Chapter 7: Problem 32

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

Expert verified

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.

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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

Step by step solution

Scenario 1: 2% Growth Rate for 20 Years

Scenario 2: 2% Growth Rate for 40 Years

Scenario 3: 4% Growth Rate for 40 Years

Scenario 4: 6% Growth Rate for 40 Years

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