Question

A price-taking firm has technology given by q=(KL)^0.25. Find the firm’s demands for capital and labor.

Answer 1

q = K0.25L0.25

q/L = MPL= 0.25K0.25L0.25 - 1

MPL= 0.25K0.25L- 0.75

q/K = 0.25K0.25 - 1 L0.25

MPK = 0.25K- 0.75L0.25  

MRTS = MPL/MPK  

=   0.25K0.25L- 0.75 /0.25K- 0.75L0.25   

= K/L

At optimal choice MRTS = w/r

K/L = w/r

K = L(w/r)

put K = L(w/r) in production function

q = K0.25L0.25

q = [L(w/r)]0.25L0.25

q = L0.25(w/r)0.25L0.25

q = L0.50(w/r)0.25

q(r/w)0.25 = L0.50

L = q2(r/w)0.5

K = L(w/r)  

K =  q2(r/w)0.5 (w/r)

= q2(w/r)1 - 0.5  

= q2(w/r)0.5

capital demand K = q2(w/r)0.5

Labor  demand L =  q2(r/w)0.5