Economics Dynamics Problems 150

Economics Dynamics Problems 150 - 134 Economic Dynamics...

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Unformatted text preview: 134 Economic Dynamics Problems (i)–(v) are all recursive equations of the first-order. The same basic form is used to solve higher-order recursive equations. Given the recursive equation yt+2 = ayt+1 + byt then this can be solved with the instructions: Mathematica RSolve[y[t+2]==ay[t+1]+by[t],y[t],t] Maple rsolve( y(t+2)=a*y(t+1)+b*y(t),y(t)); But because this is a general recursive equation the output in each case is quite involved. Mathematica’s output even more so, since it involves Binomial equations! What is revealed by the output is the need to know two initial conditions to solve such second-order recursive equations: Solving yt+2 = yt+1 + 2yt with initial conditions y(0) = 5 and y(1) = 4, we have Mathematica RSolve[{y[t+2]==y[t+1]+2y[t],y[0]==5,y[1]==4},y[t],t] with output {{y[t]->2(-1)t + 3 2t }} Maple rsolve({r(t+2)=y(t+1)+2*y(t),y(0)=5,y(1)=4},y(t)); with output 2(-1)t + 3 2t Furthermore, there is no difficulty with repeated roots, which occur in solving yt+2 = 4yt+1 − 4yt . For initial conditions y(0) = 6 and y(1) = 4, we have solutions Mathematica : {{y[t]->-21+t (-3 + 2t)}} Maple : (-4t-4)2t + 10 2t Here we see that output in the two packages need not look the same, and often does not, yet both are identical; and identical to 6(2)t – 4t(2)t which we derived in the text. Complex roots, on the other hand, are solved by giving solutions in their complex form rather than in trigonometric form. The RSolve and rsolve commands, therefore, allow a check of the following equations in this chapter. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) yt+2 = ayt+1 + byt yt+2 = yt+1 + 2yt y(0) = 5, y(1) = 4 yt+2 = 4yt+1 − 4yt yt+2 = 4yt+1 − 4yt y(0) = 6, y(1) = 4 yt+2 = 4yt+1 − 16yt yt+2 = ayt+1 − byt + c yt+2 = 4yt+1 − 16yt + 26 yt+2 = 5yt+1 − 4yt + 4 yt+2 = −yt+1 + 2yt + 12 y(0) = 4, y(1) = 5 Yt = (b + v)Yt−1 − vYt−2 + (a + G) Yt = 4.75Yt−1 + 4Yt−2 + 150 ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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