Consider the following homogenous difference equation:
xt=b0+b1xt-1
a) Iterate backwards xt can be written in terms of xt-2.
b) Now show xt can be written in terms of xt-3 and xt-4, respectively.
c) Assuming that you know a particular value of x in the past %u2013 x0 %u2013 write down the solution for xt in terms of x0.
bra
d) One of your colleagues obtained a different solution using a different method:
xt=b0/(1-b1)+bt1[x0-b0/(1-b1)]
Demonstrate using alegbra that the two solutions are identical if I b1 I < 1.
e) Show that if b1 = 1, the solution in (c) is equivalent to xt = b0*t + x0. How do you interpret the behavior of x in this case? How does the behavior of x differ from the case in (d), when | b1 |< 1?