On the Least Upper Bound of Discount Factors that are Compatible with Optimal Period-Three Cycles

In this chapter, we derive, in the standard class of optimal p growth models, the least upper bound of discount factors of future utilities for which a cyclical optimal path of period 3 may emerges.1 On the one hand, Ni s h imur a and Yano (1992) and Ni s himura, Sorger and Yano (Chapter 9) construct examples in which a cyclical optimal path of period 3 emerges for discount factors around 0.36. On the other hand, Sorger (1992, 1994) demonstrates that if such a path emerges in an optimal growth model of the standard class, the model’s discount factor cannot exceed 0.55. These results imply that the least upper bound of discount factors that can give rise to cyclical optimal paths of period 3 must lie between 0.36 and 0.55.2 We demonstrate that the least upper bound is % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qaceWGWbGbaKaacqGH9aqpdaqadaWdaeaapeGaaG4maiabgkHiTmaa % kaaapaqaa8qacaaI1aaaleqaaaGccaGLOaGaayzkaaGaai4laiaaik % daaaa!3DE3! $$\hat p = \left( {3 - \sqrt 5 } \right)/2$$ .