Question 1 Consider the situation and a mass layoff (i.e., a factory shuts down) where 1,200 people become unemployed and now begin a job search. In this case, there are two states: employed (E) and unemployed (U) with an initial vector 𝑥𝑥 가 = [𝑥 𝑥] = [0 1,200] Suppose that in any given period an unemployed person will find a job with probability 0.7 and will therefore remain unemployed with a probability of 0.3. Additionally, persons who find themselves employed in any given period may lose their job with a probability 0.1 (and will have a 0.9 probability of remaining employed). i. Set up the Markov transition matrix for the problem. ii. What will be the number of unemployed people after a) 2 periods; b) 3 periods; c) 5 periods; d) 10 periods? iii. What is the steady-state level of unemployment? Answer The Markov transition matrix for this problem can be represented as follows: |E|U| |---|---| | 0.9 | 0.3 | | 0.1 | 0.7 | When multiplied by the initial vector 𝑥᬴𝑥가 = [𝑥 𝑥] = [0 1,200], we can calculate the number of employed and unemployed people after each period. ii. What will be the number of unemployed people after a) 2 periods; b) 3 periods; c) 5 periods; d) 10 periods? a) After 2 periods, the number of employed people would be 0.9 * 1,200 = 1,080, and the number of unemployed people would be 0.1 * 1,200 + 0.3 * 1,080 = 420. b) After 3 periods, the number of employed people would be 0.9 * 420 = 378, and the number of unemployed people would be 0.1 * 420 + 0.3 * 378 = 193.8 or approximately 194. c) After 5 periods, the number of employed people would be 0.9 * 194 = 174.6 or approximately 175, and the number of unemployed people would be 0.1 * 175 + 0.3 * 174.6 = 52.38 or approximately 52. d) After 10 periods, the number of employed people would be 0.9^10 * 1,200 = 348.678, and the number of unemployed people would be 0.1 * 348.678 + 0.3 * 348.678 = 104.6034 or approximately 105. iii. What is the steady-state level of unemployment? The steady-state level of unemployment occurs when the system reaches equilibrium and the number of employed and unemployed individuals remains constant. In this case, the steady-state level of unemployment would be when the number of employed and unemployed people is the same, which would be when 0.9 * 𝑥 = 0.1 * 𝑥. Solving for 𝑥, we get 𝑥 = 9 / 109, which means that the steady-state level of unemployment will be 9 out of 109 or approximately 8.26%.