Chapter 9.2, Problem 57E

Labor force. Table 1 gives the percentage of the U.S. female population who were members of the civilian labor force in the indicated years. The following transition matrix $P$ is proposed as a model for the data, where $L$ represents females who are in the labor force and $L\text{'}$ represents females who are not in the labor force:

$\begin{array}{l}\text{ Next decade}\\ \begin{array}{cc}L& L^{\prime }\end{array}\\ \begin{array}{c}\text{Current}\\ \text{decade}\end{array}\begin{array}{c}L\\ L^{\prime }\end{array}\left[\begin{array}{cc}.92& .08\\ .2& .8\end{array}\right]=P\end{array}$

(A) Let $S_0=\left[\begin{array}{cc}.433& .567\end{array}\right]$, and find $S_1,S_2,S_3$, and $S_4$. Compute the matrices exactly and then round entries to three decimal places.

(B) Construct a new table comparing the results from part (A) with the data in Table 1.

(C) According to this transition matrix, what percentage of the U.S. female population will be in the labor force in the long run?