Question

If a country's income today is $19,000 million and its income 18 years ago was $7,490 million, what is the country's average annual growth rate in percentage terms over the last 18 years?

Answer 1

To calculate the average annual growth rate over the last 18 years, you can use the following formula:

\[ \text{Average Annual Growth Rate} = \left( \frac{ \text{Ending Value} }{ \text{Beginning Value} } \right)^{ \frac{1}{ \text{Number of Years} } } - 1 \]

Given:

Ending Value (Today's income) = $19,000 million

Beginning Value (Income 18 years ago) = $7,490 million

Number of Years = 18

Plugging in these values:

\[ \text{Average Annual Growth Rate} = \left( \frac{ $19,000 \text{ million} }{ $7,490 \text{ million} } \right)^{ \frac{1}{ 18 } } - 1 \]

\[ \text{Average Annual Growth Rate} = \left( \frac{ 19,000 }{ 7,490 } \right)^{ \frac{1}{ 18 } } - 1 \]

\[ \text{Average Annual Growth Rate} \approx 1.0893^{0.0556} - 1 \]

\[ \text{Average Annual Growth Rate} \approx 1.045 - 1 \]

\[ \text{Average Annual Growth Rate} \approx 0.045 \]

So, the average annual growth rate over the last 18 years is approximately 4.5%.

Answer 2

The average annual growth rate for a country's income from $7,490 million to $19,000 million over 18 years is calculated using the compound annual growth rate formula, resulting in an average growth rate of 5.45%. Examples also showed projected incomes and growth multiples for the United States and South Korea over 20 years and the projected increase in GDP per capita for various growth rates over 20 and 40 years.

Step-by-step explanation:

Calculating Average Annual Growth Rate

To calculate the country's average annual growth rate over 18 years from an income of $7,490 million to $19,000 million, we use the formula for compound annual growth rate (CAGR):

CAGR = [(Ending Value / Beginning Value) ^ (1 / Number of Years)] - 1

Plugging in the values, we get:

CAGR = [($19,000 million / $7,490 million) ^ (1 / 18)] - 1

CAGR = (2.53671641791 ^ 0.05555555556) - 1

CAGR = 1.054457 - 1

CAGR = 0.054457 or 5.45%

The average annual growth rate is therefore 5.45%.

Projected Incomes and Growth Multiples

Regarding the example provided:

1. In 20 years, the income of the United States which grows at 1% annually from a base of $10,000 will be $12,201.90, a multiple of 1.2.

2. South Korea, with a 4% annual growth rate, will have an income of $21,911.23 in the same period, a multiple of 2.1.

Impact of Growth Rates on GDP Per Capita

For an economy with a GDP per capita of $5,000:

At a 2% growth rate for 20 years, GDP will be $7,386.96, and for 40 years, it will be $10,948.43.

At a 4% growth rate for 40 years, it will be $21,724.52.

At a 6% growth rate for 40 years, GDP will reach $57,435.11.