# Incentives for efficient pricing mechanism in markets with non-convexities

## 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.

## Keywords

Efficient pricing mechanism Incentive compatibility Non-convexity Electricity market Quasi-equilibrium Integer relaxation## JEL Classification

C61 D41 D44 D47 D82 L94 Q41## Notes

### 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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