1. The engineering department of your firm estimated that your production function is
Q = 2 √kL
In the short run, capital is fixed at 64 units: K¯ = 64, which means that your short run production function is Qsr,. = 16√L.
a. In the shod run, you want to produce 320 units. How much labor must you hire?
b. Find the short-run cost function.
Input prices are w = 20, r = 80.
c. How much will it cost to produce 320 units in the short-run?
d. Given that the MRTSL,K =K/L, find how much it will cost to produce 320 units in the long run.
2. You own a juice bar. To make one glass of your juice of the week, you use 2 ounces of apple juice (A) and 6 ounces of grape grapefruit juice (G). A and G are measured in ounces. Apple juice costs Pa per ounce and grapefruit juice costs Pg per ounce. Please answer the following.
a. Letting Q stand for the number of eight-ounce glasses produced, write the production function for your juice of the week.
b. Find the total cost function for the juice of the week.
c. Suppose that Pa = $0.15 and Pg = $0.20. How much will it cost to produce 10 glasses of juice?
d. Given the input prices mentioned above, find the shutdown price for the juice of the week.
3. Your firm is in a perfectly competitive industry (i.e., you are a price taker). The engineering department estimated your cost function to be
TC = 50 + (Q/10)+ (Q2/10)
This cost function has marginal cost given by MC = 0.1 + (Q/5). The market price in the industry is P = 20.10.
a. Is this a short-run or long-run cost function?
b. What is the average variable cost (AVC) of producing 10 units?
c. What is the average total cost (AC) of producing 25 units?
At the current market price, how many units should your firm produce? Hint: min AVC. = 0.1.
4. You have the following daily production schedule for laborers and output
|
L
|
Q
|
|
0
|
0
|
|
1
|
14
|
|
2
|
29
|
|
3
|
40
|
|
4
|
48
|
|
5
|
54
|
|
6
|
58
|
Assume that the firm is a price taker in both inputs and outputs. Laborers are paid w = $100 per day and output can be sold for P = $20 per unit.
a. With which laborer does the Law of diminishing returns set in?
b. How many laborers would a profit-maximizing firm hire?
c. Suppose that due to a labor shortage, the wage rate in this industry rises to w = $250. How many laborers would a profit-maximizing firm hire?
5. You manufacture copper pots. Do you expect the costs of a 4 quart saucepan to be more than, equal to, or less than twice the costs of a 2 quart saucepan? Explain why (in plain English) you chose your answer.
6. Fill in the blanks on the following table of costs
|
Q
|
TC
|
FC
|
VC
|
AVC
|
MC
|
|
0
|
200
|
|
0
|
|
|
|
1
|
|
|
|
400
|
|
|
2
|
|
|
|
|
500
|
7. Ms. Simon owns a strawberry farm in Plant City. The strawberries are ripe and it is time to decide about harvesting. An average laborer can pick 16 boxes of strawberries per hour. Ms. Simon can hire as many laborers as she wants for $8 per hour.
a. Write the production function for strawberry harvesting.
b. Find the cost function for strawberry harvesting.
c. What is the minimum price per box that Ms. Simon must receive in order to warrant picking the strawberries?
8. Below are three production functions. In the space below each, write the returns to scale that each function exhibits (i.e., IRTS, DRTS or CRTS)
a. q = 2 √kL
b. q = (KL).25
c. q = KL
Which production function (a, b, or c) would have a marginal cost curve that is horizontal?
Which production function (a, b, or c) is from an industry that is a natural monopoly (i.e., it is cheaper to have one big firm than many small firms)?