We consider a setting where firms in the first stage invest in cost-reducing R&D. In the market stage, one firm sets a quantity, and another sets a price. We prove that the quantity-setting firm invests more in R&D, has a lower price, and produces higher quantity than the price-setting firm. We also consider welfare implications.
Semenov, Aggey and Tondji, Jean-Baptise, "On the Dynamic Analysis of Cournot-Bertrand Equilibria" (2019). Economics and Finance Faculty Publications and Presentations. 66.