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Chapter 12: Q4. (page 497)

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 before the other.

  1. Suppose Firm 1 is the Stackelberg leader (i.e., makes its output decisions before Firm 2). Find the reaction curves that tell each firm how much to produce in terms of the output of its competitor.
  2. How much will each firm produce, and what will its profit be?

Short Answer

Expert verified
  1. There cannot be any reaction curve for firm1 as it is leading the market. The reaction curve for firm 2 will be Q2 = 24 – 1/2Q1.
  2. Output for firm 1 will be 24 units, and for firm 2 will be 12 units. The profit for firm 1 will be $288, and for firm 2 will be $144.

Step by step solution

01

Explanation for part (a)

The reaction curve of firm 2 will be Q2=24-12Q1. There is no reaction curve for firm 1 as it sets the output decision earlier than firm 2; thus, there is nothing to react to.

02

Explanation for part (b)

Firm 1 is the leading firm, and firm 2 is the follower; thus, the firm 2 reaction curve will put in the profit function of firm 1.

The output of firm 1, and firm 2 are calculated below:

π1=53-Q1-24-Q12Q1-5Q1dπ1dQ1=53-2Q1-24+Q1-5=024-Q1=0Q1=24Q2=24-242=24-12=12

Firm 1 will produce 24 units, and firm 2 will produce 12 units.

The profit of firm 1 and firm 2 are calculated below:

role="math" localid="1644230814271" P=53-24-12=$17π1=17×24-5×24=408-120=$288π1=17×12-5×12=204-60=$144

The profit of firm 1 will be $288, and for firm 2 will be $144.

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Most popular questions from this chapter

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

  1. Calculate the profit-maximizing price and quantity for this monopolist. Also calculate its profits.
  2. 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) 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 Q1 and Q2 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, 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 that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C1 = 60Q1 and C2 = 60Q2, where Q1 is the output of Firm 1 and Q2 the output of Firm 2. Price is determined by the following demand curve P = 300 – Q where Q = Q1 + Q2.

  1. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.
  2. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm’s profit.
  3. 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?
  4. 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?

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.

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 wanted, and earn infinite profits. Marginal costs are zero.

  1. 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.)
  2. 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?
  3. 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.

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 MC = 20 + 5q.

a. Verify that the total supply curve for the five fringe firms is Qf = P - 20.

b. Find the dominant firm’s demand curve.

c. Find the profit-maximizing quantity produced and the price charged by the dominant firm, and the quantity produced and the 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 MC = 20 + 2q. How does this change your results?
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