A Nonsmooth, Nonconvex Model of Optimal Growth

This paper analyzes the nature of economic dynamics in a one-sector optimal growth model in which the technology is generally nonconvex, nondifferentiable, and discontinuous. The model also allows for irreversible investment and unbounded growth. We provide sufficient conditions for boundedness, extinction (convergence to zero), survival (boundedness away from zero), and unbounded growth. These conditions reveal that boundedness and survival are symmetrical phenomena, so are extinction and unbounded growth. Since many of the conditions are only local, it is possible that extinction occurs from small capital stocks, while unbounded growth occurs from large capital stocks. Despite such nonclassical results and nonclassical features such as nonconvexity and discontinuity, the model behaves much like a classical one as the discount factor approaches unity. In particular, we show that in most cases, if the discount factor is close to one, any optimal path from a given initial capital stock converges to a small neighborhood of what we define as the golden rule capital stock. If this stock is not finite, i.e., if sustainable consumption is maximized atinfinity, then as the discount factor approaches one, unbounded growth at least almost occurs.