our works related to this paper
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Multiple Long-Run Equilibria in a Spatial Cournot Model and Welfare Implications joint work with Takanori Ago. (1) Spatial Cournot Competition and Transportation Costs in a Circular City (ARS 2012, joint work with Noriaki Matsushima) - PowerPoint PPT PresentationTRANSCRIPT
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Multiple Long-Run Equilibria in a Spatial Cournot Model and Welfare
Implications
joint work with Takanori Ago
2012/3/4 2
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)
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Mill Pricing Model (Shopping Model)
Kichijoji
Tachikawa
Kunitachi
Kokubunji
Mitaka
Musashisakai
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Delivered Pricing Model (Shipping Model, Spatial Price Discrimination
Model)
Hokkaido
Kyusyu KansaiTokai
Tohoku
Kanto
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Delivered Pricing (Shipping) Models
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Spatial Cournot ModelConsider 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
0 1
the location of firm 1
the location of firm 2
Two firms agglomerate at the central points.similar result in . Anderson and Neven (1991).
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Location and Transport Costs
0 1
A slight increase of x1
The area for which the relocation increases the transport cost of firm 1
The area for which the relocation decreases the transport cost of firm 1
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Spatial Cournot with Circular-CityConsider 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
An increase of x2
the area for which the relocation of firm 2 decreases transport cost
the area for which the relocation of firm 2 increases transport cost
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Equilibrium Location
the output of firm 2 is large
the output of firm 2 is small
The location minimizing the transport costof firm 2.
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Equilibrium Location
the equilibrium location of firm 2
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 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 existsIn 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 CS
Y
D
0
P
Pb
Pa
Pa
Pb
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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 mixed strategy equilibria under quadratic transport cost (Shopping, quadratic transport cost (Shopping,
Bertrand)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
the locations of firm 2
( no-linear transport cost)
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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) (1)
0 1
α
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)
0
1/2
market size 1
market size α
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)
0
1/2
It costs α to across this point
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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ApplicationIn the mill pricing (shopping) location-price models, In the mill pricing (shopping) location-price models,
both linear-city and circular-city models yield both linear-city and circular-city models yield maximal differentiation. maximal differentiation.
delivered pricing model (shipping model) →linear-city delivered pricing model (shipping model) →linear-city model and circular-city model yield different model and circular-city model yield different location patterns~ We discuss this shipping model.location patterns~ We discuss this shipping model.
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Location-Quantity Model
0
1/2
Firm 1Firm 2
Firm 1
Firm 2
α =1 =1
α=0=0 1/43/4
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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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Resultsthe equilibrium location of firm 1
α0
1/2
1/4
the same outcome as the linear-city model
Ebina, Matsumura and Shimizu (2011)
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Intuition
Why discontinuous (jump)?Why multiple equilibria ?←strategic complementaritySuppose 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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ComplementarityMatsumura (2004)
01
1/2
Firm 1 Firm 2
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ComplementarityMatsumura (2004)
01
1/2
Firm 2Firm 1
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.