Bertrand and Hotelling. 2 Assume: Many Buyers Few Sellers Each firm faces downward-sloping demand because each is a large producer compared to the total

Bertrand and Bertrand and HotellingHotelling

22

Assume: Many Buyers Few Sellers

Each firm faces downward-sloping demand because each is a large producer compared to the total market size

There is no one dominant model of oligopoly… we will review several.

33

1. Bertrand Oligopoly (Homogeneous)

Assume: Firms set price* Homogeneous product Simultaneous Noncooperative

*Definition: In a Bertrand oligopoly, each firm sets its price, taking as given the price(s) set by other firm(s), so as to maximize profits.

44

Definition: Firms act simultaneously if each firm makes its strategic decision at the same time, without prior observation of the other firm's decision.

Definition: Firms act noncooperatively if they set strategy independently, without colluding with the other firm in any way

55

How will each firm set price?

Homogeneity implies that consumers will buy from the low-price seller.

Further, each firm realizes that the demand that it faces depends both on its own price and on the price set by other firms

Specifically, any firm charging a higher price than its rivals will sell no output.

Any firm charging a lower price than its rivals will obtain the entire market demand.

66

Definition: The relationship between the price charged by firm i and the demand firm i faces is firm i's residual demand

In other words, the residual demand of firm i is the market demand minus the amount of demand fulfilled by other firms in the market: Q1 = Q - Q2

77

Example: Residual Demand Curve, Price Setting

Quantity

Price

Market Demand

•Residual Demand Curve (thickenedline segments)

0

88

Assume firm always meets its residual demand (no capacity constraints)

Assume that marginal cost is constant at c per unit.

Hence, any price at least equal to c ensures non-negative profits.

99

Example: Reaction Functions, Price Setting and Homogeneous Products

Price chargedby firm 1

Price charged by firm 2 45° line

p2* •

Reaction function of firm 1

Reaction function of firm 2

p1*0

1010

Thus, each firm's profit maximizing response to the other firm's price is to undercut (as long as P > MC)

Definition: The firm's profit maximizing action as a function of the action by the rival firm is the firm's best response (or reaction) function

Example:

2 firmsBertrand competitors

Firm 1's best response function is P1=P2- eFirm 2's best response function is P2=P1- e

1111

So…

1. Firms price at marginal cost

2. Firms make zero profits

3. The number of firms is irrelevant to the price level as long as more than one firm is present: two firms is enough to replicate the perfectly competitive outcome!

1212

If we assume no capacity constraints and that all firms have the same constant average and marginal cost of c then…

For each firm's response to be a best response to the other's each firm must undercut the other as long as P> MC

Where does this stop? P = MC (!)

1313

Bertrand CompetitionBertrand Competition

Homogenous good market / perfect Homogenous good market / perfect substitutessubstitutes

Demand q=15-pDemand q=15-p Constant marginal cost MC=c=3Constant marginal cost MC=c=3 It always pays to undercutIt always pays to undercut Only equilibrium where price equals Only equilibrium where price equals

marginal costsmarginal costs Equilibrium good for consumersEquilibrium good for consumers Collusion must be ruled outCollusion must be ruled out

1414

1515

1616

1717

1818

1919

Sample result: BertrandSample result: Bertrand

0

1

2

3

4

5

6

7

8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Time

Pri

ceAverage Price Average Selling Price

Marginal Cost

“I learnt that collusion can take place in a competitive market even without any actual meeting taking place between the two parties.”

Two Firms

Fixed Partners

Two Firms

Random Partners

Five Firms

Random Partners

“Some people are undercutting bastards!!! Seriously though, it was interesting to see how the theory is shown in practise.”

2020

Hotelling’s (1929) linear cityHotelling’s (1929) linear city

Why do all vendors locate in the same Why do all vendors locate in the same spot?spot?

For instance, on High Street many For instance, on High Street many shoe shops right next to each other. shoe shops right next to each other. Why do political parties (at least in Why do political parties (at least in the US) seem to have the same the US) seem to have the same agenda?agenda?

This can be explained by firms trying This can be explained by firms trying to get the most customers.to get the most customers.

2121

Hotelling (voting version)Hotelling (voting version)

Party BParty A

If Party A shifts to the right then it gains voters.

Voters vote for the closest party.

L R

Party BParty AL R

Each has incentive to locate in the middle.

2222

Hotelling ModelHotelling Model

Party BParty AL R

Average distance for voter is ¼ total. This isn’t “efficient”!

Party BParty AL R

Most “efficient” has average distance of 1/8 total.

2323

Further considerations: Further considerations: HotellingHotelling

Firms choose location and then prices.Firms choose location and then prices. Consumers care about both distance and Consumers care about both distance and

price. price. If firms choose close together, they will If firms choose close together, they will

eliminate profits as in Bertrand eliminate profits as in Bertrand competition. competition.

If firms choose further apart, they will be If firms choose further apart, they will be able to make some profit.able to make some profit.

Thus, they choose further apart.Thus, they choose further apart.

2424

Price competition with Price competition with differentiated goodsdifferentiated goods

Prices pPrices pAA and p and pBB Zero marginal costsZero marginal costs Transport cost tTransport cost t V value to consumerV value to consumer Consumers on interval [0,1]Consumers on interval [0,1] Firms A and B at positions 0 and 1Firms A and B at positions 0 and 1 Consumer indifferent ifConsumer indifferent if

V-tx- pV-tx- pAA= V-t(1-x)- p= V-t(1-x)- pBB

Residual demand qResidual demand qAA=(p=(pBB- p- pAA+t)/2t for firm A+t)/2t for firm A Residual demand qResidual demand qBB=(p=(pAA- p- pBB+t)/2t for firm B+t)/2t for firm B

2525

Price competition with Price competition with differentiated goodsdifferentiated goods

Residual demand qResidual demand qAA=(p=(pBB- p- pAA+t)/2t for firm A+t)/2t for firm A Residual demand qResidual demand qBB=(p=(pAA- p- pBB+t)/2t for firm B+t)/2t for firm B Residual inverse demandsResidual inverse demands

ppAA=-2t q=-2t qAA +p +pBB+t, p+t, pBB=-2t q=-2t qBB +p +pAA+t+t Marginal revenues must equal MC=0Marginal revenues must equal MC=0

MRMRAA=-4t q=-4t qAA +p +pBB+t=0, MR+t=0, MRBB=-4t q=-4t qBB +p +pAA+t=0+t=0

MRMRAA=-2(p=-2(pBB- p- pAA+t)+p+t)+pBB+t=0, MR+t=0, MRBB=-2(p=-2(pAA- p- pBB+t)+p+t)+pAA+t=0+t=0

MRMRAA=2p=2pAA-p-pBB-t=0, MR-t=0, MRBB=2p=2pBB-p-pAA-t=0-t=0

ppAA=2p=2pBB-t; 4p-t; 4pBB-2t-p-2t-pBB-t=0; p-t=0; pBB=p=pAA=t=t Profits t/2Profits t/2

2626

Assume: Firms set price* Differentiated product Simultaneous Noncooperative

As before, differentiation means that lowering price below your rivals' will not result in capturing the entire market, nor will raising price mean losing the entire market so that residual demand decreases smoothly

2727

Example:

Q1 = 100 - 2P1 + P2 "Coke's demand"Q2 = 100 - 2P2 + P1 "Pepsi's demand"

MC1 = MC2 = 5

What is firm 1's residual demand when Firm 2's price is $10? $0?

Q110 = 100 - 2P1 + 10 = 110 - 2P1

Q10 = 100 - 2P1 + 0 = 100 - 2P1

2828

Example: Residual Demand, Price Setting, Differentiated ProductsEach firm maximizes profits based on its residual demand by setting MR (based on residual demand) = MC

Coke’s price

Coke’s quantityMR0

Pepsi’s price = $0 for D0 and $10 for D10

0

100

2929


Coke’s price

Coke’s quantity


0

110100

D0

D10

3030


Coke’s price

Coke’s quantityMR0

D0


0

MR10

110100

D10

3131


Coke’s price

Coke’s quantity

5

MR0

D0


0

D10

MR10

110100

3232


Coke’s price

Coke’s quantity

5

27.5

MR0

D0


0

D10

MR10

30

45 50

110100

3333

Example:

MR110 = 55 - Q1

10 = 5

Q110 = 50

P110 = 30

Therefore, firm 1's best response to a price of $10 by firm 2 is a price of $30

3434

Example: Solving for firm 1's reaction function for any arbitrary price by firm 2

P1 = 50 - Q1/2 + P2/2

MR = 50 - Q1 + P2/2

MR = MC => Q1 = 45 + P2/2

3535

And, using the demand curve, we have:

P1 = 50 + P2/2 - 45/2 - P2/4 …or…P1 = 27.5 + P2/4…reaction function

3636

Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products

Pepsi’s price (P2)

Coke’s price (P1)

P2 = 27.5 + P1/4(Pepsi’s R.F.)

27.5

3737



Coke’s price (P1)P1 = 110/3

P1 = 27.5 + P2/4 (Coke’s R.F.)

P2 = 27.5 + P1/4(Pepsi’s R.F.)

•

27.5

27.5

3838



Coke’s price (P1)P1 = 110/3

P2 = 110/3

P1 = 27.5 + P2/4 (Coke’s R.F.)

P2 = 27.5 + P1/4(Pepsi’s R.F.)

•

BertrandEquilibrium

27.5

27.5

3939

Equilibrium:

Equilibrium occurs when all firms simultaneously choose their best response to each others' actions.

Graphically, this amounts to the point where the best response functions cross...

4040

Example: Firm 1 and firm 2, continued

P1 = 27.5 + P2/4P2 = 27.5 + P1/4

Solving these two equations in two unknowns…

P1* = P2

* = 110/3

Plugging these prices into demand, we have:

Q1* = Q2

* = 190/3

1* = 2

* = 2005.55 = 4011.10

4141

Notice that

1. profits are positive in equilibrium since both prices are above marginal cost!

Even if we have no capacity constraints, and constant marginal cost, a firm cannot capture all demand by cutting price…

This blunts price-cutting incentives and means that the firms' own behavior does not mimic free entry

4242

Only if I were to let the number of firms approach infinity would price approach marginal cost.

2. Prices need not be equal in equilibrium if firms not identical (e.g. Marginal costs differ implies that prices differ)

3. The reaction functions slope upward: "aggression => aggression"

4343

Back to CournotBack to Cournot

Inverse demand P=260-Q1-Q2Inverse demand P=260-Q1-Q2 Marginal costs MC=20Marginal costs MC=20 3 possible predictions3 possible predictions Price=MC, Symmetry Q1=Q2Price=MC, Symmetry Q1=Q2

260-2Q1=20, Q1=120, P=20260-2Q1=20, Q1=120, P=20 Cournot duopoly: Cournot duopoly:

MR1=260-2Q1-Q2=20, Symmetry Q1=Q2MR1=260-2Q1-Q2=20, Symmetry Q1=Q2260-3Q1=20, Q1=80, P=100260-3Q1=20, Q1=80, P=100

Shared monopoly profits: Q=Q1+Q2Shared monopoly profits: Q=Q1+Q2MR=260-2Q=20, Q=120, Q1=Q2=60, P=140MR=260-2Q=20, Q=120, Q1=Q2=60, P=140

4444

4545

4646

Bertrand with Compliments (?!Bertrand with Compliments (?!*-)*-)

Q=15-P1-P2, MC1=1.5, MC2=1.5, MC=3Q=15-P1-P2, MC1=1.5, MC2=1.5, MC=3 Monopoly: P=15-Q, MR=15-2Q=3, Q=6, Monopoly: P=15-Q, MR=15-2Q=3, Q=6,

P=P1+P2=9, Profit (9-3)*6=36P=P1+P2=9, Profit (9-3)*6=36 Bertrand: P1=15-Q-P2, MR1=15-2Q-Bertrand: P1=15-Q-P2, MR1=15-2Q-

P2=1.5P2=1.515-2(15-P1-P2)-P2=-15+2P1+P2=1.515-2(15-P1-P2)-P2=-15+2P1+P2=1.5Symmetry P1=P2; 3P1=16.5, P1=5.5, Symmetry P1=P2; 3P1=16.5, P1=5.5,

Q=4<9Q=4<9P1+P2=11>9, both make profit P1+P2=11>9, both make profit (11-3)*4=32<36(11-3)*4=32<36Competition makes both firms and Competition makes both firms and

consumers worse off!consumers worse off!

4747

4848

4949

The Capacity GameGM

Ford

20Expand

DNE

1820

1518

1615 16

Expand

DNE

What is the equilibrium here?Where would the companies like to be?

*

5050

Mars

Venus

-1Shoot

Not Shoot

-5-1

-15-5

-10-15 -10

Shoot

Not Shoot

War

*

5151

Repeated gamesRepeated games 1. if game is repeated with same players, 1. if game is repeated with same players,

then there may be ways to enforcethen there may be ways to enforcea better solution to prisoners’ dilemmaa better solution to prisoners’ dilemma

2. suppose PD is repeated 10 times and 2. suppose PD is repeated 10 times and people know itpeople know it– then backward induction says it is a dominant then backward induction says it is a dominant

strategy to cheat everystrategy to cheat everyroundround

3. suppose that PD is repeated an 3. suppose that PD is repeated an indefinite number of timesindefinite number of times– then it may pay to cooperatethen it may pay to cooperate

4. Axelrod’s experiment: tit-for-tat4. Axelrod’s experiment: tit-for-tat

5252

Continuation payoffContinuation payoff

Your payoff is the sum of your payoff Your payoff is the sum of your payoff today plus the discounted “continuation today plus the discounted “continuation payoff”payoff”

Both depend on your choice todayBoth depend on your choice today If you get punished tomorrow for bad If you get punished tomorrow for bad

behaviour today and you value the future behaviour today and you value the future sufficiently highly, it is in your self-interest sufficiently highly, it is in your self-interest to behave well todayto behave well today

Your trade-off short run against long run Your trade-off short run against long run gains.gains.

5353

Infinitely repeated PDInfinitely repeated PD

Discounted payoff, 0<d<1 discount Discounted payoff, 0<d<1 discount factor (dfactor (d00=1)=1)

Normalized payoff: (dNormalized payoff: (d00uu00+ d+ d11uu11+ + dd22uu22+… +d+… +dttuutt+…)(1-d)+…)(1-d)

Geometric series:Geometric series: (d(d00+ d+ d11+ d+ d22+… +d+… +dtt+…)(1-d)+…)(1-d)=(d=(d00+ d+ d11+ d+ d22+… +d+… +dtt+…)+…)-(d-(d11+ d+ d22+ d+ d33+… +d+… +dt+1t+1+…)= d+…)= d00=1=1

5454

Infinitely repeated PDInfinitely repeated PD

Constant “income stream” uConstant “income stream” u00= = uu11=u=u22=… =u each period yields total =… =u each period yields total normalized income u.normalized income u.

Grim Strategy: Choose “Not shoot” Grim Strategy: Choose “Not shoot” until someone chooses “shoot”, until someone chooses “shoot”, always choose “Shoot” thereafter always choose “Shoot” thereafter

5555

Payoff if nobody shoots:Payoff if nobody shoots:

(-5d(-5d00- 5d- 5d11-5d-5d22-… -5d-… -5dtt+…)(1-d)=-5+…)(1-d)=-5

=-5(1-d)-5d=-5(1-d)-5d Maximal payoff from shooting in first Maximal payoff from shooting in first

period (-15<-10!):period (-15<-10!):

(-d(-d00-10d-10d11-10d-10d22-… -10d-… -10dtt-…)(1-d)-…)(1-d)

=-1(1-d)-10d=-1(1-d)-10d -1(1-d)-10d< -5(1-d)-5d iff 4(1-d)<5d -1(1-d)-10d< -5(1-d)-5d iff 4(1-d)<5d

or 4<9d d>4/9 or 4<9d d>4/9 0.440.44 Cooperation can be sustained if d> Cooperation can be sustained if d> 0.450.45, ,

i.e. if players weight future sufficiently i.e. if players weight future sufficiently highly. highly.

Documents

Bertrand and Hotelling. 2 Assume: Many Buyers Few Sellers Each firm faces downward-sloping demand because each is a large producer compared to the total