Question

/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.

Answer 1

q = K^1/3L^1/3

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

MPL = $\partial$ q/ $\partial$ L = [(1/3) x K^1/3] / (L^2/3)

MPK = $\partial$ q/ $\partial$ K = [(1/3) x L^1/3] / (K^2/3)

MPL/MPK = K/L = w/r

K = L x (w/r)

Substituting in production function,

q = (wL/r)^1/3L^1/3

q = (w/r)^1/3L^1/3L^1/3

q = (w/r)^1/3L^2/3

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

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

L = q^3/2 x (r/w)^1/2 [Conditional factor demand for labor]

K = [q^3/2 x (r/w)^1/2] x (w/r) = q^3/2 x (w/r)^1/2 [Conditional factor demand for capital]

(b) Total cost: C = wL + rK

C = w x [q^3/2 x (r/w)^1/2] + r x [q^3/2 x (w/r)^1/2]

C = [q^3/2 x (wr)^1/2] + [q^3/2 x (wr)^1/2]

C = q^3/2 x [(wr)^1/2 + (wr)^1/2]

C =  q^3/2 x (2wr)^1/2 [Cost function]

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

P = MC = dC/dq = (3/2) x q^1/2 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]