Imperfect Competition
Economics 302 - Microeconomic Theory II: Strategic Behavior
Instructor: Songzi Du
compiled by Shih En Lu
(Chapter 10 in Watson (2013))
Simon Fraser University
January 22, 2016
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Oligopoly
Armed with game theory, let’s return to the study of market.
What happens when the number of firms is small, but greater
than one?
Intuitively, we expect the firms to have some market power, but not
as much as a monopoly, so we expect an intermediate outcome.
Model #1: Firms compete by simultaneously setting their quantities
(Cournot model).
Model #2: Firms compete by simultaneously setting their prices
(Bertrand model).
Next week: Firms compete sequentially (Stackelberg model).
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Cournot Competition: Basic Model
Let’s start with the simplest case: two firms with the same constant
marginal cost c produce a homogeneous good with linear market
demand P(q) = a − b · q.
We look for a Nash equilibrium.
Actions: Firm 1 picks q1 ≥ 0, firm 2 picks q2 ≥ 0. Infinitely many
actions!
Let’s fix firm 2’s quantity q2 , and figure out firm 1’s best response.
Firm 1 maximizes:
(P − c)q1 = (a − b(q1 + q2 ) − c)q1
= −bq12 + (a − bq2 − c)q1
(This is strictly concave in q1 for any q2 . Is firm 1’s best response
ever a mixed strategy?)
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Cournot Competition: Basic Model (II)
First-order condition:
−2bq1 + a − bq2 − c = 0
a−c
q2
−
= q1
2b
2
Similarly, firm 2’s best response to firm 1 picking q1 is:
a−c
q1
−
= q2
2b
2
In a NE, firms best respond to each other, so both these
best-response functions (or reaction functions) must hold.
Thus we solve the system of equations, which gives:
q1 = q2 =
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a−c
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Cournot Competition: Basic Model (III)
The total market quantity is thus q1 + q2 =
2 a−c
3 b
>
1 a−c
2 b
= qm .
The price is therefore lower under a Cournot
than under a
duopoly
2
1
a
+
c;
the monopoly
monopoly. (Cournotprice is a − b 23 a−c
=
b
3
3
1 a−c
1
1
price is a − b 2 b = 2 a + 2 c).
Deadweight loss is also lower. (This conclusion changes if fixed costs
are high enough.)
You can check that the total profit under Cournot duopoly is
which is less than the monopoly profit of
1 (a−c)2
4
b .
2 (a−c)2
9
b ,
The Cournot duopoly fails to maximize the total profit: firms are not
internalizing the negative effect of their production on the other firm’s
price.
Firms could increase profit by each producing 12 qm (cooperating), but
each would have an incentive to produce more (defecting).
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Cournot Competition is a Prisoner’s Dilemma
Cooperate
Defect
Cooperate
-2, -2
-1, -5
Defect
-5, -1
-3, -3
Cournot duopoly game is dominance solvable: ISD leads to the NE
quantity (Give it a try if you want a challenge).
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Bertrand Competition
The Cournot model delivered sensible results. But do we observe
firms setting quantity or price?
Bertrand model: firms simultaneously choose price.
Let Q(p) be the demand curve.
Firm with lowest price pL sells Q(pL ) units, while firm(s) with higher
price(s) sell nothing. (Homogeneous good)
If m firms share the lowest price pL , then each sells
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1
m Q(pL )
units.
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Bertrand Competition: Simplest Case
Two firms with the same constant marginal cost c (and no fixed
cost) face downward-sloping demand that crosses c at a positive
quantity.
Suppose that each firm chooses a price in cents:
{0, 0.01, 0.02, 0.03, . . .}, and c is also in cents.
We look for pure-strategy Nash equilibria.
(There sometimes exist NE that are not in pure strategies. They are
hard to find, and we will ignore them.)
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Bertrand Competition: Simplest Case (II)
Suppose firm 2 picks p2 , what is firm 1’s best response?
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Bertrand Competition: Simplest Case (II)
Suppose firm 2 picks p2 , what is firm 1’s best response?
If p2 > c + 0.01, then firm 1 always wants to undercut him. And vice
versa.
There are two NE’s: (c, c) and (c + 0.01, c + 0.01).
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Bertrand Competition: Discussion
Does the NE seem plausible?
Some explanations:
1
2
3
Product differentiation
Capacity constraints
Collusion (after quiz)
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