Price Competition: Bertrand Introduction

1
Price Competition BertrandIntroduction

• In a wide variety of markets firms compete in
  prices
• Internet access
• Restaurants
• Consultants
• Financial services
• With monopoly setting price or quantity first
  makes no difference
• In oligopoly it matters a great deal
• nature of price competition is much more
  aggressive than quantity competition

2
Price Competition Bertrand

• In the Cournot model price is set by some market
  clearing mechanism
• Firms seem relatively passive
• An alternative approach is to assume that firms
  compete in prices this is the approach taken by
  Bertrand
• Leads to dramatically different results
• Take a simple example
• two firms producing an identical product (spring
  water?)
• firms choose the prices at which they sell their
  water
• each firm has constant marginal cost of 10
• market demand is Q 100 - 2P

Check that with this demand and these costs
the monopoly price is 30 and quantity is 40 units
3
Bertrand competition (cont.)

• We need the derived demand for each firm
• demand conditional upon the price charged by the
  other firm
• Take firm 2. Assume that firm 1 has set a price
  of 25
• if firm 2 sets a price greater than 25 she will
  sell nothing
• if firm 2 sets a price less than 25 she gets the
  whole market
• if firm 2 sets a price of exactly 25 consumers
  are indifferent between the two firms
• the market is shared, presumably 5050
• So we have the derived demand for firm 2
• q2 0 if p2 gt p1 25
• q2 100 - 2p2 if p2 lt p1 25
• q2 0.5(100 - 50) 25 if p2 p1 25

4
Bertrand competition (cont.)
Demand is not continuous. There is a jump at p2
p1

• More generally
• Suppose firm 1 sets price p1

p2

• Demand to firm 2 is

q2 0 if p2 gt p1
p1
q2 100 - 2p2 if p2 lt p1
q2 50 p2 if p2 p1

• The discontinuity in demand carries over to profit

q2
100
100 - 2p2
50 p2
5
Bertrand competition (cont.)
Firm 2s profit is
p2(p1,, p2) 0 if p2 gt p1
p2(p1,, p2) (p2 - 10)(100 - 2p2) if p2 lt p1
For whatever reason!
p2(p1,, p2) (p2 - 10)(50 - p2) if p2 p1
Clearly this depends on p1.
Suppose first that firm 1 sets a very high
price greater than the monopoly price of 30
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Bertrand competition (cont.)
What price should firm 2 set?
If p1 30, then firm 2 will only earn a
positive profit by cutting its price to 30 or
less
At p2 p1 30, firm 2 gets half of the
monopoly profit
So, if p1 falls to 30, firm 2 should just
undercut p1 a bit and get almost all the
monopoly profit
With p1 gt 30, Firm 2s profit looks like this
The monopoly price of 30
What if firm 1 prices at 30?
Firm 2s Profit
p2 lt p1
p2 p1
p2 gt p1
p1
Firm 2s Price
10
30
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Bertrand competition (cont.)
Now suppose that firm 1 sets a price less than 30
As long as p1 gt c 10, Firm 2 should aim just
to undercut firm 1
Firm 2s profit looks like this
What price should firm 2 set now?
Of course, firm 1 will then undercut firm 2 and
so on
Firm 2s Profit
p2 lt p1
Then firm 2 should also price at 10. Cutting
price below costgains the whole market but loses
money on every customer
What if firm 1 prices at 10?
p2 p1
p2 gt p1
p1
Firm 2s Price
10
30
8
Bertrand competition (cont.)

• We now have Firm 2s best response to any price
  set by firm 1
• p2 30 if p1 gt 30
• p2 p1 - something small if 10 lt p1 lt 30
• p2 10 if p1 lt 10
• We have a symmetric best response for firm 1
• p1 30 if p2 gt 30
• p1 p2 - something small if 10 lt p2 lt 30
• p1 10 if p2 lt 10

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Bertrand competition (cont.)
The best response function for firm 1
The best response function for firm 2
These best response functions look like this
p2
R1
The Bertrand equilibrium has both firms
charging marginal cost
R2
30
The equilibrium is with both firms pricing at 10
10
p1
10
30
10
Bertrand Equilibrium modifications

• The Bertrand model makes clear that competition
  in prices is very different from competition in
  quantities
• Since many firms seem to set prices (and not
  quantities) this is a challenge to the Cournot
  approach
• But the extreme version of the difference seems
  somewhat forced
• Two extensions can be considered
• impact of capacity constraints
• product differentiation

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Capacity Constraints

• For the p c equilibrium to arise, both firms
  need enough capacity to fill all demand at p c
• But when p c they each get only half the market
• So, at the p c equilibrium, there is huge
  excess capacity
• So capacity constraints may affect the
  equilibrium
• Consider an example
• daily demand for skiing on Mount Norman Q 6,000
  60P
• Q is number of lift tickets and P is price of a
  lift ticket
• two resorts Pepall with daily capacity 1,000 and
  Richards with daily capacity 1,400, both fixed
• marginal cost of lift services for both is 10

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The Example

• Is a price P c 10 an equilibrium?
• total demand is then 5,400, well in excess of
  capacity
• Suppose both resorts set P 10 both then have
  demand of 2,700
• Consider Pepall
• raising price loses some demand
• but where can they go? Richards is already above
  capacity
• so some skiers will not switch from Pepall at the
  higher price
• but then Pepall is pricing above MC and making
  profit on the skiers who remain
• so P 10 cannot be an equilibrium

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The example 2

• Assume that at any price where demand at a resort
  is greater than capacity there is efficient
  rationing
• serves skiers with the highest willingness to pay
• Then can derive residual demand
• Assume P 60
• total demand 2,400 total capacity
• so Pepall gets 1,000 skiers
• residual demand to Richards with efficient
  rationing is Q 5000 60P or P 83.33 Q/60
  in inverse form
• marginal revenue is then MR 83.33 Q/30

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The example 3

• Residual demand and MR

Price

• Suppose that Richards sets P 60. Does it want
  to change?

83.33
Demand
60

• since MR gt MC Richards does not want to raise
  price and lose skiers

MR
36.66
10
MC

• since QR 1,400 Richards is at capacity and does
  not want to reduce price

Quantity
1,400

• Same logic applies to Pepall so P 60 is a Nash
  equilibrium for this game.

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Capacity constraints again

• Logic is quite general
• firms are unlikely to choose sufficient capacity
  to serve the whole market when price equals
  marginal cost
• since they get only a fraction in equilibrium
• so capacity of each firm is less than needed to
  serve the whole market
• but then there is no incentive to cut price to
  marginal cost
• So the efficiency property of Bertrand
  equilibrium breaks down when firms are capacity
  constrained

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Product differentiation

• Original analysis also assumes that firms offer
  homogeneous products
• Creates incentives for firms to differentiate
  their products
• to generate consumer loyalty
• do not lose all demand when they price above
  their rivals
• keep the most loyal

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An Example of Product Differentiation
Coke and Pepsi are nearly identical but not
quite. As a result, the lowest priced product
does not win the entire market.
QC 63.42 - 3.98PC 2.25PP
MCC 4.96
QP 49.52 - 5.48PP 1.40PC
MCP 3.96
There are at least two methods for solving this
for PC and PP
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Bertrand and Product Differentiation
Method 1 Calculus
Profit of Coke pC (PC - 4.96)(63.42 - 3.98PC
2.25PP)
Profit of Pepsi pP (PP - 3.96)(49.52 - 5.48PP
1.40PC)
Differentiate with respect to PC and PP
respectively
Method 2 MR MC
Reorganize the demand functions
PC (15.93 0.57PP) - 0.25QC
PP (9.04 0.26PC) - 0.18QP
Calculate marginal revenue, equate to marginal
cost, solve for QC and QP and substitute in the
demand functions
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Bertrand competition and product differentiation
Both methods give the best response functions
The Bertrand equilibrium is at their intersection
PC 10.44 0.2826PP
PP
Note that these are upward sloping
RC
PP 6.49 0.1277PC
These can be solved for the equilibrium prices as
indicated
RP
8.11
B
6.49
PC
10.44
12.72
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Bertrand Competition and the Spatial Model

• An alternative approach is to use the spatial
  model
• a Main Street over which consumers are
  distributed
• supplied by two shops located at opposite ends of
  the street
• but now the shops are competitors
• each consumer buys exactly one unit of the good
  provided that its full price is less than V
• a consumer buys from the shop offering the lower
  full price
• consumers incur transport costs of t per unit
  distance in travelling to a shop
• What prices will the two shops charge?

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Bertrand and the spatial model
Xm marks the location of the marginal buyerone
who is indifferent between buying either firms
good
What if shop 1 raises its price?
Assume that shop 1 sets price p1 and shop
2 sets price p2
Price
Price
p1
p2
p1
xm
xm
Shop 1
Shop 2
All consumers to the left of xm buy from shop 1
xm moves to the left some consumers switch to
shop 2
And all consumers to the right buy from shop 2
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Bertrand and the spatial model
This is the fraction of consumers who buy from
firm 1
p1 txm p2 t(1 - xm)
How is xm determined?
?2txm p2 - p1 t
?xm(p1, p2) (p2 - p1 t)/2t
There are N consumers in total
So demand to firm 1 is D1 N(p2 - p1 t)/2t
Price
Price
p2
p1
xm
Shop 1
Shop 2
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Bertrand equilibrium
Profit to firm 1 is p1 (p1 - c)D1 N(p1 -
c)(p2 - p1 t)/2t
p1 N(p2p1 - p12 tp1 cp1 - cp2 -ct)/2t
Solve this for p1
This is the best response function for firm 1
Differentiate with respect to p1
N
?p1/ ?p1
(p2
- 2p1
t c)
0
2t
p1 (p2 t c)/2
This is the best response function for firm 2
What about firm 2? By symmetry, it has a similar
best response function.
p2 (p1 t c)/2
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Bertrand and Demand
p2
p1 (p2 t c)/2
R1
p2 (p1 t c)/2
2p2 p1 t c
R2
p2/2 3(t c)/2
c t
?? p2 t c
(c t)/2
?? p1 t c
p1
(c t)/2
c t
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Bertrand competition final remarks

• Two final points on this analysis
• t is a measure of transport costs
• it is also a measure of the value consumers place
  on getting their most preferred variety
• when t is large competition is softened
• and profit is increased
• when t is small competition is tougher
• and profit is decreased
• Locations have been taken as fixed
• suppose product design can be set by the firms
• balance business stealing temptation to be
  close
• against competition softening desire to be
  separate
• see the beach location game