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## Oligopoly Theory (2) Quantity-Setting Competition

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Title: Oligopoly Theory (2) Quantity-Setting Competition

1
Oligopoly Theory (2) Quantity-Setting Competition
Aim of this lecture (1) To understand the concept
of quantity-setting competition (2) To
understand the ideas of strategic substitutes and
complements (3) To understand the relationship
between the stability of Cournot equilibrium and
comparative statistics
2
Outline of the Second Lecture
2-1 Monopoly 2-2 Price-Setting or
Quantity-Setting 2-3 Cournot Model 2-4 Strategic
Complement and Strategic Substitute 2-5 Stability
Condition 2-6 Stability Condition and
Comparative Statistics 2-7 Stability Condition
and Uniqueness of the Equilibrium 2-8 Cournot
Limit Theorem and Perfect Competition
3
Monopoly Producer
P
D
MC
MR
0
Y
4
Equilibrium of Monopoly Producer
the superscript M denotes equilibrium values of
monopoly
P
D
PM
MC
MR
0
Y
YM
5
Marshallian View of the MarketQuantity?Price
Monopolist chooses its output and the price is
determined by the market clearing condition
P
D
PM
MC
MR
0
Y
YM
6
Walrasian View of the MarketPrice?Quantity
Monopolist chooses its output and the price is
determined by the market clearing condition.
P
D
PM
MC
MR
0
Y
YM
7
Why does not the monopolist choose both quantity
and price?
MR
P
D
PM
MC
0
Y
YM1
8
Why does not the monopolist choose both quantity
and price ?
MR
P
D
MC
PM1
0
Y
YM
9
Duopoly
Suppose that there are two or more firms in the
market The price depends on both its own output
and the rivals' outputs. The output depends on
both its own price and the rivals' prices. ?The
competition structure depends on whether firms
choose their outputs or prices. Quantity
Competition Model (The second lecture) Price
Competition Model (The third lecture) Which model
should we use?(The third lecture)
10
Cournot Duopoly
Firm 1 and firm 2 compete in a homogeneous
product market (product differentiation is fully
discussed in the 8th lecture and is also
discussed briefly in the third lecture). Each
firm i independently chooses its output Yi ?0,
8). Each firm maximizes its own profit ?i. ?i
P(Y)Yi ? Ci(Yi), P Inverse demand function, Y
Total output, Yi Firm i's output, Ci Firm i's
cost function P' lt 0, C' gt 0, C'' ? 0
(Henceforth, I assume these unless I explicitly
11
reaction function
Reaction function of firm 1R1(Y2)Given the
output of firm 2,Y2, Y1R1(Y2) implies that Y1 is
the optimal (profit-maximizing) output for firm
1. The first order condition PP'Y1 C1' ?
R1(Y2) is derived from this first order
condition. The second order condition 2P'P''Y1
- C1''lt0 Henceforth, I assume the second order
conditions unless I explicitly make contradicting
assumptions.
12
Cournot Equilibrium
Nash Equilibrium of the Cournot Model Cournot
Equilibrium Derivation of the Cournot
Equilibrium Solving PP'Y1 C1' , PP'Y2 C2'
13
Residual Demand
P
residual demand
D
MR
MC
Y2
0
Y
Y1
14
Derivation of reaction function
P
residual demand
D
MR
MC
Y2
0
Y
Y1
15
reaction curve of firm 1
Y1
reaction curve of firm 1
0
Y2
16
Question(1)Reaction function
Suppose that the inverse demand function is given
by PA -Y. Suppose that firm 1s marginal cost
c1 (ltA) is constant. Suppose that firm 1s payoff
is its profit. Derive the reaction function of
firm 1.
17
Question(2)reaction function
Suppose that the inverse demand function is given
by PA -Y. Suppose that firm 1s Cost C1
isY12/2 . Suppose that firm 1s payoff is its
profit. Derive the reaction function of firm 1.
18
Question(3)Reaction function
Suppose that the inverse demand function is given
by PA -Y. Suppose that firm 1s marginal cost
c1 is constant. Suppose that firm 1s payoff is
its revenue. Derive the reaction function of firm
1.
19
slope of the reaction curve
PP'Y1 -C1'0?dY1/dY2 - (P'P''Y1)/(2P'P''Y1 -
C1'') P'P''Y1gt0?upward sloping of the reaction
curve (strategic complements) an increase in the
rival's output increases the marginal revenue of
the firm unnatural in the context of Cournot
competition, but it is possible. P'P''Y1lt0?downwa
rd sloping of the reaction curve (strategic
substitutes) an increase in the rival's output
reduces the marginal revenue of the firm In this
course, I assume P'P''Y1lt0 unless I make
20
Question strategic substitutes, complements
Suppose that the inverse demand function is given
by PA -Y. Suppose that firm is marginal cost
ci (ltA)is constant. Suppose that firm is payoff
is its profit (i1,2). Strategies are (strategic
substitutes, strategic complements).
21
Reaction Curve of Firm 2
Y2
0
Y1
22
Reaction Curve of Firm 1
Y1
0
Y2
23
Cournot Equilibrium
reaction curve of firm 1
Y2
reaction curve of firm 2
Y2C
0
Y1
Y1C
the superscript C denotes Cournot Equilibrium
24
strategic complements case
The reaction curve of firm 1
Cournot equilibrium
Y2
The reaction curve of firm 2
Y2C
It is not natural in the context of
quantity-setting competition, but it is possible
if P'' is large
0
Y1
Y1C
25
Existence of the Equilibrium
From the definitions of the reaction function and
the Cournot equilibrium, we have R1(R2(Y1C))
Y1C , R2(R1(Y2C)) Y2C We can use the fixed
point theorem to show the existence of the
Cournot equilibrium. There exists an equilibrium
under moderate condition, either under strategic
substitutes or complements. A key property is
continuity of the reaction function.
26
Non-existence of the pure strategy equilibrium
reaction curve of firm 1
Y2
reaction curve of firm 2
0
Y1
27
Non-existence of the pure strategy equilibrium
The reaction curve of firm 1
Y2
The reaction curve of firm 2
0
Y1
28
Non-existence of the pure strategy equilibrium?
Y2
reaction curve of firm 2
reaction curve of firm 1
0
Y1
29
Non-existence of the pure strategy equilibrium?
Y2
Cournot equilibrium
reaction curve of firm 2
reaction curve of firm 1
0
Y1
30
Existence of the Equilibrium
R1(R2(Y1(1))) Y1(2) , R2(R1(Y2(1))) Y2(2)
Substituting Yi(1)YiC into the above system
yields Yi(2)YiC. What happens if we
substitute Yi(1)?YiC into the above system? ?The
discussions on the stability and the uniqueness
of the equilibrium.
31
Stability
R1(R2(Y1))-Y1C lt Y1 -Y1C R2(R1(Y2))-Y2C lt
Y2 -Y2C Starting from the non-equilibrium
point and consider the best reply dynamics ?The
distance from the equilibrium point is
decreasing ?Cournot equilibrium is stable.
32
Stable Cournot Equilibrium
Reaction Curve of Firm 1
Y2
Reaction Curve of Firm 2
R2(Y1(1))
Y2C
0
Y1
Y1(2)
Y1(1)
Y1C
33
Unstable Cournot Equilibrium
the reaction curve of firm 2
Y2
the reaction curve of firm 1
R2(Y1(1))
Y2C
0
Y1
Y1(1)
Y1(2)
Y1C
34
A sufficient condition for the stability of the
Cournot equilibrium
Ri' lt 1 The absolute value of the reaction
curve is smaller than one. One unit increase
of the rival's output changes the optimal output
of the firm less than one unit. e.g., R1' lt
1 Frim2's output is 10?Firm 1's optimal output is
5 Frim2's output is 5?Firm 1's optimal output is
7 e.g., R1' gt 1 Frim2's output is 10?Firm 1's
optimal output is 5 Frim2's output is 5?Firm 1's
optimal output is 11
35
Why do we often assume the stability condition?
Cournot Model is a One-Shot Game. It seems
nonsense to discuss the dynamic adjustment.
However, most IO papers assume this condition.
Why? (1) This condition is plausible since it is
satisfied under standard settings of cost and
demand conditions. (2) evolution, learning (3)
for comparative statistics (4) uniqueness
36
Stability condition is satisfied under moderate
conditions
(1) Ri' lt 1 is satisfied under the assumptions
of strategic substitutes, non-decreasing marginal
cost, and decreasing demand function. From the
first order condition PP'Y1 -C1'0, we have R1'
dY1/dY2 - (P'P''Y1)/(2P'P''Y1 -
C1'') strategic substitutes ( P'P''Y1lt0) ,
C1''?0, P' lt0 ?-1 ltR1'lt0 It is quite natural to
assume the stability condition.
37
The stability condition and comparative statistics
(3)So as to obtain clear results of comparative
statistics, we usually assume the stability
condition. Without the stability condition, the
results of comparative statistics often become
ambiguous. It is nonsense to derive
counter-intuitive results under the assumption
Ri' gt 1, which is not satisfied under plausible
cost and demand conditions.
38
QuestionSuppose that firm 2's MC is constant.
Consider the effect of a reduction of firm 2's
marginal cost on firm 2's reaction curve.
The reaction curve of firm 2 before the change of
the cost
A
Y2
B
0
Y1
39
The relationship between firm 2's cost and its
reaction curve
upward shift of the reaction curve of firm 2
Y2
0
Y1
40
The relationship between firm 2's cost and the
Cournot equilibrium
A decrease in the firm 2's marginal cost raises
firm 2's output and reduces firm 1's output
through strategic interaction.
Y2
firm 1's reaction curve
0
Y1
41
Unstable Cournot Equilibrium
A decrease in the firm 2's marginal cost reduces
firm 2's output and raises firm 1's output
through strategic interaction?a counter-intuitive
and nonsense result
Y2
firm 1's reaction curve
0
Y1
42
Caution
When you write a theoretical paper and face a
counter-intuitive result, you should check
whether or not the problem you formulate
satisfies the stability condition. If not, the
result is not a surprising result and most
referees may think that it is obvious.
43
The uniqueness of the equilibrium and the
stability condition
(4) If the stability condition is satisfied
globally, the equilibrium is unique (only one
equilibrium exists). We can show it by using
Contraction Mapping Theorem. (Remark) The
stability condition is sufficient, but not
necessarily condition for the uniqueness of the
equilibrium.
44
Stable Cournot Equilibrium
Y2
reaction curve of firm 1
reaction curve of firm 2
Y2C
0
Y1
Y1C
45
Does unstable case also yield the unique Cournot
equilibrium?
The reaction curve of firm 2
Y2
The reaction curve of firm 1
Y2C
0
Y1
Y1C
46
Unstable Cournot Equilibrium
Reaction Curve of Firm 2
Y2
Three equilibria exist.
Reaction Curve of Firm 1
Y2C
0
Y1
Y1C
47
Stable Cournot Equilibrium
Y2
reaction curve of firm 1
reaction curve of firm 2
Y2C
0
Y1
Y1C
48
QuestionAmong three points A, B, and C, ? yields
the largest profit of firm 2.
Y2
A
? is A, B, or C?
B
The reaction curve of firm 2
C
0
Y1
49
Question Which yields larger profit of firm 2, A
or B?
Y2
The reaction curve of firm 2
A
B
0
Y1
50
Firm 2's iso-profit curve profit
iso-profit curve of firm 2
A
All points on the iso-profit curve yield the same
profit of firm 2.
Y2
B
C
0
Y1
51
Cournot Equilibrium and Efficiency
the reaction curve of firm 1
Y2
iso-profit curve of firm 2
iso-profit curve of firm 1
Y2C
the reaction curve of firm 2
0
Y1
Y1C
Moving from the Cournot equilibrium to this point
improves both firms' payoffs (profits)
52
Welfare Implications
Each firm maximizes its own profit with respect
to its output, without considering the negative
effect on the rival. ?The output at the Cournot
equilibrium is excessive from the viewpoint of
total profits maximization. However, at the
Cournot equilibrium, PP'Y1 -C1'0, so P -C1' gt0.
?The output at the Cournot equilibrium is
insufficient from the viewpoint of total social
surplus maximization (total social surplus
maximization is achieved when P C1' C2' ).
53
Cournot Oligopoly
Firm 1, firm 2, ..., firm n compete in a
homogeneous product market. Each firm i
independently chooses its output Yi ?0, 8).
Each firm maximizes its own profit
?i. ?iP(Y)Yi?Ci(Yi), P Inverse demand function,
Y Total output, Yi Firm i's output, Ci Firm
i's cost function P' lt0, C' gt0, C'' ?0 Exactly
the same model except for the number of the firms
54
Cournot Equilibrium
Derivation of the Cournot equilibrium Solving
the system of equations PP'Y1 C1' , PP'Y2
C2',... PP'Yn Cn' If firms are symmetric
(all firm have the same cost function), the
symmetric equilibrium is derived from PP'Y1
C1' , YnY1 (or equivalently Y-1(n-1)Y1 where
Y-1 Sj ?1 Yj , total output of the rivals)
55
Symmetric Equilibrium
Y-1 Sj ?1 Yj , total output of the rivals
Y-1
the reaction curve of firm 1
n-1
0
Y1
Y1C
56
Cournot Limit Theorem
The first order condition for firm 1 PP'Y1
C1' P(1P' Y/P Y1/Y)C1' P(1-?-1Y1/Y)C1' (?
price elasticity of the demand) ??8 P ? C1' (the
world of price taker) Y1/Y?0 P ? C1' (the world
of Cournot Limit theorem) Cournot Limit Theorem
If the number of firms is sufficiently large (if
the market share of each firm is sufficiently
small), the price is sufficiently close to the
marginal cost of each firm.
57
Marginal Revenue for Small Firms
P
MR ?P if Y1 is sufficiently small MRPP' Y1
residual demand
MR
0
Y1
58
perfect competition
Price Taker The player who chooses his/her
behavior given the price exogenously. In the
context of quantity-setting competition, the firm
is a price taker if it thinks that the price
remains unchanged even if it increases the
output. In fact, unless the price elasticity of
the demand is infinity, an increase in the output
of each player reduces the price, whether the
player is small or large. The explanation that a
firm is a price taker when its size is too small
to affect the price seems ridiculous.
59
Micro Foundation of Perfect Competition
In the Cournot model, all firms are price makers
(they recognize that P'lt0). However, if the
number of the firms is sufficiently large, the
equilibrium price is approximately equal to the
perfectly competitive equilibrium price. Perfect
competition equilibrium ?Cournot equilibrium in
the large economy Perfect competition model is
an approximation of the real world when the
number of firms is sufficiently large.
60
Perfect Competition and Oligopoly
In the course, I will present 4 stories for micro
foundation of perfect competition. (1) Cournot
Limit Theorem (2nd lecture) (2) Bertrand
Competition (3rd lecture) (3) Relative
Performance Approach and Evolutionary Approach
(4th lecture) (4) Strategic Commitment Approach
(7th lecture)
61
Exercise (1)
(1) Consider a Cournot duopoly. The demand is
given by PA-Y, where A is a positive constant.
The marginal cost of each firm is c, where c is a
positive constant and Agtc. (a) Derive the
reaction function of firm1. (b) Derive R1'.
Make sure that it has downward sloping (strategic
substitutes) and that the stability condition is
satisfied. (c) Derive the output of firm 1 at
the Cournot equilibrium. (d) Compare the total
output at the Cournot equilibrium with the
monopoly output.
62
Exercise (2)
(2) Consider an n firm Cournot oligopoly. The
demand is given by PA-Y, where A is a positive
constant. The marginal cost of each firm is c,
where c is a positive constant and Agtc. (a)
Derive the output of firm 1 at the symmetric
Cournot equilibrium. (b) Derive the price at the
symmetric Cournot equilibrium. Make sure that the
price-cost margin (price minus marginal cost)
converges to 0 when n ?8.
63
Exercise (3), for the 4th lecture
(1) The demand is given by PA-Y, where A is a
positive constant. The marginal cost of firm 1 is
c1, the marginal cost of firm 2 is c2, where c1
and c2 are positive constants and Agtc1?c2. (a)
Derive the output of firm 1 and that of firm 2 at
the Cournot equilibrium. (b) Derive the
equilibrium total output at the Cournot
equilibrium. (c) Derive the consumer surplus and
total social surplus at the Cournot equilibrium