Topic 6. Product differentiation (I): patterns of price setting

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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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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
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3.1 Linear city model with linear transport costs
assumptions

• Consumers are uniformly distributed with unit
  density along a segment of L length

• 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 ? c1c2c
• Each consumer buys a single unit of the product.
• ? Alternative interpretation of the segment as a
  product characteristic

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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).

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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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3.1 Linear city model with linear transport costs
transport costs

• With linear transport costs per unit of distance

• 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)

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3.1 Linear city model with linear transport costs
demands determination
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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

x0
x1
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3.1 Linear city model with linear transport costs
firm 1 demand
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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

• Maximization problem of firm 2

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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

p1(p2)
p2(p1)
Ltc
(Ltc)/2
(Ltc)/2
Ltc
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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 tgt0 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 t0 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 gt0 ? pgtc y
  ?gt0
• Minimum product differentiation both firms
  choose the same location ? no differentiation ?
  Bertrand model with homogeneous products

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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

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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 (i1,2) takes its optimal decision of
  location and price given its rivals 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

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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

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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 p1p2c.
• DAspremont et al. (1979) shows that que abL/2
  is not a Nash equilibrium in locations ? both
  firms have an incentive to deviate from L/2 to
  set a pgtc y and in this way they would obtain
  positive profits

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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
  

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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.

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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

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3.2 Linear city model with quadratic transport
costs Obtaining the demands (II)

• Demands for firms 1 y 2

• If p1p2
• 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

• 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 agtb ? p1gtp2
• Si altb ? p2gtp1

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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 rivals 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

• DAspremont 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.

• 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
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3.2 Linear city model with quadratic transport
costs Obtaining the equilibrium in locations
(VI) strategic effect
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.

• DAspremont 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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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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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

c14.96 c23.96

• Which is the optimal price for Coca-Cola and
  Pepsi-Cola?

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4. Application Coca-Cola vs. Pepsi-Cola optimal
prices determination

• Step 1 solve the maximization problems of
  Coca-Cola and Pepsi-Cola.

• Coca-Colas maximization problem

• Pepsi-colas maximization problem

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4. Application Coca-Cola vs. Pepsi-Cola optimal
prices determination (II)

• Step 2 solve the system of reaction functions.
• p112.72 y p28.11
• Coca-Cola sets a price higher than the Pepsi-Cola
  one.

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4. Application Coca-Cola vs. Pepsi-Cola optimal
prices determination (III)

• Why Coca-Colas price is higher that Pepsi-Colas
  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) gt Pepsis-Cola
  marginal cost (3.96)
• ? Coca-Colas price gt Pepsi-Colas price

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4. Application Coca-Cola vs. Pepsi-Cola optimal
prices determination (V)

• Demand asymmetries

Q163.42 -1.73p Q249.52 -4.08p
Q163.42 - 3.98 p1 2.25 p2 Q249.52 - 5.48 p2
1.40 p1

• Graphic analysis ? normalize p1
• Q1 61.69 y Q245.44
• QQ1Q2107.13

1. Symmetric Eq. ab ? Q1Q2
2. Aymmetric Eq. agtb ? Q1gtQ2

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4. Application Coca-Cola vs. Pepsi-Cola optimal
prices determination (VI)

• The higher Coca-Colas 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-Colas 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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