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

Abstract

A nonconcave growth model may exhibit multiple optimal steady states, with a critical capital stock serving as a threshold. Optimal capital paths originating from stock levels below (above) this threshold converge to lower (higher) optimal steady states. The presence of a critical capital stock elucidates the phenomenon of history-dependent development and carries implications for achieving sustainable development. In a continuous-time model featuring a convex-concave production function, we demonstrate that: (a) the critical capital stock is continuous and strictly increasing in the discount rate; (b) its lower bound is the zero capital stock, while the upper bound lies between the stock levels associated with maximum marginal productivity and maximum average productivity; (c) at the upper bound, the critical capital stock coincides with the higher optimal steady state; and (d) the upper bound approaches the capital stock corresponding to maximum average productivity as the intertemporal elasticity of substitution approaches infinity, and converges to that of maximum marginal productivity as the intertemporal elasticity of substitution tends to 0.