Chapter 20: Problem 33

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

Short Answer

Expert verified

The future GDP per capita at different growth rates and time periods is as follows: 1. After 10 years at a 3% annual growth rate: €16,127.04 2. After 30 years at a 3% annual growth rate: €29,127.12 3. After 30 years at a 6% annual growth rate: €68,922.00

Step by step solution

Identify the variables

In this exercise, we can identify the following variables: - Present Value (PV): €12,000 (Current GDP per capita) - Time: 10 years or 30 years - Growth Rate: 3% or 6% We will use these values to calculate the future GDP per capita in the later steps.

Convert percentage to decimal

We need to convert the growth rate from percentage to decimal form: 3% = 0.03 6% = 0.06

Calculate the GDP per capita after 10 years at 3% annual growth rate

Now, we can use the compound interest formula to find the GDP per capita after 10 years of 3% growth: Future Value_A = €12,000 x (1 + 0.03)^10 Future Value_A = €12,000 x (1.03)^10 Future Value_A = €12,000 x 1.34392 Future Value_A = €16,127.04 Thus, the GDP per capita after 10 years at 3% annual growth is €16,127.04.

Calculate the GDP per capita after 30 years at 3% annual growth rate

Using the same compound interest formula, we can now find the GDP per capita after 30 years of 3% growth: Future Value_B = €12,000 x (1 + 0.03)^30 Future Value_B = €12,000 x (1.03)^30 Future Value_B = €12,000 x 2.42726 Future Value_B = €29,127.12 Thus, the GDP per capita after 30 years at 3% annual growth is €29,127.12.

Calculate the GDP per capita after 30 years at 6% annual growth rate

Lastly, let's find the GDP per capita after 30 years of 6% growth: Future Value_C = €12,000 x (1 + 0.06)^30 Future Value_C = 12,000 x (1.06)^30 Future Value_C = €12,000 x 5.7435 Future Value_C = €68,922.00 Thus, the GDP per capita after 30 years at a 6% annual growth rate is €68,922.00. To summarize: 1. GDP per capita after 10 years at 3% growth rate: €16,127.04 2. GDP per capita after 30 years at 3% growth rate: €29,127.12 3. GDP per capita after 30 years at 6% growth rate: €68,922.00