Question

An economy starts off with a GDP per capital of \$5,000. How large will the GDP per capital be if it grows at an annual rate of \(2 \%\) for 20 years? \(2 \%\) for 40 years? 4\% for 40 years? 6\% for 40 years?

Short answer

The GDP per capita after 20 years at 2% annual growth rate is approximately \$7,430.16, after 40 years at 2% annual growth rate is approximately \$11,020.57, after 40 years at 4% annual growth rate is approximately \$25,553.29, and after 40 years at 6% annual growth rate is approximately \$59,306.65.

Step-by-step solution

Convert percentage growth rate to decimal

In order to use the compound interest formula, we need to convert the annual growth rates (2%, 4%, and 6%) into decimals. To do this, divide the percentage by 100: 2% = 0.02 4% = 0.04 6% = 0.06

Calculate GDP per capita after 20 and 40 years at 2% annual growth rate

We will first calculate GDP per capita after 20 years and 40 years with a 2% annual growth rate. For 20 years: A = 5000(1 + 0.02/1)^(1*20) A = 5000(1 + 0.02)^20 A = 5000(1.02)^20 ≈ \$7,430.16 For 40 years: A = 5000(1 + 0.02/1)^(1*40) A = 5000(1 + 0.02)^40 ≈ \$11,020.57

Calculate GDP per capita after 40 years at 4% annual growth rate

Now, we calculate GDP per capita after 40 years with a 4% annual growth rate. A = 5000(1 + 0.04/1)^(1*40) A = 5000(1 + 0.04)^40 ≈ \$25,553.29

Calculate GDP per capita after 40 years at 6% annual growth rate

Finally, we will calculate GDP per capita after 40 years with a 6% annual growth rate. A = 5000(1 + 0.06/1)^(1*40) A = 5000(1 + 0.06)^40 ≈ \$59,306.65 So, the GDP per capita in various scenarios are: - After 20 years at 2% annual growth rate: \$7,430.16 - After 40 years at 2% annual growth rate: \$11,020.57 - After 40 years at 4% annual growth rate: \$25,553.29 - After 40 years at 6% annual growth rate: \$59,306.65

Key concepts

Compound Interest Formula

When dealing with the growth of an economic indicator such as GDP per capita, the compound interest formula plays a crucial role. This formula is used to calculate the accumulated amount of something over time when it grows at a consistent rate – in this case, GDP. The general form of the compound interest formula is:

\[ A = P(1 + \frac{r}{n})^{nt} \]

where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial amount of money).
• \(r\) is the annual interest rate (in decimal).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the time the money is invested or borrowed for, in years.

To apply this formula to GDP growth, we replace the principal \(P\) with the starting GDP per capita and the interest rate \(r\) with the growth rate. The concept of 'compound interest' reflects how GDP per capita increases not only on the initial amount but also on the accumulated growth from preceding years, which is pivotal for understanding economic development over the long term.

Percentage to Decimal Conversion

To use the compound interest formula, it's necessary to convert the growth rate from a percentage to a decimal because the formula requires the rate to be expressed as a fraction of 1. Here's how a percentage is converted to a decimal:

To change a percentage to a decimal, divide the percentage by 100. That means dropping the percent sign and moving the decimal point two places to the left. For instance,

• 2% becomes 0.02
• 4% becomes 0.04
• 6% becomes 0.06

This step is important for accurately applying the growth rates to GDP per capita calculations. Failing to convert percentages to decimals correctly can lead to significant errors when computing the future value of economic indicators.

Exponential Growth

The term 'exponential growth' refers to an increase where the rate becomes quicker in proportion to the growing total number or size. In the context of GDP per capita, exponential growth implies that as the economy grows, it doesn't just add a fixed amount each year; instead, it grows by a percentage, leading to a much larger increase as time goes on. This is why small differences in growth rates can lead to big differences in future GDP per capita, as evidenced by the calculated scenarios over 20 and 40 years.

Understanding the concept of exponential growth is fundamental for grasping how economies can transform over time, especially when modeling long-term economic scenarios. The compound interest formula, with its exponential function, is a mathematical representation of this kind of growth, demonstrating how populations, investments, and indeed economies can expand rapidly under the right conditions.

Economic Growth Analysis

Economic growth analysis involves examining changes in economic indicators, like GDP per capita, to understand an economy's performance over time. The calculated values of GDP per capita after 20 and 40 years at different growth rates offer vital insight into how the economy might evolve. Analysts can utilize this data to predict future economic conditions and to inform policy decisions.

Differences in growth rates exhibit significant impacts on the long-term economic health of a country; for instance, the substantial difference between 2% and 6% growth over the same period. Even small changes in the growth rate can result in large differences in GDP per capita over decades. This kind of analysis helps not only in forecasting and planning but also in comparing the developmental progress of different economies or assessing the impact of economic policies over time.