Question

Suppose a production function is given by f(K,L) = KL1/3 and that the price of capital is $10 and the price of labor is $16. The capital is fixed at the level K ̅ = 4. What is the quantity of labor that minimizes the cost of producing any given output? What is the minimum cost of producing y units of output? What are the marginal cost of production and the average total cost, average variable cost and the average fixed cost?

Answer 1

y = KL1/3

MPL = 1/3K L-2/3

r = $10

w = $16

K = 4

At optimal combination of input : MPL= w

1/3 KL-2/3 = 16

(1/3)(4) L-2/3  =16

L-2/3 = 12

L2/3 =1/12

L = (1/12)3/2

L = 0.02

The quantity of labor ,L=0.02 that minimizes the cost of producing any given output.

y = (K)(L)1/3

y = (4)(0.02)1/3

y = (4)(0.28)= 1.1

The minimum cost of producing y units of output = wL+ rK

= (16)(0.02)+ (10)(4) = (0.32+40)= $40.32

y= KL1/3

y = (4)L1/3

L = (y/4)1/3  

TC = wL+rK

= (16)(y/4)1/3 + (10)(4)

= 10.12y1/3 + 40

MC = dTC/dy = (1/3)(10.12)y-2/3

= 3.37 y-2/3

ATC = TC/y = ( 10.12y1/3 + 40)/y = 10.12y-2/3 + 40/y

AVC= TVC/y = 10.12y-2/3

AFC = FC/y = 40/y