Industrial Organization- Matilde Machado The Hotelling Model 1
Matilde Machado
4.2. Hotelling Model
The model:
1. “Linear city” is the interval [0,1]
2. Consumers are distributed uniformely along this interval.
3. There are 2 firms, located at each extreme who sell the same good. The unique difference among firms is their location.
4. c= cost of 1 unit of the good
5. 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.
6. Consumers have unit demands, they buy at most one unit of the good {0,1}
Industrial Organization- Matilde Machado The Hotelling Model 3
Graphically
0 1
1
Location of firm A Location of firm B
Mass of consumers =
1 1 0 0
1dz=z = − =1 0 1
∫
x
4.2. Hotelling Model
The transportation costs of consumer x:
Of buying from seller A are
Of buying from seller B are
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.
tx2
(1 )2
t −x
Industrial Organization- Matilde Machado The Hotelling Model 5
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
4.2. Hotelling Model
In order to derive the demands we need to derive the consumer that is just indifferent between buying from A or from B:xɶ
2 2
2 2
2 2
is defined as the location where ( ) ( ) (1 )
(1 ) 2 2
2
=
⇔ − − = − − −
⇔ + = + −
⇔ + = + + −
⇔ = − +
− +
⇔ =
x x
A B
A B
A B
B A
B A
x U A U B
s p tx s p t x
p tx p t x
p tx p t tx tx
tx p p t
p p t
x t
ɶ ɶ
ɶ
ɶ ɶ
ɶ ɶ
ɶ ɶ ɶ
ɶ
ɶ A xɶ B
Buy from A Buy from 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 7
0 A
1 B xɶ
pA
pB Total cost to
consumer x: pA+tx2
pB+t(1-x)2
The equilibrium of the Hotelling model s
Ui
i
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:
This condition is equivalent to say that s has to be high enough
2
2 0
B A
A
p p t
s p t t
− +
− − ≥
Industrial Organization- Matilde Machado The Hotelling Model 9
Once we know the indifferent consumer, we may define the demand functions of A and B.
0 0 1
1
( , ) 1 1
2 2 2
1 1
( , ) 1 1 1
2 2 2 2
− + −
= = = = = +
− −
= = = − = − + = +
∫
∫
ɶ ɶ
ɶ ɶ
ɶ
ɶ
x
x B A B A
A A B
B A A B
B A B x
x
p p t p p
D p p dz z x
t t
p p p p
D p p dz z x
t t
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).
4.2. Hotelling Model
The maximization problem of firm A is:
Because the problem is symmetric ⇒pA=pB=p*
( ) ( )
( )
A
( , ) ( , )
2
FOC: 0 1 0
2 2
2 0
2
A
A B A
A B A A A B A
p
B A
A A
B
B A A
p p t
Max p p p c D p p p c
t
p p t
p c
p t t
p t c
p p t c p
− +
Π = − = −
− +
∂Π = ⇔ − − =
∂
⇔ − + + = ⇔ = + + Firm A’s reaction curve
* *
* *
2 2 2
p t c p t c
p = + + ⇔ = + ⇔ p = +t c Note that if t=0 (no
product differentiation) we go back to Bertrand
p*=c; Π*=0
Industrial Organization- Matilde Machado The Hotelling Model 11
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 2
( , ) 1 1 ( , )
2
2
=
= =
= − = =
Π = Π = − = + − =
A A B
B A B A A B
A B
A
x
D p p x
D p p x D p p
p c D t c c x t ɶ
ɶ ɶ
ɶ
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
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 13
0 A
1 B
½ xɶ
pA=t+c pB=t+c
pA+tx2
pB+t(1-x)2
The equilibrium of the Hotelling model s
Ui
i
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 ΠA=ΠB=0.
Industrial Organization- Matilde Machado The Hotelling Model 15
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.
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 17
1. The indifferent consumer: Uxɶ( )A =U Bxɶ( )
Hence if pA=pB, A’s demand is a+(1-b-a)/2
( )
( ) ( )
( )
( ) ( )( )
( )
( )
2 2
2 2 2 2
2 2
2 2
2 2
( ) ( (1 ))
2 (1 ) 2 (1 )
2 1 (1 )
(1 ) (1 )
2 1 2 1
1 1
2 1 2 1
2 1
+ − = + − −
⇔ + + − = + + − − −
⇔ − − = − + − −
− + − −
− + − −
⇔ = =
− − − −
− − − +
⇔ = − +
− − − −
⇔ = −
− −
A B
A B
B A
B A
B A
B A
B A
p t x a p t x b
p tx ta txa p tx t b tx b
tx b a p p t b ta
p p t b a
p p t b ta
x t b a t b a
b a b a
p p
x t b a b a
p p
x t b a
ɶ ɶ
ɶ ɶ ɶ ɶ
ɶ
ɶ
ɶ
ɶ ( )
( ) ( )
1 1
2 2 1 2
− + − − −
+ = + +
− −
B A
b a p p b a
t b a a
4.2. Hotelling Model
Demands are:
( ) ( )
( ) ( )
( )
( , ) 1
2 1 2
( , ) 1 1 1
2 1 2
1
2 1 2
− − −
= = + +
− −
− − −
= − = − − −
− −
− − −
= + +
− −
B A
A A B
B A
B A B
A B
p p b a
D p p x a
t b a p p b a
D p p x a
t b a
p p b a
t b a b ɶ
ɶ
a 1-b
0 1
a (1-b-a)/2
Industrial Organization- Matilde Machado The Hotelling Model 19
Interpretation of the demand functions:
( )
captive consumers
to the left (own half of the consumers backyard) between a and 1-b
captive consumers half of the consumers to
between a and 1-b
if
( , ) 1
2
( , ) 1
2
=
= + − −
= − − +
A B
A A B
B A B
p p
D p p a b a
D p p b a b
( )
the right (own backyard)
sensitivity of the demand to price difference
if
( , ) 1
2 2 (1 )
≠
− − −
= + +
− −
A B
B A
A A B
p p
b a p p
D p p a
t b a
4.2. Hotelling Model
Graphically
0 a 1-b 1
pA
xɶ
pB pA+t(x-a)2
Industrial Organization- Matilde Machado The Hotelling Model 21
2. Finding the reaction functions
( ) ( ) ( )
( ) ( )
( )
( )
( , ) 1
2 2 (1 )
1 1
FOC: 0 0
2 2 (1 ) 2 (1 )
2 1
2 (1 ) 2 2 (1 )
1
(1 ) 2 2 (1 )
(1 )
A
A B A
A A A B A
p
A
B A
A A
A B
A B
A
b a p p
Max p c D p p p c a
t b a
b a p p
a p c
p t b a t b a
p b a p c
t b a a t b a
p b a p c
t b a a t b a
p at b a t
− −
−
Π = − = − + +
− −
− − −
∂Π = ⇔ + + + − − =
∂ − − − −
− − +
⇔ = + +
− − − −
− − +
⇔ = + +
− − − −
⇔ = − − + (1 )2
2 2
b a pB c
− − + + Firm A’s reaction
function
♦
4.2. Hotelling Model
2. Finding the reaction functions
( ) ( ) ( )
( ) ( )
( )
( , ) 1
2 2 (1 )
FOC: 0
1 1
2 2 (1 ) 2 (1 ) 0
1 2
2 2 (1 ) 0
B
B A B
B B A B B
p
B
B
A B
B
A B
b a p p
Max p c D p p p c b
t b a
p
b a p p
b p c
t b a t b a
b a p p c
b t b a
− −
−
Π = − = − + +
− −
∂Π = ⇔
∂
− − −
⇔ + + − − + − − − − =
− − − +
⇔ + + =
− −
♦
Industrial Organization- Matilde Machado The Hotelling Model 23
2. 2. Finding the reaction functions
( )
( )
( )
1 2 1 1
2 2 (1 ) 2 2 2 (1 ) 0
3 3 1 1 1
4 (1 ) 2 2 4 0
3 3 3
4 (1 ) 4 (1 ) 4 4 4
3 (1 )
3
(1 ) 1 y (1
3
B B
B
B
B
A
b a p c b a p c
b a
t b a t b a
b a
p c b a
b a
t b a
p c b a
t b a t b a
t b a b a
p c
b a
c t b a p c t b
− − − + − − +
+ + − − + + + − − =
− + − − − −
⇔ + + + + =
− −
⇔ = + + −
− − − −
+ − − −
⇔ = +
−
= + − − + = + − −
) 1
3 a b a −
+
4.2. Hotelling Model
2. 2. Finding the reaction functions
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)
* *
( , ) (1 ) 1 and ( , ) (1 ) 1
3 3
B A
b a a b
p a b c t b a − p a b c t b a −
= + − − + = + − − +
Industrial Organization- Matilde Machado The Hotelling Model 25
3. 1st period, simultaneous choice of a and b Profits are functions of (a, b) alone:
Replace and we get a function of a and b alone. Take the FOC as always with respect to a and b.
( )
( )
* * *
* * *
( , ) ( , ) ( , , ( , ), ( , )) ( , ) ( , ) ( , , ( , ), ( , ))
A
A A A B
B
B B A B
a b p a b c D a b p a b p a b a b p a b c D a b p a b p a b
Π = −
Π = −
*( , ), *( , ), *( , ), *( , )
A B A B
p a b p a b D a b D a b
4.2. Hotelling Model
3. 1st period, simultaneous choice of a and b
( )
( )
* *
* *
3 3 3 6
( , ) 1 1 1
3 2 (1 ) 2
but 2 (1 )
3 which simplifies:
( , ) 1 1 1
3 3 2
+ − − +
=
−
− − −
Π = + − − + − + +
− −
−
− = − −
− − − +
Π = − − + +
A B A
B A
A
a b b a
p p
a b b a
a b c t a b c a
t a b p p t a b b a
a b b a b a
a b t a b
( ) (3 )2
1 18
= − − − +
b a t a b
Industrial Organization- Matilde Machado The Hotelling Model 27
3. 1st period, simultaneous choice of a and b
( ) ( )
( ) ( ) ( )
( )( )
2
2
*
( , ) 1 3
18
3 2 3
( , )
FOC: 1
18 18
3 1 3 0 0
18
A a
A
b a Max a b t a b
b a b a
a b t t a b
a
t b a b a a
Π = − − − +
− + − +
∂Π = − + − −
∂
= − − + + + < ⇒ =
4.2. Hotelling Model
3. 1st period, simultaneous choice of a and b
( ) ( )
( ) ( ) ( )
( )( )
2
2
* *
( , ) 1 3
18
3 2 3
( , )
FOC: 1
18 18
3 1 3 0 0 1 1
18
B b
B
Max a b t a b b a
b a b a
a b t t a b
b
t b a b a b b
Π = − − + −
+ − + −
∂Π = − + − −
∂
= − + − + + < ⇒ = ⇔ − =
Industrial Organization- Matilde Machado The Hotelling Model 29
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.
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 1-
b=3/4. Therefore, from a social point
of view the market solution leads to
too much differentiation.
Industrial Organization- Matilde Machado The Hotelling Model 31
The social planner’s problem is:
Surplus of consumer x is:
s-t(x-a)2-pAif he buys from A
s-t(x-(1-b))2-pBif 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
4.2. Hotelling Model
To derive the social optimum we must first derive the “indifferent” consumer :
[ ]
( )
2 2
2 2
2 2 2 2
2 2
2 2
( ) ( (1 ))
( ) ( (1 ))
2 (1 ) 2(1 )
2 (1 ) 2(1 )
2 1 (1 )
(1 )(1 ) (1 )
half the distance bweteen a and 1-b
2 1 2
− − − = − − − −
⇔ − = − −
⇔ + − = + − − −
⇔ − = − − −
⇔ − − = − −
− − − + − +
⇔ = = =
− −
s t x a c s t x b c
x a x b
x a ax x b b x
a ax b b x
x b a b a
b a b a b a
x b a
ɶ ɶ
ɶ ɶ
ɶ ɶ ɶ ɶ
ɶ ɶ
ɶ ɶ
Industrial Organization- Matilde Machado The Hotelling Model 33
The planner has to max total surplus which is the same as minimize transportation costs
1
1 1
2
2 2 2 2
, 0 1 1
buy from A 2
buy from B
( ) ( ) ((1 ) ) ( (1 ))
=− + −
− + −
=
− + − + − − + − −
∫ ∫ ∫ ∫
x b a
a b
a b a x b a b
Min t a z dz t z a dz t b z dz t z b dz
ɶ
ɶ
0 a xɶ 1-b 1
4.2. Hotelling Model
1
1 1
2
2 2 2 2
, 0 1 1
buy from A 2
buy from B 1
3 3
,
0
( ) ( ) ((1 ) ) ( (1 ))
( ) ( )
3 3
=− + −
− + −
=
− + − + − − + − −
− −
⇔ − +
∫ ∫ ∫ ∫
x b a
a b
a b a b a b
x
a
a b
a
Min t a z dz t z a dz t b z dz t z b dz
a z z a
Min
ɶ
ɶ
1 1
3 3
2
1 1
2
3 3
3 3
,
(1 ) ( (1 ))
3 3
1 1 1 1
3 3 2 3 2 3
− + −
− + −
− − − −
− +
− − − −
⇔ + + +
b a b
b a b
a b
b z z b
a b a b a b
Min
Industrial Organization- Matilde Machado The Hotelling Model 35
The FOC:
,
1 1 1 1
3 3 2 3 2 3
a b
a b a b a b
Min
− − − −
+ + +
( )
( )
( )
2 2
2 2
2 2 2 2
2 2 * *
0 4 1 0 (A)
0 4 1 0 (B)
(A)-(B):
4 4 0
replacing in (A) implies that:
1 3
4 1 0 ; (1 )
4 4
a b a
a
b b a
b
a b a b a b
a a a a b
∂
= ⇔ − − − =
∂
∂
= ⇔ − − − =
∂
− = ⇔ = ⇔ =
− − − = ⇔ = − =
4.2. Hotelling Model
The basic conclusion of the Hotelling model is the principle of 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 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.
