1

Course outline II

Product differentiation

Advertising competition

Compatibility competition

Heterogeneous

goods

2

Competition on variants, locations,

and qualities

Basic idea of product differentiation

The Hotelling Model

The Schmalensee-Salop model

Competition on qualities and variations

Executive summary

3

Differentiating products in order to

overcome the Bertrand paradox

With homogeneous goods, competition can

be quite intense: Even in a market with only

two competitors, firms may face a zero-profit

situation in a Bertrand-Nash equilibrium.

Differentiating products may help to achieve

positive profits.

4

Preferences (Example: drinks)

Homogeneous preferences Diffuse preferences

Clustered preferences

calorie content

sweetness

calorie content

calorie content

sweetness

sweetness

5

Example:

product differentiation of drinks

Calorie content

Sweetness

Coca-Cola

Mineral water

Cola light

(nonalcoholic)

beer

6

Product differentiation

Horizontal product differentiation:

Some consumers prefer a good (or rather a

feature), while others prefer a different good

(or its feature).

Vertical product differentiation (quality):

A good is regarded as better than the other by

all consumers (unanimous ranking).

7

Audi

A3

Mercedes

A-Class

BMW

1 Series

Audi

A4

Mercedes

C-Class

BMW

3 Series

Audi

A8

BMW

7 Series

Mercedes

S-Class

Horizontal vs. vertical

differentiation

A

B

horizontal product differentiation within a quality class

line of

Competition

price

quality

vertical

product differentiation

(different qualities)

Audi

A6

Mercedes

E-Class

BMW

5 Series

8

Long-term and short-term action

parameters

Prices

Quantities

•Variants and locations (horizontal differentiation)

•Qualities (vertical differentiation)

•Recognition, image (image differentiation)

•Compatibility (compatibility differentiation)

9

The Hotelling Model

Linear city of length 1

Interpretation

– Competition on location: Two firms offer the

same product in different places.

– Competition on variants: Two firms offer

differentiated products in one place.

10 a 1 h a 2

10 21 aa

10

Locations or

range of variants

home turf

10

x1x2

a1 1 2 a12 a

1a ha

12 a

1a 2ah

home turf

Demand in the case of identical

prices

12

212

aaa

aaa

11

10

2

1 )( ahpt

a 1 h a 2

The consumer at h prefers producer 1’s good if:

21aht 22aht

effeff pahtpahtpp 2

2

22

2

111

Costs of transport

12

Proportionate demand with

uniform distribution

0 1

1

hh*

hppx ),( 211x p p h2 1 2 1( , ) *

The consumers are supposed to be

equally distributed over the interval

(constant density of consumers).

The consumer in h is indifferent between

good 1 and good 2.

13

The demand function

Firm 1’s demand function:

222

2

11 ahtpahtp

intensity of competition

consumers in case

of equal prices

firm 1’s

price

advantage

*:

22 12

1212 haat

ppaah

12212112

1*,,, pp

atahaappx

12

212

aaa

aaa

14

Exercise (Hotelling)

Deduce !

Calculate all equilibria in the simultaneous location

competition, if prices are given by

21212 ,,, aappx

cpp 21

15

The two-stage differentiation game

a1

a2

p1

p2 2

1

16

Solving the pricing game I

Profit functions

Reaction functions

2

2221

aatcppp R

at

ppacppp

2)(),( 12

1211

at

ppacppp

21)(),( 21

2212

2

12112

aatcppp R

17

Solving the pricing game II

Bertrand-Nash equilibrium

Output levels

Profits

When do the firms earn the same profits and

why?

aatcpaatcp BB 2

3

2,1

3

221

0)2(9

2),(;01

9

2),( 2

212

2

211 aataaaataa BB

0)2(3

10)1(

3

121 axandax BB

18

)( 21 ppR

Equilibrium in the simultaneous

competition

p 1B

p1

Bp2

Bp1

Bp2

p 2

2

12112

aatcppp R

19

Exercises (elasticity,

sequential price competition)

Find the price elasticity of demand in the case

of

Assume maximal differentiation ( ).

Find the Bertrand equilibrium in the case of

sequential price competition. Calculate the

profits.

1,0 21 aa

.121 aandpp

1

2

a 1

a 2

p 2p 1

20

Depicting the equilibria

)( 12 ppR

)( 21 ppRp 2

p 1

Bp1

tcppp

tcp

BSRBS

BS

4

5

,2

3

122

1

Prices in simultaneous

price competition

Prices in

sequential price

competition

BSp1

BSp2

Bp2

21

Exercise (Strategic trade policy)

Two firms, one domestic (d), the other foreign

(f), engage in simultaneous price competition

on a market in a third country. Assume .

The domestic government subsidizes its firm’s

exports using a unit subsidy s.

Which subsidy s maximizes domestic welfare

?scsxscsW B

d

B

d

1a

22

p f

p dB

p1

Bp2

B

dp

B

fp

)(spB

d

)(spB

f

Depicting the solution

)( f

R

d pp

)( d

R

f pp

),( spp f

R

d

23

Exercise (linear costs of transport)

Find the demand functions and the Bertrand equilibrium in

the case of and linear cost of transport, i.e. t(h-0) for

buying x1 and t(1-h) for buying x2.

1a

24

Equilibrium locations (1st stage)

Reduced profit functions:

Influence of location on profit functions:

Nash equilibrium: 1,0 21 NN aa

100since1

100since0

21

2

212

21

1

121

aafora

aa

aafora

aa

BR

BR

029

2),(

019

2),(

2

212

2

211

aataa

aataa

B

B

25

Firm 1’s reduced profit function

(1st stage)

10.80.60.40.20

19

t

49

t

39

t

29

t

2.02 a

4.02 a

6.02 a

8.02 a

0.12 a

1influence of firm 1’s choice

of location on its profit

(with several locations of

firm 2 given)

1a

26

Equilibrium outcomes

Maximal differentiation:

Prices

Output levels and profits

tandxx BBBB

2

1

2

12121

tcpp BB 21

1a

27

Lerner index (Hotelling)

Lerner index for one firm

= Lerner index for the industry (equal

costs):

1

1

tc

t

tc

c-tc

t

cp

MCp

28

Exercises (sequential choice of

location, clusters)

Which locations would you expect in the case

of sequential choice of location?

Why do firms often form clusters in reality?

a 1

p 1

p 2a 2

2

1

29

Direct and strategic effects for

accommodation

Firm 1’s reduced profit function:

? =0 >0 <0direct or profit maximizing strategic effect

demand effect prices in equilibrium of positioningof 2nd stage

(Envelope theorem)

1

2

2

1

1

1

1

1

1

1

1

1

a

p

pa

p

paa

BBB

),(),,(,,),( 212211211211 aapaapaaaa BBB

*in most cases >0

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Exception: negative direct effect

10 a 1h a 2

x1 x2

:12 pp

10 a 1h a 2

x1 x2

01

1

a

31

Direct and strategic effects for

deterrence

?* ?* =0 >0 <0

direct strategic effects

effect

1

1

1

2

1

2

2

2

1

2

1

2

a

p

pa

p

paa

BBB

),(),,(,,),( 211212212212 aapaapaaaa BBB

*in most cases <0

32

Welfare analysis

Equilibrium locations are a1=0 and a2=1

Total quantity is exogenous, at 1.

costs of transport should be minimized.

Locations a1=0.25 and a2=0.75 minimize

transportation costs (too much

differentiation).

33

1

2

3

4

5

x1

5

1a

x2

x3

x4

x5

The circular city

34

The Schmalensee-Salop model

Model for the analysis of blockade, deterrence

(limit number of variants)

Circular city of length 1

Firms are uniformly distributed

The circular city can be considered to be made

out of n linear cities.

naaa kk

11

na

1

2

1

35

The demand function I

Indifferent consumer between firm 1 and 2

*

2,1

12

1212

2

22

2

11

:22

haat

ppaah

ahtpahtp

n

nt

pp

aa

aat

pphax

1

2

1

12

22

21

12

12

21*

2,122 2,1

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The demand function II

Indifferent consumer between firm 2 and firm 3:

Firm 2’s demand function:

*

3,2

23

2323

22h

aat

ppaa

n

nt

ppaa

aat

ppahx

1

2

1

12

22

2323

23

232

*

3,22 3,2

231222 22

13,22,1

pppt

n

nxxx

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Entry and pricing decisions

The firms decide whether to enter the market:

– all firms simultaneously and equidistantly

– potential competitors midway between two established

firms.

Firms incur location costs of CF.

Game structure (note ):

Entry firm 1

:

:

Entry firm k

p 1

:

:

p n

1

:

:

k

kn

38

Solving the pricing game

Firm 2’s profit function:

Firm 2’s reaction function:

Symmetric Nash equilibrium:

231312

224

1,

n

tcpppppR

22221 ,...,,n

tcp

n

tcp

n

tcp B

n

BB

FCpppt

nn

cp

23122 2

2

11

F

B

i Cn

tn

3

39

Equilibrium number of firms

with free entry

Profit function depending on number of firms:

Entry:

F

B

i Cn

tn

3

3 2

max

max3

3

)(

:0

F

B

F

F

B

i

tCcnp

nC

tnC

n

tn

40

Lerner index (Schmalensee)

Lerner index for one firm

= Lerner index for the industry (equal

costs):

1

12

2

2

2

2

ct

n

n

tc

n

t

n

tc

cn

tc

p

MCp

41

Market equilibrium

While the price is above marginal costs, entry costs

prevent firms from realizing profits.

An equilibrium with zero profits prevails.

The lower the costs of entry the higher the number of

entering firms.

The higher the costs of transport the higher the price

and the number of entering firms.

42

Entry deterrence

1st stage:

The established firms choose the number of

variants/locations.

2nd stage:

Potential competitors decide whether to enter the

market.

3rd stage:

All firms compete in prices.

43

Entry by a potential competitor

1

2

3

E

a 1

6

a 1

6

a 1

3

a 1

3

x2

1

3

x3

1

4

x1

1

4

xz 1

6

44

Product proliferation

If there are n established firms, the potential entrant‘s profit

expectation is determined by 2n.

Limit variants or limit locations:

The established firms are able to realize positive profits

while deterring entry.

F

B

i

B

E Cn

tnn

32

2

LL nnnnn

22

maxmax

3

2

1:

F

L

C

tn

45

Linear costs of transport

Consider linear cost of transport and

keep all other assumptions of our models.

Firm 2‘s demand functions (located between firms

1 and 3):

ii ahthC )(

t

ppp

n

t

pp

nt

pp

n

t

ppaa

effeff

xxx

xahx

h

hatpahtp

hphp

2

21222

221

2221

2

*

3,22

22

*

3,2

*

3,2332

*

3,22

*

3,23

*

3,22

231

2,13,2

21

2,1

23

3,2

2332

:analogous

46

Exercise (linear costs of transport)

Calculate the price reaction function for

firm 2 and the symmetric Bertrand

equilibrium (p1=p2=...=pn).

Find the maximal number of firms and the

limit locations.

2

1

22)(

4

1:S. 312

F

L

B

R

C

tn

n

tcp

n

tcppp

47

Executive summary I

Differentiation of products gives some monopolistic power to firms.

Direct effect: If prices are fixed, “moving towards” the other firm pays in terms of sales and profits (direct effect). However: geographical nearness may enhance business (furniture shops clustered together).

Strategic effect: Prices go down because of diminished differentiation (strategic effect).

Both effects work towards deterrence.

48

Executive summary II

The more firms are in the market, the lower prices,

outputs and profits. Therefore, incumbent firms may

try to drive competitors out of business and deter

entry by product proliferation.

From the social welfare point of view, competition on locations and variants need not lead to optimal product differentiation.

49

Competition on qualities and

variants

1

1 0 h

v

• Maximal horizontal

product differen-

tiation: h position of

consumer in hori-

zontal product space.

• Quality differentia-

tion,

v consumers’

• willingness to pay

for quality.

quality

(vertical)

variation (horizontal)

:10 12 qq

50

Competition on qualities and

variations - demand function

Linear costs of transport: t(h-0) for firm 1 and

t(1-h) for firm 2

Consumer buys product 1 if

Derivation of demand curve:

t

qvph

vqhtpvqthp

22

1

1 2211

where 2112 , qqqppp

51

Competition on qualities and

variations - results

Low costs of transport

maximum quality

differentiation

1

2

p 1

p 2

q 1

q 2

91

2

32

2

2

94

1

32

1

1 01

B

B

N

B

B

N

cp

q

cp

q

t

tcp

q

t

tcp

q

B

B

N

B

B

N

21

2

2

2

21

1

1

1 11

High costs of transport

maximum (costless!)

quality

52

Executive summary

Horizontal product differentiation pays.

If horizontal product differentiation is

expensive or difficult, vertical product

differentiation may also help to avoid the

Bertrand paradox.

If horizontal product differentiation is

possible, firms will choose the maximal

quality in case of costless quality.