Answer to Question #259182 in Microeconomics for Attitude reigns

Question

Suppose a firm produces according to the production function Q = AL0.6K0.2, and faces wage rate

₵20, a rental cost of capital ₵10, and sells output at a price of ₵40.

a. Obtain and expression for the factor demand functions if A=2.

b. Compute the profit-maximizing factor demands for capital and labour if A=2.

Answer 1

a.). Q = AL^0.6K^0.2

A = 2

Q = 2L^0.6K^0.2

Input demand for labor, L = F(w,r,p)

Input demand for Capital, K = F(w,r,p)

b.). Profit (π) = TR-TC

TR = P × Q = 40(2L^0.6K^0.2)

TC = wL + rK = 20L + 40K

Profit (π) = 40(2L^0.6K^0.2) – 20L – 40K

Profit (π) = 80L^0.6K^0.2) – 20L – 40K

∂π/∂L​ = 48L^-0.4K^0.2 – 20 = 0 ...........................................(a)

∂π/∂K​ =16L^0.6K^-0.8 – 40 = 0 ............................................(b)

Solve for equation (a) for L:

48L^-0.4K^0.2 = 20

48K = 20L^0.4

2.4K = L^0.4

L = 8.92K^2.5

Plug into equation (b):

16L^0.6K^-0.8 = 40

16(8.92K^2.5)(0.6K^-0.8 )= 40

142.72K^1.5K^-0.8 = 40

142.72K1.5 = 40K^0.8

K = 0.16

L = 8.92K^2.5 = 8.92(0.16^2.5) = 0.09

Profit maximizing factor demands for labor and capital = (0.09, 0.16)