Consider a firm which has the following production function
(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 $Q.
(a) Q = 4(LK)1/2 and when Q = 16, we get 16 = 4(LK)1/2, therefore (LK)1/2 = 4, or LK = 16 (Squaring both sides)
Cost is minimied when MPL/MPK = w/r = 4/1 = 4
MPL/MPK = K/L = 4
4L = K
Substituting in production function,
L x 4L = 16
4L2 = 16
L = 2
K = 4 x 2 = 8
(b) Total cost (C) ($) = 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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