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Economics Dynamics Problems 226

# Economics Dynamics Problems 226 - 210 Economic Dynamics and...

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Unformatted text preview: 210 Economic Dynamics and we can ﬁnd a and b for t = 0 from u0 = avr + bvs where u0 is known. This can either be done by direct substitution, or using the fact that avr + bvs = vr i.e. or (4) V vs a = u0 b a = u0 b a = V−1 u0 b Write the solution ut = art vr + bst vs But we can do this whole process in one step. First we note ut = art vr + bst vs = vr = VDt vs rt 0 0 st a b a b But a = V−1 u0 b Hence ut = VDt V−1 u0 which is the result we proved above. The gain, if there is one, in doing the four steps is the need to solve for a and b. Since this can often be done by direct substitution, then the four steps involve no inverse matrix computation. Example 5.4 Let xt+1 = −8 − xt + yt yt+1 = 4 − 0.3xt + 0.9yt setting xt+1 = xt = x∗ and yt+1 = yt = y∗ for all t, the ﬁxed point is readily shown to be (x∗ , y∗ ) = (6.4, 20.8). Now consider the system in terms of deviations from equilibrium, then xt+1 − x∗ = −(xt − x∗ ) + ( yt − y∗ ) yt+1 − y∗ = −0.3(xt − x∗ ) + 0.9( yt − y∗ ) or ut = Aut−1 ...
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