Transcript Mill Pricing Model

Equilibrium Location with Elastic
Demand in Mixed Duopoly
Joint work with Minoru Kitahara
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Two Models of Spatial Competition
(1) Mill Pricing Model (Shopping Model)
Consumers pay the transport costs. Consumers go
to the firm's shop.
(2) Delivered Pricing Model (Shipping Model,
Spatial Price Discrimination Model)
Firms pay the transport costs. Firms bring the goods
to the markets.
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Mill Pricing Model
(Shopping Model)
Mitaka
Tachikawa
Kichijoji
Musashisakai
Kokubunji
Kunitachi
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Delivered Pricing Model (Shipping
Model, Spatial Price Discrimination
Model)
Hokkaido
Tohoku
Kyusyu
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Kansai
Tokai
Kanto
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Mill Pricing (Shopping) Models
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Hotelling
Duopoly Model, Fixed Price Model, Shopping Model.
Consider a linear city along the unit interval [0,1],
where firm 1 is located at x1 and firm 2 is located at x2.
Consumers are uniformly distributed along the interval.
Each consumer buys exactly one unit of the good,
which can be produced by either firm 1 or firm 2.
Each consumer buys the product from the firm that is
closer to her.
Each firm chooses its location independently.
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Hotelling
the location of firm 1
０
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firm 1's demand
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the location of firm 2
firm 2's demand
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Relocation of Firm 1
the location of firm 1
the location of firm 2
０
１
firm 1's demand
firm 2's demand
This relocation increases the demand of firm 1,
resulting in a larger profit of firm 1
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Equilibrium
Best Response of Firm 1
If the location of firm 2 is larger than 1/2, then the
location just left to it is the best reply for firm 1.
If the location of firm 2 is smaller than 1/2, then the
location just right to it is the best reply for firm 1.
→Two firms agglomerate at the central point.
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Best reply for firm 1
the optimal location of firm 1
０
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the location of firm 2
１
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Best reply for firm 1
the optimal location of firm 1
０
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the location of firm 2
１
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Equilibrium
the location of firm 1
０
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the location of firm 2
１
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Interpretation of the linear city
(1) city ~ spatial interpretation
(2) product differentiation ～ horizontal product
differentiation
(3) political preference
(3)→interpretation of minimal differentiation
～The policies of two major parties become similar.
However, following the interpretation of (1) and (2), the
model lacks the reality since consumers care about
prices as well as the locations of the firms.
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Endogenous Price
Duopoly Model, Shopping Model. Consider a linear
city along the unit interval [0,1], where firm 1 is
located at x1 and firm 2 is located at x2.
Consumers are uniformly distributed along the
interval. Each consumer buys exactly one unit of
the good, which can be produced by either firm 1
or firm 2. Each consumer buys the product from
the firm whose real price (price +transport cost) is
lower.
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One-Stage Location-Price Model
Duopoly Model, Shopping Model. Consider a linear
city along the unit interval [0,1], where firm 1 is
located at x1 and firm 2 is located at x2.
Consumers are uniformly distributed along the
interval. Each consumer buys exactly one unit of
the good, which can be produced by either firm 1
or firm 2. Each consumer buys the product from
the firm whose real price (price +transport cost) is
lower.
Each firm chooses its location and price
independently.
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One-Stage Location-Price Model
No pure strategy equilibrium exists.
Given the price of the rival, each firm has an incentive
to take a position closer to the rival's (the principle of
the Hotelling).
Given the minimal differentiation, each firm names the
price equal to its marginal cost, resulting in a zero
profit. →Each firm has an incentive for locating far
away each other. →Given the price of the rival, each
firm again has an incentive to take a position closer
to the rival's (the principle of the Hotelling).
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Two-Stage Location then Price
Model
The same structure as the previous model except for
the time structure. Each consumer buys the product
from the firm whose real price (price +transport
cost) is lower. Transport cost is proportional to (the
distance)2.~quadratic transport cost.
In the first stage, each firm chooses its location
independently.
In the second stage they faces Bertrand competition.
d'Aspremont, Gabszewics, and Thisse, (1979,
Econometrica)
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Maximal Differentiation
firm1's location
０
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firm 2's location
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Equilibrium
Maximal Differentiation
Each firm has an incentive to locate far away from the
rival so as to mitigate price competition.
A decrease in |x2-x1| increases the demand elasticity ~
price becomes more important
An increase in the demand elasticity increases the
rival's incentive for naming a lower price.
Through the strategic interaction (strategic
complements), the rival's lower price increases the
incentive for naming a lower price.→further reduction
of the rival's price
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Circular-City Model
Vickrey (1964), Salop (1979)
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Properties of Circular-City Model
(1) Symmetry ~ no central- periphery structure
→Advantage for analyzing n-firm oligopoly modes.
(2) Pure strategy equilibrium can exist when
transport cost function is linear or even concave.
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Equilibrium locations under
linear-quadratic transport cost
Both strictly
convex and
concave
transport cost
usually yield
this type of
equilibrium
the location of firm 1
De Frutos et al
(1999,2002)
the location of firm 2
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Equilibrium locations under
linear transport cost
the location of firm 1
All locations
between two
points are
equilibrium
location
These also
equilibrium locations
Kats (1995)
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the location of firm 2
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Cremer et al (1991)
mixed duopoly, linear city, mill pricing, inelastic demand,
constant marginal cost, no cost asymmetry,
quadratic transport →efficient locations
public firm’s location
０
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private firm’s location
3/4
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Welfare Maximization
inelastic demand →total outputs and so social surplus
do not depend on the price
welfare-maximization ~ transport cost minimization
public firm’s location
０ 1/8
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private firm’s location
5/8
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Welfare Maximization
inelastic demand →total outputs and so social surplus
do not depend on the price
welfare-maximization ~ transport cost minimization
public firm’s location
０ 1/8
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private firm’s location
1/2
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Equilibrium
Optimal Differentiation
A decrease in |x2-x1| increases the demand elasticity ~
price becomes more important
An increase in the demand elasticity does not matter for
the public firm’s pricing as long as the rival’s price
remains unchanged, but not for private firm’s pricing.
Through the strategic interaction (strategic
complements), the private firm's lower price increases
the incentive of the public firm for naming a lower
price.→only this indirect effect accelerates competition.
weaker incentive for product differentiation.
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Matsumura and Matsushima (2004)
(1) Cost-Reducing R&D
(2) Location Choice
(3) Bertrand Competition
mixed duopoly, linear city, mill pricing, inelastic demand,
constant marginal cost, no cost asymmetry,
quadratic transport →efficient locations, inefficient
investment (over-investment by the private firm)
Endogenous cost difference between public and private
firms.
Privatization can improve welfare because it reduces
private firm’s investment.
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Efficient Locations Given
Marginal Production Costs
Suppose that the public firm’s marginal cost is
higher than the private firm’s.
efficient location of the public firm is
０
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3/4
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R&D competition
(1) Cost-Reducing R&D
(2) Location Choice
(3) Bertrand Competition
A decrease in the private firm’s marginal production
cost (increase, decrease) x0 (location of the public firm).
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R&D competition
(1) Cost-Reducing R&D
(2) Location Choice
(3) Bertrand Competition
A decrease in the private firm’s marginal production
cost decrease x0 (location of the public firm).
→The private firm has a strategic incentive for
decreasing its marginal production cost→overinvestment.
Privatization→underinvestment
Privatization yields locational inefficiency and
underinvestment. Nevertheless, the privatization can
improve welfare.
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Kitahara and Matsumura
linear city, mill pricing, elastic demand, constant
marginal cost, no cost asymmetry,
quadratic transport →inefficient location (too
close, the private firm should be away from the
public firm.
public firm’s location private firm’s location
０
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3/4
１
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Elastic Demand and Public Firm’s
Pricing
Suppose that demand is inelastic.
Given the locations and pricing of private firm (p1),
consider the firm 0’s best reply (pricing)
p0 (>,=,<) p1.
Suppose that demand is elastic.
Given the locations and pricing of private firm (p1),
consider the firm 0’s best reply (pricing)
p0 (>,=,<) p1.
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Elastic Demand and Public Firm’s
Pricing
introducing elastic demand accelerates price
competition
The shorter distance between firms →The public firm
has a larger incentive to raise its price in order to
reduce transport distortion.
→private firm strategically chooses a closer position to
the public firm’s.
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Mixed Strategy Equilibria
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Uniqueness of the Equilibrium
Shopping, Hotelling, quadratic transport cost, uniform
distribution（standard Location-Price Model)
The unique pure strategy equilibrium location pattern
is maximal differentiation.
However, there are two pure strategy equilibria.
(x1, x2)=(0,1), (x1, x2)=(1,0)
→Mixed strategy equilibria may exist.
In fact, many (infinite) mixed strategy equilibria exist
Bester et al (1996).
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Cost Differential between Firms
Consider a production cost difference between two firms.
When the cost difference between two firms is small, the
maximal differentiation is the unique pure strategy
equilibrium location pattern.
When the cost difference between two firms is large, no
pure strategy equilibrium exists.
Suppose that firm 1 is a lower cost firm and the cost
difference is large. The best location of firm 1 is x1=x2
(minimal differentiation), while that of firm 2 is either
x2=1 or x2=0 (maximal differentiation).
Matsumura and Matsushima (2009)
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Inelastic Demand
Salop Model, Free Entry, Inelastic Demand
→Excess Entry
(exception, Matsumura and Okamura, 2006)
Salop Model, Free Entry, Elastic Demand
→Insufficient Entry takes place if demand elasticity is
large.
(Gu and Wenzel, 2009).
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Underinvestment
Private Duopoly, R&D Investment, Hotelling
→Underinvestment
If the firms can locate outside the city, the equilibrium
investment level is too high for social welfare.
(Matsumura and Matsushima, 2012).
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