In a certain country, everyone in the labor force in the year 2000 had 12 years of education. In 1950 everyone in the labor force had 4 years of education. What was the average annual growth in income per worker that was due to the increase in education? Let the return to education be 13.4% per year for the first 4 years of education, 10.1% per year for the next four, and 6.8% per year for the next four.

To find the average annual growth in income per worker due to the increase in education, we'll apply the given rates of return to the different intervals of educational attainment. We have two data points: 4 years of education in 1950 and 12 years of education in 2000. 

The problem provides the returns for three distinct intervals:

1. **First 4 years**: 13.4% per year
2. **Next 4 years (years 5–8)**: 10.1% per year
3. **Next 4 years (years 9–12)**: 6.8% per year

### Step-by-Step Calculation

1. **Calculate the increase in income for the first 4 years of education:**
   \[
   \text{Increase} = (1 + 0.134)^4 - 1
   \]
   This calculates the cumulative increase after 4 years at a 13.4% annual return.

2. **Calculate the increase in income for the next 4 years (years 5 to 8):**
   \[
   \text{Increase} = (1 + 0.101)^4 - 1
   \]
   This computes the increase after 4 more years at a 10.1% annual return.

3. **Calculate the increase in income for the next 4 years (years 9 to 12):**
   \[
   \text{Increase} = (1 + 0.068)^4 - 1
   \]
   This calculates the increase after the final 4 years at a 6.8% annual return.

4. **Combine the effects of all these intervals:**
   \[
   \text{Total increase} = (1.134^4) \times (1.101^4) \times (1.068^4)
   \]
   This product gives the cumulative factor representing the overall increase in income per worker over the 12 years of education.

5. **Find the average annual growth rate:**
   Let \( A \) be the total increase over the period (from 1950 to 2000, which is 50 years). We need to find the average annual growth rate, \( r \), over these 50 years:
   \[
   A = (1.134^4) \times (1.101^4) \times (1.068^4)
   \]
   \[
   1 + r = A^{\frac{1}{50}}
   \]
   Then:
   \[
   r = A^{\frac{1}{50}} - 1
   \]

### Detailed Solution

Let's compute each part:

1. **First interval (years 1–4)**:
   \[
   (1.134)^4 \approx 1.6488
   \]

2. **Second interval (years 5–8)**:
   \[
   (1.101)^4 \approx 1.4641
   \]

3. **Third interval (years 9–12)**:
   \[
   (1.068)^4 \approx 1.3108
   \]

Multiplying these together:
\[
A = 1.6488 \times 1.4641 \times 1.3108 \approx 3.1656
\]

Now, solve for \( r \):
\[
1 + r = (3.1656)^{\frac{1}{50}}
\]
\[
r = (3.1656)^{\frac{1}{50}} - 1
\]

Using a calculator to find the 50th root:
\[
r \approx 0.0234 \text{ or } 2.34\%
\]

### Conclusion

The average annual growth in income per worker due to the increase in education from 1950 to 2000 is approximately **2.34%**.

Would you like further details or have any questions?

### Related Questions
1. What if the return on education rates were different, such as 8% for all years? How would that affect the calculation?
2. How would changes in the time frame (e.g., from 1950 to 1990) impact the average annual growth rate?
3. What is the significance of diminishing returns on education as more years are completed?
4. How does this growth rate compare to the growth due to other factors like technology or capital investment?
5. Can this model be applied to countries with different education systems or should adjustments be made?

### Tip
When working with compounded growth rates, breaking the period into smaller intervals with consistent rates makes the calculation easier and more accurate.

Find out how to calculate the average annual growth in income per worker due to an increase in education from 4 to 12 years. With returns of 13.4%, 10.1%, and 6.8%, the overall growth rate is approximately 2.34%. Suitable for high school students learning about compounded growth.

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