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Chapter 12: Problem 13

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?

Short Answer

Expert verified
For a variety of setups, the dominant firm produced 360 units and priced them at $20 per unit, while the fringe firms would not produce anything.

Step by step solution

01

Verify the total supply curve for the five fringe firms

The marginal cost curve of each fringe firm is given as \(MC = 20 + 5q\), where \(q\) is the quantity supplied by each firm. Supply equals marginal cost for firms in perfect competition, so the supply equation for each firm is \(q = P/5 - 4\). Now, for five such firms, the total quantity supplied by the fringe firms would be \(Q_f = 5q\). This yields \(Q_f = 5(P/5 - 4) = P - 20\) as required.
02

Find the dominant firm's demand curve

The dominant firm's demand is the difference between the market demand and the fringe firms' supply. Thus, subtracting the total quantity supplied by the fringe firms from the market demand, we have \(Q_D = Q - Q_f = 400 - 2P - (P - 20) = 420 - 3P\). This is the demand curve for the dominant firm.
03

Find the profit-maximizing quantity and price

For the dominant firm, profit maximization occurs where MC equals the price derived from the demand curve. The MC is constant at 20. From the demand curve of the dominant firm which is \(Q_D = 420 - 3P\), setting \(P = MC = 20\) yields \(Q_D = 420 - 3*20 = 360\) which is the quantity produced by the dominant firm. The fringe firms collectively supply \(Q_f = P - 20 = 20 - 20 = 0\). This indicates that the firms are producing at their minimum supply point to avoid a loss.
04

Changes with 10 fringe firms

If the number of fringe firms increases to 10, the total supply curve becomes \(Q_f = 10q = 10(P/5 - 4) = 2P - 40\). Substituting this into the market demand curve to get the new demand curve of the dominant firm, we get \(Q_D = 400 - 2P - (2P - 40) = 440 - 4P\). MC still equals price, therefore, setting \(P = 20\) yields \(Q_D = 440 - 4*20 = 360\), same as before. Now, the fringe firms supply \(Q_f = 2P - 40 = 2*20 - 40 = 0\), indicating that they still produce at their minimum supply point.
05

Reduced marginal cost of the fringe firms

If the marginal cost of fringe firms reduces to \(MC = 20 + 2q\), then their supply curve becomes \(q = P/2 - 10\). This results in a total supply curve of \(Q_f = 5(P/2 - 10) = 2.5P - 50\). The demand curve of the dominant firm then shifts to \(Q_D = 400 - 2P - (2.5P - 50) = 450 - 4.5P\). MC still equals price, therefore, setting \(P = 20\) yields \(Q_D = 450 - 4.5*20 = 360\), same as before. The fringe firms now supply \(Q_f = 2.5P - 50 = 2.5*20 - 50 = 0\), indicating that they still produce at their minimum supply point.

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Most popular questions from this chapter

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 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?

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: \\[ \begin{aligned} P &=300-Q \\ \text { where } Q=Q_{1}+Q_{2} \end{aligned} \\] 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'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?

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.

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.
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