Question

2. Optimal Inputs

Your boss has given you the following production function for labor (L) and capital (K) used by your company:

q¯ = K0.2L 0.8

You want to produce q = 100 units for sale and faces prices for labor of w = 2 and capital of r = 6.

a) What is the marginal rate of technical substitution?

b) What are the optimal amounts of each input used by the firm? Round to three decimal places as needed.

c) How much does the firm spend?

Answer 1

q = K^0.2L^0.8

a) MRTS = MPL/MPK = (∂q/∂L)/(∂Q/∂K) = 0.8L^0.8-1K^0.2/0.2L^0.8K^0.2-1 = 4L^-0.2K^0.2/L^0.8K^-0.8 = 4K^0.2+0.8/L^0.2+0.8 = 4K/L
So, MRTS = 4K/L

b) Optimal amount is determined where MRTS = w/r
So, 4K/L = 2/6 = 1/3
So, L = 3*4K = 12K
So, L = 12K

q = 100 = K^0.2L^0.8 = K^0.2(12K)^0.8 = (12)^0.8K^0.2+0.8 = (12)^0.8K
So, K = 100/(12)^0.8 = 100/7.3
So, K = 13.7
L = 12K = 12*(13.7) = 164.4

K = 13.7; L = 164.4

c) Spending = wL + rK = 2(164.4) + 6(13.7) = 328.8 + 82.2 = 411
So, spending = 411