Title

Critical Capital Stock in a Continuous-Time Growth Model with a Convex-Concave Production Function

Abstract

The critical capital stock is a threshold that appears in a nonconcave aggregate growth model such that any optimal capital path from a stock level below the threshold converges to a lower steady state, whereas any optimal capital path from a stock level above the threshold converges to a higher steady state. Unlike a concave model with wealth effect, the threshold is not necessarily an optimal steady state, which makes its characterization difficult. In a continuous-time growth model with a convex-concave production function, we show that: a) the critical capital stock is continuous and strictly increasing in the discount rate; b) as the discount rate increases, it appears at the zero-stock level and disappears at a certain level between the stock levels of the maximum average productivity and the maximum marginal productivity; c) at this upper bound, it merges with the higher steady state; d) once the critical capital stock disappears, the higher steady state is no longer an optimal steady state; and e) the disappearing point can be arbitrarily close to either of the these stock levels, depending on the curvature of the utility function.

Keywords

Continuous-time growth model, Convex-concave production function, Critical capital stock

JEL Classification

C61, D90, O41

Inquiries

Ken-Ichi AKAO
School of Social Sciences, Waseda University

Takashi KAMIHIGASHI
Research Institute for Economics and Business Administration,
Kobe University
Rokkodai-cho, Nada-ku, Kobe
657-8501 Japan
Phone: +81-78-803-7036
FAX: +81-78-803-7059

Kazuo NISHIMURA
Research Institute for Economics and Business Administration,
Kobe University & KIER, Kyoto University
Rokkodai-cho, Nada-ku, Kobe
657-8501 Japan
Phone: +81-78-803-7036
FAX: +81-78-803-7059