Question

An economy starts off with a GDP per capita of \$5,000. How large will the GDP per capita be if it grows at an annual rate of \(2 \%\) for 20 years? \(2 \%\) for 40 years? \(4 \%\) for 40 years? \(6 \%\) for 40 years?

Short answer

The GDP per capita will be \$7,429.73 after 20 years with a 2% annual growth rate, \$11,041.73 after 40 years with a 2% annual growth rate, \$24,007.58 after 40 years with a 4% annual growth rate, and \$51,428.89 after 40 years with a 6% annual growth rate.

Step-by-step solution

Scenario 1: 2% Growth Rate for 20 Years

To find the GDP per capita after 20 years with a 2% annual growth rate, we will use the compound interest formula: Future Value = \(5,000 * (1 + 0.02)^{20}\) Now we just need to plug in the numbers and do the math: GDP per capita = \(5,000 * (1.02)^{20}\) GDP per capita = \(5,000 * 1.485947\ldots\) GDP per capita = \(7,429.73\) So, the GDP per capita after 20 years with a 2% annual growth rate will be \$7,429.73.

Scenario 2: 2% Growth Rate for 40 Years

We will now find the GDP per capita after 40 years with a 2% annual growth rate: GDP per capita = \(5,000 * (1.02)^{40}\) GDP per capita = \(5,000 * 2.208\ldots\) GDP per capita = \(11,041.73\) The GDP per capita after 40 years with a 2% annual growth rate will be \$11,041.73.

Scenario 3: 4% Growth Rate for 40 Years

Next, we will find the GDP per capita after 40 years with a 4% annual growth rate: GDP per capita = \(5,000 * (1.04)^{40}\) GDP per capita = \(5,000 * 4.801\ldots\) GDP per capita = \(24,007.58\) The GDP per capita after 40 years with a 4% annual growth rate will be \$24,007.58.

Scenario 4: 6% Growth Rate for 40 Years

Finally, we will find the GDP per capita after 40 years with a 6% annual growth rate: GDP per capita = \(5,000 * (1.06)^{40}\) GDP per capita = \(5,000 * 10.285\ldots\) GDP per capita = \(51,428.89\) The GDP per capita after 40 years with a 6% annual growth rate will be \$51,428.89.