In the context of a model due to Robinson, Solow and Srinivasan (the RSS model), we report results on the existence and characterization of locally optimal programs, a concept taken from theoretical physics. In particular, we propose a (new) transversality condition under which all locally optimal programs are good. An extended introduction places our theorems in the context of previous work on the existence question, including that on agreeable programs. It appears that there is no completely rational way to attack [the] problem without considering development programmes over an infinite horizon (Gale in Rev Econ Stud 34:1–8, 1967). The analysis of simple models is essential if we are to understand the corresponding situation for more complex models of the economy (Mirrlees and Stern in J Econ Theory 4:268–288, 1972). The technical convenience, for clear and quantitative results, of using an infinite time horizon is rather great (Hammond and Mirrlees in Models of economic growth, Wiley, New York, pp 283–299, 1973).