There are 15 questions in 3 major problems, 0.6 points each.
The remaining 1 point will be rewarded when you upload your answers via Blackboard correctly. When you upload, either type or write steps for each of the questions clearly. If you write on a paper, please scan it and generate one pdf (please do not upload several photos taken under the dim light) to submit. Do not forget your name on your solution file!
| π |
Potato farming, like farming of most agricultural products, is highly competitive. Price is determined by demand and supply. Based on U.S. Department of Agriculture statistics, U.S. demand for potatoes is estimated to be ππ· = 584 β 20π, where π is the farmerβs wholesale price and ππ· is the consumption of potatoes per capita. One farm π has the following cost function of producing potatoes: πΆ(ππ) = 12 + 7ππ + 3π2, where ππ is this farmβs output level. Assume that farms determine the output level to maximize their own profits and there are π identical potato farms in this market.
- You notice that under the pandemic, the market price is $25/unit for potatoes.
Q1.1. Compute the short-run output level per farm, ππ, in this market.
MR = MC
25 = 7 + 6Qi
6Qi = 18
Qi = 3
Q1.2. Compute the short-run profit per farm in this market.
Profit = TR β TC
Profit = (25×3) β (12+21+27) = $15
Q1.3. Compute the short-run farm numbers, π, in this market.
Qd = 584 β 20P
Qd = 584 β 20(25) = 84
n = Qd/Qi = 84/3 = 28
- As you have learned in class, this market will reach to an equilibrium in the long run. Assume the cost function per farm remains the same.
Q1.4. Compute the long-run output level per farm, ππ, in this market.
Q1.5. Compute the long-run market price, π, in this market.
Q1.6. Compute the long-run farm numbers, π, in this market.
Q1.7. After the pandemic, assume the market will gradually shift to the long-run equilibrium. Use a demand-supply curve diagram to explain the changes of farms, market price, and quantity supplied of this shift.
- Firms A and B make up a cartel that monopolizes the market for a scarce natural resource. The firmsβ marginal costs are ππΆπ΄ = 6 + 2ππ΄ and ππΆπ΅ = 18 + ππ΅, respectively. They seek to maximize the total profit.
- The firms have decided to limit their total output to Q = 18.
Q2.1. What outputs should the firms produce to achieve this level of output at minimum total cost? Q2.2. What is each firmβs marginal cost when they limit the total output to Q = 18?
- The market demand curve is π· = ππ β πΈ, where Q is the total output of the cartel. Q2.3. Find Firm A and Bβs output levels, respectively. (Hint: At the optimum, ππ = ππΆπ΄ = ππΆπ΅.) Q2.4. Find the cartelβs optimal price.
- A movie theater finds its consumersβ demand curve as π = 730 β 12π. (Note that this is a demand curve, not the inverse demand curve.) It can operate monopolistically because the local region has only one theater. Now the theaterβs manager faces a pure-selling problem, as the marginal cost to show a movie to one more consumer is zero. Q3.1. If the theater has unlimited space, what price should it charge to maximize profit?
Q3.2. Unfortunately, the theater has only 250 seats. Find the highest profit accordingly.
- The firm manager notices that the seniorsβ demand is lower than othersβ demand. Specifically, πΈπ =
πππ β ππ· for seniorsβ demand and πΈπ = πππ β ππ· for othersβ demand. (Hint: convert the demand curve to the inverse demand curve first.)
Q3.3. How should the manager allocate tickets among seniors and others to increase the profit? Find ππ and ππ
based on your knowledge of price discrimination.
Q3.4. Given ππ and ππ that you found in Q3.3., calculate the new profit under price discrimination and compare it with the profit in Q3.2.