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Economics Dynamics Problems 136

Economics Dynamics Problems 136 - very straightForward IF ...

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120 Economic Dynamics Figure 3.16. It is readily established that lim t →∞ y t = a b Three typical plots are shown in figure 3.16, for y 0 < a / b , y 0 = a / b and y 0 > a / b . Return to the original formulation y t + 1 y t = ay t by 2 t i.e. y t + 1 = (1 + a ) y t by 2 t It is not possible to solve this nonlinear equation, although our approximation is quite good (see exercise 6). But the equation has been much investigated by mathematicians because of its possible chaotic behaviour. 9 In carrying out this investigation it is normal to respecify the equation in its generic form x t + 1 = λ x t (1 x t ) (3.23) It is this simple recursive formulation that is often employed for investigation because it involves only a single parameter, λ . The reader is encouraged to set up this equation on a spreadsheet, which is
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Unformatted text preview: very straightForward. IF λ = 3 . 2 it is readily established that the series will, aFter a suFfcient time period, oscillate between two values: a 1 = . 799455 and a 2 = . 513045. This two-cycle is typical oF the logistic equation For a certain range oF λ . To establish the range oF λ is straightForward but algebraically tedious. Here we shall give the gist oF the solution, and leave appendices 3.1 and 3.2 to illustrate how Mathematica and Maple , respectively, can be employed to solve the tedious algebra. Let f ( x ) = λ x (1 − x ) then a two-cycle result will occur iF a = f ( f ( a )) 9 We shall investigate chaos in chapter 7....
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