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

Chung-Hui Chou

doi:10.1515/bejeap-2021-0190

Published by De Gruyter October 8, 2021

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

Chung-Hui Chou

From the journal The B.E. Journal of Economic Analysis & Policy

https://doi.org/10.1515/bejeap-2021-0190

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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.

Keywords: costs of sales forces; prisoners’ dilemma; vertical integration; vertical merger policy; vertical separation

JEL Classification: D43; L11; L13

Corresponding author: Chung-Hui Chou, Department of Finance, I-Shou University, Kaohsiung City 84001, Taiwan, ROC, E-mail: chchou@isu.edu.tw

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

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.
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.
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,

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.
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,

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.
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 .

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.

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. □

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Received: 2021-03-11

Accepted: 2021-09-16

Published Online: 2021-10-08

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

Abstract

Acknowledgments

Appendix I: Proof of Lemma 1

Appendix II: Proof of Proposition 2

Appendix III: Proof of Proposition 3

Appendix IV: Proof of Lemma 2

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

Appendix VI: Derivations of Consumers’ Surplus

Appendix VII: Proof of Proposition 5

References

Journal and Issue

Articles in the same Issue

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

Abstract

Acknowledgments

Appendix I: Proof of Lemma 1

Appendix II: Proof of Proposition 2

Appendix III: Proof of Proposition 3

Appendix IV: Proof of Lemma 2

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

Appendix VI: Derivations of Consumers’ Surplus

Appendix VII: Proof of Proposition 5

References

Journal and Issue

Articles in the same Issue

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