Question

The production of plastic bags is given by the production function q = f(L,K) = 4L^0.5K^0.5, where K is capital and L is labor.

b. Long Run Production

i) With Capital no longer fixed, solve the firm’s Cost Minimization problem, finding L and K in terms of r, w and the output level q.

ii) What is the Firm Long Run Cost Function C(q) in terms of r, w and q?

iii) What is the Marginal Cost (MC) and Average Cost (AC) of this firm in the Long Run?

Answer 1

q = 4L^0.5K^0.5

(i)

Cost is minimized when MPL/MPK = w/r

MPL = q/L = 4 x 0.5 x (K/L)^0.5

MPK = q/K = 4 x 0.5 x (L/K)^0.5

MPL/MPK = K/L = w/r

K = L x (w/r)

Substituting in production function,

q = 4 x L^0.5[L x (w/r)]^0.5

q = 4 x L^0.5 x L^0.5 x (w/r)^0.5

q = 4 x L x (w/r)^0.5

L = (q/4) x (r/w)^0.5

K = (q/4) x (r/w)^0.5 x (w/r) = (q/4) x (w/r)^0.5

(ii)

C(q) = wL + rK

C(q) = w x [(q/4) x (r/w)^0.5] + r x [(q/4) x (w/r)^0.5]

C(q) = (q/4) x [w x (r/w)^0.5 + r x (w/r)^0.5]

C(q) = (q/4) x [(wr)^0.5 + (wr)^0.5]

C(q) = (q/4) x 2 x (wr)^0.5

C(q) = (q/2) x (wr)^0.5

(iii)

MC = dC(q)/dq = (1/2) x (wr)^0.5

AC = C(q)/q = (1/2) x (wr)^0.5