A large company has two offices, with Office A housing 1300 employees and Office B having 700. Every year, there is a 20% chance that employees at Office A will need to transfer to work at Office B; likewise, there is a 10% chance that employees at Office B will need to transfer to Office A.

a. What will the respective office population sizes be in the next year? In two years?
b. What will the equilibrium populations of the two offices eventually be?

A. After the first year, we expect that
Pa = .8 * 1300 + .1 * 700 = 1110
Pb = .9 * 700 + .2 * 1300 = 890

After the second year:
Pa = .8 * 1110 + .1 * 890 = 977
Pb = .9 * 890 + .2 * 1110 = 1023

B. The series of equations this determines is as follows:
Pb = .9 Pb + .2 Pa
Pa + Pb = 2000
We can show Pa = 2000 - Pb and that:
Pb - .9 Pb = .2 Pa
.1 Pb = .2 Pa
Pb = 2 Pa
Thus Pb = 2 (2000 - Pb) = 4000 - 2Pb
3 Pb - 4000 -> Pb = 1333.3
Pa = 2000 - 1333.3 = 666.7
So Office A will eventually have approximately 667 employees, while Office B has 1333.