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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
30
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
36
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
37
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.
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