Mixed Integer Social Welfare Maximization (MI-SWM) and implications in optimal electricity pricing

In microeconomics, social welfare maximization (SWM) is formulated as a continuous problem, with the conclusion that marginal cost (MC) pricing is optimal in that it maximizes the total social welfare for consumers and producers. This is based on assumptions such as free entry/exit, no transaction costs, no externalities, etc. However, this paper reports that when the SWM problem is formulated as a mixed integer programming problem considering the constants of cost functions, start-up costs, and minimum production levels, the optimal prices can depart from marginal costs. The paper does not try to negate marginal cost pricing, but rather serves as a supplement to power economics