Optimal expenditure patterns for risky R&D projects with time-dependent returns

The basic Lucas model for risky R&D projects is revisited. New solutions for optimal expenditures are explored by exploiting the merits of the theory of differential equations. After applying the calculus of variations, a nonlinear differential equation is presented whose solution provides the optimal control for a constant conditional-completion density function and different time-dependent return models. New, exact, and approximate solutions are presented and discussed. It is found, for the class of risky R&D projects under study, that the behavior over time of the optimal expenditure is functionally similar to that of the expected return.