Question

1) Assume that cost function is C(q) = wL + rK and production function is F (K, L) = AL^aK^b. Find the amounts of capital and labor that the firm should utilize to minimize the cost of producing an output q0.

Answer 1

The cost minimization condition requires MRTS = w/r

MRTS = MPL/MPK.

MPL =

MPL = A * a L^a-1 K^b

MPK =

MPK = A * b L^a K^b-1

MPL/MPK = [A * a L^a-1 K^b] / [A * b L^a K^b-1]

MPL/MPK = a/b * K^2b/L^2a

a/b * K^2b/L^2a = w/r

a/b * K^2b * r/w = L^2a

[a/b * K^2b * r/w]^1/2a = L.....plugging this value into the prodution function, with F = q0

q0 = A [a/b * K^2b * r/w]^1/2a*a K^b

q0 = K^b [a/b * r/w]^1/2 * K^b

q0 = A K^2b

K^2b = q0/A *

K = [q0/A * ]^1/2b units.

Plugging this value of K back into the value of L, we get,

L = {a/b [q0/A ]^{1/2b * 2b} * r/w}^1/2a

L = [a/b * q0/A * b^1/2/a^1/2 * w^1/2/r^1/2 * r/w]^1/2a

L = [ * q0/A]^1/2a units.