Product Diversification, Entry

Multiple Long-Run Equilibria in a
Spatial Cournot Model and Welfare
Implications
joint work with Takanori Ago
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Our works related to this paper
(1) Spatial Cournot Competition and Transportation
Costs in a Circular City (ARS 2012, joint work with
Noriaki Matsushima)
(2) Spatial Cournot Equilibria in a Quasi-Linear City
(PiRS 2010, with Takeshi Ebina and Daisuke Shimizu)
(3) Noncooperative Shipping Cournot Duopoly with
Linear-Quadratic Transport Costs and Circular Space
(JER 2008, with Daiskuke Shimizu).
(4) Spatial Cournot Competition and Economic Welfare:
A Note (RSUE 2005, with Daisuke Shimizu)
(5) Partial Agglomeration or Dispersion in Spatial
Cournot Competition (SEJ 2005, with Takao Ohkawa
and Daisuke Shimizu)
2012/3/4
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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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Delivered Pricing (Shipping) Models
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Spatial Cournot Model
Consider a symmetric duopoly.
Transport cost is proportional to both distance and
output quantity (linear transport cost).
In the first stage, each firm chooses its location
independently.
In the second stage, each firm chooses its output
independently.
Each point has an independent market, and the
demand function is linear demand function, P=A-Y.
No consumer's arbitrage. Production cost is
normalized as zero. A is sufficiently large.
Hamilton et al (1989), Anderson and Neven (1991)
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Properties of Spatial Cournot Model
Market overlap ~ Two firms supply for all markets
Market share depends on the locations of the two
firms.
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Equilibrium Location
the location of firm 1
the location of firm 2
０
１
Two firms agglomerate at the central points.
similar result in . Anderson and Neven (1991).
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Location and Transport Costs
A slight increase of x1
０
The area for which the
relocation increases the
transport cost of firm 1
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The area for which the
relocation decreases the
transport cost of firm 1
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Spatial Cournot with Circular-City
Consider a symmetric duopoly.
Transport cost is proportional to both distance and
output quantity (linear transport cost).
In the first stage, each firm chooses its location
independently on the circle.
In the second stage, each firm chooses its output
independently.
Each point has an independent market, and the
demand function is linear demand function, P=A-Y.
No consumer's arbitrage. Production cost is
normalized as zero. A is sufficiently large.
Pal (1998)
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Equilibrium Location
Without loss of
generality. we
assume x1=0
Consider the best
reply for firm 2.
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Location and Transport Costs
the area for
which the
relocation of
firm 2
increases
transport cost
An increase of x2
the area for which the relocation of firm 2 decreases
transport cost
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Equilibrium Location
the output of firm 2 is small
The location
minimizing
the transport
cost
of firm 2.
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the output of firm
2 is large
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Equilibrium Location
the equilibrium location of firm 2
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Maximal distance
is the unique pure
strategy
equilibrium
location pattern as
long as the
transport cost is
strictly increasing
(Matsumura and
Shimizu 2006).
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Equilibrium Location
Equidistant
Location Pattern
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Equilibrium Location
Partial
Agglomeration
～Matsushima (2001)
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Equilibrium Location
a continuum of equilibria exists
～Shimizu and Matsumura (2003), Gupta et al (2004)
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Equilibrium Location
Under non-liner transport cost
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Equilibrium Location in
Under non-linear transport cost ~ Matsumura and
Matsushima
(2012)
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Equilibrium Location
a continuum of equilibria exists
In all equilibria, CS and PS, and so TS are the same.
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Equilibrium Location in
Under non-liner
transport cost,
dispersion equilibrium
yields larger PS and TS
and smaller CS than
partial agglomeration
equilibrium (Proposition
1).
Price dispersion yields
larger CS.
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CS
P
Pa
Pb
Pb
D
Pa
0
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Equilibrium Location in
Free Entry ~ Under
non-liner transport cost,
dispersion equilibrium
yields larger number of
entering firms. Thus CS
can be larger than that
in partial agglomeration
equilibrium.
(Proposition 2)
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mixed strategy equilibria under
quadratic transport cost (Shopping,
Bertrand)
the locations
of firm 1
the locations
of firm 2
non-maximal differentiation, Ishida and Matsushima,
2004.
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mixed strategy equilibria
(Shopping, Cournot)
the locations of
firm 1
（no-linear
transport cost)
the locations
of firm 2
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mixed strategy equilibria (linear
transport cost)
a continuum of
equilibria exists
～Matsumura
and Shimizu
(2008)
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Two Standard Models of Space
(1) Hotelling type Linear-City Model
(2) Salop type (or Vickery type) Circular-City Model
Linear-City has a center-periphery structure, while
every point in the Circular-City is identical.
→Circular Model is more convenient than Linear
Model for discussing symmetric except for
duopoly.
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General Model (1)
α
0
1
It costs α to transport from 0 to 1.
The transport cost from 0 to 0.9 is min(0.9t, α+0.1t).
If α＝０, this model is a circular-city model. If α >t, this
model is a linear-city model.
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General Model (2)
market size α
0
market size 1
1/2
If α =0, this model is a linear-city model. If α=1, this
model is a circular-city model.
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General Model (3)
It costs α to
across this
point
0
1/2
If α=0, this model is a circular-city model. If α>1, it is a
linear-city model. (essentially the same model as (1)).
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Application
In the mill pricing (shopping) location-price models,
both linear-city and circular-city models yield
maximal differentiation.
delivered pricing model (shipping model) →linear-city
model and circular-city model yield different
location patterns~ We discuss this shipping model.
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Location-Quantity Model
0
Firm 2
3/4
α=0
1/4
Firm 1
1/2
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Firm 1
Firm 2
α =1
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Results
・The equilibrium locations are symmetric.
・The equilibrium location pattern is discontinuous
with respect to α (A jump takes place).
・Multiple equilibria exist. Abina et al (OT 2012)
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the equilibrium
location of firm 1
Results
the same outcome as the linear-city
model
1/2
1/4
0
Ebina, Matsumura and Shimizu (2011)
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Intuition
Why discontinuous (jump)?
Why multiple equilibria？
←strategic complementarity
Suppose that firm 1 relocate form 0 to 1/2.
It increases the incentive for central location of firm
2. ～Matsumura (2004)
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Complementarity
Matsumura (2004)
Firm 2
Firm 1
0
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1/2
1
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Complementarity
Matsumura (2004)
Firm 1
0
Firm 2
1
1/2
Central location by firm 1 increases the value of
market 0 and decreases that of market 1 for firm 2→it
increases the incentive for central location by firm 2.
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