Question

A firm has the following short-run production function: Q = 30L-2L2
a. Show the range of labor when stages I, II, and III of production occur.
b. If labor costs $20 per day, and the average price of service is $10, how many workers should the firm hire?

Answer 1

a) Stage I - Between L= 0 and L where AP is maximized 

AP= $\frac{Q}{L}$= $\frac{30}{L}$ - $\frac{2L2}{L}$ = 30-2L

$\frac{dAP}{dL}$= 30 – 2L = 0 ⇒ L = $\frac{30}{2}$ = 15

Stage I from 0 to 15

Stage II - From L= 15 to L where MP = 0 

MP = $\frac{dQ}{dL}$ = 90L – 6L2 = 0 divide by 6L = 90 – L = 0 ⇒ L = 90

Stage II - from 15 to 90

Stage III - 90 onward 

b) MPL = 30-L

P$\times$MPL=10 $\times$(30-L)=300-10L=20

P$\times$ MPL = W ⇒ 300 – 10L =20 ⇒ 280 = 10L ⇒ L = 28workers