Question: [Markov Transition Matrix]

Consider the situation of a mass layoff (i.e. a firm (Markpath) goes out of business) where 2000 people become unemployed and now begin a job search. There are two states: employed (E) and unemployed (U) with an initial vector (1×2 row vector)

\({{x}_{0}}=\left[ \begin{matrix} {{E}_{0}} & {{U}_{0}} \\ \end{matrix} \right]=\left[ \begin{matrix} 0 & 2000 \\ \end{matrix} \right]\)

Suppose that in any given period

1. an unemployed person will find a job with probability \({{p}_{UE}}=0.7\) and will therefore remain unemployed with a probability \({{p}_{UU}}=0.3\).
2. persons who find themselves employed in any given period may lose their job with a probability of \({{p}_{EU}}=0.1\) (and will continue to remain employed with probability \({{p}_{EE}}=0.9\) ).

The 2×2 Markov transition matrix M can be set up as follows.

1. What will be the number of unemployed people after one period. Note
   \({{x}_{t}}M={{x}_{t+1}}\)
   where xt+1 is the distribution at time t +1 and xt is the distribution at time t.
2. What will be the number of unemployed people after two periods.
3. What will be the number of unemployed people after four periods.
4. (Bonus) What is the steady-state level of unemployment?

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