Question

The population of a city in 2000 was 500,000 while the population of the suburbs of that city in 2000 was 700,000. Suppose that demographic studies show that each year about 6% of the city's population moves to the suburbs (and 94% stays in the city), while 2% of the suburban population moves to the city (and 98% remains in the suburbs). Compute the population of the city and of the suburbs in the year 2002. For simplicity, ignore other influences on the population such as births, deaths, and migration into and out of the city/suburban region.

Answer 1

Answer:

The population of the city in 2002 is 469,280 while the population of the suburb is 730,720.

Step-by-step explanation:

• 6% of the city's population moves to the suburbs (and 94% stays in the city).
• 2% of the suburban population moves to the city (and 98% remains in the suburbs).

The migration matrix is given as:

$A= \left \begin{array}{cc} \\ C \\S \end{array} \right\left[ \begin{array}{cc} C&S\\ 0.94&0.06 \\0.02&0.98 \end{array} \right]$

The population in the  year 2000 (initial state) is given as:

$\left[ \begin{array}{cc} C&S\\ 500,000&700,000 \end{array} \right]$

Therefore, the population of the city and the suburb in 2002 (two years after) is:

$S_0A^2=\left \begin{array}{cc} [500,000&700,000]\\& \end{array} \right\left \begin{array}{cc} \end{array} \right\left[ \begin{array}{cc} 0.94&0.06 \\0.02&0.98 \end{array} \right]^2$

$A^{2} = \left[ \begin{array}{cc} 0.8848 & 0.1152 \\\\ 0.0384 & 0.9616 \end{array} \right]$

Therefore:

$S_0A^2=\left \begin{array}{cc} [500,000&700,000]\\& \end{array} \right\left \begin{array}{cc} \end{array} \right \left[ \begin{array}{cc} 0.8848 & 0.1152 \\ 0.0384 & 0.9616 \end{array} \right]\\\\=\left[ \begin{array}{cc} 500,000*0.8848+700,000*0.0384& 500,000*0.1152 +700,000*0.9616 \end{array} \right]\\\\=\left[ \begin{array}{cc} 469280& 730720 \end{array} \right]$

Therefore, the population of the city in 2002 is 469,280 while the population of the suburb is 730,720.