Transcript Topic 6.-Product differentiation: patterns of price setting (PPT

Topic 6. Product differentiation (I):
patterns of price setting
Economía Industrial Aplicada
Juan Antonio Máñez Castillejo
Departamento de Estructura Económica
Universidad de Valencia
Index
Topic 7. Product differentiation: patterns of price
setting
1. Introduction
2. Horizontal versus vertical product differentiation
3. The linear city model
3.1 Linear transport costs
3.2 Quadratic transport costs
4. Applications: Coca-Cola versus Pepsi-Cola
5. Conclussions
Departamento de Estructura Económica
2
1. Introducción
 Aim: To study an oligopoly model relaxing the homogeneous
product assumption, to analyse the effect of product
differentiation on price competition intensity and product choice.
 Main implication of the homogeneous product assumption in an
oligopoly model of price competition (à la Bertrand)
• Bertrand paradox  Price competition between two firms is
a sufficient condition to restores the competitive situation p
=c
Departamento de Estructura Económica
3
2. Horizontal and vertical product differentiation

Horizontal product differentiation: two products are
differentiated horizontally if, when they are offered at the same price
consumers do not agree on which is the preferred product.
Example: pine washing-up liquid and lemon-washing up liquid

Vertical product differentiation: two products are differentiated
vertically if, when they are offered at the same price consumers
agree on which is the preferred product.
• Example: washing-up liquid with and without product moisturizing add-up.
Departamento de Estructura Económica
4
Example
Horizontal Dif.
Opel Astra
Ford Focus
Vertical Dif.
Vertical Dif.
Opel Corsa
Ford Fiesta
Horizontal Dif.
Departamento de Estructura Económica
5
3.1 Linear city model with linear transport costs : assumptions

Consumers are uniformly distributed with unit density along a
segment of L length
0
L

Two firms (firms 1 and 2) are located along the segment

The two firms sell a product that is identical except for the location of
the firm.

The two firms have constant and identical marginal cost c  c1=c2=c

Each consumer buys a single unit of the product.
 Alternative interpretation of the segment as a product characteristic
Departamento de Estructura Económica
6
3.1 Linear city model with linear transport costs : two-stage
game
Stage 1: the two firms choose simultaneously their location (long-run
decision)
Stage 2: the two firms choose simultaneously their prices (short-run
decision)
We impose maximum product differentiation and so we focus on the
determination of the Nash equilibrium in prices (Stage 2).
F1
F2
0
L
Departamento de Estructura Económica
7
3.1 Linear city model with linear transport costs : consumers’
utility function

The utility that a consumer i located in X obtains from the purchase of
of the good of firm j is given by:
U i j  r  p j  tx ij
r: reservation price
pj: price of the product of firm j
xij.: distance (along the segment) between the location of consumer i and
the location of firm j
t: transport cost per unit of distance (or alternatively intensity of the
preference for a given product)
Departamento de Estructura Económica
8
3.1 Linear city model with linear transport costs : transport costs

With linear transport costs per unit of distance :
F1
0
x
L-x
X
F2
L
• Transport cost if the product is bought at firm 1 = tx
• Transport cost if the product is bought at firm 2 = t(L-x)

Total cost of the product = price + transport costs
• Total cost if the product is bought at firm 1 = p1+ tx
• Total cost if the product is bought at firm 2= p2+ t(L-x)
Departamento de Estructura Económica
9
3.1 Linear city model with linear transport costs : demands
determination
U X ,1  U X ,2
F1
0
d1=x
d2=L-x
X
F2
L
r  p1  tx  r  p 2  t (L  x )
p1  tx  p2  t L  x 
d1  x 
p 2  p1 L
p  p2 L
  d2  L  x  1

2t
2
2t
2
Departamento de Estructura Económica
10
3.1 Linear city model with linear transport costs : demand properties
 Price elasticity of demand

d 1 p1
p1

0
p1 d 1
p 2  p1  Lt
 Price elasticity of demand and transport costs

t

Lp1
0
2
( p 2  p1  Lt )
Departamento de Estructura Económica
11
3.1 Linear city model with linear transport costs : demands
determination
 Total cost of buying at 1 = Total cost of buying at 2
p1  tx  p2  t L  x 
p 2  t L  x 
p1  tx
p1  tx 1
p1  tx 0
p1
d2
d1
F1
0
x0
x1
x
Departamento de Estructura Económica
p2
F2
L
12
3.1 Linear city model with linear transport costs : firm 1
demand
p14
p13
p2
p12
p11
F1
0
d14
d 13
d 12
Departamento de Estructura Económica
L F
2
d11
13
3.1 Linear city model with linear transport costs : Obtaining the
Nash equilibrium in prices (I)
 Maximization problem of firm 1
 p  p1 L 
max 1  d 1  p1  c    2
   p1  c 
p1
2
t
2

d 1 p 2  2p1  c L
F .O .C .

 0
dp1
2t
2
 p1* ( p 2 ) 
p 2  Lt  c
2
 Firm 1 reaction function
 Maximization problem of firm 2
 p  p2 L 
max  2  d 2  p 2  c    1
   p2  c 
p2
2
 2t
F .O .C .
d  2 p1  2p 2  c L

 0
dp 2
2t
2
p1  Lt  c
 p 2* ( p1 ) 
2
 Firm 2 reaction function
Departamento de Estructura Económica
14
3.1 Linear city model with linear transport costs : Obtaining the
Nash equilibrium in prices (II)

Solving the system of equations given by the two reaction functions
we obtain the price equilibrium: (given locations)
p1c  p2c  Lt  c

Profits for both firms are:
1   2 
p2
1 2
Lt
2
p*1(p2)
p*2(p1)
Lt+c
(Lt+c)/2
(Lt+c)/2
Lt+c
p1
Departamento de Estructura Económica
15
3.1 Linear city model with linear transport costs : Obtaining the
Nash equilibrium in prices (I)
 Although both products are physically identical, as long as t>0
the price is greater than the marginal cost
p  c  Lt
 Why?:
• The larger is t the more differentiated are the products for the
consumers  the higher is the costs of buying in a further shop.
• The larger is t the lower in the intensity of competition between
firms 1 and 2 (for the consumers located between the two firms).
• When t=0 the products are not differentiated any more  price is
equal to marginal cost as in the Bertrand model with
homogeneous.
Departamento de Estructura Económica
16
3.1 Linear city model with linear transport costs : Analysis of the
location decisions (I)
 Two extreme cases:
• Maximum product differentiation: if t >0  p>c y >0
• Minimum product differentiation: both firms choose the same location 
no differentiation  Bertrand model with homogeneous products
p1c  p2c  c y 1  2  0
E1 y E 2
p0
E1
p1
E2
p2
p3
E1
E1 y E2
c
0
F1 y F2
L
Departamento de Estructura Económica
17
3.1 Linear city model with linear transport costs : Analysis of the
location decisions (II)
 With a gain of generality we can assume:
0
F1
F2
a
L-b
L
where a  0 , b 0 y L-a-b  0  It allows the consideration of captive
demands
 If a+b=L  minimum differentiation 
F1,F2
0
a
L-b
If a=b=0  maximum
differentiation
F1
L
a=0
F2
L
b=0
Departamento de Estructura Económica
18
3.1 Linear city model with linear transport costs : Analysis of the
location decisions (III)


Nash equilibrium in locations is the one in which firm i (i=1,2) takes its
optimal decision of location and price given its rival’s locations an price
decisions
The original result in the Hottelling model (1929): minimum differentiation.
Once prices have been chosen, both firms locate in the centre of the
segment  L/2
p1
c
p2
F1
F2
a a’
0
d1
d 1'
c
L-b
Departamento de Estructura Económica
L
19
3.1 Linear city model with linear transport costs : Analysis of the
location decisions (IV)
 This result of minimum differentiation is subject to two important
critiques: (D’ Aspremont et al., 1979)
• Critique 1: Demand discontinuity. Suppose that both firms are located
very close each other
p11
p12
p13
c
p2
p14
F1
d 11
d 12
d 13
F2
a
c
L-b
d 14
0
L
Departamento de Estructura Económica
20
3.1 Linear city model with linear transport costs : Analysis of the
location decisions (V)
 Critique 2: Suppose that both firms are located at L/2
 There is no product differentiation: each firm has an incentive to
undercut the price of the rival until p1=p2=c.
 D’Aspremont et al. (1979) shows that que a=b=L/2 is not a Nash
equilibrium in locations  both firms have an incentive to deviate from
L/2 to set a p>c y and in this way they would obtain positive profits
p1  p2
c
p11
a’
d 11
Price competition with
homogeneous products
p10  p20  c
a b L 2
d 21
0
Departamento de Estructura Económica
L
21
3.2 Linear city model with quadratic transport costs :
Assumptions

It solves the problem of the inexistence of Nash equilibrium in locations
that arises in the model with linear transport cost.

Differences with the linear transport costs model :
• Utility function
U ij  r  p j  t  x ij 
2
• We do not impose maximum product differentiation to obtain the Nash
equilibrium in prices.
0
F1
F2
a
L-b
L
where a  0 , b 0 y L-a-b 0
Departamento de Estructura Económica
22
3.2 Linear city model with quadratic transport costs :
Discontinuities in demand
 With quadratic transport costs the umbrellas that represent the total
cost of purchase are U-shaped.
0
p2
p10
p11
p12
c
d
0
1
a x L-b
L
d 11
d 12
Departamento de Estructura Económica
23
3.2 Linear city model with quadratic transport costs :
Obtaining the demands (I)
 The consumer located at X will be indifferent between consuming in
firms 1 and 2 whenever:
U X ,1  U X ,2
0
X
a
x1
d1  a  x 1
L-b
L
x2
d2  b  x 2
r  p1  tx 12  r  p2  tx 22
x1  x 2  L  a  b
Departamento de Estructura Económica
24
3.2 Linear city model with quadratic transport costs :
Obtaining the demands (II)
 Demands for firms 1 y 2
d 1  p1 , p 2   a  x 1  a 
L a b
d 2  p1 , p 2   b  x 2  b 
L a b
2
2

p 2  p1
2t L  a  b 

p1  p 2
2t L  a  b 
 If p1=p2:
• Firm 1 sells to all the consumers located at the left of its location and
firm 2 sells to all the consumers located at its right.
• Both firms share evenly the consumers located between them.
 The third term catches the sensibility of the demand to price
differentials (differences between the prices of two firms)
Departamento de Estructura Económica
25
3.2 Linear city model with quadratic transport costs :
Obtaining the equilibrium in prices and locations (II)
 Two-stage game:
• Stage 1: Firms choose locations simultaneously.
• Stage 2: Firms choose prices simultaneously.
 We solve by backwards induction:  each firm anticipates that
its location decision affects not only its demand but also price
competition intensity
•
•
To obtain the Nash equilibrium in prices given locations (a,b).
To obtain the Nash equilibrium in locations given prices.
Departamento de Estructura Económica
26
3.2 Linear city model with quadratic transport costs :
Obtaining the price equilibrium given locations (I)
 To obtain the price equilibrium, we solve the maximization
problems of firms 1 and 2:
• Maximization problem of firm 1:

L a b
p 2  p1 
Max 1  d 1  p1  c   a 

  p1  c 
p1
2
2t L  a  b  

F.O.C.
d 1
L a b
p  2 p1  c
a 
 2
0
dp1
2
2t L  a  b 
• Maximization problem of firm 2:

L a b

2
Max  2  d 2  p 2  c   a 
p2
F.O.C

p1  p 2 
  p2  c 
2t L  a  b  
d 2
L  a  b p1  2 p 2  c
b 

0
dp 2
2
2t L  a  b 
Departamento de Estructura Económica
27
3.2 Linear city model with quadratic transport costs :
Obtaining the price equilibrium given locations (II)
 To obtain the price equilibrium, we solve the system of FOCs:

a b 

3
p1c a , b   c  t L  a  b  1 



b a 

3
p 2c a , b   c  t L  a  b  1 


 Properties of the price equilibrium:
• Symmetric eq. : a=b 
p c  p1c  p2c  c  t (L  2a )  apc
• Asymmetric eq. : a  b  p1-p2 = 2/3 t(L-a-b)(a-b)
 That firm located closer the center of the segment sets a
higher price
Si a>b  p1>p2
Si a<b  p2>p1
Departamento de Estructura Económica
28
3.2 Linear city model with quadratic transport costs :
Obtaining the equilibrium in locations (I)
 In the equilibrium in locations, each firm choose location taking as
given the rival’s location:
• Firm 1 maximizes 1(a,b) choosing a and taking b as given
• Firm 2 maximizes 2(a,b) chooseli b and taking a as given
 D’Aspremont et al. (1979) shows that with quadratic transport costs
the equilibrium in location involoves maximum differentiation : both
firms are located in the ends of the segment
• Each one of the firms choose the furthest possible location from its from
its rival with the aim of differentiating the product and minimizing the
effect of a potential price reduction by the rival on its own demand
Departamento de Estructura Económica
29
3.2 Linear city model with quadratic transport costs :
Obtaining the equilibrium in locations (II)
 The reduced form of the profit functions show that the location
decision:
1 a , b    p1c (a , b )  c  d 1 a , b , p1c (a , b ), p 2c (a , b ) 
 2 a , b    p 2c (a , b )  c  d 2 a , b , p1c (a , b ), p 2c (a , b ) 
• Has an effect on firms’ demands
• Has an effect on firms’ prices
 The algebraic derivation of the Nash equilibrium in location is quite
complicated, and so we make use of a graphic analysis
 We analyze firm 1 location decision that depends on :
 Direct effect
 Strategic effect
Departamento de Estructura Económica
30
3.2 Linear city model with quadratic transport costs :
Obtaining the equilibrium in locations (III): direct effect
Direct effect: for a given pair of prices ( p1 , p 2) and a given the location of
firm 2, as firm 1 moves its location towards the location of firm 2 (i.e.
towards the center of the segment) its demand increase, and so its profis.

p2
p1
0
L
a
d1
d1’

a’
L-b
x
x’
Direct effect  minimum differentiation tendency
Departamento de Estructura Económica
31
3.2 Linear city model with quadratic transport costs :
Obtaining the equilibrium in locations (IV): strategic effect

In our two-stage game, the prices (that are chosen in the second stage)
are not given, they depend on the first-stage locations decision 
strategic effect.

a b 

3
p1c a , b   c  t L  a  b  1 



b a 

3
p 2c a , b   c  t L  a  b  1 
Strategig effect. For a given location for firm 2, as firm 1 moves its
location towards the center (i.e. closer to its rival), product differentiation
decreases  increase of price competition  price reduction 
negative effect on prices  maximum differentiation tendency
Departamento de Estructura Económica
32


3.2 Linear city model with quadratic transport costs :
Obtaining the equilibrium in locations (V): strategic effect
p2
p1
p2’
d1
d1 '
0
a
x’ x
L-b
d1
Departamento de Estructura Económica
L
33
3.2 Linear city model with quadratic transport costs :
Obtaining the equilibrium in locations (VI): strategic effect
p2
p1
p2’
d1
d1 '
0
d1
a
a’
x’ x
L-b
d1’
L
Strategic effect: maximum differentiation tendency
Departamento de Estructura Económica
34
3.2 Linear city model with quadratic transport costs :
Obtaining the equilibrium in locations (VI): strategic effect vs. direct effect
Direct effect: minimum differentiation tendency
Strategic effect: maximum differentiation tendency.




D’Aspremont et al. (1979) show analytically that, in general the
strategic effect dominates over the direct one  final result: maximum
differentiation.
Impact of t on the intensity of price competition (that determines the
strategic effect) and on the location decision:
 If t is low, each firm try to separate from its rival to avoid the strategic
effect.
 If t is high, firms locate close (each other) to take advantage of the direct
effect.
Departamento de Estructura Económica
35
4. Application: Coca-Cola vs. Pepsi-Cola
 Coca-Cola and Pepsi-Cola, the world leaders on the carbonated
colas market, sell horizintally differentiated products.
 Simplifying assumption: the relevant competition dimension is
price ( advertising)
 Laffont, Gasmi y Vuong (1992) analyse price competition
between Coca-Cola and Pepsi-Cola. They estimated using
econometric methods the following demand and marginal
costs functions.
Departamento de Estructura Económica
36
4. Application: Coca-Cola vs. Pepsi-Cola: demand and costs
functions
 Demand functions for Coca-Cola (product 1) and Pepsi-Cola
(product 2).
Q1 = 63.42 - 3.98 p1 + 2.25 p2
Q2 = 49.52 - 5.48 p2 + 1.40 p1
 Marginal costs for Coca-Cola and Pepsi-Cola
c1=4.96
c2=3.96
 Which is the optimal price for Coca-Cola and Pepsi-Cola?
Departamento de Estructura Económica
37
4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices
determination

Step 1: solve the maximization problems of Coca-Cola and Pepsi-Cola.
• Coca-Cola’s maximization problem:
Max 1  (p1  4.96)(63.49  3.98p1  2.25p2 )
p1
p1* (p2 )  10.44  0.28p2  Coca-Cola’s reaction function
• Pepsi-cola’s maximization problem:
Max 2  (p2 - 3.96)(49.52 - 5.48p2  1.40p1 )
p2
p2* (p1 )  6.49  0.127 p1  Pepsi-Cola’s reaction function
Departamento de Estructura Económica
38
4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices
determination (II)
 Step 2: solve the system of reaction functions.
p1=12.72 y p2=8.11
 Coca-Cola sets a price higher than the Pepsi-Cola one.
PPEPSI
PCOCA(pPEPSI)
P*PEPSI
PPEPSIi(pCOCA)
P*COCA
pCOCA
Departamento de Estructura Económica
39
4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices
determination (III)
 Why Coca-Cola’s price is higher that Pepsi-Cola’s one?
• Cost asymmetries
• Demand asymmetries
Departamento de Estructura Económica
40
4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices
determination (IV)
 Costs asymmetries:
• Coca-Cola marginal cost (4.96) > Pepsis-Cola marginal cost
(3.96)
 Coca-Cola’s price > Pepsi-Cola’s price
Departamento de Estructura Económica
41
4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices
determination (V)
 Demand asymmetries
Q1=63.42 - 3.98 p1+ 2.25 p2
p1= p2=p
Q1=63.42 -1.73p
Q2=49.52 -4.08p
Q2=49.52 - 5.48 p2+ 1.40 p1
 Graphic analysis  normalize p=1
• Q1= 61.69 y Q2=45.44
• Q=Q1+Q2=107.13
1. Symmetric Eq.
a=b  Q1=Q2
2. Aymmetric Eq.
a’>b  Q1>Q2
p=1
p=1
a’
L-b
Q1= 53.565
Q2= 53.565
Q1= 61.69
Q1= 45.44
a
Departamento de Estructura Económica
42
4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices
determination (VI)
 The higher Coca-Cola’s price is due to:
• Higher marginal cost (cost asymmetries)
• Demand asymmetries that favour Coca-Cola
Departamento de Estructura Económica
43
4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices
determination (VII)
 Do these asymmetries have any additional impact?  price-cost
margin
PCM1 
p1  c 1 12.72  4.96

 0.61
p1
12.72
PCM 2 
p2  c 2 8.11  3.96

 0.51
p2
3.96
 The price-cost margin of Coca-Cola is higher than the Pepsi-Cola’s
one
 Demand asymmetry in favour of Coca-Cola
 Higher market power for Coca-Cola
Departamento de Estructura Económica
44
5. Concluding Remarks
 Product differentiation solves the Bertrand paradox:
• It allows firms to set price above marginal cost
• It allows firms to obtain positive profits
 Firm will intend to differentiate their products (from those of its
competitors) as much as possible, the aim is to reduce the intensity
of price competition:
• Actual product differentiation
• Perceived product differentiation: increase consumers’ preference for
the products of the firm
Departamento de Estructura Económica
45
Departamento de Estructura Económica
46