Question

An economy starts off with a GDP per capita of 12,000 euros. How large will the GDP per capita be if it grows at an annual rate of \(3 \%\) for 10 years? \(3 \%\) for 30 years? \(6 \%\) for 30 years?

Short answer

Using the compound interest formula, we find that the GDP per capita will be: 1. 16,127.04 euros after 10 years with a 3% annual growth rate. 2. 29,127.12 euros after 30 years with a 3% annual growth rate. 3. 68,921.88 euros after 30 years with a 6% annual growth rate.

Step-by-step solution

Scenario 1: 3% growth rate for 10 years

In this case, the annual growth rate is 3%, or 0.03 as a decimal, and the number of years is 10. Applying the formula, we have: Final GDP per capita = 12,000 * (1 + 0.03) ^ 10 = 12,000 * (1.03) ^ 10 = 12,000 * 1.34392 = 16,127.04 euros So, the GDP per capita will be 16,127.04 euros after 10 years with a 3% annual growth rate.

Scenario 2: 3% growth rate for 30 years

In this case, the annual growth rate is 3%, or 0.03 as a decimal, and the number of years is 30. Applying the formula, we have: Final GDP per capita = 12,000 * (1 + 0.03) ^ 30 = 12,000 * (1.03) ^ 30 = 12,000 * 2.42726 = 29,127.12 euros So, the GDP per capita will be 29,127.12 euros after 30 years with a 3% annual growth rate.

Scenario 3: 6% growth rate for 30 years

In this case, the annual growth rate is 6%, or 0.06 as a decimal, and the number of years is 30. Applying the formula, we have: Final GDP per capita = 12,000 * (1 + 0.06) ^ 30 = 12,000 * (1.06) ^ 30 = 12,000 * 5.74349 = 68,921.88 euros So, the GDP per capita will be 68,921.88 euros after 30 years with a 6% annual growth rate.