1 Introduction
Consider the fol lowing example (from International Relations). We are interested
in the
impact of Chinese defense spending on U.S. defense spending, over time.
Graphically the problem can be re presented as :
where,
As suming ,
(arbitrary constant)
we have that:
Given an initial level of spending by the US (denoted US0)
what is the time path of USt?
The solution comes from recursive substitution (see Hamilton, Chapter 1).
Now, assume 
≠ 1 and
let:
So that,
Note: Multiplying 
by
is equivalent to adding the exponents 
Further, notice that,
implying that:
Returning to the expression for US t we have
that:
Thus the nature of the solution of the original difference
equation given in (1) is a time
sequence, which gives us the full history of USt for a given value of
C and US 0.
To see this more clearly return to the original difference equation:
First, consider the initial condition:
For t = 0 the solution should produce the initial level of spending.
So, the first condition holds.
For all other times it should be the case that:
Expanding the term on the RHS of the equation yields :
(Note:
cancels out.)
which further reduces to :
For the case of
= 1 a different
solution results.
To check this solution, substitute it into the original
difference equation:
for 
Again,
as t → 1
However, the path will be qualitatively different. For
example:
Next week. . . Linear Difference Equations
