Question

Two countries start with equal GDPs. The economy of Country A grows at an annual rate of 3 percent, whereas the economy of Country B grows at an annual rate of 4 percent. After 25 years, how much larger is Country \(\mathrm{B}^{\prime} \mathrm{s}\) economy than Country A's economy? Why is the answer not 25 percent?

Short answer

Why is the difference not 25%? Answer: The approximate percentage difference in size between the economies after 25 years is 28.10%. The difference is not 25% because the growth of each economy is exponential (compounded) rather than simple interest calculation, leading to faster accumulation of the growth as the initial GDP gets reinvested year after year.

Step-by-step solution

Identify the information given and the question

We have two countries (A and B) with equal GDPs, and their respective annual growth rates (3% and 4%). We need to find the difference in their economies after 25 years and explain why it's not a 25% difference.

Calculate the size of each economy after 25 years

To find the economies' size after 25 years, we'll use the formula for compound interest. The formula is: \(A=P(1+\frac{r}{n})^{nt}\) Where: - A is the future value of the investment/loan (the size of the economy) - P is the initial investment/loan principal (the initial GDP) - r is the annual interest rate (the annual growth rate) - n is the number of times that interest is compounded per time period - t is the time the money is invested or borrowed for, in years Since the growth is compounded yearly, n = 1. Let GDP\(_0\) be the initial GDP: For Country A: \(A_\text{A}= \text{GDP}_0 (1+0.03)^{25}\) For Country B: \(A_\text{B} = \text{GDP}_0 (1+0.04)^{25}\)

Find the difference in size between two economies

To find the difference between the two economies after 25 years, we'll subtract the size of economy A from economy B: Difference = \(A_\text{B} - A_\text{A}\) Difference = \(\text{GDP}_0 (1+0.04)^{25} - \text{GDP}_0 (1+0.03)^{25}\) We can factor out the initial GDP: Difference = \(\text{GDP}_0 [(1+0.04)^{25} - (1+0.03)^{25}]\)

Calculate the percentage difference

To find the percentage difference between the two economies, we'll divide the difference by GDP\(_0\): Percentage Difference = \(\frac{\text{Difference}}{\text{GDP}_0}\) Percentage Difference = \(\frac{\text{GDP}_0 [(1+0.04)^{25} - (1+0.03)^{25}]}{\text{GDP}_0}\) The GDP\(_0\) terms cancel out: Percentage Difference = \((1+0.04)^{25} - (1+0.03)^{25}\) Now, calculate the percentage difference: Percentage Difference ≈ 0.2810 Which is approximately 28.10% (rounded to 2 decimal places).

Explain why the difference is not 25 percent

The difference is not 25% because the growth of each economy is exponential (compounded) rather than a simple interest calculation. Compound growth accumulates at a faster rate as the interest earned on the initial GDP gets reinvested year after year. In this case, the compounded growth led to a larger difference of approximately 28.10% between Country B's and Country A's economies after 25 years.

Key concepts

GDP Comparison

Gross Domestic Product (GDP) serves as a broad measure for the economic performance of a country. When comparing the GDPs of two countries, especially over time, it's vital to consider the growth rates and how small differences can lead to significant disparities due to compound growth. In the textbook exercise, Country A and Country B start with equal GDPs, but they grow at different annual rates, 3% and 4% respectively.

Over a period of 25 years, even a seemingly small 1% difference in growth rates can result in a remarkable divergence in the sizes of their economies. This disparity is due to the exponential nature of economic growth – a concept fundamentally built on compound interest principles, that allows for gains to be made not only on the initial principal but also on the accumulated interest of prior periods. In the exercise, this is why Country B's economy ends up being approximately 28.10% larger rather than just 25% bigger than Country A's after 25 years.

Exponential Growth

Exponential growth in economics implies that a variable grows at a rate proportional to its current value, such as GDP, population, or investments. As opposed to linear growth, where a variable increases by a fixed amount each period, exponential growth increases by a percentage of the variable's current value. This means the larger the base becomes, the larger the amount it increases by each period.

In the context of our textbook problem, the annual growth rates for the economies of both countries are applied exponentially, meaning each year's growth will be a percentage of the previous year's GDP. As a result, even countries with small differences in their annual growth rates can end up with markedly different economies over time due to the compounding effect. This concept of exponential growth can often seem counterintuitive because our everyday experiences, like seeing constant speeds in car travel, more commonly align with linear rates of change.

Economic Growth Rate

The economic growth rate is a critical indicator of a country's economic health. It measures the rate at which a country's GDP increases over a specified period, typically annually. This rate is often expressed as a percentage and indicates how quickly the overall economy is expanding or contracting.

In our exercise, the countries have growth rates of 3% and 4%, respectively. These rates are not additive but are instead applied each year to the GDP of the previous year. The concept of economic growth rate is consequently tied to the principle of compounding, leading to potentially vast differences overt time. Essential to understand is that the growth rate, while it may stabilize, never acts on just the initial amount, but always on the accumulated value. Thus, even if the difference in growth rates seems negligible in the short term, over many years, that small percentage point difference expands, resulting in a significant gap in GDP after 25 years, reinforcing the powerful force of compound growth in economics.