Consider the following short-run production function (where L = variable input,  Q = output): Q = 6L^2 - 0.4L^3. Determine the marginal product function (MPL).


The marginal product of labor (MPL) represents the additional output (Q) that is produced by an additional unit of labor (L). It is the derivative of the production function \( Q(L) \) with respect to \( L \).

Given the production function:
\[
Q(L) = 6L^2 - 0.4L^3
\]

To find the marginal product of labor (MPL), we take the derivative of \( Q(L) \) with respect to \( L \):

\[
MPL = \frac{dQ}{dL} = \frac{d}{dL}(6L^2 - 0.4L^3)
\]

Applying the power rule of differentiation:

\[
MPL = 12L - 1.2L^2
\]

So, the marginal product function is:
\[
MPL = 12L - 1.2L^2
\]

Would you like further details on this? If so, I can explain more about the role of MPL in production.

Here are five related questions you might explore:

1. How does the marginal product of labor change as \( L \) increases?
2. What is the value of \( L \) at which the marginal product becomes negative?
3. How would you interpret diminishing marginal returns in this context?
4. What is the economic significance of the marginal product when it reaches zero?
5. How is the average product of labor (APL) calculated in this scenario?

**Tip:** Always interpret the marginal product in relation to how inputs affect outputs. It helps explain efficiency and productivity changes as resources vary.

Consider the following short-run production function (where L = variable input, Q = output): Q = 6L^2 - 0.4L^3. Determine the marginal product function (MPL).

Discover how to calculate the marginal product of labor (MPL) using differentiation. Given the production function Q = 6L^2 - 0.4L^3, this step-by-step guide explains how to derive the MPL, which is 12L - 1.2L^2, using calculus. Ideal for college economics or advanced high school students.

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