1. Equilibrium Location with Elastic Demand in Mixed Duopoly
Joint work with Minoru Kitahara
OT2012

2. 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.
OT2012

3. Mill Pricing Model (Shopping Model)
Mitaka
Kichijoji
Musashisakai
Tachikawa
Kokubunji
Kunitachi
OT2012

4. Delivered Pricing Model (Shipping Model, Spatial Price Discrimination Model)
Hokkaido
Tohoku
Kanto
Tokai
Kansai
Kyusyu
OT2012

5. Mill Pricing (Shopping) Models
OT2012

6. 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.
OT2012

7. Hotelling the location of firm 1 the location of firm 2 ０ １
企業１の立地
firm 1's demand
firm 2's demand
OT2012

8. Relocation of Firm 1 the location of firm 2 the location of firm 1 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
OT2012

9. 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.
OT2012

10. Best reply for firm 1 the location of firm 2
the optimal location of firm 1 ０ １
OT2012

11. Best reply for firm 1 the location of firm 2
the optimal location of firm 1 ０ １
OT2012

12. Equilibrium
the location of firm 2
the location of firm 1 ０ １
OT2012

13. 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.
OT2012

14. 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.
OT2012

15. 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.
OT2012

16. 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).
OT2012

17. 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)
OT2012

18. Maximal Differentiation
firm1's location
firm 2's location ０ １
企業１の立地
OT2012

19. 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
OT2012

20. Circular-City Model
Vickrey (1964), Salop (1979)
OT2012

21. 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.
OT2012

22. Equilibrium locations under linear-quadratic transport cost
the location of firm 1
Both strictly convex and concave transport cost usually yield this type of equilibrium
De Frutos et al (1999,2002)
transport costがstrictly convexでもstrictly concaveでも大抵こうなる
the location of firm 2
OT2012

23. 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)
the location of firm 2
OT2012

24. 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
private firm’s location
企業１の立地 １
1/4 ０
3/4
OT2012

25. 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
private firm’s location
企業１の立地 １ ０
1/8
5/8
OT2012

26. 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
private firm’s location
企業１の立地 １ ０
1/8
1/2
OT2012

27. 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.
OT2012

28. Matsumura and Matsushima (2004)
Cost-Reducing R&D
Location Choice
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.
OT2012

29. 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
企業１の立地 １
1/4 ０
3/4
OT2012

30. R&D competition Cost-Reducing R&D Location Choice Bertrand Competition
A decrease in the private firm’s marginal production cost (increase, decrease) x0 (location of the public firm).
OT2012

31. R&D competition Cost-Reducing R&D Location Choice 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.
OT2012

32. 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
企業１の立地 １
1/4 ０
3/4
OT2012

33. 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.
OT2012

34. 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.
OT2012

35. Mixed Strategy Equilibria
OT2012

36. 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).
OT2012

37. 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)
OT2012

38. 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).
OT2012

39. Underinvestment Private Duopoly, R&D Investment, Hotelling
If the firms can locate outside the city, the equilibrium investment level is too high for social welfare.
(Matsumura and Matsushima, 2012).
OT2012