Critical capital stock in a continuous time growth model with a convex-concave production function

The critical capital stock is a threshold that appears in a nonconcave growth model, such that any optimal capital path from a stock level below (above) the threshold converges to a lower (higher) steady state. It explains history-dependent development and provides an implication for the achievement of sustainable development. The threshold is rarely an optimal steady state and thus it is hard to characterize. In a continuous time growth model with a convex-concave production function, we show that: a) the critical capital stock is continuous and increasing in the discount rate; b) as the discount rate increases, the critical capital stock appears from the zero stock level and disappears at a stock level between those of the maximum average and maximum marginal productivities; c) at this upper bound, the critical capital stock coalesces with the higher optimal steady state; d) once the critical capital stock disappears, the higher steady state is no longer an optimal steady state; and e) the critical capital stock at the upper bound can be arbitrarily close to either the stock level of the maximum average productivity or that of the maximum marginal productivity, depending on the curvature of the utility function.