Answer to Question #259453 in Microeconomics for Leah

Question

4. 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.

5. In the short run, a competitive firm has a production function,

Q = f(L) = 2.6667L0.75. The output price is $8 per unit and the wage is $5 per hour. Find the short-

run labor demand curve of the firm.

Answer 1

Solution:

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 (Ï€) = TR = TC

TR = P $\times$ 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

$\frac{\partial \pi } {\partial L}$ = 48L^-0.4K^0.2 – 20 = 0                   (a)

$\frac{\partial \pi } {\partial K}$ =16L^0.6K^-0.8 – 40 = 0                   (b)

Solve for equation (a) for L:

48L^-0.4K^0.2 = 20

$\frac{48K } { L^{0.4} }$= 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

$\frac{142.72K^{1.5} } { K^{0.8} }$ = 40

142.72K^1.5 = 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)

c.). Q = f(L) = 2.6667L^0.75

MRP_L = MR $\times$ MP_L = w

MR = P = 8

W = 5

MP_L = $\frac{\partial Q} {\partial L}$= 2.000025L^-0.25

MRP_L = 8 $\times$ 2.000025L^-0.25 = 16.0002L^-0.25

16.0002L^-0.25 = w

16.0002L^-0.25 = 5

$\frac{16.0002}{L^{0.25} }$ = 5

16.0002 = 5L^0.25

L = 104.86

The short-run labor demand curve of the firm = 104.86