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Unformatted text preview: A) = Probability of moving from A to B = Probability of moving from B to A = Probability of staying in B (moving from B to B) 35 Markov Transition Matrices Let At be the number of workers in industry A at time t, while Bt is the Let while number of workers in industry B at time t. t. Call vector xt′ = [ At Bt ] the distribution of workers across industries at Call time t. t. PAA PAB Consider the following Markov transition probabilities matrix: M = PBA PBB Then the distribution of workers across industries in period t+1 will be Then t+1 given by: given PAA xt′+1 = [ At +1 Bt +1 ] = [ ( At PAA + Bt PBA ) ( At PAB + Bt PBB ) ] = [ At Bt ] × PBA PAB PBB The number The number (number of workers f workers in of workers in o in A at t)x (Number of workers in B at t)x (probability of staying time t+1 (probability t+1moving from B to A) B at time of A at in A) = = Number of workers who Number of workers who moved stayed in A at t+1 from B to A at t+1 36 Markov Transition Matrices By multiplying the initial distribution of workers by the Markov transition By probability matrix, we can find workers’ distribution by industry many periods ahead in the future. periods 2 Two periods ahead: PAA [ At +2 Bt +2 ] = [ At Bt ] PBA PAB PBB N periods ahead: PAA [ At +N Bt +N ] = [ At Bt ] PBA PAB PBB N It may happen that after a certain number of periods N 0 the vector It of workers’ distribution does not change any more, i.e. of [A t + N 0 +1 [ Bt + N 0 +1 = At + N 0 Bt + N 0 We call this unchanging vector a steady state distribution. It We corresponds to a transition matrix with identical rows (matrix corresponds M N0 ) 37...
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This note was uploaded on 09/14/2013 for the course STAT 1001 taught by Professor Kim during the Fall '11 term at Yonsei University.

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