Bertrand and Hotelling

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Bertrand and Hotelling
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Oligopoly

• 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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Price
Example Residual Demand Curve, Price Setting
Market Demand
Residual Demand Curve (thickened line segments)

Quantity
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
45 line
Price charged by firm 2
Reaction function of firm 1
Reaction function of firm 2

p2
Price charged by firm 1
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 gt 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 firms Bertrand
competitors Firm 1's best response function is
P1P2- e Firm 2's best response function is
P2P1- 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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Equilibrium
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 Pgt
MC Where does this stop? P MC (!)
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Bertrand Competition

• Homogenous good market / perfect substitutes
• Demand q15-p
• Constant marginal cost MCc3
• It always pays to undercut
• Only equilibrium where price equals marginal
  costs
• Equilibrium good for consumers
• Collusion must be ruled out

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Sample result Bertrand
Two Firms Fixed Partners
Five Firms Random Partners
Two Firms Random Partners
I learnt that collusion can take place in a
competitive market even without any actual
meeting taking place between the two parties.
Some people are undercutting bastards!!!
Seriously though, it was interesting to see how
the theory is shown in practise.
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Hotellings (1929) linear city

• Why do all vendors locate in the same spot?
• For instance, on High Street many shoe shops
  right next to each other. Why do political
  parties (at least in the US) seem to have the
  same agenda?
• This can be explained by firms trying to get the
  most customers.

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Hotelling (voting version)
Voters vote for the closest party.
R
L
Party B
Party A
If Party A shifts to the right then it gains
voters.
R
L
Party B
Party A
Each has incentive to locate in the middle.
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Hotelling Model
R
L
Party B
Party A
Average distance for voter is ¼ total. This isnt
efficient!
R
L
Party B
Party A
Most efficient has average distance of 1/8
total.
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Further considerations Hotelling

• Firms choose location and then prices.
• Consumers care about both distance and price.
• If firms choose close together, they will
  eliminate profits as in Bertrand competition.
• If firms choose further apart, they will be able
  to make some profit.
• Thus, they choose further apart.

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Price competition with differentiated goods

• Prices pA and pB
• Zero marginal costs
• Transport cost t
• V value to consumer
• Consumers on interval 0,1
• Firms A and B at positions 0 and 1
• Consumer indifferent if
• V-tx- pA V-t(1-x)- pB
• Residual demand qA(pB- pAt)/2t for firm A
• Residual demand qB(pA- pBt)/2t for firm B

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Price competition with differentiated goods

• Residual demand qA(pB- pAt)/2t for firm A
• Residual demand qB(pA- pBt)/2t for firm B
• Residual inverse demands
• pA-2t qA pBt, pB-2t qB pAt
• Marginal revenues must equal MC0
• MRA-4t qA pBt0, MRB-4t qB pAt0
• MRA-2(pB- pAt)pBt0, MRB-2(pA- pBt)pAt0
• MRA2pA-pB-t0, MRB2pB-pA-t0
• pA2pB-t 4pB-2t-pB-t0 pBpAt
• Profits t/2

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Bertrand Competition (Differentiated)
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 Products Each firm maximizes
profits based on its residual demand by setting
MR (based on residual demand) MC
Cokes price
Pepsis price 0 for D0 and 10 for D10
100
MR0
0
Cokes quantity
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Example Residual Demand, Price Setting,
Differentiated Products Each firm maximizes
profits based on its residual demand by setting
MR (based on residual demand) MC
Cokes price
110
Pepsis price 0 for D0 and 10 for D10
100
D10
D0
0
Cokes quantity
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Example Residual Demand, Price Setting,
Differentiated Products Each firm maximizes
profits based on its residual demand by setting
MR (based on residual demand) MC
Pepsis price 0 for D0 and 10 for D10
Cokes price
110
100
D10
D0
MR10
MR0
0
Cokes quantity
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Example Residual Demand, Price Setting,
Differentiated Products Each firm maximizes
profits based on its residual demand by setting
MR (based on residual demand) MC
Pepsis price 0 for D0 and 10 for D10
Cokes price
110
100
D10
D0
MR10
5
MR0
0
Cokes quantity
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Example Residual Demand, Price Setting,
Differentiated Products Each firm maximizes
profits based on its residual demand by setting
MR (based on residual demand) MC
Pepsis price 0 for D0 and 10 for D10
Cokes price
110
100
30
27.5
D10
D0
MR10
5
MR0
0
Cokes quantity
45 50
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• Example
• MR110 55 - Q110 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 gt 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 P2/4reaction
function
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Pepsis price (P2)
P2 27.5 P1/4 (Pepsis R.F.)
Example Equilibrium and Reaction Functions,
Price Setting and Differentiated Products
27.5
Cokes price (P1)
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P1 27.5 P2/4 (Cokes R.F.)
Pepsis price (P2)
P2 27.5 P1/4 (Pepsis R.F.)

Example Equilibrium and Reaction Functions,
Price Setting and Differentiated Products
27.5
Cokes price (P1)
27.5
P1 110/3
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P1 27.5 P2/4 (Cokes R.F.)
Pepsis price (P2)
P2 27.5 P1/4 (Pepsis R.F.)
Bertrand Equilibrium
P2 110/3

Example Equilibrium and Reaction Functions,
Price Setting and Differentiated Products
27.5
Cokes price (P1)
27.5
P1 110/3
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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
P2/4 P2 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 gt aggression"

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Back to Cournot

• Inverse demand P260-Q1-Q2
• Marginal costs MC20
• 3 possible predictions
• PriceMC, Symmetry Q1Q2
• 260-2Q120, Q1120, P20
• Cournot duopoly
• MR1260-2Q1-Q220, Symmetry Q1Q2
• 260-3Q120, Q180, P100
• Shared monopoly profits QQ1Q2
• MR260-2Q20, Q120, Q1Q260, P140

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Bertrand with Compliments (?!-)

• Q15-P1-P2, MC11.5, MC21.5, MC3
• Monopoly P15-Q, MR15-2Q3, Q6, PP1P29,
  Profit (9-3)636
• Bertrand P115-Q-P2, MR115-2Q-P21.5
• 15-2(15-P1-P2)-P2-152P1P21.5
• Symmetry P1P2 3P116.5, P15.5, Q4lt9
• P1P211gt9, both make profit
• (11-3)432lt36
• Competition makes both firms and consumers worse
  off!

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The Capacity Game
GM
DNE
Expand
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DNE
18
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Ford
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Expand
20
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What is the equilibrium here? Where would the
companies like to be?
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War
Mars
Not Shoot
Shoot
-5
-1
Not Shoot
-5
-15
Venus
-10

-15
Shoot
-1
-10
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Repeated games

• 1. if game is repeated with same players, then
  there may be ways to enforcea better solution to
  prisoners dilemma
• 2. suppose PD is repeated 10 times and people
  know it
• then backward induction says it is a dominant
  strategy to cheat everyround
• 3. suppose that PD is repeated an indefinite
  number of times
• then it may pay to cooperate
• 4. Axelrods experiment tit-for-tat

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Continuation payoff

• Your payoff is the sum of your payoff today plus
  the discounted continuation payoff
• Both depend on your choice today
• If you get punished tomorrow for bad behaviour
  today and you value the future sufficiently
  highly, it is in your self-interest to behave
  well today
• Your trade-off short run against long run gains.

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Infinitely repeated PD

• Discounted payoff, 0ltdlt1 discount factor (d01)
• Normalized payoff (d0u0 d1u1 d2u2
  dtut)(1-d)
• Geometric series
• (d0 d1 d2 dt)(1-d)
• (d0 d1 d2 dt)
• -(d1 d2 d3 dt1) d01

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Infinitely repeated PD

• Constant income stream u0 u1u2 u each
  period yields total normalized income u.
• Grim Strategy Choose Not shoot until someone
  chooses shoot, always choose Shoot thereafter
  

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• Payoff if nobody shoots
• (-5d0- 5d1-5d2- -5dt)(1-d)-5
• -5(1-d)-5d
• Maximal payoff from shooting in first period
  (-15lt-10!)
• (-d0-10d1-10d2- -10dt-)(1-d)
• -1(1-d)-10d
• -1(1-d)-10dlt -5(1-d)-5d iff 4(1-d)lt5d or
  4lt9d dgt4/9 ? 0.44
• Cooperation can be sustained if dgt 0.45, i.e. if
  players weight future sufficiently highly.