1. Topic 6. Product differentiation (I): patterns of price setting
Economía Industrial Aplicada
Juan Antonio Máñez Castillejo
Departamento de Estructura Económica
Universidad de Valencia

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Index
Topic 7. Product differentiation: patterns of price setting
Introduction
Horizontal versus vertical product differentiation
The linear city model
3.1 Linear transport costs
3.2 Quadratic transport costs
4. Applications: Coca-Cola versus Pepsi-Cola
5. Conclussions
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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
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4. 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.
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Example
Horizontal Dif.
Opel Astra
Ford Focus
Vertical Dif.
Vertical Dif.
Opel Corsa
Ford Fiesta
Horizontal Dif.
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6. 3.1 Linear city model with linear transport costs : assumptions
Consumers are uniformly distributed with unit density along a segment of L length 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
Hotelling (1929)
Los supuestos simplificadores no van a variar los resultados
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7. 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). L F1 F2
Reinterpretación del modelo de H. Como un juego en dos etapas
Decisión más costosa
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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:
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)
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9. 3.1 Linear city model with linear transport costs : transport costs
With linear transport costs per unit of distance : L F1 F2 x
L-x X
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)
En que incurre el consumidor si compra en la empresa X
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3.1 Linear city model with linear transport costs : demands determination L F1 F2
d1=x
d2=L-x X
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11. 3.1 Linear city model with linear transport costs : demand properties
Price elasticity of demand
Price elasticity of demand and transport costs
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3.1 Linear city model with linear transport costs : demands determination
Total cost of buying at 1 = Total cost of buying at 2 p1 p2 d1 d2 L x F1 F2 x0 x1
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13. 3.1 Linear city model with linear transport costs : firm 1 demandL
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3.1 Linear city model with linear transport costs : Obtaining the Nash equilibrium in prices (I)
Maximization problem of firm 1
 Firm 1 reaction function
Maximization problem of firm 2
 Firm 2 reaction function
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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)
Profits for both firms are: p2 p1
p*1(p2)
p*2(p1)
Lt+c
(Lt+c)/2
(Lt+c)/2
Lt+c
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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
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.
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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
E1 y E2 p0 c L
F1 y F2 p1 E1 p2 E2 p3 E1
E1 y E2
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3.1 Linear city model with linear transport costs : Analysis of the location decisions (II)
With a gain of generality we can assume: a
L-b L F1 F2
where a  0 , b 0 y L-a-b  0
 It allows the consideration of captive demands a
L-b L
F1,F2
If a+b=L  minimum differentiation L
If a=b=0  maximum differentiation
a=0 F1 F2
b=0
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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 c F1 a a’ F2
L-b c L
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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 a F1 c
L-b F2 c L
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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 c L
Price competition with homogeneous products a’
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22. 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 a
L-b L F1 F2
where a  0 , b 0 y L-a-b 0
We do not impose maximum product differentiation to obtain the Nash
equilibrium in prices.
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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. a c
L-b x L
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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: X a
L-b L x1 x2
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3.2 Linear city model with quadratic transport costs : Obtaining the demands (II)
Demands for firms 1 y 2
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)
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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.
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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:
Maximization problem of firm 2:
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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:
Properties of the price equilibrium:
Symmetric eq. : a=b 
 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
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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
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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:
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
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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 ( ) 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. a L
L-b a’ x x’ d1
d1’
Direct effect  minimum differentiation tendency
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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.
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
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3.2 Linear city model with quadratic transport costs : Obtaining the equilibrium in locations (V): strategic effect p1 a p2
L-b
p2’ x x’ L
d1
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3.2 Linear city model with quadratic transport costs : Obtaining the equilibrium in locations (VI): strategic effect p2
d1 L a p1
L-b a’
p2’ x x’
d1’
Strategic effect: maximum differentiation tendency
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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.
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36. 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.
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37. 4. Application: Coca-Cola vs. Pepsi-Cola: demand and costs functions
Demand functions for Coca-Cola (product 1) and Pepsi-Cola (product 2).
Q1 = p p2
Q2 = p 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?
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38. 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:
 Coca-Cola’s reaction function
Pepsi-cola’s maximization problem:
 Pepsi-Cola’s reaction function
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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
PCOCA(pPEPSI)
PPEPSIi(pCOCA)
P*COCA
P*PEPSI
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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
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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
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4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices determination (V)
Demand asymmetries
p1= p2=p
Q1= p p2
Q2= p p1
Q1= p
Q2= p
Graphic analysis  normalize p=1
Q1= y Q2=45.44
Q=Q1+Q2=107.13
Q1=
Q2= a
L-b
p=1 a’
1. Symmetric Eq. a=b  Q1=Q2
2. Aymmetric Eq.
a’>b  Q1>Q2
Q1= 61.69
Q1= 45.44
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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
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4. Application: Coca-Cola vs. Pepsi-Cola: optimal prices determination (VII)
Do these asymmetries have any additional impact?  price-cost margin
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
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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
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