Question

An economy starts off with a GDP per capita of \$5,000. How large will the GDP per capita 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 will be approximately: - \$7,386.20 after 20 years with a 2% growth rate - \$10,856.60 after 40 years with a 2% growth rate - \$29,567.97 after 40 years with a 4% growth rate - \$81,169.32 after 40 years with a 6% growth rate

Step-by-step solution

Scenario 1: 2% growth rate for 20 years

We have: \(P = \$5,000\), \(r = 0.02\), and \(n = 20\) We can now use the formula to find GDP per capita after 20 years: \(A = \$5,000(1 + 0.02)^{20}\) \(A \approx \$7,386.20\)

Scenario 2: 2% growth rate for 40 years

We have: \(P = \$5,000\), \(r = 0.02\), and \(n = 40\) We can now use the formula to find GDP per capita after 40 years: \(A = \$5,000(1 + 0.02)^{40}\) \(A \approx \$10,856.60\)

Scenario 3: 4% growth rate for 40 years

We have: \(P = \$5,000\), \(r = 0.04\), and \(n = 40\) We can now use the formula to find GDP per capita after 40 years: \(A = \$5,000(1 + 0.04)^{40}\) \(A \approx \$29,567.97\)

Scenario 4: 6% growth rate for 40 years

We have: \(P = \$5,000\), \(r = 0.06\), and \(n = 40\) We can now use the formula to find GDP per capita after 40 years: \(A = \$5,000(1 + 0.06)^{40}\) \(A \approx \$81,169.32\) In conclusion, the GDP per capita will be as follows: - \$7,386.20 after 20 years with a 2% growth rate - \$10,856.60 after 40 years with a 2% growth rate - \$29,567.97 after 40 years with a 4% growth rate - \$81,169.32 after 40 years with a 6% growth rate

Key concepts

GDP per Capita

Gross Domestic Product (GDP) per capita is one of the primary indicators used to gauge the health of a country's economy. It represents the average economic output each person would receive if all the GDP were evenly distributed among the population. Simplistically, it's calculated by dividing the total GDP of a country by its population.

When trying to understand the concept, it's essential to realize that a higher GDP per capita suggests that individuals in the country, on average, are better off financially and have access to more resources. However, it does not account for the distribution of income among the residents, so it's possible for a nation to have a high GDP per capita while still having significant income inequality. For students and analysts, keeping in mind the limits and the representativeness of GDP per capita is crucial for an accurate analysis of economic well-being.

Economic Growth

Economic growth is typically expressed as an increase in a country's GDP over time. It indicates how the economy is expanding and is capable of producing more goods and services compared to previous periods.

This growth is paramount for improving living standards, reducing poverty, and potentially enabling governments to invest more in public services. Most economies aim for a sustained, healthy rate of economic growth, which often requires careful fiscal and monetary policies. For example, an economy growing too quickly might lead to inflationary pressures, while one growing too slowly could signal economic stagnation or recession. Understanding economic growth in the context of GDP per capita is crucial because it tells us not just about the changes in total economic output, but how these changes may affect individual prosperity.

Compound Interest

Compound interest is a crucial concept in finance that refers to the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. This leads to the growth of a sum of money at an exponential rate over time, as the interest that accrues to the investment also earns interest.

The formula for calculating the future value of an investment due to compound interest is given by: \[ A = P(1 + r)^n \] where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount, \(r\) is the annual interest rate, and \(n\) is the number of years the money is invested for.

When applied to GDP per capita growth, this formula demonstrates how an economy's output can increase over time. The exercise provided illustrates exactly this by showing the effect of different growth rates over differing periods on the GDP per capita, using principles akin to compound interest to project future economic output.

Macroeconomics

Macroeconomics is a branch of economics that focuses on the behavior and performance of an economy as a whole. It considers various aggregate indicators such as GDP, unemployment rates, and inflation. These indicators reflect the broad economic trends and are used to formulate economic policies.

One of the central issues in macroeconomics is the study of how to generate sustainable economic growth. Macroeconomists analyze the impact of government fiscal policy and central bank monetary policy on national economies, laying the groundwork for understanding when and how an economy grows. They also consider the implications of economic decisions on GDP per capita, ensuring a comprehensive approach to economic analysis that balances overall growth with individual welfare.