Abstract
In the recent literature, it has been demonstrated that optimal capital accumulation may be chaotic; see Boldrin and Montrucchio (1986) , and Deneckere and Pelikan (1986). 1 This finding indicates, as Scheinkman (1990) discusses, that the deterministic equilibrium model of a dynamic economy may explain various complex dynamic behaviors of economic variables, and, in fact, search for such explanations has already begun (see Brock, 1986, and Scheinkman and LeBaron, 1989, for example) . In the existing literature, however, not much has yet been revealed with respect to the circumstances under which optimal accumulation exhibits complex non-linear dynamics. In particular, it has not yet been known whether or not chaotic optimal accumulation may appear in the case in which future utilities are discounted not so strongly.2
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References
These studies are followed by the recent work of Boldrin and Deneckere (1990) and Majumdar and Mitra (Chapter 3). For earlier treatments of chaotic economic behaviors, see Benhabib and Day (1982), Day (1982) and Grandmont (1985), the models of which are not of optimal growth.
It has been known that if, given a technology and a period-wise preference, the discounting of future utilities is sufficiently weak, the classical turnpike result holds, i.e., the optimal paths converges to a stationary state; see Cass and Shell (1976), Scheinkman (1976), Brock and Scheinkman (1976) and McKenzie (1983). Sorger (1990), moreover, demonstrates the existence of an upper bound of discount factors with which a given trajectory is a solution to an optimal growth model. These results, however, do not exclude the possibility that for any given degree of the discounting of future utilities, there may be a technology and a period-wise preference that result in chaotic optimal dynamics.
See Deneckere and Pelikan (1986) and Boldrin and Montrucchio (1986b) .
Boldrin and Montrucchio (1986b) discuss the possibility of chaotic optimal dynamics for values of p up to ρ = 0.25; see also Boldrin and Deneckere (1990), which focuses on topological chaos. As is well-known, topological chaos is a weaker form of chaotic dynamics (see Section 3 for an explanation).
Deneckere and Pelikan (1986), in contrast, assume that one sector has a Cobb-Douglas production function and that the other sector has a Leontief production function. In this case, a factor intensity reversal occurs.
Benhabib and Nishimura (1985) demonstrates that if and only if the sign of the cross partial derivative of a reduced form utility function is negative, the optimal transition function is downard sloping. (This result can be derived as a special case of the results obtained by Topkis (1978) and Marshall and Olkin (1979).) Because the existence of a capital intensive consumption good sector implies a negative cross partial derivative, the transition function monotone dressing. Because Benhabib and Nishimura (1985) assumes differentiability, the result of that study does not directly apply to our case, which is nondifferentiable. However, a more or less similar relationship appears in our setting, as is shown below.
Boldrin and Deneckere (1990) discusses use of the boundary imposed by the limit of capital depreciation.
0n the differentiability of an optimal transition function, see Araujo (1991), Boldrin and Montrucchio (1989) and Santos (1991).
See Section 3 for precise definitions.
The existence of cyclical optimal paths plays important roles in Deneckere and Pelikan (1986), and Boldrin and Deneckere (1990) as well. Those studies characterize a cyclical optimal path by the Euler equation. In contrast, we characterize it by support prices. This difference is due to the fact we focus on non-interior solutions in a non-differentiable model while the existing studies consider interior solutions in a differentiable model.
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Nishimura, K., Yano, M. (2000). Non-Linear Dynamics and Chaos in Optimal Growth: A Constructive Exposition. In: Optimization and Chaos. Studies in Economic Theory, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04060-7_8
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DOI: https://doi.org/10.1007/978-3-662-04060-7_8
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