The authors discuss stochastic cost-reducing R&D investments and examine efficient subsidies. They review a two-stage duopoly model in which each firm chooses R&D levels (innovation size and probability of success) in the first stage and competes a la Cournot in the second stage. They find that simple subsidies depending on the realized cost differences induce the efficient levels of R&D with respect to the innovation size and probability of success by two firms regardless of ex ante and ex post asymmetries between the two firms.
Link
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=922445