Inefficiency of sponsored research

I consider two models of sponsored research — one is the model where researchers get funded until (if ever) a research project experiences the first failure (bad news model), the other one is the model where the winner (if any) of a research race is rewarded (good news model). In either case, the researchers start working on a project of unknown quality. The quality of the project is identified with its ability to generate failures or successes, in the first and second models, respectively. The rate of arrival of success conditioned on the quality of the project is an increasing function of the total time spent on research. I find equilibria in both models and show that in case of two competing researchers, one of the researchers experiments inefficiently long, in the bad news model or in the good news model when either the winner takes all, or the laggard is punished. In the good news model where the laggard is rewarded, the equilibrium outcome is efficient.