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Social Efficiency of Entry in a Vertical Structure with Third Degree Price Discrimination

  • Junlin Chen , Arijit Mukherjee and Chenhang Zeng ORCID logo EMAIL logo
Published/Copyright: March 3, 2022
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 23 Issue 1

Abstract

We study social efficiency of entry in the presence of downstream cost asymmetry and upstream price discrimination. We show that entry is excessive when the entrants are highly inefficient, and it is insufficient when either the entrants are efficient or their inefficiency is low. The results are in sharp contrast to the existing literature considering upstream uniform pricing (Cao, H., and L. F. S. Wang. 2020. “Social Efficiency of Entry in a Vertically Related Industry Revisited.” Economics Letters 129. Art. no. 109200), as discriminatory pricing alters the relative strengths of the business-stealing, business-creation and production-(in)efficiency effects.

Keywords: excessive entry; insufficient entry; vertical market; price discrimination
JEL Classification: L13; L40
Corresponding author: Chenhang Zeng, Wenlan School of Business, Zhongnan University of Economics and Law, 182 Nanhu Ave., Wuhan 430073, China, E-mail:

Funding source: The Humanity and Social Science Planning Foundation of the Ministry of Education of China

Award Identifier / Grant number: 20YJA790001

Acknowledgments

We thank the editor, Ronald Peeters, and two anonymous reviewers for their constructive comments and suggestions that have helped to greatly improve the paper. Financial support from the Humanity and Social Science Planning Foundation of the Ministry of Education of China (Grant No. 20YJA790001) is gratefully acknowledged.

Appendix: Proofs

A The Derivation of (3)

In the second stage, the standard first order conditions are

∂ Π ( w c , w d ) ∂ w c = m q i * + m w c ∂ q i * ∂ w c + n w d ∂ q j * ∂ w c = 0 , ∂ Π ( w c , w d ) ∂ w d = m w c ∂ q i * ∂ w d + n q j * + n w d ∂ q j * ∂ w d = 0 .

Differentiating the two equations in (1) with respect to w c yields

∂ q i * ∂ w c = ( n + 1 ) p ′ ( Q * ) + p ′ ′ ( Q * ) n q j * p ′ ( Q * ) ( m + n + 1 ) p ′ ( Q * ) + Q * p ′ ′ ( Q * ) < 0 ; ∂ q j * ∂ w c = − m p ′ ( Q * ) + m p ′ ′ ( Q * ) q j * p ′ ( Q * ) ( m + n + 1 ) p ′ ( Q * ) + Q * p ′ ′ ( Q * ) > 0 .

Similarly, differentiating the two equations in (1) with respect to w d yields

∂ q i * ∂ w d = − n p ′ ( Q * ) + n p ′ ′ ( Q * ) q i * p ′ ( Q * ) ( m + n + 1 ) p ′ ( Q * ) + Q * p ′ ′ ( Q * ) > 0 ; ∂ q j * ∂ w d = ( m + 1 ) p ′ ( Q * ) + p ′ ′ ( Q * ) m q i * p ′ ( Q * ) ( m + n + 1 ) p ′ ( Q * ) + Q * p ′ ′ ( Q * ) < 0 .

Incorporating the expressions of ∂ q i * ∂ w c , ∂ q j * ∂ w c , ∂ q i * ∂ w d , ∂ q j * ∂ w d into the first order conditions leads to the equilibrium input prices in (3).

B Proof of Lemma 1

By (3), we obtain that w d * − w c * = p ′ ( Q * ) q i * − q j * . Furthermore, the equilibrium outcomes should satisfy the first order conditions in (1), which leads to w d * − w c * = c − d + p ′ ( Q * ) q j * − q i * = c − d − w d * − w c * . Therefore, we have w d * − w c * = ( c − d ) / 2 . Further, we get w d * + d − w c * + c = ( d − c ) / 2 . Hence, w d * + d − w c * + c increases with (d − c).

C Equilibrium outputs

In stage 1, incorporating (3) into (2) leads to the equilibrium quantities as functions of the number of entrants, i.e. n, which are implicitly determined by

q i * ( n ) = p ( Q * ( n ) ) − c + p ′ ( Q * ( n ) ) q i * ( n ) + Q * ( n ) + p ′ ′ ( Q * ( n ) ) m ( q i * ( n ) ) 2 + n ( q j * ( n ) ) 2 − p ′ ( Q * ( n ) ) ; q j * ( n ) = p ( Q * ( n ) ) − d + p ′ ( Q * ( n ) ) q j * ( n ) + Q * ( n ) + p ′ ′ ( Q * ( n ) ) m ( q i * ( n ) ) 2 + n ( q j * ( n ) ) 2 − p ′ ( Q * ( n ) ) ,

where Q * ( n ) = m q i * ( n ) + n q j * ( n ) .

We next verify the assumption which grantees positive quantities. As we see in (6), entrants always produce positive quantities, i.e. q j * > 0 . Furthermore, if incumbents are more efficient (i.e. c < d), they produce positive quantities. Otherwise, following (6), incumbents are active in production only when the cost difference is small such that (c − d)2 < − 4Fp′(Q*).

D Proof of Proposition 1

Simple calculations lead to

∂ SW ∂ n ∣ n = n * = ( p − c ) m ∂ q i * ( n ) ∂ n + ( p − d ) n ∂ q j * ( n ) ∂ n + ( p − d ) q j * ( n ) + P ′ ( Q * ) ( q j * ( n ) ) 2 = ( p − c ) m ∂ q i * ( n ) ∂ n + ( p − d ) n ∂ q j * ( n ) ∂ n + w d * q j * ( n ) = p − w c * − c m ∂ q i * ( n ) ∂ n + p − w d * − d n ∂ q j * ( n ) ∂ n ︸ business − stealing effect + ∂ m q i * ( n ) ∂ n w c * + ∂ n q j * ( n ) ∂ n w d * ︸ business − creation effect = p − w d * − d m ∂ q i * ( n ) ∂ n + n ∂ q j * ( n ) ∂ n ︸ pure business − stealing effect + m ( w d * + d − w c * + c ) ∂ q i * ( n ) ∂ n ︸ production − ( in ) efficiency effect + ∂ m q i * ( n ) ∂ n w c * + ∂ n q j * ( n ) ∂ n w d * ︸ business − creation effect , = p − w d * − d m ∂ q i * ( n ) ∂ n + n ∂ q j * ( n ) ∂ n ︸ pure business − stealing effect + m ( d − c ) 2 ∂ q i * ( n ) ∂ n ︸ production − ( in ) efficiency effect + ∂ m q i * ( n ) ∂ n w c * + ∂ n q j * ( n ) ∂ n w d * ︸ business − creation effect .

The second equality follows because we have P ′ ( Q * ) q j * ( n ) = − ( P ( Q * ) − d − w d * ) by (2), and the last equality follows because we have w d * + d − w c * + c = ( d − c ) / 2 in Lemma 1.

E Proof of Lemma 2

It follows from (12) that ∂ n * / ∂ m = ( Δ − 2 F ) / ( 2 F ) < 0 , and ∂ n * / ∂ Δ = m / ( 2 F ) > 0 . Furthermore, simple calculations lead to ∂ SW * / ∂ m = ( Δ − 2 F ) ( Δ − F ) / 2 and ∂ SW * / ∂ Δ = m ( Δ − 3 F / 2 ) . which yield the results in Lemma 2(ii).

F Proof of Proposition 2

The denominator of (16) is positive since a > 2 F ( 2 + m ) + ( 1 + m ) d − m c . The sign of (16) is positive if 2 F + ( c − d ) m > 0 , which reduces to d − c < 2 F / m .

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Received: 2021-05-20
Accepted: 2022-01-09
Published Online: 2022-03-03

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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  1. Frontmatter
  2. Research Articles
  3. Duty to Read vs Duty to Disclose Fine Print. Does the Market Structure Matter?
  4. Cobb-Douglas Preferences and Pollution in a Bilateral Oligopoly Market
  5. Epsilon-Efficiency in a Dynamic Partnership with Adverse Selection and Moral Hazard
  6. Management Turnover, Strategic Ambiguity and Supply Incentives
  7. Uninformed Bidding in Sequential Auctions
  8. Arrowian Social Equilibrium: Indecisiveness, Influence and Rational Social Choices under Majority Rule
  9. Family Ties and Corruption
  10. Social Efficiency of Entry in a Vertical Structure with Third Degree Price Discrimination
  11. Insufficient Entry and Consumer Search
  12. Quality Competition and Market-Share Leadership in Network Industries
  13. The Effects of Introducing Advertising in Pay TV: A Model of Asymmetric Competition between Pay TV and Free TV
  14. Redistributive Unemployment Benefit and Taxation
  15. Constrained Persuasion with Private Information
  16. A Dynamic Graph Model of Strategy Learning for Predicting Human Behavior in Repeated Games
  17. Relative Income Concerns, Dismissal, and the Use of Pay-for-Performance
  18. Delegation in Vertical Relationships: The Role of Reciprocity
  19. Step by Step Innovation without Mutually Exclusive Patenting: Implications for the Inverted U
  20. Data and Competitive Markets: Some Notes on Competition, Concentration and Welfare
  21. Notes
  22. Optimality of a Linear Decision Rule in Discrete Time AK Model
  23. Equilibrium Pricing under Concave Advertising Costs
https://doi.org/10.1515/bejte-2021-0069
Keywords for this article
excessive entry; insufficient entry; vertical market; price discrimination

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Duty to Read vs Duty to Disclose Fine Print. Does the Market Structure Matter?
  4. Cobb-Douglas Preferences and Pollution in a Bilateral Oligopoly Market
  5. Epsilon-Efficiency in a Dynamic Partnership with Adverse Selection and Moral Hazard
  6. Management Turnover, Strategic Ambiguity and Supply Incentives
  7. Uninformed Bidding in Sequential Auctions
  8. Arrowian Social Equilibrium: Indecisiveness, Influence and Rational Social Choices under Majority Rule
  9. Family Ties and Corruption
  10. Social Efficiency of Entry in a Vertical Structure with Third Degree Price Discrimination
  11. Insufficient Entry and Consumer Search
  12. Quality Competition and Market-Share Leadership in Network Industries
  13. The Effects of Introducing Advertising in Pay TV: A Model of Asymmetric Competition between Pay TV and Free TV
  14. Redistributive Unemployment Benefit and Taxation
  15. Constrained Persuasion with Private Information
  16. A Dynamic Graph Model of Strategy Learning for Predicting Human Behavior in Repeated Games
  17. Relative Income Concerns, Dismissal, and the Use of Pay-for-Performance
  18. Delegation in Vertical Relationships: The Role of Reciprocity
  19. Step by Step Innovation without Mutually Exclusive Patenting: Implications for the Inverted U
  20. Data and Competitive Markets: Some Notes on Competition, Concentration and Welfare
  21. Notes
  22. Optimality of a Linear Decision Rule in Discrete Time AK Model
  23. Equilibrium Pricing under Concave Advertising Costs
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