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Chapter 24: Problem 8

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

Expert verified
Answer: $40,984

Step by step solution

01

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 = \(20000 * (1 + (2/100))^{100}\)
02

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 = \(20000 * (1 + (1.5/100))^{100}\)
03

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 = \((20000 * (1 + (2/100))^{100}) - (20000 * (1 + (1.5/100))^{100})\)
04

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.

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