## pricing and cost

1. Suppose all firms in a monopolistically competitive industry were merged into one large firm. Would that new firm produce as many different brands? Would it produce only a single brand? Explain.

2. Consider the following duopoly. Demand is given by P = 10 – Q, where Q = Q1 + Q2. The firms’ cost functions are C1(Q1) = 4 + 2Q1 and C2(Q2) = 3 + 3Q2.

a. Suppose both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry?

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b. What is each firm’s equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms’ reaction curves, and show the equilibrium.

c. How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal, but the takeover is not?

3. A monopolist can produce at a constant average (and marginal) cost of AC = MC = 5. The firm faces a market demand curve given by Q = 53 – P.

a. Calculate the profit-maximizing price and quantity for this monopolist. Also calculate the monopolist’s profits.

b. Suppose a second firm enters the market. Let Q1 be the output of the first firm and Q2 be the output of the second. Market demand is now given by

Q1 + Q2 = 53 – P.

Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of Q1 and Q2.

c. Suppose (as in the Cournot model) each firm chooses its profit-maximizing level of output under the assumption that its competitor’s output is fixed. Find each firm’s “reaction curve” (i.e., the rule that gives its desired output in terms of its competitor’s output).

d. Calculate the Cournot equilibrium (i.e., the values of Q1 and Q2 for which both firms are doing as well as they can given their competitors’ output). What are the resulting market price and profits of each firm?

*e. Suppose there are N firms in the industry, all with the same constant marginal cost, MC = 5. Find the Cournot equilibrium. How much will each firm produce, what will be the market price, and how much profit will each firm earn? Also, show that as N becomes large the market price approaches the price that would prevail under perfect competition.

4. This exercise is a continuation of Exercise 3. We return to two firms with the same constant average and marginal cost, AC = MC = 5, facing the market demand curve

Q1 + Q2 = 53 – P. Now we will use the Stackelberg model to analyze what will happen if one of the firms makes its output decision ahead of the other one.

a. Suppose Firm 1 is the Stackelberg leader (i.e., makes its output decisions ahead of Firm 2). Find the reaction curves that tell each firm how much to produce in terms of the output of its competitor.

5. Two firms compete in selling identical widgets. They choose their output levels Q1 and Q2 simultaneously and face the demand curve

6. Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C1 = 30Q1 and C2 = 30Q2, where Q1 is the output of Firm 1 and Q2 is the output of Firm 2. Price is determined by the following demand curve:

P = 150 – Q

where Q = Q1 + Q2.

a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.

b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm’s profit.

d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement, but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm’s profits?

7. Suppose the airline industry consisted of only two firms: American and Texas Air Corp. Let the two firms have identical cost functions, C(q) = 40q. Assume the demand curve for the industry is given by P = 100 – Q, and that each firm expects the other to behave as a Cournot competitor.

a. Calculate the (Cournot-Nash) equilibrium for each firm, assuming that each chooses the output level that maximizes its profits taking its rival’s output as given. What are the profits of each firm?

c. Assuming that both firms have the original cost function, C(q) = 40q, how much should Texas Air be willing to invest to lower its marginal cost from 40 to 25, assuming that American will not follow suit? How much should American be willing to spend to reduce its marginal cost to 25, assuming that Texas Air will have marginal costs of 25 regardless of American’s actions?

*8. Demand for light bulbs can be characterized by Q = 100 – P, where Q is in millions of lights sold, and P is the price per box. There are two producers of lights: Everglow and Dimlit. They have identical cost functions:

Q = QE + QD.

a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of QE, QD, and P? What are each firm’s profits?

b. Top management in both firms is replaced. Each new manager independently recognizes the oligopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of QE, QD, and P? What are each firm’s profits?

c. Suppose the Everglow manager guesses correctly that Dimlit has a Cournot conjectural variation, so Everglow plays Stackelberg. What are the equilibrium values of QE, QD, and P? What are each firm’s profits?

d. If the managers of the two companies collude, what are the equilibrium values of QE, QD, and P? What are each firm’s profits?

9. Two firms produce luxury sheepskin auto seat covers, Western Where (WW) and B.B.B. Sheep (BBBS). Each firm has a cost function given by:

C (q) = 20q + q2

The market demand for these seat covers is represented by the inverse demand equation:

P = 2002Q,

where Q = q1 + q2 , total output.

a. If each firm acts to maximize its profits, taking its rival’s output as given (i.e., the firms behave as Cournot oligopolists), what will be the equilibrium quantities selected by each firm? What is total output, and what is the market price? What are the profits for each firm?

b. It occurs to the managers of WW and BBBS that they could do a lot better by colluding. If the two firms collude, what would be the profit-maximizing choice of output? What is the industry price? What is the output and the profit for each firm in this case?

c. The managers of these firms realize that explicit agreements to collude are illegal. Each firm must decide on its own whether to produce the Cournot quantity or the cartel quantity. to aid in making the decision, the manager of WW constructs a payoff matrix like the real one below. Fill in each box with the (profit of WW, profit of BBBS). Given this payoff matrix, what output strategy is each firm likely to pursue?

 Profit Payoff Matrix BB BS (WW profit, BBBS profit) Produce Cournot q Produce Cartel q WW Produce Cournot q 1518, 1518 1721, 1458 Produce Cartel q 1458, 1721 1620, 1620

d. Suppose WW can set its output level before BBBS does. How much will WW choose to produce in this case? How much will BBBS produce? What is the market price, and what is the profit for each firm? Is WW better off by choosing its output first? Explain why or why not.

*10. Two firms compete by choosing price. Their demand functions are Q1 = 20 – P1 + P2 and Q2 = 20 + P1 – P2

where P1 and P2 are the prices charged by each firm respectively and Q1 and Q2 are the resulting demands. (Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they want, and earn infinite profits.) Marginal costs are zero.

a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.)

b. Suppose Firm 1 sets its price first, and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be?

c. Suppose you are one of these firms, and there are three ways you could play the game: (i) Both firms set price at the same time. (ii) You set price first. (iii) Your competitor sets price first. If you could choose among these, which would you prefer? Explain why.

*11. The dominant firm model can help us understand the behavior of some cartels. Let us apply this model to the OPEC oil cartel. We shall use isoelastic curves to describe world demand W and noncartel (competitive) supply S. Reasonable numbers for the price elasticities of world demand and non-cartel supply are -1/2 and 1/2, respectively. Then, expressing W and S in millions of barrels per day (mb/d), we could write

W = 160P -1/2 and S = 3(1/3)P1/2.

Note that OPEC’s net demand is D = W – S.

a. Sketch the world demand curve W, the non-OPEC supply curve S, OPEC’s net demand curve D, and OPEC’s marginal revenue curve. For purposes of approximation, assume OPEC’s production cost is zero. Indicate OPEC’s optimal price, OPEC’s optimal production, and non-OPEC production on the diagram. Now, show on the diagram how the various curves will shift, and how OPEC’s optimal price will change if non-OPEC supply becomes more expensive because reserves of oil start running out.

b. Calculate OPEC’s optimal (profit-maximizing) price. (Hint: Because OPEC’s cost is zero, just write the expression for OPEC revenue and find the price that maximizes it.)

c. Suppose the oil-consuming countries were to unite and form a “buyers’ cartel” to gain monopsony power. What can we say, and what can’t we say, about the impact this would have on price?

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12. A lemon-growing cartel consists of four orchards. Their total cost functions are:

(TC is in hundreds of dollars, Q is in cartons per month picked and shipped.)

a. Tabulate total, average, and marginal costs for each firm for output levels between 1 and 5 cartons per month (i.e., for 1, 2, 3, 4, and 5 cartons).

b. If the cartel decided to ship 10 cartons per month and set a price of 25 per carton, how should output be allocated among the firms?

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