# Finding optimal combination of labor and capital

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 Preview

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 ...

#### Solution Summary

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

Marginal Rate of substitution

Answer the following questions based upon the information given below

Q = K^(1/2)L^(1/2), WK = $20 WL = $5

1. What relative amounts of capital and labor will be employed to

maximize output?

a. equal amounts of capital and labor

b. four times as much labor as capital

c. twice as much capital as labor

d. none of the above

2 Having a budget of $16,000, what amounts of capital and labor will be

employed?

a. 200 units of K and 200 units of L

b. 400 units of K and 1600 units of L

c. 200 units of K and 1600 units of L

d. 100 units of K and 40 units of L

e. none of the above

3. If the proper amounts of K and L are employed

a. MRS should equal WL/WK

b. MPL/MPK should equal WL/WK

c. WK/MPL should equal WL/MPK

d. all of the above

e. two of the above