Economics Dynamics Problems 119

Economics Dynamics Problems 119 - Discrete dynamic systems...

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Unformatted text preview: Discrete dynamic systems 103 Figure 3.11. or yn = n−1 ak y 0 (3.14) k=0 Hence, if ak = a for all k, then n−1 ak = an and yn = an y0 k=0 Consider an even more general case: that of the nonhomogeneous first-order equation given by yn+1 = an yn + gn a0 , g0 , y0 at n = 0 (3.15) Then y1 = a0 y0 + g0 y2 = a1 y1 + g1 = a1 (a0 y0 + g0 ) + g1 = a1 a0 y0 + a1 g0 + g1 y3 = a2 y2 + g2 = a2 (a1 a0 y0 + a1 g0 + g1 ) + g2 = a2 a1 a0 y0 + a2 a1 g0 + a2 g1 + g2 . . . with solution for yn of yn = n−1 k=0 ak y 0 + n−1 n−1 i=0 k=i+1 ak g i (3.16) ...
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