Question

Suppose there are two inputs in the production function, labor and capital, and

these two inputs are perfect substitutes. The existing technology permits 3 machines to do the work of 2

worker. So F(E,K)=2K+3E. The firm wants to produce 60 units of output. Suppose the price of capital is

$10 per machine per hour. What combination of inputs will the firm use if the wage rate is $10 or $15 or

$20 per hour? What if the firm wants to produce 90 units of the output?

Answer 1

Production function is F(E,K) = 2K + 3E

Aim of the firm is to produce 60 units of output. The price of capital is $10 per machine per hour.

At the optimal input choice, we must have MRTS = w/r

MPE/MPK = w/r

3/2 = w/10

a) At w = 10 and Y = 60

Now if w is 10, this condition becomes 3/2 = 10/10 which means MPE/MPK > w/r.

Hence marginal product of last dollar spent in hiring labor is greater than the same for capital

This suggests that firm should use only labor and production function becomes Y = 3E

For Y = 60, E will be 60/3 = 20 units and K will be 0

b) At w = 15 and Y = 60

Now if w is 15, this condition becomes 3/2 = 15/10 which means MPE/MPK = w/r.

Hence marginal product of last dollar spent in hiring labor is equal to the same for capital

This suggests that firm should use both labor and capital and production function becomes Y = 2K +

3E. For Y = 60, any combination of E and K that satisfies 60 = 2K + 3E will be optimal

c) At w = 20 and Y = 60

Now if w is 20, this condition becomes 3/2 = 20/10 which means MPE/MPK < w/r.

Hence marginal product of last dollar spent in hiring labor is less than the same for capital

This suggests that firm should use only capital and production function becomes Y = 2K

For Y = 60, E will be 0 and K will be 60/2 = 30 units

d) At w = 10, Y = 90, we have K = 0 and E = 90/10 = 9 units

e) At w = 15, any combination of E and K that satisfies 90 = 2K + 3E will be optimal

f) At w = 20 and for Y = 90, E will be 0 and K will be 90/2 = 45 units