Determining optimal prices in non-convex markets remains an unsolved challenge. Non-convex costs are critical in electricity markets, as startup costs and minimum operating levels yield a non-convex optimal value function over demand levels. While past research largely focuses on the performance of different non-convex pricing frameworks in the short-run or uses convex approximations, we determine long-run adapted resource mixes associated with each pricing framework while preserving the full extent of the non-convex operations. We frame optimal pricing in terms of social surplus achieved and transfer of consumer to producer surplus in adapted long-run market equilibria. We find that convex hull pricing achieves the highest social surplus and is also associated with the lowest transfer of consumer to producer surplus. Marginal prices determined by fixing integer variables to their optimal values in the pricing run are also associated with high social surplus and high consumer surplus when the optimality gap in the original mixed integer linear program is very small. Other pricing frameworks tend to over-compensate inframarginal units, leading to resource mixes with lower social surplus and a greater transfer of consumer surplus to producer surplus in the long-run.

在非凸市场中确定最优价格仍然是一个未解决的挑战。非凸成本在电力市场中至关重要，因为启动成本和最低运营水平会在需求水平上产生非凸最优价值函数。虽然过去的研究主要集中在短期内不同非凸定价框架的性能或使用凸近似，但我们确定与每个定价框架相关的长期适应资源组合，同时保留非凸操作的全部范围。我们根据在适应的长期市场均衡中实现的社会剩余和消费者剩余向生产者剩余的转移来构建最优定价。我们发现凸包定价实现了最高的社会剩余，并且与消费者剩余向生产者剩余的最低转移相关。当原始混合整数线性规划中的最优差距非常小时，通过在定价运行中将整数变量固定为其最优值确定的边际价格也与高社会剩余和高消费者剩余相关联。其他定价框架倾向于过度补偿超边际单位，从而导致资源混合，社会剩余较低，从长远来看，消费者剩余更多地转移到生产者剩余。