### B.5 Simultaneous Quantity Competition

```Deep Thought
To me, boxing is like a ballet, except
there’s no music, no choreography,
and the dancers hit each other. ~
Jack Handey.
(Translation: Today’s lesson teaches how to manage a
company recognizing competitors are selling substitute
products.)
BA 445 Lesson B.5 Simultaneous Quantity Competition
1
Baye “Cournot Oligopoly” (see the index)
Dixit Chapter 5
BA 445 Lesson B.5 Simultaneous Quantity Competition
2
Overview
Overview
BA 445 Lesson B.5 Simultaneous Quantity Competition
3
Overview
Cournot Duopoly has two firms controlling a large share of the market, and they
compete by simultaneously setting their output (or output capacity). Then, price
is determined by demand.
Nash Equilibrium means each player makes a best response to the strategies of
other players. It is thus a self-enforcing agreement. And it is the same as the
dominance solution of a Cournot Duopoly.
First Mover Advantage always occurs in a Stackelberg or Cournot duopoly. That
advantage can make it profitable to rush to choose output sooner, even if that
rush raises costs.
Selling Technology to a Cournot competitor is profitable if total profit increases. In
that case, there is a positive gain from the sales agreement, which is then divided
according to rules of bargaining.
Colluding with a Cournot competitor is almost always profitable. Since the
competitors produce gross substitutes, profitable collusion lowers output. But,
each cannot trust the other to collude.
BA 445 Lesson B.5 Simultaneous Quantity Competition
4
Example 1: Cournot Duopoly
Example 1: Cournot Duopoly
BA 445 Lesson B.5 Simultaneous Quantity Competition
5
Example 1: Cournot Duopoly
Overview
Cournot Duopoly has two firms controlling a large share of
the market, and they compete by simultaneously setting
their output (or output capacity). Then, price is determined
by demand.
BA 445 Lesson B.5 Simultaneous Quantity Competition
6
Example 1: Cournot Duopoly
Comment: Cournot Duopoly Games have three parts.
• Players are managers of two firms serving many consumers.
• Strategies are outputs of homogeneous products, with inverse
market demand P = a-b(Q1+Q2) if a-b(Q1+Q2) > 0, and P = 0
otherwise.
• Firm 1 chooses output Q1 > 0.
• Firm 2 chooses output Q2 > 0.
• Each chooses either simultaneously or sequentially but in
ignorance of the other’s choice.
• Payoffs are profits. When marginal costs or unit production costs of
production are constants c1 and c2, then profits are
P1 = (P- c1)Q1 and P2 = (P- c2)Q2
BA 445 Lesson B.5 Simultaneous Quantity Competition
7
Example 1: Cournot Duopoly
Question: Intel and AMD control a large share of the
consumer desktop computer microprocessor market. They
simultaneously decide on the size of manufacturing plants
for the next generation of microprocessors for consumer
desktop computers. Suppose the firms’ goods are perfect
substitutes, and market demand defines a linear inverse
demand curve P = 20 – (QI + QA), where output quantities
QI and QA are the thousands of processors produced
weekly by Intel and AMD. Suppose unit costs of production
are cI = 1.1 and cA = 1.1 for both Intel and AMD. Suppose
Intel and AMD consider any quantities QI = 1, 2, …, 9 and
QA = 1, 2, …, 9.
What quantity should Intel produce?
BA 445 Lesson B.5 Simultaneous Quantity Competition
8
Example 1: Cournot Duopoly
Answer: Intel’s quantities QI = 1, 2, …, 9 are on the rows,
and AMD’s quantities QA = 1, 2, …, 9 are on the columns in
the following normal form. For example, QI = 2 and QA = 3
generates price P = 20-5 = 15, and profits PI = (15-1.1)2 =
27.8 and PA = (15-1.1)3 = 41.7. (On an exam, I would
provide most entries.)
20
Row
1.1
1.1
1
2
3
4
5
6
7
8
9
Column
1
16.9,16.9
31.8,15.9
44.7,14.9
55.6,13.9
64.5,12.9
71.4,11.9
76.3,10.9
79.2,9.9
80.1,8.9
2
15.9,31.8
29.8,29.8
41.7,27.8
51.6,25.8
59.5,23.8
65.4,21.8
69.3,19.8
71.2,17.8
71.1,15.8
3
14.9,44.7
27.8,41.7
38.7,38.7
47.6,35.7
54.5,32.7
59.4,29.7
62.3,26.7
63.2,23.7
62.1,20.7
4
13.9,55.6
25.8,51.6
35.7,47.6
43.6,43.6
49.5,39.6
53.4,35.6
55.3,31.6
55.2,27.6
53.1,23.6
5
12.9,64.5
23.8,59.5
32.7,54.5
39.6,49.5
44.5,44.5
47.4,39.5
48.3,34.5
47.2,29.5
44.1,24.5
6
11.9,71.4
21.8,65.4
29.7,59.4
35.6,53.4
39.5,47.4
41.4,41.4
41.3,35.4
39.2,29.4
35.1,23.4
7
10.9,76.3
19.8,69.3
26.7,62.3
31.6,55.3
34.5,48.3
35.4,41.3
34.3,34.3
31.2,27.3
26.1,20.3
8
9.9,79.2
17.8,71.2
23.7,63.2
27.6,55.2
29.5,47.2
29.4,39.2
27.3,31.2
23.2,23.2
17.1,15.2
BA 445 Lesson B.5 Simultaneous Quantity Competition
9
8.9,80.1
15.8,71.1
20.7,62.1
23.6,53.1
24.5,44.1
23.4,35.1
20.3,26.1
15.2,17.1
8.1,8.1
9
Example 1: Cournot Duopoly
Strategies 1, 2, 3, and 4 are dominated for each player (by
Strategy 5). Hence, eliminate those strategies, leaving the
normal form:
20
Row
1.1
1.1
1
2
3
4
5
6
7
8
9
Column
1
16.9,16.9
31.8,15.9
44.7,14.9
55.6,13.9
64.5,12.9
71.4,11.9
76.3,10.9
79.2,9.9
80.1,8.9
2
15.9,31.8
29.8,29.8
41.7,27.8
51.6,25.8
59.5,23.8
65.4,21.8
69.3,19.8
71.2,17.8
71.1,15.8
3
14.9,44.7
27.8,41.7
38.7,38.7
47.6,35.7
54.5,32.7
59.4,29.7
62.3,26.7
63.2,23.7
62.1,20.7
4
13.9,55.6
25.8,51.6
35.7,47.6
43.6,43.6
49.5,39.6
53.4,35.6
55.3,31.6
55.2,27.6
53.1,23.6
5
12.9,64.5
23.8,59.5
32.7,54.5
39.6,49.5
44.5,44.5
47.4,39.5
48.3,34.5
47.2,29.5
44.1,24.5
6
11.9,71.4
21.8,65.4
29.7,59.4
35.6,53.4
39.5,47.4
41.4,41.4
41.3,35.4
39.2,29.4
35.1,23.4
7
10.9,76.3
19.8,69.3
26.7,62.3
31.6,55.3
34.5,48.3
35.4,41.3
34.3,34.3
31.2,27.3
26.1,20.3
8
9.9,79.2
17.8,71.2
23.7,63.2
27.6,55.2
29.5,47.2
29.4,39.2
27.3,31.2
23.2,23.2
17.1,15.2
BA 445 Lesson B.5 Simultaneous Quantity Competition
9
8.9,80.1
15.8,71.1
20.7,62.1
23.6,53.1
24.5,44.1
23.4,35.1
20.3,26.1
15.2,17.1
8.1,8.1
10
Example 1: Cournot Duopoly
Strategies 8 and 9 are now dominated for each player (by
Strategy 7). Hence, eliminate those strategies, leaving the
normal form:
20
Row
1.1
1.1
1
2
3
4
5
6
7
8
9
Column
1
16.9,16.9
31.8,15.9
44.7,14.9
55.6,13.9
64.5,12.9
71.4,11.9
76.3,10.9
79.2,9.9
80.1,8.9
2
15.9,31.8
29.8,29.8
41.7,27.8
51.6,25.8
59.5,23.8
65.4,21.8
69.3,19.8
71.2,17.8
71.1,15.8
3
14.9,44.7
27.8,41.7
38.7,38.7
47.6,35.7
54.5,32.7
59.4,29.7
62.3,26.7
63.2,23.7
62.1,20.7
4
13.9,55.6
25.8,51.6
35.7,47.6
43.6,43.6
49.5,39.6
53.4,35.6
55.3,31.6
55.2,27.6
53.1,23.6
5
12.9,64.5
23.8,59.5
32.7,54.5
39.6,49.5
44.5,44.5
47.4,39.5
48.3,34.5
47.2,29.5
44.1,24.5
6
11.9,71.4
21.8,65.4
29.7,59.4
35.6,53.4
39.5,47.4
41.4,41.4
41.3,35.4
39.2,29.4
35.1,23.4
7
10.9,76.3
19.8,69.3
26.7,62.3
31.6,55.3
34.5,48.3
35.4,41.3
34.3,34.3
31.2,27.3
26.1,20.3
8
9.9,79.2
17.8,71.2
23.7,63.2
27.6,55.2
29.5,47.2
29.4,39.2
27.3,31.2
23.2,23.2
17.1,15.2
BA 445 Lesson B.5 Simultaneous Quantity Competition
9
8.9,80.1
15.8,71.1
20.7,62.1
23.6,53.1
24.5,44.1
23.4,35.1
20.3,26.1
15.2,17.1
8.1,8.1
11
Example 1: Cournot Duopoly
Strategy 5 is now dominated for each player (by Strategy
6). Hence, eliminate that strategy, leaving the normal form:
20
Row
1.1
1.1
1
2
3
4
5
6
7
8
9
Column
1
16.9,16.9
31.8,15.9
44.7,14.9
55.6,13.9
64.5,12.9
71.4,11.9
76.3,10.9
79.2,9.9
80.1,8.9
2
15.9,31.8
29.8,29.8
41.7,27.8
51.6,25.8
59.5,23.8
65.4,21.8
69.3,19.8
71.2,17.8
71.1,15.8
3
14.9,44.7
27.8,41.7
38.7,38.7
47.6,35.7
54.5,32.7
59.4,29.7
62.3,26.7
63.2,23.7
62.1,20.7
4
13.9,55.6
25.8,51.6
35.7,47.6
43.6,43.6
49.5,39.6
53.4,35.6
55.3,31.6
55.2,27.6
53.1,23.6
5
12.9,64.5
23.8,59.5
32.7,54.5
39.6,49.5
44.5,44.5
47.4,39.5
48.3,34.5
47.2,29.5
44.1,24.5
6
11.9,71.4
21.8,65.4
29.7,59.4
35.6,53.4
39.5,47.4
41.4,41.4
41.3,35.4
39.2,29.4
35.1,23.4
7
10.9,76.3
19.8,69.3
26.7,62.3
31.6,55.3
34.5,48.3
35.4,41.3
34.3,34.3
31.2,27.3
26.1,20.3
8
9.9,79.2
17.8,71.2
23.7,63.2
27.6,55.2
29.5,47.2
29.4,39.2
27.3,31.2
23.2,23.2
17.1,15.2
BA 445 Lesson B.5 Simultaneous Quantity Competition
9
8.9,80.1
15.8,71.1
20.7,62.1
23.6,53.1
24.5,44.1
23.4,35.1
20.3,26.1
15.2,17.1
8.1,8.1
12
Example 1: Cournot Duopoly
Strategy 7 is now dominated for each player (by Strategy
6). Hence, eliminate that strategy, leaving only strategies
QI = 6 and QA = 6, and profits PI = 41.4 and PA = 41.4. As
in any game, under game theory assumptions (including
rationality), it is always best to play your strategy that is
part of a dominance solution.
20
Row
1.1
1.1
1
2
3
4
5
6
7
8
9
Column
1
16.9,16.9
31.8,15.9
44.7,14.9
55.6,13.9
64.5,12.9
71.4,11.9
76.3,10.9
79.2,9.9
80.1,8.9
2
15.9,31.8
29.8,29.8
41.7,27.8
51.6,25.8
59.5,23.8
65.4,21.8
69.3,19.8
71.2,17.8
71.1,15.8
3
14.9,44.7
27.8,41.7
38.7,38.7
47.6,35.7
54.5,32.7
59.4,29.7
62.3,26.7
63.2,23.7
62.1,20.7
4
13.9,55.6
25.8,51.6
35.7,47.6
43.6,43.6
49.5,39.6
53.4,35.6
55.3,31.6
55.2,27.6
53.1,23.6
5
12.9,64.5
23.8,59.5
32.7,54.5
39.6,49.5
44.5,44.5
47.4,39.5
48.3,34.5
47.2,29.5
44.1,24.5
6
11.9,71.4
21.8,65.4
29.7,59.4
35.6,53.4
39.5,47.4
41.4,41.4
41.3,35.4
39.2,29.4
35.1,23.4
7
10.9,76.3
19.8,69.3
26.7,62.3
31.6,55.3
34.5,48.3
35.4,41.3
34.3,34.3
31.2,27.3
26.1,20.3
8
9.9,79.2
17.8,71.2
23.7,63.2
27.6,55.2
29.5,47.2
29.4,39.2
27.3,31.2
23.2,23.2
17.1,15.2
BA 445 Lesson B.5 Simultaneous Quantity Competition
9
8.9,80.1
15.8,71.1
20.7,62.1
23.6,53.1
24.5,44.1
23.4,35.1
20.3,26.1
15.2,17.1
8.1,8.1
13
Example 1: Cournot Duopoly
Comment: The dominance solution QI = 6 and QA = 6, with
profits PI = 41.4 and PA = 41.4, is the only Nash
Equilibrium. A Nash Equilibrium means QI = 6 is Intel’s best
response to QA = 6, and QA = 6 is AMD’s best response to
QI = 6.
20
Row
1.1
1.1
1
2
3
4
5
6
7
8
9
Column
1
16.9,16.9
31.8,15.9
44.7,14.9
55.6,13.9
64.5,12.9
71.4,11.9
76.3,10.9
79.2,9.9
80.1,8.9
2
15.9,31.8
29.8,29.8
41.7,27.8
51.6,25.8
59.5,23.8
65.4,21.8
69.3,19.8
71.2,17.8
71.1,15.8
3
14.9,44.7
27.8,41.7
38.7,38.7
47.6,35.7
54.5,32.7
59.4,29.7
62.3,26.7
63.2,23.7
62.1,20.7
4
13.9,55.6
25.8,51.6
35.7,47.6
43.6,43.6
49.5,39.6
53.4,35.6
55.3,31.6
55.2,27.6
53.1,23.6
5
12.9,64.5
23.8,59.5
32.7,54.5
39.6,49.5
44.5,44.5
47.4,39.5
48.3,34.5
47.2,29.5
44.1,24.5
6
11.9,71.4
21.8,65.4
29.7,59.4
35.6,53.4
39.5,47.4
41.4,41.4
41.3,35.4
39.2,29.4
35.1,23.4
7
10.9,76.3
19.8,69.3
26.7,62.3
31.6,55.3
34.5,48.3
35.4,41.3
34.3,34.3
31.2,27.3
26.1,20.3
8
9.9,79.2
17.8,71.2
23.7,63.2
27.6,55.2
29.5,47.2
29.4,39.2
27.3,31.2
23.2,23.2
17.1,15.2
BA 445 Lesson B.5 Simultaneous Quantity Competition
9
8.9,80.1
15.8,71.1
20.7,62.1
23.6,53.1
24.5,44.1
23.4,35.1
20.3,26.1
15.2,17.1
8.1,8.1
14
Example 2: Nash Equilibrium
Example 2: Nash Equilibrium
BA 445 Lesson B.5 Simultaneous Quantity Competition
15
Example 2: Nash Equilibrium
Overview
Nash Equilibrium means each player makes a best
response to the strategies of other players. It is thus a selfenforcing agreement. And it is the same as the
dominance solution of a Cournot Duopoly.
BA 445 Lesson B.5 Simultaneous Quantity Competition
16
Example 2: Nash Equilibrium
Comment: Although Cournot Duopoly Games have dominance
solutions even when quantities can be continuous variables (including
fractions), it is hard go through the entire sequence of reasoning like in
Example 1. It turns out, however, that the unique dominance solution
of a Cournot Duopoly Game is also the unique Nash Equilibrium of the
Game. And finding a Nash Equilibrium is relatively simple.
A Nash Equilibrium of any game with two or more players means each
player is assumed to know the chosen strategies of the other players,
and each player chooses a best response to those chosen strategies -- that is, no player has anything to gain by changing only his or her own
strategy unilaterally.
BA 445 Lesson B.5 Simultaneous Quantity Competition
17
Example 2: Nash Equilibrium
Question: Coke and Pepsi control a large share of the soft
drink market. Consumers find the two products to be
indistinguishable. The inverse market demand for soft
drinks is P = 3-Q (in dollars). You are a manager of Pepsi.
Your unit cost of production is \$2, and the unit cost of Coke
is \$1. Suppose you choose your output of soft drinks a few
hours before Coke but Coke does not know your output
before they decide their own output.
How many soft drinks should you produce?
BA 445 Lesson B.5 Simultaneous Quantity Competition
18
Example 2: Nash Equilibrium
Answer: You are Firm 1 in a Cournot Duopoly Game with
inverse demand P = 3 - (Q1+Q2) and marginal costs c1 =
MC1 = 2 and c2 = MC2 = 1.
Find the Nash Equilibrium to the Cournot Duopoly Game
(which turns out to be the dominance solution).
BA 445 Lesson B.5 Simultaneous Quantity Competition
19
Example 2: Nash Equilibrium
Each firm correctly deduces the other firm’s output, so each
firm chooses it’s output as a best response to the other
firm’s output.
Given Q1, Firm 2 computes revenue and marginal revenue
R2 = (3 – (Q1 + Q2)) Q2
MR2 = dR2 /dQ2 = 3 – Q1 – 2Q2
Hence, equate marginal cost to marginal revenue
1 = MC2 = MR2 = 3 – Q1 – 2Q2
to determine the optimal reaction
Q2 = r2 (Q1) = 1 – .5Q1
BA 445 Lesson B.5 Simultaneous Quantity Competition
20
Example 2: Nash Equilibrium
Given Q2, Firm 1 computes revenue and marginal revenue
R1 = (3 – (Q1 + Q2)) Q1
MR1 = dR1 /dQ1 = 3 – 2Q1 – Q2
Hence, equate marginal cost to marginal revenue
2 = MC1 = MR1 = 3 – 2Q1 – Q2
to determine the optimal reaction
Q1 = r1 (Q2) = .5– .5Q2
BA 445 Lesson B.5 Simultaneous Quantity Competition
21
Example 2: Nash Equilibrium
Complete solution for P = 3 - (Q1+Q2), MC1 = 2, MC2 = 1.
• Solve Q2 = 1 – .5Q1 and Q1 = .5– .5Q2 for Q1 = 0 and Q2
=1
• P = 3 - (Q1+Q2) = 2
• Firm 1 profit P1 = (P - c1) Q1 = (2 - 2)0 = 0
• Firm 2 profit P2 = (P - c2) Q2 = (2 - 1)1 = 1
BA 445 Lesson B.5 Simultaneous Quantity Competition
22
Example 2: Nash Equilibrium
Comment: Given any inverse demand
P = a - b(Q1+Q2)
Firm 1’s revenue and marginal revenue
R1 = (a – b(Q1 + Q2)) Q1
MR1 = dR1 /dQ1 = a – 2bQ1 – bQ2
That is, MR1 is the inverse demand P = a - bQ1 - bQ2 with
double the coefficient of Q1
Firm 2’s revenue and marginal revenue
R2 = (a – b(Q1 + Q2)) Q2
MR2 = dR2 /dQ2 = a – bQ1 – 2bQ2
That is, MR2 is the inverse demand P = a - bQ1 - bQ2 with
double the coefficient of Q2
BA 445 Lesson B.5 Simultaneous Quantity Competition
23
BA 445 Lesson B.5 Simultaneous Quantity Competition
24
Overview
First Mover Advantage always occurs in a Stackelberg or
Cournot duopoly. That advantage can make it profitable to
rush to choose output sooner, even if that rush raises
costs.
BA 445 Lesson B.5 Simultaneous Quantity Competition
25
Comment: If the unit production costs are the same c for
two firms in a duopoly with inverse demand P = a –
b(Q1+Q2), then profits are
• Pi = (a – c)2/(9b) if Firm i is a Cournot competitor
• P1 = (a – c)2/(8b) if Firm 1 is a Stackelberg leader
• P2 = (a – c)2/(16b) if Firm 2 is a Stackelberg follower
So the Stackelberg leader has more profit than a Cournot
competitor, who in turn has more profit than a Stackelberg
follower.
In particular, a firm can find it profitable to become the first
mover or avoid being a follower by rushing to set up an
assembly line, even if it means increasing marginal costs of
production.
BA 445 Lesson B.5 Simultaneous Quantity Competition
26
Question: PetroChina and Sinopec control a large share of Chinese oil
production. The inverse market demand for Chinese oil is P = 3-Q (in
yuan) and both firms produce at a unit cost of 1 yuan. You are a
manager of PetroChina, and have a decision to make about competing
with Sinopec in Siberia, where the inverse market demand for Chinese
oil is P = 3-Q (in rubles).
Option A. Sinopec sets up its refineries and distribution networks now,
and you set up later. And both produce at a unit cost of 1 ruble.
Option B. You hurry set up your refineries and distribution networks at
the same time as Sinopec. Your hurry means your unit costs are 1.1
rubles, while Sinopec’s unit costs remain 1.
Which Option is better for PetroChina?
BA 445 Lesson B.5 Simultaneous Quantity Competition
27
Answer: In Option A, you are the follower in a Stackelberg
Duopoly with inverse demand P = 3 - (Q1+Q2) and
marginal costs c1 = MC1 = 1 and c2 = MC2 = 1. In Option B,
you are Firm 1 in a Cournot Duopoly with inverse demand
P = 3 - (Q1+Q2) and marginal costs c1 = MC1 = 1.1 and c2 =
MC2 = 1.
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Option A: Starting from the end, given Q1, Firm 2 computes
revenue and marginal revenue
R2 = (3 – (Q1 + Q2)) Q2
MR2 = dR2 /dQ2 = 3 – Q1 – 2Q2
Hence, equate marginal cost to marginal revenue
1 = MC2 = MR2 = 3 – Q1 – 2Q2
to determine the optimal reaction
Q2 = r2 (Q1) = 1 – .5Q1
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Rolling back to the beginning,
• The Stackelberg leader uses the reaction function r2 (Q1)
to determine its revenue
 R1 = (3 – Q1 – r2 (Q1) )) Q1
 R1 = (3 – Q1 – (1 – .5Q1)) Q1
 R1 = (2 – .5Q1) Q1
and its profit-maximizing output level:




1 = c1 = dR1/dQ1
1 = d/dQ1 (2 – .5Q1) Q1
1 = 2 – Q1
Q1 = 1
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Complete Stackelberg solution for c1 = MC1 = 1 and c2 =
MC2 = 1:
• Q1 = 1
• Q2 = r2 (Q1) = 1 – .5Q1 = 1 – .5(1) = .5
• P = 3 - (Q1+Q2) = 1.5
• Firm 1 profit P1 = (P - c1) Q1 = (1.5 - 1)1 = 0.5
• Firm 2 profit P2 = (P - c2) Q2 = (1.5 - 1).5 = 0.25
BA 445 Lesson B.5 Simultaneous Quantity Competition
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In Option B, you are Firm 1 in a Cournot Duopoly with
inverse demand P = 3 - (Q1+Q2) and marginal costs c1 =
MC1 = 1.1 and c2 = MC2 = 1.
Find the Nash Equilibrium to the Cournot Duopoly Game
(which turns out to be the dominance solution).
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Each firm correctly deduces the other firm’s output, so each
firm chooses it’s output as a best response to the other
firm’s output.
Given Q1, Firm 2 computes revenue and marginal revenue
R2 = (3 – (Q1 + Q2)) Q2
MR2 = dR2 /dQ2 = 3 – Q1 – 2Q2
Hence, equate marginal cost to marginal revenue
1 = MC2 = MR2 = 3 – Q1 – 2Q2
to determine the optimal reaction
Q2 = r2 (Q1) = 1 – .5Q1
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Given Q2, Firm 1 computes revenue and marginal revenue
R1 = (3 – (Q1 + Q2)) Q1
MR1 = dR1 /dQ1 = 3 – 2Q1 – Q2
Hence, equate marginal cost to marginal revenue
1.1 = MC1 = MR1 = 3 – 2Q1 – Q2
to determine the optimal reaction
Q1 = r1 (Q2) = .95– .5Q2
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Complete solution for P = 3 - (Q1+Q2), MC1 = 1.1, MC2 = 1.
• Solve Q2 = 1 – .5Q1 and Q1 = .95– .5Q2 for Q1 = .6 and
Q2 = .7
• P = 3 - (Q1+Q2) = 1.7
• Firm 1 profit P1 = (P - c1) Q1 = (1.7 - 1.1).6 = 0.36
• Firm 2 profit P2 = (P - c2) Q2 = (1.7 - 1).7 = 0.49
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Option B is thus best for PetroChina since PetroChina
profits (as a Stackelberg follower) are 0.25 in Option A,
while PetroChina profits (as a Cournot Duopolist) are 0.36
in Option B.
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Comment: In this particular case, PetroChina increased
production cost hurt profits less than profits increase
because of eliminating the second mover disadvantage. In
other problems, increased production cost hurt profits more
than profits increase because of eliminating the second
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Example 4: Selling Technology
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Overview
Selling Technology to a Cournot competitor is profitable if
total profit increases. In that case, there is a positive gain
from the sales agreement, which is then divided according
to rules of bargaining.
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Question: Nvidia and the ATI subsidiary of Advanced Micro
Devices control a large share of the mainstream graphics
card market. You are a manager of Nvidia, and you and
ATI both expect to produce the next generation of graphics
card in October of next year. Your graphics cards and ATI’s
graphics cards are indistinguishable to consumers. The
inverse market demand for graphics cards 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 produce graphics
cards, which reduces your unit cost to \$1. Should you
keep that procedure to yourself? Or is it better to sell that
secret to ATI so that both you and ATI can produce at unit
cost equal to \$1?
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Answer: If you do not sell your technology, you are Firm 1
in a Cournot Duopoly with inverse demand P = 4 - (Q1+Q2)
and marginal costs are c1 = MC1 = 1 and c2 = MC2 = 2; if
you do sell, marginal costs are c1 = MC1 = 1 and c2 = MC2
= 1.
Find the Nash Equilibrium to each Cournot Duopoly Game
(which turns out to be the dominance solution).
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
If Nvidia does not sell its technology, given Q1, Firm 2
computes revenue and marginal revenue
R2 = (4 – (Q1 + Q2)) Q2
MR2 = dR2 /dQ2 = 4 – Q1 – 2Q2
Hence, equate marginal cost to marginal revenue
2 = MC2 = MR2 = 4 – Q1 – 2Q2
to determine the optimal reaction
Q2 = r2 (Q1) = 1 – .5Q1
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Given Q2, Firm 1 computes revenue and marginal revenue
R1 = (4 – (Q1 + Q2)) Q1
MR1 = dR1 /dQ1 = 4 – 2Q1 – Q2
Hence, equate marginal cost to marginal revenue
1 = MC1 = MR1 = 4 – 2Q1 – Q2
to determine the optimal reaction
Q1 = r1 (Q2) = 1.5– .5Q2
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Complete solution for P = 4 - (Q1+Q2), MC1 = 1, MC2 = 2.
• Solve Q2 = 1 – .5Q1 and Q1 = 1.5– .5Q2 for Q1 = 1 1/3
and Q2 = 1/3
• P = 4 - (Q1+Q2) = 2 1/3
• Firm 1 profit P1 = (P - c1) Q1 = (2 1/3 - 1)(1 1/3)= 1 7/9
• Firm 2 profit P2 = (P - c2) Q2 = (2 1/3 - 2)(1/3) = 1/9
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
If Nvidia does sell its technology, given Q1, Firm 2
computes revenue and marginal revenue
R2 = (4 – (Q1 + Q2)) Q2
MR2 = dR2 /dQ2 = 4 – Q1 – 2Q2
Hence, equate marginal cost to marginal revenue
1 = MC2 = MR2 = 4 – Q1 – 2Q2
to determine the optimal reaction
Q2 = r2 (Q1) = 1.5 – .5Q1
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Given Q2, Firm 1 computes revenue and marginal revenue
R1 = (4 – (Q1 + Q2)) Q1
MR1 = dR1 /dQ1 = 4 – 2Q1 – Q2
Hence, equate marginal cost to marginal revenue
1 = MC1 = MR1 = 4 – 2Q1 – Q2
to determine the optimal reaction
Q1 = r1 (Q2) = 1.5– .5Q2
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Complete solution for P = 4 - (Q1+Q2), MC1 = 1, MC2 = 1.
• Solve Q2 = 1.5 – .5Q1 and Q1 = 1.5– .5Q2 for Q1 = 1 and
Q2 = 1
• P = 4 - (Q1+Q2) = 2
• Firm 1 profit P1 = (P - c1) Q1 = (2 - 1)1 = 1
• Firm 2 profit P2 = (P - c2) Q2 = (2 - 1)1 = 1
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 4: Selling Technology
Selling technology and reducing c2 = 2 to c2 = 1 has to
effects:
• Firm 1’s profit reduces from P1 = 1 7/9 to P1 = 1
• Firm 2’s profit increases from P2 = 1/9 to P2 = 1
Nvidia should sell the technology because doing so
increases total profit from production from 1 8/9 to 2, so
there is 1/9 gains from trade to be divided between the two
firms according to the rules of the resulting bargaining
game. For example, if Nvidia can make a credible take-itor-leave-it offer of 1/9 minus a pittance to ATI, then Nvidia
captures most of those gains.
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 5: Colluding
Example 5: Colluding
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 5: Colluding
Overview
Colluding with a Cournot competitor is almost always
profitable. Since the competitors produce gross
substitutes, profitable collusion lowers output. But, each
cannot trust the other to collude.
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 5: Colluding
Question: TV Azteca and Televisa control a large share of the Mexican
multimedia market. As a manager of TV Azteca, you choose the
(Lo que callamos las mujeres, Ventaneando, Hechos, Venga la Alegria,
…) to air 1 hour before your competitor, but Televisa does not have any
demand for broadcast hours is Q = 13 - P; TV Azteca’s costs are
C1(Q1) = Q1; and Televisa’s costs are C2(Q2) = Q2.
Would it be mutually profitable for the companies to collude by setting
TV Azteca’s and Televisa’s outputs to 3 and 3. Can TV Azteca trust
Televisa to collude? Can Televisa trust TV Azteca to collude?
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 5: Colluding
Answer: You are Firm 1 in a Cournot Duopoly with demand
Q = 13 - P, inverse demand P = 13- (Q1+Q2), C1 (Q1) = Q1
and C2 (Q2) = Q2, and marginal costs c1 = MC1 = 1 and c2 =
MC2 = 1.
Find the Nash Equilibrium to the Cournot Duopoly Game
(which turns out to be the dominance solution), and
compare the Nash Equilibrium to the collusive proposal of
quantities 3 and 3.
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 5: Colluding
Given Q1, Firm 2 computes revenue and marginal revenue
R2 = (13 – (Q1 + Q2)) Q2
MR2 = dR2 /dQ2 = 13 – Q1 – 2Q2
Hence, equate marginal cost to marginal revenue
1 = MC2 = MR2 = 13 – Q1 – 2Q2
to determine the optimal reaction
Q2 = r2 (Q1) = 6 – .5Q1
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 5: Colluding
Given Q2, Firm 1 computes revenue and marginal revenue
R1 = (13 – (Q1 + Q2)) Q1
MR1 = dR1 /dQ1 = 13 – 2Q1 – Q2
Hence, equate marginal cost to marginal revenue
1 = MC1 = MR1 = 13 – 2Q1 – Q2
to determine the optimal reaction
Q1 = r1 (Q2) = 6 – .5Q2
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 5: Colluding
Complete Nash equilibrium for non-colluding firms with P =
13 - (Q1+Q2), MC1 = 1, MC2 = 1:
• Solve Q2 = 6 – .5Q1 and Q1 = 6 – .5Q2 for Q1 = 4 and Q2
=4
• P = 13 - (Q1+Q2) = 5
• Firm 1 profit P1 = (P - c1) Q1 = (5 - 1)4 = 16
• Firm 2 profit P2 = (P - c2) Q2 = (5 - 1)4 = 16
Collusive proposal of quantities Q1 = 3 and Q2 = 3:
• P = 13 - (Q1+Q2) = 7
• Firm 1 profit P1 = (P - c1) Q1 = (7 - 1)3 = 18
• Firm 2 profit P2 = (P - c2) Q2 = (7 - 1)3 = 18
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Example 5: Colluding
The collusive proposal of quantities Q1 = 3 and Q2 = 3 is
thus mutually profitable for TV Azteca’s and Televisa. But
TV Azteca cannot trust Televisa to collude since Televisa’s
best response to TV Azteca’s Q1 = 3 is Q2 = r2 (3) = 6 –
.5(3) = 4.5, not the collusive proposal Q2 = 3.
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Summary
Summary
BA 445 Lesson B.5 Simultaneous Quantity Competition
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Summary
Complete solution to a Cournot Duopoly Game with inverse demand P
= a - bQ and constant marginal costs c1 = MC1 and c2 = MC2:
• Reaction Q1 = r1 (Q2) = (a - c1)/2b – .5Q2
• Reaction Q2 = r2 (Q1) = (a - c2)/2b – .5Q1
• Solution Q1 = 2(a - c1)/3b – (a - c2)/3b
• Solution Q2 = 2(a - c2)/3b – (a - c1)/3b
• P = a - b(Q1+Q2)
• Firm 1 profit P1 = (P - c1) Q1
• Firm 2 profit P2 = (P - c2) Q2
Tip: Use those formulas to double check your computations. However,
computations as in the answers to Examples 1 through 5 are required
for full credit on exam and homework questions.
BA 445 Lesson B.5 Simultaneous Quantity Competition
58
Review Questions
Review Questions
 You should try to answer some of the review
questions (see the online syllabus) before the next
class.
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