q = K1/3L1/3
(a) Cost is minimized when (MPL/MPK) = w/r
MPL =
q/
L
= [(1/3) x K1/3] / (L2/3)
MPK =
q/
K
= [(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]
/3 y13 6. Suppose that a firm has a production function given by: q-Ks a) Derive...
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