Chiang/Wainwright: Fundamental Methods of Mathematical Economics
(b)
Instructor’s Manual
The characteristic equation is b3 − 2b2 − 5b/4 − 1/4 = 0, which can be written as (b − 1/2)(b2 − 3b/2 + 1/2) = 0. The first factor gives the root 1/2; the second gives
the roots 1, 1/2,. Since the two roots are repeated, we must write yc = A1 (1/2)t + A2 t(1/2)t + A3 .
6.
(a)
Since n = 2, a0 = 1, a1 = 1/2 and a2 = −1/2, we have ¯ ¯ ¯ ¯ ¯ 1 0 −1/2 1/2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1/2 ¯ 1 1 0 −1/2¯ −1/2¯ 3 ¯=0 ¯ ¯ ¯ ∆1 = ¯ ¯ ¯ = 4 > 0, but ∆2 = ¯ ¯ ¯ ¯−1/2 ¯ −1/2 0 1 1/2 1 ¯ ¯ ¯ ¯ ¯ 1/2 −1/2 0 1 ¯
Thus the time path is not convergent. (b)
Since a0 = 1, a1 = 0 and a2 = −1/9, we have ¯ ¯ ¯ ¯ ¯ 1 0 −1/9 0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 6400 ¯ ¯ 1 ¯ 0 1 0 −1/9 −1/9 ¯= ¯ = 80 ; ∆2 = ¯ ∆1 = ¯¯ ¯ ¯ ¯ 81 ¯−1/9 ¯−1/9 0 1 0 ¯ 6561 1 ¯ ¯ ¯ ¯ ¯ ¯ 0 −1/9 0 0 ¯
The time path is convergent.
7. Since n = 3, there are three determinants as follows: ¯ ¯ ¯1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 a3 ¯ ¯a ¯ ∆2 = ¯ 1 ∆1 = ¯¯ ¯ ¯ ¯a3 1 ¯ ¯a3 ¯ ¯ ¯a2 and
¯ ¯ ¯1 ¯ ¯ ¯a1 ¯ ¯ ¯a2 ∆3 = ¯¯ ¯a3 ¯ ¯ ¯a2 ¯ ¯ ¯a1
0
a3
1
0
0
1
a3
0
0
0
a3
a2
1
0
0
a3
a1
1
0
0
0
0
1
a1
a3
0
0
1
a2
a3
0
0
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¯ ¯ a1 ¯ ¯ ¯ a2 ¯ ¯ ¯ a3 ¯ ¯ ¯ a2 ¯ ¯ ¯ a1 ¯ ¯ ¯ 1¯
¯ ¯ a2 ¯ ¯ ¯ a3 ¯ ¯ ¯ a1 ¯ ¯ ¯ 1¯