Question

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

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

Key concepts

Compound Interest Formula

The compound interest formula is a powerful tool used to calculate the future value of an investment or economic measure, such as GDP, over time. In essence, it helps us understand how the value of an amount grows when interest adds to the principal in multiple periods. The formula is expressed mathematically as:

\[FV = PV \times (1 + r)^n\]
Where:

• \(FV\) is the Future Value.
• \(PV\) is the Present Value - the starting amount.
• \(r\) is the annual growth rate (in decimal form).
• \(n\) represents the number of years.

Using this formula, when we plug in the values, it calculates how much GDP per capita will grow over a specified period. The formula accounts for not only the initial principal amount but also the interest accrued over each year. Hence, it reflects exponential growth, which is crucial when understanding long-term economic projections.

Economic Growth Rate

The economic growth rate indicates how much an economy's GDP per capita changes over a specific period, often annually. This rate is usually expressed as a percentage, showing how much the economy is expanding or contracting.

To utilize this rate in mathematical calculations, it is crucial to convert it from a percentage to a decimal. For instance, a 3% growth rate translates to 0.03 in decimal form. This conversion is essential for plugging into the compound interest formula, as it ensures the growth rate is factored into calculations correctly.

Understanding the economic growth rate helps policymakers and economists assess an economy's health. It provides a snapshot of whether an economy is growing and, if so, at what pace. This insight can inform policy decisions and economic forecasts, making it a fundamental concept in economics.

Future Value Calculation

Future value calculation is a method used to determine what a present amount of money or economic measure will be worth at a future date, considering a particular growth rate. It is essential in planning and forecasting economic outcomes, such as GDP, investments, or savings.

The future value is determined by applying the compound interest formula, which takes into account the present amount, expected growth rate, and the time frame of growth. For example, if an economy's GDP per capita starts at €12,000 with a growth rate of 6% over 30 years, the future GDP per capita can be calculated as:

\[Future \ Value = 12,000 \times (1 + 0.06)^{30}\]
In our case from the exercise, this calculation yields a future GDP per capita of €68,922.00. This concept allows us to predict how current values will change over time, helping in economic planning and strategic decisions.