8. Suppose that, in any given month, the probability that an unemployed person will find employment is 0.25, while the probability that they remain unemployed is 0.75. On the other

hand, each month the probability that an employed person will lose their job is 0.05, while the probability that they remain employed is 0.95. The number of people in the labor force is fixed at 1000 (ut + et = 1000 at all times t). \text { a. Let the state vector at time } t \text { be defined as } x_{t} \equiv\left[\begin{array}{l} e_{t} \\ u_{t} \end{array}\right], \text { where u denotes the number of } people in the labor force who are unemployed and e denotes the number employed. What \text { is the Markov transition matrix, } \mathrm{M}, \text { for which } x_{t+1}=M x_{t} ? b. If half of the labor force is unemployed at time 0, how many people will be employed at time 1? c. Write down the characteristic equation for M, and use it to find both of M's eigenvalues. d. What do the eigenvalues tell you about the steady state for this process? e. What is the steady-state vector, x*, for this process?