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?

Understanding the compound interest formula is crucial when calculating the growth of investments or, in this exercise, the future value of an economy's GDP per capita. The formula is expressed as
\(FV = PV \times (1 + r)^n\),
where

• \(FV\) represents the future value,
• \(PV\) is the present value or initial amount,
• \(r\) is the interest rate per period (in decimal form), and
• \(n\) is the number of periods.

To apply this formula, convert the percentage growth rate into a decimal by dividing it by 100. This formula shows how an initial amount grows over time with the power of compounding, meaning that interest is earned on the initial principal and on the accumulated interest from previous periods. This effect makes the compound interest formula a powerful tool for predicting economic growth over the long run.
The key takeaway is that as time passes (increasing \(n\)), and/or as the growth rate (\(r\)) increases, the future value rises exponentially, which highlights the profound impact of compounding over longer periods.

The economic growth rate is a measure of the increase in a country's economic output over time, commonly represented by its GDP (Gross Domestic Product). For individual well-being, economists often look at GDP per capita, which divides GDP by the population to assess the average economic output per person.
When an economy grows at a certain percentage rate, such as the given \(2\%\), \(4\%\), or \(6\%\), it signifies how much the quantity of goods and services produced—and consequently, the standard of living—is increasing annually for the average citizen.

Evaluating Long-Term Growth

In our exercise, the multi-year growth illustrates the compounding effect; not only does the GDP increase due to the growth of the previous year's output, but it also expands further with every subsequent year's growth laid on top of the last, similar to how interest compounds in a bank account. This exercise smartly uses the compound interest formula to simplify the complex real-world dynamics into a clear mathematical model, projecting the future value of GDP per capita.

The future value calculation provides a way to estimate how much an investment will be worth after a certain number of periods, accounting for compound interest. This is critical in planning for economic growth, retirement, or evaluating the growth potential of various assets.
In the context of GDP per capita growth, the future value tells us what the economic output per person may look like down the line if the economy continues to grow at a consistent rate.

Real-world Implications

While the formula gives a mathematical projection, it's important to recognize that actual economic conditions are subject to many variables like inflation, changes in technology, political stability, and policy decisions. Therefore, these future value calculations are best used as a guide rather than a definitive forecast.
Ultimately, whether for an individual's savings or an entire country's economy, understanding the future value calculation empowers us to plan with foresight and appreciate the potential outcomes of sustained growth or investment.