Question

I have found the Markov Process to be

.5 .2 .3
.1 .7 .2
.4 .4 .2

that state 1 is Industry, that state 2 is Small Business, and that state 3 is Self-Employed.
How do you find the long-run distribution of people among these various types of employment?

Answer 1

Find the eigenvector that corresponds to an eigenvalue of 1

Answer 2

It's also \[\lim_{N \rightarrow \infty}P ^{N}\]
where P is that transition matrix. Here is an approximation that might be useful to you in checking your results.

0.28070181, 0.49122801, 0.22807019
0.28070172, 0.49122812, 0.22807017
0.28070177, 0.49122805, 0.22807018

I used the Python numpy library.

Answer 3

\[\left[\frac{16}{57},\frac{28}{57},\frac{13}{57}\right]\]