Economics Problem; Help!?

Question

I was hoping someone could show me a walkthrough of this one:

The short-run production function of a competitive firm is given by f(L)= 6L ^2/3, where L is the amount of labor it uses. The cost per unit of labor is w=6 and the price per unit of output is p=3

A.) How many units of labor will the firm hire? ___

How much output will it produce? _____

If the firm has no other costs, how much will it's total

profits be? _____

B.) Suppose that the wage of labor falls to 4, and the price of output remains at "p". Will the firm increase it's output at the new price? ____ Explain why.

Answer 1

I would need to know what "f" is. Is it a constant? I'd like to help, but I can't with the data you have offered.

Answer 2

Q=6L^(2/3)

L^(2/3)=Q/6

L=(Q/6)^(3/2)

TC=w*L=w * (Q/6)^(3/2)

MC=(TR)' = w/4 * √(Q/6)

TR= p * Q

MR=(TR)'=p

Optimal equilibrium then MC=MR

w/4 * √(Q/6) = p

√(Q/6) = 4p/w

Q/6=16 (p/w)²

Q=96*(p/w)²

Profit=TR-TC

A:

w=6

p=3

p/w=3/6=1/2

Q=96*(1/2)²=96/4=24

L=(24/6)^(3/2)=4^(3/2)=8

TC=w*L=6*8=48

TR=p*Q=3*24=72

Profit=72-48=24

B:

w=4

p=3

p/w=3/4

Q=96*(3/4)²=96*9/16=6*9=54

L=(54/6)^(3/2) = (54/6)^(3/2) = 9^(3/2) =√729=27

TC=w*L=4*27=108

TR=p*Q=3*54=162

Profit=162-108=54