Lecture Notes
Oligopoly
Oligopoly = a few competitors > 1All have impact on others reactions and decisionsFor 1/3 => lots of competitors but all so small (don’t
care)For 2 => only 1 firm, no competitorsOligopoly unique in that each firm must react to
competitorsHow?
P.C. M.C. Oligopoly Monopoly (1) (3) (4)
(2)
Multiple ways => oligopoly is difficult to model because there is not just 1 model but multiple models. Duopoly: 2 firms (simpler)
Price leader/follower Q leader/follower Collusion No followers but simultaneous decisions => how? Sequential games, cooperate games, simultaneous
games Plus many more
Do 2 types of models firstCournot: each firm chooses yi given belief
about yi => EQ where beliefs true. Stackelberg- Q leader, Q follower model =>
dominant firmThese are actually similar modelsDo the basics of both models first
Find Reaction Function = each firm’s optimal profit given other firms’ y decision
Look at monopoly profit for 1 firm Same if 2 firms 1 & 2 so thatY = Y1 + Y2 but Y2 = 0 Pm
Ym
Ym Y
DMRMC
profit
Y
But suppose Y2 > 0 => D for 1 decreases => 1 is the residual claimant
For D1 Y2=Z-EYm is optimalAssuming Y2=0Y1* is optimalassuming Y2 = Z-E
Z
π
Y
D1
P
Y
Y1* Ym
MR
MRm
Dm
MC
E
π|Y2=0π|Y2=Z-E
Get a whole series of profit curves for firm 1 given firm 2’s output y2
Now construct iso profit curves—hold profit constant and change Y1 & Y2
That is find Y1 & Y2 for a given profit levelStart with m = monopoly profit level
Y1
Y2 ↑
$
What is firm 1’s Q (y1) = when 1 is a monopoly? = Ym => = m
Iso w/ = m is where? At Ym => simple pointY2 = 0 if increase or decrease Y1 => decrease
Y2’
Y2
Ym Y1Y2’ =m
Now suppose Y2 increases => profit decreases what happens to optimal Y1 w/ an increase in Y2?
Proved before that it decreases but so does 1.
If Y2 = Y2’ => firm 1 chooses Y1’Firm 1 can choose a different level of Y1 than
Y1’ but either increase or decrease Y1 will decrease profit given Y2=Y2’
A firm’s reaction function (ridge line) shows that firm’s max profit given other firm’s y
Do the same for Firm 2: Looks like…
Y2
Ym Y1
1 increase
2 increase
Firm 1’s reaction Fn Firm 2’s reaction Fn
Book shows that for P = a-b yY2 = (a- b Y1) / 2bY1= (a – b Y2)/2bAssuming MC = 0 if MC =C (constant
Y2= (a- b Y1 – C) / 2bY1= (a – b Y2 – C) / 2b
Gets more complicated if MC = F (Y1)Common to assume MC = Constant
Stackelberg: assume firm 1 is the dominant leader and firm 2 is the follower . Get the following equilibrium:
Y2 always on Firm 2’s reaction FnChooses Y2 | Y1Firm 1 chooses Y1 to get on highest iso profit
given Firm 2 being on R2.=> tangency b/w 1’s iso profit and R2
Ym R1
R2Y2
Y1
Cournot each firm on its reactionFn and eq. only if expectations about other firm nextPoint A not Cournot Eq.
At A 1 would move to B; at B 2 would move to C and so on until eventually get to D where R1 and R2 cross and
E2 (Y1) = Y1E1 (Y2) = Y2
Y2
Ym Y1R2R
1
A
CD
B
Get A= Stackelburg EQ. B= Cournot EQ.
Note: can show that at A Y1 =Ym
B
Ym Y1R1
A
R2
Define Nash EQ= Both partiesChoosing optimally to max profit given info
they haveCournot = Nash EqStackelberg = not Nash Eq1’s higher than 2 =>return to not being a
reaction function
Now suppose both oligopoly players realize this simultaneously and play Stackelburg
Y1 = Y2 = Ym => Y = Ym + Ym = the competitive outputa = b = 0 => by playing “smart” both firms get less
profit.Stackelburg bluff if 1 or 2 can convince the other other
player that he will stay at Qm no matter what => the other players rational move is to go to his reaction function since profit increases if he does so. Both want to be this player. Y2
Ym
Ym Y1
Esc EssEcc
Ecs
Collusion implies joint max (i.e. on Ym-Ym) but here there is a problem with cheating
If at E on collusive agreement A or B can get to a lower hill (more ) by increasing q a little => incentives to cheat.
If they both cheat then they are both worse off.
Bertrand Model: the only difference between Cournot and Bertrand model is that price (not qty) is used as the choice variable. Let Q = D(P) be the demand function.The problem is to max 1 = D(P1) P1 – C (P1) P2
= P2Everything is a function of price
MC
Qm Qc
P
Q
Look at Eq.If P > MC => 1 firm can increase Q (increase ) by
decreasing P slightly the other will follow and P decreases to MC
If P=MC and 1 firm increases P, Q decreases to o so no increase in
=> P = MC is an equilibriumNow what if MC is upward sloping? Then what is the
equilibrium? EQ does not exist because it is always optimum to
change price. If P = MC incentive is to increase pricei.e. if 1 firm increases P => at old price consumers want
to buy more of other firms price also => both increase price
(z) if p= monopoly price (and each firm has ½ Qm) => >MC and by decreasing price can get more => both decrease price until P=MC
Game Theory Applied to OligopolyGame theory: method of analyzing outcomes of
choices made by people who are interdependent. Define:
Players: those making choices Strategies: the possible choices to achieve ____ Payoffs: returns to different choices Payoff matrix: shows how different choices affect
payoffs
For example: A owns A house with his value of $60k. B values House at $80K and has $70K => Possibility of exchange
(1) Cooperative solution: reach an agreement over price and exchange occurs
(2) non-cooperative solution => no exchange
Q: What does each party get in both cases? For 2: A gets house = $60k; B gets $70,000 => total of $130K For 1: A gets $70,000 and B gets house - $80K => total of
$150k => Cooperative surplus = $20,000 What would the payoff matrix look like?
1st: what are the decisions? Suppose bargain hard v. soft
If you bargain hard and other soft=> assume you get A4 surplus
If both bargain soft => split surplus If both bargain hard=> no exchange and no surplus H S
H
S
60K70K
60K90K
70K80K 70K
80K
Note: Reach cooperative solution as long as not H1, H.
Q: What will the _____ be?Look at A’s choiceIf B choose S => A better off with HIf B choose H => A indiff. for H & S=> A choose H=> B choose H=> H1 H = Eq. Essentially this is a dominant strategy game
(not quite because of the ind.)
Look at the 2 games from the book1st: Dominant Strategy Game
Both firms better off with choosing High Q
Low Q High QLow Q
High Q
2010
309
2017 25
18
2nd– What should A choose?
Clearly depends on what B choosesA: Low Q => Low Q; High Q => High QBut B has a dominant strategy = High Q => A also chooses
High QNash Eq: each player chooses best one given stratgey
chosen by other.Now look at the prisoner’s dilemma: Common/classic game
and widely applicable to many other situations including firm decisions.
Low Q High QLow Q
High Q
2022
309
2017 25
18
2 people arrested for whatever (book uses drug dealing)Suppose choices presented = confess, don’t confess
with payoff matrix =
2 points: both have a dominant strategy= confess => both confessCooperative surplus available, that is D1 D = cooperation
and better off there then at the actual EQ. Why don’t they? Not just because believe other will
confess, more than that because better off to confess regardless of what the other does.
ConfessDon’tC
D
10 yr10 yr
15 yr1
yr
15 yr1 yr 2 yr
2 yr
How to deal with Prisoner’s Dilemma? Mob does it by charging payoff matrix to
increase penatly if confess
Now don’t confess = Nash Eq and dominant strategy => D1 D
C DC
D
deathdeath
15 yrdeath
15 yrdeath 2 yr
2 yr
Mixed vs. Pure StrategyPure: make choice and stick with itMixed: make a choice some % of the time. Where %
of sum to 1 for all possible choicesExample:
First, with mixed strategies always Nash Eq => each party chooses optimal prob. given the other parties prob.
Second, not always Nas Eq w/ Pure Strategies
0,0 0, -1
1, 0 -1, 3
Left RightTop
bottom
____ above 1st with Pure StrategyIf B = L => A = BIf B = R => A = TIf A = T => B = LIf A = B => B = RNotice how no eq. existsWith mixed strategy, can show Nash Eq=
A P(T) = ¾ P(B) = ¼B P(L) = ½ P(R) = ½ How?
Look at expected values with prob. Define EU
Let p = prob. A plays top1-p = prob A plays bottomq= prob B plays left1-q= prob B plays right
=> each party’s EU depends on other party’s choice of their prob.
For example: If A plays Top => EUA
1 = 0q + 0(1-q) If A plays bottom => EQA
2 = 1q + -1 (1-q) If B plays left => EUB
1 = 0P + 0(1-P) If B plays right =>EUB
2 = -1P + 3(1-P) What is the EQ con
Q occurs when no change in behaviorIf for A EUA
1 = EUA2 => no reason for A to
change behavior if > or 0 = q – (1-q) if L < decrease in P 0 = q – 1 + q 1 = 2q or q = ½ (1-q) = ½
Same for B EUB1 = EUB
2 => or 0 = -P + (1-P) 0= -P + 3 – 3P 4P =3 P= ¾(1-P) =1/4
This is the Nash EQ. in two mixed strategy games
Prisoner’s Dilemma in CartelsWhere Q = do we cheat on the cartel
agreement to ___ Q?
If both comply D1 d => acting like a monopoly and each earn ½ monopoly profit
EQ is the same with prisoner’s dilemma C1 C and for same reasons.
Cheat Don’t CheatC
D
10 10
525
525 20
20
Conclude Cartels only form/stable if cartel can enforce punishment of cheating
Must be done so that cheating decreases profits. Perhaps they fine cheaters…
i.e. lose $20 if caught cheatingDon’t cheat is dominant strategy for both D1 D = Nash Eq.
C DC
D
-10 -10
55
55 20
20
Look at expected values with prob. Define EU
Let p = prob. A plays top1-p = prob A plays bottomq= prob B plays left1-q= prob B plays right
=> each party’s EU depends on other party’s choice of their prob.
For example: If A plays Top => EUA
1 = 0q + 0(1-q) If A plays bottom => EQA
2 = 1q + -1 (1-q) If B plays left => EUB
1 = 0P + 0(1-P) If B plays right =>EUB
2 = -1P + 3(1-P) What is the EQ con
Look at expected values with prob. Define EU
Let p = prob. A plays top1-p = prob A plays bottomq= prob B plays left1-q= prob B plays right
=> each party’s EU depends on other party’s choice of their prob.
For example: If A plays Top => EUA
1 = 0q + 0(1-q) If A plays bottom => EQA
2 = 1q + -1 (1-q) If B plays left => EUB
1 = 0P + 0(1-P) If B plays right =>EUB
2 = -1P + 3(1-P) What is the EQ con
Repeated GamesSuppose game is played multiple timesGo back to original Artesia/ Utopia Cartel Game Suppose Utopia (U) uses tit for tat strategy—
choose D in a given week as long as Artesia (A) chooses D in the previous week.
Also assume that A knows U following tit for tat => A will not cheat If A follows D => U = D and get…
Period A U1 20 202 20 203 20 204 20 20
Which is better for A then either of the 2 strategies=> not sure to get collusion but more likely
because 1. firms can enforce with a tit for tat strategy 2. other firms can easily identify a tit for tat strategy
Sequential GamesSo far been doing simultaneous games
Notice 2 Nash EQ w/ Pure Strategy simultaneous Game 1) T, L 2) B, R Neither party can make themselves better of from these
given other’s choice But, 1) is not a reasonable solution…why?
A knowing matrix will never choose T => before choices knows if he chooses B => B will choose R
A increases profit
1, 9 1, 9
0, 0 2, 1
Left RightTop
bottom
Put in extensive form to show sequential decision making
A
2 Possible solutions to this game1st- A choose B => B chooses R2nd- B convinces A that if A chooses B => B will
choose L => A chooses T How? By locking himself in to choosing L all the time via
3rd party, contracts, etc.
T
B
R
L
R
L
1, 9
1, 90,0
2,1
=> Like sequence of choices, now the game becomes..
B
And A chooses TNote: not necessary for the sequence to
change, just look in ______=> would look like…
Other branches are gone
L A T
B
1, 9
0, 0
AT
B
L
B L
1, 9
0, 0