I am struggling with part B of this problem. I understand Markov chains and transition matrices but I'm stuck on where to start. Maybe it is just the wording of the problem. Can anybody point me in the right direction?
I have figured out that the transition matrix is
$$ \begin{bmatrix} 0.95 & 0.06 & 0 \\ 0.04 & 0.9 & 0.1 \\ 0.01 & 0.04 & 0.9 \end{bmatrix} $$
Let $ x $ be the vector with the fraction of population in each country (ordered consistent with how you ordered the entries in the matrix).
Then the question asks "Is there a vector x (distribution of the population) so that $ A x = x $ (where $ A $ is the transition matrix)?" (Is there a vector such that if everyone migrates, you again end up with exactly the same distribution of the matrix?)
Hmmm, if you look carefully at this, you will recognize this as an eigenvalue problem $ A x = \lambda x $ except that $ \lambda = 1 $... What does this mean? I'll let you ponder this, and maybe will give another hint later.