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Vertical Contract and Competition Intensity in Hotelling’s Model

  • Xingtang Wang , Jie Li and Leonard F.S. Wang EMAIL logo
Published/Copyright: September 1, 2017
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 19 Issue 1

Abstract

This paper explores the impact of intensity of rivalry in downstream market on the equilibrium locations of the downstream firms under a vertical market structure á la Hotelling. We find that: (i) the presence of upstream firms softens the spatial competition in downstream market; (ii) minimum differentiation cannot be achieved as the equilibrium outcome and the equilibrium product differentiation is insufficient relative to socially optimum; (iii) social welfare is higher with a higher weight attached to intensity of rivalry, which is different from the non-monotonic relationship under the horizontal market case; (iv) the equilibrium product differentiation is independent of bargaining power under the two-part tariff contracts, which is different from Brekke and Straume (2004) under linear pricing.

Keywords: spatial competition; vertical structure; maximum (minimum) differentiation; intensity of rivalry
JEL Classification: D43; L13; L22

Funding statement: Jie Li acknowledges support from the key project of the National Natural Science Foundation of China (71333007), National Social Science Fund of China (15BJL087), Guangdong Provincial Natural Science Foundation (2014A030313395), and Fundamental Research Funds for the Central Universities (15JNYH001).

Appendix

∂Fi∂xi=L(λ,β,xi,xj) = N1−N2−N3+N42(5−3λ)2(−1+λ2)N1=(5−3λ)[β(3−λ)(2−λ)(1+λ)−7λ+(7−λ)λ2−3]N2=3[1−β(2−λ)−λ](1−λ)2(1+λ)xi2N3=[1−β(2−λ)−λ](1−λ)2(1+λ)(2−xj)xjN4=9−β(2−λ)(1+λ)(9−5λ)+5λ[1−(3−λ)λ]−[1−β(2−λ)−λ](1−λ2)xj

Since the downstream firms are symmetric, substituting xi=xj=x into L(λ,β,xi,xj), we can get l(λ,β,x).

l(λ,β,x)=3−β(3−λ)(2−λ)(1+λ)+M2(5−3λ)(1−λ2)where M=λ[7−(7−λ)λ]+4x(1−λ)[1+λ−λ2−β(2−λ)(1+λ)].

We find that when −1<λ⩽λ^(β,x),l(λ,β,x)⩽0; when λ^(β,x)<λ<1, l(λ,β,x)>0. As the expression of l(λ,β,x) is too complex, we use Figure 1 to graph λ^(β,x).

Figure 1 The graph of ˆλ(β,x).
Figure 1

The graph of ˆλ(β,x).

References

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Published Online: 2017-09-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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  4. On Decay Centrality
  5. Sequential Auctions with Decreasing Reserve Prices
  6. The core of a strategic game
  7. Targeted Advertising on Competing Platforms
  8. Representation in Multi-Issue Delegated Bargaining
  9. Endogenous Mergers in Markets with Vertically Differentiated Products
  10. Standards of Proof and Civil Litigation: A Game-Theoretic Analysis
  11. Retained Earnings, Interest Rates and Lending Relationship
  12. Uniform Price Auctions with Asymmetric Bidders
  13. Conformity and Influence
  14. Sellouts, Beliefs, and Bandwagon Behavior
  15. Notes
  16. Eco-Firms and the Sequential Adoption of Environmental Corporate Social Responsibility in the Managerial Delegation
  17. Vertical Contract and Competition Intensity in Hotelling’s Model
  18. Constrained Allocation of Projects to Heterogeneous Workers with Preferences over Peers
  19. Irrelevance of the Strategic Variable in the Case of Relative Performance Maximization
  20. Critical Efficiencies as Upward Pricing Pressure with Feedback Effects
  21. On the Openness of Unique Pure-Strategy Nash Equilibrium
  22. Forecast Dispersion in Finite-Player Forecasting Games
https://doi.org/10.1515/bejte-2017-0048
Keywords for this article
spatial competition; vertical structure; maximum (minimum) differentiation; intensity of rivalry

Articles in the same Issue

  1. Articles
  2. Privatizing Multi-subsidiary Public Firm in Location Model
  3. Efficient Combinatorial Allocations: Individual Rationality versus Stability
  4. On Decay Centrality
  5. Sequential Auctions with Decreasing Reserve Prices
  6. The core of a strategic game
  7. Targeted Advertising on Competing Platforms
  8. Representation in Multi-Issue Delegated Bargaining
  9. Endogenous Mergers in Markets with Vertically Differentiated Products
  10. Standards of Proof and Civil Litigation: A Game-Theoretic Analysis
  11. Retained Earnings, Interest Rates and Lending Relationship
  12. Uniform Price Auctions with Asymmetric Bidders
  13. Conformity and Influence
  14. Sellouts, Beliefs, and Bandwagon Behavior
  15. Notes
  16. Eco-Firms and the Sequential Adoption of Environmental Corporate Social Responsibility in the Managerial Delegation
  17. Vertical Contract and Competition Intensity in Hotelling’s Model
  18. Constrained Allocation of Projects to Heterogeneous Workers with Preferences over Peers
  19. Irrelevance of the Strategic Variable in the Case of Relative Performance Maximization
  20. Critical Efficiencies as Upward Pricing Pressure with Feedback Effects
  21. On the Openness of Unique Pure-Strategy Nash Equilibrium
  22. Forecast Dispersion in Finite-Player Forecasting Games
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