In a certain community, currently, 10% of the people are H (high income), 60% are M (medium income), and 30% are L (low income). Studies show that for the children of H parents, 50% also become H, 40% become M, and 10% become L. Of the children of M parents, 40% become H, 30% become M, and 30% become L. Of the children of L parents, 30% become H, 50% become M, and 20% become L.
Use the above information to answer parts (a)-(c).
a. Set up the stochastic matrix that displays these transitions. Label completely.
b. Use the matrix from part (a) to determine what percentage of the children after 2 generations will have high incomes. You may use a calculator, but need to explicitly show all the matrices involved.
c. In the long run, what percentage of the population will have low incomes? Round to three decimal places. Show all the details.