Final answer:
The per capita income of a country with a real growth rate of 6% and a population growth rate of 1% will take approximately 14 years to double. After 30 years, the per capita income would grow to about $4,321.90.
Step-by-step explanation:
To calculate how long it will take for a country's per capita income to double with a real growth rate of 6% and a population growth rate of 1%, we need to use the rule of 70, a simplified way to determine the doubling time of a number. The rule states that you can approximate the number of years it will take for a number to double by dividing 70 by the annual growth rate. However, since we have both a growth rate and a population growth rate, we need to use the effective growth rate, which is the growth rate minus the population growth rate.
The effective growth rate for per capita income is 6% - 1% = 5%. Using the rule of 70, we divide 70 by the effective growth rate:
70 / 5 = 14 years to double the per capita income.
As for the income level after 30 years with the real growth rate of 5% (after accounting for population growth), we can use the formula for compound growth, which is:
Future Value = Present Value * (1 + growth rate)^number of periods
After 30 years, the per capita income would be $1,000 * (1 + 0.05)^30. This calculation gives us Future Value = $1,000 * (1.05)^30 = $1,000 * 4.3219 = $4,321.90.
Therefore, after 30 years, the per capita income would be approximately $4,321.90.