Accessible Unlicensed Requires Authentication Published by De Gruyter October 8, 2021

Costs of Sales Forces, Substitution between Competing Products, and Vertical Integration Decisions

Chung-Hui Chou ORCID logo

Abstract

This paper analyzes duopolistic firms’ vertical integration decisions with considering costs of sales forces and sales delegation under vertical integration. The main contribution of our research is showing that full vertical integration (separation) is more common when competing products are highly (weakly) substitutable. Second, contrary to conventional wisdom, an asymmetric vertical structure may not only be an equilibrium outcome but may also be optimal for consumers’ surplus in spite of yielding higher retail prices than those arising under full vertical integration. We also examine the impacts of vertical structures on welfare which have vertical merger policy relevance. First, when products are weakly substitutable, keeping vertical merger costs low may induce full vertical integration to be an equilibrium outcome which optimizes consumers’ surplus and social welfare simultaneously. Second, imposing a vertical merger tax increasing with substitution between products on firms may induce firms’ vertical integration decisions to be optimal for social welfare.

JEL Classification: D43; L11; L13

Corresponding author: Chung-Hui Chou, Department of Finance, I-Shou University, Kaohsiung City 84001, Taiwan, ROC, E-mail:

Acknowledgments

I declare that: (1) no support, financial or otherwise, has been received from any organization that may have an interest in the submitted work; and (2) there are no other relationships or activities that could appear to have influenced the submitted work. I appreciate the editor, Till Requate and an anonymous referee for their helpful and insightful suggestions which improve this paper significantly.

Appendix I: Proof of Lemma 1

  1. (1)

    If R < Π 1 * I , I Π 1 * S , I , then Π 1 * I , I R > Π 1 * S , I . Similarly, Π 2 * I , I R > Π 2 * I , S . Hence, (I, I) is the Nash Equilibrium.

  2. (2)

    If Π 1 * I , I Π 1 * S , I < R < Π 1 * I , S Π 1 * S , S , then Π 1 * I , S R > Π 1 * S , S and Π 1 * S , I > Π 1 * I , I R . Hence, (I, S) and (S, I) are Nash Equilibria.

  3. (3)

    If R > Π 1 * I , S Π 1 * S , S , then Π 1 * S , S > Π 1 * I , S R . Similarly, Π 2 * S , S > Π 2 * S , I R . Hence, (S, S) is the Nash Equilibrium.

This completes the proof. □

Appendix II: Proof of Proposition 2

The first and third part can be derived directly by Lemma 1.

The proof of the second part is stated as follows.

From (8a) and (13b),

Π 1 * ( I , I ) Π 1 * ( S , I ) = ( λ 1 + a ) 2 2 λ 2 2 ( 512 896 λ 2 + 480 λ 4 72 λ 6 λ 8 ) ( 4 2 λ λ 2 ) 2 ( 32 32 λ 2 + 7 λ 4 ) 2 R 1 ( λ ) .

From (8b) and (18b),

Π 2 * S , I Π 2 * S , S = ( λ 1 + a ) 2 2 λ 2 2 ( 2 + λ ) ( 512 1152 λ 2 + 900 λ 4 287 λ 6 + 32 λ 8 ) ( 2 λ ) ( 4 λ 2 λ 2 ) 2 ( 32 32 λ 2 + 7 λ 4 ) 2 R 2 ( λ ) .

Hence,

R 2 ( λ ) R 1 ( λ ) = 4 λ ( λ 1 + a ) 2 ( 2 λ 2 ) 4 Φ ( λ ) ( 2 λ ) ( 4 λ 2 λ 2 ) 2 ( 4 2 λ λ 2 ) 2 ( 32 32 λ 2 + 7 λ 4 ) 2 > 0 if λ < 0.9494 < 0 if λ > 0.9494

Φ λ = 512 576 λ 736 λ 2 + 824 λ 3 + 356 λ 4 360 λ 5 78 λ 6 + 49 λ 7 + 7 λ 8 .

From the above results,

  1. (1)

    If λ < 0.9494, then Π 2 * I , I Π 2 * S , I < R < Π 1 * I , S Π 1 * S , S .This implies that

    Π 1 * S , S < Π 1 * I , S R .

    Π 2 * I , I R < Π 1 * S , I .

    Hence, if λ < 0.9494 and costs of sales forces are moderate, i.e. R 1(λ) < R < R 2(λ), then S , I and I , S are Nash equilibria. In other words, vertical integration and vertical separation coexist in a market with symmetric firms.

  2. (2)

    If λ > 0.9494, then Π 1 * I , S Π 1 * S , S < R < Π 2 * I , I Π 2 * S , I .This implies

    Π 2 * I , I R > Π 2 * S , I .

    Π 1 * I , S R < Π 1 * S , S .

    Hence, if λ > 0.9494 and costs of sales forces are moderate, i.e. R 2(λ) < R < R 1(λ), then S , S and I , I are Nash equilibria.

To sum up the above findings,

  1. (1)

    If λ < 0.9494 and costs of sales forces are moderate, i.e. min R 1 λ , R 2 λ < R < max R 1 λ , R 2 λ , an asymmetric vertical structure is the unique equilibrium outcome.

  2. (2)

    If λ > 0.9494 and costs of sales forces are moderate, i.e. min R 1 λ , R 2 λ < R < max R 1 λ , R 2 λ , then firms may choose full vertical integration or separation symmetrically in equilibrium.

This completes the proof. □

Appendix III: Proof of Proposition 3

From (13b) and (18b),

Π 1 * I , I R Π 1 * S , S = ( λ 1 + a ) 2 2 λ 2 2 ( 16 24 λ 2 λ 2 + 9 λ 3 ) ( 2 λ ) 4 2 λ λ 2 2 ( 4 λ 2 λ 2 ) 2 R .

( λ 1 + a ) 2 2 λ 2 2 ( 16 24 λ 2 9 λ 3 ) ( 2 λ ) 4 2 λ λ 2 2 ( 4 λ 2 λ 2 ) 2 > 0 if λ < 0.8133 < 0 if λ > 0.8133 .

In the following analysis,

R 3 ( λ ) ( λ 1 + a ) 2 2 λ 2 2 ( 16 24 λ 2 λ 2 + 9 λ 3 ) ( 2 λ ) 4 2 λ λ 2 2 ( 4 λ 2 λ 2 ) 2 .

From footnote 9, R 1 λ > R 3 λ .

  1. (1)

    If λ < 0.8133 and R > R 3(λ), then Π 1 * I , I R < Π 1 * S , S which was pointed out in Proposition 1.

If R 3(λ) < R < R 1(λ), then firms choose full vertical integration in equilibrium, and Π 1 * I , I R < Π 1 * S , S .

This means that when λ < 0.8133 and R 3(λ) < R < R 1(λ), firms choose full vertical integration in equilibrium and face the prisoners’ dilemma.

  1. (2)

    If λ > 0.8133, then Π 1 * I , I R < Π 1 * S , S which was pointed out in Proposition 1.

If λ > 0.8133 and R < R 1(λ), then firms choose full vertical integration in equilibrium, and Π 1 * I , I R < Π 1 * S , S .

This means that when λ > 0.8133 and R < R 1(λ), firms choose full vertical integration in equilibrium and face the prisoners’ dilemma.

This completes the proof. □

Appendix IV: Proof of Lemma 2

From (19a), (19b), and (19c),

c s * I , I c s * ( S , S ) = 4 ( 1 λ ) ( λ 1 + a ) 2 ( 2 λ 2 ) 3 ( 6 4 λ 2 λ 2 + λ 3 ) ( 2 λ ) 2 ( 4 λ 2 λ 2 ) 2 ( 4 2 λ λ 2 ) 2 > 0 ,

c s * S , I c s * S , S = 2 ( λ 1 + a ) 2 ( 2 λ 2 ) 3 G ( λ ) 2 λ 2 4 λ 2 λ 2 2 ( 32 32 λ 2 + 7 λ 4 ) 2 > 0 ,

c s * S , I c s * I , I = 2 ( λ 1 + a ) 2 ( 2 λ 2 ) 3 J ( λ ) 4 2 λ λ 2 2 ( 32 32 λ 2 + 7 λ 4 ) 2 < 0   if    λ < 0.9254 > 0   if    λ > 0.9254 ,

Here,

G λ = 384 64 λ 888 λ 2 + 96 λ 3 + 766 λ 4 46 λ 5 306 λ 6 + 7 λ 7 + 57 λ 8 4 λ 10 .

J λ = 96 + 64 λ + 168 λ 2 96 λ 3 94 λ 4 + 46 λ 5 + 18 λ 6 7 λ 7 λ 8 .

From (20a), (20b), and (20c),

w * I , I w * S , S = 4 ( λ 1 + a ) 2 ( 2 λ 2 ) 2 F ( λ ) ( 2 λ ) 2 ( 4 2 λ λ 2 ) 2 ( 4 λ 2 λ 2 ) 2 > 0   if    λ < 0.6790 < 0   if    λ > 0.6790 .

w * S , I w * S , S = 2 ( λ 1 + a ) 2 ( 2 λ 2 ) 2 H ( λ ) 2 λ 2 4 λ 2 λ 2 2 ( 32 32 λ 2 + 7 λ 4 ) 2 > 0 if λ < 0.7183 < 0 if λ > 0.7183

w * S , I w * I , I = 2 ( λ 1 + a ) 2 ( 2 λ 2 ) 2 K ( λ ) ( 4 2 λ λ 2 ) 2 ( 32 32 λ 2 + 7 λ 4 ) 2 < 0 if λ < 0 . 6175 > 0 if λ > 0 . 6175

Here,

F λ = 20 44 λ + 10 λ 2 + 24 λ 3 9 λ 4 3 λ 5 + λ 6 .

H λ = 1280 896 λ 3600 λ 2 + 1664 λ 3 + 3884 λ 4 1124 λ 5 2054 λ 6 + 328 λ 7 + 567 λ 8 35 λ 9 77 λ 10 + 4 λ 12

K λ = 320 + 384 λ + 720 λ 2 640 λ 3 556 λ 4 + 372 λ 5 + 178 λ 6 88 λ 7 23 λ 8 + 7 λ 9 + λ 10 .

This completes the proof. □

Appendix V: The Proof of R 3(λ) < R 2(λ)

Π 1 * I , I Π 1 * S , S = ( λ 1 + a ) 2 2 λ 2 2 ( 16 24 λ 2 λ 2 + 9 λ 3 ) ( 2 λ ) 4 2 λ λ 2 2 ( 4 λ 2 λ 2 ) 2 R 3 ( λ ) .

Π 1 * I , S Π 1 * S , S = ( λ 1 + a ) 2 ( 2 λ 2 ) 2 ( 2 + λ ) ( 512 1152 λ 2 + 900 λ 4 287 λ 6 + 32 λ 8 ) ( 2 λ ) ( 4 λ 2 λ 2 ) 2 ( 32 32 λ 2 + 7 λ 4 ) 2 R 2 ( λ ) .

Hence,

R 3 ( λ ) R 2 ( λ ) = 8 λ ( λ 1 + a ) 2 ( λ 2 2 ) 3 ( 32 + 4 λ 32 λ 2 4 λ 3 + 7 λ 4 + λ 5 ) ( 4 2 λ λ 2 ) 2 ( 32 32 λ 2 + 7 λ 4 ) 2 < 0 .

Therefore, R 3(λ) < R 2(λ).

From the above derivations, if R > R 2 λ , then Π 1 * I , I R < Π 1 * S , S .

This means that firms cannot face the prisoners’ dilemma when full vertical separation is an equilibrium outcome.

This completes the proof. □

Appendix VI: Derivations of Consumers’ Surplus

c s * I , I = [ a p i * ( I , I ) + λ p i * ( I , I ) ] 2 = [ a ( 1 λ ) p I * ( I , I ) ] 2 .

c s * S , S = [ a p i * ( S , S ) + λ p i * ( S , S ) ] 2 = [ a ( 1 λ ) p i * ( S , S ) ] 2 .

c s * S , I = 1 2 a p 1 * S , I + λ p 2 * S , I 2 + 1 2 [ a p 2 * ( S , I ) + λ p 1 * ( S , I ) ] 2 .

Inserting the retail prices derived in Section 3 into the above expressions yields consumers’ surplus under three possible vertical structures stated as (19a), (19b), and (19c).

This completes the proof. □

Appendix VII: Proof of Proposition 5

From (9a), (9b), and (13a),

p 1 * ( S , I ) p 1 * ( I , I ) = 2 λ 1 + a ( 2 λ ) ( 2 + λ ) 2 λ 2 2 4 2 λ λ 2 32 32 λ 2 + 7 λ 4 > 0 .

p 2 * ( S , I ) p 2 * ( I , I ) = 4 λ 1 + a λ 2 λ 2 2 4 2 λ λ 2 32 32 λ 2 + 7 λ 4 > 0 .

This completes the proof. □

References

Baik, K., and D. Lee. 2019. “Decisions of Duopoly Firms on Sharing Information on Their Delegation Contracts.” Review of Industrial Organization 57: 145–65. https://doi.org/10.1007/s11151-019-09732-3. Search in Google Scholar

Barcena-Ruiz, J., and N. Olaizola. 2006. “Cost-Saving Production Technologies, and Strategic Delegation.” Australian Economic Papers 45: 141–57. https://doi.org/10.1111/j.1467-8454.2006.00283.x. Search in Google Scholar

Bonanno, G., and J. Vickers. 1988. “Vertical Separation.” The Journal of Industrial Economics 36: 257–65. https://doi.org/10.2307/2098466. Search in Google Scholar

Buehler, S., and A. Schmutzler. 2005. “Asymmetric Vertical Integration.” The B.E. Journal of Theoretical Economics 5: 1–27. https://doi.org/10.2202/1534-5963.1164. Search in Google Scholar

Caillaud, B., and P. Rey. 1994. “Strategic Aspects of Vertical Delegation.” European Economic Review 39: 421–31. Search in Google Scholar

Chakrabarty, D., A. Chaudhuri, and C. Spell. 2002. “Information Structure and Contractual Choice in Franchising.” Journal of Institutional and Theoretical Economics 158: 638–63. https://doi.org/10.1628/0932456022975204. Search in Google Scholar

Chou, C. 2014. “Strategic Delegation and Vertical Integration.” Managerial and Decision Economics 35: 580–6. https://doi.org/10.1002/mde.2675. Search in Google Scholar

Cyrenne, P. 1994. “Vertical Integration versus Vertical Separation: An Equilibrium Model.” Review of Industrial Organization 9: 311–22. Search in Google Scholar

Fanti, L., and N. Meccheri. 2015. “On the Cournot-Bertrand Profit Differential and the Structure of Unionization in a Managerial Duopoly.” Australian Economic Papers 54: 266–87. https://doi.org/10.1111/1467-8454.12054. Search in Google Scholar

Fershtman, C., and K. Judd. 1987. “Equilibrium Incentives in Oligopoly.” The American Economic Review 77: 927–40. Search in Google Scholar

Gal-Or, E. 1999. “Vertical Integration or Separation of Sales Function as Implied by Competitive Forces.” International Journal of Industrial Organization 17: 641–62. https://doi.org/10.1016/s0167-7187(97)00056-8. Search in Google Scholar

Jansen, J. 2003. “Coexistence of Strategic Vertical Separation and Integration.” International Journal of Industrial Organization 21: 699–716. https://doi.org/10.1016/s0167-7187(02)00146-7. Search in Google Scholar

Kopel, M., and C. Riegler. 2006. “R&D in a Strategic Delegation Game Revisited: A Note.” Managerial and Decision Economics 27: 605–12. https://doi.org/10.1002/mde.1271. Search in Google Scholar

Liao, P. 2010. “Strategic Delegation under Unionised Duopoly: Who Will Bargain with Unions.” Australian Economic Papers 49: 276–88. https://doi.org/10.1111/j.1467-8454.2010.00402.x. Search in Google Scholar

Liao, P. 2014. “Strategic Delegation of Multiple Tasks.” Australian Economic Papers 53: 77–96. https://doi.org/10.1111/1467-8454.12022. Search in Google Scholar

Nakamura, Y. 2017. “Choosing Price or Quantity? The Role of Delegation and Network Externalities in a Mixed Duopoly.” Australian Economic Papers 56: 174–200. https://doi.org/10.1111/1467-8454.12079. Search in Google Scholar

Salinger, M. 1988. “Vertical Mergers and Market Foreclosure.” Quarterly Journal of Economics 103: 345–56. https://doi.org/10.2307/1885117. Search in Google Scholar

Scott, F. 1995. “Franchising vs. Company Ownership as a Decision Variable of the Firm.” Review of Industrial Organization 10: 69–81. https://doi.org/10.1007/bf01024260. Search in Google Scholar

Sklivas, S. 1987. “The Strategic Choice of Managerial Incentives.” The RAND Journal of Economics 18: 452–8. https://doi.org/10.2307/2555609. Search in Google Scholar

Zhang, J., and Z. Zhang. 1997. “R&D in a Strategic Delegation Game.” Managerial and Decision Economics 18: 391–8. https://doi.org/10.1002/(sici)1099-1468(199708)18:5<391::aid-mde837>3.0.co;2-1. Search in Google Scholar

Received: 2021-03-11
Accepted: 2021-09-16
Published Online: 2021-10-08

© 2021 Walter de Gruyter GmbH, Berlin/Boston