Industrial Org Homework

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ECP 4403: Industrial Organization

Homework Set 3 (Chapters 9-11) Due: March 30, Monday

Mathematical Hint: If f(x) = ax2 + bx + c, where a, b and c are constants, then Δf/Δx = 2ax + b.

General Hint: It is often helpful (and sometimes required) to draw diagrams to accompany your analysis. Show all of your work to maximize credit. Workspace is left between parts and at the end of the question for these purposes. Although the problems have many parts, they are not always cumulative. If you get stuck, do not give up entirely on the problem, just go on to the next part.

1. Fred and Barney are fishermen who operate fishing ships out of Nantucket. They own similar ships that can each carry a day’s catch of 800 pounds of cod. Marginal costs of fishing are constant at $4 per pound (for simplicity, assume they have no fixed -- or sunk -- costs). For the current discussion, assume that Fred and Barney are the only fishermen and that the aggregate demand for Nantucket cod is characterized by the demand function Q = 1600 - 100P, where P is the price per pound of cod and Q is in pounds of cod. This corresponds to an inverse demand function of P = 16 - .01Q, where Q = qF + qB. Finally, the fish market operates in the following manner: fishermen go out in the morning and catch fish, then they return to the dock and drop the catch on the dock. Customers gather around and a market-clearing price is announced, at which point sales commence.

(a) What is Fred’s residual (inverse) demand curve if his catch size is qF and Barney’s is qB?

(b) What is Fred’s profit as a function of his catch qF and Barney’s catch qB?

(d) What is Barney’s residual (inverse) demand curve if his catch size is qB and Fred’s is qF?

(e) What is Barney’s profit as a function of his catch qB and Fred’s catch qF?

(f) If Barney expects Fred to catch qF, what is Barney’s best response function BRB(qF)?

(g) Using the fact that each fisherman’s equilibrium catch is a best response to the other’s, solve for the equilibrium catch sizes qBCN and qFCN .

(h) What will be the resulting price?

(i) Show that each fisherman’s Cournot duopoly profits will be $1600.

2. Consider question 1. Suppose that Fred wakes up earlier than Barney and decides how many pounds of cod to catch and calls the dock manager to tell him how many pounds of cod to bring to dock. When Barney wakes up, he realizes that Fred’s output is already announced, and he has to decide how many pounds of cod to catch accordingly.

(a) Write Barney’s best response function (Hint: Similar to what you found in question 1).

(b) What is Fred’s profit as a function of his catch qF?

(d) What will be the resulting price?

(e) Find each fisherman’s Stackelberg duopoly profits.

3. It was late on Saturday afternoon. As Barney's men pulled up the net for the last time that day he couldn't believe his eyes: there were bluefish in the net. He had accidentally stumbled upon an area populated by bluefish. Barney knew that as soon as he started bringing in bluefish, Fred would find out. There being no sense in keeping the word to himself, he told Fred over their traditional drink at Greta's. Fred popped a cheese ball in his mouth and settled back to consider the news. "Bluefish, you say. What if you specialize in hauling in blues and I'll stick to the cod? That should give us a little market power."

Barney thought about Fred's suggestion. It seemed fair. But with two types of fish in the market, what would happen to prices? Fred and Barney went over to the Nantucket Bait Shop and Fishing Museum that an old fish auctioneer, Leon Walrus, now ran. Walrus had some old copies of the Nantucket Fisherman's Daily, a local trade paper. The copies were from ten years before, when blues were locally available, before they seemingly disappeared from the area. The newspaper carried a story about a study of the demand for different fish by an economist at the local university. Focusing on cod and blue, the following (inverse) demand functions had been estimated: Pb = 16 -.01qb -.005qc, Pc = 16 -.01qc -.005qb, where the b subscript refers to bluefish and the c subscript refers to cod. Assume that Fred fishes for cod (so Pc = PF and qc = qF) and Barney fishes for bluefish (so Pb = PB and qb = qB), and that they choose their catch sizes simultaneously and non-cooperatively (i.e., they act as Cournot duopolists). Assume there is a constant marginal and average cost per pound of either fish of $4.

(a) What is Fred’s profit as a function of his output qF and Barney’s output qB? What is Fred’s best response function?

(b) What is Barney’s profit as a function of his output qB and Fred’s output qF? What is Barney’s best response function?

(d) What are the equilibrium prices for cod and bluefish PC CN and Pb CN

(e) What are the equilibrium profits πFCN and πBCN, for Fred and Barney, respectively?

4. Idaho City, Ohio, is a town built along a strip of road that is 1 mile long. There are two sporting goods stores in town. Fin N Feather (hereafter FNF) is at the West end of town and Outdoor Adventures (hereafter OA) is located at the East end of town. Both stores sell tents of equal quality, but FNF has a constant marginal and average cost of $50 per tent while OA (a national chain that can take advantage of quantity discounts offered by tent suppliers) has a constant marginal and average cost of $40 per tent. Idaho City is a real camping town. There are 5000 families in Idaho City that are currently in the market for a new tent. Each family’s maximum willingness to pay for a tent is $120. In addition to the price at the store, each family bears a cost of $20 per mile to transport the tent home. If pF is the price of a tent at FNF and pO is the price of a tent at OA, then the marginal consumer (who is indifferent about which store to buy at) is located at a distance xm from the West end of town (see diagram below).

(West) FNF 0______________________________________1 OA (East)

5000 families spread evenly across the mile

(a) What is the value of xm as a function of pF and pO? Customers to the left of xm(pF,pO) represent demand at FNF; remaining customers patronize OA.

(b) What is FNF’s profit as a function of pF and pO? What is it’s best response function?

(d) Find the Bertrand-Nash equilibrium prices pF BNand pOBN.

(e) What is the location of the marginal consumer? How much does this family pay for a tent (include all costs)?

(f) What are the Bertrand-Nash equilibrium profits for FNF and OA?

5. Consider the street in Idaho City, with 5000 families looking for a new tent. Each family’s maximum willingness to pay for a tent is $120. In addition to the price at the store, each family bears a cost of $20 per mile to transport the tent home. Suppose there is a single tent seller on this street, Camping Gears, CG, and CG is planning to open new branches in this camping town along this street. CG has a constant unit cost of $40 per tent, and the fixed cost of opening a new branch is $2000.

(a) If CG wants to set a price low enough to serve all 5000 families, what is the highest price it can set if CG will have 1 branch? 2 branches? 3 branches? What is the highest price it can set if CG will have n branches?

(b) Write CG’s profit function with n branches and find the optimal number of branches if CG wants to serve all families.

(c) What is the total transportation cost paid by all customers when CG has only one branch? 2 branches? 3 branches? What is the total transportation cost paid by all customers when CG has n branches?

(d) Write the cost minimization problem for the sum of the total transportation and set-up costs. Solve for the optimal number of branches that maximizes total surplus.