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/3 y13 6. Suppose that a firm has a production function given by: q-Ks a) Derive the conditional factor demands. b) Derive the cost function. c) Derive the supply function. d) Derive the input demand functions.

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Answer #1

q = K1/3L1/3

(a) Cost is minimized when (MPL/MPK) = w/r

MPL = \partial q/\partialL = [(1/3) x K1/3] / (L2/3)

MPK = \partial q/\partialK = [(1/3) x L1/3] / (K2/3)

MPL/MPK = K/L = w/r

K = L x (w/r)

Substituting in production function,

q = (wL/r)1/3L1/3

q = (w/r)1/3L1/3L1/3

q = (w/r)1/3L2/3

L2/3 = q x (r/w)1/3

L = [q x (r/w)1/3]3/2

L = q3/2 x (r/w)1/2 [Conditional factor demand for labor]

K = [q3/2 x (r/w)1/2] x (w/r) = q3/2 x (w/r)1/2 [Conditional factor demand for capital]

(b) Total cost: C = wL + rK

C = w x [q3/2 x (r/w)1/2] + r x [q3/2 x (w/r)1/2]

C = [q3/2 x (wr)1/2] + [q3/2 x (wr)1/2]

C = q3/2 x [(wr)1/2 + (wr)1/2]

C =  q3/2 x (2wr)1/2 [Cost function]

(c) Firm's supply function is its Marginal cost (MC) function.

P = MC = dC/dq = (3/2) x q1/2 x (2wr)1/2 [Supply function]

(d) Substituting K = L x (w/r) into the cost function,

C = wL + r x [L x (w/r)]

C = L x [w + (w/r)]

C = L x [(wr + w)/r]

C = L x [w x (r + 1)/r]

L = (rC) / [w x (r + 1)] [Input demand for labor]

K = [(rC) / [w x (r + 1)/r] x (w/r) = (wC) / [w x (r + 1)/r] [Input demand for capital]

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