STABLE OPTIMAL CYCLES WITH SMALL DISCOUNTING IN A TWO-SECTOR DISCRETE-TIME MODEL: A NON-BIFURCATION APPROACH  By HARUTAKA TAKAHASHI Meiji Gakuin University, Tokyo This paper presents a standard two-sector optimal growth model with general neo- classical production functions: strictly quasi-concave, twice continuously differentiable homogeneous of degree 1 functions. Instead of applying the standard local bifurcation theory, I exploit two well established properties in Turnpike TheoryÐ``simple dynamics'' and the Neighbourhood TurnpikeÐand, combining both results, I demon- strate that there exists an interval of the discount factor near 1 such that a corresponding optimal steady state is totally unstable and an optimal path converges asymptotically to a two-period cycle for a chosen discount factor in it. JEL Classi®cation Numbers: O21, O41. 1. Introduction McKenzie (1983, 1984) and Scheinkman (1976) have proved that any optimal path converges asymptotically to a corresponding unique optimal steady state path (OSS) when a representative consumer does not discount the future heavily; in other words, the discount factor is suf®ciently close to 1. This property is often referred to as the Turnpike Property. Furthermore, in Takahashi (1985, 1992) the Turnpike Property is also estab- lished for a general neoclassical optimal growth model with many capital goods. Other recent studies, however, have yielded different and somewhat contradictory results. McKenzie (1983) scrutinizes a generalized version of Weitzman's example reported by Samuelson (1973) and ®nds that, under a certain value of a technical parameter, there is a cyclic path regardless of the discount factor. Benhabib and Nishimura (1985) present a similar example in which the strict concavity is allowed and give suf®cient conditions for the existence of optimal cycles of period 2 for a discrete- time optimal growth model. Nishimura and Yano (1995) have constructed the two-sector model with Cobb±Douglas production functions (i.e. no factor intensity reversal), and have demonstrated that, when the technology parameters are properly chosen for any value of the discount factor of future utility and capital depreciates completely within a period, cyclical optimal paths of period 2 will appear. In a continuous-time optimal growth model, Benhabib and Nishimura (1979a) have demonstrated that, by applying the Hopf bifurcation theorem, there exist optimal growth paths consisting of persistent cycles (see also Benhabib and Rustichini, 1990, and Nishimura and Takahashi, 1992). Applying curvature conditions on the indirect utility function, Venditti (1997) has demonstrated the existence of an optimal cycle by the Hopf bifurcation theorem in a multi-sector continuous-time model. In a multi-sector discrete-time model, under the symmetric  The paper was presented at the conference, ``Dynamic Equilibria, Expectations and Indeterminacies'', held at University of Paris on 14±16 June 1999. I would like to thank Jess Benhabib, Gerhard Sorger, Alain Venditti and a referee of the Journal for their valuable comments. ± 328 ± # Japanese Economic Association 2001. The Japanese Economic Review Vol. 52, No. 3, September 2001 Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK.