An R&D Race with Learning and Forgetting

I develop a model of an R&D race in which firms learn and forget. The firm which makes a discovery first is awarded a prize. Firms compete to be the first by investing in R&D. As a by-product of its R&D effort, a firm accumulates knowledge. This knowledge stock is valuable even if success is not immediate. On the other hand, over time a firm's knowledge base depreciates. Unlike traditional models of symmetric R&D races, my model does not inherit the memorylessness property of the exponential distribution. Unlike models of multi-stage races where the stages or experience levels are mere labels, knowledge is productive in my model. I show that learning and forgetting shape firms' equilibrium payoffs and strategies and that many findings of the previous literature are reversed in this more general setting. The resulting patterns of strategic interactions appear to be consistent with both anecdotal evidence and empirical research on R&D races. The model does not in general allow for an analytical solution, and I employ projection methods to solve the partial differential equation that characterizes a firm's value function. Projection methods approximate the value function by a high-order polynomial. Special considerations arise since I need not only a good approximation of the value function but also good approximations of its derivatives to compute the Nash equilibrium in feedback strategies. An accuracy check indicates that the approximations yield a good description of the equilibrium payoffs and strategies. This suggests that projection techniques are promising tools for the analysis of differential games.