Question

A cost minimizing firm’s production function is Q=2KL. The price of labor, w, is currently $4, and the price of capital, r, is currently $1. At the firm’s current level of output, it has total costs of $160. Input prices change such that the wage rate is now 8 times the rental rate. The firm adjusts its input combination, but leaves total output unchanged.

Answer the questions below as you solve for the cost - minimizing input combination after the price changes.

1. Solve for the initial input combination and the initial level of output (before the price change). Hint: you must maximize output subject to a “total cost” constraint...think utility optimization.

Answer 1

Q = 2KL

Initial Total cost (C) ($) = wL + rK

160 = 4L + K

Cost is minimized when MPL/MPK = w/r = $4/$1 = 4

MPL = $\dpi{100} \partial$ Q/$\dpi{100} \partial$L = 2K

MPK = $\dpi{100} \partial$ Q/$\dpi{100} \partial$K = 2L

MPL/MPK = K/L = 4

K = 4L

Substituting in total cost function,

160 = 4L + 4L

8L = 160

L = 20

K = 4 x 20 = 80