Question

Use the age transition matrix L and age distribution vector X1 to find the age distribution vectors X2 and X3. 0 2 0 0 0 LE 2 X = 16 16 16 0 0 32 8 X2 4 #1 64 4 X3 = It 1 11 x Then find a stable age distribution vector. 8 4 X=0

Answer 1

ta Here, given the transition matrix and age distri- bution vectors are L Y2 oo 02 su Now, 22 zlay 2 o 16 Flip th 0 O 2o 16 16 06 16 16 2 0 174 ) ( ine, x2 32 8

and x3=Lazz O 0 4 ines a 16 ( ) 2 distribution 2 o x3 = Again, to obtain the stable age vector, find the eigenvector of L- corresponding to its positive eigenvalues. Now, for eigen values of L, det (L-11) --0 ire, Y2 -> o Y4 - lie, -xox2 + 20 ie, &(²1) - i-e d = -1,0, 1 Soither positiewe eigenvalue of L 13,421 Now, det 1-() be.thie eigen vector of 1#he (2-1.I) O)(O)- -x +2y i) 쓸 - - Z =0 Liani ) are see that equation is and cus represent Samae equation Now, let y=46, then from (i) and cing x=84, zat So, the eigen vector of L corresponding to A=1 is ting, the stable age distribution vector is O 2 1/27 ie, -O - awe get ) t xat (5)