unit iii: monopoly & oligopoly monopoly oligopoly strategic competition 7/28
TRANSCRIPT
UNIT III: MONOPOLY & OLIGOPOLY
• Monopoly• Oligopoly• Strategic Competition
7/28
Market Structure
Perfect Comp Oligopoly Monopoly
No. of Firms infinite (>)2 1
Output MR = MC = P ??? MR = MC < P
Profit No ? Yes
Efficiency Yes ? ???
Oligopoly
We have no general theory of oligopoly. Rather, there are a variety of models, differing in assumptions about strategic behavior and information conditions.
All the models feature a tension between:
– Collusion: maximize joint profits– Competition: capture a larger share of the pie
Duopoly Models
• Cournot Duopoly
• Nash Equilibrium
• Leader/Follower Model
• Price Competition
Duopoly Models
• Cournot Duopoly
• Nash Equilibrium
• Stackelberg Duopoly
• Bertrand Duopoly
Monopoly
Cyberstax is the only supplier of Vidiot, a hot new computer game. The market for Vidiot is characterized by the following demand and cost conditions:
P = 30 - 1/6Q TC = 40 + 8Q
Monopoly
P = 30 - 1/6Q TC = 40 + 8Q
MR = 30 - 1/3Q MC = 8 => Q* = 66
P* = 19
= TR – TC
= PQ – (40 + 8Q)
= (19)(66) – 40 -(8)(66)
= 686
$
30
P* = 19
Q* = 66 180 Q
MC = 8
MR D
Duopoly
Megacorp is thinking of moving into the Vidiot business with a clone which is indistinguishable from the original. It has access to the same production technology, reflected in the following total cost function:
TC2 = 40 + 8q2
Will Megacorp enter the market?
What is its profit maximizing level of output?
Duopoly
If Megacorp (Firm 2) takes Cyberstax’s (Firm 1) output as given, its residual demand curve is
P = 30 - 1/6QQ = q1+ q2; q1 = 66
P = 30 - 1/6(q1+ q2)
P = 19 - 1/6q2
$
30
19
q2 = 0
q1 = 66 180 Q
Duopoly
P = 19 - 1/6q2 TC2 = 40 + 8q2
MR2 = 19 - 1/3q2 = MC2 = 8 => q2* = 33 q1* = 66
P = 30 – 1/6(q1 + q2)
P* = $13.50 2 = 141.5
Before entry, P* = 19; 1 = 686
Now, 1’ = 323 ow, C‘ = 297
q1*+q2* = 99 180 Q
$
30
19
13.5
q2 = 0
MC2 = 8
Duopoly
What will happen now that Cyberstax knows there is a competitor? Will it change its level of output?
How will Megacorp respond? Where will this process end?
Cournot Duopoly
Reaction curves (or best response curves) show each firm’s profit maximizing level of output as a function of the other firm’s output.
q1
qm
q2 q2
R1: q1* = f(q2)
Cournot Duopoly
To find R1, set MR = MC. Now, MR is a (-) function not only of q1 but also of q2:
q1
66
132 q2
P = 30 - 1/6(q1+q2)
TR1 = Pq1 = [30 - 1/6(q1+q2)]q1
= 30q1 - 1/6q12 - 1/6q2q1
MR1 = 30 -1/3q1 - 1/6q2 = MC = 8
R1: q1* = 66 – 1/2q2
Cournot Duopoly
The outcome (q1*, q2*) is an equilibrium in the following sense: neither firm can increase its profits by changing its behavior unilaterally.
q1
q1* = 44
q2* = 44 q2
R2: q2* = 66 - 1/2q1
R1: q1* = 66 - 1/2q2
For the case of identical firms
Nash Equilibrium
q1
q1* = 44
q2* = 44 q2
R2: q2* = 66 - 1/2q1
R1: q1* = 66 - 1/2q2
For the case of identical firms
A Nash Equilibrium is a pair of “best responses,” such that q1* is a best response to q2* and q2* is a best response to q1*.
Nash Equilibrium
q1
q1*
q2* q2
Is this the best they can do?
If Firm 1 reduces its output while Firm 2 continues to
produce q2*, the price rises and Firm 2’s profits
increase.
A Nash Equilibrium is a pair of “best responses,” such that q1* is a best response to q2* and q2* is a best response to q1*.
Nash Equilibrium
q1
q1*
q2* q2
Is this the best they can do?
If Firm 2 reduces its output while Firm 1 continues to
produce q1*, the price rises and Firm 1’s profits
increase.
A Nash Equilibrium is a pair of “best responses,” such that q1* is a best response to q2* and q2* is a best response to q1*.
Nash Equilibrium
q1
q1*
q2* q2
Is this the best they can do?
If they can agree to restrict output, there are a range of
outcomes to the SW that make both firms better off.
A Nash Equilibrium is a pair of “best responses,” such that q1* is a best response to q2* and q2* is a best response to q1*.
Stackelberg Duopoly
Firm 1 is the dominant firm, or Leader, (e.g., GM) and moves first. Firm 2 is the subordinate firm, or Follower.
q1
q2
R2
Firm 1 gets to search along Firm 2’s reaction curve to find the point that maximizes Firm 1’s profits.1 = ?
1 = ?
Stackelberg Duopoly
Firm 1 is the dominant firm, or Leader, (e.g., GM) and moves first. Firm 2 is the subordinate firm, or Follower.
q2* = 33 q2
R2
q1
q1* = 66
MR1 = MC1
TR1 = Pq1 = [30-1/6(q1+q2*)]q1
Find q2* from R2: q2* = 66 - 1/2q1
= [30-1/6(q1+66-1/2q1)]q1
= 30q1-1/6q12 -11q1+1/12q1
2
MR1 = 19 -1/6q1 = MC1 = 8
q1* = 66; q2* = 33
Stackelberg Duopoly
Firm 1 is the dominant firm, or Leader, (e.g., GM) and moves first. Firm 2 is the subordinate firm, or Follower.
q2* = 33 q2
Firm 1 has a first mover advantage: by committing itself to produce q1, it constraints Firm 2’s output decision.
Firm 1 can employ excess capacity to deter entry by a potential rival.
R2
q1
q1* = 66
Bertrand Duopoly
Under Bertrand duopoly, firms compete on the basis of price, not quantity (as in Cournot and Stackelberg).
P
q1
If P1 > P2 => q1 = 0
If P1 = P2 => q1 = q2 = ½ Q
If P1 < P2 => q2 = 0
Assumptions:
Bertrand Duopoly
Under Bertrand duopoly, firms compete on the basis of price, not quantity (as in Cournot and Stackelberg).
P
P2
q1
If P1 > P2 => q1 = 0
If P1 = P2 => q1 = q2 = ½ Q
If P1 < P2 => q2 = 0
d1
Assumptions:
Bertrand Duopoly
Under Bertrand duopoly, firms compete on the basis of price, not quantity (as in Cournot and Stackelberg).
P
P2
q1
Eventually, price will be competed down to the perfect competition level.
Not very interesting model (so far).
d1
Duopoly Models
If we compare these results, we see that qualitatively different outcomes arise out of the finer-grained assumptions of the models:
P
15.3
13.5
8
c
88 99 132 Q
Cournot
Stackelberg
Bertrand
P = 30 - 1/6Q
TC = 40 + 8q
Duopoly Models
If we compare these results, we see that qualitatively different outcomes arise out of the finer-grained assumptions of the models:
Cournot
Stackelberg
Bertrand
Qc < Qs < Qb
Pc > Ps > Pb
1s > 1c >1b
2c > 2s >2b
P
Pc
Ps
Pb=Ppc
c
Qc Qs Qb = Qpc Q
Duopoly ModelsSummary
Oligopolistic markets are underdetermined by theory. Outcomes depend upon specific assumptions about strategic behavior.
Nash Equilibrium is strategically stable or self-enforcing, b/c no single firm can increase its profits by deviating.
In general, we observe a tension between– Collusion: maximize joint profits– Competition: capture a larger share of the pie
Game Theory
• Game Trees and Matrices
• Games of Chance v. Strategy• The Prisoner’s Dilemma• Dominance Reasoning• Best Response and Nash
Equilibrium• Mixed Strategies
Games of Chance
Buy Don’t Buy
(1000) (-1) (0) (0)
Player 1
Chance
You are offered a fair gamble to purchase a lottery ticket that pays $1000, if your number is drawn. The ticket costs $1.
What would you do?
Games of Chance
Buy Don’t Buy
(1000) (-1) (0) (0)
Player 1
Chance
You are offered a fair gamble to purchase a lottery ticket that pays $1000, if your number is drawn. The ticket costs $1.
The chance of your number being chosen is independent of your decision to buy the ticket.
Games of Strategy
Buy Don’t Buy
(1000,-1000) (-1,1) (0,0) (0,0)
Player 1
Player 2
Player 2 chooses the winning number.
What are Player 2’s payoffs?
Games of Strategy
Advertise Don’t
Advertise
A D A D
(10,5) (15,0) (6,8) (20,2)
Firm 1
Firm 2
Duopolists deciding to advertise. Firm 1 moves first. Firm 2 observes Firm 1’s choice and then makes its own choice.
How should the game be played?
Profits are in ( )
Games of Strategy
Advertise Don’t
Advertise
A D A D
(10,5) (15,0) (6,8) (20,2)
Firm 1
Firm 2
Duopolists deciding to advertise. Firm 1 moves first. Firm 2 observes Firm 1’s choice and then makes its own choice.
How should the game be played?
Backwards-induction
Games of Strategy
Advertise Don’t
Advertise
A D A D
(10,5) (15,0) (6,8) (20,2)
Firm 1
Firm 2
Duopolists deciding to advertise. The 2 firms move simultaneously. (Firm 2 does not see Firm 1’s choice.)
Imperfect Information.
Information set
Matrix Games
Advertise Don’t
Advertise
A D A D
(10,5) (15,0) (6,8) (20,2)
Firm 1
Firm 2
10, 5 15, 0
6, 8 20, 2
A D
A
D
Matrix Games
Advertise Don’t
Advertise
A D A D
(10,5) (15,0) (6,8) (20,2)
Firm 1
Firm 2
10, 5 15, 0
6, 8 20, 2
A D
A
D
Games of Strategy
• Games of strategy require at least two players.
• Players choose strategies and get payoffs. Chance is not a player!
• In games of chance, uncertainty is probabilistic, random, subject to statistical regularities.
• In games of strategy, uncertainty is not random; rather it results from the choice of another strategic actor.
• Thus, game theory is to games of strategy as probability theory is to games of chance.
A Brief History of Game Theory
Minimax Theorem 1928
Theory of Games & Economic Behavior 1944
Nash Equilibrium 1950
Prisoner’s Dilemma 1950
The Evolution of Cooperation 1984
Nobel Prize: Harsanyi, Selten & Nash 1994
The Prisoner’s Dilemma
In years in jail Player 2
Confess Don’t
Confess
Player 1
Don’t
-10, -10 0, -20
-20, 0 -1, -1
The pair of dominant strategies (Confess, Confess)is a Nash Eq.
GAME 1.
The Prisoner’s Dilemma
Each player has a dominant strategy. Yet the outcome (-10, -10) is pareto inefficient.
Is this a result of imperfect information? What would happen if the players could communicate?
What would happen if the game were repeated? A finite number of times? An infinite or unknown number of times?
What would happen if rather than 2, there were many players?
DominanceDefinition
Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s).
T1 T2 T3 T1 T2 T3
0,2 4,3 3,3
4,0 5,4 5,6 3,5 3,5 2,3
0,2 4,3 3,3
4,0 5,4 5,3 3,5 3,5 2,3
S1
S2
S3
S1
S2
S3
Sure Thing Principle: If you have a dominant strategy, use it!
DominanceDefinition
Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s).
T1 T2 T3 T1 T2 T3
0,2 4,3 3,3
4,0 5,4 5,6 3,5 3,5 2,3
0,2 4,3 3,3
4,0 5,4 5,3 3,5 3,5 2,3
S1
S2
S3
S1
S2
S3
Sure Thing Principle: If you have a dominant strategy, use it!
(S2,T3)(S2,T2)
Nash EquilibriumDefinitions
Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s.
-3 0 -10
-1 5 2
-2 -4 0
0,4 4,0 5,3
4,0 0,4 5,3 3,5 3,5 6,6
S1
S2
S3
S1
S2
S3
T1 T2 T3
Nash Equilibrium: a set of best response strategies (one for
each player), (s*, t*) such that s* is a best
response to t* and t* is a b.r. to s*.
(S3,T3)
Nash Equilibrium
-3 0 -10
-1 5 2
-2 -4 0
4,4 2,3 1,5
3,2 1,1 0,0 5,1 0,0 3,3
S1
S2
S3
S1
S2
S3
T1 T2 T3Nash equilibrium need not be Efficient.
Nash Equilibrium
-3 0 -10
-1 5 2
-2 -4 0
1,1 0,0 0,0
0,0 1,1 0,0 0,0 0,0 1,1
S1
S2
S3
S1
S2
S3
T1 T2 T3Nash equilibrium need not be unique.
A COORDINATION PROBLEM
Nash Equilibrium
-3 0 -10
-1 5 2
-2 -4 0
1,1 0,0 0,0
0,0 1,1 0,0 0,0 0,0 3,3
S1
S2
S3
S1
S2
S3
T1 T2 T3Multiple and Inefficient Nash Equilibria.
Nash Equilibrium
-3 0 -10
-1 5 2
-2 -4 0
1,1 0,0 0,-100
0,0 1,1 0,0 -100,0 0,0 3,3
S1
S2
S3
S1
S2
S3
T1 T2 T3Multiple and Inefficient Nash Equilibria.
Is it always advisable to play a NE strategy?
What do we need to know about the other player?
Button-Button
Left Right
L R L R
(-2,2) (4,-4) (2,-2) (-1,1)
Player 1
Player 2
Player 1 hides a button in his Left or Right hand. Player 2 observes Player 1’s choice and then picks either Left or Right.
How should the game be played?
GAME 2.
Button-Button
Left Right
L R L R
(-2,2) (4,-4) (2,-2) (-1,1)
Player 1
Player 2
Player 1 should hide the button in his Right hand. Player 2 should picks Right.
GAME 2.
Button-Button
Left Right
L R L R
(-2,2) (4,-4) (2,-2) (-1,1)
Player 1
Player 2
What happens if Player 2 cannot observe Player 1’s choice?
GAME 2.
Button-Button
Left Right
L R L R
(-2,2) (4,-4) (2,-2) (-1,1)
Player 1
Player 2
-2, 2 4, -4
2, -2 -1, 1
L R
L
R
GAME 2.
Mixed Strategies
-2, 2 4, -4
2, -2 -1, 1
Definition
Mixed Strategy: A mixed strategy is a probability distribution over all strategies available to a player.
Let (p, 1-p) = prob. Player 1 chooses L, R.(q, 1-q) = prob. Player 2 chooses L, R.
L R
L
R
GAME 2.
Mixed Strategies
-2, 2 4, -4
2, -2 -1, 1
Then the expected payoff to Player 1:
EP1(L) = -2(q) + 4(1-q) = 4 – 6qEP1(R) = 2(q) – 1(1-q) = -1 + 3q
Then if q < 5/9, Player 1’s best response is to always play L (p = 1)
L R
L
R
(p)
(1-p)
(q) (1-q)
GAME 2.
q
LEFT 1
5/9
RIGHT 0
0 1 p
p*(q)
Button-Button
Player 1’s best response function.
GAME 2.
Mixed Strategies
-2, 2 4, -4
2, -2 -1, 1
Then the expected payoff to Player 1:
EP1(L) = -2(q) + 4(1-q) EP1(R) = 2(q) – 1(1-q)
(Equalizers) q* = 5/9 and for Player 2:
p* = 1/3 EP2(L) = -2(p) + 2(1-p) EP2(R) = 4(p) – 1(1-p)
L R
L
R
(p)
(1-p)
(q) (1-q)
NE = {(1/3), (5/9)}
GAME 2.
q
LEFT 1
5/9
RIGHT 0
0 1/3 1 p
q*(p)
p*(q)
NE = {(1/3), (5/9)}
Button-Button
GAME 2.
2x2 Game
T1 T2
1. Prisoner’s Dilemma
2. Button – Button
3. Stag Hunt
4. Chicken
5. Battle of Sexes
S1
S2
x1,x2 w1, w2
z1,z2 y1, y2
Stag Hunt
T1 T2
S1
S2
5,5 0,3
3,0 1,1
also Assurance Game
NE = {(S1,T1), (S2,T2)}
GAME 3.
Chicken
T1 T2
S1
S2
3,3 1,5
5,1 0,0
also Hawk/Dove
NE = {(S1,T2), (S2,T1)}
GAME 4.
Battle of the Sexes
T1 T2
S1
S2
5,3 0,0
0,0 3,5
NE = {(S1,T1), (S2,T2)}
GAME 5.
P2
5
3
0
0 3 5 P1
GAME 5.NE = {(1, 1); (0, 0); (5/8, 3/8)}
(0,0)
(5/8,3/8)
(1,1)
Battle of the Sexes
Existence of Nash Equilibrium
Prisoner’s Dilemma Battle of the Sexes Button-Button
GAME 1. GAME 5. (Also 3, 4) GAME 2.
0 1 0 1 0 1 p
q
1
0
There can be (i) a single pure-strategy NE; (ii) a single mixed-strategy NE; or (iii) two pure-strategy NEs plus a single mixed-strategy NE (for x=z; y=w).
Next Time
8/4 Repeated Games
Decision under Uncertainty
Pindyck, Chs 5, 13.
Besanko, Chs 14-16