We show that the monotonicity property of optimal paths (or, equivalently, the uniform boundedness of the marginal propensity of consumption by unity) is a necessary condition for local (as well as for global) optimality, and is also suﬀicient for local optimality, but not for global optimality. We also show that the well-known properties of the value function — continuity and monotonicity — are suﬀicient (along with the above conditions) to guarantee global optimality. In other words, if at any stock level, a local non-global maximizer is selected, a discontinuity in the value function will be observed. We suggest that the previous literature on this problem has not distinguished between local and global maxima, and consequently has not attempted to derive conditions that uniquely characterize global optimality. This is the major aim of this paper, and we hope to have provided some insight towards a systematic approach to non-convex dynamic optimization.
Amir, Rabah, "A Characterization of Globally Optimal Paths in the Non-Classical Growth Model" (1985). Cowles Foundation Discussion Papers. 995.