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Chapter 12: Problem 6

Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by \(C_{1}=60 Q_{1}\) and \(C_{2}=60 Q_{2}\), where \(Q_{1}\) is the output of Firm 1 and \(Q_{2}\) the output of Firm 2. Price is determined by the following demand curve: $$P=300-Q$$ where \(Q=Q_{1}+Q_{2}\) 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. c. Suppose Firm 1 were the only firm in the industry. How would market output and Firm \(1^{\prime}\) s profit differ from that found in part (b) above? 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?

Short Answer

Expert verified
a) Cournot-Nash equilibrium is Q1=Q2=100, price is 100 and each firm's profit is 4000. b) In the cartel, each firm produces 120 widgets and makes no profit. c) As a monopoly, Firm 1 produces 240 widgets and makes no profit. d) If Firm 2 cheats by increasing production, its profits will increase while Firm 1's profits will decrease.

Step by step solution

01

Cournot-Nash Equilibrium

In the Cournot-Nash Equilibrium, each firm maximizes its profit given the other's output. Since both firms are identical, they have symmetrical reactions. So, Firm 1's and Firm 2's output reaction functions are \(Q_{1}=Q_{2}=(300-Q)/2\). Solving these equations simultaneously gives \(Q_{1}=Q_{2}=100\). The price at this equilibrium is \(P=300-Q=300-200=100\). The profit of each firm is \(\Pi=(P*C)-Cost=(100*100)-(60*100)=4000\).
02

Cartel Profits

When the two firms form a cartel to maximize joint profits, they act as a monopoly and produce where marginal cost equals marginal revenue. The total quantity produced by the cartel is \(Q=300-P\). Solving for Q gives \(Q=240\) and, because the firms are identical, they each produce half: \(Q_{1}=Q_{2}=120\). The price at this output level is \(P=300-Q=60\). The profit of each firm is \(\Pi=(P*C)-Cost=(60*120)-(60*120)=0\). Here, profits are zero because the firms are essentially producing at cost in a perfectly competitive market.
03

Monopoly Profits

If Firm 1 were the only firm in the industry, it would act as a monopoly and produce where marginal cost equals marginal revenue. Solving for Q gives \(Q=240\). The price at this output level is \(P=300-Q=60\). The profit of the firm is \(\Pi=(P*C)-Cost=(60*240)-(60*240)=0\). Again, profits are zero because the firm is essentially producing at cost in a perfectly competitive market.
04

Cheating within the Cartel

If Firm 1 abides by the agreement but Firm 2 increases production, the new joint quantity is higher and consequently, the price is lower. While the exact quantity Firm 2 will produce depends on a number of factors including the demand elasticity and the cost structure, it is clear that Firm 2's profit will increase while Firm 1's profit will decrease.

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Most popular questions from this chapter

Suppose the airline industry consisted of only two firms: American and Texas Air Corp. Let the two firms have identical cost functions, \(C(q)=40 q\). Assume that 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 when taking its rival's output as given. What are the profits of each firm? b. What would be the equilibrium quantity if Texas Air had constant marginal and average costs of \(\$ 25\) and American had constant marginal and average costs of \(\$ 40 ?\) c. Assuming that both firms have the original cost function, \(C(q)=40 q,\) 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?

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Suppose that two competing firms, \(A\) and \(B\), produce a homogeneous good. Both firms have a marginal cost of \(\mathrm{MC}=\$ 50 .\) Describe what would happen to output and price in each of the following situations if the firms are at (i) Cournot equilibrium, (ii) collusive equilibrium, and (iii) Bertrand equilibrium. a. Because Firm \(A\) must increase wages, its \(\mathrm{MC}\) increases to \(\$ 80\). b. The marginal cost of both firms increases. c. The demand curve shifts to the right.

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 \(\mathrm{WW}\) and \(\mathrm{BBBS}\) 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 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.

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.
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