Question

A firm’s production process is represented by y= L^2/3 K^1/3. The price of Labor, w is $2 and the price of capital, r, is $27.

(a) Write down the firm’s cost minimization problem

(b) What is the firm’s MRTS?

(c) What are the firm’s cost minimizing levels of labor and capital (these will both be functions of y)?

(d) What is the firm’s cost curve (ie, derive C(y))?

(e) If the firm chooses output y= 450, what are the firms optimal levels of K and L? What is the firm’s (minimum) cost?

Answer 1

y = L^2/3K^1/3

Total cost (C) = wL + rK = 2L + 27K

(a) The firm's cost minimizing problem is

Minimize C = 2L + 27K

Subject to: y = L^2/3K^1/3

(b) MRTS = MPL/MPK

MPL = y/L = (2/3) x (K/L)^1/3

MPK = y/K = (1/3) x (L/K)^2/3

MRTS = 2 x (K/L) = 2K/L

(c) Cost is minimized when MRTS = 2K/L = w/r = 2/27

2L = 54K

L = 27K

Substituting in production function,

y = (27K)^2/3(K)^1/3 = (27)^2/3K^2/3K^1/3 = 9K

K = y/9

L = 27 x (y/9) = 3y

(d) Substituting in cost function,

C ($) = 2L + 27K = 2 x (3y) + 27 x (y/9) = 6y + 3y

C = 9y

(e) When y = 450,

L = 3 x 450 = 1350

K = 450/9 = 50

C ($) = (2 x 1350) + (27 x 50) = 2700 + 1350 = 4050