Industrial Organization- Matilde Machado The Hotelling Model 1 4.2. Hotelling Model Matilde Machado.

Industrial Organization- Matilde Machado The Hotelling Model 1

4.2. Hotelling Model

Matilde Machado


4.2. Hotelling ModelThe 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 good5. 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}


4.2. Hotelling ModelGraphically

0 1

1

Location of firm ALocation of firm B

Mass of consumers =

11

00

1 1 0 1dz z

x


4.2. Hotelling ModelThe 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.

2tx 21t x


4.2. Hotelling ModelWe 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 ModelIn order to derive the demands we need to derive

the consumer that is just indifferent between buying from A or from B:

x

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 tx

t

% %%

% %

% %

% % %

%

% A BxBuy 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.



0

A

1

B

x

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 ModelWe 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

02

B AA

p p ts p t

t


4.2. Hotelling ModelOnce we know the indifferent consumer, we may

define the demand functions of A and B.

00

11

1( , ) 1

2 2 2

1 1( , ) 1 1 1

2 2 2 2

%%

%%

%

%

xx B A B A

A A B

B A A BB A B x

x

p p t p pD p p dz z x

t t

p p p pD 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 ModelThe maximization problem of firm A is:

Because the problem is symmetric pA=pB=p*

A

( , ) ( , )2

1FOC: 0 0

2 2

2 02

A

A B AA B A A A B A

p

B AA

A

BB A A

p p tMax p p p c D p p p c

t

p p tp c

p t t

p t cp p t c p

Firm A’s reaction

curve

* ** *

2 2 2

p t c p t cp p t c

Note that if t=0 (no product

differentiation) we go back to Bertrand

p*=c; *=0


4.2. Hotelling ModelOnce the equilibrium prices are determined, we may

determine the other equilibrium quantities:

*

* * *

* * * * *

* * * * *

1 (the indifferent consumer is in the middle because prices are equal)

21

( , )2

1( , ) 1 ( , )

2

2

A A B

B A B A A B

A BA

x

D p p x

D p p x D p p

tp c D t c c 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


4.2. Hotelling ModelObservations: 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



0

A

1

B

½x%

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 ModelHow 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.


4.2. Hotelling ModelGeneral 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 ModelSecond period: Denote by a the location of A Denote by (1-b) 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


4.2. Hotelling Model1. The indifferent consumer: ( ) ( )x xU A U B

Hence if pA=pB, A’s demand is a+(1-b-a)/2

2 2

2 2 2 2

2 2

2 22 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 AB 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 ap p t b tax

t b a t b a

b a b ap px

t b a b a

p px

t b a

% %

% % % %

%

%

%

%

1 1

2 2 1 2

B A

b a b ap pa

t b a


4.2. Hotelling ModelDemands are:

1( , )

2 1 2

1( , ) 1 1

2 1 2

1

2 1 2

B AA A B

B AB A B

A B

b ap pD p p x a

t b a

b ap pD p p x a

t b a

p p b ab

t b a

%

%

a 1-b0 1

a (1-b-a)/2


4.2. Hotelling ModelInterpretation of the demand functions:

{

captive consumers to the left (own half of the consumersbackyard) between a and 1-b

captive consumers tohalf of the consumers

between a and 1-b

if

1( , )

2

1( , )

2

A B

A A B

B A B

p p

b aD p p a

b aD p p b

144424443

1442443{

the right (own backyard)

sensitivity of the demandto price difference

if

1( , )

2 2 (1 )

A B

B AA A B

p p

b a p pD p p a

t b a14444244443


4.2. Hotelling ModelGraphically

0 a 1-b 1

pA

x

pB

pA+t(x-a)2

Firm A’s captive market

Firm B’s captive market


4.2. Hotelling Model2. Finding the reaction functions

1( , )

2 2 (1 )

1 1FOC: 0 0

2 2 (1 ) 2 (1 )

12

2 (1 ) 2 2 (1 )

1

(1 ) 2 2 (1 )

(1 )

A

A B AA A A B A

p

AB A

AA

A B

A B

A

b a p pMax p c D p p p c a

t b a

b a p pa p c

p t b a t b a

b ap p ca

t b a t b a

b ap p ca

t b a t b a

tp at b a

21

2 2B

b a p c

Firm A’s reaction function


4.2. Hotelling Model2. Finding the reaction functions

1( , )

2 2 (1 )

FOC: 0

1 10

2 2 (1 ) 2 (1 )

1 20

2 2 (1 )

B

B A BB B A B B

p

B

B

A BB

A B

b a p pMax p c D p p p c b

t b a

p

b a p pb p c

t b a t b a

b a p p cb

t b a


4.2. Hotelling Model2. 2. Finding the reaction functions

1 2 1 10

2 2 (1 ) 2 2 2 (1 )

13 3 1 10

4 (1 ) 2 2 4

3 3 3

4 (1 ) 4 (1 ) 4 4 4

3 (1 )

3

(1 ) 1 y (13

B B

B

B

B

A

b a p c b a p cb a

t b a t b a

b ap c b ab a

t b a

p c b a

t b a t b a

t b a b ap c

b ac t b a p c t b

) 13

a ba


4.2. Hotelling Model2. 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 ) 13 3B A

b a a bp a b c t b a p a b c t b a


4.2. Hotelling Model3. 1st period, simultaneous choice of a and bProfits 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.

* * *

* * *

( , ) ( , ) ( , , ( , ), ( , ))

( , ) ( , ) ( , , ( , ), ( , ))

AA A A B

BB 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

* * * *( , ), ( , ), ( , ), ( , )A B A Bp a b p a b D a b D a b


4.2. Hotelling Model3. 1st period, simultaneous choice of a and b

* *

* *

3363

1( , ) 1 1

3 2 (1 ) 2

but 2 (1 )3

which simplifies:

1( , ) 1 1

3 3 2

A B A

B A

A

b aa b

p pa b b aa b c t a b c a

t a b

b ap p t a b

a b b a b aa b t a b

144 214444244443 2

31

18

b at a b

4444 4444443



2

2

*

3( , ) 1

18

3 2 3( , )FOC: 1

18 18

3 1 3 0 018

A

a

A

b aMax a b t a b

b a b aa bt t a b

at

b a b a a



2

2

* *

3( , ) 1

18

3 2 3( , )FOC: 1

18 18

3 1 3 0 0 1 118

B

b

B

b aMax a b t a b

b a b aa bt t a b

bt

b a b a b b


4.2. Hotelling ModelConclusion: 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.



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.


4.2. Hotelling ModelThe social planner’s problem is:

Surplus of consumer x is:s-t(x-a)2-pA if he buys from As-t(x-(1-b))2-pB if he buys from B

For each consumer, the seller’s profit ispA-c firm ApB-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 As-t(x-(1-b))2-pB+pB-c= s-t(x-(1-b))2-c if he buys from B



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 ax

b a

% %

% %

% % % %

% %

%

%



The planner has to max total surplus which is the same as minimize transportation costs

11 12

2 2 2 2

,10 1

2buy from Abuy from B

( ) ( ) ((1 ) ) ( (1 ))

b a

xa b

a bb aa bx

Min t a z dz t z a dz t b z dz t z b dz

%

%14444444444444244444444444443144444444444444444424444444444444444443

0 a 1-b 1x%



11 12

2 2 2 2

,10 1

2buy from Abuy from B

13 3

,0

( ) ( ) ((1 ) ) ( (1 ))

( ) ( )

3 3

b a

xa b

a bb aa bx

a

a ba

Min t a z dz t z a dz t b z dz t z b dz

a z z aMin

%

%14444444444444244444444444443144444444444444444424444444444444444443

1 13 32

11

2

3 33 3

,

(1 ) ( (1 ))

3 3

1 1 1 1

3 3 2 3 2 3

b ab

b ab

a b

b z z b

a b a b a bMin



The FOC:

3 33 3

,

1 1 1 1

3 3 2 3 2 3a b

a b a b a bMin

22

22

2 2 2 2

22 * *

0 4 1 0 (A)

0 4 1 0 (B)

(A)-(B):

4 4 0

replacing in (A) implies that:

1 34 1 0 ;(1 )

4 4

a b aa

b b ab

a b a b a b

a a a a b


4.2. Hotelling ModelThe 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.

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Industrial Organization- Matilde Machado The Hotelling Model 1 4.2. Hotelling Model Matilde Machado.

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