Can rivalry in R&D be harmful under supply function competition?

We consider a duopoly with cost asymmetry and demand uncertainty and show that rivalry in (process) R&D can be extremely harmful to both firms even in the presence of spillovers if the firms produce under supply function competition. However, if they produce under Cournot competition, firms can earn higher profits in the presence of R&D rivalry than what they could earn when they (mutually) invested in no R&D. We also show that R&D rivalry always works to widen the efficiency gap between the firms under Cournot competition, and also under supply function competition if R&D spillovers are present. On the other hand, consumers' welfare is affected by R&D rivalry only under supply function competition. Under this competition, consumers always prefer the presence of R&D rivalry with or without spillovers to the absence of R&D rivalry; however, they obtain their highest surplus under R&D rivalry when there are no R&D spillovers.