Abstract
Consider a standard aggregative dynamic optimization framework (Ω, u, δ), where Ω is the transition possibility (technology) set, u is a (reduced form) utility function defined on this set, and 0 < δ < 1 a discount factor. Can an optimal program in this framework exhibit a period-three cycle?
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After completing this chapter, I learned that essentially the same result has been independently obtained by Nishimura and Yano (Chapter 12). Their maintained assumptions on the dynamic optimization model differ somewhat from those used in this chapter. Their approach in obtaining the basic result is quite different from mine.
This is the notion of chaos, which is referred to in the literature as “Li-Yorke chaos”. An alternate definition, proposed in the work of Li et al. (1982) involves the existence of (i) infinitely many periodic points of different periods and (ii) an uncountable scrambled set. They have shown that if f has a periodic point of period which is not a power of 2, then f is chaotic in this sense. We focus, in this chapter, on Li-Yorke chaos but our discussion in Section 7 relates to the alternate definition. Both definitions involve topological chaos, which is quite distinct from the notion of ergodic chaos, an important concept in the study of complex dynamic behavior. We do not discuss the concept of ergodic chaos in this chapter.
An observation about the set of maintained assumptions (A.1)-(A.7) is in order at this point. We are concerned with providing an exact discount factor restriction for period-three cycles as indicated in (1.1) in the introduction. This involves two parts, the “necessity” part (i) and the “sufficiency” part (ii). As a general principle, with more assumptions, the sufficiency part becomes harder to establish, the necessity part may become easier. Thus, the choice of the set of maintained assumptions can become an important one. I have chosen to keep the assumptions as close as possible to “standard” ones used in the literature on intertemporal allocation theory. The reader will note that this has made some assumptions (like the “free-disposal” assumption (A.4) and the “monotonicity” assumption (A.7)) superfluous in providing the “necessity” results (Theorems 5.1 and 6.1). They have, however, made the “sufficiency” part somewhat harder to establish.
The fuss about making a distinction between and ζ should perhaps be explained. Clearly, ζ can be a “maximum sustainable stock”, while cannot. Thus, Y = [0, ζ] is a somewhat larger closed interval than the state space X = [0, ζ], where the important dynamics will take place. We wanted to make assumption (A.7) on the monotone nature of the utility function in “standard” form; that is, without restricting x, x’ in any way. But this created problems in the construction of the example in Section 6.2. So we settled for (A.7) in its present form, noting thereby that the monotone restriction on u can be maintained on a larger closed interval than the state space X, while preserving the example in Section 6.2. In fact, the reader will note that the method followed in constructing the example allows us to maintain the monotone restrictions on u on any large finite interval [0, ε] containing the state space, by suitable modification in the definition of u in the example.
The strict concavity of the value function is essential to this study (as it is in the literature discussed in Section 1). Assumption (A.5) is sufficient for this, but is not necessary (see Montrucchio and Sorger (1994) for a useful discussion). We have still maintained (A.5), since we view, for this model, the utility function, u, as a “primitive,” and the value function, V, as a “derived” concept.
In fact this is all that is needed in proving Theorems 5.1 and 6.1. The non-negativity of prices and the necessity of the transversality condition for optimal programs are not needed for that. However, given our maintained set of assumptions (see footnote 4), these properties do follow easily and are noted to keep our exposition closely related to the literature on “price characterizations” of optimal programs.
Although Proposition 4.2 has not been explicitly noted in the literature, Gerhard Sorger has used it in his oral presentations on the subject, to show how just three observations on an (assumed) optimal program might yield a rather good upper bound on the discount factor.
Sorger (1992a) proves a result similar to Proposition 7.1. However, his set of maintained assumptions on a dynamic optimization model is different from ours. It is not clear to us whether Proposition 7.1 is valid if the upper-semicontinuity of the utility function in (A.6) is replaced by continuity.
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Mitra, T. (2000). An Exact Discount Factor Restriction for Period-Three Cycles in Dynamic Optimization Models. In: Optimization and Chaos. Studies in Economic Theory, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04060-7_11
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DOI: https://doi.org/10.1007/978-3-662-04060-7_11
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