Abstract
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 \(\hat p = \left( {3 - \sqrt 5 } \right)/2\).
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The focus in the early literature on optimal growth is on the convergence of an optimal path to a stationary state. In that literature, it has been proved that an optimal path converges to a stationary state if the utility function is fixed and if the discounting of future utilities is sufficiently weak (see Brock and Scheinkman, 1976, Cass and Shell, 1976, Scheinkman, 1976, and McKenzie, 1983). In this chapter, we are concerned with the existence of a utility function that gives rise to an optimal cyclical path of period 3 for a given discount factor of future utilities.
In relation to these results, Nishimura and Yano (Chapter 8) prove that for values of the discount factor p arbitrarily close to 1, ergodically chaotic optimal paths can emerge that are generated by a unimodal, expansive dynamical system. In that result, it is shown that part of the graph of the optimal dynamical system lies on a von Neumann facet containing the stationary state and that any optimal path is confined in a small neighborhood of the facet. In this respect, the result is closely related to the neighborhood turnpike theorem of McKenzie (1983), which implies that any optimal paths converge into a neighborhood of the von Neumann facet. See also the result of Nishimura, Sorger, and Yano (Chapter 9), is ρ = (3 — /2. The importance of finding this least upper bound stems from the following two facts: (i) given the long-run real interest rate per annum, the discount factor is positively correlated to the length of an individual period of the model;3 (ii) the existence of a cyclical path of period 3 is a fundamental criterion for the emergence of complex non-linear dynamics, in particular of cyclical paths of any other periodicity (Sarkovskii, (1964), and Li and Yorke, (1975)). which extends Nishimura and Yano (Chapter 8) for the case in which the von Neumann facet is trivial.
See Deneckere and Pelikan (1986), Boldrin and Montrucchio (1986), Boldrin and Deneckere (1990), Nishimura and Yano (1994) and Majumdar and Mitra (1994). These studes are concerned with deterministic economic flutuations. For broader issues on deterministic and indeterministic fluctuations, see, for example Shell (1977) and Cass and Shell (1983).
E t6 and F t6 are defined as follows.
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Nishimura, K., Yano, M. (2000). On the Least Upper Bound of Discount Factors that are Compatible with Optimal Period-Three Cycles. In: Optimization and Chaos. Studies in Economic Theory, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04060-7_12
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DOI: https://doi.org/10.1007/978-3-662-04060-7_12
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