Our works related to this paper

OT2012 1

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)

OT 2012 3

Mill Pricing Model (Shopping Model)

Kichijoji

Tachikawa

Kunitachi

Kokubunji

Mitaka

Musashisakai

OT 2012 4

Delivered Pricing Model (Shipping Model, Spatial Price Discrimination

Model)

Hokkaido

Kyusyu KansaiTokai

Tohoku

Kanto

OT 2012 5

Delivered Pricing (Shipping) Models

OT 2012 6

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)

OT 2012 7

Properties of Spatial Cournot Model

Market overlap ~ Two firms supply for all markets

Market share depends on the locations of the two firms.

OT 2012 8

Equilibrium Location

０１

the location of firm 1

the location of firm 2

Two firms agglomerate at the central points.similar result in . Anderson and Neven (1991).

OT 2012 9

Location and Transport Costs

０１

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

OT 2012 10

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)

OT 2012 11

Equilibrium Location Without loss of generality. we assume x1=0

Consider the best reply for firm 2.

OT 2012 12

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

OT 2012 13


the output of firm 2 is large

the output of firm 2 is small

The location minimizing the transport costof firm 2.

OT 2012 14


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).

OT 2012 15

Equilibrium Location Equilibrium Location

Equidistant Location Pattern

OT 2012 16


Partial Agglomeration～ Matsushima (2001)

OT 2012 17


a continuum of equilibria exists～ Shimizu and Matsumura (2003), Gupta et al (2004)

OT 2012 18


Under non-liner transport cost

OT 2012 19

Equilibrium Location in

Under non-linear transport cost ~ Matsumura and Matsushima (2012)

OT 2012 20


a continuum of equilibria existsIn all equilibria, CS and PS, and so TS are the same.

OT 2012 21


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.

OT2012 22

CS CS

Y

D

0

P

Pb

Pa

Pa

Pb

OT 2012 23


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)

OT 2012 24

mixed strategy equilibria under mixed strategy equilibria under quadratic transport cost (Shopping, quadratic transport cost (Shopping,

Bertrand)Bertrand)

the locations of firm 1


non-maximal differentiation, Ishida and Matsushima, 2004.

OT 2012 25

mixed strategy equilibria (Shopping, Cournot)



（ no-linear transport cost)

OT 2012 26

mixed strategy equilibria (linear

transport cost) a continuum of equilibria exists～ Matsumura and Shimizu (2008)

OT 2012 27

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.

OT 2012 28

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.

OT 2012 29

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.

OT 2012 30

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)).

OT 2012 31

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.

OT 2012 32

Location-Quantity Model

0

1/2

Firm 1Firm 2

Firm 1

Firm 2

α =1 =1

α=0=0 1/43/4

OT 2012 33

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)

OT 2012 34

Resultsthe equilibrium location of firm 1

α0

1/2

1/4

the same outcome as the linear-city model

Ebina, Matsumura and Shimizu (2011)

OT 2012 35

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)

OT 2012 36

ComplementarityMatsumura (2004)

01

1/2

Firm 1 Firm 2

OT 2012 37

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.

Our works related to this paper

Documents

Transcript of Our works related to this paper