In this paper, we postulate a general class of price competition models with Mixed Multi Nomial Logit demand functions under affine cost functions. We characterize the equilibrium behavior of this class of models starting with the case where each product in the market is sold by a separate, independent firm. Here we identify a natural upper bound for the price levels. Assuming the prices are restricted by these bounds, we show that a pure Nash equilibrium exists in the interior of this feasible price region and that the set of equilibria is given by the solutions of the system of First Order Condition equations. This provides a justification for the many structural estimation methods which require that equilibria correspond with the solutions to this system of FOC equations. We provide an example that shows that a condition like ours is necessary for the existence of a Nash equilibrium. We show that the upper bound for the prices can be assumed, without loss of generality, under an intuitive, widely applicable and easily verified condition. We also provide conditions when the Nash equilibrium is unique and computable with a simple computational scheme. The above results are generalized to the general multi-product case where several products may be sold by the same firm.