The present state of convex programming theory for infinite horizon free endpoint economic models is not entirely satisfactory. Roughly speaking, classical duality principles can be shown to apply tof inite subsections of an optimal trajectory and this avoids classical inefficiencies of the finite horizon variety. But it has never been completely clear how to avoid the kind of non-optimality which results from piling up too much “left over” capital in the limit. While certain rule of thumb “transversality conditions” have been proposed by analogy with finite horizon models, they have not in general been put on a rigorous footing and it is not clear which of them are valid under what circumstances. In this paper a rigorous treatment of the subject is undertaken. Under a set of general axioms, a certain limiting transversality condition in conjunction with other duality conditions is shown to be necessary and sufficient for infinite horizon optimality.
Following closely the approach to optimal economic growth taken in the work of Frank Ramsey (1928), a highly simplified two-sector model is presented in which the “overhead capital” sector exhibits increasing returns to scale. Basic properties of the optimal growth path are discussed and the optimal policy is explicitly demonstrated for a special case. From an economic standpoint, the model might be relevant in bearing on some issues of development programming. Mathematically, this kind of a model has an interesting structure because it is a combination of convex and concave sub-problems.