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# A new shop has opened and has the following production function: Q=E0.25 K0.5 The price of output is \$P, the wage rate is \$w, and the rental rate of...

1. A new shop has opened and has the following production function:
Q=E0..25 K0.5
The price of output is \$P, the wage rate is \$w, and the rental rate of capital is \$r. Hint, "as a function" means an equation that only has the parameters listed ie as a function of P, w, r, means no E or K in the right hand side of the equation.

a. Find the short-run labor demand equation as a function of P, w, r and K.

b. Find the long-run labor demand equation as a function of P, w, and r.

c. Find the long-run capital demand equation as a function of P, w, and r.

d. Find the long run production function as a function of P, w, r.

e. Find the long-run the elasticity. You do not need to type this answer or the mathematics. Leave a blank spot on your answer sheet and hand write your math.

2. A new shop has opened and has the following production function:
Q=E0..35 K0.6
The price of output is \$20, the wage rate is \$11, and the rental rate for capital is \$8 per unit.

a. Find the short-run profit maximizing level of labor demand if capital is fixed at 40 units.

b. Find the long-run profit maximizing 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 will submit for grading.

d. If wages change to \$10, which effect is stronger for capital, substitution or scale?

3. A new shop has opened and has the following production function:
Q=E4/5 K1/7
The price of output is \$22, 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 20 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?

4. A new shop has opened and has the following production function:
Q=2E3/4 K2/5
The price of output is \$15, the wage rate is \$8, and the rental rate for capital is \$6 per unit.

a. Find the short-run profit maximizing level of labor demand if capital is fixed at 20 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.

5. 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. (potentially hard) What is the long-run quantity produced? Don’t assume Q=50.

6. Graphically show the long-run demand for labor if the price of capital increases.

7. (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? (This is an unasked question you should automatically do.)

d. Graph parts a-c on one graph.

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Here is the answer... View the full answer 1. A new shop has opened and has the following production function:
Q=E0..25 K0.5
The price of output is \$P, the wage rate is \$w, and the rental rate of capital is \$r. Hint, &quot;as a...

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