To solve these problems, we will use the Rule of 70 to compute the doubling times and the formula for compound interest to determine the future GDP per capita.

Doubling Time

The Rule of 70 is a way to estimate the number of years it takes for a quantity to double at a constant growth rate. It is calculated as:

$\text{Doubling Time} = \frac{70}{\text{Growth Rate (\%)}}$

1. Doubling Time for France’s per capita real GDP: $\text{Doubling Time} = \frac{70}{1.9} \approx 36.84 \text{ years}$

2. Doubling Time for Korea’s per capita real GDP: $\text{Doubling Time} = \frac{70}{4.2} \approx 16.67 \text{ years}$

Future GDP per capita in 2045

To calculate the future GDP per capita, we use the compound interest formula:

$P(t) = P_0 \left(1 + \frac{r}{100}\right)^t$

where:

• $P(t)$ is the future GDP per capita
• $P_0$ is the current GDP per capita
• $r$ is the annual growth rate
• $t$ is the number of years

3. France’s per capita real GDP in 2045:

From 2003 to 2045 is 42 years. Therefore, $t = 42$, $P_0 = 28,900$, and $r = 1.9$.

$P(2045) = 28,900 \left(1 + \frac{1.9}{100}\right)^{42}$ $P(2045) = 28,900 \left(1.019\right)^{42}$

Calculating the exponent: $1.019^{42} \approx 2.076$

Therefore, $P(2045) \approx 28,900 \times 2.076 \approx 60,022.84$

4. Korea’s per capita real GDP in 2045:

From 2003 to 2045 is 42 years. Therefore, $t = 42$, $P_0 = 12,700$, and $r = 4.2$.

$P(2045) = 12,700 \left(1 + \frac{4.2}{100}\right)^{42}$ $P(2045) = 12,700 \left(1.042\right)^{42}$

Calculating the exponent: $1.042^{42} \approx 5.11$

Therefore, $P(2045) \approx 12,700 \times 5.11 \approx 64,897$

Summary

• Doubling Time for France: 36.84 years
• Doubling Time for Korea: 16.67 years
• France’s per capita real GDP in 2045: $60,022.84
• **Korea’s