4.2. Hotelling Model
Matilde Machado
Industrial Organization- Matilde Machado
The Hotelling Model
1
4.2. Hotelling Model
The model:
1.
2.
3.
4.
5.
6.
“Linear city” is the interval [0,1]
Consumers are distributed uniformely along this interval.
There are 2 firms, located at each extreme who sell the
same good. The unique difference among firms is their
location.
c= cost of 1 unit of the good
t= transportation cost by unit of distance squared. This
cost is up to the consumer to pay. If a consumer is at a
distance d to one of the sellers, its transportation cost is
td2 . This cost represents the value of time, gasoline, or
adaptation to a product, etc.
Consumers have unit demands, they buy at most one
unit of the good {0,1}
Industrial Organization- Matilde Machado
The Hotelling Model
2
4.2. Hotelling Model
Graphically
Mass of
consumers =
1
1dz  z 0  1  0  1
1
1
0
0
1
x
Location of firm B
Location of firm A
Industrial Organization- Matilde Machado
The Hotelling Model
3
4.2. Hotelling Model
The transportation costs of consumer x:

Of buying from seller A are tx 2
2

Of buying from seller B are t 1  x 



s ≡ gross consumer surplus - (i.e. its maximum
willingness to pay for the good)
Let’s assume s is sufficiently large for all
consumers to be willing to buy (this situation is
referred to as “the market is covered”). The
utility of each consumer is given by:
U = s-p-td2 where p is the price paid.
Industrial Organization- Matilde Machado
The Hotelling Model
4
4.2. Hotelling Model
We first take the locations of the sellers as given
(afterwards we are going to determine them
endogenously) and assume firms compete in
prices.
1. Derive the demand curves for each of the sellers
2. The price optimization problem given the
demands
Industrial Organization- Matilde Machado
The Hotelling Model
5
4.2. Hotelling Model
In order to derive the demands we need to derive
the consumer x that is just indifferent between
buying from A or from B:
x%is defined as the location where U %x ( A)  U x%( B)
 s  p A  tx%2  s  pB  t (1  x%) 2
 p A  tx%2  pB  t (1  x%) 2
 p A  tx%2  pB  t  tx%2  2tx%
 2tx% pB  p A  t
 x%
Buy from A
pB  p A  t
2t
A
Buy from B
x
B
If (pB=pA) then the indifferent consumer is at half the distance between A and B. If
(pB-pA)↑ the indifferent consumers moves to the right, that is the demand for firm A
increases and the demand for firm B decreases.
Industrial Organization- Matilde Machado
The Hotelling Model
6
4.2. Hotelling Model
s
Ui
Total cost to
consumer x: pA+tx2
pB+t(1-x)2
pA
pB
0
i
1
x
A
B
The equilibrium of the Hotelling model
Industrial Organization- Matilde Machado
The Hotelling Model
7
4.2. Hotelling Model
We say the market is covered if all consumers buy.
Since the consumer with the lowest utility is the
indifferent consumer (because it is the one who
is further away from any of the sellers), we may
say that the market is covered if the indifferent
consumer buys i.e. if:
 p  pA  t 
s  pA  t  B
 0
2t


2
This condition is equivalent to say that s has to be
high enough
Industrial Organization- Matilde Machado
The Hotelling Model
8
4.2. Hotelling Model
Once we know the indifferent consumer, we may
define the demand functions of A and B.
%
x
DA ( p A , pB )   1dz  z 0  x%
0
x%
pB  p A  t pB  p A 1


2t
2t
2
1
 p  p A 1   p A  pB 1 
DB ( p A , pB )   1dz  z x%  1  x% 1   B
 
 
2
t
2
2
t
2

 
%
x
1
Demand of firm A depends positively on the difference (pB-pA) and
negatively on the transportation costs. If firms set the same prices
pB=pA then transportation costs do not matter as long as the
market is covered, firms split the market equally (and the indifferent
consumer is located in the middle of the interval ½).
Industrial Organization- Matilde Machado
The Hotelling Model
9
4.2. Hotelling Model
The maximization problem of firm A is:
Max  A ( p A , pB )   p A  c  DA ( p A , pB )   p A  c 
pA
pB  p A  t
2t
p  pA  t 1
 A
FOC:
0 B
  pA  c   0
p A
2t
2t
 pB  2 p A  t  c  0  p A 
pB  t  c
2
Firm A’s
reaction
curve
Because the problem is symmetric pA=pB=p*
p*  t  c
p* t  c
p 


 p*  t  c
2
2
2
*
Industrial Organization- Matilde Machado
The Hotelling Model
Note that if t=0 (no
product
differentiation) we
go back to Bertrand
p*=c; *=0
10
4.2. Hotelling Model
Once the equilibrium prices are determined, we may
determine the other equilibrium quantities:
1
(the indifferent consumer is in the middle because prices are equal)
2
1
DA ( p*A , pB* )  x%* 
2
1
DB ( p*A , pB* )  1  x%*   DA ( p*A , pB* )
2
t
 A*   B*   p*  c  DA*   t  c  c  x%* 
2
x%* 
Note: The higher is t , the more differentiated are the goods from the point of
view of the consumers, the highest is the market power (the closest consumers
are more captive since it is more expensive to turn to the competition) which
allows the firms to increase prices and therefore profits. When t=0 (no
differentiation) we go back to Bertrand
Industrial Organization- Matilde Machado
The Hotelling Model
11
4.2. Hotelling Model
Observations:
 Each firm serves half the market D*A=D*B=1/2
 The Bertrand paradox disapears (note that firms
compete in prices) pA=pB>c
 An increase in t implies more product
differentiation. Therefore, firms compete less
vigorously (set higher prices) and obtain higher
profits.
 t=0 back to Bertrand
Industrial Organization- Matilde Machado
The Hotelling Model
12
4.2. Hotelling Model
s
Ui
pA+tx2
pB+t(1-x)2
pA=t+c
pB=t+c
0
i
A
½
1
x%
B
The equilibrium of the Hotelling model
Industrial Organization- Matilde Machado
The Hotelling Model
13
4.2. Hotelling Model
How do prices change if the locations of A and B
change?
 If A=0 and B=1 there is maximum differentiation
 Si A=B, there is no differentiation, all consumers
will buy from the seller with the lowest price,
back to Bertrand,
pA=pB=c y A=B=0.
Industrial Organization- Matilde Machado
The Hotelling Model
14
4.2. Hotelling Model
General Case– Endogenous locations:
2 periods:
 In the first period, firms choose location
 In the second period firms compete in prices given their
locations
We solve the game backwards, starting from the
second period.
Industrial Organization- Matilde Machado
The Hotelling Model
15
4.2. Hotelling Model
Second period:
 Denote by a [0,1] the location of A
 Denote by (1-b) [0,1] the location of B
Note: Maximum differentiation is obtained with a=0;
and 1-b=1 (i.e. b=0)
Minimum differentiation (perfect substitutes)
is obtained with a=1-b a+b=1
Industrial Organization- Matilde Machado
The Hotelling Model
16
4.2. Hotelling Model
1. The indifferent consumer:
U x ( A)  U x ( B)
p A  t ( x% a) 2  pB  t ( x% (1  b)) 2
%  pB  tx%2  t (1  b) 2  2tx%(1  b)
 p A  tx%2  ta 2  2txa
 2tx%1  b  a   pB  p A  t (1  b) 2  ta 2
2
2
pB  p A  t (1  b) 2  ta 2 pB  p A  t  (1  b)  a 
 x%

2t 1  b  a 
2t 1  b  a 
 x%
1  b  a 1  b  a 
pB  p A

2t 1  b  a 
2 1  b  a 
 x%
1 b  a
1 b  a


pB  p A
pB  p A



a
2t 1  b  a 
2
2t 1  b  a 
2
Hence if pA=pB, A’s demand is a+(1-b-a)/2
Industrial Organization- Matilde Machado
The Hotelling Model
17
4.2. Hotelling Model
Demands are:
DA ( p A , pB )  x%
1 b  a

pB  p A

a
2t 1  b  a 
2
DB ( p A , pB )  1  x% 1 

p A  pB
1 b  a

b
2t 1  b  a 
2
a
0
1  b  a   a
pB  p A

2t 1  b  a 
2
(1-b-a)/2
a
Industrial Organization- Matilde Machado
1-b
The Hotelling Model
1
18
4.2. Hotelling Model
Interpretation of the demand functions:
if p A  pB
DA ( p A , p B ) 

a{
captive consumers
to the left (own
backyard)
DB ( p A , pB ) 
DA ( p A , p B
144422 4443
half of the consumers
between a and 1-b
1 b  a

b{
captive consumers
14422 443
half of the consumers
between a and 1-b
if p A  pB
1  b  a 
to the right (own
backyard)
1 b  a

)a

2
pB  p A
2t (1  b44443
 a)
144442
sensitivity of the demand
to price difference
Industrial Organization- Matilde Machado
The Hotelling Model
19
4.2. Hotelling Model
Graphically
pA+t(x-a)2
pA
pB
0
a
x
Firm A’s
captive market
Industrial Organization- Matilde Machado
1-b
1
Firm B’s
captive market
The Hotelling Model
20
4.2. Hotelling Model
2. Finding the reaction functions
1 b  a


A

Max    p A  c  DA ( p A , pB )   p A  c   a 
p

A
2
pB  p A 

2t (1  b  a) 
1 b  a



pB  p A
1
 A
  pA  c   

0 a
FOC:
0
2t (1  b  a)
2
p A
 2t (1  b  a) 
1 b  a

pB  c
2 pA

a

2t (1  b  a)
2
2t (1  b  a)
1 b  a

pB  c
pA

a

2t (1  b  a)
2
t (1  b  a)
t 1  b  a 
p c
 B
 p A  at (1  b  a) 
2
2
2
Industrial Organization- Matilde Machado
The Hotelling Model

Firm A’s reaction
function
21
4.2. Hotelling Model
2. Finding the reaction functions
1 b  a


p A  pB 
Max    pB  c  DB ( p A , pB )   pB  c   b 


pB
2
2
t
(1

b

a
)


 B
FOC:
0
pB
B
1 b  a

 b

2


p A  pB
1
  pB  c   
0
2t (1  b  a)
 2t (1  b  a) 
1  b  a  p A  2 pB  c

 b

0
2
2t (1  b  a )

Industrial Organization- Matilde Machado
The Hotelling Model
22
4.2. Hotelling Model
2. 2. Finding the reaction functions
b
1  b  a  
2
2 pB  c
1
1 b  a
pB  c 
 a 

0

2t (1  b  a ) 2 
2
2t (1  b  a ) 
1 b  a 1

3 pB  3c
1 b  a

b
 a
0
4t (1  b  a )
2
2
4
3 pB
3c
b 3 a


  
4t (1  b  a ) 4t (1  b  a ) 4 4 4
t  3  b  a  (1  b  a )
 pB  c 
3
 ba
 ab
 c  t (1  b  a )  1 
y
p

c

t
(1

b

a
)
A

1 

3 
3 


Industrial Organization- Matilde Machado
The Hotelling Model
23
4.2. Hotelling Model
2. 2. Finding the reaction functions
 ba
 a b 
*
pB* (a, b)  c  t (1  b  a) 1 
and
p
(
a
,
b
)

c

t
(1

b

a
)
A

1 

3 
3 


Note that prices are maximum when differentiation is
maximum (a=b=0; pA=pB=c+t) and minimum
when there is no differentiation (a+b=1 (same
location) and pA=pB=c)
Industrial Organization- Matilde Machado
The Hotelling Model
24
4.2. Hotelling Model
3. 1st period, simultaneous choice of a and b
Profits are functions of (a, b) alone:
 A (a, b)   p*A (a, b)  c  DA (a, b, p*A (a, b), pB* (a, b))
 B (a, b)   pB* (a, b)  c  DB (a, b, p*A (a, b), pB* (a, b))
*
A
*
B
*
A
*
B
b)
Replace p (a, b), p (a, b), D (a, b), D (a,and
we get a function of a and b
alone. Take the FOC as always with respect to a and b.
Industrial Organization- Matilde Machado
The Hotelling Model
25
4.2. Hotelling Model
3. 1st period, simultaneous choice of a and b
*
*


1 b  a
 a  b    pB  p A
 (a, b)   c  t 1  a  b  1 

 a
  c
3    2t (1  a  b)
2



A
ba
but pB*  p*A  2t (1  a  b) 

3


which simplifies:





2

3

b

a




a

b
b

a
1

b

a

 
  t 1  a  b 
 A (a, b)   t 1  a  b  1 



3
3
2
18


144
444
4
2
444444
3
1444
42
444
4
3



3 b  a
3 a  b 





6

 3 


Industrial Organization- Matilde Machado
The Hotelling Model
26
4.2. Hotelling Model
3. 1st period, simultaneous choice of a and b
Max  (a, b)  t 1  a  b 
A
a
3  b  a 
2
18
3  b  a
2 3  b  a 

 (a, b)
FOC:
 t
 t 1  a  b 
a
18
18
t
   3  b  a 1  b  3a   0  a*  0
18
A
Industrial Organization- Matilde Machado
2
The Hotelling Model
27
4.2. Hotelling Model
3. 1st period, simultaneous choice of a and b
3  b  a

B
Max  (a, b)  t 1  a  b 
b
18
2
3  b  a
2 3  b  a 

 (a, b)
FOC:
 t
 t 1  a  b 
b
18
18
t
   3  b  a 1  3b  a   0  b*  0  1  b*  1
18
B
Industrial Organization- Matilde Machado
2
The Hotelling Model
28
4.2. Hotelling Model
Conclusion: Firms choose to be in the
extremes i.e. they choose maximum
differentiation.
For firm A, for example, an increase in a (movement
to the right):
 Has a positive effect because it moves towards
where the demand is (demand effect)
 Has a negative effect (competition effect)
 If transportation costs are quadratic, the
competition effect is stronger than the demand
effectand firms prefer maximum differentiation.
Industrial Organization- Matilde Machado
The Hotelling Model
29
4.2. Hotelling Model
The social optimum solution is the one
that minimizes costs (or maximizes
utility) and it would be a=1/4 and 1b=3/4. Therefore, from a social point
of view the market solution leads to
too much differentiation.
Industrial Organization- Matilde Machado
The Hotelling Model
30
4.2. Hotelling Model
The social planner’s problem is:



Surplus of consumer x is:
 s-t(x-a)2-pA if he buys from A
 s-t(x-(1-b))2-pB if he buys from B
For each consumer, the seller’s profit is
 pA-c firm A
 pB-c firm B
Prices are therefore pure transfers between consumers and sellers
(note that here it is important the assumption that the market is
covered that is that s is sufficiently high), the total surplus
associated with a given consumer x is:
 s-t(x-a)2-pA+pA-c= s-t(x-a)2-c if he buys from A
 s-t(x-(1-b))2-pB+pB-c= s-t(x-(1-b))2-c if he buys from B
Industrial Organization- Matilde Machado
The Hotelling Model
31
4.2. Hotelling Model
To derive the social optimum we must first derive
the “indifferent” consumer :
s  t ( x% a ) 2  c  s  t ( x% (1  b)) 2  c
 ( x% a) 2  ( x% (1  b)) 2
 x%2  a 2  2ax% x%2  (1  b) 2  2(1  b) x%
 a 2  2ax% (1  b) 2  2(1  b) x%
 2 x%[1  b  a ]  (1  b) 2  a 2
 x%
(1  b  a)(1  b  a) (1  b  a)

 half the distance bweteen a and 1-b
2 1  b  a 
2
Industrial Organization- Matilde Machado
The Hotelling Model
32
4.2. Hotelling Model
The planner has to max total surplus which is
the same as minimize transportation costs
%
x
a
1b  a
2
1b
1
Min  t (a  z ) dz   t ( z  a) dz   t ((1  b)  z ) dz   t ( z  (1  b)) 2 dz
a ,b
0
a
1b
1444444444444
42 4444444444444
3 %x 1b2 a
buy from A
14444444444444444442 4444444444444444443
2
2
2
buy from B
0
Industrial Organization- Matilde Machado
a
x%
1-b
The Hotelling Model
1
33
4.2. Hotelling Model
a
%
x
1b  a
2
1b
1
Min  t (a  z ) 2 dz   t ( z  a) 2 dz   t ((1  b)  z ) 2 dz   t ( z  (1  b)) 2 dz
a ,b
0
a
1b
1444444444444
42 4444444444444
3 %x 1b2 a
buy from A
14444444444444444442 4444444444444444443
buy from B

3 a
3
(
a

z
)
(
z

a
)
 Min  

a ,b 
3 0
3

1b  a
2
a
(1  b  z )

3
3 1b
1b  a
2
( z  (1  b))

3



1b

3 1
 a 3 1  1  b  a 3 1  1  b  a 3 b 3 
 Min   
  
  
a ,b
2  3
2 
3 
 3 3 
Industrial Organization- Matilde Machado
The Hotelling Model
34
4.2. Hotelling Model
 a 3 1  1  b  a 3 1  1  b  a 3 b 3 
Min   
  
  
a ,b
2  3
2 
3 
 3 3 
The FOC:
2

2

0

4
a

1

b

a
 0 (A)


 a

   0  4b 2  1  b  a 2  0 (B)
 b
(A)-(B):
4a 2  4b 2  0  a 2  b 2  a  b
replacing in (A) implies that:
4a  1  a  a 
2
Industrial Organization- Matilde Machado
2
1
3
*
 0  a  ;(1  b ) 
4
4
*
The Hotelling Model
35
4.2. Hotelling Model
The basic conclusion of the Hotelling model is the principle
fo differentiation: firms want to differentiate as much as
possible in order to soften the price competition.
It may happen that some forces will lead firms to locate in
the the same location, usually the center (minimum
differentiation):
1) Firms may want to locate where demand is (i.e. in the
center)
2) In the case of no price competition (for example if prices
are regulated) firms may want to locate in the center
and split the market 50-50.
Industrial Organization- Matilde Machado
The Hotelling Model
36