Solution to Exercise 2

a) The transition matrix is $A = \left[ \begin{array}{rrr} 0.5&0.2&0.1 \\ 0.3 &0.6&0.4 \\ 0.2&0.2&0.5 \\ \end{array} \right]$

Let $x^{(0)} = \begin{array}{r} 1\\ 0\\ 0\\ \end{array}$, then $x^{(1) } =A x^{(0)} = \begin{array}{r} 0.5\\ 0.3\\ 0.2\\ \end{array}$

And $x^{(2) } =A x^{(3)}= \begin{array}{r} 0.3300\\ 0.4100\\ 0.2600\\ \end{array}$

Hence the probability that the grandchild of a family of low income to be in high income level is 0.26 .

b) Since $A$ is a regular matrix, we need to find $A^n$ for large $n$. Here is some powers of $A$ computed by using MATLAB:

$A^{10}= \left[ \begin{array}{rrr} 0.24490804170000 & 0.24490213680000 & 0.2... ....28571259860000 &0.28571259860000 & 0.28571850350000 \\ \end{array} \right]$

and $A^{100}= \left[ \begin{array}{rrr} 0.24489795918367 & 0.24489795918367 & 0.... ...0.28571428571429 &0.28571428571429 & 0.28571428571429\\ \end{array} \right]$

This suggest that the steady-state vector can be approximate by

$A^{100}= \left[ \begin{array}{r} 0.245 \\ 0.469 \\ 0.288 \\ \end{array} \right]$

therefore, in long run, $29 \%$ of the population will be in in low income level.