4.2.Hotelling Model (VERY GOOD)

8/10/2019 4.2.Hotelling Model (VERY GOOD)

1/18

1

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 theirlocation.

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 adistance d to one of the sellers, its transportation cost istd 2 . This cost represents the value of time, gasoline, oradaptation to a product, etc.

6. Consumers have unit demands, they buy at most oneunit of the good {0,1}


2/18

2


4.2. Hotelling ModelGraphically

0 1

1

Location of firm ALocation of firm B

Mass ofconsumers =

11

00

1 1 0 1dz z= = =

x


4.2. Hotelling ModelThe transportation costs of consumer x:

Of buying from seller A areOf buying from seller B are

s gross consumer surplus - (i.e. its maximumwillingness to pay for the good)Lets assume s is sufficiently large for allconsumers to be willing to buy (this situation isreferred to as the market is covered). Theutility of each consumer is given by:U = s -p-td 2 where p is the price paid.

2tx( )21t x


3/18

3


4.2. Hotelling ModelWe first take the locations of the sellers as given

(afterwards we are going to determine themendogenously) and assume firms compete inprices.

1. Derive the demand curves for each of the sellers

2. The price optimization problem given thedemands


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

the consumer that is just indifferent betweenbuying 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 t x

t

A B x

Buy from A Buy from B

If (p B =p A ) then the indifferent consumer is at half the distance between A and B. If(p B -p A ) the indifferent consumers moves to the right, that is the demand for firm A

increases and the demand for firm B decreases.


4/18

4



0

A

1

B

x

p Ap B

Total cost toconsumer x: p A+tx 2

p B +t(1-x) 2

The equilibrium of the Hotelling model

s

U i

i


4.2. Hotelling ModelWe say the market is covered if all consumers buy.

Since the consumer with the lowest utility is theindifferent consumer (because it is the one whois further away from any of the sellers), we maysay that the market is covered if the indifferentconsumer buys i.e. if:

This condition is equivalent to say that s has to behigh enough

2

02

B A A

p p t s p t

t

+


5/18

5


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

+ = = = = = +

= = = = + = +

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 (p B -p A ) andnegatively on the transportation costs. If firms set the same prices

pB=pA then transportation costs do not matter as long as themarket is covered, firms split the market equally (and the indifferent

consumer is located in the middle of the interval ).


4.2. Hotelling ModelThe maximization problem of firm A is:

Because the problem is symmetric pA=p B=p*

( ) ( )

( )A

( , ) ( , )2

1FOC: 0 0

2 2

2 02

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

+ = =

+= =

+ + + + = = Firm As

reactioncurve

* ** *

2 2 2 p t c p t c

p p t c+ + +

= = = +Note that if t=0 (no

productdifferentiation) we

go back to Bertrandp*=c; *=0


6/18

6


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 B A

x

D p p x

D p p x D p p

t p c D t c c x

Note: The higher is t , the more differentiated are the goods from the point ofview 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) whichallows 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 firmscompete in prices) p A=p B>c

An increase in t implies more productdifferentiation. Therefore, firms compete lessvigorously (set higher prices) and obtain higherprofits.t=0 back to Bertrand


7/18

7



0

A

1

B x

p A=t+c p B =t+c

p A+tx 2

p B +t(1-x) 2

The equilibrium of the Hotelling model

s

U i

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 differentiationSi A=B, there is no differentiation, all consumers

will buy from the seller with the lowest price,back to Bertrand,pA=p B=c y A= B=0.


8/18

8


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 theirlocations

We solve the game backwards, starting from thesecond period.


4.2. Hotelling ModelSecond period:

Denote by a [0,1] the location of ADenote 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


9/18


10/18

10


4.2. Hotelling ModelInterpretation of the demand functions:

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

captive consumerstohalf of the consumers

between a and 1-b

if

1( , )

2

1( , )

2

=

= +

= +

A B

A A B

B A B

p p

b a D p p a

b a D p p b

( )

the right (ownbackyard)

sensitivity of the demandto 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 ModelGraphically

0 a 1-b 1

p A

x

p B

p A+t(x-a) 2

Firm Ascaptive market

Firm Bscaptive market


11/18

11


4.2. Hotelling Model2. Finding the reaction functions

( ) ( ) ( )

( )( )

( )

( )

1( , )

2 2 (1 )

1 1FOC: 0 0

2 2 (1 ) 2 (1 )

122 (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 pa p c

p t b a t b a

b a p p ca

t b a t b a

b a p p cat b a t b a

t p at b a

= = + +

= + + + = +

= + +

+ = + +

= + ( )2

1

2 2 Bb a p c ++

Firm As reactionfunction


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 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 pb p c

t b a t b a

b a p p cb

t b a

= = + +

=

+ + + =

+ + + =


12/18

12


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 34 (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 a p c b ab a

t b a

p c b at b a t b a

t b a b a p c

b ac t b a p c t b

+ ++ + + + + =

+ + + + + =

= + +

+ = +

= + + = + ) 1

3a b

a +


4.2. Hotelling Model2. 2. Finding the reaction functions

Note that prices are maximum when differentiation ismaximum (a=b=0; p A=p B=c+t) and minimumwhen there is no differentiation (a+b=1 (samelocation) and p A=p B=c)

* *( , ) (1 ) 1 and ( , ) (1 ) 13 3 B A

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

= + + = + +


13/18


14/18

14


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

( ) ( )

( ) ( ) ( )

( ) ( )

2

2

*

3( , ) 1

18

3 2 3( , )FOC: 1

18 18

3 1 3 0 018

A

a

A

b a Max a b t a b

b a b aa bt t a b

at

b a b a a

+ =

+ += +

= + + + < =


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

( ) ( )

( )( )

( )

( )( )

2

2

* *

3( , ) 1

183 2 3( , )

FOC: 118 18

3 1 3 0 0 1 118

B

b

B

b a Max a b t a b

b a b aa bt t a b

bt

b a b a b b

+ =

+ + = +

= + + + < = =


15/18

15


4.2. Hotelling ModelConclusion: Firms choose to be in the

extremes i.e. they choose maximumdifferentiation.

For firm A, for example, an increase in a (movementto the right):Has a positive effect because it moves towardswhere the demand is (demand effect)Has a negative effect (competition effect)If transportation costs are quadratic, thecompetition effect is stronger than the demandeffectand firms prefer maximum differentiation.



The social optimum solution is the onethat minimizes costs (or maximizesutility) and it would be a=1/4 and 1-b=3/4. Therefore, from a social pointof view the market solution leads totoo much differentiation.


16/18

16


4.2. Hotelling ModelThe social planners 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 sellers 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 iscovered that is that s is sufficiently high), the total surplusassociated with a given consumer x is:

s-t(x-a) 2-pA+p A-c= s-t(x-a) 2-c if he buys from As-t(x-(1-b)) 2-pB+p B-c= s-t(x-(1-b)) 2-c if he buys from B


4.2. Hotelling ModelTo 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


17/18

17


4.2. Hotelling ModelThe 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 b x

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

0 a 1-b 1 x



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 b x

a

a ba

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

a z z a Min

1 13 32

11

2

3 33 3

,

(1 ) ( (1 ))3 3

1 1 1 13 3 2 3 2 3

+

+

+

+ + +

b ab

b ab

a b

b z z b

a b a b a b Min


18/18



The FOC:

3 33 3

,

1 1 1 13 3 2 3 2 3a ba b a b a b

Min

+ + +

( )

( )

( )

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

of differentiation: firms want to differentiate as much aspossible 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 (minimumdifferentiation):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 centerand split the market 50-50.

Documents

4.2.Hotelling Model (VERY GOOD)