Sweden has real GDP per capita of $50,000, while Chile has real GDP per capita of $25,000. If real GDP per capita in Sweden grows at 2% and Chile's real GDP per capita grows at 4%, 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 the two nations will converge, we can use the formula for the time it takes for an investment to double, adjusting it for the growth rates and starting values of Sweden and Chile. The formula for the doubling time using the rule of 70 is $T = \frac{70}{r}$, where $T$ is the time in years and $r$ is the annual growth rate. However, in this context, we need to find when the GDP per capita of Chile equals that of Sweden, starting from different bases and growing at different rates.Let's denote Sweden's GDP per capita by $S$ and Chile's GDP per capita by $C$, with their respective growth rates as $r_S$ and $r_C$. We are given:- $S_0 = \$50,000$, $r_S = 2\%$- $C_0 = \$25,000$, $r_C = 4\%$We want to find $t$ such that $S_t = C_t$, where $S_t = S_0 \times (1 + r_S)^t$ and $C_t = C_0 \times (1 + r_C)^t$.Rearranging the equation gives us a formula to solve for $t$, but in practice, we can use the rule of 72 (a simplified version of the rule of 70 for easier math with these percentages) or directly calculate using logarithms for precise calculation. The exact calculation involves solving the equation $50,000 \times (1.02)^t = 25,000 \times (1.04)^t$, which simplifies to $(1.02)^t / (1.04)^t = 1/2$, or $(1.02/1.04)^t = 1/2$. Taking the natural logarithm of both sides allows us to solve for $t$.The correct method involves logarithms and growth rates, leading to the calculation that shows it will take approximately 35 years for the real GDP per capita of Chile to equal that of Sweden, assuming constant growth rates for both countries.

Sweden Has Real GDP Per Capita of $50,000, While Chile