Question

Over the next 100 years, real GDP per capita in Groland is expected to grow at an average annual rate of \(2.0 \%\). In Sloland, however, growth is expected to be somewhat slower, at an average annual growth rate of \(1.5 \%\) If both countries have a real GDP per capita today of \(\$ 20,000,\) how will their real GDP per capita differ in 100 years? [Hint: A country that has a real GDP today of \(\$ x\) and grows at \(y \%\) per year will achieve a real GDP of \(\$ x\) \(\times(1+(y / 100))^{2}\) in \(z\) years. We assume that \(\left.0 \leq y<10 .\right]\)

Short answer

Answer: $40,984

Step-by-step solution

Find Groland's real GDP per capita after 100 years

Using the formula provided in the hint, we can calculate Groland's real GDP per capita after 100 years: Real GDP per capita = \(x * (1 + (y/100))^z\) \(x = \$20,000\), \(y = 2\%\), and \(z = 100\) years. Plug in the values to get: Real GDP per capita = \(20000 * (1 + (2/100))^{100}\)

Find Sloland's real GDP per capita after 100 years

Using the same formula, we can calculate Sloland's real GDP per capita after 100 years: \(x = \$20,000\), \(y = 1.5\%\), and \(z = 100\) years. Plug in the values to get: Real GDP per capita = \(20000 * (1 + (1.5/100))^{100}\)

Calculate the real GDP per capita difference

Now that we have the real GDP per capita for both countries, we can find the difference between the two: Difference = Groland's real GDP per capita - Sloland's real GDP per capita Difference = \((20000 * (1 + (2/100))^{100}) - (20000 * (1 + (1.5/100))^{100})\)

Simplify the expression and find the difference

Simplify the expression and calculate the difference: Difference = \((20000 * (1.02)^{100}) - (20000 * (1.015)^{100})\) Difference = \(20000 * ((1.02)^{100} - (1.015)^{100})\) Now, find the numerical value of the difference: Difference ≈ \(20000 * (7.2444 - 5.1952)\) Difference ≈ \(20000 * 2.0492\) Difference ≈ \(40,984\) Hence, after 100 years, the real GDP per capita of Groland will be approximately $40,984 greater than that of Sloland.

Key concepts

Economic growth rate

An economic growth rate is a measure that indicates how much a country's economy increases over a certain period, typically a year. It reflects the percentage change in the country's Gross Domestic Product (GDP). GDP represents the total value of all goods and services a country produces within a given time frame.

In the context of Groland and Sloland, both countries are starting with a real GDP per capita of $20,000. The economic growth rate for Groland is 2.0% per year, while Sloland's is 1.5% per year. This means that each year, Groland's GDP per capita increases by 2% of its current value, and Sloland's by 1.5%.

This percentage growth is crucial as it determines the economic health and potential of a country. Higher growth rates suggest better improvements in living standards and economic stability. Conversely, a slower rate can indicate economic challenges.

Annual growth rate calculation

Calculating the annual growth rate involves understanding how to project future GDP after a number of years. Using the example from the exercise, let’s delve into how this is calculated.

The given formula is \( x \times (1 + \frac{y}{100})^z \), where:

• \( x \) is the initial real GDP per capita ($20,000).
• \( y \) is the annual growth rate (2% for Groland, 1.5% for Sloland).
• \( z \) is the number of years (100 years).

For Groland, you substitute the values into the formula: \( 20000 \times (1 + \frac{2}{100})^{100} \). For Sloland: \( 20000 \times (1 + \frac{1.5}{100})^{100} \).

This method uses the power of exponential growth, capitalizing on the compound interest nature of GDP growth, which can lead to large differences over long periods.

Long-term economic projection

Long-term economic projection involves estimating a country’s economic status into the future, based on current trends. It is a critical process for policymakers and planners as it helps in setting economic goals and policy measures.

In the case of Groland and Sloland, a 100-year projection of real GDP per capita shows significant divergence due to their differing growth rates. After a century, even a slight difference in annual growth, like 0.5% (as seen between Groland and Sloland), can result in a substantial economic gap.

Long-term projections take into account various factors such as technological advancements, population growth, and fiscal policies. The results can offer insights into which strategies might be beneficial to sustain or boost future economic growth and prosperity, highlighting the importance of stable and strategic economic planning.