Bertrand and Hotelling
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.
2
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.
3
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
4
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.
5
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
•
0
Residual Demand Curve (thickened
line segments)
Quantity
7
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.
8
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*
0
•
p1*
Price charged
by firm 1 9
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 firms
Bertrand competitors
Firm 1's best response function is P1=P2- e
Firm 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!
11
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 (!)
12
Bertrand Competition







Homogenous good market / perfect substitutes
Demand q=15-p
Constant marginal cost MC=c=3
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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14
15
16
17
18
Sample result: Bertrand
Average Price
Average Selling Price
8
7
6
Price
5
4
Marginal Cost
3
2
Two Firms
1
Five Firms
Two Firms
Fixed Partners Random Partners
Random Partners
3
21
0
1
5
7
9
11
13
15
17
19
23
25
27
29
Time
“I “Some
learnt that
collusion
can take place
in a
people
are undercutting
bastards!!!
competitive
market even
any actual
Seriously though,
it waswithout
interesting
to see how the
meeting
place
between the two parties.”
theorytaking
is shown
in practise.”
19
Hotelling’s (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.

20
Hotelling (voting version)
Voters vote for the closest party.
R
L
Party A
Party B
If Party A shifts to the right then it gains voters.
R
L
Party A
Party B
Each has incentive to locate in the middle.
21
Hotelling Model
R
L
Party A
Party B
Average distance for voter is ¼ total. This isn’t “efficient”!
R
L
Party A
Party B
Most “efficient” has average distance of 1/8 total.
22
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.

23
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- pA+t)/2t for firm A
Residual demand qB=(pA- pB+t)/2t for firm B
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Price competition with
differentiated goods
Residual demand qA=(pB- pA+t)/2t for firm A
 Residual demand qB=(pA- pB+t)/2t for firm B
 Residual inverse demands

pA=-2t qA +pB+t, pB=-2t qB +pA+t

Marginal revenues must equal MC=0
MRA=-4t qA +pB+t=0, MRB=-4t qB +pA+t=0
MRA=-2(pB- pA+t)+pB+t=0, MRB=-2(pA- pB+t)+pA+t=0
MRA=2pA-pB-t=0, MRB=2pB-pA-t=0
pA=2pB-t; 4pB-2t-pB-t=0; pB=pA=t

Profits t/2
25
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
Coke’s price
100
0
Pepsi’s price = $0 for D0 and $10 for D10
MR0
Coke’s quantity
28
Example: Residual Demand, Price Setting, Differentiated Products
Each firm maximizes profits based on its residual demand by
setting MR (based on residual demand) = MC
Coke’s price
110
100
Pepsi’s price = $0 for D0 and $10 for D10
D10
D0
0
Coke’s quantity
29
Example: Residual Demand, Price Setting, Differentiated Products
Each firm maximizes profits based on its residual demand by
setting MR (based on residual demand) = MC
Pepsi’s price = $0 for D0 and $10 for D10
Coke’s price
110
100
MR10
0
MR0
D10
D0
Coke’s quantity
30
Example: Residual Demand, Price Setting, Differentiated Products
Each firm maximizes profits based on its residual demand by
setting MR (based on residual demand) = MC
Pepsi’s price = $0 for D0 and $10 for D10
Coke’s price
110
100
D10
5
0
MR10
MR0
D0
Coke’s quantity
31
Example: Residual Demand, Price Setting, Differentiated Products
Each firm maximizes profits based on its residual demand by
setting MR (based on residual demand) = MC
Pepsi’s price = $0 for D0 and $10 for D10
Coke’s price
110
100
30
27.5
D10
MR10
5
0
MR0
45
50
D0
Coke’s quantity
32
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
33
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
34
And, using the demand curve, we have:
P1 = 50 + P2/2 - 45/2 - P2/4 …or…
P1 = 27.5 + P2/4…reaction function
35
Pepsi’s price (P2)
P2 = 27.5 + P1/4
(Pepsi’s R.F.)
27.5
Example: Equilibrium
and Reaction Functions,
Price Setting and
Differentiated Products
Coke’s price (P1)
36
P1 = 27.5 + P2/4 (Coke’s R.F.)
Pepsi’s price (P2)
P2 = 27.5 + P1/4
(Pepsi’s R.F.)
•
27.5
27.5
P1 = 110/3
Example: Equilibrium
and Reaction Functions,
Price Setting and
Differentiated Products
Coke’s price (P1)
37
P1 = 27.5 + P2/4 (Coke’s R.F.)
Pepsi’s price (P2)
Bertrand
Equilibrium
•
P2 =
110/3
27.5
27.5
P1 = 110/3
P2 = 27.5 + P1/4
(Pepsi’s R.F.)
Example: Equilibrium
and Reaction Functions,
Price Setting and
Differentiated Products
Coke’s price (P1)
38
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...
39
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
40
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
41
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"
42
Back to Cournot
Inverse demand P=260-Q1-Q2
 Marginal costs MC=20
 3 possible predictions
 Price=MC, Symmetry Q1=Q2

260-2Q1=20, Q1=120, P=20

Cournot duopoly:
MR1=260-2Q1-Q2=20, Symmetry Q1=Q2
260-3Q1=20, Q1=80, P=100

Shared monopoly profits: Q=Q1+Q2
MR=260-2Q=20, Q=120, Q1=Q2=60, P=140
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44
45
Bertrand with Compliments (?!*-)
Q=15-P1-P2, MC1=1.5, MC2=1.5, MC=3
 Monopoly: P=15-Q, MR=15-2Q=3, Q=6,
P=P1+P2=9, Profit (9-3)*6=36
 Bertrand: P1=15-Q-P2, MR1=15-2QP2=1.5

15-2(15-P1-P2)-P2=-15+2P1+P2=1.5
Symmetry P1=P2; 3P1=16.5, P1=5.5, Q=4<9
P1+P2=11>9, both make profit
(11-3)*4=32<36
Competition makes both firms and consumers
worse off!
46
47
48
The Capacity Game
GM
DNE
DNE
Ford
Expand
Expand
18
18
20
20
15
15
*
16
16
What is the equilibrium here?
Where would the companies like to be?
49
War
Mars
Not Shoot Shoot
Not Shoot
Venus
Shoot
-5
-5
-1
-15
-15
*
-1
-10
-10
50
Repeated games
1. if game is repeated with same players, then
there may be ways to enforce
a 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 every
round

3. suppose that PD is repeated an indefinite
number of times
– then it may pay to cooperate

4. Axelrod’s experiment: tit-for-tat
51
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.

52
Infinitely repeated PD
Discounted payoff, 0<d<1 discount factor
(d0=1)
 Normalized payoff: (d0u0+ d1u1+ d2u2+…
+dtut+…)(1-d)
 Geometric series:
(d0+ d1+ d2+… +dt+…)(1-d)
=(d0+ d1+ d2+… +dt+…)
-(d1+ d2+ d3+… +dt+1+…)= d0=1

53
Infinitely repeated PD
Constant “income stream” u0= u1=u2=…
=u each period yields total normalized
income u.
 Grim Strategy: Choose “Not shoot” until
someone chooses “shoot”, always choose
“Shoot” thereafter

54
Payoff if nobody shoots:
(-5d0- 5d1-5d2-… -5dt+…)(1-d)=-5
=-5(1-d)-5d
 Maximal payoff from shooting in first
period (-15<-10!):
(-d0-10d1-10d2-… -10dt-…)(1-d)
=-1(1-d)-10d
 -1(1-d)-10d< -5(1-d)-5d iff
4(1d)<5d or 4<9d d>4/9  0.44
 Cooperation can be sustained if d> 0.45,
i.e. if players weight future sufficiently
highly.

55