# Labor Economics & profit maximization

Labor Economics and Profit Maximization

1. A new shop has opened and has the following production function:
f(E,K) = E0..6 K0.35
The price of output is \$22, the wage rate is \$8, and the rental rate for capital is \$10 per unit.

a. Find the short-run profit maximization level of labor demand if capital is fixed at 400 units.

b. Find the long-run profit maximization level of labor demand and capital demand.

c. Find the long-run the elasticity. You do not need to show your math on the answer page you submit for grading.

d. If wages change to \$10, which effect is stronger for capital, substitution or scale?
2. A shop has the following production function:
f(E,K) = 2E3/5 K1/7
The price of output is \$20, the wage rate is \$7, and the rental rate for capital is \$5 per unit.

a. Find the short-run profit maximizing level of labor demand if capital is fixed at 30 units.

b. Find the long-run profit maximizing level of labor demand and capital demand.

c. If wages change to \$9, which effect is stronger for capital, substitution or scale?

3. A shop has opened and has the following production function:
f(E,K) = 3E3/5 K3/4
The price of output is \$18, the wage rate is \$8, and the rental rate for capital is \$9 per unit.

a. Find the short-run profit maximizing level of labor demand if capital is fixed at 25 units.

b. Find the long-run profit maximizing level of labor demand and capital demand. Do not state Q=K=E=infinity since increasing returns to scale. Instead maximize the function and find optimal Q, E, and K.

c. Find the long-run the elasticity. You do not need to show your math on the answer page you will submit for grading.

d. Interpret the elasticity from part c. Does it make sense? Why or why not?

e. In class it was stated that increasing returns to scale long run results are Q=K=E=infinity, prove this is true. Hint look at your answer from a and b.
4. A firm uses 1 capital for every 5 workers to produce 1 output. The price of output is \$75, the wage rate is \$8, and the rental rate for capital is \$2 per unit. Assume capital is not a fixed cost.

a. What is the profit maximizing level of labor demand if K=5?

b. What is the short-run quantity produced?

c. What is the profit maximizing level of labor demand if Q=50?

d. What is the long-run quantity produced? Don’t assume Q=50.
5. Graphically show the long-run demand for labor if the price of capital increases.
6. (attempt even if not covered) Juice and More faces a perfectly elastic demand for drinks at the price of \$5 per drink. They also face an upward-sloping marginal cost of labor curve given by:
MCE= 3+0.3E
and
SE=25w – 5

Each hour of labor produces 12 drinks. The cost of producing each drink is \$2.

a. How many workers should Juice and More hire to maximize profits?

b. What wage will Juice and More pay?

c. What are Juice and More’s profits?

d. Graph parts a-c on one graph.
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