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I love going down to the elementary school, watching all the kids jump and shout, but they don’t know I’m using blanks. ~ Jack Handey.

(Translation: Today’s lesson teaches when it is important to you that your opponents know your actions so you can manipulate their reactions.)

BA 210 Lesson II.3 Sequential Quantity Competition

- Lesson II.1 Strategic Bargaining
- Lesson II.2 Bargaining and Impatience
- Lesson II.3 Sequential Quantity Competition
- Example 1: Stackelberg Duopoly
- Example 2: First Mover Advantage
- Example 3: Selling Technology
- Example 4: Colluding
- Example 5: Merging
- Summary
- Review Questions

BA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg Duopoly

Example 1: Stackelberg DuopolyBA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg Duopoly

- Comment: Stackelberg Duopoly Games have three parts.
- Players are managers of two firms serving many consumers.
- Firm 1 is the leader, and acts before Firm 2, the follower.
- Strategies are outputs of homogeneous products (perfect substitutes), so they sell at the same price P.
- Firm 1 chooses output Q1> 0.
- Firm 2 knows Firm 1’s Q1> 0 before he chooses his own.
- Payoffs are profits. When unit production costs are constants c1 and c2, then profits are
- P1 = (P- c1)Q1 and P2 = (P- c2)Q2

BA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg Duopoly

Payoffs for the two players often listed in a table. The strategies are listed as rows for Player 1, and columns for Player 2. The combination of strategies by both players determines a cell in the payoff table, and that cell specifies the payoffs to the players, with Player 1 listed first. For example, if Player 1 chooses Q1=3 and Player 2 chooses Q1=1, then in the payoff table below 12,5 specifies payoff 12 to Player 1 and 5 to Player 2.

BA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg Duopoly

Question: You are a manager of Marvel Comics and you compete directly with DC Comics selling comic books. Consumers find the two products to be indistinguishable. The inverse market demand for comic books is P = 11-Q (in dollars). Your unit costs of production are $3, and the unit costs of DC Comics are $2. Compute profits when you produce 1 units and DC produces 4 units. Suppose profits from other combinations of production are in the table below:

BA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg Duopoly

Question (continued): Suppose you choose your output of comic books to be either 1, 3, or 4 before DC Comics, and DC Comics knows your output before they decide their own output of either 1, 3, or 4.

How many comic books should you produce?

BA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg Duopoly

Answer: You are the leader in a Stackelberg Duopoly Game with inverse demand P = 11 - (Q1+Q2) and unit costs c1 = 3 and c2 = 2. If you produce Q1=1 units and DC produces Q2=4 units, then total output is Q1+Q2=5, so price is P = 11 - (Q1+Q2)=6, and profits are

P1 = (P- c1)Q1 = (6-3)1 = 3 and P2 = (P- c2)Q2 = (6-2)4 = 16, which we write 3,16 to complete the profit table.

BA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg Duopoly

Find the rollback solution to the Stackelberg Duopoly Game. Starting from the end of the game,

- if Marvel has chosen Q1=1, then DC will respond with Q2=4, and so profits are 3 for Marvel and 16 for DC
- if Q1=3, then Q2=3, and so profits are 6 and 9
- if Q1=4, then Q2=3, and so profits are 4 and 6

BA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg Duopoly

So Marvel should produce 3 comic books, which leads DC Comics to produce 3 and generate profit of 6 for Marvel.

BA 210 Lesson II.3 Sequential Quantity Competition

Example 2: First Mover Advantage

Example 2: First Mover AdvantageBA 210 Lesson II.3 Sequential Quantity Competition

Example 2: First Mover Advantage

Comment: If the unit production costs are the same for the leader and the follower in a Stackelberg duopoly, then the leader produces more and makes more profit.

In particular, a firm can find it profitable to become the first mover by rushing to set up an assembly line, even if it means increasing unit costs of production.

BA 210 Lesson II.3 Sequential Quantity Competition

Example 2: First Mover Advantage

Question: You are the manager of Kleenex and you compete directly with Puffs selling facial tissues in America. Consumers find the two products to be indistinguishable. The inverse market demand for facial tissues is P = 11-Q (in dollars) in America and both firms produce at a unit cost of $2. You have a decision to make about competing with Puffs in New Zealand, where the inverse market demand for facial tissues is P = 11-Q (in dollars), and both you and Puffs can choose an output quantity 1, 3, or 6. You must choose the option that is best for Kleenex.

BA 210 Lesson II.3 Sequential Quantity Competition

Example 2: First Mover Advantage

Option A. Puffs sets up its factories and distribution networks now, and you set up later. And both produce at a unit cost of $2, resulting in the first profit table:

Option B. You hurry set up your factories and distribution networks now, and Puffs sets up later. Your hurry means your unit costs are $3, while Puffs unit costs remain $2 resulting in the second profit table:

BA 210 Lesson II.3 Sequential Quantity Competition

Example 2: First Mover Advantage

Answer: In Option A, you are the follower in a Stackelberg Duopoly. Puffs anticipates your reactions on the right, and so chooses to produce 6, you react with 1 and so you earn 2.

In Option B, you are the leader in a Stackelberg Duopoly. You anticipate Puffs’ reactions on the right, and so choose to produce 6, Puffs reacts with 1 and so you earn 6.

BA 210 Lesson II.3 Sequential Quantity Competition

Example 2: First Mover Advantage

Option B is thus best for Kleenex since Kleenex profits (as a follower) are 2 in Option A, while Kleenex profits (as the leader) are 6 in Option B.

BA 210 Lesson II.3 Sequential Quantity Competition

Example 2: First Mover Advantage

Comment: In this particular case, Kleenex increased production cost hurt profits less than profits increase because of the first mover advantage, so it is worth being the first mover.

In other problems, increased production cost hurt profits more than profits increase because of the first mover advantage, so it is not worth being the first mover.

BA 210 Lesson II.3 Sequential Quantity Competition

Example 3: Selling Technology

BA 210 Lesson II.3 Sequential Quantity Competition

Question: You are a manager of Home Depot and your only significant competitor in the retail home improvement market is Lowes. You expect to open the first home improvement store in the Conejo Valley, and Lowes will follow a month later. Your lumber and Lowes’s lumber are indistinguishable to consumers. The inverse market demand for lumber is P = 4-Q (in dollars) and both firms used to produce at a unit cost of $2. However, you just found a better way to treat lumber, which reduces your unit cost to $1. Should you keep that procedure to yourself? Or is it better to sell that secret to Lowes so that both you and Lowes can produce at unit cost equal to $1?

To answer the question, suppose both Home Depot and Lowes separately choose to produce either 0, or 1, or 3 units of lumber.

BA 210 Lesson II.3 Sequential Quantity Competition

Answer: You are the leader in a Stackelberg Duopoly with inverse demand P = 4-(Q1+Q2). Compare the rollback solution with unit costs c1 = 1 and c2 = 2, to the solution with c1 = 1 and c2 = 1.

First, compute the payoff table and rollback solution for unit costs c1 = 1 and c2 = 2:

BA 210 Lesson II.3 Sequential Quantity Competition

You are the leader in a Stackelberg Duopoly. You anticipate Lowes’ reactions below, and so choose to produce 1, Lowes reacts with 1 and so you earn 1 or 2 and Lowes earns 0.

BA 210 Lesson II.3 Sequential Quantity Competition

Next, compute the payoff table and rollback solution for unit costs c1 = 1 and c2 = 1:

BA 210 Lesson II.3 Sequential Quantity Competition

You are the leader in a Stackelberg Duopoly. You anticipate Lowes’ reactions below, and so choose to produce 1, Lowes reacts with 1 and so you earn 1 and Lowes earns 1.

BA 210 Lesson II.3 Sequential Quantity Competition

Selling technology and reducing c2 = 2 to c2 = 1 has to effects:

- Firm 1’s profit reduces from P1 = 1 or 2 to P1 = 1 for sure
- Firm 2’s profit increases from P2 = 0 to P2 = 1

In particular, selling technology increases total profit P1+P2 from the uncertain result of 1 or 2 to the certainty of 2. Selling technology is thus a bargaining problem between Home Depot and Lowes, and the division of the positive gain from selling technology is determined by the rules of bargaining.

For example, if Home Depot can make a take-it-or-leave-it offer to Lowes, then Lowes should accept anything as being better than nothing. After deducing that, Home Depot’s best acceptable offer to Lowes leaves Lowes with a pittance of the gains, which means Lowes pays the full 1 unit of profit for the technology.

BA 210 Lesson II.3 Sequential Quantity Competition

Comment: The demand for a product is sometimes presented in standard form, like Q = 10 - 2P. That should be inverted, to P = 5 - 0.5Q, to facilitate duopoly calculations.

BA 210 Lesson II.3 Sequential Quantity Competition

Question: The market for razor blades consists of two firms: Gillette and Wilkinson Sword/Schick. As the manager of Gillette, you enjoy a patented technology that permits your company to produce razor blades more quickly. You use that advantage to be first to choose your profit-maximizing output level in the market, and your competitor knows your output before choosing their own output. The demand for razor blades is Q = 13 - P; Gillette’s unit costs are 1; and Wilkinson’s unit costs are 1. Both firms separately produce either 0, 3, or 4 units.

Compute Gillette’s profit, and compute Wilkinson’s profit. Ignoring antitrust law considerations, would it be mutually profitable for the companies to collude by changing Gillette’s and Wilkinson’s outputs to 3 each. Can Gillette trust Wilkinson?

BA 210 Lesson II.3 Sequential Quantity Competition

Answer: You are the leader in a Stackelberg Duopoly with inverse demand P = 13-(Q1+Q2). Compare the rollback solution with unit costs c1 = 1 and c2 = 1 with the collusive proposal of quantities 4 and 2.

First, compute the payoff table and rollback solution:

BA 210 Lesson II.3 Sequential Quantity Competition

You are the leader in a Stackelberg Duopoly. You anticipate Wilkinson’ reactions below, and so choose to produce 4, Wilkinson reacts with 4 and so you earn 16 and Wilkinson earns 16.

The collusive proposal of quantity 3 for each is thus mutually profitable for the companies. But Gillette cannot trust Wilkinson since Wilkinson’s best

response to Gillette’s

Q1 = 3 is Q2 = 4, not 3.

BA 210 Lesson II.3 Sequential Quantity Competition

Question: The market for commercial large jet aircraft consists of two firms: Airbus and Boeing. As the manager of Boeing, you enjoy a patented technology that permits your company to produce jets more quickly and at a lower cost than Airbus. You use that advantage to be first to choose your profit-maximizing output level in the market. The demand for jets is Q = 9 - P; Boeing’s unit costs are 1 and Airbus’s unit costs are 2.

Compute Boeing’s profit, and compute Airbus’s profit. Would it be profitable for the two companies to merge?

Suppose the firms separately produce quantities 0, 3, or 5 units, and if merged, the two can each choose from those quantities, and they can both use the Boeing technology with unit cost 1.

BA 210 Lesson II.3 Sequential Quantity Competition

Answer: You are the leader in a Stackelberg Duopoly with inverse demand P = 9-(Q1+Q2). Compare the rollback solution with unit costs c1 = 1 and c2 = 2 with the with the monopoly solution.

First, compute the payoff table and rollback solution:

BA 210 Lesson II.3 Sequential Quantity Competition

You are the leader in a Stackelberg Duopoly. You anticipate Airbus’ reactions below, and so choose to produce 5, Airbus reacts with 0 and so you earn 15 and Airbus earns 0.

BA 210 Lesson II.3 Sequential Quantity Competition

The monopoly proposal reduces Airbus unit cost to 1, and so generates a new profit table. But there are no cells in that new payoff table that increase total profit P1+P2 from the Stackelberg value of P1+P2=15+0. So in this case, it would not be profitable for the two companies to merge.

BA 210 Lesson II.3 Sequential Quantity Competition

- Payoff table entry to any Duopoly Game with inverse demand P = a -bQ and constant unit costs c1 and c2:
- P= a - b(Q1+Q2)
- Firm 1 profit P1 = (P - c1) Q1
- Firm 2 profit P2 = (P - c2) Q2

BA 210 Lesson II.3 Sequential Quantity Competition

- Review Questions
- You should try to answer some of the following questions before the next class.
- You will not turn in your answers, but students may request to discuss their answers to begin the next class.
- Your upcoming Exam 2 and cumulative Final Exam will contain some similar questions, so you should eventually consider every review question before taking your exams.

BA 210 Lesson II.3 Sequential Quantity Competition

BA 210 Introduction to Microeconomics

End of Lesson II.3BA 210 Lesson II.3 Sequential Quantity Competition

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