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I am wondering whether there is a model of oligopolies in which we have some intermediate case of Bertrand and Cournot competition. What I do not mean by "intermediate" is the mixed Bertrand - Cournot game in which one firm sets a price and the other sets a quantity. What I also do not mean (I think) is monopolistic competition in which goods are not perfect substitutes.

Rather, I mean a model in which there are degrees of commitment to prices and quantities. For example, a cookie producer can softly commit to a cookie amount by only purchasing a certain number of ovens and softly commit to a price by printing the price on the cookie boxes. At some extra cost, it is however possible to later scramble to rent an extra oven or to reprint price labels. I am wondering whether this has been modeled anywhere. Ideal would be a parametric version with the parameter indicating whether we are in the pure Bertrand or in the Cournot case.

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    $\begingroup$ Not exactly what you are looking for, but "A Generalized Oligopoly Model" (Watt (2002)) has a generalized model that nests Cournot and Stackelberg. $\endgroup$ – BKay Apr 2 at 15:08
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    $\begingroup$ I've seen a 2-stage model where firms choose capacities in stage 1 and set prices in stage 2. The second stage is like a Bertrand and, given equilibrium behavior in stage 2, the first stage problem is Cournot-like. I remember the punch line being "Bertrand + capacity constraint = Cournot", but not much else... $\endgroup$ – Herr K. Apr 2 at 18:57
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    $\begingroup$ @HerrK. I am guessing you are thinking of Kreps-Scheinkmann. $\endgroup$ – Giskard Apr 2 at 19:22
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    $\begingroup$ @Bertrand Kind sir, having risen from the grave, would you mind posting answers as answers so we can vote on them? If you feel inclined to contribute more, copying the relevant parts (from the abstracts?) would also be welcome. $\endgroup$ – Giskard Apr 2 at 21:12
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    $\begingroup$ Also see Montez & Schutz (wp 2019) - "All-Pay Oligopolies: Price Competition with Unobservable Inventory Choices" (@ the Kreps-Scheinkman point "Bertrand + capacity constraint = Cournot" which relies on the inventory being observable) $\endgroup$ – Bayesian Apr 3 at 14:57
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These papers could be interesting to you. First, a classical contribution:
Singh Nirvikar and Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, vol. 15(4), pages 546-554.
And these two interesting papers, using the concept of competition toughness to reconcile Cournot and Bertrand:
d'Aspremont, Claude & Dos Santos Ferreira, Rodolphe, 2009. "Price-quantity competition with varying toughness," Games and Economic Behavior, 65, 62-82.
d'Aspremont Claude and Rodolphe Dos Santos Ferreira & Louis-André Gérard-Varet, 2007. "Competition For Market Share Or For Market Size: Oligopolistic Equilibria With Varying Competitive Toughness," International Economic Review, 48, 761-784.

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