4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Product differentiation
Horizontal differentiation
4820–4
Linear city
Circular city
Geir B. Asheim
Advertising
Department of Economics, University of Oslo
ECON4820
Spring 2010
Last modified: 2010.02.16
The extent of the market — Different products or
differentiated variants of the same product
4820–4
Product
different. I
Geir B.
Asheim
How far does the market extend?
Which firms compete with each other?
What is an industry?
Introdution
Questions
Outline
Bertrand
model
Linear city
Circular city
Advertising
Products are not homogeneous
Exceptions: Gasoline, electricity
Still, some products are more equal to each other than to
other products in the economy. These products constitute
an industry with differentiated products.
But where do we draw the line? Examples:
beer vs. soda?
soda vs. milk?
beer vs. milk?
Two kinds of product differentiation
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Questions
Outline
Bertrand
model
1 Horizontal differentiation
Ex: Different location of stores
Different time slots for airline departures
Consumers differ in their preferences over the product’s
characteristics (as color, taste, location of outlet).
The optimal choice for identical prices depends on consumer
preferences. Covered here.
Linear city
Circular city
Advertising
2 Vertical differentiation
Ex: Cars of different quality and in different price segments
Products differ in some characteristic (called quality ) in
which all consumers agree what is best.
For equal prices, all prefer the variant with higher quality.
The optimal choice is the same for everyone.
For different prices, consumer make different choice (e.g.
due to differences in income). Covered next time.
Horizontal product differentiation
Positive questions
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Questions
Outline
Bertrand
model
Linear city
Circular city
Advertising
Pricing behavior for given number of firms and given
locations. How does product differentiation influence the
firms’ price competition?
Location choice for given number of firms. How do firms
differentiate their products to weaken price competition?
Entry decisions.
Horizontal product differentiation
Normative questions
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Questions
Outline
Bertrand
model
Is the product variation too large in equilibrium?
Too much product differentiation?
Linear city
Circular city
Advertising
Are there too many variants in equilibrium?
Too many firms enter?
Outline
4820–4
Product
different. I
Geir B.
Asheim
Bertrand model with differentiated products
Introdution
Questions
Outline
Bertrand
model
Linear city
Spatial competition
The linear city
The circular city
Circular city
Advertising
Advertising and informational product differentiation
Informative?
Persuasive?
Bertrand model with differentiated products (1)
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Structure: (1) Prices set (2) Demand determines quantities
Firms 1 and 2 have constant unit cost c1 and c2 and set prices
p1 and p2 simultaneously. Sales are given by Di (pi , pj ), which is
the continuous demand function for firm i.
Bertrand
model
max(pi − ci )Di (pi , pj )
pi
Linear city
Circular city
Advertising
Why does a continuous demand function mean that products are
differentiated?
Result
Let c1 = c2 = c. If D1 (p1 , p2 ) > 0 when p1 ≤ min{c, p2 } and
D2 (p2 , p1 ) > 0 when p2 ≤ min{c, p1 }, then in any Nash
equilibrium (p1∗ , p2∗ ) we have that (p1∗ , p2∗ ) (c, c).
Bertrand model with differentiated products (2)
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Linear city
Circular city
Advertising
Proof.
To be shown: (p1 , p2 ) is not a Nash equilibrium if
min{p1 , p2 } ≤ c.
Case 1: min{p1 , p2 } < c. At least one firm i has negative profit.
Non-negative profit by setting pi ≥ c.
Case 2: min{p1 , p2 } = c. If pi = c and pj ≥ c, then by
assumption Di (pi , pj ) > 0, and firm i’s profit is zero.
Since Di (·, ·) is continuous, firm i can attain positive profit by
setting pi = c + ε for ε > 0 sufficiently small.
Hence, differentiated products solve the Bertrand paradox,
in the sense that it leads to price exceeding marginal cost.
The linear city model
with linear transportation costs (1)
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Following Hotelling (1929), consider a linear city of length 1 with
consumers uniformly distributed along this interval. Also, assume
that firm 1 is located at 0 and firm 2 is located at 1.
For a consumer located at x (∈ [0, 1]) the cost to buy from firm
1 is p1 + tx and the cost to buy from firm 2 is p2 + t(1 − x).
Every consumer buys one unit from a firm with lowest cost.
Linear city
Linear transp.
costs
Quadratic
transp. costs
Circular city
Advertising
Indifferent consumer x̃: p1 + tx̃ = p2 + t(1 − x̃), which implies
p2 − p1 + t
p1 − p2 + t
and 1 − x̃ =
2t
2t
⎧
⎪
if pi ≤ pj − t
⎨1
pj −pi +t
Di (pi , pj ) =
if pi ∈ (pj − t, pj + t)
2t
⎪
⎩
0
if pi ≥ pj + t
x̃ =
The linear city model
with linear transportation costs (2)
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Linear city
Linear transp.
costs
Quadratic
transp. costs
Circular city
Advertising
max
(pi − c)
pi ∈[pj −t,pj +t]
FOC if pi ∈ (pj − t, pj + t):
⎧
⎪
⎨ pj − t
pj +c+t
pi =
2
⎪
⎩
pj + t
pj − pi + t
2t
pj + c + t − 2pi = 0
if pj ≥ c + 3t
if pj ∈ (c − t, c + 3t)
if pj ≤ c − t
Unique Nash equilibrium, (p1c , p2c ) satisfies:
p1c = p2c = c + t. Hence, price exceeds c.
Do firms maximize product differentiate to relax price
competition? Problematic to discuss choice of product
differentiation with linear transportation costs. Why?
What product differentiation maximizes welfare?
The linear city model
with quadratic transportation costs (2)
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Linear city
Linear transp.
costs
Quadratic
transp. costs
Circular city
Assume that firm 1 is located at a and firm 2 is located at 1 − b,
where 0 ≤ a < 1 − b ≤ 1.
For a consumer located at x (∈ [0, 1]) the cost to buy from firm
1 is p1 + tx 2 and the cost to buy from firm 2 is p2 + t(1 − x)2 .
Structure:
(1) Firms choose locations a and b
(2) Given locations, firms choose prices p1 and p2
(3) Demand determines quantities
Advertising
Maximal product differentiation to weaken price competition:
a = 0 and b = 0
Less product differentiation would have maximized welfare:
a = 14 and b = 14
What about entry? Better to analyze in the circular city model.
The circular city model and firm entry
Presentation of model
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Linear city
Structure:
(1) Firms choose whether to enter;
locate symmetrically around a circle (length 1)
(2) Given entry and locations, firms choose prices;
we assume maximal differentiation
(3) Demand determines quantities
Circular city
Equilibrium
price
Equilibrium
entry
Socially opt.
entry
Advertising
f : Fixed cost of entry
(pi − c)Di − f : Firm i’s (ex ante) profit
Assume linear transportation costs.
Ask: How many firms will enter?
The circular city model and firm entry
Equilibrium price
4820–4
Product
different. I
Demand for i’s product, given that the competitors’ price is p.
Assume that firm i only competes with his nearest competitors.
Geir B.
Asheim
Indifferent consumer x̃: pi + tx̃ = p + t( n1 − x̃), which implies:
Introdution
p − pi +
Di (pi , pj ) = 2x̃ =
t
Bertrand
model
t
n
Linear city
p − pi +
max(pi − c)
pi
t
Circular city
Equilibrium
price
Equilibrium
entry
Socially opt.
entry
Advertising
FOC if pi ∈ (p − nt , pj + nt ): p + c +
t
n
t
n
−f
− 2pi = 0
Find the symmetric equilibrium by setting pi = p:
p=c+
t
n
The profit margin (p − c) is reduced if greater entry (larger n)
The circular city model and firm entry
Equilibrium entry
4820–4
Product
different. I
Geir B.
Asheim
How many firms will enter? Demand for each:
maximize n subject to (p − c) n1 − f =
nc ≤
Introdution
Bertrand
model
and nc + 1 >
p c+
Circular city
Advertising
t
f
c
Linear city
Equilibrium
price
Equilibrium
entry
Socially opt.
entry
√
1
n
t
n2
−f ≥0
t
f
tf
Higher transport costs (larger t) weakens price competition,
increases price, and leads to greater entry (larger n).
In equilibrium:
Total cost of entry:
Total transportation cost:
Total cost:
t
ff
=
√
tf
√
t
1
t 4 f = 4 tf
√
5
4 tf
The circular city model and firm entry
Socially optimal entry
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Consumption is given (p > c does not lead to efficiency loss).
Hence, social planner wishes to minimize the sum of
entry costs and transportation costs:
2n1
t
min nf + t 2n
xdx = min nf + 4n
n
Linear city
Circular city
Equilibrium
price
Equilibrium
entry
Socially opt.
entry
Advertising
n=
1
2
n
0
t
f
= 12 nc
Too many firms in equilibrium
Private motivation for entry: Business stealing
Social motivation for entry: Saving transportation costs
In social optimum:
Total cost of entry:
Total transportation cost:
t
1
2
4 12
t
ff
t
f
Total cost:
=
1
2
=
1
2
√
√
√
tf
tf
tf
The circular city model and firm entry
Discussion
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Maximal product differentiation is assumed.
Can this be endogenized through a 3–stage game
→ Economides (1984)
Simultaneous entry is assumed
Linear city
Circular city
Equilibrium
price
Equilibrium
entry
Socially opt.
entry
Advertising
What about sequential entry?
→ Prescott & Visscher (1977)
Each firm produces only one brand (one product variant)
What if each firm can produce multiple brands? Then they
can attach their competitors from more than one direction.
Leading to multi-brand monopoly
rather than one-brand oligopoly?
Maximal or minimal differentiation?
Arguments for less differentiation
4820–4
Product
different. I
Locating where demand is.
Geir B.
Asheim
Introdution
Bertrand
model
Linear city
Circular city
Equilibrium
price
Equilibrium
entry
Socially opt.
entry
Positive externalities between firms.
Ex.: A particular type of stores is located in the same street.
The consumers know where to go and can compare more
easily products that are differentiated from other
characteristics than location.
No price competition (Hotelling, 1929).
Advertising
Ex.: Commercial TV companies wishing to maximize their
audience, transmit similar programs in the same time slot.
Political parties approach the center (median voter).
Related to the positive demand effect.
Advertising
Informational product differentiation
4820–4
Product
different. I
Geir B.
Asheim
Informative: focus here
Persuasive: shifting consumers’ preferences
Introdution
Bertrand
model
Linear city
Circular city
Advertising
Model
Equilibrium
Consider a linear city of length 1 with consumers uniformly
distributed along this interval. Also, assume that firm 1 is
located at 0 and firm 2 is located at 1.
For a consumer located at x (∈ [0, 1]) the cost to buy from firm
1 is p1 + tx if informed of firm 1 and the cost to buy from firm 2
is p2 + t(1 − x) if informed of firm 2. A consumer becomes
informed of firm i by receiving advertising from firm i.
Fraction of consumers receiving advertising from firm i: ϕi
Advertising costs: Ai = A(ϕi ) = 2a ϕ2i
Advertising
Model of informational product differentiation
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Linear city
Circular city
Advertising
Model
Equilibrium
Potential market for firm 1: ϕ1
Out of these consumers,
a fraction 1 − ϕ2 have not received information from firm 2
The rest, a fraction
both firms
ϕ2 out of ϕ1 , 1knowp2about
−p1 D1 = ϕ1 (1 − ϕ2 ) + ϕ2 2 + 2t
Firms simultaneously & independently choose advertising & price
1 p2 −p1 a 2
max(p1 − c)ϕ1 (1 − ϕ2 ) + ϕ2 2 + 2t
− 2 ϕ1
p1 ,ϕ1
FOCs p1 : ϕ1 (1 − ϕ2 ) + ϕ2 12 +
⇒ p1 =
p2 +c−t
2
+
t
ϕ2
p2 −p1 2t
1 ϕ2
=0
− (p1 − c) ϕ2t
−p1 ϕ1 : (p1 − c) (1 − ϕ2 ) + ϕ2 12 + p22t
− aϕ1 = 0
−p1
⇒ ϕ1 = p1a−c (1 − ϕ2 ) + ϕ2 12 + p22t
Advertising
Equilibrium
4820–4
Product
different. I
Geir B.
Asheim
Introdution
Bertrand
model
Linear city
Circular city
Find the symmetric equilibrium
by setting p1 = p2 = p and ϕ1 = ϕ2 = ϕ:
p=
+
t
ϕ
√
⇒ p = c + t ϕ − 1 = c + t 2a
=
c
+
2at
t
p−c ϕ
ϕ
t 2
ϕ = a (1 − ϕ) + 2 = a ϕ − 1 1 − 2
⇒ ϕ=
Advertising
Model
Equilibrium
p+c−t
2
2
1+
q2
(which requires t ≤ 2a)
2a
t
Π = 2a ϕ2 = ∂p
∂a
> 0,
∂ϕ
∂a
1+
q2a
2a
t
< 0,
2
∂Π
∂t
> 0,
∂Π
∂a
> 0.
An increase in advertising costs increases firms’ profits
A direct negative effect.
An indirect positive effect: a ↑ ⇒ ϕ ↓ ⇒ p ↑