Question

Consider the following short run production function: Q=6L²-0.4L³ 

1. a) Find the value of L that maximizes output

b) Find the value of L that maximizes marginal product

c) Find the value of L that maximizes average product

Answer 1

(a) Let's first find the marginal product:

$$MP=\dfrac{d}{dL}(Q)=\dfrac{d}{dL}(6L^2-0.4L^3)=12L-1.2L^2.$$

Then. we can find the value of L that maximizes output:

$$MP=0,$$ $$12L-1.2L^2=0,$$ $$L=10.$$

b) Let's take the derivative from MP with respect to L:

$$\dfrac{d}{dL}(MP)=\dfrac{d}{dL}(12L-1.2L^2)=12-2.4L.$$

Then, we can find the value of L that maximizes marginal product:

$$12-2.4L=0,$$ $$L=5.$$

c) Let's find the AP:

$$AP=\dfrac{Q}{L}=\dfrac{6L^2-0.4L^3}{L}=6L-0.4L^2.$$

Let's take the derivative from AP with respect to L:

$$\dfrac{d}{dL}(AP)=\dfrac{d}{dL}(6L-0.4L^2)=6-0.8L.$$

Then, we can find the value of L that maximizes average product:

$$6-0.8L=0,$$ $$L=7.5$$