Q. 88

Question

Economic Mobility; The relative income of a child (low, medium, or high) generally depends on the relative income of the child’s parents. The matrix P, given by

$P a r e n t' s i n c o m e L M H P = [\begin{array}{lll} 0.4 0.2 0.1 \\ 0.5 0.6 0.5 \\ 0.1 0.2 0.4 \end{array}] \begin{aligned} L \\ M \\ H \end{aligned} C h i l d' s i n c o m e$

is called a left stochastic transition matrix. For example, the entry $P_{21} = 0.5$ means that $50 %$ of the children of low relative income parents will transition to the medium level of income. The diagonal entry $P_{i i}$ represents the percent of children who remain in the same income level as their parents. Assuming that the transition matrix is valid from one generation to the next, compute and interpret $P^{2}$ .

Step-by-Step Solution

Verified

Answer

The matrix $P^{2}$ shows the dependency of relative income of a grandchild with respect to his grandparents.

1Step 1. Given data

According to the economic mobility report, the relative income of a child generally depends on their parents' income and can be represented by $P$

$P = [\begin{array}{lll} 0.4 0.2 0.1 \\ 0.5 0.6 0.5 \\ 0.1 0.2 0.4 \end{array}]$

2Step 2. Compute the matrix P 2

Now let us compute the matrix $P^{2}$

$P^{2} = [\begin{array}{lll} 0.4 & 0.2 & 0.1 \\ 0.5 & 0.6 & 0.5 \\ 0.1 & 0.2 & 0.4 \end{array}] [\begin{array}{lll} 0.4 & 0.2 & 0.1 \\ 0.5 & 0.6 & 0.5 \\ 0.1 & 0.2 & 0.4 \end{array}] P^{2} = [\begin{array}{lll} 0.27 0.22 0.18 \\ 0.55 0.56 0.55 \\ 0.18 0.22 0.27 \end{array}]$

Therefore, this matrix shows the dependency of relative income of a grandchild with respect to his grandparents.

Q. 87

Q. 89

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