Question

Consider a firm which has the following production function
Q=f(L,K)=4?LK

(MPL=2?(K/L) and MPK=2?(L/K).

(a) If the wage w= $4 and the rent of capital r=$1, what is the least expensive way to produce 16 units of output? (That is, what is the cost-minimizing input bundle (combination) given that Q=16?)

(b) What is the minimum cost of producing 16 units?

(c) Show that for any level of output Q, the minimum cost of producing Q is $Q.

Answer #1

(a) Q = 4(LK)^{1/2} and when Q = 16, we get 16 =
4(LK)^{1/2}, therefore (LK)^{1/2} = 4, or LK = 16
(Squaring both sides)

Cost is minimied when MPL/MPK = w/r = 4/1 = 4

MPL/MPK = K/L = 4

4L = K

Substituting in production function,

L x 4L = 16

4L^{2} = 16

L^{2}= 4

L = 2

K = 4 x 2 = 8

(b) Total cost (C) ($) = wL + rK = 4 x 2 + 1 x 8 = 8 + 8 = 16

(c) MPL/MPK = K/L = 4

4L = K

Substituting in generalized production function,

Q = 4 x (L x 4L)^{1/2} = 4 x 2 x L = 8L

L = Q/8

K = 4L = 4Q/8 = Q/2

Substituting in total cost function,

C ($) = 4L + K = (4Q/8) + (Q/2) = (Q/2) + (Q/2)

C ($) = Q

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