A general model of oligopoly endogenizing Cournot, Bertrand, Stackelberg, and Allaz-Vila

In many industries, firms pre-order input and forward sell output prior to the actual production period. It is known that forward buying input induces a "Cournot-Stackelberg endogeneity" (both Cournot and Stackelberg outcomes may result in equilibrium) and forward selling output induces a convergence to the Bertrand solution. I analyze the generalized model where firms pre-order input and forward sell output. First, I analyze oligopolists producing homogenous goods, generalize the Cournot-Stackelberg endogeneity to oligopoly, and show that it additionally includes Bertrand in the generalized model. This shows that the "mode of competition" between firms may be entirely endogenous. Second, I consider heterogenous goods in duopolies, which generalizes existing results on forward sales of output, and derive the outcome set in general duopolies. This set does not contain the Bertrand solution anymore, but it is well-defined and shows that forward sales increase welfare also when goods are complements