Bertrand-Edgeworth equilibrium with a large number of firms

We examine a model of price competition where the firms simultaneously decide on both price and quantity, and are free to supply less than the quantity demanded. We demonstrate that if the tie-breaking rule is `non-manipulable', then, for a large class of rationing rules, there is a unique equilibrium in pure strategies whenever the number of firms is large enough. We then show that the `folk theorem' of perfect competition holds. Finally, we examine if the results go through when the firms are asymmetric, or produce to order.