We model a pay-as-bid electricity market as equilibrium problem with equilibrium constraints (EPEC) with an Independent System Operator (ISO) playing a central role. We find analytic solution to the ISO problem assuming quadratic bid functions of producers. Then, we derive also the best response of a producer corresponding to its optimal bid function. Finally, we provide a full characterization of Nash equilibria in such an electricity market. We show that two distinct classes of equilibria arise, corresponding to monopolistic or fully competitive market. For both situations we provide analytic formulae for the respective market prices.