Question

Suppose there is a country in which 40% of people are below age 20,
40% are between ages 20 and 40, and 20% are between ages 40 and
60. Assume no one lives above age 60 and the fertility rate is 4. The
initial population size is 100.
Compute the proportion of young, adult, and old after 20, 40, 60,
80, and 100 years.
Calculate the population growth rate over each period of 20 years.
(c) Assume a country with the same population composition but
with a fertility rate of 2 as a consequence of family planning
policies. Do the same exercise as in (a) and (b) above. What is
the effect of family planning policies after 20, 40, 60, 80, and
100 years on i) population growth and ii) the composition of the
population?

Answer 1

Since, the fertility rate is the same for all the age group so, the relative proportion of young, adult and old in the total population shall remain the same. Formula for projecting future population is following

Future Population = (Population in Current Year) × e^rt

e = natural logarithm base of 2.71828

r = Fertility rate divided by 100

t = Year for which population projection is needed

People	Population	Fertility Rate	N1 Future Year	Population after 20 Years	%age Share
Young	40	4	20	89	40%
Adult	40	4	20	89	40%
Old	20	4	20	45	20%
Total	100			223	100%
People	Population	Fertility Rate	N2 Future Year	Population after 40 Years	%age Share
Young	40	4	40	198	40%
Adult	40	4	40	198	40%
Old	20	4	40	99	20%
Total	100			495	100%
People	Population	Fertility Rate	N3 Future Year	Population after 60 Years	%age Share
Young	40	4	60	441	40%
Adult	40	4	60	441	40%
Old	20	4	60	220	20%
Total	100			1102	100%
People	Population	Fertility Rate	N4 Future Year	Population after 80 Years	%age Share
Young	40	4	80	981	40%
Adult	40	4	80	981	40%
Old	20	4	80	491	20%
Total	100			2453	100%
People	Population	Fertility Rate	N5 Future Year	Population after 100 Years	%age Share
Young	40	4	100	2184	40%
Adult	40	4	100	2184	40%
Old	20	4	100	1092	20%
Total	100			5460	100%

When Fertility rate is 2 then future population projection declines, however, the population composition will remain the same.

People	Population	Fertility Rate	N1 Future Year	Population after 20 Years	%age Share
Young	40	2	20	60	40%
Adult	40	2	20	60	40%
Old	20	2	20	30	20%
Total	100			149	100%
People	Population	Fertility Rate	N2 Future Year	Population after 40 Years	%age Share
Young	40	2	40	89	40%
Adult	40	2	40	89	40%
Old	20	2	40	45	20%
Total	100			223	100%
People	Population	Fertility Rate	N3 Future Year	Population after 60 Years	%age Share
Young	40	2	60	133	40%
Adult	40	2	60	133	40%
Old	20	2	60	66	20%
Total	100			332	100%
People	Population	Fertility Rate	N4 Future Year	Population after 80 Years	%age Share
Young	40	2	80	198	40%
Adult	40	2	80	198	40%
Old	20	2	80	99	20%
Total	100			495	100%
People	Population	Fertility Rate	N5 Future Year	Population after 100 Years	%age Share
Young	40	2	100	296	40%
Adult	40	2	100	296	40%
Old	20	2	100	148	20%
Total	100			739	100%