South Korea has real GDP per capita of $25,000, while England has real GDP per capita of $50,000. If real GDP per capita in South Korea grows at 7% and England's real GDP per capita grows at 3.5%, how long will it take for real GDP per capita in the two nations to converge?

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Quizplus · Accepted Answer

To find when the real GDP per capita of South Korea and England will converge, we use the formula for the time it takes for an investment to double, adjusting it for the growth rates and starting values. The formula for the doubling time is derived from the rule of 72, but in this case, we are looking for the time it takes for South Korea's GDP per capita to catch up to England's, not just double. Given:- South Korea's starting GDP per capita = $25,000, growing at 7% per year.- England's starting GDP per capita = $50,000, growing at 3.5% per year.We can set up the equation as follows, where $T$ is the time in years for the GDPs per capita to equalize:$$25,000 \times (1 + 0.07)^T = 50,000 \times (1 + 0.035)^T$$To solve for $T$, we can simplify the equation by dividing both sides by 25,000 and then taking the natural logarithm of both sides to solve for $T$. However, without going through the logarithmic steps here, we can estimate the time based on the rule of 70 (a variation of the rule of 72 used for more precise calculations in economics), which states that the doubling time of an investment can be approximated by dividing 70 by the annual growth rate.The key insight is that South Korea's GDP per capita is growing faster than England's, allowing it to catch up over time. Given the growth rates, South Korea's GDP per capita is growing at a rate that will allow it to double faster than England's GDP per capita can double. By calculating or estimating the time it takes for the gap to close, considering the growth rates provided, it would take approximately 20 years for South Korea's GDP per capita to converge with that of England, assuming constant annual growth rates for both countries. This is a simplified approach to illustrate the concept, as the actual calculation would involve more complex logarithmic functions to solve for $T$.

South Korea Has Real GDP Per Capita of $25,000, While