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Incentives for efficient pricing mechanism in markets with non-convexities

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Abstract

This paper examines the incentives for efficient pricing mechanism in markets with non-convexities. The wholesale electricity market is a prominent example. Ideally, an efficient pricing mechanism produces market signals that reflect costs and scarcities, incents price-taking behavior and yields sufficient revenues to attract new investment. However, under non-convex conditions, there is no assurance that these goals can be fully achieved, and market equilibrium may not even exist. Previous studies on markets with convexities have been focusing on the revenue sufficiency problem. Positive results on incentives are relatively scarce. This paper is intended to fill the gap. With non-convexities, quasi-equilibrium entails solving separately a non-convex allocation model and a convexified pricing model with solution support payments in settlement. We consider three convex relaxation methods, including Lagrangian dualization, convex-hull relaxation and integer relaxation (Integer relaxation refers to a convex relaxation of mixed integer programing problem in which the integer variables are linearized). We show that quasi-equilibrium pricing is dominant strategy incentive compatible in the limit and the total side payment divided by the total surplus approaches zero when the market size (e.g., measured by the number of consumers) increases to infinity. In essence, the quasi-equilibrium pricing mechanism extends efficient pricing principles from a convex market environment to one that is non-convex in ways that preserve economic efficiency, incentive compatibility and revenue sufficiency. These results are illustrated in the context of wholesale electricity markets. Since 2014, price formation issues have been vigorously debated in the U.S. including FERC’s conferences and proceedings with comments from academics, policy and business communities across ISO/RTO regions. Convex-hull pricing is generally considered an ideal solution but it remains computationally prohibitive. In this paper, we identify conditions under which the integer relaxation method can produce close and sometimes even exact approximations to convex-hull pricing. In April 2019, FERC authorized the use of integer relaxation as a just and reasonable pricing method for fast-start units in PJM’s energy markets.

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Notes

  1. Under convex conditions, the average cost (as well as the incremental cost) of production does not decline when the output increases and does not rise when the output decreases.

  2. Arrow and Debreu 1954; Arrow and Hahn (1971).

  3. As is well known, Adam Smith created the famous metaphor of “invisible hand” with the conjecture that the invisible hand will guide efficient exchange of scarce resources through competition in the marketplace. The late Stanford University Professor Kenneth J. Arrow, a winner of Economics Nobel Prize in 1972, offered fundamental insights in social choice theory laying the foundation for fundamental theorems of welfare economics setting out the precise conditions under which the “invisible hand” conjecture holds true so that the general competitive equilibrium can achieve efficient outcome and maximize social welfare.

  4. Dantzig (1948).

  5. Vickrey (1962).

  6. Schweppee et al. (1980), Hogan (1992).

  7. See O’Neill et al. (2001).

  8. See Chao and Huntington (1998).

  9. Chao (2018) Challenges for Getting the Prices Right in PJM’s Wholesale Electricity Markets, Harvard Energy Policy Seminar, March 26. Retrieved from https://sites.hks.harvard.edu/m-rcbg/cepr/HKS%20Energy%20Policy%20Seminar%20-%20Chao%2020180326.pdf. Chao (2019) Electricity market reform to enhance the energy and reserve pricing mechanism: Observations from PJM, Energy Systems Workshop at Isaac Newton Institute, University of Cambridge, January 7, 2019. Retrieved from http://www.newton.ac.uk/files/seminar/20190107160017001-1481148.pdf.

  10. Unlike linear pricing which maintains the same price per unit, nonlinear pricing includes multi-part pricing, price menu with differentiated options, and a variety of price schedules with nonlinear structures.

  11. See Ring (1995), Hogan and Ring (2003), Sioshansi et al. (2008), and Gribik et al. (2007).

  12. PJM (2019a, b).

  13. Under the assumption of a quasi-linear utility function with a numeraire good denoted by \(x_{i}^{0}\) and maximum expenditure by \(m_{i}\), a consumer’s problem of utility maximization subject to budget constraint is equivalent to the maximization of the consumer’s surplus absent the budget constraint:

    \(\mathop {\hbox{Max} }\limits_{{x_{i}^{0} \in R,\varvec{x}_{i} \in R^{K} }} \left\{ {u_{i} \left( {\varvec{x}_{i} } \right) + x_{i}^{0} |\varvec{p} \cdot \varvec{x}_{i} + x_{i}^{0} = m_{i} } \right\} = \mathop {\hbox{max} }\limits_{{\varvec{x}_{i} \in R^{K} }} u_{i} \left( {\varvec{x}_{i} } \right) - \varvec{p} \cdot \varvec{x}_{i} + m_{i}\)

  14. The real-time market clearing model includes multiple periods and look ahead. Prices are calculated for a time window containing the day of the real-time market on a rolling basis as day progresses.

  15. The demand is defined as net of behind-the-meter generation or self-scheduled generation. Self-scheduling reveals participant’s willingness to behave as a pure price taker. Self-scheduled units are not included in the economic dispatch choices and therefore are not treated as dispatch choices that set prices in the pricing model.

  16. Chao and Peck (1996).

  17. PJM (2019a, b).

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Acknowledgements

The author is very grateful to Stu Bresler, Yonghong Chen, Richard Cottle, Robert Entriken, Anthony Giacomoni, Paul Gribik, William Hogan, Adam Keech, Alberto Lamadrid, Javad Lavaei, Richard O’Neill, Shmuel Oren, Asanga Perera, Congcong Wang, Peter Whitman, Jim Wilson, and Robert Wilson for their helpful comments, and to participants at the Harvard Electricity Policy Group (HEPG) workshop on January 26, 2018, Energy Policy Seminar at Harvard Kennedy School on March 26, 2018, the CRRI Eastern Conferences on June 7, 2018, and Energy Systems Workshop at Isaac Newton Institute, University of Cambridge, on January 7, 2019, for helpful discussions. The views and any remaining errors remain those of the author.

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Chao, Hp. Incentives for efficient pricing mechanism in markets with non-convexities. J Regul Econ 56, 33–58 (2019). https://doi.org/10.1007/s11149-019-09385-w

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  • DOI: https://doi.org/10.1007/s11149-019-09385-w

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