Question

The production function is: Q = 4L1/2K1/2 when Q=200, what is the marginal cost

Answer 1

We have MPL = 4*(1/2)(K^1/2)(L^-1/2) & MPK = 4*(1/2)(L^1/2)(K^-1/2).

This gives MRTS = MPL/MPK K/L.

Wage rental ratio is given by w/r. At the optimum input mix, MRTS = w/r or K/L = w/r which then becomes K = wL/r

The production function is now Q = 4L^(1/2) (wL/r)^(1/2)

Q = 4L * (w/r)^1/2

Labor demand L* = 0.25Q*(r/w)^(1/2) and capital demand K* = 0.25Q*(w/r)^(1/2)

Cost function is C = wL + rK

C = w*0.25Q(r/w)^(1/2) + r*0.25Q(w/r)^(1/2)

C = 0.25Q(wr)^(1/2) + 0.25Q(wr)^(1/2)

C = 0.5Q(wr)^(1/2)

Marginal cost MC = 0.5(wr)^(1/2)

Hence marginal cost is a fixed value of 0.5(wr)^(1/2) independent of output.