Question

A firm's technology is represented by the production function q = (KL)^1/3

In the short run, K is fixed at 64 = 4³

What is the firm's short run production function?

Find the short run conditional factor demand for L.

What is the short run cost function?

What is the shut down price?

Answer 1

Profit Function = Revenue-Cost= PQ-C(Q)=P*(KL)^(1/3)-(wL+rK)

K=64 therefore Profit is =P*(64L)^(1/3)-(wL+rK)=4P(L^1/3)-wL-rK

Short run production function=q=4L^(1/3)

We need to maximise Profit with respect to change in labour (L)

d(Profit)/dL=4/3*PL^(-2/3)-w=0

4/3PL^(-2/3)=w

L^(-2/3)=3w/4P

L^(2/3)=4P/3w

L=[(4/3)*(P/w)]^(3/2).....Conditional Demand Function

Short Run Cost Function is

C=wL+r(64)=w*[(4/3)*(P/w)]^(3/2)+64r

C=1/(w^(1/2))*(4/3*P)^(3/2)+64r

SHutDown Function when Profit <=0

Revenue<Cost

P*4L^(1/3)<wL+r64

P=(wL+64r)/4L^(1/3)=wL^(2/3)/4+16r*L^(-1/3)...Shut Down Price