Economists are studying a town of 40,000 family units. For the purpose of analysis they group each family into one of three cohorts depending upon their taxable income. For convenience we refer to these as the lower cohort, middle cohort, and upper cohort. Each year 6% of those in the lower rise into the middle, and 12% of those in the middle fall down to the lower. Meanwhile 11% of those in the middle move to the upper, and 11% of the upper move down to the middle. Lastly, 4% of the upper cohort drop all the way to the lower. We can assume that these changes occur approximately continuously throughout the year due to shifting economic and social conditions. At the time that the analysis begins the distribution of families into the cohorts is 22000, 13000, 5000). Let L), M(O), U(t) describe the number of family units in the respective cohort at time t measured in years with t - 0 denoting the first year considered in the study.

Write a system of equations and initial conditions which defined L(t), M(1), U(). Express this system in matrix form. Then use a computer algebra system to obtain a solution. Determine what the distribution of families into cohorts will be if these conditions persist indefinitely.