Question

Say that the average worker in Canada has a productivity level of \(\$ 30\) per hour while the average worker in the United Kingdom has a productivity level of \(\$ 25\) per hour (both measured in U.S. dollars). Over the next five years, say that worker productivity in Canada grows at \(1 \%\) per year while worker productivity in the UK grows \(3 \%\) per year. After five years, who will have the higher productivity level, and by how much?

Short answer

After five years, the productivity level in Canada will be \(\$30 * (1 + 0.01)^5 \approx \$32.16\), and the productivity level in the UK will be \(\$25 * (1 + 0.03)^5 \approx \$28.92\). Canada will have a higher productivity level by approximately \(\$32.16 - \$28.92 \approx \$3.24\).

Step-by-step solution

Calculate the future productivity levels

To calculate the productivity levels after five years, we'll apply the formula for compound interest: \(Future\ productivity = Current\ productivity * (1 + growth\ rate) ^{years}\).

Determine productivity in Canada after five years

For Canada, we have a current productivity of \(\$30\) and a growth rate of \(1 \%\) per year. Converting the percentage to a decimal, we get \(growth\ rate = 0.01\). Plugging these values into the formula, we get: \(Future\ productivity\ of\ Canada = 30 * (1 + 0.01)^5\) Calculate the future productivity of Canada.

Determine productivity in the UK after five years

For the UK, we have a current productivity of \(\$25\) and a growth rate of \(3 \%\) per year. Converting the percentage to a decimal, we get \(growth\ rate = 0.03\). Plugging these values into the formula, we get: \(Future\ productivity\ of\ UK = 25 * (1 + 0.03)^5\) Calculate the future productivity of the UK.

Compare productivity levels and find the difference

Once we have calculated the future productivity levels for both countries, we need to compare them to find which country will have a higher productivity level and by how much. To do this, subtract the lower productivity level from the higher productivity level. If the future productivity level of Canada is higher than the future productivity level of the UK, find the difference: \(Difference = Future\ productivity\ of\ Canada - Future\ productivity\ of\ UK\) If the future productivity level of the UK is higher than the future productivity level of Canada, find the difference: \(Difference = Future\ productivity\ of\ UK - Future\ productivity\ of\ Canada\) Finally, state which country will have a higher productivity level after five years and by how much.

Key concepts

Compound Interest in Economics

In the realm of economics, compound interest plays a crucial role, especially when forecasting the future growth of investments or productivity. It's the concept that not only the initial principal earns interest, but the accumulated interest from previous periods also earns interest as time goes on. This effect can cause wealth to grow exponentially, and it is precisely the formula used in determining the future productivity levels of workers in Canada and the UK from our exercise.

To understand compound interest in our context, imagine a worker's productivity as an ‘investment’ in the economy, with each hour of work contributing a certain economic value. When productivity grows, it's akin to earning 'interest' on this value. A higher rate of productivity growth, therefore, acts like a higher interest rate in a bank account, leading to a greater increase in overall economic value over time. This compounding effect is what can potentially lead to one country surpassing another in productivity levels, despite starting at a lower initial value.

Productivity Measurement

Measuring productivity is essential for comparing the economic health and potential growth of different countries. Productivity, often defined as output per hour worked, gives an indication of how efficiently a country is using its labor resources. By quantifying the average amount an average worker contributes to the economy per hour, we gain insights into the workforce’s efficiency and a nation's potential for economic expansion.

The exercise provided uses a simple numerical value to represent productivity, allowing us to apply mathematical formulas such as the compound interest formula for our projection. It's critical to understand that productivity measurement can be complex in reality, involving many variables such as technology, education, economic policies, and labor market conditions. The simplicity of the exercise nevertheless offers a clear picture of how to compare and project productivity growth in differing economic environments.

Economic Growth Comparison

Comparing economic growth between two different economies is a nuanced exercise that involves several factors, one of which is certainly productivity growth. The exercise we're examining showcases a simplified version of this comparison, focusing solely on the productivity growth rates of workers in two different countries. In the global economy, consistent productivity growth is a strong indicator of a country’s economic health and its ability to improve living standards over time.

Through the exercise, we can observe how even small differences in annual productivity growth rates can lead to significant discrepancies over time due to the power of compounding. A country with a lower initial productivity level but a higher growth rate can catch up and even surpass another country with a higher initial productivity but a lower growth rate. This serves as an enlightening model on how sustained and compounded productivity improvements are essential for long-term economic prosperity.