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Unformatted text preview: 134 Economic Dynamics
Problems (i)–(v) are all recursive equations of the ﬁrstorder. The same basic form
is used to solve higherorder 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 secondorder 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 : (4t4)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.
 Fall '11
 Dr.Gwartney
 Economics

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