers 15656, C.E.P.R. Discussion Papers.
    13. Cumpston, Anne & Khezr, Peyman, 2020. "Multi-Unit Auctions: A Survey of Theoretical Literature," MPRA Paper 101336, University Library of Munich, Germany.
    14. 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.
    15. Wittwer, Milena, 2020. "Interconnected pay-as-bid auctions," Games and Economic Behavior, Elsevier, vol. 121(C), pages 506-530.
    16. Burkett, Justin & Woodward, Kyle, 2020. "Uniform price auctions with a last accepted bid pricing rule," Journal of Economic Theory, Elsevier, vol. 185(C).

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    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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