Question

for a firm with Cobb-Douglas production function q = f (k, L) = k ^ (1/2)...

for a firm with Cobb-Douglas production function

q = f (k, L) = k ^ (1/2) L ^ (1/2)

calculate the total, average and marginal cost.

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Answer #1

q = k1/2L1/2

Total cost (C) = wL + rK

Total cost is minimized when MPL/MPk = w/r

MPL = q/L = (1/2) x (k/L)1/2

MPk = q/k = (1/2) x (L/k)1/2

MPL/MPk = k/L = w/r

wL = rK

Therefore,

L = rK/w

k = wL/r

Substituting in production function,

q = L1/2(wL/r)1/2 = L1/2L1/2(wL/r)1/2 = L x (w/r)1/2

L = q x (r/w)1/2

k = wL/r = (w/r) x q x (r/w)1/2 = q x (w/r)1/2

Substituting in total cost function,

C = wL + rK = w x q x (r/w)1/2 + r x q x (w/r)1/2 = q x (wr)1/2 + q x (wr)1/2 = 2q x (wr)1/2 [Total cost]

Average cost = C/q = 2 x (wr)1/2

Marginal cost = dC/dq = 2 x (wr)1/2

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