Dudey, Marc (1992), "Dynamic Edgeworth-Bertrand Competition." Quarterly Journal of Economics, 107(4), 1461-1477.

"A popular criticism of the duopoly models developed by Bertrand [1883] and Edgeworth [1897] is that their predictions are inconsistent with observed duopoly behavior. In Bertrand's model, 
profit-maximizing sellers with the same constant cost technology choose prices; in the only equilibrium the sellers set prices equal to marginal cost and earn zero profit. Edgeworth showed that, when capacity constraints are introduced into Bertrand's model, a pure strategy equilibrium will not generally exist.' Unfortunately, casual observation suggests that duopolists earn positive profits and that duopoly pricing behavior is relatively stable. 
This paper develops a simple, multiperiod variant of the Edgeworth model that is not susceptible to this criticism. In the model, price-setting and capacity-constrained duopolists meet consumers with unit demands and a common reservation value; by contrast with the static paradigm, the consumers come to market at different times, and the duopolists may change their prices at any time. The model has a (subgame perfect) equilibrium in pure strategies and unique equilibrium payoffs for any specification of the duopolists' capacities. Moreover, so long as at least one seller cannot supply the entire market, both sellers earn positive profit, and at least one of the duopolists can sell all the units it is able to produce."