Some inequalities related to the analysis of electricity auctions

Most balancing markets of electric power are organized as uniform-price auctions. In 2001, the balancing market of England and Wales switched to a pay-as-bid auction with the intention of reducing wholesale electricity prices. Numerical simulations of an electricity auction model have indicated that this should lead to decreased average prices. In this work we prove two inequalities which give an analytic proof of this claim in the same model.