To calculate the average annual growth in income per worker due to the increase in education, we can apply the concept of the return to education, which is 10% per year in this case. Here's the step-by-step breakdown:

Given Information:

1. Return to education: 10% per additional year of education.
2. Years of education in 1950: 4 years.
3. Years of education in 2000: 12 years.
4. Period: 2000 - 1950 = 50 years.

Step 1: Calculate the total increase in education

The increase in the average number of years of education from 1950 to 2000 is: $12 \, \text{years} - 4 \, \text{years} = 8 \, \text{years}.$

Step 2: Calculate the increase in income due to education

With a 10% return per year of education, the total increase in income due to the additional 8 years of education is: $(1 + 0.10)^{8} = (1.10)^{8}.$ Now, calculate $(1.10)^8$: $1.10^8 \approx 2.1436.$ This means that income per worker increased by approximately 2.1436 times due to the additional 8 years of education.

Step 3: Calculate the average annual growth rate

We now need to find the average annual growth rate over the 50-year period (from 1950 to 2000). The formula for the average annual growth rate $g$ is: $(1 + g)^{50} = 2.1436.$ To solve for $g$, we take the 50th root of 2.1436: $1 + g = 2.1436^{\frac{1}{50}}.$ Using a calculator: $2.1436^{\frac{1}{50}} \approx 1.0147.$ Thus: $g \approx 1.0147 - 1 = 0.0147 \, \text{or} \, 1.47\% \, \text{per year}.$

Conclusion:

The average annual growth in income per worker due to the increase in education is approximately 1.47% per year.

Would you like further details or have any questions?

Here are 5 related questions to expand this topic:

1. How would a different return to education (e.g., 8%) affect the growth rate?
2. How does the return to education vary across different countries or industries?
3. How does the length of the time period impact the average annual growth rate?
4. What happens if the return to education is not constant over time?
5. How can technological changes also contribute to income growth aside from education?

Tip: The power rule for exponents is very helpful when calculating compound growth over time. You can use logarithms for more complex cases.