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