Bertrand and Hotelling. 2 Assume: Many Buyers Few Sellers Each firm faces downward-sloping demand...
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Transcript of Bertrand and Hotelling. 2 Assume: Many Buyers Few Sellers Each firm faces downward-sloping demand...
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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.
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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.
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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
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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.
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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
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Example: Residual Demand Curve, Price Setting
Quantity
Price
Market Demand
•Residual Demand Curve (thickenedline segments)
0
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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.
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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
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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
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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!
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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 (!)
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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
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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.”
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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.
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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 quantity
Pepsi’s price = $0 for D0 and $10 for D10
0
110100
D0
D10
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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
D0
Pepsi’s price = $0 for D0 and $10 for D10
0
MR10
110100
D10
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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 quantity
5
MR0
D0
Pepsi’s price = $0 for D0 and $10 for D10
0
D10
MR10
110100
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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 quantity
5
27.5
MR0
D0
Pepsi’s price = $0 for D0 and $10 for D10
0
D10
MR10
30
45 50
110100
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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
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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
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And, using the demand curve, we have:
P1 = 50 + P2/2 - 45/2 - P2/4 …or…P1 = 27.5 + P2/4…reaction function
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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
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Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products
Pepsi’s price (P2)
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
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Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products
Pepsi’s price (P2)
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
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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...
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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
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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
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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"
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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
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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!
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The Capacity GameGM
Ford
20Expand
DNE
1820
1518
1615 16
Expand
DNE
What is the equilibrium here?Where would the companies like to be?
*
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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
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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.
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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
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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
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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.