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# Finding optimal combination of labor and capital

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Crew Brew produces a popular brand of beer in its mini-brewery located on a small river in Kentucky. It uses a special formula, combined with the fresh water from the local stream, to produce a drink popular with local folks and tourists who visit during the summer fishing season, and autumn deer hunting season. The production function of Crew follows the formula: Q = 50K + 50L, where Q = barrels of beer, K = units of capital, and L = units of labor.
a) Suppose that capital can be purchased for \$8 per unit, and labor costs \$6 per unit. What is the optimal combination of inputs for the firm to employ?
b) Suppose that the cost of inputs changes to \$7 for a unit of capital, and \$9 for a unit of labor. What is the new optimal combination of inputs?
c) Explain the results of a. and b.

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Solution determines the optimal combination of labor and capital in the given case.

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a) Suppose that capital can be purchased for \$8 per unit, and labor costs \$6 per unit. What is the optimal combination of inputs for the firm to employ?

Q = 50K + 50L
Price of one unit of capital =c=\$8
Price of one unit of labor =w=\$6

Marginal Product of capital=MPK=dQ/dK=50

Marginal Product of labor=MPL=dQ/dL=50

MPK/c=50/8=6.25
MPL/w=50/6=8.33

Since (MPK/c) is less than (MPL/w) for all values of K and L, he should use labor units only.

Optimal combination of capital and labor ...

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• BEng (Hons) , Birla Institute of Technology and Science, India
• MSc (Hons) , Birla Institute of Technology and Science, India
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