# Two sets of problems: Bertrand with differentiated products

```Two sets of problems:
• How does product differentiation influence
the firms’ price competition?
Bertrand model where demanded quantity is reduced
in own price and increased in the other’s price
• How do firms differentiate their products
to weaken price competion?
Types of differentiation: Horizontal differentiation
Vertical differentiation
Bertrand with differentiated products
Each firm’s demand increases in the other’s price:
Di ( pi , p j )  a  bpi  dp j
If production costs equal zero, then profit equals:
 i ( pi , p j )  a  bpi  dp j  pi
 i ( pi , p j ) / pi  a  2bpi  dp j  0
a  dp j
Best response fn:
pi  f i ( p j ) 
2b
Best response fns are
2

 2 i
d

i
increasing; prices are f i ( p j )  

0
2
pi p j pi
 2b
strateg. complements:
FOC:
Bertrand with production costs
Nash equilibrium in the Bertrand model
p2
Are there prices for the two firms so that no firm
will regret its own choice when getting to know
the other firm’s choice?
1's best resonse fn: p1  ( a  dp2 ) / 2b
2's best response fn: p2  (a  dp1 ) / 2b
p1b  p2b  a /( 2b  d )
q1b  q2b  ab / (2b  d )
 1 ( p1b , p2b )   2 ( p1b , p2b )  a 2 b /(2b  d ) 2
p1
• Best response curves have positive slope
p2
• The ”reaction story”
p2
is stabile.
• Increased c1 shifts 1’s
curve outwards.
• Increased c1 leads to inpp1 1
creased p1 and increased p2.
• Increased c1 is an indirect advantage for both.
 1 ( p1 , p2 )  a  bp1  dp2 ( p1  c1 )
p2
Firm 1’s best
response fn
Firm 2’s best
response fn
 2 ( p1 , p2 )  a  bp2  dp1 ( p2  c2 )
FOC: p1   a  dp2  bc1  / 2b
p2  a  dp1  bc2  / 2b
2b( a  bc1 )  d ( a  bc2 )
b
p1 
( 2b  d )( 2b  d )
2b( a  bc2 )  d ( a  bc1 )
b
p2 
( 2b  d )( 2b  d )
What happens when firm 1’s
costs increase?
p1
• Horizontal product differentiation
Eks: Different locations of stores.
Different time slots for airline departures.
The optimal choice for identical prices
depends on the consumer taste.
• Vertical product price differentiation
Ex: Hyundai, BMW
For identical prices, all wish to own a BMW. The optimal choice for identical prices is same for everyone.
For different prices, consumer make different choices.
1
Choice of location, followed by price comp.
Horizontal differentiation: Linear city
0
A consumer
Store 1
• Stage 1: Simultaneously firms decide where to
locate (a og b).
• Stage 2: Given location, firms compete in price.
1
Store 2
x
a
b
An SPE is found by determining the Nash equilibrium in stage 2 for all locations.
• Consumers distributed uniformly between 0 and 1.
• Two stores located a og b from the end points.
When firms choose locations in stage 1, they take
into account the consequences in stage 2.
Motivation: Show how firms in stage 1 choose
maximal differentiation to weaken price comp.
• With quadratic transport costs, consumer at x
will buy from store1 if and only if
p1  t ( x  a ) 2  p2  t (1  b  x ) 2
Nash equilibrium in prices in stage 2
Demand
For given firm locations, a og b, and prices, p1 og p2,
where does the indifferent consumer live?
p  t( ~
x  a ) 2  p  t (1  b  ~
x )2
1
2
p  p1  t (1  a  b)[2a  (1  a  b)]
~
x 2
2t (1  a  b)
Consumers to the left of ~
1  a  2b
p2  p1
~
2
2
p1 Dt 1 ~
x( p1, p22a)~
xxaap2  t ~
x 2(1  b) ~
x  (1  b) 2 
2
2t (1  a  b)
~ 2 from
Consumers to ptheright
 a 2 ) firm 2.
p  tof
~
x  2 ~ 1 1 a  b
p1  p2
D2 ( p1 , p2 )  1  x 2bt (1  a  b) 
2
2t (1  a  b)
 2 ( a, b)  D2 ( a , b, p1c ( a , b), p2c ( a, b))( p2c ( a, b)  c )
0

 p
 1  D1 D1 p  c
D


( p1  c )   D1  1 ( p1c  c ) 
a  a p2 a 
p1

 a

 1 ( a , b)
0
a

p1  12 c  p2  t (1  a  b)(1  a  b) 
p2  12 c  p1  t (1  b  a )(1  b  a ) 
a  b
p1c ( a, b)  c  t (1  a  b)1 

3 

ba
p2c ( a , b)  c  t (1  b  a )1 

3 

What happens when firm 1 gets
close to 0; i.e. reduces a?
p1
Firm 1’s best
response fn
Firm 2’s best
response fn
Welfare analysis
 1 ( a , b)  D1 ( a, b, p1c ( a , b), p2c ( a , b))( p1c ( a, b)  c )
Demand
effect
 2  D2 ( p1 , p2 )( p2  c )
p2
Consumption is
Therefore: Welfare max. by min. transport costs.
Location choice in stage 1
c
2
 1  D1 ( p1 , p2 )( p1  c )
Store 2
Store 1
1
c
1
Strategic
effect
a  14
b  14
Transport costs in equilibrium
0

 Max differentiation
Differentiation to weaken price competition.
a0
b0
a  0
and b  0 
2 0 tx dx  x 0 
Transport costs in social optimum  a  14 and b  14 
1
2
2
2t
3
3
1
2
t
12
4 0 tx 2 dx  43t x 3 0  48t
1
4
1
4
2
```