On the Minimum Rate of Impatience for Complicated Optimal Growth Paths
The occurrence of complex dynamics in economic models has received wide attention by the economic profession during the last decade (see Boldrin and Woodford (1990) for a recent survey). One of the most interesting and surprising results obtained so far is the indeterminacy theorem of Boldrin and Montrucchio (1986b), which implies that virtually every dynamical behavior is fully compatible with the standard assumptions of decreasing returns, competitive markets, and perfect foresight. Deneckere and Pelikan (1986) have used a related approach to derive similar results for the special case of one-dimensional dynamics. Whereas the analysis in Boldrin and Montrucchio (1986b) and Deneckere and Pelikan (1986) is restricted to optimal growth models formulated in discrete time, an analogous result holds also for the continuous time case (see Montrucchio (1988) and Sorger (1990)). All of these indeterminacy theorems have been proved by a constructive approach which requires a sufficiently high rate of impatience on the side of the decision maker. In particular, to construct optimal growth models exhibiting some well known chaotic maps (like the logistic map, or the Henon map) as optimal policy functions, one needs time preference rates of more than 400%.
KeywordsCapital Stock Discount Factor Bellman Equation Policy Function Discrete Time Case
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- 1.The author is not aware of any published proof of the Bellman equation under the specific assumptions made in this chapter, that is, infinite horizon, no differentiability assumptions, convexity of T, and concavity of V(.), etc. Related results can be found, for example, in Clarke (1983), Chapter 3 or in Vinter and Lewis (1980). A proof under the present assumptions can be obtained from the author upon request.Google Scholar