In the short run, the capital $K$ is fixed at a value of 3. Hence, we will substitute $K = 3$ into the original production function $Q = 5KL + 2K^2L^2 - 0.1K^3L^3$ to obtain the short-run production function, $Q_{\text{SR}}$.

Step 1: Short-Run Production Function

Substitute $K = 3$ into the production function:

$Q_{\text{SR}} = 5(3)L + 2(3^2)L^2 - 0.1(3^3)L^3$

$Q_{\text{SR}} = 15L + 18L^2 - 0.9L^3$

So, the short-run production function is:

$Q_{\text{SR}} = 15L + 18L^2 - 0.9L^3$

Step 2: Average Product of Labor (APL)

The average product of labor is given by the ratio of total output $Q$ to the amount of labor $L$:

$\text{APL} = \frac{Q_{\text{SR}}}{L}$

Substitute $Q_{\text{SR}} = 15L + 18L^2 - 0.9L^3$:

$\text{APL} = \frac{15L + 18L^2 - 0.9L^3}{L}$

Simplify the expression:

$\text{APL} = 15 + 18L - 0.9L^2$

So, the average product of labor is:

$\text{APL} = 15 + 18L - 0.9L^2$

Summary:

• The short-run production function is: $Q_{\text{SR}} = 15L + 18L^2 - 0.9L^3$
• The average product of labor is: $\text{APL} = 15 + 18L - 0.9L^2$

Would you like to explore more about the marginal products, the concept of diminishing returns, or any other related topics? Here are 5 additional questions to deepen your understanding:

1. How do you compute the marginal product of labor (MPL) and marginal product of capital (MPK)?
2. How can you find the point where marginal returns to labor become negative?
3. What happens to the total output when $L$ increases in the short run?
4. How would the short-run production function change if $K$ were fixed at a different value?
5. How do you calculate the marginal cost using the short-run production function?

Tip: Always remember that the marginal product is the derivative of the production function with respect to the relevant input (labor or capital).