A firm can manufacture a product according to the production function: Q = F(K,L

Question

A firm can manufacture a product according to the production function:

Q = F(K,L) = K3/4L1/4.

a. Calculate the average product of labor, APL, when the level of capital is fixed at 81 units and the firm uses 16 units of labor.

Instruction: Round your responses to 3 decimal places.

What is the average product of labor when the firm uses 256 units of labor?



b. Find an expression for the marginal product of labor, MPL, when the amount of capital is fixed at 81 units.

Instruction: The second response is the exponent on L in the expression. Round your responses to 2 decimal places.

MPL = ______*L ^-0.75

Then, illustrate that the marginal product of labor depends on the amount of labor hired by calculating the marginal product of labor for 16 and 81 units of labor.

Instruction: Round your responses to 3 decimal places.

MPL when L = 16: _______

MPL when L = 81: _______

c. Suppose capital is fixed at 81 units. If the firm can sell its output at a price of $200 per unit and can hire labor at $50 per unit, how many units of labor should the firm hire in order to maximize profits?

Instruction: Enter your response as a whole number.

________

Explanation / Answer

Q = F(K,L) = K3/4L1/4.

(a)

(i) K = 81, L = 16

Q = (81)3/4(16)1/4

= 27 x 2 = 54

So, APL = Q / L = 54 / 16

= 3.375

(ii) K = 81, L = 256

Q = (81)3/4(256)1/4

= 27 x 4 = 108

So, APL = Q / L = 108 / 256

= 0.422

(b) K = 81

Q = K3/4L1/4 = (81)3/4L1/4 = 27L1/4

MPL = dQ / dL = 27 x (1/4) / L3/4 = 6.75 x L-0.75

(i) When L = 16, MPL = 6.75 x (16)3/4 = 6.75 x 8 = 54

(ii) When L = 81, MPL = 6.75 x (81)3/4 = 6.75 x 27 = 182.25

(c) K = 81

Q = 27L1/4

MPL = dQ / dL = 27 x (1/4) / L3/4

The firm would hire labor upto that point when

MPL x Price = Wage rate

Or,

27 x (1/4) / L3/4 x $200 = $50

27 x (1/4) / L3/4 = $50 / $200 = 1/4

L3/4 = 27

L = (27)4/3 = 81 units

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