Question

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

Short answer

The GDP per capita will be approximately 16,127 euros after 10 years of growth at a 3% annual rate, 29,127 euros after 30 years of growth at a 3% annual rate, and 68,882 euros after 30 years of growth at a 6% annual rate.

Step-by-step solution

Part 1: GDP per capita growth at 3% for 10 years

First, let's find the GDP per capita if it grows at a rate of 3% for 10 years. We will plug our values into the formula: GDP_final = 12000 * (1 + 0.03) ^ 10 Now carry out the calculations: GDP_final = 12000 * (1.03) ^ 10 GDP_final = 12000 * 1.34392 GDP_final ≈ 16127 euros The GDP per capita will be approximately 16,127 euros after 10 years of growth at a 3% annual rate.

Part 2: GDP per capita growth at 3% for 30 years

Now, let's find the GDP per capita if it grows at a rate of 3% for 30 years. We will use the same formula and input appropriate values: GDP_final = 12000 * (1 + 0.03) ^ 30 Now carry out the calculations: GDP_final = 12000 * (1.03) ^ 30 GDP_final = 12000 * 2.42726 GDP_final ≈ 29127 euros The GDP per capita will be approximately 29,127 euros after 30 years of growth at a 3% annual rate.

Part 3: GDP per capita growth at 6% for 30 years

Finally, let's find the GDP per capita if it grows at a rate of 6% for 30 years. We will use the same formula and input appropriate values: GDP_final = 12000 * (1 + 0.06) ^ 30 Now carry out the calculations: GDP_final = 12000 * (1.06) ^ 30 GDP_final = 12000 * 5.74349 GDP_final ≈ 68882 euros The GDP per capita will be approximately 68,882 euros after 30 years of growth at a 6% annual rate.

Key concepts

Understanding the Economic Growth Rate

When economists refer to the economic growth rate, they are often describing the pace at which a nation's economic output is expanding or contracting over time. It's an essential indicator that reflects the health of an economy, guiding policymakers and investors in their decisions.

Specifically, the rate of economic growth is typically measured as the percentage increase in Gross Domestic Product (GDP)—which is the total market value of all finished goods and services produced in a country during a specific period. Several factors can influence this rate, including technological advancements, increases in the workforce, education, and capital investment.

Demystifying the GDP Per Capita Formula

The GDP per capita is a measure that divides a country's GDP by its total population, offering a per-person average that can be a more telling indicator of the relative economic performance of a nation, especially when comparing economies of different sizes.

GDP per capita formula:

\[ \text{GDP per capita} = \frac{\text{Total GDP}}{\text{Total population}} \]

Key advantages of using GDP per capita include its ability to provide insight into the standard of living and economic well-being of a country's average individual, allowing for more nuanced comparisons across countries or regions than total GDP alone. However, it doesn't account for income inequality or non-market transactions and is not a comprehensive measure of an economy's health.

Calculating the Annual Growth Rate

To calculate the annual growth rate of any financial metric, such as GDP per capita, analysts use a common formula that takes into account the starting value, the ending value, and the number of time periods.

Annual growth rate calculation:

\[ \text{Growth Rate} = \left(\frac{\text{Ending Value}}{\text{Starting Value}}\right)^{\frac{1}{n}} - 1 \]

Where the ending value is the final amount, the starting value is the initial amount, and 'n' stands for the number of years. Applied to GDP per capita growth, this formula gives a clear depiction of how the average economic output per person is expected to change annually over a given period, presumes a constant growth rate during that period. Understanding this concept allows individuals to forecast future economic conditions and make informed decisions.