Question

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.

Answer #1

q = k^{1/2}L^{1/2}

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 = L^{1/2}(wL/r)^{1/2} =
L^{1/2}L^{1/2}(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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