Mann Iterations with Power Means
In this paper we analyze a recurrence , where is a weighted power mean of ,…., . Such an iteration scheme has been proposed to model a class of non-linear forward-looking economic models ( the state today is affected by tomorrow’ s expectation ) under bounded rationality; the agents employ a recursive learning rule to update beliefs using weighted power means of the past states. A proposition on the convergence of the dynamical system with memory, proven with a general weighted power mean, generalizes some results given in the literature, where only the arithmetic mean is considered. A power weighted mean with exponentially decreasing weights decreasing is proposed to simulate a fading memory. In this case the iteration scheme with memory is reduced to an equivalent two-dimensional autonomous map whose possible kinds of asymptotic behaviors are the same as those of a one-dimensional map. By this general technique it is proved, for a function f which maps a compact interval into itself, that the presence of a long memory has a stabilizing effect, in the sense that with a sufficiently strong memory convergence to a steady state is obtained even for an otherwise oscillating, or chaotic, dynamical system. In the appendix is considered an economic example from an overlapping generation models which leads to a harmonic mean.
|Date of creation:||2007|
|Date of revision:||2007|
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- Balasko, Yves & Royer, Daniel, 1996. "Stability of Competitive Equilibrium with Respect to Recursive and Learning Processes," Journal of Economic Theory, Elsevier, vol. 68(2), pages 319-348, February.
- Benassy, Jean-Pascal & Blad, Michael C., 1989. "On learning and rational expectations in an overlapping generations model," Journal of Economic Dynamics and Control, Elsevier, vol. 13(3), pages 379-400, July.
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