# Production Planning: Specialty Steel

Specialty Steel has carefully measured production in its new plant to determine whether it is technically efficient in production. It has found that the production function of the firm is represented by the following equation

Q = 20K^(1/2)L^(1/2)

where Q is output level, K is capital and L is labor.

The firm currently owns 100 units of capital equipment and employs 16 units of labor. The inputs are hired in perfectly competitive markets, and the firm faces input costs as follows: The price of labor (w) is $10 per unit and the rental price of capital (r) is $1.25 per unit.

You have been hired as a consultant to assist Specialty in increasing profitability.

a. Calculate the following:

i. Marginal product of labor.

ii. Marginal product of capital.

iii. Marginal rate of technical substitution.

b. What do you recommend about production planning? (In other words, is the firm employing the efficient input combination? Why or why not? What adjustment should take place in terms of the input mix for the firm to minimize production cost?)

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#### Solution Preview

a. Calculate the following:

i. Marginal product of labor.

Q = 20*K^0.5*L^0.5

Marginal product of labor=MPL= dQ/dL

= 0.5*20*K^0.5*L^(0.5-1)

= 10*K^0.5*L^(-0.5)

Put K = 100 and L = 16

MPL = 10*100^0.5*16^(-0.5) = 25

ii. Marginal product of capital.

Q = 20*K^0.5*L^0.5

Marginal product of ...

#### Solution Summary

Solution describes the steps to calculate marginal product of labor, marginal product of capital and marginal rate of technical substitution. It also checks if the firm is operating with optimal input combination at the given point.