Dynamic Optimization on a Non-Convex Feasible Set: Some General Results for Non-Smooth Technologies

In the theory of optimal intertemporal allocation, the assumption of a convex feasible set has played a dominant role. In recent years, several contributions have focused on the implications for this theory, when the feasible set does not have the convexity property. (See, in particular, Skiba (1978), Majumdar and Mitra (1982, 1983), Dechert and Nishimura (1983), Majumdar and Nermuth (1982), and the much earlier insightful paper by Clark (1971)). These contributions have not only clarified the qualitative differences in the theory in convex and non-convex models, but they have also led to the development of new analytical techniques which have made some issues in the earlier theory in convex models simpler to address (see, for example, Mitra (1983)). However, most of the contributions mentioned above have focused on particular types of non-convex feasible sets; that is, those generated by an S-shaped production function, exhibiting an initial phase of increasing returns, with diminishing returns setting in eventually. Majumdar and Nermuth (1982) work with a more general production function than this, but even there, for the development of the asymptotic stability theory of optimal programs, they have to impose some structure on the type of "non-concavities"