Question

Production function of a firm has the following form: 

Q= 48L2 – 4L3 , where L stands for amounts of labor and Q represents level of output

Find the level of output at a point where marginal product reaches its maximum.

Answer 1

Given the production function, the marginal product of labor is equal to

$MPL=\dfrac{\Delta Q}{\Delta L}=96L-12L^2$

The slope of the marginal product is equal to

$\dfrac{d MPL}{d L}=96-24L$

When marginal product is maximized, the slope of the marginal product is equal to zero. Therefore

$96-24L=0\\[0.3cm] 24L=96\\[0.3cm] L=\dfrac{96}{24}\\[0.3cm] L=4$

Therefore, the marginal product is maximized at L=4 units.

The level of output at the point where the marginal product is maximized is equal to

$Q=48(4^2)-4(4^3)\\[0.3cm] Q=512$