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DA7- GDP Growth Rates

Statement

The rate of economic growth per capita in France from 1996 to 2000 was 1.9% per year, while in Korea over the same period it was 4.2%. Per capita real GDP was $28,900 in France in 2003, and $12,700 in Korea. Assume the growth rates for each country remain the same.

  1. Compute the doubling time for France’s per capita real GDP.
  2. Compute the doubling time for Korea’s per capita real GDP.
  3. What will France’s per capita real GDP be in 2045?
  4. What will Korea’s per capita real GDP be in 2045?

Answer

Introduction

The Rule of 72 is a simple way to determine the approximate doubling time for a variable that grows at a given percentage rate (exponential growth). The Rule of 72 states that the doubling time is approximately equal to 72 divided by the growth rate, stated as a whole number (Rittenberg & Tregarthen, 2009). According to the formula:

\[ \text{Doubling Time} = \frac{72}{\text{Growth Rate}} \]

The growth function for real GDP depends on the real GDP of the previous year and the growth rate; that is, the total real GDP in a year is the real GDP of the previous year multiplied by the growth rate, and then added to the real GDP of the previous year. The growth rate is the percentage increase in real GDP from one year to the next. According to the formula (Smith, 2009):

\[ Y_{t} = Y_{t-1} + (Y_{t-1} \times r) \\ Y_{t} = Y_{0}(1 + r)^{t} \]

Where:

  • \(Y_{t}\) = real GDP in year t.
  • \(Y_{t-1}\) = real GDP in the previous year.
  • \(r\) = growth rate.
  • \(Y_{0}\) = initial real GDP.

1. Compute the doubling time for France’s per capita real GDP

Given that the growth rate for France is 1.9% per year, we can calculate the doubling time using the Rule of 72:

\[ dt = \frac{72}{1.9} \approx 37.89 \]

Therefore, the real GDP per capita in France will double approximately every 38 years, that is, the real GDP per capita will double in 2041 (2003 + 38).

2. Compute the doubling time for Korea’s per capita real GDP

Given that the growth rate for Korea is 4.2% per year, we can calculate the doubling time using the Rule of 72:

\[ dt = \frac{72}{4.2} \approx 17.14 \]

Therefore, the real GDP per capita in Korea will double approximately every 17 years, that is, the real GDP per capita will double in 2020 (2003 + 17).

3. What will France’s per capita real GDP be in 2045?

Let’s find the the growth function for France’s real GDP per capita, knowing that the initial real GDP per capita was $28,900 in 2003 and the growth rate is 1.9% per year:

\[ Y_{t} = 28900(1 + 0.019)^{t} \]

The year 2045 is 42 years after 2003. Therefore, we substitute \(t = 42\) into the formula:

\[ \begin{aligned} Y_{42} &= 28900(1 + 0.019)^{42} \\ &\approx 28900(1.019)^{42} \\ &\approx 28900(2.2045) \\ &\approx 63710.05 \end{aligned} \]

Therefore, France’s per capita real GDP in 2045 will be approximately $63,710.

4. What will Korea’s per capita real GDP be in 2045?

Let’s find the the growth function for Korea’s real GDP per capita, knowing that the initial real GDP per capita was $12,700 in 2003 and the growth rate is 4.2% per year:

\[ Y_{t} = 12700(1 + 0.042)^{t} \]

The year 2045 is 42 years after 2003. Therefore, we substitute \(t = 42\) into the formula:

\[ \begin{aligned} Y_{42} &= 12700(1 + 0.042)^{42} \\ &\approx 12700(1.042)^{42} \\ &\approx 12700(5.62916) \\ &\approx 71490.332 \end{aligned} \]

Therefore, Korea’s per capita real GDP in 2045 will be approximately $71,490.

Conclusion

In 2003, France had almost 2.3 times the per capita real GDP of Korea. However, Korea’s growth rate was 2.2 times that of France. As a result, in 4 decades, Korea’s per capita real GDP will surpass that of France. This is an important example of how growth rates can significantly impact the economic development of countries over long periods.

References