To determine the returns to scale for each of the production functions, we look at the exponents of $K$ (capital) and $L$ (labor) in each of the functions. Specifically:

• If the sum of the exponents of $K$ and $L$ is greater than 1, the function exhibits increasing returns to scale (IRS).
• If the sum is equal to 1, the function has constant returns to scale (CRS).
• If the sum is less than 1, it indicates decreasing returns to scale (DRS).

Now, let’s analyze each function:

1. $Q(K, L) = 8K^{1.2}L^{0.4}$

   • Sum of exponents: $1.2 + 0.4 = 1.6$ (greater than 1)
   • This corresponds to increasing returns to scale.

2. $Q(K, L) = 5K^{0.2}L^{0.8}$

   • Sum of exponents: $0.2 + 0.8 = 1.0$ (equal to 1)
   • This corresponds to constant returns to scale.

3. $Q(K, L) = 3K^{0.3}L^{0.6}$

   • Sum of exponents: $0.3 + 0.6 = 0.9$ (less than 1)
   • This corresponds to decreasing returns to scale.

Thus, the correct matches are:

• $Q(K, L) = 8K^{1.2}L^{0.4}$ → C. Increasing returns to scale
• $Q(K, L) = 5K^{0.2}L^{0.8}$ → B. Constant returns to scale
• $Q(K, L) = 3K^{0.3}L^{0.6}$ → A. Decreasing returns to scale

Would you like more details on returns to scale? Here are some related questions:

1. How do we calculate returns to scale for more complex functions?
2. Can returns to scale vary depending on input proportions?
3. What real-world factors lead to increasing returns to scale?
4. How do returns to scale affect business decisions?
5. How do we interpret returns to scale in long-term production planning?

Tip: The concept of returns to scale is crucial for understanding how changes in production inputs affect output.