Question

2. Suppose that a firm's production function is given by Q = KL(MPK = L and MPL = K), where Q is the quantity of output, K is units of capital, and L is units of labor. The price per unit of labor and capital are $30 and $20, respectively.

(a) How many units of labor and capital should the firm use if it wants to minimize the cost of producing 600 units of output?

(b) Suppose that the firm experiences a technological change, transforming its production function to Q = 1.23KL (MPK = L and MPL = K). What is the minimum cost of producing 600 units of output?

Answer 1

Q = KL

Wage = 30

Rent or price per unit of capital = 20

a) A firm minimizes its cost at the point where marginal rate of technical substitution is equal to the ratio of the prices of inputs.

So, marginal rate of technical substitution = marginal product of labor / marginal product of capital = wage / rent

K / L = 30 / 20

K / L = 1.5

K = 1.5L

Q = KL

Putting the value of Q = 600 and K = 1.5L in the above equation,

600 = 1.5L x L

600 = 1.5L²

L² = 600/1.5 = 400

L = 20 is the answer.

K = 1.5L

K = 1.5 x 20 = 30 is the answer.

So, 20 units of labor and 30 units of capital should be used by firm to minimize the cost of producing 600 units of output.

b)

A firm minimizes its cost at the point where marginal rate of technical substitution is equal to the ratio of the prices of inputs.

So, marginal rate of technical substitution = marginal product of labor / marginal product of capital = wage / rent

K / L = 30 / 20

K / L = 1.5

K = 1.5L

Q = 1.23KL

Putting the value of Q = 600 and K = 1.5L in the above equation,

600 = 1.23 x 1.5L x L

L² = 600 / (1.23 x 1.5)

L² = 325.2

L = 18 approximately or rounded off.

K = 1.5L

K = 1.5 x 18 = 27

Total cost = (wage x units of labor used) + (rent x units of capital used)

= (30 x 18) + (20 x 27)

= 540 + 540 = 1080 is the answer.