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Chapter 12: Problem 9

Demand for light bulbs can be characterized by \(Q=100-P,\) where \(Q\) is in millions of boxes of lights sold and \(P\) is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: \\[ \begin{array}{c} C_{i}=10 Q_{i}+\frac{1}{2} Q_{i}^{2}(i=E, D) \\ Q=Q_{E}+Q_{D} \end{array} \\] a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of \(Q_{E}, Q_{D},\) 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 \(Q_{E}\) \(Q_{D},\) and \(P ?\) What are each firm's profits? c. Suppose the Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow plays Stackelberg. What are the equilibrium values of \(Q_{E}\) \(Q_{D},\) and \(P ?\) What are each firm's profits? d. If the managers of the two companies collude, what are the equilibrium values of \(Q_{E}, Q_{D},\) and \(P ?\) What are each firm's profits?

Short Answer

Expert verified
Under perfect competition, the equilibrium values are \(Q_{E}=Q_{D}=30, P=40\) and each firm's profits are 800. Under Cournot competition, the equilibrium values are \(Q_{E}=Q_{D}=20, P=60\) and each firm's profits are 600. Under Stackelberg, with Everglow as the leader, the equilibrium values are \(Q_{E}=25, Q_{D}=12.5, P=62.5\), with profits of respectively 756.25 and 281.25. Under collusion, the equilibrium values are \(Q_{E}=Q_{D}=20, P=60\) and each firm's profits are 600.

Step by step solution

01

Perfect Competition

Under perfect competition, each firm takes the market price as given and equates marginal cost to price. The marginal cost for each firm \(i\) is \(MC_{i}=10+Q_{i}\). We equate this to the price derived from the market demand: \(10+Q_{i}=100-Q\), where \(Q=Q_{E}+Q_{D}\). Solving these equations, we get \(Q_{E}=Q_{D}=30\) and \(P=40\). Profits for each firm are \(\pi = P*Q_{i} - C_{i} = 800\).
02

Cournot competition

Under Cournot, each firm assumes the output of the competitor is fixed and chooses its own output to maximize its profit. We first write down the profit function for each firm: \(\pi_{i} = (100 - Q)*Q_{i} - (10*Q_{i} + 0.5*Q_{i}^{2})\). Taking first order conditions and solving the subsequent system of equations yields \(Q_{E}=Q_{D}=20\) and \(P=60\). Profits for each firm are \(\pi = 600\).
03

Stackelberg competition

Under Stackelberg, one firm (Everglow) becomes the leader and assumes Dimlit will adjust its output according to the output of Everglow. We first determine the reaction function of Dimlit (firm 2) from the Cournot setup, and then maximize Everglow's profit with respect to this reaction function. Solving the appropriate equations yields \(Q_{E}=25\), \(Q_{D}=12.5\), and \(P=62.5\). Profits for each firm are \(\pi_{E} = 756.25\) and \(\pi_{D} = 281.25\).
04

Collusion

Under collusion, the two firms act as a single monopoly maximizing its joint profit. The total marginal cost is \(MC = MC_{E} + MC_{D} = 20 + Q\), equating this to the price we find \(Q=40\) and \(P=60\). Given that the firms share the total output, we have \(Q_{E}=Q_{D}=20\). Profits for each firm are \(\pi = 600\).

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Most popular questions from this chapter

A monopolist can produce at a constant average (and marginal) cost of \(\mathrm{AC}=\mathrm{MC}=\$ 5 .\) It faces a market demand curve given by \(Q=53-P\) a. Calculate the profit-maximizing price and quantity for this monopolist. Also calculate its profits. b. Suppose a second firm enters the market. Let \(Q_{1}\) be the output of the first firm and \(Q_{2}\) be the output of the second. Market demand is now given by \\[ Q_{1}+Q_{2}=53-P \\] Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of \(Q_{1}\) and \(Q_{2}\) c. Suppose (as in the Cournot model) that each firm chooses its profit- maximizing level of output on 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 \(Q_{1}\) and \(Q_{2}\) for which each firm is doing as well as it can given its competitor's 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, \(\mathrm{MC}=\$ 5 .\) 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.

Two firms compete by choosing price. Their demand functions are \\[ Q_{1}=20-P_{1}+P_{2} \\] and \\[ Q_{2}=20+P_{1}-P_{2} \\] where \(P_{1}\) and \(P_{2}\) are the prices charged by each firm, respectively, and \(Q_{1}\) and \(Q_{2}\) 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 wanted, 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 that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

Consider two firms facing the demand curve \(P=50-5 Q\), where \(Q=Q_{1}+Q_{2}\). The firms' cost functions are \(C_{1}\left(Q_{1}\right)=20+10 Q_{1}\) and \(C_{2}\left(Q_{2}\right)=10+12 Q_{2}\) 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? 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 a takeover is not?

Suppose the market for tennis shoes has one dominant firm and five fringe firms. The market demand is \(Q=400-2 P .\) The dominant firm has a constant marginal cost of \(20 .\) The fringe firms each have a marginal cost of \(\mathrm{MC}=20+5 q\) a. Verify that the total supply curve for the five fringe firms is \(Q_{f}=P-20\) b. Find the dominant firm's demand curve. c. Find the profit-maximizing quantity produced and price charged by the dominant firm, and the quantity produced and price charged by each of the fringe firms. d. Suppose there are 10 fringe firms instead of five. How does this change your results? e. Suppose there continue to be five fringe firms but that each manages to reduce its marginal cost to \(\mathrm{MC}=20+2 q\). How does this change your results?

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)=30 q+1.5 q^{2} \\] The market demand for these seat covers is represented by the inverse demand equation \\[ P=300-3 Q \\] where \(Q=q_{1}+q_{2},\) 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 will be the profit-maximizing choice of output? The industry price? 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 \(\mathrm{WW}\) constructs a payoff matrix like the one below. Fill in each box with the profit of \(\mathrm{WW}\) and the profit of BBBS. Given this payoff matrix, what output strategy is each firm likely to pursue? 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.
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